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Analysis of multiplicative noise in laser amplifiers and its effect on laser linewidth

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Analysis of multiplicative noise in laser amplifiers and its effect on laser linewidth

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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zeeb Road, Atm Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A N A L Y S IS O F M U L T IP L IC A T IV E N O IS E IN L A S E R A M P L IF IE R S A N D IT S E F F E C T O N L A S E R L IN E V V T D T H by R o n a ld T h o m a s Logan. Jr. ■\ D is s e rta tio n Presented to th e F A C U L T Y O F T H E G R A D U .\T E S C H O O L U N IV E R S IT Y O F S O U T H E R N C A L IF O R N IA In P a rtia l F u lfillm e n t o f th e R e q u ire m e n ts fo r th e Degree D O C T O R O F P H IL O S O P H Y (E le c tric a l E n g in e e rin g ) D e ce m b e r 1991 © 1 9 9 7 R o n a ld T h o m a s Logan. J r. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D ill N tunber: 9835069 UMI Microform 9835069 Copyright 1998, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNTVERSmr PARK LOS ANGELES, CALflFORNIA 90007 Tnis dissertation, written by Ronald Thomas Logan, Jr. under the direction of hks Dissertation Committee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of re quirements for the degree of DOCTOR OF PHILOSOPHY Dean of Craduata Studies Date 4.537. DISSERTATION COMMITTEE Chairperson .... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To X ico le Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A ckn ow ledgm ents I am g ra te fu l to e ve ryo ne w ho helped me th ro u g h o u t th is p ro je c t, and fo r th e en v iro n m e n t in w h ic h to c o n d u c t th e research. I especially w ish to th a n k m y advisor. Professor E lsa G a rm ire . fo r her guidan ce and forbearance w ith a "n o n -tra d itio n a l gra d u a te s tu d e n t, and D r. L u te M a le k i o f .JPL for his c o lla b o ra tio n and constant su p p o rt o f th e research. I also th a n k th e rest o f th e m em bers o f th e Frequency S tandards L a b o ra to ry at .JPL fo r th e ir tim e , expertise, e n co u ra g e m e n t, and frie n d ship. T h is w o rk was g e ne ro u sly fu n d e d by th e .Jet P ro p u lsio n L a b o ra to ry and GS.AF Rom e L a b o ra to ry : I e s p e c ia lly th a n k B ria n H endrickson o f R o m e L a b o ra to ry for his su p p o rt o f th e p ro je c t. F in a lly . 1 th a n k N icole, m y w ife, fo r b e in g m y best frie n d and e n co u ra g in g m e to fin is h , and m y parents and fa m ily fo r th e ir s u p p o rt. Ill Reproduced witfi permission of tfie copyrigfit owner. Furtfier reproduction profiibited witfiout permission. C ontents A cknow ledgm ents iii List of figures vi A b stract viii 1 Intro d u ctio n 1 1.1 M o t iv a t io n .......................................................................................................................... 2 1.2 ,A.pproach.... ......................................................................................................................... I 1.2.1 .\d d itiv e and M u ltip lic a tiv e N oise S o u r c e s ........................................ 7 1.2.2 P a rtia l-W a v e M o d e l w ith M u ltip lic a tiv e N o is e ................................. 8 1.2.3 Laser as a N o ise -D rive n R esonant . A m p lif ie r .................................... 9 1.3 Thesis O rg a n iz a tio n ...................................................................................................... 13 2 Partial-W ave Analysis of a R esonant O ptical Am plifier w ith M ul tiplicative Noise 16 2.1 B a ckg ro u n d and O v e r v ie w ........................................................................................ 16 2.2 R eview o f S ta n d a rd L in e a r Feedback T h e o r y .................................................. 18 2.3 L in e a r Feedback System T im e -D o m a in .Analysis w ith M u ltip lic a tiv e N o is e ............................................................................ 23 2.3.1 S m a ll-a n g le regim e: lin e a r a p p ro x im a tio n for ideal noiseless in p u t f i e l d .............................................................................................................. 31 2.3.2 S m a ll-a n g le a p p ro x im a tio n w ith noisy in p u t field .............................3-1 2.4 O u tp u t F ie ld Pow er S p e ctru m fo r S m all-.A ngle M o d e l....................................36 2.4.1 L im its o f sm all-angle a n a ly s is .......................................................................42 2.4.2 E quivalen ce to P henom e nological M o d e l ................................................45 2.4.3 S u m m a ry o f sm a ll-a n g le r e s u l t s ...................................................................46 2.5 Large-angle r e g im e ............................................................................................................47 2.5.1 N oise m o d e l............................................................................................................48 2.5.2 N u m e ric a l r e s u lt s .................................................................................................48 2.5.3 S im u la tio n a lg o r it h m .........................................................................................69 2.6 S u m m a ry o f p a rtia l-w a v e m odel r e s u lt s ..................................................................69 2.7 .A ppendix: D e riv a tio n o f sm a ll-a n g le th e o ry w ith in p u t field flu c tu a tio n s 71 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 Effect of m ultiplicative noise on laser linew idth in th e sm all-angle regim e 74 .3.1 R esonant a m p lifie r lin e w id th w ith m u ltip lic a tiv e n o is e ............................................................................................................ To 3.2 .A p p lic a tio n o f tim e -v a ry in g p a rtia l-w a v e m o d e l to laser lin e w id th . . SI 3.2.1 M o d e l o f a laser as a n o ise -d rive n resonant a m p lif ie r ......................... 82 3.3 S u m m a r y ............................................................................................................................... 8 6 4 O ptical phase fluctuations due to electron d ensity fluctuations in a sem iconductor laser 88 4.1 I n t r o d u c t io n ........................................................................................................................ 8 8 4.2 E le c tro n D e n s ity to O p tic a l Phase C o n v e r s io n ...................................................92 4.3 E le c tro n D e n s ity F lu c tu a tio n s in a S e m ic o n d u c to r....................................................................................................................... 96 4.3.1 M ean-S quare D e n s ity F lu c t u a t io n s ............................................................97 4.3.2 Pow er S p e c tra l D e n s ity o f F lu c tu a tio n s ..................................................101 4.4 D iscussion and S u m m a r y ................................................................................................ I l l 5 Sem iconductor laser linew idth 114 .3.1 O nset o f lin e w id th r e b ro a d e n in g ................................................................................. 113 3.2 S m a ll-a n g le lin e w id t h ........................................................................................................117 3.3 S u m m a r y .............................................................................................................................. 1 20 6 C onclusion 121 6 .1 D iscussion o f R e s u lt s ........................................................................................................121 6.2 C o m p a riso n w ith O th e r R e s u lt s .................................................................................124 6.2.1 L in e w id th o f se m ic o n d u c to r le is e rs ............................................................. 124 6.2.2 In te n s ity p ro b a b ility d is trib u tio n ............................................................. 123 6.3 F u rth e r re s e a rc h ................................................................................................................... 126 6.3.1 Laser lin e w id th e x p e r im e n ts .........................................................................126 6.3.2 In te n s ity noise in large-angle re g im e ......................................................... 129 6.3.3 1 / / noise ...............................................................................................................129 6.4 C lo s in g ......................................................................................................................................131 R eferences 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List o f Figures 1.1 P o w e r-in d e p e n d e n t lin e w id th o f a s e m ic o n d u c to r laser diode. (D a ta fro m K . K o b a ya sh i and I. M ito , P roc. O F C /IO O C '87. paper W 'C l. Reno. N evada, US.A.. Jan. 1987................................................................................. 5 1 .2 L in e w id th re b ro a d e n in g o f a s e m ic o n d u c to r laser d io d e . (D a ta fro m H . Y a m a z a k i, M . Y am aguch i, M . K ita m u ra , IE E E P h o to n ic Tech. L e tt. 6 . 341-343 (1 9 9 4 )................................................................................................. 6 1.3 D e lay-fee db ack m od el for an o p tic a l c a v ity ............................................................. 10 1.4 P a rtia l wave a d d itio n to produce in sta n ta n e o u s o u tp u t fie ld ......................... I I 2 .1 D e la y -ty p e feedback system m o d e l fo r a resonant o p tic a l a m p lifie r. . 2 0 2.2 P ow er tra n s m is s io n fu n c tio n o f resonant o p tic a l c a v ity fo r p, = pi = 0.9 and = 1.......................................................................................................................22 2.3 Real and im a g in a ry parts o f m u ltip lic a tiv e noise.................................................49 2.4 .M u ltip lic a tiv e noise in th e c o m p le x plane fo r a ll th e tim e s in F ig u re 2.3 .50 2.5 O u tp u t fie ld in co m p le x plane fo r net gain Ko = 0.9. M = 50...................... 54 2 . 6 O u tp u t fie ld in co m p le x plane fo r net gain Ko = 0.95. M = 1 0 0 . . . . 54 2.7 O u tp u t fie ld in c o m p le x plane fo r net gain Ko = 0.995. M = 1000. . . 55 2.8 O u tp u t fie ld in co m p le x plane fo r net gain Ko = 0.999. M = 5000. . . 55 2.9 O u tp u t in te n s ity versus tim e fo r net gain Ko = 0.9. .1/ = 50..........................57 2.10 O u tp u t in te n s ity versus tim e fo r n e t g ain Ko = 0.95. M = 1 0 0 ..................... 57 2 .1 1 O u tp u t in te n s ity versus tim e fo r net gain Ko = 0.995. M = 1 0 0 0 . . . . 58 2.12 O u tp u t in te n s ity versus tim e fo r net gain AT, = 0.999. M = 5000. . . . 58 2.13 O u tp u t phase versus tim e fo r net g ain Ko — 0.9, M = 50 .................................59 2.14 O u tp u t phase versus tim e fo r net g ain Ko — 0.95. M = 1 0 0 ............................ 59 2.15 O u tp u t phase versus tim e for net gain AT, = 0.995. .V/ = 1000................ 60 2.16 O u tp u t phase versus tim e for net g a in A „ = 0.999. .V/ = 5000................ 60 2.17 In te n s ity h is to g ra m for net g ain Ko = 0.9. .V/ = 50. S olid curve: G aussian f i t ............................................................................................................................. 61 2.18 In te n s ity h is to g ra m fo r net gain Ko = 0 .9 5 .4 / = 100. S olid curve: G aussian f i t ............................................................................................................................. 61 2.19 In te n s ity h is to g ra m fo r net gain Ko = 0.995. M = 1000. S olid curve: G aussian f i t ............................................................................................................................. 62 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.20 In te n s ity h is to g ra m fo r net gain R'o = 0.999. \ I = 5000. S o lid curve: G aussian f i t ............................................................................................................................. 62 2.21 O u tp u t fie ld pow er s p e c tru m for net gain R'o = 0.9............................................ 64 2 .2 2 O u tp u t fie ld pow er s p e c tru m fo r net ga in R'o = 0.91..........................................64 2.2.3 O u tp u t fie ld p ow er s p e c tru m fo r net ga in R'o = 0.93..........................................65 2.24 O u tp u t fie ld p ow er s p e c tru m fo r net ga in R'o = 0.95..........................................65 2.25 O u tp u t fie ld in c o m p le x plane fo r net g a in R'o = 0.999. M = 5000. m u ltip lic a tiv e phase noise o n ly ...................................................................................... 6 6 2.26 O u tp u t in te n s ity versus tim e for net g a in Ro = 0.999. M = 5000. m u ltip lic a tiv e phase noise o n ly ...................................................................................... 6 6 2.27 O u tp u t phase versus tim e fo r net gain R'o = 0.999. M = 5000. m u lti p lic a tiv e phase noise o n ly .................................................................................................67 2.28 In te n s ity h is to g ra m fo r net gain R'o = 0.999. M = 5000. m u ltip lic a tiv e phase noise o n ly .................................................................................................................... 67 2.29 O u tp u t fie ld pow er s p e c tru m for net g a in R'o = 0.999. M = 5000. m u ltip lic a tiv e phase noise o n ly ...................................................................................... 68 4.1 S ingle q u a n tu m -w e ll laser s tru c tu re considered in th e analysis o f elec tro n d e n s ity flu c tu a tio n s ...................................................................................................90 4.2 C a lc u la te d pow er s p e ctra l d e n sity o f phase flu c tu a tio n s due to elec tro n d e n s ity flu c tu a tio n s .................................................................................................110 6.1 T h e o re tic a l re su lts fo r in te n s ity p ro b a b ility d is trib u tio n fro m Fokker- P lan ck a na lysis o f m u ltip lic a tiv e noise in a h e liu m -n e o n laser, (fro m S. Z h u . P h ysica l R eview .A 47. 2405-2408 ( 1 9 9 3 ) ) ............................................ 127 6 .2 E x p e rim e n ta l re su lts fo r in te n s ity p ro b a b ility d is trib u tio n fro m F okker- P la nck an alysis o f m u ltip lic a tiv e noise in a h e liu m -n e o n laser, (d a ta fro m M . R. \b u n g . S. S ingh. O p t. L e tt. 13. 21-23 ( 1 9 8 8 ) ) .............................128 VII Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A b stract A general fo rm a lis m is developed fo r a n a lyzin g th e o u tp u t fie ld flu c tu a tio n s o f a resonant o p tic a l a m p lifie r th a t has ra n d o m gain and phase flu c tu a tio n s . B y gener a liz in g th e p a rtia l-w a v e m o d e l ty p ic a lly used to analyze F a b ry -P e ro t o p tic a l ca vitie s to tre a t th e case o f tim e -v a ry in g ro u n d -trip gain and phase, an in tu itiv e physical p ic tu re for th e effect o f m u ltip lic a tiv e noise in the o p tic a l resonant a m p lifie r is ob tained . T im e -v a ry in g c o m p le x gain c o n s titu te s a multiplicative g a in and phase noise source th a t is tra n s fo rm e d to th e o u tp u t fie ld in a d iffe re n t w ay fro m additive noise sources, such as spontaneous em ission. N e x t, by m o d e lin g a laser as a noise-driven resonant o p tic a l a m p lifie r, it is show n th a t m u ltip lic a tiv e noise generates an a d d i tio n a l p o w e r-ind e pe nd ent te rm in th e Schaw low -Tow nes fo rm u la fo r th e lin e w id th o f the laser, and a re -b ro a d e n in g o f th e lin e w id th at h ig h o u tp u t p ow er levels, as is ty p i c a lly observed in sin g le -m o d e se m ic o n d u c to r lasers. T h e fo rm a lis m developed allow s ca lcu la tio n o f th e o u tp u t fie ld pow er sp e ctra o f a m p litu d e and phase flu c tu a tio n s due to m u ltip lic a tiv e noise. .A d e ta ile d analysis o f electron d e n s ity flu c tu a tio n s in a se m ico nd ucto r laser g a in m e d iu m is also oerform ed. fro m w h ich th e pow er spectral de n sity o f m u ltip lic a tiv e phase noise due to electron n u m b e r-d e n s ity flu c tu a tio n s in the gain m e d iu m is fo u n d to have a 1/ /- lik e character. T h is re s u lt is th e n used in the m u ltip lic a tiv e laser noise m odel to o b ta in an e stim a te o f th e m in im u m lin e w id th o f a q u a n tu m -w e ll s e m ic o n d u c to r laser. .A lthough the m u ltip lic a tiv e noise analysis is applied here to resonant o p tic a l a m p lifie rs and lasers, th e tim e -v a ry in g p a rtia l wave fo rm a lis m is general and m ay be a p p lica b le to th e analysis o f o th e r types o f feedback system s p e rtu rb e d by m u ltip lic a tiv e noise, such as e le c tro n ic o scilla to rs and frequency sta n d a rd s, b io lo g ica l feedback system s, and m e ch a n ica l o s c illa tin g system s. Vlll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 1 Introduction S ingle -m o de s e m ic o n d u c to r laser diodes ty p ic a lly e x h ib it lin e w id th s a tu ra tio n and re b ro a d e n in g at high o u tp u t pow er levels. T h is b e h a v io r is d e le te rio u s fo r a p p lic a tions re q u irin g a h ig h degree o f sp e ctra l p u rity , such as o p tic a l h e te ro d yn e m ic ro w a v e signal g e n e ra tio n , fib e r-o p tic in te rfe ro m e tric sensing, coherent c o m m u n ic a tio n s , and a to m ic spectroscop y. V ariou s m echanism s have been advanced to e x p la in th e on set o f lin e w id th s a tu ra tio n and rebroade ning, b u t th e q u e stio n re m a in s w h e th e r lin e w id th s a tu ra tio n is an in trin s ic lim ita tio n , o r s im p ly due to e x trin s ic "te c h n i ca l" sources o f noise th a t can. in p rin c ip le , be reduced b y im p ro v e d processing or a lte rn a tiv e d e v ic e geom etries. To in v e s tig a te th is q u e stio n , in th is d is s e rta tio n , a generalize d p a rtia l-w a v e m o d e l is developed to a n alyze th e o u tp u t held flu c tu a tio n s o f a resonant o p tic a l a m p lifie r p e rtu rb e d b y tim e -v a ry in g m u ltip lic a tiv e gain and phase flu c tu a tio n s . N e x t, a lin e a r m odel o f a laser as a n o ise -d rive n resonant o p tic a l a m p lih e r is developed using th e results o f th e tim e -v a ry in g p a rtia l wave analysis. B y v ie w in g th e laser as a noise- d riv e n resonant a m p lih e r. and in c lu d in g th e results o f th e generalized p a rtia l-w a v e analysis, a m o d e l is o b ta in e d fo r th e lin e w id th o f th e laser w ith m u ltip lic a tiv e noise as a fu n c tio n o f o u tp u t pow er. It is show n th a t m u ltip lic a tiv e noise leads to an a d d i tio n a l p o w e r-in d e p e n d e n t regim e in th e S chaw low -T ow nes fo rm u la fo r th e lin e w id th o f a laser a t in te rm e d ia te pow er levels, follow ed b y lin e w id th re b ro a d e n in g as th e pow er is increased fu rth e r. F ro m th is re s u lt, an e s tim a te d value fo r th e m in im u m p o w e r-in d e p e n d e n t lin e w id th o f a single-m ode q u a n tu m -w e ll s e m ic o n d u c to r laser o f Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 k H z to 15 k H z . du e to in trin s ic e le ctro n n u m b e r-d e n s ity flu c tu a tio n s , is o b ta in e d . T h is value is in reasonable agreem ent w ith e x p e rim e n ta l observations. T h e laser m o d e l de velo p ed here p ro vid e s an in tu itiv e p ic tu re o f a com m on m echanism th a t can acco u n t fo r b o th lin e w id th s a tu ra tio n and re b ro a d e n in g in lasers as a n a tu ra l consequence o f th e presence o f a m u ltip lic a tiv e noise process. 1.1 M o tiv a tio n In th e ir o rig in a l o p tic a l m aser paper o f 1958. S chaw low and Townes [4] d e rive d th e inverse-po w e r fo rm u la fo r laser lin e w id th . w h ic h m a y be w ritte n as , At/ .V2 àu= {àu^)--^ — — ------- — ( 1. 1 ) r -v j — I w here 6 1/^ is th e "c o ld c a v ity " lin e w id th d e te rm in e d b y the losses, h is P la n ck's c o n s ta n t, is th e o s c illa tio n frequency. P is th e o u tp u t power, and .\%. are th e p o p u la tio n s o f th e u p p e r and low er states o f th e a to m ic tra n s itio n responsible for th e o p tic a l g a in . T h is expression is v a lid fo r a laser be lo w o scilla tio n th re sh o ld , and is reduced by a fa c to r o f tw o above th re sh o ld by d a m p in g o f a m p litu d e flu c tu a tio n s due to gain s a tu ra tio n [5]. T h e fin ite lin e w id th is due to the spontaneously e m itte d photon s th a t ra n d o m ly p e rtu rb th e phase o f th e laser fie ld . Spontaneous em ission pow er is p ro p o rtio n a l to th e g ain, w h ich is e sse n tia lly cla m p e d at th e th re sh o ld level. H ow ever, as th e laser m ode a m p litu d e increases above th re sh o ld , th e c o n trib u tio n o f th e sp onta n e ou s em ission to th e to ta l o u tp u t becom es p ro p o rtio n a lly less, and th e lin e w id th decreases. In m ost lasers, th e observed lin e w id th is ty p ic a lly several orders o f m a g n itu d e la rg e r th a n th e S chaw low -T ow nes lin e w id th p re d ic tio n , due to e x trin s ic sources o f noise such as m ir r o r v ib ra tio n [3]. H ow ever, in se m ic o n d u c to r lasers, th e com bined effects o f la rg e o u tp u t c o u p lin g (i.e .. low re fle c tiv ity facet m irro rs ) and a m p litu d e - phase c o u p lin g th ro u g h th e c a rrie r d e n s ity [6 . 7] y ie ld p re d icte d q u a n tu m lin e w id th s th a t are re a d ily m e a su ra b le fo r ty p ic a l o u tp u t pow er levels. Due to th e ir sm a ll size and th e ease o f in je c tio n -c u rre n t p u m p in g and m o d u la tio n , there is considerab le co m m e rc ia l and s c ie n tific in te re s t in th e fu rth e r d e ve lo p m e n t o f these devices to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. achieve th e low est possible lin e w id th s a t lo w cost for a p p lic a tio n in c o m m u n ic a tio n s system s, m icro w a ve signal generatio n, o p tic a l spectroscopy, and o p tic a l sensing. F rom th e S chaw low -Tow nes theory, one m ig h t expect th a t th e lin e w id th o f a single-m o de s e m ico n d u c to r laser could be m ade a rb itra rily n a rro w by h ig h -p o w e r o p e ra tio n , since th e e x trin s ic sources o f in s ta b ility such as m irro r v ib ra tio n s and dye stre a m flu c tu a tio n s th a t plague large-scale gas and dye lasers are absent or ne g lig ib le . H ow ever, as o p tic a l pow er is increased, th e lin e w id th o f sin gle-m ode d is trib u te d -fe e d b a c k (D E B ) s e m ico n d u cto r lasers has been fo u n d to e x h ib it a pow er- indep e n d e n t flo o r o f several M H z [S. 10] as show n in F ig u re 1 . 1 . .\s th e o p tic a l power is increased fu rth e r, th e lin e w id th e v e n tu a lly rebroadens [11. 12] F ig u re 1.2. T h is b e h a vio r is not p re d ic te d by th e m o d ifie d S chaw low -Tow nes re la tio n , and lim its th e usefulness o f s e m ic o n d u c to r lasers in a p p lic a tio n s re q u irin g h ig h coherence. To achieve th e low est lin e w id th s w ith se m ic o n d u c to r lasers, e x te rn a l lin e -n a rro w in g ca vitie s are ty p ic a lly em ployed, a t the expense o f gre a te r cost a n d c o m p le x ity . It appears th a t th e po w e r-inde pendent lin e w id th c h a ra c te ris tic o f s e m ic o n d u c to r lasers was firs t re p o rte d by W e lfo rd and .VIooradian in 1982 [8 ]. and was a ttrib u te d to re fra c tiv e in d e x flu c tu a tio n s due to e le c tro n n u m b e r-d e n s ity flu c tu a tio n s . T he change in re fra c tiv e in d e x was assum ed to lead to a change in th e o p tic a l le n g th o f th e laser c a v ity , a n d hence to a d ire c t m o d u la tio n o f th e laser em ission frequency. In th a t w o rk, th e tra n s fo rm a tio n o f re fra c tiv e in d e x va ria tio n s Sn to flu c tu a tio n s o f th e laser freq ue ncy Su was derived fro m th e phenom enological re la tio n Su Sn u n T h e re fra c tiv e in d e x flu c tu a tio n s were in c lu d e d in a ro o t-m e a n -sq u a re sense, and de rive d fro m th e rm o d y n a m ic co n sid e ra tio n s re g a rd in g the n u m b e r-d e n s ity flu c tu ations o f th e e le ctro n s containe d in th e a c tiv e m e d iu m . T h e re was no re p o rt o f re b roa de ning o f th e lin e w id th at high pow ers. T h e ir m odel p ro vid e s a basic m ech anism fo r th e o rig in o f th e p o w e r-in d e p e n d e n t lin e w id th . b u t does not p re d ic t the onset o f lin e w id th rebroadening. In la te r w o rk, e sse n tia lly the same ph e n o m e n o lo g ica l m o d e l has been em ployed ;i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to e x p la in the p o w e r-in d e p e n d e n t lin e w id th using o th e r sources o f re fra c tiv e index flu c tu a tio n such as in je c tio n -c u rre n t flu c tu a tio n s . 1/ /-n o is e o f in d e te rm in a te o rig in [10]. and s p a tia lly -d e p e n d e n t te m p e ra tu re and c a rrie r flu c tu a tio n s [13]. Separate m echanism s have been suggested to e x p la in lin e w id th re b ro a d e n in g , such as sp a tia l hole b u rn in g [ 1 1 ] a n d excess c a rrie r d e n sity in th e co n fin e m e n t regions o f q u a n tu m w ell lasers [12] at h ig h in je c tio n . These m echanism s m ay be o p e ra tiv e in various com b in a tio n s in any p a rtic u la r laser. .At present, however, th e re appears to be no u n ifie d m odel capable o f e x p la in in g b o th th e pow er-inde pendent lin e w id th and lin e w id th rebroad e ning in s in g le -m o d e se m ico n d u cto r lasers, and w h e th e r these phenom ena are in trin s ic p ro p e rtie s o f lasers, o r o f e x trin s ic o rig in . T h e re fo re , a p rin c ip a l goal o f th is d is s e rta tio n is to p ro v id e a u n ifie d th e o ry o f s e m ico n d u cto r laser lin e w id th th a t e xp la in s th e in ve rse -p o w e r. p o w e r-in d e p e n d e n t. and lin e w id th -re b ro a d e n in g regim es as a consequence o f a single in trin s ic noise source. F in a lly , it is n o te d th a t in th e pow er-inde pendent lin e w id th re g im e o f o p e ra tio n , th e frequency flu c tu a tio n pow er sp e ctru m o f se m ico n d u cto r lasers has been observed to have a 1/ / p o w e r s p e c tra l d ensity. T herefore, a fu rth e r m o tiv a tio n o f th e present w o rk is to u n d e rs ta n d b e tte r th e o rig in s o f the u b iq u ito u s 1/ / in te n s ity and fre quency noise o f s e m ic o n d u c to r lasers: .Are 1/ / flu c tu a tio n s due to m o d u la tio n o f the laser o u tp u t b y e x trin s ic noise sources such as I / / c a rrie r-d e n s ity o r th e rm a l flu c tu a tio n s , o r is th e re an in trin s ic m echanism for g e n e ra tio n o f 1 / / noise in the lasing process? T h is d is s e rta tio n w ill not p ro vid e a d e fin itiv e answ er to th e I / / noise question, b u t th e analysis suggests a p o te n tia l in trin s ic I / / noise m echanism in se m ico n d u cto r lasers. 1.2 A p proach P revious theories o f laser noise ty p ic a lly s ta rt fro m the cissum ption o f a sin g le eigen- m o d e o f the laser c a v ity , and proceed to p e rfo rm a sm a ll-sig n a l p e rtu rb a tio n analysis o f th a t m ode. T h e re fo re , by d e fin itio n , large departures fro m th e c e n tra l eigenfre- quency can no t be p re d ic te d by such a m odel. In re a lity, h ow ever, laser frequencies e x h ib it lo n g -te rm flu c tu a tio n s , and w ander aw ay fro m th e ir s ta rtin g places, ty p ic a lly 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N •5 -a I .1 1 0 -1 9- 8- 7 - 6- 4 - 3- 1 - I I I I I I ‘ ^ 1 I I I I T I I I r I I I I I I I 5 6 7 8 9 ' 2 3 4 1 -----1 — I— I— r 5 6 7 8 9 10 i / V i Figure 1.1: Power-independent linewidth of a semiconductor laser diode. (Data from K. Kobayashi and I. Mito, Proc. OFC/IOOC '87, paper W Cl, Reno, Nevada, USA, January 1987. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4H 3 - N X I 2^ S C 1 -A 0 - i I ' I I 0.00 “I— r -I 1 -------1 -------1 -------1 -------r “I “ 'I Î 0.05 0.10 0.15 Inverse of Output Power [1 / mW] 0.20 Figure 1.2; Linewidth rebroadening of a semiconductor laser diode. (Data from H. Yamazaki, M. Yamaguchi, M. Kitamura, IEEE Photonic Tech. Lett. 6, 341-343,1994.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w ith a lo w -fre q u e n c y pow er s p e c tru m p ro p o rtio n a l to 1/ / . W h ile th e eigenm od e ap proach has been e x tre m e ly successful in p re d ic tin g th e sm a ll-sig n a l h ig h -fre q u e n cy b e h a v io r o f th e laser o u tp u t flu c tu a tio n s , such as re la x a tio n o s c illa tio n s in sem icon d u c to r lasers [6 . 7]. th e basic assum ption o f an eigenm ode renders th is approach u n s u ita b le to e x p la in th e low -frequen cy, lo n g -te rm b e h a vio r o f th e laser in te n s ity and fre q u e n cy flu c tu a tio n s . In th e present w o rk, in o rd e r to in ve stig a te th e low frequency flu c tu a tio n s and lo n g -te rm e v o lu tio n o f the laser m ode, the a s s u m p tio n o f an eigenm ode is abandoned. T h e o u tp u t fie ld o f a resonant o p tic a l c a v ity w ith ga in is instead co m p u te d at d iscrete tim e in s ta n ts as a sum o f partial waves by a c a re fu l a nalysis o f th e in te ra c tio n o f the o p tic a l fields in a c a v ity th a t is p e rtu rb e d by m u ltip lic a tiv e flu c tu a tio n s . T h e phase o f th e o u tp u t fie ld is free to execute a r b itr a r ily large excursions fro m its s ta rtin g p o in t, o ver a r b itr a r y tim e scales, so th a t lo w -fre q u e n cy flu c tu a tio n s are n o t e xcluded a p rio i. as in th e eigenm od e approach. U sing th is "tim e -v a ry in g p a rtia l-w a v e m o d e l" we re s tric t o u r a tte n tio n to low -frequen cy flu c tu a tio n s o f phase and a m p litu d e o f th e laser fie ld . T h e response tim e o f the m o d e l is lim ite d to tim e s g re a te r th a n the ro u n d -trip tim e o f th e laser. T h is m odel c o m p le m e n ts th e previous w o rk on high- frecjuency noise b e h a v io r o f lasers. Since th e h ig h -fre cp ie n cy noise is w e ll-u n d e rs to o d , the em phasis o f th e present w o rk is to e lu c id a te th e m echanism s fo r lo w -fre q u e n cy flu c tu a tio n s o f th e laser fie ld . 1 .2.1 A d d itiv e and M u ltip lica tiv e N o ise Sources In lasers, a d d itiv e noise arises fro m photons w h ic h are added ra n d o m ly to th e fie ld by spontaneous e m issio n . M u ltip lic a tiv e noise arises in an o p tic a l feedback syste m w hen a m u ltip lic a tiv e fa c to r a ffe c tin g the field, such as g ain. loss, o r ro u n d -trip phase, flu c tu a te s in tim e . In a d d itio n to the in trin s ic a d d itiv e noise due to spontaneous em ission, in trin s ic sources o f m u ltip lic a tiv e ga in and phase flu c tu a tio n w ill be present in th e co m p o n e n ts o f a laser system at some level. In gas and dye laser system s, fo r exa m p le , th e net gain and o p tic a l le n g th o f the laser c a v itv can flu c tu a te due to dve-stream w id th flu c tu a tio n s , m irro r v ib ra tio n s . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p u m p pow er flu c tu a tio n s , o r o th e r e n v iro n m e n ta l fa cto rs. In these types o f laser system s, it is these eitringic sources o f m u ltip lic a tiv e noise th a t la rg e ly d e te rm in e the observed lin e w id th [3]. and not intrinsic a d d itiv e q u a n tu m noise due to spon taneous e m ission, as p re dicted by th e S chaw low -T ow nes fo rm u la o f e q u a tio n (1 .1 ). In p ra c tice , special efforts m u st be m ade to iso la te gas lasers fro m e n v iro n m e n ta l p e rtu rb a tio n s in o rd e r to m easure th e c[u a n tu m -n o ise lim ite d lin e w id th . M u ltip lic a tiv e noise is no t considered in th e d e riv a tio n o f th e S chaw low -T ow nes lin e w id th fo rm u la o f equatio n (1.1). .A.s discussed p re vio u sly, th e effects o f m u lti p lic a tiv e noise can be m odeled to firs t o rd e r by th e p h eno m e nological e q u a tio n ( 1.2 ). T h is e q u a tio n lin k s th e c a v ity ce n te r fre q u e n cy u to th e in stantan eous value o f the re fra c tiv e in d e x, b u t does n o t specify a tim e scale fo r th e c a v ity fre q u e n cy to respond to the in d e x changes. Since noise is a ra n d o m , tra n s ie n t phenom ena encom passing m any tim e scales, it is im p o rta n t to u n d e rs ta n d th e regions o f v a lid ity o f th is m odel. In th e present w o rk , th e p a rtia l-w a v e a p p ro a ch includes th e effects o f th e m u ltip lic a tiv e noise on th e o u tp u t fie ld o f th e resonant a m p lifie r in a fu n d a m e n ta l way. B y p e rfo rm in g th is analysis, th e p h e n o m e n o lo g ica l re s u lt o f equ a tio n ( 1.2 ) is recovered, b ut th e regions o f its a p p lic a b ility are revealed. 1.2.2 P artial-W ave M o d el w ith M u ltip lica tiv e N o ise T he tim e -v a ry in g p a rtia l-w a v e m odel p ro vid e s a s im p le ph ysica l p ic tu re fo r th e ef fect o f m u ltip lic a tiv e noise in an o p tic a l resonant a m p lifie r. T h e p rin c ip a l result o f th e tim e -v a ry in g p a rtia l-w a v e m o d e l develop ed here is th a t it id e n tifie s m u lti p lic a tiv e flu c tu a tio n s to be a possible m e ch a n ism fo r th e phenom ena o f b o th pow er- in d epe nd en t lin e w id th and lin e w id th -re b ro a d e n in g in lasers. In th e tim e -v a ry in g p a rtia l-w a v e a n a lysis, th e resonant a m p lifie r is m o d e le d as a d e la y -ty p e feedback system as show n in F ig u re 1.3. w ith s lo w ly -v a ry in g c o m p le x valued in p u t fie ld a m p litu d e R(t). o u tp u t fie ld C(t). fo rw a rd gain 6 '( /) . reverse gain H(t) = G(t). and m irro r a m p litu d e r e fle c tiv ity p and tra n s m is s iv ity 7 . T h e net ro u n d -trip loop g a in is always assum ed to be less th a n u n ity , and th e fo rw a rd and reverse gains are assumed to be p e rtu rb e d by noise. T h e o u tp u t fie ld at d iscre te Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tim e in s ta n ts is view ed as a sum o f p a rtia l waves, as depicted in F ig u re I.-l. In th is m o d e l, th e "o ld e s t" p a rtia l waves have tra v e le d m o re tim es th ro u g h th e c a v ity , so th a t th e y c a rry the "m e m o ry " o f th e m u ltip lic a tiv e noise events fro m e a rlie r tim e s. T h e n u m b e r o f p a rtia l waves needed to c o m p u te th e o u tp u t is d e te rm in e d b y th e net ro u n d -trip gain. Since th e net g a in is assum ed to be less th a n u n ity , th e n u m b e r o f p a rtia l waves is fin ite . In a resonant o p tic a l a m p lifie r, tim e -v a ry in g phase and g ain c o n s titu te m ulti plicative noise sources whose effects a re tra n s fo rm e d to the o u tp u t fie ld in a d iffe re n t w ay fro m additive noise sources, w h ic h m a y be tre a te d as a d d itio n a l in p u ts to th e syste m . W h e n no m u ltip lic a tiv e noise is present, th e w id th o f th e c a v ity resonance peak decreases as the net g ain, a n d hence th e c a v ity m e m o ry tim e , is increased. H ow ever, in th e presence o f m u ltip lic a tiv e noise, th e o ld e r p a rtia l waves a c c u m u la te p ro g re ssive ly larger ra n d o m phase flu c tu a tio n s , le a d in g to in creasing o u tp u t fie ld flu c tu a tio n s . For high enough net ro u n d -tr ip gain, th e m u ltip lic a tiv e noise-indu ced flu c tu a tio n s w ill d o m in a te th e o u tp u t fie ld flu c tu a tio n s , and th e lin e w id th w ill not decrease w ith fu rth e r increases in net g a in . .As th e g ain is increased fu rth e r, th e la te r p a rtia l w aves a ccu m u la te so m uch phase th a t th e y begin to d e s tru c tiv e ly in te rfe re w ith th e e a rlie r p a rtia l waves, le a d in g to large o u tp u t flu c tu a tio n s . 1 .2 .3 L aser as a N o ise -D r iv e n R eso n a n t A m plifier T h e tim e -v a ry in g p a rtia l-w a v e m o d e l is a p p lie d to th e laser by tre a tin g th e laser as a resonant a m p lifie r w ith an in p u t s ig n a l equal to th e spontaneous em ission noise in th e la s in g m ode. T h is analysis is d e ta ile d fu lly in C h a p te r '■ ] . \ s th e net ro u n d - tr ip g a in (i.e .. single-pass ro u n d -trip g a in tim e s losses) o f the resonant a m p lifie r is increased fro m zero, th e o u tp u t fie ld a m p litu d e increases due to a m p lific a tio n o f th e spontaneous em ission fie ld . .At o r above th re s h o ld , the p o p u la tio n in ve rsio n and th e re fo re , the spontaneous e m issio n pow er, is assum ed to be cla m p e d . .Above th re s h o ld , th e gain is "s a tu ra te d " because o f e xte n sive s tim u la te d em ission. L 'n d e r these c o n d itio n s , we assum e th a t th e g a in m a y be h e u ris tic a lly d e te rm in e d in te rm s o f th e s te a d y -s ta te o u tp u t pow er d iv id e d b y th e spontaneous em ission pow er. 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Input Field R(œ) Mirror Transmission Loss Forward Gain H(û}) Y Output Field ► C(o)) Mirror Reflection Loss Reverse Gain F ig u re 1.3: D elay-feedback m o d e l fo r an o p tic a l ca vity. 1 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. G(t) P 1 À % time ■ m F igure 1 .4 : P a rtia l wave a d d itio n to produce in sta n ta n e o u s o u tp u t field. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In th is m o d e l o f th e laser as a noise-driven resonant a m p lifie r, th e ste a d y-sta te laser o u tp u t pow er P fro m b o th m irro rs is re la te d to th e in te rn a l spontaneous em is sion noise pow er P^p as w here R is th e pow er re fle ctio n coefficient o f th e c a v ity m irro rs , so th a t R — p - . r = * * = 1 — /? is th e pow er tra n sm issio n co e fficie n t o f th e m irro rs , and R~(Pp is th e pow er gain per ro u n d trip . T h e q u a n tity I — R~Gp is defined as th e "s a tu ra tio n a m p lific a tio n '' o f th e resonant a m p lifie r; I - r ^G; = -1T ^ (1.4) W e ec|uate th e ro u n d -trip power gain R^G^ w ith A j . th e square o f th e ro u n d -trip a m p litu d e gain o f th e resonant a m p lifie r fro m th e lin e a r a n a lysis o f C h a p te r '2. W hen h'o ~ I. as it is in a laser, th is expression m ay be w ritte n I — ko = T ( 1-Ô ) W e o b ta in th is expression for th e sa tu ra te d a m p litu d e gain o f th e laser in C h a p te r 3. T h u s, in th is laser m o d e l, changes in th e o u tp u t pow er above th re sh o ld are s im p ly re la te d to changes in th e s a tu ra te d net gain, w h ic h is alw ays less th a n u n ity . So. we m ay m ode l th e laser in term s o f th e h e u ris tic a lly d e te rm in e d s a tu ra te d net gain. w h ich is s im p ly re la te d to th e m easurable q u a n tity o f o u tp u t pow er b y e q u a tio n ( 1.5). T h e te rm s "n e t gain " a nd "o u tp u t pow er " w ill th e re fo re be used in te rch a n g e a b ly th ro u g h o u t th is d is s e rta tio n . T he effect o f m u ltip lic a tiv e noise in th e laser is then d e te rm in e d using th e s a tu ra te d net gain as a p a ra m e te r in th e p a rtia l-w a v e analysis develop ed in C h a p te r 2. It w ill be show n th a t th e regim e o f p o w e r-in d e p e n d e n t lin e w id th corresponds to th e s itu a tio n w hen th e m u ltip lic a tiv e noise-induced flu c tu a tio n s d o m in a te th e o u tp u t fie ld flu c tu a tio n s , a nd th e lin e w id th does n o t decrease w ith fu rth e r increases in o u tp u t pow er. T h e lin e w id th rebroadening regim e correspond s to th e h ig h -n e t-g a in 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s itu a tio n , w hen th e la te r p a rtia l waves a c c u m u la te so m uch phase th a t th e y begin to d e s tru c tiv e ly in te rfe re w ith th e e a rlie r p a rtia l waves, le a d in g to in cre a sin g o u tp u t flu c tu a tio n s w ith increasing o u tp u t pow er. T h u s, th e tim e -v a ry in g p a rtia l-w a v e m o d e l p re d ic ts pow e r-ind e pendent lin e w id th and lin e w id th re b ro a d e n in g in lasers to be d iffe re n t m a n ife sta tio n s o f th e sam e m u ltip lic a tiv e noise source in d iffe re n t o p e ra tin g regim es. I f a p a rtic u la r m u ltip lic a tiv e noise source is an in trin s ic fe a tu re o f a laser system , th e n the tim e -v a ry in g p a rtia l-w a v e m odel p re d icts th a t p ow er- in d e p e n d e n t lin e w id th and lin e w id th re b ro a d e n in g w ill be in trin s ic phenom ena. W e w ill d e riv e a general fo rm u la in C h a p te r 3 fo r laser lin e w id th th a t includes th e effects o f m u ltip lic a tiv e noise, fro m w h ich th e o rig in a l S chaw low -Tow nes lin e w id th fo rm u la is recovered as th e special case o f no m u ltip lic a tiv e noise. 1.3 T h esis O rganization T h e d is s e rta tio n is organized as follow s: B e g in n in g in C h a p te r 2. th e g e n e ra lize d p a rtia l-w a v e m od el is derived fo r th e o u tp u t e le c tric fie ld o f a resonant o p tic a l a m p li fie r w ith m u ltip lic a tiv e noise a c tin g on a n o isy in p u t field. d is c re te -tim e fo rm u la fo r th e s lo w ly -v a ry in g com plex envelope o f th e o u tp u t field is o b ta in e d , ecpiation (2 .1 5 ). T w o regim es o f the o u tp u t flu c tu a tio n s are th e n exa m in e d : s m a ll-a n g le and large-an g le. T h e s m a ll-a n g le a p p ro x im a tio n yie ld s e q u a tio n (2 .3 9 ). w h ich has th e fo rm o f a fln ite -im p u ls e -re s p o n s e filte r th a t lin e a rly tra n s fo rm s th e m u ltip lic a tiv e noise and in p u t fie ld noise processes to th e o u tp u t fie ld . F ro m th is expression, th e im p u ls e responses and frequency responses o f th e resonant a m p lifie r fo r m u ltip lic a tiv e a n d in p u t fie ld flu c tu a tio n s are o b ta in e d , fro m w h ic h th e o u tp u t fie ld flu c tu a tio n p o w e r s p e c tru m is d e rive d , e qu atio n (2.51). From th is expression, it is seen th a t fo r low net g a in and a fixe d level o f m u ltip lic a tiv e noise, th e lin e w id th is d e te rm in e d b y th e in p u t fie ld flu c tu a tio n s , and lin e a rly decreases w ith in cre a sin g pow er, as th e S chaw low - Tow nes th e o ry p re d icts. ,A.t a c ritic a l value o f net gain, th e o u tp u t flu c tu a tio n p ow e r s p e c tru m becomes d o m in a te d b y m u ltip lic a tiv e noise as shown in e q u a tio n (2 .5 5 ). and rem a ins constant w ith in cre a sin g pow er. T h is is th e p o w e r-in d e p e n d e n t 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. lin e w id th regim e. .A.t a second, higher c ritic a l value o f net ga in even closer to u n ity , th e m e m o ry o f th e c a v ity has increased so th a t th e sm a ll-a n g le c o n d itio n is exceeded fo r th e oldest p a rtia l waves. In th is "la rg e -a n g le " re g im e , th e m a p p in g o f th e m u ltip lic a tiv e flu c tu a tio n s to the o u tp u t field is n o n -lin e a r. N u m e ric a l s im u la tio n re su lts for th e la rg e-an g le regim e illu s tra te in cre a sin g a m p litu d e and phase flu c tu a tio n s , as the o ld e r p a rtia l waves accum ulate enough ra n d o m phase to cause d e s tru c tiv e in te r ference (F ig u re s 2.5. 2.6. 2.7. 2.8). W ith in cre a sin g net ga in , the o u tp u t in te n s ity flu c tu a tio n s increase, causing the p ro b a b ility d is trib u tio n o f th e o u tp u t in te n s ity in th e large angle regim e to m ove to low er values. F ig u re 2.20. and the pow er s p e c tru m to broaden . F ig u re 2.24. T h is is the m e ch a n ism fo r th e lin e w id th re broade ning. In C h a p te r 3. the results o f th e p a rtia l wave m odel are used to d e riv e the lin e w id th o f a resonant a m p lifie r. In th e sm a ll-a n g le regim e, th e o u tp u t fie ld pow er sp e c tru m o f th e resonant a m p lifie r is fo u n d to have a L o re n tzia n lineshape. and has a lin e w id th th a t is lin e a rly related to th e s tre n g th o f a m u ltip lic a tiv e noise source th a t has a w h ite p ow er spe ctru m , s im ila r to W e lfo rd and M o o ra d ia n 's pheno m e nological m o d e l. In th e large-angle regim e, th e lin e w id th is p re d ic te d to broaden, w h ic h is not p re d ic te d by W elford and M o o ra d ia n 's a nalysis, w h ich uses equatio n (1 .2 ). T h e resonant a m p lifie r analysis is then a p p lie d to th e laser by m o d e lin g it as a resonant a m p lifie r d riv e n by its ow n spontaneous em ission noise. In th e presence o f m u l tip lic a tiv e noise, th e lin e w id th o f the laser is show n to have a p o w er-inde pendent lin e w id th m in im u m for th e sm a ll-a n g le re g im e th a t rebroadens fo llo w in g th e onset o f th e la rg e-an g le regim e. T h e S chaw low -Tow nes lin e w id th fo rm u la is recovered as th e lim itin g case o f no m u ltip lic a tiv e noise. C h a p te r 4 co ntain s a d e ta ile d analysis o f th e e le ctro n n u m b e r-d e n sity flu c tu a tio n s in th e c o n fin e m e n t regions o f a q u a n tu m -w e ll s e m ic o n d u c to r laser, fro m w h ic h the pow er s p e c tru m o f induced m u ltip lic a tiv e pha.se flu c tu a tio n s is derived. E le c tro n n u m b e r-d e n s ity flu c tu a tio n s represent an in trin s ic source o f m u ltip lic a tiv e phase noise in se m ic o n d u c to r lasers, and are show n to have a I / /- lik e pow er sp e ctra l d ensity. In C h a p te r 5. these results are used w ith the laser m odel o f C h a p te r 3 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to e s tim a te th e lin e w id th fo r a ty p ic a l se m ico n d u cto r q u a n tu m -w e ll laser diode. T h e fin a l C h a p te r sum m arize s a ll o f th e results and offers p o s s ib ilitie s fo r fu rth e r e x p e rim e n ta l and th e o re tic a l w o rk. T h e p a rtia l-w a v e th e o ry d e n ve d in C hapters 2 and 3 is general and m a \' be used w ith o th e r sources o f m u ltip lic a tiv e noise or w ith o th e r types o f lasers. .Also, th e th e o ry is n o t lim ite d o n ly to lasers: th e general concepts m ay be a p p lic a b le to o th e r types o f o s c illa tin g feedback system s such as e le ctro n ic o s c illa to rs and fre q u e n cy standa rd s, b io lo g ica l o s c illa to rs , and m echanical oscilla to rs. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Partial-W ave A nalysis of a Resonant O ptical Am plifier w ith M ultiplicative N oise In th is C h a p te r, the s ta n d a rd lin e a r analysis o f a resonant o p tic a l a m p lifie r m o d eled as a d e la y-typ e feedback syste m is generalized to tre a t tim e -v a ry in g gain and phase p e rtu rb a tio n s . T h e g a in and phase coefficients m u ltip ly th e co m p le x a m p litu d e o f th e field as it passes th ro u g h th e a m p lify in g m e d iu m , so th a t gain and phasf fluctuations c o n s titu te sources o f multiplicative noise. F irs t, to fra m e the discussion and define te rm in o lo g y , th e basic analysis o f a resonant a m p lifie r as a lin e a r d e la y -ty p e feedf^ack syste m without m u ltip lic a tiv e noise [14] w ill be review ed. N e x t, a d isc re te -tim e fo rm a lis m is developed including m u ltip lic a tiv e phase and gain flu c tu a tio n s o f th e a m p lify in g m e d iu m between th e c a v ity m irro rs . T h e re su lt is a m odel fo r th e e v o lu tio n o f th e o u tp u t field o f a resonant a m p lifie r th a t d isp la ys p o w e r-in d e p e n d e n t-lin e w id th and rebroade ning w ith in c re a s in g g a in , as a d ire c t consequence o f m u ltip lic a tiv e noise. 2.1 B ackground and O verview In th e sta n d a rd analysis o f a resonant c a v ity , th e tra n s m itte d o u tp u t fie ld is com p u te d fro m an in fin ite sum o f p a rtia lly -re fle c te d fields inside th e c a v ity , o r " p a rtia l w aves." as illu s tra te d in F ig u re 1.4. For a p e rfe c tly stable c a v ity w ith n e t ro u n d -trip 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. g a in less th an u n ity , an a n a ly tic fo rm fo r th is s u m m a tio n m a y be o b ta in e d . C a v ity resonances o ccu r at d is c re te frequencies fo r w h ich a ll o f th e p a rtia l waves add con s tru c tiv e ly . and co rre sp o n d to inverse m u ltip le s o f th e ro u n d -trip tim e for lig h t in th e c a v ity . T h e n u m b e r o f p a rtia l waves in th e c a v ity at an y in s ta n t is re la te d to the net ro u n d -trip g ain, and d e te rm in e s th e storage tim e and lin e w id th o f the cavity. In response to a s u ffic ie n tly slow v a ria tio n o f the c a v ity le n g th o r re fra c tiv e index, th e resonance frequencies w ill change lin e a rly acco rd in g to th e phenom enological m o d e l discussed in th e p re vio u s c h a p te r, e q u a tio n (1 .2 ). T h is m o d e l is valid if th e change in th e c a v ity le n g th o ccurs on a tim e scale th a t is long com pared to th e storage tim e o f th e c a v ity . In th is case, we m a y e n visio n th a t th e re fra ctive in d e x varies so s lo w ly th a t a ll o f th e p a rtia l waves in th e ca vity' a t any in sta n t ha\e experienced th e sam e valu e o f th e re fra c tiv e in d e x, so th e c a v ity resonance frequency s m o o th ly follow s th e re fra c tiv e in d e x flu c tu a tio n s . H ow ever, in th e presence of a ra p id ly flu c tu a tin g re fra c tiv e in d e x on tim e scales s h o rte r th a n th e c a v ity storage tim e , th e p a rtia l waves in th e re so n a to r a t any in s ta n t w ill have ra n d o m ly -v a ry in g phase and a m p litu d e , so th e usual a n a ly tic s u m m a tio n is not a p p lic a b le . Such rapid flu c tu a tio n s o f g a in and phase in th e re so n a to r c o n s titu te m u ltip lic a tiv e noise. In the present a nalysis, th e effect o f m u ltip lic a tiv e noise on th e resonant a m p lifie r is inve stig ate d by c o m p u tin g th e o u tp u t fie ld at d iscrete in s ta n ts o f tim e as a random phasor s u m m a tio n o f a ll th e p a rtia l waves co n ta in e d in th e c a v ity . In th e presence o f m u ltip lic a tiv e noise, at a n y in s ta n t o f tim e , o ld e r p a rtia l waves in the ca vity have experienced a d iffe re n t c u m u la tiv e gain and phase th a n th e m o re recent p a rtia l waves. T he re s u ltin g spread o f phase and a m p litu d e o f th e p a rtia l waves determ ines th e o u tp u t field flu c tu a tio n s . T h e case o f an ideal noiseless in p u t fie ld is exam ined firs t to a pp re cia te th e effe ct o f th e m u ltip lic a tiv e noise on th e tra n s m itte d field. T h e n th e m ore re a lis tic case o f an in p u t fie ld w ith phase and a m p litu d e noise is tre a te d . T w o regim es are e x a m in e d : a s m a ll-a n g le lin e a r regim e, and a large-angle regim e. In th e sm all-a ng le re g im e , th e flu c tu a tin g c o m p le x phase o f a ll th e p a rtia l waves is less th a n ~ /2 radians a b o u t th e m ean value. In th is regim e, expressions fo r the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o u tp u t fie ld a m p litu d e and phase are a n a ly tic a lly d e riv e d in te rm s o f th e pow er spectra o f th e m u ltip lic a tiv e gain flu c tu a tio n s and a d d itiv e in p u t flu c tu a tio n s , and the net ro u n d -trip g ain. It is seen th a t the resonant a m p lifie r acts like a lin e a r fin ite -im p u ls e -re s p o n s e filte r in tra n s fo rm in g th e m u ltip lic a tiv e flu c tu a tio n s and th e in p u t fie ld flu c tu a tio n s to th e o u tp u t field. In th e sm a ll-a n g le regim e, the im p u lse responses o f th e resonant a m p lifie r for b o th m u ltip lic a tiv e noise and in p u t fie ld noise are d e riv e d a n a ly tic a lly and found to decay e x p o n e n tia lly , le a d in g to L o re n tzia n term s in th e o u tp u t fie ld pow er spectrum . . \ t h ig h net g ain, th e m u ltip lic a tiv e noise d o m in a te s th e o u tp u t field flu c tu a tio n s and lim its th e o u tp u t pow er spectrum to a m in im u m w id th . It is seen th a t th e s m a ll-a n g le reg im e corresponds to the p h e n o m e n o lo g ica l m odel o f e quatio n ( 1.2 ). in w h ic h th e resonant c a v ity acts as a perfect in te g ra to r th a t tra n sfo rm s o p tic a l phase flu c tu a tio n s in th e c a v ity to frequency flu c tu a tio n s o f th e c a v ity resonance frecjuency. For th e case o f la rg e -a m p litu d e m u ltip lic a tiv e noise, o r s u ffic ie n tly high net gain, the lim its o f th e sm a ll-a n g le a p p ro x im a tio n are exceeded, and th e a n a ly tic s o lu tio n no lo n g e r o b ta in s . To in ve stig a te this regim e, n u m e ric a l s im u la tio n s are pe rfo rm e d using s im u la te d noise tim e series on a co m p u te r. It is fo u n d th a t th e pow er tra n s m ission o f th e c a v ity w ith m u ltip lic a tiv e phase flu c tu a tio n s grow s progressively m ore e rra tic w ith increasing net gain. The o ld e r p a rtia l waves a ccu m u la te enough phase to in te rfe re d e s tru c tiv e ly , causing discontinuous ju m p s o f th e o u tp u t phase, and fast, sp iky o u tp u t a m p litu d e flu c tu a tio n s. .A,lso. th e p ro b a b ility d is trib u tio n o f th e o u t p u t in te n s ity becom es m ore lik e th a t o f a th e rm a l source, and th e power s p e ctru m becom es broadened. In th is regim e, the im p u lse response is no lo nger e x p o n e n tia lly decaying , b u t becom es e rra tic and enhanced at long tim e s , re s u ltin g in enhancem ent o f th e lo w -fre q u e n c y p o rtio n o f the o u tp u t field pow er s p e c tru m . 2.2 R ev iew o f Standard Linear Feedback T heory To b e g in th e d e ve lo p m e n t, th e standard analysis o f a s im p le d e la y -ty p e feedback system w ith lin e a r, tim e -in v a ria n t com ponents w ill be review ed, and then a p p lie d to I S Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a resonant o p tic a l a m p lifie r s im ila r to th e approach fo u n d in Siegm an [14]. C o n sid e r the feedback system m odel o f an o p tic a l c a v ity illu s tra te d in F ig u re 2.1. T h e in p u t, o u tp u t, and in te rn a l states are e le c tric fie ld s w ith sinusoida l tim e dependence. T he in p u t fie ld R in c id e n t fro m th e le ft o f F ig u re 2.1 is assumed to be a sin u so id o f frequency a,' and a m p litu d e A r . R(^') = A r e'"'‘. I 2 . 1 1 T h e o u tp u t fie ld C ( ^ ’ ) e x itin g to the rig h t also has th e same sinusoidal dependence. It is desired to co m p u te th e tra n sfe r c h a ra c te ris tic s o f th e feedback system fro m the in p u t sta te R(^') to the o u tp u t state C ( . T h e system has an in p u t coupler 1 on th e le ft o f F igure 2.1. and an o u tp u t co u p le r 2 on th e rig h t. These couplers have b ra n c h in g ra tio p i,2 fo r th e refle cte d a m p litu d e , and ~ i.j for the tra n s m itte d a m p litu d e . T h e couplers are assum ed to be lossless, so th a t * ,'2 + Plj — T he syste m has co m p le x forw ard gain coefficient G'(u,’ ) and reverse gain coefficient A c ru c ia l a ssu m p tio n is th a t th e net ro u n d - tr ip gain is less th a n u n ity , i.e.. < 1- T h e n , the o u tp u t fie ld can be co m p u te d as a ve cto r a d d itio n o f p a rtia l waves. T h ro u g h o u t th e re m a in d e r o f the analysis, th e e x p lic it sinusoidal tim e dependence o f w ill be d ropped to s im p lify th e n o ta tio n , and R. C w ill be used instead o f R(u:). C lu,’ ). etc. For a pure sinusoida l in p u t field R th a t has been a p p lie d for an in fin ite tim e , rlie ste a d y-sta te o u tp u t field C is th e s u m m a tio n o f an in fin ite num ber o f p a rtia l waves w hich m ay be w ritte n G = (~i'r2 )GR + {~!ii2 ){piP2 )G^ H R + (-n~.2 )[p\p2 )'G^H-R + { l \ l 2 ){p\p2 Ÿ G ^ R ------ = (■'n72)Tr/?[I -(- p \ p 2 G H [p\p2GHŸ [p\p2GHŸ (2.2) T h e te rm o u ts id e the square brackets represents th e transm ission o f the in p u t and o u tp u t couplers 71 and 7 2 . and th e fo rw a rd g a in G fo r the firs t one-w ay pass o f the fie ld th ro u g h th e ca vity. T h e term s in s id e th e brackets represent th e successive feedback te rm s w h ich experience a net ro u n d -trip g ain p\p 2 GH on each tr ip th ro u g h 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Input Field Forward gain with time-variation Output Field NIirror Trans. Loss Mirror Reflection Loss Reverse gain with time-variation Figure 2.1: Delay-type feedback system model for a resonant optical amplifier with time-varying gain as multiplicative noise source. 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. th e c a v ity . F or net g ain P1P2 G H < I. th e q u a n tity inside the square b ra cke ts m a y be a n a ly tic a lly su m m e d , so th a t th e o u tp u t m a y be w ritte n i - ' ( m , g h y Eequation (2.3) has th e fo rm o f th e . \ ir y fo rm u la o b ta in e d v ia th e s ta n d a rd tre a t m e n t o f a resonant o p tic a l a m p lifie r fo u n d in o p tics te x ts [14]. Id e n tify in g p\. p 2 and as th e m irro r a m p litu d e re fle c tio n and tra n sm issio n coefficients, re s p e c tiv e ly , and s e ttin g G = H = as th e g ain and phase experienced b y a fie ld fo r a one-w ay pass th ro u g h th e c a v ity o f le n g th / and in d e x o f re fra c tio n n. w ith r th e speed o f lig h t in vacu um , th e fa m ilia r tra n sm issio n ch a ra c te ris tic fo r a F a b ry -P e ro t resonant a m p lifie r is o b ta in e d : T h e p o w e r tra n s m is s io n fu n c tio n is th e m a g n itu d e -sq u a re d o f th is expression: Pout(’ ^') ^ ____________ P,n(^') \ + (pipoGI)- - ' 2 pip 2 G'lcos('2 u:nl/c) T h e resonant a m p lifie r pow er tra n sm issio n fu n c tio n is p lo tte d in F ig u re 2.2 fo r facet re fle c tiv itie s pi = p, = 0.9 and Go = 1. T h e system has tra n sm issio n peaks at resonance frequencies co rre sp o n d in g to in te g e r m u ltip le s m = 1. 2 . • o f th e inverse ro u n d -trip tim e th ro u g h th e c a v ity r = ' 2 nl/c: 2 nm = • ( 2 .6 ) r E q u a tio n (2.3) is th e general fo rm o f th e in p u t-to -o u tp u t tra n s fe r fu n c tio n for a d e la y -ty p e feedback system as a fu n c tio n o f frequency. T h is expression is v a lid fo r a syste m w ith no m u ltip lic a tiv e noise, a c tin g on sinusoidal in p u t fields. .\s th e feedback fra c tio n is increased, the w id th s o f th e tra n sm issio n peaks g ive n by- e q u a tio n (2 .5) decrease. In th e lim it o f net g a in approach ing u n ity , th e h a lf-p o w e r a m p lific a tio n b a n d w id th o f a resonance peak is given a p p ro x im a te ly by 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Transmitted o .ô Optical Power (relative to input) O .-l 15 20 5 10 Frequency (arbitrary linear scale) Figure 2.2: Power transmission function of resonant optical cavity versus frequency for pi = p2 = 0.9 and Go = 1. 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. . (2 .7 ) v W Æ So. as th e net ro u n d -trip g ain approaches u n ity , th e b a n d w id th o f th e resonant a m p lifie r approaches zero. 2.3 Linear Feedback S y stem T im e-D o m a in A n alysis w ith M u ltip lica tiv e N oise In th is section, a general fo rm a lis m w ill be developed fo r d e riv in g th e system o u tp u t fie ld C fo r th e resonant a m p lifie r o f F ig u re 2.1 w hen th e fo rw a rd and reverse gains are p e rtu rb e d by m u ltip lic a tiv e noise. T h e m u ltip lic a tiv e noise m a y be intrinsic o r extrinsic in o rig in , as discussed p re vio u sly. In th e general case th a t G and H have n o n -n e g lig ib le gain and phase flu c tu a tio n s , th e clo se d -fo rm s u m m a tio n le a d in g to e q u a tio n (2.3) can no t be a p p lie d . Instead, th e p a rtia l waves m u st be e x p lic itly su m m e d to o b ta in th e o u tp u t at each desired tim e in s ta n t t. and th e a m p litu d e and phase o f th e o u tp u t fie ld at th e o p tic a l fre q u e n cy u,'m- w ill flu c tu a te in tim e . T h is analysis w ill co m p u te th e s lo w ly -v a ry in g co m p le x envelope o f th e o u tp u t field at a c a v ity resonance o r C(u,'m-t). It is desired to analyze th e flu c tu a tio n s o f th e o u tp u t a m p litu d e and phase o f C(o,’rji. t ) on tim e scales th a t are lo n g com pared to th e p e rio d o f th e o p tic a l fre q u e n cy u.'m- .\s w ill be described, th is can be c o n v e n ie n tly done by m o v in g to a d iscre te - tim e p ic tu re o f the system , b re a kin g up tim e in to d iscre te in cre m e n ts equal to th e tim e de la y for one ro u n d -trip th ro u g h th e c a v ity , r . N ear the c o m p le tio n o f th e present w o rk , a s im ila r approach was discovered in R eference [17] in w h ic h Z -d o m a in techniq u es fro m d ig ita l system s th e o ry w ere used to o b ta in the tra n s m is s io n and re fle c tio n ch a ra c te ris tic s o f F a b ry -P e ro t étalons versus th e o p tic a l in p u t frequency. In th a t w o rk , the Z -d o m a in analysis was used to s im p lify th e d iffic u lt p ro b le m o f c o m p u tin g the tra n sm issio n and re fle c tio n ch a ra c te ris tic s o f c o m p lic a te d m u lti-la y e r s tru c tu re s . In th e preaent w o rk, a s im ila r d is c re te -tim e a p p iu a ch is used on a tw o - m irro r F a b ry -P e ro t c a v ity , b u t e xte n d e d to th e case o f tim e -v a ry in g g a in coefficients 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to c o m p u te th e noise p ro p e rtie s o f th e o u tp u t field. N o w . th e resonant a m p lifie r s tru c tu re o f F ig u re 2.1 w ill be a n a lyze d near a single resonance freq ue ncy w h e n th e g ain coefficients G and H are p e rtu rb e d by m u ltip lic a tiv e gain and phase noise. To keep th e a n a lysis s im p le and th e results tra n s p a re n t, in itia lly it w ill be assum ed th a t th e in p u t s ta te /? is a p u re sinusoid w ith co n sta n t a m p litu d e A r . as defin e d in e q u a tio n (2 .1 ). T h e in p u t fre q u e n cy is equal to a resonance fre q u e n cy d e fin e d by th e c a v ity le n g th a n d n o m in a l re fra c tiv e in d e x . L a te r, th e m ore re a lis tic case o f an in p u t fie ld w ith a d d itiv e noise w ill be tre a te d . T h e in p u t and o u tp u t co u p le rs 1 and 2 are assum ed to be id e n tic a l, so th a t P\ = p 2 = P and 7 i = 7 > = 7 . It is assum ed th a t th e c a v ity le n g th / is tim e - in v a ria n t. as is th e n o m in a l re fra c tiv e in d e x ri. .All phase o r g a in flu c tu a tio n s are described by co m p le x zero-m ean ra n d o m variables Sg{t) a n d Sh{t). T h e tim e -v a ry in g fo rw a rd and reverse gain c o e fficie n ts due to m u ltip lic a tiv e noise are th e n m odeled as = H, ^ H(t). (2.9) w here Go- Hj represent th e s ta tic fo rw a rd and reverse loss o r g a in . is th e s ta tic one-w ay phase s h ift a dded to a signal at fre q u e n cy u-Vi passing th ro u g h th e c a v ity . .At a loop resonance u-'m. th e s ta tic phase s h ift w ill s a tis fy th e c o n d itio n / c = 2 ~m radians, w ith m an in te g e r. Sg(t) and Sh(t) are c o m p le x ra n d o m variables re prese ntin g th e tim e -v a ry in g phase and gain a t fre q u e n c y These m ay be w ritte n Sg(t) = Sg'{t) -t- iSg"(t) (2 . 1 0 ) Sh{t) = Sh'{t) + iSh"{t) (2 . 1 1 ) w here th e real pa rts Sg'{t) and Sh'(t) represent phase flu c tu a tio n s , and th e im a g in a ry pa rts Sg"{t) and 5h"(t) represent g a in flu c tu a tio n s . T h e s ta tis tic a l d e s c rip tio n and 2-1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. any c o rre la tio n o f th e real and im a g in a ry p a rts w ill depend in d e ta il on th e physics o f th e g a in m e d iu m . T h e tra n s fo rm a tio n fro m a c o n tin u o u s -tim e re p re s e n ta tio n to th e d is c re te -tim e re p re s e n ta tio n is acco m plishe d by co n sid e rin g th e s ta tic phase s h ifts as delay elem ents o f d u ra tio n r / 2 . We th e n represent th e c o n tin u o u s -tim e ga in va ria tio n s o f G{t) a n d H{t) as a d is c re te -tim e ra n d o m sequence o f c o m p le x values Gr and Ht separated b y tim e -in te rv a ls r , and w ith average gains Go and Ho'- G t = G o e “ ’ ^ ' = G o G f ( - - I ' d ) H i = H o = H o H t - ( 2 . I d ) (T h ro u g h o u t th e rest o f th e analysis, d is c re te -tim e variables w ill be d e n o te d w ith a su b s c rip te d tim e in d e x , e.g.. ,V(. to d is tin g u is h th e m fro m c o n tin u o u s -tim e q u a n ti ties: X(t)). In th is d is c re te -tim e p ic tu re . C (a ,y ,.0 is assum ed to be a sinusoid at loop resonance fre q u e n cy a,v„. w ith s lo w ly -v a ry in g a m p litu d e and phase flu c tu a tio n s , w h ich w ill be referred to as C f .\o w . we a p p ly th is d is c re te -tim e d e fin itio n o f th e gain to th e lin e a r analysis o f the p re vio u s se ctio n to o b ta in an expression fo r th e tim e -v a ry in g o u tp u t held f , . .\t tim e in c re m e n t t. th e fo rw a rd ga in coefficient is Gt. th e reverse g a in is Hi. and the in p u t s ta te is /?(. For n o ta tio n a l c la rity , it is u n d e rs to o d th ro u g h o u t th a t tim e index t — j refers to tim e t — j r . since tim e has been q u a n tiz e d in u n its o f th e ro u n d -trip tim e in th e c a v ity r . T h e o u tp u t s ta te is th e n w ritte n G = ~,^GiRi + -,^p-GiHi.iG i.iRi., +~;~p^Gt Ht-iGt-i Ht-zGt-i Rt - 2 + ••• = 'i^Gt[Rt p~Ht^iGt-i R (-i -\-p^Ht-iGt.-i Ht..2 G t- 2 Rt - 2 + ■ • •] (2 . 1-1 ) It is assum ed th a t th e average fo rw a rd and reverse gains and s ta tic ro u n d -trip phase are id e n tic a l, so th a t Go = Ho. T h is a s s u m p tio n is e q u iva le n t to re q u irin g the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. average loss (o r g a in ) a n d tim e -o f-flig h t to be equal for b o th d ire c tio n s o f p ro p a g a tio n in th e o p tic a l c a v ity . T h is is a good a ssum ption for a F a b ry -P e ro t c a v ity , since th e lig h t passes th ro u g h th e sam e physical m e d iu m in b o th d ire c tio n s , b u t m ay not be va lid for o th e r c a v ity g eom etries. It is fu rth e r assumed th a t th e gain flu c tu a tio n rate is slow com pared to th e tim e -o f-flig h t r fo r a ro u n d -trip in th e c a v ity . T h e n , it is va lid to assum e th e fo rw a rd and reverse gain flu c tu a tio n s are equal over any round- tr ip tim e in te rv a l r . so th e ro u n d -trip gain m ay be w r itte n GtH^ = H^. D ehning the tim e -v a ry in g net ro u n d -trip gain to be p^Hf. th e o u tp u t fie ld at tim e t is given by th e in fin ite series: G = V -G [G -h p-Hl,Rt-, -H + ■•■]• (2.15) To analyze th e c h a ra c te ris tic s o f th e gain coefficient flu c tu a tio n s on the o u tp u t held, hrst the case o f an ideal noiseless in p u t held w ill be e xa m in e d . T h e in p u t state is taken to be a co n sta n t /?< = /? for a ll tim es t. so it m ay be factored out o f equatio n (2.15). T h is is th e m a in tric k to the analysis, a n d enables us to express the o u tp u t held G as a co m pact in h n ite sum o f p a rtia l waves in the c a v ity at any given tim e in s ta n t t: G = '-fRR[[ + p-Hl, + ( / G t , G t j + ( / G t i G t . G t , ) + •■■]. (2.16) T h e a m p litu d e s o f th e successive te rm s in this series are d e te rm in e d by increasing powers o f th e s ta tic n e t ro u n d -trip gain coefficient p~H^ = A5,. (i.e .. the j — 1st te rm is p ro p o rtio n a l to A’j ) . In p rin c ip le , th is series contains an in h n ite n u m b e r o f te rm s representin g an in h n ite n u m b e r o f p a rtia l waves stored in th e ca vity. To m ake a n a ly tic a l progress, we shall a p p ro x im a te th e series o f e q u a tio n (2.16) by a h n ite n u m b e r o f te rm s. T he physical ju s tih c a tio n fo r th is a p p ro x im a tio n is th a t th e a m p litu d e o f successive term s (i.e .. p a rtia l waves) o f e q u a tio n (2.16) decreases p ro p o rtio n a l to A 'j. T h e reader is re m in d e d th a t we have a lre a d y assum ed th a t th e net ro u n d -trip gain A'^ < 1 . T herefore, fo r som e j g re a te r th a n som e large value M . th e oldest p a rtia l waves 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w ill c o n trib u te n e g lig ib ly to th e o u tp u t fie ld C. W e s h a ll th e re fo re assum e th a t the o u tp u t fie ld o f th e c a v ity at a n y tim e t can be a p p ro x im a te d as a ra n d o m phasor sum o f a fin ite n u m b e r o f \ I p a rtia l waves. W e w ill discuss how M is d e te rm in e d below. S u b s titu tin g th e fu ll fo rm o f th e d iscre te g ain co e fficie n ts g ive n by equat ion ( - . l- l) fo r the Ht te rm s in e q u a tio n (2.16) yields: Ct = ~ -Ho I -) [Y j Each successive te rm o f th is expression represents th e c o m p le x a m p litu d e o f an in d iv id u a l p a rtia l w ave th a t has tra ve le d an increcising n u m b e r o f tim e s th ro u g h the ca vity, and has th u s e xp e rie n ce d th e ra n d o m gain and phase flu c tu a tio n s Sht over an increasing tim e span. For th e case o f no g ain o r phase flu c tu a tio n s , (i.e .. Sht = 0 for a ll t.) th is expression m a y be su m m e d a n a ly tic a lly , an d collapses to th e w e ll-kn o w n .Airy fo rm u la fo r th e F a b ry -P e ro t tra n sm issio n d e rive d p re v io u s ly as ec|uation (2.-5). B u t in th e prese nt fo rm , each p a rtia l wave has a d iffe re n t ra n d o m gain and phase, due to th e m u ltip lic a tiv e c o m p le x flu c tu a tio n s Sht. We note th a t th e successive te rm s o f e q u a tio n (2.16) c o n ta in p ro d u c ts o f th e net ro u n d -trip g a in A j . In th e lim it o f no m u ltip lic a tiv e noise (i.e .. Ht = H^. A', = A', for all t). th e o u tp u t fie ld m a y be w ritte n : \r c, = c = rffoRJ2 ^'i- (2. IS ) J=0 To o b ta in a m o re c o m p a c t re p re se n ta tio n o f (2.16) we m ake use o f th e fact th a t the fin ite s u m m a tio n o f A 'j fro m j = 0 to j = M is given b y j =0 ^ Now, the series m a y be s u m m e d a n a ly tic a lly to o b ta in : 1 _ C = 7 ' / / o A - -----------------------------------------------------------(2 .2 0 ) I - ho Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As M becomes in fin ite . — > 1. and we o b ta in th e previous re su lt e q u iva le n t to e q u a tio n (2.3): c Æ . 1 - ho P h ysica lly. M is th e n u m b e r o f p a rtia l waves in th e c a v ity at any in s ta n t o f tim e , and M increases w ith o u t b o u n d as th e net gain approaches u n ity . T h e reader is again re m in d e d th a t th e o u tp u t fie ld C't is represented as a d is c re te -tim e sequence, each elem ent o f w h ic h is c o m p ris e d o f a s u m m a tio n o f a large n u m b e r ( .V/) o f p a rtia l waves. U p to now . th e in p u t sig n a l R is assum ed to be a co n sta n t. Since, in th is m o d e l, we re q u ire th e net g a in to be less th a n u n ity , th e largest p a rtia l wave is th e m ost recent. T h is is e q u a l to th e in p u t signal R tim e s th e firs t h a lf-ro u n d -trip g ain H . tim es th e in p u t and o u tp u t c o u p lin g losses y i ' j : a ll o ld e r p a rtia l waves have p rogressively s m a lle r a m p litu d e s , p ro p o rtio n a l to In p rin c ip le , th e re are an in fin ite n u m b e r o f p a rtia l waves. T h e question o f d e te rm in in g M then reduct's to d e c id in g how m a n y p a rtia l waves m u st be su m m e d to o b ta in convergence o f th e o u tp u t to w ith in a specified va lu e o f th e re su lt g ive n by th e a n a ly tic s u m m a tio n o f e q u a tio n (2 .2 1 ). T h is is e q u iva le n t to c o u n tin g how m a n y ro u n d -trip s it takes fo r th e o u tp u t to b u ild up to C fro m th e tim e th e in p u t R is firs t a p p lie d to th e a m p lifie r o p e ra tin g w ith net gain h'o- W e sh a ll d e te rm in e th e n u m b e r M + I o f te rm s re ta in e d in the s u m m a tio n o f e q u a tio n (2.17) b y re q u irin g th e d iffe re n ce betw een th e in fin ite sum o f e q u a tio n (2.16) and th e .1/ 4- 1 -te rm sum o f e q u a tio n (2.17) to be less th a n a sm all fra c tio n e o f th e va lu e o f th e fu ll in fin ite sum . U sing e q u a tio n s (2.20) and (2 .2 1 ). we m a y w rite th is c o n d itio n as: ^ ^ ^ < ( f - r - ( 2.2 2) 1 — ho / \ 1 — ho J \ 1 — ho T h e n , so lvin g fo r .V/ in te rm s o f h'o and e yie ld s th e c o n d itio n on \ f : M > f:n{e) (:n{h'o) (2.23) 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For e x a m p le , i f = 0.9 and e = 0 .0 1 . then fro m e q u a tio n (2.23) we re q u ire .1/ > 44 p a rtia l waves fo r th e sum o f e q u a tio n (2.17) to be w ith in e o f th e value o f th e fu ll in fin ite su m o f e q u a tio n (2.16). It was found b y c o m p u te r s im u la tio n s th a t a value o f e = 0 .0 1 . o r a d iffe re n ce betw een th e fin ite sum and th e in fin ite sum o f less th a n 1 p e rce n t, p ro du ce s re sults th a t are in d is tin g u is h a b le in te rm s o f noise ch a ra c te ris tic s fro m re s u lts c a lc u la te d using s m a lle r values o f e. T h e ph ysica l reason fo r th is is th a t th e . \/ th p a r tia l w ave a m p litu d e is o n ly = (O.O)"*"* = 0.0097 tim e s th e a m p litu d e o f th e firs t p a r tia l w ave, and a ll subsequent waves are s m a lle r s till. T h e re fo re , p a rtia l waves o ld e r th a n th e .\/- th wave can be ignored w ith o u t a ffe ctin g th e o u tp u t by m ore th a n one p e rc e n t. It is noted th a t th e exact value o f e chosen does not have a very stro n g e ffect on M. since .1/ o n ly depends lo g a rith m ic a lly on e. In th e re m a in d e r o f th is d is s e rta tio n , e = 0 .0 1 w ill be used in a ll ca lc u la tio n s . F ro m e q u a tio n (2.17) it is seen th a t the phases o f th e successive p a rtia l waves are c u m u la tiv e sum s o f th e m u ltip lic a tiv e noise process Sht over in cre a sin g tim es. If Sht is a w h ite G aussian co m p le x noise process w ith zero m ean and variance rrl. then th e c o m p le x phases o f th e successive te rm s represent a ra n d o m w a lk process. T he va ria n ce o f a ra n d o m -w a lk process grows lin e a rly w ith tim e , so th e com plex phase o f th e 4 / t h p a rtia l wave. 2 (J’/q _ [ + Sht - 2 + • • • + Sht-\[). w ill have variance aif = 2 M a i [2 0 ]. For low levels o f net gain a n d /o r m u ltip lic a tiv e noise, the o u tp u t fie ld flu c tu a tio n s are d e te rm in e d by th e filte re d in p u t field flu c tu a tio n s , as given by th e analysis w ith o u t m u ltip lic a tiv e noise le a d in g to e q u a tio n (2 .7 ). W e denote th is as th e clas sical re g im e . W h e n m u ltip lic a tiv e noise is present, tw o regim es m a y be id e n tifie d w ith respect to th e phase-spreading o f the p a rtia l waves in th e o u tp u t fie ld su m m a tio n o f e q u a tio n (2.17 ). w h ich sh a ll be denoted th e sm a ll-a n g le and large-angle regim es. T h e s m a ll-a n g le regim e corresponds to re la tiv e ly weak m u ltip lic a tiv e noise a n d /o r lo w n e t ro u n d -trip gain, such th a t th e ro o t-m e a n -sq u a re phase o f th e last (i.e .. ( M + 1 ) s t) p a rtia l wave is less th a n ~/2. In th is case, th e e x p o n e n tia ls in e q u a tio n (2 .1 7 ) m a y be lin e a rize d by th e sm a ll-a n g le a p p ro x im a tio n , and th e su m m a tio n fo r Ct m a y be o b ta in e d a n a ly tic a lly , as w ill be show n in th e n e x t section. 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In th is re g im e , th e lin e w id th in itia lly decreases w ith in cre a sin g net gain, w hen the in p u t fie ld flu c tu a tio n s are d o m in a n t. E v e n tu a lly , th e m u ltip lic a tiv e noise begins to d o m in a te th e o u tp u t flu c tu a tio n s , and th e lin e w id th sa tu ra te s at a constant value. T h e la rg e-an g le re g im e corresponds to th e case w hen th e th e rm s com plex phase variance o f th e o ld e r p a rtia l waves exceeds t / 2 . In th is case, th e sm all-angle ap p ro x im a tio n does n o t a p p ly, and th e fu ll phasor expression o f e q u a tio n (2.17) m ust be used to c o m p u te th e o u tp u t field Ct at each tim e in c re m e n t t. .\s w ill be show n, for a large m u ltip lic a tiv e noise variance a n d /o r h ig h net ro u n d -trip gain A/, 1. the ra n d o m -w a lk in g phase flu c tu a tio n s o f th e la te r p a rtia l waves m ay becom e large enough to cause d e s tru c tiv e interference w ith e a rlie r p a rtia l waves, causing large flu c tu a tio n s o f th e o u tp u t field a m p litu d e and phase. .Also, th e ra n d o m -w a lk in g gain flu c tu a tio n s o f th e la te r p a rtia l waves w ill e v e n tu a fly cause sig n ifica n t d e v ia tio n s fro m th e e x p o n e n tia l decay o f th e p a rtia l waves g ive n b y th e coefficients /v /. T he tra n s itio n fro m th e sm all-angle to large-angle regim es is m arked by th e onset o f ra p id , large flu c tu a tio n s o f the o u tp u t fie ld a m p litu d e and phase. These re su lts are c o n tra ry to the expected b e h a v io r o f decreasing lin e w id th w ith in crea sin g net g a in fro m e qu atio n (2.7) fo r th e s ta n d a rd analysis w ith o u t m u ltip lic a tiv e noise. F rom th e fo llo w in g analysis, it is seen th a t th e s m a ll- and large-angle regim es are n a tu ra l consequences o f th e presence o f m u ltip lic a tiv e noise in the op tic a l re son ator. T h e classical result o f decreasing lin e w id th versus increasing net gain is seen to be a lim itin g case o f th e generalized p a rtia l wave m odel in c lu d in g m u ltip lic a tiv e noise. T h e tim e in te rv a l M r is p ro p o rtio n a l to th e m e m o ry o f th e feedback system . In th is m o d e l, th e net g ain is always less th a n u n ity , so th e n u m b e r o f p a rtia l waves and th e m e m o ry o f th e system is alw ays fin ite . H ow ever, th is poses an in te re s tin g q uestio n re g a rd in g th e m e m o ry o f a resonant a m p lify in g system operated above the th re s h o ld o f self-sustained o s c illa tio n : does an o p tic a l resonator o p erated in the se lf-su sta in e d o s c illa tin g regim e, (i.e .. a laser.) have an in fin ite m e m o ry fo r th e m u ltip lic a tiv e noise? T h e p o s s ib ility o f u n ity gain is set aside fo r th e m o m e n t, since its tre a tm e n t fa lls o u ts id e the scope o f th e present lin e a r m o d e l. The im p lic a tio n s ;io Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o f u n ity ga in w ill be re v is ite d in th e C onclusion . 2.3.1 S m a ll-a n g le regim e: linear a p p r o x im a tio n for id eal n o iseless in p u t field In th is se ctio n , th e case o f weak m u ltip lic a tiv e noise in th e F a b ry -P e ro t resonant a m p lifie r w ill be tre a te d using a lin e a r a p p ro x im a tio n . T h e lin e a r a p p ro x im a tio n is va lid w hen a ll o f th e p a rtia l waves in the c a v ity have a c c u m u la te d less th a n Ü.1 radians. T h e la st te rm o f e q u a tio n (2.17) represents th e "o ld e s t" p a rtia l w ave th a t has tra ve le d M + I ro u n d -trip s in th e ca vity. W h e n a m u ltip lic a tiv e noise source w ith a w h ite p o w e r s p e c tra l d e n s ity p e rtu rb s th e c a v ity , th e c o m p le x phase o f th e .V/-th p a rtia l w ave at tim e t has e xecuted an .V/ - f 1-step ra n d o m w a lk re la tiv e to its s ta rtin g phase o f zero at tim e t — .V/. T h erefore, th e s ta n d a rd d e v ia tio n o f th e phase o f th e .V /-th p a rtia l wave is p ro p o rtio n a l to \/.V / 4- 1. and so a ll p a rtia l waxes I < M have s ta n d a rd d e v ia tio n o f phase p ro p o rtio n a l to x / T + l . T h e sm a ll-a n g le regim e is defined as th e case when th e standard d e v ia tio n o f th e c o m p le x phase for th e last (i.e. . l/ - t h ) p a rtia l wave is less th a n ~ / 2 ra d ia n s, o r y/2 XI(TH<'-^ (2 .2 1 ) w here M is d e fin e d by e q u a tio n (2.23) and as before, cr^ is th e s ta n d a rd d e v ia tio n o f th e phase fo r a on e -w a y pass th ro u g h th e ca vity. P h y s ic a lly , th is im p lie s th a t w hen th e a ccu m u la te d phase angle o f the oldest p a rtia l w ave exceeds ~/'l. th e y begin to d e s tru c tiv e ly in te rfe re w ith e a rlie r p a rtia l waves. We m a y e s tim a te an u p p e r lim it on net gain in th e s m a ll-a n g le re g im e by in s e rtin g th e value o f . \ / fro m e q u a tio n (2.23) in to eq u a tio n (2 .2 4 ). and th e n s o lv in g fo r I\.. to o b ta in : h'o < (2.25) T h is m eans th a t h'o m u s t be s m a lle r th a n th is value in o rd e r th a t th e o ld e st p a rtia l wave re ta in e d in th e fin ite sum does n o t exceed the lim its o f th e s m a ll-a n g le c rite rio n . For net gain la rg e r th a n th is value o f h'o. the resonant a m p lifie r is th e n o p e ra tin g 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in th e large-ang le regim e. VVe no te th a t th is lim it on h'o depends on th e d e fin itio n o f M in te rm s o f e. T h is is th e n u m b e r o f p a rtia l waves re q u ire d fo r th e c a lc u la te d o u tp u t fie ld C to be w ith in a fa c to r e o f its fin a l a n a ly tic a lly -d e te rm in e d va lu e (i.e .. i f no noise was pre sen t.) H ow ever, as discussed p re vio u sly, for t = 0.01. a ll o f th e p a rtia l waves g re a te r th a n M c o n trib u te less th a n I percent to th e o u tp u t fie ld . .A p p ly in g th e lin e a r a p p ro x im a tio n fo r th e e xp o n e n tia l (e-^ % I -(- x . a p p lic a b le w hen X < O .I). we now lin e a rize th e expression fo r th e flu c tu a tin g o u tp u t fie ld so th a t e q u a tio n (2.17) m ay be re w ritte n ; C t ~ ~i~ Ho( I -f ) ) ^ [1 P~ (1 "T 2 i(Sht_i)) -f ( I (Sh c_>)) + f)~ .\[ H~'^^ {]. + 2i{Sht-i 4- + - - - ) )]. (2.26) S u b s titu tin g h'o = p^Ho in to th is expression yie ld s: C t = ~~ ho{ I 4- ) ) /^ [l ho ( I 4- 2i(Sht-i)) 4- A j (1 4- 2 i{Sht-i 4- Sht - 2 )) 4- • • • 4- h '^ (1 + 2i{5ht-i + Sht - 2 4- • • • 4- ()7p_.v/))]- (2 .2 / ) E x p a n d in g th is expression w ill y ie ld m a n y te rm s o f 0{Sh-). like {Sli’ ^Sht-k). w h ich are sm a ll com pared to 0(Sh). and so m a y be discarded, leaving ^ ( = ~~ hIoR[{ 1 4- ho 4- A ‘ 4- • • • 4- A ) 4- (1 4- ho 4- h~ 4- • • • 4- h^^ )(iSht) {ho + A * 4- • • • 4- h o ‘ ){2iSht-i) 4- ( A ‘ 4- A ^ 4- • ■ • 4- h ^ ' )(2iSht-2) + ■■■ + h^^'{2 iSht-M)\. (2.28) T h e te rm p ro p o rtio n a l to Sht is due to th e phase flu c tu a tio n enco u n te re d in th e first " h a lf ro u n d -trip " th ro u g h th e c a v ity . .A ll o th e r te rm s represent " fu ll ro u n d -trip s " th ro u g h th e c a v ity , hence th e factors o f 2 appea r in te rm s Sht-i.Sht- 2 . ' ' ' etc. T h e s u m m a tio n s o f pow ers o f A", m a y be p e rfo rm e d a n a ly tic a lly u sin g e q u a tio n (2 .1 9 ). T h e coefficients in e q u a tio n (2.28) are th e n designated as w here th e c o e ffic ie n t o f th e firs t te rm is oq. d e fin e d as 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. {.M + l ) - l I _ r-.\[ + l a o= = I 12.29) j=o I — ho I — ho w here th e fin a l a p p ro x im a tio n is valid since h'^^ << I. Subsequent te rm s k- > I m a v be w ritte n t—I (Ik = «0 - X I j = 0 1 - I - A ! I — ho I — ho A ! - A ? ^ + ' (2..-5Ü) I — ho N ow . s u b s titu tin g th e coefficients ak in to e q u a tio n (2.28). fo r th e present special case o f a noiseless in p u t R. th e o u tp u t fie ld at an y tim e in s ta n t t is seen to be a w e ig h te d s u m m a tio n o f th e m u ltip lic a tiv e noise sam ples ht over th e tim e in te rv a l t — .\/ to ti C t — ' ~ Ho R[cio c iq iSht I'lSht— 1 4 - • • • + c i \fi‘ lSht- ' = '-H o R \r «o( 1 + làht) + X2 ((k i'2 Sht-k k= 1 12.31 S u b s titu tin g e q u a tio n (2.30) fo r the «*..• and fa c to rin g o u t the 1 / ( 1 — A '„) te rm . W( o b ta in : C t = 1 + i 6 ht + - h'o^' ^ i2 Sht-k t= l k=l ■M (2.32) r H o R r I — ho T h e last te rm in th is expression m ay be ignored, since h '^ ’ < < 1 . and th e variance o f th e s u m m a tio n o f Sht-k is fin ite . D ro p p in g th is te rm and e xp a n d in g th e su m m a tio n , we fin a lly o b ta in : c = [ l + iSht + iho 2 Sht.i + iK;2Sht.2 + • • • + ih'^^2 Sht.M] . (2.-33) T h is expression represents th e tra n s fo rm a tio n o f th e sm a ll-a n g le m u ltip lic a tiv e phase a nd g a in flu c tu a tio n s to th e o u tp u t fie ld a t every tim e in s ta n t t for a noiseless 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sinusoida l in p u t fie ld R. T h e firs t te rm is th e s ta tic g a in o f th e resonant a m p lifie r as o b ta in e d fro m th e sta n d a rd analysis leading to e q u a tio n (2.3). T he te rm in brackets represents th e effect o f th e m u ltip lic a tiv e noise source Sht on the o u tp u t fie ld . W hen Sht = 0 , (i.e .. no m u ltip lic a tiv e noise), th e re s u lt o f th e sta n d a rd analysis, e q u a tio n (2.3). is recovered. R ecall fro m e q u a tio n (2.11) th a t Sht is a co m p le x-va lu e d random series w hose real and im a g in a ry parts represent th e tim e -v a ry in g g ain and phase, re sp e ctive ly, o f th e a m p lify in g m e d iu m . P h y s ic a lly , th e real-valued co efficients u < . . d e te rm in e th e c o n trib u tio n o f th e m u ltip lic a tiv e phase and gain flu c tu a tio n at the previous tim e in c re m e n t t — k to th e o u tp u t at tim e in c re m e n t t. Since th e net gain \Ko\ < 1 . th e a t m a g n itu d e s decrease e x p o n e n tia lly w ith increasing k in e q u a tio n (2.33). T h u s , noise events fu rth e r rem oved in tim e fro m th e tim e in c re m e n t t have a d im in is h in g effect on th e o u tp u t state. In th is sm a ll-a n g le a p p ro x im a tio n , noise events o c c u rrin g at tim e s p rio r to t — ,\f have no effect on the o u tp u t, w h ich is ju s t an a n a ly tic s ta te m e n t o f the fin ite m e m o ry tim e o f th e system . W e w ill re tu rn la te r to th is expression to co m p u te the pow er s p e c tra l d e n s ity o f th e o u tp u t field flu c tu a tio n s . 2.3.2 S m a ll-a n g le ap p roxim ation w ith n o isy input field In th e a n a lysis thus fa r. flu c tu a tio n s o f the in p u t fie ld R were not considered. T lu ' preceding a n a lysis has e lu c id a te d the effect o f th e m u ltip lic a tiv e flu c tu a tio n s on the o u tp u t fie ld . In th is section, the m ore re a lis tic case o f a noisy in p u t fie ld incid ent on th e re sonant a m p lifie r w ith m u ltip lic a tiv e noise is tre a te d . In g en eral, any real in p u t field / ? = / ? , w ill have a tim e -v a ry in g a m p litu d e and phase. T o tre a t th is p o s s ib ility , th e in p u t state is now defined in the c o n tin u o u s -tim e p ic tu re as a sinu soid w ith m ean frequency c o in c id e n t w ith a c a v ity resonance b u t p e rtu rb e d by a m p litu d e noise SA{t) and phase noise So{t) R(t) = (Am 4 -& 4 (< )) (2.34) .Assum ing Sot < 0 .1 ra d ia n , th e sm all-angle a p p ro x im a tio n can again be em ployed, and the tim e -v a ry in g in p u t fie ld R{t) m ay be m a n ip u la te d in to th e d is c re te -tim e 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fo rm : f-h /fn ( Rt ~ (-4/Î + ( 1 + = (A.fi + i.AfiSot 4- <^.4( + i6AtSot)£ % .4fi + iSot + e'"""' = .4 fie ‘" " " '( l +Ô '/?,) Rc = Ro(].+SRt). (2 .35) w here th e te rm o f 0{6A)(So) was neglected, since it is sm a ll co m p a re d to th e te rm s o f 0((!).4) and 0(So). and th e noise sources SA and So are assum ed to be u n co rre la te d . N ow . th is e xp re ssio n fo r th e flu c tu a tin g in p u t held is in s e rte d in to th e fu ll fo rm o f ecjuation (2 .17) to o b ta in th e o u tp u t at every tim e in c re m e n t t due to b o th in p u t held h u c tu a tio n s a n d m u ltip lic a tiv e gain h u ctu a tio n s: n = A, Ht-iGt-i Ht-iGt-iRt-i + • • • = ',‘ Gt[Rt A p ~ H t -iG t- iR t- i Ap^Ht-\Gt-i Ht-zGt-zRt-i + • • •] = ~ ~ G t[R t A R t - i R t - i A h t - i Rt-2 R t - i A ■ ■ ■ ]■ I 2.3b I U sing m a n ip u la tio n s s im ila r to those used to d e rive e q u a tio n (2.33) (th e d e ta ile d algebra is re le g a te d to an .A p p e n d ix) a s im p lih e d expression fo r th e co m p le x o u tp u t held a m p litu d e is o b ta in e d : Gt — — [1 + iSht + Ixoi'lSht^i + h.^i'lSht-2 + • • • + i'2Sht-{_\[-i) l - H o R , 1 - A'o ' + ( 1 — Ro) (SRt A l\oSRt-i A [\^SRt-2 + RoSRt-^ + ■ • • + SRt-.\[)]. (2.3< ) T h e powers o f m u ltip ly in g th e SR te rm s m ay be dehned as a set o f coefhcients bk'- 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. bk = A 'f . (2 .;i8 ) T h e fu ll expression fo r th e o u tp u t fie ld can now be w ritte n m ore c o m p a c tly as s u m m a tio n s o f th e tw o noise processes Sht and SR t in term s o f th e co e fficie n ts and bk'- Ct = r H , R a o (I + c^7?()+ ^ bkSRt-i \J c = l / \ t = 0 C 2 .;W) T h is expression represents a lin e a r a p p ro x im a tio n to the fu ll fo rm o f e q u a tio n (2.17) fo r th e s lo w ly -v a ry in g c o m p le x envelop e o f th e o u tp u t held C't in c lu d in g b o th in p u t held h u c tu a tio n s and m u ltip lic a tiv e phase and gain h u c tu a tio n s . T h is lin e a r a p p ro x im a tio n is va lid in rh e s m a ll-a n g le regim e as dehned by e q u a tio n (2.24). 2.4 O u tp u t F ield P ow er S p ectru m for S m all-A n gle M od el In th is section, an a n a ly tic e xpression is o b ta in e d for th e pow er s p e c tru m o f the resonant a m p lih e r o u tp u t h e ld in th e sm a ll-a n g le regim e. It w ill be show n th a t the tim e -v a ry in g p a rtia l wave m o d e l in th e sm a ll-a n g le regim e yie ld s th e sam e results as th e phe no m e nological lin e w id th m o d e l discussed previously, e q u a tio n ( 1 .2 ). In the fo llo w in g c h a p te r, the clo s e d -fo rm expression fo r th e sm a ll-a n g le pow er sp e ctru m w ill be used to d erive a g e n e ra lize d expression fo r th e lin e w id th o f a laser in c lu d in g m u ltip lic a tiv e h u c tu a tio n s . E ([n a tio n (2.39) has th e fo rm o f a h n ite -im p u lse -re sp o n se h lte r o p e ra tin g on " in p u t” processes Sht and SRt. For lin e a r system s in general, th e F o u rie r tra n s fo rm o f th e tim e im pulse-response y ie ld s th e fre q u e n cy response o f th e system [IS ]. For an in p u t held noise source m o d e le d by a G a u s s ia n -d is trib u te d ra n d o m process, the pow er s p e c tru m o f th e o u tp u t h e ld is th e p ro d u c t o f th e in p u t noise po w e r sp e c tru m and th e m a g n itu d e -s q u a re d o f th e syste m frequency response [19]. So. i f th e im p u lse responses, net ro u n d -trip g a in h'o. and th e a n a ly tic a l fo rm o f th e pow er s p e c tra o f the 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m u ltip lic a tiv e flu c tu a tio n s Sh{f) and th e in p u t fie ld flu c tu a tio n s S n ( f) are know n, th e n th e o u tp u t pow er sp e ctru m Sc{f) m a y be o b ta in e d a n a ly tic a lly . T h e noise processes Sht and SRt are each described m a th e m a tic a lly as a series o f co m p le x-va lu e d ra n d o m num bers on th e u n it disc in th e co m p le x plane. T h e y are co m p le x-va lu e d sca lin g factors th a t m u ltip ly th e m a g n itu d e s o f th e s ta tic gain H„ and th e field a m p litu d e R^. respectively. T h e re fo re , th e ir pow er s p e ctra Sh(f) and Rnif) represent flu c tu a tio n s re la tiv e to u n ity gain and 2 ~ radians o f phase. T h e im pulse-responses o f the resonant a m p lifie r to a d e lta -fu n c tio n o f m u lti p lic a tiv e noise Sht o r in p u t noise SRt are defined to be SCk.t and SCr j . respectively. C on sid e rin g firs t th e effect o f m u ltip lic a tiv e noise o n ly , we set SRt = 0. T h e im pulse- response SCk.t is th e decaying o u tp u t field response given by e q u a tio n (2.39) due to a d e lta -fu n c tio n m u ltip lic a tiv e im pulse in Sht a p p lie d to th e system at tim e t = Ü: SC'h.t = 'i'HoR Hcit lit f-ln -H o R ' \ ( I — A o) 2 n - H , R lit w here f = |fn ( A'^ )| is th e decay co n sta n t, u, is th e u n it step fu n c tio n , w ith iit = 0 for t < 0 . and u, = I fo r f > 0 . and it is assum ed th a t / takes o n ly in te g e r values (i.e .. tim e is n o rm a liz e d to increm ents o f r). In d e riv in g th is expression, the first h a lf-ro u n d -trip {k = 0 te rm ) was taken to be equal to a fu ll ro u n d -trip th ro u g h the ca vity. For net g a in such th a t M > > 1. th is is a reasonable a p p ro x im a tio n , and s im p lifie s the analysis. For exam ple, w hen .\/ > 100 co rrespond ing to Ko > 0.9.5. th is in tro d u ce s a s m a ll e rro r o f less th a n I percent in th e o u tp u t sum . The response to an in p u t field im p u lse in SRt is o b ta in e d s im ila rly as th e response o f e q u a tio n (2.39) to a d e lta -fu n c tio n im p u lse in SRt a p p lie d to th e system at tim e t = 0 w hen th e m u ltip lic a tiv e noise source Sht = 0 : SCn.t = 't^HoR bt lit Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = r H o R Kl u, = 'i~HoR U(. (-.4 1 ) T h e F o u rie r tra n s fo rm s o f th e im p u ls e responses SC'h.t and S C r i y ie ld th e co r respon din g fre q u e n cy responses Fh.{f) and F r ( / ) . w here / is th e F o u rie r frequency offset fro m th e o p tic a l m ode frequency. B o th im p u lse responses represent decaying e xp o n e n tia ls , le a d in g to L o re n tzia n frequency responses. T h e frequency response o f th e c a v ity to th e m u ltip lic a tiv e noise process is and th e response to th e in p u t held flu c tu a tio n s is ttW w here o-v. = F / r is th e ro llo ff fre q u e n cy o f th e L o re n tzia n response, and th e F o u rie r frequency / is th e offset in Hz fro m th e o p tic a l frequency ujm- These expressions m ay also be cast in te rm s o f th e o u tp u t pow er, as w ill now be show n. W e m o de l th e to ta l o u tp u t po w e r sp e c tru m as being co m p rise d o f a D C te rm due to th e o p tic a l c a rrie r at the c a v ity resonance frequency plus tw o flu c tu a tin g te rm s due to th e m u ltip lic a tiv e noise and in p u t held h u c tu a tio n s . re sp e ctive ly. T h e DC te rm was d e riv e d p re vio u sly in e q u a tio n (2..5) and is given by T h e q u a n tity 5 c (0 ) is p ro p o rtio n a l to th e o p tic a l o u tp u t pow er at th e peak o f th e resonant a m p lih e r transm ission. T h e q u a n tity R~ is p ro p o rtio n a l to th e average o p tic a l in p u t pow er, so we m ay w rite ^ ^ = (l^ ) W e m a y now use th is fo rm to w rite equ a tio n s (2.42) and (2.4.3) in te rm s o f power. We m ake use o f an a p p ro x im a tio n w h e n h'o % 1 th a t F = \f:n(Ko)\ ~ I — /W . W e can th e re fo re w rite 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - r y For h ig h -re fle c tiv ity m irro rs such th a t /? % 1. th e n th e single-pass gain w ill also be close to u n ity , i.e.. % I. .Also, since ~ ‘ = 1 — p* = I — /?. th e co rn e r fre q u e n cy m a y be w ritte n ; ‘ (2 .4 7 ) o u t W e sh a ll see s h o rtly th a t th is fo rm leads to th e co rre ct lin e a r decrease o f th e c a v ity b a n d w id th w ith in cre a sin g o u tp u t pow er th a t we e xpect fro m th e s ta n d a rd a n a lysis o f a resonant c a v ity discussed previously. .M aking these s u b s titu tio n s yie ld s th e m u ltip lic a tiv e flu c tu a tio n response to be: and th e response to th e in p u t field flu c tu a tio n s to be FrU) ^ ^ (2.491 P.r W e observe th a t th e m a g n itu d e o f th e m u ltip lic a tiv e noise flu c tu a tio n s p e c tru m increases lin e a rly w ith th e o u tp u t pow er, whereas th e in p u t flu c tu a tio n s p e c tru m increases as th e s q u a re -ro o t o f pow er. It is assum ed th a t th e co m p le x-va lu e d noise processes Sht and SR- arise fro m in d e p e n d e n t p h ysica l processes, and are th e re fo re s ta tis tic a lly u n c o rre la te d . T h e pow er s p e c tra due to each noise process can th e n be added to o b ta in th e to ta l o u tp u t flu c tu a tio n pow er sp e ctru m . T h e te rm due to m u ltip lic a tiv e noise is th e p ro d u c t o f th e c o m p le x pow er sp e ctra l d e n s ity Sk(f) and th e m a g n itu d e -sq u a re d o f th e fre q u e n cy response Fk{f ). S im ila rly , th e c o n trib u tio n due to th e in p u t noise is th e p ro d u c t o f th e in p u t fie ld noise co m p le x pow er s p e c tra l d e n s ity Spt{f) and th e m a g n itu d e -s q u a re d o f th e frequency response Fpt{f). W e have p re v io u s ly defin e d th e noise processes Sht and SRt in th e sm a ll-a n g le analysis to be zero-m ean ra n d o m processes w ith respect to th e u n it circ le In th e co m p le x p la n e , so th e ir pow er s p e c tra l d e n sitie s are u n itle ss. ;39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T herefore, th e to ta l o u tp u t fie ld pow er sp e ctru m is c o m p ris e d o f th e D C o p tic a l pow er, plus te rm s due to b o th th e in p u t fie ld flu c tu a tio n s and th e m u ltip lic a tiv e g ain and phase flu c tu a tio n s : (2.50) S u b s titu tin g e q u a tio n s (2 .4 2 ) and (2.43) for th e fre q u e n cy responses Fhif) and Ffiif) yie lds th e to ta l o u tp u t fie ld pow er sp e ctru m : ^ c ( / ) = + 1 - A ' J i-H oR ' 1 + i - ( i - A ' j r / i + = 5 c (0 ) I + 5 h ( / ) + + Skif) W e m ay also w rite th is in te rm s p ro p o rtio n a l to the in p u t and o u tp u t pow er and th e o th e r a p p ro x im a tio n s m a d e above to o b ta in the e q u iv a le n t fo rm : M ) P o u t 1 + SRif) + ‘ + ( r S ) f e ? < .!/) 2.52) These are general expressions fo r th e o u tp u t field flu c tu a tio n pow er s p e c tru m o f a resonant a m p lifie r in c lu d in g in p u t fie ld noise and m u ltip lic a tiv e noise. These expressions represent th e c u lm in a tio n o f th e sm a ll-a n g le analysis. W e w ill now discuss the b e h a v io r o f these expressions in various lim its . .Also, we w ill a p p ly th e m in th e next c h a p te r to d e riv e th e lin e w id th o f a laser m od e le d as a resonant a m p lifie r th a t has its ow n spontaneous e m issio n noise as in p u t. C onsider now th e second te rm o f th is expression, w h ic h is due to th e in p u t field noise SR. For lo w ga in a n d /o r lo w -le v e l m u ltip lic a tiv e noise, th e sig n a l-to -n o ise ra tio fo r F ourier frequencies less th a n w ill be th e same as th e in p u t sig n a l-to -n o ise ra tio . P h ysica lly, th is im p lie s th a t th e o u tp u t field phase and a m p litu d e e x a c tly 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Follow th e in p u t fie ld phase and a m p litu d e flu c tu a tio n s fo r flu c tu a tio n rates less th a n th e c a v ity b a n d w id th w h ic h is in tu itiv e ly e xpected . For flu c tu a tio n rates fa ste r th a n th e in p u t fie ld flu c tu a tio n s are filte re d by th e c a v ity , and so are ro lled o ff as I / / ^ . .\s th e o u tp u t pow er is increased, th e c a v ity c o rn e r frequency o-y from e q u a tio n (2.47) decreases, as . ^. y % (( 1 — R ) j r ) y j P o u t - Since th e o u tp u t power s p e c tru m depends on th is im p lie s a lin e a r decrease in th e w id th o f th e power s p e c tru m w ith increasing o u tp u t pow er. T h e resonant a m p lifie r th e re fo re produces a n a rro w b a n d o u tp u t signal o f in cre a sin g s p e ctra l p u rity fro m a broadba nd noise in p u t as th e o u tp u t pow er is increased. T h is is th e expected b e h a v io r o f a resonant a m p lifie r as o b ta in e d fro m th e s ta n d a rd analysis leading to e q u a tio n (2.5). and is th e basis o f the lin e a r m odel o f th e laser as a noise-driven resonant a m p lifie r, to w h ic h we w ill re tu rn later. N ow consider th e last te rm o f e q u a tio n (2.51). T h is te rm due to m u ltip lic a tiv e noise is new . and is n ot o b ta in e d fro m th e s ta n d a rd analysis o f th e resonant a m p lifie r. In th e lim it o f h ig h net gain such th a t A’o — > I. the last te rm o f e q u a tio n (2.51) o ve rw h e lm s the te rm due to in p u t fie ld noise. .A fter this happens, th e signal-to-noise ra tio fo r F ou rie r frequencies less th a n a.y w ill decrease w ith fu rth e r increases in the net g a in , due to th e effect o f th e m u ltip lic a tiv e noise. In th e lim it o f high net gain, e q u a tio n (2.51) becomes ^ ’c ( / ) U y - n ~ L I + 0 5 ^ ' ( 1 - / C ) ^ 4 ( 2 7 r / T ) ^ ^ , ( / ) l ( 2 .o.f) / In th e lim it as h'o — > I. th e co rn e r fre q u e n cy u,y — > ■ 0. and th e last te rm due to m u ltip lic a tiv e noise overw helm s th e firs t te rm . T he o u tp u t pow er sp e ctru m then becom es S c ( / ) I a - . . „ = S c (0 ) ( l + (2.5-t) w h ic h is in d e pe nd ent o f the o u tp u t pow er. .As w ill be shown in th e n e xt ch a p te r, this effect produces a p o w e r-in d e p e n d e n t lin e w id th fo r a laser as a n a tu ra l consequence 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o f th e presence o f m u ltip lic a tiv e noise. T h e re la tiv e flu c tu a tio n s o f the o u tp u t held are s im p ly S c ( / ) k - „ ^ , = p — - ( 2 .-WI (-rf)- In th is case, the o u tp u t fie ld flu c tu a tio n pow er s p e c tru m approaches a constant value d e te rm in e d by th e m u ltip lic a tiv e noise a m p litu d e and th e ro u n d -trip c a v ity tim e . 2 .4 .1 L im its o f sm all-an gle an a ly sis T h e reader is re m in d e d th a t th is analysis o n ly holds fo r net ga in k'o such th a t the s m a ll-a n g le c o n d itio n as defined in e q u a tio n (2.25) is sa tisfie d . I f this c o n d itio n is exceeded, the n th e a ssu m p tio n s o f the sm a ll-a n g le an a lysis are no longer v a lid , and the large-angle reg im e , to be discussed la te r in th is c h a p te r, is entered. It w o u ld be useful to p re d ict th e g a in and pow er level at th e onset o f th e large angle regim e, but th is w o u ld require know ledg e o f th e m u ltip lic a tiv e noise variance. 1 /n fo rtu n a te ly . it is not possible to d e fine th is c o n d itio n in te rm s o f in p u t and o u tp u t pow er, since th e level o f m u ltip lic a tiv e noise is an a rb itra ry q u a n tity th a t is indepe ndent o f the gain and pow er in general. In p ra ctice , th e le ve l o f m u ltip lic a tiv e noise present in th e system m ay be d iffic u lt, if not im p o ssib le , to m easure. It is therefore useful to express th e sm a ll-a n g le c o n d itio n in term s o f m easurable p a ra m e te rs, such as in p u t pow er, o u tp u t pow er, etc. W e m a y o b ta in an h e u ris tic re la tio n fo r th e m u ltip lic a tiv e noise m a g n itu d e in term s o f m easurable q u a n titie s as follow s. W h e n o b s e rv in g th e o u tp u t pow er spec tru m o f th e resonant a m p lifie r, as the o u tp u t pow er is increased, th e o u tp u t pow er s p e c tru m w ill firs t decrease w ith increasing pow er u n til th e m u ltip lic a tiv e noise te rm becom es d o m in a n t. W e m ay th e re fo re d e te rm in e a m easured value o f in p u t and o u t p u t pow er at w hich th e tw o noise term s on th e rig h t-h a n d -s id e o f eq u a tio n (2.52) are o f e q ual m a g n itu d e . At these values o f in p u t and o u tp u t pow er, w hich we shall c a ll P'^ and th e fo llo w in g re la tio n is o b ta in e d betw een th e m u ltip lic a tiv e and a d d itiv e noise processes by e q u a tin g b o th te rm s on th e rig h t-h a n d -s id e o f e q u a tio n (2.52), and fa c to rin g o u t th e frequency dependence: 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2 .Ô6 I a-R ï^ PU T h is expression m a y be solved fo r Sh. to o b ta in an e s tim a te o f th e o f m u ltip lic a tiv e noise level in te rm s o f th e m easurable q u a n titie s o f in p u t a n d o u tp u t pow er, m irro r re fle c tiv ity , and a d d itiv e noise m a g n itu d e : ^ ---- ■ {2 ..1 1 ) w here G ' = P U !Pout the m easured value o f re so n a n t a m p lifie r e xte rn a l g a in , defined as th e o u tp u t pow er d iv id e d by th e in p u t p o w e r, at th e p o in t w here th e m u ltip lic a tiv e a nd in p u t noise processes are d e te rm in e d to be e q u a l. .As discussed p re vio u s ly , th is o p e ra tin g p o in t ro u g h ly correspond s to th e onset o f th e pow er- in d e p e n d e n t lin e w id th regim e. In o rd e r to p re d ic t th e onset o f th e large-ang le re g im e using th e c rite rio n o f ec[uation (2 .2 5 ). we need the variance o f th e m u ltip lic a tiv e noise process. I f we assum e th a t Sh and S r are b o th zero-m ean G aussian w h ite noise processes, th e n in te g ra tin g th e ir pow er spectra over a ll F o u rie r frequencie s yie ld s th e m ean-square values o f th e noise processes. .Also, we assum ed e a rlie r th a t th e noise processes 6 R and Sh were n o rm a liz e d to the u n it c irc le in th e c o m p le x p la n e , so th e ir m ean-square values are equal to th e ir variances, i.e.. / : R{f)df = o-l. (2.58) Jo In te g ra tin g b o th sides o f e q ua tio n (2.57) over fre q u e n c y y ie ld s th e fo llo w in g re la tio n betw een th e variances o f th e m u ltip lic a tiv e and a d d itiv e noise processes: ■ (2.59) N ow . th is e m p iric a lly -d e riv e d expression for th e m u ltip lic a tiv e noise variance af^ m a y be s u b s titu te d in to th e expression fo r th e sm a ll-a n g le c o n d itio n , g ive n by e q u a tio n (2.25). to o b ta in an a p p ro x im a te value fo r th e net g a in a t th e onset o f th e large-angle regim e: 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A . . . < < ,2.601 T h is expression gives th e m a x im u m va lu e o f th e resonant a m p lifie r net ro u n d - tr ip g a in A ’max fo r w hich th e s m a ll-a n g le an a lysis is v a lid , in te rm s o f in p u t and o u tp u t pow er, m irro r re fle c tiv ity , and th e va ria n ce o f th e a d d itiv e noise o f th e in p u t fie ld . S R . For net gain la rg e r th a n h'max- th e resonant a m p lifie r w ill be o p e ra tin g in th e large angle regim e, to be discussed in d e ta il la te r. O f course, to use e q u a tio n (2 .6 0 ). th e in p u t and o u tp u t pow ers a t th e onset o f th e p o w e r-in d e p e n d e n t lin e w id th re g im e . P'^ and P^ut- m ust be m easured. F o r net gain larger th a n R'mnx- th e oldest p a rtia l-w a v e s in th e a m p lifie r w ill have a c q u ire d rm s phase la rg e r th a n tt/'I. and th e re fo re begin to d e s tru c tiv e ly in te rfe re w ith th e e a rlie r p a rtia l waves. .-\.s w ill be show n la te r, th is leads to large flu c tu a tio n s o f th e to ta l o u tp u t fie ld , a n d re su lts in lin e w id th re b road e n ing w ith in cre a sin g o u tp u t pow er. W e re ca ll th a t e was chosen e a rlie r to be e = 0.01. w hich im p lie s (ne = —4.6. T h e n , fo r to ta l g ain G' > 10. th e e x p o n e n tia l m a y be lin e a rize d so th a t Rmax ~ 1 — 4.6<r^ — ' (2.61 ) T h is value o f m ay be used to d e riv e o u tp u t pow er. o f th e resonant a m p lifie r a t th e onset o f th e large-angle re g im e . To do th is , we insert th e expression fo r A max in to e q u a tio n (2.45) fo r th e o u tp u t pow er to o b ta in : p, < ' out.m ax Z: I , r ' I ' iri V 1 - A max / T h is expression allow s us to p re d ic t th e m a x im u m o u tp u t pow er o b ta in a b le fro m th e resonant a m p lifie r before th e onset o f th e la rg e -a n g le regim e, in te rm s o f m e a su ra b le q u a n titie s . 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 .4 .2 E q u ivalen ce to P h e n o m en o lo g ica l M o d el It is s tra ig h tfo rw a rd to show th a t e q u a tio n (2.55) for th e o u tp u t fie ld flu c tu a tio n pow er s p e c tru m is equivalent to th e p h e no m e nological m odel used b y W e lfo rd and M o o ra d ia n fo r th e case of m u ltip lic a tiv e phase noise, given by e q u a tio n (1 .2 ). .As s u m in g th a t th e instantaneous fre q u e n cy flu c tu a tio n Su and in d e x o f re fra c tio n Sn o f th e ph e n o m e n o lo g ica l m odel are tim e -v a ry in g q u a n titie s Su(t) a n d Sn(t). then p e rfo rm in g a F o u rie r tra n s fo rm o f e q u a tio n (1.2) and ta k in g the m a g n itu d e -sq u a re d o f b o th sides o b ta in s u- w here 5 ^ ( /) is th e power spectral d e n s ity o f frequency flu c tu a tio n s in u n its o f [H z’ /H z ], and Sn(f) is the pow er s p e c tra l d e n s ity o f re fra c tiv e in d e x flu c tu a tio n s in u n its o f [I/Hz]. B u t in general, th e pow er sp e ctru m o f frequency flu c tu a tio n s is re la te d to th e pow er spectrum o f phase flu c tu a tio n s by a tim e d e riv a tiv e [27]. w h ich is a fa c to r o f p in te rm s o f p o w e r-sp e ctra l densities, i.e.. & ( / ) = / ' & ( / ) ( 2.(71) So. e q u a tio n (2.63) m ay be re w ritte n as / - & ( / ) & ( / ) (2 .6-1 ) u- and re a rra n g e d to y ie ld ■ % (/) = (2 .6 6 ) n - N ow . re c o g n iz in g th a t ujn is an in te g e r m u ltip le o f th e inverse ro u n d -trip tim e I / r . th is can be re -w ritte n in term s o f a c o n sta n t .4 « .6 7 ) w here .4 is a p ro p o rtio n a lity constant betw een Sn{f) and M a k in g th e s u b s ti tu tio n Shif) = ASnif). then it is seen th a t e q u a tio n (2.67) is e q u iv a le n t to e q u a tio n (2.55) fo r th e pow er spectral d e n s ity o f phase flu c tu a tio n s o b ta in e d fro m th e sm all 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. angle m o de l in th e lim it, o f h ig h net gain. T h u s, th e sm a ll-a n g le m o d e l o f th e phase flu c tu a tio n s o f th e resonant a m p lifie r due to m u ltip lic a tiv e noise in th e lim it o f high net gain is e q u iva le n t to th e p h e n o m e n o lo g ica l m o d e l o f e q u a tio n ( 1 .2 ) . 2 .4 .3 S u m m ary o f sm a ll-a n g le resu lts In th is section, it was show n how th e feedback m echanism o f th e resonant a m p lifie r converts m u ltip lic a tiv e phase and gain flu c tu a tio n s to o u tp u t fie ld flu c tu a tio n s , in th e lim it o f sm a ll-a n g le phase flu c tu a tio n s . T h e conversion o f th e in p u t field flu c tu a tio n s to o u tp u t flu c tu a tio n s was also d e rive d . T h e a c tio n o f th e c a v ity on the m u ltip lic a tiv e noise has s im ila r frequency response as for th e in p u t field noise, but th e m a g n itu d e increases w ith increasing net gain. T herefore, fo r h ig h gain, the m u l tip lic a tiv e noise c o n trib u tio n to th e o u tp u t fie ld do m in a te s th e o u tp u t field power s p e c tru m . T he o u tp u t p o w e r is s im p ly re la te d to th e net g ain th ro u g h e q u a tio n (2.4-5), so the concepts o f n e t gain and o u tp u t pow er m ay be used interchang eably. .As th e net g a in (an d o u tp u t pow er) is increased, th e ro ll-o ff fre q u e n c y o f the reso na n t a m p lifie r m u ltip lic a tiv e fre q u e n cy response m oves to low er frequencies. For fre quencies above th e ro ll-o ff frequency, th e o p tic a l c a v ity in te g ra te s th e m u ltip lic a tiv e flu c tu a tio n s : for frequencies below th e ro ll-o ff, th e o u tp u t fie ld is lin e a rly m o d u la te d by th e m u ltip lic a tiv e noise. So, as th e net g ain is increased fro m zero, the c a v ity approaches a perfect in te g ra to r o f th e m u ltip lic a tiv e noise. P h y s ic a lly , this lim itin g case o f the sm all angle m o d e l represents th e in te g ra tio n o f th e phase flu ctu a tio n s b y th e c a v ity to becom e in sta n ta n e o u s fre q u e n cy flu c tu a tio n s o f th e o u tp u t field. In th is regim e, th e s m a ll-a n g le m o d e l is e q u iva le n t to the ph e n o m e n o lo g ica l m odel used by W elford and M o o ra d ia n to m o d e l th e effect o f m u ltip lic a tiv e noise fro m e le c tro n -n u m b e r-d e n s ity flu c tu a tio n s in a laser d io d e , e q u a tio n ( 1 .2 ), in w hich the d ire c t p ro p o rtio n a lity o f m u ltip lic a tiv e phase flu c tu a tio n s to o u tp u t fie ld frequency flu c tu a tio n s was assum ed. .As w ill be discussed in th e next c h a p te r, the laser can be m odeled as a resonant a m p lifie r d riv e n b y its ow n spontaneous em ission noise, o p e ra tin g w ith v e ry h ig h net gain. .Also, it w ill be show n in th e n e xt ch a p te r th a t th e constant p o w e r spectrum due 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to th e m u ltip lic a tiv e noise a t h ig h g a in leads to a m in im u m lin e w id th o f th e resonant a m p lifie r. T h is is in m a rk e d c o n tra s t to th e p re d ic tio n o f th e s ta n d a rd analysis [14j o f e q u a tio n (2 .5 ). w h ic h p re d ic ts decreasing lin e w id th w ith in c re a s in g o u tp u t power. I f th e m u ltip lic a tiv e noise arises fro m in trin s ic processes in th e laser, th e sm all- angle ana lysis p re d ic ts th a t th is m in im u m lin e w id th w ill be a fu n d a m e n ta l lim it. T h e s m a ll-a n g le tim e -v a ry in g p a rtia l-w a v e m odel generalizes th e s ta n d a rd analysis fo r resonant a m p lifie rs to in c lu d e v a ria tio n o f th e gain and in d e x o f th e a m p lify in g m e d iu m . T h e p re vio u s re su lts o f th e sta n d a rd analysis th e re fo re represent th e special case o f an id e a l re sona nt a m p lifie r w ith no m u ltip lic a tiv e noise. 2.5 L arge-angle regim e F ro m th e p re v io u s sections, it was seen th a t for h igh net g a in , th e resonant a m p lifie r o u tp u t fie ld flu c tu a tio n s becom e d o m in a te d by th e m u ltip lic a tiv e noise. .A .s the net g a in is increased fu rth e r s till, th e sm all-angle c o n d itio n fo r th e phase o f the M- th p a rtia l w ave, e q u a tio n (2 .2 4 ). w ill be exceeded, and th e s m a ll-a n g le linearized analysis o f th e last section no longer applies. In th is "la rg e -a n g le " re g im e , the fu ll phasor fo rm o f th e o u tp u t fie ld g ive n b y equation (2.17) m u s t be used to com pute' th e o u tp u t fie ld flu c tu a tio n s at each in sta n t o f tim e . In th e la rg e -a n g le regim e, as th e old e st p a rtia l waves a c c u m u la te m ore than ~/2 radians o f rm s phase, th e y begin to d e s tru c tiv e ly in te rfe re w ith e a rlie r p a rtia l waves, le ading to la rg e a m p litu d e and phase flu c tu a tio n s o f th e to ta l o u tp u t fie ld C f In th is se ctio n , th e o u tp u t fie ld tim e series o f th e resonant a m p lifie r in the large- angle re g im e is c o m p u te d fo r a noiseless in p u t fie ld R o f u n it a m p litu d e at fixed c a v ity resonance fre q u e n cy fo r various values o f th e net ro u n d -trip gain. Since it was show n in th e p re v io u s analysis th a t the m u ltip lic a tiv e noise is th e d o m inant c o n trib u tio n to th e o u tp u t fie ld flu c tu a tio n s in the h ig h -g a in lim it o f th e sm all-angle re g im e , it is assum ed in th e fo llo w in g analysis th a t th e m u ltip lic a tiv e noise is also th e d o m in a n t noise source in th e large-angle regim e. T h e re fo re , in th e fo llo w in g a n a lysis, th e in p u t fie ld noise is neglected. Reproduced witfi permission of ttie copyrigfit owner. Furtfier reproduction profiibited witfiout permission. 2 .5 .1 N o ise m o d el T h e re a l and im a g in a ry p a rts o f th e m u ltip lic a tiv e noise source Sht are w ritte n in te rm s o f a co m m o n noise source Srit re p re se n tin g flu c tu a tio n s o f th e re fra c tiv e index: Sh(t) = Sh'(t) + iSh"(t) T h e re ade r is re m in d e d th a t th e m u ltip lic a tiv e noise process Sh(t) appears in the e x p o n e n tia l te rm s o f th e p a rtia l-w a v e analysis, and so has u n its o f phase. T he co n sta n t J relates th e changes in th e real and im a g in a ry p a rts o f th e re fra c tiv e in d e x, an d is a m easure o f th e a m o u n t o f a m p litu d e -p h a se c o u p lin g fo r th e o p tic a l fie ld in th e gain m e d iu m . T h e real and im a g in a ry pa rts o f th e m u ltip lic a tiv e noise are show n as a fu n c tio n o f tim e in F ig u re 2.3. and in th e co m p le x plane w ith tim e as a p a ra m e te r in F ig u re 2.4. P h a s e -a m p litu d e c o u p lin g v ia th e p o p u la tio n d e n s ity is re sponsible fo r broad e n in g o f th e s e m ic o n d u cto r laser lin e w id th above the S chaw low -Tow nes p re d ic tio n at a g ive n pow er level, and J is u su a lly referred to as the lin e w id th enhancem ent fa c to r [6 ]. [7]. In s e m ic o n d u c to r lasers, d % .5 is ty p ic a l. T h e re fra c tiv e in d e x in a s e m ic o n d u c to r laser ga in m e d iu m is ty p ic a lly m odeled as a lin e a r fu n c tio n o f the c a rrie r d e n s ity [22]. T h e re m ay be m a n y possible m echanism s to m o d u la te th e re fra c tiv e in d e x due to c a rrie r d e n s ity m o d u la tio n in a p a rtic u la r laser. For exam ple, changes in th e e le ctro n d e n s ity due to in je c tio n c u rre n t noise, th e rm a l flu c tu a tio n s , or e le c tro n -h o le re c o m b in a tio n processes w ill cause co rre la te d changes in th e gain and phase o f th e a m p lify in g m e d iu m v ia 3. 2 .5 .2 N u m erica l resu lts T h e fo llo w in g re sults are c a lc u la tio n s o f th e effect o f th e G a u s s ia n -d is trib u te d zero- m ean ra n d o m m u ltip lic a tiv e noise source in th e large angle re g im e for th e cases o f co rre la te d gain and phase flu c tu a tio n s , and for phase flu c tu a tio n s only. T h e system m odele d is a resonant c a v ity w ith a h x e d -a m p litu d e . ideal noiseless in p u t fie ld , th a t 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. '-0 D 0 1000 2000 3000 t (round trips) 4000 50 60x10 4000 1000 2000 3000 t (round trips) F ig u re 2.3: Real and im a g in a ry pa rts o f m u ltip lic a tiv e noise. 4!) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60x10 40 20 0 -20 -40 0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 Re (6h) F ig u re 2.4: M u ltip lic a tiv e noise in th e c o m p le x p la n e fo r a ll th e tim es in F ig u re 2.3. •50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is p e rtu rb e d b y a m u ltip lic a tiv e noise source as described above. T o illu s tra te th e effect o f in cre a sin g net gain on th e syste m , th e level o f m u ltip lic a tiv e noise is chosen to be a r tific ia lly h ig h , so th a t its effects w ill be e v id e n t w ith re la tiv e ly low values o f net ro u n d -trip g a in , com pared to th e levels o f net gain ty p ic a l o f a s e m ic o n d u c to r laser. T h is a p p ro a ch is taken because th e n u m b e r o f p a rtia l waves re q u ire d to c o m p u te th e value o f th e o u tp u t field C\ fo r a se m ico n d u cto r laser at each in s ta n t o f tim e is on th e o rd e r o f .V/ ~ 10^ to 10' o r g re a te r. T o co m p u te th e p o w e r s p e c tru m o f th e o u tp u t fie ld , a tim e series o f at least several thousan d p o in ts is re q u ire d . 1 hese re q u ire m e n ts m ake th e c o m p u te r tim e re q u ire d to co m p u te these p lo ts p ro h ib itiv e ly large fo r th e case o f a s e m ico n d u cto r laser. H ow ever, we can o b ta in a q u a lita tiv e u n d e rs ta n d in g o f th e be ha vio r o f th e resonant a m p lifie r o u tp u t fie ld in th e large angle re g im e by increa sin g the level o f m u ltip lic a tiv e noise, w h ic h reduces th e net gain at th e onset o f th e large angle re g im e . T h is reduces th e n u m b e r o f p a rtia l waves re q u ire d fo r convergence o f the o u tp u t fie ld to a m anageable n u m b e r th a t is am en able to c a lc u la tio n on a personal c o m p u te r. For th e present n u m e ric a l a nalysis, we sh a ll a d ju s t the m a g n itu d e o f th e m u lti p lic a tiv e noise s tre n g th so th a t o u r m o d e l system is o p e ra tin g Just a t th e onset o f th e large -a ng le re g im e fo r net gain o f Ko = 0.9. T h e n as th e net g a in is increased to l\o = 0.95. Ko — 0.995. and Ko — 0.999. th e system w ill be o p e ra tin g deeper in the large-angle re gim e. For net gain o f Ko = 0.9. th e num ber o f p a rtia l waves needed to achieve 1 p e rce n t convergence o f th e o u tp u t s u m m a tio n is c o m p u te d fro m equa tio n (2.23) to be a p p ro x im a te ly 50. F s in g e q u a tio n (2.24). th e s ta n d a rd d e v ia tio n (sq u a re -ro o t o f th e variance) o f th e phase fo r th e last (i.e .. .V /th ) p a rtia l wave for th e c o n d itio n s 1\q = 0.9 and ct/, = O.I ra d ia n s is co m p u te d to be ~ \ / 100 (0.1 ) = 1 radian. (2.69) T h is im p lie s th a t th e system is in th e tra n s itio n regim e betw een th e sm a ll-a n g le and larg e-an g le regim es w hen Ko = 0.9 and cT h . = 0 . 1 ra d ia n s, a n d in th e large- angle re g im e fo r a ll o f th e higher net g a in values tested. T h is is co n s is te n t w ith the a ssu m p tio n th a t th e onset o f th e la rge-ang le re g im e occurs w hen th e rm s phase o f 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the oldest p a rtia l waves is e q u a l to t / 2 radian. T h e onset o f th e large angle regim e w ill be cast in te rm s usable fo r laser design and o p e ra tio n (i.e .. m irro r re fle c tiv ity , net g ain, noise m a g n itu d e ) in th e n e xt C hapter. . A . S w ill be show n in th e n e x t chapter, th e co rre la te d g a in and phase noise is a p p lica b le to m o d e lin g th e effect o f e le ctro n -d e n sity flu c tu a tio n s in a se m ico n d u cto r laser gain m e d iu m , w hereas th e case o f phase noise o n ly is a p p lic a b le to m odeling m irro r p o s itio n flu c tu a tio n s o r m o d u la tio n by an in tra -c a v ity id e a l phase-shifter. T he results o f th e an a lysis e lu c id a te the general features o f th e resonant a m p lifie r o u tp u t fie ld b e h a vio r in th e large-ang le regim e. T he reader is re m in d e d th a t these levels o f net gain and noise do n o t correspond to th e case o f a ty p ic a l se m ico n d u cto r laser, b u t are chosen fo r convenience o f c a lcu la tio n , and to illu s tra te th e general behavio r o f th e resonant o p tic a l c a v ity versus increasing net gain in th e large-angle regim e. C o m p u te r e va lu a tio n s o f e q u a tio n (2.17) were p e rfo rm e d over th re e decades o f net ro u n d -trip gain: = 0.9. 0.9-5. 0.995. and 0.999. T h e reader is re m in d e d that the net gain and o u tp u t pow er are related by e q u a tio n (2 .4 5 ). For each value o f [\o. an .V -elem ent c o m p le x v e c to r o f the flu c tu a tin g o u tp u t fie ld C\ is produced by d ire c t c o m p u te r e v a lu a tio n o f equatio n (2.17) for each in s ta n t o f tim e t. The m in im u m n u m b e r o f p a rtia l waves recjuired fo r convergence o f th e o u tp u t fie ld sum m a tio n e q u a tio n (2.17) at each value o f net gain is c o m p u te d fro m e q u a tio n (2.25). P hysically, e q u a tio n (2.25) gives th e num ber o f p a rtia l waves (i.e .. passes th ro u g h the c a v ity ) re q u ire d fo r th e o u tp u t to be w ith in a ce rta in fra c tio n o f th e fin a l an swer. In th is analysis, we re q u ire th e com puted o u tp u t fie ld a m p litu d e to be w ith in a fra c tio n t = 0.01 o f th e value o b ta in e d a n a ly tic a lly when no m u ltip lic a tiv e noise is present. For the net g a in values considered above, equ a tio n (2.25) s tip u la te s th e to ta l n u m b e r M o f p a rtia l-w a v e s used in the calcu la tio n s to be: M — 50 fo r AT, = 0.9. -\[ = 100 fo r a ; = 0.95. M = 1000 fo r A", = 0.995. and M = 5000 fo r A ', = 0.999. I f th e n u m b e r o f p a rtia l waves in the o u tp u t field s u m m a tio n is equal to or greater th a n th a t re q u ire d b y e q u a tio n (2.25). th e n the dependence o f th e o u tp u t field c h a ra cte ristics on net g a in is e x a c tly related to the dependence on th e o u tp u t 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. pow er. T h a t is. th e value o f M chosen is irre le v a n t i f .V/ is g re a te r th a n th e m in im u m value set by e q u a tio n (2 .23 ). T herefore, th e re is no need to consider m ore p a rtia l waves th a n th e m in im u m value specified by e q u a tio n (2 .2 3 ). and choosing a larger n u m b e r o n ly increases th e c o m p u ta tio n tim e w ith o u t c h a n g in g th e results. T h e com plex m u ltip lic a tiv e noise process Sht is s im u la te d b y a co m p u te r-g e n e ra te d G a u s s ia n -d is trib u te d zero-m ean ra n d o m tim e series w ith s ta n d a rd d e v ia tio n o f cr^» = O .I. as shown in F ig u re 2.3. T h e c o m p u te r produces a series o f ra n d o m values fro m a G aussian d is trib u tio n such th a t th e rm s value o f an in fin ite n u m b e r o f such values w o u ld be ah'. T h e co m p u te d results fo r net ro u n d -trip g a in o f A'o = 0.9. 0.95. 0.995. and 0.999 are now discussed. T o see th e q u a lita tiv e effect o f the in cre a sin g net gain on the o u tp u t, the real and im a g in a ry parts o f th e o u tp u t e le c tric fie ld C\ are displayed p a ra m e tric a lly in th e c o m p le x plane in Figures 2.5 th ro u g h 2.8. fo r increasing values o f h'o- In these p lo ts, each o f the .V dots on th e c o m p le x p lane represents the co m p u te d p o sitio n o f th e tip o f th e o u tp u t fie ld ve cto r C f F ig u re 2.5 illu s tra te s th e case o f net ro u n d -trip g a in h'o = 0.9. The tip o f th e e le c tric fie ld ve cto r describes an arc w ith m a x im u m phase e xcursion o f a p p ro x im a te ly + /- 0.3 radians, b u t w ith re la tiv e ly con sta nt a m p litu d e . .As th e gain is increased to = 0.95. 0.995. and 0.999. illu s tra te d in F igures 2.6 - 2.8. the m a x im u m phase e xc u rs io n increases and th e a m p litu d e e x h ib its prog ressively larger flu c tu a tio n s w h ic h a ppea r as a s w irlin g p a tte rn in the co m p le x plane. I f th e re were no m u ltip lic a tiv e noise, a ll o f the dots w o u ld fa ll on one p o in t on th e real axis, d e te rm in e d by th e s ta tic g ain 1/(1 — /v',J. as in d ic a te d on the re sp e ctive plo ts. T he sm a ll-a n g le re g im e tre a te d p re vio u sly represents sm a ll excursions a b o u t th is p o in t. .At th e h ig h e st net g a in values, the o u tp u t field v e cto r s w irls a ro u n d th e com plex plane, and its phase and m a g n itu d e change d ra s tic a lly . T h is is m ost e vid e n t in F igures 2.7 and 2.8. T h e instantaneous in te n s ity (m a g n itu d e -sq u a re d o f th e e le c tric fie ld ) versus tim e and phase versus tim e fo r each value o f net g ain are p lo tte d in F ig u re s 2.9 th ro u g h 2.12 and Figures 2.13 th ro u g h 2.16. respectively. .As th e n e t g a in increases, th e in te rfe re n ce betw een th e p a rtia l waves becomes m ore c o m p le te , so th a t the o u tp u t 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -10 -15 10 15 -15 -10 0 Re (E) Figure 2.5: Output rteid in eoniplex plane for net gain A = 0.0. M = 50. 30 20 10 0 -10 -20 -30 i=- 30 J_______I_______L -20 -10 0 Re (E) 10 20 30 F ig u re 2.6: O u tp u t fie ld in com plex plane fo r net g a in A'„ = 0.95. M = 100. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Re (E) -300 300 F ig u re 2.7: O u tp u t tiel<i in c o u ip le x plane to r net gain A 'j = 0 .9 9 5 .3 / = 1000 . -1500 •1000 0 Re (E) 1000 F ig u re 2.8: O u tp u t fie ld in co m p le x plane for net gain A'_, = 0.999. .V/ = 5000. 0 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in te n s ity and phase flu c tu a tio n s increase. .At th e low er gain values, th e m ean value o f th e in te n s ity flu c tu a tio n s is close to th e value expected w ith o u t m u ltip lic a tiv e noise. H ow ever, as the gain is increased, th e in te n s ity flu c tu a tio n s increase g re a tly. T h e in te n s ity not o n ly becomes m ore d e e p ly m o d u la te d , b u t its m ean value moves to low er values, com pared to th e value o f th e in te n s ity w ith o u t m u ltip lic a tiv e noise. This b e h a vio r is m ore apparent fro m th e p ro b a b ility d is trib u tio n o f th e in te n s ity , discussed below . .As th e net gain is increased, the phase flu c tu a tio n a m p litu d e also increases u n til eventua lly, the phase "w ra p s a ro u n d " m ore th a n ~ ra d ia n s. T h e p ro b a b ility d is trib u tio n o f th e o u tp u t in te n s ity versus in cre a sin g net gain is p lo tte d in Figures 2.17 th ro u g h 2.20. .Also p lo tte d are G aussian c u rv e -fits to th e n u m e ric a l d a ta (solid line s), w h ich illu s tra te the tre n d o f the h is to g ra m s . In F ig u re 2.17 a t A'a = 0.9. the p ro b a b ility d is trib u tio n has developed a " t a il" s tre tc h in g to low er values o f in te n sity. .As th e net g ain increases, th is ta il becom es m o re pro nounced. and the peak value o f th e p ro b a b ility d is trib u tio n m oves to low er values. .At th e highest gain o f /v'o = 0.999. F ig u re 2.20. th e peak o f th e in te n s ity p ro b a b ility d is trib u tio n has m oved s u b s ta n tia lly away fro m its e xp e cte d value w ith o u t m u ltip lic a tiv e noise. In Figures 2.21 th ro u g h 2.24. th e c o m p u te d pow er spectra o f th e o u tp u t held versus increasing net gain are p lo tte d fo r th e case o f net gains A'^ = 0.9. A ’„ = 0.91. ho = 0.93. and A^ = 0.9.5. It is seen th a t th e noise sidebands o f th e pow er spectra increase as th e net gain is increased, in d ic a tin g increasing lin e w id th . T h e phase "w ra p s a ro u n d " more th a n rr radians fo r th e h ig h e r gain cases, (i.e .. h'o > 0.95). w h ich introd uces an a m b ig u ity th a t th e F F T ro u tin e can not resolve. These in creasing phase excursions correspond to ra n d o m frequency m o d u la t ion o f th e o p tic a l c a rrie r, so th a t the lin e w id th fo r th e h ig h e r gain cases increases fu rth e r s till. T h is is not s u rp ris in g , given th e in c re a sin g ly large in te n s ity and phase flu c tu a tio n s o f th e o u tp u t fie ld as the net gain is increased. These in te n s ity m o d u la tio n s w ill appear as sidebands around the o p tic a l c a rrie r frequency, w hich e ffe c tiv e ly broadens the lin e w id th . H owever, it is c o n tra ry to th e b e h a vio r p re d icte d b y th e classical m odel o f resonant a m p lifie r lin e w id th versus o u tp u t pow er. In th e absence o f m u ltip lic a - 5G Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140- 1 20- 73 c 100- CJ r- GO- 3 r ' . GO- C 40 - 20- 0 - 500 1000 1500 t (round-trips) 2000 F ig u re 2.9: O utput; iiite r iiir y versus tim e for net gain = 0.9. A / = 50. 600 — 500- d 400- 300- CL 2 0 0 - 1 00 - 2000 1500 1000 0 500 t (round-trips) F ig u re 2.10: O u tp u t in te n s ity versus tim e fo r net gain A'o = 0.95. A / = 100. ■ 1 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60x10' 5 0 - ■ T j r* 3 0 - 2 0 - 500 1000 1500 t (round-trips) 2000 F ig u re 2 .11 : O ur p ut in te n s ity versus rim e for ner gain A .j = 0.99Ô. .V/ = 1000. 1.4x10 e n r* a 0 . 8 - 0.6 - C L o 0.4 - 0 . 2 - 0.0 8000 4000 6000 0 2000 t (round-trips) F ig u re 2.12: O u tp u t in te n s ity versus tim e for net gain K j = 0.999. A / = 5000. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -4 — -2 - 0 - O 2 - 1500 2000 500 1000 0 t (round-trips) F ig u re 2.13; O u rp u t pha^e versus rim e for ner gain I\.^ = Ü.9. M = 50. -2 — 0 - 2 - 500 1000 1500 2000 0 t (round-trips) F igure 2.14: O u tp u r phase versus tim e for net gain Ko = 0.95. M = 100. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -4 — -2 — 0 - 2 - 2000 500 1000 0 1500 t (round-trips) F ig u re 2.15: O u tp u r pliage versus rim e tor net gain K , = Ü.995. .V/ -- 1000. -4 — 2 - 8000 2000 4000 6000 0 t (round-trips) F ig u re 2.16: O u tp u t phase versus rim e for net gain = 0.999. M = .5000. 6 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 - 30- C (C 80 120 0 40 60 1 00 1 40 20 Intensity F ig u re 2.17: Inren.siry hi.<ri;u:r<iru tor net gain /v'., - Ü.!). .1/ = 50. S o lid curve: G aus sian fit. Xi < z £ 1 0 - 0 400 500 600 200 300 0 100 Intensity F ig u re 2 .IS: In te n s ity h isto g ra m tor net gain AT, = 0.95. .V/ = 100 . S olid curve: G aussian fit. 0 1 Reproduced witti permission of ttie copyrigfit owner. Furtfier reproduction profiibited witfiout permission. 2 0 - c ? r Z > > 5 - 0 -i 50 40 60x10 20 30 1 0 0 Intensity F ig u re 2.19: In te n s ity h isto g ra m tor net gain A \ - 0.995. .U — 1000. S o lid cu rve : G aussian tit. -5 I 2 0 - 0 - I 1.4x10 0.8 Intensity 0.2 0.6 0.0 0.4 F ig u re 2.20: In te n s ity h isto g ra m tor net g ain A 'j = 0.999, M = 5000. S olid cu rve : G aussian fit. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tiv e noise, th e o u tp u t pow er sp e ctru m w o u ld be a d e lta -fu n c tio ii. since no in p u t noise source was in c lu d e d in th e s im u la tio n s. T h e d isp la ye d pow er sp e ctra are the m a g n itu d e -s q u a re d o f a fa s t-F o u rie r-tra n s fo rm ( F F T ) o f th e c o m p le x tim e series C, fo r th e o u tp u t e le c tric fie ld . It is n o te d th a t in a ll o f these results, th e va ria n ce o f th e m u ltip lic a tiv e noise tim e -se rie s was k e p t c o n s ta n t, and o n ly th e net g a in was changed. W h ile such a large value o f m u ltip lic a tiv e noise m ay not ty p ic a lly be e n co u n te re d in practice, these s im u la tio n re su lts serve to illu s tra te th e effects o f increasing net gain on th ^ o u tp u t fie ld . B y using a re la tiv e ly large value o f m u ltip lic a tiv e noise, it was po.ssibie to s tu d y th e q u a lita tiv e b e h a vio r o f the o u tp u t fie ld flu c tu a tio n s versus net gain, w h ile using a m anagea ble n u m b e r o f p a rtia l waves. F ro m these n u m e ric a l results, it is a p p a re n t th a t fo r o u tp u t pow er at o r above th e onset o f th e large-ang le regim e, an a ctu a l s e m ic o n d u c to r laser p e rtu rb e d by m u ltip lic a tiv e noise w ill e x h ib it increased in te n s ity and phase flu c tu a tio n s , a ta il in th e in te n s ity p ro b a b ility d is trib u tio n , and lin e w id th re b ro a d e n in g . In th e fo llo w in g F ig u re s 2.25 th ro u g h 2.29 th e c o m p le x fie ld , in te n s ity , phase, in te n s ity h is to g ra m , and pow er sp e ctru m are p lo tte d fo r th e case o f p u re phase noise at the hig hest net g a in Ko = 0.999. T he phase noise is generated fro m th e real part o f th e noise series d isp la y e d in F ig u re 2.3. T h e im a g in a ry p a rt, re p re se n tin g gain flu c tu a tio n s , has been set to zero. It is seen fro m F ig u re 2.25 th a t th e o u tp u t field has s im ila r c h a ra c te ris tic s as in the case o f c o rre la te d gain and phase noise. T iie in te n s ity a nd phase versus tim e e x h ib it s im ila r deep m o d u la tio n , as in F igures 2 .2 fi and 2.27. and th e in te n s ity h isto g ra m in d ica te s a h ig h e r p ro b a b ility o f a low in te n sity, as show n in F ig u re 2.28. T h e pow er sp e ctru m also broadens, as in F ig u re 2.29. Thus, in the large angle re g im e , pure m u ltip lic a tiv e phase noise is co n ve rte d to in te n s ity and phase noise a t th e o u tp u t. For a fixed level o f in tra -c a v ity m u ltip lic a tiv e phase noise, th e o u tp u t in te n s ity and phase flu c tu a tio n s becom e p ro g re ssive ly larger as th e net ga in is increased. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 _ c 1 o' r“ C 1 o ' u o CL c/1 1 0 '' 3 c 1 o ' C - 1 o '' -100x10 -50 0 50 Frequency (Hz^FSR) 1 00 F ig u re 2.‘ J1: O u t[)u t tie lii pow er sp e c tru m tor net gain I\„ - Ü.9. ■100x10 -50 0 50 Frequency (Hz*FSR) 1 00 F ig u re 2.22: O u tp u t fie ld pow er s p e c tru m fo r net gain = 0.91. 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0*^ ,_^ tc 1 o ’ ’ r - c 1 0^ u 3 • 3 m 1 0 3 •4 c 1 0 ■100x10 -50 0 50 Frequency (Hz'^FSR) 1 00 Fisure 2.23: Ourput KeKl power it)e><:rrviin tor net sain I\j = 0.93. 1 0 ° ^^ bc _c 1 o ' B i_ 1 o'^ 3 3 D- 3 cn 1 0 3 -4 O 1 0 O h _-5 1 0 -100x10 -50 0 50 Frequency (Hz*FSR) 100 F ig u re 2.24: O u tp u t fie ld pow er s p e c tru m for net g ain K j = 0.95. bo Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -1000 Re (E) F ig u re 2.25: O u r p iit fie lii in coinpie:-: plane for net gain A ’ , = 0.999. .1/ = 5000. m u ltip lic a tiv e phase noise only. 1.4x10 1.0 - f- 0 . 8 - Cu 0 . 6 - O 0.4 - 0 . 2 - 0.0 6000 8000 2000 4000 0 t (round-trips) F ig u re 2.26: O u tp u t in te n s ity versus tim e for net gain K , — 0.999, M = 5000, m u ltip lic a tiv e phase noise only. 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -4 — 2000 4000 6000 t (round-trips) 8000 Figure 2.21 : Output :U ui.>e \ 'U<u> titiir for uet gain h = ().y!)9. .\/ = 5000. multi plicative pha.'e noise uiily. 200-1 1 50 - 1 0 0 - Z j > rz 50 - 0.4 0.6 0.8 Intensitv 0.2 1.4x10 0.0 Figure 2.25: Intensity liistogran: fur net uain l\ = O.t'lM'. .1/ = 5000, multiplicative phase noise only. 07 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -100x10 -50 0 50 Frequency (Hz’ ^FSR) 100 F ig u re 2.2!): O u tp u t field pow tT s p fc iru in for net gain /O = 0.999. A / = 5000. m u ltip lic a tiv e phase noise on ly. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 .5 .3 S im u lation a lg o rith m T h e d e ta ils o f th e s im u la tio n a lg o rith m used to p roduce th e plots o f th is se ctio n for each value o f th e net gain R'j are as fo llo w s; F irs t, a real-valued ra n d o m noise v e c to r Srit o f le n g th 2.V is generated. T h e n th e c o m p le x v e c to r 6ht o f le n g th 2 .V is g enerate d a cco rd in g to e q u a tio n (2.68). T h e c u m u la tiv e su m o f Sht is then c o m p u te d , re s u ltin g in a n o th e r v e cto r o f le n g th 2.V th a t represents a co m p le x ra n d o m w a lk process. T h e indices o f th is ve cto r are th e n reversed, so th a t th e ra n d o m -w a lk begins w ith th e last (2 .V th ) ele m e n t and moves to e a rlie r tim e s. T h is v e c to r is called ir. T h e n , th e o u tp u t fie ld C( is c o m p u te d fo r each o f th e .V tim e in c re m e n ts t by s u m m in g th e M + 1 p a rtia l waves whose co m p le x phases are g ive n b y th e successive values o f a\ T h u s, th e firs t sam ple o f O is c o m p u te d u sin g e q u a tio n (2.17) by s ta rtin g at th e m id d le o f th e v e c to r œ ( i.e.. u’ v - i )• and u sin g th e p re vio u s \ f values. T h e co m p le x phase o f th e firs t p a rtia l wave at tim e in c re m e n t t is given by — u’(+.v_2. and th e last p a rtia l wave has co m p le x phase — u’f+ .v_ i_ (.v/+ i). T h e n , th e instantan eous in te n s ity and phase o f Ct are c o m p u te d . T h e m a g n itu d e -sq u a re d o f C't is p ro p o rtio n a l to th e in te n s ity o f the o u tp u t fie ld a t tim e t. T h e p ro b a b ility d is trib u tio n (i.e .. h is to g ra m ) o f th e in te n s ity is also c o m p u te d . F in a lly , a H a n n in g (raised-cosine) w in d o w in g fu n c tio n is a p p lie d to th e tim e series Ct and a c o m p le x fa s t-F o u rie r-tra n s fo rm ( F F T ) is p e rfo rm e d on th is tim e series. T h e m a g n itu d e squared o f th e F F T is th e c o m p u te d pow er s p e c tru m o f the o u tp u t fie ld . 2.6 Sum m ary o f p artial-w ave m od el resu lts In th is ch a p te r, a fo rm a lis m was developed fo r c o m p u tin g the o u tp u t fie ld flu c tu a tio n s o f a resonant o p tic a l a m p lifie r p e rtu rb e d by m u ltip lic a tiv e phase and gain flu c tu a tio n s . T h e fo rm a lis m d evelop ed is general, and m ay be a p p lie d to o th e r reso n a n t feedback system s. .-\n a n a ly tic s o lu tio n was o b ta in e d fo r th e case o f re la tiv e ly low levels o f net gain and m u ltip lic a tiv e noise, a n d it was fou n d th a t the po w e r spec tr a l d e n s ity o f a m p litu d e and phase flu c tu a tio n s o f th e resonant a m p lifie r e v e n tu a lly becom es d o m in a te d by m u ltip lic a tiv e noise as th e g a in is increased. It was show n 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. th a t in th e re g im e w he re th e m u ltip lic a tiv e noise is d o m in a n t, th e sm a ll-a n g ie m o d e l is e q u iva le n t to th e ph e n o m e n o lo g ica l m o d e l e m p lo ye d b y W e lfo rd and M o o ra d ia n to e x p la in th e p o w e r-in d e p e n d e n t lin e w id th o f s e m ic o n d u c to r lasers. .A.S th e g a in is increased fu rth e r, fo r a h xe d level o f m u ltip lic a tiv e noise, th e lin e a r a n a ly tic m o d e l e v e n tu a lly breaks d o w n . T h e o ld e r p a rtia l waves a c c u m u la te so m uch phase th a t th e y begin to d e s tru c tiv e ly in te rfe re w ith th e recent p a rtia l waves, le a d in g to larg e a m p litu d e and phase flu c tu a tio n s o f th e o u tp u t field. In th is large-angle re g im e , c o m p u te r s im u la tio n s are used to c o m p u te th e o u tp u t fie ld . .As th e net g a in is increased, th e in te n s ity and phase flu c tu a tio n s o f the o u tp u t held grow p ro g re s s iv e ly la rg e r, th e peak o f th e in te n s ity p ro b a b ility d is trib u tio n m oves to low er values, and th e pow er sp e c tru m broadens. In th e fo llo w in g c h a p te r, these results fo r th e o u tp u t pow er sp e ctra l d e n s ity o f phase noise fro m th e s m a ll-a n g le analysis w ill be used to c o m p u te th e lin e w id th o f th e resonant a m p lih e r. T h e n , by v ie w in g th e laser as a n o ise -d rive n resonant a m p lih e r. a general e xpre ssion fo r laser lin e w id th in c lu d in g th e effects o f m u ltip lic a tiv e noise w ill be o b ta in e d . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.7 A p p en d ix: D erivation o f sm all-an gle th eory w ith in p u t field fiu ctu ation s In th is section, th e d e ta ile d d e riv a tio n o f e q u a tio n (2.37) fo r th e co m p le x o u tp u t field a m p litu d e is presented w hen b o th m u ltip lic a tiv e noise and a n o isy in p u t fie ld are considered. .\s w ill be seen, the fin a l re s u lt o f e q u a tio n (2.37) follow s in a s tra ig h tfo rw a rd m a n n e r fro m th e lin e a riz a tio n th a t is assum ed in th e sm a ll-a n g le m odel. B e g in n in g w ith e q u a tio n (2.36). th e o u tp u t held Ct a t tim e t is g ive n by Ct = ~i~C jt\R t 4- /v(_i/? (-! 4- A(_iKt-iR-t-i 4 * • • •]. (2 .(0 ) C o n side ring th e gains Gt and Kt as dehned p re vio u sly, th is can be w ritte n Ct = [Rt 4- Ko -h K l I y^'3 g — 1 2 ) ^ (2.71) M a k in g the s m a ll-a n g le a p p ro x im a tio n fo r th e e xp o n e n tia ls, we o b ta in C't = 'C Ho{\. — iSht) [/?( - f Ao ( 1 — 'liSht-\ )Rt-i 4" A ■ ( I — '2i{Sht-i 4- Sht-z) )Rt-i 4- A^ ( I — 2i(Sht-i -f àht-2 4- Sht-3))Rt-3 4- • • • 4- A^^ ( 1 — 2i(Sht-i 4- Sht-2 4- Sht-s 4- • • • 4- Sht-\[ ) ) A(_.v/]. (2.72) S u b s titu tin g e q u a tio n (2.35) fo r the noisy in p u t e le c tric held in th e sm a ll-a n g le a p p ro x im a tio n , th e o u tp u t state becomes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ct = y H o i l — iSht) [/?o({ SRt) Ko [I — iiSht—i)Ro( 1 + ÔR t - i ) + K j ( 1 — 2i(Sht-i + 6ht-z))Ro( 1 + SRt-z) + A j ( 1 — 2i{Sht-i + Sht-2 + Sht- 3 ))Ro( 1 + SRt-:i) + ■ • • + Ko^ ( 1 — 2i{Sht-i + Sht-2 + Sht-3 -(-••• + Sht-,\f))Ro( 1 + < ! > (2.7:]) F a cto rin g o u t th e n o n -flu c tu a tin g p a rt o f th e in p u t. Ro. and e x p a n d in g yields th e to ta l o u tp u t fie ld for th e case o f flu c tu a tin g in p u t and feedback: Ct = HoRo{(-- iSht) [ I ho ( 1 — 2iSht-i) + K~ ( 1 — 2i{Sht-i -h Sht-2 )) -f- A g ( 1 — 2i(Shf-i -h Sht- 2 " f Sht- 3 )) ho^ ( 1 — 2i{Sht-i - f Sht-2 + Sht-3 -H • • • -h Sht-\i ) ) -j- SRt -h Ko (1 — 2 iSht-i)SRt—i 4- A ^ ( I — 2i(Sht-i S h t- 2 ))SRt-> + A ^ (1 — 2i{Sht-i i- Sht-2 Sht-3))SRt-3 -f • • • 4- K^^ ( I — 2i{Sht-i -f Sht-2 + Sht-3 -h • • • -f Sht-\[))SRt-\i]. (•2.711 T h e firs t set o f te rm s are recognized to be e q u a l to th e previous e q u a tio n (2.28) for th e case o f no in p u t flu c tu a tio n s . T h e second set o f te rm s can be s im p lifie d by re cogn izin g th a t te rm s o f O(ShSR) m ay be n eglecte d com pared to te rm s o f 0(Sh) o r 0 {S R ). since it is assum ed th a t Sh and SR a re in d e p e n d e n t, u n c o rre la te d noise sources w ith a m p litu d e m uch less th a n one. D o in g th is , and c o lle c tin g te rm s w ith co m m o n tim e indices vie ld s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ct = 'I'HoRo [( 1 + Ao + A g + ■ • • + R'o^) — ( 1 + Ao + A ^ + ■ • ■ + )(iSht] — ( A j + A ^ + • ■ • -f- h^^){2iSht-i) - ( A'^ + A'^ + . . . + A'-'^( ■•Ul (•2.75) + SRt + k o S R t - i 4- k ^ S R t - 2 + k ^ S R t s + • • • + k ^ ^ S R t - \ T h e o u tp u t fie ld is now seen to be th e sum o f a dc gain te rm and tw o sets o f te rm s th a t represent w eighted sum s o f e a rlie r sam ples o f th e tw o in d e p e n d e n t noise processes Sht and SRt. F in a lly , s u b s titu tin g th e d e fin itio n s o f th e co e fficie n ts for th e s u m m a tio n s o f AT, fro m equ a tio n s (2 .2 9 ) and (2.30). we o b ta in e q u a tio n (2.37) fo r th e resonant a m p lifie r co m p le x o u tp u t fie ld a m p litu d e w hen b o th m u ltip lic a tiv e noise and in p u t fie ld flu c tu a tio n s are present: = T T T : [I — iSht — koi'2Sht-i — k~i2Sht-2 — • • • — k ^ ^ l2Sht^^\f_l^ + ( I - A 'J (SR, + K ,SR t_i + k':SRt_2 + [Ç,SRt_:, + • • - + K ^ 'S R ,_ „ )]. (2.7b I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 3 Effect o f m ultiplicative noise on laser linew idth in the sm all-angle regim e In th is C h a p te r, th e re s u lts o f th e tim e -v a ry in g p a rtia l-w a v e analysis in th e sm a ll- angle re g im e are a p p lie d to a lin e a r m odel o f a laser to d e rive an a n a ly tic expression for th e laser lin e w id th versus o u tp u t pow er in th e presence o f m u ltip lic a tiv e noise. T h e laser is tre a te d as a resonant a m p lifie r th a t is d riv e n by an ecjuivalent in p u t noise fie ld c o rre sp o n d in g to th e spontaneous em ission in th e a c tiv e m e d iu m [4], [2-5]. [14]. T h e a n a ly tic expressions for th e o u tp u t flu c tu a tio n pow er s p e ctra fro m the s m a ll-a n g le m u ltip lic a tiv e noise analysis o f C h a p te r 2 are used to d e rive a generalized fo rm u la fo r th e o u tp u t lin e w id th o f the resonant a m p lifie r th a t in cludes th e effects o f m u ltip lic a tiv e noise. T h is generalized expression fo r th e resonant a m p lifie r lin e w id th is th e n used to o b ta in an expression for laser lin e w id th versus o u tp u t pow er th a t e x h ib its th e inve rse-p ow e r dependence o f the S chaw low -Tow nes fo rm u la at low power levels, as w ell as a p o w e r-in d e p e n d e n t ch a ra cte r at h igh pow er levels. It is shown th a t th e S chaw low -T ow nes lin e w id th fo rm u la is recovered as th e lim itin g case o f no m u ltip lic a tiv e noise. T h e laser o u tp u t p ow e r level correspond ing to th e onset o f th e large-angle regim e is also c a lc u la te d . .-\s seen fro m th e n u m e rica l s im u la tio n s o f th e last ch a p te r for th e la rg e -a n g le regim e, th e o u tp u t in te n s ity and phase flu c tu a tio n s increase w ith in cre a sin g o u tp u t pow er, le a d in g to increasing lin e w id th o f th e resonant a m p lifie r. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In th e m o d e l o f th e laser as a n o ise -d rive n resonant a m p lifie r, th is im p lie s th a t the laser lin e w id th increases w ith fu rth e r increases in o u tp u t po w e r a fte r th e onset o f th e large-angle regim e. T h e th e o ry o f laser lin e w id th presented here p re d ic ts a range o f o u tp u t pow er corresp ond in g to a m in im u m lin e w id th fo r a laser p e rtu rb e d by m u ltip lic a tiv e noise. W hen a s u ffic ie n tly h ig h level o f m u ltip lic a tiv e noise is present , th e m in im u m lin e w id th does not necessarily occur a t th e highest o u tp u t pow er level o f th e laser, as p re d icte d by the S chaw low -Tow nes lin e w id th m odel. These results are in good q u a lita tiv e agreem ent w ith the results observed fo r single-m ode s e m ic o n d u c to r diode lasers, w h ich e x h ib it p o w e r-in d e p e n d e n t lin e w id th and lin e w id th re b ro a d e n in g w ith in creasing o u tp u t pow er, as discussed in C h a p te r I. In C h a p te r 5. th e results d e rive d in th e present ch a p te r and th e n e xt w ill be used to o b ta in an e s tim a te o f th e m in i m u m lin e w id th o f a q u a n tu m -w e ll se m ico n d u cto r d is trib u te d -fe e d b a c k laser th a t is in good agreem ent w ith p u b lis h e d e x p e rim e n ta l results. 3.1 R esonant am plifier lin ew id th w ith m u ltip lica tiv e noise T he tra n sm issio n b a n d w id th o f a resonant a m p lifie r w ith o u t m u ltip lic a tiv e noise decreases w ith increasing g a in , as seen fro m the sta n d a rd a n a lysis le a d in g to equa tio n (2.7). In the absence o f m u ltip lic a tiv e noise, a w id e b a n d n o isy in p u t signal tra n s m itte d th ro u g h a resonant a m p lifie r ju s t equals th e tra n s m is s io n b a n d w id th given by e q u a tio n (2.7 ). and becom es progressively m o re n a rro w as th e gain is in creased. In th e previous c h a p te r, a n a ly tic expressions fo r th e po w e r sp e ctra o f the resonant a m p lifie r o u tp u t fie ld a m p litu d e and phase flu c tu a tio n s w ere o b ta in e d in the sm a ll-a n g le regim e. T h e observed lin e w id th o f a signal tra n s m itte d th ro u g h the resonant a m p lifie r arises fro m th e a m p litu d e and phase flu c tu a tio n s o f the o u tp u t e le ctric field. In the presence o f m u ltip lic a tiv e noise, we shall assum e th a t th e to ta l observed resonant a m p lifie r lin e w id th is co m p rise d o f th e s ta tic p a rt g ive n b y e q u a tio n (2.7). plus a te rm due to m u ltip lic a tiv e noise. W e assume th e o u tp u t lin e w id th has the 7-5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fu n c tio n a l fo rm : = à u + 6u\,niiU- (-1-U T h e firs t te rm on th e rig h t is th e lin e w id th fro m th e sta n d a rd analysis w ith o u t m u ltip lic a tiv e noise [14]. discussed in the previous ch a p te r, given by e q u a tio n (2.7) VVe recall th a t th e net ro u n d -trip gain o f th e resonant a m p lifie r is defined as /v"o = pipnGl- T h e reader is re m in d e d th a t the present lin e a r m odel assumes th e net ro u n d -trip g a in h'o is less th a n u n ity . W hen k'o ~ I. we m ay w rite Chd) T T T Som e o th e r useful re la tio n s h ip s are: 2 ~ t - - 7T l ~ n l .\s th e net ga in is increased, th e o u tp u t pow er increases, and th e s ta tic lin e w id th o f eq u a tio n (3..3) decreases u n til th e m u ltip lic a tiv e noise lim ite d lin e w id th is reached. W hen th e m u ltip lic a tiv e lin e w id th flo o r is reached, th e lin e w id th th e n becomes constant w ith fu rth e r increases in o u tp u t pow er. W e n e xt c a lcu la te th e c o n trib u tio n to th e resonant a m p lifie r o u tp u t fie ld lin e w id th due to m u ltip lic a tiv e noise. by co nside rin g the phase flu c tu a tio n s im p a rte d to a m o n o c h ro m a tic laser in p u t field w ith fre qu en cy c o in cid e n t w ith a tra n sm issio n peak. B y th e W ie n e r-K h in c h in T h e o re m , the pow er s p e c tru m o f a s ta tio n a ry noise process is th e F o u rie r tra n s fo rm o f its a u to -c o rre la tio n fu n c tio n [26]. T h e resonant a m p lifie r o u tp u t fie ld linesh ap e is d e te rm in e d by th e pow er sp e ctru m o f th e o u tp u t field. For th e case o f m u ltip lic a tiv e noise w ith a w h ite pow er sp e ctra l d e n sity, the o u tp u t e le c tric fie ld a u to c o rre la tio n fu n c tio n can be d e rive d a n a ly tic a lly fro m th e results o f th e previous s m a ll-a n g le analysis. T h e o u tp u t fie ld lineshape is th e n o b ta in e d as th e F o u rie r tra n s fo rm o f th e fie ld a u to -c o rre la tio n fu n c tio n . 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For th e case o f a laser in p u t fie ld to th e resonant a m p lifie r p e rtu rb e d b y s m a ll- angle m u ltip lic a tiv e phase noise, th e a m p litu d e flu c tu a tio n s o f th e o u tp u t fie ld are sm a ll c o m p a re d to th e phase flu c tu a tio n s , so th e fie ld a u to -c o rre la tio n fu n c tio n is d e te rm in e d p r im a r ily by th e phase flu c tu a tio n s . B u t in th e large-angle re g im e , large a m p litu d e and phase flu c tu a tio n s o ccu r even fo r p u re m u ltip lic a tiv e phase noise, as seen in th e pre viou s ch ap ter. T h e re fo re , in th e large-ang le regim e, th e fie ld a u to c o rre la tio n fu n c tio n and lin e w id th are d e te rm in e d b y b o th th e phase and a m p litu d e flu c tu a tio n s . W e w ill consider o n ly th e s m a ll-a n g le re g im e in th e fo llo w in g analysis o f th e m in im u m resonant a m p lifie r lin e w id th . It is assum ed th a t th e la rg e -a n g le regim e w ill lead to increasing o u tp u t lin e w id th w ith increasing o u tp u t p o w e r. F ro m th e previou s sm a ll-a n g le a n a lysis, i f th e m u ltip lic a tiv e noise p o w e r spec tru m Sk(f] is pure phase noise (u n its o f [r a d '/H z ]) . th e o u tp u t flu c tu a tio n pow er s p e c tru m Scif) w ill also be pure phase flu c tu a tio n s . In th e lim it o f n e t gain ap p ro a c h in g u n ity for a G a u s s ia n -d is trib u te d noise source w ith variance such th a t th e s m a ll-a n g le a p p ro x im a tio n applies, it was show n p re v io u s ly th a t th e o u tp u t flu c tu a tio n po w e r s p e c tru m due to m u ltip lic a tiv e noise is give n by s * ( / ) (rad V H z|. (:i.ô) T h e in sta n ta n e o u s frequency flu c tu a tio n s o f th e fie ld are th e firs t fim e -d e riv a tiv t' o f th e in sta n ta n e o u s phase flu c tu a tio n s . T h e pow er s p e c tru m o f in sta n ta n e o u s fre quency flu c tu a tio n s o f th e o u tp u t fie ld , w h ic h we d enote as Sc.^if)- can th e re fo re be re la te d to th e pow er sp e ctru m o f phase flu c tu a tio n s as [27] [H z^/H z]. CTfil S u b s titu tin g e q u a tio n (3.5) for th e pow er s p e c tru m o f o u tp u t phase flu c tu a tio n s . ■ S ’c ( / ) . th e pow er s p e c tru m o f th e o u tp u t fre q u e n cy flu c tu a tio n s fo r n e t g ain /v’^ a p p ro a c h in g u n ity m a y be w ritte n T h e fa c to rs o f / * cancel, leaving Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S c.uif) = S h(f) [H z V H z ]. {:i.S) ( ~ t )- T h e o u tp u t fre q u e n cy flu c tu a tio n pow er s p e c tru m Sc.„(f). thus has th e sam e frequenc} dependence as th e m u ltip lic a tiv e noise source phase flu c tu a tio n pow er sp e c tru m S ’hif)- T h is is a n o th e r w ay o f s ta tin g th e e quivale nce o f the tim e -v a ry in g p a rtia l- wave m o d e l in th e lim it o f h igh net g a in w ith th e p h eno m e nological m odel discussed in th e p re vio u s c h a p te r, in w h ich th e o u tp u t fre q u e n cy flu c tu a tio n s are assum ed to be d ire c tly p ro p o rtio n a l to th e in tra -c a v ity phase flu c tu a tio n s . W e now d e riv e th e a u to c o rre la tio n fu n c tio n o f th e o u tp u t field o f th e resonant a m p lifie r p e rtu rb e d by m u ltip lic a tiv e phase noise, fro m w h ich th e o u tp u t fie ld pow er s p e c tru m is o b ta in e d th ro u g h the F o u rie r tra n s fo rm . F o llo w in g the develop m e nt o f P e te rm a n n [21]. th e fie ld a u to c o rre la tio n fu n c tio n m a y be derived in te rm s o f the fre q u e n cy (o r phase) flu c tu a tio n s o f th e fie ld w hen th e a m p litu d e flu c tu a tio n s are n e g lig ib le . Since we have eissumed a laser in p u t fie ld , th e a m p litu d e flu c tu a tio n s are suppressed due to gain s a tu ra tio n and th e o u tp u t fie ld is w e ll-a p p ro x im a te d as a c o n s ta n t-a m p litu d e fie ld w ith tim e -v a ry in g phase [14]. Since we are u ltim a te ly in te re s te d in a p p ly in g th is analysis to th e p ro b le m o f laser lin e w id th . th is a p p ro x i m a tio n is ju s tifie d . H ow ever, it is noted th a t it m ay n o t be a p p lica b le to o th e r in p u t signals. .A ssum ing a laser in p u t field, the resonant a m p lifie r o u tp u t field a u to c o rre la tio n fu n c tio n m a y be w ritte n [2 1 ]: (C(nr-(f - n) = ( .1 .1 ) ) w here ( P ) is th e average o u tp u t in te n s ity , (o ) is th e m ean frequency offset fro m th e m ode fre q u e n cy and (A o ^ ) is th e m ean square phase flu c tu a tio n w ith u n its o f [rad^]. ( A o * ) is re la te d to th e o u tp u t pow er s p e c tru m o f th e frequency flu c tu a tio n s S c A f ) as s i n ^ ( - / r ) r f T y 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. So. we m a y o b ta in th e a u to c o rre la tio n fu n c tio n o f the o u tp u t fie ld in te rm s o f the o u tp u t fre q u e n cy flu c tu a tio n p ow er s p e ctru m , w h ich is d e riv e d a n a ly tic a lly from th e s m a ll angle analysis o f th e p re vio u s chapter. T he pow er d e n s ity s p e c tru m o f the o u tp u t fie ld is the n o b ta in e d b y F o u rie r tra n s fo rm in g e q u a tio n (3 .9 ): W cif) = {C{t)C-{t - T ) ) e - ‘^ ^ '^ d T (3.111 w here a,’ = 2 ~ f is th e o p tic a l ra d ia n frequency offset fro m th e m o d e frequency These expressions re la tin g th e pow er d e n sity sp e ctru m o f th e e le c tric fie ld to the fre q u e n cy flu c tu a tio n pow er s p e c tru m are va lid o n ly w hen th e m u ltip lic a tiv e phase flu c tu a tio n s e x h ib it a G aussian p ro b a b ility d is trib u tio n . N ow . to e stim a te th e o u tp u t fie ld lin e w id th o f th e resonant a m p lifie r due to m u l tip lic a tiv e noise, consider th e sp e cia l case w hen th e pow er s p e c tru m o f m u ltip lic a tiv e flu c tu a tio n s Sk(f) is w h ite . E q u a tio n (3.8) then im p lie s th a t th e p ow er sp e ctru m o f o u tp u t fre que ncy flu c tu a tio n s . S c.uif)- is also w h ite . In th is case. S c.u(f ) = .S'c..,(0) is a c o n s ta n t, and th e in te g ra l fo r th e m ean-square phase flu c tu a tio n in equation ( 3.10) reduces to ( A o ‘ ) = | r | N’c.^/fO) (3.12) w here N'c.j/(0 ) is th e value o f th e o u tp u t fie ld fre q u e n c y -flu c tu a tio n p ow er spectrum at zero frecjuency. N ow . th e a u to c o rre la tio n fu n c tio n o f th e fie ld a m p litu d e m ay he w ritte n [2 1 ] {C(t)C'(t - T)) = (P )c<‘( o ) T ) ^ ( ^ ) (3.13) w here G = ~/Sc.u{0) is d e fin e d as th e coherence len g th o f th e fie ld . S u b s titu tin g th is expression in to e q u a tio n (3 .1 1 ) and p e rfo rm in g the F o u rie r tra n s fo rm yields the o u tp u t fie ld pow er sp e c tru m . rriiê ^ iiT w h ich has a L o re n tzia n line shap e centered a round the o p tic a l fre q u e n c y Thus, th e o u tp u t s p e ctru m due to m u ltip lic a tiv e noise has h a lf-p o w e r lin e w id th 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. àiy\mult = — = 3 2 [H z]. (3.1.5) T h is is th e c o n trib u tio n to th e resonant a m p lifie r o u tp u t lin e w id th due to m u lti p lic a tiv e phase noise w ith a w h ite power sp e ctra l d e n sity. T h e to ta l lin e w id th o f th e resonant a m p lifie r can fin a lly be expressed in the lim it o f net gain ho a p p ro a ch in g u n ity as th e sum o f th e s ta n d a rd and m u ltip lic a tiv e c o n trib u tio n s = ---------- — T - - j';i(0 ). (3.16) ~ r T h e firs t te rm , fro m th e s ta n d a rd analysis, decreases w ith in cre a sin g net gain A ,, whereas th e second te rm due to m u ltip lic a tiv e noise is a c o n s ta n t in th e lim it — > 1. . \ noise source th a t has a w h ite pow er s p e c tru m u p to in fin ite frequencies has a d e lta -fu n c tio n a u to c o rre la tio n fu n c tio n , w h ich is a useful m a th e m a tic a l m odel, b u t is no t p h y s ic a lly re a lis tic . .A m ore re a lis tic m o d e l fo r a noise process has an e x p o n e n tia l tim e -c o rre la tio n p ro p o rtio n a l to w ith c o rre la tio n tim e I / o . The F o u rie r tra n s fo rm o f an e x p o n e n tia lly -c o rre la te d noise source y ie ld s a Lorentzian pow er sp e ctra l d e n sity, w h ich is w h ite up to a F o u rie r fre q u e n cy o f a p p ro x im a te ly J = a . and th e n ro lls o ff as I / / " fo r higher frequencies. T h e n , fo r p ra c tic a l purposes, a w h ite noise source m a y be m odeled as h a v in g a c o rre la tio n tim e I / o th a t is shorter th a n a ll tim e scales o f in te re s t. T h is im p lie s th a t e q u a tio n (3.14) ju s t d erived for th e m u ltip lic a tiv e lin e w id th is va lid if th e c o rre la tio n tim e I / o o f th e frequency flu c tu a tio n noise process is m uch sh o rte r th a n th e coherence tim e o f th e o u tp u t field. Since th e coherence tim e is p ro p o rtio n a l to th e m a g n itu d e o f th e frequency flu c tu a tio n pow er s p e c tru m at zero frequency (fo r w h ite fre q u e n cy noise), we m ay w rite th is c o n d itio n as a >> ^ = Su\muit [H z]. (3.17) In th e present analysis, it is assum ed th a t th e ro llo ff fre q u e n cy a o f th e m u ltip lic a tiv e noise is a t least te n tim e s th e m a g n itu d e o f th e fre q u e n cy flu c tu a tio n pow er sp e ctru m 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. at zero fre q u e n cy, in o rd e r for e q u a tio n (3.14) fo r th e lineshape o f th e o u tp u t fie ld to be v a lid . W h e n Q < (i.e .. Sii(f) has a colored pow er s p e c tru m ), then the ap p ro x im a tio n s used to d e riv e e quatio ns (3.14) and (3.1.5) are no longer va lid and th e lin e sh a p e o f th e o u tp u t fie ld pow er s p e c tru m w ill n o t necessarily be L o re n tzia n . T h e lin e sh a p e m u s t th e n be c o m p u te d fro m th e F o u rie r tra n s fo rm o f th e a u to -c o rre la tio n fu n c tio n o f th e e le c tric held . T he e le c tric held a u to -c o rre la tio n fu n c tio n m ust hrst be d e riv e d in te rm s o f th e frequency h u c tu a tio n p o w e r s p e c tru m , b y p e rfo rm in g th e in te g ra tio n in d ic a te d in e q u a tio n (3.10) to o b ta in th e m ean-square phase. 3.2 A p p lica tio n o f tim e-v a ry in g partial-w ave m od el to laser lin ew id th In th is s e c tio n , th e re su lts o f the preceding s m a ll-a n g le analysis fo r th e effect o f m u ltip lic a tiv e noise in a resonant a m p lih e r w ill be a p p lie d to a lin e a r m odel o f a laser to d e riv e an a n a ly tic expression fo r th e laser lin e w id th . .Also, th e onset o f th e la rg e -a n g le re g im e w ill be co m p u te d in te rm s o f laser c a v ity param eters. T h e laser w ill be con sid ered to be a resonant a m p lih e r d riv e n b y an e q u iva le n t in p u t noise source c o rre s p o n d in g to th e spontaneous em ission in th e a c tiv e m e d iu m as o rig in a lly a n a lyze d b y S ch aw lo w and Townes [ 1]. and now c o m m o n ly found in sta n d a rd te x ts on lasers [14]. [15]. [16]. In th e absence o f m u ltip lic a tiv e noise, th is analysis leads to th e S ch a w lo w -T ow n e s fo rm u la th a t p re d icts th e in ve rse -p o w e r dependence o f th e laser lin e w id th . In th is m o d e l, th e presence o f th e co n sta n t a m p litu d e spontaneous em ission noise in p u t held causes th e net ro u n d -trip g ain to s a tu ra te a t a value s lig h tly less th a n u n ity , re s u ltin g in a h n ite lin e w id th fo r th e laser. T h e ra tio o f th e o u tp u t pow er to th e in p u t spontaneous em ission noise p o w e r d e te rm in e s th e value o f th e net ro u n d -trip g a in fo r a g ive n o u tp u t pow er level. In th is m o d e l, we consider o n ly net gains A" < 1. T h e w id e -b a n d spontaneous e m issio n noise o f th e o p tic a l gain m e d iu m is a m p lih e d w ith in th e n a rro w bandpass o f th e resonant a m p lih e r. y ie ld in g th e Iciser o u tp u t. T h u s , fo r h igh g a in , th e laser o u tp u t held m a y be view ed as a 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. wave o f c o n s ta n t a m p litu d e and frequency, on w h ic h s m a ll s ta tis tic a l a m p litu d e and phase flu c tu a tio n s due to the spontan eously e m itte d photon s are superim pose d. T h e m ore e xa ct "m o d ifie d Schaw low -Tow nes m o d e l" [5] takes in to acco u n t the d a m p in g effect o f th e s a tu ra te d gain above th re s h o ld on th e a m p litu d e flu c tu a tio n s , w h ich results in a fa c to r o f tw o decrease in th e lin e w id th above th re sh o ld com pared to th e S chaw low -T ow ne s fo rm u la . .A lthough th e lin e a r laser m odel considered here does not e x p lic itly in c lu d e d a m p in g o f the a m p litu d e flu c tu a tio n s , its s im p lic ity c le a rly illu s tra te s th e basic physics responsible fo r th e lin e w id th versus pow er c h a ra c te ris tic , and th e effects o f th e m u ltip lic a tiv e noise. In b o th th e o rig in a l and the m o d ifie d S chaw low -T ow nes laser m odels, the lin e w id th at a given o u tp u t p ow er level is seen to a rise fro m th e additive noise source o f th e sp o n ta n e o u sly e m itte d photons w hich add ra n d o m a m p litu d e and phase com ponents to the o u tp u t fie ld . Since the net gain is e s s e n tia lly u n ity above th e th re sh o ld o f laser o s c illa tio n , th e spontan eous em ission pow er level re m a in s cla m p e d at its sa tu ra te d th re s h o ld value. T h u s , th e a d d itiv e ra n d o m phase c o n trib u tio n o f th e spontan e ously e m itte d p h o to n s to th e o u tp u t field becom es less s ig n ific a n t as th e o u tp u t pow er in creases. re s u ltin g in an increasing s ig n a l-to -n o ise ra tio and decreasing lin e w id th . T h u s, by th e S chaw low -Tow nes m odel, it is p re d ic te d th a t th e lin e w id th o f a laser can be m ade a r b itr a r ily sm a ll by increasing th e o p tic a l o u tp u t pow er level. H ow ever, in th e presence o f m u ltip lic a tiv e noise, th e resonant a m p lifie r lin e w id th is lim ite d to a m in im u m value, as discussed p re vio u sly. In th is section, it w ill be shown th a t th is effect also d e te rm in e s the m in im u m lin e w id th o f th e laser, and th a t the o rig in a l S chaw low -T ow ne s lin e w id th m odel s tr ic tly a p plies o n ly in the lim it o f an ideal c a v ity w ith no m u ltip lic a tiv e noise. 3.2.1 M o d e l o f a laser as a n o ise-d riv en reso n a n t am plifier T he S ch aw lo w -T o w ne s fo rm u la m a t be o b ta in e d b y d e riv in g th e sa tu ra te d gain o f the resonant a m p lifie r fo r steady-state o p e ra tio n in te rm s o f the in p u t spontaneous em ission p o w e r an d th e o u tp u t power. T h e n , th is value o f th e s a tu ra te d gain is in serted in to th e fo rm u la fo r the resonant a m p lifie r lin e w id th . e q u a tio n (3 .3 ). y ie ld in g 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. th e S chaw low -T ow nes fo rm u la . In th e analysis th a t follow s, we d e riv e th e s a tu ra te d gain o f th e spontaneous em ission n o ise -d rive n resonant a m p lifie r in s im ila r fashion to th e s ta n d a rd tre a tm e n t [16]. B u t th e n we use instead th e generalized m odel for th e to ta l resonant a m p lifie r lin e w id th th a t includes th e c o n trib u tio n due to m u lti p lic a tiv e noise in the sm a ll-a n g le regim e, e q u a tio n (3.16). th a t was derived in the previou s se ctio n . T h is leads to an a d d itio n a l p ow er-inde pendent te rm in th e laser lin e w id th m o d e l. VVe firs t re vie w the m ode l o f th e laser as a noise-driven resonant a m p lifie r fo l lo w in g th e de ve lop m e nt in th e te x tb o o k o f V erdeyen [16], to d e riv e th e in trin s ic q u a n tu m lin e w id th o f th e laser in th e absence o f m u ltip lic a tiv e noise. VVe assume a lin e a r tw o -m irro r F a b ry-P e ro t c a v ity w ith equal re fle c tiv itie s . In th is m o d e l, the ste a d y -s ta te o u tp u t pow er P fro m b o th m irro rs is related to th e in te rn a l spontaneous em ission noise pow er P^p as w here R is th e pow er re fle ctio n co e fficie n t o f th e c a v ity m irro rs , so th a t R = p -. T = '•■ = 1 — /? is the pow er tra n sm issio n co e fficie n t o f th e m irro rs , and R^Ci^ is th e pow er g ain per rou nd tr ip . T h e q u a n tity I — R'G p is defined as th e "s a tu ra tio n a m p lific a tio n " o f the resonant a m p lifie r: I - R^G I = - 1 T ^ ( 3 .i t , VVe eq uate th e ro u n d -trip pow er g ain R~G~^ w ith A j . the square o f th e ro u n d -trip a m p litu d e g a in o f the resonant a m p lifie r fro m th e lin e a r analysis o f C h a p te r 2 . For self-sustained laser o s c illa tio n . Ko ~ 1 . so we m a y w rite Ko = I — (i. w here < < 1 so th a t I - R K ;l = I - = [ - (I - 5)- = I - (I - 2J) = 2( 1 - ^) = 2( 1 - Ko). (3.20) So we set I — R^G^ = 2(1 — R’o) in e q u a tio n (3.19) and o b ta in an expression fo r the s a tu ra te d a m p litu d e gain o f th e laser l - K o = T ^ (3.21) S3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For an o p tic a l g a in m e d iu m in an o p tic a l c a v ity w ith u pper e n e rg y level ZTj and lo w e r energy level betw een w h ic h th e o p tic a l gain tra n s itio n occurs, the spontaneous em ission pow er p e r c a v ity m ode o f th e E 2 — £"1 tra n s itio n can be w ritte n |I6 ].[1 5 ] P,p - n — ' —^ { G p - I) (d.2'2i 1 hi/ S 2 2 T .V, - ,Vi w here h is P la n ck's c o n s ta n t, u = ( E 2 — Ei)lh is th e o p tic a l fre q u e n cy o f th e tra n s itio n . T is th e ro u n d -trip tim e o f lig h t in th e c a v ity , and .Vb. . \ \ are th e p o p u la tio n o f th e u p p e r and low e r energy levels, re sp e ctive ly. (F o r s e m ic o n d u c to r lasers, the e v a lu a tio n o f .\*2 and . \ \ is n ot tr iv ia l: we w ill re tu rn to th is la te r in C h a p te r -5. 1 .-\s the net g a in R 'C ^ approaches u n ity , th e q u a n tity I — R~Gp approaches zero, and ve ry sm a ll changes in Gp w ill have a large effect on th e o u tp u t pow er le ve l. So for R 'G p a p p ro ach in g u n ity (i.e.. near th re s h o ld ), th e ab so lu te m a g n itu d e o f th e gain Gp changes very lit t le ove r a w id e range o f o u tp u t pow er, and is e s s e n tia lly equal to th e c a v ity losses. Since th e spontaneous em ission pow er is p ro p o rtio n a l to the gain Gp. it rem ains e sse n tia lly c o n s ta n t over a w id e range o f o u tp u t p o w e r levels. In s e rtin g th is in to th e expression fo r th e sa tu ra te d gain we o b ta in 1 — ~ i G p — I) . (d .2 .J) I h i/T .V2 P I t \ 2 - -V, .Assum ing ideal lossless h ig h -re fle c tiv ity m irro rs . £ = p - % 1 and T = \ — R. .Also, since th e g a in RGp ~ I. we m ay w rite ( G'p — I) % (l — R). M a k in g these s u b s titu tio n s yie ld s fo r th e s a tu ra te d a m p litu d e g a in o f th e n o ise -d rive n resonant a m p lifie r laser m odel. T h e generalized fo rm o f th e resonant a m p lifie r lin e w id th th a t in c lu d e s m u lti p lic a tiv e noise was d e riv e d p re v io u s ly as e q u a tio n (3.16): = -----------— + 3- 2 £ 0 ( 0 ) (3.25) 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w here 5 ^ (0 ) [ra d ^ /H z ] is th e m a g n itu d e o f th e w h ite m u ltip lic a tiv e phase noise pow er s p e c tra l d e n s ity at zero frequency. S u b s titu tin g th e s a tu ra te d g a in fro m e q u a tio n (3.24 ) fo r 1 — Ko yie ld s an expression fo r th e laser lin e w id th as a fu n c tio n o f th e o u tp u t pow er: VVe recognize the firs t te rm in parentheses as th e tra n s m is s io n b a n d w id th o f th e c a v ity w ith no gain m e d iu m , and equate th is w ith th e "c o ld -c a v ity " b a n d w id th I 1 — R)/~~ = 'iSi/o = 'lUjnjQ. w here i^ > th e o p tic a l fre q u e n cy o f th e c a v ity m ode, a n d Q is th e q u a lity fa c to r o f th e c a v ity . It is n oted th a t is e q u a l to l / 'p . o r th e inverse o f th e p h o to n life tim e o f th e c a v ity w ith th e g a in m e d iu m re m o ve d , w h ich is o n e -h a lf o f th e fu ll tra n s m is s io n b a n d w id th o f th e co ld c a v ity . S u b s titu tin g th is in to th e last expression o b ta in s -r.-------^ + T—5— 7 -"'o(O)- (3 .2 /) r V 2 — V 1 2ïï"^r- T h e firs t te rm o f th is expression is e q u iva le n t to th e o rig in a l fo rm u la derived b y S chaw low and Tow nes fo r th e lin e w id th o f th e laser, w h ich is c o rre c t up to th re s h o ld , and d isplays th e fa m ilia r inverse-pow e r dependence. T h e second te rm is seen to be inde pe n de nt o f th e o u tp u t pow er, and represents an a d d itio n a l lin e w id th c o n trib u tio n due to the m u ltip lic a tiv e phase noise in th e resonant c a v ity , as discussed p re v io u s ly . T h is expression is v a lid fo r values o f net g ain and m u ltip lic a tiv e noise va ria n ce such th a t the s m a ll-a n g le a p p ro x im a tio n is va lid . In th e id e a l case o f no m u ltip lic a tiv e noise, (i.e .. So(f) = 0). th e second te rm is zero, a nd th e S chaw low -T ow nes lin e w id th fo rm u la is recovered. H ow ever, i f m u lti p lic a tiv e noise is an in trin s ic p ro p e rty o f th e m a te ria ls c o m p ris in g th e laser c a v ity , th e laser is p re d ic te d to have a m in im u m in trin s ic lin e w id th . W e s h a ll discuss one in trin s ic noise m e cha nism fo r se m ic o n d u c to r q u a n tu m -w e ll lasers, e le c tro n n u m b e r- d e n s ity flu c tu a tio n s , in C h a p te r 4. S o Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3 S um m ary In th is c h a p te r, th e resu lts o f the tim e -v a ry in g p a rtia l-w a v e analysis for a resonant o p tic a l a m p lifie r w ere a p p lie d to derive expressions fo r th e lin e w id th o f a resonant a m p lifie r and a laser. T h e fin a l re su lt, e q u a tio n (3.27), represents a generalized version o f th e S chaw low -T ow nes lin e w id th fo rm u la th a t includes the effects o f m u l tip lic a tiv e noise. T h e o rig in a l Schaw low -Tow nes lin e w id th fo rm u la is recovered as th e special case o f no m u ltip lic a tiv e noise. T h is expression p re d ic ts th a t the laser lin e w id th should in itia lly decrease lin e a rly w ith in cre a sin g o u tp u t pow er, b u t w ill fin a lly reach a m in im u m value th a t is pro p o rtio n a l to th e m u ltip lic a tiv e noise s tre n g th a n d in d e p e n d e n t o f fu rth e r increases in o u tp u t pow er. I f th e gain is increased yet fu rth e r past th e onset o f th e large-angle regim e as d efin e d by e q u a tio n (2.24). then e q u a tio n (3.27) is no longer va lid . In th e large-an g le re g im e , th e lin e w id th broadens w ith in creasing net gain, co rrespond ing to th e large a m p litu d e and phase flu c tu a tio n s seen fro m th e n u m e rica l analysis o f th e p re v io u s c h a p te r. T h e th re e regim es o f lin e w id th versus o u tp u t pow er p re d icte d by th is m odel are c |u a lita tiv e ly s im ila r to w h a t is observed in sin g le -m o d e se m ico n d u cto r d iode lasers as discussed in th e In tro d u c tio n : decreasing lin e w id th w ith increasing pow er at low pow er levels, a p o w e r-ind e pendent lin e w id th flo o r at h ig h e r power levels, and an e v e n tu a l lin e w id th re-b ro a d e n in g for th e highest po w e r levels. T h is analysis the re fo re provide s a possible u n ifie d m odel to e xp la in th e p o w e r-inde pendent lin e w id th and lin e w id th re b ro a d e n in g ty p ic a lly observed in laser diodes [ 1 0 ] as a m em ory-effect process th a t can be caused by a single source o f m u ltip lic a tiv e phase and gain flu c tu a tio n s . 8 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In th e n ext ch a p te r, th e pow er sp e c tru m o f m u ltip lic a tiv e phase noise d u e to e le ctro n d e n s ity flu c tu a tio n s in a se m ico n d u cto r w ill be ca lcu la te d . T h is represents a fu n d a m e n ta l noise source in se m ico n d u cto r lasers, a risin g fro m s im ila r o rig in s as .Johnson noise in e le ctro n ic c irc u its . In C h a p te r 5. the results o f th e present ch a p te r w ill be used in c o n ju n c tio n w ith th e e le c tro n -d e n s ity -flu c tn a tio n c a lc u la tio n o f C h a p te r 4 to e s tim a te th e m in im u m fu n d a m e n ta l lin e w id th o f a ty p ic a l q u a n tu m - w ell s e m ic o n d u c to r laser. S7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 O ptical phase fluctuations due to electron density fluctuations in a sem iconductor laser 4.1 In trod u ction Local tim e -v a ry in g e le ctro n d e n s ity flu c tu a tio n s o ccu r in the e le c tro n gas fillin g a se m ic o n d u c to r laser c a v ity due to th e th e rm a l m o tio n o f the e le ctro n s. These local flu c tu a tio n s in c a rrie r d e n s ity cause co rre s p o n d in g flu c tu a tio n s in th e o p tic a l in d e x o f re fra c tio n o f th e s e m ic o n d u c to r m e d iu m , w h ic h m odulates the phase o f a lig h t beam p ro p a g a tin g th ro u g h the m e d iu m . T h e th e rm a l m o tio n o f the e le ctro n s im p lie s th a t in any fixe d v o lu m e T at co n sta n t te m p e ra tu re T . the electron n u m b e r w ill va ry in tim e [29]. [31]. w ith th e m a g n itu d e o f th e flu c tu a tio n s being in v e rs e ly p ro p o rtio n a l to th e vo lu m e . In a regenera tive syste m such as a laser diode, th is e ffe ct represents a m u ltip lic a tiv e noise te rm th a t sh o u ld be considered in the a n a lysis o f th e laser o u tp u t phase flu c tu a tio n s . In s o lid -s ta te s e m ic o n d u c to r plasm as, th e s c a tte rin g o f lig h t by e le c tro n n u m b e r- d ensity. sp in -d e n s ity . and energy d e n s ity flu c tu a tio n s has been s tu d ie d p re v io u s ly [32]. [33]. [34]. [35]. P re vio usly. W e lfo rd and M o o ra d ia n [8 ] e s tim a te d th e re fra c tiv e in d e x flu c tu a tio n s in a s e m ic o n d u c to r laser ga in m e d iu m fro m th e m e a n -sq u a re elec tro n n u m b e r-d e n s ity flu c tu a tio n s , as d e riv e d fro m th e rm o d y n a m ic c o n s id e ra tio n s , and used th is value to e x p la in th e ir o b se rva tio n s o f p o w e r-in d e p e n d e n t lin e w id th . 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. H ow e ve r, th e ir analysis d id n o t y ie ld in fo rm a tio n on th e pow er s p e c tra l d e n s ity o f th e m u ltip lic a tiv e phase flu c tu a tio n s . Lang. V ah ala. and Y a riv [13] la te r ca lc u la te d the d e ta ile d pow er s p e c tra o f th e in te n s ity and freq uen cy flu c tu a tio n s in a b u lk se m ico n d u cto r laser d io d e b y s o lv in g th e s p a tia lly v a ry in g e q u a tio n s o f m o tio n fo r the c a rrie r d e n sity, te m p e ra tu re d is tr ib u tio n . and o p tic a l fie ld . T h e y show ed th a t because th e e le ctro n d e n s ity in th e b u lk a c tiv e region o f th e laser d io d e is c la m p e d above th re sh o ld at a p p ro x im a te ly th e ro u n d -trip loss level due to s tim u la te d em ission, the c a rrie r-d e n s ity flu c tu a tio n s in th e a c tiv e region do n o t c o n trib u te s ig n ific a n tly to o p tic a l phase flu c tu a tio n s o f th e o u tp u t fie ld . T h e ir an alysis showed in ste a d th a t s p a tia lly -v a ry in g te m p e ra tu re flu c tu a tio n s m ay c o n trib u te s ig n ific a n tly to th e low -frequen cy p o rtio n o f b o th th e in te n s ity and freq ue ncy flu c tu a tio n sp e ctra , since these flu c tu a tio n s are not s im i la rly d a m p e d . T h e d o m in a n t sources o f te m p e ra tu re flu c tu a tio n s w ere assum ed to be n o n -ra d ia tiv e re c o m b in a tio n and o p tic a l a b so rp tio n events. T h e y assum ed th a t te m p e ra tu re flu c tu a tio n s w ere co upled to th e o p tic a l field a m p litu d e v ia gain de pendence on te m p e ra tu re , and to th e o p tic a l phase v ia re fra c tiv e in d e x dependence' on te m p e ra tu re . T h e ir analysis d iffe re d fro m th e present tre a tm e n t, since th e y w ere c o n sid e rin g th e case o f a b u lk laser d io d e , in w h ich th e a c tiv e region and o p tic a l m o d e volum es s u b s ta n tia lly o ve rla p p e d . H ow ever, in q u a n tu m -w e ll laser d io d e s tru c tu re s , a s ig n if ic a n t p o rtio n o f th e o p tic a l m ode ty p ic a lly lies outside o f th e a c tiv e region o f th e q u a n tu m w ell stack, in th e o p tic a l co n fin e m e n t region. T he tra n s p a re n t o p tic a l con fin e m e n t region does n o t s u p p o rt s tim u la te d em ission o r a b s o rp tio n , so th e e le ctro n d e n s ity here is not c la m p e d as it is in th e a c tiv e region o f th e q u a n tu m w ells. F ig u re 4.1 in d ica te s th e s tru c tu re o f a ty p ic a l q u a n tu m w ell laser considered in th is analysis. S ince th e ca rrie rs are in je c te d th ro u g h th e confinem en t region on th e w ay to th e q u a n tu m w ells, th is re g ion ty p ic a lly has a h ig h density o f electrons in th e co n d u c tio n band whose d e n s ity flu c tu a tio n s m a y m o d u la te the phase o f th e o p tic a l m ode. T h e re fo re , th e e le c tro n d e n s ity flu c tu a tio n s in the confinem en t re gion m a y be ex- 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. cladding region transparent optical confinement region 0.2 pm single quantum well optical mode active region, 6.5 nm thick InGaAsP waveguide and active region 200 nm k InP InP J L single quantum well 6.5 nm Figure 4.1: S ingle-quantum -w eU laser s tru ctu re considered in the analysis o f electron d en sity flu ctu atio n s. 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. pected to play a s ig n ific a n t role as a source o f m u ltip lic a tiv e noise in q u a n tu m well lasers. W ei ford and M o o ra d ia n 's [S] e s tim a te o f the m ean-square re fra c tiv e index flu c tu a tio n s represents th e in te g ra l o f th e pow er-spectral d e n s ity o ve r a ll frequencies. H ow ever, th e ir e s tim a te is based on th e rm o d y n a m ic co n sid e ra tio n s, and gives no in fo rm a tio n about th e po w e r s p e c tru m o f the flu c tu a tio n s . T h e d e ta ile d shape o f th e a c tu a l power s p e c tru m depends on th e dynam ics o f th e e le c tro n m o tio n in the plasm a. .As was seen in th e previous chapters, the shape o f th e pow er sp e ctru m o f th e m u ltip lic a tiv e phase flu c tu a tio n s w ill d e te rm in e th e pow er sp e c tru m o f the o p tic a l frequency flu c tu a tio n s o f th e laser. T herefore, to c a lc u la te th e pow er spec tru m o f m u ltip lic a tiv e phase noise due to electron d e n s ity flu c tu a tio n s , th e d e tailed m o tio n o f the electrons m u s t be considered. G le n n [I] has ca lcu la te d th e pow er sp e ctra o f phase flu c tu a tio n s fo r lig h t propa g a tin g in an o p tic a l fib e r d u e to d e n s ity and te m p e ra tu re flu c tu a tio n s . G lenn char acterizes the t h e rm o d y n a m ic phase flu c tu a tio n s as th e o p tic a l e q u iv a le n t o f e le ctrica l .Johnson noise in resistors, a n d has show n th a t in even a o n e -m e te r le n g th o f o p ti cal fib e r, these phase flu c tu a tio n s can be larger th a n th e s ta n d a rd q u a n tu m lim it. These re fra c tiv e in d e x flu c tu a tio n s are a d ire ct result o f th e F lu c tu a tio n -D is s ip a tio n T h e o re m [2]. and th e re fo re represent a fu n d a m e n ta l noise source in o p tic a l system s. In th is chapter, th e m e th o d o f G le n n is adapted to c a lcu la te th e pow er sp e ctra l den s ity o f o p tic a l phase flu c tu a tio n s in d u ce d by electron d e n s ity flu c tu a tio n s in a solid sta te plasm a ty p ic a l o f th e c o n fin e m e n t region in a q u a n tu m -w e ll se m ico n d u cto r laser diode. .As w ill be show n , fo r th e c a rrie r d e n sity ty p ic a lly re q u ire d to achieve o s c illa tio n th re sh old in a s e m ic o n d u c to r laser, the effect o f u n d a m p e d ele ctro n den s ity flu c tu a tio n s in th e n o n -la sin g tra n s p a re n t confinem en t re gion on th e phase o f a lig h t beam passing th ro u g h th e se m ico n d u cto r is not n e g lig ib le . .Also, since the m ean-square value o f th e e le c tro n d e n s ity flu c tu a tio n s scales in ve rse ly w ith the vo l um e, its effect is m ost e v id e n t in p h y s ic a lly sm all s tru c tu re s such as se m ico n d u cto r lasers, as com pared to large-scale gas o r solid-state lasers. T h e ch ap ter is organized as follow s: F irs t, the m echanism fo r phase m o d u la tio n 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o f lig h t in a s e m ico n d u cto r due to e le ctro n d e n s ity flu c tu a tio n s is review ed. T he m ean-square flu c tu a tio n s o f e le c tro n d e n s ity in an e le c tro n p la sm a are th e n com p u te d fro m s ta tis tic a l m echanics co n sid e ra tio n s. F in a lly , th e m e th o d o f G le n n is a d a p te d fo r c o m p u tin g th e pow er s p e c tru m o f th e e le c tro n d e n s ity flu c tu a tio n s in co n fin e m e n t regions o f a q u a n tu m -w e ll s e m ic o n d u c to r laser. F lu c tu a tio n s in th e ac tiv e regio n are assum ed to be d a m p e d above th re s h o ld due to s tim u la te d em ission, and are the re fore n e g lig ib le [13]. T h e tim e a u to c o rre la tio n fu n c tio n o f th e e le ctro n d e n s ity is d e te rm in e d fo r each s p a tia l m ode (i.e .. w a ve n u m b e r) o f th e d e n s ity va ria tio n s. based on th e a ssu m p tio n th a t th e e le ctro n m o tio n is de scrib e d by a d iffu s io n e q u a tio n . T h e pow er s p e c tra l d e n s ity o f th e e le c tro n flu c tu a tio n s for each m ode is o b ta in e d by ta k in g th e F o u rie r tra n s fo rm o f th e m o d a l a u to c o rre la tio n fu n c tio n . T h e to ta l pow er s p e c tra l d e n s ity is o b ta in e d by in te g ra tin g th e m o d a l pow er spec tra l d e n s ity over a ll s p a tia l m odes. T h e a b so lu te m a g n itu d e o f the to ta l pow er s p e c tru m is o b ta in e d v ia th e n o rm a liz a tio n c o n d itio n th a t th e in te g ra l o f th e pow er sp e ctra l d e n s ity over a ll F o u rie r frequencies m u st be e q u a l to th e m ean square value o f th e flu c tu a tio n s as d e te rm in e d fro m s ta tis tic a l m echanics co n sid e ra tio n s. From th is proced ure , an e s tim a te o f th e pow er s p e c tru m o f m u ltip lic a tiv e flu c tu a tio n s due to e le ctro n d e n s ity flu c tu a tio n s is o b ta in e d th a t is s u ita b le fo r s u b s titu tio n in to th e th e o re tic a l m o del for laser lin e w id th developed in th e p re vio u s chapters. In th e fo llo w in g ch a p te r, th e pow er s p e c tru m d e rive d in th e present ch a p te r w ill be used to e s tim a te th e effect on th e lin e w id th o f a ty p ic a l sin g le -m o d e q u a n tu m -w e ll se m ic o n d u c to r laser. 4.2 E lectron D e n sity to O p tical P h a se C onversion VVe now re vie w some m echanism s fo r phase m o d u la tio n o f a lig h t beam in a q u a n tu m - w e ll s e m ic o n d u c to r laser due to a change in th e lo ca l e le c tro n d e n sity. T h e e le ctro n d e n s ity flu c tu a tio n s w ill affect th e lig h t d iffe re n tly in th e q u a n tu m -w e ll activée re gion, co m p a re d to the tra n s p a re n t o p tic a l co n fin e m e n t region. In th e activée gain region o f a se m ico n d u c to r laser, a change in th e c a rrie r d e n s ity causes a c o rre sp o n d in g change in th e net gain due to in te r-b a n d tra n s itio n s . .As discussed p re v io u s ly , flu c tu a tio n s 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. o f th e c a rrie r d e n s ity in th e a c tiv e region are h e a v ily d a m p e d w hen th e device is o p e ra te d above th e th re sh o ld o f lasing. because th e g ain is clam ped at th e th re sh o ld value. H ow ever, we note th a t i f th e device is o p e ra te d as an o p tic a l a m p lifie r, th e e le c tro n d e n s ity flu c tu a tio n s in th e a c tiv e re g io n m ay n o t be dam ped. In ge ne ra l, a lig h t beam o f w a ve le n g th p ro p a g a tin g th ro u g h a se m ico n d u c to r slab o f le n g th L w ill a cq u ire phase p ro p o rtio n a l to th e local re fra c tiv e index a cco rd in g to o(n) = - — ^ — - [radians] (-1. 1 ) Ac w here fi{n) is th e re fra c tiv e in d e x o f th e m e d iu m th a t is assum ed to be a fu n c tio n o f th e e le c tro n density, n. w h ich has u n its o f cm "'^. VVe w ill now consider in tu rn the effects o f c a rrie r d e n sity flu c tu a tio n s in th e a c tiv e region and th e co n fin e m e n t region. In th e a c tiv e region, th e ra tio o f th e changes in th e real and im a g in a ry pa rts o f th e re fra c tiv e in d e x due to e le ctro n d e n s ity is defined as th e lin e w id th enhancem ent fa c to r [2 2 ]: dgjdn w here A '„= 2 - / A ^ is the free-space w a ve n u m b e r. g is th e g a in , and n is th e mean value o f th e c a rrie r density. In v e rtin g th is e q u a tio n , th e re fra c tiv e index dependence on th e e le c tro n d e n s ity flu c tu a tio n dn in th e a c tiv e region m a y be o b ta in e d : To firs t o rd e r, the change in g ain above th re sh o ld w ith c a rrie r d e n s ity m ay be assum ed to be lin e a r according to ^ = a^{n — nth) w here nth is the c a rrie r d e n sity re q u ire d to achieve lasing th re s h o ld , and — dg/dn and has ty p ic a l m easured value in a b u lk g a in m e d iu m o f 2.5 x 10"^^ cm ^ [22] and fo r a single 6 n m InG a.As q u a n tu m -w e ll w ith 1 percent s tra in is on th e o rd e r o f 10"^^ cm ^ [40]. So. i f th e c a rrie r d e n s ity flu c tu a tio n s in a given vo lu m e o f se m ic o n d u c to r are kn o w n , the o p tic a l phase 9.1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. flu c tu a tio n s im pressed upon a lig h t beam can be o b ta in e d by c o m b in in g e quatio ns (4.1) an d (4 .3 ) to o b ta in = (- U A J \ 2k T h is m a y be s im p lifie d to y ie ld : ^C^fictive — dn., (4.5) T h e m ean-square phase flu c tu a tio n s are re la te d to th e m ean-square e le c tro n d e n sity flu c tu a tio n s in th e a ctive region ( A n ) - as k Cl ,j ( A o . . , ...)■ -= (^ ( A n ) T ( 4.Ü ) T h is fo rm o f th e e quatio n is v a lid fo r th e s m a ll range o f w avelengths w h ic h have g ain w ith dependence on c a rrie r d e n s ity g iv e n by a^. T h e w aveleng th dependence o f th e phase is im p lie d in th e p ro d u c t 3c a^. W e co n sid e r n ext the tra n sp a re n t c o n fin e m e n t re g io n , in w hich th e b a n d gap is la rg e r th a n th e p h o to n energy at th e laser fre q u e n cy so th a t th e e lectrons in th e con d u c tio n ban d sca tte r photons v ia in tra -b a n d processes, i.e., fre e -ca rrie r s c a tte rin g . These free c a rrie rs c o n trib u te a te rm Ae^ to th e to ta l p e r m ittiv ity o f th e con fin e m e n t region g ive n by [39] ^ (4.7) w here q is th e e le ctro n ic charge. m ‘ is th e e ffe c tiv e mass o f th e e le ctro n s in tin ' c o n d u c tio n b a n d , and uj is th e o p tic a l frequency. T h e to ta l d ie le c tric p e r m ittiv ity is the sum o f th is te rm and th e b a ckg ro u n d d ie le c tric p e r m ittiv ity , ct,. w h ic h is equal to th e sq u a r‘d o f the index o f re fra c tio n in th e c o n fin e m e n t region = E = + A cg. (4.8) Since Ae^ is ty p ic a lly m uch less th a n u n ity , th e to ta l re fra c tiv e in d e x in the co n fin e m e n t region m ay be w ritte n as 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I , 1 Ate i / — ------— % 4- (4.9) V ^ 6 - M » - u’here is th e b a ckg ro u n d value o f th e in d e x o f re fra ctio n in th e c o n h n e n ie n t region. VVe assum e th a t th e change o f re fra c tiv e in d e x due to a change in th e e le c tro n density n m a y be o b ta in e d by d iffe re n tia tin g e q u a tio n (4.9) w ith respect to n. w h ich yields df^conf _ 1 ^ ( A e J _ -q - _ q - \ l 1 ~ 7:--------------1;-----------— T — ---------------------T - ( - t . i u ) on 2f.li, on fitjm'u:-^ A -fihm 'c- For G aA s and In P . fit, % 3.5, m ' = O.OTtUg [39]. and at = l.ô ô ^ a n . we o b ta in àficonf/dn % 2.2 X I0 ~ ’ ‘ [cm'^j. T he phase flu c tu a tio n s in th e c o n fin e m e n t region are th e n g iven by ^Oconj = dn % - ^ dn. (4.11) Ao on 'lfLt,m'c^ T h is re su lts in a m ean-square phase flu c tu a tio n o f 4 \2 r 2 (A o c o ./)- % v ( A n ) T (4.12) It is in te re s tin g to n o te th a t the phase flu c tu a tio n s are in v e rs e ly p ro p o rtio n a l to the square o f th e rest energy o f a reduced-m ass e lectron in th e c o n d u c tio n band. T h e re fo re , in m a te ria ls w ith low e le ctro n effective mass, w h ich is co n siste n t w ith high-speed m o d u la tio n c a p a b ility , th e in trin s ic phase flu c tu a tio n s due to electron d e n s ity flu c tu a tio n s in th e confinem en t region w ill be g re a te r th a n in m a te ria ls w ith h ig h e r e le c tro n e ffe c tiv e mass. Fhe to ta l o p tic a l phase flu c tu a tio n s in th e se m ico n d u cto r m e d iu m o ccu p ie d by the o p tic a l m ode w ill be the sum o f th e a c tiv e region c o n trib u tio n and th e con fin e m e n t regio n c o n trib u tio n . T he phase noise c o n trib u tio n o f each region is pro p o rtio n a l to its re la tiv e cross-sectional area tim e s th e o p tic a l fie ld a m p litu d e in the region. T h is m a y be w ritte n in te rm s o f th e n o rm a lize d o v e rla p in te g ra l o f the o p tic a l m ode w ith th e a c tiv e region. F : dOtot = ^dOactive + (1 ~ F)dO con/m f (4.13) 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. W h e n th e o p tic a l m o de is t ig h tly g u id e d in th e a c tiv e re g io n , i.e .. F % I. w h ich is th e case w ith m o st b u lk in d e x -g u id e d s e m ico n d u cto r lasers, th e n th e c a rrie r d e n sity flu c tu a tio n s w ill no t a ffect th e to ta l phase s ig n ific a n tly above th re s h o ld , due to the d a m p in g effect discussed p re v io u s ly . H ow ever, in q u a n tu m -w e ll lasers, th e a ctive q u a n tu m -w e ll g a in region o n ly o ve rla p s a v e ry sm a ll p o rtio n o f th e to ta l fie ld cross- section. so th a t r < < I. In th is case, th e to ta l phase flu c tu a tio n s a re d o m in a te d by the second te rm o f e q u a tio n (4 .1 3 ). due to the u n d a m p e d d e n s ity flu c tu a tio n s in th e confinem en t regio n. W e now tu r n to th e c a lc u la tio n o f th e m e a n -sq u a re e le ctro n d e n sity flu c tu a tio n s in th e c o n fin e m e n t re g io n , and th e c o rre s p o n d in g p o w e r sp e ctra l density. 4.3 E lectron D e n s ity F lu ctu a tio n s in a S em icon d u ctor In th is section, th e pow er s p e c tra l d e n s ity o f phase flu c tu a tio n s d u e to lo c a l e le ctro n de n sity flu c tu a tio n s in a s e m ic o n d u c to r is ca lcu la te d . T h e p ro c e d u re used is along the same lines as reference [1]. as fo llo w s: F irs t, the m e a n-squa re flu c tu a tio n o f the electron d e n s ity is d e riv e d fro m th e rm o d y n a m ic co n sid e ra tio n s. T h e m ean-square value is a c tu a lly th e s p a tia l average o f th e flu c tu a tin g q u a n tity o v e r th e vo lu m e under co n sid e ra tio n , and re presents th e w id th o f a G aussian d is tr ib u tio n centered at th e m ean value o f th e d e n s ity . T h e d e ta ile d d yn a m ics o f th e c o n s titu e n ts o f the electro n gas m u s t be c o n sid e re d to d e te rm in e the b e h a v io r o f th e syste m in tim e . To th is end. th e tim e a u to c o rre la tio n fu n c tio n o f th e e le c tro n d e n s ity is d e te rm in e d for each s p a tia l m ode (i.e .. w a v e n u m b e r) o f th e d e n s ity v a ria tio n s , based on the assum ption th a t th e e le c tro n m o tio n is described by a d iffu s io n e q u a tio n . For a s ta tio n a ry process, th e a u to c o rre la tio n fu n c tio n and p ow er s p e c tra l d e n s ity fo rm a F ourier tra n s fo rm p a ir. T h e p o w e r s p e c tra l d e n s ity o f th e e le c tro n flu c tu a tio n s fo r each m ode is th u s th e F o u rie r tra n s fo rm o f each m o d a l a u to c o rre la tio n fu n c tio n . T h e to ta l pow er s p e c tra l d e n s ity is o b ta in e d b y in te g ra tin g th e m o d a l p o w e r sp e ctra l d e n sity over a ll m o d a l w a v e n u m b e rs. T h e absolute m a g n itu d e o f th e to ta l pow er sp e ctru m is o b ta in e d v ia th e n o rm a liz a tio n c o n d itio n th a t th e in te g ra l o f th e pow er 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sp e c tra l d e n s ity over a ll F o u rie r frequencies m u s t be equal to th e m ean square value o f th e flu c tu a tio n s as d e te rm in e d fro m th e rm o d y n a m ic co n sid e ra tio n s. 4 .3 .1 M ea n -S q u a re D e n sity F lu ctu a tio n s W e now c a lc u la te th e m ean-square e le c tro n d e n s ity flu c tu a tio n s in th e tra n s p a re n t o p tic a l co n fin e m e n t region fro m th e rm o d y n a m ic considerations. B o ltz m a n n s ta tis tics (as opposed to F e rm i-D ira c s ta tis tic s ) m a y be used to describe a gas o f id e n tic a l p a rtic le s p ro v id e d th e m ean se p a ra tio n betw een the p a rticle s is less th a n th e ir de- B ro g lie w a ve le n g th [31]. In a s e m ic o n d u c to r laser, the m a x im u m c a rrie r d e n s ity in th e a c tiv e region is bo un d ed b y th e value re q u ire d to achieve th re s h o ld . .A d d itio n a l in je c te d ca rrie rs above th is level re co m b in e v ia s tim u la te d em ission to s u p p o rt the p h o to n p o p u la tio n in th e la sin g m ode. T h e c a rrie r density in th e s u rro u n d in g tra n s pa re n t o p tic a l co n fin e m e n t region is n o t s im ila rly clam ped, and increases lin e a rly w ith incre asing c u rre n t. T h e c a rrie r d e n s ity in th e tra n s p a re n t o p tic a l confinem ent region m a y be ca lcu la te d fro m c u rre n t co n se rva tio n as fo llo w s. S e ttin g the c u rre n t flo w in g th ro u g h th e co n fin e m e n t region Iconf eq ual to th e c u rre n t flow ing in to th e a c tiv e re gion I.ictu.-,-. and assum ing no s ig n ific a n t c a rrie r leakage, we have ^tot'll — Iconf — fictive. — T Istim (4.1-1) w here Itotai is th e to ta l c u rre n t. Ith is th e cu rre n t re q u ire d to reach th e lasing th re s h o ld , and Istim is th e p o rtio n o f c u rre n t th a t goes in to s tim u la te d em ission. W ritin g th e th re s h o ld c u rre n t in te rm s o f c a rrie r density, c a v ity v o lu m e , and c a rrie r life tim e , we o b ta in : r a c tiv e _ Ith = ----------------- (4.1.-)) T e w here and rith are th e v o lu m e and c a rrie r density a t th re s h o ld in th e a c tiv e region, and is th e c a rrie r life tim e fo r e le ctro n s in the a c tiv e region a t th re s h o ld . S im ila rly , th e c u rre n t in th e tra n s p a re n t co n fin e m e n t region can be w ritte n : 9/ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = 14.l,i I w here l ’ on/ and n ^ o n / are the vo lu m e and c a rrie r d e n s ity at th re sh o ld in th e tra n s p a re n t con fin em en t region. T h e n , we m ay re w rite e q u a tio n (4.14) to o b ta in ^ ^ c o n f ^ c o n f Q ^ a c t i v e ^ t h r , , —. — I s t i m • Tç Tf, S o lv in g fo r r i c o n f yield s ^ IC flL T , r ( I tO l ^con f — »- ^th y w ■ ^ stiTTi' ( 4 . 1 b ) ^ conf Ç ^conf W ritin g the c u rre n t responsible fo r s tim u la te d em ission in te rm s o f the to ta l c u rre n t m in u s th e th re s h o ld c u rre n t o b ta in s r K - o n f = - p " ( A 4 p ( f t o t a l ~ I t h ) - 1 4 . 1 9 ) ^ ron/ Q* con f a nd a fte r some s tra ig h tfo rw a rd m a n ip u la tio n s , we o b ta in a s im p le expression for th e c a rrie r d e n s ity above th re s h o ld in th e tra n sp a re n t c o n fin e m e n t region: I l i T f l L P i t o t a l , , , 'Icon! = -p " t h —.----- (4.JU) *con/ ^ t h In th e ty p ic a l q u a n tu m w e ll laser considered here, th e a c tiv e region com prises 4 percent o f the cross-sectional area o f th e o p tic a l m ode. T h e re fo re , in the re m a in d e r o f th is analysis, we w ill m ake the a ssu m p tio n th a t n „ n / ~ O .O dn,,^///(/.. _\t th re sh o ld , a q u a n tu m w ell has a ty p ic a l areal d e n sity o f a p p ro x im a te ly 10'^ [c m ~ '] [40]. For th e q u a n tu m w e ll thickness o f 6.5 n m considered here, th is im p lie s a volum e d e n s ity at th re s h o ld o f ~ 1 5 x 10'^ cm~'^. T h e c a rrie r d e n s ity in th e q u a n tu m wells w ill be cla m p e d at the th re s h o ld value. H ow ever, un d e r ty p ic a l o p e ra tin g co n d itio n s for hig h -p o w e r s e m ic o n d u c to r lasers, th e m a x im u m o p e ra tin g c u rre n t m ay be five to te n tim e s the th re sh o ld value, so th e c a rrie r d e n s ity in th e tra n sp a re n t non-lasing co n fin e m e n t region w ill increase by th is fa cto r. .A ssum ing a m a x im u m o p e ra tin g c u rre n t o f l O I t h - t hen we m a y c o m p u te th e range o f c a rrie r d e n s ity in the confinem ent region over th e o p e ra tin g range o f I t h to l O I t h fro m e q u a tio n (4.20): 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 X 10“ ^ < < 6 X 10‘ \ (4.21) For B o ltz m a n n s ta tis tic s to be a p p lic a b le to th e c a rrie r d e n sity analysis, th e m a x im u m p a rtic le d e n s ity in th e co n fin e m e n t region m u st be below the level given by w here m is th e mass o f th e p a rticle s, k is B o ltz m a n n 's c o n s ta n t. T is the te m p e ra tu re , and h is P la n ck's c o n s ta n t [31]. For a gas o f e le ctro n s in a b u lk se m ico n d u cto r at room te m p e ra tu re , th e m a x im u m d e n s ity is c a lc u la te d fro m e quatio n (4.22) to be a p p ro x im a te ly 10'^ cm ~^. T h e m a x im u m e le c tro n d e n s ity in th e confinem ent region ca lcu la te d fro m e q u a tio n (4.21) is a p p ro x im a te ly 6 x 1 0 '' c m " ', so the c o n d itio n o f (4.22) is satisfie d, a n d e le c tro n gas in th e co n fin e m e n t regions o f the se m ico n d u cto r laser m ay be tre a te d using B o ltz m a n n s ta tis tic s . N ow we w ill firs t c a lc u la te th e m ean square value o f th e e le ctro n d e n sity flu c tu a tio n s. In a give n fix e d v o lu m e V o f a la rg e r v o lu m e c o n ta in in g th e electron gas. th e absolu te n u m b e r o f e le ctro n s at any in s ta n t in tim e w ill flu c tu a te about its mean value .\ due to th e th e rm a l m o tio n o f th e e le ctro n s. T h e vo lu m e \ we consider is defined by th e reg ion o f tra n s p a re n t s e m ic o n d u c to r in te rc e p te d b y the guided o p tic a l wave, w h ich is ju s t th e v o lu m e o f th e co n fin e m e n t region. T h e electron d e n sity in the fixed vo lu m e , d e n o te d n = .V /V . m ay be d iffe re n tia te d w ith respect to vo lu m e to y ie ld dn .V n ÿ ÿ = - ÿ ï = “ F - T h is m a y be rea rra n g e d , squared, and tim e -a ve ra g e d to o b ta in the m ean-square d e n sity flu c tu a tio n s ( A n ) ^ in te rm s o f th e m ean square flu c tu a tio n s o f the vo lu m e occupied by th e .V p a rtic le s (A n)-^ = ( - j ( A V ) T (4.24) 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It was assum ed th a t th e v o lu m e V’ o f th e e le c tro n gas in te rc e p te d by th e guided o p tic a l w ave is s m a lle r th a n a la rg e r v o lu m e th a t c o n ta in s th e e le ctro n gas. Even th o u g h th e c o n fin e m e n t s tru c tu re o f th e s e m ic o n d u c to r laser o fte n co n ta in s the e le c tro n gas a n d th e o p tic a l fie ld in th e sam e p h y s ic a l v o lu m e , th e e le ctro n gas is s im u lta n e o u s ly b e in g de p le te d by spontaneous and s tim u la te d re c o m b in a tio n in th e a c tiv e re g io n an d replenishe d b y th e in je c tio n c u rre n t. T h e c u rre n t flow in a fo rw a rd -b ia s e d p -n J u n c tio n is p re d o m in a n tly d u e to d iffu s io n [.30]. So. we assume th a t th e e le c tro n s in th e co n fin e m e n t region are c o u p le d to a large e x te rn a l reservoir o f charge th a t c o n s titu te s a th e rm a l b a th . It m a y be show n [29] th a t th e v o lu m e flu c tu a tio n s o f a fixe d n u m b e r o f p a rticle s in c o n ta c t w ith a th e rm a l b a th at co n sta n t te m p e ra tu re T are = ( 1 ^ ) ^ - H .2Ô I For a clcLssical gas o b e yin g B o ltz m a n n s ta tis tic s , th e d e riv a tiv e d V j d P m a y be o b ta in e d b y d iffe re n tia tin g th e ideal gas law w ith respect to V’: dP - S k T d v = S u b s titu tin g e q u a tio n (4.26) in to e q u a tio n (4.2.5). a n d th e re su lt in to e q u a tio n (4.24) y ie ld s th e m ean square d e n sity flu c tu a tio n ( A n ) ^ ] i- = (-J ^ ) ^ r T h u s, we see th a t th e m ean square d e n s ity flu c tu a tio n o f th e n u m b e r o f p a rtic le s in a s m a ll v o lu m e I o f th e gas is ju s t th e m ean va lu e o f th e p a rtic le d e n s ity d iv id e d by th e v o lu m e o f th e region. T h e re fo re , th e a b s o lu te m a g n itu d e o f th e m ean-scjuare d e n s ity flu c tu a tio n s is m ore pro n o u n ce d w hen a s m a ll v o lu m e is considered. T h e v o lu m e o f th e tra n s p a re n t o p tic a l c o n fin e m e n t re g io n o f th e ty p ic a l sem i c o n d u c to r laser considered here o f c a v ity le n g th L = 250 /zm . a n d la te ra l d im ensions o f 0.2 p m a n d 2 p m . is V’ = L0“ '° cm ^. S u b s titu tin g th e e le c tro n d e n s ity in the o p tic a l c o n fin e m e n t region as ca lc u la te d in e q u a tio n (4.21 ) in to e q u a tio n (4 .2 7 ). the 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m ean square flu c tu a tio n o f c a rrie r d e n s ity in the co n fin e m e n t region is e stim a te d to be 6 X cm "*’ < (An^on/)'^ < 6 x I0 “ ‘ c m “ ^. Since th e d e n s ity flu c tu a tio n s in th e q u a n tu m w e ll a c tiv e region are dam ped above th re s h o ld , th e y are n e g lig ib le co m p a re d to th e e le c tro n d e n sity flu c tu a tio n s in the co n fin e m e n t region. T herefore, th e to ta l phase flu c tu a tio n s due to e le c tro n d e n sity flu c tu a tio n s g ive n by e q u a tio n (4.13) m a y be a p p ro x im a te d as ÔOtot ~ (1 f)^^co n /in e ^ ^^confine' (4.2t>) X o w . in s e rtin g th e range o f (A u c o n /)" ju s t co m p u te d in to e q u a tio n (4.12). we o b ta in th e m ean square phase flu c tu a tio n s fo r a l.ô 5 ;m i-w a ve le n g th g u id e d wave passing th ro u g h a ty p ic a l q u a n tu m -w e ll s e m ico n d u cto r laser due to e le c tro n d e n sity flu c tu a tio n s in th e tra n s p a re n t o p tic a l co n fin e m e n t region to be in th e range: .3.2 X IQ -s < (A o )'^ < 3.2 x 10"* [ra d "). (4.29) T h e re w ill be a s im ila r range o f flu c tu a tio n due to holes, b u t reduced b y th e ra tio [rrie/mh)~. w h ich for In P is (0 .0 7 /0 .4 0 )' % 0.03. T h u s, th e m ean-scpiare electron d e n s ity flu c tu a tio n s are a p p ro x im a te ly 33 tim e s greater th a n th e h o le -d e n s ity flu c tu a tio n s . so we w ill ign ore th e hole d e n s ity flu c tu a tio n s . T h e m ean-square d e n sity flu c tu a tio n s ca lcu la te d above represent the in te g ra te d phase noise o ve r a ll flu c tu a tio n frequencies. W h e th e r th is is a s ig n ific a n t level o f phase flu c tu a tio n depends on th e shape o f th e pow er sp ectral d e n s ity o f th e e le ctro n and hole d e n s ity -in d u c e d phase flu c tu a tio n s co m p a re d to the w h ite pow er sp e ctru m o f th e shot noise lim it, o r o th e r sources o f noise. In the n e xt section, we w ill c a lc u la te th e pow er spectral d ensitie s o f the e le c tro n d e n s ity flu c tu a tio n s and the in d u ce d phase flu c tu a tio n s . 4 .3 .2 P ow er S p ectra l D en sity o f F lu c tu a tio n s T h e p ow er sp e ctra l d e n s ity o f electron d e n s ity flu c tu a tio n s depends on th e d yn a m ics o f th e e le ctro n s. W e a d a p t G lenn's approach [ 1] for th e c a lc u la tio n o f th e pow er s p e ctra l d e n s ity o f re fra c tiv e index flu c tu a tio n s in an o p tic a l fib e r to th e present 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p ro b le m o f th e e le ctro n d e n s ity-in d u ce d re fra c tiv e in d e x flu c tu a tio n s in a sem icon d u c to r picism a. W e are in te re ste d in kn o w in g th e phase flu c tu a tio n s im p a rte d to a g u id e d wave as it passes th ro u g h th e p a rtic u la r vo lu m e o f s e m ico n d u cto r th a t com prises th e laser c a v ity and c o n ta in s a classical gas o f e le ctro n s and holes. W e w ill p e rfo rm th e c a lc u la tio n for th e e le ctro n d e n s ity flu c tu a tio n s , b u t th e sam e analysis applies also to th e holes, w ith th e s u b s titu tio n o f th e a p p ro p ria te co nstants fo r the holes. T h e e le c tro n m o tio n is co n stra in e d in th e lo n g itu d in a l d im e n sio n along th e op tic a l axis (z -a x is ) by th e le n g th o f the se m ico n d u cto r laser c ry s ta l. W e w ill assume th a t th e ele ctro ns are confined in th e transverse d im e n sio n (y -a x is ) p e rp e n d ic u la r to th e c u rre n t flow by th e device s tru c tu re , so th a t th e e le ctro n s m ay o n ly m ove in to or o u t o f th e o p tic a l m ode vo lu m e in th e d im ension p a ra lle l to th e c u rre n t flow along th e x-a x is. T h e re fo re , in th is analysis, we assume th a t n u m b e r d e n s ity flu c tu a tio n s in th e v o lu m e occupie d b y th e guided wave can o n ly o c c u r due to th e rm a l m o tio n o f ca rrie rs e n te rin g o r le a vin g th e beam region along th e c u rre n t flow axis, (th e x -a x is .) T h e c u rre n t in a fo rw a rd -b ia se d diode is p re d o m in a n tly due to d iffu s io n [30]. T h u s, we m ay th in k o f th e ca rriers as d iffu s in g th ro u g h a re c ta n g u la r tu b e o f cross- section equal to th e laser diode c a v ity le n g th L tim e s th e w id th o f th e a c tive region along th e y-a xis. T h is g e o m e try defines th e sp a tia l m odes o f th e e le ctro n gas. w hich w ill affect th e shape o f th e pow er sp e ctru m , as w ill be seen below . T h e electrons have a net v e lo c ity p ro p o rtio n a l to th e d iffu sio n c u rre n t in th e d ire c tio n o f th e cu rre n t flow , b u t th e y also execute ra n d o m m o tio n s in a ll d ire c tio n s w ith ve lo citie s p ro p o r tio n a l to th e te m p e ra tu re . It is these ra n d o m m o tio n s th a t cause th e instantan e ous d e n s ity flu c tu a tio n s in th e v o lu m e occupied by th e o p tic a l field. O nce th e ele ctro ns are in th e a ctive region o f th e laaer above th re s h o ld , th e ir n u m b e r-d e n s ity is c la m p e d at th e gain-equals-loss level by s tim u la te d em ission. .A.S discussed p re v io u s ly , the d e n s ity flu c tu a tio n s in th e a c tiv e region are s tro n g ly da m p e d and do not c o n trib u te s ig n ific a n tly to the pha.se flu c tu a tio n s o f th e o u tp u t lig h t [13]. T h e re fo re , we shall o n ly consider flu c tu a tio n s o f th e e le ctro n d e n s ity in th e n o n -la sin g tra n s p a re n t confinem en t regions. In q u a n tu m w ell lasers, th e con- 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fin e m e n t fa c to r fo r th e o p tic a l fie ld in th e q u a n tu m -w e ll a c tiv e re g io n is ty p ic a lly o n ly a fe w pe rce nt o f th e o v e ra ll beam d ia m e te r [2 2 ]. so th e u n d a m p e d c a rrie r den s ity flu c tu a tio n s in th e co n fin e m e n t region com prise a p o te n tia lly la rg e source o f m u ltip lic a tiv e phase noise, co m p a re d to laser diodes w ith b u lk a c tiv e regions. W e assum e th a t th e ca rrie rs in th e tra n s p a re n t co n fin e m e n t re gion o n ly in te ra c t w ith th e la s in g fie ld v ia c o n d u rtio n -b a n d s c a tte rin g . T h e present a n a lysis differs in th is respect fro m th e w o rk o f Lang. V a h a la and Y a riv [13] discussed p re vio u sly, in w h ic h th e c a rrie r p o p u la tio n in th e a c tiv e region was coupled to th e p hoton p o p u la tio n b y a set o f ra te equa tio n s. In fa c t, it is th e p h o to n -e le c tro n p o p u la tio n c o u p lin g in th e a c tiv e region th a t results in th e s tro n g d a m p in g o f th e e le ctro n d e n s ity flu c tu a tio n s above th re s h o ld in th e w o rk o f Lang et. al. In th e present p ro b le m , th e flu c tu a tio n s o f th e c a rrie r d e n s ity in th e tra n s p a re n t c o n fin e m e n t region c o n s titu te a m u ltip lic a tiv e noise source w ith ra n d o m flu c tu a tio n s th a t are in d e p e n d e n t o f th e p h o to n n u m b e r flu c tu a tio n s in th e laser. T h e effects o f th is noise source on th e la sin g fie ld m ay be ca lcu la te d v ia th e lin e a r m odel for m u ltip lic a tiv e noise in resonant a m p lifie rs developed in th e p re vio u s c h a p te rs. It is n oted th a t th e m e th o d fo r c a lc u la tio n o f the pow er sp e ctru m d e scrib e d b elow is not c o n s tra in e d to e le c tro n d e n s ity flu c tu a tio n s , b u t could be m o d ifie d to c a lc u la te th e effect o f th e rm a l flu c tu a tio n s , phonon flu c tu a tio n s , etc. To b e g in th e c a lc u la tio n o f th e pow er sp e ctra l d e n s ity o f th e flu c tu a tio n s , we assum e th a t th e local e le c tro n d e n s ity n = rio Sn(r.t) is a fu n c tio n o f sp a tia l c o o rd in a te s r and tim e t. has m ean value and satisfies a d iffu s io n e q u a tio n De V ^ n = (4.30) at w here De = -^ie ----- q cm ^ (4.31) s is th e e le c tro n d iffu s iv ity a nd /t, is th e e le c tro n m o b ility . T h is e q u a tio n has s o lu tio n o f th e fo rm A n = A noe'<Tr + ^0 (4.32) 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w here = i3^ D^. and 3 is th e s p a tia l m ode n u m b e r o f th e e le ctro n d e n s ity flu c tu ations. T h e re w ill be an analogous e q u a tio n fo r th e holes, w ith th e s u b s titu tio n o f the d iffu s io n co e ffic ie n t fo r holes. N ow . to c a lc u la te th e to ta l pow er spectral d e n s ity o f th e phase flu c tu a tio n s for the guided w ave, fo llo w in g th e procedure o f G le n n [ I ] , we firs t average th e so lu tio n (4.-32) over th e v o lu m e o f in te re s t: in th is case, it is th e vo lu m e o f th e confinem en t region th a t is in te rc e p te d b y th e g u id e d wave as discussed above. T h e n , we co m p u te th e tim e -a u to c o rre la tio n fu n c tio n o f the vo lum e-averag ed s o lu tio n fo r each sp a tia l m ode 3 o f th e e le c tro n d e n s ity flu c tu a tio n s . T h e pow er sp e ctra l d e n s ity o f flu c tu a tio n s due to each s p a tia l m ode is the F o u rie r tra n s fo rm o f its a u to c o rre la tio n fu n c tio n . W e o b ta in th e to ta l pow er sp e ctra l d e n s ity fo r th e e le c tro n d e n s ity in the volum e V b y in te g ra tin g th e m o d a l pow er s p e ctra l d e n s ity o ve r a ll s p a tia l m odes. T he e le ctro n d e n s ity p ow er s p e c tru m is converted to th e o p tic a l phase flu c tu a tio n s power s p e c tru m using e q u a tio n (4.11). F in a lly , th e to ta l pow er s p e c tra l d e n s ity is n o rm a lize d b y re q u irin g th a t its in te g ra l over a ll F o u rie r frequencies be equal to th e m ean-square phase flu c tu a tio n s ca lcu la te d fro m th e rm o d y n a m ic co n sid e ra tio n s, given by e q u a tio n (4.1 2). T h is c a lc u la tio n yie ld s th e o p tic a l phase flu c tu a tio n s due to the flu c tu a tio n s o f th e e le c tro n d e n sity in th e v o lu m e V on a tim e scale long com pared to th e o p tic a l tra n s it tim e o f the m e d iu m . T h e re fo re , we m ay th in k o f the gu ide d w ave as c o n tin u o u s ly "s a m p lin g " th e s lo w ly -v a ry in g re fra c tiv e index flu c tu a tio n s in th e v o lu m e I ’ . P roceeding w ith th is a p p roach , th e sp a tia l average o f th e s o lu tio n (4.32) over th e volu m e V’ is A n = — / Ange'^^'^dre = A n ^ I V Jv e - '" ' (4.33) where, fo r convenie nce, we have defined / to be an in te g ra l o f th e s p a tia l v a ria tio n over the v o lu m e o f th e g u id e d o p tic a l wave, given by / = i / (4.34) V Jv 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T h e tim e -a u to c o rre la tio n fu n c tio n fo r each sp a tia l m ode 3 is defined as 1 a j(-)|r-+ o c = ^ / A n ( 0 A n ( t - r) clt. (4.35) I Jo W e assum e th a t the tim e response o f th e electron d e n sity flu c tu a tio n s a t a given p o in t in space is an e x p o n e n tia lly deca yin g ra n d o m su p e rp o sitio n : i.e.. an in te g ra l over a ll m odes 3. It m ay be show n [36] th a t an e x p o n e n tia lly decaying process w ith decay c o n sta n t k = 3~ D, has a tim e a u to c o rre la tio n fu n c tio n o f the fo rm a{r) = (4.36) th a t has co rrespon d ing L o re n tz ia n pow er sp e ctra l density = 7 U + ^-0 ■ ' T h u s, th e a u to c o rre la tio n fu n c tio n o f th e e le ctro n density flu c tu a tio n s in th e vo lu m e \ for each s p a tia l m ode 3 is a j ( r ) = ( A n J * (4.38) T h e q u a n tity (A u ^ )* is th e m ean-square d e n s ity flu c tu a tio n per vo lu m e o f w a ve n u m ber space. T h e volum e o ccu p ie d in w a ve n u m b e r space is ju s t ( 2 ~ ) '^ /r . so (A n ^ ) - = ^ 3 j • (4.39) B y th e W e in e r-K h in c h in T h e o re m , th e pow er spectral d e n sity o f a ra n d o m pro cess is th e F o u rie r tra n s fo rm o f its a u to c o rre la tio n fu n ctio n . So. th e pow er sp e ctra l d e n sity o f flu c tu a tio n s due to each m ode is given bv = J ^ ( a j( r ) ) = ( A n j ' / / • H ) . (4.40) w here J-(g) denotes the F o u rie r tra n s fo rm o f th e fu n c tio n g ir). S u b s titu tin g equa tio n (4 .37 ) fo r th e p o w er-sp ectral d e n sity, we o b ta in = 1 / / ■ ,.U1, 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S u b s titu tin g fo r / and I ' and in te g ra tin g over 3. vve o b ta in th e expression for th e to ta l pow er s p e c tra l d e n sity fo r th e e le ctro n d e n s ity flu c tu a tio n s : ^ I J , 1 : jr T 7 T O N ow . to o b ta in the to ta l pow er s p e ctra l d e n s ity o f th e e le ctro n d e n s ity flu c tu a tio n s in th e v o lu m e V’. we m ust p e rfo rm th e in te g ra tio n s in f and r' over th e real-space v o lu m e o f th e beam . V. and th e n in te g ra te over w avenum be r space, (i.e .. •i-space. ) T h e w a ven um be r in te g ra tio n is bound ed b y th e possible sp a tia l m odes o f the electron s th a t c o n trib u te to n u m b e r-d e n s ity flu c tu a tio n s . In G lenn's tre a tm e n t o f th e d e n s ity and te m p e ra tu re flu c tu a tio n s in an o p tic a l fib e r [I], he assum ed th a t the g u id in g core region o f the fib e r is em bedded in a c la d d in g m e d iu m o f in fin ite e x te n t, so th a t th e re is no d is c o n tin u ity presented by th e co re -cla d d in g d ie le c tric in te rfa ce for th e th e rm a l or acoustic flu c tu a tio n s . T h e re fo re , th e possible occupied m odes in w a ve n u m b e r space in G le n n 's tre a tm e n t fo rm a c o n tin u u m , and th e J-space in te g ra tio n is c a rrie d o u t over an a rb itra ry large vo lu m e . In th e present p ro b le m , we m ay s im p lify th e analysis som ew hat by n o tin g th a t th e electron s are confined in th e lo n g itu d in a l d im e n sio n by th e physical le n g th o f th e c ry s ta l. L. and in one transverse d im e n sio n b y th e device s tru c tu re w h ich we assume is designed to prevent c a rrie r leakage o u ts id e o f th e a c tive region. These dim ensio ns are on th e o rde r o f 2 5 0 ^m and 2 ^ m . re sp e ctive ly, and we assume th a t th e ca rrie rs can never m ove o u tsid e o f th e o p tic a l m ode v o lu m e in these tw o dim ensions. Fherefore. c a rrie r n u m b e r de nsity flu c tu a tio n s w ith in th e vo lu m e o f the o p tic a l m ode can o ccu r o n ly due to th e rm a l m o tio n o f th e e le ctro n s along the axis o f c u rre n t flow . T h is m eans th a t th e th re e -d im e n sio n a l in te g ra tio n in w avenum ber-space is reduced to one d im e n sio n . T he o n ly s p a tia l m odes th a t need to be considered are those co rre s p o n d in g to electron m o tio n along th e c u rre n t flow axis, w hich we have designa ted as th e j-a x is . Since we o n ly consider s p a tia l m odes along th e x axis, th e ve cto r p ro d u c t 3 ■ r becomes J^-x and 3 ■ r' becomes J ^ x '. T h e real-space in te g ra tio n s are ca rrie d o u t over th e v o lu m e o f th e o p tic a l c a v ity , fro m —Xo/ 2 to -t-Xo/2 . —y o l- to -|-yo/2 . and 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. — L j'l to + L /'l. T h e expression for th e to ta l pow er s p e c tra l d e n s ity fo r th e electron d e n s ity flu c tu a tio n s th e n becomes . £ a 2 -2ü .L J2 n /_L‘'"/-L ''"'A V -''3 rT T j^ ^ T h e fa c to r .4^ is a n o rm a liz a tio n constant needed because th e w a ve n u m b e r in te g ra tio n is n ot c a rrie d o u t over th re e dim ensions: it w ill be d e te rm in e d la te r. T he in te g ra tio n s over y. y' an d r . z' y ie ld factors o f y l and L~. re sp e ctive ly. T h e integrals over X and x ' b o th y ie ld fa ctors o f sin( J x ^ /2 ) /( J /2 ) : F u rth e r s tra ig h tfo rw a rd s u b s titu tio n s and s im p lific a tio n y ie ld s E v a lu a tin g th e J -in te g ra l. we fin a lly o b ta in the p o w e r-s p e c tru m o f in d e x flu c tu a tio n s due to e lc c tro n -d e n s itv flu c tu a tio n s n.4, ID w here we have d e fin e d \ = Xo\J^I'lD ^ and F ( v ) = I - (c o s (\) 4- s in ( \) ) ( s in h ( \) - c o s h { \ ) ) . (4.-17) W e m ay also o b ta in th e po w er sp e ctra l d e n sity fo r holes. Fn./ifLi;). using th e a p p ro p ri ate d iffu sio n c o e ffic ie n t fo r holes Dh. and m a k in g th e s u b s titu tio n \ = Xa\J^j'2Dh- The phase flu c tu a tio n po w e r sp e ctru m due to e le c tro n d e n s ity flu c tu a tio n s in the confinem e n t re gion is fin a lly given by: lO i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. all I To o b ta in an e s tim a te o f th e m a g n itu d e o f th e phase flu c tu a tio n power spectra, we shall assum e th e fo llo w in g ty p ic a l se m ic o n d u c to r laser d io d e param eters (fo r In P ): the c a v ity le n g th is = 250 x lO""* [cm ], the th ickness o f th e o p tic a l confinem en t region a lo n g th e axis o f th e c u rre n t flo w is = 0 . 2 x 1 0 ~ ' [cm ], single q u a n tu m w ell o f th ic k n e s s 6.5 n m . th e w id th o f th e c a v ity (tra n sve rse to th e current flow ) is ijo = 2 X 1 0 ~‘‘ [cm ], th e c a rrie r d e n s ity in the c o n fin e m e n t region corresponding to in je c tio n c u rre n t in th e range Ith to 10/,/. is given b y e q u a tio n (4.21) as 6 x 10“ ’ c m " ’ < riconf < 6 x 10*' [c m " ’ ], e le c tro n m o b ility = 4000 [cm ‘ /\'- s ]. hole m o b ility fih = 650 [c m '/V -s ] [39]. e le c tro n reduced mass m " = 0.07/7?^. and hole reduced m ass m l = 0.40m ^. For these param eters, we o b ta in electron d iffu s iv ity = H ^ k g T /q = — 103.5 [cm ^/s]. and hole d iffu s iv ity Dh = 16.S [cm */s]. In th e e va lu a tio n o f e q u a tio n (4.48). we also assume e le c tro n charge ^ = 4 .8 x 1 0 "'“ esu. and o p tic a l w a ve le n g th A = 1.55 x lO""* [cm ]. R e c a llin g th e n o rm a liz a tio n re q u ire m e n t, the in te g ra te d phase noise pow er spec tru m over a ll F o u rie r frequencies fro m a ,- = 0 to = oc m ust be equal to th e m ean-square values o b ta in e d fro m th e rm o d y n a m ic co n sid e ra tio n s. T h is was derived p re vio u sly as e q u a tio n (4.29). T h e value for e le ctro n s is in th e range 3.2 x 1 0"“ < (A o )'- < 3.2 X 1 0 "* [ra d '] fo r in je c tio n cu rre n t in th e range to lOIth- N u m e rica l in te g ra tio n o f e q u a tio n (4.48) over uj y ie ld s the re q u ire d value o f th e n o rm a liz a tio n constant to be = 5 x 1 0'. T he p ow er s p e c tra l d e n s ity o f o p tic a l phase flu c tu a tio n s fro m e q u a tio n (4.48) for the th re s h o ld c a rrie r d e n s ity in th e co n fin e m e n t re g io n g ive n by e quatio n (4.21) o f i^conf = 6 x 1 0 '“ [c m " ’ ], and th e o th e r param eters g ive n above, is p lo tte d in F ig u re 4.2 versus fre q u e n c y / = ujI'Itt. T h is p lo t represents th e phase noise power sp e ctru m in u n its o f [ra d ^ /H z ] th a t w o u ld be a cq u ire d by an id e a l noiseless m o n o ch ro m a tic o p tic a l s ig n a l a fte r a single pass th ro u g h th e tra n s p a re n t o p tic a l confinem en t region. 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T h e m a g n itu d e o f th e pow er s p e c tra l d e n s ity scales lin e a rly w ith th e c a rrie r d e n s ity nconf- w h ich is lin e a rly p ro p o rtio n a l to th e in je c tio n c u rre n t. From F ig u re 4.2, we see th a t th e pow er sp e ctru m o f phase flu c tu a tio n s has an inverse pow er-law fre que ncy dependence p ro p o rtio n a l to w ith 0.5 < a < 1.5. .\s seen in th e previous c h a p te r, a m u ltip lic a tiv e phase noise source w ith such a pow er-law dependence w ill p ro d u ce a p o w e r-sp e ctra l-d e n sity o f fre q u e n cy flu c tu atio n s w ith a s im ila r 1 / / ^ fre q u e n cy dependence. T h is is s im ila r to th e ty p e o f frequency flu c tu a tio n pow er s p e c tra ty p ic a lly observed in s e m ic o n d u c to r lasers. In the lo w -freq ue n cy lim it. \ < < I. or a,’ < < 'ID ^/xi. th e te rm F[\) in e q u a tio n (4.48) reduces to So. in th e low -frequency lim it, e q u a tio n (4.48) can be a p p ro x im a te d as > w ( = ? ) ' A i In th e h igh -fre qu en cy lim it, a.’ > > 'ID^/xf^. F{\) in e q u a tio n (4.48) becom es I. so (4.48) can be a p p ro x im a te d as For the s e m ic o n d u c to r laser d io d e param eters assum ed above, th e cro ss-o \e r frequency fro m 1/ / ^ ''* to frequency dependence is = \De\/~x-^ % 82.4 G H z. T here fore, fo r frequencies less th a n % 0.1 = 8.2 G H z . th e pow er sp e ctru m o f ele ctro n d e n s ity in d u c e d phase flu c tu a tio n s is w e ll-a p p ro x im a te d by a 1/ / ' / - pow er law . For e le c tro n d e n s ity in th e confinem en t region in th e range given by e q u a tio n (4.21). we m a y w rite th is as q X iq -iü Q y in-15 — [rad^/Hzj. (4.51 ) B y com parison, th e q u a n tu m noise-induced lim it o f phase flu c tu a tio n s has a w h ite pow er sp e ctra l d e n s ity g ive n a p p ro x im a te ly by h i/lP % 10"^'’ [ra d ^ /H z ] fo r a P = I m W pow er level, and 10“ ^* [ra d ^ /H z ] for a P = 100 mVV pow er level. T herefore, at a m a x im u m pow er level o f 100 m \V . th e e le ctro n d e n s ity flu c tu a tio n - induce d phase noise is p re d ic te d to exceed the shot noise lim it o f phase u n c e rta in ty 109 Reproduced witfi permission of tfie copyrigfit owner. Furtfier reproduction profiibited witfiout permission. [dBrad/Hz] - 1 5 0 - 1 5 0 - 1 8 0 - 1 9 0 -2 0 0 11 12 5 7 g 9 10 1 T 3 4 6 0 logio (frequency) Figure 4.2; Calculated power spectral density of optical phase fluctuations due to electron density fluctuations. 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fo r frequencies less th a n / < 100 M H z fo r a single pass th ro u g h th e q u a n tu m - w e ll se m ic o n d u c to r laser o f le n g th L = 250//m considered in th is analysis. In the n e x t ch apter, th e fre q u e n cy flu c tu a tio n pow er s p e c tru m and lin e w id th due to the e le ctro n d e n s ity flu c tu a tio n noise w ill be ca lcu la te d . For th e ty p ic a l q u a n tu m -w e ll laser s tru c tu re considered , it w ill be show n in the n e x t c h a p te r th a t th e e le ctro n d e n s ity flu c tu a tio n s cause th e lin e w id th to have a th e o re tic a l m in im u m value in the range o f 5.6 k H z to 17.6 k H z . 4.4 D iscu ssio n and Sum m ary In th is ch a p te r, th e pow er s p e c tru m o f phase flu c tu a tio n s in d u c e d b y e le ctro n d e n s ity flu c tu a tio n s in th e tra n s p a re n t o p tic a l co n fin e m e n t re g io n o f a q u a n tu m -w e ll sem i c o n d u c to r laser was c a lc u la te d . These flu c tu a tio n s are in trin s ic in the sense th a t th e y result fro m th e u n a v o id a b le d iffu s iv e m o tio n o f th e c a rrie rs in th e co n fin e m e n t region due to th e ir th e rm a l ve lo citie s. T h e e le ctro n d e n s ity noise is enhanced at low frequencies due to th e la ck o f a d a m p in g m echanism fo r th e d iffu s iv e m o tio n o f th e electrons. T h e la ck o f d a m p in g is s im ila r to th e enh a n ce d lo w -fre q u e n cy pow er s p e ctru m o f fre q u e n cy and in te n s ity noise due to te m p e ra tu re -flu c tu a tio n -in d u c e d phase noise c a lc u la te d by La ng, et al. [13]. W h ile e le c tro n -d e n s ity flu c tu a tio n s are h e a v ily dam p ed in th e a c tiv e lasing region o f th e q u a n tu m w e lls because th e c a rrie r d e n s ity is cla m p e d a t its th re s h o ld level by s tim u la te d e m issio n , the th e rm a lly - induced n u m b e r-d e n s ity flu c tu a tio n s o f th e ca rrie rs in th e tra n s p a re n t co n fin e m e n t regions are not s im ila rly d a m p e d , and can th erefore im p a rt s ig n ific a n t phase flu c tu a tio n s to th e o p tic a l fie ld . In th e tra n s p a re n t o p tic a l c o n fin e m e n t region o f a ty p ic a l q u a n tu m -w e ll Iciser. it was show n th a t th e e le ctro n d e n s ity flu c tu a tio n -in d u c e d noise can be s u b s ta n tia lly h ig h e r th a n th e shot noise level a t lo w frequencies. T h is an a l ysis suggests th a t s e m ic o n d u c to r d io d e lasers w ith b u lk a c tiv e regions sh o u ld have low er c a rrie r-d e n s ity -in d u c e d phase noise th a n q u a n tu m -w e ll lasers w ith n o n -la sin g o p tic a l co n fin e m e n t regions in w h ic h a h ig h d e n sity o f c a rrie rs are present. T h is is because th e c a rrie r-d e n s ity flu c tu a tio n s in th e a ctive re g io n o f a s e m ic o n d u c to r laser are s tro n g ly d a m p e d due to s tim u la te d em ission. I l l Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It was show n p re v io u s ly th a t a l / / ‘^-typ e m u ltip lic a tiv e phase flu c tu a tio n pow er s p e c tru m w ill p ro d u ce c o rre sp o n d in g 1/ / ‘^-typ e pow er s p e c tra fo r th e o u tp u t fre quency flu c tu a tio n pow er spectra. T h is analysis in d ic a te s th a t c a rrie r d e n s ity flu c tu a tio n s in th e co n fin e m e n t region o f a q u a n tu m -w e ll s e m ic o n d u c to r laser has a 1/ /'^ c h a ra c te r, an d can cause enhanced frequency flu c tu a tio n s at low -frequen cies. leading to enhanced lin e w id th . lin e w id th s a tu ra tio n , and re b ro a d e n in g . T h is effect also represents a new e x p la n a tio n fo r th e u b iq u ito u s 1/ fre q u e n cy noise ty p ic a l o f s e m ic o n d u c to r lasers. It w ill ty p ic a lly be th e case th a t c a rrie r-d e n s ity flu c tu a tio n s w ill be ju s t one o f several noise m echanism s a c tin g s im u lta n e o u s ly in a p a rtic u la r device. H ow ever, th e rm a lly -in d u c e d c a rrie r-d e n s ity flu c tu a tio n s in th e co n fin e m e n t regions represent an in trin s ic noise source th a t w ill lim it th e laser lin e w id th w hen a ll o th e r noise sources are a d e q u a te ly suppressed. T h e pow er s p e c tru m o f e le c tro n -d e n s ity flu c tu a tio n s scales lin e a rly w ith th e car rie r d e n s ity in th e c o n fin e m e n t region. T herefore, as in je c tio n c u rre n t is increased, th e c a rrie r-d e n s ity -in d u c e d phase flu c tu a tio n s w ill also increase. T h is m a y be a pos sible m e ch a n ism fo r lin e w id th re broade ning in q u a n tu m -w e ll s e m ic o n d u c to r lasers at hig h p ow er levels. T h e p ow er s p e c tru m o f phase flu c tu a tio n s was fo u n d to scale in v e rs e ly w ith the square o f th e e le ctro n reduced mass. m “ . T he re fo re , c o n fin e m e n t s tru c tu re s w ith h ig h e r e le c tro n e ffe c tiv e masses are expected to e x h ib it less c a rrie r-d e n s ity -in d u c e d phase noise, due to th e reduced th e rm a l velocities o f th e c a rrie rs . T h e phase flu c tu a tio n pow er sp e c tru m also scales in v e rs e ly as th e square o f the o p tic a l m ode thickne ss p a ra lle l to th e c u rre n t-flo w axis. i.e. as l / j ^ . T he re d u c tio n in noise w ith in crea sing thickness o f the laser m e d iu m re su lts because the m u ltip lic a tiv e noise generators add in p a ra lle l as th e v o lu m e o f th e a c tiv e region increases. It has been fo u n d th a t m u ltip lic a tiv e noise sources cancel each o th e r w hen added in p a ra lle l. T h is re s u lt is in q u a lita tiv e agreem ent w ith th e b e h a v io r o f m u ltip lic a tiv e 1/ / noise in o th e r system s. For e xa m p le . I / / noise o f a p a ra lle l c o m b in a tio n o f m atched tra n s is to rs w ith s im ila r noise a m p litu d e s has less I / / noise th a n a sin g le transis- 112 Reproduced witfi permission of tfie copyrigfit owner. Furtfier reproduction profiibited witfiout permission. to r. .A.Iso. o s c illa to rs e m p lo y in g la rg e r-v o lu m e q u a rtz resonators have been found to e x h ib it lo w e r I / / freq ue ncy noise [38] th a n o s c illa to rs w ith s m a lle r resonators. .-\nd m u ltip lic a tiv e phase noise sources in a phased a rra y antenna syste m cancel each o th e r to y ie ld an o ve ra ll low er m u ltip lic a tiv e phase noise flo o r fo r th e a rra y co m p a re d to a single a ntenn a elem ent o f th e sam e to ta l area [37]. T h e phase flu c tu a tio n pow er s p e c tru m is also seen to be p ro p o rtio n a l to the square o f th e c a v ity length. T h is is because th e o p tic a l field integrates th e ra n d o m in d e x p e rtu rb a tio n s as it traverses th e le n g th o f th e a c tive region, re s u ltin g in a ra n d o m w a lk o f phase. T h e m ean-square value o f a ra n d o m w a lk process increases as th e square o f th e num b er o f steps in th e w a lk , w h ich is p ro p o rtio n a l to th e c a v ity le n g th . T hese g e o m e trica l dependencies o f th e phase noise pow er spect ru m at firs t glance suggest th a t it w ould be advantageous to design th e laser to be as s h o rt and as th ic k as possible, to m in im iz e th e effects o f m u ltip lic a tiv e c a rrie r d e n s ity noise. H ow ever, these g e o m etrical factors m u st be considered together w ith th e g ain and th re s h o ld dependence on th e c a v ity g e o m e try, since the laser th re s h o ld increases w ith decreasing c a v ity le n g th . .\n increased laser th re sh o ld w ill p ro d u ce increased c a rrie r d e n sity, and hence a h ig h e r level o f m u ltip lic a tiv e noise. In th e n e x t c h a p te r, th e c a rrie r n u m b e r-d e n sity -in d u ce d phase noise ca lcu la te d here w ill be used in the th e o ry o f laser lin e w id th fro m C h a p te r 3 to e s tim a te th e pow er s p e ctru m o f frecptenc\- flu c tu a tio n s and the in trin s ic lin e w id th o f a ty p ic a l q u a n tu m -w e ll s e m ic o n d u c to r laser. 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5 Sem iconductor laser line w idth As discussed in th e In tro d u c tio n , d is trib u te d -fe e d b a c k single-frequ ency laser diodes ty p ic a lly e x h ib it a pow er-independent lin e w id th a n d /o r lin e w id th re b ro a d e n in g w ith increasing o p tic a l power [8 ], [10], T h e m o d e l o f e q u a tio n (3.26) th a t was d e rive d fro m th e tim e -v a ry in g p a rtia l-w a ve a n a lysis in c lu d in g m u ltip lic a tiv e noise in th e sm a ll-a n g le regim e predicts a s im ila r p o w e r-in d e p e n d e n t lin e w id th b e h a v io r, w h ile the n u m e ric a l analysis in th e large-ang le re g im e p re d ic ts lin e w id th re b ro a d e n in g as a consequence o f d e stru c tive in te rfe re n ce o f p a rtia l waves in th e laser c a v ity at high o u tp u t pow er. It was show n th a t th e tim e -v a ry in g p a rtia l-w a v e a n a lysis in the sm a ll-a n g le regim e is analogous to th e p h e n o m e n o lo g ica l expression o f e q u a tio n ( 1.2 ) for th e pow er-inde pendent lin e w id th used b y W ei fo rd and M o o ra d ia n [8 ]. B u t th e analysis o f chapters 2 and 3 also reveals th a t th e sam e source o f m u ltip lic a tiv e noise can be responsible for bo th a p o w e r-in d e p e n d e n t lin e w id th and lin e w id th re b ro a d e n ing, co rre sp o n d in g to the effect o f m u ltip lic a tiv e noise in the sm a ll- and large-ang le regim es, respectively. .Also, in C h a p te r 4, it was show n th a t the m u ltip lic a tiv e noise source o f e le ctro n -d e n sity flu c tu a tio n s in th e co n fin e m e n t region o f a c|u a n tu m w ell laser has m a g n itu d e p ro p o rtio n a l to th e in je c tio n c u rre n t. .An im p lic a tio n o f these th e o re tic a l re su lts is th a t a p a rtic u la r laser m a y e x h ib it a region o f o u tp u t pow er c o rre sp o n d in g to m in im u m lin e w id th . T h is is indeed observed in som e q u a n tu m -w e ll lasers. T h e re fo re , in a p p lica tio n s th a t re q u ire a m in im u m lin e w id th , an o p tim u m range o f o u tp u t pow er m ay exist th a t w ill depend on th e m u ltip lic a tiv e noise s tre n g th . .Also, th e re su lts o f the e le c tro n -d e n s ity fiu c- 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tu a tio n an a lysis suggest th a t p a rtic u la r laser c a v ity g e o m e trie s a n d h ig h e r ca rrie r e ffe ctiv e m ass values m a y be e x p e c te d to y ie ld th e n a rro w e s t-lin e w id th laser diodes. In th is c h a p te r, th e m in im u m lin e w id th o f a ty p ic a l s in g le -m o d e q u a n tu m -w e ll s e m ic o n d u c to r laser w ill be e s tim a te d c o n sid e rin g th e effects o f th e rm a lly -in d u c e d e le ctro n d e n s ity flu c tu a tio n s in th e c o n fin e m e n t re g io n o f th e o p tic a l c a v ity th a t were analyzed in th e p re vio u s ch a p te r. E le c tro n d e n s ity flu c tu a tio n s cause correspond ing changes in th e re fra c tiv e in d e x o f th e tra n s p a re n t c o n fin e m e n t re g io n , and c o n s titu te a fu n d a m e n ta l source o f m u ltip lic a tiv e phase noise. In th is c h a p te r, we consider a ty p ic a l sin g le -m o d e In G a .\s P q u a n tu m -w e ll s e m ic o n d u c to r laser o f le n g th / = 'iôO ^m o p e ra tin g a t A = l.o o ^ m . w h ic h im p lie s a c a v ity ro u n d -trip tim e o f r % 5.7 picoseconds. W e assum e th e e ffe c tiv e pow er re fle c tiv ity o f th e d is trib u te d -fe e d b a c k c a v ity m irro rs is 0.32. and m u ltip lic a tiv e phase flu c tu a tio n s due to e le c tro n d e n sity flu c tu a tio n s in th e tra n s p a re n t n o n -la s in g c o n fin e m e n t region are pre se n t, w ith power sp e ctra l d e n s ity as ca lc u la te d in C h a p te r 4. .As discussed in C h a p te r 4. th e ca lcu la te d m u ltip lic a tiv e phase noise pow er s p e c tru m is a p p ro x im a te d at lo w frec[uencies at 1 0 /,/. using e q u a tio n (4.51) as S 'o if) = ^ [ra d ^ /H z ] at c a rrie r d e n s ity o f 0 x 1 0 *' c m “ '^ in th e c o n fin e m e n t region, co rre sp o n d in g to 1.5 x 10*^ c m " ^ in th e c|uantum w ell a c tiv e re gion . T h e m ean-squa re m u ltip lic a tiv e phase noise co rre sp o n d in g to th is c a rrie r d e n s ity was c a lc u la te d in e q u a tio n (4.29) to be (A o )^ = = 3.2 x 10 rad" fo r a sin g le pass th ro u g h th e laser c a v ity . 5.1 O n set o f lin ew id th reb road en in g F irs t, we w ill e s tim a te th e o u tp u t pow er c o rre sp o n d in g to th e onset o f th e large- angle re g im e in o u r m odel o f th e laser as a n o is e -d riv e n resonant a m p lifie r. T h is w ill d e te rm in e b o th th e region o f v a lid ity o f th e s m a ll-a n g le lin e w id th m o d e l, and th e pow er level at w h ic h lin e w id th re b ro a d e n in g is p re d ic te d to beg in . F rom the previou s an a lysis le a d in g to e q u a tio n (2.25). th e net g a in c o rre s p o n d in g to th e onset o f th e la rg e-an g le re gim e is e s tim a te d to be A'o > (5.1) 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w here we have assum ed e = 0.01. T h e reader is re m in d e d th a t th is gain corresponds to th e c o n d itio n w h e n th e s ta n d a rd d e v ia tio n o f th e phase fo r th e .V /-th p a rtia l wave ju s t equals - /'I . w h ere M is th e n u m b e r o f p a rtia l waves re q u ire d fo r th e o u tp u t field s u m m a tio n to converge to w ith in a fa cto r o f e = 0.01 o f its fin a l value. For net gains exceeding th is va lu e, th e o ld e r p a rtia l waves begin to d e s tru c tiv e ly in te rfe re w ith th e e a rlie r p a rtia l w aves, le a d in g to large o u tp u t a m p litu d e a n d phase flu c tu a tio n s . T h e net g ain aa a fu n c tio n o f o u tp u t pow er fo r th e laser u n d e r co n sid e ra tio n is ca lcu la te d fro m e q u a tio n (3.24) to be ' ^ P [W a tts ]' S u b s titu tin g (5.2) in to (5.1) and so lvin g fo r P yie ld s th e e s tim a te d o u tp u t power co rresp on d in g to th e onset o f th e large-angle regim e: " ^ For ty p ic a l values o f m u ltip lic a tiv e noise, th e q u a n tity in th e e x p o n e n tia l is m uch less th a n u n ity , a n d since fo r x « 1. 1 — % —x . we m a y a p p ro x im a te th is expression as For th e m u ltip lic a tiv e phase va ria n ce given above, crj = 3.2 x 10 rad^. we e stim a te th a t the large-angle re g im e occurs fo r P > 79 m illiw a tts . T h is pow er level represents an e stim a te d u p p e r-lim it to th e o p e ra tin g pow er o f a D F B laser d iode p rio r to th e onset o f the large-angle re g im e and is co n sid e ra b ly hig h e r th a n th e p o w e r levels observed e .xp e rim e n ta lly fo r th e onset o f lin e w id th re broaden ing in som e laser diodes. T h e preceding c a lc u la tio n assum ed a constant c a rrie r d e n sity in th e c o n fin e m e n t region o f n^onf ~ 6 x 1 0 '' c m " " at an o p e ra tin g c u rre n t o f lOPh. W e n o te th a t th e power level p re d ic te d b y e q u a tio n (5.4) fo r th e onset o f th e large a n g le re g im e increases lin e a rly as th e c a rrie r d e n s ity decreases. In a p a rtic u la r laser d io d e , s p a tia l h o le -b u rn in g and s p e c tra l h o le -b u rn in g m ay p la y m ore s ig n ific a n t roles in lin e w id th re b ro a d e n in g at lo w e r p ow er levels th a n c a rrie r d e n s ity flu c tu a tio n s in th e tra n sp a re n t o p tic a l c o n fin e m e n t regions. B u t in 116 Reproduced witfi permission of tfie copyrigfit owner. Furtfier reproduction profiibited witfiout permission. th e absence o f h o le -b u rn in g effects, th e c a rrie r d e n sity noise is e xp e cte d to cause lin e w id th re b roa de ning a t high pow er levels. 5.2 S m all-an gle lin ew id th For pow er levels less th a n th e value p re d ic te d by e q u a tio n (5 .4 ). we assum e th a t the sm all-a ng le analysis is v a lid . W e expect th a t the laser lin e w id th is th e n th e sum o f th e s ta n d a rd S chaw low -T ow nes te rm w ith I/P -d e p e n d e n c e . and a m u ltip lic a tiv e te rm due to e le ctro n d e n s ity flu c tu a tio n s , as in the m odel o f e q u a tio n (3.26) d e rive d pre vio u sly: du = dl'\s-T + ' .. (O ..-)) 2 -^ T - T h is m ode l was based on th e a ssu m p tio n o f a w h ite pow er s p e c tru m o f m u l tip lic a tiv e phase flu c tu a tio n s . H ow ever, it was shown in C h a p te r 4 th a t th e m u l tip lic a tiv e phase noise p ow e r sp e c tru m due to e le c tro n -d e n s ity flu c tu a tio n s has an inverse pow er-law fre q u e n cy dependence p ro p o rtio n a l to 1 //'C w ith 0.5 < a < 1.5. Since e qu atio n (5.5) assum es a w h ite pow er sp e ctru m o f fre q u e n cy flu c tu a tio n s , th is presents a d iffic u lty . T o e x a c tly c a lc u la te th e lin e w id th . th e a u to -c o rre la tio n fu n c tio n o f the o p tic a l fie ld m u st be ca lcu la te d from th e fre q u e n cy flu c tu a tio n pow er sp e ctru m , as discussed in C h a p te r 3. For an a rb itra ry pow er s p e c tru m , a closed- fo rm s o lu tio n is n ot ava ila b le . H ow ever, in th e case o f I / / fre q u e n cy flu c tu a tio n s , th e re s u ltin g lineshape is G aussian, fo r w h ich an a p p ro x im a te a n a ly tic fo rm u la is available [21]. So. to e s tim a te th e lin e w id th o f the laser, we a p p ro x im a te the low - frequency lim it o f th e po w er sp e ctru m o f phase flu c tu a tio n s due to c a rrie r d e n s ity flu c tu a tio n s d e rive d in C h a p te r 4 as e q u a tio n 4.49 as a I / / pow er law : T h is form u n d e re stim a te s th e noise, how ever, since a 1 / / pow er s p e c tru m ro lls o ff fa ste r than l/\/7- T h is y ie ld s a fre q u e n cy flu c tu a tio n pow er s p e c tra l d e n sity given by 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (é^Ïvk^f ' " ' ' / " ' I T o use th is 1 / / pow er law to ca lc u la te th e lin e w id th . we define S c .J f) = ~ (H zV H zl (Ô.S) so th a t \ 1 V 1 (ttt-)- 647r^/i^ \m ~ c-J \/2D l T h is m a y be s im p lifie d by s u b s titu tin g r = 2iJi,L/c to y ie ld ( 0 . 9 ) A q 1 ‘'^r^con (Ô.10) 16^^ m - c ( 2 £ ) e )*''■'V T h e a p p ro x im a te fo rm fo r th e G aussian h a lf-p o w e r lin e w id th due to 1 / / fre q u e n cy flu c tu a tio n s is g ive n by [2 1 ]: w here d e te rm in e s th e m a g n itu d e o f th e L / / fre q u e n cy noise. a,v is th e low er fre q u e n cy lim it o f th e m easurem ent system co rre sp o n d in g to l / 2 ~ r,„ in w h ich r„, is th e m ea su rem e nt in te rv a l, and G is the coherence tim e o f th e laser. S u b s titu tin g e q u a tio n (5.10) in to e q u a tio n (5.11). th e fu ll fo rm o f th e lin e w id th m a y be w ritte n : ■Assuming a c o n fin e m e n t region c a rrie r d e n s ity at th re s h o ld o f = 6 x 10"^ c m “ ’ . and = IQ— e q u a tio n (5.12) y ie ld s a lin e w id th o f S u \i/f % 5.6 kH z. .At high o u tp u t pow er w ith a c o n fin e m e n t region c a rrie r d e n s ity o f riconf = 6 x 1Q‘ ‘ c m " h th e lin e w id th is increased by a fa c to r o f \/TÔ to Si/hfj % 17.6 k H z. These values are in reasonable agreem ent w ith th e p u b lish e d m in im u m observed lin e w id th s o f 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a p p ro x im a te ly 4 kH z for D F B s e m ic o n d u c to r lasers in w h ich s tra in and v a ria tio n in g ra tin g p itc h Wcis em ployed to suppress s p a tia l h o le -b u rn in g [28]. T h e c a lc u la te d lin e w id th values fro m e q u a tio n (5.12) are 10 to 10 0 tim e s s m a lle r th a n ty p ic a l observed values fo r c o m m e rc ia lly -a v a ila b le h ig h -p o w e r D F B lasers, in w h ic h m o d e -h o p p in g and s p a tia l hole b u rn in g m ay p la y a s ig n ific a n t role. T o pro duce th e lin e w id th s on th e o rd e r o f 1 M H z to 10 M H z ty p ic a lly observed in co m m e rc ia l D F B lasers, the level o f m u ltip lic a tiv e noise m u st be m uch h ig h e r th a n th a t due to e le c tro n d e n s ity flu c tu a tio n s . E ven w hen e le c tro n -d e n s ity flu c tu a tio n s are th e p re d o m in a n t m u ltip lic a tiv e noise source, th e lin e w id th o f a p a rtic u la r laser w ill also d ep en d in d e ta il on th e ra tio o f c o n fin e m e n t region to a c tiv e region m o d e v o l um e. a n d th e e ffe ctive mass o f th e c a rrie rs in th e co n fin e m e n t region, as discussed in C h a p te r 4. For p o w e r-la w noise sources, th e a c tu a l m easured lin e w id th depends on th e m ea su re m e n t in te rv a l, as seen fro m th e dependence o f e q u a tio n (5.12) on th e m easure m e n t tim e th ro u g h a,';. T h is is a m a n ife s ta tio n o f th e n o n -s ta tio n a rity o f th e 1/ / noise process. T h e a ssu m p tio n o f u.’(C- ~ 1 0 '^ im p lie s a m easurem ent low er frecpiencv lim it o f u-’(/2 rr = lO ~ M V /2 = 0.5 H z. o r a m easurem ent sweep tim e o f ~ 2 sec onds. i f we ta k e 6u = 1 kH z. For th is reason, it is c ritic a l to know th e d e ta ils o f th e m easurem en t a p p a ra tu s b a n d w id th w hen c o m p a rin g th e results o f laser lin e w id th m easurem e nts i f 1/ / noise is p re d o m in a n t. W e n o te th a t the e le c tro n -d e n s ity flu c tu a tio n pow er sp e c tru m c a lc u la te d in C h a p te r 4 is p ro p o rtio n a l to l / \ / J . b u t in th e lin e w id th c a lc u la tio n o f e q u a tio n (5.12). th e e le c tro n d e n sity flu c tu a tio n pow er s p e c tru m was a p p ro x im a te d as 1/ / . H ow ever, th e value o f m in im u m lin e w id th p re d ic te d by the s m a ll-a n g le m o d e l due to e le c tro n -d e n s ity flu c tu a tio n s in th e c o n fin e m e n t region is a p p ro x im a te ly e q u a l to th e low est re p o rte d lin e w id th results o f 3.6 k H z [28] fo r D F B lasers w ith suppression o f s p a tia l hole b u rn in g 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.3 Sum m ary It Wcis show n th a t m u ltip lic a tiv e o p tic a l phase flu c tu a tio n s in th e co n fin e m e n t region o f a q u a n tu m -w e ll s e m ic o n d u cto r laser are capable o f p ro d u cin g a p o w er-inde pendent iin e w id th in reasonable agreem ent w ith p u b lish e d e x p e rim e n ta l values, and lin e w id th re b ro a d e n in g . Since e le c tro n d e n s ity flu c tu a tio n s arise from th e sam e co n sid e ra tio n s responsible fo r Johnson noise in e le c tro n ic co m p o n e n ts [1]. th is source o f m u ltip lic a tiv e noise represents a fu n d a m e n ta l lim it to th e m in im u m lin e w id th o f a laser. W hen th is lim it is reached, fu rth e r increases in o u tp u t pow er w ill n o t a ffo rd decreased lin e w id th . In fact, th e lin e w id th is p re d ic te d to rebroaden if th e o u tp u t pow er and net g a in is increased past th e p o in t th a t th e large-angle regim e is entered. These ca lcu la tio n s im p ly th a t th e sam e m u ltip lic a tiv e noise m e ch a n ism is ca pable o f p ro d u cin g b o th phenom ena o f p o w e r-in d e p e n d e n t lin e w id th and lin e w id th re b ro a d e n in g in a s e m ic o n d u c to r laser. T h e tra n s itio n from th e p o w e r-in d e p e n d e n t lin e w id th to th e lin e w id th re -b ro a d e n in g regim e corresponds to th e tra n s itio n fro m th e s m a ll-a n g le to large-ang le regim es. V arious m echanism s have been advanced to e x p la in th e o rig in o f th e p o w e r-in d e p e n d e n t lin e w id th and lin e w id th re broad e ning. in c lu d in g th e rm a lly -in d u c e d e le ctro n n u m b e r-d e n s ity flu c tu a tio n s [32], spa tia l hole b u rn in g [ I I ] . and s p a tia lly -d e p e n d e n t te m p e ra tu re and c a rrie r flu c tu a tio n s [13]. These o r o th e r e x trin s ic m echanism s, such as in je c tio n -c u rre n t (i.e . power- s u p p ly ) flu c tu a tio n s , m a y be o p e ra tiv e in various c o m b in a tio n s in any p a rtic u la r laser. W h a te v e r th e source, m u ltip lic a tiv e phase and gain flu c tu a tio n s in th e laser c a v ity w ill cause a m in im u m lin e w id th fo r th e o u tp u t field em ission s p e c tru m . 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 Conclusion In th is ch a p te r, the results o f th e d is s e rta tio n w ill be su m m arized and th e im p lic a tio n s fo r laser o p e ra tio n discussed. C o m p a riso n s o f these results to o th e r th e o re tic a l and e x p e rim e n ta l w o rk w ill be m ade. F in a lly , suggestions fo r fu tu re e x p e rim e n ta l and th e o re tic a l w ork w ill be offered. 6.1 D iscu ssion o f R esu lts T h e focus o f th is d is s e rta tio n has been on g a in in g an u n d e rsta n d in g o f a u n ifie d m e ch a n ism responsible fo r p o w e r-in d e p e n d e n t lin e w id th and lin e w id th re b ro a d e n in g in single-m ode s e m ico n d u cto r lasers. T h e In tro d u c tio n gave an o v e rv ie w o f th e discrepancies between th e o ry and e.xperim ents fo r low -frequen cy noise a n d lin e w id th in s e m ic o n d u c to r lasers. In C h a p te r 2. a m o d ifie d p a rtia l wave analysis w h ic h in co rp o ra te d th e effects o f m u ltip lic a tiv e noise was derived. T h is analysis is a general fo rm a lis m fo r tre a tin g th e e v o lu tio n o f th e c o m p le x a m p litu d e o f th e e le c tric field in a resonant a m p lifie r s u b je cte d to m u ltip lic a tiv e noise. T he p a rtia l w ave analysis pro vid e s an in tu itiv e ph ysica l p ic tu re fo r th e effect o f m u ltip lic a tiv e noise in reso n a n t o p tic a l a m p lifie rs and lasers, .\p p ly in g th e p a rtia l wave results to th e m odel o f th e laser as a n oise -d riven resonant a m p lifie r in C h a p te r 3. it was show n th a t th e c h a ra c te ris tic s o f pow e r-ind e pendent lin e w id th and lin e w id th re b ro a d e n in g th a t are ty p ic a lly observed in sin g le-m ode s e m ic o n d u c to r lasers can be e x p la in e d by th e ac tio n o f one com m on m u ltip lic a tiv e noise source. It was found th a t in th e presence o f 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m u ltip lic a tiv e noise, th e c ru c ia l fa cto rs th a t d e te rm in e th e linew idth-versus-p ovver dependence o f th e laser are th e net ro u n d -trip gain o f th e laser ca vity, and the m u ltip lic a tiv e noise a m p litu d e . .A . sm a ll-a n g le lin e a r re g im e and a large-angle n o n -lin e a r re g im e were id e n tifie d . T h e sm a ll-a n g le regim e is show n to be analogous to p re vio u s tre a tm e n ts in w h ich the laser fre que ncy flu c tu a tio n s are e x p la in e d as th e re su lt o f a lin e a r m o d u la tio n o f the real p a rt o f th e re fra c tiv e in d e x b y a m u ltip lic a tiv e noise source. T h e re fra c tiv e in d e x flu c tu a tio n s are lin e a rly tra n s fo rm e d to flu c tu a tio n s o f th e laser center frequency. T h e large-angle regim e, how ever, appears to be new. and is n o t p re d ic te d by a lin e a r noise analysis. T h e m o d ifie d p a rtia l-w a v e fo rm a lis m was then a p p lie d to th e m odel o f a laser as a n o ise -d rive n resonant a m p lifie r. For a g ive n fixed va lu e o f m u ltip lic a tiv e noise, it was show n th a t the laser lin e w id th w ill in itia lly decrease as th e net g ain is increased, as p re d ic te d by th e s ta n d a rd S chaw low -T ow nes m odel. B u t beyond a c ritic a l level o f net g a in , th e phenom enon o f p o w e r-in d e p e n d e n t lin e w id th occurs, due to th e effect o f m u ltip lic a tiv e noise in th e sm a ll-a n g le regim e o f th e p a rtia l wave m odel. .After a second c ritic a l value o f net g ain is reached, lin e w id th re b ro a d e n in g is p re d icte d to o c c u r due to the onset o f th e n o n -lin e a r large-angle regim e. In th is regim e, the p a rtia l-w a v e m odel p re d ic ts th a t th e laser lin e w id th sh o u ld increase w ith fu rth e r increases in net ro u n d -trip g a in and o u tp u t pow er. T h e loss o f coherence is due to the a c c u m u la tio n o f m u ltip lic a tiv e phase and a m p litu d e flu c tu a tio n s by th e o p tic a l field in th e c a v ity th a t e v e n tu a lly leads to d e s tru c tiv e in te rfe re n c e betw een th e p a rtia l waves. T h is phenom enon is fu n d a m e n ta lly a large-angle e ffe ct, and so has a p p a re n tly not been p re d ic te d by p re vio u s lin e a r s m a ll-sig n a l analyses o f laser dynam ics. In C h a p te r 4. a d e ta ile d a n a lysis o f c a rrie r n u m b e r-d e n s ity flu c tu a tio n s in a s e m ic o n d u c to r laser was presented. T h e pow er s p e c tru m o f phase flu c tu a tio n s in duced by th e n u m b e r-d e n s ity flu c tu a tio n s in the c o n fin e m e n t region o f a ty p ic a l q u a n tu m -w e ll laser s tru c tu re was fo u n d to have a I / / " p o w e r-la w frequency depen dence. w ith 0.5 < Q < 1.5. In th is a n a lysis, it was assum ed th a t th e o p tic a l phase o f th e lasing field is m o d u la te d due to free-electron in tra -b a n d s c a tte rin g by the 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ca rrie rs in th e n o n -la s in g confinem en t region. In th e a c tiv e region, c a rrie r-d e n s ity flu c tu a tio n s are d a m p e d due to s tim u la te d e m issio n , b u t in q u a n tu m -w e ll lasers, th e a c tiv e g a in reg ion o n ly overlaps a few percent o f th e o p tic a l m ode. T h e m a g n itu d e o f th e c a rrie r-d e n s ity induced I / / " * m u ltip lic a tiv e phase noise was show n to be in versely p ro p o rtio n a l to th e square o f th e c a rrie r e ffe c tiv e mass as w ell as th e c a v ity g e o m e try. In s e rtin g these results in to th e p a rtia l-w a v e laser th e o ry o f C h a p te r I yie ld s a laser fre q u e n c y flu c tu a tio n pow er s p e c tru m w ith a s im ila r 1/ / ^ p o " er-la w dependence. E le c tro n d e n sity flu c tu a tio n s th e re fo re c o n s titu te an in trin s ic ty p e o f fre q u e n cy noise m echan ism in q u a n tu m -w e ll s e m ic o n d u c to r lasers, a risin g fro m th e sam e p h y s ic a l m e ch a n ism as .Johnson noise in re sisto rs. T h is ty p e o f c a rrie r-d e n s ity - in d u ce d m u ltip lic a tiv e noise is expected to affect th e lin e w id th o f any laser in w hich a s u b s ta n tia l p o rtio n o f the o p tic a l m ode lies o u ts id e th e a c tiv e lasing region in an area w ith h ig h c a rrie r density. O f course, o th e r e x trin s ic sources o f l/ / " ^ noise m a y be present in a p a rtic u la r de vice a t a h ig h e r le ve l th a n c a rrie r n u m b e r-d e n s ity flu c tu a tio n s , such as p u m p c u rre n t- s u p p ly flu c tu a tio n s , e x te rn a l te m p e ra tu re flu c tu a tio n s , o p tic a l b a ck-re fle ctio n s. etc. H ow ever, i f a ll e x trin s ic noise sources are a d e q u a te ly suppressed, th is th e o ry p re d icts a re sid u a l fu n d a m e n ta l level o f 1/ / “ noise w ill be present, due to the u n a vo id a b le th e rm a l e le c tro n d e n s ity flu c tu a tio n s in th e c o n fin e m e n t region o f a q u a n tu m -w e ll laser. T h e noise s p e c tru m is influenced by th e b a n d -s tru c tu re and g e o m e try o f th e laser c a v ity . T h e analysis suggests th a t in o rd e r to o b ta in th e m in im u m fu n d a m e n ta l lin e w id th fo r a q u a n tu m -w e ll s e m ic o n d u c to r laser, th e ca rrie rs should have the largest possible e ffe c tiv e mass in the co n fin e m e n t re g io n , and th a t it m ay be possible to o p tim iz e th e laser c a v ity design in lig h t o f th e g e o m e tric consideratio ns. In c h a p te r n. th e re sults o f chapters 3 and 4 are co m b in e d to e s tim a te th e m in i m u m lin e w id th a n d th e pow er level co rre sp o n d in g to th e onset o f lin e w id th re bro a d e n in g in a ty p ic a l q u a n tu m -w e ll s e m ico n d u cto r laser d io d e . T h e m a in im p o rta n t conclusions o f th is d is s e rta tio n are th e fo llo w in g : I) . . \ new th e o ry o f laser lin e w id th was presented , in w h ic h lin e w id th s a tu ra tio n and re b ro a d e n in g are shown to be a n a tu ra l consequence o f th e presence o f m u lti- 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p lic a tiv e noise. In th e c o n te x t o f the m o d ifie d p a rtia l-w a v e m o d e l developed here to tre a t m u ltip lic a tiv e noise, the o rig in a l S chaw low -Tow nes in ve rse -p o w e r lin e w id th fo rm u la was show n to be th e lim itin g case o f no m u ltip lic a tiv e noise. T h e effects o f m u ltip lic a tiv e noise w ill be m ost pronounced in s m a ll s tru c tu re s lik e sem icon d u c to r lasers, in w h ic h we should expect to see a m in im u m o f th e p o w e r-lin e w id th c h a ra c te ris tic a t an o p tim u m power level. 2 ). .\n analysis o f th e c o n trib u tio n o f th e rm a lly -in d u c e d c a rrie r d e n s ity flu c tu a tio n s in th e co n fin e m e n t region o f a ty p ic a l q u a n tu m -w e ll s e m ic o n d u c to r laser yie ld s an e s tim a te o f 5.6 k H z to 17.6 kH z fo r th e m in im u m lin e w id th . T h is is in reasonable agreem en t w ith reported results fo r m in im u m lin e w id th s o f D F B lasers in w h ic h s p a tia l h o le -b u rn in g is suppressed [28]. T h e c a rrie r d e n s ity noise also rep resents an in trin s ic source o f 1/ / ^ frequency noise in q u a n tu m -w e ll s e m ico n d u cto r lasers. C a rrie r d e n s ity noise is expected to p la y a ro le in an y laser s tru c tu re in w hich a s ig n ific a n t p o rtio n o f th e o p tic a l m ode lies o u ts id e th e a c tiv e region in an area o f high c a rrie r d e n sity. T h e dependence o f the c a rrie r d e n s ity noise on c a rrie r effe ctive mass and c a v ity g e o m e try suggests possible stra te g ie s to o b ta in re d u c e d -lin e w id th laser diodes. 6.2 C om parison w ith O ther R esu lts In th is se ctio n , th e p re d ic tio n s o f th e p a rtia l-w a v e m o d e l w ill be co m p a re d to o th e r th e o re tic a l and e x p e rim e n ta l results. 6 .2 .1 L in ew id th o f sem icon d u ctor la sers T h e m easured p o w e r-in d e p e n d e n t lin e w id th and lin e w id th re b ro a d e n in g o f a m u lti- q u a n tu m -w e ll se p a ra te -c o n fin e m e n t-h e te ro stru ctu re (M Q W -S C H ) D F B laser was show n in th e In tro d u c tio n as F igure 1.2 [41]. F ro m th e fig u re , it is seen th a t th e lin e w id th in itia lly decreases w ith increasing pow er, th e n is e s se n tia lly pow er- indep e nde nt fro m P = 10 m W to a p p ro x im a te ly P = 33 m \V . T h e m in im u m lin e w id th o f 800 kH z occurs at an o u tp u t pow er level o f a p p ro x im a te ly 20 m \V . F or P > 33 mVV. th e lin e w id th rebroadens. T h e e s tim a te d lin e w id th o f fro m the 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p a rtia l w ave m o d e l o f 5.6 kH z to 17.6 k H z fo r a ty p ic a l q u a n tu m -w e ll sem icon d u c to r laaer c a lc u la te d in ch apter 5 is s u b s ta n tia lly less th a n these observations. T h e a u th o rs o f th is s tu d y [41] a ttrib u te th e lin e w id th re b ro a d e n in g to shot noise due to spontaneous re c o m b in a tio n in th e S C H regions. H ow ever, o th e r a u th o rs have achieved lin e w id th s o f several k ilo h e rtz [28] b y ta k in g steps to reduce s p a tia l hole b u rn in g th ro u g h th e in tro d u c tio n o f c o rru g a tio n -p itc h -m o d u la te d g ra tin g s , and com pressive s tra in in th e q u a n tu m w ells. T h e re fo re , it appears th a t suppression o f s p a tia l hole b u rn in g is c ritic a l to a c h ie vin g th e m in im u m lin e w id th s in D F B lasers. W h e n s p a tia l hole b u rn in g is a dequ ately suppressed, the m easured lin e w id th results are co n siste n t w ith th e p re d ictio n s o f th e p a r tia l w ave m o d e l w ith in trin s ic e le ctro n - d e n s ity flu c tu a tio n s . H ow ever, it is noted th a t th e m easured results o f 3.6 kH z were at th e level o f th e m eeisurem ent system re s o lu tio n in [28]. T h e p a rtia l wave m o d e l also p re d ic ts th a t lin e w id th rebroadening s h o u ld n o t o ccu r fo r o u tp u t powers u p to a b o u t 80 m illiw a tts . 6 .2 .2 In te n sity p rob ability d istr ib u tio n In th is section, we discuss th e broadening o f th e in te n s ity p ro b a b ility d is trib u tio n in th e large-a ng le re gim e seen fro m th e n u m e ric a l c o m p u ta tio n s in c h a p te r 2. Some pre vious stu d ie s o f m u ltip lic a tiv e noise in lasers have p re d ic te d a "ta iT ’ in th e in te n s ity p ro b a b ility d is trib u tio n e xte n d in g to low er in te n s itie s , w hen th e laser is o p e ra te d in th e presence o f m u ltip lic a tiv e noise [23]. In th is aspect, th e n u m e ric a l results o f the tim e -v a ry in g p a rtia l-w a v e m odel presented in ch a p te r 2 are s im ila r to p re d ic tio n s o f o th e r m odels o f m u ltip lic a tiv e noise in lasers th a t proceed fro m a F o kke r-P la n ck analysis o f th e ra te eq ua tio ns. In these analyses, th e m ean value and h ig h e r m o m e n ts o f th e p h o to n n u m b e r are derived as a fu n c tio n o f o u tp u t pow er and m u ltip lic a tiv e noise s tre n g th . In th e large-angle reg im e , the ca lc u la te d ta il o f th e in te n s ity p ro b a b ility d is trib u tio n , and s h ift in m ean in te n s ity to low er values, is in agreem ent w ith a th e o re tic a l F o k k e r-P la n c k analysis [23] and e x p e rim e n ta l results [24] o b ta in e d fo r a h e liu m - neon laser w ith m u ltip lic a tiv e noise, as show n in F igures 6.1 and 6.2. T h is s h ift o f 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the p ro b a b ility d is tr ib u tio n to lo w e r values w ith increéising g a in is c o n tra ry to the b e h a vio r e xp e cte d fo r th e resonant a m p lifie r w ith o u t m u ltip lic a tiv e noise. In the co n te x t o f laser o p e ra tio n , th is re su lt im p lie s th a t th e m ean value o f th e in te n s ity p ro b a b ility d is tr ib u tio n w ill in itia lly grow cis th e ga in is firs t increased, b u t even tu a lly w ill reach a m a x im u m value, and th e n begin to decrease. T h is agreem ent w ith th e F o k k e r-P la n c k analysis therefore is a d d itio n a l c o n firm a tio n o f th e v a lid ity o f th e tim e -v a ry in g p a rtia l-w a v e m odel. .\ls o , th e p h y s ic a l m e ch a n ism responsible fo r th e b ro a d e n in g o f th e in te n s ity p ro b a b ility d is trib u tio n is re a d ily apparent in the p a rtia l-w a v e m o d e l (i.e ., d e s tru c tiv e in te rfe re n ce o f p a rtia l w aves,) whereas the m a th e m a tic a l s to c h a s tic ca lcu lu s tre a tm e n t o f th e F o k k e r-P la n c k an a lysis renders the p h y s ic a l m e ch a n ism fo r th e enhanced in te n s ity flu c tu a tio n s less a p paren t. 6.3 F urther research In th is se ctio n , several ideas fo r fu rth e r th e o re tic a l and e x p e rim e n ta l in ve stig a tio n s are offered. 6.3.1 L aser lin ew id th ex p erim en ts T h e m o st im m e d ia te need is to c a re fu lly v e rify th e th e o re tic a l p re d ic tio n s o f the p a rtia l w ave m o d e l in e x p e rim e n ts w ith devices o f k n o w n p a ra m e te rs. T he m ost obviou s tests w o u ld be on h ig h -p o w e r D F B q u a n tu m -w e ll laser diodes. T h e results o f an e x p e riin e n t c o m p a rin g th e lin e w id th o f th e sam e q u a n tu m -w e ll laser for both o p tic a l and e le c tric a l p u m p in g c o u ld va lid a te th e th e o ry th a t lin e w id th broadening is due to e le c tro n d e n s ity flu c tu a tio n s in th e c o n fin e m e n t regions. O th e r solid- sta te Icisers m ig h t be tested, such as the N d :Y A G laser system . D io d e -p u m p e d Nd:Y.A.G lasers are a va ila b le w ith p ie zo-electric in tra -c a v ity fre q u e n c y -m o d u la tio n elem ents. T h is ty p e o f device co u ld be m o d u la te d w ith w h ite noise to sim u la te m u ltip lic a tiv e phase flu c tu a tio n s , to v e rify th e p re d ic tio n s o f th e p a rtia l-w a v e m odel fo r th e p o w e r-in d e p e n d e n t lin e w id th and lin e w id th -re b ro a d e n in g . .<\lso, th e th e o ry could be a p p lie d to d iffe re n t sp e cific lasers o r o th e r e le c tro n ic o s c illa tin g system s, such as e le c tro n ic fre q u e n c y standards. 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.30- Q = 0.0 0.25- 0.2 0 - 0.15- 0.1 0 - 0.05- 0.30-| 0.25- 0.2 0- Q = 3.15 0.15- 0.1 0 - 0.05- 0.00 0 10 20 30 0.30-1 Q = 0.20 0.25- 0.2 0 - 0.15- 0.10- 0.05- 0.30- 0 = 0.77 0.25- 0.2 0 - 0.15- 0.1 0- 0.05- Figure 6.1: Theoretical results for intensity probablility distribution from Fokker-Planck analysis of multiplicative noise in a helium-neon laser. (Data from S. Zhu, Physical Review A 47, 2405-2408, 1987.) 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.0 — 2.5 - 2.0-1 „ A 1.5 ^ I V 1.0 ^ 0 .5 - 0.0 - ▲ ♦ 0.1 • Q = 0 ♦ Q = 0.66 ▲ Q = 1.15 ■ Q = 2.13 ▲ A A ^ ♦ ♦ ♦ ♦ % # . A ♦ A ♦ A ♦ T T T m r r 1 0 <i> Figure 6.2: Experimental results for intensity probablility distribution from Fokker-Planck analysis of multiplicative noise in a helium-neon laser. (Data from M.R. Young, S. Singh, Optics Lett. 13, 21-23,1988.) 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 .3 .2 In te n sity n o ise in l&rge-angle re g im e T h e p ro b a b ility d is tr ib u tio n o f in te n s ity in th e la rge-ang le re g im e co u ld be m easured versus n et g a in and c o m p a re d to th e m odel. T h is c o u ld be done in th e sta n d a rd w ay w ith a fast d e te c to r and a fa st-sa m p lin g oscilloscope to b u ild up a h isto g ra m o f th e laser in te n s ity . T h is co u ld be done w ith b o th s e m ic o n d u c to r laser diodes and w ith a Nd:Y.A.G laaer m o d u la te d by noise. T h e e rra tic in te n s ity pulsations in the lin e w id th b ro a d e n in g reg im e co u ld also be observed w ith a s tre a k cam era. .A.lso. th e s im ila r ity betw een th e large-ang le regim e and th e “ coherence-collapse” regim e o f s tro n g feedback in s e m ic o n d u c to r lasers is an a p p e a lin g th e o re tic a l and e xp e rim e n ta l to p ic o f stu d y . .A.t firs t glance, it appears th a t since th e large-ang le re g im e represents a non lin e a r n o ise -d rive n syste m , th e flu c tu a tio n s in th e la rg e -a n g le re g im e m ay represent a fo rm o f o p tic a l chaos. W h ile th is hypothesis requires m o re in v e s tig a tio n , if it is tru e , it m a y be possible to use th e recently-deve loped te ch n iq u e s o f "chaos c o n tro l" [42] to reduce th e in te n s ity flu c tu a tio n s and lin e w id th in th is regim e. 6 .3 .3 1 / / n o ise W h ile th e p a rtia l-w a v e m o d e l provides a p lausible m e ch a n ism fo r pow er-inde pendent lin e w id th and lin e w id th re b ro a d e n in g , it does n o t p re d ic t 1 / / flu c tu a tio n s as a con sequence o f laser a c tio n . T h e m u ltip lic a tiv e flu c tu a tio n s due to e le ctro n -d e n sity flu c tu a tio n s w ere p re d ic te d to have a low -frequen cy e n h a n ce m e n t o f th e pow er spec tru m , and th e re fo re cause a correspond ing lo w -fre q u e n cy e n h a n ce m e n t o f the pow er s p e c tru m o f fre q u e n cy flu c tu a tio n s in the sm a ll-a n g le re g im e o f th e p a rtia l-w a ve m o d e l. H ow ever, th is is n o t due to laser a ctio n , b u t ra th e r th e c a rrie r density flu c tu a tio n s w ith a l/ / " - n o is e . I t w o u ld be far m ore in te re s tin g to fin d a m echanism in th e lasin g process cap ab le o f p ro d u c in g I / / noise. .A.recchi has show n [43] th a t th e power sp e ctru m o f a n o n -lin e a r d yn a m ic a l sys te m w ith m o re th a n one ba sin o f a ttra c tio n becomes 1 / / in th e presence o f noise, p ro vid e d th a t th e b o u n d a ry betw een th e basins is fra c ta l. T h e re fo re , th e large-angle regim e o f th e p a rtia l-w a v e m o d e l m ay pro vid e a m e ch a n ism w h e re b y m u ltip lic a tiv e 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. noise w ith a w h ite p o w e r-spe ctral d e n s ity is tra n s fo rm e d to 1/ / noise th ro u g h th e process A re c c h i has d ub be d "hyperchaos.” A fra c ta l b o u n d a ry e xists between th e so lu tio n s o f an e q u a tio n w ith th re e o r m ore ro o ts , such as = 1 . I f we consider th e m o d e -d e fin in g e q u a tio n o f a m u lti-m o d e laser s i n Ç ^ ) = 0 (6.1) it is ob vio us th a t th is eq ua tio n has m a n y ro o ts . In th e presence o f m u ltip lic a tiv e noise, th e la s in g m ode m ay * ‘h o p ” betw een these s o lu tio n s in a chaotic fashion, g e n e ra tin g 1/ / noise. T h is is w h a t happens in s p a tia l hole b u rn in g , b u t it does n o t appea r th a t s p a tia l hole b u rn in g has been in v e s tig a te d fro m th e p o in t o f view o f 1 / / noise g e n e ra tio n . I f we lo o k a t th e p ro b le m o f ge n e ra tin g 1 / / noise in ste a d fro m a purely m a th e m a tic a l p o in t o f vie w , it is apparent th a t a syste m w ith an im p u ls e response th a t decays as l / \ / F e x h ib its a I / / frequency response o f its pow er spectrum . T h is is because a 1/ / pow er sp e ctru m corresponds to a syste m w ith an a m p litu d e frequency response p ro p o rtio n a l to l/\/7> w h ich has inverse F o u rie r tra n s fo rm p ro p o rtio n a l to l / \ / î . T h e re fo re , if a system w ith a l/v T - im p u ls e response is d riv e n b y noise w ith a w h ite po w e r s p e c tra l density, th e o u tp u t po w e r s p e c tru m w ill have 1/ / frequency dependence. It has been show n th a t a h a lf-o rd e r in te g ra tio n o f an in p u t process represents a I / ' / i tim e im p u ls e response [44]. A n e x a m p le o f a syste m th a t produces th is response is an in fin ite tran sm ission lin e : th e v o lta g e p ro d u ce d a t th e in p u t o f th e lin e in response to a s te p -in p u t o f c u rre n t decays as i / \ / t . It seems possible to m ake an analog y betw een an in fin ite tra n sm issio n lin e and a laser o p e ra te d w ith u n ity n e t gain. F ro m th e p a rtia l-w a ve m o d e l, u n ity n e t ro u n d -trip g a in im p lie s an in fin ite n u m b e r o f p a rtia l waves, and hence an in fin ite ly long m e m o ry tim e o f th e c a v ity for m u ltip lic a tiv e noise events. T h is m ust be fu rth e r e xp lo re d th e o re tic a lly . I f it can be show n th a t th e c a v ity produces a l / \ / i tim e im p u ls e response to m u ltip lic a tiv e noise, th e n th is w o u ld represent an in trin s ic source o f 1/ / noise in self-sustained o s c illa tin g system s. 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I t hcis also been show n th a t th e observed lin e w id th o f a tu n a b le F a b ry-P e ro t o p tic a l c a v ity in the “ speed o f lig h t lim ite d re g im e ” broadens as y /J , w here / is the ra te o f th e c a v ity tu n in g sweep [45], [46]. T h is im p lie s th a t th e observed storage tim e o f th e c a v ity is p ro p o rtio n a l to 1 / y/J. T h e “ speed o f lig h t lim ite d regim e” is ch a ra cte rize d by large a m p litu d e flu c tu a tio n s o f th e c a v ity o u tp u t, a risin g from d e s tru c tiv e in te rferen ce o f th e stored and tra n s m itte d fields w h e n th e sweep rate is faster th a n th e in trin s ic storage tim e o f th e c a v ity as d e te rm in e d b y th e c a v ity losses. It thus appears th a t th e re is s im ila rity betw een th e “ speed o f lig h t lim ite d regim e " and th e large-angle regim e o f th e tim e -v a ry in g p a rtia l-w a v e a n a lysis. I f th is is indeed tru e , th is im p lie s th a t th e response o f th e o p tic a l resonator to m u ltip lic a tiv e noise in th e large-angle regim e is a su p e rp o s itio n o f tim e constants th a t are p ro p o rtio n a l to l / y / J , w here / is th e m u ltip lic a tiv e noise flu c tu a tio n fre q u e n cy. T h u s, each increasing m u ltip lic a tiv e flu c tu a tio n fre q u e n cy e ffe ctive ly causes a decreased Q for the ca v ity . T h e higher flu c tu a tio n frequencies are th e re fo re ro lle d -o ff less th a n in the sm a ll angle regim e, w here th e y ro ll o ff as 1/P , as seen fro m th e a n a lysis o f chapter 2. T h e re fo re , it seems p ro m is in g to th e o re tic a lly and e x p e rim e n ta lly in v e s tig a te the s itu a tio n o f a w h ite m u ltip lic a tiv e noise source m o d u la tin g th e c a v ity le n g th in the large-angle regim e. I f th e preceding hyp o th e sis is co rre ct, th is m a y represent an in trin s ic source o f 1/ / noise in resonant system s. 6.4 C losing T h e tim e -v a ry in g p a rtia l-w a v e m odel d e rive d here provides a u n ifie d p h ysica l m odel for pow er-in de p en de nt lin e w id th and lin e w id th rebroade ning in lasers due to m u lti p lic a tiv e noise. T h e analysis o f e le ctro n d e n s ity flu c tu a tio n s in se m ico n d u cto rs in dicates th a t th is process m a y be responsible fo r 1/ / “ noise in s e m ic o n d u c to r lasers, w ith 0.5 < a < 1.5. A lso, th is process m a y be o p e ra tive in o th e r typ e s o f sem i co n d u c to r devices such as tra n sisto rs, diodes, etc. A lth o u g h th e p a rtia l-w a v e m odel does n o t ye t p ro vid e a clear e x p la n a tio n o f an in trin s ic la s e r-d y n a m ic a l process for the g e ne ratio n o f 1/ / noise, several p ro m is in g avenues o f fu rth e r in v e s tig a tio n have becom e a p p a re n t in th e course o f th is w o rk. 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bibliography [1] [2] [3 [4 [5 [6 [8 [9 [10 [11 [12 [13 [14 [15 VV. H . G le n n , IE E E J o u rn a l o f Q u a n tu m E le c tro n ic s 25, 1218-1224 (1989). H . B . C a lle n , R. F . G reene, P hysical R eview 8 6 , pp. 702-710. (1952). R. D re ve r. et a l.. A p p lie d Physics B P h o to p h ysics A n d Laser C h e m is try 31. 97-105 (1983). A . L. Schavvlow, C . H . Townes, P hysical R e vie w 1 1 2 , 1940-1949 (1958). M . 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Asset Metadata

Creator Logan, Ronald Thomas, Jr (author)

Core Title Analysis of multiplicative noise in laser amplifiers and its effect on laser linewidth

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, electronics and electrical,OAI-PMH Harvest,physics, optics

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Garmire, Elsa (committee chair), Maleki, Lute (committee chair), [illegible] (committee member), Maleki, Lute (committee member), Willner, A.E. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c17-315331

Unique identifier UC11353919

Identifier 9835069.pdf (filename),usctheses-c17-315331 (legacy record id)

Legacy Identifier 9835069.pdf

Dmrecord 315331

Document Type Dissertation

Rights Logan, Ronald Thomas, Jr.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA