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Modulation response and linewidth properties of microcavity lasers

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Modulation response and linewidth properties of microcavity lasers

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Content MODULATION RESPONSE AND LINEWIDTH PROPERTIES OF
MICROCAVITY LASERS
by
Mahmood Bagheri
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
December 2008
Copyright 2008 Mahmood Bagheri
Dedication
to my Father,
whom without his constant help and support, I would not be close
to where I am now
ii
Acknowledgments
There are lots of people I would like to thank for a huge variety of reasons.
Firstly, I would like to thank my Ph.D supervisor, Dr. John D. O’Brien. I could
not have imagined having a better advisor and mentor for my Ph.D, and without
his common-sense, knowledge, perceptiveness and patience I would never have
finished. I enjoyed the freedom he provided in his group to explore new ideas,
and his research group in USC was a scientifically challenging environment that
was really fun to work at. It was a great opportunity for me to have him as my
Ph.D adviser.
I humbly acknowledge my dissertation and qualifying exam committee
members help and assistance; Dr. Steier, Dr. Dapkus, Dr. Lidar and Dr.
Hashemi. It was my privilege to benefit from a group of world-known scientist
and researchers during my Ph.D at USC. I have always enjoyed Dr. Dapkus’s
meticulous advice and remarks, and Dr. Steier insight and wisdom has always
been inspirational.
I feel indebted to a large group of people at USC because of their help and
support during my graduate study. I would like to thank Dr. Kian Kaviani who
served as my first mentor at USC and the United States. I thank my previous
and current fellow colleagues Dr. Shih, Dr. Cao, Dr. Marshall, Dr. Akhavan, Dr.
Stapleton, Dr. S. Cockerham, Dr. Kuang, Dr. Yang, Dr. Shafiiha, Stephen Farrel,
Raymond Sarkissian, Ling Lu and Adam mock.
iii
It was a fun experience living my graduate studies at USC because of a group
fantastic individuals that I had the honor to share friendship with. I thank them
all.
And at last I would like to thank my family members for their constant help
and support. I thank my fianc´ ee, Arezou, for her constant support and love
during writing this thesis. My dad has been a constant source of support and
inspiration from the very beginning of childhood. I greatly owe my success as a
Ph.D to his help and love.
iv
Table of Contents
Dedication ii
Acknowledgments iii
List of Figures vii
Preface xi
Chapter 1: Introduction 1
1.1 Interconnects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Optical Interconnects . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Photodiode Noise . . . . . . . . . . . . . . . . . . . . . . . . 8
Chapter 2: Optical Microcavity 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Microdisk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Photonic Crystal Defect Cavity . . . . . . . . . . . . . . . . . . . . 15
2.4 Nano-Fabrication of Photonic Devices . . . . . . . . . . . . . . . . 19
2.5 Optical Characterization of Microcavity Lasers . . . . . . . . . . . 20
Chapter 3: Spontaneous Emission in Semiconductor Microcavity Lasers 26
3.1 Interaction of Cavity Optical Mode with the Active Material . . . 27
3.2 Spontaneous Emission in Free Space . . . . . . . . . . . . . . . . . 28
3.3 Purcell Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Characteristics of Semiconductor Microcavity Lasers . . . . . . . 39
Chapter 4: Semiconductor Laser Diode Noise 41
4.1 Langevian Noise Forces . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Noise Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Spontaneous Emission and Laser Linewidth . . . . . . . . . . . . 50
4.4 Microcavity Laser Linewidth . . . . . . . . . . . . . . . . . . . . . 55
4.4.1 Photonic Crystal Laser Linewidth . . . . . . . . . . . . . . 56
4.4.2 Microdisk Laser Linewidth . . . . . . . . . . . . . . . . . . 59
4.4.3 Laser Linewidth and Microcavity Eects . . . . . . . . . . 61
4.4.4 Evaluation of R and I . . . . . . . . . . . . . . . . . . . . . . 63
v
4.5 Laser Linewidth Measurement Techniques . . . . . . . . . . . . . 68
4.5.1 Optical Heterodyne and Optical Self-Delayed-Homodyne 69
Chapter 5: Modulation Response Properties of Semiconductor Microcav-
ity Lasers 73
5.1 Semiconductor Laser Rate Equations . . . . . . . . . . . . . . . . . 74
5.2 Modulation Response Measurement of Microcavity Lasers . . . . 79
5.3 Gain Compression in Semiconductor Lasers . . . . . . . . . . . . . 84
5.4 Frequency Modulation of Semiconductor Lasers . . . . . . . . . . 88
5.5 Measurement of Frequency Chirping Characteristics of Semicon-
ductor Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.5.1 Dispersive Fiber Transfer Function . . . . . . . . . . . . . . 91
5.5.2 Optical Coherent Discriminator Method . . . . . . . . . . . 95
References 99
Appendix A: Scattering Matrix Approach to Resonant Mode and Q Value
Calculation of Microdisk 108
Appendix B: Master Equation 114
B.1 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
B.2 Density Matrix Operator Elements of a Two Level System Inter-
acting with Its Reservoir . . . . . . . . . . . . . . . . . . . . . . . . 117
vi
List of Figures
1.1 Imbalance between the internal CPU and the memory bus band-
width for dierent computer architectures [LAP
+
04] . . . . . . . . 2
1.2 Receiver sensitivity at 10 Gb/s (2.0-V supply) and 12.5 Gb/s (1.8 V ,
2.0 V , full power) [IEE06] . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 (a) Schematic drwaing of a microdisk coupled to a silicon waveg-
uide [CRRR
+
07], (b) SEM image of a microdisk coupled to an InP
bus waveguide [CDCD03]. . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Photodetection process . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Calculated BER of a photodetector vs. the detected photocurrent
at 20 Gbps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 (a) Top view schematic image of a microdisk cavity, (b) A microdisk
cavity at cross section view . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Calculated quality factors of microdisks with the S-matrix method
corresponding to the modes with resonant wavelength, = 1589:0
nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Top view schematic image of a defect photonic crystal cavity with
37 missing air holes (D
4
) . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 (a) Calculated quality factor of a suspended membrane D
3
defect
cavity, (b) Calculated quality factor of a D
3
defect cavity on sap-
phire substrate (Courtesy of Wan Kuang, USC) . . . . . . . . . . . 18
2.5 (a) Calculated quality factor of a suspended membrane D
4
defect
cavity sapphire substrate (Courtesy of Wan Kuang, USC) . . . . . 18
2.6 (a) Scanning electron microscope image of a microdisk at 45

view,
(b) Top view SEM image of a fabricated microdisk . . . . . . . . . 20
2.7 (a) Scanning electron microscope image of a D
4
photonic crystal
defect cavity at 45

view, (b) Top view SEM image of the same
device as (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
vii
2.8 Schematic illustration of measurement setup for optical charac-
terization of microcavity lasers . . . . . . . . . . . . . . . . . . . . 22
2.9 (a) collected output power vs, the incident optical power curve
for a D
4
photonic crystal defect cavity, (b) Lasing spectrum of the
photonic crystal laser in (a) . . . . . . . . . . . . . . . . . . . . . . 24
2.10 (a) Incident threshold pump power (circles) and lasing wave-
length (squares) vs. the photonic crystal lattice constant of the
laser array , (b) Threshold pump power vs. wavelength (circles)
and its overlap with quantum well photoluminescence spectrum 24
2.11 (a) collected output power vs, the incident optical power curve
for a 2m radius microdisk cavity, (b) Lasing spectrum of the
microdisk laser in (a) . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.12 (a) collected output power vs, the incident optical power curve
D
4
photonic crystal defect laser at dierent operating temperatures 25
4.1 The change of complex field amplitude(t) during a short time T. 52
4.2 (a) a D
4
photonic crystal defect cavity laser linewidth evolution
versus the collected output power, (b) Laser Lineshape profile
taken on an electrical spectrum analyzer with a Lorentzian curve
fit, with a extracted linewidth of 575 MHz . . . . . . . . . . . . . . 57
4.3 D
4
and D
5
photonic crystal defect cavity lasers linewidth evolu-
tion versus the collected output power (laser linewidth of the cav-
ity with larger mode volume (D
5
) in this case, satrated at higher
output power and narrower linewidth values) . . . . . . . . . . . 58
4.4 Dependence of linewidth enhancement factor,, on the laser oper-
ating point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5 (a) D
4
photonic crystal defect cavity laser linewidth evolution
versus the collected output power, (b) D
4
photonic crystal defect
cavity laser linewidth evolution versus the collected output power
normalized to 1 +
2
dependence on laser operating point . . . . 59
4.6 (a) Lasing spectrum of three dierent microdisks with slightly
dierent radii with the spontaneous emission spectrum from the
QW active region overlaid, (b) laser linewidth versus the incident
pump power inverse for cavities in figure 4.6(a) . . . . . . . . . . 60
4.7 Dispersion of linewidth enhancement factor, versus the photon
energy after Yamanaka et al. [YYY
+
93] . . . . . . . . . . . . . . . . 61
viii
4.8 (a) Linewidth vs. the eective injected current inverse for a 2:0m
radius microdisk (squares) and a 3:8m radius microdisk (circles),
(b) detected photocurrent power spectrum of the minimum mea-
sured linewidth for the 3:8m microdisk in figure 4.8(a) and the
Lorentzian fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.9 (a) Linewidth vs. the eective injected current inverse for a 1:8m
radius microdisk, (b)detected photocurrent power spectrum of
the minimum linewidth in figure 4.9(a) and the Lorentzian fit. . . 63
4.10 Band-to-band radiative transitions: stimulated absorption, stim-
ulated emission, and spontaneous emission. (All rates are defined
per unit volume.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.11 Measured microdisk laser linewidth versus the eective injected
current inverse for three dierent microdisks with dierent radii
lasing at
lasing
= 1581nm . . . . . . . . . . . . . . . . . . . . . . . . 65
4.12 Collected output power versus the incident pump power for three
dierent microdisks with dierent radii lasing at
lasing
= 1581nm 66
4.13 Measured ratio between the measured and estimated slope of the
linewidth versus the inverse of the current curve. . . . . . . . . . 67
4.14 Estimated Purcell factor based on measured quality factors and
calculated optical mode volumes . . . . . . . . . . . . . . . . . . . 68
4.15 Setup for interfering two optical fields . . . . . . . . . . . . . . . . 69
4.16 Lineshape of a laser measured with the self-delayed homodyne
method with the Lorentzian curve fit . . . . . . . . . . . . . . . . . 72
5.1 (a) Output power vs. the injected current density, (b) output
power of a laser (scattered) in response to a time varying input
current(solid line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Schematic illustration of measurement setup for high-speed char-
acterization of microcavity lasers . . . . . . . . . . . . . . . . . . . 80
5.3 (a)VCSEL mounted directly on SMA socket (best frequency per-
formance) (b) Intensity modulation response of high-bandwidth
VCSEL ( Courtesy of ULM Photonics). . . . . . . . . . . . . . . . . 81
5.4 small signal modulation response of a 3:2m diameter cavity (D
4
)
at dierent bias levels with the curvefit applied to the data at two
times threshold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
ix
5.5 (a) Relaxation Oscillation frequency squared versus the incident
input power normalized to the threshold power, (b)damping of
laser versus the relaxation oscillation frequency squared . . . . . 83
5.6 (a) Relaxation Oscillation frequency squared versus the collected
output power, and (b)damping versus the relaxation oscillation
frequency squared of a D
4
photonic crystal defect cavity laser . . 84
5.7 Relaxation Oscillation frequency squared versus the collected out-
put power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.8 (a) Relaxation Oscillation frequency squared versus the collected
output power, and (b)damping versus the relaxation oscillation
frequency squared of a 1:92m microdisk laser . . . . . . . . . . . 86
5.9 Photon density inside two cavity geometries with V
opt
= 2(=n)
3
and V
opt
= 100(=n)
3
versus the cavity quality factor at 100W out-
put power, the shaded area shows the boundaries for the region
in whichS
0
1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.10 Measured transfer function of a 25 km long dispersive optical
fiber using an externally modulated laser . . . . . . . . . . . . . . 93
5.11 Measured transfer function of a 25 km long dispersive optical
fiber using a D
4
photonic crystal defect cavity laser . . . . . . . . . 94
5.12 Measured transfer function of a 25 km long dispersive optical
fiber using a D
4
photonic crystal defect cavity laser . . . . . . . . . 95
5.13 (a) Optical discriminator setup used to measure the AM and FM
response of microcavity lasers, (b) the quadrature points . . . . . 96
5.14 (a) Intensity modulation of a microdisk laser measured with inter-
ferometer setup, (b) frequency modulation of a microdisk laser
measured with the interferometer setup . . . . . . . . . . . . . . . 98
A.1 Schematic representation of a microdisk waveguide structure . . 108
A.2 (a) Wigner time delay spectrum of mode HE
7;1
of a 1m microdisk
on Sapphire substrate, (b) zoomed in view of a the peak labeled
(1) in (a) with a Lorentzian curve fit (solid curve) to extract the
quality factor and the center wavelength . . . . . . . . . . . . . . . 113
B.1 System interacting with its reservoir . . . . . . . . . . . . . . . . . 114
x
Preface
Integrated optics is the ultimate solution for reducing the manufacturing and
operational costs of optical circuits. The growing demand for high-bandwidth
communication between integrated circuits has also initiated the interest in chip-
scale integrated optical circuits. Microcavity lasers as an indispensable part of
such circuits, have been the subject of intensive research during the past two
decades. There have been a lot of speculations on the dynamic properties of
these lasers in the past, but to date, there has not been a comprehensive study
and analysis of those properties on these lasers. Lack of experimental data
providing enough feedback to the theories, has prevented the existing theories
to mature. The goal of this thesis is to understand the modulation response and
linewidth properties of these lasers through combining our experiments with
the theory.
Chapter 2 reviews the optical microcavities investigated for this thesis. Their
fabrication process is briefly presented, and their optical properties are reviewed.
Chapter 3 discusses the optical microcavity physics, and how microcavities
can modify the electronic states lifetime. A derivation of this phenomenon
based on the Wigner-Weiskopft theory of spontaneous emission is given and it
is compared with a more detailed analysis based on system-reservoir theory of
dissipation.
xi
In chapter 4, a single mode laser rate equation is presented based on a semi-
classical approach. Then Langevian noise sources are introduced to account
for fluctuations of the laser field. In this chapter, the laser linewidth is derived
and microcavity linewidth measurements are then presented. In the end, the
eect of increased spontaneous emission rate on the linewidth of lasers in micro-
cavities are investigated. It is shown that the increased spontaneous emission
rates results in further broadening of the spectral width than what is expected
in conventional lasers.
Chapter 5 presents the data on small signal intensity modulation response
properties of microcavities. It is shown that these cavities are influenced by
gain nonlinearities that ultimately limits the maximum achievable modulation
bandwidth. The fiber transfer function technique is then applied to these cavities
to extract their linewidth enhancement factors. It is shown that by including
the carrier dynamics outside the quantum wells, reasonable agreement can be
obtained between the theory and experiment.
xii
Chapter 1
Introduction
Optical interconnects are the dominant solution in long haul data communica-
tions. The invention of low loss optical fibers and single mode lasers, as well as
fast detectors and dense wavelength division multiplexing technology helped
the fast growth of this technology to the point where the optical networks per-
formance doubled every nine months. After revolutionizing the long distance
communication networks such as telephony systems and wide area networks,
optical networks became the prevailing solution in shorter distance networks
such as metropolitan area networks (MAN) and local area networks (LAN) over
the past decade. Now, these interconnects are expected to be used in even
shorter distance networks and soon to be used in rack-to-rack, chip-to-chip and
even on-chip data communication circuits.
The ever increasing demand for high bit rate data services has been the
driving force behind deployment of optical interconnects in long haul to mid-
distance data communication. Today, however, maintaining the overall need for
higher data rates and for decrease in operational costs and increase in revenues
is the force behind using new intelligent optical networks. Photonic integrated
circuits as an emerging technology, will help the data service and the internet
companies to keep up with these demands by consuming power more eciently
and utilizing more automatic decision making units. Photonic integrated circuits
also will pave the way for optical interconnects to play a more important role in
rack-to-rack, chip-to-chip and on-chip interconnects.
1
1.1 Interconnects
Dierent generations of CMOS technology have revolutionized the comput-
ing technology base units. Although the overall performance of systems have
improved over the years, various aspects of the systems have evolved at dif-
ferent rates. This is best indicated by the slow improvement of memory access
time ( 5-10% per year) as opposed to the processor performance per chip (
40% per year). Another disparity exists between the on-chip and o-chip data
bandwidth [BIK
+
05]. Figure 1.1 shows the imbalance between the internal CPU
and the memory bus bandwidth for dierent computer architectures [LAP
+
04].
While implementing dierent levels of high speed cache memory improves the
overall system performance, it is expected that the imbalance as shown in figure
1.1 limits the aggregate data throughput.
0.1
1
10
100
1000

P3-733
P2-450
P1-233
P1-200
P1-133
P1-100
P1-66
i486Dx-33
i486Dx-25
i386Dx-16
P4-1500
P4-2000
P4-3000
P4-3200

External Memory Bandwidth
Internal CPU Bandwidth
Bus Bandwidth (Gb/s)
itanium-2
Figure 1.1: Imbalance between the internal CPU and the memory bus bandwidth
for dierent computer architectures [LAP
+
04]
For CMOS technologies smaller than 200 nm, the o-chip data rate is mostly
limited by the RC time constant of the electrical interconnects [WLJ
+
06]. Switch-
ing from the traditional Al to less resistive Cu stripes and from SiO
2
to low
dielectric constant materials () for the electrical interconnects was to avoid this
2
problem. The current technology, however, is at the point where other solutions
beyond Cu and low are required to close the gap between the on-chip and o-
chip data bandwidth. The diculties for electrical interconnects resulting from
technology scaling imposes a more significant limitation on the performance of
the electrical interconnects because the RC time constant per unit length is begin-
ning to increase hyperbolically with decreasing the interconnect cross-section.
The increase in RC time constant of the electrical interconnects will limit the
maximum distance a bit of data can propagate in a clock cycle limiting the clock
speed of a processor. The interconnect delay performance limitation shows
clearly the inadequacy of continuing to scale the conventional metal/dielectric
system to meet future interconnect requirements [ITR].
The need for interconnect concepts beyond the conventional metal/dielectric
system that has served the industry for the first four decades of its existence has
been brought on by the continued increase of frequency and power of ICs. The
electrical interconnects suer very high signal attenuation at frequencies higher
than 5 GHz. Although equalizer circuits have been proposed that compensate
for the attenuation, these circuits occupy chip area and consume power which
is not preferable from a design point of view. It is also hard to isolate dierent
signal channels when they are operating at high data rates, because the lines will
radiate the electromagnetic waves into the environment, and this radiation will
interfere with the signals on the data channels adjacent to the channel. Therefore
cross talk imposes a serious problem for electrical interconnects carrying data at
multi giga bit per second . As a result, speed and cross talk can not be optimized
at the same time in electrical interconnects.
In the past two decades, there has been dramatic improvement in the per-
formance of optical interconnects in long distance networks, and they have
replaced electrical cables in these networks. Optical interconnects have good
3
prospects to overcome the challenges that their electrical counterparts are fac-
ing in chip-scale communication schemes. Optical interconnects do not suer
high attenuation at high frequencies, and are immune to cross talk at any given
frequency. Optical interconnects are considered a possible option for replacing
the conductor/dielectric system in backplane, chip-to-chip, and on-chip data
communication.
1.2 Optical Interconnects
The optical approach can be implemented in many dierent ways. However,
most of research in both academia and industrial research labs are today focused
on the so called “silicon photonics“ approach. This approach aims to benefit
from the mature silicon processing techniques and fabs that have served the
microelectronics industry for more than four decades and ultimately is more
conveniently integrated with CMOS circuits. This approach suers from the
fact that silicon is a not an ecient optical emitter. There are choices that need
to be made here, where to have o-chip emitters or have more friendly optical
emitters like derivatives of InP material hybridly integrated with the rest of the
optical circuit. There is also an in interdisciplinary eort between people who
do the architecture design and VLSI design to decide which signals to include
in the optical circuits which signals are transmitted through the conventional
electrical interconnect to minimize the latency and power consumption of the
entire circuit. In the case of optical interconnects, it is easy to assume that
this solution will meet speed requirements because the signal travels at ”the
speed of light” and has relatively large bandwidth. However, to define the
total interconnect system for this approach it is necessary to consider the delays
associated with rise and fall times of the optical emitters and detectors, the speed
of light in the transmitting medium, losses in the optical waveguides (if used),
4
the signal noise due to coupling between waveguides, and a myriad of other
details [ITR].
For the optical interconnects to replace the electrical interconnects, they not
only have to oer at least what the electrical interconnects provide, but also they
have to be cost eective. MAUI project [LAP
+
04] which is collaboration between
Agilent Technologies and professor Anthony Levi at the University of SOuthern
California is aiming to provide optical data at cost of 1$/Gbs.
Photonic integrated circuits (PICs) on a planar lightwave platform are one
of the most promising solutions that can potentially outperform the electrical
interconnects and drop the manufacturing and maintenance cost of those cir-
cuits. There are dierent pieces of optics that need to be integrated on a chip to
make such circuits. The two most important elements of any optical intercon-
nect that largely determine the performance of the circuit are the transmitter (a
Laser in this case) and the receiver (photodetector).
Transmitting data at high data rates and retrieving high bit-rate data are the
most basic functionality of PICs, therefore, high speed optical elements are the
elements that can make PICs look attractive. On the transmitting end, although a
laser source with an external modulator is used commonly in long haul commu-
nication networks, it is a less preferable alternative to directly modulate lasers in
PICs, because it adds to the complexity and manufacturing cost of the circuits.
For implementations requiring on-die emitters, a high eciency, high switching
rate laser source, monolithically integrated into Si CMOS, (at low cost) needs to
be developed.
Low power, high eciency, small size optical detectors monolithically inte-
grated into Si CMOS (at low cost) is another component that needs to be devel-
oped. As will be shown in section 1.2.1, design of the photodetection stage needs
careful attention in order to minimize the sources the contribute to the noise.
5
These noise sources will ultimately limit the speed at which these circuits will
operate at a fixed optical power. In 2006, The IBM group [IEE06] demonstrated
a 17 GBs optical receiver that can be integrated with electronic components on
a chip. Most of today state-of-the-art photodetector technology satisfies the
projected performance by ITRS. Figure 1.2 shows the sensitivity performance
of the fabricated optical receiver by IBM group. A sensitivity of -12.5 dBm is
measured for this receiver. This receiver is the fastest and lowest consuming
power on-chip detector to date.

Figure 1.2: Receiver sensitivity at 10 Gb/s (2.0-V supply) and 12.5 Gb/s (1.8 V , 2.0
V , full power) [IEE06]
As is shown in figure1.2, and from other calculations ( see appendix 1.2.1),
there needs to be a certain amount of optical power incident on the optical
receiver so that data communication is immune to detector noise. It can be
concluded that the research direction in design of optical sources for photonic
integrated circuits should be directed towards chip-scale high speed lasers with
reasonable output power. There is an independent research direction in profes-
sor John O’Brien’s groups at the University of Southern California that aims at
optimizing the output coupling of on-chip laser cavities to photonic waveguides
[Con07].
6
So far, there are successful on-chip demonstration of microdisk lasers cou-
pled to waveguide in planar lightwave platform [CDCD03, CRRR
+
07]. The
latter report, also has integrated photodetectors on the same die on a sili-
con substrate. This looks very promising for optical interconnect technology.
Figure1.3(a) shows a schematic drawing of an InP-based microdisk coupled to
a silicon bus waveguide on an SOI wafer capabale of room temperature CW
operation [CRRR
+
07], and figure 1.3(b) shows an scanning electron micrograph
of a fabricated microdisk coupled to an InP bus waveguide [CDCD03].
(b)

(a)
Figure 1.3: (a) Schematic drwaing of a microdisk coupled to a silicon waveguide
[CRRR
+
07], (b) SEM image of a microdisk coupled to an InP bus waveguide
[CDCD03].
Photonic crystal and photonic bandgap engineering is a newer technology
that gives more freedom in design of electromagnetic radiation properties of
light. It has shown promising potential for the components for photonic inte-
grated circuits in planar lightwave platform and providing high integration
density on chip. This technology is still relatively immature and as yet there
has not been any demonstration of an electrically injected photonic crystal laser
cavity that can operate under CW conditions at room temperature with good
waveguide-cavity coupling.
7
The size, power dissipation, bandwidth, and noise of lasers are impor-
tant metrics in evaluating the lasers cavity compatibility for these applications.
Although the small size of lasers helps to lower the lasing threshold power, the
small number of optical modes, resulting from decreased optical mode volume,
overlapping with the active medium can have significant eect on their noise
performance as the fraction of spontaneous emission from the active medium
coupled into the optical mode is significantly larger than the conventional bulky
lasers. The maximum modulation bandwidth of these devices must exceed
the projected bandwidth for the interconnects by the International Technology
Roadmap for Semiconductor (ITRS), in order for the optical interconnect to be a
viable solution. The turn on delay and turn o fall time of these lasers should
also be less than 180-270 ps for the optical interconnect advantageous to using
electrical interconnects.
1.2.1 Photodiode Noise
The sensitivity of a photodetector is essentially the amount of light required so
that the signal-to-noise ratio (SNR) is 1:1. There are many units used to measure
sensitivity and the appropriate choice depends of type of detector, the applica-
tion, whether or not it is desirable to normalize application parameters such as
measurement bandwidth or detector geometry. In high speed telecommunica-
tion applications the noise equivalent- power, NEP , is used as a figure of merit
for the weakest optical signal that can be detected.
The ”signal” of a photodetector fundamentally depends on its quantum
eciency, , that is defined as the percentage of photons that generate electron-
hole pairs. The external quantum eciency is the percentage of incident photons
that generate electron-hole pairs collected at the electrodes of the detector. This
is reduced from the internal quantum eciency by the photons that either reflect
8
from the detector surface or transmit through the detector. If P
opt
is the average
optical power incident on the photodetector then I
p
, the average photocurrent is
expressed by:
I
p
=
qP
opt
h
(1.1)
where q is the electron charge and is the frequency of the optical beam incident
on the photodetector, and h is Planck’s constant. A more practical way to
characterize the response of a photodetector to incident light is by defining
photoresponsitivity, R, of a photodetector as:
R =

1:24
(1.2)
where is the wavelength of the incident optical beam.
Thermal
Noise
Amplifier
Load
Resistor
Background
Dark
Optical
Signal
Background
Photodiode
Signal
Figure 1.4: Photodetection process
There are several sources of noise in photodetectors. These include shot noise
from the detector dark current, shot noise from the photocurrent, and Johnson
noise from thermal fluctuations in the detector impedance. Figure 1.4 shows a
simple block diagram of a photodetector with noise sources.
”Shot Noise” is caused by fluctuations in the current that are due to the
discreteness of the charge carriers and to the random electronic emission. Shot
9
noise is white noise and it is not frequency dependent. The spectral density
function of shot noise is [Chu95]
S( f ) = 2q (1.3)
where is the average photocurrent. The power of shot noise within
frequency interval f and f + f associated with the average current is
denoted by
 S( f ) f = 2q f (1.4)
Shot noise is caused by current from background radiation, I
b
and the current
from dark current, I
d
. Dark current is due to the thermal generation of electron-
hole pairs in the depletion region of the p-n junction of a photodetector. The
photocurrent also contributes to the total shot noise of a photodiode that is
= 2q(I
p
+ I
d
+ I
b
)B (1.5)
where B is the bandwidth of the system under investigation. The random
thermal fluctuations of charge carriers in electrical conductors at equilibrium
leads to a thermal noise current which is known as thermal noise or Johnson
(Nyquist Noise). The power spectral density of the thermal noise across a resistor
R at temperature T is
S( f ) =
4
R
h f
e
(h f=k
B
T)
1
(1.6)
where k
B
is Boltzmann’s constant. At low frequencies ( h f k
B
T, the above
spectrum can be simplified to
S( f ) =
4k
B
T
R
(1.7)
10
and the thermal noise power is described by
=
Z
f+ f
f
S( f )d f =
4k
B
T f
R
(1.8)
The power signal-to-noise ratio can be expressed as
(S=N)
power
=
i
2
p
 +
(1.9)
In digital system where the information is carried in a binary pulse code
modulation format, system performance is characterized by the probability of
making an error, or bit error rate (BER). BER is related to the signal-to-noise ratio
through
BER =
1
2
er f c(
p
(S=N)
power
2
p
2
) (1.10)
10 20 30 40 50 60 70 80 90 100
1E-14
1E-12
1E-10
1E-8
1E-6
1E-4
0.01
1
Calculated BER
Data Communication
Data Link
BER
i
p
( A)
Figure 1.5: Calculated BER of a photodetector vs. the detected photocurrent at
20 Gbps
For the photodetector, noise at the output of the photodetector is dominated
by the thermal noise of the terminating resistance (usually 50
). In most
applications however, this noise is much less than that added by a subsequent
amplifier stage.
11
Figure 1.5 shows the calculated BER for a photodetector followed by an
amplifier stage that has a noise figure of 6 dB working at 20 Gbps. It can be
inferred from the figure that to achieve BER of 10
9
which is the acceptable error
for long distance communication systems, the detected photocurrent should be
70A. If the responsitivity of the detector is R = 0.8 A/W, then the incident optical
power needs to be at least P
min
= 90W before the BER becomes unacceptable.
12
Chapter 2
Optical Microcavity
2.1 Introduction
Semiconductor optical microcavities have attracted a great deal of interest dur-
ing the last two decades. They are interesting for providing a platform for
physics experiments and for their potential role in the future technological
devices. Their relatively small mode volume and extremely high quality fac-
tors [Vah03] has helped physicists demonstrate some peculiar features of inter-
action of light and matter e.g. optical Kerr eect [RV05], radiation pressure
[RKCV05], and enhanced and inhibited spontaneous emission[VFS
+
03, Yab87]
in these microstructures.
Dierent configurations have been proposed and realized for the microcavity
structures. Optical whispering gallery modes[Vah03], and the photonic double-
heterostructure bound state modes[SNAA05] have been the most successful
platforms in bringing the losses down to a point where optical quality factors in
excess of 5 10
5
have been realized.
The extremely high quality factor combined with the small optical mode
volume of the microcavity modifies the density of states of photons inside the
cavity significantly compared to the density of states in free space. This signif-
icant change in density of states is the key for some of the QED experiments in
microcavities, but as it will be pointed out in the next chapters, large circulating
optical power inside the cavities with high quality factor above threshold causes
degradation of laser performance due to gain nonlinearities. Hence, lasers with
13
reasonable output coupling loss are more preferable for practical applications.
On the other hand, a minimum quality factor is required so that the active
medium can compensate loss to achieve lasing threshold.
This thesis focuses on microdisk and photonic crystal defect cavity lasers
bonded to sapphire substrate. To date, these two cavity geometries are the only
two microcavity geometries on high index substrate that lase under continuous
wave operation condition at room temperature with moderate quality factor.
2.2 Microdisk
Microdisk cavities are formed by a disk-like dielectric material with refractive
index of n
1
which is surrounded by an environment with lower refractive index
material. Microdisk cavities confine light by total internal reflection at the inter-
face of the disk with the lower refractive index environment. Quality factors of
greater than one million have been achieved in microdisks. Fig. 2.1(a),(b) show
a schematic top view and cross section view representation of the microdisk
cavity geometry, respectively. In this geometry, to confine light by total internal
reflection the refractive indices should satisfy n
1
> n
2
; n
3
.
(b)
n
2
n
1
n
2
n
1
n
3
n
2
n
1
(a)
Figure 2.1: (a) Top view schematic image of a microdisk cavity, (b) A microdisk
cavity at cross section view
14
Because of the simple geometry of the microdisk, approximate analytical
solutions has been introduced to calculate the optical mode properties of these
microcavities. All these approximate solutions are based on separating the out-
of-plane field distribution and the in-plane field distribution under the eective
index method approximation. As a result of this simplification, the quality
factors of microdisks obtained with these methods are overestimated. A new
approach, based on scattering matrix method, developed by Lou et al., takes into
account the out-of-plane radiation modes, and the results are in close agreement
with three dimensional finite dierence time domain calculations [LHG
+
06].
The details of this approach are given in appendix A.
The scattering matrix approach due to Lou et al., was adopted to calculate
the optical mode properties of microdisk cavities on a sapphire substrate. Fig2.2
shows the calculated quality factor of the optical mode with a 1589 nm resonant
wavelength for microdisks with dierent radii. The quality factor of the resonant
optical mode of these microdisks has a R
5:2
dependence. This calculation is
done for microdisks with perfectly smooth sidewalls. In practice, however,
sidewall roughness, originating from fabrication imperfection, dominates the
loss of microdisks. Quality factors on the order of a few million has been
demonstrated experimentally in this geometry.
2.3 Photonic Crystal Defect Cavity
Photonic crystals are a regular array of materials with dierent refractive index.
Analogous to the electrons, the periodic potential results in opening a photonic
bandgap where no electromagnetic wave can propagate. The freedom pho-
tonic crystals oer in engineering the properties of electromagnetic radiation
has brought about design of novel optical devices [Nod06]. By introducing a
defect in an otherwise periodic photonic crystal, localized defect states form in
15
1.0 1.2 1.4 1.6
10
5
10
6
10
7
Q
Radius( m)
Figure 2.2: Calculated quality factors of microdisks with the S-matrix method
corresponding to the modes with resonant wavelength, = 1589:0 nm.
these artificial crystals. This approach has been successfully used to form high
quality factor and small mode volume optical cavities.
The photonic crystal cavities used in this study were defect cavities in a trian-
gular two-dimensional lattice with 37 holes missing air holes(D
4
). Fig.2.3 shows
a schematic representation of the D
4
cavity geometry. Although room temper-
ature CW lasing operation of membrane photonic crystal and microdisk micro-
cavities with very high quality factors have been demonstrated [NKB07], any
implementation of these microcavities in practical system applications requires
the cavity to have good output coupling loss to achieve high slope eciencies.
However, the quantum well active medium fails to compensate for the optical
losses in membrane structures with quality factors less then a few thousands
at room temperature under CW operating condition due to high thermal resis-
tance of the device, therefore heating and the consequent drop in gain prevents
the devices from reaching threshold. In the cavities on a sapphire substrate,
the lower high index substrate cladding allows room temperature CW lasing
operation of these small size cavities through dissipating heat into the substrate.
16
However, the drawback of this approach is that the mode leaks into the substrate,
lowering the overall quality factor of the cavity.

Figure 2.3: Top view schematic image of a defect photonic crystal cavity with 37
missing air holes (D
4
)
Fig.2.4(a) shows the calculated quality factor of optical modes of a suspended
membrane cavity with 19 missing air holes (D
3
) vs. their respective normalized
resonant frequency. Fig.2.4(b) shows the calculated quality factor for a D
3
cavity
on a sapphire substrate. The significant drop in quality factor from the mem-
brane structure to the cavities on sapphire substrate can prevent the QW gain
medium from completely compensating the optical losses in D
3
defect cavities
in order to reach a lasing threshold. Although, the quality factor of D
4
defect
cavities on sapphire substrate decrease from their value in suspended mem-
brane geometry, the QW gain medium can provide enough gain to compensate
loss, and reach threshold. The larger optical mode volume of the D
4
cavity in
comparison with the D
3
cavity, causes the optical mode to be less spread in
the momentum space, therefore it overlaps less with the radiation modes of
the structure, making it less susceptible to lower optical cladding layer refrac-
tive index. Fig.2.5 shows the calculated quality factor of a D
4
defect cavity on
sapphire substrate. The figures of merit for a photonic crystal lattice are its
17
lattice constant and the filling factor which is characterized by the normalized
coecient r=a.
0.28 0.30 0.32 0.34 0.36 0.38
0
2000
4000
6000
8000
10000
12000
14000
(a)
0.26 0.28 0.30 0.32 0.34
0
2000
4000
6000
8000
10000
12000
14000
Q
a/
Q
a/
(b)
Figure 2.4: (a) Calculated quality factor of a suspended membrane D
3
defect
cavity, (b) Calculated quality factor of a D
3
defect cavity on sapphire substrate
(Courtesy of Wan Kuang, USC)
0.24 0.26 0.28 0.30 0.32 0.34
0
1000
2000
3000
4000
5000
Q
a/
Figure 2.5: (a) Calculated quality factor of a suspended membrane D
4
defect
cavity sapphire substrate (Courtesy of Wan Kuang, USC)
The optical mode properties of defect cavity photonic crystal lasers, includ-
ing their quality factor, farfield radiation, mode identification etc., have been
investigated extensively both theoretically and experimentally in professor John
18
O’Brien’s group at USC [Cao04, Shi06]. Part of this thesis is devoted to studying
dynamic properties of photonic crystal defect cavity lasers.
2.4 Nano-Fabrication of Photonic Devices
In this section, the fabrication process used to make these devices is reviewed
briefly. The basic approach to fabricating these devices consisted of bonding an
epitaxial layer structure containing an active region in a waveguide epitaxial side
down to a sapphire substrate to dissipate heat in laser cavities more eciently
than the air. This approach was used previously to demonstrate microdisk
cavity lasers operating under CW operating conditions at room temperature
[TLL
+
67]. The substrate was subsequently removed and the photonic devices
were then patterned. Here the epitaxial layer structure was a 240nm thick
InGaAsP layer containing four InGaAsP strained quantum wells designed to
emit near 1.55m at room temperature. This was deposited by metal-organic
chemical vapor deposition (MOCVD) on an InP substrate in professor Dapkus’s
Compound Semiconductor Laboratory (CSL) at USC. The bonding was done to
a 350m thick sapphire substrate at 500 C in a H
2
chamber. The InP substrate
was removed in a wet chemical etch.
To pattern the semiconductor membrane, a 70 nm thick silicon nitride layer
was deposited and spin coated with 3% polymethylmethacrylate (PMMA) resist.
This resist was patterned with electron beam lithography. The electron beam
dosage was varied in order to vary device size around some average value. The
PMMA resist was then developed with in 1:3 mixture of Methyl Isobutyl Ketone
(MIBK) and Isopropyl alcohol (IPA). The patterns were then transfered into sili-
con nitride layer with a reactive ion etch. The PMMA layer was removed using
remover PG solution at 70

C followed by an O
2
ash step. This patterned nitride
19
(a)

(b) (b)
Figure 2.6: (a) Scanning electron microscope image of a microdisk at 45

view,
(b) Top view SEM image of a fabricated microdisk
layer was then used as a mask to transfer the patterns into the semiconduc-
tor membrane in an electron cyclotron resonance (ECR) etch. A comprehensive
review of the process can be found in Dr. Min-Hsiung Shih’s PhD. thesis [Shi06].
Figure 2.6(a) shows a scanning electron microscope (SEM) image of a fabricated
microdisk at a 45

view, and figure 2.6(b) is a top view SEM image of the final
fabricated device.
Figure 2.7(a) is a tilted view SEM image of the final structure of a D
4
photonic
crystal defect cavity at 45

angle, and figure 2.7(b) shows the top view SEM
image of the same device as figure 2.7(a).
2.5 Optical Characterization of Microcavity Lasers
Optical characterization of microcavities was performed using the optical mea-
surement setup as illustrated by figure 2.8. The pump laser beam was gener-
ated by a 100mW 850nm edge-emitting Fabry-Perot semiconductor laser (Power
Technology, IQ1H85/5557) with a corrected circular output. The pump beam was
20
(a)

(b)
Figure 2.7: (a) Scanning electron microscope image of a D
4
photonic crystal
defect cavity at 45

view, (b) Top view SEM image of the same device as (a)
then expanded and passed through a polarizer that was used to control the inten-
sity of the pump beam. A 50% beam splitter was used to split the pump intensity
into half. One half of the power was incident on a PIN silicon photodetector
(New Focus 1621) and the resulting current was displayed on an oscilloscope to
monitor the pumping optical power. The other half of the pumping laser beam
power was focused onto microcavity devices by a microscope objective lens (
Mitutoyo NIR 100x long working distance lens). The precise alignment between
the pump laser spot and the microcavity laser was achieved by overlapping the
microcavity position with pumping spot position on TV monitor receiving video
signal from the CCD camera, as illustrated in figure 2.8.
The out-of-plane radiation from the photonic crystal microcavities was col-
limated by the same microscope objective lens. Light within its numerical aper-
ture was coupled into an optical fiber after passing through a long wave pass
dichroic beamsplitter (CVI, LWP-45-Rs850- Ts1550-PW-1025-C). The long wave
21
Flipper
Flipper
Oscilloscope
CCD
camera
Monitor
850 nm pump
laser
Optical isolator
Beam
Expander
Polarizer
50%
beam splitter
White light
source
Motorized
XYZ stage
Si Detector
Fiber Coupler
dichromatic mirror
100X
objectiv e lens
Flipper
Optical Specrum
Analy zer (OSA)
Pulse generator
Figure 2.8: Schematic illustration of measurement setup for optical characteri-
zation of microcavity lasers
pass dichroic beamsplitter was designed to transmit 1.5m band signal (Trans-
mission 95%), and to reject most of the 850nm band signal at 45 incident angle
regardless of the polarization of the light. The cut o wavelength is 1025nm. The
optical signal collected into the optical fiber was then analyzed with an optical
spectrum analyzer (HP 70951B).
The fabricated microcavities were mounted on a copper bar that was temper-
ature controlled with a TEC cooler stage. The lasing characteristics of microcavi-
ties with quantum well active medium were measured both at room temperature
and elevated temperatures.
Arrays of photonic crystal cavities with varying lattice constant within the
array was fabricated as was mentioned in section 2.4. D
4
and D
5
photonic
22
crystal defect cavities were tested. Figure 2.9(a) and (b) show the collected out-
put power vs. the incident power curve and the optical lasing spectrum of a
D
4
microcavity laser, respectively. Figure 2.10(a) shows the measured incident
threshold pump power and lasing wavelength of an array of D
4
photonic defect
cavities with dierent lattice constant and constant r=a across the array. As is
shown in figure 2.10(b) the overlap between the lasing wavelength and quantum
well gain results in a large variation of the lasing threshold pump power. From
figure 2.10(b), it can be inferred that the quantum well gain peak for pumping
powers between 1-10mW is around 1590nm for this epi-layer. For characteri-
zation of noise and modulation properties of microcavity lasers, low threshold
laser cavity with emission wavelength in optical communication C band (1525-
1565nm) and L band (1570-1610nm) range is required, so that commercially
available Erbium Doped Fiber Amplifiers (EDFA) could be used to boost the
optical signal.
Microdisk cavities with radii varying between 1:54:5m fabricated on same
wafer as the photonic crystal defect cavities were also tested. Figure 2.11(a) and
(b) show the collected output power vs. the incident power curve and optical
lasing spectrum of a 1:90m radius microdisk cavity laser.
The temperature characteristics of photonic crystal defect lasers and
microdisk lasers was also investigated. Figure 2.12 shows the measured out-
put power of a D
4
photonic crystal defect cavity at dierent temperatures. The
temperature dependence of semiconductor lasers is usually modeled as [CC95a]
I
th
= I
0
e
T=T
0
(2.1)
where T
0
is the overall temperature characteristic of the laser. A T
0
value of
15K was found for both photonic crystal defect and microdisk cavities. This
value of T
0
is smaller than the reported values of T
0
for this material system in
23
0 1 2 3 4 5 6 7 8 9
0
20
40
60
80
1.560 1.565 1.570 1.575 1.580 1.585
1E-10
1E-9
1E-8
Collected Output Power (W)
Wavelength ( m)
(b)
Collected Output Power (a.u.)
Incident Power (mW )
(a)
Figure 2.9: (a) collected output power vs, the incident optical power curve for
a D
4
photonic crystal defect cavity, (b) Lasing spectrum of the photonic crystal
laser in (a)
1300 1400 1500 1600
0.00E+000
5.00E-012
1.00E-011
0
2
4
6
8
10
12

P
th
(mW)
Wavelength (nm)
Output Power (W)

340 360 380 400 420
0
2
4
6
8
10
12
1400
1450
1500
1550
1600
1650
Lasing Wavelength (nm)
Lattice Constant (nm)
P
th
(mW)

(b) (a)
Figure 2.10: (a) Incident threshold pump power (circles) and lasing wavelength
(squares) vs. the photonic crystal lattice constant of the laser array , (b) Thresh-
old pump power vs. wavelength (circles) and its overlap with quantum well
photoluminescence spectrum
the literature with much larger optical cavities [Mat96]. This is because of the
increased threshold carrier density of microcavity lasers due to an overall lower
quality factor and the carriers lost to the surface recombination [AGP
+
00].
24
0 1 2 3 4 5
0
20
40
60
80
100
120
140
160
180
Collected Output Power (a.u.)
Incident Power (mW )
(b)
1.54 1.56 1.58 1.60 1.62
1E-11
1E-10
1E-9
1E-8
1E-7
Collected Optical Power (W)
Wavelength ( m)
(a)
Figure 2.11: (a) collected output power vs, the incident optical power curve for
a 2m radius microdisk cavity, (b) Lasing spectrum of the microdisk laser in (a)
0 1 2 3 4 5
0
200
400
600
800
1000
40
0
C
35
0
C
30
0
C
25
0
C
19
0
C
Output Power (a.u.)
Incident Power (mW )
Figure 2.12: (a) collected output power vs, the incident optical power curve D
4
photonic crystal defect laser at dierent operating temperatures
25
Chapter 3
Spontaneous Emission in
Semiconductor Microcavity Lasers
Introduction
The semiconductor laser diode, since its invention in early 1960’s, has come to
dominate the laser field, becoming a multi-billion dollar industry. Semiconduc-
tor laser diodes have wide applications ranging from optical storage devices like
CD/DVDs, telecommunications, TV displays, pumps for other solid state lasers
and a myriad of other applications.Laser diodes owe their success to their small
size, high energy conversion eciency, high reliability, and ease of use. They
operate by simply passing a current through them at current and voltage levels
which are compatible with those of integrated circuits. The light emission from
the laser diodes can be modulated by switching pump current with switching
times of less than 100 ps. Laser diodes can also be mass-produced with the same
processing techniques used for electronic integrated circuits, and can be inte-
grated with such circuits [Yar84]. The possibility of integrating optical elements
with high performance electronic circuits in VLSI design has drawn significant
attention during the past few years to improve the over all performance of the
circuits by eliminating the distance-bandwidth limitation and is a direction that
considered to boost the semiconductor laser industry. Present-day long haul
telecommunication device technology (based on InP materials), however, is not
well suited to the requirements of optical data communication between and
26
within computers because the computer environment is much more demand-
ing. It imposes a higher ambient temperature on the devices, and requires denser
packaging and smaller power dissipation per device, as well as a high degree of
parallelism [HZK
+
90].
The advancement in semiconductor nanofabrication techniques and expitax-
ial growth of semiconductor materials in the past two decades helped researchers
develop optical cavities with dimensions that are on the order the wavelength
of the light. Microcavity resonators provide relatively low-cost, ecient and
high-density optoelectronic devices over a wide range of the spectrum from the
near infra red to well into the visible. Therefore, they fit in any optical integrated
platform. In these devices, as the physical dimensions become comparable to the
optical wavelength new mechanisms dominate the operation of these devices,
ie., the presence of the microcavity significantly modifies the dynamical interac-
tion of matter with vacuum field fluctuations, leading to spontaneous emission
[Pur46]. In addition to being attractive for studying the fundamental laws of
physics, optical microcavities hold technological promise for constructing novel
kinds of light-emitting devices. This chapter deals with the analysis and appli-
cations of microcavities as lasers.
3.1 Interaction of Cavity Optical Mode with the
Active Material
In world war II, Purcell [Pur46] noticed the enhancement of the rate of sponta-
neous emission from atoms in cavities with sizes on the order of emitted elec-
tromagnetic field wavelength at radio frequencies, and formulated the enhance-
ment rate, today known as Purcell factor. It took about 35 years before Klepner
[Kle81] showed the possibility of inhibition of spontaneous emission of atoms
27
at radio frequencies. In late 1980’s however, advances in the semiconductor
processing technology and fabrication of optical wavelength-size cavities made
control of spontaneous emission in the optical domain possible. There is strong
motivation for spontaneous emission rate control in modern semiconductor
devices, where it can play a detrimental role in the performance of bright LEDs,
semiconductor lasers and fast electronic transistors [Yab87].
Spontaneous emission in its primitive description is defined as the irre-
versible decay of population from its excited state to the ground state. The
irreversible decay process is a consequence of interaction of the excited popula-
tion with a continuum of optical modes. It can be shown that in the presence a
single mode optical field, the excited population decays into the ground state,
but flops back into the excited state sinusoidally in time. This phenomena,
known as Rabi flopping, will not be pursued in this thesis, because the case of
a semiconductor microcavity laser with quantum well active medium at room
temperature is not in the strong coupling condition regime [YSH
+
04, HBW
+
07].
The interaction of the excitation in a quantum well material with the cavity opti-
cal mode at room temperature is in the weak coupling regime. In this regime,
first-order perturbation theory which leads to Fermi’s golden rule can predict
appropriate behavior of the system to some extent [GM95]. Using this approach
it is shown that spontaneous emission is not an intrinsic property of an isolated
atom, but is a property of the emitter coupled to the electromagnetic radiation
field environment [Pur46].
3.2 Spontaneous Emission in Free Space
In this section, to better understand the dynamics of the interaction of an excited
population with optical fields, the interaction of a two level system with a single
mode optical field is first considered, and then by including a continuum of
28
optical fields the decay constant of the excited population is derived. A first
order perturbation analysis is applied to arrive at Fermi’s golden rule. Then the
Weisskopf-Wigner theory of spontaneous emission is presented and it is shown
that the Fermi’s golden rule provides a good approximation to the spontaneous
decay of carriers.
The total system Hamiltonian that describes the coupled two-level system,
field, and their interaction is described by the Jaynes-Cummings Hamiltonian:
H = H
E
+ H
F
+ H
I
(3.1)
where H
E
, H
F
, and H
I
are the two-level system, elecromagnetic field, and the
coupled system interaction Hamiltonians, respectively. In this formalism, the
optical field Hamiltonian is described in quantized form. The details of second
quantization can be found in [CTDRG89]. The two-level system and the optical
field Hamiltonians are expressed as
H
E
=~!
e
je>< ej +~!
g
jg>< gj = E
0
+~!
0

z
(3.2)
H
F
=~!
c
(a
y
a +
1
2
) (3.3)
where~!
e
,~!
g
are the energies of the excited and ground state level of the two
level system, respectively, and !
0
= !
e
!
g
.
z
= (je >< ejjg >< gj)=2 is
a Pauli spin operator, and has been introduced for convenience. a, and a
y
are
the annihilation and creation operators for the optical field and they obey the
boson commutation relation [a; a
y
] = 1. The two level system-field interaction
Hamiltonian H
I
in the dipole approximation is represented as:
H
I
=er:E (3.4)
29
where er =}

jg>< ej + h:c:

is the atomic dipole moment operator and
E(~ r) =E

~ f (~ r)a + h:c:

(3.5)
is the electric field. Here~ is the polarization of the optical mode, and f (~ r) is its
spatial distribution, which is the solution to the classical Maxwell’s equations.
In second quantized formalism, the interaction Hamiltonian in equation 3.4
becomes
H
I
=~

a + a
y

g(~ r
a
)

+ g

(~ r
a
)
+

(3.6)
where
+
=jg >< ej = (
+
)
y
are the Pauli spin flip operators. The coupling
parameter g(~ r) is the so called vacuum Rabi frequency
~g(~ r
a
) =E f (~ r
a
)(}:~ ) (3.7)
where~ r
a
is the spatial position of the two level system. The rotating wave approx-
imation then casts the interaction Hamiltonian, equation 3.6, in the following
form
H
I
=~

g(~ r
a
)a
y

+ h:c:

(3.8)
If originally, the unperturbed system is in thejei
j0i state, wherej0i is the vacuum
state of the optical field, the interaction Hamiltonian, equation 3.8, couples this
state only to the statejgi
j1i. This property states that the total number of
excitations is conserved in the closed atom-field system. This also means that
30
the Jaynes-Cummings problem reduces to a two-level system problem that can
be solved exactly.
j (0)i =jei
j0i =je; 0i (3.9)
The atom field system remains in the one excitation manifold for all time. There-
fore the time evolution of the system is expressed by a superposition of single
excitation states. On resonance,!
0
=!
c
, it can be shown that the time dependent
state of the system is given [GM95]
j (t)i = cos(gt)je; 0i i sin(gt)jg; 1i (3.10)
resulting in an oscillatory behavior for the probability for the atom to be in the
ground state
P
g
(t) =jhgj (t)ij
2
= sin
2
(gt) (3.11)
This simple oscillatory behavior of the probability for the atom to be in the
ground state is a consequence of presence of only a single optical mode. In this
picture, once a photon is emitted, it can be subsequently reabsorbed by the atom,
moving the atom into its excited state. In the presence of multi-mode optical
fields, however, once the excitation in the atomic system diuses into the large
phase space of the optical fields, the excitation does not return to the atomic
system coherently, making spontaneous emission an irreversible process. In the
case of multi-mode, the total atom-field interaction Hamiltonian takes the form
H = H
0
+
X
k
~!
k
a
y
k
a
k
+
X
k
~

g
k
(~ r
a
)a
y
k

+ h:c:

(3.12)
31
Again considering only a single excitation, the state of the atom-field evolution
is a superposition of the single excitation states of the total atom-field system
j (t)i = a(t)e
i!
0
t
je; 0i +
X
k
b
k
(t)e
i!
k
t
jg; 1
k
i (3.13)
with a(0) = 1; b
k
= 0. The equations of motion for a(t) and b
k
(t) can be derived
from Shr¨ oringer equation
i~
d
dt
j (t)i = Hj (t)i (3.14)
which results in
da(t)
dt
=i
X
k
g
k
e
i(!
k
!
0
)t
b
k
(t) (3.15)
db
k
(t)
dt
=ig

k
e
i(!
k
!
0
)t
a(t) (3.16)
If the time variations of the atomic system population probability amplitude,
a(t), is much slower than the variations of the optical field modes occupation
probability amplitudes,in the first order in perturbation theory, the a(t) can be
replaced with its value at time t = 0 in equation 3.16, a(t) 1 for short enough
times,
db
k
(t)
dt
=ig

k
e
i(!
k
!
0
)t
a(0) (3.17)
b
k
(t) = ig

k
sin [(!
k
!
0
)t=2]
(!
k
!
0
)=2
e
(!
k
!
0
)t=2
(3.18)
The probability for the atom to be in the excited state at time t is then
P
e
(t) = 1
X
k
jb
k
(t)j
2
= 1
X
k
jg
k
j
2
sin
2
[(!
k
!
0
)t=2]
(!
k
!
0
)
2
=4
(3.19)
32
Now the sum over optical modes, k, is transformed into an integral over a
continuum of modes, using the density of the optical states in free space[MI99],
X
k
f (k) =
V
(2)
3
Z
d
3
k f (k) =
V
(2c)
3
Z
d!
k
!
2
k
Z
1
1
d(cos())
Z
2
0
d f (!
k
) (3.20)
then equation 3.19 becomes
P
e
(t) = 1
V
(2c)
3
Z
d
3
kjg
k
j
2
sin
2
[(!
k
!
0
)t=2]
(!
k
!
0
)
2
=4
= 1
V
(2c)
3
Z
d!
k
!
2
k
Z
1
1
d(cos())
Z
0
2djg(!
k
;)j
2
sin
2
[(!
k
!
0
)t=2]
(!
k
!
0
)
2
=4
(3.21)
Since the Rabi frequency (g), depends on the dot product, equation 3.7, it is
expressed as a explicit function of. It can be shown that g(!;) for an atomic
dipole interacting with a free running optical field with two possible polariza-
tions can be expressed as following [MI99]
jg(!;)j
2
=jE (!)} sin()=~j
2
(3.22)
for optical fields in free space, the electric field amplitude per photon is
jE (!)j =
r
~!
2
0
V
(3.23)
Replacing equations 3.22 and 3.23 into equation 3.21 becomes
P
e
(t) = 1
1
2
0
~(2c)
3
Z
d!
k
!
3
k
Z
1
1
d(cos()) sin
2
()
Z
0
2dj}j
2
sin
2
[(!
k
!
0
)t=2]
(!
k
!
0
)
2
=4
= 1
1
6
0

2
~c
3
Z
d!
k
!
3
k
j}j
2
sin
2
[(!
k
!
0
)t=2]
(!
k
!
0
)
2
=4
(3.24)
33
The sin
2
[(!
k
!
0
)t=2]=(!
k
!
0
)
2
=4 term is then approximated by a delta
function, for large times. In this limit,
lim
t!1
sin
2
[(!
k
!
0
)t=2]
(!
k
!
0
)
2
=4
= 2t!
k
!
0
(3.25)
therefore, the excited atomic population coupled to an optical field in free space
decays to the ground state at a rate

f
=
dP
e
dt
=
!
3
0
j}j
2
3
0
~c
3
(3.26)
This result which is known as Fermi’s golden rule, predicts that the excited
atomic population decays into the ground state at the rate
f
given in equa-
tion 3.26. Although this simple derivation explains features of spontaneous
emission, it is valid only for such short times that the excited state has not yet
depleted significantly. Weisskopf-Wigner theory of spontaneous emission pro-
vides a more accurate description of spontaneous emission. In this method,
equation 3.16 is solved by integrating both sides, then the result is substituted
in equation 3.15 to get
da(t)
dt
=
X
k
jg
k
j
2
Z
t
0
dt
0
e
i(!
k
!
0
)(tt
0
)
a(t
0
) (3.27)
changing the sum over optical modes into an integral with the same recipe as
stated before, the following can be derived
da(t)
dt
=
1
6
2
~c
3
Z
d!
k
!
3
k
j}j
2
Z
t
0
dt
0
e
i(!
k
!
0
)(tt
0
)
a(t
0
) (3.28)
34
If the time variations of a(t
0
) are slow compared to the exponential in the integral,
a(t
0
) can be evaluated at time t
0
= t and can be factored out of the integral. The
remaining exponential term in the integral can be cast into the form,
lim
t!1
Z
t
0
dt
0
e
i(!
k
!
0
)(tt
0
)
=(!
k
!
0
)P

i
!
k
!
0

(3.29)
The validity of evaluating the integral for t!1 is due to the assumption that
the exponential in the integral oscillates faster than the time scales which the
excited state of the atomic system becomes depleted. The principal value term
corresponds to a frequency shift, which is a reminiscent of the Lamb shift, and
will be ignored here. The in equation 3.29 function term gives
da(t)
dt
=

f
2
a(t) (3.30)
where
f
is given by equation 3.26. This result is consistent with the results of
Fermi’s golden rule. The strong dependence of the decay rate on the emission
frequency comes from the density of the optical modes at the emission frequency.
3.3 Purcell Factor
Let’s consider an atom that is at rest in an optical cavity whose free spectral
range is large so that only one of modes, with frequency!
c
, is nearly resonant
with the transition. As in section 3.2, the subsystem consisting of the atom, the
optical mode and their coupling is described by the Hamiltonian,equation 3.1,
H
S
= H
E
+ H
F
+ H
I
The evolution of this subsystem is described by the Hamiltonian given above,
subject to two other dissipative processes. The first one is the coupling of the
35
associated atomic system to the electromagnetic radiation background, which
results in irreversible decay of the atomic excited population. The second dis-
sipation is due to cavity optical loss that results in escape of photons from the
cavity [GM95]. The quantum dynamics of open quantum system cannot, in
general, be described in terms of unitary time evolution, and in many cases their
dynamics is more conveniently formulated by an appropriate equation of motion
for its density matrix, a quantum master equation [BP02]. A general derivation
of a system coupled to reservoir master equation is given in appendix B.
In this section we consider a small subsystem expressed by the Hamiltonian
given in equation 3.31. Using a system of reference that is rotating at the cavity
mode frequency, this Hamiltonian can be rewritten in the following form [GM95]
H
S
=~
z
+ H
I
(3.31)
where
H
I
=~

g(~ r
a
)a
y

+ g

(~ r
a
)a
+

; (3.32)
and = !
c
!
0
is the atom-cavity detuning. Following the master equation
derivation given in appendix B it can be shown that the atom-cavity mode
master equation for the density matrix operator is given by
d
S
dt
=
i
~

H
S

S

S
H
S

0
2
(
+

S
+
S

+

2

S

+
)

0
2
(a
y
a
S
+
S
a
y
a 2a
S
a
y
); (3.33)
where is the photon escape rate in the optical cavity, and
0
is the spontaneous
decay of the atom excited state due to its coupling to background radiation fields
or any other scattering or dephasing of the excited population.
36
We assume that at time t = 0, the optical field is in its vacuum state and there
is a single excitation in atomic system. Therefore, the system can be described as
the superposition of the product statesj2
m
ijf0gi,j1
m
ij1
j
i, andj1
m
ijf0gi, wherej2
m
i
andj1
m
i are the atomic upper and lower states, respectively.jf0gi is the optical
field vacuum state andj1
p
i denotes state of the optical field in which there is
one photon in the optical mode.j1
m
ijf0gi corresponds to a case where there is no
excitation in the system of interest. For convenience, these states are written in
the form
j2
m
ijf0gi =j0i; (3.34)
j1
m
ij1
j
i =jmji; (3.35)
j1
m
ijf0gi =j0
0
mi: (3.36)
The elements of the density matrix operator are derived by sandwiching both
sides of equation B.7 between the states of interest. We are primarily interested
in the dynamics of
00
. Following the prescription described in appendix B, it
can be shown the spontaneous decay rate of
00
due to its interaction with the
optical mode and its environment is written in the form,
d
00
dt
=
"
jgj
2

0
+

2
+ (
0
+)
2
=4

0
#

00
(3.37)
In this derivation, the terms that do not conserve the excitation number in the
system have been dropped. The first term on the right hand side of equation 3.37
37
corresponds to spontaneous decay of the excited population that leads to emis-
sion of a photon, therefore the spontaneous emission rate of the system can be
written in the form

0
=

2jgj
2
+
0
!
1
1 + 2(2=(
0
+))
2
(3.38)
Expressingjgj in terms of the free space spontaneous emission rate
f
of
equation 3.26 for a case where standing wave quantization has been used, it can
be shown that at zero detuning from the atomic transition frequency and cavity
resonance frequency = 0 [GM95], this rate can be expressed as

max
=
3
4
2

(
0
=n)
3
V
p
!
!
c
+
0
: (3.39)
If the dephasing and scattering mechanisms are considered to be negligible

0
0, and using the fact that the quality factor of an optical mode is expressed
by
Q =!
c
= (3.40)
then equation 3.39 reduces to the well known Purcell factor

max
=
3Q
4
2

(
0
=n)
3
V
p
!
; (3.41)
This expression equation 3.41, predicts that for a transition wavelength com-
parable to the dimensions of optical cavity and for suciently high quality
factor optical modes, there is considerable amount of enhancement in the spon-
taneous emission rate compared to its free space value. However, as indicated
by equation 3.39 in calculating the eective quality factor, mechanism leading
to broadening the atomic transitions should be taken into account.
38
3.4 Characteristics of Semiconductor Microcavity
Lasers
In conventional semiconductor lasers, due to the isotropic radiation pattern
of spontaneous emission, the broader spontaneous emission spectral linewidth
compared to the resonant mode of the optical cavity, and the large active volume
size, a substantial fraction of the spontaneously emitted photons are coupled to
radiation modes and nonlasing resonant modes of the optical cavity. Therefore, a
small fraction of the total spontaneous emission of the active medium is coupled
into the single lasing mode ( one photon in every 10
5
photons). This coupling
eciency (the ratio of the number of spontaneously emitted photons coupled
to the lasing mode to the total number of spontaneously emitted photons) can
increase dramatically in microcavity lasers and give rise to significant changes
in the operation characteristics of these lasers compared to conventional lasers
with large active medium. Spontaneous emission coupling factors as high as 0:1
and:4 have been reported in the literature for microdisk and photonic crystal
microcavity lasers [FUB01, NKB07].
The intrinsic origin of a clear laser threshold is the loss caused by recom-
bination processes other than spontaneous emission into the lasing mode; the
spontaneous radiative recombination of carriers into nonlasing modes as well
as nonradiative recombination processes present themselves as a net loss of car-
riers for the lasing process. In microcavity lasers, an increased spontaneous
emission coupling coecient and the increased spontaneous emission rate help
make the radiative recombination into the single lasing mode dominate both
radiative and nonradiative loss mechanisms. One advantage of the increased
spontaneous emission coupling coecient in the microcavity lasers, is that little
39
power is carried away by nonlasing optical modes[BKY94]. This results in a lin-
ear increase of the emitted optical power with increasing pump rate. The ideal
structure where all the spontaneous carrier recombination result in photons in
the lasing mode is called a thresholdless laser [KJC95].
40
Chapter 4
Semiconductor Laser Diode Noise
Introduction
Noise properties of lasers have been the subject of great interest from the early
days after the laser invention. Schawlow and Townes [ST58], in their first paper
proposing the laser, derived a formula for laser linewidth which is a measure of
average rate at which phase fluctuations happen in a laser oscillator. The laser as
a source of coherent radiation is a relatively noisy medium. The main source of
noise is the inevitable spontaneous decay of the active medium polarization into
the lasing mode. The output of the laser is aected by this noise in two ways.
First, the spontaneous emission causes the amplitude of the laser to fluctuate
around an average value. However, this fluctuation becomes negligible when
the lasers are operating far above threshold. The second eect of the quantum
noise is to cause the phase of the laser output beam to wander randomly. This
phase fluctuation is the main source of the finite spectral width of lasers.
The classical and quantum descriptions of laser noise have both been very
successful in describing the noise properties of lasers. It took some time after
the first formulations of laser noise properties by Lax [HL67] and others, before
Gerhardt et al. [GWG72] measured the extremely narrow linewidth of He-Ne
lasers and showed close agreement between the earlier theories and experiment.
The availability of single mode semiconductor lasers operating continuous
wave at room temperature drew a lot of attention to the noise properties due to
its technological importance. The first systematic measurement of linewidth
41
of semiconductor lasers by Fleming and Moradian [FM81] revealed a laser
linewidth that was 50 times larger from the modified Schawlow-Townes formula
developed earlier by Lax. Henry [Hen82] and Vahala and Yariv [VY83a, VY83b]
explained the discrepancy by relating the change in resonant cavity frequency to
the material gain. This resulted in introducing 1 +
2
in the modified Schawlow-
Townes formula, where is ratio of the changes in the real and the imaginary
parts of the refractive index with the changes in the number of carriers. This
coecient was present in earlier works by Lax, but the correction factor is unity
for resonant cavities tuned to the center of a transition line of an active medium,
and therefore the correction was neglected for gas lasers. Henry noted that the
situation in a semiconductor is that of a detuned oscillator.
In this chapter, first the Langevian noise equations for a semiconductor is pre-
sented. Next, noise properties of semiconductor laser based on these equations
are derived. The linewidth of semiconductor lasers is also presented
4.1 Langevian Noise Forces
Open quantum and classical systems experience fluctuation and dissipation (
spontaneous emission in the case of a laser) through interacting with a reservoir.
In most noise treatments, however the reservoir is eliminated and the frequency
shifts and the dissipation induced by the reservoir are incorporated into the
mean equations of motion provided that suitable noise sources, with the correct
moments are added [Lax66].
Thus, if a =fa
1
; a
2
;:::g is some set of system operators, and
d
dt
D
a

E
=
D
A

(a)
E
(4.1)
42
are the mean equations of motion including frequency shifts and the damping,
then
da

dt
= A

(a) + F

(a; t) (4.2)
describes the behavior of the system, provided that the operators F

have the
correct statistical properties. The A’s are chosen such thathFi = 0, wherehi
stands for statistical average. The role of the Langevian force is to account
for how the statistical distribution of the variables a(t) change in time. The
Langevian noise forces are assumed to be a Markovian process,
hF

(t)F

0(t
0
)i = 2D

0(t t
0
) (4.3)
where D

0 is called the diusion coecient.
In the next section, a set of Langevian equations is derived for the amplitude
and phase of a semiconductor laser, and laser noise properties are investigated
based on these equations.
4.2 Noise Equations
With proper noise sources included, equation 4.2 is a valid description of the
evolution of the system. Theoretical treatments of laser noise can be broadly
grouped in two categories based on how fast the variables in equation 4.2 are
evolving in time. In the first category, if variables a
s+1
, ..., a
n
vary more rapidly
than a
1
, a
2
, ..., a
s
, then a set of equation can be derived for the usually slower
variables by eliminating the more rapidly variables. The price to be paid dealing
with fewer equations is that now the equations for the slower varying variables
includes integrals over the history and nonwhite noise. However, if solutions
43
at frequencies smaller than the decay constant of the rapid variables are sought,
it is quite legitimate to treat the slow variables as Markovian and the corre-
sponding noise sources as white. This approach has been used extensively in
semiconductor laser noise theory [VY83a] by eliminating the dynamics of car-
riers adiabatically. However, this method is valid only for systems with two
sets of variables with significantly dierent time scales and for describing the
low frequency behavior of the system noise. This adiabatic elimination must be
avoided when extending noise calculations to high frequencies and analyzing
systems with variables with time scales of the same order.
McCumber [McC66] noted that the prediction of relaxation oscillations in
the intensity noise spectrum of lasers is contingent upon keeping the carrier
dynamics in the rate equations. Vahala and Yariv [VY83b], and Henry [Hen86b]
also were able to explain the satellite peaks far out in the wings of the lineshape
spectrum of semiconductor laser observed earlier by Daino et al. [DSTP83]. In
this section a semiclassical derivation of noise in semiconductor laser is given
following Vahala and Yariv [VY83b].
The starting point of this analysis is adding a randomly fluctuating polariza-
tion~ p(~ r; t), due to spontaneous emission, to the Maxwell’s equations:
r
~
E(~ r; t) =@
t
~
H(~ r; t) (4.4)
r
~
H(~ r; t) =( +@
t
)
~
E(~ r; t) +@
t
h
~
P(~ r; t) +~ p(~ r; t)
i
(4.5)
is the magnetic permeability, is the medium conductivity, and is the dielec-
tric constant of the medium. Solving for
!
E (
!
r; t) yields
h
r
2
@
t
@
2
t
i
~
E(~ r; t) =@
2
t
h
~
P(~ r; t) +~ p(~ r; t)
i
(4.6)
44
wherer(r
~
E(~ r; t)) 0 is assumed. Now it is assumed that~ e
n
(~ r) are the solutions to
the equation 4.6 when the driving forces (
~
P(~ r; t);~ p(~ r; t)) and the losses () are set to
zero, and they form a complete orthonormal basis. Equation 4.6 is multiplied by
e
n
(~ r

) and integrated over the whole volume, to arrive at the following equation
¨
E
n
+
1

p
˙
E
n
+!
2
n
E
n
=
1

2
(
¨
P
n
+ ¨ p
n
) (4.7)
where E
n
, P
n
, and p
n
are the expansion coecients of
~
E(~ r; t),
~
P(~ r; t) and~ p(~ r; t) in
terms of~ e
n
(~ r; t).
~
E(~ r; t) =
X
n
E
n
(t)~ e
n
(~ r; t) (4.8)
~
P(~ r; t) =
X
n
P
n
(t)~ e
n
(~ r; t) (4.9)
~ p(~ r; t) =
X
n
p
n
(t)~ e
n
(~ r; t) (4.10)
where
p
= is the photon lifetime, is the refractive index of material.
Gain and refractive index are function of carrier density, so carrier density
should be considered as a dynamical variable and an equation for its evolution in
time is required. The standard rate equation for carrier density in semiconductor
material with a Langevian noise force term added to the equation describe the
behavior of the carrier density
dn
dt
=g(n)p
n

s
+ E +# (4.11)
where n is the carrier density, g(n) is optical gain, p is photon density,
s
is the
carrier lifetime, E is the pumping rate, and# is a Langevian noise force associated
with the carriers.
45
Under single mode lasing conditions, the projection of the active medium
polarization onto the n
th
spatial mode,~ e
n
(~ r), P
n
can be expressed as
P
n
=
0
(n)E
n
(4.12)
where is the spatial overlap between the field and polarization. The optical
gain g(n) is related to the imaginary part of (
i
)
g(n) =
!
m

i
(n)

2
(4.13)
where!
m
is the lasing frequency. Using equation 4.13 and the following expres-
sion for photon density p
p =

0

2
2~!
m
jE
n
j
2
(4.14)
the equation 4.11 can be rewritten into the follawing form
dn
dt
=

0
2~

i
(n)jE
n
j
2

n

s
+ E +#: (4.15)
The Langevian force term driving the electric field, ¨ p
n
in equation 4.7 can be
written in the following form for convenience
e
i!
m
t

1

0

2
¨ p
n
(4.16)
where is defined as a slowly varying complex amplitude of the second deriva-
tive in time of the polarization fluctuation p
n
(t). Using equations 4.12 and 4.16,
equation 4.7 is rewritten in the form
d
2
dt
2
"
1 +
(n)

2
!
E
n
#
+
1

p
d
dt
E
n
+!
2
n
E
n
= e
i!
m
t
(4.17)
46
Equations 4.17 and 4.11 are then linearized by expanding n, E
n
around their
average value
E
n
= [A
0
+(t)] e
i[!
m
t+(t)]
(4.18)
n! n
0
+ n (4.19)

r
(n) =
r
(n
0
) +
r
n (4.20)

i
(n) =
i
(n
0
) +
i
n (4.21)
jE
n
j
2
A
2
0
+ 2A
0
(t) (4.22)
where
r
and
i
are the first order Taylor series expansion of
r
and
i
respec-
tively about the biasing point n
0
. Keeping only first order terms in the expansion
coecients, equations 4.17 and 4.11 are linearized in the following form respec-
tively,
2i!
m
(
˙
+ iA
0
˙
) +
2i!
m
A
0

2
˙ n
!
2
m
A
0

2
n
+

!
2
n
!
2
m
+ i
!
m

p

!
2
m

2
(n
0
)
!
A
0
= e
i
(4.23)
˙ n +

0

i
A
2
0
2~
n +

0

i
(n
0
)A
0
~
+
n

s
+

0

i
(n
0
)A
2
0
2~
+
n

s
E =# (4.24)
The second order derivatives in time,
¨
, ¨ n,
¨
have been neglected, becase their
variation in time is slow compared to the lasing optical frequency. The field
amplitude,, phase,, and carrier, n fluctuations as well as the Langevian noise
47
forces have zero mean value, therefore averaging equations 4.23 and 4.24 results
in the operating point values of carrier and field amplitude

i
(n
0
) =

2
!
m

p
(4.25)
!
2
m
=
!
2
n
1 +

r
(n
0
)

2
(4.26)
p
0
=
p

E
n
0

s

(4.27)
Using the operating point values obtained in equations (4.25)–(4.27), equa-
tions 4.23 and 4.24 can be written as follows
˙ +

r

2
˙ n
!
m

i
2
2
n =

i
2!
m
A
0
(4.28)
˙
+

i

2
˙ n +
!
m

r
2
2
n =

r
2!
m
A
0
(4.29)
˙ n +
1

R
n +
2
2
!
2
R
!
m

i
=# (4.30)
where the following definitions have been made for convenience:
!
2
R

0
A
2
0
!
m

i

i
(n
0
)
2~
2
= g(n
0
)g
0
(n
0
)p
0
(4.31)
1

R

1

s
+

0
A
2
0

i
2~
=
1

s
+ g
0
(n
0
)p
0
(4.32)

A
0
(4.33)
where equations 4.31 and 4.32 for!
R
and
R
are the familiar equations for the
relaxation oscillation frequency and the damping associated with it.
Starting from here, one can calculate the correlation functions of the above
equations, and arrive at spectral density functions for the field amplitude and
phase fluctuations. This approach that is based on small signal analysis has
48
been used widely for calculating the noise properties of semiconductor lasers,
citeYariv1,Yariv2. In next section, the linewidth properties of semiconductor
lasers are derived based on the Langevian noise forces but without going into a
small signal analysis.
To find the correlation functions of field fluctuations, equations (4.28)–(4.30)
are solved by taking the driving noise forces,
r
,
i
, and# to be deterministic
functions. This allows application of a Laplace transform to solve the system of
dierential equations
(t) =
1
2!
m
A
0
Z
t
0
Delta
i
()exp

1
2
R
(t)

cos(t)d
+
1

Z
t
0

i
()
4!
m
A
0

R
+
!
m

i
#()
2
2
!
exp

1
2
R
(t)

sin(t)d (4.34)
n(t) =
Z
t
0
#()exp

1

R
(t)

cos(t)d

1

Z
t
0

!
2
R

2

i
()
!
2
m
A
0

i
+
#()
2
R
!
exp

1

R
(t)

sin(t)d (4.35)
(t) =
Z
t
0

r
2!
m
A
0
d
!
m

r
2
2
Z
t
0
n()d (4.36)

!
2
R

1
4
2
R
!
(4.37)
In equations 4.28 and 4.29 the second terms containing the derivative of carrier
density with respect to time ( ˙ n) are neglected because!
m
n(t) ˙ n and
r
=
i

has a value between 1 and 10. Equations 4.34 to 4.36 are then used to calculate the
correlation functions of the carrier, field amplitude, and field phase fluctuations
49
hn(t
1
)n(t
2
)i,

(t
1
)(t
2
)

, and
D
(t
1
)(t
2
)
E
[VY83b]. The following Langevian force
correlation forms help to simplify the averages
h
i
(t +)
i
(t)i =h
r
(t +)
r
(t)i = W(t) (4.38)
h
i
(t +)#(t)i = W
1
(t) (4.39)
h
r
(t +)#(t)i = 0 (4.40)
h#(t +)#(t)i = W
2
(t) (4.41)
4.3 Spontaneous Emission and Laser Linewidth
Spectral purity of lasers has been the subject of interest from the early days of its
invention. Schawlow and Townes [ST58] calculated the fundamental quantum
limit of a laser oscillation due to spontaneous emission. The spontaneously
emitted photons into the lasing mode add randomly in phase to the coherent
laser oscillation which results in a finite linewidth given by

osc
= (4h=P)()
2
(4.42)
where is the half-width at half-maximum intensity of the optical mode, P is the
resonant mode optical power and is the resonant mode frequency. However, as
pointed out by Lax, this equation holds below threshold only. Above threshold,
the linewidth is given by half this value. In this section, the relations for laser
linewidth and its dependence of spontaneous emission rate of photons in the
active medium will be established.
The optical field of the laser can be described as follows
E(x; t) = B
h
(t)(x) +(t)

(x)

i
(4.43)
50
where (t) and (x) give the temporal and spatial dependence of the optical
mode, respectively. B is a normalization constant, and it is chosen such that
the average optical intensityhIi equals the number of photons in the mode
[Hen86b]. The spatial dependence of the field is the solution to equation 4.6
when the losses () and the driving forces (P and p) are set to zero. Thus, as was
mentioned previously (see section 4.2), the optical mode behavior is described
by three real variables =
0
+i
00
and number of the injected carriers that govern
the dynamics of active medium gain, N.
The Langevian rate equation for(t) is expressed as [Hen86a]
˙
=

i!
0
+ i
G
2
(1 i)

+ F

(t) (4.44)
where!
0
is the cavity resonance frequency at threshold and G is the net gain
(s
1
). , the linewidth enhancement factor. The diusion coecient relating F

and F

are expressed as
2D

= 2D

= 0 (4.45)
2D

= R (4.46)
where R is the spontaneous emission rate. Figure 4.1 shows that during a short
time T, the Langevian force (F

(t)) changes the complex field amplitude by
R
T
0
F

(t)dt. Applying the law of cosines to the triangle in figure 4.1
I + I =j(t)j
2
+hj
Z
T
0
F

(t)dtj
2
i 2Re
"
h(t)
Z
T
0
F
(t)
dt cos(
)i
#
= I +
Z
T
0
Z
T
0
hF

(t)F

(t
0
)idtdt
0
2Re
"
(t)
Z
T
0
hF
(t)
i cos(
)dt
#
I = RT (4.47)
51

’’
( ) t T
( ) t
1/ 2
| ( ) | t I
1/ 2
| ( ) | ( ) t T I I
dt t F
T
) (
0

’
Figure 4.1: The change of complex field amplitude(t) during a short time T.
where
is the angle between the(t) and
R
T
0
F

(t)dt vectors. As can be seen, the
change in I, during a short time (T ) is expressed by RT. Therefore, it is reasonable
to refer to R as the spontaneous emission rate into the optical mode. The same
result can be inferred by requiring that the optical mode be in equilibrium with
the semiconductor material [Hen86a].
The linewidth of a laser has a dierent behavior below, near and above
threshold. The linewidth of a laser is defined as the half-width at half-maximum
spectral density (W

(!))of the Fourier transform of the field amplitude(!)
(t) =
1
p
2
Z
1
0
(!)e
i!t
d! (4.48)
Below threshold G can be considered constant, and(!) can be obtained by
writing equation 4.44 in terms of its Fourier components
(!) =
F

(!)
i (!
0
+G=2!) G=2
(4.49)
52
With the Langevian force diusion coecients given in equation 4.46, the spec-
tral density of a semiconductor laser below threshold is calculated
W

(!)(!
0
!) =h

(!
0
)(!)i
R(!
0
!)
(!
0
+G=2!)
2
+ (G=2)
2
(4.50)
The laser spectrum below threshold has a Lorentzian lineshape with a linewidth
given by
=
G
2
=
R
2I
(4.51)
in the last step G =R=I has been used [Hen91].
Above threshold, the amplitude fluctuations of the laser field are stabilized
in the oscillation which results in a factor of 2 reduction in the linewidth. The
remaining spectral width are due to phase fluctuations. Therefore, it is more
convenient to express the field in terms of the phase of the field,(t), and the
field intensity, I(t), instead of the complex amplitude,(t)
(t) =
p
I(t)e
i(!
0
t(t))
(4.52)
Then the Langevian rate equations for the new variables can be calculated
using the general procedure for transforming Langevian equations resulting in
[Hen91]
˙
I = GI + R + F
I
(t) (4.53)
˙
=

2
G + F

(t) (4.54)
53
with the diusion coecients
2D
II
= 2RI; 2D

=
R
2I
; 2D
I
= 0: (4.55)
Assuming that the amplitude fluctuations are negligible, the field amplitude
correlation functionh(t)

(0)i can be expressed as
h(t)

(0)i = Ie
i!
0
t
he
i(t)
i (4.56)
Since (t) =(t)(0) has a Gaussian probability distribution [Hen91], there-
fore
he
i(t)
i = e
h(t)
2
i=2
(4.57)
Thus, to evaluate the field autocorrelation function, the mean square of (t)
needs to be calculated. At low frequencies, the fluctuations of the field intensity
can be neglected, therefore, equation 4.53 can be solved for G
G =
F
I
(t)
I
(4.58)
where the term proportional to R has been neglected, because it is a constant,
and it results in a frequency shift that can be neglected. The following equation
can be derived for
˙
, by substituting equation 4.58 in equation 4.54
˙
= F

(t)

2I
F
I
(t) (4.59)
54
The phase change during a time t, is given by integrating this equation
=(t)(0)
=
Z
t
0
F

(t
0
)dt
0

2I
Z
t
0
F
I
(t
0
)dt
0
(4.60)
Now the mean square of (t) can be expressed in terms of F

and F
I
h()
2
i =h
"Z
t
0
F

(t
0
)dt
0

2I
Z
t
0
F
I
(t
0
)dt
0
#"Z
t
0
F

(t
00
)dt
00

2I
Z
t
0
F
I
(t
00
)dt
00
#
i
=
Z
t
0
Z
t
0
hF

(t
0
)F

(t
00
)idt
0
dt
00

I
Z
t
0
Z
t
0
hF
I
(t
0
)F

(t
00
)idt
0
dt
00
+ (

2I
)
2
Z
t
0
Z
t
0
hF
I
(t
0
)F
I
(t
00
)idt
0
dt
00
=
R
2I
(1 +
2
)t (4.61)
Thus the field autocorrelation function of the laser above threshold decays
exponentially. Therefore, the field amplitude spectral density of the laser emis-
sion above threshold has a Lorentzian profile with its linewidth given by
=
R
4I
(1 +
2
) (4.62)
This is dierent from the equation below threshold by a factor of (1 +
2
)=2. The
1 +
2
is due to coupling of phase and amplitude brought about by the feedback
mechanisms that stabilizes the intensity fluctuations, and the 1=2 factor is due
to the stabilization of amplitude fluctuations.
4.4 Microcavity Laser Linewidth
As was shown in previous sections 4.2 and 4.3, the laser noise properties are
dependent on the rate of spontaneous emission into the lasing mode. As it was
55
shown in section 3.3, a microcavity laser can significantly enhance the sponta-
neous emission rate into the lasing mode by modifying the density of optical
states at the lasing mode frequency. In practical applications, the noise proper-
ties of these lasers play an important role in evaluating the overall performance
of the system. So far, there has been a lot of theoretical investigation on the noise
performance of these lasers [AG91, BKY92]. However, there has not been a
complete experimental verification of these theories. As was mentioned before,
see section 3.3, there are experimental demonstrations that confirm the change
in the spontaneous emission lifetime of carriers inside microcavities, but to date,
there is no report that experimentally shows how microcavity eects change
the noise properties of lasers operating above threshold. Lack of stable micro-
cavity lasers working continuous wave (CW) above threshold has prevented
researchers from investigating the microcavity eects on laser performance. In
this section, the linewidth data for both photonic crystal and microdisk lasers
working CW at room temperature will be presented.
4.4.1 Photonic Crystal Laser Linewidth
The self-delayed optical homodyne technique, see section 4.5, was used to mea-
sure the linewidth of these microcavities [BSW
+
06]. Figure 4.2(a) shows a typical
evolution of photonic crystal laser linewidth versus the output power.
At low output powers, the laser is below threshold, and laser linewidth
follows equation 4.51. As the output power increases, the term in equation 4.62
starts to contribute to the linewidth, therefore the linewidth of the laser increases
as the output power increases. Finally, the term contribution saturates, and
thereafter the linewidth of the laser follows the usual P
1
dependence [AG91].
The increase in laser linewidth close to threshold is not a microcavity eect
phenomenon [BKY92] and is present in lasers with large linewidth enhancement
56
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
-50
-48
-46
-44
-42
S
i
(f) (dBm)
f (GHz)
= 575 –8.5 (MHz)
(b)
0.04 0.06 0.08 0.10 0.12
580
600
620
640
660
680
700
(MHz)
1 / P
Collected
( nW
-1
)
(a)
Figure 4.2: (a) a D
4
photonic crystal defect cavity laser linewidth evolution versus
the collected output power, (b) Laser Lineshape profile taken on an electrical
spectrum analyzer with a Lorentzian curve fit, with a extracted linewidth of 575
MHz
factor. Toano et al. has measured this eect in DFB laser cavities and VCSELs
[Tof97, FTL06]. Although this eect is present in all laser cavity geometries, this
can eect is more pronounced in microcavities. Unlike laser cavities with large
active mode volumes, due to the increased spontaneous emission coupling into
microcavity lasing mode, the transition between below and above threshold is a
soft transition, resulting in a more gradual saturation of the term contribution.
The linewidth of the photonic crystal lasers follows the P
1
dependence
until at high bias points, it starts rebroadening. The mechanisms that lead to
rebroadening of laser linewidth are not very well understood, however, gain
compression mechanisms contribute to the rebroadening and saturation of laser
linewidth. As will be shown in section 5.4, in the presence of gain nonlinearities,
the linewidth enhancement factor, , depends on the laser operating point.
Therefore, at high output powers, the linewidth of the laser deviated from the
P
1
behavior and rebroadens. Figure 4.3 shows the measured linewidth versus
the collected output power inverse for a D
4
and a D
5
cavity lasing around

lasing
= 1570nm. Both D
4
and D
5
cavities show linewidth rebroadening at high
57
output powers, and D
5
cavities saturates at higher output powers and slightly
narrower linewidth.
0.00 0.04 0.08 0.12 0.16
0
2
4
6
8
10
12
(GHz)
1/P
collected
(nW
-1
)
D
4
Linewidth
D
5
Linewidth
Figure 4.3: D
4
and D
5
photonic crystal defect cavity lasers linewidth evolution
versus the collected output power (laser linewidth of the cavity with larger
mode volume (D
5
) in this case, satrated at higher output power and narrower
linewidth values)
Figure 4.4 shows the measured linewidth of photonic crystal defect cavity
laser at dierent bias points. As will be discussed in section 5.5.1 the typical
normal fiber transfer functions used to extract laser parameters do not provide
a reasonable fit across all frequency ranges. We used a complex cross over
frequency term ,!
g
(see section 5.5.1)in the curve fit procedure applied to extract
from measurement that makes the model unphysical. However, the linear
dependence of on the laser operating point is as expected. The extrapolated
dependence of , on the laser operating point is then used to normalize the
linewidth of lasers. Figure 4.5(a) shows the measured linewidth of D
4
photonic
crystal defect cavity laser versus the inverse of the collected output power, and
figure 4.5 shows the evolution of the laser linewidth after the dependence of
on the laser operating point has been normalized. Figure 4.5 shows that the
normalization can successfully remove the rebroadening of laser linewidth at
high output power.
58
1.5 2.0 2.5 3.0 3.5
-3.0
-2.5
-2.0
-1.5

P
Incident
/P
th
Figure 4.4: Dependence of linewidth enhancement factor,, on the laser operat-
ing point
(b)
0.00 0.04 0.08 0.12 0.16
0
1
2
3
4
5
(GHz)
1 / P
Colleced
(nW
-1
)
0.00 0.04 0.08 0.12 0.16
0
2
4
6
8
10
12
(GHz)
1 / P
Colleced
(nW
-1
)
(a)
Figure 4.5: (a) D
4
photonic crystal defect cavity laser linewidth evolution versus
the collected output power, (b) D
4
photonic crystal defect cavity laser linewidth
evolution versus the collected output power normalized to 1 +
2
dependence
on laser operating point
4.4.2 Microdisk Laser Linewidth
Previously, there has been microdisk laser linewidth reports with linewidth a
few GHz wide [TL01]. In this section, a systematic analysis of microdisk laser
linewidth will be presented, and the narrowest linewidth of a microcavity laser
measured to date, will be presented. In the next section, the microdisk laser
linewidth behavior versus the cavity geometry will be analyzed.
59
Figure 4.6(a) shows the lasing spectra characteristics of three dierent
microdisk with slightly dierent radii that lase at dierent wavelengths (
lasing
=
1576nm;
lasing
= 1592nm, and
lasing
= 1606nm). Figure 4.6(b) is the linewidth
of the microdisks versus the incident input power inverse. The linewidth of
all of the cavities saturate at the same photon density ( P P
th
N
photon
) and
the slope of the linear part of the curves increases for cavities with longer las-
ing wavelength. The increase in the slope of the linear portion of the curves
(0:454GHz:mW for
lasing
= 1576nm, 0:493GHz:mW for
lasing
= 1592nm, and
0:634GHz:mW for
lasing
= 1606nm laser cavities)is consistent with the increase
in linewidth enhancement factor of strained quantum well active material with
the lasing wavelength, figure 4.7.
0.0 0.4 0.8 1.2 1.6 2.0
0.4
0.6
0.8
1.0

lasing
= 1576 nm

lasing
= 1592 nm

lasing
= 1606 nm
(GHz)
1 / (P - P
th
) (mW
-1
)
1.45 1.50 1.55 1.60 1.65
0.01
0.1
1
0.0
0.2
0.4
0.6
0.8
1.0
Normalized Intensity (a.u.)
Normalized Intensity (a.u.)
Wavelength ( m)

(b) (a)
Figure 4.6: (a) Lasing spectrum of three dierent microdisks with slightly dif-
ferent radii with the spontaneous emission spectrum from the QW active region
overlaid, (b) laser linewidth versus the incident pump power inverse for cavities
in figure 4.6(a)
Figure 4.8(a) shows linewidth versus the eective current inverse of a 2:0m
radius microdisk (squares) and a 3:8m radius microdisk laser (circles). The
linewidth of microdisk lasers with larger radius shows the rebroadening at
60
Figure 4.7: Dispersion of linewidth enhancement factor, versus the photon
energy after Yamanaka et al. [YYY
+
93]
higher output power, which is consistent with the onset of gain nonlineari-
ties eects. Figure 4.8(b) shows the narrowest semiconductor microdisk laser
linewidth measured to date. This linewidth is on the same order of other bulky
semiconductor laser linewidth, and about 25 times larger than the narrowest
measured linewidth (3:6MHz) for VCSEL laser cavities [SMV
+
01].
Figure 4.9(a),(b) shows the narrowest laser linewidth measured for a
microdisk laser. This device has a 1:8m radius. This laser operates at

lasing
= 1565nm. This corresponds to a small linewidth enhancement factor,
therefore the measured linewidth is narrower than the cavities with similar
radius but lasing at a longer wavelength.
4.4.3 Laser Linewidth and Microcavity Eects
As was previously derived, see section 3.3, the Purcell eect in microcavities
changes the spontaneous emission lifetime of carriers inside the cavity. Previ-
ously, an enhancement of carrier spontaneous emission rate has been measured
in semiconductor active material [FXY
+
02, GSG
+
98, VFS
+
03, EFW
+
05, BSN
+
04,
BS03]. To date, there has not been any report, showing the eect of increased
61
0 5 10 15
100
200
300
400
(GHz)
(I-I
th
)
-1
(mA
-1
)
(b)
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-95
-90
-85
-80
-75
-70
-65
-60
S
i
(f) (dB)
f (MHz)
Measured Lineshape
Lorenzian fit
= 85 MHz
(a)
Figure 4.8: (a) Linewidth vs. the eective injected current inverse for a 2:0m
radius microdisk (squares) and a 3:8m radius microdisk (circles), (b) detected
photocurrent power spectrum of the minimum measured linewidth for the
3:8m microdisk in figure 4.8(a) and the Lorentzian fit.
spontaneous emission rate on the laser noise properties above threshold. In
this section, the eect of enhanced spontaneous emission rate on the laser noise
properties in microcavities is analyzed by investigating laser linewidth behavior
as a measure of the laser noise performance.
To investigate the eects of the modification of the spontaneous emission rate
into the lasing mode, microdisk linewidth behavior for cavities with dierent
radius was investigated. As was shown in equation 4.62, above threshold the
laser spectral width decreases linearly as a function of the inverse of number of
photons in the cavity optical mode. However, to be able to make a comparison
with experimental data more readily measurable cavity parameters can be used.
62
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-70
-65
-60
-55
-50
S
i
(f)(dB)
f (GHz)
= 140 MHz
(b)
0 5 10 15 20 25
100
150
200
250
300
350
400
(MHz)
(I - I
th
)
-1
(mA
-1
)
(a)
Figure 4.9: (a) Linewidth vs. the eective injected current inverse for a 1:8m
radius microdisk, (b)detected photocurrent power spectrum of the minimum
linewidth in figure 4.9(a) and the Lorentzian fit.
4.4.4 Evaluation of R and I
Above threshold, the number of photons in the optical mode of the cavity (I) is
related to the current by relations that can be easily derived from the conservation
of energy[CC95a]
I =
i

p
(C C
th
) (4.63)
where
i
,
p
, and C C
th
are the internal quantum eciency, the photon lifetime,
and the electron current above threshold (in number of carriers per second),
respectively. To determine the spontaneous emission rate into the cavity optical
mode, R, the band-to-band radiative transitions in active semiconductor material
needs to be considered, as shown in figure 4.10. The rates at which the three
radiative processes occur depend on the density of photons and density of
available state pairs. By establishing the relations between these radiative rates,
it can be shown that the spontaneous emission rate into the optical mode is
[CC95b]
63
v-f
R
sp
R
21
2
E
V
E
C
1
2
E
V
E
C
1
2
E
V
E
C
1
R
12
v-f
Figure 4.10: Band-to-band radiative transitions: stimulated absorption, stim-
ulated emission, and spontaneous emission. (All rates are defined per unit
volume.)
R
spj
threshold
= v
g
g
th
n
sp
(4.64)
where is optical mode confinement factor, g is the material gain, v
g
is the optical
mode groups velocity, and n
sp
is the inversion factor. v
g
g = 1=
p
has been used
to simplify the relations.
Substituting equations 4.63 and 4.64 in the equation for the laser linewidth
above threshold, equation 4.62 simplifies to
=

v
g
g
th

2
n
sp
4
1
C C
th
(1 +
2
) (4.65)
Figure 4.11 shows the measured linewidth vs. the inverse of the current
above threshold for three microdisks lasing near 1581 nm with radii 1.40m,
2.25m, 3.25m, respectively. As indicated by equation 4.65, the slope of these
curves can be estimated by measuring the modal gain for each optical mode.
The modal gain can be calculated using the threshold pump power, and known
gain-current relations for the quantum well active material[SKY
+
06, Mat96].
g = G
0
ln
J
J
tr
(4.66)
64
0 5 10 15 20 25 30
200
400
600
800 1.40 m
2.25 m
3.25 m

(MHz)
(I-I
th
)
-1
(mA)
-1
Figure 4.11: Measured microdisk laser linewidth versus the eective injected
current inverse for three dierent microdisks with dierent radii lasing at
lasing
=
1581nm
G
0
, and J
tr
are extracted from measurements carried out on broad area laser.
was then calculated using a finite dierence algorithm for a slab waveguide
containing four 10nm thick quantum wells at the center of the slab waveguide.
Its value was evaluated to be = 0:17. Figure 4.12 shows the measured collected
output power versus the incident pump power for these three microdisk lasers.
Figure 4.13 shows the ratio between the measured slope and

v
g
g
th

2
n
sp
4
,
that is a representative of (1 +
2
). A large linewidth enhancement factor () of
6:0 does not compensate for the dierence. We attribute the dierence to the
increase in laser spontaneous emission rate into the optical cavity mode due to
the Purcell eect.
As was shown in section 3.3, microcavities increase or inhibit the spontaneous
emission rate of the active medium by modifying the density of optical states.
65
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0
20
40
60
80
100
1.40 m
2.25 m
3.25 m

Collected Output Power (a.u.)
Incident Power (mW)
Figure 4.12: Collected output power versus the incident pump power for three
dierent microdisks with dierent radii lasing at
lasing
= 1581nm
This eect is characterized by the Purcell factor, F, and it depends on the ratio of
the quality factor to the optical mode volume,
F =
3(=n)
3
Q
4
2
V
e f f
(4.67)
which can be a large quantity. However this value is smaller when the dephas-
ing of carriers due to other scattering mechanisms is taken into account. We
attributed the ratio between the measured slope and the estimated slopes to be
due to Purcell factor. Therefore equation 4.64 is modified to
R
spj
threshold
= Fv
g
g
th
n
sp
(4.68)
Figure 4.14 show the estimated Purcell factors (F) for the microdisk lasers in
figures 4.11 and 4.12. The optical mode volume of the microdisks was calculated
from a simple eective index method. This method provides enough accuracy
for calculating the resonant wavelength and the field distribution of the optical
66
1.0 1.5 2.0 2.5 3.0 3.5
0
50
100
150
200
250
300
Radius ( m)

F(1+
2
)
Figure 4.13: Measured ratio between the measured and estimated slope of the
linewidth versus the inverse of the current curve.
mode, but typically overestimates the quality factors [LHG
+
06]. Then using the
measured optical quality factors and calculated V
opt
, we estimated the expected
Purcell factors using equation 4.67.
The Purcell factors calculated in figure 4.14 are then used to extract the
linewidth enhancement factor from the data points in figure 4.13. Such a
procedure results in = 3:8. That value is within the range of values we expect
for this material system.
This is the first experimental demonstration of the microcavity eects on
noise properties of microcavity lasers. This phenomenon has practical signif-
icance in applications where these microcavities are used as source for trans-
mitting data. In such application the intensity fluctuations of the optical source
( spontaneously emitted photons in this case) limit the maximum achievable
signal-to-noise ratio.
67
1.0 1.5 2.0 2.5 3.0 3.5
4
6
8
10
12
14
Purcell Factor, F

Radius ( m)
Figure 4.14: Estimated Purcell factor based on measured quality factors and
calculated optical mode volumes
4.5 Laser Linewidth Measurement Techniques
Optical phase noise and frequency fluctuations have substantial impact on the
optical power spectrum and the quality of data communication. Therefore, it
is of great practical importance to be able to characterize a laser’s phase noise
and frequency chirp in practical applications. Although, laser intensity noise is
relatively easy to characterize, phase noise characterization is complicated due
to the fact that it should be converted to intensity fluctuations before it can be
detected with any photodetector. Optical mixing and interferometric techniques
are commonly used to convert phase noise into intensity fluctuations, detectable
by photodetectors.
Typical grating based optical spectrum analyzers can not resolve the fine
structures of a laser line due to limitation in resolution ( 1Å). Other monochro-
mator based techniques such as scanning Fabry-Perots are also limited in res-
olution [DSN90]. In this section optical-heterodyne and optical-self-delayed-
homodyne linewidth measurement techniques are briefly reviewed [Der98].
68
4.5.1 Optical Heterodyne and Optical Self-Delayed-Homodyne
As was mentioned previously, interfering two optical fields is one of the tech-
niques used in measuring the linewidth of lasers. In optical-heterodyne method,
as shown in figure 4.15, the laser field under study, E
S
is mixed with the emission
from a known local oscillator laser, E
LO
. In this measurement scheme, the central
Figure 4.15: Setup for interfering two optical fields
frequency of the LO laser is tuned close to the signal laser frequency to allow
the mixing product to fall within the bandwidth of detection electronics. The
optical spectrum analyzer(OSA) can be used for course wavelength tuning. The
coupler combines the two fields, delivering partial power to each output port.
One port is directed to a photodetector (PD) which detects the interference beat
tone. The resulting electrical signal after the photodetector can be expressed as
follows
i(t) =<
h
P
S
(t) + P
LO
+ 2
p
P
S
(t)P
LO
cos
h
(
s

LO
)t + (t)
ii
(4.69)
where P
S
(t) is the laser under test optical power, P
LO
(t) is reference laser power,

S
and
LO
are the average frequencies of the laser under test and reference laser
respectively, (t) is the instantaneous phase dierence between the laser and
LO, and< is the photodetector responsivity. The electrical spectrum analyzer
measures the electrical power spectrum of the photodetector current.
69
S
i
( f )<
2

S
d
( f ) + 2 [S
LO
()
S
S
()]

(4.70)
S
d
( f ) is the direct detection terms that results from each laser optical power. The
second term represents the heterodyne signal that contains all the phase noise
information. There is an extra spectral component that is centered at 2(
S

LO
),
which can be filtered if (
S

LO
) is much larger than the spectral spread of the
laser under study, therefore it is neglected in equation 4.70. Now, if the local
oscillator phase noise is small compared to the test laser, the beat tone will be
broadened primarily by the phase noise of the test laser ((t)
S
(t)). The
lineshape of the laser under test is replicated at frequencies within the operating
range of electronic equipments.
Optical heterodyne technique allows oers high sensitivity, and very high
spectral resolution ( limited by the LO linewidths), and allows measuring of
asymmetries in lineshape, but at the expense of a low frequency jitter, narrow
linewidth tunable local oscillator laser.
Optical self-delayed heterodyne/homodyne provides an alternative method
for measuring the linewidth of laser which is simpler and less restrictive than
heterodyne method. In this technique, optical field from the laser under test
is split in half, one half is delayed with respect to the other, the two halves
are combined together, and then photodetected. This technique is simpler to
implement compared to the optical heterodyne method and does not require
an external laser. The interpretation of linewidth from this method is more
complicated, but if the optical delay () in one arm of the interferometer is long
compared to coherence time of the laser(
laser
1), then the power spectrum
70
of the photodetector current displayed on the electrical spectrum analyzer is
given as
S
i
( f ) S
dc
( f ) + S
shot
( f ) + S
mix
( f ) (4.71)
S
mix
( f )

f
2
+ ()
2
(4.72)
The linewidth of the laser is half in equation 4.71. This method is able to
measure narrow linewidth lasers with long optical delays and is less sensitive
to slow wavelength drifts compared to the optical heterodyne method, but it is
unable to measure asymmetries in the lineshape.
In our experiments, the light from the microcavity lasers, after collection into
a single-mode fiber, was amplified using an EDFA. A 360 pm FWHM bandpass
filter was then used to suppress ASE-ASE beat noise. The amplified and filtered
optical signal was 50:50 split using a fiber splitter, delayed in one arm using
a 2.2km long single-mode fiber, and recombined in another 50:50 splitter. The
light was then photodetected using a 25GHz photodiode (New Focus 1434).
The spectrum of the electrical current was observed on a HP8563E spectrum
analyzer. figure 4.16 shows a measured lineshape profile recorded on electrical
spectrum analyzer with a Lorentzian curve fit applied to the measured data, to
extract its linewidth.
71
-300 -200 -100 0 100 200 300
0
50
100
150
200
250
S
i
(f) ( W)
f (MHz)
Figure 4.16: Lineshape of a laser measured with the self-delayed homodyne
method with the Lorentzian curve fit
72
Chapter 5
Modulation Response Properties of
Semiconductor Microcavity Lasers
Introduction
The laser is one of the most important elements of any optical interconnect sys-
tem. 1:3m and 1:5m emission wavelengths are the standard technologies used
in long haul data communication, because they correspond to the minimum
dispersion and minimum loss region of the typical single mode silica fibers,
respectively. These two wavelength regions are also potential candidates for
on-chip data communication systems because of the available mature semicon-
ductor material growth and fabrication technologies at these two regions, as well
as the possibility of routing these two wavelength regions on a CMOS compati-
ble platform ( their photon energies are below the bandgap of silicon so they are
not absorbed in waveguides or other photonic elements based on silicon). This
spectral region corresponds to a frequency range around 200 THz, which means
that optics can carry information at data rates up to a few THz. This information
can be modulated onto a laser beam either directly or through an external opti-
cal modulator. Although optical modulators with high modulation eciencies
and high modulation speeds have been demonstrated [Liu07, CFC
+
97], direct
modulation of laser beam is simpler to implement and more cost eective for
most of applications. Such applications require lasers with high slope eciency
73
to obtain high link gain, low laser noise to keep the link noise figure low and
high modulation bandwidth.
The microcavity semiconductor lasers are the potential sources for driving
chip-scale optical networks. As mentioned previously, see chapter 3, micro-
cavity lasers modify the spontaneous decay of carriers. The shortening of the
spontaneous emission decay of carriers can have significant impacts on dynamic
properties of lasers and have been the subject of great interest due to its prac-
tical implications [YMB91]. The short radiative lifetime of carriers promises
large modulation bandwidth, both below and above threshold. Below thresh-
old, the modulation speed is limited by the spontaneous emission lifetime. The
substantially shortened spontaneous emission lifetime in a well designed and
fabricated microcavity leads to a wider modulation bandwidth light emitting
diodes (LEDs)[ZBQ01], and above threshold, the extremely short photon life-
time in an appropriately designed microcavity accompanied with the extremely
short cavity length, and enhanced spontaneous emission rate will result in fast
modulation response [YB89].
This chapter will present the data on modulation response properties of
microdisk and photonic crystal laser cavities.
5.1 Semiconductor Laser Rate Equations
The basic description of a single mode laser dynamics involves a pair of
rate equations governing the photon and carrier densities inside the laser
medium[Jr.93]
dN
dt
=
J
ed

N

s
vg(N)S; (5.1)
dS
dt
= v
g
g(N)S
S

p
+R
s
p (5.2)
74
where N is carrier density, S is photon density in the optical cavity mode, is the
optical confinement factor, J is pump current density, d is thickness of the active
region,
s
is total carrier recombination lifetime,
p
is the photon lifetime, g(N)
is the optical gain as a function of carrier density, v is the optical mode group
velocity, is the fraction of spontaneous emission coupled to the lasing mode,
R
sp
is spontaneous emission rate, and e is the electron charge.
This set of coupled dierential equations describe both the static and dynamic
properties of diode lasers. Figure 5.1(a) shows an output power vs. the injected
current density of a laser obtained by solving equations 5.1 and 5.2 with param-
eters extracted from experiment, and figure 5.1(b) shows the output power of
the laser (scattered) in response to a modulated input current (solid curve). The
appearance of a clear threshold, figure 5.1(a), and the ringing of the output
power above threshold, figure 5.1(b), are distinct features of lasing action.
200 400 600 800 1000 1200
5
10
15
20
25
(b)
0 200 400 600 800 1000 1200 1400
0
10
20
30
40
50
60
70
80
0
1
2
3

P
out
(a.u.)
Time (ps)

(I - I
th
)/I
th
Output Power (a.u.)
J (A/cm
2
)
(a)
Figure 5.1: (a) Output power vs. the injected current density, (b) output power
of a laser (scattered) in response to a time varying input current(solid line)
Small signal analysis is a standard method for analyzing dynamical behav-
ior of complex systems. Equations 5.1 and 5.2 are a set of coupled dierential
equations that govern the behavior of the lasers as a function of the injected
75
current. Similar to the procedure used in section 4.2 to analyze the noise proper-
ties of single-mode lasers, the small signal analysis assumes that the dynamical
changes in the carrier density and photon density are small with respect to their
steady state values. This assumption helps to simplify the dierential equations,
and analytical solution are then obtained for the small changes in carrier and
photon densities. For a small sinusoidal change in injected carrier density,
j(t) = Re
h
j(!)e
i!t
i
(5.3)
small signal modulations of the form
n(t) = Re
h
n(!)e
i!t
i
(5.4)
s(t) = Re
h
s(!)e
i!t
i
(5.5)
are sought in response to the change in current density. Then, the total injected
current density, carrier density, and photon density have the following form,
J
t
ot = J
0
+ j(t) = J
0
+ Re
h
j(!)e
i!t
i
(5.6)
N
t
ot = N
0
+ n(t) = N
0
+ Re
h
n(!)e
i!t
i
(5.7)
S
t
ot = S
0
+ s(t) = S
0
+ Re
h
s(!)e
i!t
i
(5.8)
76
where N
0
and S
0
are the steady state solutions of equations 5.1 and 5.2 to the DC
excitation, J
0
, when J
0
> J
th
, and Re stands for the real part of the argument. Sub-
stituting equations (5.6)–(5.8) into equations 5.1 and 5.2, the following equation
are derived for the dynamical variables, s(!) and n(!),
s(!) =

v
g
g
0
S
0
qd
!
j(!)
!
2
R
!
2
+ i!
(5.9)
n(!) =

i!
v
g
g
0
S
0
!
s(!) (5.10)
where!
R
and
in equations 5.11 and 5.12 are defined
!
2
R

v
g
g
0
S
0

p
(5.11)

1

s
+ v
g
g
0
S
0
=
1

s
+
p
!
2
R
(5.12)
and are called laser relaxation oscillation frequency and damping term, respec-
tively. It is worth noticing that equations 5.11 and 5.12 are similar to those found
in equations 4.31 and 4.32. The small signal analysis for deriving equations 5.9
and 5.10, implicitly assumes that gain nonlinearities are negligible and material
gain can be modeled as
g(N) = g
0
+ g
0
(N N
0
) (5.13)
where g
0
= g(N
0
), and g
0
=

@g=@N

N
0
is called dierential gain, and is an
important parameter in the high-speed operation of semiconductor lasers. The
expected large dierential gain in quantum confined structures such as quantum
wells and quantum dots due to their modified density of electronic states com-
pared to bulk material attracted a lot of attention during the early days of these
77
structures. It was later realized, however, that the enhancement of gain com-
pression mechanisms in these quantum confined structures reduces dierential
gain and the maximum achievable modulation bandwidth [Lau93]. Therefore,
directly modulated lasers with quantum well active region show little or no
improvement in their modulation bandwidth compared to the lasers with bulk
active region. Gain compression eects are taken into account in the small signal
analysis, presented above by including a phenomenological gain compression
term into material gain [Chu95]
g(N) =
g
0
+ g
0
(N N
0
)
1 +S
0
(5.14)
The consequence of gain reduction due to gain compression on the modula-
tion response of laser is then modeled as a reduction in relaxation modulation
frequency and an increase in damping term [OHLP87, NFI
+
92]
!
2
R

v
g
g
0
S
0

p
(1 +S
0
)
(5.15)

1

s
+
v
g
g
0
S
0
1 +S
0
+
S
0
1 +S
0
=
1

s
+
K
4
2
!
2
R
(5.16)
K = 4
2

p
+

v
g
g
0
!
(5.17)
where a figure of merit denoted by K has been introduced that is a measure of
maximum 3-dB modulation bandwidth of semiconductor lasers [OHLP87]
( f
3dB
)
max
=
2
p
2
K
(5.18)
Semiconductor lasers with a quantum well providing gain, also are influ-
enced by the transport of carrier across the optical waveguides. This eect
appears as a simple pole in the intensity modulation transfer function of lasers
[NIGB92, NFI
+
92]. In our experiments, since the waveguide thickness is thin
78
(240nm) and the cavities are optically pumped, the transport eects can be
neglected.
5.2 Modulation Response Measurement of Micro-
cavity Lasers
Figure 5.2 is a schematic representation of the setup used in measuring the
modulation response of the microcavity lasers. In order to provide small-signal
modulation the microcavity lasers with small-signal modulation input beam,
emission from a high-bandwidth direct-current-modulated multi-mode VCSEL
( ULM850-10-TN-U46TPP) with emission wavelength at 850nm was spatially
overlapped with the CW edge-emitting pump and applied to the device under
test, as shown in figure 5.2. To reduce the electrical parasitics on the modulation
response of the VCSEL, the VCSEL chip is directly mounted on a SMA connector
socket as shown in figure 5.3(a). Figure 5.3(b) shows the S21 measurement of
the VCSEL laser.
The collected out-of-plane emission of the microcavity lasers was coupled
into a single mode fiber, amplified with an EDFA, and filtered with a 360pm
optical bandpass filter to suppress the ASE noise from the EDFA. The ampli-
fied and filtered optical signal was photodetected with a 25GHz photodetector
(New Focus 1421) and then preamplified using a 26.5GHz broadband microwave
amplifier (Miteq AFS44-00102650-42-10P-44) in order to boost the electrical sig-
nal. A HP8510C network analyzer was used to provide the modulating signal
to the VCSEL and to measure the amplified output of the photodiode. In this
set-up, the measurement bandwidth was limited to about 15GHz by the roll-o
of the VCSEL modulation bandwidth beginning at about 8GHz as shown in
figure 5.3(b).
79
Photodetector
New Focus(1621)
Optical isolator
Flipper
Flipper
Oscilloscope
CCD
camera
Monitor
850 nm pump
laser
Optical isolator
Beam
Expander
Polarizer
50%
beam splitter
White light
source
Motorized
XYZ stage
Fiber Coupler
dichromatic mirror
100X
objectiv e lens
Flipper
10Gbps VCSEL
Bias T
Flipper Flipper
Flipper
Single mode
fiber
Network
Analy zer
Figure 5.2: Schematic illustration of measurement setup for high-speed charac-
terization of microcavity lasers
In operation the VCSEL pump was biased at 9mA to achieve the highest
available modulation bandwidth. The measured frequency response of the
microcavity laser was then normalized to the measured response of the VCSEL
to calibrate out the frequency response of the modulating VCSEL, photodetector,
microwave amplifier, and all other elements of the measurement system other
than the microcavity laser under study. Figure 5.4 shows the dierential modu-
lation response representing the small-signal response of a typical D
4
laser cavity
operating at several bias levels. The laser relaxation oscillation frequency and
damping term, equations 5.11 and 5.12, are extracted by fitting the measured
curve to equation 5.9. A curvefit applied to the data taken at two times laser
threshold is also demonstrated in figure 5.4 [BSW
+
06].
Figure 5.5(a) shows the extracted relaxation oscillation frequency squared of
a D
4
(squares) and a D
5
(circles) photonic crystal laser cavity, lasing at 1572nm.
The relaxation oscillation frequency squared shows a linear dependence on the
80
(b)
SMA Socket
VCSEL Chip
Short Wire-Bond
(a)
Figure 5.3: (a)VCSEL mounted directly on SMA socket (best frequency perfor-
mance) (b) Intensity modulation response of high-bandwidth VCSEL ( Courtesy
of ULM Photonics).
incident input power as expected from equation 5.11. The extracted damping
term, equation 5.9, is plotted against the relaxation oscillation frequency squared
for both lasers in figure 5.5(b). Based on equation 5.18, the slopes of the curves
in figure 5.5 (0:303 0:02ns for D
4
, and 0:242 0:04ns for D
5
) indicate a possible
maximum -3dB bandwidth of around 30GHz for both lasers. This -3dB band-
width is achieved when lasers are biased ten times above threshold. These lasers
could not operate far above threshold before their output power rolls o due to
heating.
Although the data points in figure 5.5(a) shows a linear dependence of the
relaxation oscillation frequency squared against the incident input power, a more
detailed study with more data points reveals deviations from linear behavior.
As indicated by equation 5.15, the relaxation oscillation frequency becomes a
nonlinear function of output power ( or equivalently the incident pump power)
in the presence of gain nonlinearities. Figure 5.6 shows a measured relaxation
oscillation frequency curve squared versus the output power. The nonlinear
81
5 10 15
15
20
25
30
35
40
45
1.3xThreshold
2.0xThreshold
2.5xThreshold
Curvefit
Power(dB)
Frequency(GHz)
Figure 5.4: small signal modulation response of a 3:2m diameter cavity (D
4
) at
dierent bias levels with the curvefit applied to the data at two times threshold.
behavior of the curve in figure 5.6 signifies the presence of gain compression
in the active material. Gain nonlinearities will limit the maximum acceptable
quality factor of microcavities in communication applications where a minimum
output power is required to establish a data link with certain signal-to-noise ratio
values.
Figure 5.7 presents the measured 1:92m radius microdisk laser relaxation
oscillation frequency versus the collected output power. The presence of gain
compression in these microcavities also impairs the frequency response of the
laser by reducing the maximum relaxation oscillation frequency.
In order to extend the measurement bandwidth, an externally modulated
laser (Alcatel 1948 FBG) emitting at 1410 nm ( above the quantum well bandgap
,but below the lasing wavelength of the microcavity lasers) was used to provide
the small signal modulation pump. Using this measurement setup, modulation
bandwidth in excess of 17 GHz was demonstrated for microdisk lasers. Fig-
ure 5.8(a) shows the measured small signal intensity modulation response of
82
0 1 2 3 4 5
0
20
40
60
80
f
R
2
(GHz
2
)
(P
incident
-P
th
)/P
th
D
5
data
D
4
data
0 10 20 30 40 50 60 70
0
5
10
15
20 D
5
data
(ns
-1
)
f
R
2
(GHz
2
)
D
4
data
(b) (a)
Figure 5.5: (a) Relaxation Oscillation frequency squared versus the incident
input power normalized to the threshold power, (b)damping of laser versus the
relaxation oscillation frequency squared
a typical 1:9m radius microdisk laser cavity operating at dierent bias pow-
ers measured with the 1410nm externally modulated pump. The discontinuity
at 2.5 GHz is due to the pump beam modulating circuit. As shown in fig-
ure 5.8(a), a maximum -3dB modulation bandwidth of 17.5 GHz is achieved for
a microdisk laser. Figure 5.8(b) is a plot of behavior of the extracted laser damp-
ing term against the extracted laser relaxation oscillation frequency squared,
which shows a linear behavior as expected. The slope of this curve (0:45ns) indi-
cates a possible maximum -3dB bandwidth of 21 GHz. The previously reported
-3dB modulation bandwidth for microdisk lasers on a sapphire substrate was 2
GHz, limited by the detection circuit [TL01].
In microcavity lasers, because of the small optical mode volume and high
photon density in the cavity even at fairly low output powers, gain nonlinearities
play an important role in preventing reaching high modulation bandwidth.
Figure 5.9 shows the photon density inside an edge emitting laser cavity with
V
optical
= 100(=n)
3
and a microcavity cavity with V
optical
= 2(=n)
3
at 100W
output power. Photon density for fixed output power is inversely proportional
83
(b)
0 10 20 30 40 50 60 70
0
5
10
15
20
25
30
35
(ns
-1
)
f
R
2
( GHz
2
)
0 20 40 60 80 100 120
0
5
10
15
20
25
30
35
40
45
50
55
f
R
2
(GHz
2
)
Collected Output Power (nW)
= 0.0039
(a)
Figure 5.6: (a) Relaxation Oscillation frequency squared versus the collected out-
put power, and (b)damping versus the relaxation oscillation frequency squared
of a D
4
photonic crystal defect cavity laser
to the optical mode volume. To maintain a fixed output power, the photon
density inside the smaller cavities is larger than the photon density in the cavities
with larger optical mode volume.
P
out
=
0
~!
S
0
V
opt

p
(5.19)
where the optical eciency,
0
is the fraction of photons that escape the cavity
as useful output to the total photons lost, and is assumed to be unity for both
cavities in figure 5.9 for convenience.
5.3 Gain Compression in Semiconductor Lasers
Semiconductor lasers with quantum confined active medium such as quantum
wells and quantum dots were expected to have superior characteristics such as
threshold current, dierential gain, and-parameter ( linewidth enhancement
factor) compared to the their counterparts with bulk active medium. All these
84
(b)
0 20 40 60 80 100 120
0
10
20
30
40
50
60
70
80
90
(ns
-1
)
f
R
2
(GHz
2
)
0 5 10 15 20
0
20
40
60
80
100
120
f
R
2
(GHz
2
)
Collected Output Power (nW)
(a)
Figure 5.7: Relaxation Oscillation frequency squared versus the collected output
power.
were expected due to the modified density of electronic states that is achieved
by engineering the edges of conduction and valance bands in the semiconductor
material.
Another expected benefit from a quantum confined active medium in con-
nection with high-speed operation of semiconductor lasers was the achievement
of very high maximum modulation speed. As equation 5.18 indicated, the max-
imum modulation bandwidth achieved in semiconductor lasers increases by
lowering the K factor, K/ (@g=@n)
1
. So the increased dierential gain in quan-
tum well lasers would have increased the maximum achievable modulation
bandwidth. An overall improvement of 3-4 times in modulation bandwidth
was predicted for quantum well lasers and the extra factor of 2-3 improve-
ment in modulation bandwidth for strained layer quantum wells would have
resulted in modulation speeds in 100GHz regime [Jr.93]. However the increase
in gain nonlinearities in quantum well structures impairs the benefits gained by
increased dierential gain, K/ =(@g=@n), where is the nonlinear gain coef-
ficient. The other drawback of using quantum wells is the faster degradation
85
0 50 100 150
0
20
40
60
80
f
R
2
(ns
-1
)
0 5 10 15 20
-45
-40
-35
-30
-25
Power (dB)
Frequency (GHz)
(a) (b)
Figure 5.8: (a) Relaxation Oscillation frequency squared versus the collected out-
put power, and (b)damping versus the relaxation oscillation frequency squared
of a 1:92m microdisk laser
of dierential gain with increased output power as indicated by equation 5.14
[TA91].
There are dierent mechanisms leading to gain compression in semicon-
ductor active medium. Spectral hole burning, carrier heating, and spatial hole
burning are the dominant processes leading to an intensity dependent gain in
semiconductor material.
Spectral hole burning corresponds to a dip in the carrier energy distribu-
tion centered around the lasing wavelength [GDG
+
95]. Above threshold, the
stimulated emission consumes the electron and hole populations at a fast rate
causing the carrier density to be fixed at its threshold value. The carrier den-
sity at energy levels other than the lasing wavelength is not fixed, because the
collisions that are responsible for replenishing the lasing transition, feed the
carriers at a finite intraband relaxation timee
in
. This causes the total carrier
density to increase slightly above threshold [AA94], leaving a dip at the lasing
86
10
2
10
3
10
4
10
5
10
6
10
13
10
14
10
15
10
16
10
17
10
18
S
0
(cm
-3
)
Quality Factor
V
opt
=2( /n)
3
V
opt
=100( /n)
3
Figure 5.9: Photon density inside two cavity geometries with V
opt
= 2(=n)
3
and
V
opt
= 100(=n)
3
versus the cavity quality factor at 100W output power, the
shaded area shows the boundaries for the region in whichS
0
1
wavelength. The width of the dip is approximately 1=
in
and its depth is pro-
portional to
2
in
. The hole burning eect gets smaller for structures with small
intraband relaxation times[Yam94]. Although normally, semiconductor lasers
are operated in a regime where the output intensities are below the level at which
hole burning eects can dominate, but small-mode-volume optical microcavi-
ties can lie within the range where spectral hole burning be responsible for gain
nonlinearities (see figure 5.9).
The other process that is of equal importance in semiconductor gain non-
linearities is carrier heating. Gain nonlinearities due to carrier heating has
its maximum at shorter wavelengths than the lasing mode of the cavity
[WUM
+
91, GDG
+
95]. There are mainly two mechanism that contribute to the
changes carrier temperature. Stimulated emission removes the cool carriers
close to the bottom of the conduction and valence bands, increasing the overall
carrier population temperature. Stimulated absorption can increase or decrease
the carrier temperature depending the absorbed photon energy. Free carrier
87
absorption is the other process that serves to change the carrier population
almost always increases the carrier temperature [KI87]. There are other mecha-
nisms such as Auger recombination, well-barrier hole burning, and transverse
spatial hole burning [WS96, RSK
+
91] that are proposed to contribute to the car-
rier temperature change in semiconductor active material. The temperature
change in hole and electron population is dierent and small. However, due the
sensitivity of gain to the carrier temperature, the temperature change causes an
appreciable change in gain.
5.4 Frequency Modulation of Semiconductor Lasers
As was shown previously, equation 5.10, current density modulation brings
about a change in carrier density of a semiconductor laser. The change in carrier
density N, aects the optical gain g, but it also aects the refractive index of
the active region, (d=dN2 10
20
cm
3
). The consequence of change in
refractive index of active material, is a change in the lasing wavelength (chirp).
The possibility of direct frequency modulation in semiconductor lasers is
of practical importance in applications such as coherent optical transmission
systems. Also the presence of sources with large amount of chirp causes rapid
pulse spreading in high-speed digital optical networks leading to intersym-
bol interference. Semiconductor laser chirp is commonly characterized by two
parameters, the linewidth enhancement factor,, and the cross-over frequency
between the adiabatic chirp and transient chirp,!
g
, which follows from a rate
equation analysis of the laser[KL86, KB84, Pet91].
=

th

th

0
=

th

th

0
=N

00
=N

00
=
1
4
v
g
g (5.20)
88
where , ( =
0
+ i
00
), and N are the frequency excursions from its value
at threshold, the refractive index of semiconductor material, and the carrier
density, respectively. , the linewidth enhancement factor, is defined

0
=N

00
=N
(5.21)
and the optical gain is related to the imaginary part of the refractive index
g =
4

0

00
(5.22)
The rate equation for the photon density in the optical mode, equation 5.2,
considering the gain nonlinearities can be written in the following form,
dS
dt
=
vg(N)S
1 +S

S

p
+R
s
p (5.23)
The frequency excursion around the threshold point g
0
and S
0
is obtained by
allowing changes in the form of g = g
0
+ g and S = S
0
+ S. Then equation 5.2
can be rearranged to express gain changes around its operating point in terms
of the output power,
v
g
g = (1 +
p
P(t))
"
1
P(t)
dP
dt
+

p
v
g
g(N)P
(1 +
p
P)
2
#
= (1 +
p
P(t))
"
d
dt
lnP(t) +!
g
P
#
(5.24)
Equation 5.24 is now substituted in equation 5.20 to form a relation between
the laser chirping () and the output power.
=
1
4
(1 +
p
P(t))
"
d
dt
lnP(t) +!
g
P
#
=
1
4

e f f
"
d
dt
lnP(t) +!
g
P
#
(5.25)
89
where
e f f
(1 +
p
P(t)). It is worth noticing that the equations derived in
[KL86, Pet91] are valid for the cases where nonlinear gain forms a symmetric
spectrum around the lasing wavelength, therefore Kramers-Kronig relations
ensure that symmetric features in the imaginary part of refractive index (
00
)
will not contribute to the chirping of lasers. In cases, such as nonlinear gain due
to spectral home burning, the frequency chirping is given by [KL86]
=
1
4

"
d
dt
lnP(t) +!
g
P
#
(5.26)
the first term is referred to as transient chirp, which causes significant amount of
chirp during zero-one and one-zero transitions in digital optical systems.The sec-
ond term is referred to as adiabatic chirp, which causes frequency shifts between
the high- and low- power points in optical waveform.
5.5 Measurement of Frequency Chirping Character-
istics of Semiconductor Lasers
Since in an optical dispersive medium, phase or frequency modulations can
be converted into intensity modulation or in digital optical links, frequency
chirping leads to pulse spreading, it is important to characterize the amount of
coupling of intensity modulation to frequency modulation in a directly modu-
lated laser. There are dierent methods to characterize the frequency chirping of
lasers, and the most convenient method is to convert the frequency modulation
(FM) of optical beam to intensity modulation. Optical coherent discriminator
and propagation through dispersive fiber will be discussed here.
90
5.5.1 Dispersive Fiber Transfer Function
If the modulated output of a directly modulated laser is transmitted through
a single-mode fiber, the transmission of the optical field is described by the
propagation factor e
jL
, with the phase constant. Since the lasers are operated
near 1:5m regime, the optical fibers are dispersive in this wavelength range. The
phase constant,(!), then can be Taylor expanded around the lasing frequency
(!
th
= 2
th
)[Pet88].
(!) =
0
+(!!
th
) +
1
2
(
d
d!
)(!!
th
)
2
+::: (5.27)
with representing the delay per unit length, = d=d!, and the dispersion
term
d
d!
=
d
d
d
d!
=
d
d

2
2c
(5.28)
If the input and output optical fields of the fiber are expressed in terms of their
complex slowly varying electric field amplitudes
E
in;out
(t) =
p
P
in;out
(t)e
j(!
th
t+(t))
(5.29)
with the corresponding Fourier transforms,
˜
E
in;out
. Then the output electric field
is related to the input electric field through
˜
E
out
(!) = e
jL
˜
E
in
(!) = e
j
[

0
+(!!
th
)+
1
2
(
d
d!
)(!!
th
)
2
]
L
˜
E
in
(!) (5.30)
91
then it can be shown that the small-signal optical intensity and frequency mod-
ulations in the output of the fiber are described by the following matrix relations
to their values in the input of the fiber [WP92],
0
B
B
B
B
B
B
B
B
B
@

˜
P
out
( f )
˜
˙

out
( f )
1
C
C
C
C
C
C
C
C
C
A
=
0
B
B
B
B
B
B
B
B
B
B
B
B
@
cos

2
2
D
2
f
2

j
< S> sin

2
2
D
2
f
2

f
j
f sin

2
2
D
2
f
2

< S>
cos

2
2
D
2
f
2

1
C
C
C
C
C
C
C
C
C
C
C
C
A
0
B
B
B
B
B
B
B
B
B
@

˜
P
in
( f )
˜
˙

in
( f )
1
C
C
C
C
C
C
C
C
C
A
(5.31)
where D is defined [Pet88]
D =
r
d
d
L

2
2c
(5.32)
The output power modulations then can be described in terms of the input
power and frequency modulations as

˜
P
out
( f ) =
˜
P
in
( f ) cos

2
2
D
2
f
2

+ j
˜
˙

in
( f )< S> sin

2
2
D
2
f
2

f
=
2
6
6
6
6
4
cos

2
2
D
2
f
2

+ j
˜
˙

in
( f )

˜
P
in
( f )
< S> sin

2
2
D
2
f
2

f
3
7
7
7
7
5

˜
P
in
( f ) (5.33)
The fiber intensity modulation transfer function, equation 5.33, can be used
to characterize the intensity to frequency modulation characteristics of optical
sources. Figure 5.10 shows a 25 km long fiber transfer function with an optical
Mach-Zender as an optical intensity modulator. The intensity to frequency
conversion in the Mach-Zender modulator is characterized by a chirp factor,.
The solid line is the curve fit applied to the measured data to extract the chirping
factor of the modulator.
This method can be used the characterize the frequency modulation
imparted by intensity modulation in directly modulated semiconductor lasers
92
0 5 10 15 20 25
-40
-30
-20
-10
0
|H(f)|(dB)
Frequency (GHz)
| | = 0.735
Figure 5.10: Measured transfer function of a 25 km long dispersive optical fiber
using an externally modulated laser
[WP92, RBMH94]. Equations 5.25 and 5.26 show the relation between the fre-
quency modulation imparted by the intensity modulation of a directly modu-
lated semiconductor laser. Therefore, equation 5.33 can be cast in the following
form

˜
P
out
( f ) =
"
cos

2
2
D
2
f
2

1 j
!
g
2 f
!
sin

2
2
D
2
f
2

#

˜
P
in
( f ) (5.34)
The measurement setup and procedure are the same as in figure 5.2, with the
exception that before photodetection stage, the coupled light from the micro-
cavity lasers passes through a single mode dispersive optical fiber with dierent
lengths. The optical intensity modulation after the optical fiber,
˜
P
out
( f ) mea-
sured on a Network analyzer (HP 8510C) is then calibrated to the optical signal
modulation input to the optical fiber
˜
P
in
( f ). Figure 5.11 shows modulation
response of a 25 km long single mode optical fiber using a D
4
photonic crystal
defect cavity laser modulated output power.
93
2 4 6 8 10 12 14
-25
-20
-15
-10
-5
0
5
|H(f)| (dB)
Frequency (GHz)
Figure 5.11: Measured transfer function of a 25 km long dispersive optical fiber
using a D
4
photonic crystal defect cavity laser
The parameters,!
g
, and D (dispersion coecient of optical fibers) in equa-
tion 5.33 were fitted to the measured data using photonic crystal defect cavity
and microdisk lasers at varying operating points as the optical source feeding
power through the optical fiber. A least square curve fitting procedure was used
in Matlab to carry out the fitting procedure. The fitting procedure produced
poor agreement between the theory and the experiment. This discrepancy is
attributed to the fact that the simple laser model in equation 5.33 does not
describe all the details of FM response of lasers with quantum well active region
[PMY98]. Figure 5.12 shows the measured response of a 25 km long optical fiber
using a D
4
photonic crystal defect cavity laser, and a typical fit performed using
equation 5.34. Although at low frequencies, the measured response fits very
well to equation 5.34, at higher frequencies the response deviates significantly
from the theory. There have been more complete models that have been put
forth to take into account the eects of the modulation of the 3 D unconfined
carriers in the barrier layers of quantum well structures on the intensity and
94
frequency modulations of semiconductor lasers [NCB
+
97, RdRC
+
95]. Including
these models into the equation 5.34 results in more complicated fiber transfer
function. The measured data fits very well to such functions, however this
procedure fails to predict laser parameters independently.
2 4 6 8 10 12 14
-25
-20
-15
-10
-5
0
5
|H(f)| (dB)
Frequency (GHz)
Figure 5.12: Measured transfer function of a 25 km long dispersive optical fiber
using a D
4
photonic crystal defect cavity laser
5.5.2 Optical Coherent Discriminator Method
In this method a Mach-Zender modulator biased at the quadrature point is used
to convert the frequency chirping of the optical source into intensity modulation
[SCCH92]. Figure 5.13 shows a schematic representation of the interferometer
setup used for this measurement. The complex optical electric field of the
modulated laser source is expressed as
E(t) =
p
P(t)e
j(2
0
t+(t))
(5.35)
95
where
0
is the average laser frequency and(t) describes the frequency or phase
excursions away from the optical carrier. A sinusoidally modulated laser optical
power then is expressed
P(t) = P
0
h
1 + Re
n
˜ m( f )e
j2 f t
oi
(5.36)
and the optical phase modulation is given by
(t) = Re
h
˜
( f )e
i2 f t
i
(5.37)
Sending the modulated optical beam through the optical FM discriminator
(b)
I
+
q
I
d
I
0

2
0
2I
0
I
+
q
(a)
Figure 5.13: (a) Optical discriminator setup used to measure the AM and FM
response of microcavity lasers, (b) the quadrature points
setup, the resulting photodetected current on a high-speed detector, I
d
, is express
as
I
d
/ P(t) + P(t
0
) + 2
p
P(t)P(t
0
) cos((t) + 2
0

0
) (5.38)
(t) =(t)(t
0
) (5.39)
96
The phase dierence, equation 5.39, contains the information on frequency chirp-
ing of the optical source. When the interferometer is biased at positive and nega-
tive slope points of the transmission curve, figure 5.13(b), the complex quantities
˜
I
+
q
( f ) and
˜
I

q
( f ), are the phasor representation of the measured photocurrent at
frequency f . The optical phase modulation on the input optical signal can be
found by subtracting the measured photocurrent at the two biasing points.
˜
I
PM
( f ) =
˜
I
+
q
( f )
˜
I

q
( f ) 4jI
0
sin(2 f
0
)e
j2 f
0 ˜
( f ) (5.40)
The approximation sign is due to the fact that only the first term in Taylor
series expansion of sin((t)) is kept. The corresponding frequency modulation
equation can be derived from equation 5.40, using the following relation
˜
( f ) =
˜ ( f )
j f
(5.41)
then
˜
I
FM
( f ) = 4
0
I
0
sinc( f
0
)e
j2 f
0
˜ ( f ) (5.42)
If the dierential delay ,
0
, of the setup is such that f
0
1 for the frequency
range of interest, then sinc( f
0
) 1, and then measured dierential current, is
proportional to the frequency modulation transfer function of the optical source
under test. The frequency modulation response of microdisk lasers was mea-
sured using the method mentioned above. Figure 5.14(a) shows the measured
amplitude modulation of the microdisk laser, and figure 5.14 shows the fre-
quency modulation of the microdisk imparted by the amplitude modulation.
97
(b)
2 4 6 8 10 12 14
-60
-50
-40
-30
Power (dBm)
Frequency (GHz)
f
r
= 5.54 GHz
2 4 6 8 10 12 14
-20
-15
-10
-5
0
5
Power (dBm)
Frequency(GHz)
f
r
= 5.58 GHz
(a)
Figure 5.14: (a) Intensity modulation of a microdisk laser measured with inter-
ferometer setup, (b) frequency modulation of a microdisk laser measured with
the interferometer setup
98
References
[AA94] Masahiro Asada and Alfred . R. Adams. Handbook of Semiconductor
Lasers and Photonic Integrated Circuits. Chapman & Hall, London,
1994.
[AG91] Govind P . Agrawal and George R. Gray. Intensity and phase noise in
microcavity surface emitting semiconductor lasers. Applied Physics
Letters, 59(4):399–401, 1991.
[AGP
+
00] L. V . Asryan, N. A. Gun’ko, A. S. Polkovnikov, G. G. Zegrya,
R. A. Suris P . K Lau, and T Makino. Threshold characteristics
of ingaasp/inp multiple quantum well lasers. Semiconductor Science
Technology, 15(12):1131–1140, December 2000.
[BIK
+
05] A. F. Benner, M. Ignatowski, J. A. Kash, D. M. Kuchta, and M. B.
Ritter. Exploitation of optical interconnects in future server archi-
tectures. IBM Journal of Research and Developement, 49(4/5):755–775,
2005.
[BKY92] Gunnar Bj¨ ork, Anders Karlsson, and Yoshihisa Yamamoto. On the
linewidth of microcavity lasers. Applied Physics Letters, 60(3):304–
306, 1992.
[BKY94] Gunnar Bjørk, Anders Karlsson, and Yoshihisa Yamamoto. Defini-
tion of a laser threshold. Physical Review A, 50(2):1675–1680, 1994.
[BP02] Heinz-Peter Breuer and Francesco Petruccione. The Theory of Open
Quantum Systems. Oxford University Press, USA, 2002.
[BS03] T. Baba and D. Sano. Low-threshold lasing and purcell eect in
microdisk lasers at room temperature. Selected Topics in Quantum
Electronics, IEEE Journal of, 9(5):1340–1346, Sept.-Oct. 2003.
[BSN
+
04] Toshihiko Baba, Daisuke Sano, Kengo Nozaki, Kyoji Inoshita,
Yusuke Kuroki, and Fumio Koyama. Observation of fast
spontaneous emission decay in gainasp photonic crystal point
defect nanocavity at room temperature. Applied Physics Letters,
85(18):3989–3991, 2004.
99
[BSW
+
06] Mahmood Bagheri, M. H. Shih, Zhi-Jian Wei, S. J. Choi, John D.
OBrien, P . Daniel Dapkus, and W. K. Marshall. Linewidth and mod-
ulation response of two-dimensional microcavity photonic crystal
lattice defect lasers. IEEE Photonics Technology Letters, 18(10):1161–
1163, May 2006.
[Cao04] Jiang-Rong ”Frank” Cao. Photonic Crystal Microcavity Lasers and
Laser Arrays. PhD thesis, University of Southern California, 2004.
[CC95a] Larry A. Coldren and Scott W. Corzine. Diode Lasers and Photonic
Integrated Circuits, chapter Dynamic Eects. John Wiley and Son,
Inc., 1995.
[CC95b] Larry A. Coldren and Scott W. Corzine. Diode Lasers and Photonic
Integrated Circuits, chapter Gain and Current Relations. John Wiley
and Son, Inc., 1995.
[CDCD03] S. J. Choi, K. D. Djordjev, S. J. Choi, and P . D. Dapkus. Microdisk
lasers vertically coupled to output waveguides. IEEE Photonic Tech-
nology Letters, 10(15):1330–1132, 2003.
[CFC
+
97] Datong Chen, Harold R. Fetterman, Antao Chen, William H. Steier,
Larry R. Dalton, Wenshen Wang, and Yongqiang Shi. Demonstra-
tion of 110 GHz electro-optic polymer modulators. Applied Physics
Letters, 70(25):3335–3337, 1997.
[Chu95] Shun Lien Chuang. Physics of Optoelectronic Devices. John Wiley &
Sons Inc., 1995.
[Con07] Conference on Laser and ElectroOptics(CLEO). 60 microWatts of
Fiber-Coupled Peak Output Power from an Edge-Emitting Photonic Crys-
tal Heterostructure Laser, Baltimore, Maryland, 2007.
[CRRR
+
07] J. Van Campenhout, P . Rojo-Romeo, P . Regreny, C. Seassal, D. Van
Thourhout, S. Verstuyft, L. Di Cioccio, J.-M. Fedeli, C. Lagahe,
and R. Baets. Electrically pumped inp-based microdisk lasers inte-
grated with a nanophotonic silicon-on-insulator waveguide circuit.
OPTICS EXPRESS, 15(11):6744–6749, 2007.
[CTDRG89] Claude Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert Gryn-
berg. Photons and Atoms, Introduction Quantum Electrodynamics. John
Wiley and Sons Inc., 1989.
[CTLDL92] Claude Cohen-Tannoudji, Frank Laloe, Bernard Diu, and Franck
Laloe. Quantum Mechanics, volume I. John Wiley and Sons Inc.,
1992.
100
[Der98] D. Derickson. Fiber Optic Test and Measurement. Upper Saddle River,
N.J., Prentice Hall PTR, 1998.
[DSN90] M. O. Van Deventer, P . Spano, and S. K. Nielsen. Comparison of
DFB laser linewidth measurement techniques results from cost 215
round robin. Electronics Letters, 26(24):2018–2020, 22 Nov 1990.
[DSTP83] B. Daino, P . Spano, M. Tamburrini, and S. Piazzolla. Phase noise
and spectral line shape in semiconductor lasers. IEEE Journal of
Quantum Electronics, QE-19(3):266–270, March 1983.
[EFW
+
05] Dirk Englund, David Fattal, Edo Waks, Glenn Solomon,
Bingyang Zhang, Toshihiro Nakaoka, Yasuhiko Arakawa, Yoshi-
hisa Yamamoto, and Jelena Vuckovic. Controlling the spontaneous
emission rate of single quantum dots in a 2d photonic crystal. Phys-
ical Review Letters, 95(013904), 2005.
[FM81] Mark W. Fleming and Aram Mooradian. Fundamental line broad-
ening of single-mode (GaAl)As diode lasers. Appllied Physic Letters,
38:511–513, 1981.
[FTL06] T. Fordell, Z. Toano, and . M. Lindberg. A vertical-cavity surface-
emitting laser at threshold. IEEE Photonics Technology Letters,
18(21):2263–2265, Nov.1, 2006.
[FUB01] M. Fujita, R. Ushigome, and T. Baba. Large spontaneous emission
factor of 0.1 in a microdisk injection laser. IEEE Photonic Technology
Letters, 13(5):403–405, 2001.
[FXY
+
02] W. Fang, J. Y. Xu, A. Yamilov, H. Cao, Y. Ma, S. T. Ho, and G. S.
Solomon. Large enhancement of spontaneous emission rates of inas
quantum dots in gaas microdisks. Opt. Lett., 27(11):948–950, 2002.
[GDG
+
95] Franc ¸ois Girardin, Guang-Hua Duan, Philippe Gallion, Anne Tal-
neau, and Abdallah Ougazzaden. Experimental investigation of the
relative importance of carrier heating and spectral-hole-burning on
nonlinear gain in bulk and strained multi-quantum-well 1.55m
lasers. Applied Physics Letters, 67(6):771–773, 1995.
[GM95] Elena V . Goldstein and Pierre Meystre. Spontaneous Emission and
Laser Oscillation in Microcavities, chapter Spontaneous Emission in
Optical Cavities: A Tutorial Review. CRC Press, 1995.
[GSG
+
98] J. M. G´ erard, B. Sermage, B. Gayral, B. Legrand, E. Costard,
and V . Thierry-Mieg. Enhanced spontaneous emission by quan-
tum boxes in a monolithic optical microcavity. Phys. Rev. Lett.,
81(5):1110–1113, Aug 1998.
101
[GWG72] H. Gerhardt, H. Welling, and A. G ¨ uttner. Measurements of the
laser linewidth due to quantum phase and quantum amplitude
noise above and below threshold. Zeitschrift f¨ ur Physik, 253:113–
126, 1972.
[HBW
+
07] K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atature,
S. Gulde, S. Falt, E. L. Hu, and A. Imamoglu. Quantum nature
of a strongly coupled single quantum dot-cavity system. Nature,
445:896–899, 2007.
[Hen82] Charles Henry. Theory of the linewidth of semiconductor lasers.
IEEE Journal of Quantum Electronics, 18:259–264, 1982.
[Hen86a] C. Henry. Theory of spontaneous emission noise in open resonators
and its application to lasers and optical amplifiers. Journal of Light-
wave Technology, 4(3):288–297, Mar 1986.
[Hen86b] Charles H. Henry. Phase noise in semiconductor lasers. Journal of
Lightwave Technology, LT-4(3):298–311, 1986.
[Hen91] C. Henry. Coherence, Amplification, and Quantum Eects in Semi-
conductor Lasers, chapter Linewidth Broadening in Semiconductor
Lasers. John Wiley & Sons, Inc., 1991.
[HL67] Robert D. Hempstead and Melvin Lax. Classical noise. VI. noise in
self-sustained oscillator near threshold. Physical Review, 161:350–
361, 1967.
[HZK
+
90] Ch. S. Harder, B. J. Van Zeghbroeck, M. P . Kesler, H. P . Meier,
P . Vettiger, D. J. Webb, and P . Wolf. High-speed GaAs/AlGaAs
optoelectronic devices for computer applications. IBM Journal of
Research and Development, 34:568–584, 1990.
[IEE06] IEEE/OSA Optical Fiber Communication (OFC). A 17-Gb/s low-
power optical receiver using a Ge-on-SOI photodiode with a 0.13-m
CMOS IC, Anahiem, California, 2006.
[ITR] International Technology Roadmap for Semiconductor, 2005 edition.
[Jr.93] Peter S. Zory Jr. Quantum Well Lasers. Academic Press, Inc., 1993.
[KB84] T.L. Koch and J.E. Bowers. Nature of wavelength chirping in
directly modulated semiconductor lasers. Electronics Letters, 20(25-
26):1038–1040, 1984.
[KI87] M. P . Kesler and E. P . Ippen. Subpicosecond gain dynamics in
GaAlAs laser diodes. Applied Physics Letters, 51:17651767, 1987.
102
[KJC95] S. W. Koch, F. Jahnke, and W. W. Chow. Physics of semiconductor
microcavity lasers. Semiconductor Science Technology, 10:739–751,
1995.
[KL86] T. L. Koch and R. A. Linke. Eect of nonlinear gain reduction on
semiconductor laser wavelength chirping. Applied Physics Letters,
48(10):613–615, 1986.
[Kle81] Daniel Kleppner. Inhibited spontaneous emission. Phys. Rev. Lett.,
47(4):233–236, Jul 1981.
[LAP
+
04] Brian E. Lemo, Mohammed E. Ali, George Panotopoulos, Gra-
ham, M. Flower, B. Madhavan, A. F. J. Levi, and David W.
Dolfi. Maui: Enabling fiber-to-the-processor with parallel multi-
wavelength optical interconnects. Journal OF Lightwave Technology,
22:2043–2054, 2004.
[Lau93] Kam. Y. Lau. Quantum Well Lasers, chapter Dynamics of Quantum
Well Lasers, pages 217–272. Academic Press, Inc., 1993.
[Lax66] Melvin Lax. Quantum noise. iv. quantum theory of noise sources.
Physcal Review, 145(1):110–129, May 1966.
[LHG
+
06] Xian-Shu Luo, Yong-Zhen Huang, Wei-Hua Guo, Qin Chen, Miao-
Qing Wang, and Li-Juan Yu. Investigation of mode characteris-
tics for microdisk resonators by s-matrix and three-dimensional
finite-dierence time-domain technique. Journal of Optical Society of
America B, 23(B):1068–1073, 2006.
[Lid06] Daniel Lidar. Theory of Open Quantum Systems, Class Notes. Univer-
sity of Southern California, 2006.
[Liu07] High-Speed Silicon Modulator for Future VLSI Interconnect. Optical
Society of America, 2007.
[Mat96] Dapkus P .D. Mathur, A. Fabrication, characterization and analy-
sis of low threshold current density 1.55-m-strained quantum-well
lasers. Quantum Electronics, IEEE Journal of, 32(2):222–226, Feb 1996.
[McC66] D. E. McCumber. Intensity fluctuations in the output of cw laser
oscillators. i. Physcal Review, 141(1):306–322, Jan 1966.
[MI99] P . Meytre and M. Sargent III. Elements of Quantum Optics. Springer,
1999.
[NCB
+
97] C.-Z. Ning, W.W. Chow, D.J. Bossert, R.A. Indik, and J.V . Moloney.
Influences of unconfined states on the optical properties of
quantum-well structures. IEEE Journal of Selected Topics in Quan-
tum Electronics, 3(2):129–135, Apr 1997.
103
[NFI
+
92] R. Nagarajan, T. Fukushima, M. Ishikawa, J. E. Bowers, R. S. Geels,
and L. A. Coldren. Transport limits in high-speed quantum-well
lasers: experiment and theory. IEEE Photonics Technology Letters,
4(2):121–123, 1992.
[NIGB92] R. Nagarajan, M. Ishikawa, T. Fukushima R. Geels, and J. E. Bowers.
High speed quantum well lasers and carrier transport eects. IEEE
Journal of Quantum Electronics, 28(10):1990–2008, 1992.
[NKB07] Kengo Nozaki, Shota Kita, and Toshihiko Baba. Room temperature
continuous wave operation and controlled spontaneous emission
in ultrasmall photonic crystal nanolaser. Optics Express, 15:7506–
7514, 2007.
[Nod06] Susumu Noda. Recent progress and future prospects of two- and
three-dimensional photonic crystals. Journal of Lightwave Technology,
24(12):4554–4567, 2006.
[OHLP87] R. Olshansky, P . Hill, V . Lanzisera, and W. Powazinik. Frequency
response of 1.3m InGaAsP high speed semiconductor lasers. IEEE
Journal of Quantum Electronics, 23(9):1410–1418, 1987.
[Pet88] K. Petermann. Laser Diode Modulation and Noise, chapter Noise Char-
acteristics of Solitaroy Laser Diodes. Kluwer Academic Publishers,
1988.
[Pet91] K. Petermann. Laser Diode Modulation and Noise, chapter Frequency
Modulation Characteristics of Semiconductor Lasers. Kluwer Sci-
entific Publishers, 1991.
[PMY98] E. Peral, W. K. Marshall, and A. Yariv. Precise measurement of
semiconductor laser chirp using eect of propagation in dispersive
fiber and application to simulation of transmission through fiber
gratings. Journal of Lightwave Technology, 16(10):1874–1880, Oct 1998.
[Pur46] E. M. Purcell. Spontaneous emission probablities at radio frequen-
cies. Physics Review, 69:681, 1946.
[RBMH94] A. Røyset, L. Bjerkan, D. Myhre, and L. Hafskjaer. Use of dispersive
optical fibre for characterisation of chirp in semiconductor lasers.
Electronics Letters, 30(9):710–712, 29 Apr 1994.
[RdRC
+
95] R. F. S. Ribeiro, J. R. F. da Rocha, A. V . T. Cartaxo, H. J. A. da Silva,
B. Franz, and B. Wedding. Fm response of quantum-well lasers tak-
ing into account carrier transport eects. IEEE Photonics Technology
Letters, 7(8):857–859, Aug 1995.
104
[RKCV05] H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala.
Radiation-pressure-driven micro-mechanical oscillator. Optics
Express, 13(14):5293–5301, 2005.
[RSK
+
91] W. Rideout, W.F. Sharfin, E.S. Koteles, M.O. Vassell, and B. Elman.
Well-barrier hole burning in quantum well lasers. IEEE Photonics
Technology Letters, 3(9):784–786, Sept. 1991.
[RV05] H. Rohksari and K. J. Vahala. Observation of kerr nonlinearity in
microcavities at room temperature. Optics Express, 30(4):427–429,
2005.
[SCCH92] W.V . Sorin, K.W. Chang, G.A. Conrad, and P .R. Hernday. Frequency
domain analysis of an optical fm discriminator. Journal of Lightwave
Technology, 10(6):787–793, Jun 1992.
[Shi06] Min-Hsiung Shih. Photonic Crystal Devices for Photonic Integrated
Circuits. PhD thesis, University of Southern California, 2006.
[SKY
+
06] M.H. Shih, Wan Kuang, Tian Yang, M. Bagheri, Zhi-Jian Wei, Sang-
Jun Choi, Ling Lu, J.D. O’Brien, and P .D. Dapkus. Experimental
characterization of the optical loss of sapphire-bonded photonic
crystal laser cavities. Photonics Technology Letters, IEEE, 18(3):535–
537, Feb. 1, 2006.
[SMV
+
01] P . Signoret, F. Marin, S. Viciani, G. Belleville, M. Myara, J. P . Tour-
renc, B. Orsal, A. Plais, F. Gaborit, and J. Jacquet. 3.6-mhz linewidth
1:55m monomode vertical-cavity surface-emitting laser. IEEE
Photonics Technology Letters, 13(4):269–271, April 2001.
[SNAA05] B.-S. Song, S. Noda, T. Asano, and Y. Akahane. Ultra-high-Q pho-
tonic double-heterostructure nanocavity. Nature Materials, 4:207–
210, March 2005.
[ST58] A. L. Schawlow and C. H. Townes. Infrared and optical masers.
Physical Review, 112(6):1940–1949, Dec 1958.
[TA91] T. Takahashi and Y. Arakawa. Nonlinear gain eects in quantum
well, quantum well wire, and quantum well box lasers. IEEE Journal
of Quantum Electronics, 27(6):1824–1829, June 1991.
[TL01] S. M. K. Thiyagarajan and A. F. J. Levi. Dynamic behavior of scaled
microdisk lasers. Solid-State Electronics, 45:1821–1826, 2001.
[TLL
+
67] S. M. K. Thiyagarajan, A. F. J. Levi, C. K. Lin, I. Kim, P . D. Dap-
kus, and S. J. Pearton. Continuous room-temperature operation
of optically pumped InGaAs/InGaAsP microdisk lasers. Electronics
Letters, 29, 1666-1667.
105
[Tof97] Zeno Toano. Investigation of threshold transition in semiconduc-
tor lasers. IEEE Journal of Selected Topics in Quantum Electronics,
3(2):485–490, 1997.
[Uji95] Kikuo Ujihara. Spontaneous Emission and Laser Oscillation in Micro-
cavities, chapter Eects of Atomic Broadening on Spontaneous
Emission in an Optical Microcavity. CRC Press, 1995.
[Vah03] K. J. Vahala. Optical microcavities. Nature, 424(6959):839–846, 2003.
[VFS
+
03] J. Vuckovic, D. Fattal, C. Santori, G. S. Solomon, and Y. Yamamoto.
Enhanced single-photon emission from a quantum. dot in a micro-
post microcavity. Applied Physics Letters, 82(15):2374–76, 2003.
[VY83a] Kerry Vahala and Amnon Yariv. Semiclassical theory of noise in
semiconductor lasers- part I. IEEE Journal of Quantum Electronics,
QE-19(6):1096–1101, 1983.
[VY83b] Kerry Vahala and Amnon Yariv. Semiclassical theory of noise in
semiconductor lasers- part II. IEEE Journal of Quantum Electronics,
QE-19(6):1102–1109, 1983.
[WLJ
+
06] K. Wada, J. F. Liu, S. Jongthammanurak, D. D. Cannon, D. T. Daniel-
son, D. H. Ahn, S. Akiyama, M. Popovic, D. R. Lim, K. K. Lee,
H. C. Luan, Y. Ishikawa, X. Duan, J. Michel, H. A. Huas, and L. C.
Kimerling. Optical Interconnects The Silicon Approach, chapter Si
Microphotonics for Optical Interconnection. Springer, 2006.
[WP92] J. Wang and K. Petermann. Small signal analysis for dispersive opti-
cal fiber communication systems. Journal of Lightwave Technology,
10(1):96–100, Jan 1992.
[WS96] Jian Wang and Heinz Schweizer. Nonlinear gain suppression
in quantum-well lasers due to carrier heating: The roles of car-
rier energy relaxation, electron–hole interaction, and auger eectg.
IEEE Photonic Technology Letters, 8:1441–1443, 1996.
[WUM
+
91] M. Willatzen, A. Uskov, J. Mrk, H. Olesen, B. Tromborg, and A.-P .
Jauho. Nonlinear gain suppression in semiconductor lasers due to
carrier heating. IEEE Photonic Technology Letters, 3:606–609, 1991.
[Yab87] Eli Yablonovitch. Inhibited spontaneous emission in solid-state
physics and electronics. Phys. Rev. Lett., 58(20):2059–2062, May
1987.
[Yam94] Minoru Yamada. Handbook of Semiconductor Lasers and Photonic Inte-
grated Circuits. Chapman & Hall, London, 1994.
106
[Yar84] Amnon Yariv. The beginning of integrated optoelectronic circuits.
IEEE Transactions on Electron Devices, Ed-31:1656–1661, 1984.
[YB89] H. Yokoyama and S. D. Brorson. Rate equation analysis of micro-
cavity lasers. Journal of Applied Physics, 66(10):4801–4805, 1989.
[YMB91] Y. Yamamoto, S. Machida, and G. Bj¨ ork. Microcavity semiconductor
laser with enhanced spontaneous emission. Phys. Rev. A, 44(1):657–
668, Jul 1991.
[YSH
+
04] T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs,
G. Rupper, C. Ell, and O. B. Shchekin D. G. Deppe. Vacuum rabi
splitting with a single quantum dot in a photonic crystal nanocavity.
Nature, 432:200–203, 2004.
[YYY
+
93] T. Yamanaka, Y. Yoshikuni, K. Yokoyama, W. Lui, and S. Seki. The-
oretical study on enhanced dierential gain and extremely reduced
linewidth enhancement factor in quantum-well lasers. IEEE Journal
of Quantum Electronics, 29(6):1609–1616, Jun 1993.
[ZBQ01] Weidong Zhou, Pallab Bhattacharya, and Omar Qasaimeh. InP-
based cylindrical microcavity light-emitting diodes. IEEE Journal of
Quantum Electronics, 37:48–54, 2001.
107
Appendix A
Scattering Matrix Approach to
Resonant Mode and Q Value
Calculation of Microdisk
One of the most powerful and versatile numerical techniques used in photonic
simulation is the finite-dierence time domain method. A disadvantage of
this technique is that smooth geometry of the cavity has to be mapped onto
a discrete grid with a very small lattice constant. This makes this technique
inecient in terms of both computational power and memory. Other numerical
methods are required in analysis of photonic devices that can be fast, eective
and numerically accurate to analyze photonic devices.
The scattering matrix is a powerful tool for investigating the response of a
physical system to an incoming wave. It has been widely used in the analysis
of waveguides and resonant cavities in optics in cases where an analytical or
semi-analytical solution was available.
z
n
4
n
2
n
1
n
1
n
3
2R
L
d
Figure A.1: Schematic representation of a microdisk waveguide structure
108
Figure A.1 shows a schematic representation of the microdisk wavegiude
system. R, d, and n
1
are the raduis, thickness, and refrective index of the
microdisk, and n
2
, n
3
, and n
4
are the refractive indices of the upper and lower
confining layers and the external region

2
.

1
is assumed to be finite along
z direction, and its length is taken to be L. A periodic boundary condition
is applied along this direction to take into account the infinite extents along
z. The electromagnetic field of an optical wave associated with the microdisk
waveguide system in the cylindrical coordinate system is as follows:
8
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
:
H
z
= k
2
0
(z)g +
@
2
g
@z
2
;
H

=
@
2
g
@@z
+
k
0

@ f
@
;
H

=
@
2
g
@@z
k
0
@ f
@
(A.1)
8
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
:
E
z
= i!
0
( f +
1
k
2
0
(z)
@
2
f
@z
2
)k
0
;
E

= i!
0
(
k
0
k
2
0
(z)
@
2
@@z
+
@g
@
);
E

= i!
0
(
k
0
k
2
0
(z)
@
2
@@z

@g
@
)
(A.2)
The slab waveguide is assumed to support only the first order TE and TM
modes, so the field components tangential to the circumference of the disk can
be expressed as:
E
z
= Ag
z
(z)J

()J
1

(
R); (A.3)
H
z
= B f
z
(z)J

(
)J
1

(
R); (A.4)
E

= A
i

2

g
0
z
(z)J

()J
1

(R) + B
ik

f
z
(z)J
0

(
)J
1

(
R); (A.5)
H

= A
ik

g
z
(z)(z)J
0

()J
1

(R) + B
i

2

f
0
z
(z)J

(
)J
1

(
R) (A.6)
109
where J

(x) is J-type Bessel function of order, k is the wavenumber in vacuum,z
is the relative permittivity in region

1
along z direction, and prime denotes
derivative with respect to the arguments. The wavefunctions g
z
(z) and f
z
(z)
are the fundamental TM and TE modes of the slab waveguide, satisfying the
following equations:
d
dz
1
(z)
d(z)g
z
(z)
dz
+ k
2
(z)g
z
(z) =
2
g
z
(z); (A.7)
d
2
f
z
(z)
dz
2
+ k
2
(z)g
z
(z) =
2
g
z
(z) (A.8)
Due to the periodic boundary condition along the z direction, the following
set of functions forms a complete orthonormal set long z in

1
jmi =
exp(ip
m
z)
p
L
; p
m
=
2m
L
; m = 0;1;2;::: (A.9)
Using the completeness of the basisjmi, equation A.3-A.6 are multiplied byhmj,
and then integrated from 0 to L to arrive at the following
e
z;m
=hmjE
z
i = Ag
m
J

()J
1

(R); (A.10)
h
z;m
=hmjH
z
i = B f
m
J

(
)J
1

(
R); (A.11)
e
;m
=hmjE

i = A
i

2

J

()J
1

(R)(ip
m
g
m
)
+ B
ik

J
0

(
)J
1

(
R) f
m
; (A.12)
h
;m
=hmjH

i = A
ik

J
0

()J
1

(R)s
m
+ B
i

2

J

(
)J
1

(
R)(ip
m
f
m
) (A.13)
s
m
=
1
X
n=1
hmj(z)jnig
n
(A.14)
110
In region

2
the fields also can be expressed as a sum of incident and outward
traveling TE and TM modes
e
z;m
= C
in
m
H
(1)

(q
m
)
H
(1)

(q
m
R)
+ C
out
m
H
(2)

(q
m
)
H
(2)

(q
m
R)
; (A.15)
h
z;m
= D
in
m
H
(1)

(q
m
)
H
(1)

(q
m
R)
+ D
out
m
H
(2)

(q
m
)
H
(2)

(q
m
R)
; (A.16)
e
;m
=
p
m
q
2
m

C
in
m
H
(1)

(q
m
)
H
(1)

(q
m
R)
+ C
out
m
H
(2)

(q
m
)
H
(2)

(q
m
R)

+
ik
q
m

D
in
m
H
(1)
0

(q
m
)
H
(1)

(q
m
R)
+ D
out
m
H
(2)
0

(q
m
)
H
(2)

(q
m
R)

; (A.17)
h
;m
=
ik
q
m

C
in
m
H
(1)
0

(q
m
)
H
(1)

(q
m
R)
+ C
out
m
H
(2)
0

(q
m
)
H
(2)

(q
m
R)

+
ik
q
m

D
in
m
H
(1)

(q
m
)
H
(1)

(q
m
R)
+ D
out
m
H
(2)

(q
m
)
H
(2)

(q
m
R)

(A.18)
where H
(1)

(x) and H
(2)

are the first- and the second-kind Hanckel functions
of order , respectively, and q
m
=
p
k
2
p
2
m
. Enforcing the continuity of the
tangential components of electric and magnetic field at the circumference of the
disk, relations between coecients A, B, C
in
m
, C
out
m
, D
in
m
and D
out
m
are established.
After eliminating the coecients A and B, the corresponding scattering matrix
can be set up as
0
B
B
B
B
B
B
B
B
B
@
C
out
i
D
out
i
1
C
C
C
C
C
C
C
C
C
A
=
0
B
B
B
B
B
B
B
B
B
@
O
11
(i; j) O
12
(i; j)
O
21
(i; j) O
22
(i; j)
1
C
C
C
C
C
C
C
C
C
A
0
B
B
B
B
B
B
B
B
B
@
C
in
j
D
in
j
1
C
C
C
C
C
C
C
C
C
A
= S(k)
0
B
B
B
B
B
B
B
B
B
@
C
in
j
D
in
j
1
C
C
C
C
C
C
C
C
C
A
(A.19)
Since only real valued q
m
s form traveling waves in

2
, m assumes values such
thatjp
m
j < k. Then i,j are integers that run between -kL=2 and kL=2. The
scattering matrix is unitary and its determinant can be expressed as a phase,
111
det(S(k))=exp(i). The resonant modes of the cavity can be found from the
Wigner delay time(k), that is defines as
(k) =
2d
dk
(A.20)
The total phase of det(S(k)) goes through 2 around the resonant mode, and
Lorentzian peaks would apear in(k) spectrum. The resonant mode wavelength
can be calculated from the central wavenumber of the Lorentzian peak, and its
quality factor Q is the ration between k
center
and the FWHM k of the peak
=
2
k
center
;Q =
k
center
k
(A.21)
Microdisks with the vertical structure consisting of a 240 nm thick InGaAsP
membrane sandwiched between air and sapphire as top and bottom cladding
layers with the refractive indices of air, InGaAsP , and sapphire taken to be 1, 3.4
and 1.7 respectively, are considered in this section. Equations A.7,A.8 are solved
numerically to find g
z
(z) and f
z
(z), and then the coecients g
m
, f
m
, and s
m
were
calculated using the appropriate integrals. These coecients were then used
to construct the appropriate scattering matrix S(k) using equation A.19. Figure
A.2(a) shows the calculated Wigner delay time for Mode HE
7;1
of a 1m radius
microdisk, and figure A.2(b) shows a zoomed in view of the peak marked (1) in
figure A.2(a) with a Lorentzian curve fit (solid curve) to extract the quality factor
and the center wavelength. The calculated quality factor of this mode is 28,000.
Luo et al.[LHG
+
06] compares this method with the more standard finite
dierence time domain (FDTD) method, and finds very close agreement between
these two methods.
112
40.45 40.50 40.55 40.60 40.65
0
1x10
6
2x10
6
3x10
6
4x10
6
5x10
6
6x10
6
7x10
6
8x10
6
(a)
40.528 40.529 40.530 40.531 40.532 40.533 40.534
5.0x10
6
1.0x10
7
1.5x10
7
2.0x10
7
2.5x10
7
3.0x10
7
3.5x10
7

(k)
k ( 2 / )
k
cen ter
= 40.5 cm
-1

Q = 28,000

k ( 2 / )

(k)
(1)
(b)
Figure A.2: (a) Wigner time delay spectrum of mode HE
7;1
of a 1m microdisk
on Sapphire substrate, (b) zoomed in view of a the peak labeled (1) in (a) with
a Lorentzian curve fit (solid curve) to extract the quality factor and the center
wavelength
113
Appendix B
Master Equation
B.1 Master Equation
In section 3.3 a master equation for a two level system coupled to an electromag-
netic mode in the presence of optical mode loss and carrier loss due dephasing
and other mechanisms was used to derive the Purcell factor. In this section,
a brief derivation of the dynamics of a system coupled to a reservoir is given
using the language of density matrix operators [CTLDL92]. This approach is
then used to derive the spontaneous decay of excited atomic levels to the ground
state when the coupling of the carrier to its reservoir results in broadening of
the atomic transition spectrum. The system and its reservoir are depicted in
figure B.1. H
S
, H
R
, and H
SR
correspond to the Hamiltonian of the system, the
reservoir, and the interaction Hamiltonian between the system and the reservoir.
is the density matrix of the system, and W is the density matrix of the coupled
system + reservoir [Uji95].
H
S-R Reservoir
H
R
System
H
S
W
Figure B.1: System interacting with its reservoir
114
The system is described by H
S
=
P
i
~!
i
a
y
i
a
i
, where a
y
i
and a
i
are the creation
and annihilation operators of the i
th
state of the system, respectively. ~
P
k
V
k
B
k
where V
k
= a
y
i
a
j
and B
k
= B
ij
is the interaction Hamiltonian between the system
and its reservoir. The operator V
k
describes the change in the state of the system
from j to i, and the operator B
k
is the operator from the reservoir that causes the
transition of the element of the system from state j to i.
Time evolution of the total system+reservoir in interaction picture (
˜
W) is
governed by the following equation of motion [CTDRG89]
d
dt
˜
W =
i
~
[
˜
H
SR
;
˜
W] (B.1)
where
˜
W = e
(iH
0
t=~)
We
(iH
0
t=~)
(B.2)
and
˜
H
SR
= e
(iH
0
t=~)
H
SR
e
(iH
0
t=~)
=~
X
k
V
k
(t)B
k
(t)
=~
X
i;j
a
y
i
a
j
e
(i!
ij
t)
B
ij
(t) =~
X
k
V
k
B
k
(t)e
(i
k
)
(B.3)
and H
0
= H
S
+ H
R
is the free motion Hamiltonian.
Equation B.1 is solved by successive integration, and keeping only to the
second order in
˜
H
SR
˜
W(t) = W
0

i
~
Z
t
0
dt
0
h
˜
H
SR
; W
0
i
dt
0
+

i
~

2
Z
t
0
dt
0
Z
t
0
dt
00
h
˜
H
SR
(t
0
);
h
˜
H
SR
(t
00
); W
0
ii
(B.4)
115
The system density matrix,, is obtained by tracing over the reservoir states
˜ (t) = Tr
R
˜
W. Then equation B.4 becomes
˜ (t) =(0)
Z
t
0
dt
0
Z
t
0
dt
00
X
k;k
0
fV
k
(t
0
)V
k
0(t
00
)(0)Tr
R

B
k
(t
0
)B
k
0(t
00
)
R
(0)

+ V
k
(t
0
)(0)V
k
0(t
00
)Tr
R

B
k
(t
0
)
R
(0)B
k
0(t
00
)

+ V
k
(t
00
)(0)V
k
0(t
0
)Tr
R

B
k
(t
00
)
R
(0)B
k
0(t
0
)

(0)V
k
(t
00
)V
k
0(t
0
)Tr
R

R
(0)B
k
(t
00
)B
k
0(t
0
)

g
(B.5)
In this derivation, the second term in equation B.4 is assumed to be zero.
This is justified because this term results in constant shift in system energy
levels [Lid06]. After a coarse-grained time derivative of the density operator,
, and Born-Markov assumption [BP02, Uji95], the following equation can be
derived for the time derivative of the density operator at a general time t,
d(t)
dt
=
X
k;k
0
[ V
k
V
k
0 ˜ (t)A
kk
0 V
k
˜ (t)V
k
0A
0
kk
0
V
0
k
˜ (t)V
k
A
kk
0 + ˜ (t)V
k
0V
k
A
0
kk
0
]; (B.6)
where A
0
kk
0
are constants that depend on the correlations of reservoir operators B
k
,
and are a function of reservoir temperature. Going back to using the Schr¨ odinger
picture, the equation of motion for the density matrix operator is
d
dt
=
i
~

H
S
;

+

@
@t
!
incoh;atom
; (B.7)
116
where

@
@t
!
incoh;atom
=
X
i
X
j
f
h
a
y
i
a
j

a
y
j
a
i

a
y
j
a
j

i
A
jiij
+
h
a
y
i
a
j

a
y
j
a
i

a
y
j
a
j
i
A

jiij
g (B.8)
B.2 Density Matrix Operator Elements of a Two
Level System Interacting with Its Reservoir
As was mentioned in section B.1, the equation of motion of a two level atom
coupled to an optical mode in a cavity interacting with the atom reservoir is
given by
d
dt
=
i
~

H
S
;

+

@
@t
!
incoh;atom
; (B.9)
where
H
s
=
X
mj
~!
j
a
y
j
a
j
+~!
mj
a
y
m2
a
m2
+~
X
mj

g
12mj
a
y
j
a
y
m1
a
m2
+ g
21mj
a
y
m2
a
m1
a
j

; (B.10)
and

@
@t
!
incoh;atom
=
X
mi
X
mj
f
h
a
y
mi
a
mj

a
y
mj
a
mi

a
y
mj
a
mj

i
A
jiijm
+
h
a
y
mi
a
mj

a
y
mj
a
mi

a
y
mj
a
mj
i
A

jiijm
g (B.11)
117
The optical field operators a
y
j
and a
j
create or annihilate photons inside the
j
th
mode with following rules
a
y
j
jni
j
=
p
n + 1jn + 1i
j
; (n 0) (B.12)
a
j
jni
j
=
p
njn 1i
j
; (n 1) (B.13)
a
y
j
j0i
j
= 0; (B.14)
a
y
j
a
j
jni = njni: (n 0) (B.15)
The creation and annihilation of the upper and lower states, a
y
m1;2
and a
m1;2
, of
the atomic system also operate on the atomic system states
a
y
mi
a
mj
jk
m
i =
jk
ji
m
i; (i; j; k = 1; 2) (B.16)
Since in section B.1, only single excitations in the system are considered, the
elements of the density matrix operator are those which are formed between
those states with one or zero excitation
00
,
mjmj
,
0mj
,
mj0
, and
0
0
m0
0
m
.
d
00
dt
=
i
~
h0j[H
s
;]j0i +h0j

@
@t
!
incoh;atom
j0i (B.17)

i
~
h0j(H
s
H
s
)j0i = ih0j
X
mj

g
12mj
a
y
mj
a
y
m1
a
m2
+ g
21mj
a
y
m2
a
m1
a
j

g
12mj
a
y
mj
a
y
m1
a
m2
+ g
21mj
a
y
m2
a
m1
a
j

j0i
=i
X
mj

g
21mj
hmjjj0i g
12mj
h0jjmji

= i
X
mj

g
12mj

0mj
g
21mj

mj0

(B.18)
118
and
h0j

@
@t
!
incoh;atom
j0i =
X
i
X
j
h0jf
h
a
y
mi
a
mj

a
y
mj
a
mi

a
y
mj
a
mj

i
A
jiijm
+
h
a
y
mi
a
mj

a
y
mj
a
mi

a
y
mj
a
mj
i
A

jiijm
gj0i
=
X
i
X
j
f
h
h0ja
y
mi
a
mj

a
y
mj
a
mi
j0i

h0ja
y
mj
a
mj

j0i
i
A
jiijm
+
h
h0ja
y
mi
a
mj

a
y
mj
a
mi
j0i

h0j

a
y
mj
a
mj
j0i
i
A

jiimj
g
=
X
i
X
j
f
h

hf0gjhj
m
j
i2

i2
jj
m
ijf0gi

hf0gjhj
m
j
j2

j0i
i
A
jiijm
+
h
h0ja
y
mi
a
mj

i2
jj
m
ijf0gi

h0j

j2
jj
m
ijf0gi
i
A

jiijm
g
=
00
A
2222m
+
0
0
m0
0
m
A
1221m

00
(A
2222m
+ A
2112m
)
+
00
A

2222m
+
0
0
m0
0
m
A

1221m

00
(A

2222m
+ A

2112m
)
=
00
(A
2112m
+ A

2112m
) +
0
0
m0
0
m
(A
1221m
+ A

1221m
) (B.19)
substituting equations B.18 and B.19 in equation B.17
d
00
dt
=i
X
j

g
12mj

0mj
g
21mj

mj0

00
(A
2112m
+ A

2112m
)
+
0
0
m0
0
m
(A
1221m
+ A

1221m
) (B.20)
Similarly it can shown that
d
0mj
dt
= i(!
m
!
j
)
0mj
+ ig
21mj

00

0mj
(A

1111m
+ A
2222m
+ A

1221m
+ A
2112
); (B.21)
d
mj0
dt
= i
(
d
0mj
dt
)

(B.22)
d
0
0
m0
0
m
dt
=2ReA
2112m

00
2ReA
1221m

0
0
m0
0
m
(B.23)
119
W
21m
= (A
2112m
+ A

2112m), W
12m
= (A
1221m
+ A

1221m),
m
= Re(A

1111m
+
A2222 + A

1221m
+ A2112), and

m
= Im(A

1111m
+ A2222 + A

1221m
+ A2112) are the
damping rate, upward relaxation rate, dephasing rate and the frequency shift
(Lamb shift), respectively. The upward relaxation rate is usually slower than the
damping rate and can be ignored in equation B.20. Under these assumptions,
the density matrix operator elements can be cast in the form
d
00
dt
= i
X
j

g
12mj

0mj
g
21mj

mj0

W
21m

00
(B.24)
d
0mj
dt
= (i
mj

m
)
0mj
+ ig
21mj

00
(B.25)
d
mj0
dt
=i(
mj
+
m
)
0mj
ig
12mj

00
(B.26)
where
mj
=!
j
!
m

m
. Equations B.25 and B.26 are solved for
mj0
(t) and

0mj
(t)

0mj
(t) = ig
21mj
Z
t
0
dt
0

00
(t
0
)e
(i
mj

m
)(tt
0
)
(B.27)

mj0
(t) =ig
12mj
Z
t
0
dt
0

00
(t
0
)e
(i
mj
+
m
)(tt
0
)
(B.28)
(B.29)
Equations B.27 and B.28 are then substituted in equation B.24
d
00
dt
=
X
j

jg
12j
j
2
Z
t
0
dt
0

00
(t
0
)e
(i
j

)(tt
0
)
!

X
j

jg
12j
j
2
Z
t
0
dt
0

00
(t
0
)e
(i
j
+
)(tt
0
)
!
W
21m

00
(B.30)
In equation B.30, the subscript m is dropped for convenience. Now if the
Markov assumption holds for this system, in equation B.30 exp
t
0
decays faster
120
than
00
(t
0
). Therefore,
00
(t
0
) can be taken out of integral and t!1 in the
integral. Then equation B.30 simplifies
d
00
dt
=
X
j

jg
12j
j
2

00
(t)
Z
t
0
dt
0
e
(i
j

)(tt
0
)
!

X
j

jg
12j
j
2

00
(t)
Z
t
0
dt
0
e
(i
j
+
)(tt
0
)
!
W
21m

00
=
00
X
j
jg
12j
j
2

1
(i
j

)
+
1
(i
j

)
!
W
21m

00
=
00
X
j
jg
12j
j
2
(t)
0
B
B
B
B
B
@
2
(
2
j
+
2
)
1
C
C
C
C
C
A
W
21m

00
=
00
X
j
2jg
12j
j
2
(t)L(
j
;
) W
21m

00
(B.31)
where L is normalized Lorentzian function
L(
j
;
) =

=
(
2
j
+
2
)
: (B.32)
The total decay rate of the atomic state A
t
is then
A
t
=
X
j
2jg
12j
j
2
(t)L(
j
;
) + W
21m
: (B.33)
The first term in equation B.33, gives the radiative decay rate of the atom due to
its interaction with optical field. If the reservoir is not present, W
21
,
and the
frequency shifts are zero, and L(
j
;
) approaches a-function, and the analysis
reduces to Fermi’s golden rule.
121

Abstract (if available)

Abstract Integrated optics is the ultimate solution for reducing the manufacturing and operational costs of optical circuits. The growing demand for high-bandwidth communication between integrated circuits has also initiated the interest in chipscale integrated optical circuits. Microcavity lasers as an indispensable part of such circuits, have been the subject of intensive research during the past two decades. There have been a lot of speculations on the dynamic properties of these lasers in the past, but to date, there has not been a comprehensive study and analysis of those properties on these lasers. Lack of experimental data providing enough feedback to the theories, has prevented the existing theories to mature. The goal of this thesis is to understand the modulation response and linewidth properties of these lasers through combining our experiments with the theory.

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Asset Metadata

Creator Bagheri, Mahmood (author)

Core Title Modulation response and linewidth properties of microcavity lasers

School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 10/01/2008

Defense Date 08/14/2008

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

Tag laser noise,microcavity,modulation response,OAI-PMH Harvest,quantum-well laser,semiconductor laser

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor O'Brien, John D. (committee chair), Dapkus, P. Daniel (committee member), Hashemi, Hossein (committee member), Lidar, Daniel (committee member), Steier, William H. (committee member)

Creator Email mahmood_bagheri@yahoo.com,mbagheri@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1618

Unique identifier UC149946

Identifier etd-Bagheri-2362 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-116442 (legacy record id),usctheses-m1618 (legacy record id)

Legacy Identifier etd-Bagheri-2362.pdf

Dmrecord 116442

Document Type Dissertation

Rights Bagheri, Mahmood

Type texts

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

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu