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Computer simulation and control for production of an ideal diffraction pattern in optical fibers

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Computer simulation and control for production of an ideal diffraction pattern in optical fibers

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Content COMPUTER SIMULATION AND CONTROL FOR PRODUCTION OF AN IDEAL DIFFRACTION PATTERN IN OPTICAL FIBERS Copyright 2002 by Rahul Kamath A Thesis Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (APPLIED MATHEMATICS) December 2002 Rahul Kamath R eproduced with perm ission o f the copyright owner. Further reproduction prohibited without perm ission. UMI Number: 1414843 UMI UMI Microform 1414843 Copyright 2003 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. UNIVERSITY O F S O U T H E R N CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 8 0 0 0 7 This thesis, written by K a h A T H , under the direction of his. Thesis Committee, and approved by all its members, has been pre sented to and accepted by the Dean of The Graduate School, in partial fulfillment of the requirements for the degree of H a sxek a p. S c r e iN C E THE2JS COMM! R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Dedication To my parents, Gopalkrishna and Gowri Kamath and to my loving brother Yogesh Kamath R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Acknowledgements First and foremost, I would like to thank Dr. Gary Rosen for making this thesis an interesting and rewarding one. I received truly remarkable assistance and guidance from him during the course of this thesis. I would also like to sincerely thank Dr. Robert Sacker and Dr. Alan Schumitzky for serving as members of my thesis committee, along with Dr. Rosen. I would like to dedicate this thesis to my parents, Gopalkrishna Kamath and Gowri Kamath and to my loving brother Yogesh Kamath. This would not have been possible without their love and support. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Table of Contents Dedication Acknowledgements List of Figures Abstract 1 Background 1.1 Optical Fibers 1.2 Fiber Gratings 1.3 History Of Fiber Gratings 2 The Simulation Process 2.1 Experimental Setup 2.2 Computer Simulation 3 The Control Process 3.1 Procedure For The One-Dimensional Case 3.2 The Two-Dimensional Case 4 Conclusions Bibliography Appendices A Source Code A.l Simulation Module-General case A.2 Simulation Module-Special case A.3 Control Module in one dimension R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. V List of Figures 1.1 Optical Fiber 1 1.2 Fiber Bragg grating writing system with a phase mask 5 2.1 Production of diffraction pattern 9 2.2 Using Interpolation 12 3.1 Ideal diffraction pattern 15 3.2 Skewed diffraction patterns 16 3.3 Various diffraction patterns IB 3.4 Skewness as a function of position 20 3.5 Behavior of parameters and input and output variables 27 3.6 Different measures (when fiber movement is fixed in transverse 29 direction) 3.7 Different measures (when fiber movement is fixed in longitudinal 30 direction) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Abstract The purpose of this thesis is to develop a control algorithm that produces diffraction patterns ideal for writing gratings on optical fibers. We make use of computer simulation and control to achieve this goal. A moving optical fiber is impinged by a laser beam and a diffraction pattern is produced on the charge-coupled device. The intensity distribution of the diffraction pattern is traced out as a function of position along movement of the fiber. The ideal diffraction pattern is realized when the fiber is in the center of the beam. Depending on fiber position, the control process decides on the direction of movement towards the ideal diffraction pattern. The least-squares parameter estimation algorithm is implemented so that the moving fiber is in the center of the beam and the ideal diffraction pattern is obtained. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 1 Chapter 1 Background 1.1 Optical fibers An optical fiber is a transparent thread of glass or plastic that can transmit light. The structure of an optical fiber can be depicted [5], as in Figure 1.1. Coating Core Cladding Figure 1.1 Optical Fiber R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2 The innermost part of the fiber is called the core. It carries the light. The core is surrounded by an outer cladding. It helps to keep the optical signal within the core. The coating ensures protection from abrasion and moisture. An optical fiber transmits light on the basis of total internal reflection. The core is commonly made from a combination of silica and germania in order to increase its refractive index. The cladding is made of silica. The difference in refractive indices of the cladding and the core ensures that light travels through the core by constantly reflecting from the cladding. Almost no light is lost on each reflection and light entering the fiber at one end eventually exits at the other end. Optical fibers transmit data in the form of pulses of light. They have a large bandwidth which means that they have the capacity to carry a large amount of information in a fixed amount of time. Additional benefits include low signal loss over long distances due to non-absorption of light by the cladding from the core and low sensitivity to temperature changes. These advantages have made them very useful in communication systems and micro-surgeries. An optical fiber that is designed to carry a single ray or mode of light is called a single mode optical fiber. It is used in long distance signal R eproduced with perm ission o f the copyright owner. Further reproduction prohibited without perm ission. 3 transmission. An optical fiber that is designed for transmission of multiple rays or modes of light concurrently is called a multimode optical fiber. It is used for relatively short distances since the modes tend to disperse over longer lengths. 1.2 Fiber gratings Fiber gratings are produced in the optical fiber by exposure to intense ultraviolet light having a specific interference pattern. This results in a permanent variation of the refractive index in the core of the fiber, giving rise to gratings. Gratings are generally used to make modulators, band-pass filters, sensors and other optical devices. Depending on the wavelength of incident light and spacing of grating, there are two important types of gratings: Fiber Bragg gratings and long-period gratings. A fiber Bragg grating is typically a grating with spacing of the order of 0.5 microns and reflects back an incident beam of light. It can reflect a pre determined narrow or broad range of wavelengths of light incident on the grating, while passing all other wavelengths of light. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 4 A long-period grating is commonly a grating with spacing in the range of 5 to 500 microns and transmits light from the core to the cladding. 1.3 History of fiber gratings In 1978 [2], K. O. Hill and co-workers at the Center for Communications Research, Ottawa, discovered that irradiating a optical fiber with a laser permanently changed the refractive index of the fiber’s core and created a grating in the fiber. B. K. Garside’s group at McMaster University [3] demonstrated that the mechanism by which visible light imprints the grating is a two-photon effect. The fiber’s photosensitivity arose from the breaking of a molecular bond that required more energy than a single photon of visible light could provide. This meant that UV light was useful in forming gratings in fibers. G. Meltz, W. W. Morey and co-workers at the United Technologies Research Center, Hartford Connecticut, showed that side-on ultraviolet illumination of the fiber by a striped pattern was effective in imprinting gratings rapidly. Moreover, by generating such patterns from the interference of waves from a suitable ultraviolet laser, Bragg gratings with reflectance at any desired wavelength could be fabricated [4], R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 5 Of late, fiber Bragg gratings are fabricated by forming periodic modulations in the refractive index of optical fibers. These modulations are typically created by exposing the fiber to an intense interference pattern created by passing UV light through a phase mask as seen in Figure 1.2. Long-period gratings are usually written by using an amplitude mask. They can also be written by simply moving a laser beam along the fiber and turning it on and off. UV Laser Beam Phase mask Fiber Figure 1.2 Fiber Bragg grating writing system with a phase mask In this thesis we replicate by computer simulation the process of writing gratings in an optical fiber by moving the fiber and exposing it to a laser beam. We also attempt to design a computer-controller in order to produce pure and uncontaminated spectra that would in turn generate high-quality R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. gratings. We use computer simulation and control since it faster and cheaper than the actual experiment. 6 is significantly R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 7 Chapter 2 The Simulation Process 2.1 Experimental Setup In this section, we discuss how the process of movement of an optical fiber exposed to a laser beam for writing gratings is conducted in a laboratory setting. Definition 2.1.1 A CCD or a charge-coupled device is a light-sensitive integrated circuit or chip that stores and displays the data for an image by converting each pixel of the image into an electrical charge whose intensity is related to a color in the spectrum. We use the approach of writing the grating point by point through the side of the optical fiber using a UV laser source. The point-by-point writing is particularly flexible because the spectral response of the grating can be R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 8 precisely controlled. The spectral response of the grating largely depends on the period of the grating and the quality of the grating. Variation in grating quality or period can result in non-uniformity in spectral response. It is therefore desirable to have a wider grating response to eliminate the loss of optical signal. As compared to movement of the fiber in one direction, movement of the fiber in both the transverse and longitudinal directions gives rise to two different grating periods and covers a wider spectral range. The optical fiber is placed on a computer-controlled motorized translation stage. A UV laser beam is placed on one side and a CCD on the other to measure the grating response. The beam is flashed at regular intervals and the fiber is moved in both the transverse and longitudinal directions. The motor stops at each point and a single laser pulse impinges on the fiber. The portion of the fiber in front of the beam is translated and a diffraction pattern is produced on the CCD. This is depicted in Figure 2.1. After a fixed interval of time, the UV light is blocked by a shutter and the fiber is moved to produce the next diffraction pattern. The intensity distribution of the diffraction pattern is traced out as a function of position along the movement of the fiber. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 9 Laser Beam Transverse displacement Optical Fiber Computer-controlled motorized translation stage Longitudinal displacement Figure 2.1 Production of diffraction pattern 2.2 Computer Simulation The actual experiment should preferably replicate the intensity distribution generated by an ideal diffraction pattern at all the points. This can be achieved if the optical fiber is in the center of the laser beam. But because of the movement of the fiber, it is not always possible to get this optimal value. However, computer technology enables us to simulate the diffraction pattern on the CCD and design a controller that helps in replicating the intensity distribution generated by an ideal diffraction pattern without extra R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 10 resonances. The purpose of simulation is to use it in designing this computer- controller. We therefore replicate the experiment by using computer simulation and control. In this section, we explain the simulation of diffraction patterns with the aid of a computer depending on the position of the fiber when the beam hits it. The assumption is that we have actual data from experiments conducted in the laboratory. We make use of two-dimensional interpolation to calculate the values between data points and thereby generate the CCD patterns. From experiments conducted in a laboratory setting, a data set consisting of the intensity distribution as a function of position of the fiber has already been defined. We assume that the fiber moves from -5 microns to 5 microns (in steps of 0.5 microns) in the longitudinal (X) direction and -500 microns to 500 microns (in steps of 50 microns) in the transverse (Y) direction. The computer program is able to pull out required values already present in the data set. In addition, given any two-dimensional co-ordinate (that is not a part of the defined data set or whose data value is unknown) in the space S={(x,y): -5<=x<=5, -500<=y<=500}, the program makes use of the known data points and calculates the value of that particular co-ordinate by using R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 11 interpolation. So the data values at any particular two-dimensional co-ordinate in the space S can be determined. The computer-simulated program decides whether to use the known data set or to use interpolation. The data set is defined between the integers 1 and 21 for both the x and y co-ordinates. The integers 1 and 21 are attained by scaling the minimum value -5 and maximum value 5 from the space S. For a given pair of co-ordinates, the program checks whether each co-ordinate is an integer. If both the co-ordinates are integers, then the program picks up the appropriate value from the data set; otherwise, it uses interpolation. Let (x,y) be the position and xrange and yrange be the maximum values taken by the x-coordinate and y-coordinate respectively (1 is the minimum value taken by both the x and y co-ordinates). We determine LX, the nearest integer less than or equal to x, RX, the nearest integer greater than or equal to x, LY, the nearest integer less than or equal to y and RY, the nearest integer greater than or equal to y. LX = min (max (1, floor (x)), xrange) RX = max (min (range, ceil (x)), 1) LY = min (max (1, floor (y)), yrange) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 12 RY = max (min (range, ceil (y)), 1) where floor (x) is the value of the nearest integer less than or equal to x and ceil (x) is the value of the nearest integer greater than or equal to x. We compare LX with RX and LY with RY. If LX is equal to RX and LY is equal to RY simultaneously, it means that both x and y are integers. The data value is known in this case and the program pulls out this value from the defined data set. If either LX is not equal to RX or LY is not equal to RY, then one of x and y is not an integer. In this case, the data value is unknown and the program uses interpolation to calculate the data value. Suppose the value of the intensity of each of the 2048 pixels at four points A(xl, yl), B(x2, y2), C(x3, y3) and D(x4, y4) is known. If the data value is to be calculated at the point P (x,y), which is located somewhere in the area enclosed by the points A, B, C and D, as shown in Figure 2.2, the known data values at the four points are used. A (x l, yl) D (x4, y4) C (x3, y3) B (x2, y2) Figure 2.2 Using Interpolation R eproduced with perm ission o f the copyright owner. Further reproduction prohibited without perm ission. 13 To begin with, the distance of point P from each of the points (with the known data values) A, B, C and D, is calculated, using the distance formula. Distance of point P from point A dl = «J(x-xl)2 + (y - yl)2 Distance of point P from point B d2= -J(x - x2)2 + (y - y2)2 Distance of point P from point C d3 = yj(x - x3)2 + (y - y3)2 Distance of point P from point D d4= y](x-x4)2 + (y - y4)2 Now weights are allocated to data values of each of these neighboring points of P, based on their proximity to P. The closer the point is to P, the higher is the weight allocated to the data value of that point. The weights are calculated in the following manner: Let a l = d2 + d3 + d4 a2 = dl + d3 + d4 a3 = dl + d2 + d4 cc4 = dl + d2 + d3 a = a l + a2 + a3 + a4 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 14 Then the weights are given by a lia , a lia , a lia , and a4/a. The weights add up to one. We denote the data value at point P as D (x, y), data value at point A as D (xl, yl), data value at point B as D (x2, y2), data value at point C as D (x3, y3) and data value at point D as D (x4, y4). Then D (x, y)= (cd/cc)*D (xl, yl)+ (a2/a)*D (x2, y2)+ (a3/a)*D (x3, y3)+ (a4/a)*D (x4, y4) Thus by using interpolation, any intermediate unknown value in the space S can be found. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 15 Chapter 3 The Control Process 3.1 Procedure for the one-dimensional case The main purpose of a controller is to keep the moving optical fiber in the center of the laser beam so that an ideal diffraction pattern is obtained at all points. When the fiber is in the center of the beam, the diffraction pattern generated is as shown in Figure 3.1. The peculiar bimodal shape occurs due to diffraction or bending of rays from the laser beam hitting the fiber. 3000 2500 a 1 5 0 0 1000 500 1000 1200 1400 1600 1800 2000 Pixel Laser Figure 3.1 Ideal diffraction pattern R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 16 The generated diffraction pattern is significantly impacted by the position of the fiber with respect to the laser beam. If the fiber is not in the center, but skewed to either side of the beam, the diffraction patterns are also skewed as shown in Figure 3.2. 2000 1800 1600 1400 1 1200 1 1000 e “ 800 600 400 200 1500 2000 500 1000 Pixel 2000 1800 1600 £.1400 §1200 < u C1000 800 600 400 200 0 , 2000 500 1500 1000 Pixel Figure 3.2 Skewed diffraction patterns Fiber Laser Beam Fiber Laser Beam R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 17 Various diffraction patterns are possible depending on the position and angle of the fiber with respect to the laser beam (although this is subject to sizes of the fiber and the beam as well). Some of the possible diffraction patterns are shown in Figure 3.3. The controller ensures that the ideal diffraction pattern (as shown in Figure 3.1) can be produced at all positions. In the one-dimensional case, that is, when the fiber is moving in just one direction, the controller generates the pulses required for the change in movement and decides the direction of movement. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Intensity Intensity Intensity 18 (A h f A Y % 5 0 0 10 0 0 15 0 0 12 0 0 10 0 0 8 0 0 6 0 0 4 0 0 2 00 0 0 500 10 0 0 15 0 0 2 0 0 0 Pixel 12 0 0 1 000 8 0 0 6 0 0 4 0 0 2 0 0 0 5 0 0 1 0 0 0 15 0 0 0 2 0 0 0 Pixel Figure 3.3 Various diffraction patterns R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 19 The first step in the control process is to choose the ideal (optimal) diffraction pattern. This can be done manually by going through all the data values generated by actual experiment. After choosing the ideal diffraction pattern, we try to control the generated diffraction patterns and keep them close to this ideal pattern. Given any diffraction pattern, we have to figure out which way to move in order to get closer to the ideal pattern. A unique measure is needed that will enable us to determine direction of movement. In the one-dimensional case, this is given by the skewness. Definition 3.1.1 The skewness, y , of a distribution is defined as the ratio of the third moment juz about the mean E(X) and the third power of the standard deviation a . ; th _ E{(X -E (X ))3 } S'3 [E{ (X - E(X))2 }]3/2 From symmetry of the graph in Figure 3.1, we see that skewness of the optimal diffraction pattern should be ideally equal to zero. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 20 In order to calculate the skewness in the one-dimensional case, we consider the graph of a diffraction pattern as a probability distribution function. We scale the area under this graph to one. Then skewness is calculated for different fiber positions between -10 and 10 microns. It is seen to be a monotone function of position as depicted in Figure 3.4. Skewness Linear approximation "vL * • » , co-l -10 Position of the fiber (in microns) Figure 3.4 Skewness as a function of position Also, as seen from the graph, it is possible to fit a straight line to the skewness curve and express skewness as a linear function of position. So, skewness can be written as y = ax + b where y - skewness of the diffraction pattern x - position of the fiber R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 21 a - slope and b - y-intercept. In order to design a computer-controller that would control a generated diffraction pattern and keep it close to the ideal one, we use the recursive least squares method with exponential data weighting [1]. It is necessary to use a simple model whose algorithm will calculate the parameter estimates quickly since these estimates need to be recursively and frequently updated. Also, in this method, the estimated parameters converge to the true parameters in a short duration of time. In addition, when exponential data weighting is used, greater weighting is attached to more recent data since it is considered to be more informative than past data. Finally, since we are estimating the parameter vector 6 on-line, the parameters become necessarily time-varying. As can be seen from Figure 3.4, skewness of a diffraction pattern satisfies the linear model y(t) - d^xit) + 02. This can also be written in vector notation as y(t) = W - i f e where R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 22 # = [#! 02J , 6X representing the slope and 02 representing the y-intercept, < p(t) = [jc(0 l]r , x(t) representing the position of the fiber at time t and y(t) represents the skewness of the diffraction pattern at time t . We begin by randomly generating a diffraction pattern. The assumption in the one-dimensional case is that the fiber is moving from -10 microns to 10 microns and that the position of the fiber x(t) lies in this range. The algorithm we use results from the following optimization problem: At time t = k, y(k) = <p(k-\)T0(k) is the estimate of the true skewness y(k) and 0(k) is the current estimate of the true parameter 9. We now want to determine 0(k) to minimize a simple error function Vk(0) = f Jp k~ i[ y (i)-^ (i-l)T0(k)]2 i=1 Here p is a forgetting or deweighting factor, 0 < p <1. With this parameter, we can emphasize new data and discount old data. /\ We determine the values of 0x{t) and 02(t) as follows: We set the partial derivative of Vk (0) to zero: R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. ~ = - l)[y(i) - < 2 > (i - D1 d m = 0 a o i = 1 which can be written as 23 - 1 ) ^ 0 - 1)TW - D < = i i= i The estimated parameter 0(k) can be calculated as 0{k) = i= i ~ i i=l where y{k -1 ) 1 = ^ p k ~ 1) $(i ~ l)7 i= i k - 1 i= i y(k - 1)"1 = pyik - 2)-1 + (j> (k -1 ) < j){k - 1)1 We now calculate y(k - 1) making use of the matrix inversion lemma. (2) Lemma 3.1.1 If A, C, and C +DA B are nonsingular square matrices, then (A + BCD)~X = A 1 - A~X B(C~X + DA~X B)~X DA~X Proof: Let M - A + BCD and N = B C . R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2 4 Then I = M~X M = M~X (A + BCD) = M~X (A + ND) = M~X A + M~X ND Post-multiplying by A-1, we get IA~X = M~1 AA~1 + M~X NDA~X A~l - M ~ x +M ~X NDA~X (a) A~x - M~x = M~X NDA~X (b) Post-multiplying (a) with N gives A~X N - M~X N + M ~X NDA~X N A~X N = M~X N(I + DA~X N) M~X N = A~X N(I + DA~X N)~X Post-multiplying by DA~X , we get M ~X NDA~X = A~X N (I + DA~~lN)~~x DA~l (c) Substituting (b) into the left-hand side of (c), we get A~x - M ~x = A~X N(1 + DA~X N)~X DA'1 M~x = A~x - A~X N{I + DA~X N)~X DA~X Plugging back the values of M and N , we have (A + BCD)~X = A"1 - A~X BC(I + DA~lB C y xDA~x Using (XYyx =Y~xX~x with X = (T 1 and Y = I + DA~X B C , we get (A + BCD)~X = A-1 - A_ 1 J5[(7 + DA~X BC)C~X ]_ 1 DA~X (A + B C D yx = A-1 - A~X B[C~X + DA~X BCC~X ]~X DA~X R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2 5 (A + BCDY1 = A"1 - A~lB[C~l + DA-1 B y 1 DA This proves the lemma. Taking A = py(k - 2) 1, B - 0{k - 1), C - — and D = p0(k - l)7 in the P above lemma, we have py{k - 2Yl + 0(k - l)— p0(k - l)r P = - y ( k - 2) - — y(k - 2)0(k - l)[lp + P P p 0 (k -iy - y ( k - 2 ) 0 ( k - I ) P T 1 P0 { k -\)‘ - y { k -2 ) P Noting that the left hand side of the above equation is y(k - 1) from (2), we have y(k-D y(k - 2) ■ y(k - 2)0(k - l).0(k - l)r yjk - 2) p + 0 ( k -l) ry (k -2 )0 (k -l) (3) From (1) 6(k) = y { k - 1) f Jpk -iy(iW-l) ( = i = y ( k - 1) k - l p ^ p ^ y i i M i -1 ) + y(k)0(k - 1) 1 = 1 y(k -1 )[py(k - 2) 6{k -1 ) + y(k)<f>(k - 1)] {from (1)} = 7(k - 1 )l(y(k - 1)-1 - < f> (k -1 )0{k - l)r )0(k -1 ) + -1)] {from (2)} = 0{k -1 ) + y(k - 1 )0{k - l)[y(£) - < p(k - 1)7 0(k - 1)] R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2 6 = 0 (k -l) + y(k-2)</>(k-l) y(k)-<p{k-l)T 0 (k-\) _p + </>(k-l)Ty (k -2 )0 (k -l) (4) [since y(k -1 )0(k - 1) = y (k -2 )0 (k -l) p + <j>(k-iy y(k-2)tj>(k-I) ] We need to implement equations (3) and (4). In order to perform these, we need an initial value for y(-l) (it should be a positive definite matrix) and an initial estimate for 0(0). Then we linearly approximate the position, where the ideal diffraction pattern is achieved, by using the following: xl - xik) + yo p , - y(k) 0, where yo p t represents the skewness of the ideal diffraction pattern y(k) represents the skewness of the diffraction pattern generated by the linear model at time t = k x(k) represents the position of the fiber at time t = k and 6X represents the latest parameter value generated by the algorithm. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2 7 At time t = k + 1, equations (3) and (4) are used recursively with approximation xx as input for the vector (p. This recursive method gives us a reasonable approximation for the position where the ideal diffraction pattern is achieved. As seen from Figure 3.5, the parameter estimates stay constant after just a few iterations and the optimal value is approached fast. 10 8 I 6 14 b b G Z s 4 -10 c l D M C O 0 10 20 30 40 50 + + + • +*+++• +♦+++++■*+ ++++-H -H-+ •+++++++++ +++ + + + + -H - ++ 1 0 20 30 40 50 Simulation Intervals Simulation Intervals 5 4 3 ^ 2 5 - •4--H 4++4-44- -+ 4-H -M +++■ + + - H -H - + 4 + - -H 10 20 30 40 Simulation Intervals 50 5 4 3 « 2 is 1 1 I 0 S a * -1 -2 -3 - 4 -5 -A + + + + + + 1 0 20 30 40 Simulation Intervals 50 Figure 3.5 Behavior of parameters and input and output variables R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2 8 3.2 The two-dimensional case We extend the concept of one-dimensional control process to the two- dimensional case. In the one-dimensional case, a monotone measure (skewness) is determined and used in the control process to get close to the ideal diffraction pattern. In the two-dimensional case, we attempt to determine a monotone measure in each direction (transverse and longitudinal). We fix the fiber movement in each direction and observe the first few moments. As seen from Figure 3.6, when fiber movement is fixed in the transverse direction, skewness is a reasonable monotone measure. However, we could not determine a monotone measure when fiber movement is fixed in the longitudinal direction (Figure 3.7). The control process is more complicated in the two-dimensional case. We were unable to find a monotone measure that enables us to determine direction of movement and thereby control the data values. However, we are optimistic that this may be able to be achieved by the use of Artificial Neural Networks. We are currently looking into this possibility. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 2 9 1100 1000 900 800 700 600 500 -5 -4 -3 500 •| 460 03 I 420 " 2 380 I 340 300 1.5 1 0.5 t 0 < D 05 -0.5 -1.5 -2 - 1 0 1 2 Position (microns) -5 -4 - 3 - 2 - 1 0 1 2 Position (microns) -5 -4 -3 -2 - 1 0 1 2 Position (microns) -5 -4 -3 -2 -1 0 1 2 3 4 5 Position (microns) Figure 3.6 Different measures (when fiber movement is fixed in transverse direction) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 3 0 a _ o '> a Q T ) E 3 T 3 1100 1000 900 800 700 600 50(?500-400-300-200-100 0 100 200 300 400 500 Position (microns) 500 r 460 420 380 340 300, -500-400-300-200-100 0 100 200 300 400 500 Position (microns) 1.5 r i c n | 0 . 5 M 0 00 -0.5 - 1 -1.5, -500-400-300-200-100 0 100 200 300 400 500 Position (microns) 4 3.5 c n 3 0 2.5 2, 1.5 1 0.5 o< 00-100 0 100 200 300 400 500 Position (microns) Figure 3.7 Different measures (when fiber movement is fixed in longitudinal direction) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 31 Chapter 4 Conclusions We gave a brief overview and background of optical fibers and gratings. We discussed the experimental procedure for fiber movement when it is exposed to a laser beam in the two-dimensional case. We followed computer simulation to perform the same operation. We made use of experimental data and interpolation to generate different data sets and diffraction patterns. We stressed the importance of obtaining an optimal diffraction pattern at all positions. This can be achieved when the fiber is in the center of the beam. We designed a computer control process so that this can be achieved. We explained a particular control process in the one-dimensional case. We determined that skewness was a monotonic function of fiber position and fitted a linear model to the skewness curve. We implemented the least-squares parameter estimation algorithm recursively as part of the control process to achieve the ideal diffraction pattern. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 3 2 We indicated the need for further research in the two-dimensional case. Unlike the one-dimensional case, we could not find a monotone measure that could be used for control. However, we believe that it is possible to generalize the one-dimensional control process to the two-dimensional case. The use of Artificial Neural Networks may be a very interesting approach to consider. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 33 Bibliography [1] Graham C. Goodwin and Kwai Sang Sin, “Adaptive Filtering Prediction and Control”, Prentice Hall, Englewood Cliffs, NJ, 1984. [2] K. O. Hill, Y. Fujii, D. C. Johnson and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: Application to reflection filter fabrication,” Appl. Phys. Lett. 32 (10), 647-649 (1978). [3] D. W. K. Lam and B. K. Garside, “Characterization of single-mode optical fiber filters,” Appl. Opt. 20 (3), 440 (1981). [4] G. Meltz, W. W. Morey, and W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823-825 (1989). [5] Norma Thorsen, “Fiber Optics and the Telecommunications Explosion”, Upper Saddle River NJ, Prentice Hall PTR 1998. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 3 4 Appendix A Source Code A.l Simulation module - General case All the code has been written in Matlab. This program simulates the behavior of an optical fiber when exposed to a laser beam. The fiber moves both transversely and longitudinally. The fiber movement is from -5 microns to 5 microns in the longitudinal direction in jumps of 0.25 microns and from -500 microns to 500 microns in the transverse direction in jumps of 25 microns. clear % Loading the database ‘ newdata’ load newdata figure (1) hold off szd = size (D); R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. range = szd (2); orig = floor ((range + l)/2); xint = 0.25; yint = 25; xO = 0; yO = 0; % Scaling xOi = orig + xO/xint; yOi = orig + yO/yint; % Plotting the background with the laser beam ‘off’ ccd = background; plot (ccd,V) axis ([0, 2048,0, 3000]); xlabel ('Pixel') ylabel ('Intensity') hold on % Plotting the data values with the laser beam ‘ on’ disp ('Hit Enter to turn on laser') pause elf ccd = justlaser; R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 3 6 plot (ccd, V) axis ([0,2048, 0, 3000]); xlabel ('Pixel') ylabel ('Intensity') hold on % Plotting the data values with the optical fiber installed disp ('Hit Enter to install fiber') pause elf LX = min (max (1, floor (xOi)), range); RX = max (min (range, ceil (xOi)), 1); LY = min (max (1, floor (yOi)), range); RY = max (min (range, ceil (yOi)), 1); if ((RX = LX)& (RY = LY)) % Plotting the values from the database for z = 1:1:2048 ccd (z,l)= D(LX, LY, z); end else % Calculating the data values by interpolation and plotting them dl = sqrt ((xOi - LX)A 2 + (yOi - LY)A 2); R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 3 7 d2 = sqrt ((xOi - RX)A 2 + (yOi - LY)A 2); d3 = sqrt ((xOi - RX)A 2 + (yOi - RY)A 2); d4 = sqrt ((xOi - LX)A 2 + (yOi - RY)A 2); alpha 1 = d2+d3+d4; alpha2 = dl+d3+d4; alpha3 = dl+d2+d4; alpha4 = dl+d2+d3; alpha = alphal+alpha2+alpha3+alpha4; for z=l: 1:2048 ccd (z,l)= (alphal/alpha)*D (LX, LY, z)+ (alpha2/alpha)*D (RX, LY, z) + (alpha3/alpha)*D (RX, RY, z)+ (alpha4/alpha)*D (LX, RY, z); end end plot (ccd, 'b') axis ([0, 2048, 0, 3000]); xlabel ('Pixel') ylabel ('Intensity') hold on % Beginning o f simulation disp ('Hit Enter to begin simulation') pause R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 38 y = -500; for x = -5:0.25:5 if y = -500 for y= -500:25:500 xi = orig + x/xint; yi = orig + y/yint; LX = min (max (1, floor (xi)), range); RX = max (min (range, ceil (xi)), 1); LY = min (max (1, floor (yi)), range); RY = max (min (range, ceil (yi)), 1); if ((RX == LX)& (RY == LY)) % Plotting the values from the database for z = 1:1:2048 ccd (z,l)= D (LX, LY, z); end else % Calculating the data values by interpolation and plotting them dl = sqrt ((xOi - LX)A 2 + (yOi - LY)A 2); d2 = sqrt ((xOi - RX)A 2 + (yOi - LY)A 2); d3 = sqrt ((xOi - RX)A 2 + (yOi - RY)A 2); d4 = sqrt ((xOi - LX)A 2 + (yOi - RY)A 2); R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 39 alphal = d2 + d3 + d4; alpha2 = dl + d3 + d4; alpha3 = d l + d2 + d4; alpha4 = dl + d2 + d3; alpha = alphal + alpha2 + alpha3 + alpha4; for z = 1:1:2048 ccd (z,l)= (alpha 1/alpha)*D (LX, LY, z)+ (alpha2/alpha)*D RX, LY, z) + (alpha3/alpha)*D (RX, RY, z)+ (alpha4/alpha)*D (LX, RY, z); end end figure (1) elf plot (ccd, 'b') hold on xlabel ('Pixel') ylabel ('Intensity') axis ([0, 2048, 0, 3000]); hold on pause(1) end else R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 4 0 for y = 500:-25:-500 xi = orig + x/xint; yi = orig + y/yint; LX = min (max (1, floor (xi)), range); RX = max (min (range, ceil (xi)), 1); LY = min (max (1, floor (yi)), range); RY = max (min (range, ceil (yi)), 1); if ((RX = LX)& (RY = LY)) % Plotting the values from the database for z = 1:1:2048 ccd (z, 1)= D (LX, LY, z); end else % Calculating the data values by interpolation and plotting them dl = sqrt ((xOi - LX)A 2 + (yOi - LY )A 2); d2 = sqrt ((xOi - RX)A 2 + (yOi - LY)A 2); d3 = sqrt ((xOi - RX)A 2 + (yOi - RY )A 2); d4 = sqrt ((xOi - LX)A 2 + (yOi - RY )A 2); alphal = d2 + d3 + d4; alpha2 = dl + d3 + d4; alpha3 = dl + d2 + d4; R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 41 alpha4 = dl + d2 + d3; alpha = alphal + alpha2 + alpha3 + alpha4; for z = 1:1:2048 ccd (z,l)= (alphal/alpha)*D (LX, LY, z)+ (alpha2/alpha)*D (RX, LY, z) + (alpha3/alpha)* D (RX, RY, z)+ (alpha4/alpha)*D (LX, RY, z); end end figure (1) elf plot (ccd, 'b') hold on xlabel ('Pixel') ylabel ('Intensity') axis ([0, 2048, 0, 3000]); hold on pause(1) end end end figure (1) hold off R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 4 2 % End of simulation disp ('End of Simulation') A.2. Simulation module - Special case Data values have been calculated by actual experiment and have been stored in the database. The values are for points between (-5, -500) and (5, 500) in jumps of 0.50 microns in the longitudinal direction and jumps of 500 microns in the transverse direction. This program pulls out the data value from the database for any given point. clear x = input ('Enter x-coordinate (between -5 and 5)') y = input ('Enter y-coordinate (between -500 and 500)') % Converts the number to a string strx = num2str (x, '%-12.6f) stry = num2str (y, '%-12.6f) % Concatenates the two strings t = strcat (strx, stry, '.txt') % Opens the file corresponding to the string name fid = fopen (t); a = fscanf (fid, '%g', [1 inf]); R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 43 a = a'; % Closes the file fclose (fid); % Plotting the data values from the file plot (a) xlabel ('Pixel') ylabel ('Intensity') axis ([0, 2048,0, 2000]); pause (1) A.3. Control module in one dimension This program makes use of the recursive least-squares algorithm with exponential data weighting and enables us to move closer to the optimal data value set. This helps us to control the output. % Inputs from the main program function x = motinpl (ccd, i, xO, bkgrd, jstlsr, simints, opt) % y = ax + b, Least Squares Method, y = skewness, x l = xO + (yo p t-y0)/ a % Declaration of variables global u R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 4 4 global y global thetal global theta2 global gammaO global WL global WR global nl global skopt aO = 1.0; bO = 0.0; gammaO 1 = 2.0; gamma02 = 2.0; bf = 0.90; freqct = 10; dispf = 3; rho = 0.9; if (i = = 0) u= []; y = [ ]; thetal = [a0]; R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 4 5 theta2 = [bO]; gammaO = diag ([gammaOl, gamma02]); z = 0 : 2047; area = trapz (opt - bkgrd); mu = floor (trapz ((opt - bkgrd)'.*z)/ area); radius = floor (0.8*min (mu, 2048 - mu)); WL = mu - radius; WR = mu + radius; nl = WR - WL +1; Wopt = opt (WL:WR); Wbkgrd = bkgrd (WL:WR); TWopt = fft (Wopt - bf*Wbkgrd); TWopt (freqct + 1 : nl - freqct)= 0*TWopt (freqct + 1 : n l - freqct); FWopt = real (ifft (TWopt)); z = 0 : nl - 1; areaopt = trapz (FWopt); muopt = trapz (FWopt'.*z)/ areaopt; sdopt = sqrt (trapz (FWopt'.* ((z - muopt).A 2))/ areaopt); skopt = trapz (FWopt'.* ((z - muopt).A 3))/ (areaopt* (sdoptA 3)); R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. figure (2) elf axis ([0, simints, -10, 10]) grid on xlabel ('Simulation Intervals') ylabel ('Input x Position (mu-m)') hold on figure (3) elf simdisp = floor (simints / dispf); plot (dispf* (0 : simdisp), ones (1, simdisp + l)*skopt, '.r') axis ([0, simints, -5, 5]) grid on xlabel ('Simulation Intervals') ylabel ('Skewness') hold on figure (4) elf axis ([0, simints, -5, 5]) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 7 grid on xlabel ('Simulation Intervals') ylabel ('Parameter a') hold on figure (5) elf axis ([0, simints, -5, 5]) grid on xlabel ('Simulation Intervals') ylabel ('Parameter b') hold on pause end Weed = ccd (WL : WR); Wbkgrd = bkgrd (WL:WR); TWccd = fft (Weed - bf*Wbkgrd); TWccd (freqct + 1 : n l - freqct)= 0*TWccd (freqct + 1 : nl - freqct); FWccd = real (ifft (TWccd)); Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. area = trapz (FWccd); mu = trapz (FWccd'.*z)/ area; sd = sqrt (trapz (FWccd1 .* ((z - mu).A 2))/ area); sk = trapz (FWccd1 .* ((z - mu).A 3))/ (area* (sdA 3)); yO = sk; u = [u, xO]; y = [y, yO]; figure (2) plot ([i], u (i + 1), '+g') figure (3) plot ([i], y (i+1), ’ +bT ) figure (4) plot ([i], [thetal (i+1)], '+c') figure (5) plot ([i], [theta2 (i+1)], '+m') Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 9 phi = [u (i+1), 1]’ ; % Updating the parameters a and b gamma 1 = (rhoA -l)* (gammaO - (gammaO*phi*phi'*gammaO) / (rho + phi'* gammaO*phi)); theta = [thetal (i + 1), theta2 (i + 1)]' + gammaO*phi*((y (i + 1)- thetal (i + l)*u (i + 1)- theta2 (i + 1)))/ (rho + phi'*gammaO*phi); % Calculating the position x x = u (i + 1)+ (skopt - y (i + 1))/ theta (1); thetal = [thetal, theta (1)]; theta2 = [theta2, theta (2)]; gammaO = gammal; if (i = = simints) figure (2) hold off figure (3) hold off figure (4) hold off figure (5) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 5 0 hold off end R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

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Asset Metadata

Creator Kamath, Rahul Gopalkrishna (author)

Core Title Computer simulation and control for production of an ideal diffraction pattern in optical fibers

School Graduate School

Degree Master of Science

Degree Program Applied Mathematics

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

Tag engineering, electronics and electrical,Mathematics,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Rosen, Gary (committee chair), Sacker, Robert (committee member), Schumitzky, Alan (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-298328

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Identifier 1414843.pdf (filename),usctheses-c16-298328 (legacy record id)

Legacy Identifier 1414843.pdf

Dmrecord 298328

Document Type Thesis

Rights Kamath, Rahul Gopalkrishna

Type texts

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

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

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