Extended Depth-of-focus in a Laser Scanning System Employing a Synthesized Difference-of-Gaussians Pupil by Alexander Kourakos Thesis submitted to the faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering Dr. Ting-Chung Poon, Chair Dr. Gary S. Brown Dr. Guy J. Indebetouw May 1999 Blacksburg, Virginia Keywords: extended depth of field, optical scanning microscopy, pupil synthesis Copyright c 1999 Alexander Kourakos Extended Depth-of-focus in a Laser Scanning System Employing a Synthesized Difference-of-Gaussians Pupil by Alexander Kourakos Abstract Traditional laser scanning systems, such as those used for microscopy, typically image objects of finite thickness. If the depth-of-focus of such systems is low, as is the case when a simple clear pupil is used, the object must be very thin or the image will be distorted. Several methods have been developed to deal with this problem. A microscope with a thin annular pupil has a very high depth-of-focus and can image the entire thickness of a sample, but most of the laser light is blocked, and the image shows poor contrast and high noise. In confocal laser microscopy, the depth-of-focus problem is eliminated by using a small aperture to discard information from all but one thin plane of the sample. However, such a system requires scanning passes at many different depths to yield an image of the entire thickness of the sample, which is a time-consuming process and is highly sensitive to registration errors. In this thesis, a novel type of scanning system is considered. The sample is simultaneously scanned with a combination of two Gaussian laser beams of different widths and slightly different temporal frequencies. Information from scanning with the two beams is recorded with a photodetector, separated electronically, and processed to form an image. This image is similar to one formed by a system using a difference-of-Gaussians pupil, except no light has been blocked or wasted. Also, the entire sample can be scanned in one pass. The depth- of-focus characteristics of this synthesized difference-of-Gaussians pupil are examined and compared with those of well-known functions such as the circular, annular, and conventional difference-of-Gaussians pupils. Acknowledgments First I would like to thank my advisor, Dr. Ting-Chung Poon, for introducing me to this research topic, and more importantly, for his suggestions, guidance, and patience. I would also like to thank Dr. Gary Brown and Dr. Guy Indebetouw for serving on my committee, and for the insights they’ve given me both in and out of the classroom. I want to thank my mother and father for their understanding and encouragement. No matter what paths I take in life, they are always by my side. I’d like to thank Doug Mauer of Biz Net Technologies for being patient and not making too big a fuss when I basically disappeared from my job for a few months to finish up my thesis. Finally, I’d like to extend heartfelt thanks to Kathleen Rio. With her support and caring, impossible things were made possible. She is a wonderful friend and companion, and I dedicate this thesis to her. This work was financially supported by a James A. Shannon Director’s Award from the National Institutes of Health. ii Contents 1 Introduction 1 1.1GeneralIntroduction............................... 1 1.2Motivation..................................... 2 1.3OrganizationofThesis.............................. 2 2 Background 4 2.1MathematicalPreliminaries........................... 4 2.2SimpleLaserScanningSystem.......................... 4 3 Depth-of-focus Concepts 8 3.1GeneralizedPupilFunction........................... 8 3.2AnalysisofPupilFunctions........................... 11 3.2.1 PSFandMTFofPupilFunctions.................... 12 3.2.2 AmbiguityFunction........................... 18 3.2.3 StrehlRatio................................ 20 3.2.4 Hopkin’sCriterion............................ 21 3.3Post-processingWithCorrectiveFilter..................... 23 3.4ImagingSimulations............................... 26 iii CONTENTS iv 4 Synthesized Difference-of-Gaussians Pupil 32 4.1Implementation.................................. 32 4.2EffectsofMisfocus................................ 35 4.2.1 FindingOptimumParameterValues.................. 36 4.2.2 PSFandMTFResults.......................... 37 4.2.3 StrehlRatio................................ 43 4.2.4 Hopkin’sCriterion............................ 45 4.3ImagingSimulations............................... 45 4.4Signal-to-noiseConsiderations.......................... 45 5 Conclusions 53 A Strehl Ratio Derivations 55 A.1Difference-of-Gaussians.............................. 55 A.2SynthesizedDifference-of-Gaussians....................... 56 B Program Listings 57 B.1 p.m — PupilFunctions............................. 58 B.2 plot_otf_2d.m — PlotMTF......................... 59 B.3 plot_otf_2d_w20.m — Plot MTF vs. W20 .................. 61 B.4 strehl.m — StrehlRatio............................ 63 B.5 find_w20_max_strehl.m —FindMaxW20 withStrehlRatio........ 64 B.6 hopkins.m — Hopkin’sCriterion....................... 65 B.7 hopkins_dog.m — Hopkin’sCriterion(Difference-of-Gaussians)...... 67 B.8 find_w20_max_hopkins.m —FindMaxW20 with Hopkin’s Criterion . 69 B.9 find_best_wb.m — Find Optimal wb (Difference-of-Gaussians)....... 70 B.10 spoke.m — SpokePattern........................... 71 B.11 plot_spoke.m — PlotImageofSpokePattern................ 72 B.12 plot_otf_2d_sdog.m — PlotMTF(SDoG)................. 74 B.13 plot_otf_2d_w20_sdog.m — Plot MTF vs. W20 (SDoG).......... 76 CONTENTS v B.14 strehl_sdog.m — StrehlRatio(SDoG).................... 78 B.15 find_w20_max_strehl_sdog.m —FindMaxW20 with Strehl Ratio (SDoG) 80 B.16 hopkins_sdog.m — Hopkin’sCriterion(SDoG)............... 81 B.17 find_best_wb_sdog.m — Find Optimal wb (SDoG)............. 83 B.18 find_best_beta.m — Find Optimal β (SDoG)................ 84 B.19 find_w20_max_hopkins_sdog.m —FindMaxW20 with Hopkin’s Criterion (SDoG)...................................... 85 B.20 plot_spoke_sdog.m — PlotImageofSpokePattern(SDoG)........ 86 Bibliography 89 Vita 91 List of Figures 2.1Simplifieddiagramoflaserscanningsystemoptics............... 5 3.1Diagramclarifyingthedefinitionoftheaberrationfunction.......... 9 3.2 Plots of circular pupil function. ......................... 12 3.3 Plots of annular pupil function. ......................... 13 3.4 Plots of Gaussian pupil function. ..................... 13 3.5 Plots of difference-of-Gaussians pupil function. ................. 14 3.6 Plots of PSF vs. W20/λ forcircularpupil..................... 15 3.7 Plots of MTF vs. W20/λ forcircularpupil.................... 15 3.8 Plots of PSF vs. W20/λ forannularpupil..................... 16 3.9 Plots of MTF vs. W20/λ forannularpupil.................... 16 3.10 Plots of PSF vs. W20/λ forGaussianpupil.................... 17 3.11 Plots of MTF vs. W20/λ forGaussianpupil................... 17 3.12 Plots of PSF vs. W20/λ fordifference-of-Gaussianspupil............ 18 3.13 Plots of MTF vs. W20/λ fordifference-of-Gaussianspupil............ 19 3.14PlotsofStrehlratioforvariouspupils...................... 22 3.15 Plots of minimum (u, v; W20)/ (u, v;0) vs. W20/λ for various pupils. 24 |H H | 3.16 Plot of minimum (u, v; W20)/ (u, v;0) vs. wb for difference-of-Gaussians pupil.........................................|H H | 25 3.17Spokepattern.................................... 27 3.18Spokepatternimagedwithmisfocusedcircularpupil.............. 28 3.19Spokepatternimagedwithmisfocusedannularpupil.............. 29 3.20SpokepatternimagedwithmisfocusedGaussianpupil............. 30 vi LIST OF FIGURES vii 3.21Spokepatternimagedwithmisfocuseddifference-of-Gaussianspupil...... 31 4.1SystemtocreatetheSDoGpupil......................... 33 4.2 Plot of minimum (u, v; W20)/ (u, v;0) vs. wb for SDoG pupil (β =1).. 36 |H H | 4.3 Plot of minimum (u, v; W20)/ (u, v;0) vs. β forSDoGpupil........ 37 |H H | 4.4 Plot of minimum (u, v; W20)/ (u, v;0) vs. wb for SDoG pupil (β =1.08). 38 |H H | 4.5 Plots of PSF vs. W20/λ for SDoG pupil (β =1)................. 38 4.6 Plots of MTF vs. W20/λ for SDoG pupil (β =1)................ 39 4.7 Plots of PSF vs. W20/λ for SDoG pupil (β =1.08)............... 39 4.8 Plots of MTF vs. W20/λ for SDoG pupil (β =1.08)............... 40 4.9 Plots of extremely misfocused PSF and MTF for SDoG pupil (β =1)..... 41 4.10 Plots of extremely misfocused PSF and MTF for SDoG pupil (β =1.08). 41 4.11 Plots of PSF vs. W20/λ for SDoG pupil (β = βw)................ 42 4.12 Plots of MTF vs. W20/λ for SDoG pupil (β = βw)................ 42 4.13PlotsofStrehlratioforSDoGpupil....................... 43 4.14 Plots of minimum (u, v; W20)/ (u, v;0) vs. W20/λ forSDoGpupil..... 44 |H H | 4.15 Spoke pattern imaged with misfocused SDoG pupil (β =1,W20 = λ/3). 46 4.16 Spoke pattern imaged with misfocused SDoG pupil (β =1.08, W20 = λ/3). 47 4.17 Spoke pattern imaged with misfocused SDoG pupil (β =1,W20 = λ/2). 48 4.18 Spoke pattern imaged with misfocused SDoG pupil (β =1.08, W20 = λ/2). 49 4.19 Spoke pattern imaged with misfocused SDoG pupil (β =1,W20 = λ)..... 50 4.20 Spoke pattern imaged with misfocused SDoG pupil (β =1.08, W20 = λ). 51 List of Tables 3.1 Maximum W20 valuesforvariouspupils,foundusingStrehlratio....... 21 3.2 Maximum W20 values for various pupils, found by applying Hopkin’s criterion. 23 4.1 Maximum W20 valuesforSDoGpupil,foundusingStrehlratio........ 44 4.2 Maximum W20 values for SDoG pupil, found by applying Hopkin’s criterion. 44 5.1