OPTIMAL BEAM FORMING FOR LASER BEAM PROPAGATION ...

AcknowledgementFirst and foremost, I would like to thank my advisor Dr. Timothy Schulz, forhis support, guidance, and encouragement in the Ph.D program. He showedme the way to approach research problems and taught me how to express myideas. I am deeply impressed by his inspiring and encouraging way to guideme to a deeper understanding of knowledge. I am grateful for his advise andcomments in this dissertation.I also would like to thank the rest of my thesis committee: Dr. MichaelRoggemann, Dr. Jeffrey Burl, and Dr. Mark Gockenbach, for their kindhelp and valuable comments on this work. Thanks also go to my friends andcolleagues, who give me help and have fun with me. I really enjoy the lifewith them in the beautiful Keweenaw Peninsula.Last, I wish to thank my parents, my sister, and my brother, for theircontinued support and encouragement. I am also very grateful to my wifeLing, for her love and patience during the Ph.D period.v

ContentsAbstractAcknowledgementList of FiguresList of Tablesivvxivxvi1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Research Goals . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Dissertation Organization . . . . . . . . . . . . . . . . . . . . 72 Background 82.1 Reasons for Optical Disturbance in Random Medium . . . . . 82.2 Waves in Random Media . . . . . . . . . . . . . . . . . . . . . 112.3 Turbulence Effects on Beam Propagation . . . . . . . . . . . . 122.4 Adaptive Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Mathematical Models 163.1 Scalar Diffraction in Homogeneous Medium . . . . . . . . . . 163.2 Optical Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 19vi

3.2.1 Paraxial waves and Gaussian beam . . . . . . . . . . . 193.2.2 Hermite-Gaussian beams . . . . . . . . . . . . . . . . . 213.2.3 Laguerre-Gaussian beams . . . . . . . . . . . . . . . . 223.3 Optical Coherence . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 Mutual coherence function . . . . . . . . . . . . . . . . 253.3.2 Propagation of coherence . . . . . . . . . . . . . . . . . 283.3.3 Coherent mode representation . . . . . . . . . . . . . . 293.4 Statistical Model of Atmospheric Turbulence . . . . . . . . . . 303.4.1 Kolmogorov turbulence model . . . . . . . . . . . . . . 313.4.2 Power spectrum of refractive index fluctuations . . . . 323.5 Wave Propagation through Turbulence . . . . . . . . . . . . . 343.5.1 Inhomogeneous wave equation . . . . . . . . . . . . . . 343.5.2 Born approximation . . . . . . . . . . . . . . . . . . . 363.5.3 Rytov approximation . . . . . . . . . . . . . . . . . . . 373.5.4 Extended Huygens-Fresnel principle . . . . . . . . . . . 393.5.5 Turbulence layer model . . . . . . . . . . . . . . . . . . 413.6 Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . 423.6.1 Maximization of the integrated intensity . . . . . . . . 433.6.2 Minimization of the scintillation index . . . . . . . . . 434 Optimal Beam to Maximize the Integrated Intensity 454.1 Optimal Beam in Free Space . . . . . . . . . . . . . . . . . . . 494.2 Optimal Beam to Maximize the Average Integrated Intensity . 504.3 Averaged Kernel in Turbulence . . . . . . . . . . . . . . . . . 52vii

4.3.1 Averaged kernel for the phase screen model . . . . . . 524.3.2 Averaged kernel for the extended Huygens-Fresnel principlemodel . . . . . . . . . . . . . . . . . . . . . . . . 564.4 Performance of the Optimal Beams . . . . . . . . . . . . . . . 604.4.1 One dimensional experiment . . . . . . . . . . . . . . . 614.4.2 Two dimensional experiment . . . . . . . . . . . . . . . 644.4.3 Performance analysis . . . . . . . . . . . . . . . . . . . 645 Optimal Beam to Minimize the Scintillation Index 675.1 Optimal Pupil Plane Mutual Coherence . . . . . . . . . . . . . 685.2 Optimal Beams in Random Phase Screen Model . . . . . . . . 715.2.1 First moment of the integrated intensity . . . . . . . . 725.2.2 Second moment of the integrated intensity . . . . . . . 725.3 Optimal Beams in the extended Huygens-Fresnel principle model 735.3.1 First moment of the integrated intensity . . . . . . . . 745.3.2 Second moment of the integrated intensity . . . . . . . 755.4 Selection of coherent modes . . . . . . . . . . . . . . . . . . . 775.4.1 Scintillation reduction by using Hermite-Gaussian beams 785.4.2 Scintillation reduction by using Laguerre-Gaussian beams 815.5 Monte-Carlo Simulation . . . . . . . . . . . . . . . . . . . . . 855.5.1 Phase screen generation . . . . . . . . . . . . . . . . . 865.5.2 Pupil plan phase screen simulation . . . . . . . . . . . 885.5.3 Multiple phase screen simulation . . . . . . . . . . . . 925.6 Aperture Averaging of Scintillation . . . . . . . . . . . . . . . 94viii

6 Summary 996.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Appendices 102A Glossary of Symbols 102Bibliography 105ix

List of Figures3.1 The scalar diffraction of optical wave from the aperture planeto the target plane. . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Hermite Gaussian beams f l,m (x, y) with l = 0, 1, 2, 3 for columnsfrom left to right and m = 0, 1, 2, 3 for rows from top to bottom . 233.3 Laguerre Gaussian beams f p,m (r, θ) with p = 0, 1, 2, 3 for rowsfrom top to bottom and m = 0, 1, 2, 3 for columns from left toright. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Propagation of optical coherence. . . . . . . . . . . . . . . . . 283.5 Turbulence layer propagation model: (a) Pupil plane phasescreen model. (b) Multiple phase screen model. . . . . . . . . 424.1 Free space diffraction of focused field from circular aperture:λ = 10 −6 , d = 10 4 , D = 0.2, all units in meters. (a) Transmittingaperture a(⃗u). (b) Normalized amplitude of the receivedfield g(⃗v). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46x

4.2 Airy pattern: (a)Normalized Airy pattern slice, (b)Fractionalintegrated intensity I/I 0 vs. normalized target plane radius(rD)/(λd) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3 Kernel H 0 (u, u ′ ) for one dimensional propagation in free spacewith rectangular weighting function. . . . . . . . . . . . . . . 514.4 Averaged kernel 〈H(u, u ′ )〉 for one dimensional propagationthrough turbulence with rectangular weighting function andr 0 = 10 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.5 Weighting function changes: (a) Original weighting function.(b) New weighting function in turbulence. . . . . . . . . . . . 564.6 Optimal beams and integrated intensity for one dimensionalpropagation in free space and random medium: (a) Optimalbeam ˆf(u) at transmitter aperture, (b) Fractional integratedintensity I/I 0 vs. receiver plane normalized radius (rD)/(λd). 634.7 Optimal beams and integrated intensity for two dimensionalpropagation in free space and random medium: (a) A centerslice of the optimal beam ˆf(⃗u) at transmitter aperture, (b)Fractional integrated intensity I/I 0 vs.receiver plane normalizedradius (rD)/(λd) for focused beam and optimal beam. 655.1 Amplitude of one dimensional Hermite-Gaussian beams f m (x). 79xi

5.2 Scintillation index for transmitting Hermite-Gaussian modes:(a) Scintillation index for single modes.(b) Reduction ofthe optimal scintillation index when the number of Hermite-Gaussian modes increased. . . . . . . . . . . . . . . . . . . . . 815.3 One dimensional Laguerre-Gaussian beams f p,m (x) with m = 0. 835.4 Scintillation index by transmitting Laguerre-Gaussian modes:(a) Scintillation index by transmitting single modes. (b) Reductionof the optimal scintillation index when the number ofLaguerre-Gaussian modes increased . . . . . . . . . . . . . . . 845.5 Theoretical and Monte-Carlo simulation of the phase structurefunction D φ (∆u) and 〈exp[φ(u 1 ) − φ(u 2 )]〉 of the randomphase screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.6 Monte-Carlo simulation by transmitting single one dimensionalHermite-Gaussian modes f m (x): (a) Scintillation index, (b)average integrated intensity. . . . . . . . . . . . . . . . . . . . 885.7 Monte-Carlo simulation by transmitting the optimal coherencecomposed of multiple one dimensional Hermite-Gaussianmodes f m (x): (a) Scintillation index, (b) average integratedintensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.8 Monte-Carlo simulation by transmitting single one dimensionalLaguerre-Gaussian modes f m,0 (x): (a) Scintillation index, (b)average integrated intensity. . . . . . . . . . . . . . . . . . . . 90xii

5.9 Monte-Carlo simulation by transmitting the optimal coherencecomposed of multiple one dimensional Laguerre-Gaussianmodes f m,0 (x): (a) Scintillation index, (b) average integratedintensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.10 Monte-Carlo simulated performance of transmitting single twodimensional Hermite-Gaussian beam: (a) Scintillation index,(b) average integrated intensity. . . . . . . . . . . . . . . . . . 915.11 Monte-Carlo simulated performance of transmitting the optimalpupil plane coherence composed of multiple two dimensionalHermite-Gaussian beams: (a) Scintillation index, (b)average integrated intensity. . . . . . . . . . . . . . . . . . . . 925.12 Monte-Carlo simulated performance of transmitting single twodimensional Laguerre-Gaussian beams: (a) scintillation index,(b) average integrated intensity. . . . . . . . . . . . . . . . . . 935.13 Monte-Carlo simulated performance of the optimal coherencecomposed of multiple two dimensional Laguerre-Gaussian modes:(a) scintillation index, (b) associated average integrated intensity.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.14 Monte-Carlo simulated performance of transmitting single twodimensional Hermite-Gaussian beams through multiple phasescreens: (a) scintillation index, (b) average integrated intensity. 95xiii

5.15 Monte-Carlo simulated performance of transmitting optimalpartial coherence synthesized by two dimensional Hermite-Gaussian beams through multiple phase screens: (a) scintillationindex, (b) average integrated intensity. . . . . . . . . . . . 955.16 Monte-Carlo simulated performance of transmitting single twodimensional Laguerre-Gaussian beams through multiple phasescreens: (a) scintillation index, (b) average integrated intensity. 965.17 Monte-Carlo simulated performance of transmitting optimalpartial coherence synthesized by two dimensional Laguerre-Gaussian beams through multiple phase screens: (a) scintillationindex, (b) average integrated intensity. . . . . . . . . . . . 965.18 Aperture averaging effect in the pupil plane phase screen model:(a) Scintillation index vs. the normalized receiver radius forsingle Hermite Gaussian modes f m (x).(b) Optimal scintillationindex vs.the normalized receiver radius for optimalbeams synthesized by N Hermite-Gaussian modes. . . . . . . . 98xiv

List of Tables4.1 Physical parameters used in the one dimensional propagationsimulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1 Propagation parameters used in the one dimensional simulation. 795.2 Fractional integrated intensity ξ m in free space, fractional averageintegrated intensity µ m through turbulence, and the scintillationindex S m of the received fields by transmitting singleHermite-Gaussian modes f m (u). . . . . . . . . . . . . . . . . . 805.3 Mutual intensity 〈I m I n 〉 by using Hermite Gaussian beams. . 805.4 Optimal weighting α m , scintillation index S I , and the fractionalaverage integrated intensity µ when the first N Hermite-Gaussian modes f m (x) are used to generate the optimal pupilplane partially coherence. . . . . . . . . . . . . . . . . . . . . 825.5 Fractional integrated intensity ξ p in free space, fractional averageintegrated intensity µ p through turbulence, and the scintillationindex S p of the received fields by transmitting singleLaguerre-Gaussian modes f p,0 (x). . . . . . . . . . . . . . . . . 84xv

5.6 Optimal weighting α p , scintillation index S I , and the fractionalaverage integrated intensity µ when the first N Laguerre-Gaussian modes f p,0 (u) are used to generate the optimal pupilplane partially coherence. . . . . . . . . . . . . . . . . . . . . 855.7 Mean square error ɛ 2 of the Monte-Carlo simulation for differentperformance values: µ m , S m and 〈I m I n 〉. . . . . . . . . . . 895.8 Parameters used in the Monte-Carlo simulation of laser beamspropagate through multiple random phase screens. . . . . . . . 94xvi

Chapter 1Introduction1.1 MotivationPropagation of optical waves through random media such as the atmosphere,the ocean, and biological matter, is very important in many applications suchas optical communications and astronomical imaging. In these situations, thetemporally and spatially varying media change the amplitude and phase ofthe propagating optical fields in both time and space. These effects can causeattenuation or loss of the transmitted information and energy.Because of their directionality and high energy concentration, laser beamsare widely used in modern optical communications, remote sensing, and laserweapon systems. The performance of all these systems is limited by turbulenceeffects. Following is an introduction to some important applications ofwave propagation through random medium.1

Free space optical communicationsBecause of the increasing need for bandwidth and high data rates , opticalcommunication and network systems continue to be more and more important.Free space optical communications using laser beams have been developedfor line-of-sight information transmission, such as in satellite communications.Systems with infrared, visible, and ultraviolet carrier frequenciesare all included in optical communications. There are many advantages ofoptical communications compared with an RF system, such as wider bandwidthand higher power density in an extremely narrow beam [27]. But thedetrimental effects of the propagation media form the main drawback of thesystem, because optical wavelengths are commensurate with molecule andparticle sizes. Even in clear sky situations, laser beams are subject to turbulence,scattering and absorption through the atmosphere. In bad weathersuch as fog, rain and snow, the received optical power can be decreased andthe bit error rate increased significantly.Laser radar systemLaser radar, or lidar, is a short wave extension of radar, which uses radiationat wavelengths that are 10, 000 to 100, 000 times shorter than those used byconventional radar. Radiation scattered by a target is collected and processedto yield information about the target and the path to the target. By usinghigher frequency laser beams, higher accuracy and resolving power can beapproached with lidar systems. Phase and amplitude fluctuations of the laser2

eam due to turbulent media can, however, disrupt lidar’s functions such astarget tracking, imaging and ranging.Astronomical imagingAstronomers have known for many years that the Earth’s atmospheric propagationpath causes phenomena such as image jitter and star twinkling. Thephase distortions induced by propagation in turbulence cause degradationproblems in imaging systems such as image blurring and image jitter whichboth contribute to a loss of resolution. This is the main reason that the HubbleSpace Telescope was placed in space. Long exposure images from groundbasedtelescopes are severely blurred due to the much broader point spreadfunction (PSF), a result of the average effects of turbulence. The Earth’sturbulent atmosphere can also degrade radio astronomy systems. Althoughlonger wavelength and different receivers (radio-telescopes) are used in radioastronomy, atmospheric turbulence has similar effects on optical and radioastronomy because both light and radio waves can be corrupted by randommedia.Laser guide starReference beacons are needed in adaptive optics systems because wavefrontcompensation requires measurement of the turbulent propagation path. Twomain requirements for the reference source are that it have enough brightnessand it is located within the isoplanatic patch of the desired view angle [33].Because there are few natural stars or objects that satisfy these requirements,3

man-made sources such as laser guide stars are developed to provide therequired references. In the laser guide star system, high power laser beams areused to generate an artificial beacon in the sky by Rayleigh/Raman scatteringor by sodium resonance scattering.Due to turbulent propagation paths,the generated laser beacon is subject to problems, such as broadening andreduced brightness.Remote sensingRemote sensing is now the principal technology used to observe, measure, andinterpret an object without coming into direct contact with it. By utilizingsensing instruments, radiation of different wavelengths reflected or emittedfrom distant objects is detected and measured. Spatially distributed informationabout the object, such as spectral and physical properties, are acquiredor measured within the sensing instrument. Laser beams are used in remotesensing to probe the properties of turbulent plasmas, hydrodynamic flowsand the earth’s atmosphere. Knowledge of meteorological parameters suchas wind velocity and the strength of turbulence has become increasinglyimportant in the study of optical waves propagation through atmosphericturbulence [66].Diffraction tomographyWave propagation through random media can also find applications in thethree-dimensional reconstruction of absorbing and scattering objects. Theaccuracy of geometrical-radiation based computed tomography suffers from4

the effects of refraction and diffraction. When the sizes of inhomogeneitiesin the object become comparable to or smaller than the wavelength, it is notappropriate to use geometric propagation based ray theory.Tomographicreconstruction approaches based on Fourier diffraction theory are developedwhen the interaction of an object and a field is described by the inhomogeneouswave equation. This method is the so called diffraction tomography.In diffraction tomography, acoustic and electromagnetic waves do not travelalong straight rays and the projections are not line integrals [40]. Diffractionand scattering of optical fields in the inhomogeneous medium must beinvolved.1.2 Research GoalsIt is well known that the propagation of waves in homogeneous media canbe described by the wave equation, which is a second order linear differentialequation. Optical fields in any position inside a homogeneous medium canbe determined by the source fields in the aperture and the wave equation.However, for the problem of wave propagation in an inhomogeneous medium,there is no direct analytic solutions for the wave equation. In practice, approximateformalisms of the homogeneous medium wave propagation arewidely used in the cases of weak inhomogeneities. Born and Rytov approximationsare well known approximate methods [37]. Many other propagationmodels have been developed recently, including linear system representationsand simulation schemes [23, 24].5

Historically, most of the original work on optical propagation deals withuniform plane or spherical waves. Recently, however, the propagation of otherbeams, such as truncated Gaussian beams, through random media have beenwidely studied [47]. But the problem of what type of optical beam profile willperform better through random media is seldom studied. By saying “performbetter”, we mean that the optical beam is less sensitive to turbulence effectssuch as beam spreading and scintillation. To be more precise, specific criteriaare needed to evaluate the beam performance.In this research, we will study the statistical properties of the opticalbeams after propagation through turbulent media. The effects of both sourcecoherence and random media are investigated. The goal of this research is tocompensate the random media effects in beam propagation by changing thetransmitted optical fields or choosing the optical source coherence for certainperformance metrics. Compensated beams are expected to concentrate morepower in the target region or reduce the beam scintillation effectively.In the study of the optimal beam to maximize the weighted integrated intensity,we derive a similarity relationship between pupil-plane phase screenand extended Huygens-Fresnel model, and demonstrate the limited utilityof maximizing the average integrated intensity. In the study of the optimalbeam to minimize the scintillation index, we derive the first- and second-ordermoments for the integrated intensity of multiple coherent modes. Hermite-Gaussian and Laguerre-Gaussian modes are used as the coherent modes tosynthesize an optimal partially coherent beam. The optimal beams demon-6

strate evident reduction of scintillation index, and prove to be insensitive tothe aperture averaging effect.1.3 Dissertation OrganizationThe rest of this dissertation is organized as follows.In Chapter 2, I willdescribe the background of laser beam propagation through random media.Mathematic models of beam propagation, statistical random media, and optimizationcriterions are introduced in Chapter 3.The optimal beams tomaximize the concentrated power and to minimize the scintillation index areinvestigated in Chapter 4 and Chapter 5 respectively. A summary and theconclusions of this research are given in Chapter 6.7

Chapter 2BackgroundBecause of its importance in many applications, wave propagation throughrandom media has been studied by many researchers [73, 37]. In this chapter,we give an overview of the turbulence effects on beam propagation.2.1 Reasons for Optical Disturbance in RandomMediumRandom media may be grouped into three categories: random scatters, randomcontinua, and rough surfaces [37]. Random scatters are a random distributionof many discrete scatters such as rain, fog, aerosols and bloodcells.Random continuum refers to a medium whose index of refractionvaries randomly and continuously in time and space, such as troposphericand ionospheric turbulence, planetary atmospheres, solar corona, clear airturbulence and biological media.The interaction of light with random media causes absorption, scattering,and distortions induced by random fluctuations of the refractive index. These8

are the most important causes of all types of random media effects in wavepropagation and may happen all at once in one propagation case.AbsorptionWhen laser radiation is propagated in an absorbing medium, heating of themedium takes place and causes temperature gradients in the media. Thiseffect will in turn cause media density changes and result in refractive indexgradients being formed across the beam. These gradients act as a distributedlens which distorts the beam and decreases the peak intensity at thefocal point [39]. This self-induced distortion and spreading of the beam iscalled thermal blooming, which is commonly met in high power laser beampropagation [70, 54].ScatteringScattering is the direction redistribution phenomenon caused by rough surfaceor small particles in the media. Rayleigh scattering, Mie scattering, andnonselective scattering are common situations associated with different sizesof the scatter.The scattering effect of particles of matter was first studied by LordRayleigh, who studied the single scattering of a plane wave with a dielectricsphere in a vacuum and deduced that the color of the sky is the resultof stronger scattering in the radiation of shorter wavelengths [55, 46].Rayleigh’s scattering theory was extended by Mie, who considered light scatteringby spheres of different sizes, comparable with the wavelength of light,9

and included the effect of absorption in his study [51]. Schuster studied theradiation in foggy atmospheres and initiated the radiative transport theory[37]. He is the first to consider multiple scattering, since single scatteringtheory becomes inadequate when the particle density is high.TurbulenceOptical turbulence is a kind of random continuum with continually varyingrefractive index.People knew very early that star twinkling at nightwas caused by atmosphere turbulence, the density fluctuations caused by atmospherictemperature fluctuations. The induced random variations of theindex-of-refraction produces phase distortion of the wavefront and continuesto distort the amplitude for further propagation.Turbulence effects suchas image blurring, laser beam spreading and scintillation can degrade theoptical system performance severely.Random optical turbulence can be described by its statistical properties,which are important in the study of wave propagation.Kolmogorov proposeda widely accepted model based on the mechanical structure of theturbulence [44, 45]. From this turbulence model, power spectra of refractiveindex fluctuations are developed and modified with a wide range of usefulnessin inhomogeneous wave propagation.In this study, we will focus on the turbulence effects on optical beampropagation. It is assumed that atmospheric turbulence is a non-dissipativemedium for the propagating wave, which means that absorption and ther-10

mal blooming can be ignored and there is no energy lost due to scatteringturbulence eddies. Accordingly, optical effects arising out of absorption andscattering will not be discussed here. The approaches in this study treat theturbulence and its induced effects, such as the index of refractive fluctuations,as a continuum.2.2 Waves in Random MediaThe problem of wave propagation through random media has been studiedby many researchers and a great deal of progress has been made in the pastseveral decades [12, 73, 72]. Many propagation models and compensationschemes have been proposed and applied. Most of these schemes can be usedin both image compensation and adaptive beam control.Optical field propagation through turbulence can be described by the inhomogeneouswave equation with a randomly changing index of refraction.Because there is no analytic solution of the inhomogeneous wave equation,many approximate solutions are developed by using the small perturbationmethod. Earliest studies of this problem employed the geometric optics approximation[9], which is of very limited utility. A new method based on theRytov approximation was developed by Tatarski in the late 1950’s [73, 12]and has been adopted as the basic tool in this area by most researchers[61, 36].The Rytov method has proved to agree with experimental data pretty wellin weak turbulence cases. But in strong turbulence and long distance prop-11

agation cases, the experimental data does not match the predictions madeusing the Rytov approximation [32, 35]. New theories has been developed forthese situations, such as the diagrammatic techniques [11, 18, 17], transportmethods [22], coherence theory [8, 52], Markov approximation [74] and severalother approaches. Linear system representations of the inhomogeneouspropagation were also developed [49].The properties of wave propagation in random media and many importantresults obtained in recent history have been summarized in numerous reviewsand books [23, 24, 37, 25, 30, 59, 80, 7, 1, 2].2.3 Turbulence Effects on Beam PropagationFor laser beams propagating in turbulence, the cumulative effect of the refractiveindex fluctuation can cause significant wavefront and intensity distortions.Beam wander, beam spreading and scintillation are the main beamdistortions resulting from turbulent media.Beam wanderWhen the optical beam deviates from its original path, this is called beamwander. The phase front distortion induced by the random fluctuations inphase velocity alters the direction of beam propagation. The statistics of thisangle of arrival fluctuations have been widely studied [17, 36]. Measured dataof beam wander under different conditions show that it is nearly independentof wavelength [80, 13, 43].This random displacement of the beam spot12

centroid is caused by large eddies in the turbulence and the detected intensityfor a detector appears to follow a Rayleigh distribution of probability in somecases [23].Beam spreadingBeam spreading means that the optical beam size spreads beyond the dimensionspredicted by diffraction theory, which is sometimes also called beambroadening. In the absence of turbulence, a laser beam exiting from an apertureof diameter D would, in the far field, have an angular spread θ of orderλ/D. The beam spreading is induced by small scale turbulence eddies inthe propagation media and is usually wavelength dependent. Beam width orbeam radius is commonly used as a measure of the beam spreading [19].Beam spreading and beam wander are the beam counterparts to imageblurring and image dancing. Beam wander is important for beam pointingand tracking considerations, while the short-time broadening is importantfor pulse propagation and high-energy laser systems [7].ScintillationPropagation of distorted wave fronts causes beam intensity fluctuations.Random deflection and interference between different portions of the wavefrontlead to an internal breaking up of the beam spot into smaller “hotspots”. This intensity fluctuation is described as scintillation, which includesboth the temporal and spatial variations in received intensity within the receiveraperture. The scintillation phenomenon can be easily observed with13

the naked eye: bright stars close to the horizon twinkle strongly and changecolor on a time scale of seconds. The temporal frequency of the intensityfluctuations in atmospheric turbulence recorded at a fixed point within thebeam usually varies between 1 and 100 Hz.Due to the importance of scintillation in optical detection, previous researchhas led to extensive theoretical calculations [73, 36] and experimentaldata [42, 20]. Most of the theoretical studies are based on the Rytov methodand use the scintillation index as a measurement. Then scintillation indexis defined as the normalized variance of the intensity fluctuations, which is afunction of the optical receiver size. As the diameter of the receiving apertureis increased, the magnitude of the fluctuation in the received power decreases.This is the so called aperture averaging effect which has been widely studiedby many researchers [26, 43, 34, 14, 3].2.4 Adaptive OpticsBecause turbulence effects severely degrade the system performance in applicationssuch as imaging and optical communications, many compensationschemes based on adaptive optics have been proposed in recent history. Theconcept of adaptive optics was first proposed by Babcock in the 1950s as apossible method of compensating atmospheric turbulence in telescope imaging[4].The core of adaptive optics is to measure the aberrations of anincoming wavefront and then cancel these in real time by introducing anelectronically controlled phase shift.In the late 1960s, the first adaptive14

optics systems for laser beam control were developed to compensate for atmosphericturbulence in real time. As well as applications in astronomicalimaging, adaptive optics has also been widely studied and used in retinalimaging and vision correction. Many surveys and reviews in this area havebeen summarized [5, 58, 75, 33].There are three main components in a conventional adaptive optics system:wavefront sensing, electronic controller, and wavefront compensator.Turbulence properties can be measured from the distorted wavefront by usingwavefront sensing devices such as the shear-interferometer and Hartmannwavefront sensors. According to this wavefront measurement, the controllersends out actuator positioning control signals to wavefront compensatorssuch as deformable mirrors. The whole process is expected to perform ata higher rate than the rate of turbulence changing since this is a real timecompensation.Adaptive optics also have important applications in laser beam propagationthrough turbulence, such as the target-in-the-loop (TIL) problem. InTIL, laser beams are projected toward a target through random media andpropagated back after surface scattering at the target.This problem hasbeen involved in many applications such as laser target designation, remotesensing, and laser weapon systems [77]. In this dissertation, we will focus onthis laser beam projection problem.15

Chapter 3Mathematical ModelsIn this chapter, we describe the basic mathematic models that are used inthis research for beam propagation, optical coherence, and atmosphere turbulence.Some terms and optimization metrics are also defined for beamperformance measurement.3.1 Scalar Diffraction in Homogeneous MediumThe scalar wave function f(⃗r, t) propagating in a homogeneous medium mustsatisfy the scalar wave equation [10]∇ 2 f(⃗r, t) − 1 c 2 ∂ 2 f(⃗r, t)∂t 2 = 0, (3.1)where ⃗r = (x, y, z) is the position vector, ∇ 2 is the Laplacian operator:∇ 2 = ∂2∂x 2 + ∂2∂y 2 + ∂2∂z 2 , (3.2)16

YYouXθvXAZZ = 0Z = dFigure 3.1: The scalar diffraction of optical wave from the aperture plane tothe target plane.and c is the propagation speed of the wave in the medium. For monochromaticwaves, the complex wave function f(⃗r, t) can be written in the formf(⃗r, t) = f(⃗r)e j2πνt , (3.3)where f(⃗r) is the complex amplitude and satisfies the Helmholtz equation:(∇ 2 + k 2 )f(⃗r) = 0, (3.4)where k = 2π/λ is the wave number. For homogeneous media, the wavelengthand corresponding wave number is constant. For propagation in inhomogeneousmedium, the wave number can be expressed as a function ofthe position k(⃗r).Consider the wave propagation from aperture plane z = 0 to plane z = das shown in Figure 3.1. By applying the integral theorem of Helmholtz andKirchhoff, the wave diffraction by an aperture A in an infinite opaque screen17

yields the Huygens-Fresnel diffraction [31]∫g(⃗v) =Ah(⃗v, ⃗u)f(⃗u)d⃗u, (3.5)where ⃗u = (x, y) and ⃗v = (x, y) are position vectors in the transverse planesz = 0 and z = d, f(⃗u) is the optical field at the aperture A and h(⃗v, ⃗u) isthe system impulse response [31]h(⃗v, ⃗u) = 1 exp(jkr)cos θ, (3.6)jλ rwhere r = √ ‖⃗u + ⃗v‖ 2 + d 2 is the distance from point (⃗u, 0) to point (⃗v, d).The scalar diffraction can also be analyzed in the Fourier domain. Variousspatial Fourier components of the complex field can be identified as planewaves traveling in different directions [31].It has been shown that wavepropagation is a linear shift invariant system when the direction cosines (α, β)satisfyα 2 + β 2 < 1, (3.7)whereα = λν x , β = λν y . (3.8)Then the propagation of angular spectrum can be described as a linear spatialfilter with the transfer function⎧⎪⎨ exp[jkd √ ]1 − (λν x ) 2 − (λν y ) 2 νx 2 + νy 2 < 1 λH(ν x , ν y ) =2⎪⎩ 0 otherwise, (3.9)where ν x and ν y are the spatial frequencies.18

When the propagation distance d is much greater than the maximumlinear dimension of both the aperture A and the observe region R, and onlya finite region around the z axis is of interest, the following approximationscan be usedcos θ ≈ 1, (3.10)1r≈ 1 d , (3.11)exp(jkr) ≈ exp [ jk(d + ‖⃗v − ⃗u‖ 2 /(2d)) ] . (3.12)Then we get the Green’s function of Fresnel diffraction as3.2 Optical Beamsh(⃗v, ⃗u) ≃ ejkdjλd ej π λd ‖⃗v−⃗u‖2 . (3.13)3.2.1 Paraxial waves and Gaussian beamWe are interested primarily in the paraxial waves because they form beamlikeoptical waves. The complex amplitude of a paraxial wave can be expressedasf(⃗u, z) = a(⃗u, z)e jkz , (3.14)where a(⃗u, z) is a slowly varying function and ⃗u = (x, y) is the positionvector in plane of constant z. Substituting the paraxial wave function intothe Helmholtz equation, and making the assumption that∂a∂z ≪ ka,∂ 2 a∂ 2 z ≪ k2 a, (3.15)19

we find that paraxial waves satisfy the following paraxial Helmhotz equation∇ 2 T a − j2k ∂a∂z= 0, (3.16)where ∇ 2 T = ∂2 /∂x 2 + ∂ 2 /∂y 2 is the transverse Laplacian operator.There are many solutions of the paraxial Helmhotz equation. A commonexample is a uniform spherical wave diverging from the origin with complexamplitudea(⃗u, z) = 1 ( )z exp −jk ‖⃗u‖2 , (3.17)2zwhere ‖⃗u‖ 2 = x 2 + y 2 . Replacing the radius of curvature z by the complexradius q(z) = z + jz 0 in the above equation, we get the common Gaussianbeam:a(⃗u, z) = 1 ( )q(z) exp −jk ‖⃗u‖2 , (3.18)2q(z)where z 0 is the Rayleigh range. By separating the real and imaginary partof 1/q(z) asthe Gaussian beam can be expressed as [60]:1q(z) = 1R(z) − j λπw 2 (z) , (3.19)u(⃗u, z) = 1 ( )]w 0jz 0 w(z) exp − ‖⃗u‖2 exp[−jkz − jk ‖⃗u‖2w 2 (z)2R(z) + jζ(z) , (3.20)20

wherew(z) = w 0[1 +( zz 0) 2] 1/2, (3.21)R(z) = z[1 +( z0) ] 2, (3.22)zζ(z) = tan −1 z z 0, (3.23)w 0 =( ) 1/2 λz0. (3.24)πThe Gaussian beam is the lowest order solution in an infinite solutionfamily of the free space paraxial wave equation. There are higher order solutionssets for the paraxial Helmhotz equation, such as the Hermite-Gaussianbeams in rectangular coordinates and the Laguerre-Gaussian beam in cylindricalcoordinates.3.2.2 Hermite-Gaussian beamsThe Hermite-Gaussian beams are the most widely used solution set of theparaxial wave equation in rectangular coordinates. The complex amplitudefor Hermite-Gaussian beams can be expressed as [67]( ) ( )w 0 √2f l,m (x, y, z) = a l,mw(z) H x √2 ylH m (3.25)w(z) w(z)]× exp[− r2w 2 (z) − jkr22R(z) − jkz + j(l + m + 1)ζ(z) ,where r = √ x 2 + y 2 , H m is the Hermite polynomial of order m:H m+1 (x) = 2xH m (x) − 2mH m−1 (x), (3.26)21

withH 0 (x) = 1, and H 1 (x) = 2x. (3.27)A plot of the transverse amplitude of the Hermit-Gaussian beams is shownin Figure 3.2, for l = 0, 1, 2, 3 and m = 0, 1, 2, 3.3.2.3 Laguerre-Gaussian beamsThe set of Laguerre-Gaussian beams is another complete solution of theparaxial wave equation expanded in cylindrical coordinates. The complexamplitude of Laguerre-Gaussian modes take the form [67]√2p ! exp [j(2p + m + 1)(ζ(z) − ζ 0 )]f p,m (r, θ, z) =(1 + δ 0m )π(m + p)!w(z)( √ ) m ( )]2r 2r× L m 2p exp[−jk r2w(z) w 2 (z)2q(z) + imθ , (3.28)whereL m p (x) =p∑k=0(p + m)!(m + k)!(p − k)!k! (−x)k (3.29)is the generalized Laguerre polynomials and 1/q(z) = √ 2/w(z). In the abovesolution, the integer p ≥ 0 is the radial index and the integer m is theazimuthal mode index. A plot of the transverse distribution of the Laguerre-Gaussian beams is shown in Figure 3.3, for p = 0, 1, 2, 3 and m = 0, 1, 2, 3.22

Figure 3.2: Hermite Gaussian beams f l,m (x, y) with l = 0, 1, 2, 3 for columnsfrom left to right and m = 0, 1, 2, 3 for rows from top to bottom .23

Figure 3.3: Laguerre Gaussian beams f p,m (r, θ) with p = 0, 1, 2, 3 for rowsfrom top to bottom and m = 0, 1, 2, 3 for columns from left to right.24

3.3 Optical CoherenceThe optical coherence theory is used to describe the statistical fluctuationsof an optical field by using coherence functions, and is very important inthe study of optical beam propagation through turbulence. Basic conceptsand methods of coherent optics are introduced in this section. We follow thenotations in [50].3.3.1 Mutual coherence functionPerhaps the most important concept in statistical optics is the mutual coherence,which is defined as the cross correlation function of two optical fieldsΓ(⃗r 1 , ⃗r 2 ; t 1 , t 2 ) = 〈f ∗ (⃗r 1 , t 1 )f(⃗r 2 , t 2 )〉 , (3.30)where ∗ denotes the complex conjugate and 〈·〉 denotes ensemble average.For stationary fields, the mutual coherence becomesΓ(⃗r 1 , ⃗r 2 ; τ) = 〈f ∗ (⃗r 1 , t)f(⃗r 2 , t + τ)〉 . (3.31)There are two types of coherence, temporal coherence characterized by Γ(⃗r, ⃗r, τ)and spatial coherence characterized by Γ(⃗r 1 , ⃗r 2 , 0). The average field intensityis expressed in terms of the coherence asi(⃗r, t) = Γ(⃗r, ⃗r, 0) = 〈f ∗ (⃗r, t)f(⃗r, t)〉 . (3.32)From the relationship between correlation functions of complex analytic randomprocesses, the cross correlation function Γ(⃗r 1 , ⃗r 2 ; τ) of an analytic func-25

tion f(⃗r, t) is also an analytic function and its real and imaginary part forma Hilbert transform pair [50]:andIm Γ(⃗r 1 , ⃗r 2 ; τ) = 1 π P ∫ ∞−∞Re Γ(⃗r 1 , ⃗r 2 ; τ) = − 1 π P ∫ ∞−∞Re Γ(⃗r 1 , ⃗r 2 ; τ ′ )dτ ′ , (3.33)τ ′ − τIm Γ(⃗r 1 , ⃗r 2 ; τ ′ )dτ ′ , (3.34)τ ′ − τwhere P denotes the Cauchy principle value of the integrals at τ ′ = τ.A field’s mutual intensity is defined as its correlation function at a timelag of zero:J(⃗r 1 , ⃗r 2 ) = Γ(⃗r 1 , ⃗r 2 , 0). (3.35)To measure a field’s degree of coherence, the complex degree of coherence isdefined as the normalized mutual coherence functionγ(⃗r 1 , ⃗r 2 ; τ) =Γ(⃗r 1 , ⃗r 2 ; τ). (3.36)[Γ(⃗r 1 , ⃗r 1 ; 0)] 1/2 1/2[Γ(⃗r 2 , ⃗r 2 ; 0)]It can be seen that0 ≤ | γ(⃗r 1 , ⃗r 2 ; τ)| ≤ 1. (3.37)Mutual coherence can also be measured in the frequency domain. Thecross spectral density (or cross power spectrum) is defined as the correlationof the spectral amplitudes at frequency ν:W (⃗r 1 , ⃗r 2 ; ν) = 〈F ∗ (⃗r 1 , ν)F (⃗r 2 , ν)〉 , (3.38)26

where F (⃗r, ν) is the Fourier transform of field f(⃗r, t):f(⃗r, t) =∫ ∞0F ∗ (⃗r, ν)e −j2πνt dν. (3.39)Then the cross spectral density and the mutual coherence function form aFourier transform pairΓ(⃗r 1 , ⃗r 2 , τ) =W (⃗r 1 , ⃗r 2 , ν) =∫ ∞0∫ ∞−∞W (⃗r 1 , ⃗r 2 ; ν)e −j2πντ dν, (3.40)Γ(⃗r 1 , ⃗r 2 ; τ)e j2πντ dτ. (3.41)The spectral density (or power spectrum) S(⃗r, ν) of light is a special case ofthe cross spectral density W (⃗r 1 , ⃗r 2 , ν) at ⃗r 1 = ⃗r 2S(⃗r, ν) = W (⃗r, ⃗r, ν). (3.42)The spectral degree of coherence is defined as the normalized cross spectraldensity functionµ(⃗r 1 , ⃗r 2 ; ν) =W (⃗r 1 , ⃗r 2 ; ν). (3.43)[W (⃗r 1 , ⃗r 1 ; ν)] 1/2 [W (⃗r 2 , ⃗r 2 ; ν)]1/2The degree of coherence can be measured by the value of |γ(⃗r 1 , ⃗r 2 ; τ)|.Fully coherent and fully incoherent are two special cases that occur whenthe value of |γ(⃗r 1 , ⃗r 2 ; τ)| approaches its two extremes: 1 and 0, respectively.Other states of coherence are called partially coherent.27

YYu 1v 1u 2oXXAv 2ZZ = 0Z = dFigure 3.4: Propagation of optical coherence.3.3.2 Propagation of coherenceTo express the state changes of the optical coherence along propagation, thewave equations for mutual coherence propagation in free space were developedby Wolf [50]∇ 2 1Γ(⃗r 1 , ⃗r 2 ; τ) = 1 ∂ 2c 2 ∂τ Γ(⃗r 1, ⃗r 2 2 ; τ) (3.44)∇ 2 2Γ(⃗r 1 , ⃗r 2 ; τ) = 1 ∂ 2c 2 ∂τ Γ(⃗r 1, ⃗r 2 2 ; τ), (3.45)where ∇ 2 1 and ∇ 2 2 are the Laplacian operators taken with respect to ⃗r 1 and⃗r 2 . Taking the Fourier transform of the above equations yields the Helmholtzequations for monochromatic waves(∇ 2 1 + k 2 )W (⃗r 1 , ⃗r 2 ; ν) = 0, (3.46)(∇ 2 2 + k 2 )W (⃗r 1 , ⃗r 2 ; ν) = 0. (3.47)Consider the propagation scheme as shown in Figure 3.4. For propagation28

from finite surfaces with small diffraction angle, the mutual coherence andthe cross spectral density in the far field can be derived asW (⃗v 1 , ⃗v 2 ; ν)Γ(⃗v 1 , ⃗v 2 ; τ)≈≈∫∫W (⃗u 1 , ⃗u 2 ; ν)h ∗ (⃗v 1 , ⃗u 1 ; ν)h(⃗v 2 , ⃗u 2 ; ν)d⃗u 1 d⃗u 2 ,(3.48)∫∫A 2 Γ(⃗v 1 , ⃗v 2 ; τ)h ∗ (⃗v 1 , ⃗u 1 )h(⃗v 2 , ⃗u 2 )d⃗u 1 d⃗u 2 . (3.49)A 2where h(⃗v, ⃗u) is the impulse response function of wave propagation and τ =(d 1 − d 2 )/c. When τ = 0, Zernike’s propagation law [50] for the mutualintensity is derived as∫∫J(⃗v 1 , ⃗v 2 ) = J(⃗u 1 , ⃗u 2 )h ∗ (⃗v 1 , ⃗u 1 )h(⃗v 2 , ⃗u 2 )d⃗u 1 d⃗u 2 .A 2 (3.50)For a planar and spatially incoherent source at A, assuming that Fresnelapproximation is valid, the mutual intensity in the far field can be derivedasJ(⃗v 1 , ⃗v 2 ) =( ) 2 ∫1e −j π λd (|⃗v 2| 2 −|⃗v 1 | 2 )i(⃗u)e −jk(⃗v 2−⃗v 1 )·⃗u d⃗u. (3.51)λdAThis is the so called van Citter-Zernike theorem [30].3.3.3 Coherent mode representationWe know that for a stationary optical field f(⃗r, t) in some finite closed domainD in free space, its mutual coherence function Γ(⃗r 1 , ⃗r 2 , τ) and the crossspectral density function W (⃗r 1 , ⃗r 2 , ν) form Fourier transform pairsW (⃗r 1 , ⃗r 2 , ν) =∫ ∞−∞Γ(⃗r 1 , ⃗r 2 , τ)e 2πντ dτ. (3.52)29

It has been shown that the cross-spectral density function W (⃗r 1 , ⃗r 2 , ν) isa Hilbert-Schmidt, Hermitian and non-negative definite kernel [50], whichmeans that it has the Mercer expansion as [16]W (⃗r 1 , ⃗r 2 , ν) = ∑ nα n (ν)ψ ∗ n(⃗r 1 , ν)ψ n (⃗r 2 , ν), (3.53)where the functions ψ n (⃗r, ν) are the eigenfunctions and α n (ν) are the eigenvaluesof the integral equation∫W (⃗r 1 , ⃗r 2 , ν)ψ n (⃗r 1 , ν)d⃗r 1 = α n (ν)ψ n (⃗r 2 , ν). (3.54)DBecause the kernel W (⃗r 1 , ⃗r 2 , ν) is Hermitian and non-negative definite, all itseigenvalues α n (ν) are real and non-negative and the eigenfunctions ψ n (⃗r, ν)form an orthonormal set.3.4 Statistical Model of Atmospheric TurbulenceBecause of its nonlinear, non-closure (more variables than equations) andrandom nature [7], the turbulence phenomenon is very difficult to describemathematically and can best be investigated using statistical methods. Inthis section, we introduce the most commonly used statistical models foratmospheric turbulence.30

3.4.1 Kolmogorov turbulence modelKolmogorov investigated the mechanical structure of turbulence, and proposeda widely accepted model on the basis of a set of hypotheses [44, 45]. InKolmogorov’s model, the energy dissipation in fluid medium is from sourceto large structures, and then to small structures. For example, in the atmosphericturbulence, energy moves from the sun heating, through largescale disturbances with outer scale L 0 , which then break down into smallerand smaller structures. These disturbances with different scale sizes are modeledas ”eddies” with statistically varying index of refraction. The continuingkinetic energy transformation (energy cascade) stops and the kinetic energydissipates into heat when the scale size of eddies approach the inner scale l 0 ,where the Reynolds number drops below its critical value and the turbulencedies away.For eddies with scale size l within the inertial subrange (ie., l 0 ≤ l ≤ L 0 ),Kolmogorov showed that the structure function of turbulence conforms to auniversal ”two-third law”:D f (τ) = C 2 f τ 2/3 , (3.55)where the structure function D f (τ) of a random process f(t) was introducedby Kolmogorov asD f (τ) = 〈 [f(t + τ) − f(t)] 2〉 . (3.56)The structure function D f (τ) can filter the slow fluctuations of the random31

process. The fluctuations about the varying mean f(t + τ) − f(t) can bemodeled as a stationary process (when τ is not too large) even though f(t)is not. This is referred to as possessing stationary increments in the timedomain, or locally homogeneous in the spatial domain. For locally homogenousrandom processes, the spectrum is related to the structure function bythe generalized Wiener-Khinchine theorem∫ ∞D f (⃗r) = 2 Φ f (⃗κ) [1 − cos (⃗κ · ⃗r)] d⃗κ, (3.57)−∞where ⃗κ is the vector wave number. For isotropic processes we haveD f (r) = 8π∫ ∞0[Φ f (κ) 1 − sin(κr) ]κ 2 dκ, (3.58)κrandΦ f (κ) = 14π 2 κ 2 ∫ ∞0sin(κr)κrddr[r 2 dD ]f(r)dr. (3.59)dr3.4.2 Power spectrum of refractive index fluctuationsAs we have discussed, refractive index fluctuations induced by solar heatingare the main reason for atmospheric turbulence effects on optical beampropagation. The refractive index of the atmosphere can be expressed as [53]n = 1 + a d (λ) P dT + a w(λ) P wT , (3.60)where a d (λ) is a wavelength dependent function and P d is artial pressure (inmbar) of the dry air. a w (λ) and P w are those for water vapor, and T isthe absolute temperature in Kelvin. According to Kolmogorov’s theory, the32

structure function of refractive index can be expressed asD n (r) = C 2 n r 2/3 , (3.61)where C 2 n is the structure constant of the index of refraction. This is thecritical parameter for describing optical turbulence. Substituting this modelinto the Winer-Khinchine theorems yields the spectrum within the inertialrange asΦ n (κ) = 5 ∫ L018π C2 nκ r −1/3 sin(κ)rdr. (3.62)l 0In the case that 2π/L 0 ≪ κ ≪ 2π/l 0 , with the integration limits approximatedfrom 0 to ∞, we obtain the Kolmogorov spectrum of the refractiveindex fluctuationsΦ n (κ) = 0.033 C 2 nκ −11/3 . (3.63)To resolve the deficiency with large wave numbers in the above form, Tatarski[73] modified it as)Φ n (κ) = 0.033Cnκ 2 −11/3 exp(− κ2, (3.64)κ 2 mwhere κ m = 5.92/l 0 is an inner scale related constant. However the singularityat κ = 0 is still there. This problem is solved in the modified von Karmanspectrum with improved range of usefulnessΦ n (κ) =0.033C2 n(κ 2 + κ 2 0) 11/6 e−κ2 /κ 2 m , (3.65)33

where κ 0 = 2π/L 0 and κ m = 5.92/l 0 . It should be noted that these modificationsare for computational, but not physical, reasons.3.5 Wave Propagation through TurbulenceTurbulence causes spatial and temporal random fluctuations in the index ofrefraction which distorts the Green’s function for a propagation field.Tomodel the propagation of optical waves through turbulence, researchers developedmany methods to provide appropriate statistical descriptions of thepropagated field. In this section, we will introduce the models that are usedin our investigation of the optimal beam through random media.3.5.1 Inhomogeneous wave equationAs mentioned in Section 3.1, scalar field propagation in homogeneous mediais governed by the Helmholtz equation(∇ 2 + k 2 )f(⃗r) = 0. (3.66)For propagation in inhomogeneous media, the wave number k is a functionof the position ⃗r, and the above equation can be modified to form the inhomogeneouswave equation as(∇ 2 + k 2 (⃗r))f(⃗r) = 0. (3.67)34

The wave number k(⃗r) can be expressed as a scalar function of the refractiveindex n(⃗r)k(⃗r) = k 0 n(⃗r), (3.68)where k 0 represents the average wave number of the medium and n(⃗r) is therefractive index of the medium given by√µ(⃗r)ɛ(⃗r)n(⃗r) = , (3.69)µ 0 ɛ 0where µ and ɛ are the magnetic permeability and dielectric constant of themedium, and µ 0 and ɛ 0 are corresponding average values.The refractiveindex n(⃗r) is a random variable and can be written in the average and fluctuationterms asn(⃗r) = 〈n〉 + n s (⃗r). (3.70)In general cases such as propagation in the atmosphere, we can assume that〈n〉 = 1 . Then the wave equation becomes∇ 2 f(⃗r) + k 2 0 [1 + δ n (⃗r)]f(⃗r) = 0, (3.71)whereδ n (⃗r) = 2n s (⃗r) + n 2 s(⃗r). (3.72)It is difficult to solve this inhomogeneous wave equation analytically. Severalapproximations have been developed for certain turbulence cases andsolutions for these approximations can be derived.35

3.5.2 Born approximationIn the Born approximation, the optical field f(⃗v) is expressed as a sum ofthe field in free space, f 0 (⃗v), and the perturbation term, f s (⃗v),f(⃗v) = f 0 (⃗v) + f s (⃗v). (3.73)Substituting the above representation into the wave equation (3.71), we have(∇ 2 + k 2 0)f s (⃗v) = −k 2 0δn(⃗v) f(⃗v). (3.74)This equation can not be solved directly, but the solution can be written interms of the Green’s function as∫f s (⃗v) = k0δ 2 n (⃗u)G(⃗v − ⃗u)f(⃗u)d⃗u, (3.75)where G(⃗v −⃗u) is the Green’s function of the wave equation (3.74), a solutionof the differential equation(∇ 2 + k 2 0)G(⃗v, ⃗u) = −δ(⃗v − ⃗u). (3.76)It can be shown that the Green’s function is only a function of the difference(⃗v − ⃗u).When the perturbation f s (⃗v) is small compared with f 0 (⃗v), theintegral (3.75) can be approximated as∫f s (⃗v) ≃ f B (⃗v) = k0δ 2 n (⃗u)G(⃗v − ⃗u)f 0 (⃗u)d⃗u. (3.77)36

An iterative method can then be taken to approach the final solution andthe ith order Born field can be derived as∫f (i+1)B(⃗v) =k 2 0δ n (⃗u)G(⃗v − ⃗u)[f (i)B (⃗u) + f 0(⃗u)]d⃗u. (3.78)In weak turbulence case, the first order Born approximation is valid and iswidely taken because of its simplicity in calculation.3.5.3 Rytov approximationThe Rytov method states that the optical fields in inhomogeneous media canbe represented in the complex exponential form asf(⃗r) = e ψ(⃗r) , (3.79)Substituting this into the inhomogeneous wave equation (3.71), we obtain anonlinear first order differential equation for ∇ψ, which is called the Riccatiequation:∇ 2 ψ + ∇ψ · ∇ψ + k 2 (1 + δ n ) = 0. (3.80)The complex phase ψ can be written asψ = ψ 0 + ψ s , (3.81)wheref 0 (⃗r) = e ψ 0(⃗r)(3.82)is the solution of the wave equation in the absence of turbulence and ψ saccounts for the random fluctuations. The Riccati equation (3.5.3) can be37

ewritten as∇ 2 ψ s + 2∇ψ 0 · ∇ψ s + (∇ψ s ) 2 + k 2 0δ n = 0. (3.83)The solution of the fluctuation part ψ s can be found to be [37]ψ s (⃗r) = 1 ∫f 0 (⃗r)G(⃗r − ⃗u) [ (∇ψ s ) 2 + k 2 δ n (⃗u) ] f 0 (⃗u)d⃗u. (3.84)As in the Born approximation, this equation can be solved by iteration,yielding a sequence of solutions. In weak turbulence cases, assuming that(∇ψ s ) 2 ≪ k 2 δ n (⃗u), (3.85)the first Rytov solution is derived:ψ s (⃗r) = 1 ∫f 0 (⃗r)G(⃗r − ⃗u)k 2 δ n (⃗u)f 0 (⃗u)d⃗u. (3.86)It has been shown that the conditions for the Rytov approximation to bevalid are less restrictive than that for Born approximation [41]. For smallperturbations, the Rytov approximation provides nearly the same results asthe Born approximation.The wave fields can also be expressed in the form asf(⃗r) = a(⃗r)e jφ(⃗r) , (3.87)andf 0 (⃗r) = a 0 (⃗r)e jφ 0(⃗r) . (3.88)38

We then get( ) aψ s (⃗r) = χ + jφ s = ln + j(φ − φ 0 ), (3.89)a 0where χ denotes the log-amplitude fluctuations and φ s is the phase fluctuations.It will be seen that the following statistical properties of χ and φ s areimportant:B χ (⃗r 1 , ⃗r 2 ) = 〈χ(⃗r 1 )χ(⃗r 2 )〉 (3.90)B φ (⃗r 1 , ⃗r 2 ) = 〈φ s (⃗r 1 )φ s (⃗r 2 )〉 , (3.91)andB χφ (⃗r 1 , ⃗r 2 ) = 〈χ(⃗r 1 )φ s (⃗r 2 )〉 . (3.92)These properties can be obtained by using the autocorrelation function ofthe index of refraction of the turbulent mediaB n (⃗r 1 , ⃗r 2 ) = 〈n(⃗r 1 )n(⃗r 2 )〉 , (3.93)or the corresponding power spectrum of the index of refraction fluctuations.Since it simplifies the calculations and also provides another point of view,the spectral method is commonly used in practice.3.5.4 Extended Huygens-Fresnel principleIn strong turbulence and long path propagation cases, the Born and Rytovapproximations cannot be used. The extended Huygens-Fresnel (EHF) principleproposed by Lutomirski and Yura [49] demonstrated that the solution39

of the inhomogeneous wave equation can be expressed in a linear system representation.In contrast with previous work, the properties of the randommedium appear only in the mutual coherence function of a spherical wave.Applying the extended Huygens-Fresnel principle, the received field atdistance d is expressed as∫g(⃗v) =Ah(⃗v, ⃗u)f(⃗u)d⃗u, (3.94)where h(⃗v, ⃗u) is the spatial impulse response of the system. Reciprocity ofthe response [64] implies that h(⃗v, ⃗u) is symmetric with respect to the sourcepoint and the observer point. Therefore, a point source at ⃗u will produce at⃗v the same effect that a point source of equal intensity at ⃗v will produce at⃗u. The impulse response in random media takes the formwhereh(⃗v, ⃗u) = 1 (jλd exp j 2π ) [exp j π ]λd λd ‖⃗v − ⃗u‖2 + ψ(⃗v, ⃗u) , (3.95)ψ(⃗v, ⃗u) = χ(⃗v, ⃗u) + jφ(⃗v, ⃗u) (3.96)is the random part of the complex phase of a spherical wave propagating inthe random medium from point (⃗u, 0) to point (⃗v, d). Then the intensity attarget position ⃗v can be derived asi(⃗v) = |g(⃗v)| 2( ) 2 ∫∫ 1=expλd A 2[j π (‖⃗v − ⃗u2 ‖ 2 − ‖⃗v − ⃗u 1 ‖ 2)]λd× exp[ψ ∗ (⃗v, ⃗u 1 ) + ψ(⃗v, ⃗u 2 )]f ∗ (⃗u 1 )f(⃗u 2 )d⃗u 1 d⃗u 2 . (3.97)40

It can be seen that the ensemble average 〈exp[ψ ∗ (⃗v, ⃗u 1 ) + ψ(⃗v, ⃗u 2 )]〉 is neededto compute the mean irradiance 〈i(⃗v)〉 from a finite diffracting aperture.Similarly, higher moments of i(⃗v) can be computed from a knowledge ofterms of the form 〈exp[ψ ∗ (⃗v 1 , ⃗u 1 ) + ψ(⃗v 1 , ⃗u 2 ) + ψ ∗ (⃗v 2 , ⃗u 3 ) + ψ(⃗v 2 , ⃗u 4 )]〉, etc.3.5.5 Turbulence layer modelIn some applications, the random media can be modeled by thin turbulencelayers between the transmitter and receiver. These turbulence layers can bemodeled as thin screens, and the field that passes through each screen canbe expressed asf(⃗u) = f 0 (⃗u) t s (⃗u), (3.98)where f 0 (⃗u) is the wave function before the phase screen and t s (⃗u) is thescreen transmittance.In general, t s (⃗u) includes random changes in bothphase and amplitude. Sometimes only phase perturbations are considered.Then we have the random phase screen model asf(⃗u) = f 0 (⃗u) e jφ(⃗u) , (3.99)where φ(⃗u) is the random phase change. Because of its mathematical simplification,the random phase screen model is widely used in areas such aswave propagation through the ionosphere, scattering from rough surfaces,and optical communications between the Earth and satellites.The turbulence layer model is a geometrical optics model for wave propagation.It is assumed that there is no refraction inside the very thin tur-41

AZAZ(a)(b)Figure 3.5: Turbulence layer propagation model: (a) Pupil plane phase screenmodel. (b) Multiple phase screen model.bulence layer. The turbulence effect on wave propagation can be simulatedby the pupil plane phase screen model or the multiple phase screens model.In the pupil plane phase screen model, the turbulence of the whole propagationpath is modeled as a random phase screen at the transmitter pupilplane, as shown in Figure 3.5 (a). The pupil plane phase screen model is asimplified propagation model that is only appropriate for certain cases. Formore complicated situations, multiple phase screens are used to simulate theturbulence effect as shown in Figure 3.5 (b).3.6 Performance MetricsTo overcome the random media effects such as beam spread and intensityfluctuations, compensation schemes are investigated by modifying the transmittedfields.Various optimization criteria have been defined to evaluatethe optical beam performance. In this study, two optimization criteria areintroduced to measure the beam spread and scintillation effect.42

3.6.1 Maximization of the integrated intensityFor this criterion, the goal is to concentrate more power or energy at thereceiver region through the propagation media.We define the integratedintensity at the receiver as the performance metric∫I = w(⃗v)| g(⃗v)| 2 d⃗v, (3.100)where w(⃗v) is the weighting function at the receiver region, and g(⃗v) is thereceived field at the target diffracted from the transmitted source field f(⃗u)at pupil aperture A∫g(⃗v) = h(⃗v, ⃗u)f(⃗u)d⃗u. (3.101)AAccordingly, the optimal beam is defined to be the input field that maximizesthis integrated intensity for a particular weighting function w(⃗v):ˆf(⃗u) = arg max I, (3.102)‖f‖ 2 =I 0where ‖f‖ 2 = I 0 is the energy constraint for the source field:∫‖f‖ 2 =A|f(⃗u)| 2 d⃗u. (3.103)3.6.2 Minimization of the scintillation indexTo reduce the random fluctuations of the received fields, the “scintillationindex” has been widely used in laser beam compensation [2]. The scintillationindex of the received field has been defined as the normalized variance of43

intensity fluctuations:S I = var[I](E[I]) = E[I2 ]− 1, (3.104)2 (E[I])2where I is the integrated intensity at the receiver as defined in equation(3.100). Accordingly, the optimal beam has been defined as the transmittedoptical field to minimize this scintillation index:ˆf(⃗u) = arg min‖f‖ 2 =I 0S I . (3.105)There are several other metrics for laser beam performance measurementin random medium, such as beam width and on axis Strehl ratio. In thisstudy, we will use the two criteria we introduced and discuss the optimalbeams that corespond to them respectively.44

Chapter 4Optimal Beam to Maximize theIntegrated IntensityIn many optical applications, the goal is to concentrate the greatest percentageof the transmitted energy or power on the receiver. For example inlaser weapon systems high power laser beams are used to disable or destroymilitary targets at far distances. In free space laser communication systemsusing direct detection, the detection process can be considered as a simpleenergy collection process in the receiver photodetector. In situations withrandom propagation media, system performance is degraded by the turbulenceeffect. In this chapter, we will discuss beams that are used to maximizethe integrated energy or power through turbulence.As shown in equation (3.102), the optimal beam is defined as the transmittedfield f(⃗u) that maximizes the integrated weighted intensity I in thereceiver region for a particular weighting function w(⃗v),ˆf(⃗u) = arg max I, (4.1)‖f‖ 2 =I 045

−0.2−0.2−0.1−0.1(meter)0(meter)00.10.1−0.2 −0.1 0 0.1(meter)−0.2 −0.1 0 0.1(meter)(a)(b)Figure 4.1: Free space diffraction of focused field from circular aperture:λ = 10 −6 , d = 10 4 , D = 0.2, all units in meters. (a) Transmitting aperturea(⃗u). (b) Normalized amplitude of the received field g(⃗v).where the weighted integrated intensity I has been defined in equation (3.100)as∫I =w(⃗v)| g(⃗v)| 2 d⃗v. (4.2)When the Fresnel approximation is valid, the received field g(⃗v) can be derivedas∫g(⃗v) =where h(⃗v, ⃗u) is the propagation impulse functionAh(⃗v, ⃗u)f(⃗u)d⃗u, (4.3)h(⃗v, ⃗u) = 1 (jλd exp j 2π )λ d exp(j π )λd ‖⃗v − ⃗u‖2 . (4.4)For example, consider the free space diffraction from a circular aperture46

A with aperture function⎧⎪⎨ 1 ‖⃗u‖ ≤ D/2a(⃗u) =⎪⎩ 0 ‖⃗u‖ > D/2, (4.5)where D is the diameter of the aperture. It is well known that when a focusedbeam is transmitted,f(⃗u) = exp(−j π )λd ‖⃗u‖2 , (4.6)the intensity of the received field is the Airy pattern|g(⃗v)| 2 ==( ) 2 ∣∫(1 ∣∣∣a(⃗u) exp −j 2π )2λdλd ⃗v · ⃗u d⃗u∣( ) 2 ∣ ( )∣ 1 ∣∣∣ ⃗v ∣∣∣2A , (4.7)λd λdwhere A(⃗ν) is the Fourier transform of the aperture function a(⃗u)( ) 2 D 2J 1 (πD|⃗ν|)A(⃗ν) =. (4.8)2 D|⃗ν|Illustrations of the transmitting aperture and the received field are shown inFigure 4.1.To measure the integrated intensity inside a circular region R in thereceiver, we define the weighting function as⎧⎪⎨ 1 ‖⃗v‖ ≤ rw(⃗v) =⎪⎩ 0 ‖⃗v‖ > r, (4.9)where r is the radius of R. Then the integrated intensity inside the receiver47

10.90.80.70.60.50.40.30.20.10−4 −2 0 2 4normalized radius(a)10.90.80.70.60.50.40.30.20.100 1 2 3 4normalized radius(b)Figure 4.2: Airy pattern: (a)Normalized Airy pattern slice, (b)Fractionalintegrated intensity I/I 0 vs. normalized target plane radius (rD)/(λd)region R can be expressed as∫I = w(⃗v)i(⃗v)d⃗v=∫ 2π ∫ r00D 2 J 2 1 [π(D/λd)|ρ|]4|ρ| 2 ρdρdθ. (4.10)The normalized integrated intensity can be derived as [50](I= 1 − J02 π Dr ) (− J12 π Dr ), (4.11)I 0 λd λdwhere J 0 and J 1 are the first kind of Bessel functions of order 0 and 1, andI 0 is the total transmitted intensity∫I 0 =a(⃗u)|f(⃗u)| 2 d⃗u = πD24 . (4.12)A plot of the Airy pattern center slice and the integrated intensity is shownin Figure 4.248

4.1 Optimal Beam in Free SpaceThe maximization of the fractional integrated intensity at the receiver is aradiation transportation problem that has been widely studied as a case ofan “apodization” problem [71, 48]. Apodization theory is concerned withthe determination of the distribution of light over the exit pupil of an opticalsystem required in order to achieve a desired distribution of illuminance overa given plane in the image field [68]. Most often the goal is suppression of theside lobes of the diffraction pattern. It is sometimes also called “tapering”,a mathematical technique used to reduce the Gibbs phenomenon “ringing”that is produced in a spectrum obtained from a truncated interferogram[81]. A comprehensive review of the apodization problem has been given inreference [38].The integrated intensity at the receiver can be expressed as∫I = w(⃗v)|g(⃗v)| 2 d⃗v∫∫2= w(⃗v)∣ h(⃗v, ⃗u)f(⃗u)d⃗u∣ d⃗vA∫∫= f ∗ (⃗u 1 )H(⃗u 1 , ⃗u 2 )f(⃗u 2 )d⃗u 1 d⃗u 2 , (4.13)where∫H(⃗u 1 , ⃗u 2 ) = a ∗ (⃗u 1 )w(⃗v)h ∗ (⃗v, ⃗u 1 )h(⃗v, ⃗u 2 )d⃗v a(⃗u 2 ). (4.14)This optimal beam has the same mathematical formalization as the apodizationproblem discussed in [68]. Two apodization problems in free space are49

discussed and complete solutions are given. Slepian [68] shows that the solutionis the largest eigenvalue’s eigenfunction for the integral equation∫λ ξ(⃗u 1 ) = H(⃗u 1 , ⃗u 2 ) ξ(⃗u 2 )d⃗u 2 . (4.15)In free space propagation with the Fresnel approximation, the kernel H 0 (⃗u 1 , ⃗u 2 )takes the form∫H 0 (⃗u 1 , ⃗u 2 ) = a ∗ (⃗u 1 )= e−j π λd (‖⃗u 1‖ 2 −‖⃗u 2 ‖ 2 )=( 1λdh ∗ (⃗v, ⃗u 1 )h(⃗v, ⃗u 2 )w(⃗v)d⃗v a(⃗u 2 ) (4.16)∫a ∗ 2π−j(⃗u(λd) 2 1 )a(⃗u 2 ) w(⃗v)e λd (⃗u 2−⃗u 1 )·⃗v d⃗v) 2e −j π λd (‖⃗u 1‖ 2 −‖⃗u 2 ‖ 2) a ∗ (⃗u 1 )a(⃗u 2 )W (⃗ν)| ⃗u ⃗ν= 2 −⃗u 1 ,λdwhere W (⃗ν) is the spatial spectrum of the weighting function w(⃗v) evaluatedat frequency ⃗ν. With circular transmitter aperture A and circle binaryweighting function at the receiver region R, the optimal beam is proven bySlepian [69, 68] to be the Prolate spheroidal function, which is the principleeigenfunction of H 0 (⃗u 1 , ⃗u 2 ) associated with the largest eigenvalue. A plotof H 0 (u, u ′ ) is shown in Figure 4.3 for one dimensional propagation withrectangular weighting function.4.2 Optimal Beam to Maximize the AverageIntegrated IntensityWhen a beam propagates through random media such as the atmosphericturbulence, the impulse response h(⃗v, ⃗u) and the kernel H(⃗u 1 , ⃗u 2 ) are random.50

−0.132.5−0.052u’ (meter)01.510.050.50.1−0.1 −0.05 0 0.05 0.1u (meter)0−0.5Figure 4.3: Kernel H 0 (u, u ′ ) for one dimensional propagation in free spacewith rectangular weighting function.Accordingly, eigenvalues and eigenfunctions of the kernel are subject to randomperturbations. To evaluate the turbulence effects on the optimal beam,statistical properties of the eigenvalues and eigenfunctions of the perturbedkernel H(⃗u 1 , ⃗u 2 ) are needed.Unfortunately, it is not easy to get analyticresults of these statistical properties for a randomly perturbed kernel. Toovercome the difficulties existing in the above problem, we investigate theoptimal beams that can compensate the beam spread statistically within along exposure time.In the case of propagation in random media, the average weighted intensityover turbulence at the target plane can be shown to be:∫∫〈I〉 = f ∗ (⃗u 1 ) 〈H(⃗u 1 , ⃗u 2 )〉 f(⃗u 2 )d⃗u 1 d⃗u 2 , (4.17)51

where 〈H(⃗u 1 , ⃗u 2 )〉 is the averaged kernel over turbulence:∫〈H(⃗u 1 , ⃗u 2 )〉 = a ∗ (⃗u 1 )w(⃗v) 〈h ∗ (⃗v, ⃗u 1 )h(⃗v, ⃗u 2 )〉 d⃗v a(⃗u 2 ). (4.18)Then the optimal beam can be defined to be the transmitted field that maximizesthe average integrated intensity 〈I〉:ˆf(⃗u) = arg max‖f‖ 2 =I 0〈I〉 . (4.19)It is evident that the optimal beam is the largest eigenvalue’s eigenfunctionfor the averaged Hermitian kernel 〈H(⃗u 1 , ⃗u 2 )〉. The eigenvalues and eigenvectorsof 〈H(⃗u 1 , ⃗u 2 )〉 depend on the statistical properties of the randommedia.4.3 Averaged Kernel in TurbulenceIt is easy to see that the averaged kernel 〈H(⃗u 1 , ⃗u 2 )〉 over turbulence is importantin this problem.In this section, we will discuss how the kernelH(⃗u 1 , ⃗u 2 ) has been changed in turbulence and the corresponding effects onits eigenvalues and eigenvectors.4.3.1 Averaged kernel for the phase screen modelConsider the geometry of a thin turbulence layer near the pupil and take thestandard assumption that the refractive index is a Gaussian random process.The optical length variations cause Gaussian phase perturbations at the pupil52

plane as∫ z+∆zφ(⃗u) = k n(⃗u, z)dz, (4.20)zwhere k is the optical wave number and n(⃗u, z) is the index of refractionvariation as a function of positions. The transmittance of such a phase screentakes the formt s (⃗u) = exp [jφ(⃗u)] , (4.21)where the random phase shift φ(⃗u) introduced at position ⃗u is assumed tobe wide sense stationary (WSS). The averaged kernel in this case follows〈H(⃗u 1 , ⃗u 2 )〉 = H 0 (⃗u 1 , ⃗u 2 ) 〈 e j∆φ(⃗u 1,⃗u 2 ) 〉 , (4.22)where H 0 (⃗u 1 , ⃗u 2 ) is the free space kernel and∆φ(⃗u 1 , ⃗u 2 ) = φ(⃗u 1 ) − φ(⃗u 2 ) (4.23)is the phase difference between two positions ⃗u 1 and ⃗u 2 at the pupil plane.The random phase φ(⃗u) can be modeled as a zero mean Gaussian randomprocess [58]. Then the phase difference ∆φ(⃗u 1 , ⃗u 2 ) is also a zero meanGaussian process with following statistical properties〈∆φ(⃗u 1 , ⃗u 2 )〉 = 0, (4.24)andσ 2 ∆φ = 〈 [φ(⃗u 1 ) − φ(⃗u 2 )] 2〉= D φ (⃗u 1 , ⃗u 2 ), (4.25)53

where D φ (⃗u 1 , ⃗u 2 ) is the structure function of the random process φ(⃗u). ForGaussian random phase, we have〈 〉 [ej∆φ(⃗u 1 ,⃗u 2 )= exp − 1 ]2 D φ(⃗u 1 , ⃗u 2 ) . (4.26)By using the Kolmogorov power spectral density to characterize the index ofrefraction fluctuations, the structure function takes the form( ) 5/3 ‖⃗u1 − ⃗u 2 ‖D φ (⃗u 1 , ⃗u 2 ) = 6.88, (4.27)r 0where r 0 is the Fried parameter of the turbulence [58]. So the averaged kerneltakes the form〈H(⃗u 1 , ⃗u 2 )〉 = H 0 (⃗u 1 , ⃗u 2 ) 〈 e 〉 j∆φ(⃗u 1,⃗u 2 )(4.28)= e−j π λd (‖⃗u 1‖ 2 −‖⃗u 2( )‖ 2 )(λd) 2 a ∗ ⃗u1 − ⃗u 2(⃗u 1 )a(⃗u 2 )We − 1 2 D φ(⃗u 1 −⃗u 2 ) .λdThe principle eigenvector of this kernel is the optimal beam to maximize 〈I〉.An image of the averaged kernel 〈H(u, u ′ )〉 for one dimensional propagationis shown in Figure 4.4.Notice that both the structure function D φ (⃗u 1 , ⃗u 2 ) and the spectrumW (⃗u 1 , ⃗u 2 ) are functions of ∆u = ‖⃗u 1 − ⃗u 2 ‖.Therefore, we define a newweighting function in the frequency domain as( ) [⃗u1 − ⃗u 2W n = exp − 1 ]λd2 D φ(⃗u 1 − ⃗u 2 ) W( ⃗u1 − ⃗u 2λd). (4.29)Then we use the inverse Fourier transform to obtain the new weighting func-54

−0.132.5−0.052u’ (meter)00.051.510.50.10−0.1 −0.05 0 0.05 0.1u (meter)Figure 4.4: Averaged kernel 〈H(u, u ′ )〉 for one dimensional propagationthrough turbulence with rectangular weighting function and r 0 = 10 cm.tion in the space domain[w n (⃗v) = F{exp−1 − 1 ]2 D φ(⃗u 1 − ⃗u 2 ) WThe averaged kernel can then be expressed as( ⃗u1 − ⃗u 2λd)}. (4.30)〈H(⃗u 1 , ⃗u 2 )〉 = e−j π λd (‖⃗u 1‖ 2 −‖⃗u 2( )‖ 2 )(λd) 2 a ∗ ⃗u1 − ⃗u 2(⃗u 1 )a(⃗u 2 )W n . (4.31)λdand the average integrated intensity can be expressed as∫〈I〉 =w n (⃗v)|g(⃗v)| 2 d⃗v, (4.32)where g(⃗v) is the received field in free space. An example of the changes ofweighting function is shown in Figure 4.5.Equation (4.32) indicates that the average integrated intensity weightedby w(⃗v) at target through turbulence equals the integrated intensity throughfree space weighted by another function w n (⃗v). So the optimal beam with55

−10.9−10.4(meter)0.500.80.70.60.50.4(meter)−0.500.350.30.250.20.50.30.50.150.20.110.110.05−1 −0.5 0 0.5 1(meter)(a)−1 −0.5 0 0.5 1(meter)(b)Figure 4.5: Weighting function changes: (a) Original weighting function. (b)New weighting function in turbulence.this new weighting function in free space should also be the optimal beamto maximize the average integrated intensity through turbulence with theoriginal weighting function w(⃗v).It is easy to see qualitatively from this new weighting function how muchpower is lost inside the receiver due to turbulence compared with free spacepropagation. The definition of w n (⃗v) has also been used in the numericalcomputation of the optimal beam to simplify the simulation complexity inthis study. By using w n (⃗v), there is no more need to generate random phasescreens and propagate optical beams through them in the iterative algorithm.4.3.2 Averaged kernel for the extended Huygens-Fresnelprinciple modelFrom Section 3.5.4 we know that according to the extended Huygens-Fresnelprinciple, the impulse function of propagation through turbulence takes the56

formh(⃗v, ⃗u) = 1 (jλd exp j 2π ) (λ d exp j π )λd ‖⃗v − ⃗u‖2 + ψ(⃗v, ⃗u) , (4.33)where ψ(⃗v, ⃗u) is the random part of the complex phase.So the averagedkernel becomes∫〈H(⃗u 1 , ⃗u 2 )〉 = a ∗ (⃗u 1 )w(⃗v) 〈h ∗ (⃗v, ⃗u 1 )h(⃗v, ⃗u 2 )〉 d⃗v a(⃗u 2 )= a ∗ 1(⃗u 1 )a(⃗u 2 )π (λd) 2 e−j λd (‖⃗u 1‖ 2 −‖⃗u 2 ‖ 2 )(4.34)∫× w(⃗v)e j 2πλd ⃗v·(⃗u 1−⃗u 2 ) 〈exp[ψ ∗ (⃗v, ⃗u 1 ) + ψ(⃗v, ⃗u 2 )]〉 d⃗v.The second order moment of the extended Huygens Fresnel principle hadbeen studied in [82] and the modulation transfer function (MTF) of theturbulence was shown to beM(⃗v 1 , ⃗v 2 , ⃗u 1 , ⃗u 2 ) = 〈exp [ψ(⃗v 1 , ⃗u 1 ) + ψ ∗ (⃗v 2 , ⃗u 2 )]〉 (4.35)= exp { − [〈 |ψ 1 | 2〉 − 〈ψ 1 (⃗v 1 , ⃗u 1 )ψ ∗ 1(⃗v 2 , ⃗u 2 )〉 ]} ,where ψ 1 = f 1 /f 0 , f 0 is the field in the absence of turbulence and f 1 is thefirst order Born approximation, and〈|ψ1 | 2〉 ∫= 2πk 2 dΦ n (⃗κ)d⃗κ. (4.36)This expression is valid through terms of the second order in refractiveindex perturbation n s .Experimental evidence and theoretical argumentsindicate that ψ is a Gaussian random variable, which means that the above57

equation is correct to all orders in n s . For a spherical wave with paraxialapproximation and local homogeneous random media, Yura [82] shows that∫〈ψ 1 (⃗v 1 , ⃗u 1 )ψ1(⃗v ∗ 2 , ⃗u 2 )〉 = 4π 2 k 2 d⃗κΦ n (⃗κ)d⃗κ∫ 1where J 0 is the first kind of Bessel function of zero order and0J 0 [|t∆v + (1 − t)∆u|⃗κ] dt,(4.37)∆u = ‖⃗u 1 − ⃗u 2 ‖, ∆v = ‖⃗v 1 − ⃗v 2 ‖. (4.38)Then the MTF can be expressed asM(∆v, ∆u) = exp { − [〈 |ψ 1 | 2〉 − 〈ψ 1 (⃗v, ⃗u 1 )ψ1(⃗v, ∗ ⃗u 2 )〉 ]} (4.39){ ∫ 1 ∫}= exp −4π 2 k 2 d dt ⃗κΦ n (⃗κ) [1 − J 0 (|t∆v + (1 − t)∆u|⃗κ)] d⃗κ .0For Kolmogorov spectrum characterized turbulence with ∆v ≪ L 0 , it hasbeen shown that [82][M(∆v, ∆u) = exp − 2.91 ∫ 1]2 C2 nk 2 d dt|t∆v + (1 − t)∆u| 5/3 (4.40)0⎧⎡⎨( ) ( ) ⎤⎫= exp⎩ −2.91 2 C2 nk 2 d ⎣ 3 ∆v 8/3 − ∆u 8/3 ⎬⎦8 |∆v − ∆u| ⎭ .In the case of intensity fluctuations of a point at the target ∆v = 0, thisequation becomesM(∆v = 0, ∆u) = exp ( −0.545C 2 nk 2 d∆u 5/3) . (4.41)58

The transverse coherence length has been defined asρ 0 = (0.545k 2 C 2 nd) −3/5 , (4.42)so thatM(∆v = 0, ∆u) = exp[−( ∆uρ 0) 5/3]. (4.43)Finally, we get the averaged kernel for the extended Huygens-Fresnelpropagation as〈H(⃗u 1 , ⃗u 2 )〉 = H 0 (⃗u 1 , ⃗u 2 ) exp[−( ∆uρ 0) 5/3]= a ∗ 1(⃗u 1 )a(⃗u 2 )π (λd) 2 e−j λd (‖⃗u 1‖ 2 −‖⃗u 2 ‖ 2 )(4.44)∫ [× w(⃗v) exp j 2π] [ ( ) ] 5/3 ∆uλd ⃗v · (⃗u 1 − ⃗u 2 ) exp −d⃗v.ρ 0Notice that the averaged kernels in different propagation models havevery similar form. The extended Huygens-Fresnel principle includes randomeffects from both the transmitter and receiver sides for strong turbulencecases. But the second order moment of the field perturbation in this modeldoes not differ much from the one in the phase screen model. This is becauseonly perturbations in the far field have evident effects on the second ordermoment of the received perturbation.Another reason is that, because ofthe homogeneous and isotropic random media assumption, receiver regioneffects vanish when ⃗v 1 = ⃗v 2 . So when we deal with the second order momentof the field perturbation in 〈H(⃗u 1 , ⃗u 2 )〉, we get similar results for the extendedHuygens-Fresnel principle and the phase screen model.59

This result is not unexpected because most of the intensity fluctuationsare induced by turbulence far from the receiver aperture.Near apertureturbulence contributes mainly to the phase distortions but not the field amplitude(∆v = 0). So the average kernel 〈H(⃗u 1 , ⃗u 2 )〉 is affected mainly by thesecond order statistics of turbulence far from the receiver.The MTF of propagation in the extended Huygens-Fresnel principle modelhas also been expressed in the structure function form as [6, 24]M(∆v, ∆u) = 〈exp[ψ ∗ (⃗v 1 , ⃗u 1 ) + ψ(⃗v 2 , ⃗u 2 )]〉= exp[−D ψ (∆v, ∆u)/2], (4.45)where⎡( ) ( ) ⎤D ψ (∆v, ∆u) = 2.91Cnk 2 2 d ⎣ 3 ∆v 8/3 − ∆u 8/3 ⎦ (4.46)8 |∆v − ∆u|is the structure function of the complex phase ψ(⃗v, ⃗u).We will use thisexpression later to be consistent with the discussion in random phase screenmodel.4.4 Performance of the Optimal BeamsTo maximize the average integrated intensity, the optimal beam is the largesteigenvalue’s eigenvector of 〈H(⃗u 1 , ⃗u 2 )〉, which establishes an upper boundfor the performance of different compensation schemes.We will computethe optimal beams and the corresponding maximum integrated intensitiesfor different propagation cases in this section. Because the two turbulence60

models – extended Heygens-Fresnel principle and phase screen model – havesimilar results on the second moment of the received field, we will use theFried parameter to characterize the turbulence strength in our experiments.4.4.1 One dimensional experimentWe begin with the optimal beam in the one dimensional case because ofcomputational simplicity. To evaluate the performance of the optimal beams,we introduce one dimensional free space propagation first.Consider theFresnel diffraction from a one dimensional rectangular slit aperture A,⎧⎪⎨ 1 |u| D/2a(u) =, (4.47)⎪⎩ 0, |u| > D/2where D is the width of the slit. The free space received field at distance dcan be expressed aswhere∫g(v) =Ah(v, u)f(u)du, (4.48)h(v, u) = √ 1 exp(j 2π )jλd λ d exp(j π )λd |v − u|2(4.49)is the impulse function of one dimensional Fresnel propagation. When thefocused beam is transmitted from the pupil,f(u) = exp(−j π )λd |u|2 , (4.50)61

the received intensity is|g(v)| 2 = 1λd= 1λd∫ (∣ a(u) exp −j 2π )∣ ∣∣∣2λd uv (∣ v)∣ ∣∣2∣A , (4.51)λdwhere A(ν) is the Fourier transform of the aperture function a(u)A(ν) = D sinc(Dν). (4.52)The sinc function in the above equation is defined as⎧⎪⎨ sin(πu)u ≠ 0πusinc(u) =. (4.53)⎪⎩ 1 u = 0For a rectangular receiver region R:⎧⎪⎨ 1 |v| rw(v) =⎪⎩ 0 |v| > r, (4.54)and the integrated intensity can be derived asI =∫= 2D2λdw(v)|g(v)| 2 dv∫ r0sinc 2 ( Dvλd)dv. (4.55)As mentioned before, the optimal beams in free space are the prolate functions.In the case of one dimensional propagation, we can compute theoptimal beams and integrated intensities through the eigen decomposition ofthe kernel H(u 1 , u 2 ) directly. A plot of the optimal beams and correspond-62

21.8free spaceturbulence10.91.60.81.40.71.20.610.50.80.40.60.30.40.20−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25Normalized radius(a)0.2Focused beam in free space0.1Optimal beam in free spaceOptimal beam through turbFocused beam through turb00 0.5 1 1.5 2 2.5 3 3.5 4Normalized radius(b)Figure 4.6: Optimal beams and integrated intensity for one dimensional propagationin free space and random medium: (a) Optimal beam ˆf(u) at transmitteraperture, (b) Fractional integrated intensity I/I 0 vs. receiver planenormalized radius (rD)/(λd).ing integrated intensities of a one dimensional propagation experiment areshown in Figure 4.6. The physical parameters used in this one dimensionalpropagation experiment are shown in Table 4.1.Physical parameter Symbol Valuewave length λ 1.0 µmpropagation distance d 20 kmtransmitter diameter D 0.5 mFried parameter r 0 10 cmTable 4.1: Physical parameters used in the one dimensional propagationsimulation.63

4.4.2 Two dimensional experimentAs we have shown, an Airy pattern is formed at the target for a focusedbeam transmitted from a circular aperture in free space. In the two dimensionalcase, it is difficult to do direct eigen decomposition of the four dimensionalHermitian kernel H(⃗u 1 , ⃗u 2 ). To compute the principle eigen beam ofH(⃗u 1 , ⃗u 2 ), an iterative algorithm developed by Schulz [62] is used here. In thepropagation through random medium, a new weighting function w n (⃗v) areused in this algorithm and Monte-Carlo method is avoided. We set up thetwo dimensional propagation experiment by using similar parameters as inthe one dimensional case, with aperture diameter D = 0.4m and propagationdistance d = 20km. Plots of the optimal beams and the average integratedintensities in free space and through turbulence are shown in Figure 4.7.4.4.3 Performance analysisWe have seen that experiments in the one dimensional and two dimensionalcases show very similar results. In free space propagation, the focused beamperforms well with small target radius and the optimal beam makes improvementswith large radius. In the case of propagation through random media,the average integrated intensity of the optimal beam through turbulence doesnot have evident improvement compared with other beams, which means thatthere is not much improvement space for the optical system. This can beexplained from the kernel changes we have shown before.Compare the averaged kernel in Figure 4.4 with the one in Figure 4.364

2.5free spaceturbulence10.920.80.71.50.60.510.40.30.50−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2Normalized radius(a)0.2Focused beam in free space0.1Optimal beam in free spaceOptimal beam through turbFocused beam through turb00 0.5 1 1.5 2 2.5 3 3.5 4Normalized radius(b)Figure 4.7: Optimal beams and integrated intensity for two dimensionalpropagation in free space and random medium: (a) A center slice of the optimalbeam ˆf(⃗u) at transmitter aperture, (b) Fractional integrated intensityI/I 0 vs. receiver plane normalized radius (rD)/(λd) for focused beam andoptimal beam.for free space, we can see easily that the averaged kernel after turbulence ismore like the identity matrix. This explains intuitively why the optimal beamdoes not improve the integrated intensity greatly. For the identity matrix,any vector can be its eigenvector with all the same eigenvalues equal to 1. Sofor 〈H(⃗u 1 , ⃗u 2 )〉, which is close to the identity matrix, its eigenvalues are closein value. In our consideration, because the summation of all the eigenvaluesof the kernel H(⃗u 1 , ⃗u 2 ) indicates the whole transmitted energy/power, evenlydistributed eigenvalues implies that all the eigenvectors have nearly the sameperformance. So there is no beam that can perform much better than others.This result indicates that, to compensate the average integrated intensityby changing the transmitted field does not work well since there is not much65

improvement space for it. Short term compensation schemes, compensatingover a short exposure time, is preferred to maximize the integrated intensityin real time. Adaptive optics can be used to do the compensation in thepupil or the image plane.66

Chapter 5Optimal Beam to Minimize theScintillation IndexLaser beam scintillation is an important random-media effect that degradesoptical system performance.The scintillation index is commonly used tomeasure the severity of the disturbance. Whereas fully coherent beams areextremely sensitive to the random media, recent investigations show that partiallycoherent beams can be less sensitive to the random media [28, 56, 65].In [63], the optimal beam to minimize the scintillation index has been provento be partially coherent in general cases. The coherent modes and associatedintegrated intensities depend on the second and forth order moments of thepropagation Green’s function h(⃗v, ⃗u).67

5.1 Optimal Pupil Plane Mutual CoherenceAs shown in equation (3.104), the scintillation index has been defined as thenormalized variance of the received integrated intensityS I =σ2 I(µ I ) = 〈I2 〉22− 1, (5.1)〈I 〉where 〈·〉 is the average over turbulence and I denotes the integrated intensityat the target plane. Consider the propagation of a source field from apertureA to receiver region R. The (short term average) intensity of the receivedfield at R can be expressed as∫∫i(⃗v) = h ∗ (⃗v, ⃗u 1 )J(⃗u 1 , ⃗u 2 )h(⃗v, ⃗u 2 )d⃗u 1 d⃗u 2 , (5.2)A 2where J(⃗u 1 , ⃗u 2 ) = 〈f ∗ (⃗u 1 )f(⃗u 2 )〉 is the mutual intensity of the transmittedfield in the pupil plane. According to the analysis in section 3.3.3, J(⃗u 1 , ⃗u 2 )has the coherent mode expansion in the aperture region asJ(⃗u 1 , ⃗u 2 ) = ∑ kα k ψ k (⃗u 1 )ψ ∗ k(⃗u 2 ), (⃗u 1 , ⃗u 2 ) ∈ A 2 , (5.3)where Ψ = {ψ k (⃗u)} are the orthonormal coherent modes over aperture regionA, and {α k } are nonnegative real numbers associated with the integratedintensity of single modes. When a single coherent mode ψ k (⃗u) propagates tothe target, the received field is∫ζ k (⃗v) = h(⃗v, ⃗u)ψ k (⃗u)d⃗u, (5.4)A68

and the corresponding received integrated intensity of this single mode is∫I k =∫w(⃗v)|ζ k (⃗v)| 2 d⃗v =∫w(⃗v)∣Ah(⃗v, ⃗u)ψ k (⃗u)d⃗u∣2d⃗v. (5.5)Because ψ k (⃗v) is normalized, 0 I k 1. The total received intensity fromthe optical source with mutual intensity J(⃗u 1 , ⃗u 2 ) isI =∫= ∑ k∫∫w(⃗v) h ∗ (⃗v, ⃗u 1 )J(⃗u 1 , ⃗u 2 )h(⃗v, ⃗u 2 )d⃗u 1 d⃗u 2 d⃗v∫A 2α k w(⃗v) |ζ k (⃗v)| 2 d⃗v = ∑ α k I k . (5.6)kThe statistical properties of the received intensities {I k } are important inderiving the statistics of the total received intensity I. The average receivedintensity of a single transmitted mode can be expressed as∫∫µ k = 〈I k 〉 = ψk(⃗u ∗ 1 ) 〈H(⃗u 1 , ⃗u 2 )〉 ψ k (⃗u 2 )d⃗u 1 d⃗u 2 , (5.7)A 2where∫〈H(⃗u 1 , ⃗u 2 )〉 =w(⃗v) 〈h ∗ (⃗v, ⃗u 1 )h(⃗v, ⃗u 2 )〉 d⃗v (5.8)is the averaged kernel over turbulence. Then we can express the mean of theintegrated intensity at the receiver as〈I〉 = ∑ kα k µ k . (5.9)To calculate the scintillation index, the second order moments of the received69

intensity are needed:〈 〉 I2− 〈I 〉 2 = ∑ ∑α k R kl α l , (5.10)klwhereR kl = 〈I k I l 〉 − µ k µ l . (5.11)It has been shown in [63] that the optimal orthonormal coherence ˆΨ tominimize the scintillation index can be expressed asˆΨ = arg max ∑ k∑lµ k (Ψ)R −1kl (Ψ)µ l(Ψ), (5.12)with associated integrated intensities ˆα k asˆα k = ∑ lR −1kl ( ˆΨ)µ l ( ˆΨ), (5.13)where R −1klis defined as:∑l⎧⎪⎨R −1kl (Ψ)R lm(Ψ) =⎪⎩1 k = m0 k ≠ l(5.14)Notice that we have assumed all ˆα k are non-negative here. In the case thereare possible negative ˆα k values, a constrained optimization is used to approachthe optimal modes.It can be seen from the above equations that the optimal pupil planecoherence to minimize the scintillation index depends on the first and secondmoments of the received intensity I through turbulence. Further discussionrequires more information about the wave propagation in turbulence. In the70

following two sections, we will discuss the first and second moments of theintegrated intensity in two different models: the pupil plane phase screenmodel and the extended Huygens-Fresnel model.5.2 Optimal Beams in Random Phase ScreenModelConsider wave propagation for the pupil plane random phase screen modelintroduced in section 3.5.5. The received optical field can be expressed as∫f(⃗v) = h(⃗v, ⃗u)f(⃗u)e jφ(⃗u) d⃗u. (5.15)AFor Gaussian distributed random process φ(⃗u), we have noted that the averagedkernel over turbulence takes the form〈H(⃗u 1 , ⃗u 2 )〉 = H 0 (⃗u 1 , ⃗u 2 ) 〈 e 〉j[φ(⃗u 1)−φ(⃗u 2 )]= H 0 (⃗u 1 , ⃗u 2 )e − 1 2 D φ(⃗u 1 ,⃗u 2 ) , (5.16)where H 0 (⃗u 1 , ⃗u 2 ) is the kernel in free space, and D φ is the phase structurefunction. For local stationary and isotropic random media with refractiveindex fluctuation characterized by the Kolmogorov power spectral density,we have shown before thatwhere r 0 is the Fried parameter.( ) 5/3 ‖⃗u1 − ⃗u 2 ‖D φ (⃗u 1 , ⃗u 2 ) = 6.88, (5.17)r 071

5.2.1 First moment of the integrated intensityThe average received intensity for transmitting a single coherent mode ψ k (⃗u)can be derived as∫∫ ∫µ k = 〈I k 〉 = ψk(⃗u ∗ 1 ) w(⃗v) 〈H(⃗u 1 , ⃗u 2 )〉 d⃗v ψ k (⃗u 2 )d⃗u 1 d⃗u 2 (5.18)A( ) 2 2 ∫∫ ( ) [1=ξ ∗ ⃗u1 − ⃗u 2λdk(⃗u 1 )Wexp − 1 ]A λd2 D φ(⃗u 1 , ⃗u 2 ) ξ k (⃗u 2 )d⃗u 1 d⃗u 2 ,2whereξ k (⃗u) = ψ k (⃗u)e j π λd ‖⃗u‖2 . (5.19)As we have shown in Chapter 4, µ k depends on the second correlation ofdisturbed propagation Green’s function.5.2.2 Second moment of the integrated intensityThe second order moment of the integrated intensity at the receiver can bederived as〈I k I l 〉 ===∫∫∫∫ψk(⃗u ∗ 1 )ψl ∗ (⃗u 3 )ψ k (⃗u 2 )ψ l (⃗u 4 ) 〈H(⃗u 1 , ⃗u 2 )H(⃗u 3 , ⃗u 3 )〉 d⃗u 1 d⃗u 2 d⃗u 3 d⃗u 4A∫∫∫∫4 ψk(⃗u ∗ 1 )ψl ∗ (⃗u 3 )ψ k (⃗u 2 )ψ l (⃗u 4 )H 0 (⃗u 1 , ⃗u 2 )H 0 (⃗u 3 , ⃗u 4 )A 4× 〈exp {j[φ(⃗u 1 ) − φ(⃗u 2 ) + φ(⃗u 3 ) − φ(⃗u 4 )]}〉 d⃗u 1 d⃗u 2 d⃗u 3 d⃗u 4( ) 4 ∫∫∫∫( ) ( )1ξ ∗λdk(⃗u 1 )ξ k (⃗u 2 )ξl ∗ ⃗u1 − ⃗u 2 ⃗u3 − ⃗u 4(⃗u 3 )ξ l (⃗u 4 )WWA λdλd4×e[− D(⃗u 1 −⃗u 2 )2 − D(⃗u 3 −⃗u 4 )2 +Γ(⃗u 1 −⃗u 4 )+Γ(⃗u 2 −⃗u 3 )−Γ(⃗u 1 −⃗u 3 )−Γ(⃗u 2 −⃗u 4 )]d⃗u 1 d⃗u 2 d⃗u 3 d⃗u 4(5.20)72

where Γ(⃗u 1 , ⃗u 2 ) = 〈φ(⃗u 1 )φ(⃗u 2 )〉 is the autocorrelation function of randomphase. For WSS Gaussian process φ(⃗u) with zero mean, we haveThen we get〈I k I l 〉 =Γ(⃗u 1 , ⃗u 2 ) = Γ(0) − D(⃗u 1 − ⃗u 2 ). (5.21)2( ) 4 ∫∫∫∫ 1ξλdk(⃗u ∗ 1 )ξl ∗ (⃗u 3 )ξ k (⃗u 2 )ξ l (⃗u 4 )WA 4( ⃗u1 − ⃗u 2λd)W( ) ⃗u3 − ⃗u 4×e − 1 2 [D(⃗u 1−⃗u 2 )+D(⃗u 3 −⃗u 4 )+D(⃗u 1 −⃗u 4 )+D(⃗u 2 −⃗u 3 )−D(⃗u 1 −⃗u 3 )−D(⃗v 2 −⃗u 4 )] d⃗u 1 d⃗u 2 d⃗u 3 d⃗u 4λd(5.22)The above equations about the first and second moments of the integrated intensitycan be used to compute the optimal coherent mode set (Ψ = {ψ k (⃗u)})and the associated weights ({α k }) according to equation (5.12) and equation(5.13). The performance of the partially coherent source will be evaluatedlater.5.3 Optimal Beams in the extended Huygens-Fresnel principle modelAs we discussed in Section 3.5.4, the received field at distance d under theextended Huygens-Fresnel principle can be expressed asg(⃗v) = 1 (jλd exp j 2π ) ∫ (f(⃗u) exp j π )λd Aλd ‖⃗v − ⃗u‖2 + ψ(⃗v, ⃗u) d⃗u, (5.23)where ψ(⃗v, ⃗u) is the random part of the complex phase. Second and fourthorder moments of this random part are required in the computation of the73

optimal pupil plane coherence.It was shown that the simulation resultsassuming Gaussian distribution of ψ(⃗v, ⃗u) in this extended Huygens-Fresnelprinciple model agreed with experimental data [6].5.3.1 First moment of the integrated intensityThe first moment of the received intensity after turbulence propagation canbe expressed as∫∫〈I〉 = f ∗ (⃗u 1 ) 〈H(⃗u 1 , ⃗u 2 )〉 f(⃗u 2 )d⃗u 1 d⃗u 2 ,A 2 (5.24)where the averaged kernel has been derived in Section 4.3.2 and takes theform〈H(⃗u 1 , ⃗u 2 )〉 = H 0 (⃗u 1 , ⃗u 2 ) 〈exp[ψ ∗ (⃗v, ⃗u 1 ) + ψ(⃗v, ⃗u 2 )]〉[= H 0 (⃗u 1 , ⃗u 2 ) exp − 1 ]2 D ψ(∆v, ∆u) . (5.25)In local homogeneous turbulence cases with Kolmogorov spectrum of therefractive index fluctuation, we have shown in section 4.3.2 that the structurefunction of complex phase ψ(⃗u, ⃗u) takes the form⎡( ) ( ) ⎤D ψ (∆v, ∆u) = 2.91Cnk 2 2 d ⎣ 3 ∆v 8/3 − ∆u 8/3 ⎦ . (5.26)8 |∆v − ∆u|For the propagation of a single coherent mode ψ k (⃗u), the average received74

intensity can be expressed as∫∫ ∫µ k = 〈I k 〉 = ψk(⃗u ∗ 1 ) w(⃗v) 〈H(⃗u 1 , ⃗u 2 )〉 d⃗v ψ k (⃗u 2 )d⃗u 1 d⃗u 2 (5.27)A( ) 2 2 ∫∫ ( ) [1=ξ ∗ ⃗u1 − ⃗u 2λdk(⃗u 1 )Wexp − 1 ]A λd2 D ψ(∆v = 0, ∆u) ξ k (⃗u 2 )d⃗ud⃗u 2 ,2whereξ k (⃗u) = ψ k (⃗u)e j π λd ‖⃗u‖2 . (5.28)5.3.2 Second moment of the integrated intensityThe second order moment of the received intensity from two transmittedcoherent modes can be derived as〈I k I l 〉 =∫∫∫∫∫∫ψk(⃗u ∗ 1 )ψl ∗ (⃗u 3 )ψ k (⃗u 2 )ψ l (⃗u 4 ) w(⃗v 1 )h ∗ (⃗v 1 , ⃗u 1 )h(⃗v 1 , ⃗u 2 )(5.29)A 4 R 2×Γ 4 (⃗v 1 , ⃗v 2 , ⃗u 1 , ⃗u 2 , ⃗u 3 , ⃗u 4 )w(⃗v 2 )h ∗ (⃗v 2 , ⃗u 3 )h(⃗v 2 , ⃗u 4 )d⃗v 1 d⃗v 2 d⃗u 1 d⃗u 2 d⃗u 3 d⃗u 4where Γ 4 (⃗v 1 , ⃗v 2 , ⃗u 1 , ⃗u 2 , ⃗u 3 , ⃗u 4 ) is the fourth moment of the turbulence partΓ 4 (⃗v 1 , ⃗v 2 ; ⃗u 1 , ⃗u 2 , ⃗u 3 , ⃗u 4 )= 〈exp [ψ ∗ (⃗v 1 , ⃗u 1 ) + ψ(⃗v 1 , ⃗u 2 ) + ψ ∗ (⃗v 2 , ⃗u 3 ) + ψ(⃗v 2 , ⃗u 4 )]〉= exp(− 1 )2 C . (5.30)75

The term C in the above equation has been derived as [24]:C = D ψ (⃗v 1 , ⃗v 1 , ⃗u 1 , ⃗u 2 ) + D ψ (⃗v 1 , ⃗v 2 , ⃗u 1 , ⃗u 4 ) + D ψ (⃗v 1 , ⃗v 2 , ⃗u 2 , ⃗u 3 )+D ψ (⃗v 2 , ⃗v 2 , ⃗u 3 , ⃗u 4 ) + D ψ (⃗v 1 , ⃗v 2 , ⃗u 2 , ⃗u 4 ) + D ψ (⃗v 1 , ⃗v 2 , ⃗u 1 , ⃗u 3 )−4B χ (⃗v 1 , ⃗v 2 , ⃗u 2 , ⃗u 4 ) − 4B χ (⃗v 1 , ⃗v 2 , ⃗u 1 , ⃗u 3 )−j2D χφ (⃗v 1 , ⃗v 2 , ⃗u 2 , ⃗u 4 ) + j2D χφ (⃗v 1 , ⃗v 2 , ⃗u 1 , ⃗u 3 ), (5.31)whereD χφ (⃗v 1 , ⃗v 2 , ⃗u 1 , ⃗u 2 ) = 〈[χ(⃗v 1 , ⃗u 1 ) − χ(⃗v 2 , ⃗u 2 )][φ(⃗v 1 , ⃗u 1 ) − φ(⃗v 2 , ⃗u 2 )]〉 ,B χ (⃗v 1 , ⃗v 2 , ⃗u 1 , ⃗u 2 ) = 〈χ(⃗v 1 , ⃗u 1 )χ(⃗v 2 , ⃗u 2 )〉 − 〈χ〉 2 . (5.32)In strong and moderately strong turbulence, it has been shown that theB χ and D χφ terms can be ignored [21]. So finally, we getC ≃ D ψ (⃗v 1 , ⃗v 1 , ⃗u 1 , ⃗u 2 ) + D ψ (⃗v 1 , ⃗v 2 , ⃗u 1 , ⃗u 4 ) + D ψ (⃗v 1 , ⃗v 2 , ⃗u 2 , ⃗u 3 )+D ψ (⃗v 2 , ⃗v 2 , ⃗u 3 , ⃗u 4 ) + D ψ (⃗v 1 , ⃗v 2 , ⃗u 2 , ⃗u 4 ) + D ψ (⃗v 1 , ⃗v 2 , ⃗u 1 , ⃗u 3 ),(5.33)where D ψ is the structure function in equation (5.26). It is clear that thesecond moment of the received intensity 〈I k I l 〉 can be evaluated numericallyfrom the above results. But the computation complexity exceeds the powerof our workstations.76

5.4 Selection of coherent modesThe optimal coherent modes of J(⃗u 1 , ⃗u 2 ) to minimize the scintillation indexhave been shown in [63] to be the fields maximizing the quadratic form inequation (5.12) asˆΨ = argmaxorthonormal Ψ∑ ∑klµ k (Ψ)R −1kl (Ψ)µ l(Ψ).For different turbulence cases, the optimal coherence modes Ψ = {ψ k (⃗u)}and associated weights {α k } vary with time. In practice, it is difficult to getthese coherent modes for certain turbulence cases. Also, it is not reasonableto generate sets of orthogonal modes for different desired partial coherence.Fortunately, many laser modes generated by the laser resonator are beam likesolutions of the paraxial wave equation. So it is natural to consider usingthese commonly used modes sets as the eigen modes set of J(⃗u 1 , ⃗u 2 ).There are several complete orthogonal solution families of the paraxialwave equation that are sometimes called the eigen modes of the PWE(Paraxial Wave Equation). These solution sets, such as the Hermite-Gaussianbeams and the Laguerre-Gaussian beams introduced in Section 3.2, can beused as the coherent modes to generate partially coherent sources.77

5.4.1 Scintillation reduction by using Hermite-GaussianbeamsAccording to Section 3.2, the field amplitude of one dimensional Hermite-Gaussian beam at plane z = 0 can be expressed as( 2f m (u) =πw 2 0) 1/41√2m m! H m(√ ) )2uexp(− u2, (5.34)w 0 w02where H m (u) is the Hermite polynomial of order m, and w 0 is the spot size.The functions are normalized to represent a fixed amount of total beam powerin all the modes∫A|f m (u)| 2 du = 1, m = 0, 1, 2, · · · . (5.35)Plots of the amplitude of one dimensional Hermite-Gaussian beams f m (x)are shown in Figure 5.1. To focus the beams at the target plane z = d,a quadratic phase is included in the Hermite-Gaussian beam to form thecoherent modesψ m (u) = f m (u) exp(−j π )λd |u|2 . (5.36)We set up a one dimensional propagation simulation with rectangulartransmitter A and receiver region R (weighting function w) in random phasescreen turbulence model to study the scintillation reduction by using partiallycoherent beams. The propagation parameters are shown in Table 5.1.When a single normalized coherent mode f m (x) is transmitted to thetarget, the integrated intensity I m through free space, the average integrated78

m = 0m = 1m = 2222000−2−2−2−0.2 0 0.2m = 3−0.2 0 0.2m = 4−0.2 0 0.2m = 5222000−2−2−2−0.2 0 0.2m = 6−0.2 0 0.2m = 7−0.2 0 0.2m = 8222000−2−2−2−0.2 0 0.2x (meter)−0.2 0 0.2x (meter)−0.2 0 0.2x (meter)Figure 5.1: Amplitude of one dimensional Hermite-Gaussian beams f m (x).Physical parameter Symbol Valuebeam waist w 0 0.05 mwave length λ 1.0 µmpropagation distance d 20 kmtransmitter diameter D 0.5 mreceiver radius r w 0.1 mFried parameter r 0 10 cmTable 5.1: Propagation parameters used in the one dimensional simulation.79

m I m µ m S m0 0.9249 0.6419 0.13031 0.6317 0.5513 0.06552 0.3138 0.4792 0.04953 0.2997 0.4222 0.05424 0.3125 0.3771 0.05555 0.2128 0.3411 0.05436 0.2408 0.3120 0.05547 0.2144 0.2878 0.05438 0.1816 0.2667 0.05159 0.2105 0.2463 0.0500Table 5.2: Fractional integrated intensity ξ m in free space, fractional averageintegrated intensity µ m through turbulence, and the scintillation index S mof the received fields by transmitting single Hermite-Gaussian modes f m (u).intensity µ m through turbulence and the scintillation index of the receivedfields have been shown in Table 5.2.From the results in the previous section, we can derive the minimum scintillationindex and corresponding optimal weighting α m numerically whenmultiple Hermite-Gaussian modes are used to produce the optimal partiallym 0 1 2 3 4 5 6 70 0.4657 0.3694 0.3065 0.2631 0.2325 0.2102 0.1931 0.17921 0.3694 0.3239 0.2681 0.2300 0.2037 0.1837 0.1678 0.15492 0.3065 0.2681 0.2410 0.2078 0.1812 0.1615 0.1469 0.13553 0.2631 0.2300 0.2078 0.1880 0.1640 0.1454 0.1314 0.12024 0.2325 0.2037 0.1812 0.1640 0.1501 0.1334 0.1193 0.10875 0.2102 0.1837 0.1615 0.1454 0.1334 0.1227 0.1106 0.10016 0.1931 0.1678 0.1469 0.1314 0.1193 0.1106 0.1027 0.09337 0.1792 0.1549 0.1355 0.1202 0.1087 0.1001 0.0933 0.0873Table 5.3: Mutual intensity 〈I m I n 〉 by using Hermite Gaussian beams.80

Scintillation Index0.140.130.120.110.10.090.080.070.060.050.040 1 2 3 4 5 6 7 8 9Modes index(a)Optimal Scintillation Index0.140.120.10.080.060.040.0201 2 3 4 5 6 7 8 9 10Number of modes used(b)Figure 5.2: Scintillation index for transmitting Hermite-Gaussian modes: (a)Scintillation index for single modes. (b) Reduction of the optimal scintillationindex when the number of Hermite-Gaussian modes increased.coherence in the pupil plane. An evident reduction of the optimal scintillationindex is shown in Figure 5.2 and the corresponding optimal parametersare shown in Table 5.4. At the same time, we note that the average integratedintensity by transmitting this optimal coherence drops down with thescintillation index reduction.5.4.2 Scintillation reduction by using Laguerre-GaussianbeamsThe Laguerre-Gaussian beam has been introduced in section 3.2. The correspondingone dimensional beam amplitude at transmitter plane z = 0 canbe expressed asf p,m (r, θ) = L m p( 2r2w 2 0) ( √2rw 0) mexp(− 2r2w 2 0), (5.37)81

N = 1 2 3 4 5 6 7 8α 0 1.0000 0.1775 0.1472 0.1344 0.1305 0.1173 0.1032 0.0918α 1 0.8225 0.2185 0.2292 0.1544 0.1136 0.1040 0.1006α 2 0.6343 0.0778 0.1442 0.1496 0.1309 0.0999α 3 0.5586 0.1132 0.1413 0.0959 0.1135α 4 0.4577 0.0259 0.1665 0.1176α 5 0.4522 0.0000 0.0913α 6 0.3995 0.0548α 7 0.3305µ I 0.6419 0.5674 0.5189 0.4858 0.4584 0.4333 0.4142 0.3979S I 0.1303 0.0611 0.0317 0.0201 0.0134 0.0092 0.0068 0.0052Table 5.4: Optimal weighting α m , scintillation index S I , and the fractionalaverage integrated intensity µ when the first N Hermite-Gaussian modesf m (x) are used to generate the optimal pupil plane partially coherence.where L m pis the generalized Laguerre polynomials. Plots of the normalizedone dimensional Laguerre-Gaussian beam amplitude with m = 0 have beenshown in Figure 5.3.We setup the same one dimensional propagation situation as which isused in the previous section with parameters in Table 5.1. When a singlenormalized coherent mode f p,0 (x) propagated to the target, the integratedintensity ξ p through free space, the average integrated intensity µ p throughturbulence and the scintillation index S Ipof the received fields are shown inTable 5.5.An evident reduction of the optimal scintillation index has been shown inFigure 5.4 and the corresponding optimal parameters are shown in Table 5.6.As in the case using Hermite-Gaussian modes, we noticed that the averageintegrated intensity drops down with the scintillation index reduction.82

p = 0p = 1p = 2420−2420−2420−2−0.2 0 0.2xp = 3−0.2 0 0.2xp = 4−0.2 0 0.2xp = 5420−2420−2420−2−0.2 0 0.2xp = 6−0.2 0 0.2xp = 7−0.2 0 0.2xp = 8420−2−0.2 0 0.2x (meter)420−2−0.2 0 0.2x (meter)420−2−0.2 0 0.2x (meter)Figure 5.3: One dimensional Laguerre-Gaussian beams f p,m (x) with m = 0.83

p ξ p µ p S p0 0.8981 0.6296 0.12851 0.3743 0.4291 0.03502 0.2810 0.3062 0.07453 0.1343 0.2353 0.07554 0.1564 0.1920 0.05745 0.1601 0.1633 0.04276 0.1157 0.1416 0.03767 0.1129 0.1210 0.03758 0.1030 0.0992 0.03879 0.1007 0.0832 0.0358Table 5.5: Fractional integrated intensity ξ p in free space, fractional averageintegrated intensity µ p through turbulence, and the scintillation index S p ofthe received fields by transmitting single Laguerre-Gaussian modes f p,0 (x).0.160.14Optimal Scintillation Index0.140.12Scintillation Index0.120.10.080.060.040.020 1 2 3 4 5 6 7 8 9Modes index(a)Minimum scintillation index0.10.080.060.040.0201 2 3 4 5 6 7 8 9 10Number of modes used(b)Figure 5.4: Scintillation index by transmitting Laguerre-Gaussian modes:(a) Scintillation index by transmitting single modes. (b) Reduction of theoptimal scintillation index when the number of Laguerre-Gaussian modesincreased84

N 1 2 3 4 5 6 7 8α 0 1.0000 0.1177 0.2403 0.2019 0.1579 0.1116 0.0935 0.0840α 1 0.8823 0.0718 0.0000 0.0367 0.0904 0.0835 0.0732α 2 0.6879 0.3174 0.1838 0.0924 0.0910 0.0822α 3 0.4807 0.2203 0.1473 0.0577 0.0727α 4 0.4013 0.0555 0.1783 0.0844α 5 0.5029 0.0000 0.0623α 6 0.4960 0.2121α 7 0.3291µ I 0.6296 0.4527 0.3927 0.3374 0.3003 0.2648 0.2406 0.2228S I 0.1285 0.0313 0.0195 0.0103 0.0077 0.0058 0.0048 0.0042Table 5.6: Optimal weighting α p , scintillation index S I , and the fractionalaverage integrated intensity µ when the first N Laguerre-Gaussian modesf p,0 (u) are used to generate the optimal pupil plane partially coherence.The minimum scintillation index S I and corresponding optimal weightingα p are computed numerically when multiple Laguerre-Gaussian modes areused to produce the optimal partially coherence in the pupil plane.5.5 Monte-Carlo SimulationBecause the multiple integral expressions in equation (5.20) and equation(5.29) can not be solved in closed form, analytical calculation of higher orderstatistics of the integrated intensity becomes very difficult. In our case, thenumerical computation of the second moment of the integrated intensity isbeyond the capacity of today’s high speed computers, especially for two dimensionalpropagation cases. So we turn to Monte-Carlo simulation, which iswidely used in the problem of optical beam propagation through turbulence.In this simulation study, we generate random realizations of the phase85

screen in the pupil plane according to the structure function D φ (⃗u 1 , ⃗u 2 ). Bycalculating the integrated intensity of different input beams for each realizationof the pupil phase and accumulating the results, statistical properties(first and second moments) of the received intensity can be obtained.To measure the accuracy of the Monte-Carlo simulation, we define a criterion,the mean square error ɛ 2 , as followsɛ 2 =∑ ( ) 2V m −V tV t, (5.38)(M × N)where V m is the performance estimated from Monte-Carlo simulation andV t is the theoretical calculation result. The summation is taken over allelements in the matrices V M×N and the error is measured per array elementsby dividing (M × N), the number of elements in the array V .We set up the propagation simulation by taking the same physical parametersthat we used in Table 5.1, so that the Monte-Carlo simulation resultscan be compared with the theoretical calculation results that we obtained inthe previous section.5.5.1 Phase screen generationThere are many approaches that can be used to generate the random phasescreens with proper statistical properties. A variety of phase screen generatorshave been introduced in the literature [58, 15, 29]. One simple approachbased on linear combination of orthonormal basis functions gives good performancein terms of statistics matching. We use this approach in our simulation86

120100TheoreticalSimulation1.21TheoreticalSimulation800.8D(u,u)6040−D(∆ u)/2e0.60.40.22000−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5∆ u(a)−0.2−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5∆ u(b)Figure 5.5: Theoretical and Monte-Carlo simulation of the phase structurefunction D φ (∆u) and 〈exp[φ(u 1 ) − φ(u 2 )]〉 of the random phase screento generate random phase screens with the structure function expressed as( ) 5/3 |u1 − u 2 |D φ (u 1 , u 2 ) = 6.88,r 0where r 0 is the Fried parameter. More details about this approach can befound in Section 3.7 of reference [58].It is straight forward to see that the accuracy of the final simulationresults depend on the accuracy of the random phase screens generated. Thestructure function D φ (u 1 , u 2 ) of the random phase and 〈exp[φ(u 1 ) − φ(u 2 )]〉are taken as performance metrics and the mean square error can be calculatedas in equation (5.38). Taking the computation across a data set of 1,000random draws, the theoretical and Monte-Carlo simulation results are shownin Figure 5.5 and the square errors are shown in Table 5.7.87

Scintillation index of single mode0.40.350.30.250.20.150.1r 0= 0.05 mr 0= 0.1 mr 0= 0.15 mIntegrated intensity of single mode10.90.80.70.60.50.4free spacer 0= 0.05 mr 0= 0.1 mr 0= 0.15 m0.050.300 1 2 3 4 5 6 7m(a)0.20 1 2 3 4 5 6 7m(b)Figure 5.6: Monte-Carlo simulation by transmitting single one dimensionalHermite-Gaussian modes f m (x): (a) Scintillation index, (b) average integratedintensity.5.5.2 Pupil plan phase screen simulationMonte-Carlo simulation for the pupil plane phase screen model is taken firstsince the theoretical calculation results in one dimensional propagation areavailable which can be used for comparison with the simulation results.One-dimensional Monte-Carlo simulationBy transmitting a single one dimensional Hermite-Gaussian mode f m (u)through the simulated random phase screens, the average integrated intensityand the scintillation index at the receiver are shown in Figure 5.6.When multiple Hermite-Gaussian modes are used to synthesize the optimalpupil plane coherence and propagate through the turbulence, the averageintegrated intensity and the scintillation index are shown in Figure 5.7. Evidentreduction of the scintillation index can be seen.88

Scintillation Index0.40.350.30.250.20.150.10.05r 0= 0.05 mr 0= 0.1 mr 0= 0.15 mAverage integrated intensity for optimal beam0.80.750.70.650.60.550.50.450.40.35r 0= 0.05 mr 0= 0.1 mr 0= 0.15 m01 2 3 4 5 6 7 8Number of modes used(a)1 2 3 4 5 6 7 8Number of modes used(b)Figure 5.7: Monte-Carlo simulation by transmitting the optimal coherencecomposed of multiple one dimensional Hermite-Gaussian modes f m (x): (a)Scintillation index, (b) average integrated intensity.D φ µ m S m 〈I m I n 〉ɛ 2 1.4835 × 10 −4 4.4090 × 10 −5 2.1388 × 10 −4 7.8846 × 10 −4Table 5.7: Mean square error ɛ 2 of the Monte-Carlo simulation for differentperformance values: µ m , S m and 〈I m I n 〉.Compare the above simulation results with the theoretical calculationsin Section 5.4.1, the mean square error of the fractional average integratedintensity µ m , of the scintillation index of single transmitted mode S m , of theaverage mutual received intensity array are shown in Table 5.7.The above results show that Monte-Carlo simulation matches the theoreticalcalculations very well for the first and second moments of the integratedintensities.The simulation results, achieved by transmitting one dimensional Laguerre-Gaussian beams and the performance of the optimal coherence composed ofmultiple Laguerre-Gaussian modes, are shown in Figure 5.8 and Figure 5.989

0.4r 0= 0.050.8r 0= 0.050.35r 0= 0.1r 0= 0.150.7r 0= 0.1r 0= 0.15Scintillation index0.30.250.20.150.1Average integrated intensity0.60.50.40.30.050.201 2 3 4 5 6 7 8m(a)0.11 2 3 4 5 6 7 8m(b)Figure 5.8: Monte-Carlo simulation by transmitting single one dimensionalLaguerre-Gaussian modes f m,0 (x): (a) Scintillation index, (b) average integratedintensity.0.4r 0= 0.050.8r 0= 0.050.35r 0= 0.1r 0= 0.150.7r 0= 0.1r 0= 0.15Scintillation Index0.30.250.20.150.1Average integrated intesity0.60.50.40.30.050.201 2 3 4 5 6 7 8Number of modes used(a)0.11 2 3 4 5 6 7 8Number of modes used(b)Figure 5.9: Monte-Carlo simulation by transmitting the optimal coherencecomposed of multiple one dimensional Laguerre-Gaussian modes f m,0 (x): (a)Scintillation index, (b) average integrated intensity.90

Scintillation index of single mode0.90.80.70.60.50.40.30.20.1r 0= 0.05 mr 0= 0.1 mr 0= 0.15 mIntegrated intensity of single mode0.60.50.40.30.20.1r 0= 0.05 mr 0= 0.10 mr 0= 0.15 m00 1 2 3 4 5 6 7 8m(a)00 1 2 3 4 5 6 7 8m(b)Figure 5.10: Monte-Carlo simulated performance of transmitting single twodimensional Hermite-Gaussian beam: (a) Scintillation index, (b) averageintegrated intensity.Two-dimensional Monte-Carlo simulationBecause the Monte-Carlo simulation in one dimensional case showed goodperformance in terms of matching theoretical calculations well, it is naturalto extend it to two dimensional cases. Without changing the physical propagationparameters, the first and second moments of the integrated intensity,the minimum scintillation index, and the associated optimal weighting areshown in the following figures and table. As in the one dimensional case, evidentreduction of the scintillation index can be seen when the optimal pupilplane coherence generated by multiple beams are propagated to the target.But the average integrated intensity drops down also with the increasing ofthe number of coherent modes used.Monte-Carlo simulation by transmitting two dimensional Hermite-Gaussianbeams and the optimal pupil plane coherence composed of Hermite-Gaussian91

0.90.80.7r 0= 0.05 mr 0= 0.1 mr 0= 0.15 m0.5r 0= 0.05 mr 0= 0.1 mr 0= 0.15 mScintillation Index0.60.50.40.3Integrated intensity0.40.30.20.20.10.101 2 3 4 5 6 7 8 9Number of modes used(a)01 2 3 4 5 6 7 8 9Number of modes used(b)Figure 5.11: Monte-Carlo simulated performance of transmitting the optimalpupil plane coherence composed of multiple two dimensional Hermite-Gaussian beams: (a) Scintillation index, (b) average integrated intensity.modes are shown in Figure 5.10 and Figure 5.11. Simulation results of transmittingtwo dimensional Laguerre-Gaussian beams and the optimal pupilplane coherence composed of Laguerre-Gaussian modes are shown in Figure5.12 and Figure 5.13.5.5.3 Multiple phase screen simulationWe set up the Monte-Carlo simulation by using multiple phase screens tosimulate the turbulent propagation path. The von Karman spectrum is usedand four phase screens are set equally distanced from the transmitter pupilto the target.In this simulation, the FFT method are used to generatemultiple phase screens following von Karman turbulence spectrum.Fourrandom phase screens are put in the propagation path from pupil to receiverwith equal distance. Physical parameters used in the simulation are shown92

0.90.80.7r 0= 0.05 mr 0= 0.1 mr 0= 0.15 m0.5r 0= 0.05 mr 0= 0.1 mr 0= 0.15 m0.60.4Scintillation index0.50.40.3Integrated intensity0.30.20.20.10.100 1 2 3 4 5 6 7 8(a)00 1 2 3 4 5 6 7 8(b)Figure 5.12: Monte-Carlo simulated performance of transmitting single twodimensional Laguerre-Gaussian beams: (a) scintillation index, (b) averageintegrated intensity.0.90.80.7r 0= 0.05 mr 0= 0.1 mr 0= 0.15 m0.5r 0= 0.05 mr 0= 0.1 mr 0= 0.15 mScintillation Index0.60.50.40.3Integrated intensity0.40.30.20.20.10.101 2 3 4 5 6 7 8 9Number of modes used(a)01 2 3 4 5 6 7 8 9Number of modes used(b)Figure 5.13: Monte-Carlo simulated performance of the optimal coherencecomposed of multiple two dimensional Laguerre-Gaussian modes: (a) scintillationindex, (b) associated average integrated intensity.93

Physical parameter Symbol Value(s)transmitter diameter D 0.5 mpropagation distance d 20 kmwave length λ 1.0 µmbeam waist w 0 5 cmreceiver radius r 8 cminner scale l 0 0 mouter scale L 0 10 mstructure constant Cn 2 2, 5, 9 × 10 −16 m −2/3Table 5.8: Parameters used in the Monte-Carlo simulation of laser beamspropagate through multiple random phase screens.in Table 5.8.By using two dimensional Hermite-Gaussian beams, the scintillation indexand average integrated intensity by transmitting single coherent modesare shown in Figure 5.14. Corresponding performances by transmitting theoptimal partial coherence are shown in Figure 5.15. It is easy to see that weget similar results as what we achieved in the pupil plane phase screen model.Monte-Carlo simulation results by using two dimensional Laguerre-Gaussianbeams set are shown in Figure 5.16 and Figure 5.17.5.6 Aperture Averaging of ScintillationIt is noticed that when the receiver aperture has a finite size R rather than apoint receiver, the intensity fluctuations S I (R) drop because of the averagingprocess over the aperture. This phenomenon is called aperture averaging, andhas been studied by many researchers [73, 26, 79, 57]. Aperture averagingeffect (also called aperture smoothing) can help us to understand the well94

0.6C n2 = 2x10−16C n2 = 5x10−16C n2 = 9x10−160.50.450.4C n2 = 2x10−16C n2 = 5x10−16C n2 = 9x10−16Scintillation index0.50.40.30.20.100 1 2 3 4 5 6 7 8Coherent mode index(a)Average integrated intensity0.350.30.250.20.150.10.0500 1 2 3 4 5 6 7 8Coherent mode index(b)Figure 5.14: Monte-Carlo simulated performance of transmitting single twodimensional Hermite-Gaussian beams through multiple phase screens: (a)scintillation index, (b) average integrated intensity.0.6C n2 = 2x10−16C n2 = 5x10−16C n2 = 9x10−160.50.450.4C n2 = 2x10−16C n2 = 5x10−16C n2 = 9x10−16Scintillation index0.50.40.30.20.101 2 3 4 5 6 7 8 9Number of modes used(a)Average integrated intensity0.350.30.250.20.150.10.0501 2 3 4 5 6 7 8 9Number of modes used(b)Figure 5.15: Monte-Carlo simulated performance of transmitting optimalpartial coherence synthesized by two dimensional Hermite-Gaussian beamsthrough multiple phase screens: (a) scintillation index, (b) average integratedintensity.95

0.6C n2 = 2x10−16C n2 = 5x10−16C n2 = 9x10−160.50.450.4C n2 = 2x10−16C n2 = 5x10−16C n2 = 9x10−16Scintillation index0.50.40.30.20.100 1 2 3 4 5 6 7 8Coherent mode index(a)Average integrated intensity0.350.30.250.20.150.10.0500 1 2 3 4 5 6 7 8Coherent mode index(b)Figure 5.16: Monte-Carlo simulated performance of transmitting single twodimensional Laguerre-Gaussian beams through multiple phase screens: (a)scintillation index, (b) average integrated intensity.0.6C n2 = 2x10−16C n2 = 5x10−16C n2 = 9x10−160.50.450.4C n2 = 2x10−16C n2 = 5x10−16C n2 = 9x10−16Scintillation index0.50.40.30.20.101 2 3 4 5 6 7 8 9Number of modes used(a)Average integrated intensity0.350.30.250.20.150.10.0501 2 3 4 5 6 7 8 9Number of modes used(b)Figure 5.17: Monte-Carlo simulated performance of transmitting optimalpartial coherence synthesized by two dimensional Laguerre-Gaussian beamsthrough multiple phase screens: (a) scintillation index, (b) average integratedintensity.96

known fact that planets scintillate less than stars when viewed at the samezenith angle [80].The ratioA(R) = S I(R)S I (0)(5.39)is defined as the aperture averaging factor [73], which is used to measurethe intensity fluctuation decrease as the receiver aperture size increases. Inthis section, we will show how the aperture averaging process affects thescintillation index reduction when the optimal pupil plane coherence are usedin this section.A simulation of one dimensional propagation in the random phase screenmodel was set up with the same physical parameters as were used in theprevious two sections. When single Hermite-Gaussian modes are transmitted,the scintillation index changes vs. the receiver size that has been shown inFigure 5.18 (a). It can be seen that there are evident aperture averagingeffects for every coherent mode when the receiver radius increased.When multiple Hermite-Gaussian modes are used to synthesize the optimalpupil plane coherence, the changes of scintillation index vs. the receiversize has been shown in Figure 5.18 (b). It can be seen that the apertureaveraging effect is evident when only one or two Hermite modes are used tosynthesize the optimal coherence. But when the number of coherent modesis increased, the scintillation index does not change much with the aperturesize, which means there is not much aperture averaging effect. It should benoticed that for certain aperture sizes, the scintillation index values in the97

0.70.6m=1m=2m=3m=40.450.40.35N=1N=2N=3N=4Minimum scintillation index0.50.40.30.2Minimum scintillation index0.30.250.20.150.10.10.0500 0.5 1 1.5 2 2.5 3 3.5 4radius of receiver(a)00 0.5 1 1.5 2 2.5 3 3.5 4normalized radius(b)Figure 5.18: Aperture averaging effect in the pupil plane phase screen model:(a) Scintillation index vs. the normalized receiver radius for single HermiteGaussian modes f m (x). (b) Optimal scintillation index vs. the normalizedreceiver radius for optimal beams synthesized by N Hermite-Gaussian modes.figure are the minimum one since the optimal pupil plane coherence are used.So we can conclude that by using the optimal pupil plane coherence, increasingreceiver aperture size will not help to cancel optical field scintillation.98

Chapter 6Summary6.1 ConclusionsIn this work, we investigated the optimal pupil plane field for two differentperformance metrics to compensate for the turbulence effects on laser beampropagation through turbulence.To compensate for laser beam spreading, the optimal beam to maximizethe averaged integrated intensity has been investigated and shown to be fullycoherent. In this study,1. We have derived the averaged kernel 〈H(u, u ′ )〉 over turbulence fortwo different turbulence models: the turbulence layer model and theextended Huygens-Fresnel principle model. A similarity relationshipbetween the two models has been shown.2. An equivalence between the average integrated intensity in turbulencepropagation and the integrated intensity in free space propagation witha new weighting function has been established.99

3. We have shown that the optimal coherent beam to maximize the averageintegrated intensity 〈I〉 does not provide evident improvement.Mathematical analysis demonstrates that the eigenvalues of 〈H(u, u ′ )〉distributed much more evenly because of the turbulence induced changes.There is not one beam that can perform much better than others.Short-term compensation is thus suggested.The optimal pupil plane coherence to reduce the beam scintillation hasbeen shown to be partially coherent in general cases. In this study,1. We have derived the first and second moments of the received intensityby transmitting coherent modes in two turbulence models: the turbulencelayer model and the extended Huygens-Fresnel principle model.2. We have used the Hermite-Gaussian and Laguerre-Gaussian beams asthe coherent beam set to synthesize the optimal partially coherentsource. We derived the optimal weights α k for coherent modes ψ k (u)of the optimal pupil plane mutual intensity J(u, u ′ ) and have shownevident reduction of the scintillation index when optimal partially coherentsources are used. We also showed the trade-off effect, decreasingof the average integrated intensity.3. We have shown the aperture averaging effects of the optimal beams.Comparison with the coherent beam showed the decreasing of apertureaveraging when optimal partially coherent beams were used. We100

showed that increasing the receiver aperture size will not help whenoptimal pupil plane coherence are used.6.2 Future workIt has been shown that partially coherent sources can effectively reduce thelaser beam scintillation. But an important issue in practice is that a schemeto generate the partially coherent beams with desired optimal coherence isstill not available.In multi-mode laser cases, the lack of phase correlations between themodes can be used to generate partially coherent fields. Another possibilityis to overlap multiple beams with independent phases. Recently, a newscheme [76] is proposed based on random phase modulation of a coherentbeam source. In this method, the coherent beam passes through a phasediffusor with a limited number of spatial correlation zones. Along with theinherent time averaging of the photo-detection processes, a beam with partiallycoherent properties can be produced. In another approach [78], thenon-linear effect in a single mode fiber is used to reduce the temporal coherenceof the transmitted beam, and its high spatial coherence is then reduced.Although multiple schemes have been proposed to generate partially coherentbeams, there is one important unsolved problem with the existingscheme: it is difficult to control the generated coherence.To achieve thedesired pupil plane partial coherence is still a difficult problem.We willinvestigate this problem in the future.101

Appendix AGlossary of SymbolsAa(x)cC 2 nDD f (x)df(u)H n (x)Hh(v, u)II 0i(x)jJ(r 1 , r 2 )kl 0L 0Transmitter pupilAperture functionSpeed of lightStructure constant of refractive indexTransmitter aperture diameterStructure function of fluctuation fPropagation distanceOptical field functionHermite polynomial of order nHermitian kernelLinear system spatial impulse functionIntegrated intensityIntegrated intensity of the transmitted fieldField intensity= √ −1Mutual intensityOptical wave numberInner scale of turbulenceOuter scale of turbulence102

L m p (x) Laguerre polynomialn Index of refractionn sRrFluctuation of the index of refractionReceiver region= ‖⃗r‖⃗r Position vector = (x, y, z)r 0TtFried parameterTemperatureTime⃗u Position vector in transmitter plane = (x, y)⃗v Position vector in receiver plane = (x, y)w(v) Weighting function at receiverx Distance in the plane transverse to direction of propagationy Distance in the plane transverse to direction of propagationz Distance in direction of propagationα Coefficients of the coherent modesχ Log amplitudeΓ 2Γ 4γκλSeconde moment of the fieldFourth moment of the fieldComplex degree of coherenceWave number of index-of-refraction fluctuationsWave lengthµ Mean of the integrated intensityν Spacial frequencyω Radian frequency = 2πfΦ nφψρ 0Spectrum of the refractive index fluctuationsPhase perturbation of wave fieldComplex phase perturbation or coherent modesTransverse coherence distance103

τ Time shift = t 2 − t 1θ Propagation angle104

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