Laser Beam Propagation through Random Media, Second Edition (SPIE Press Book)

Preface to Second Edition xi

Preface to First Edition xv

Symbols and Notation xix

1 Prologue 3

1.1 Introduction 4

1.2 Historical Background of Light 5

1.3 Optical Wave Models 8

1.4 Atmospheric Effects 9

1.5 Application Areas 15

1.6 A Brief Review of Communication Systems 22

1.7 Summary and Overview of the Book 26

References 32

2 Random Processes and Random Fields 35

2.1 Introduction 36

2.2 Probabilistic Description of Random Process 37

2.3 Ensemble Averages 38

2.4 Time Averages and Ergodicity 41

2.5 Power Spectral Density Functions 42

2.6 Random Fields 45

2.7 Summary and Discussion 49

2.8 Worked Examples 51

Problems 53

References 56

3 Optical Turbulence in the Atmosphere 57

3.1 Introduction 58

3.2 Kolmogorov Theory of Turbulence 58

3.3 Power Spectrum Models for Refractive-Index Fluctuations 66

3.4 Atmospheric Temporal Statistics 72

3.5 Summary and Discussion 73

3.6 Worked Examples 74

Problems 77

References 80

4 Free-Space Propagation of Gaussian-Beam Waves 83

4.1 Introduction 84

4.2 Paraxial Wave Equation 85

4.3 Optical Wave Models 87

4.4 Diffractive Properties of Gaussian-Beam Waves 91

4.5 Geometrical Interpretations?Part I 96

4.6 Geometrical Interpretations?Part II 99

4.7 Higher-Order Gaussian-Beam Modes 101

4.8 ABCD Ray-Matrix Representations 107

4.9 Single Element Optical System 112

4.10 Summary and Discussion 118

4.11 Worked Examples 122

Problems 127

References 133

5 Classical Theory for Propagation Through Random Media 135

5.1 Introduction 137

5.2 Stochastic Wave Equation 137

5.3 Born Approximation 141

5.4 Rytov Approximation 143

5.5 Linear Systems Analogy 151

5.6 Rytov Approximation for ABCD Optical Systems 152

5.7 Classical Distribution Models 154

5.8 Other Methods of Analysis 157

5.9 Extended Rytov Theory 159

5.10 Summary and Discussion 165

5.11 Worked Examples 167

Problems 170

References 177

6 Second-Order Statistics: Weak Fluctuation Theory 179

6.1 Introduction 181

6.2 Basic Concepts 182

6.3 Mutual Coherence Function 187

6.4 Spatial Coherence Radius 192

6.5 Angle-of-Arrival Fluctuations 199

6.6 Beam Wander 201

6.7 Angular and Temporal Frequency Spectra 206

6.8 Slant Paths 207

6.9 Summary and Discussion 210

6.10 Worked Examples 214

Problems 219

References 227

7 Second-Order Statistics: Strong Fluctuation Theory 229

7.1 Introduction 230

7.2 Parabolic Equation Method 231

7.3 Extended Huygens-Fresnel Principle 234

7.4 Method of Effective Beam Parameters 241

7.5 Summary and Discussion 247

7.6 Worked Examples 250

Problems 252

References 255

8 Fourth-Order Statistics: Weak Fluctuation Theory 257

8.1 Introduction 259

8.2 Scintillation Index 261

8.3 Beam Wander and Scintillation 269

8.4 Covariance Function of Irradiance 279

8.5 Temporal Spectrum of Irradiance 282

8.6 Phase Fluctuations 288

8.7 Slant Paths 299

8.8 Summary and Discussion 302

8.9 Worked Examples 308

Problems 313

References 318

9 Fourth-Order Statistics: Strong Fluctuation Theory 321

9.1 Introduction 322

9.2 Modeling Optical Scintillation 323

9.3 Asymptotic Theory 330

9.4 Scintillation Theory: Plane Wave Model 333

9.5 Scintillation Theory: Spherical Wave Model 341

9.6 Scintillation Theory: Gaussian-Beam Wave Model 349

9.7 Covariance Function of Irradiance 361

9.8 Temporal Spectrum of Irradiance 364

9.9 Distribution Models for the Irradiance 367

9.10 Gamma-Gamma Distribution 369

9.11 Summary and Discussion 379

9.12 Worked Examples 384

Problems 387

References 390

10 Propagation Through Complex Paraxial ABCD Optical Systems 495

10.1 Introduction 396

10.2 Single Element Optical System 396

10.3 Aperture Averaging 409

10.4 Optical Systems with Several Optical Elements 425

10.5 Summary and Discussion 430

10.6 Worked Examples 431

Problems 433

References 438

11 Free-Space Optical Communication Systems 441

11.1 Introduction 442

11.2 Direct Detection Optical Receivers 444

11.3 Fade Statistics?Part I 449

11.4 Fade Statistics?Part II 457

11.5 Spatial Diversity Receivers 465

11.6 Summary and Discussion 471

11.7 Worked Examples 472

Problems 474

References 475

12 Laser Satellite Communication Systems 477

12.1 Introduction 478

12.2 Atmospheric Channels 479

12.3 Background 484

12.4 Second-Order Statistics 490

12.5 Irradiance Statistics: Downlink Channel 493

12.6 Irradiance Statistics: Uplink Channel 500

12.7 Fade Statistics: Downlink Channels 510

12.8 Fade Statistics: Uplink Channels 516

12.9 Summary and Discussion 520

12.10 Worked Examples 524

Problems 526

References 529

13 Double-Passage Problems: Laser Radar Systems 533

13.1 Introduction 534

13.2 Laser Radar Configuration 536

13.3 Modeling the Backscattered Wave 540

13.4 Finite Smooth Target?Part I 547

13.5 Finite Smooth Target?Part II 559

13.6 Finite Smooth Reflector?Part III 570

13.7 Unresolved (Point) Target 575

13.8 Diffuse Target 589

13.9 Summary and Discussion 596

13.10 Worked Examples 597

Problems 599

References 604

14 Imaging Systems Analysis 607

14.1 Introduction 608

14.2 Coherent Imaging Systems 610

14.3 Incoherent Imaging Systems 614

14.4 Laser Imaging Radar 624

14.5 Zernike Polynomials 628

14.6 Summary and Discussion 636

14.7 Worked Examples 637

Problems 639

References 642

15 Propagation Through Random Phase Screens 647

15.1 Introduction 648

15.2 Random Phase Screen Models 649

15.3 Mutual Coherence Function 653

15.4 Scintillation Index and Covariance Function 656

15.5 Multiple Phase Screens 659

15.6 Summary and Discussion 662

Problems 664

References 666

16 Partially Coherent Beams 667

16.1 Introduction 668

16.2 Basic Beam Parameters 669

16.3 Mutual Coherence Function?Part I 671

16.4 Mutual Coherence Function?Part II 673

16.5 Scintillation Index?Part I 680

16.6 Scintillation Index?Part II 684

16.7 FSO Communication Systems 689

16.8 Ladar Model in Free Space 693

16.9 Ladar Model in Optical Turbulence 699

16.10 Summary and Discussion 704

16.11 Worked Examples 704

Problems 706

References 710

17 Other Beam Shapes 713

17.1 Introduction 714

17.2 Beam Spreading: Higher-Order Gaussian Beams 714

17.3 Annular Beam 720

17.4 Other Beams 729

17.5 Summary and Discussion 733

Problems 734

References 736

18 Pulse Propagation 737

18.1 Introduction 737

18.2 Background 738

18.3 Two-Frequency Mutual Coherence Function 740

18.4 Four-Frequency Cross-Coherence Function 746

18.5 Summary and Discussion 749

Problems 750

References 752

Appendix I: Special Functions 755

Appendix II: Integral Table 763

Appendix III: Tables of Beam Statistics 765

Index 775

PREFACE to Second Edition

Since publication of the first edition of this text in 1998 there have been several new and important developments in the theory of beam wave propagation through a random medium that we have incorporated into this second edition. Also, there were some topics excluded in the first edition that are now included. Nonetheless, we recognize that the general field of wave propagation through random media has grown in the last several years beyond what we can adequately cover in this one volume. For that reason, the reader should not consider this text an exhaustive treatment of propagation through turbulence.

One specific change in notation introduced here is the use of sigma_R^2 for the Rytov variance in place of sigma_1^2 (except in Chapter 13) to avoid confusion of the latter with the scintillation index sigma_I^2. Other changes/additions that now appear include the following:

• more worked examples and expanded sets of exercise problems
• models for the scintillation index under moderate-to-strong irradiance fluctuations
• models for aperture averaging based on ABCD ray matrices
• beam wander and its effects on scintillation
• theory of partial coherence of the source
• models of rough targets (other than Lambertian) for ladar applications
• phase fluctuations
• analysis of other beam shapes
• expanded analysis of free-space optical communication systems
• expanded imaging systems analysis

Foremost among the new theoretical developments is the extension of the Rytov theory from regimes of weak irradiance fluctuations into moderate-to-strong fluctuation regimes. Although much of this theory has been published in a companion text by the authors and C. Y. Hopen, called LASER BEAM SCINTILLATION WITH APPLICATIONS (SPIE Press, 2001), we present it here in a somewhat more complete treatment along with the standard Rytov theory that formed the basis for the first edition. Another topic in this second edition concerns the effects of beam wander on the scintillation index associated with an untracked beam. Conventional theory predicts that the on-axis scintillation associated with a focused beam along a horizontal path and that for a collimated beam on an uplink path to space will experience a substantial reduction (by orders of magnitude) as transmitter beam size increases provided there is limited beam wander. In the case of an untracked beam, however, the predicted reduction in scintillation will not occur. Also included in this second edition is a treatment of phase fluctuations, incorporating the phase variance, structure function, covariance, and temporal power spectrum. Among other topics, we introduce models for and discuss the role of partial coherence (spatially) of the source beam in reducing scintillation for example, in a free-space optical communication system. The same partial- coherence model can also be employed to describe the reflected radiation from a rough target like that which occurs in many laser radar applications.

In preparing this second edition, each chapter of the first edition was carefully examined for clarity and content, and most chapters have had some alteration in such cases the material is either broadened or simply rearranged, or both. As a consequence, the second edition has expanded the original twelve chapters of the first edition into eighteen chapters divided into three fundamental areas:

Part I: Basic Theory 1 PROLOGUE contains a brief discussion of fundamental concepts and application areas. It is basically the same as in the first edition, but now contains updated information on some of the application areas.

2 RANDOM PROCESSES AND RANDOM FIELDS contains a brief introduction to random processes and random fields. Only minor changes appear from first edition.

3 OPTICAL TURBULENCE IN THE ATMOSPHERE introduces Kolmogorov theory and various spectral models. Only minor changes appear from first edition.

4 FREE SPACE PROPAGATION OF GAUSSIAN-BEAM WAVES the introduction of higher- order Gaussian beam modes has been expanded from first edition and we have also moved the free-space propagation through optical elements by the use of ABCD ray matrices to this chapter.

5 CLASSICAL THEORY FOR PROPAGATION THROUGH RANDOM MEDIA introduces the Rytov approximation and other basic theories of wave propagation through random media. The treatment of Rytov theory for ABCD optical systems now appears in this chapter as well as the extended version of the Rytov theory that permits its use in regimes of strong irradiance fluctuations.

6 SECOND-ORDER STATISTICS: WEAK FLUCTUATION THEORY the second edition expands Chapter 6 from the first edition into Chapters 6 and 7. The discussion concerning the second-order field moment (mutual coherence function) is restricted to weak fluctuations but includes a new treatment of beam wander and slant path formulations in addition to the original horizontal path treatment.

7 SECOND-ORDER STATISTICS: STRONG FLUCTUATION THEORY the parabolic equation method and extended Huygens-Fresnel principle are introduced as theories used for calculating the mutual coherence function under strong irradiance fluctuations. The method of effective beam parameters is also introduced for calculating the spatial coherence radius of a beam and the variance of beam wander displacements.

8 FOURTH-ORDER STATISTICS: WEAK FLUCTUATION THEORY the second edition expands Chapter 7 from the first edition into Chapters 8 and 9. Here we discuss scintillation models and the effect of beam wander on scintillation of both collimated and focused beams. Other new topics included here are a discussion of phase fluctuations and scintillation along a slant path.

9 FOURTH-ORDER STATISTICS: STRONG FLUCTUATION THEORY scintillation models for plane waves, spherical waves, and Gaussian-beam waves are separately developed based on the extended Rytov theory for the strong fluctuation regime. The gamma-gamma distribution for irradiance fluctuations is also introduced in this chapter, illustrating how the parameters of this model are completely determined by atmospheric conditions (refractive-index structure constant, inner scale, and outer scale).

10 PROPAGATION THROUGH COMPLEX PARAXIAL ABCD OPTICAL SYSTEMS the propagation of a Gaussian beam wave through complex paraxial ABCD optical systems in the presence of atmospheric turbulence is featured here. In particular, we use the ABCD method to calculate the effect of a large-aperture receiver (aperture averaging) on the irradiance flux variance in the plane of a detector.

Part II: Applications

11 FREE SPACE OPTICAL COMMUNICATION SYSTEMS here we examine the impact of scintillation on free-space optical communication systems that operate along a horizontal path. Various fade statistics are introduced, including the probability of fade and mean fade time.

12 LASER SATELLITE COMMUNICATION SYSTEMS we extend the treatment from Chapter 11 to examine laser satellite communication systems. Various second-order and fourth-order statistics are developed. Beam-wander-induced scintillation caused in an untracked uplink collimated beam is discussed in detail and several comparisons with recent simulation results are included.

13 DOUBLE-PASSAGE PROBLEMS: LASER RADAR SYSTEMS the double-pass propagation problem associated with a laser radar system is treated here, which includes some new models developed since the first edition was published.

14 IMAGING SYSTEMS ANALYSIS a brief treatment on performance measures of imaging systems is presented. Both coherent and incoherent systems are discussed. We also introduce the Zernike polynomials and related filter functions used in adaptive optics systems.

Part III: Related Topics

15 PROPAGATION THROUGH RANDOM PHASE SCREENS the propagation of a beam wave through a random phase screen is taken up here, calculating the statistical quantities introduced in Chapters 6 and 8. The phase screen model also forms the basis for developing the (spatially) partial coherent beam analysis in Chapter 16.

16 PARTIALLY COHERENT BEAMS the notion of transmitter aperture averaging is presented for a partially coherent source and its impact on a free-space optical communication system. The same idea is used to model a rough target in a laser radar system.

17 OTHER BEAM SHAPES here we examine a few effects of atmospheric turbulence on higher-order Gaussian beams and annular beam shapes.

18 PULSE PROPAGATION this chapter briefly covers some aspects (beam spreading and scintillation) on the propagation of ultra-short pulses.

The second edition contains three appendices at the end of the book: (I) a review of properties associated with some of the special functions; (II) a short table of integrals for easy reference purposes; and (III) tables of tractable formulas for the wave structure function, spatial coherence radius, and scintillation index as predicted by various theories and atmospheric spectrum models.

Last, we value the constructive comments made by several users of the first edition that helped to guide us in developing this second edition.

Larry C. Andrews
Ronald L. Phillips

Orlando, FL