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Printed in the United States of America. Second printing 2007.

## Introduction to the Series

Welcome to the * SPIE Field Guides*! This volume is one of the first in a new series of publications written directly for the practicing engineer or scientist. Many textbooks and professional reference books cover optical principles and techniques in depth. The aim of the

*is to distill this information, providing readers with a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena, including definitions and descriptions, key equations, illustrations, application examples, design considerations, and additional resources. A significant effort will be made to provide a consistent notation and style between volumes in the series.*

**SPIE Field Guides**Each * SPIE Field Guide* addresses a major field of optical science and technology. The concept of these

*is a format-intensive presentation based on figures and equations supplemented by concise explanations. In most cases, this modular approach places a single topic on a page, and provides full coverage of that topic on that page. Highlights, insights and rules of thumb are displayed in sidebars to the main text. The appendices at the end of each*

**Field Guides***provide additional information such as related material outside the main scope of the volume, key mathematical relationships and alternative methods. While complete in their coverage, the concise presentation may not be appropriate for those new to the field.*

**Field Guide**The * SPIE Field Guides* are intended to be living documents. The modular page-based presentation format allows them to be easily updated and expanded. We are interested in your suggestions for new

*topics as well as what material should be added to an individual volume to make these Field Guides more useful to you. Please contact us at fieldguides@SPIE.org.*

**Field Guide**## The Field Guide series

Keep information at your fingertips with all of the titles in the Field Guide Series:

* Field Guide to Geometrical Optics,* John E. Greivenkamp (FG01)

* Field Guide to Atmospheric Optics,* Larry C. Andrews (FG02)

* Field Guide to Adaptive Optics,* Robert K. Tyson and Benjamin W. Frazier (FG03)

* Field Guide to Visual and Ophthalmic Optics,* Jim Schwiegerling (FG04)

* Field Guide to Polarization,* Edward Collett (FG05)

* Field Guide to Optical Lithography,* Chris A. Mack (FG06)

* Field Guide to Optical Thin Films,* Ronald R. Willey (FG07)

* Field Guide to Spectroscopy,* David W. Ball (FG08)

* Field Guide to Infrared Systems,* Arnold Daniels (FG09)

* Field Guide to Interferometric Optical Testing,* Eric P. Goodwin and James C. Wyant (FG10)

## Field Guide to Atmospheric Optics

The material in this * Field Guide* is a condensed version of similar material found in two textbooks:

*by L. C. Andrews and R. L. Phillips and*

**Laser Beam Propagation through Random Media***by L. C. Andrews, R. L. Phillips, and C. Y. Hopen. Both books are SPIE Press publications.*

**Laser Beam Scintillation with Applications**Topics chosen for this volume include a review of classical Kolmogorov turbulence theory, Gaussian-beam waves in free space, and atmospheric effects on a propagating optical wave. These atmospheric effects have great importance in a variety of applications like imaging, free space optical communications, laser radar, and remote sensing. Specifically, we present tractable mathematical models from which the practitioner can readily determine beam spreading, beam wander, spatial coherence radius (Fried’s parameter), angle-of-arrival fluc-tuations, scintillation, aperture averaging effects, fade probabilities, bit error rates, and enhanced backscatter effects, among others.

Notation used in this field guide is largely based on common usage found in propagation studies but may be different from that commonly used in related areas. For example, the symbol * “I”* is used here for irradiance. In the radiometry community, the symbol

*is commonly used for irradiance (W/m*

**“E”**^{2}) and the symbol

*is reserved for intensity (W/sr).*

**“I”**The foundational material for atmospheric optics has generally been widely dispersed throughout the journal literature over many years, making it difficult for researchers to update their knowledge and for newcomers to the field to compile and understand this difficult subject area. It is hoped that this * Field Guide* will serve a useful purpose for practicing engineers and scientists who wish to have access to such material in a single concise presentation.

## Table of Contents

**Glossary of Symbols x**

**Atmospheric Structure 1**

Atmospheric Structure 1

Atmospheric Structure with Altitude 2

Absorption and Scattering 3

Transmittance, Optical Depth, and Visibility 4

Meteorological Phenomena 5

**Kolmogorov Theory of Turbulence 6**

Kolmogorov Theory of Turbulence 6

Classical Turbulence 7

Velocity Fluctuations 8

Temperature Fluctuations 9

Optical Turbulence 10

Structure Parameter and Inner Scale 11

${C}_{n}^{2}$ Profile Models 12

Power Spectrum Models 13

**Optical Wave Models in Free Space 14**

Optical Wave Models in Free Space 14

Par axial Wave Equation 15

Plane Wave and Spherical Wave Models 16

Gaussian-Beam Wave at Transmitter 17

Gaussian-Beam Wave at Receiver 18

Hermite-Gaussian Beam Wave 19

Laguerre-Gaussian Beam Wave 20

Example 21

**Atmospheric Propagation: Second-Order Statistics 22**

Atmospheric Propagation: Second-Order Statistics 22

Rytov Approximation 23

Extended Huygens-Fresnel Principle 24

Parabolic Equation Method 25

Mean Irradiance and Beam Spreading 26

Beam Wander 27

Spatial Coherence Radius: Plane Wave 28

Spatial Coherence Radius: Spherical Wave 29

Spatial Coherence Radius: Gaussian-Beam Wave 30

Fried’s Parameter and the Phase Structure Function 31

Angle-of-Arrival and Image Jitter 32

Example 33

Example 34

**Atmospheric Propagation: Fourth-Order Statistics 35**

Atmospheric Propagation: Fourth-Order Statistics 35

Rytov Approximation: Fourth-Order Specializations 36

Scintillation Index: Theory 37

Scintillation Index: Plane Wave 38

Scintillation Index: Spherical Wave 39

Scintillation Index: Gaussian-Beam Wave 40

Covariance Function: Plane Wave 41

Temporal Power Spectrum: Plane Wave 42

Aperture Averaging: Plane Wave 43

Aperture Averaging: Spherical Wave 44

Example 45

**Imaging Systems and Adaptive Optics 46**

Imaging Systems and Adaptive Optics 46

Fried’s Atmospheric Parameter and Greenwood’s Time Constant 47

Point Spread Function and Modulation Transfer Function 48

Spatial Resolution 49

Strehl Ratio and Image Resolving Power 50

Isoplanatic Angle and Point-Ahead Angle 51

Zernike Polynomials and Wave Front Representation 52

Zernike Polynomials for Atmospheric Imaging 53

Modal Expansion and Aperture Filter Functions 54

Zernike Tilt, Piston, and Angle-of Arrival Jitter 55

**Free Space Optical Communication Systems 56**

Free Space Optical Communication Systems 56

Direct Detection System 57

Threshold Detection 58

Signal-to-Noise Ratio: Direct Detection 59

Bit Error Rate 60

Coherent Detection System 61

Signal-to-Noise Ratio: Coherent Detection 62

Probability of Fade: Lognormal Model 63

Probability of Fade: Gamma-Gamma Model 64

Lasersatcom: Mean Irradiance and Beam Spreading 65

Lasersatcom: Uplink Scintillation under Weak Fluctuations 66

Lasersatcom: Downlink Scintillation under Weak Fluctuations 67

Lasersatcom: General Theory for Uplink/Downlink Scintillation 68

Lasersatcom: General Theory for Downlink Covariance and Correlation Width 69

**Laser Radar and Optical Remote Sensing 70**

Laser Radar and Optical Remote Sensing 70

Basic Radar Principles 71

Statistical Characteristics of Echo Beam 72

Enhanced Backscatter: Spherical Wave 73

Enhanced Backscatter: Gaussian-Beam Wave 74

Spatial Coherence 75

Scintillation Index: Spherical Wave and Point Target 76

Scintillation Index: Gaussian-Beam Wave and Point Target 77

Scintillation Index: Smooth Target 78

Scintillation Index: Diffuse Target—I 79

Scintillation Index: Diffuse Target—II 80

**Appendix 81**

Equation Summary 81

Notes 87

**Bibliography 89**

**Index 91**

#### Glossary of Symbols

**A**

Aperture averaging factor

**b _{I}(ρ, L)**

Normalized covariance function of irradiance

BER

Bit error rate

* B_{I}*(

**r**

_{1},

**r**

_{2},

*),*

**L***(ρ)*

**B**_{I}Covariance function of irradiance

* ${B}_{\mathit{I}}^{\mathit{iR}}$* (

**r**,

*)*

**L**Correlation function associated with amplitude enhancement of reflected wave

c

Speed of light (= 3 × 10^{8} m/s)

${C}_{n}^{2}$, ${C}_{v}^{2}\left(h\right)$

Refractive-index structure parameter

${C}_{v}^{2}$

Velocity structure parameter

${C}_{T}^{2}$

Temperature structure parameter

CNR

Carrier-to-noise ratio

**d**

Normalized lens diameter $\left(=\sqrt{k{D}^{2}/4L}\right)$

**D**

Diameter of collecting lens

DOC

Modulus of the complex degree of coherence

* D*(

**r**

_{1},

**r**

_{2},

*),*

**L***(ρ,*

**D***)*

**L**Wave structure function

* D_{s}*(ρ,

*)*

**L**Phase structure function

* D_{n}*(

*)*

**R**Index of refraction structure function

**D _{RR}(R)**

Velocity structure function

**D _{T}(R)**

Temperature structure function

**E**

Electric field

erf(* x*), erfc(

*)*

**x**Error functions

EBS

Enhanced backscatter

EG

Equal gain coherent detection scheme

EO

Electro-optics

**F**

Phase front radius of curvature of beam at receiver

FAR

False alarm rate

**F _{G}**

Effective focal length of Gaussian lens

**F _{T}**

Fade level (in dB) below the mean on-axis irradiance

**F _{0}**

Phase front radius of curvature of beam at transmitter

_{p}F_{q}

Generalized hypergeometric function

* G*(

**s, r**;

*)*

**L**Green’s function

**h**

Altitude

**h _{0}**

Altitude of transmitter/receiver

**H**

Altitude of receiver/transmitter

**H _{n}(x)**

Hermite polynomial of degree **n**

* H*−

**V**_{5/7}

Hufnagle-Valley **C**^{2}*_{n}*(

*) model*

**h****i _{IF}**

Intermediate frequency (IF) signal current

**i _{S}**

Signal current in a receiver

**i _{N}**

Shot noise current in a receiver

**i _{T}**

Current threshold in a receiver

**I**^{0}(**r**, * L*)

Irradiance of beam in free space

* I*(

**r**,

*)*

**L**Irradiance of beam in random medium

* I_{v}*(

*)*

**x**Modified Bessel function of order **v**

* J_{v}*(

*)*

**x**Bessel function of order **v**

**k**

Wave number of beam wave (= 2π/λ)

* K_{v}*(

*)*

**x**Modified Bessel function of order **v**

**l _{0}**

Inner scale of turbulence

**L**

Propagation path length

LO

Local oscillator

**L**_{0}

Outer scale of turbulence

**L**_{n}^{(m)}(* x*)

Associated Laguerre polynomial

MCF

Mutual coherence function

MTF

Modulation transfer function

* n*(

**R**)

Index of refraction

**n**_{1}(**R**), **n**_{1}(**r**, * z*)

Random fluctuation in index of refraction

OTF

Optical transfer function

**p**

Transverse vector between two observation points

* p_{I}*(

*)*

**I**Probability density function of irradiance

PSF

Point spread function

Pr_{d}

Probability of detection

Pr_{fa}

Probability of false alarm

Pr(* E*)

Probability of error

**P _{s}**

Signal power

**q**

Strength of turbulence parameter (= * L/k*${\rho}_{\mathit{0}}^{\mathit{2}}$)

**Q _{l}**

Nondimensional inner-scale parameter (= * L*${\kappa}_{\mathit{l}}^{\mathit{2}}$/

*)*

**k****Q _{0}**

Nondimensional outer-scale parameter (= * L*${\kappa}_{\mathit{0}}^{\mathit{2}}$/

*)*

**k****r**

Transverse position of observation point

r_{0}

Atmospheric coherence length (Fried’s parameter)

**R**

Position vector in three dimensions

**Re**

Reynolds number

* S*(

**r**,

*)*

**L**Random phase

* S_{I}*(ω)

Power spectral density of irradiance

SNR

Signal-to-noise ratio

SR

Strehl ratio

TEM_{mn}

Transverse electromagnetic mode

**U**_{0}(**r**, * z*)

Complex amplitude of the field in free space

* U*(

**r**,

*)*

**z**Complex amplitude of the field in random medium

**W _{0}**

Beam radius at transmitter

**W**

Beam radius in free space at receiver

**W _{B}**

Beam radius in free space at the waist

**W _{e}**

Effective beam radius in random medium at receiver

**W _{G}**

Radius of Gaussian lens

**W _{R}**

Radius of target (reflector) surface

WSF

Wave structure function

**z _{B}**

Distance to beam waist from transmitter

${Z}_{n}^{\mathrm{m}}$(* r*, θ),

*[*

**Z**_{i}*]*

**m, n**Zernike polynomials

α(* L*)

Extinction coefficient

α, β

Parameters of the gamma-gamma distribution

${\beta}_{2}^{0}$

Rytov variance for a spherical wave

Γ(* x*)

Gamma function

Γ_{2} (**r**_{1}, **r**_{2}, * L*)

Mutual coherence function

Γ_{4}(**r**_{1}, **r**_{2}, **r**_{3}, **r**_{4}, * L*)

Fourth-order moment of the field

θ_{0}

Isoplanatic angle

Θ_{0}

Beam curvature parameter at transmitter

Θ

Beam curvature parameter at receiver

κ

Scalar spatial wave number

**κ _{l}**

Inner-scale wave number parameter (= 3.3/**l**_{0})

κ_{m}

Inner-scale wave number parameter (= 5.92/**l**_{0})

κ_{0}

Outer-scale wave number parameter (= 2π/**L**_{0})

λ

Wavelength

Λ_{0}

Fresnel ratio of beam at transmitter

Λ

Fresnel ratio of beam at receiver

ρ

Scalar separation distance between two observation points

ρ_{c}

Correlation width of irradiance fluctuations

ρ_{0}

Transverse spatial coherence radius

${\sigma}_{1}^{2}$

Rytov variance for a plane wave

${\sigma}_{B}^{2}$

Rytov variance for a Gaussian-beam wave

${\sigma}_{\text{diff}}^{2}$

Scintillation associated with reflected wave from a diffuse target

${\sigma}_{I}^{2}$

Scintillation index (normalized irradiance variance)

${\sigma}_{I}^{2}\left(D\right)$

Irradiance flux variance for a collecting aperture of diameter **D**

${\sigma}_{2}^{n}$

Total noise power in detector current

Φ*_{n}*(κ)

Three-dimensional spatial power spectrum of refractive index

Φ*_{RR}*(κ)

Three-dimensional spatial power spectrum of velocity

Φ*_{T}*(κ)

Three-dimensional spatial power spectrum of temperature

χ(**r**, * L*)

Random log-amplitude

ψ_{1}(**r**, * L*), ψ

_{2}(

**r**,

*)*

**L**Complex phase perturbations of Rytov approximation

ψ_{1}(**r**, **s**), ψ_{2}(**r**, **s**)

Complex phase perturbations of extended Huygens-Fresnel principle

Ω_{f}

Focusing parameter for geometric focus

Ω_{G}

Fresnel ratio characterizing radius of Gaussian lens

Ω_{R}

Fresnel ratio characterizing radius of reflector target

ζ

Zenith angle

∇^{2}

Laplacian operator

∇^{2}_{T1}

Laplacian operator with respect to **r**_{1}

∇^{2}_{T2}

Laplacian operator with respect to **r**_{2}

〈 〉

Ensemble average