Statistical Optics 2nd Edition by Joseph Goodman ISBN 1119009456 978-1119009450

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ISBN 10: 1119009456
ISBN 13: 978-1119009450
Author: Joseph Goodman

This book discusses statistical methods that are useful for treating problems in modern optics, and the application of these methods to solving a variety of such problems

This book covers a variety of statistical problems in optics, including both theory and applications.  The text covers the necessary background in statistics, statistical properties of light waves of various types, the theory of partial coherence and its applications, imaging with partially coherent light, atmospheric degradations of images, and noise limitations in the detection of light. New topics have been introduced in the second edition, including:

• Analysis of the Vander Pol oscillator model of laser light
• Coverage on coherence tomography and coherence multiplexing of fiber sensors
• An expansion of the chapter on imaging with partially coherent light, including several new examples
• An expanded section on speckle and its properties
• New sections on the cross-spectrum and bispectrum techniques for obtaining images free from atmospheric distortions
• A new section on imaging through atmospheric turbulence using coherent light
• The addition of the effects of “read noise” to the discussions of limitations encountered in detecting very weak optical signals
• A number of new problems and many new references have been added 

Statistical Optics, Second Edition is written for researchers and engineering students interested in optics, physicists and chemists, as well as graduate level courses in a University Engineering or Physics Department.

Statistical Optics 2nd Table of contents:

1 Introduction

1.1 Deterministic Versus Statistical Phenomena and Models

1.2 Statistical Phenomena in Optics

Figure 1.1 An optical imaging system.

1.3 An Outline of the Book

2 Random Variables

2.1 Definitions of Probability and Random Variables

2.2 Distribution Functions and Density Functions

Figure 2.1 Examples of distribution functions. (a) Discrete random variable; (b) continuous random variable; and (c) mixed random variable.

Figure 2.2 Typical probability density functions for (a) discrete, (b) continuous, and (c) mixed random variables. For the discrete and mixed cases, the labels and heights of the delta functions are meant to represent their areas.

2.3 Extension to Two or More Joint Random Variables

2.4 Statistical Averages

2.4.1 Moments of a Random Variable

2.4.2 Joint Moments of Random Variables

2.4.3 Characteristic Functions and Moment-Generating Functions

2.5 Transformations of Random Variables

2.5.1 General Transformations

Figure 2.3 The shaded sections on the u-axis are the regions for which the random variable Z will be less than or equal to the particular value of z shown.

Figure 2.4 The transformation z = au2. The gray horizontal line on the u-axis bounded by is the region Lz.

2.5.2 Monotonic Transformations

Figure 2.5 Example of a one-to-one probability transformation.

Figure 2.6 Plots of (a) the probability density before transformation, (b) the transformation law, and (c) the probability density after transformation.

2.5.3 Multivariate Transformations

2.6 Sums of Real Random Variables

2.6.1 Two Methods for Finding pZ(z)

Figure 2.7 The shaded and cross-hatched region represents the area within which Z ≤ z.

2.6.2 Independent Random Variables

2.6.3 The Central Limit Theorem

2.7 Gaussian Random Variables

2.7.1 Definitions

Figure 2.8 The Gaussian probability density function.

2.7.2 Special Properties of Gaussian Random Variables

Figure 2.9 Contours of constant probability density for a joint Gaussian density with and (a) ρ = 0, (b) ρ = 0.5, and (c) ρ = 0.999. The density functions have been normalized to have value unity at the origin.

Two Uncorrelated Jointly Gaussian Random Variables Are Also Statistically Independent

The Sum of Two Statistically Independent Jointly Gaussian Random Variables Is Itself Gaussian

The Sum of Two Dependent (Correlated) Jointly Gaussian Random Variables Is Itself Gaussian

Any Linear Combination of Jointly Gaussian Random Variables, Dependent Or Independent, Is A Gaussian Random Variable

For Jointly Gaussian Random Variables U1, U2, …, Un, Joint Moments of Order Higher Than Two Can Always Be Expressed in Terms of the First- And Second-Order Moments

2.8 Complex-Valued Random Variables

2.8.1 General Descriptions

2.8.2 Complex Gaussian Random Variables

Figure 2.10 Contours of constant probability density in the complex plane for a circular complex Gaussian random variable.

2.8.3 The Complex Gaussian Moment Theorem

2.9 Random Phasor Sums

2.9.1 Initial Assumptions

Figure 2.11 Random phasor sum.

2.9.2 Calculations of Means, Variances, and the Correlation Coefficient

2.9.3 Statistics of the Length and Phase

Figure 2.12 Rayleigh probability density function.

2.9.4 Constant Phasor Plus a Random Phasor Sum

Figure 2.13 Sum of a constant phasor of length s and a random phasor sum.

Figure 2.14 Rician density function. The parameter k represents the ratio s/σ.

Figure 2.15 Probability density function of the phase θ of the sum of a constant phasor and a random phasor sum, as a function of the parameter k = s/σ.

2.9.5 Strong Constant Phasor Plus a Weak Random Phasor Sum

Figure 2.16 Large constant phasor with length s plus a small random phasor sum.

2.10 Poisson Random Variables

Problems

3 Random Processes

3.1 Definition and Description of a Random Process

Figure 3.1 An ensemble of sample functions, where t1 and t2 are the parameter values for which the joint density function pU(u1, u2; t1, t2) is specified.

3.2 Stationarity and Ergodicity

Figure 3.2 Sample functions of a nonstationary process.

Figure 3.3 A stationary process that is nonergodic.

Figure 3.4 The hierarchy of classes of random processes.

3.3 Spectral Analysis of Random Processes

3.3.1 Spectral Densities of a Known Function

3.3.2 Spectral Densities of a Random Process

3.3.3 Energy and Power Spectral Densities for Linearly Filtered Random Processes

3.4 Autocorrelation Functions and the Wiener–Khinchin Theorem

3.4.1 Definitions and Properties

3.4.2 Relationship to the Power Spectral Density

3.4.3 An Example Calculation

Figure 3.5 Sample function of a random telegraph wave.

Figure 3.6 Autocorrelation function and power spectral density of a random telegraph wave.

3.4.4 Autocovariance Functions and Structure Functions

3.5 Cross-Correlation Functions and Cross-Spectral Densities

Figure 3.7 Transformation of cross-spectral density under linear filtering.

3.6 Gaussian Random Processes

3.6.1 Definition

3.6.2 Linearly Filtered Gaussian Random Processes

3.6.3 Wide-Sense Stationarity and Strict Stationarity

3.6.4 Fourth- and Higher-Order Moments

3.7 Poisson Impulse Processes

3.7.1 Definition

Figure 3.8 (a) A sample function of a Poisson impulse process, together with (b) the corresponding rate function.

3.7.2 Derivation of Poisson Statistics from Fundamental Hypotheses

Figure 3.9 Filtered Poisson impulse process. (a) Filter response when a single impulse lies in the time interval (t1, t2). (b) Typical sample function when (c) Typical response when

3.7.3 Derivation of Poisson Statistics from Random Event Times

3.7.4 Energy and Power Spectral Densities of Poisson Processes

Energy Spectral Density, Deterministic Rate Function

Power Spectral Density, Deterministic Rate Function

3.7.5 Doubly Stochastic Poisson Processes

Figure 3.10 Region of integration.

3.7.6 Spectral Densities of Linearly Filtered Poisson Impulse Processes

3.8 Random Processes Derived from Analytic Signals

3.8.1 Representation of a Monochromatic Signal by a Complex Signal

Figure 3.11 Fourier spectra of (a) a monochromatic real-valued signal and (b) its complex representation.

3.8.2 Representation of a Nonmonochromatic Signal by a Complex Signal

Figure 3.12 Construction of an analytic signal from a real signal.

3.8.3 Complex Envelopes or Time-Varying Phasors

Figure 3.13 Power spectrum of a narrowband signal.

3.8.4 The Analytic Signal as a Complex-Valued Random Process

3.9 The Circular Complex Gaussian Random Process

3.10 The Karhunen–Loève Expansion

Problems

Figure 3-3p

4 Some First-Order Statistical Properties of Light

4.1 Propagation of Light

4.1.1 Monochromatic Light

Figure 4.1 Propagation geometry.

4.1.2 Nonmonochromatic Light

4.1.3 Narrowband Light

4.1.4 Intensity or Irradiance

4.2 Thermal Light

4.2.1 Polarized Thermal Light

Figure 4.2 Probability density function of the instantaneous intensity of polarized thermal light.

4.2.2 Unpolarized Thermal Light

Figure 4.3 Probability density function of the instantaneous intensity of unpolarized thermal light.

4.3 Partially Polarized Thermal Light

4.3.1 Passage of Narrowband Light Through Polarization-Sensitive Systems

Figure 4.4 Old (X, Y) and new (X′, Y′) coordinate systems after rotation by angle θ.

4.3.2 The Coherency Matrix

4.3.3 The Degree of Polarization

4.3.4 First-Order Statistics of the Instantaneous Intensity

Figure 4.5 Probability density function of the instantaneous intensity of a thermal source with degree of polarization

4.4 Single-Mode Laser Light

4.4.1 An Ideal Oscillation

Figure 4.6 Probability density functions of (a) amplitude and (b) intensity for a perfectly monochromatic wave of unknown phase.

4.4.2 Oscillation with a Random Instantaneous Frequency

4.4.3 The Van der Pol Oscillator Model

The Van der Pol Oscillator Equation

Linearizing the Equation

Output Noise Power and Noise Bandwidth

In-Phase and Quadrature Components of the Noise Output

Statistics of the Total Output Field and Intensity

4.4.4 A More Complete Solution for Laser Output Intensity Statistics

Figure 4.7 Normalized average output intensity of a single-mode laser as a function of the pump parameter w.

Figure 4.8 Normalized standard deviation of output intensity of a single-mode laser as a function of pump parameter w.

Figure 4.9 Risken’s solution for the probability density function of the output intensity of a single-mode laser oscillator for various pump parameters w.

4.5 Multimode Laser Light

4.5.1 Amplitude Statistics

Figure 4.10 Probability density function of the amplitude of a wave consisting of N equal-strength, independent sinusoidal modes. The total average intensity is held constant and equal to unity. A Gaussian curve is indistinguishable from the curve N = 5 on this plot.

4.5.2 Intensity Statistics

Figure 4.11 Probability density functions of intensity when N independent modes of identical strength are added. The total average intensity is assumed to be unity in all cases. When N = ∞, the probability density function is a negative-exponential density.

Figure 4.12 Ratio of standard deviation σI to mean for the intensity of light emitted by a laser oscillating in N independent, equal-strength modes.

4.6 Pseudothermal Light Produced by Passing Laser Light Through a Changing Diffuser

Figure 4.13 Pseudothermal light produced by a laser and a moving diffuser.

Problems

5 Temporal and Spatial Coherence of Opticalwaves

5.1 Temporal Coherence

5.1.1 Interferometers that Measure Temporal Coherence

Figure 5.1 The Michelson interferometer, including the source S, the lenses L1 and L2, mirrors M1 and M2, beam splitter BS, compensator C, and detector D.

Figure 5.2 Normalized intensity incident on detector D vs. normalized mirror displacement The spectrum shape has been assumed to be Gaussian, centered at for this plot. The envelope of the fringe pattern is drawn dotted.

Figure 5.3 Michelson interferometer based on single-mode fiber.

5.1.2 The Role of the Autocorrelation Function in Predicting the Interferogram

5.1.3 Relationship Between the Interferogram and the Power Spectral Density of the Light

Figure 5.4 Normalized power spectral densities for three spectral line shapes.

Figure 5.5 Visibility vs. Δντ for three spectral line shapes.

5.1.4 Fourier Transform Spectroscopy

Figure 5.6 Typical mid-infrared interferogram plotted with two different scales. The vertical axis represents detected intensity, and the horizontal axis represents optical path difference. The maximum path difference is 0.125 cm.

Figure 5.7 The Fourier transform of Fig. 5.6, representing the spectrum of the source. The vertical axis represents power spectral density, and the horizontal axis represents optical wavenumber, 2π/λ, in inverse centimeters.

Figure 5.8 The fiber-based Michelson interferometer for use in OCT. The axial scanning mirror changes the path length delay in the reference arm to select axial depth and the transverse scanning mirror selects the transverse coordinates to be imaged.

5.1.5 Optical Coherence Tomography

Figure 5.9 Image of hamster skin and underlying tissue obtained by time-domain OCT.

Figure 5.10 Diagrammatic representation of a Fourier domain OCT system.

5.1.6 Coherence Multiplexing

Figure 5.11 Coherence multiplexed sensor system.

5.2 Spatial Coherence

5.2.1 Young’s Experiment

Figure 5.12 Young’s interference experiment.

5.2.2 Mathematical Description of the Experiment

Figure 5.13 Physical explanation for loss of fringe visibility at large pinhole spacings: (a) small pinhole spacing and (b) large pinhole spacing.

5.2.3 Some Geometrical Considerations

Figure 5.14 Interference geometry for Young’s experiment.

Figure 5.15 Geometric properties of the fringes.

5.2.4 Interference Under Quasimonochromatic Conditions

Figure 5.16 Fringe patterns obtained for various values of the complex coherence factor (I(1) assumed equal to I(2)).

5.2.5 Cross-Spectral Density and the Spectral Degree of Coherence

Table 5.1 Names and Definitions of Various Measures of Coherence

5.2.6 Summary of the Various Measures of Coherence

5.2.7 Effects of Finite Pinhole Size

Figure 5.17 Effects of finite pinhole size. (a) Geometry of the experiment. (b) Partially overlapping diffraction patterns. The diffraction patterns of the individual pinholes are shown with a dashed line, while the interference between the light from the two pinholes is shown with a solid line. μ(P1, P2) has been assumed to be unity.

Figure 5.18 Optical system for interference experiment.

Figure 5.19 Interference pattern produced by the modified system. The diffraction patterns of both pinholes overlap and are indicated by the dashed line, while the interference pattern is shown with a solid line. μ(P1, P2) has been assumed to be unity.

5.3 Separability of Spatial and Temporal Coherence Effects

Figure 5.20 Measurement of the mutual coherence function of light transmitted by a moving diffuser.

5.4 Propagation of Mutual Coherence

5.4.1 Solution Based on the Huygens–Fresnel Principle

Figure 5.21 Geometry for propagation of mutual coherence, where θ1 and θ2 represent, respectively, the angle between the line joining P1 to Q and the surface normal at P1 and the corresponding angle for the line joining P2 to Q.

5.4.2 Wave Equations Governing Propagation of Mutual Coherence

5.4.3 Propagation of Cross-Spectral Density

5.5 Special Forms of the Mutual Coherence Function

5.5.1 A Coherent Field

5.5.2 An Incoherent Field

5.5.3 A Schell-Model Field

5.5.4 A Quasihomogeneous Field

5.5.5 Expansion of the Mutual Intensity Function in Coherent Modes

5.6 Diffraction of Partially Coherent Light by a Transmitting Structure

Figure 5.22 Geometry for diffraction calculation.

5.6.1 Effect of a Thin Transmitting Structure on Mutual Intensity

5.6.2 Calculation of the Observed Intensity Pattern

5.6.3 Discussion

5.6.4 An Example

Figure 5.23 Log plot of the normalized intensity Î in the diffraction pattern of a circular aperture for various states of illumination coherence. A Gaussian-shaped complex coherence factor with 1/e radius rc and a circular aperture with radius r0 are assumed. The intensity is normalized to maintain total constant energy. The parameter α represents r0/rc. Small values of α imply highly coherent illumination, and large values of α imply low coherence illumination.

5.7 The Van Cittert–Zernike Theorem

5.7.1 Mathematical Derivation of the Theorem

Figure 5.24 Geometry for derivation of the Van Cittert–Zernike theorem.

5.7.2 Discussion

5.7.3 An Example

Figure 5.25 The complex coherence factor vs. normalized spacing, under the assumption that ψ = 0.

5.8 A Generalized Van Cittert–Zernike Theorem

Figure 5.26 Normalized intensity distributions in the fringe patterns produced by two circular pinholes separated in a direction parallel to the x-axis. Coordinates in the fringe plane are assumed to be (α, β). The coordinate α has been normalized so that for a value of α = 1.22, the diffraction pattern of an individual pinhole falls to its first zero. The pinhole separation is held constant at value Δx and the diameter D of the incoherent source is changed. In the four cases shown, the diameter of the source is (i) D → 0 (a point source), (ii) (iii) and (iv) The total power emitted by the source has been held constant.

Figure 5.27 Fourier transform relations for the generalized Van Cittert–Zernike theorem. The (ξ, η) plane is on the left and the (x, y) plane is on the right.

5.9 Ensemble-Average Coherence

Problems

Figure 5-4p

Figure 5-5p

Figure 5-7p

Figure 5-9p

Figure 5-10p

Figure 5-14p

6 Some Problems Involving Higher-Order Coherence

6.1 Statistical Properties of the Integrated Intensity of Thermal or Pseudothermal Light

6.1.1 Mean and Variance of the Integrated Intensity

6.1.2 Approximate Form of the Probability Density Function of Integrated Intensity

Figure 6.1 Plots of vs. T/τc, exact solutions for Gaussian, Lorentzian and rectangular spectral profiles.

Figure 6.2 Approximation of (a) a smoothly varying instantaneous intensity I(t) by (b) a “boxcar” approximation.

Figure 6.3 Approximate probability density function of the integrated intensity of a polarized thermal wave for various values of

6.1.3 “Exact” Solution for the Probability Density Function of Integrated Intensity

Figure 6.4 Plot of for the case of unpolarized thermal light and various values of

Figure 6.5 Plot of the first five eigenvalues obtained from a discretized version of Eq. (6.1-54).

Figure 6.6 Approximate (dotted line) and “exact” (solid line) probability density functions for integrated intensity for a rectangular power spectral density and (a) c = 0.25, T/τc = 0.16; (b) c = 1.0, T/τc = 0.64; (c) c = 4.0, T/τc = 2.54; (d) c = 8.0, T/τc = 5.10.

6.2 Statistical Properties of Mutual Intensity with Finite Measurement Time

6.2.1 Moments of the Real and Imaginary Parts of J12(T)

Figure 6.7 Normalized variances of the real and imaginary parts of J12(T) as a function of μ12.

Figure 6.8 Probability clouds for J12(T) when μ12 = 0 and μ12 = 0.99. Both plots are for T/τc = 20.

6.3 Classical Analysis of the Intensity Interferometer

6.3.1 Amplitude versus Intensity Interferometry

Figure 6.9 Intensity interferometer. PM, photomultiplier.

6.3.2 Ideal Output of the Intensity Interferometer

6.3.3 “Classical” or “Self”-Noise at the Interferometer Output

Problems

7 Effects of Partial Coherence in Imaging Systems

7.1 Preliminaries

7.1.1 Passage of Partially Coherent Light through a Thin Transmitting Structure

Figure 7.1 A thin transmitting object. The x-axis points into the paper. n1 and n2 represent refractive indices. d(x, y) is the thickness of the object at coordinates (x, y) and d0 is the normal distance between the two parallel bounding planes.

7.1.2 Hopkins’ Formula

Figure 7.2 Geometry for calculation of Hopkins’ formula.

7.1.3 Focal Plane to Focal Plane Coherence Relationships

Figure 7.3 A generic optical system.

7.1.4 A Generic Optical Imaging System

7.2 Space-Domain Calculation of Image Intensity

7.2.1 An Approach to Calculate the Mutual Intensity Incident on the Object

Figure 7.4 Illumination optics.

7.2.2 Zernike’s Approximation

7.2.3 Critical Illumination and Köhler’s Illumination

7.3 Frequency Domain Calculation of the Image Intensity Spectrum

7.3.1 Mutual Intensity Relations in the Frequency Domain

7.3.2 The Transmission Cross-Coefficient

Figure 7.5 Up to multiplicative constants, the transmission cross-coefficient for arguments (νU, νV) and (s, t) is the area of overlap of the three circles shown. rc is the radius of the condenser lens and ri is the radius of the imaging lens. The circles are drawn in frequency space; hence, the dimensions 1/length for their radii.

7.4 The Incoherent and Coherent Limits

7.4.1 The Incoherent Case

7.4.2 The Coherent Case

7.4.3 When is an Optical Imaging System Fully Coherent or Fully Incoherent?

7.5 Some Examples

7.5.1 The Image of Two Closely Spaced Points

Figure 7.6 A telecentric imaging system. All lenses are assumed to have the same focal length f.

Figure 7.7 Normalized image intensity distributions for an object consisting of two pinholes separated by the Rayleigh resolution limit for various values of the complex degree of coherence of the light illuminating the two pinholes.

7.5.2 The Image of an Amplitude Step

Figure 7.8 Normalized intensity in the image of a step object for various coherence conditions. σ represents the ratio of the NA of the condenser lens to the NA of the first imaging lens, as viewed from the object.

7.5.3 The Image of a π-Radian Phase Step

Figure 7.9 Normalized intensity distribution along the u-axis in the image of a π-radian phase edge for σ = 0.1, 1, and 10. Again, σ = NAc/NAi.

7.5.4 The Image of a Sinusoidal Amplitude Object

7.6 Image Formation as an Interferometric Process

Figure 7.10 Apparent transfer functions for the intensity terms (a) cos (2πν0u) and (b) cos (4πν0u) as functions of the parameter σ = NAc/NAi. The parameter νc is the coherent cutoff frequency, where w is the half width of the imaging pupil.

7.6.1 An Imaging System as an Interferometer

Figure 7.11 Region of integration for calculating the spectrum of image intensity at frequency (νU, νV). The two circles have radius rp, the radius of the exit pupil. The center of the upper circle is displaced from the center of the lower circle by vector distance and the separation of the two pinholes (shown as small black circles at the ends of the heavy line) is also this same vector spacing.

Figure 7.12 Entrance and exit pupils.

7.6.2 The Case of an Incoherent Object

7.6.3 Gathering Image Information with Interferometers

Figure 7.13 Fizeau stellar interferometer.

7.6.4 The Michelson Stellar Interferometer

Figure 7.14 Michelson stellar interferometer.

7.6.5 The Importance of Phase Information

Figure 7.15 Illustration of the importance of phase information. (a) The original object intensity distribution (a rect function). (b) The complex coherence factor corresponding to that object intensity (a sinc function). (c) The modulus of the sinc function. (d) The image obtained by inverse transforming the modulus information rather than the full complex coherence factor.

Figure 7.16 Determining the separation Δ of two small sources from the autocorrelation function of their combined intensity distribution. (a) The object intensity distribution. (b) The autocorrelation function of the object intensity distribution.

7.6.6 Phase Retrieval in One Dimension

Figure 7.17 Special case in which full image recovery is possible from the autocorrelation function of the object intensity distribution. (a) Object intensity distribution. (b) Autocorrelation function of the object intensity distribution.

7.6.7 Phase Retrieval in Two Dimensions —Iterative Phase Retrieval

Figure 7.18 Illustration of the results of applying iterative phase-retrieval algorithms. (a) Original object (a simulated spacecraft), (b) modulus of the Fourier spectrum of the object, and (c) image recovered by using iterative algorithms.

7.7 The Speckle Effect in Imaging

Figure 7.19 Speckle in coherent imaging. (a) Object illuminated with incoherent light—no speckle discernible. (b) Object illuminated with coherent light—speckle very noticeable. (c) Magnified image of one character in the coherent case.

7.7.1 The Origin and First-Order Statistics of Speckle

Figure 7.20 Speckle formation in the image of a rough object.

7.7.2 Ensemble-Average Van Cittert–Zernike Theorem

7.7.3 The Power Spectral Density of Image Speckle

Figure 7.21 Cross section of the power spectral density of a speckle pattern resulting from an image of a uniformly bright rough surface using an imaging system having an unobstructed square exit pupil of width L.

7.7.4 Speckle Suppression

Polarization Diversity

Dynamic Diffusers

Angle Diversity

Wavelength Diversity

Problems

Figure 7-4p

8 Imaging Through Randomly Inhomogeneous Media

8.1 Effects of Thin Random Screens on Image Quality

8.1.1 Assumptions and Simplifications

Figure 8.1 Optical system assumed in the random screen analysis.

8.1.2 The Average Optical Transfer Function

8.1.3 The Average Point-Spread Function

8.2 Random-Phase Screens

8.2.1 General Formulation

8.2.2 The Gaussian Random-Phase Screen

Figure 8.2 Typical behavior of the structure function of phase for a wide-sense stationary case.

Figure 8.3 Typical OTFs for a system with a random-phase screen: (a) diffraction-limited OTF; (b) average OTF of the screen; (c) average OTF of the system. Note

Figure 8.4 Typical average system point-spread function for various phase variances. σ2(1) < σ2(2) < σ2(3) < σ2(4). Note the gradual reduction of the narrow core of the PSF as the phase variance increases.

8.2.3 Limiting Forms for the Average OTF and the Average PSF for Large Phase Variance

8.3 The Earth’s Atmosphere as a Thick Phase Screen

Figure 8.5 Imaging geometry.

8.3.1 Definitions and Notation

Refractive Index Dependencies on Space, Time, and Wavelength

Correlation Function and Power Spectral Density of the Refractive Index Fluctuations

Structure Function of Refractive Index

Autocorrelation Function and Power Spectrum of Refractive Index in a Plane

8.3.2 Atmospheric Model

Temperature Inhomogeneities Lead to Refractive Index Inhomogeneities

Power Spectral Density of the Refractive Index Fluctuations

Figure 8.6 Normalized power spectrum of the refractive index fluctuations: the Kolmogorov spectrum with the Tatarski and von Kármán modifications. Inner scale of 2 mm and outer scale of 10 m are assumed for illustration purposes.

The Refractive Index Structure Function

Figure 8.7 Refractive index structure function corresponding to the Kolmogorov spectrum.

8.4 Electromagnetic Wave Propagation Through the Inhomogeneous Atmosphere

8.4.1 Wave Equation in an Inhomogeneous Transparent Medium

8.4.2 The Born Approximation

8.4.3 The Rytov Approximation

8.4.4 Intensity Statistics

Figure 8.8 Log-normal probability density function for intensity for various values of σχ.

8.5 The Long-Exposure OTF

8.5.1 Long-Exposure OTF in Terms of the Wave Structure Function

Figure 8.9 (a) Long- and (b) short-exposure photographs of the star Lambda Cratis.

Figure 8.10 Imaging a distant point source through the atmosphere. The smooth curve on the right is a long-time-average intensity distribution in the image, while the atmospheric inhomogeneities and optical ray paths shown are at a fixed instantaneous time.

8.5.2 Near-Field Calculation of the Wave Structure Function

Figure 8.11 Geometry for phase structure function calculation.

Figure 8.12 Value of h[z/r] as a function of z/r. The horizontal line represents the constant 2.91, which is the asymptotic value of the integral as z/r → ∞.

Figure 8.13 Long-exposure atmospheric transfer functions for various values of (shown by solid curves) and diffraction-limited transfer functions for pupil sizes of 2 cm, 40 cm, and 5 m (shown by dashed curves). and z = 100 m assumed.

Figure 8.14 Long-time-average atmospheric MTFs. The dotted curves result from using the constant 2.91 in Eq. 8.5-39 and the solid curves from using the full expression from Eq. 8.5-38. Three different values of are assumed. The distance is 100m and the wavelength is 500nm. The values along the horizontal axis are expressed in cycles per milliradian.

8.5.3 Effects of Smooth Variations of the Refractive Index Structure Constant

Figure 8.15 Average profile of the height dependence of the structure constant h is the height above the ground in meters. Calculated from the “Hufnagel-Valley” model ([211]).

8.5.4 The Atmospheric Coherence Diameter r0

Figure 8.16 Normalized bandwidth in angular-frequency space vs. the ratio of the pupil diameter D0 of the optics to the coherence diameter r0 of the atmosphere. The dashed lines represent asymptotes for the resolution when D/r0 ≪ and ≫ than unity.

8.5.5 Structure Function for a Spherical Wave

Figure 8.17 Geometry for spherical wave propagation within the atmosphere.

8.5.6 Extension to Longer Propagation Paths—Log-Amplitude and Phase Filter Functions

Figure 8.18 Propagation geometry for filter function analysis. (x, y, z) is a particular point in the collecting aperture.

Figure 8.19 Filter functions for refractive index in a single plane of the turbulent medium and a single transverse wave number κt as a function of distance z − z′ in the medium.

Figure 8.20 Log-amplitude and phase filter functions for an extended turbulent medium, dependence on wavenumber, κt for a fixed z.

Figure 8.21 Log-amplitude and phase filter functions as a function of z for a fixed κt.

8.6 The Short-Exposure OTF

8.6.1 Long versus Short Exposures

Figure 8.22 Areas (shaded) on the exit pupil that influence the spatial frequency

8.6.2 Calculation of the Average Short-Exposure OTF

Figure 8.23 Diffraction-limited OTF and three OTFs that are products of the diffraction-limited OTF and the average atmospheric OTF for the cases α = 0, 1/2, and 1. The ratio of D0 to r0 has been taken to be 10.

Figure 8.24 Normalized bandwidths achieved for a diffraction-limited system of diameter D0 when atmospheric turbulence is present with a coherence diameter r0. α = 0 corresponds to the long-exposure case, while α = 0.5 and α = 1.0 correspond to short-exposure cases. The normalizing constant is equal to (r0/λ)2.

8.7 Stellar Speckle Interferometry

8.7.1 Principles of the Method

Figure 8.25 Average energy spectrum produced from 120 short-exposure images of the double star 9 Pupis, after compensation for the speckle interferometry transfer function.

8.7.2 Heuristic Analysis of the Method

Figure 8.26 Mean-squared MTF for a short exposure with a system having a circular pupil and for a ratio r0/D0 = 0.14. ν0 is the cutoff frequency of the diffraction-limited OTF.

8.7.3 Simulation

Figure 8.27 Simulation results showing the squared magnitude of the diffraction-limited OTF the squared magnitude of the product of the diffraction-limited OTF with the long-exposure atmospheric OTF, and the speckle interferometer transfer function, The symbol ν0 represents the diffraction-limited cutoff frequency.

8.7.4 A More Complete Analysis

Figure 8.28 The average squared magnitude of the system MTF for an imaging system with a circular pupil in the presence of atmospheric turbulence. The dotted curve is the square of the diffraction-limited MTF, while the solid curves correspond to three different values of r0/D0: 0.95, 0.14, and 0.03.

8.8 The Cross-Spectrum or Knox–Thompson Technique

8.8.1 The Cross-Spectrum Transfer Function

8.8.2 Constraints on

8.8.3 Simulation

Figure 8.29 Cross-spectrum transfer functions as a function of normalized spatial frequency. Three different frequency offsets Δν are used. Part (a) shows as a function of Δν/ν0 when Δν = 0 pixels and 4 pixels and the slice through is in the direction orthogonal to the separation Δν. Part (b) shows the corresponding results when the slice is in the same direction as Δν. Parts (c) and (d) are similar to parts (a) and (b) except that the separations Δν are 0 and 16 pixels.

8.8.4 Recovering Object Spectral Phase Information from the Cross-Spectrum

8.9 The Bispectrum Technique

8.9.1 The Bispectrum Transfer Function

8.9.2 Recovering Full Object Information from the Bispectrum

8.10 Adaptive Optics

Figure 8.30 Block diagram of an adaptive-optics system. Solid lines represent light paths, dashed lines represent electronic connections.

Figure 8.31 Two types of wavefront sensors. (a) The shearing interferometer. (b) The Shack–Hartmann sensor.

8.11 Generality of the Theoretical Results

8.12 Imaging Laser-Illuminated Objects through a Turbulent Atmosphere

Figure 8.32 Laser illuminator, satellite, and 2D detector array, separated by atmospheric turbulence.

Problems

Figure 8-4p Random checkerboard absorbing screen.

9 Fundamental Limits in Photoelectric Detection of Light

9.1 The Semiclassical Model for Photoelectric Detection

9.2 Effects of Random Fluctuations of the Classical Intensity

9.2.1 Photocount Statistics for Well-Stabilized, Single-Mode Laser Light

Figure 9.1 Probability masses associated with the Poisson distribution for

9.2.2 Photocount Statistics for Polarized Thermal Light

Counting Interval Much Shorter Than the Coherence Time

Arbitrary Counting Interval

Figure 9.2 Probability masses associated with the Bose–Einstein distribution for

9.2.3 Polarization Effects

Figure 9.3 Probability masses associated P(K) for partially polarized light with and 1, and

9.2.4 Effects of Incomplete Spatial Coherence

9.3 The Degeneracy Parameter

9.3.1 Fluctuations of Photocounts

9.3.2 The Degeneracy Parameter for Blackbody Radiation

Figure 9.4 Contours of constant degeneracy parameters in the and Tk plane. The visible and microwave portions of the spectrum are shown in gray.

9.3.3 Read Noise

Figure 9.5 Probability density functions of the continuous count measurement c. The average number of photocounts has been chosen to be 100. In part (a), the read noise standard deviation is 0.1, while in part (b) it is 10. The increase in width of the probability density function of c with increasing read noise is evident. In computing these curves, 1000 terms in the sum of Eq. 9.3-24 were used.

9.4 Noise Limitations of the Amplitude Interferometer at Low Light Levels

9.4.1 The Measurement System and the Quantities to Be Measured

Figure 9.6 Detection and estimation system assumed for the amplitude interferometer. and are estimates of the fringe visibility and phase.

9.4.2 Statistical Properties of the Count Vector

9.4.3 The Discrete Fourier Transform as an Estimation Tool

9.4.4 Accuracy of the Visibility and Phase Estimates

Figure 9.7 Phasor diagram for noisy fringe estimation.

9.4.5 Amplitude Interferometer Example

9.5 Noise Limitations of the Intensity Interferometer at Low Light Levels

9.5.1 The Counting Version of the Intensity Interferometer

Figure 9.8 Counting version of the intensity interferometer.

9.5.2 The Expected Value of the Count-Fluctuation Product and Its Relation to Fringe Visibility

9.5.3 The Signal-to-Noise Ratio Associated with the Visibility Estimate

9.5.4 Intensity Interferometer Example

9.6 Noise Limitations in Stellar Speckle Interferometry

9.6.1 A Continuous Model for the Detection Process

9.6.2 The Spectral Density of the Detected Image

Figure 9.9 Example of a rate function and the corresponding compound Poisson impulse process.

9.6.3 Fluctuations of the Estimate of Image Spectral Density

Figure 9.10 (a) Normalized energy spectral density of the image intensity and (b) corresponding energy spectral density of the detected image for

Figure 9.11 Energy spectral density estimate for a sinusoidal image intensity. The solid line represents the mean, and the shaded area represents the standard deviation of the estimate at each frequency.

9.6.4 Signal-to-Noise Ratio for Stellar Speckle Interferometry

9.6.5 Discussion of the Results

Figure 9.12 Typical single-image rms signal-to-noise ratio for speckle interferometry

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