General Theory of Light Propagation and Imaging Through the Atmosphere

Preface
References
Contents
In Memoriam
About the Author
1 History of the Telescope and Its Remarkable Contribution to Scientific Discovery (and the 400-Year Journey from Galileo to a Rigorous General Theory of Imaging Through Earth's Turbulent Atmosphere)
1.1 Telescope Imaging Through Earth’s Turbulent Atmosphere
1.2 Kolmogorov Theory (Mid-1960s)
1.3 Origins of the New General Theory
1.4 Publication of the New General Theory (1989/90)
1.5 Definitive Confirmation of Cores in Star Images
1.5.1 The UKIRT 3.8-m Instrument and Star Image Cores
1.6 Horace Babcock and Adaptive Optics
1.7 Pivotal Equation Yielded by the New General Theory
1.8 Future Direction of Ground-Based Observational Astronomy
1.9 Final Destination!
References
2 Introduction
2.1 Principal Cause of Differences Between Kolmogorov Theory and the New General Theory
2.1.1 Visible and IR Star Images for Large Turbulence Structure
2.1.2 Visible and IR Star Images for Small Turbulence Structure
2.2 Significant Features of the New General Theory
2.3 Book Content Preview
2.4 Kolmogorov Theory
2.4.1 Kolmogorov Theory and Its Damage Legacy
2.5 The New General Theory
References
3 Terms, Definitions, and Theoretical Foundations
3.1 Air Refractive Index
3.1.1 Air Temperature and Altitude
3.1.2 Air Pressure and Altitude
3.1.3 Integrated Optical Path Difference Over the Entire Atmospheric Depth
3.1.4 Effect of Humidity
3.1.5 Effect of Dispersion
3.1.6 Random Variables Associated with Atmospheric Turbulence
3.1.7 Astronomical Refraction
3.1.8 Atmospheric Extinction
3.2 Point-Objects
3.3 The Electromagnetic Spectrum
3.4 Quasi-Monochromatic Light
3.5 Amplitude and Phase of Light Waves Disrupted by Turbulence
3.6 The Atmosphere Considered as a Stochastic Process
3.6.1 Spatial and Temporal Stationarity and the Ensemble Average
3.6.2 Standard Error and Standard Deviation
3.6.3 Autocovariance and Autocorrelation Functions, the Variance, and Rms
3.6.4 The Atmospheric Refractive Index Field
3.7 Scalar Diffraction Theory
3.7.1 Scalar Diffraction Theory Applied to Atmospheric Propagation
3.7.2 Scalar Diffraction Theory Applied to Telescope Imaging
3.7.3 Monochromatic Light Fields
3.7.4 Analytic Signal
3.7.5 Complex Amplitude
3.7.6 Intensity
3.7.7 Irradiance
3.7.8 Polychromatic Light Fields
3.8 Coherence Terminology
3.9 Free-Space Propagation
3.9.1 Maxwell’s Electromagnetic Wave Equations
3.9.2 Helmholtz Equation
3.9.3 Solutions for Infinitely Extensive Plane Waves
3.10 Mathematical Notations and Quantity Dimensions
References
4 Diffraction
4.1 Diffraction by an Aperture
4.1.1 Fresnel Number
4.1.2 Fresnel–Kirchoff Diffraction Formula
4.1.3 Fresnel Near-Field Diffraction
4.1.4 Stationary Phase
4.1.5 Fraunhofer Far-Field Diffraction
4.2 Optical System Terminology
4.2.1 Telescopes, Telescope Objectives, and Eyepieces
4.2.2 Aperture Stops, Pupils, Conjugate Distances, Focal Lengths, and F/Numbers
4.2.3 Light Rays and Ray Terminology
4.2.4 Objects at Finite Distances
4.2.5 Objects at Infinite Distances
4.2.6 Pupil Functions
4.3 The Amplitude Point Spread Function
4.3.1 For Diffraction-Limited Telescopes with Circular Apertures
4.4 The Intensity Point Spread Function
4.4.1 The Airy Pattern
4.5 Strehl Intensity
4.5.1 Expressed in Terms of Rms Wavefront Error
4.5.2 For Circularly Symmetric Images
4.6 Rayleigh Resolution Criterion
4.7 Images of Extended Objects
4.7.1 Superposition Property
4.7.2 Nonlinear Optical Phenomena
4.7.3 Isoplanaticity
4.7.4 Convolution Integrals
4.7.5 Images of Coherently Illuminated Extended Objects
4.7.6 Images of Incoherently Illuminated Extended Objects
4.7.7 Images of Partially Coherently Illuminated Extended Objects
4.8 Images of Two-Point Objects
4.8.1 Incoherently Illuminated Two-Point Objects
4.8.2 Coherently Illuminated Two-Point Objects
4.9 Stellar Speckle Patterns
4.10 Effect of Central Obstruction on Telescope Point Spread Functions
4.11 Mathematical Notation Used in This Chapter
References
5 Wave Propagation After Scattering by a Thin Atmospheric Layer
5.1 Characterizing Atmospheric Paths and Telescopes by MTFs and OTFs
5.2 The Atmospheric Refractive Index
5.3 Wave Propagation in the Geometrical Optics Region
5.3.1 Optical Path Difference
5.3.2 Phase Angle of the Exiting Wave
5.3.3 Complex Amplitude of the Exiting Wave
5.3.4 The Two-Point Two-Wavelength Correlation Function for Exiting Waves
5.3.5 Complex Coherence Factor for Exiting Waves
5.3.6 Illustrative Plots of the Complex Coherence Factor
5.3.7 Illustrative Plots of the Two-Point Two-Wavelength Correlation Function
5.4 Near-Field Propagation of the Complex Amplitude
5.5 Near-Field Propagation of the Two-Point Two-Wavelength Correlation Function
5.5.1 Cases Where the Function Conserves
5.5.2 General Case of Non-Conservation of the Function
5.6 Near-Field Propagation of the Complex Coherence Factor
5.7 Development of Scintillation After Light Scattering by a Thin Layer
5.7.1 Dependence of Scintillation on Turbulence Scale Sizes in the Layer
5.7.2 Dependence of Scintillation on the Various Controlling Parameters
5.7.3 Effective Fresnel Numbers for Atmospheric Paths
5.8 Mathematical Notation Used in This Chapter
References
6 Wave Propagation Over Extended Atmospheric Paths
6.1 Atmospheric MTF Expressions Developed by Hufnagel and Stanley
6.1.1 Hufnagel and Stanley’s General Expression for the Atmospheric MTF
6.1.2 Hufnagel and Stanley’s Kolmogorov-Based Expression for the Atmospheric MTF
6.2 Layered Model Representations of Extended Atmospheric Paths
6.2.1 Two Equivalent Random Phase Screen Atmospheric Path Models
6.2.2 Properties of the Phase Screens in the Uncorrelated Random Phase Screen Path Model
6.2.3 Effect of Individual Random Phase Screens on Transmitted Light Waves
6.3 General Expression for the Two-Point Two-Wavelength Correlation Function
6.3.1 Case of Isotropic Turbulence
6.3.2 The Functional Form When ρ( ξ, ) is Gaussian
6.4 General Expression for the Atmospheric MTF
6.4.1 Case of Isotropic Turbulence
6.4.2 Functional Forms When ρ( ξ, ) is Gaussian
6.4.3 Comparison of the General Expression to that of Hufnagel and Stanley
6.5 Equivalent Phase Screen Representation of an Atmospheric Path
6.5.1 Relationship Between ρ(ξ, η) and the Refractive Index Structure Function, DN
6.5.2 Location of the Equivalent Phase Screen in the Atmospheric Path
6.5.3 Complex Amplitude Properties Arising from an Equivalent Phase Screen
6.5.4 Properties of the OPD Fluctuation Created by an Equivalent Phase Screen
6.6 General Expressions for M and S that Include Dispersion
6.7 Mathematical Notation Used in This Chapter
References
7 Properties of Point-Object Images Formed by Telescopes
7.1 Long- and Short-Exposure Images of Point-Objects
7.2 Telescope Coordinate Systems
7.3 The Complex Amplitude in an Instantaneous Point-Object Image
7.4 Telescope OTFs and MTFs
7.4.1 Telescope OTF and MTF for Incoherent Illumination
7.4.2 Amplitude Transfer Function of a Telescope for Coherent Illumination
7.5 Two-Point Two-Wavelength Correlation Function of the Complex Amplitudes in the Image
7.5.1 Characterizing the Influence of the Telescope Optics
7.5.2 Unit-Normalized Form of the Function
7.5.3 The Function at a Single Point in the Image
7.5.4 The Spectral Correlation Function at the Center of a Point-Object Image
7.6 Complex Coherence Factor of the Complex Amplitude in the Image
7.7 Average Intensity Envelopes for Point-Object Images
7.8 Statistics of the Complex Amplitude in Point-Object Images Formed by Large Telescopes
7.8.1 Reed’s Theorem for Gaussian-Distributed Complex Random Variables
7.8.2 Unit-Normalized Two-Point Two-Wavelength Correlation Function of the Image Intensities
7.8.3 Two-Wavelength Correlation Function of the Intensity at a Single Point in the Image
7.8.4 Two-Wavelength Correlation Function of the Complex Amplitude at a Single Point in the Image
7.9 OTF for an Entire End-To-End Imaging Path
7.9.1 OTF for an Entire End-To-End Imaging Path for Space Telescopes
7.9.2 OTF and Intensity PSF for a Diffraction-Limited Telescope with Circular Aperture
7.10 Mathematical Notation Used in This Chapter
References
8 Atmospheric Path Characterization
8.1 Obtaining the Atmospheric MTF from Point-Object Images
8.1.1 For Large Diffraction-Limited Telescopes
8.1.2 Long- and Short-Exposure Atmospheric MTFs
8.1.3 Effective End-to-End OTF for a Telescope Equipped with Adaptive Optics
8.1.4 Atmospheric MTF Plots and Corresponding Intensity Envelopes
8.2 Measurement of the rms OPD Fluctuation
8.2.1 Measurement for the Case σ/λge0.4 Using Two Narrowband Filters
8.2.2 Measurement for the Case σ/λ ≥ 0.4 Using a Broadband Filter
8.2.3 Actual Field Measurements of σ
8.2.4 Telescope Aberrations Do not Affect the Measured σ Values
8.2.5 Convergence of σ as λ2to λ1
8.2.6 Measurement of σ for the Case σ/λ<0.4
8.2.7 Measurement of Residual OPD Fluctuation for an AO-Equipped Telescope
8.2.8 Dependence of σ on Zenith Angle
8.3 Obtaining the Autocorrelation Function of the OPD Fluctuation
8.3.1 Obtaining the Function for an AO-Equipped Telescope
8.3.2 Significance of the Average Turbulence Structure Size
8.3.3 Path Characterization During Daytime
8.4 The Wavefront Structure Function
8.4.1 Wavefront Structure Function for Isotropic Turbulence
8.4.2 Wavefront Structure Function for AO-Equipped Telescopes
8.5 Refractive Index Structure Function
8.5.1 Refractive Index Structure Function for Isotropic Turbulence
8.6 Power Spectral Density of the Turbulence Structure
8.6.1 Power Spectral Density for Isotropic Turbulence
8.6.2 Power Spectrum of Residual OPD Fluctuation for AO-Equipped Telescopes
8.6.3 Volume Contained Under the Power Spectrum
References
9 The Average Intensity Envelope of an Unresolved Star Image
9.1 Average Intensity Envelope for the Image of an Unresolved Star
9.2 Average Intensity Envelope for Small Diffraction-Limited Telescopes with Circular Apertures
9.2.1 Case of Isotropic Turbulence
9.3 Average Envelopes for Small Diffraction-Limited Telescopes with Circular Apertures
9.4 Seeing Disk Envelopes Formed by Large Telescopes
9.4.1 Case (1): Speckle Images for the Case σ/λ ge0.4
9.4.2 Case (2): Core and Halo Images for the Case σ/λ <0.4
9.5 Mathematical Notation Used in This Chapter
References
10 Core and Halo Star Images Formed by Large Telescopes
10.1 Core and Halo Image Structure
10.1.1 Core Strength
10.1.2 Core Shape
10.1.3 Halo Strength
10.1.4 Halo Shape
10.1.5 Characteristics of Core and Halo Images
10.2 Circularly Symmetric Core and Halo Image Envelopes Formed by Large Telescopes
10.2.1 Circularly Symmetric Telescope Point-Spread Functions
10.2.2 Normalization of 10.8
10.2.3 Numerical Accuracy of 10.8
10.3 Theoretical and Observed Core and Halo Structure
10.3.1 Core Dependence on Wavelength
10.3.2 Core Dependence on Seeing
10.3.3 Core Dependence on Telescope Size
10.3.4 Core Dependence on Telescope Aberrations
10.3.5 Image Cores Obtained in the Near-IR by the 4-m Mayall Telescope
10.4 The Optimum Wavelength
10.5 Irradiance at Image Center as a Function of Wavelength
10.6 Effect of Telescope Aberrations on Image Cores
10.6.1 The Effect on Irradiance at Core Center
10.6.2 The Effect on the Optimum Wavelength
10.7 Instantaneous Core Location for Diffraction-Limited Telescopes
10.7.1 Angular Jitter of the Image Centroid
10.7.2 Instantaneous Core Location for Telescopes with Rectangular Apertures
10.7.3 Instantaneous Core Location for Telescopes with Square Apertures
10.7.4 Variance of Core Excursions for Rectangular Aperture Telescopes
10.7.5 rms Core Excursions for Rectangular Aperture Telescopes
10.7.6 rms Core Excursions for Telescopes with Circular Apertures
10.8 Griffin’s Naked-Eye Core Observations
10.9 Cores at Near-IR Wavelengths
10.10 Adaptive Optics
10.10.1 Tip–Tilt Correction
10.10.2 Active Optics
10.10.3 Laser Guide Stars
10.11 Speckle Imaging
10.11.1 Lucky Imaging
10.11.2 Speckle Interferometry
10.12 Mathematical Notation Used in This Chapter
References
11 Statistical Properties of Stellar Speckle Patterns
11.1 Probability Density Function of the OPD Fluctuation
11.2 Probability Density Function of the Phase
11.2.1 PDF of Phase in the Primary Phase Range
11.3 Star Image Characteristics Dependence on the Phase PDF
11.4 Circular Gaussian Speckle in Star Images
11.4.1 Stellar Speckle with Circular Gaussian Statistics
11.4.2 First-Order Statistics of the Complex Amplitude
11.4.3 First-Order Statistics of the Intensity and Phase
11.4.4 Moments of the Intensity
11.4.5 The Intensity PDF
11.4.6 Speckle Contrast
11.4.7 Signal-To-Noise Ratio
11.4.8 Reduced Speckle as the Sum of Uncorrelated Gaussian Patterns
11.4.9 Gaussian Speckle in Star Images Formed by Large Telescopes
11.4.10 Second-Order Statistics of the Complex Amplitude
11.5 Statistical Properties of Polychromatic Speckle Patterns
11.5.1 Autocovariance Function of the Integrated Polychromatic Intensity
11.5.2 The Spectral Correlation Function
11.6 Speckle Reduction Applied to Stellar Speckle Patterns
11.6.1 Speckle Reduction by Wavelength Integration
11.6.2 Effective Number of Uncorrelated Speckle Patterns in the Integrated Pattern
11.6.3 Aperture-Averaged (or Pixel-Averaged) Speckle
11.6.4 Time-Averaged Speckle
11.6.5 Multiple Speckle Reduction Mechanisms Acting Simultaneously
11.6.6 Intensity PDF for a Reduced Speckle Pattern
11.7 Stellar Speckle Statistics When Cores Are Present in the Image
11.7.1 The Core and Halo Light Fractions in Star Images
11.7.2 Probability Density Function of the Complex Amplitude
11.7.3 Probability Density Function of the Intensity and Phase
11.7.4 Second Moment of the Intensity and Variance
11.7.5 Contrast and Signal-To-Noise Ratio
11.8 Speckle Statistics for Light of Arbitrary State of Polarization
11.8.1 Speckle Statistics for Depolarizing Telescopes
11.8.2 Partially Polarized Speckle Fields and the Degree of Polarization
11.8.3 Intensity PDFs for Depolarizing and Non-Depolarizing Telescopes
11.8.4 Moments of the Intensity for Partially Polarized Speckle
11.8.5 Intensity Variance for Partially Polarized Speckle
11.8.6 Contrast and S/N Ratio for Partially Polarized Speckle
11.8.7 Summary of the Polarization Dependence of Speckle Statistics
11.9 Mathematical Notation Used in This Chapter
References
12 Star Image Dependence on Turbulence Structure Size
12.1 The Autocorrelation Function of the OPD Fluctuation
12.2 Generation of Instantaneous Wavefront Realizations
12.2.1 Smoothing the Intermediate Wavefront to Obtain the Final Wavefront
12.3 Wavefront Realizations for Small and Large Turbulence Structure
12.3.1 Instantaneous Star Image Realizations at Visible Wavelengths
12.3.2 Instantaneous Star Image Realizations at Near-IR Wavelengths
12.3.3 Conclusions About the Average Turbulence Structure Size
12.3.4 Correlation Between Speckles at Closely Spaced Wavelengths
12.4 Atmospheric MTFs for Small and Large Turbulence
12.5 Mathematical Notation Used in This Chapter
References
13 Approximation of Star Images Formed by Large Telescopes
13.1 Gaussian Approximations for Unresolved Star Images
13.1.1 General Properties of Gaussian Functions
13.1.2 Gaussian Approximations for the Telescope PSF and the Image Core
13.1.3 Gaussian Approximations for Halo-Only Images
13.1.4 Gaussian Approximations for Core and Halo Images
13.1.5 Anticipated Accuracy of Gaussian Star Image Approximations
13.1.6 Normalization of Gaussian Star Image Approximations
13.1.7 Expressing AC, AH, BC, and BH in Terms of the Telescope and Seeing Parameters
13.2 Obtaining the Seeing Parameters and Telescope Strehl from Gaussian Image Approximations
13.2.1 Example Calculation of SI(λ), σ, and wo from ACH, BC, and BH
13.2.2 Calculation of σ and wo for Reflector Telescopes
13.2.3 Calculation of Residual Phase Error After AO Image Correction
13.2.4 Obtaining SI, σ, woα, and woβ from Asymmetric Core–Halo Images
13.2.5 Maintaining Detailed Seeing Logs While Observing
13.3 Comparing Intensity Envelopes Formed by Two Different Telescopes
13.4 Optimum Wavelength for Maximum Irradiance at Image Center
13.4.1 Optimum Wavelength for Extremely Large AO-Equipped Telescopes
13.4.2 Irradiance at Image Center for the Optimum Wavelength
13.5 MTFs Corresponding to Gaussian Core and Halo Image Envelops
13.5.1 The Cutoff Frequency for Gaussian MTF Approximations
13.6 Mathematical Notation Used in This Chapter
Reference
14 Telescope Resolution and Optical Tolerance Specifications
14.1 Telescope Resolution Criteria
14.1.1 Rayleigh Criterion
14.1.2 Dawes Criterion
14.1.3 Sparrow Criterion
14.2 Effect of Central Obstructions on Resolution
14.3 Effect of Mild Aberrations on Resolution
14.4 Resolution Provided by Gaussian Approximation to the Airy Pattern
14.5 Resolution for Images Displaying Core and Halo Structure
14.5.1 Calculating the Just-Resolved Separation
14.5.2 Just-Resolved Separation for Core-Dominated Images
14.5.3 Just-Resolved Separation for Halo-Dominated Images
14.5.4 Relative Angular Widths of Cores and Halos
14.6 Resolution of AO-Equipped Telescopes
14.6.1 Halo-Dominated Images
14.6.2 Emergence of Cores in AO-Corrected Images
14.6.3 Strehl Intensity Limit Imposed by Uncorrected Scintillation
14.7 Irradiance in Center of Star Images Formed by Large Telescopes
14.7.1 Telescope Resolution and the Intensity in Center of a Star Image
14.7.2 Diffraction-Limited Imaging and Imaging at the Optimum Wavelength
14.8 Optical Tolerances for Large Ground-Based Telescopes
14.8.1 Optical Tolerances for Resolving Image Cores
14.8.2 Stability of Multiple-Segment Primary Mirrors
14.9 Defocus Tolerances for Large Telescopes
14.9.1 Allowance for Delivering Diffraction-Limited Image Cores
14.9.2 Allowance for Delivering Substantially Ideal Halo-Only Images
14.9.3 Ratio of Depth of Focus Allowances for Resolving Cores and Halos
14.10 Resolution Obtained by the Keck 10-m Telescopes
14.10.1 Obtaining Diffraction-Limited Images at Visible Wavelengths
14.11 Resolution Possibilities with Future ELT Instruments
14.12 Apparent Star Image Size and Its Dependence on Star Brightness
14.12.1 Use of Binary Stars to Estimate Limiting Detectable Magnitude
14.13 Resolution Obtained from Speckle Imaging
14.13.1 The Star Test
14.13.2 Optical Transfer Function Tests
14.13.3 Interferometric Tests
14.13.4 The Hartmann Wavefront Test
14.14 Mathematical Notation Used in This Chapter
References
15 Laboratory Simulation of Images Formed by Large Telescopes
15.1 Choice of Detector in the Optical Simulator
15.2 Choice of Scale Factors in the Optical Simulator
15.2.1 Scaling Equations for Image Simulation at the Same Wavelength
15.2.2 Scaling When the Images Are Simulated at a Different Wavelength
15.3 Extended Incoherent Illumination and Image Simulation
15.3.1 Incoherent and Partially Coherent Illumination
15.4 Space Shuttle Image Simulations
15.4.1 Parameter Values Used for Simulating the Space Shuttle Images
15.5 Practical Aspects of Illumination Used in Optical Simulators
15.6 Simulating Images of Actively Illuminated Targets
15.6.1 Illumination and Imaging of Actively Illuminated Targets
15.6.2 Additional Scaling Requirements When Active Illumination Is Used
15.6.3 Simulation of Speckle Reduction Mechanisms
15.7 Mathematical Notation Used in This Chapter
References
16 Laser Beam Propagation and Path Characterization
16.1 OPD Line Integrals for Convergent Beam Paths
16.1.1 Phase Screen Stack Representation of Convergent Beam Paths
16.2 Autocorrelation Function of the Integrated OPD Fluctuation for Convergent Paths
16.2.1 OPD Autocorrelation Function Width for Convergent Beam Paths
16.3 OPD Autocorrelation Function for Paths Inside the Telescope
16.4 OPD Autocorrelation Function for Telescope Coude Paths
16.5 Integrated OPD Fluctuation for Entire End-To-End Beam Paths
16.6 Reducing End-to-End Integrated OPD Fluctuation by Use of AO
16.7 Integrated OPD Fluctuation for an End-to-End Beam Path
16.8 Reversibility of Light and Path Characterization Options
16.9 Characterizing High Energy Laser (HEL) Beam Paths
16.9.1 Optimum Wavelength for Maximum HEL Irradiance at the Target
16.9.2 Top-Level Feasibility Analysis of HEL Weapon Systems
16.9.3 Final Recourse When Design Changes Fail to Deliver Performance
16.9.4 Example Calculation for Hypothetical HEL Weapon
16.10 Optimum Wavelengths for Laser Communication Systems
16.11 Mathematical Notation Used in This Chapter
References
17 Atmospheric Isoplanatic Angle
17.1 Isoplanatic Angle Background
17.2 Calculating Isoplanatic Angle
17.2.1 Effect of Zenith Angle
17.2.2 Isoplanatic Angles for Kolmogorov Turbulence
17.2.3 Isoplanatic Angles for Non-Kolmogorov Turbulence
17.3 Why Stars Twinkle but not Planets?
17.3.1 Eye Sensitivity to Twinkling
17.3.2 Minimum Angular Size for Planets to Cease Twinkling
17.3.3 Planetary Twinkling and Estimating Atmospheric Isoplanatic Angle
17.4 Use of Natural Stars for Stabilizing Images in Large Telescopes
17.4.1 Estimating the Location Where the Core Attains Maximum Intensity
17.4.2 Radiometry of Reference Star Cores
17.5 Use of Natural Stars as Reference Objects for AO Image Correction
17.5.1 Radiometry of Natural Stars Used as AO Reference Objects
17.6 Sky Coverage When Natural Stars Are Used as Reference Objects
17.6.1 Sky Coverage for Image Stabilization at Near-IR Wavelengths
17.6.2 Sky Coverage for Image Stabilization at Visible Wavelengths
17.6.3 Coverage for AO Correction at Near-IR Wavelengths
17.6.4 Coverage for AO Correction at Visible Wavelengths
17.6.5 Coverage Using Natural Reference Stars with ELT Instruments
17.7 Mathematical Notation Used in This Chapter
References
18 Extremely Large Telescopes (ELTs): Imaging Performance and Imaging Path Characterization
18.1 Modelling the Intensity PSF as the Best-Fit Sum of Three Gaussian Functions
18.2 Residual OPD Fluctuation in AO-Corrected ELT Imaging Wave Fronts
18.3 Focusing ELT Instruments
18.4 Characterizing End-To-End Imaging Paths for ELT Instruments
18.5 Calculating the Intensity PSFs at Other Visible and IR Wavelengths
18.6 Applications of the Analytic Equation Package to AO-Equipped ELTs
18.7 Analysis and Characterization of AO-Equipped ELT Imaging Performance
18.7.1 The Ubiquitous Core and Halo Star Image
18.7.2 Normalized Forms of the Three-Gaussian Function Star Image Approximation
18.7.3 Fourier Transform Relationships Between the Intensity PSF and the Residual OPD Fluctuation
18.7.4 Inverse Fourier Transform Relationships
18.7.5 Calculation of Star Image Intensity PSFs at Other Wavelengths
18.8 Representative Intensity PSFs Delivered by AO-Assisted ELTs
18.8.1 Intensity Attained in the Center of the Average Short-Exposure PSF
18.8.2 Sweet-Spot Wavelength Regions for ELT Instruments
18.9 Resolution Versus Wavelength for the E-ELT Instrument
18.10 Detection of Faint Objects
18.10.1 Detection of Exoplanets
18.10.2 Star Cluster Images Formed by the 25 m GMT Instrument at Visible and IR Wavelengths
18.11 Mathematical Notation Used in This Chapter
References
Appendix A James Clerk Maxwell and the Electromagnetic Field Equations
Appendix B Coherence Terminology
Appendix C Turbulence Outer-Scale Limits Measured by Coulman et al.
Appendix D Optical Path Characterization Using Scintillometry
Appendix E Radiometry of the Sun and Stars
Appendix F Intensity Correlation Coefficient Estimates and Photon Noise Compensation
Appendix G Image Core Correspondence from Roger F. Griffin
Appendix H Light Scattering by Spherical Turbulence Structures
Appendix I A Critique of Kolmogorov Theory as Applied to Atmospheric Turbulence Modeling
References
Appendix J The Importance of HEL Wavelength Choice IN HEL Weapon Systems
Appendix K Three-Gaussian Function Least Mean Squares Best-Fit Approximation of Star Images
Appendix L Acronyms
Index