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*Understanding Surface Scatter Phenomena: A Linear Systems Formulation* deals with optical phenomena that continue to be an important issue in diverse areas of science and engineering in the 21st century. Scattering effects from microtopographic surface roughness are merely nonparaxial diffraction phenomena. After reviewing the historical background of surface scatter theory, this book describes how integrating sound radiometric principles with scalar diffraction theory results in the development of a linear systems formulation of nonparaxial scalar diffraction theory, which then becomes the basis of the generalized Harvey–Shack (GHS) surface scatter theory characterized by a two parameter family of surface transfer functions. This GHS surface scatter theory produces accurate results for rougher surfaces than the classical Rayleigh–Rice theory and (due to a more general obliquity factor) for larger incident and scattered angles than either the Beckmann–Kirchhoff or Rayleigh–Rice theories. The transfer function characterization of scattering surfaces can be readily incorporated into the traditional linear systems formulation of image formation, thus allowing a systems engineering analysis of image quality as degraded by diffraction effects, geometrical aberrations, surface scatter effects, and a variety of other miscellaneous error sources. This allows us to derive the optical fabrication tolerances necessary to satisfy a specific image quality requirement, which further enables the integration of optical fabrication and metrology into the optical design process.

Pages: 252

ISBN: 9781510627871

Volume: PM306

### Table of Contents

*Preface**Acknowledgments**List of Acronyms***1 Introduction and Overview**- 1.1 A Linear Systems Formulation of Surface Scatter Theory
- 1.2 Motivation for this Text
- 1.3 Organization of the Text
- References
**2 Technical Background**- 2.1 Surface Characteristics
- 2.1.1 Mid-spatial-frequency surface irregularities
- 2.1.2 Bandlimited, or relevant, surface roughness
- 2.2 Specular and Diffuse Reflectance (Total Integrated Scatter)
- 2.3 The Bidirectional Reflectance Distribution Function (BRDF)
- 2.4 Vector Analysis and Direction Cosine Space
- 2.4.1 The direction cosines of a vector
- 2.4.2 Direction cosine space and the direction cosine diagram
- 2.4.3 Relationship between direction cosine diagram and spherical coordinates
- 2.4.4 Sign convention for diffraction gratings and surface scatter phenomena
- 2.5 Mathematical Description of Optical Wave Fields
- 2.5.1 The complex amplitude of an electromagnetic field
- 2.5.2 Plane wave fields
- 2.5.3 Spherical wave fields
- 2.5.4 The angular spectrum of plane waves
- 2.5.5 The Poynting vector and irradiance
- 2.6 Radiometric Quantities and their Relationship to Complex Amplitude
- 2.6.1 The solid angle
- 2.6.2 Definitions and terminology
- 2.6.3 The fundamental theorem of radiometry
- 2.6.4 Lambertian sources and Lambert's cosine law
- 2.6.5 Radiometry of an imaging system and the brightness theorem
- 2.6.6 Numerical aperture and focal ratio (F-number)
- 2.6.7 The cosine-to-the-fourth illumination fall-off
- References
**3 Historical Background of Surface Scatter Theory**- 3.1 Rayleigh–Rice (RR) Surface Scatter Theory
- 3.2 Beckmann–Kirchhoff (BK) Surface Scatter Theory
- 3.3 The Original Harvey–Shack (OHS) Surface Scatter Theory
- 3.4 The Modified Harvey–Shack (MHS) Surface Scatter Theory
- 3.5 References and Notes
- Notes and References
**4 A Modified Beckmann–Kirchhoff Surface Scatter Model***James E. Harvey, Andrey Kyrwonos, and Cynthia L. Vernold*- 4.1 Nonintuitive Surface Scatter Measurements
- 4.2 Qualitative Explanation of Nonintuitive Scatter Data
- 4.3 Empirical Modification of the Classical BK Scatter Theory .
- 4.4 Experimental Validation: Rough Surfaces at Large Angles
- 4.5 Comparison with Rayleigh–Rice Theory for Smooth Surfaces
- References
**5 The Generalized Harvey–Shack (GHS) Surface Scatter Theory***Andrey Krywonos, James E. Harvey, and Narak Choi*- 5.1 Evolution of a Linear Systems Formulation of Surface Scatter Theory
- 5.1.1 Inadequate state-of-the-art in surface scatter analysis (1997)
- 5.1.2 Roadmap to successful image degradation analysis (1997)
- 5.1.3 Nonparaxial scalar diffraction theory (1999)
- 5.1.4 Derivation of the GHS surface transfer function (2011)
- 5.2 Numerical Calculations of Scatter for Rough Surfaces
- 5.2.1 Surfaces characterized with a Gaussian PSD
- 5.2.2 Surfaces characterized by an inverse power law PSD
- 5.2.3 Surfaces characterized with more-general PSDs
- 5.3 Smooth Surface Approximation of the GHS Surface Scatter Theory
- 5.4 Comparison of the GHS
_{Smooth}and the RR Theories - 5.4.1 Comparison of GHS and RR scattered intensity predictions
- 5.4.2 The inverse scattering problem: predicting PSDs from BRDFs
- 5.4.3 Predicting BRDFs from surface metrology data
- 5.5 Inherent Angular Limitation of the Rayleigh–Rice Theory
- 5.5.1 Searching for limiting assumptions
- 5.5.2 GHS
_{Smooth}obliquity factor: more general than that of RR theory - 5.5.3 GHS
_{Smooth}obliquity factor: direct result of sinusoidal grating behavior - 5.6 Predicting BRDFs for Arbitrary Wavelength and Angle of Incidence for a Moderately Rough Surface
- 5.7 Summary
- References and Notes
**6 Numerical Validation of the GHS Surface Scatter theory***Narak Choi and James E. Harvey*- 6.1 Surfaces with Gaussian Statistics
- 6.2 Surfaces with an Inverse Power Law PSD
- 6.3 Summary
- References
**7 Empirical Modeling of Rough Surfaces and Subsurface Scatter**- 7.1 The Helmholtz Reciprocity Theorem
- 7.2 Example 1: Rough Ground Glass with Oblique Incident Angles
- 7.3 Example 2: Modeling Subsurface Scatter, Material #1
- 7.4 Example 3: Modeling Subsurface Scatter, Material #2
- 7.5 Concluding Remarks
- References
**8 Integrating Optical Fab and Met into the Optical Design Process***James E. Harvey and Narak Choi*- 8.1 Surface Scatter in the Presence of Aberrations
- 8.1.1 The geometrical PSF for multi-element imaging systems
- 8.1.2 Image degradation due to the surface scatter PSF
- 8.1.3 The PSF due to scattering in the presence of aberrations
- 8.1.4 The PSF due to scattering in the presence of aberrations
- 8.1.5 Summary and Conclusions
- 8.2 A Systems Engineering Analysis of Image Quality
- 8.3 Deriving Opt Fab Tolerances to meet Image Quality Requirements
- References and Notes
*Index*

## Preface

The material in this book was first developed as the seventh and final chapter in a manuscript entitled *Diffraction for Engineers, Including Surface Scatter Phenomena*. However, that chapter on a rather specialized application of diffraction theory grew to over 200 pages. The topic of surface scatter phenomena is probably of limited interest to many people looking for a book on diffraction, and the small community of optical engineers specializing in surface scatter and stray light may rather have access to that material without having to purchase a 400-page book on diffraction. It was thus decided to strip off that final chapter and offer it as a separate book entitled *Understanding Surface Scatter Phenomena: A Linear Systems Formulation*.

Scattering effects from microtopographic surface roughness are merely nonparaxial diffraction phenomena resulting from random phase variations in the reflected or transmitted wavefront. The Rayleigh–Rice, Beckmann–Kirchhoff, or Harvey–Shack surface scatter theories are commonly used to predict surface scatter effects. Smooth-surface approximations and/or moderate-angle limitations have severely reduced the range of applicability of each of the above theoretical treatments.

The true nature of most physical phenomena, including the propagation of light, becomes evident when simple elegant theories and mathematical models conform to experimental observations. Often the simple nature of some natural phenomenon is obscured by applying a complex theory to model an inappropriate physical quantity in some cumbersome coordinate system or parameter space. By integrating sound radiometric principles with scalar diffraction theory, it has been shown that diffracted radiance (not irradiance or intensity) is the natural quantity that exhibits shift invariance with respect to incident angle if formulated in terms of the direction cosines of the incident and diffracted angles. Thus, simple Fourier techniques can be used to predict a variety of wide-angle diffraction phenomena. These include the redistribution of radiant energy from evanescent diffraction orders into propagating ones, and the calculation of diffraction grating efficiencies with accuracy usually thought to require rigorous electromagnetic theory.

Since a random rough surface can be modeled as a superposition of sinusoidal reflection gratings of different frequencies, amplitudes, orientations, and phases, the resulting scattered light distribution is merely the superposition of a myriad of diffraction grating orders. The above linear systems formulation of nonparaxial scalar diffraction theory was thus applied to surface scatter phenomena and resulted first in an empirically modified Beckmann–Kirchhoff surface scatter model, then a generalized Harvey–Shack (GHS) surface scatter theory characterized by a two-parameter family of surface transfer functions. This GHS surface scatter theory produces accurate results for rougher surfaces than the Rayleigh–Rice theory and for larger incident and scattered angles than either the classical Beckmann–or Rayleigh–Rice theories. The GHS theory also agrees with Rayleigh–Rice predictions within their domain of applicability for smooth surfaces and moderately large scattering angles (up to 50 or 60 deg). These new developments simplify the analysis and understanding of nonintuitive scattering behavior from rough surfaces for large incident and scattering angles.

The transfer function characterization of scattering surfaces can be readily incorporated into the linear systems formulation of image formation, thus allowing a systems engineering analysis of image quality as degraded by diffraction effects, geometrical aberrations, surface scatter effects, and a variety of other miscellaneous error sources. This allows us to derive the optical fabrication tolerances necessary to satisfy a specific image quality requirement, which further enables the integration of optical fabrication and metrology into the optical design process.
The fact that scattered radiance (not irradiance or intensity) is shift invariant with respect to changes in incident angle *only when expressed in terms of the direction cosines of the propagation vectors* makes a strong case for always displaying scatter measurements as bidirectional scatter distribution function (BSDF) (i.e., radiance), not angled-resolved scatter (i.e., intensity), as a function of *β – β _{o}*. Even BSDF measurements from unknown materials exhibiting subsurface scatter frequently produce simple, elegant behavior that allows empirical parametric modeling when plotted in this format.

Before becoming an associate professor at CREOL—The College of Optics and Photonics at the University of Central Florida, I spent over fifteen years in industry working on real-life optical engineering problems in major DoD and NASA programs. Many of those problems were adequately modeled by the simple and direct application of the linear systems theory and Fourier techniques I learned in Jack Gaskill's Fourier Optics course (and Joe Goodman's textbook entitled Introduction to Fourier Optics) at the University of Arizona. However, a frequent frustration concerned the fact that many of those real-life problems did not satisfy the paraxial assumption in the conventional linear systems formulation of scalar diffraction theory.

My goal in writing this book is to present the above recent developments and understanding made possible by a linear systems formulation of surface scatter phenomena to my students as well as to practicing optical engineers. Integrating the subject of radiometry (a neglected step-child in the field of physics) into mainstream optics was particularly satisfying to me (and I hope to my former mentor, Professor Emeritus William L. Wolfe at the Optical Sciences Center at the University of Arizona from whom I learned the fundamentals of radiometry).

**James E. Harvey**
June 2019

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