Narak Choi

Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles.

Scattering effects from microtopographic surface roughness are merely nonparaxial diffraction phenomena resulting from random phase variations in the reflected or transmitted wavefront. Rayleigh-Rice, Beckmann-Kirchhoff. or Harvey-Shack surface scatter theories are commonly used to predict surface scatter effects. Smooth-surface and/or paraxial approximations have severely limited the range of applicability of each of the above theoretical treatments. A recent linear systems formulation of nonparaxial scalar diffraction theory applied to surface scatter phenomena resulted first in an empirically modified Beckmann-Kirchhoff surface scatter model, then a generalized Harvey-Shack theory that produces accurate results for rougher surfaces than the Rayleigh-Rice theory and for larger incident and scattered angles than the classical Beckmann-Kirchhoff and the original Harvey-Shack theories. These new developments simplify the analysis and understanding of nonintuitive scattering behavior from rough surfaces illuminated at arbitrary incident angles.

Calculating BRDFs from surface PSDs for moderately rough optical surfaces

Image degradation due to scattered radiation is a serious problem in many short wavelength (X-ray/EUV) imaging systems. Most currently-available image analysis codes require the scatter behavior (BRDF data) as input in order to calculate the image quality from such systems. This BRDF data is difficult to measure and rarely available for the operational wavelengths of interest. Since the smooth-surface approximation is often not satisfied at these short wavelengths, the classical Rayleigh-Rice expression that indicates the BRDF is directly proportional to the surface PSD cannot be used to calculate BRDFs from surface metrology data for even slightly rough surfaces. We discuss the implementation of an FFTLog numerical Hankel transform algorithm that enables the practical use of the computationally intensive Generalized Harvey-Shack (GHS) surface scatter theory. The FFTLog Hankel transform algorithm is validated over the large dynamic range of relevant spatial frequencies required for short wavelength imaging applications, and BRDFs are calculated and displayed for increasingly short wavelengths that violate the smooth surface approximation implicit in the Rayleigh-Rice surface scatter theory.

Image degradation due to scattering effects in two-mirror telescopes

Image degradation due to scattered radiation is a serious problem in many short-wavelength (x-ray and EUV) imaging systems. Most currently available image analysis codes require the scattering behavior [data on the bidirectional scattering distribution function (BSDF)] as input in order to calculate the image quality from such systems. Predicting image degradation due to scattering effects is typically quite computation-intensive. If using a conventional optical design and analysis code, each geometrically traced ray spawns hundreds of scattered rays randomly distributed and weighted according to the input BSDF. These scattered rays must then be traced through the system to the focal plane using nonsequential ray-tracing techniques. For multielement imaging systems even the scattered rays spawn more scattered rays at each additional surface encountered in the system. In this paper we describe a generalization of Petersons analytical treatment of in-field stray light in multielement imaging systems. In particular, we remove the smooth-surface limitation that ignores the scattered-scattered radiation, which can be quite large for EUV wavelengths even for state-of-the-art optical surfaces. Predictions of image degradation for a two-mirror EUV telescope with the generalized Peterson model are then numerically validated with the much more computation-intensive ZEMAX® and ASAP® codes.

Scattering from moderately rough interfaces between two arbitrary media

The generalized Harvey-Shack (GHS) surface scatter theory has been shown to accurately predict the BRDF produced by moderately rough mirror surfaces from surface metrology data. The predicted BRDF also holds for both large incident and scattering angles. Furthermore, it provides good agreement with the classical Rayleigh-Rice theory for those surfaces that satisfy the smooth-surface criterion. The two-dimensional band-limited portion of the surface PSD contributing to scattered radiation is discussed and illustrated for arbitrary incident angles, and the corresponding relevant roughness necessary to calculate the total integrated scatter (TIS) is determined. It is shown that BRDF data measured with a large incident angle can be used to expand the range of surface roughness for which the inverse scattering problem can be solved; i.e., for which the surface PSD can be calculated from measured BRDF data. This PSD and the GHS surface scatter theory can then be used to calculate the BRDF of that surface for arbitrary incident angles and wavelengths that do not satisfy the smooth-surface criterion. Finally, the surface transfer function characterizing both the BTDF and the BRDF of a moderately rough interface separating two media of arbitrary refractive index is derived in preparation for modeling the scattering of structured thin film solar cells.

Numerical validation of the generalized Harvey–Shack surface scatter theory

Abstract. The generalized Harvey–Shack (GHS) surface scatter theory is numerically compared to the classical small perturbation method, the Kirchhoff approximation method, and the rigorous method of moments for one-dimensional ideally conducting surfaces whose surface power spectral density function is Gaussian or exhibits an inverse power law (fractal) behavior. In spite of its simple analytic form, our numerical comparison shows that the new GHS theory is valid (with reasonable accuracy) over a broader range of surface parameter space than either of the two classical surface scatter theories.

Image degradation due to surface scatter in the presence of aberrations

Image analysis in the presence of surface scatter due to residual optical fabrication errors is often perceived to be complicated, nonintuitive, and achieved only by computationally intensive nonsequential ray tracing with commercial optical analysis codes such as ASAP, Zemax, Code V, TracePro, or FRED. However, we show that surface scatter can be treated very similarly to conventional wavefront aberrations. For multielement imaging systems degraded by both surface scatter and aberrations, the composite point spread function is obtained in explicit analytic form in terms of convolutions of the geometrical point spread function and scaled bidirectional scattering distribution functions of the individual surfaces of the imaging system. The approximations and assumptions in this formulation are discussed, and the result is compared to the irradiance distribution obtained using commercial software for the case of a two-mirror telescope operating at an extreme ultraviolet wavelength. The two results are virtually identical.

New capabilities for predicting image degradation from optical surface metrology data

Image degradation due to scattered radiation form residual optical fabrication errors is a serious problem in many short wavelengths imaging system. Most currently-available image analysis codes require the bidirectional scattering distribution function (BSDF) data as an input in order to calculate the image quality from such systems. This BSDF data is difficult to measure and rarely available for the operational wavelengths of interest. Since the smooth-surface approximation is often not satisfied at these short wavelengths, the classical Rayleigh-Rice expression that indicates the BSDF is directly proportional to the surface PSD cannot be used to calculate BSDFs from surface metrology data for even slightly rough surfaces. An FFTLog numerical Hankel transform algorithm enables the practical use of the computationally intensive Generalized Harvey-Shack surface scatter theory to calculate BRDFs for increasingly short wavelengths that violate the smooth surface approximation implicit in the Rayleigh-Rice surface scatter theory. A generalized Peterson analytical scatter model is then used to make accurate image quality predictions. The generalized Peterson model is numerically validated by both ASAP and ZEMAX.

Effect of surface scatter upon the MTF of the solar ultra violet imager (SUVI) telescope

The solar UV imager (SUVI) is an extreme ultraviolet instrument that will fly on the Geostationary Operational Environmental Satellite (GOES)-R and-S platforms, as part of NOAA’s space weather monitoring fleet. It will provide important information on solar activity and the effects of the Sun on the earth and the near-earth environment. This instrument will image the full solar disc in 6 EUV wavebands between 303.8 Å and 93.9 Å. A generalized Cassegrain telescope configuration is employed where six mirror sectors utilize multilayer coatings optimized for the six wavelengths of interest. An aperture shutter is used to select the appropriate sector for observations at a particular wavelength. A thinned, back-illuminated CCD sensor with 21μm (2.5 arcsec) pixels resides in the telescope focal plane. The modulation transfer function (MTF) is usually considered to be the image quality criterion of choice for applications where fine detail in extended images needs to be specified or evaluated. However, the contractual image quality requirement was specified in terms of fractional ensquared energy for a variety of different wavelengths and square sizes. In this paper we will calculate and present MTF plots (as degraded by diffraction, geometrical aberrations, surface scatter effects and detector effects) for each of the SUVI wavelengths of interest. Surface scatter due to residual optical fabrication errors is a major factor limiting the performance at the shorter wavelengths, and the large detector size severely limits the performance at all SUVI wavelengths.

Domain of validity of the equation for total integrated scatter (TIS)

The analytical expression for total integrated scatter (defined as diffuse reflectance divided by total reflectance) has been around for almost six decades TIS = 1 - exp[-(4π cosθi σ/λ)2]. Most surface scatter analysts now realize that the expression is ambiguous unless spatial frequency band-limits are specified for the rms roughness, σ, in the expression. However, there still exists uncertainty about the domain of validity of the expression with regard to both surface characteristics and incident angle. In this paper we will quantitatively illustrate this domain of validity for both Gaussian and fractal one-dimensional surfaces as determined by the rigorous integral equation method (method of moments) of electromagnetic theory. Two dimensional error maps will be used to illustrate the domain of validity as a function of surface characteristics and incident angle. Graphical illustrations comparing the TIS predictions of several approximate surface scatter theories will also be presented.

Comparison of the domain of validity of several approximate surface scatter theories

A new generalized Harvey-Shack (GHS) surface scatter theory is numerically compared to the classical small perturbation method (SPM), the Kirchhoff approximation method (KM) and the rigorous method of moment (MoM) for one-dimensional ideally conducting surfaces whose surface power spectral density function is Gaussian or abc-function. In spite of its simple analytic form, our numerical comparison shows that the new GHS theory is valid (with reasonable accuracy) over a broader range of surface parameter space than either of the two classical surface scatter theories.