**Publications**(91)

With the widespread availability of electromagnetic (vector) analysis codes for describing the diffraction of electromagnetic waves by periodic grating structures, the insight and understanding of nonparaxial parametric diffraction grating behavior afforded by approximate methods (i.e., scalar diffraction theory) is being ignored in the education of most optical engineers today. We show that the linear systems formulation of nonparaxial scalar diffraction theory enables the development of a scalar parametric diffraction grating model [for transverse electric (TE) polarization] for sinusoidal reflection gratings with arbitrary groove depths and arbitrary nonparaxial incident and diffracted angles. This scalar parametric analysis is remarkably accurate as it includes the ability to redistribute the energy from evanescent orders into the propagating ones, thus allowing the calculation of nonparaxial diffraction efficiencies to be predicted with an accuracy usually thought to require rigorous electromagnetic theory. These scalar parametric predictions of diffraction efficiency are compared to paraxial scalar and rigorous electromagnetic (vector) predictions for a variety of nonparaxial diffraction grating configurations, thus providing quantitative limits of applicability of nonparaxial scalar diffraction theory to sinusoidal reflection gratings as a function of the grating period-to-wavelength ratio (λ /

With the wide-spread availability of rigorous electromagnetic (vector) analysis codes for describing the diffraction of electromagnetic waves by specific periodic grating structures, the insight and understanding of nonparaxial parametric diffraction grating behavior afforded by approximate methods (i.e., scalar diffraction theory) is being ignored in the education of most optical engineers today. Elementary diffraction grating behavior is reviewed, the importance of maintaining consistency in the sign convention for the planar diffraction grating equation is emphasized, and the advantages of discussing “conical” diffraction grating behavior in terms of the direction cosines of the incident and diffracted angles are demonstrated. Paraxial grating behavior for coarse gratings (

^{5}. We discuss the “pinwheel pupil” advantages to spectroscopy, image processing, and observatory operations. We show that, segment fabrication of curved-sided mirrors is not more difficult than fabrication of hexagonal mirror segments. . This is the report of quantitative study of Fraunhofer (far field) diffraction patterns produced by three different topologies or architectures of mirror segmentation, when illuminated by a plane wave of monochromatic white-light. A plot, in angular units of the intensity as a function of azimuth, Phi

*, within annular rings at different FOVs, centered on the system axis of the diffraction pattern will be presented. The advantages of the segmented pinwheel pupil is discussed.*

_{f}^{+10}or better is required for exoplanet exploration. We will discuss the optical fabrication tolerances necessary to minimize narrowangle forward scatter and their relative effects upon direct imaging coronagraph instruments used to characterize terrestrial exoplanets.

_{Smooth}surface scatter theory results in an expression for the BRDF in terms of the surface PSD that is very similar to that provided by the rigorous Rayleigh-Rice (RR) vector perturbation theory. However it contains correction factors for two extreme situations not shared by the RR theory: (i) large incident or scattered angles that result in some portion of the scattered radiance distribution falling outside of the unit circle in direction cosine space, and (ii) the situation where the relevant rms surface roughness, σrel, is less than the total intrinsic rms roughness of the scattering surface. Also, the RR obliquity factor has been discovered to be an approximation of the more general GHS

_{Smooth}obliquity factor due to a little-known (or long-forgotten) implicit assumption in the RR theory that the surface autocovariance length is longer than the wavelength of the scattered radiation. This assumption allowed retaining only quadratic terms and lower in the series expansion for the cosine function, and results in reducing the validity of RR predictions for scattering angles greater than 60°. This inaccurate obliquity factor in the RR theory is also the cause of a complementary unrealistic “hook” at the high spatial frequency end of the predicted surface PSD when performing the inverse scattering problem. Furthermore, if we empirically substitute the polarization reflectance, Q, from the RR expression for the scalar reflectance, R, in the GHS

_{Smooth}expression, it inherits all of the polarization capabilities of the rigorous RR vector perturbation theory.

*TIS*= 1 - exp

**[**-(4π cosθ

*σ/λ)*

_{i}^{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.

*ABC*or

*K*-correlation power spectral density (PSD) functions have been modeled. These parametric TIS predictions provide insight and understanding regarding optical fabrication tolerances necessary to satisfy specific optical performance requirements.

*diffracted radiance*is the fundamental quantity predicted by scalar diffraction theory, combined with the observation that radiance (not irradiance or intensity) is shift-invariant in direction cosine space, has lead to the development of a generalized linear systems formulation of non-paraxial scalar diffraction theory. Thus simple Fourier techniques can now be used to predict a variety of wide-angle diffraction phenomena. These include: (1) the redistribution of radiant energy from evanescent diffracted orders to propagating ones, (2) the angular broadening (and apparent shifting) of wide-angle diffracted orders, and (3) diffraction efficiencies predicted with an accuracy usually thought to require rigorous electromagnetic theory. In addition, this new insight and understanding has led to an empirically modified Brckmann-Kirchhoff surface scatter model that is more accurate than the classical Beckmann-Kirchhoff theory in predicting scatter effects at large incident and scattered angles, without the smooth-surface limitation of the Rayleigh-Rice vector perturbation surface scatter theory. This new understanding of non-paraxial diffraction phenomena is becoming increasingly important in the design and analysis of novel optical systems containing nano-structures.

*analysis*of the final system performance. Aplanatic optical designs (corrected for spherical aberration and coma) are widely considered to be superior to non-aplanatic designs. However, there is little merit in an aplanatic design for wide field applications because one needs to optimize some-field-weighted average measure of resolution over the desired operational field of-view (

*OFOV*). Furthermore, when used with a mosaic detector array in the focal plane, detector effects eliminate the advantage of the aplanatic design even at small field angles. For wide fields of view, the focal plane is frequently despaced to balance field curvature with defocus thus obtaining better overall performance. We will demonstrate that including detector effects in the design process results in a different optimal (non-aplanatic) design for each

*OFOV*that is even superior to an optimally despaced aplanatic design.

_{o}format). During the 1980s this STF was generalized to include: (1) the effects of small-angle scatter caused by 'mid' spatial frequency surface irregularities which span the gap between the traditional 'figure' and 'finish' errors, and (2) the extremely large incident angles inherent to grazing incidence Wolter Type I x-ray telescopes. Since no explicit smooth surface approximation is imposed, this STF can be utilized to predict the scattering behavior of rough surfaces not accurately modeled by vector perturbations techniques considered to be more rigorous by many investigators. Also, the scattering function is normalized by the total reflectance of the surface. Hence, the dominant polarization effects are included (in spite of the fact that this is basically a scalar treatment) by using the Fresnel reflectance coefficients for the desired polarization. In this paper it is emphasized that scattered radiance (not irradiance or intensity) is shift- invariant in direction cosine space paper to explain some non- intuitive scattering behavior reported in the literature.

**Proceedings Volume Editor**(3)

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