Cynthia L. Vernold

Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory.

Most authors include a paraxial (small-angle) limitation in their discussion of diffracted wave fields. This paraxial limitation severely limits the conditions under which diffraction behavior is adequately described. A linear systems approach to modeling nonparaxial scalar diffraction theory is developed by normalization of the spatial variables by the wavelength of light and by recognition that the reciprocal variables in Fourier transform space are the direction cosines of the propagation vectors of the resulting angular spectrum of plane waves. It is then shown that wide-angle scalar diffraction phenomena are shift invariant with respect to changes in the incident angle only in direction cosine space. Furthermore, it is the diffracted radiance (not the intensity or the irradiance) that is shift invariant in direction cosine space. This realization greatly extends the range of parameters over which simple Fourier techniques can be used to make accurate calculations concerning wide-angle diffraction phenomena. Diffraction-grating behavior and surface-scattering effects are two diffraction phenomena that are not limited to the paraxial region and benefit greatly from this new development.

Description of Diffraction Grating Behavior in Direction Cosine Space

It is well known that the angular separation of non-paraxial diffracted orders from a linear grating varies drastically with incident angle. Furthermore, for oblique incident angles (conical diffraction), it is rather cumbersome both analytically and graphically to describe the number and angular position of the various propagating orders. One can readily demonstrate that wide-angle diffraction phenomena (including conical diffraction from gratings) are shift-invariant with respect to incident angle in direction cosine space. Only when the grating equation is expressed in terms of the direction cosines of the propagation vectors of the incident beam and the diffracted orders can we apply the Fourier techniques resulting from linear systems theory. This formulation has proven extremely useful for small-angle diffraction phenomena and in modern, image formation theory. New insight and an intuitive understanding of diffraction grating behavior results from a simple direction cosine diagram.

Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles

Surface scattering effects are merely diffraction phenomena resulting from random phase variations induced on the reflected wave- front by microtopographic surface features. The Rayleigh-Rice and Beckmann-Kirchhoff theories are commonly used to predict surface scat- tering behavior. However, the Rayleigh-Rice vector perturbation theory is limited to smooth surfaces, and the classical Beckmann-Kirchhoff theory contains a paraxial assumption that confines its applicability to small incident and scattering angles. The recent development of a linear sys- tems formulation of nonparaxial scalar diffraction phenomena, indicating that diffracted radiance is a fundamental quantity predicted by scalar diffraction theory, has led to a reexamination of the classical Beckmann- Kirchhoff scattering theory. We demonstrate an empirically modified Beckmann-Kirchhoff scattering model that accurately predicts nonintui- tive experimental scattering data for rough surfaces at large incident and large scattering angles, yet also agrees with Rayleigh-Rice predictions within their domain of applicability for smooth surfaces.

Surface scatter effects in grazing incidence x-ray telescopes

Non-intuitive surface scatter effects resulting from practical optical fabrication tolerances frequently dominate both diffraction effects and geometrical aberrations in high resolution grazing incidence x-ray telescopes. The resulting reduction optical performance due to scattering is a strong function of x-ray energy, residual surface characteristics, incident angle, and the optical performance criterion appropriate to the application. A simple Fourier treatment of surface scatter phenomena, based upon a non-paraxial scalar diffraction theory, is referenced and utilized to produce parametric performance predictions that provide physical insight and understanding into the surface scatter phenomenon and its effect upon image quality. The optical prescription for the Solar X-ray Imager will be used as an example.

Understanding surface scatter effects in grazing incidence x-ray synchrotron applications

Non-intuitive surface scatter effects resulting from practical optical fabrication tolerances frequently dominate both diffraction effects and geometrical aberrations in high resolution grazing incidence x-ray synchrotron applications. The resulting reduction optical performance due to scattering is a strong function of x-ray energy, residual surface characteristics, incident angle, and the optical performance criterion appropriate to the application. A simple Fourier treatment of surface scatter phenomena, based upon a non-paraxial scalar diffraction theory, is referenced and utilized to produce parametric performance predictions that provide physical insight and understanding into the surface scatter phenomenon and its effect upon optical performance in x-ray synchrotron applications.

Comparison of Harvey-Shack scatter theory with experimental measurements

Rayleigh-Rice or Beckmann-Kirchoff theories are commonly used to predict scatter results. However, in order to apply these theories in practice, inherent assumptions must be made that either limit the roughness of the surface under test or limit the predictions to small, paraxial incident and scatter angles. Various published reports show experimental scatter results and diffraction efficiencies that do not agree with these theories. One possible explanation for these discrepancies is that there is some confusion between whether the data being plotted is intensity or radiance. The quantity intensity is usually measured in the laboratory, not radiance. Using the Harvey-Shack theory, a Fourier linear systems theory based on using a surface transfer function, we show excellent agreement between experimental results and theoretical predictions. This holds true for scatter from rough surfaces as well as large scatter angles and angles of incidence.