A rigorous coupled-wave analysis for metallic surface-relief gratings is presented. This approach allows an arbitrary complex permittivity to be used for the material and thus avoids the infinite conductivity (perfect-conductor) approximation. Both TE and TM polarizations and arbitrary angles of incidence are treated. Diffraction characteristics for rectangular-groove gold gratings with equal groove and ridge widths are presented for free-space wavelengths of 0.5, 1.0 and 10.0 μm for all diffracted orders as a function of period, groove depth, polarization, and angle of incidence. Results include the following: (1) TM-polarization diffraction characteristics vary more rapidly than do those for TE polarization, (2) 95% first-order diffraction efficiency occurs for TM polarization at 10.0 μm, (3) 50% absorption of incident power occurs at 0.5 μm, and (5) the perfect-conductor approximation is not valid for TM polarization at any of the wavelengths and is not valid for TE polarization at 0.5 μm.

Rigorous coupled-wave analysis of metallic surface-relief gratings

Publisher Summary This chapter describes the rigorous vector theories of diffraction gratings. The diffraction grating is a valuable instrument for scientific research. It is mainly used for spectroscopy and filtering. Even though the holographic grating has not eliminated the classical, ruled grating, which remains the most suitable for important applications, it has permitted a considerable extension of the use of gratings for industrial or scientific purposes such as wavelength selectors for tunable lasers, selective surfaces for solar energy, masks for photolithography, beam sampling mirrors for high power lasers, spectrometers in extreme ultraviolet (UV) or x-ray regions for Space Optics. In addition, one can consider that rigorous vector theories of gratings are able to solve almost all the classical problems of gratings encountered by manufacturers and users.

I Rigorous Vector Theories of Diffraction Gratings

Publisher Summary This chapter describes analytical techniques for multiple scattering from rough surfaces. All real surfaces are rough. The degree of roughness depends on both the geometry and the wavelength of the incident probe. The types of rough surfaces are generally classified into periodic rough surfaces such as diffraction gratings and nonperiodic surface variability, which includes random rough surfaces. The chapter discusses theoretical treatment of the latter class of surfaces, although, as can be seen, much of the development is in terms of a stochastic surface and stochastic equations, which in principle hold true for an arbitrary non-stochastic surface as well. In addition, it comprises of a mathematically oriented review of rough surface scattering theory. The basic concepts of vertical and horizontal scales of roughness are introduced and a very succinct review of the kinds of surface statistics required in subsequent analyses is provided. The chapter also presents a brief discussion of other recent multiple scattering approaches.

I Analytical Techniques for Multiple Scattering from Rough Surfaces

The spectral amplitudes of the scattered and transmitted fields for plane-wave incidence on an arbitrary rough interface in one dimension are derived exactly and simply by using Green’s theorem. Results are stated in terms of integrals on values of the field and its normal derivative on the interface. The energy constraint is derived, and individual energy contributions in each region are also related to the surface-field values. The latter contributions can be calculated from coupled linear equations that are also derived using Green’s theorem. The interface separates media of different but constant densities and sound speeds (acoustics) or different dielectrics (electromagnetics).

Exact spectral formalism for rough-surface scattering

A Monte Carlo soil reflectance model has been developed to study the effect of macroscopic surface irregularities larger than the wavelength of incident flux. The model treats incoherent multiple scattering from Lambertian facets distributed on a periodic surface. Resulting bidirectional reflectance distribution functions are non-Lambertian and compare well with experimental trends reported in the literature. Examples showing the coupling of the Monte Carlo soil model to an adding bidirectional canopy of reflectance model are also given.

A Monte Carlo Reflectance Model for Soil Surfaces with Three-Dimensional Structure

We extend our exact analysis for plane wave scattering from a sinusoidal surface to include the case of an arbitrary periodic surface. Both Dirichlet (transverse electric polarization) and Neumann (transverse magnetic polarization) boundary conditions are considered. The method uses Green's theorem and our previous idea of expanding the surface fields in Fourier series multiplied by the Kirchhoff or physical optics approximation. The expansion coefficients solve a set of linear equations. The field amplitudes of the upgoing scattered (or evanescent) waves (valid above the highest surface excursion) are expressed as a summation over these expansion coefficients multiplied by integrals over the surface function. The Rayleigh hypothesis is not invoked. Some examples are presented. For an analytic surface, steepest descent methods yield the asymptotic values of the amplitudes. Using this and other asymptotic results, the convergence of the scattered wave expansion is studied as it is analytically continued into the surface wells, and a simple and explicit confirmation of the conditions under which the Rayleigh hypothesis is valid is presented as well as new results for other examples. The periodic surface examples include a sinusoid, an echelette, a quadratic surface, a log cosine surface, a vortexlike surface, a cycloid, a trapezoid, a full-wave rectified surface, and a surface of semicircular cylinders (bosses). The method is general and applies to a very broad class of physical problems.

J. A. DeSanto

Papers

Scattering from a perfectly reflecting arbitrary periodic surface: An exact theory