Chebyshev spectral methods for radiative transfer

In this paper, we present a numerical method for solving the radiative transfer equation that models electromagnetic wave propagation in a constant background, plane-parallel medium containing randomly distributed, identically sized, dielectric spheres. Applications of this study include optical waves in media such as biological tissue and fog as well as millimeter waves in rain. In these problems, the scattering is highly anisotropic, which is problematic for standard methods that approximate the integral term of the radiative transfer equation first to yield an associated system of differential equations. In our method, we use a Chebyshev spectral approximation of the spatial part of the 4 £ 1 vector of Stokes parameters. This spectral approximation then yields a coupled, linear system of integral equations that has a bordered, block sparsity structure that can be efficiently solved using a deflated block elimination method. By readjusting the focus of this numerical method to the integral operators instead of the derivative operators, we find that we can effectively study highly anisotropic scattering media. We present some examples of Mie resonant scattering in which a circularly polarized plane wave propagates and scatters in a plane-parallel medium containing randomly distributed, identically sized, dielectric spheres whose radii are comparable to the wavelength. c ° 1999 Academic Press Key Words: radiative transfer equation; Chebyshev spectral methods; deflated block elimination method; Mie resonance scattering