Refraction in Strongly Absorbing Media
The direction of ray propagation may be described via the loci of points having constant phase or amplitude. In a nonabsorbing medium, one which may be described by a real-valued refractive index, changes in the ray intensity are caused only by the convergence or divergence of the wavefront, so the surfaces in which the phase is constant are parallel to the surfaces in which the amplitude is constant.
In a weakly absorbing medium, in which the attenuation of the power transmitted by each ray decays over a length scale much larger than the wavelength, it may be assumed that the surfaces of constant amplitude are still parallel to the surfaces of constant phase. The directions of the reflected and refracted rays at material discontinuities, as well as the Fresnel coefficients that govern their intensity, may be computed using the real parts of the refractive indices on either side of the boundary. However, as the imaginary part of the refractive index increases in magnitude, this assumption may begin to present a considerable source of error because the surfaces of constant amplitude and surfaces of constant phase are no longer parallel.
In the Settings window for the Geometrical Optics interface, select the Use corrections for strongly absorbing media check box in the physics interface Intensity Computation section to modify Snell’s law and the Fresnel equations to accurately model refraction between media with complex refractive indices. This check box is available when Intensity computation is set to Compute intensity or Compute intensity and power.
Following Chang and others in Ref. 14, the wave vector in an absorbing medium is treated as a bivector with complex components,
where k is the wave vector, k0 is the wave number in free space, and e and f are unit vectors with real-valued components indicating the normal direction to the surfaces of constant phase and surfaces of constant amplitude, respectively. The real-valued quantities N and K, sometimes called the apparent refractive indices, are related to the complex refractive index by the relations
where n − iκ is the complex refractive index of the medium. From these relations the apparent refractive indices can be computed as long as the dot product e ⋅ f is known. To store information about the value of this dot product, auxiliary dependent variables for the components of f are stored when the Use corrections for strongly absorbing media check box is selected.
At material discontinuities, the normal vectors to the surfaces of constant amplitude and phase follow a modified form of Snell’s law that uses the apparent refractive indices,
where θ and ψ are the acute angles between the surface normal and the normal vectors to the surfaces of constant phase and surfaces of constant amplitude, respectively. The real part of the apparent refractive index in the second domain is a root of the quartic equation
where Ns = N1sin θ1, Ks = K1sin ψ1, and the angle ϕ is the azimuthal angle between the normal to surfaces of constant amplitude and the plane of incidence. This equation is obviously quadratic in and can thus be solved using the quadratic formula.
The reinitialized unit normal vectors to the surfaces of constant amplitude and phase are
whereas for the reflected ray the reinitialized unit normal vectors are