Attenuation Within Domains
Rays can gradually lose energy as they propagate through a domain in which the refractive index is complex. The convention of the Geometrical Optics interface is that a lossy medium has refractive index given by n − iκ. Given the electric field E0 at one location, the electric field of a ray with infinite wavefront radii of curvature as it propagates through a homogeneous, lossy medium is
where L is the optical path length between the points at which E and E0 are measured and k0 is the wave number in free space. In weakly absorbing media, N = n, K = κ, and α = 0. If the Use corrections for strongly absorbing media check box is selected, N is the real part of the apparent refractive index, K is the complex part of the apparent refractive index, and α is the angle between the surfaces of constant amplitude and surfaces of constant phase; both of these concepts are explained in Refraction in Strongly Absorbing Media. The ray intensity and power are both proportional to the square of the electric field norm, so these quantities change according to the relations
When computing the ray intensity in absorbing media, the principal radii of curvature of the wavefront must also be considered. The two contributing factors are accounted for by allocating auxiliary dependent variables for the initial Stokes parameters following the most recent reinitialization of the wave vector, which are expressed as solutions to first-order differential equations of the form
(3-3)
where c is the speed of light in the medium and si,0 is value of one of the Stokes parameters of the ray before the effects of changes in the principal radii of curvature have been applied.
The Geometrical Optics interface defines an additional dependent variable A (dimensionless) for the path integral of the attenuation coefficient,
(3-4)
Despite requiring an extra degree of freedom per ray, the advantage of Equation 3-4 over Equation 3-3 is that the dependent variables in Equation 3-4 all vary linearly within a homogeneous medium. This makes Equation 3-4 much less prone to numerical stiffness; that is, it remains more accurate when the solver takes long time steps, compared to Equation 3-3, potentially reducing solution time and improving accuracy.