Effective refractive index for determining ray propagation in an absorbing dielectric particle

The ray-tracing technique can be employed to simulate the scattering of light by a dielectric particle whose characteristic dimension is much larger than the incident wavelength. When a scattering particle is absorptive, a localized electromagnetic wave refracted into the scatterer is inhomogeneous, which requires the use of an effective refractive index to determine the propagation direction of the refracted ray. The effective refractive index for the first-order reflection-refraction event (i.e., the case for the ray-transmission from air into a particle) has been previously derived by the authors. In this study, we further develop recurrence formulae for the effective refractive indices associated with higher-order reflection-refraction events when the ray-transmission is from a particle to air. It is shown from the new formulae that effective refractive indices in this case depend upon ray history. Numerical results indicate that the real and imaginary parts of the effective refractive index are larger and smaller, respectively, than the real and imaginary parts of the inherent complex refractive index of an absorbing particle. Furthermore, if the particle faces associated with two sequential internal reflections are parallel to each other, the corresponding effective refractive indices are the same.