# Modeling radiative transfer in a two-layer, vertically inhomogeneous medium in local thermodynamic equilibrium

## Background

I'd like to determine the power observed by an antenna due to the blackbody radiation emitted by the medium in which it is immersed. I'd like to take into account absorption, emission, and scattering effects. I know that I can achieve this by integrating the spectral radiance over some volume surrounding the antenna which I determine to be sufficient, and I know also that I can determine the component of the spectral radiance contributed by each point in the medium (taking into account the aforementioned effects) using the radiative transfer (RTE) equation.

The medium in question, however, has two properties which have made determining the appropriate form (and solution thereof) of the RTE difficult:

1. Firstly, the medium comprises two layers: an opaque layer, and a semi-transparent (as in glass, ice, etc) layer, as depicted in the attached figure. Associated with each is some depth-dependent temperature profile, $$T(Z)$$.
2. Secondly, the semi-transparent layer has an index of refraction which is a non-negligible function of depth, $$n(z)$$. Consequently, light takes a curved path through the medium, and there are regions within the medium (in this case, the region above a cone surrounding the antenna, as depicted in red) for which light from that region cannot reach the antenna.

## My Thinking

For both layers, we may assume local thermodynamic equilibrium, which should hopefully simplify the solution. Ignoring the aforementioned complications, Wikipedia tells me that the solution to the RTE in this case is given by:

$$I_\nu(s) = I_\nu(s_0)e^{-\tau_\nu(s_0,s)} + \int_{s_0}^s B_\nu(T(s'))\alpha_\nu(s')e^{-\tau_\nu(s',s)}ds'$$

...where $$I_\nu$$ denotes the spectral radiance, $$B_\nu$$ the black body spectral radiance, $$T$$ the temperature (a function of position), $$\alpha_\nu$$ the absorption coefficient, and $$\tau_\nu$$ the optical depth between positions $$s$$ and $$s'$$.

I know that the blackbody spectra is dependent upon the index of refraction, however I'm struggling to find a reference that gives the exact form. If I remember correctly, the blackbody spectral radiance is proportional to $$n^2$$, hence:

$$B_{\nu, n(z)} = B_\nu \times n(z)^2$$

...and:

$$I_{\nu(s), n(z)} = I_\nu(s_0)e^{-\tau_\nu(s_0,s)} + \int_{s_0}^s B_\nu(T(s'))n(z)^2\alpha_\nu(s')e^{-\tau_\nu(s',s)}ds'$$

The absorption coefficient is also dependent upon the index of refraction (or, more accurately, the imaginary component of the complex refractive index), however this is something we simply measure directly.

It isn't clear to me whether this is sufficient to take into account the curved path light takes through the medium. I know that determining the path taken by light in such a medium, or the time taken to do so, involves variational calculus and minimization according to Fermat's principle.

It also isn't clear to me how to take into account the two layers, and the reflection associated with their interface. Naively, I would assume that I could solve separately the contribution of each layer, and then compute the transmission and reflection using the Fresnel equations.

The body of RTE literature is vast, and I'm struggling to sort through it all to find something both applicable to my problem, and sufficiently accessible to me as an upper-level undergraduate student.

How can I take into account the curved path light takes due to the varying index of refraction? How do I take into account the two layers, and their interface?

If an analytic solution is impractical and/or impossible here, I'm open to answers describing numerical (e.g. Monte Carlo) solutions.