# Fresnel Scattering

This test validates the scuff-transmission application module of the scuff-em code suite by using it to study the textbook case of Fresnel scattering: the transmission and reflection of plane waves at a dielectric interface.

## Exact solution

The situation considered here is that of a plane wave impinging from below on a dielectric half-space (relative permittivity $\epsilon=10$) filling the region $z>0$. (More details on the setup for scuff-transmission calculations may be found in the document Computation of reflection and transmission coefficients in scuff-em.)

For this case, the transmission and reflection coefficients for the TE and TM polarizations read

where $\theta$ is the incident angle ($\theta=0$ for normal incidence), $n=\sqrt{\epsilon}$ is the index of refraction, and

## scuff-em solution

The transmission and reflection coefficients for the $\epsilon=10$ dielectric half-space problem may be computed using scuff-transmission as follows:

% scuff-transmission --geometry E10HalfSpace_40.scuffgeo --Omega 1.0 --ThetaMin 0.0 --ThetaMax 88.0 --ThetaPoints 20"

Here the file E10HalfSpace_40.scuffgeo describes the scuff-em" geometry (it refers to a mesh file named Square_40.msh) and the command-line arguments ask for a calculation at angular frequency $\omega=1\cdot 3\times 10^{14}$ rad/sec and at 20 incident angles in the range $0\le \theta\le 88$ degrees.

## Comparison

Running the above command yields the file E10HalfSpace_40.transmission. Plotting in gnuplot yields a comparison of scuff-transmission data (point) to theoretical predictions (curves):

Here is the gnuplot script that I use to produce this plot: PlotFresnelData.gp.