# 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 ) filling the region . (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 is the incident angle ( for normal incidence), is the index of refraction, and

## scuff-em solution

The transmission and reflection coefficients for the 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 rad/sec
and at 20 incident angles in the range 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.