# Analytical Solution: EM Scattering from Hemispherical Indentation in Plane

I have a problem of trying to calculate the losses that occur when a beam of light reflects from the surface of a partially transparent medium that has an open cavity. The circumference of the cavity opening and its depth are approximately the same size as the wavelength of the illuminating light (so no low or high frequency approximations). The problem is set up similarly to the image in the paper at the link below, except in my case some of the light would transmit into the material since it is partially transparent.

https://www.researchgate.net/figure/The-geometry-of-the-cavity_fig1_266061837

However, I was hoping that I could model this as a hemispherical indentation and find an analytical solution that would be similar to the Rayleigh or Mie scattering cross section that I could use to estimate the deviation from the Fresnel reflection of a perfectly flat surface. So far, I have not found anything and all of the open cavity scattering problems that I have seen involve an indentation, usually not a hemisphere, in a conducting (perfectly reflecting) medium. Is there an analytical solution to such a problem? If not, how to go about estimating the losses?