The anisotropic Beer-Lambert law in \(\beta\)-Ga\(_{2}\)O\(_{3}\): Spectral and polarization dependent absorption and photoresponsivity

Due to its low symmetry, \(\beta\)-Ga\(_{2}\)O\(_{3}\) exhibits a strongly anisotropic optical response. As a result, the absorption spectra change with the polarization state of the incoming photons. To understand this phenomenon, here we calculate the complete electromagnetic wave equation solutions as a function of linear polarization angle and photon energy for \(\beta\)-Ga\(_{2}\)O\(_{3}\) using its previously measured complex dielectric function tensor. The significant off-diagonal terms in this tensor can result in a non-exponential decay in the photon flux, indicating that the Beer-Lambert law is not generally valid in this anisotropic material. However, for above-band-gap spectral regions which depend on crystallographic orientations (> 5.8 eV (001 plane),>5.2 eV (010 plane)) an effective absorption coefficient well approximates the photon flux decay with depth. On the other hand, near the optical absorption edge (4.9 - 5.8 eV (001 plane),4.65 - 5.2 eV (010 plane)) the photon flux decay exhibits a sum of two exponential decays, such that two effective absorption coefficients are necessary to model the loss behavior versus the absorption depth. This behavior manifests from the presence of dichroism in \(\beta\)-Ga\(_{2}\)O\(_{3}\). A single effective absorption coefficient can only be recovered for this energy range by augmenting the isotropic Beer-Lambert law with a critical penetration depth and polarization dependence. Using these results, we calculate the polarization-dependent photoresponsivity spectra for light polarized along different crystallographic directions.