Analytical Fresnel laws for curved dielectric interfaces

Fresnel laws and the corresponding Fresnel reflection and transmission coefficients provide the quantitative information of the amount of reflected and transmitted (refracted) light in dependence on its angle of incidence. They are at the core of ray optics at planar interfaces. However, the well-known Fresnel formulae do not hold at curved interfaces and deviations are appreciable when the radius of curvature becomes of the order of several wavelengths of the incident light. This is of particular importance for optical microcavities that play a significant role in many modern research fields. Their convexly curved interfaces modify Fresnel's law in a characteristic manner. Most notably, the onset of total internal reflection is shifted to angles larger than critical incidence (Martina and Henning 2002 Phys. Rev. E 65 045603). Here, we derive analytical Fresnel formulae for the opposite type of interface curvature, namely concavely curved refractive index boundaries, that have not been available so far. The accessibility of curvature-dependent Fresnel coefficients facilitates the analytical, ray-optics based description of light in complex mesoscopic optical structures that will be important in future nano- and microphotonic applications.