Abstract
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.
Highlights
The Fresnel equations, derived by Augustine-Jean Fresnel in 1823, quantify the amount of the reflected, R, and transmitted, T = 1 − R, intensity of a plane wave incident under a certain angle of incidence χ at a planar interface between two isotropic optical media of refractive indices n1 and n2 [1]
For internal reflection configurations (n > 1, reflection at the optically thinner medium) total internal reflection occurs above the critical angle of incidence given by ccr = arcsin1 n
To be taken into account when the radii a of curvature of the interfaces become as small as several dozens or even several wavelengths λ of the incident light to ensure a reliable description of the reflection and transmission process
Summary
To be taken into account when the radii a of curvature of the interfaces become as small as several dozens or even several wavelengths λ of the incident light to ensure a reliable description of the reflection and transmission process This applies for example to optical microcavities with typical sizes of a few dozens micrometers across operated at infrared light [2, 3]. The interface curvature affects, partly via the change in the Fresnel coefficients, the direction of the far field emission of the microcavities [4, 5] as well as semiclassical corrections to the ray picture [6, 7,8,9,10,11,12] This implies in particular to deviations from Snell’s law as a result of the so-called Fresnel-filtering effect [13]. They will provide the basis for a reliable ray-based description of photonic devices with convex or concave interfaces or complex
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