Lower bounds on the absorption probability of beam splitters

We derive a lower limit to the amount of absorptive loss present in passive linear optical devices such as a beam splitter. We choose a particularly simple beam splitter geometry, a single planar slab surrounded by vacuum, which already reveals the important features of the theory. It is shown that, using general causality requirements and statistical arguments, the lower bound depends on the frequency of the incident light and the transverse resonance frequency of a suitably chosen single-resonance model only. For symmetric beam splitters and reasonable assumptions on the resonance frequency $\omega_T$, the lower absorption bound is $p_{\min}\approx 10^{-6}(\omega/\omega_T)^4$.