Causality and Kramers-Kronig relations for waveguides

Starting from the condition that optical signals propagate causally, we derive Kramers-Kronig relations for waveguides. For hollow waveguides with perfectly conductive walls, the modes propagate causally and Kramers-Kronig relations between the real and imaginary part of the mode indices exist. For dielectric waveguides, there exists a Kramers-Kronig type relation between the real mode index of a guided mode and the imaginary mode indices associated with the evanescent modes. For weakly guiding waveguides, the Kramers-Kronig relations are particularly simple, as the modal dispersion is determined solely from the profile of the corresponding mode field.