For a simple Fabry-Perot cavity formed by a dielectric slab, the quasi-normal mode frequencies (i.e. $\omega=i \frac{c}{2nL}\ln{r^2}$, where $nL$ is the optical path difference between ends of the cavity, $c$ is the speed of light, and $r$ is the reflection coefficient off of one edge), when projected onto the real axis, correspond precisely to points of zero reflection (i.e., something like $\omega=\frac{m\pi c}{nL},\forall m\in\mathbb{Z}$). For some reason, I had the intuition that this generalised to the resonances of more general structures (specifically structures formed by varying a real susceptibility) in 1D, I guess maybe informed by the insertion of defects to allow transmission through photonic crystals, but I realise that I actually have little/no proof of this being the case.
I'm fairly confident that this result doesn't hold in higher dimensions, but am uncertain whether or not this is a known result in 1D wave equations, at least.
Is there some sort of general result I can use to this effect, or is hoping for some such result just wishful thinking, and I should just knuckle down and compute the real parts of my QNMs properly.
And, as a follow-up, my intuition for resonance in 1D was built around the idea that, at least in the 1D lossless case (I'm assuming that measuring 'straight through' transmission in 3D acts similarly to lossy systems in 1D - with channels coupling the light out in other directions just acting like any other channel for loss, which would peak when $\omega$ is near the real part of a QNM frequency), QMNs contribute perfect transmission peaks with a width in frequency space given by the decay rate. Presumably they do still contribute features with width scaling like their decay rate, but if my idea above is wrong, what feature-structure, if any, would you expect from such a resonance?