This Fourier Transfer Function properties?.

[monchosoad]

Hi Everyone. I want to know what do you think about this Transfer function:

$T(\omega) = \frac{e^{i\omega\tau}}{1-\rho e^{i2\omega\tau}}$

If$\tau$ is Real, this function is "good and pretty"(?), because it has a "nice" representation in time with its inverse transform: a delay plus a series of deltas displaced in time in $n\tau$ times $\rho^n$.
But the question is: What about if $\tau$ is Complex?? Is there a magical property that I'm missing to deal with this?

Sorry if i was mistaken on the election of the Forum. I don't know if this question may be in "Physics" because $T(\omega)$ is the Fabry-Perot Transfer Function, or in Engineering. And sorry for my English.

Thanks!

 

[blue_leaf77]

The simplest case of a FP resonator is that when the medium in between the mirrors is just vacuum, in that case $\tau$ is a constant with respect to frequency. By invoking that $\tau$ be complex (which physically means the intermediate medium is absorbing), this implies that this quantity must also be a function of frequency - there is no occurrence in reality where the index of refraction contained in $\tau$ is a constant complex number as this would violate Kramers-Kronig relation. Furthermore, there is no closed form of the dependence of refractive index on frequency as this is medium specific, so if $\tau$ is complex which also means $\tau$ is frequency dependent, the inverse FT of $T(\omega)$ won't be as nice as it is for free space case and I believe its analytical expression does not exist.