Stochastic excitation method for calculating the resolvent band structure of periodic media and waveguides

We introduce a stochastic excitation method for calculating the dispersion relation for waves propagating in periodic media or along waveguides and subject to material loss or radiation damping. Instead of looking for an explicit or implicit functional relation between frequency $\ensuremath{\omega}$ and wave number $k$, as is usually done, we consider a mapping of the resolvent set in the dispersion space $(\ensuremath{\omega},k)$. Bands appear as the trace of Lorentzian responses containing local information on propagation loss in both time and space domains. For illustration purposes, the method is applied to a lossy sonic crystal, a radiating surface phononic crystal, and a radiating optical waveguide. The resolvent band structure can be obtained for any system described by a time-harmonic wave equation.