A. Tip, J. M. Combes, A. Moroz
Bloch decomposition and band structure for absorptive photonic crystals
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ABSTRACT.  We show that dielectric photonic crystals do not have band gaps 
in frequency regions where absorption takes place, i.e. where the 
frequency-dependent electric permeability 
$\varepsilon (\QTR{bf}{x},\omega )$ has a non-zero imaginary part. 
Under these circumstances real eigenvalues of the Helmholtz operator 
in the Bloch-decomposed formalism are absent. We find, using a suitable 
analytic continuation procedure, that the former change into resonances, 
i.e., complex numbers depending on $\QTR{bf}{k}$, the wave vector from 
the first Brillouin zone, thus leading to complex bands in the lower 
half plane.