A new treatment of lossy periodic waveguides

Abstract
The conventional treatment of propagation coefficients in lossy periodic waveguides suffers from certain major defects. It gives no information about the effects of the losses on the phase-change coefficient, it breaks down when the frequency approaches the edge of a pass-band from within and it does not work at all in a stop-band. A new treatment is described which removes all these defects. It is based on the following result: the propagation coefficient of a mode in a lossy guide at the frequency ω is equal to the propagation coefficient of the corresponding mode in the lossless guide at the frequency ω(1 — j/2Qc). Here, Qc is the “complex Q-factor” of the mode at the frequency ω. It is given by an explicit formula which holds good at all frequencies when the losses are small. When ω lies within a pass-band Qc is equal to ω times the mean energy stored in a period of the guide divided by the complex power dissipated in the same period. When ω lies in a stop-band Qc is equal to the analytic continuation of its values in the pass-bands.
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