Light propagation in photonic crystal with gain: Applicability of the negative loss approximation

The applicability of the concept of permittivity with Im ?_gain < 0 to describe the light propagation in a metamaterial system with gain is discussed using the example of a 1D photonic crystal containing gain layers. It is shown that this approach is in agreement with the principle of causality, unless lasing is present. Though the lasing process itself requires nonlinear analysis, the lasing threshold can be determined by linear (negative loss) approximation. Connecting the onset of lasing with the passage of the transfer function pole into the upper half-plane of the complex frequency, we show that (i) if the pump frequency lies in a pass band then an increase in the number N of elementary cells will sooner or later lead to lasing; (ii) if the frequency of the pump lies in the band gap, then lasing at band gap frequencies may occur in a sample with a low N before the band gap has been formed. Nevertheless, lasing will be necessarily suppressed by a further increase in N. In any case, for sufficiently large number of layers due to a finite line width of ?_gain(ω), lasing appears in the pass band even if the pump frequency is in the band gap.