Quasi-periodic complex-coupled distributed-feedback structures with an exponential-like gradient of coupling

We investigate theoretically the linear properties of quasi-periodic complex-coupled distributed-feedback (DFB) structures in which the coupling constants experience a longitudinal variation along the propagation axis according to an exponential-like relationship (exp, sh, ch, ···). The results of the simulations are in good accordance with what could be expected at first sight from the particular form taken in this case by the coupled-wave equations. In particular, the gradient of coupling can provide a shift, either of the apparent average absorption (respectively, gain) or of the detuning from Bragg resonance, depending upon its exact form, Moreover, we show that this shift is asymmetric with respect to the direction of propagation, leading to a nonreciprocal behavior in reflection. We extend our analysis up to the laser modes at threshold