Variational technique for nonuniform gratings and distributed-feedback lasers

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Abstract
Gratings with a nonuniform period or modulation depth have many interesting properties. For each grating there are characteristic frequencies for which the grating is reflectionless and other frequencies for which the grating is resonant (lasing modes in a distributed-feedback structure). The imaginary part of the resonant frequency gives the gain threshold required for lasing. The reflection spectrum of an arbitrary grating or accurate approximations to this spectrum can be reconstructed from these characteristic frequencies. A variational result for these characteristic frequencies is presented, and a numerical technique based on the variational result is developed that efficiently and accurately produces these frequencies and gain thresholds and also the fields within the grating. Some analytical results are also presented.