Resonance mode expansions and exact solutions for nonuniform gratings

Resonance modes play an important part in understanding linear nonuniform gratings, analogous to the role played by waveguide modes in waveguide theory. Using resonance mode expansions, exact expressions are obtained for the fields, the grating profile, and the reflection and transmission spectra for a large class of nonuniform linear gratings. The method can deal with linear gratings that couple a pair of either copropagating or contrapropagating modes. The formalism covers the effects of gain and loss (in the small signal limit), chirp, taper, and birefringence. The exact solutions can be used to investigate designs for grating structures. Two detailed example applications of the technique are presented here: an exact solution for a grating that supports only a single resonance mode, and an exact solution for a grating that has nonreciprocal reflective properties from its two ends.