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We consider linear propagation through shallow, nonuniform gratings, such as those written in the core of photosensitive optical fibers. Though, of course, the coupled-mode equations for such gratings are well known, they are often derived heuristically. Here we present a rigorous derivation and include effects that are second order in the grating parameters. While the resulting coupled-mode equations can easily be solved numerically, such a calculation often does not give direct insight into the qualitative nature of the response. Here we present a new way of looking at nonuniform gratings that immediately does yield such insight and, as well, provides a convenient starting point for approximate treatments such as WKB analysis. Our approach, which is completely within the context of coupled-mode theory, makes use of an effective-medium description, in which one replaces the (in general, nonuniform) grating by a medium with a frequency-dependent refractive index distribution but without a grating.

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 limit͒, 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. ͓S1063-651X͑96͒04909-4͔

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