Second-order coupled-mode equations for spatially periodic media*

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Abstract
A new pair of second-order coupled-mode equations are derived, solved, and applied to the case of reflection and transmission by a spatially modulated slab medium. The derivation is compared with those that arrive at first-order coupled-mode equations. The solutions of the second-order coupled-mode equations are facilitated by reducing them to two uncoupled Helmholtz equations. The solutions of the boundary-value problem for a slab medium are analytical and expressed in closed form. They are confirmed by the rigorous modal theory in the limit of small modulation strength. At the Bragg angle of incidence, energy conservation is shown to be preserved for all modulation strengths.

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