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We study the diffraction produced by a $PT$-symmetric volume Bragg gratingthat combines modulation of refractive index and gain/loss of the sameperiodicity with a quarter-period shift between them. Such a complex gratinghas a directional coupling between the different diffraction orders, whichallows us to find an analytic solution for the first three orders of the fullMaxwell equations without resorting to the paraxial approximation. This isimportant, because only with the full equations can the boundary conditions,allowing for reflections, be properly implemented. Using our solution weanalyze the properties of such a grating in a wide variety of configurations.

Analytic Solution for $PT$-Symmetric Volume Gratings