Lop-sided diffraction by absorbing crystals

Diffraction of atoms by a particular absorbing 'crystal of light', that has been studied experimentally, is described by the complex potential iV(0)(exp(iK x)-1). The diffracted beam intensities I-n can be calculated exactly, as a function of (dimensionless) potential strength sigma, angle of incidence alpha, and crystal thickness zeta. Only the beams n greater than or equal to 0 have nonzero intensity; this 'lop-sidedness' is a dramatic violation of Friedel's law. The nth beam is strong at Bragg angles alpha = -1/2, -1,..., -(n-1/2), and the peaks get sharper with increasing zeta; they correspond to degeneracies of the (non-Hermitian) governing matrix. For normal incidence (alpha = 0), the I-n are periodic in zeta, and as n increases they approach a self-similar function of zeta.