Optical lattices with PT symmetry are not transparent

In diffraction of a plane wave by a non-Hermitian PT symmetric optical lattice, the sum of the Bragg beam intensities need not be conserved, even though the gain and loss are equally distributed: the evolution is not unitary. Instead, different sums are conserved, in which the intensities are weighted with real numbers (positive or negative); several such sum rules are derived. Two-beam diffraction from a refractive index of the form constant -a cos x + ib sin x is studied in detail; the sum rule depends on the balance between the (real) Hermitian parameter a and the (real) anti-Hermitian parameter b.