Conical diffraction from an N-crystal cascade

The transmitted field is calculated for a cascade of N biaxial crystals, with their optic axes parallel but with arbitrary orientations about this axis, and arbitrary strengths. The focused pattern is the superposition of 2(N-1) single-crystal concentric conical diffraction patterns, whose ring radii are combinations of those from the individual crystals in the cascade. Explicit expressions are given for the general case of arbitrary incident polarization and for N = 1-4, and for the mean intensity for a cascade with random orientations and strengths and arbitrary N. For circular incident polarization, the ring pattern is rotationally symmetric for all N, but for N > 1 and away from focus the patterns for left and right circular incident polarizations are different.