The geometry underlying photorefractive two-beam coupling

We cast the theory of photorefractive two-beam coupling in geometrical terms and derive a closed form solution to the problem of two-beam coupling with time-dependent fields that evolve in such a way as to give rise to a stationary index of a refraction grating. The geometrical approach treats the optical field amplitudes in terms of vectors whose spatial dynamics is governed by a Hamiltonian. A density matrix formed from the optical fieldvectors is shown to have particularly simple dynamics. We derive a formula for a transfer matrix that accommodates complex coupling coefficients and that can be used to calculate the instantaneous output of the photorefractive grating given any input. The geometrical picture of photorefractive two-beam coupling is pedagogically developed through several specific examples.