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Physical Review Letters

Information Dynamics of Photorefractive Two-Beam Coupling

Dana Z. Anderson, Roger W. Brockett, Nathan Nuttall

Published Feb 15, 1999• 17 citations

Abstract

An operator approach to two-beam coupling in photorefractive media for which each beam is composed of information that varies in both space and time reduces the problem to a single differential equation: $d\ensuremath{\rho}/dz\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}[[{\ensuremath{\sigma}}_{0},\ensuremath{\rho}],\ensuremath{\rho}]/4$. $[A,B]\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}AB\ensuremath{-}BA$ is the commutator of two operators, $\ensuremath{\rho}$ is a density operator that embodies the state of the optical field, and ${\ensuremath{\sigma}}_{0}$ is a coupling operator. A solution exists in closed form for specific cases and is amenable to numerical integration in general. The evolution of $\ensuremath{\rho}$ is unitary and approaches a block diagonal in a representation that diagonalizes ${\ensuremath{\sigma}}_{0}$.

DOI: 10.1103/physrevlett.82.1418

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An operator approach to two-beam coupling in photorefractive media for which each beam is composed of information that varies in both space and time reduces the problem to a single differential equation:

d\ensuremath{\rho}/dz\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}[[{\ensuremath{\sigma}}_{0},\ensuremath{\rho}],\ensuremath{\rho}]/4

.

[A,B]\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}AB\ensuremath{-}BA

is the commutator of two operators,

\ensuremath{\rho}

is a density operator that embodies the state of the optical field, and

{\ensuremath{\sigma}}_{0}

is a coupling operator. A solution exists in closed form for specific cases and is amenable to numerical integration in general. The evolution of

\ensuremath{\rho}

is unitary and approaches a block diagonal in a representation that diagonalizes

{\ensuremath{\sigma}}_{0}

.

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.

A theoretical analysis is presented of two-wave mixing in photorefractive materials in which the complex amplitude of a weak signal beam has slowly varying but otherwise arbitrary spatial dependence and arbitrary time variation. The set of coupled partial differential equations that couple the dynamics of the optical fields to the photorefractive medium is solved analytically to give the complex amplitude of the transmitted signal in the undepleted pump regime. The results are compared with known results for the transient two-wave mixing of plane waves.

In a biased photorefractive crystal, the process of two one-dimensional waves mixing, i.e., the dynamical evolution of both pump beam and signal beam, is traced by numerically solving the coupled-wave equation. Direct simulations show that the propagation and stability of the two beams are completely determined by the system parameters, such as the external bias field, the intensity and the beam waist of the pump beam. By adjusting these parameters, one can control the state of two Gaussian waves mixing. The numerical results are helpful for performing a two-wave mixing experiment.

While the interaction of optical beams in photorefractive crystals can often be accurately described in terms of 1-D plane-wave approximation, 2-D models are sometimes more suitable, as suggested by recent work.1 Here we discuss some aspects of photorefractive interaction of finite size optical beams. We have solved the system of partial differential equations governing two- and four-wave mixing numerically by using iterative procedures. Simulations of beam amplification in two-beam coupling show strong beam deformation even without any spatial information imposed on them. Spatially nonuniform energy transfer is responsible for this effect. The pump beam loses energy during propagation and at the end of its passage through the beam intersection region (which was completely inside the crystal) it is strongly depleted and cannot contribute significantly to beam amplification. As the pump beam crosses through the probe, it amplifies most strongly that side of the probe that it meets first. By the time the pump has reached the far side of the probe, it is strongly depleted and less effective at probe amplification. In the case of phase conjugation through four-wave mixing, we found that the conjugate beam is strongly deformed only when the coupling constant is negative (corresponding to deamplification of the probe input beam). For positive coupling constants the output conjugate beam is a good replica of the input probe beam.