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Journal of Optics A Pure and Applied Optics

The geometry underlying photorefractive two-beam coupling

Valeria Damiao, Dana Z. Anderson

Published Oct 27, 2003• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1464-4258/5/6/023

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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.
We show that a certain phase feedback between output and input changes dramatically the characteristics of photorefractive two-beam coupling. The diffraction efficiency of the dynamic phase grating, $\ensuremath{\eta}$, reaches 1 or 0 (depending on the feedback sign) within a finite time for a wide range of experimental parameters. The spatial-temporal behavior of the wave amplitudes is strongly nonlinear and the grating fringes for $\ensuremath{\eta}=1$ or 000 are bended but not tilted. The theory elucidates a number of empirical observations and reveals the great potential of active stabilization for applications.
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 present a closed-form solution for the two-wave coupling process, resulting in a solution with an interesting part, the arbitrary function Lambda(t), not being considered in the previous works. In the photorefractive medium, it has been shown that a moving grating will produce a phase shift in the coupling constant. To our knowledge, this has not been derived directly from the grating dynamic equations. We will also show that the temporal dynamics does not have any asymptotically stable equilibrium points of an exponential form. As an example, we have developed resonance conditions resulting in a new expression for the frequency detuning of a resonator with a photorefractive gain medium.
Light-induced change in the index of refraction of a Kerr medium is well known.1 Illumination with a spatially periodic light intensity leads to the formation of an index grating in the Kerr medium. We make a comparative study of the response time and power requirement for index grating formation in photorefractive and Kerr media. We examine the conditions in which the index grating in the Kerr medium will be spatially shifted from the light intensity that induces it. Two-beam coupling via Bragg diffraction from the nonlocal index grating is studied.2 The coupling constant is found to depend on the relaxation mechanism of the atoms in the Kerr media and the frequency detunings of the two beams from the line center.
We present an eight-dimensional matrix formulation of vector four-wave mixing. We study the case of a nonoptically active photorefractive for the transmission grating. The coupled equations are represented in a group formalism and irreducible forms are found. We show that when cross-polarization coupling is neglected, the system of equations is governed by two coupled SU(2) groups. We develop a theory that allows for pump depletion and electric-field-induced bulk birefringence. Analytical solutions are found when cubic 4¯3m materials are considered in two standard experimental geometries, i.e., electric fields perpendicular to the (110) crystal faces and electric fields perpendicular to the (001) crystal faces. Results are presented that show the effects of linear and circular polarized beams, and the use of these materials for vector phase conjugation is discussed. Solution routes to other geometries and noncubic photorefractive materials are also presented.
A theory of the two-wave mixing in a photorefractive medium placed in a ring cavity is developped following the model first derived by Yariv and Kwong [1]: a steady refractive index involving the photorefractive effect is used as a non-linear source-term (PNL) in the Maxwell equations, and the cavity field is expanded on the transverse cavity modes. In this model we take into account the intensity change of both the pump and resonator beams inside the crystal due to the mutual power exchange: we have developed a multimode model which gives in the long-time limit the z-dependance of the fields (z is the optical axis) for a passage in the crystal, using the expansion in terms of the complete set of the cavity transverse modes. [2]