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Physical Review A

Coupled-mode calculation ofK/2 spatial subharmonics in photorefractive moving gratings

Sugie Shim, Chong Hoon Kwak

Published Jul 1, 1996• 4 citations
Publisher

Abstract

The frequency detuning curves of the spatial subharmonic intensity of photorefractive moving gratings are calculated using a simple theory based on coupled-mode analysis of wave equations combined with Kukhtarev's materials equations. The optimum detuning frequency and the intensity distribution of K/2 spatial subharmonics against detuning frequency are calculated both analytically and numerically by varying the external electric field and the modulation of two pump beams. It is found that the results of our theoretical calculations for the frequency detuning curves of subharmonic intensity agree with the experimental data very well. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.863

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The frequency detuning curves of the spatial subharmonic intensity of photorefractive moving gratings are calculated using a simple theory based on coupled-mode analysis of wave equations combined with Kukhtarev's materials equations. The optimum detuning frequency and the intensity distribution of K/2 spatial subharmonics against detuning frequency are calculated both analytically and numerically by varying the external electric field and the modulation of two pump beams. It is found that the results of our theoretical calculations for the frequency detuning curves of subharmonic intensity agree with the experimental data very well. \textcopyright{} 1996 The American Physical Society.
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.
We present an analysis of what is called parametric oscillation in photorefractive media. It is shown theoretically that when a running grating and a constant electric field are applied to a photorefractive medium it is possible for two secondary gratings to grow spontaneously. These theoretical predictions are subsequently verified experimentally in the crystal Bi12SiO20, and excellent qualitative agreement is obtained. In previous subharmonic experiments, only grating vectors K/2, K/3, and K /4 were observed. Now, however, a whole continuum of grating vectors has been observed. Thus subharmonic generation in photorefractive materials is simply a special case of parametric oscillation in which the two secondary gratings are identical.
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.
It has been shown experimentally1 that a subharmonic beam may appear spontaneously between two pump beams without detuning either of the beams when a field varying sinusoidally or in the form of a square wave is applied across the crystal. That work has now been extended to cover the rise of the subharmonics as a function of time and their dependence under stationary conditions on the ratio between the pump intensities. The experiments were done on photorefractive bismuth silicate crystals. A simple theory is found to describe well the repetition frequency at which the subharmonics appear.
Two-wave mixing in photorefractive crystals has been shown to combine coherently a modulated optical signal beam with a strong pump optical field.’ For a phase-modulated signal beam, the induced index of refraction phase grating inside the crystal yields a coupling of the optical fields of the pump and signal so that subsequent intensity detection results in apparent intensity modulation of the transmitted signal or pump beams. This provides an easy way to detect phase differences between two optical beams. Some of the advantages the above scheme offers over commonly used coherent optical phase detection schemes are: no loss of 3 dB in the signal beam intensity at the 50/50 beam splitter (used to combine signal and local oscillator fields); automatic alignment of the index of refraction grating with the two incident optical wavefronts; insensitivity to slow random changes in the phases of both beams. Experimental results are given, and a new definition for the resulting mixing gain for the phase-modulated beam is introduced, through which optimal conditions for input beam intensity ratios can be found. Applications to coherent homodyne detection are discussed.
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.