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Physical Review A

Frequency shift in an oscillator with photorefractive gain

Laurent Dambly, H. Zeghlache

Published Mar 1, 1993• 15 citations
Publisher

Abstract

A self-consistent theory is used to derive the frequency shift presented by the oscillating mode in a cavity that contains a photorefractive crystal. A long-time-limit theory has been used to get the steady characteristics of the cavity field. The problem of the weak-field approximation is avoided. This approximation is useless in the case studied because the interaction along the crystal is so strong that one has to take in account the intensity exchange occurring inside the material between the pump and oscillating beams (two-wave mixing). We have fully analyzed the influence of the transverse nature of the beams on the intensity exchange in the two-wave mixing situation. In the photorefractive resonator case, the frequency shift of the oscillating beam is expressed in terms of a ratio that presents poles. The dependence on the problem parameters of these poles is studied and a perturbative method is developed; it is based on the smallness of the transverse overlap between the pump and cavity modes. An analytical expression for this frequency shift is obtained.

DOI
10.1103/physreva.47.2264

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It is known that two-wave coupling in photorefractive crystal may result in an intensity-dependent phase shift as well as energy transfer between the two waves without phase crosstalk. In a sense, this kind of phase shift is similar to the intensity-dependent refractive-index effects in Kerr media. Consequently, optical bistability can be established in an optical cavity with a photorefractive crystal. We formulate the model and get the expected bistable transmission with numerical simulations. Several cases, including that of the codirectional coupling and the contradirectional coupling, are considered. The effects of the photorefractive crystal’s distribution absorption and the spatial phase shift between the index grating and the light interference pattern in the crystal on the nonlinear transmission characteristics have also been studied. It is interesting to note that the output of the pumping wave is dependent on the cavity intensity and thus shows bistable behavior with respect to the cavity input as well.

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We compare weak-field theory to strong field numerical results for the bistable ring resonator with saturable photorefractive gain and loss.1,2 Our weak-field model, which only considers the time dependence of the gratings and neglects the spatial dependence,3 captures an approximate picture of the two-crystal resonator. The basic model consists of two coupled complex equations for the gain and loss gratings. An analytic equation is derived for the self-pulsing period and amplitude for the case of zero detuning and near threshold gain. In this case, the system is analogous to a Van der Pol oscillator. We solve for the stationary points of the system and discuss their stability conditions. We also present computer simulation of the nonzero detuning case.

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A unidirectional photorefractive ring resonator oscillating at 633 nm

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We have constructed an optical resonator using a photorefractive crystal as the gain medium. The orientation of the crystal and the angle between the signal and the pump beam are chosen to obtain optimal two-wave coupling. The average output power of the resonator is proportional to the power of the pump beam for a small cavity loss, but this linear relation is broken when the cavity loss is increased. The transverse beam pattern of the signal can be controlled by intracavity apertures.

American Journal of Physics•Jul 1, 1998•4 citations
A self-consistent theory is used to derive the frequency shift presented by the oscillating mode in a cavity that contains a photorefractive crystal. A long-time-limit theory has been used to get the steady characteristics of the cavity field. The problem of the weak-field approximation is avoided. This approximation is useless in the case studied because the interaction along the crystal is so strong that one has to take in account the intensity exchange occurring inside the material between the pump and oscillating beams (two-wave mixing). We have fully analyzed the influence of the transverse nature of the beams on the intensity exchange in the two-wave mixing situation. In the photorefractive resonator case, the frequency shift of the oscillating beam is expressed in terms of a ratio that presents poles. The dependence on the problem parameters of these poles is studied and a perturbative method is developed; it is based on the smallness of the transverse overlap between the pump and cavity modes. An analytical expression for this frequency shift is obtained.
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]
A theoretical analysis of the oscillator with a photorefractive gain is presented. The steady multimode oscillation is considered. In such a system, the cavity light beam undergoes a frequency ``pulling'' due to the nonlinear interaction between light and matter. This frequency shift is evaluated with respect to the empty-cavity frequency. We show that it can be as large as necessary depending on the experimental parameters. This can be a physical interpretation for the periodic (and chaotic) oscillation of several transverse family modes already observed in these setups.
Photorefractive resonators exhibit an extremely small frequency difference (Deltaomega/omega~10(-15)) between the oscillating and pumping beams. The observed frequency difference is proportional to cavity-length detuning. This dependence is explained by a photorefractive phase shift that is due to slightly nondegenerate two-wave mixing that compensates for cavity detuning and satisfies the round-trip phase condition for steady-state oscillation. The measured onset or threshold of oscillation as a function of photorefractive gain and intensity agrees with theory.
It is known that two-wave coupling in photorefractive crystal may result in an intensity-dependent phase shift as well as energy transfer between the two waves without phase crosstalk. In a sense, this kind of phase shift is similar to the intensity-dependent refractive-index effects in Kerr media. Consequently, optical bistability can be established in an optical cavity with a photorefractive crystal. We formulate the model and get the expected bistable transmission with numerical simulations. Several cases, including that of the codirectional coupling and the contradirectional coupling, are considered. The effects of the photorefractive crystal’s distribution absorption and the spatial phase shift between the index grating and the light interference pattern in the crystal on the nonlinear transmission characteristics have also been studied. It is interesting to note that the output of the pumping wave is dependent on the cavity intensity and thus shows bistable behavior with respect to the cavity input as well.
We compare weak-field theory to strong field numerical results for the bistable ring resonator with saturable photorefractive gain and loss.1,2 Our weak-field model, which only considers the time dependence of the gratings and neglects the spatial dependence,3 captures an approximate picture of the two-crystal resonator. The basic model consists of two coupled complex equations for the gain and loss gratings. An analytic equation is derived for the self-pulsing period and amplitude for the case of zero detuning and near threshold gain. In this case, the system is analogous to a Van der Pol oscillator. We solve for the stationary points of the system and discuss their stability conditions. We also present computer simulation of the nonzero detuning case.