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Stability of multiple solutions in photorefractive four-wave mixing with external electrical field
Abstract
The stability of multi-branched solutions of the nonlinear differential equations describing four-wave mixing for the case where an electric field applied to a photorefractive medium (such as barium titanate) is investigated, and details of the method of analysis are given. It is found that although the steady-state solutions may exhibit hysteresis, such behaviour is probably not practically realizable since in all of such cases investigated, one or more of the branches is unstable. It is also found that for high applied fields, there is no stable solution except for very small pump beam intensity ratios.
This review paper contains three parts. The first part covers the photorefractive (PR) effect and the processes [e.g. two-wave mixing (TWM) and four-wave mixing (FWM)] which are associated with the recording of dynamic volume holograms. This part reviews the PR effect, popular PR crystals and their performances in dynamic holography, general aspects of hologram formation, coupled wave formulations of dynamic holograms through degenerate two wave mixing (DTWM) and degenerate four-wave mixings (DFWM), and performances in obtaining analytical or numerical solutions of DTWM and DFWM equations. The second part presents a brief summary of our recent investigations in this area. The third discusses some suggestions for future research activities and outlines the current status and progress.
The steady-state coupled wave equations describing anisotropic photorefractive four-wave mixing with absorption in cubic crystals are solved for the general case where the coupling constant is complex. The effects of absorption on the behaviour of the multivalued solutions that occur under these conditions are examined, and the consequences for the dynamical behaviour are discussed.
A set of coupled power equations that is applicable to the general case of a quasi-monochromatic light source is derived from the set of coupled amplitude equations for single-mode single polarization (SMSP). The spectral densities of Lorentzian-shaped and Gaussian-shaped distributions are characterized for the quasi-monochromatic light source. Our final results show that the power coupling coefficients are related not only to the birefringence of the SMSP fibers but also to the leaky-mode loss and the spectral half-width of the light source.
The kinetics of the phase conjugate beam generated by standard four-wave mixing in a BaTiO3 crystal has been investigated by means of an accurate numerical simulation. The transmission grating configuration has been considered and depletion of pumps was allowed. Special care has been taken in using suitable parameters for this material and consequently for the coupling coefficients γ and grating response times T. The influence on the reflectivity of the beam intensity ratios and coupling coefficients has been analysed. Oscillating as well as monotonic increasing kinetics has been found depending on the values of the coupling coefficients and the pump ratio. A comparison between the undepleted and depleted pump cases is performed finding that reflectivity oscillations are more pronounced for higher coupling coefficients and when the pumps were not depleted. Some comments on the origin of the oscillations will be presented.
We investigate the bistable and multistable response of the phase conjugate reflectivity (PCR) for the case of a linear photorefractive (PR) oscillator in presence of the moving grating operation. Numerical evaluation of stable multiple solutions of the PCR shows enhancement, bistable, multistable and self-oscillating behaviours with respect to the control parameters like the frequency-shift (Ω) due to grating motion, reflectivities (M 1 and M 2) of cavity mirrors, the coupling strength (of complex gL and real g 0L values) and the PR phase shift (PR). If the different branches are assumed to be stable, multistable but isolated branching behaviour with an increased PCR can be predicted. Results are interpreted physically by considering the competitions between (i) the strengths of two moving transmission gratings that arise from A1–A4 (forward oscillating pump-incident) and A2–A3 (backward oscillating pump-phase conjugate) interferences and (ii) the applied frequency detuning that causes the gratings to move and the self-induced cavity detuning that appears when the gratings are stationary. Numerical evidence of the results is presented graphically.
The optimal grating shift for four-wave mixing differs from that of two-wave mixing. With temperature control of this shift, the phase-conjugate reflectivity is increased to 400% in negative readout geometry. In the range of strong coupling the phase-shift control leads to bistable behavior of the four-wave mixing reflectivity.
Optical bistability based on the theory of photorefractive nearly degenerate four-wave mixing (DFWM) is considered for the case of transmission geometry with orthogonally polarized pump beams. The coupled wave equations are solved to obtain a nonlinear algebraic equation. This equation is being solved iteratively using Muller's method. The effect of different parameters pump ratio (r), probe ratio (q), coupling strength ( Gamma ) and frequency shift ( Omega ) on the phase conjugate reflectivity (PCR) have been investigated and the results are reported.
We establish possible regions of bistability (BIST), multistability (MUST), hysteresis, self-oscillation (SO) and enhancement features of the phase-conjugate reflectivity (PCR) by use of moving-grating operation (MGO) in the orthogonally polarized pump four-wave mixing (OPP-FWM) geometry employing a photorefractive (PR) crystal (usually BaTiO3). Numerical evaluation of such PCR features has been presented as a function of parameters such as coupling strength (of complex gL and real g
0
L values), frequency-shift () owing to grating motion, the PR phase shift (PR), and pump (r) and probe (q) intensity ratios. PCR results of this geometry are compared with those obtained from the regular (i.e. parallel-polarized pump) four-wave mixing (REG-FWM) geometry assuming the same set of parameters in both geometries. We find that the OPP geometry leads to a drastic PCR enhancement together with the appearance of a rich variety of multibranched solutions exhibiting BIST and MUST features of the PCR, particularly in the range 0.5 3 with values g
0L 10, r=1 and q 0.1 that are practically realizable. Nevertheless, this geometry sometimes offers the possibility of a PCR jump (from one stable state to the other), suggesting hysteresis caused by changing only slightly. In the REG geometry, however, the PCR enhancement is less prominent, and also the BIST/MUST features rarely exist unless g
0
L becomes sufficiently large, say 20 or more.
Optical multistability based on the theory of four-wave mixing (FWM) is considered for the case of reflection geometry in photorefractive crystals. The resultant nonlinear algebraic equation obtained from the initial nonlinear coupled differential equations, is solved numerically by using a successive bisection method which can predict repeated zeros (roots) as well as discontinuities (jumps). We show that apart from the effects of photorefractive phase shift, the strong nonlinear coupling itself is sufficient to bring about multivalued branching behaviour of the phase conjugate reflectivity (PCR). As the coupling strength increases, the solutions become richer and more complex. We find that for certain values of pump and probe ratios and owing to the nature of strong nonlinear coupling the appearance of multiple solutions has a close connection with the phenomenon of bistability and self-oscillation.
Large-amplitude oscillations in the phase conjugate intensify have been observed when one of the pumping beams of an externally pumped photorefractive phase conjugate mirror deviates slightly from the Bragg angle. The Fourier transform of the experimental time series data exhibits the broad frequency spread characteristics of a chaotic signal. A phase space plot is constructed, and is found to exhibit some of the features associated with strange attractors. The fractal dimension of the attractor is determined to be approximately 5.7, indicating chaotic behavior. The number of dynamical dimensions for this attractor is found from Takens' theorem to be approximately 13, considerably greater than what is predicted by the standard scalar four-wave mixing equations, and an explanation for such high dimensionality is presented in terms of a vector field description of photorefractive four-wave mixing. Additionally, evidence is presented to support the conclusion that the chaotic behavior results from the off-Bragg detuning of one of the pumping beams.
The stability of the steady-state solutions of four-wave mixing in photorefractive crystals has been studied. Two complementary methods have been used: linear stability analysis and time integration of the nonlinear equations. A critical dependence of the stability on the phase of the perturbation has been revealed. The existence of two stable solutions for certain values of some parameters has been found. Also, regimes of unstable behavior in cases with a unique steady-state solution for large values of the pump ratio have been found.
We show that the standard model of four-wave mixing in a photorefractive crystal predicts the appearance of deterministic chaos. In this model there is a single (transmission) grating and no external or internal (intracavity) feedback. The intensity of the phase-conjugate wave is found to exhibit a period-doubling route to chaos on variation of the intensity of the probe beam and the linear absorption coefficient. The crucial elements in obtaining chaotic behavior are operation above the threshold for self-oscillation and the presence of an external electric field, which causes a shift in the optical frequency of the phase-conjugate wave.
The development of a theory of four-wave mixing in photo-refractive crystals is described. This theory is solved in the undepleted pumps approximation with linear absorption and without using the undepleted pumps approximation for negligible absorption. Both the transmission and reflection gratings are treated individually. The results are used to analyze several photorefractive phase conjugate mirrors, yielding reflectivities and thresholds. The use of photorefractive crystals as optical distortion correction elements and experimental demonstrations of several of the passive phase conjugate mirrors are described.
The coupled wave equations for four-wave mixing (FWM) in a cubic, non-optically-active photorefractive crystal are solved for the case of orthogonally polarized pumping beams. For certain values of the pump and probe beam intensity ratios, the solutions exhibit a multi-valued branching behavior. If the different branches are assumed to be stable, bistability of the phase conjugate reflectivity is predicted. Furthermore, it is shown that this solution is valid for the case where the pump beams are parallel-polarized if the geometry of the four-wave mixing configuration is modified slightly. Our results are therefore applicable to crystals of other symmetry groups, such as barium titanate, using the modified FWM configuration.
This paper presents a review of photorefractive effects in electro-optic materials and their applications. Materials properties giving rise to increased photo-sensitivities are discussed and compared with experimental data for a series of newer materials. The formation of volume holograms in these crystals, and accompanying optical effects such as beam coupling and phase conjugation, are discussed. Theoretical results derived from charge transport parameters are compared with our original data obtained in “dynamic holography” experiments with Bi12GeO20, Bi12SiO 20 and KNbO3:Fe2+. The optimum parameters for applications such as volume hologram storage, coherent light amplification, optical phase conjugation etc. have been determined, and possible applications are discussed.
Volume phase hologram recording in pure LiNbO3 crystals with applied electric field E ≈ 5 kV/cm is accompanied by considerable intensity redistribution of the writing beams by transient energy transfer. Practically the whole intensity of the donor beam may be transferred into the weak acceptor beam with initial intensity 3% from total incident intensity. This effect originates from transient phase mismatch of the recorded phase grating and fringe pattern in the applied electric field when the interacting beams have different intensities.
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