Study On Nonlinear Phenomenon Of Beam Propagation In Photorefractive Medium

Two kinds of photorefractive medium are studied, which are logarithmicallyphotorefractive medium and semiconductor photorefractive medium. The nonlinear quantumtunneling and the self-trapping phenomenon of beam propagation in the two kinds ofphotorefractive medium are studied. In this paper, two forms of nonlinear quantum tunnelingare studied, which are the nonlinear Landau-Zener tunneling and the nonlinear Rosen-Zenertransition. The difference of the two forms of quantum tunneling is that the former is a changein the energy gap, which the latter is fixed.The study on the nonlinear quantum tunneling and the self-trapping phenomenon ofbeam propagation in photorefractive medium need to use the two-level model. The nonlinearterms of the logarithmically and semiconductor photorefractive medium are very complx, soit is very difficult to deduce the two-level model and before there is not relevant report.According to the dimensionless paraxial evolution equation of electric field amplitude ofbeam propagation in the photorefractive medium, we deduce the two-level model of the twokinds of photorefractive medium and get the matrix form of the Hamiltonian. The classicalcanonical equation of Hamilton and the corresponding classical Hamiltonian are derivedbased on the two-level model.The phase spaces of logarithmically and semiconductor photorefractive medium areanalyzed. According to the classical canonical equation of Hamilton, we solve the fixed pointsof the phase space and analyze the stability of Hamilton-Jacoby matrix. We find there are twoellipse fixed points in the photorefractive medium and when certain conditions are satisfied,one of the ellipse fixed point becomes hyperbolic fixed point, which two ellipse fixed pointsare appeared in the upper and lower of. We calculate the critical value of topology changesand find two kinds of self-trapping:1) Both the population difference and the relative phase inenergy levels oscillate near an equilibrium point in the phase space;2) The populationdifference in energy levels oscillate near an equilibrium point while the relative phaseincreases monotonously. We numerically solve the classical canonical equations of Hamiltonby the C programming language to gain the corresponding data and make the space phasediagram of Hamiltonian, using Origin software. We analyze the nonlinear Rosen-Zener transition in the photorefractive medium through the space phase diagram and verify the frontresult.For the logarithmically photorefractive medium, we plus periodic modulation in thecoupling coefficient to analyze the impact to the nonlinear Rosen-Zener transition and theself-trapping phenomenon. We find in the high frequency modulation, compared with withoutmodulationthe, effective coupling constant is only more than a factor of0.5and the systemdoes not have specific threshold value in low frequency modulation and intermediatefrequency modulation. For the low frequency modulation, the system must be in the nonlinearRosen-Zener transition on a large scale, when the coupling coefficient is greater than0.5359.No matter how big the coupling coefficient is, there is always some range to make the systembe in the the self trapping on a small scale.For the semiconductor photorefractive medium, we plus periodic modulation in theenergy gap to analyze the impact to the nonlinear Rosen-Zener transition and the self-trappingphenomenon. We find in the high frequency modulation, compared with withoutmodulationthe, the effective coupling constant is only more than a constant term of zero-orderBessel function. Because the zero-order Bessel function is less than1, when the nonlinearintensity is weak, the self-trapping phenomenon can also occur, at the same time we cancontrol the self-trapping phenomenon by adjusting the ratio of the modulation intensity andmodulation frequency; In low-frequency modulation, only when the nonlinear intensity isgreater than a certain value associated with modulation strength, the self-trappingphenomenon occurs. It makes the self-trapping phenomenon not more likely to occur. Thesituation of intermediate frequency modulation is more complicated, for a while as atunneling, while for the self trapping.The effect of quantum fluctuation on the nonlinear tunneling and the self trapping of thelogarithmically photorefractive medium is studied. We find that it is necessarily tunneling invery large time scale, regardless of whether the modulation or what kind of modulation. Whenthe coupling coefficient is small, the layout of the rate difference infinite approacoaches0. Inthe time scale is not too big, for the high frequency modulation and without modulationthe,there is not the critical value between self trapping and tunneling but the critical area, and thecritical area is near the critical value of the two kinds of self trapping in the mean-field approximation. For the low-frequency modulation and intermediate frequency modulation,when the coupling coefficient is increased, the layout of the rate difference is not unlimitedapproaches0, but from the oscillation of+1down to-1, then an increase. When the couplingcoefficient increases to a certain extent, it can no longer as self trapping. In this process, thelow frequency modulation is changed gradually, and the intermediate frequency modulation ismutant.