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S.Bugaychuk(1), G.Mandula(2), L.Kovács(2), R.A.Rupp(3)
(1)Institute of Physics, Prospect Nauki 46, Kiev 03028, Ukraine, e-mail: bugaich@iop.kiev.ua (2)Research Institute for Solid State Physics and Optics, Konkoly-Thege M.út 29-33, H-1121 Budapest, Hungary, e-mail: mandula@szfki.hu (3)University of Vienna, Institute for Experimental Physics, Strudlhofgasse 4, A-1090 Vienna, Austria, e-mail: rrupp@ap.univie.ac.at There are a lot of well-known applications of the sine-Gordon equations including the field theory, differential geometry, and the Josephson junction. In the present work, we report that the problem of optical dynamic holography in a medium with a non-local response can be reduced to a nonlinear equation of the sine-Gordon family [1,2]. Namely, in the transmission geometry it is a non-local sine-Gordon equation that contains an additional term of the first order derivation on the spatial coordinate. (In the reflection geometry it is a similar non-local tanh-Gordon equation.)The function of the equation is the integral under spatial distribution of the dynamic grating amplitude. By numerical simulation, we have found two types of solutions for the grating amplitude spatial profile in dependence of the input conditions. They are (i) a single localized state, and (ii) auto-oscillations between two localized states. In the steady state, the equation has localized solutions in the form of the secance hyperbolic function for the grating amplitude profile. The parameters of this profile (the maximum amplitude, the localization degree, the position of the maximum) are determined completely by the input intensity ratio, input wave phase difference and the photorefractive gain of the crystal. This result shows possibilities of all-optical control of the grating amplitude distribution and this way, the output wave characteristics, since the output intensities are found as a sinus function from the integral under the grating amplitude profile over the crystal boundary [3]. In a medium with complex response (containing both local and non-local components), we obtain solutions separately for the grating amplitude and for the grating phase. The solution for the grating amplitude distribution depends on both components in a transient case. It depends only on the non-local gain constant in the steady state because in this case the grating phase is assumed to be constant. We perform the results of the first experiments made with the help of the optical topography set-up in photorefractive crystals that prove a non-homogeneous distribution of the grating amplitude profile inside the crystal volume and a control of this distribution by changing input intensity ratio. We obtain experimentally auto-oscillations of the output intensities in the four-wave mixing experiment in accordance with the predictions of our non-local sine-Gordon equation. In addition to the traditional amplification of optical signals, the provided researches open a view for many new potential applications of the dynamic holography, which utilize the non-local response. For example, they can be: formation localized gratings to increase optical information density; all-optical control of output wave and time characteristics versus input intensities and phases of the beams; optimization of the parameters at the optical phase-conjugation; creation of auto-oscillations. As well as an experimental measurement of the distribution of the grating amplitude strength becomes an important parameter for material characterization by the holographic technique. In particular, with the help of this additional parameter it is possible to calculate separately the local component of the response and the non-local one both in the steady state and in the transient process. [1] S. Bugaychuk, L. Kovács, G. Mandula, K. Polgár, R. A. Rupp, "Nonuniform dynamic gratings in photorefractive media with nonlocal response", Phys. Rev. E 67 (4), 046603 (2003). [2] M. Jeganathan, M. C. Bashaw, and L. Hesselink, "Evolution and propagation of grating envelopes during erasure in bulk photorefractive media", J. Opt. Soc. Am. B 12, 1370 (1995). [3] S. Bugaychuk, R. A. Rupp, G. Mandula, L. Kovács, "Soliton profile of the dynamic grating amplitude and its alteration in photorefractive wave mixing", TOPS Proc. "Photorefractive Effects, Materials, and Devices" 87, 404 (2003). |
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