1. The influence of backward Stokes on quasi-phase matched multiwave SRS in nonlinear periodical structures Victor G. Bespalov, Russian Research Center "S. I. Vavilov State Optical Institute" Nikolai S. Makarov, Saint-Petersburg State Institute of Fine Mechanics and Optics (Technical University)

2. Abstract In 1962 N. Bloembergen proposed the quasi-phase matched interaction in media with periodic variations in the second order nonlinearity (  (2) ) along longitudinal coordinate for efficient second harmonic generation [1] and then it was experimentally demonstrated. We offer to use the similar approach for anti-Stokes radiation generation in the medium with variable parameters of the third order Raman nonlinearity along longitudinal coordinate [2]. In previous publications we showed that our proposal provides the increasing of anti-Stokes SRS conversion efficiency. For each set of medium parameters we determined the optimum ratio between input pump and Stokes waves intensities at which the efficiency of energy conversion in anti-Stokes wave is maximal. We also describe the analytical model of four-wave quasi-phase matched SRS in silica fiber, where the efficiency of high-order Stokes and anti-Stokes components generation is very low. In this paper we analyze the influence of backward SRS on realization of quasi-phase matched conditions at multiwave SRS. We received a system of differential equations described multiwave backward and forward SRS, where E j  are complex amplitudes of interacting waves ( (+) corresponds forward components and (-) corresponds backward components), g j  are Raman gain coefficients, q  are complex amplitudes of phonon wave,  j =  0 +j ,  0 is the frequency of pump wave,  is the Raman shift frequency,  j i are the wave mismatchings and T 2  are the dephasing times. By numerical simulation we analyzed the influence of backward SRS on realization of quasi-phase matched conditions at multiwave SRS in hydrogen and barium nitrate. In our calculations we used systems up to 42 equations (pump, 10 Stokes and 10 anti-Stokes, all backward and forward). Our calculations have showed that for high precision it is required to take into account at least the generation of 3 Stokes and 3 anti-Stokes backward and forward components. We determined that at backward SRS the optimal ratio between input Stokes and pump waves intensities is changed. It is connected with changing of phase-matching conditions for interacting waves on layers input and output due to existence of backward waves. We discovered that taking into account the process of backward SRS results on slightly increasing of medium length. The obtained results of numerical simulations are promising for development and optimization of new effective nonlinear frequency up-converter devices. The influence of backward Stokes on quasi-phase matched multiwave SRS in nonlinear periodical structures; Munich, 22 – 27 June 2003 Makarov N.S., mak_nick@newmail.ruBespalov V.G., bespalov@admiral.ru

3. Principle of quasi-phase matching Raman active medium Nonlinearity  (2) Nonlinearity  (3) H2H2 H2H2 H2H2 H2H2  (3)  0  (3) =0 z I 2w LкLк d 31 c-axis LкLк The influence of backward Stokes on quasi-phase matched multiwave SRS in nonlinear periodical structures; Munich, 22 – 27 June 2003 Makarov N.S., mak_nick@newmail.ruBespalov V.G., bespalov@admiral.ru

4. Principle of quasi-phase matching at SRS Generalized phase on active layers input do not practically change, that in a final result provides a realization of quasi- phase matching conditions , rad  (3)  0  (3) =0 The influence of backward Stokes on quasi-phase matched multiwave SRS in nonlinear periodical structures; Munich, 22 – 27 June 2003 Makarov N.S., mak_nick@newmail.ruBespalov V.G., bespalov@admiral.ru

5. System of forward and backward multiwawe SRS equations  j i – wave mismatching, g j ± – steady-state Raman gain coefficient,  j – frequencies of interacting waves, E j ± – complex wave amplitudes The influence of backward Stokes on quasi-phase matched multiwave SRS in nonlinear periodical structures; Munich, 22 – 27 June 2003 Makarov N.S., mak_nick@newmail.ruBespalov V.G., bespalov@admiral.ru

6. Model verification: waves profiles at different input pump intensities (left – input pump and right – output pump) The influence of backward Stokes on quasi-phase matched multiwave SRS in nonlinear periodical structures; Munich, 22 – 27 June 2003 Makarov N.S., mak_nick@newmail.ruBespalov V.G., bespalov@admiral.ru t, ns I, GW/cm 2

7. The influence of backward Stokes on quasi-phase matched multiwave SRS in nonlinear periodical structures; Munich, 22 – 27 June 2003 Makarov N.S., mak_nick@newmail.ruBespalov V.G., bespalov@admiral.ru Model verification: waves profiles at different input pump intensities (left – output forward Stokes and right – output backward Stokes) t, ns I, GW/cm 2

8. Barium nitrate Hydrogen Raman gain dispersion The influence of backward Stokes on quasi-phase matched multiwave SRS in nonlinear periodical structures; Munich, 22 – 27 June 2003 Makarov N.S., mak_nick@newmail.ruBespalov V.G., bespalov@admiral.ru

9. Influence of high SRS components on calculations precision For best calculation accuracy it is necessary to take into account at least the generation of 3 Stokes and 3 anti-Stokes SRS components The influence of backward Stokes on quasi-phase matched multiwave SRS in nonlinear periodical structures; Munich, 22 – 27 June 2003 Makarov N.S., mak_nick@newmail.ruBespalov V.G., bespalov@admiral.ru

10. The influence of backward SRS on QPM realization (active layers length) The influence of backward Stokes on quasi-phase matched multiwave SRS in nonlinear periodical structures; Munich, 22 – 27 June 2003 Makarov N.S., mak_nick@newmail.ruBespalov V.G., bespalov@admiral.ru

11. The influence of backward Stokes on quasi-phase matched multiwave SRS in nonlinear periodical structures; Munich, 22 – 27 June 2003 Makarov N.S., mak_nick@newmail.ruBespalov V.G., bespalov@admiral.ru The influence of backward SRS on QPM realization (passive layers length)

12. Conclusions Our model of forward and backward multiwave SRS is quality and quantity compared with experimental results For best accuracy of QPM SRS simulations it is necessary to take into account the dispersion of Raman gain coefficient For studying of multiwave SRS influence on QPM structure realization it is necessary to take into account the generation at least of 3 Stokes and 3 anti-Stokes SRS components The influence of backward SRS on QPM structure realization results in the small difference between layers length of optimal QPM structure and small decreasing of resulting anti-Stokes conversion efficiency (~25% at backward and forward SRS, ~30% at forward SRS) The oscillations of optimal layers length are partially connected with backward SRS influence and with insufficient precision of layers length determination due to high computational complexity of this task The influence of backward Stokes on quasi-phase matched multiwave SRS in nonlinear periodical structures; Munich, 22 – 27 June 2003 Makarov N.S., mak_nick@newmail.ruBespalov V.G., bespalov@admiral.ru

13. References Armstrong J.A., Bloembergen N., Ducuing J., Pershan P.S. // Phys. Rev., 1962, 127, pp. 1918-1939. Bespalov V.G., Makarov N.S. Quasi-phase matching generation of blue coherent radiation at stimulated Raman scattering // Optics Communications 2002, 203 (3-6), pp. 413-420. Maier M., Kaiser W., Giordmaine J.A. Backward stimulated Raman scattering // Phys. Rev., 1969, V. 177, №2, pp. 580-599. Raijun Chu, Morton Kanefsky, Joel Falk Numerical study of transient stimulated Brillouin scattering // J. Appl. Phys., 1992, V. 71, №10, pp. 4653-4658. Zaporozhchenko R.G., Kilin S.Ya, Bespalov V.G., Stasel’ko D.I. Formation of the spectra of backward stimulated Raman scattering from the quantum noise of polarization of a scattering medium // Opt.&Spectr., 1999, V. 86, №4, pp. 632-639. Bischel W.K., Dyer M.J. Wavelength dependence of the absolute Raman gain coefficient for the Q(1) transmission in H2 // J. Opt. Soc. Am. B, 1985, V. 3, pp. 677- 682. The influence of backward Stokes on quasi-phase matched multiwave SRS in nonlinear periodical structures; Munich, 22 – 27 June 2003 Makarov N.S., mak_nick@newmail.ruBespalov V.G., bespalov@admiral.ru