Cascaded induced lattices in quadratic nonlinear medium

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1 Cascaded induced lattices in quadratic noinear medium Olga V. Borovkova* a, Valery E. Lobanov a, natoly P. Sukhorukov a, and nna K. Sukhorukova b a Faculty of Physics, Lomonosov Moscow State University, Leninskie Gori, Moscow 999; b Russian State Geological Prospecting University, Mikluho-Maklaya st.,, Moscow 787 BSTRCT We first study features of optical periodic lattices generated with mismatched cascade three-wave interaction in quadratically noinear media. We elaborate the theory of parametric waveguides arrays induced by two crossing pump waves together with exited sum wave. s the signal wave spreads due to the diffraction the induced lattice appears and its transverse dimension increases. Note the parametric periodic grating becomes apparent since launching any signal beam. It s likely to be the leading peculiarity of cascaded induced lattices. Parametric inhomogeneity depends on wave vector mismatch sign, and its modulation depth can be controlled by pump beam intensity. We observed a transformation from the discrete diffraction into the waveguiding of one or several signal beams with the increasing of pump intensity. The discrete diffraction dynamics dependence on pump intensity, spatial period, and signal beam tilting is analyzed when one or few central waveguides are exited at the input. t the certain incidence diffractioess propagation of signal beam takes place. The similar discrete diffraction effects in D cascaded lattices with various pump structure geometries have been studied. The additional degree of freedom gives novel properties to the effect of discrete diffraction. Keywords: cascade parametric process, induced waveguides, discrete diffraction. INTRODUCTION The modern progress tendencies of the telecommunication systems make very high requirements to the data transfer rate. In this case the operation speed plays a key role. The electronic and optoelectronic switching systems are used in the modern data transfer channels. But all of them are inappropriate for the ultrafast switching implementation (from several THz upwards). Such processing speed could oy be reached by using all-optical switching devices. The diminutiveness and tunability should also be mentioned as the advantages of such systems. set of coupled tunnel waveguides is acceptable for the all-optical switching device implementation. The various effects which are absent in common homogeneous noinear media could be observed in coupled waveguides. The most interesting of them is the phenomenon of anisotropic diffraction when the properties of the diffraction depend on beam propagation direction. Such sets of waveguides or periodic structures are created in crystals by lithography, and material modification, etc -. But the artificial waveguides have unchangeable properties therefore it is difficult to work with them. Instead of the artificial arrays the induced periodic lattices could be used. They are produced by modulation of linear or noinear part of the refractive index and therefore the parameters of such arrays could be controlled easily by the varying of the pump parameters. Induced lattices are usually created in media with cubic noinearity, such as photorefractive and nematic liquid crystals 4-7. However mentioned crystals have extremely large relaxation time. That s why the medium with an electronic noinearity as quadratic one is preferable for ultrafast all-optical switching. The effective third order noinearity is created by a cascade three-wave interaction in such crystals 8 in which the intense pump, signal and sum waves participate. In this article the formation of the cascaded induced lattice is described and the dynamics of the anisotropic discrete diffraction is investigated. It is interesting to note that parametric periodic grating becomes apparent since launching any signal beam. It s likely to be the leading peculiarity of cascaded induced lattices. It is shown that while pump intensity borovkovaolga@yahoo.co.uk Photon Management III, edited by John T. Sheridan, Frank Wyrowski Proc. of SPIE Vol. 6994, 69940, (008) X/08/$8 doi: 0.7/ SPIE Digital Library -- Subscriber rchive Copy Proc. of SPIE Vol

2 increases discrete diffraction turns into waveguiding and then signal beam propagates without changing its amplitude envelope. We analyze the discrete diffraction phenomenon depending on pump intensity, spatial period, and signal beam tilting when one or few central waveguides are excited at the input. t the certain incidence the diffractioess propagation of signal beam occurs 9. FORMTION ND EVOLUTION OF CSCDED INDUCED LTTICES We study the noncollinear three-wave interaction of waves in quadratic crystal. ω is the pump wave frequency, ω is the signal wave frequency and ω = ω + ω relates to the sum wave. The intensity of the pump beam is much higher then the intensity of signal and sum waves. Consequently the feedback of weak signal and sum waves to the pump could be neglected. The evolution of wave complex envelopes ( x, y, z), =, during their propagation along the z axis (scaled to the certain length L) is described by the equations: z z + i D * + i D = i γ = 0, (), () + i z D + i k = i γ, () where - Laplacian in transverse coordinates x and y scaled to the width of the pump lattice site a ; D = L / k a diffraction coefficient; k wave number; γ = β I 0 L normalized coefficient of noinearity; β dimensional coefficient of noinearity; I 0 pump intensity at the input; γ + γ = γ ; = k + k k wave vector mismatch along z axis. k z z z First of all we consider the phenomenon of discrete diffraction in D cascade induced lattice. There are two steps of the cascaded lattice formation. t the first stage the bulk lattice is created by the interference of two waves crossed at an angle of φ: i k ϕ z ϕ. (4) ( x) = cos( ) exp 0 k x If the following condition k >> γ is satisfied (it means that the value of the wave vector mismatch is large), the 0 cascade three-wave interaction makes the medium optically inhomogeneous for the signal wave and leads to the periodic modulation of the refractive index at the signal frequency: The negative magnitude of the mismatch ( < 0) positive magnitude ( > 0) n = 4[ γ γ / ( k k)] cos ( k ϕ x). (5) 0 k corresponds to the focusing cascaded noinearity ( n > 0) k corresponds to the defocusing cascaded noinearity ( n < 0) ; and the. The area nearby the maximum of the refractive index is analogously to the single waveguide. Therefore the induced lattice could be used as the set of coupled waveguides. Proc. of SPIE Vol

3 We choose the lattice period Λ = π /( k ϕ) so that several periods of the lattice are equal to the width of the crystal and therefore the stationary wave is formed in the transverse location along X axis. Such stationary wave is stable and not experienced by the diffraction distortion. Then a narrow signal beam and the pump beam in their intersection area excite the sum wave. t the cascade interaction self-action the process of the signal wave propagation is described by the single equation for signal wave envelope taking into account the induced modulation of the refractive index: z + i D = i k n ( x). (6) The leading peculiarity of cascaded induced lattices is the fact that the parametric periodic grating becomes apparent since launching any signal beam. It can be explained by the fact that the first step in the cascade concerns with the generation of the sum wave = ( γ / k ) ( x). nd ust after it both the sum wave and the pump change the signal wave refraction index, see (4, 6). DISCRETE DIFFRCTION ND WVEGUIDING IN D INDUCED LTTICE We consider the process when central waveguides of the lattice are excited by the Gaussian signal beam: ( x) = exp( x / ). (7) 0 a t the average depth of the refractive index modulation the discrete diffraction of the signal beam is observed (Fig. a). While pump beam intensity increases the depth of the lattice modulation rises and the signal wave propagates through the central waveguides without spreading, with the constant amplitude profile (Fig. b). It may be named as the waveguiding of the signal beam. Let s consider it in details p ooi a). Fig.. The discrete diffraction of the signal beam (a) degenerates into the waveguiding (b) in D cascade induced lattices as the pump wave intensity becomes four times more. 0.08: b). For the sake of simplicity we will consider that the signal beam excites oy one central waveguide and is localized in it. We consider two pump beams to describe the spatial mode. We analyze beams with the rectangle amplitude profile: Proc. of SPIE Vol

4 Λ a Λ+ a ( x) = 0 = const < x < Λ a Λ+ a ( x) = < x < x >,, 0, 0,, where a is the width of each of the pump beams. The fundamental waveguide mode localized between two sites of the induced lattice has the amplitude profile that is described by the following formula: = 0 cos( q D x), (9) where q is the noinearity induced shift of the signal wave number determined by the dispersion equation γγ 0 Λ a q = cos q D. (0) k If 0 = 0 (there is no pump wave), q = 0 i.e. the medium is homogeneous and the mode is delocalized (9). When the pump wave amplitude is increasing the depth of the induced lattice modulation (5) is increasing also and shift of the wave number q is tending to its limit q π 4D lim = ( Λ a) (8). () For instance, the diffraction coefficient of the medium is D = 0, 5, the coefficient of the noinearity is γγ 0 > 0, k the induced lattice has period Λ = and one site width a = 0, 5. The limiting value of the wave number shift is q lim = π 9,87. The result of the numerical modeling of the signal waveguiding for these values of parameters is shown on the Fig. b. DISCRETE DIFFRCTION OF TILTED SIGNL BEM Heretofore we have assumed that the signal beam propagates along axis z, i.e. the signal beam wave vector is parallel to axis z. But there are also intriguing effects when the signal beam enters the lattice at the angle of θ to the axis z: x + θ ( x) = 0 exp ik x. () a If one compares it with the well-known sets of coupled artificial waveguides it becomes obvious that the induced lattice is anisotropic. It means that the longitudinal component of the wave vector k z depends on the transverse one k x, in particular k z cos ( k x Λ). If the signal beam enters the lattice at the angle of θ to the axis z then k x = k θ. In this case the coefficient of the discrete diffraction is determined by the following relation 6 ( ) D = D 0 cos k θ Λ, () where D 0 is the coefficient of the discrete diffraction when the signal beam propagates along axis z. Consequently the discrete diffraction disappears, D = 0, if = ± π k Λ. (4) ( ) θ Proc. of SPIE Vol

5 It corresponds to the diffractioess propagation of the tilted signal beam, when its transverse dimension is constant (Fig. ). It is interesting to note that the direction of the signal wave propagation does not coincide with its wavefront incline at the input and depends on the refractive index modulation depth x Fig.. Diffractioess propagation of the signal beam. z The dependence of the magnitude of the diffraction spreading of the signal beam on the wavefront incidence angle at the input is presented on the Fig.. One can see that the minimum of the diffraction spreading is in the agreement with the theoretical condition (4). relative (Ci spreadinç P c) C-), C, C) (C b ---- N) C) -J cc (C I I. I I I I of the signal beam. Fig.. The dependence of the magnitude of the diffraction spreading of the signal beam on the wavefront incidence angle at the input. Proc. of SPIE Vol

6 D CSCDE INDUCED LTTICES ll effects described above correspond to plane geometry. The numerical modeling of the discrete diffraction in bulk cascaded induced lattices is also of great interest. The additional degree of freedom gives novel properties to the effect of discrete diffraction. We have performed numerical modeling of D bulk lattice formed by the interference of four waves crossed in pairs at the angle of φ and symmetric relative to the axis z: ϕ i k ϕ z. (5) ( x, y) = 4 cos( ) cos( ) exp 0 k x kϕy Similar to the planar lattice (6) due to the cascade interaction the periodic modulation of the pump wave amplitude leads to the periodic modulation of the refractive index at the signal wave frequency: γ γ 6 0 cos ( kϕx) cos ( k ϕy). (6) k k n = If we excite central waveguides of D lattice the discrete diffraction in the bulk array is observed, see Fig. 4a. s in the D case if pump intensity increases the discrete diffraction turns into the waveguiding of the signal beam (Fig. 4b). Such effects in D arrays require greater value of the pump intensity then in D lattices. '' l(} v 6 a). 4 b). Fig. 4. The intensity profiles at the output in the case of discrete diffraction (a) and waveguiding (b) in D cascade induced arrays. Proc. of SPIE Vol

7 We have made the following estimations for the parameters of optical beams required for the experimental research of the described effects. The crystal of LiNbO could be chosen as the noinear medium. Its length is about 4 cm. s a source of pump laser beam with the width a =0 µm and the wave length λ =.06 µm is suitable. The ratios between waves frequencies are following: ω = ω, ω = ω. Thus the diffraction length is L d = cm; the pump power density at the axis is I 0 =680 MW/cm ; the noinear length is L =γ 0 L= mm. CONCLUSION The periodic lattices cascaded induced in quadratic medium have all properties of ordinary artificial optical periodic structures. It was revealed theoretically and numerically that the effects of discrete diffraction, diffractioess propagation and waveguiding of the signal beam could be observed in such lattices. Induced structures have undoubted advantages as the tunability of their parameters. The modulation depth and the period of the structure could be easily changed by the varying of the amplitude and the incidence angle of the pump beams. The phenomenon of discrete diffraction can be applied for the multiplexing of the signal in photonic devices. CKNOWLEDGEMENTS This work is partially supported by the Russian Foundation for Basic Research (proects no and ) and by the Russian Federation President s grants ( Leading Scientific Schools no. NSh and Support of Young Scientists no. MK-497.). Olga Borovkova and Valery Lobanov also acknowledge the financial support of The Dynasty Foundation. REFERENCES [] [] [] [4] [5] [6] [7] [8] [9] Lan, S., Del Re, E., Chen, Z., Shih, M., and Segev, M., Directional coupler with soliton-induced waveguides, Opt. Lett. 4, (999). Sukhorukov,.P., Chuprakov, D.., Optical Spatial Structures in a quadratically noinear medium, Laser Physics 5(4), (005). Guo,., Henry, M., Salamo, G.J., Segev, M., and Wood, G.L., Fixing multiple waveguide induced by photorefractive solitons: directional couplers and beam splitters, Opt. Lett. 6(6), (00). Petter, J., Schroder, J., Trager, G., and Denz, C., Optical control of arrays of photorefractive screening solitons, Opt. Lett. 8(6), (00). Martin, H., Eugenieva, E.D., Chen, Z., and Christodoulides, D.N., Discrete solitons and soliton-induced dislocations in partially-coherent photonic lattices, Phys. Rev. Lett. 9(), (004). Eisenberg, H.E., Silberberg, Y., Morandotti, R., Boyd,.R., and itchison, J.S., Discrete spatial optical solitons in waveguide arrays, Phys. Rev Lett. 8(6), 8-86 (998). Fleische, J.W., Segev, M., Efremidis, N.K., and Christodoulides, D.N., Observation of two-dimensional discrete solitons in optically induced noinear photonic lattices, Nature 4, (00). Lobanov, V.E., Sukhorukov,.P., Parametric reflection of wave beams under mismatched three-frequency interaction, BRS 69(), (005). Pertsch, T., Zentgraf, T., Peschel, U., Brauer,., and Lederer F., nomalous refraction and diffraction in discrete optical systems, Phys. Rev. Lett. 88(9), (00). Proc. of SPIE Vol