Modulational instability of optical beams in photorefractive media in the presence of two wave and four wave mixing effects
aa r X i v : . [ n li n . PS ] J un Modulational instability of optical beams inphotorefractive media in the presence of two waveand four wave mixing effects
C P Jisha, V C Kuriakose, K Porsezian and B Kalithasan Department of Physics, Cochin University of Science and Technology, Kochi-682 022 Department of Physics, Pondicherry University, Pondicherry - 605014E-mail: [email protected]
Abstract.
Modulational instability in a photorefractive medium is studied in thepresence of two wave mixing. We then propose and derive a model for forward fourwave mixing in the photorefractive medium and investigate the modulational instabilityinduced by four wave mixing effects. By using the standard linear stability analysisthe instability gain is obtained. In both the cases, the geometry is such that the effectof self-phase-modulation self-focusing is suppressed and only the holographic focusingnonlinearity is acting. odulational instability in photorefractives
1. Introduction
A soliton is a localized wave that propagates without any change through a nonlinearmedium. Such a localized wave forms when the dispersion or diffraction associatedwith the finite size of the wave is balanced by the nonlinear change of the properties ofthe medium induced by the wave itself [1]. Solitons are ubiquitous in nature and havebeen identified in various physical systems like fluids, plasmas, solids, matter waves andclassical field theory. Such self-trapped beams are the building blocks in future ultra-fast all-optical devices. They can be used to create reconfigurable optical circuits thatguide other light signals. Circuits with complex functionality and all-optical switchingor processing can then be achieved through the evolution and interaction of one or moresolitons [2]. It is possible to compress circuitry into a compact space with many circuitssharing the same physical location. Furthermore, certain photosensitive materials offerthe potential for erasing one light written device and replacing it by another. Hence,we have the building blocks for dense reconfigurable virtual circuitry. Due to thedevelopment of materials with stronger nonlinearities the optical power needed to createsuch virtual circuits has been reduced to the milliwatt and even microwatt level, bringingthe concept nearer to practical implementation. In a recent experimental study [3],optical spatial screening solitons have been observed for the first time in strontiumbarium niobate (SBN). Soliton states of this sort are known to occur when the processof diffraction is exactly balanced by light-induced photorefractive (PR) waveguiding [4].Modulational instability (MI) is a process in which tiny phase and amplitudeperturbations that are always present in a wide input beam grow exponentially duringpropagation under the interplay between diffraction (in spatial domain) or dispersion(in temporal domain) and nonlinearity. Instabilities and chaos can occur in manytypes of nonlinear physical systems. Optical instabilities can be classified as temporaland spatial instabilities depending on whether the electromagnetic wave is modulatedtemporally or spatially after it passes through the medium. Temporal instability hasbeen studied by various authors [5, 6, 7], and the first experimental observation of MI ina dielectric material was in 1986 [8]. The temporal MI occurs as an interplay betweenself-phase modulation and group velocity dispersion. In optical fibers, MI occurs inthe negative dispersion region and is responsible for the breakup into solitons. In thespatial domain, diffraction plays the role of dispersion. When both diffraction anddispersion are present simultaneously, it results in spatio-temproral MI [9]. Recently,Wen et al. investigated MI in negative refractive index materials [10]. In a rather loosecontext, MI can be considered as a precursor of self-trapped beam formation. In thespatial domain, MI manifests itself as filamentation of a broad optical beam throughthe spontaneous growth of spatial-frequency sidebands. The MI is a destabilizationmechanism for plane waves. It leads to delocalization in momentum space and, in turn,to localization in position space and the formation of self-trapped structures. DuringMI, small amplitude and phase perturbations tend to grow exponentially as a result ofthe combined effects of nonlinearity and diffraction. This results in the disintegration of odulational instability in photorefractives
3a large-diameter optical beam during propagation. Such instabilities have been widelystudied in photorefractive media [11, 12, 13]. Assanto et al. [14] investigated transverseMI in undoped nematic liquid crystals, a highly nonlocal material system encompassinga reorientational nonlinear response. They observed the one-dimensional development oftransverse patterns which eventually lead to beam breakup, filamentation, and spatialsoliton formation, using both spatially coherent and partially incoherent excitations.The MI phenomena has been previously observed in various media like Kerr media[15], electrical circuits [16], plasmas [17], parametric band gap systems [5], quasi-phase-matching gratings [18], discrete dissipative systems [19] and in PR polymers [20]. Thetransverse instability of counterpropagating waves in PR media is studied by Saffmanet al [21]. From the above investigations, it is clear that the study of MI in a mediumis both of fundamental as well as of technological importance.In this work, we investigate the modulational instability induced in aphotorefractive medium by two wave mixing and the forward four wave mixing occurringin the PR material. The photorefractive medium is a well suited medium for thepractical implementation of virtual circuits as self-trapping of beams can be observedin this medium at very low laser powers. Four wave mixing in the phase conjugategeometry is widely used in photorefractive materials for various applications. It can beused to implement several different computing functions [22], optical interconnects [23],matrix addition [24], and optical correlator [25]. Nonlinear solutions for photorefractivevectorial two-beam coupling and for forward phase conjugation in photorefractivecrystals have been found in [26]. Recently Jia et al. [27] experimentally demonstrateddegenerate forward four wave mixing effects in a self-defocusing PR medium, in both oneand two transverse dimensions. They observed the nonlinear evolution of new modes asa function of propagation distance, in both the near-field and far-field (Fourier space)regions.The motivation behind studying MI in such a geometry is due to the recent proposalby Cohen et al. [28] of a new kind of spatial solitons, known as the holographic (HL)solitons. They are formed when the broadening tendency of diffraction is balanced byphase modulation that is due to Bragg diffraction from the induced grating. Holographicsolitons are solely supported by cross-phase modulation arising from the inducedgrating, not involving self-phase modulation at all. In 2006 [29], they showed thatthe nonlinearity in periodically poled photovoltaic photorefractives can be solely of thecross phase modulation type. The effects of self-phase modulation and asymmetricenergy exchange, which exist in homogeneously poled photovoltaic photorefractives,can be considerably suppressed by the periodic poling. They demonstrated numericallythat periodically poled photovoltaic photorefractives can support Thirring-type (solitonswhich exist only by virtue of cross phase modulation) (holographic) solitons. HL solitonsin PR dissipative medium was studied by Liu [30]. Existence of HL solitons in agrating mediated waveguide was studied by Freedman et al. [31]. Salgueiro et al.[32] studied the composite spatial solitons supported by mutual beam focusing in aKerr-like nonlinear medium in the absence of the self-action effects. They predicted the odulational instability in photorefractives . Na . Sr . Ba . Nb O crystal in which twocoherent laser beams, a signal beam, as well as a strong and uniform pump beam at 532nm are coupled to each other via two-wave mixing [33].This paper is organized as follows. The basic propagation equation for the twowave mixing geometry is presented in Section 2 and the modulational instability inthis geometry is studied. In section 3 the governing equations for the forward four wavemixing geometry is presented. The system is studied without using the undepelted pumpbeam approximation. The standard linear stability analysis for the coupled equationsis carried out and the gain spectrum is obtained. Section 4 concludes the paper.
2. Two wave mixing geometry
In this section, we present the derivation of the model equation for the two wave mixinggeometry [34, 35] and study the MI in it. Consider the interaction of two laser beamsinside a photorefractive medium. Stationary interference pattern is formed, if the twobeams are of the same frequency.Let the electric field of the two beams be written as E j = A j exp[ i ( ωt − k j . r )] , for j = 1 , . (1)Here, A and A are the amplitudes, ω is the angular frequency, and k and k are thewave vectors.The medium is assumed to be isotropic and both beams are polarized perpendicularto the plane of incidence. The total intensity of the beams I = | E | = | E + E | , (2)can be expressed as I = | A | + | A | + A A ∗ exp[ i K . r ] + A A ∗ exp[ − i K . r ] , (3)where K = k − k . The magnitude of the vector K is 2 π/ Λ where Λ is the period of the fringe pattern.Equation 3 represents a spatial variation of optical energy in the photorefractivemedium. Such an intensity pattern will generate and redistribute charge carriers. As aresult, a space charge field is created in the medium. This field induces an index volumegrating via the Pockels effect. In general, the index grating will have a spatial phaseshift relative to the interference pattern.The index of refraction including the fundamental component of the intensity-induced gratings can be written as n = n + (cid:20) n A ∗ A I exp[ iφ ] exp[ − i K . r ] + cc (cid:21) , (4) odulational instability in photorefractives n is the index of refraction when no light is present, φ is real and n is a realand positive number. The wave equation reduces to the Helmholtz equation for highlymonochromatic waves like laser beams which can be viewed as a superposition of manymonochromatic plane waves with almost identical wave vectors ∇ E + ω n E/c = 0 , (5)where E = E + E .Let E j = A j ( x, y, z ) exp[ iωt − iβ j z ] , where A ( x, y, z ) is the complex amplitude that depends on position, β and β arethe z component of the wave vectors k and k inside the medium respectively and z ismeasured along the central direction of propagation. We solve for the steady state sothat A j is taken to be time independent, so that equation (5) becomes (cid:16) ∂ ∂x + ∂ ∂y + ∂ ∂z + ω n /c (cid:17) E = 0 . (6)Substituting equation (4) in equation (6) and solving by neglecting the second order term n and using the fact that for highly directional monochromatic waves with a >> λ , ∂ A/∂z can be neglected, we get2 iβ ∂A ∂z = ∇ ⊥ A + ω n n c I e − iφ A ∗ A A , (7)and 2 iβ ∂A ∂z = ∇ ⊥ A + ω n n c I e − iφ A ∗ A A . (8)For the case when the two laser beams enter the medium from the same side at z = 0 β = β = k cos θ = 2 πλ n cos θ. where, 2 θ is the angle between the beams inside the medium. Simplifying we obtain i ∂A ∂z = 12 k cos θ ∇ ⊥ A + Γ2 I A | A | , (9 a ) i ∂A ∂z = 12 k cos θ ∇ ⊥ A + Γ2 I A | A | , (9 b )where Γ = 2 πn λ cos θ e − iφ (10)is the complex coupling coefficient. A similar model was used to study the existenceof Holographic solitons [28]. A prerequisite for obtaining Holographic focusing is thatthe induced grating be in phase with the intensity grating. If the induced grating isshifted by π , then the grating leads to holographic defocusing. If the grating is ± π/ odulational instability in photorefractives φ = 0 for the existence of brightsolitons. This gives the coupling constant asΓ = 2 πn λ cos θ . (11)The next step is to study the propagation of a broad optical beam through a PRmedium. We study MI of a one dimensional broad optical beam. Hence the y dependentterm in the transverse Laplacian in equation (9 a ) can be neglected. For a broad opticalbeam, the diffraction term in equation (9 a ) can be set to zero giving the steady statesolutions as A j = √ P exp h − i Γ P I z i , (12)for j = 1 , A j = ( √ P + a j ) exp h − i Γ P I z i . (13)Substituting in (9 a ) and neglecting the quadratic and higher order terms in a j , theperturbations a and a are found to satisfy the following linearized set of two coupledequations : i ∂a ∂z = 12 k cos θ ∂ a ∂x + Γ P I ( a + a ∗ ) , (14) i ∂a ∂z = 12 k cos θ ∂ a ∂x + Γ P I ( a + a ∗ ) . (15)It is important to note that the evolution of the perturbations depend solely onthe cross phase modulation. To solve equations (14) and (15), we assume that theperturbations be composed of two side bands : a j ( x, z ) = U j ( z ) exp[ iκx ] + V j ( z ) exp[ − iκx ] . (16)The substitution of equation (16) in (14) and (15) results in a set of fourhomogeneous equations in U , U , V and V as ∂U ∂z = i κ k cos θ U − i Γ P I ( U + V ∗ ) , (17) ∂V ∗ ∂z = − i κ k cos θ V ∗ + i Γ P I ( U + V ∗ ) , (18) ∂U ∂z = i κ k cos θ U − i Γ P I ( U + V ∗ ) , (19) ∂V ∗ ∂z = − i κ k cos θ U + i Γ P I ( U + V ∗ ) . (20)This set has a nontrivial solution only if the determinant of the coefficient matrixvanishes. The eigenvalues of the system are obtained asΛ ± = ± A (cid:16) − I κ − A Γ P κ I (cid:17) / , (21)Λ ± = ± A (cid:16) − I κ + 2 A Γ P κ I (cid:17) / , (22) odulational instability in photorefractives -15 -10 -5 5 10 15 Κ H Κ L Figure 1.
Growth rate as a function of spatial frequency for the two wave mixinggeometry. where A = k cos θ . The plane wave solution is stable if perturbations at any wavenumber κ do not grow with propagation. MI gain will exist only when Re [Λ] >
0. Thiscondition is satisfied only by the second set of eigenvalues. The gain associated withthe system is given by G = |ℜ (Λ ± ) | . (23)A typical plot of the gain spectrum is given in figure (1). We consider the case ofBaTiO crystal with a space charge field of 10 V/m, electro-optic coefficient r =1640 × − m/V and refractive index n = 2 . − . Figure 2 gives the variation of gain with the angle θ . The gain increases withdecrease in angle.
3. Forward four wave mixing geometry
In the two wave mixing case, two coherent beams interfere inside a photorefractivemedium and produce a volume index grating. In the case of optical phase conjugation,using four-wave mixing, a third beam is incident at the Bragg angle from the oppositeside and a fourth beam is generated. We now consider the situation in which the thirdbeam is incident from the front at the Bragg angle and a diffracted beam is generated.That is we consider the interaction of four beams in a PR medium in the forwardgeometry (see figure (3)). We assume that all the beams have the same frequency ω .We propose a model for the observation of MI in a PR medium induced by four wavemixing. The method proceeds by first writing the four coupled equations for the presentgeometry. Of the four beams, let beams A and A be the pump beam, A be the signalbeam and A be the generated beam. Beam A is coherent with beam A and beam odulational instability in photorefractives Κ H Κ L Figure 2.
Variation of gain with respect to the angle between the two beams. Thelowermost curve is for θ = π/
16, the dotted curve is for π/ π/ A A A A z = 0 z = L Figure 3.
Forward four wave mixing in photorefractive media in the transmissiongeometry. odulational instability in photorefractives A is coherent with beam A . Thus the index grating consists of two contributions : A ∗ A and A ∗ A . The index of refraction including the fundamental component of theintensity-induced gratings can thus be written as n = n + (cid:20) n ( A ∗ A + A ∗ A )2 I exp[ iφ ] exp[ − i K . r ] + cc (cid:21) , (24)where k − k = k − k = K .Following the above procedure for the two wave mixing case, we get the followingfour coupled equations for the present geometry: i ∂A ∂z = 12 k cos θ ∇ ⊥ A + Γ2 I ( A A ∗ + A A ∗ ) A , (25 a ) i ∂A ∂z = 12 k cos θ ∇ ⊥ A + Γ2 I ( A ∗ A + A ∗ A ) A , (25 b ) i ∂A ∂z = 12 k cos θ ∇ ⊥ A + Γ2 I ( A A ∗ + A A ∗ ) A , (25 c ) i ∂A ∂z = 12 k cos θ ∇ ⊥ A + Γ2 I ( A ∗ A + A ∗ A ) A , (25 d )where I = I + I + I + I .The model permits plane wave solutions of the form A j ( x, z ) = √ P exp[ − i Γ P I z ].The next step is to carry out a Linear Stability Analysis of the plane wave solutions.For this the plane wave solution is perturbed as A j = ( √ P + a j ( x, z )) exp h − i Γ P I z i , (26)where a j is a small complex perturbation. Inserting this into the coupled equation (3)and linearizing around the solution yields the equations for the perturbations: i ∂a ∂z = − Γ2 I a + 12 k cos θ ∂ a ∂x + Γ2 I (2 a + a ∗ + a + a ) , (27) i ∂a ∂z = − Γ2 I a + 12 k cos θ ∂ a ∂x + Γ2 I (2 a + a ∗ + a ∗ + a ) , (28) i ∂a ∂z = − Γ2 I a + 12 k cos θ ∂ a ∂x + Γ2 I ( a + a ∗ + 2 a + a ∗ ) , (29) i ∂a ∂z = − Γ2 I a + 12 k cos θ ∂ a ∂x + Γ2 I ( a ∗ + a + 2 a + a ∗ ) . (30)Now, we assume that the spatial perturbation a ( x, z ) is composed of two side bandplane waves, i.e a j ( x, z ) = U j ( z ) exp[ iκx ] + V j ( z ) exp[ − iκx ] . (31)Substituting, we get eight homogeneous equations as ∂U ∂z = − i ( − Γ P I − κ k cos θ ) U − i Γ P I (2 U + V ∗ + U + V ∗ ) , (32 a ) ∂U ∂z = − i ( − Γ P I − κ k cos θ ) U − i Γ P I (2 U + V ∗ + V ∗ + U ) , (32 b ) odulational instability in photorefractives ∂U ∂z = − i ( − Γ P I − κ k cos θ ) U − i Γ P I ( U + V ∗ + 2 U + V ∗ ) , (32 c ) ∂U ∂z = − i ( − Γ P I − κ k cos θ ) U − i Γ P I ( V ∗ + U + 2 U + V ∗ ) , (32 d ) ∂V ∗ ∂z = − i ( − Γ P I − κ k cos θ ) V ∗ + i Γ P I (2 V ∗ + U + V ∗ + U ) , (32 e ) ∂V ∗ ∂z = − i ( − Γ P I − κ k cos θ ) V ∗ + i Γ P I (2 V ∗ + U + U + V ∗ ) , (32 f ) ∂V ∗ ∂z = − i ( − Γ P I − κ k cos θ ) V ∗ − i Γ P I ( V ∗ + U + 2 V ∗ + U ) , (32 g ) ∂V ∗ ∂z = − i ( − Γ P I − κ k cos θ ) V ∗ − i Γ P I ( U + V ∗ + 2 V ∗ + U ) . (32 h )The obtained eight coupled equations can be written in the compact matrix formas, ∂ z X = M X , where M is an 8x8 matrix with X = [ U U U U V ∗ V ∗ V ∗ V ∗ ] T .This system has a nontrivial solution only if the determinant of the matrix vanishes.The real part of the eigenvalues of the stability matrix of equation (3) gives the gainassociated with the system [36]. Out of the eight roots of the system, only the root withmaximum positive value contributes to the MI gain. Here only the eigenvalue given byΛ ± = ± √− κ I + 4 κ Γ k cos θP √ I k cos θ (33 a )contributes to the MI of the system.We note that the gain vanishes for all values of κ greater than κ max = 4 k Γ P/I .Defining γ = Γ P/I , we can rewrite the gain as Gγ = r (cid:16) κκ max (cid:17) (cid:16) − (cid:16) κκ max (cid:17) (cid:17) . (34)The variation of the gain coefficient in the forward four wave mixing processwith respect to the spatial perturbation is plotted in figure (4). Such instabilities areuseful for pattern formation. A transverse modulation instability of a single beam orcounterpropagating beams is a general mechanism that leads to pattern formation innonlinear optics [37, 38]. We expect that similar results will be obtained using thepresent geometry.
4. Conclusion
We first studied MI occurring in a PR medium in the two wave mixing geometry andfurther modeled the forward four wave mixing occurring in the PR medium and studiedMI in this geometry. In both the cases, the geometry is such that only the holographicfocusing nonlinearity is acting. MI does not rely on self-phase-modulation self-focusingbut results only by virtue of the competition between induced periodic modulation ofthe refractive index and diffraction of the beam. The MI gain spectrum is obtained for odulational instability in photorefractives -40 -20 20 40 Κ H Κ L €€€€€€€€€€€€€€Γ Figure 4.
A typical plot showing the gain spectrum of the system in the forward fourwave mixing process with respect to the spatial perturbation κ . both two wave mixing and forward four wave mixing geometry. Such instabilities willbe useful for pattern formation. Photorefractive materials are attractive for the studiesof pattern formation as their slow time constant gives the possibility of observing thespatiotemporal dynamics of the system in real time. This also reduces the demands onexperimental equipment where speed is often a crucial parameter. Another advantage ofphotorefractive pattern formation is that patterns can be observed using optical powersof tens of mW. In contrast, pattern formation using other nonlinearities require opticalpowers in the order of 1W. Acknowledgments
J.C.P and V.C.K wish to acknowledge DST (Research Grant) for financial support.J.C.P also acknowledges CSIR for the award of senior research fellowship. KP wishes tothank DST, DAE-BRNS, IFCPAR, CSIR and DST-Ramanna fellowships for financialhelp through projects.
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