aa r X i v : . [ phy s i c s . op ti c s ] D ec Collision and fusion of counterpropagatingmicron-sized optical beams in non-uniformly biasedphotorefractive crystals
A. Ciattoni [email protected] Nazionale delle Ricerche, CASTI Regional Lab 67100 L’Aquila, ItalyDipartimento di Fisica, Universit ` a dell’Aquila, 67100 L’Aquila, Italy A. Marini
Dipartimento di Fisica, Universit ` a dell’Aquila, 67100 L’Aquila, Italy C. Rizza
Dipartimento di Ingegneria Elettrica e dell’Informazione, Universit ` a dell’Aquila, Montelucodi Roio 67100 L’Aquila, Italy E. DelRe
Dipartimento di Ingegneria Elettrica e dell’Informazione, Universit ` a dell’Aquila, Montelucodi Roio 67100 L’Aquila, Italy
1e theoretically investigate collision of optical beams travelling in oppositedirections through a centrosymmetric photorefractive crystal biased by aspatially non-uniform voltage. We analytically predict the fusion of coun-terpropagating solitons in conditions in which the applied voltage is rapidlymodulated along the propagation axis, so that self-bending is suppressed bythe ”restoring symmetry” mechanism. Moreover, when the applied voltage isslowly modulated, we predict that the modified self-bending allows conditionsin which the two beams fuse together, forming a curved light-channel splice.c (cid:13)
OCIS codes: straight line, triggers their mutuallongitudinally nonlocal interaction [4, 5]. In the situation where the two beams are mutuallyincoherent, we show that soliton fusion occurs by analytically proving the existence of a two-parameter family of fully overlapping counterpropagating solitons. In conditions where theapplied voltage is slowly modulated, we numerically identify the conditions for exploiting thelongitudinal wiggling beam profile [9] to achieve a robust fusion of two counterpropagatingbeams impinging on the opposite crystal facets. The results indicate a self-adaptive mergingof two channels, an effect that has a potential photonic application in minimizing opticalpower loss associated to diffraction and misalignment in optical fiber splicing.Consider a photorefractive crystal whose facets x = − L x and x = L x have a set of3lectrodes that deliver the periodical potential profiles − V cos( κ v x ) and V cos( κ v x ), re-spectively, as shown in Fig.(1). In Ref. [9], it has been proved that, if I ( x, z ) is the opticalintensity of the light travelling along the z -axis through the crystal, the photorefractivelyinduced refractive index change is δn = α ( I + I b ) " ψ cos( κ v z ) + χ ∂I∂x , (1)where χ = K B T /q , α = − (1 / n gǫ ( ǫ r − ( T is the crystal temperature, q is the electroncharge, ǫ r is the relative dielectric constant at the given T , n is the uniform refractiveindex background, g is the significant quadratic electro-optic coefficient), I b is the intensityof a reference background uniform illumination, and ψ = V I b / [ L x cosh( κ v L x )]. In the TEconfiguration, the complex amplitude of the monochromatic (at frequency ω ) optical electricfield E ( x, z ) satisfies the Helmholtz equation ( ∂ xx + ∂ zz ) E + k ( n + δn ) E = 0 where k = ω/c and δn is given by Eq.(1). In order to describe head-on collision of two beamscounterpropagating along the z -axis we set E ( x, z ) = exp( ikz ) A + ( x, z ) + exp( − ikz ) A − ( x, z )where k = k n and A + and A − are the slowly-varying amplitudes of the forward andbackward propagating beams, respectively. Inserting this expression for E into the Helmholtzequation, in the paraxial approximation and noting that for κ v << k (i.e. the period ofvoltage modulation is much greater than the optical wavelength) light cannot be Bragg-matched with the periodic refractive index profile, A ± satisfy the coupled parabolic equations[ ± i∂ z + (1 / k ) ∂ xx ] A ± = − ( k/n ) δnA ± . We here focus our attention on the relevant case oftwo mutually incoherent counterpropagating beams for which the total optical intensity is4iven by I = | A + | + | A − | . Since Eq.(1) is not an even function of x if the intensity I iseven, the two beams A ± do not propagate on a straight line: they experience the effect ofself-bending in the same lateral direction, so that the effect of their interaction is generallylimited by the smallness of the overlapping region. In order to maximize the overlap of thetwo beams we take κ v ≫ π/L d (the situation corresponding to the geometry illustrated inFig.(1a)) where L d is the longitudinal scale characterizing the propagation of A ± ( L d typicallycoincides with the optical diffraction length). In these conditions the optical beams are notable to follow the rapid voltage oscillation and the averaged fields do not experience self-bending, and do not wiggle (the mechanism of ”restoring symmetry” discussed in Ref. [9]).In this regime (i.e. κ v ≫ π/L d ) it is possible to set A ± = √ I b V ± + δA ± , where V ± are thoseparts of the fields having a longitudinal scale of variation L d , and δA ± are longitudinallyrapidly varying, on a scale 2 π/κ v , and they are uniformly in the condition | δA ± | ≪ √ I b | V ± | .This self-consistent decomposition of the fields into a slowly varying mean-field componentand a rapidly oscillating and small correction allows us to derive a set of equations for V ± (following a procedure very close to that reported in Ref. [9]) that are i ∂V + ∂ζ + ∂ V + ∂ξ = 12 + γ " ∂∂ξ (cid:16) | V + | + | V − | (cid:17) [1 + | V + | + | V − | ] V + , − i ∂V − ∂ζ + ∂ V − ∂ξ = 12 + γ " ∂∂ξ (cid:16) | V + | + | V − | (cid:17) [1 + | V + | + | V − | ] V − (2)where we have also introduced dimensionless variables according to ξ = k | ψ/I b | q | α | /n x , ζ = k ( | α | /n )( ψ/I b ) z and γ = 2 k χ | α | /n . Note that, as expected, the two beams V ± | V + | and | V − | are transversally even functions of ξ , is transversallyeven as well so that no self-bending occurs. Equations (2) admit of the solution V + ( ξ, ζ ) = cos Φ exp " i a ξ − i a − β ! ζ v ( ξ − aζ ) ,V − ( ξ, ζ ) = sin Φ exp " − i a ξ + i a − β ! ζ v ( ξ − aζ ) (3)for any values of the real parameters Φ and a if the function v ( τ ) satisfies the equation d vdτ = βv + + γ (cid:16) dv dτ (cid:17) (1 + v ) v. (4)Note that Eq.(4) coincides with the equation describing solitons propagating through themedium in the presence of the ”restoring symmetry” mechanism, as discussed in Ref. [9], sothat the fields in Eqs.(3) constitute a two-parameter family of counterpropagating solitons.It is worth stressing that the two solitons of each pair do not suffer self-bending, are fullyoverlapping and therefore Eqs.(3) describe fusion of solitons counterpropagating along astraight line. The parameter Φ sets the mutual power content of the two solitons in such away that | V + | + | V − | = | v | , whereas the parameter a (subjected to the restriction a ≪ z -axis.In order to check the above analytical results and to extend our investigation to the offaxis interaction configuration (see Fig.(1b)), we have integrated the full time-dependent pho-torefractive nonlinear optical model [10]. In our numerical approach, at each instant of time,we evaluate the electric field distribution induced by the boundary applied voltage and the6hotoinduced charge solving the ( x, z ) electro-static Poisson equation, and the correspond-ing optical field distribution determined by the electro-optic response through the parabolicequation [11]. We have chosen a crystal bulk (layer) of potassium lithium tantalate niobate(KLTN) ( n = 2 .
4) of thickness 2 L x = 2 × µm and length L z = 1000 µm . In order toinvestigate fusion of coaxial counterpropagating beams in the fast modulated regime (withelectrode modulation period 2 π/κ v = 200 µm ), we have chosen γ = 0 . A + ( x,
0) = q I b / f ( ξ ) at z = 0 and A − ( x, L ) = q I b / f ( ξ ) at z = L where f ( ξ ) is a real Gaussian profile centered at ξ = 0 ( ξ is the same dimensionless spatialvariable as in Eqs.(2)). We have performed various numerical simulations varying both Gaus-sian width and amplitude together with the applied voltage thus determining their valueswhich maximize the overlap between the forward and backward propagating field profile at z = 0, so to observe the formation of a stable and straight optical channel (see Fig.(1a)).In Fig.(2a) we have plotted the FWHM, σ , of f ( ξ ) as a function of f = f (0) (plottedas stars) corresponding to coaxial beam fusion and, for comparison purposes, we have alsoreported the theoretical soliton existence curve (solid line) derived by Eq.(4) (see Ref. [9]).The good qualitative agreement between the two different situations indicates that the fusionmechanism is robust and feasible.If the applied voltage is slowly modulated, a form of modified self bending occurs since theoptical beams are able to adiabatically follow the electrode modulation [8, 9]. This propertycan be profitably exploited to design a configuration where two beams, impinging onto the7rystal facets z = 0 and z = L along two parallel propagation directions, are made to form asingle and curved light channel within the crystal bulk. In the circumstance of the geometrydepicted in Fig.(1b), for example, the applied voltage is reversed one time along the z − axisso that beams fusion is possible since the optical beams bend toward negative and positive x direction in the z < µm and z > µm crystal regions, respectively. We have investigatedoff axis beam fusion in the configuration as in Fig.(1b) by means of the above discussednumerical scheme by launching two identical but shifted Gaussian profiles (i.e. by setting A + ( x,
0) = A exp[( x − d/ ) / (2 s )] at z = 0 and A − ( x, L ) = A exp[( x + d/ ) / (2 s )]at z = L , where A = 2 √ I b and s = 3 µm ) for various different applied voltages V andmutual beam displacement d thus determining their values which maximize the overlapbetween the forward and backward propagating field profile at z = 0. The result of thesecalculations are reported in Fig.(2b) from which we note that fusion can be attained evenfor counterpropagating beams whose mutual distance d is much greater than their commonwidth. This result suggest a feasible way for obtaining self-adaptive optical fiber splicing.8 eferences
1. E. DelRe, M. Segev, D. Christodoulides, B. Crosignani, and G. Salamo, Photorefrac-tive solitons in
Photorefractive Materials and Their Applications 1 , P. Gunter and J. P.Huignard (eds.) (Springer-Verlag, New York, 2006)2. M.I. Carvalho, S.R. Singh, D.N. Christodoulides, Opt. Commun. 120, 311 (1995)3. D. Kip, C. Herden, and M. Wesner, Ferroelectrics 135, 274278 (2002).4. C. Rotschild, O. Cohen, O. Manela, T. Carmon and M. Segev, J. Opt. Soc. Am. B ,1354 (2004).5. E. Del Re, A. Ciattoni, B. Crosignani and P. Di Porto, J. Nonlin. Opt. Phys. Mat. , 1(1999).6. O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, Opt. Lett. 27, 2013(2002).7. In the case of the Kerr nonlinearity, this coupling mechanism is generally referred toas ”holografic focusing”, see, e.g., O. Cohen, R. Uzdin, T. Carmon, J. W. Flescher, M.Segev and S. Odoulov, Phys. Rev. Lett. 89, 133901-1 (2002)8. A. Ciattoni, E. DelRe, A. Marini and C. Rizza, Opt. Express , 16867 (2008).9. A. Ciattoni, E. DelRe, C. Rizza and A. Marini, Opt. Lett. , 2110 (2008).10. L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Pho-torefractive Materials (Oxford Press, 1996).11. E. DelRe, A. Ciattoni and E. Palange, Phys. Rev. E , 017601 (2006).92. B. Crosignani, A. Degasperis, E. DelRe, P. Di Porto and A. J. Agranat, Phys. Rev. Lett. , 1664 (1999) 10ist of Figure Captions • Figure 1: Geometry of the collision between counterpropagating optical beams (re-ported as shaded regions around x = 0) through a non-uniformly biased photorefrac-tive crystal layer (black and gray stripes are here electrodes at opposite potentials).(1a) Fusion of two coaxial counterpropagating solitons in the fast modulated regime.(1b) Merging of two off axis counterpropagating beams into a single optical channeldue to modified self-bending in the slowly modulated regime. • Figure 2: (2a) Intensity full width at half maximum FWHM σ as a function of the peakamplitude f of the Gaussian input counterpropagating beam profiles allowing beamfusion (stars) and corresponding theoretical existence curve (solid line) associated withcounterpropagating soliton fusion (evaluated from Eq.(4)). (2b) Voltage V as a functionof the displacement d between the two beams required to form an optimal fused splicealong a curved trajectory. 11ig. 1. Geometry of the collision between counterpropagating optical beams (reported asshaded regions around x = 0) through a non-uniformly biased photorefractive crystal layer(black and gray stripes are here electrodes at opposite potentials). (1a) Fusion of two coax-ial counterpropagating solitons in the fast modulated regime. (1b) Merging of two off axiscounterpropagating beams into a single optical channel due to modified self-bending in theslowly modulated regime. γ = 0.2f σ µ m) V ( V o lt ) (a) (b) Fig. 2. (2a) Intensity full width at half maximum FWHM σ as a function of the peak ampli-tude f of the Gaussian input counterpropagating beam profiles allowing beam fusion (stars)and corresponding theoretical existence curve (solid line) associated with counterpropagat-ing soliton fusion (evaluated from Eq.(4)). (2b) Voltage V as a function of the displacement dd