Analytic structure of the four-wave mixing model in photorefractive materials

28 December 2007 16:20 WSPC - Proceedings T rim Size: 9in x 6in 2008˙Scicli˙arXiv˙fourw ave 1 Analytic structure of the four-wa v e mixing mo del in photorefractiv e materials ∗ Robert Con te Servic e de physique de l’´ etat c ondens´ e (CNRS URA 2464) CEA–Saclay, F–911 91 Gif-sur-Y vette Ced ex, F r anc e E-mail: R ob ert.Conte@ c e a.fr Sv etlana Buga yc huk Institute of Physics of the National A c ademy of Sciences of Ukr aine 46 Pr osp e ct Nauki, Kie v-39, UA 0 3039, Ukr aine E-mail: bugaich@iop.kiev.ua 28 December 2007 In order to later find explicit analyt ic solutions, w e in vestigate the singulari t y structure of a fundamen tal model of nonlinear optics, the four-wa v e mixing model in one space v ariable z . This structure i s quite similar, and this i s not a surpr ise, to that of the c ubic c omplex Ginzburg-Landau equation. The main result is t hat, in order t o be single v alued, time-dep enden t sol utions should de- pend on the space-time co ordinates through the r educ ed v ariable ξ = √ z e − t/τ , in whic h τ is the relaxa tion time. Keywor ds : four-wa v e mixing mo del; Painlev ´ e test; P A CS 2001 02.30.J r,42.65 AMS MSC 2000 35A20, 35C05, 1. The four-w a ve mixing mo del Dynamic holography relies on the ability o f the refra ctiv e index of a non- linear media to lo cally change under the action of light, i.e. under the so- called photor efractive r esponse. The most conv entional sc heme for dynamic hologra phic a pplications consists in the for mation of a fr inge interference pattern b et ween the in ter a cting w av es (the lig h t grating ), whic h creates the refractive index g r ating. During the wav e in teractio n, these ar e the sa me wa ves which create the g rating patter n and diffra ct on this grating . There ∗ W a v es and stabilit y in cont inuous media, 30 Jun e-7 July 2007 , Scicli ( Rg), Italy . 28 December 2007 16:20 WSPC - Proceedings T rim Size: 9in x 6i n 2008˙Scicli˙arXiv˙fourw ave 2 exist man y applications bas ed on dynamic ho lography with pho torefractive crystals 1 . In the four- w av e mixing model, the grating, i.e. the modula tio n of the refractive index induced by the light interference pattern, is cr eated b y tw o pairs of co-pro pagating w av es, 1–2 a nd 3–4 . Let us denote A 1 , A 2 , A 3 , A 4 the complex amplitudes of the four wa ves, ∆ ε the complex amplitude of the index grating, τ the c hara cteristic time of the evolution, I 0 the total int ensity of ligh t (a real p ositiv e constan t), γ the photorefractive coupling (a c o mplex constant). The four-wav e mixing mo del is the system of five complex partia l differential equations ob ey ed by these fiv e complex a mpli- tudes 2 (bar denotes complex conjugation)    ∂ z A 1 = − i ∆ εA 2 , ∂ z A 2 = i ∆ εA 1 , ∂ z A 3 = − i ∆ ε A 4 , ∂ z A 4 = i ∆ εA 3 ,  ∂ t + 1 τ  ∆ ε = γ I 0  A 1 A 2 + A 3 A 4  . (1) The corre s pondence o f notation with Ref. 3 is ( A 1 , A 2 , A 3 , A 4 , Q, γ ) of Ref. = ( A 1 , A 4 , A 3 , A 2 , − i ∆ ε, iγ ) here . (2) A t present time, no analytic solution is known to this system, except a stationary solution in which the gra ting amplitude has a pulse profile 1 , ∆ ε = K sec h k ( z − z 0 ) , ( K, k ) = real constants . (3) The purpose of this short article is to inv estigate the structure of the sin- gularities of the time-dep enden t solutions ( ∂ t 6 = 0) in order to find, in a forthcoming pap er, clo sed form analytic solutions by making suitable as- sumptions dictated by the singular it y structure. The first four equations (1), which do not depend on ∂ t , admit six quadratic first integrals. Two of them, | A 1 | 2 + | A 2 | 2 = f 2 1 ( t ) , | A 3 | 2 + | A 4 | 2 = f 2 2 ( t ) , (4) allow us to repr esen t the four complex amplitudes A j ( z , t ) with tw o rea l functions f 1 , f 2 of t and six real functions θ 1 , θ 2 , ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 of ( z , t ), A 1 = f 1 sin θ 1 e iϕ 1 , A 2 = f 1 cos θ 1 e iϕ 2 , A 3 = f 2 cos θ 2 e iϕ 3 , A 4 = f 2 sin θ 2 e iϕ 4 . (5) The four other quadratic first integrals are pairwis e complex co njugate ( A 1 A 3 + A 2 A 4 = f 3 ( t ) , A 1 A 3 + A 2 A 4 = f 3 ( t ) , A 1 A 4 − A 2 A 3 = f 4 ( t ) , A 1 A 4 − A 2 A 3 = f 4 ( t ) , (6) and constra ine d by one relation, − f 2 1 f 2 2 + | f 3 | 2 + | f 4 | 2 = 0 . (7) 28 December 2007 16:20 WSPC - Proceedings T rim Size: 9in x 6i n 2008˙Scicli˙arXiv˙fourw ave 3 2. Lo ca l s ingularit y analysis of the FWM mo del The dep enden t v a riables A j , ∆ ε generica lly prese n t mov able singularities, i.e. singular ities whose loca tion in the co mplex do main depends on the ini- tial conditions. The study of the b ehaviour of A j , ∆ ε near these mov able singularities is a prerequisite to the pos sible obtention of closed form ana- lytic solutions to the nonlinear system (1), so let us first p erform it. Such a study is made b y performing the s uc c e ssiv e steps of the s o -called Painlev´ e test , a pr ocedur e expla ined in detail in Ref. 4 , 5 . The mov able singularities o f a P DE la y on a manifold r epresen ted by the equation 6 ϕ ( z , t ) − ϕ 0 = 0 , (8) in whic h ϕ is an arbitra r y function of the indep enden t v a r iables, and ϕ 0 an arbitrar y mov able consta n t. In o rder to expr ess the b eha viour o f the ten dep enden t v a riables A j , A j , ∆ ε, ∆ ¯ ε , it is co nvenien t to intro duce an expansion v ar iable χ ( z , t ) 4 which v anis hes as ϕ − ϕ 0 and which is defined by its gradient χ z = 1 + S 2 χ 2 , χ t = − C + C z χ − 1 2 ( C S + C z z ) χ 2 . (9) with the cross-der iv ative condition X ≡ S t + C z zz + 2 C z S + C S z = 0 . (10) Since they will b e needed later, we giv e the dependence of χ, S, C on ϕ , χ = ϕ − ϕ 0 ϕ z − ϕ z z 2 ϕ z ( ϕ − ϕ 0 ) =  ϕ z ϕ − ϕ 0 − ϕ z z 2 ϕ z  − 1 , ϕ z 6 = 0 , (11) S = { ϕ ; z } = ϕ z zz ϕ z − 3 2  ϕ z z ϕ z  2 , C = − ϕ t /ϕ z . (12) 2.1. L e ading b ehaviour The first step is to determine all possible families of nonch ar a cteristic (i.e. ϕ x ϕ t 6 = 0) mov able singularities , i.e. a ll the leading b eha viours χ → 0 :  A k ∼ a k χ p k , A k ∼ b k χ P k , a k b k 6 = 0 , k = 1 , 2 , 3 , 4 , ∆ ε ∼ q 0 χ α , ∆ ¯ ε ∼ r 0 χ β , q 0 r 0 6 = 0 , (13) in whic h the ex ponents p k , P k , α, β are not all positive integers. If one as- sumes that the t wo ter ms A 1 A 4 and A 3 A 2 hav e the same singular it y order, 28 December 2007 16:20 WSPC - Proceedings T rim Size: 9in x 6i n 2008˙Scicli˙arXiv˙fourw ave 4 the ten exp onen ts p k , P k , α, β ob ey the t welv e linear equa tions,                  p 1 − 1 = p 2 + α, P 2 − 1 = P 1 + α, p 4 − 1 = p 3 + α, P 3 − 1 = P 4 + α, P 1 − 1 = P 2 + β , p 2 − 1 = p 1 + β , P 4 − 1 = P 3 + β , p 3 − 1 = p 4 + β , α − 1 = p 1 + P 2 = p 4 + P 3 , β − 1 = P 1 + p 2 = P 4 + p 3 , (14) and the ten co efficien ts a k , b k , q 0 , r 0 ob ey the ten no nlinear equations        p 1 a 1 = − q 0 a 2 , p 2 a 2 = − r 0 a 1 , p 4 a 4 = − q 0 a 3 , p 3 a 3 = − r 0 a 4 , P 1 b 1 = r 0 b 2 , P 2 b 2 = q 0 b 1 , P 4 b 4 = r 0 b 3 , P 3 b 3 = q 0 b 4 , C αq 0 = − ( γ /I 0 )( a 1 b 2 + a 4 b 3 ) , C β r 0 = − (¯ γ /I 0 )( b 1 a 2 + b 4 a 3 ) . (15) Therefore the five squar e d moduli | A k | 2 , | ∆ ε | 2 behave lik e double p oles, p k + P k = − 2 , k = 1 , 2 , 3 , 4 , α + β = − 2 , (16) and the linear system (14) is solved as            p 1 = − 1 + s 1 + δ, P 1 = − 1 − s 1 − δ, p 2 = − 1 + s 1 − δ, P 2 = − 1 − s 1 + δ, p 4 = − 1 + s 2 + δ, P 4 = − 1 − s 2 − δ, p 3 = − 1 + s 2 − δ, P 3 = − 1 − s 2 + δ, α = − 1 + 2 δ, β = − 1 − 2 δ, (17) in which δ, s 1 , s 2 are to be determined by the nonlinear sys tem (15). The fir st set of rela tio ns − q 0 r 0 = p 1 p 2 = p 4 p 3 = P 1 P 2 = P 4 P 3 , (18) imply s 1 = 0 , s 2 = 0 , − q 0 r 0 = 1 − δ 2 . (19) The se c ond set of rela tions a 1 a 2 = − a 4 a 3 = − b 2 b 1 = b 3 b 4 = iq 0 1 − δ = 1 + δ ir 0 (20) is solved as  a 1 = a 12 λ, b 2 = − b 12 λ, a 4 = a 34 λ, b 3 = b 34 λ, iq 0 = ( 1 − δ ) λ 2 , a 2 = a 12 λ − 1 , b 1 = b 12 λ − 1 , a 3 = − a 34 λ − 1 , b 4 = b 34 λ − 1 , ir 0 = ( 1 + δ ) λ − 2 , (21) in whic h a 12 , a 34 , b 12 , b 34 , λ are nonzero fun ctions of ( z , t ) to b e determined. 28 December 2007 16:20 WSPC - Proceedings T rim Size: 9in x 6i n 2008˙Scicli˙arXiv˙fourw ave 5 Finally , the tw o remaining equations      a 12 b 12 − a 34 b 34 + I 0 C iγ (1 − δ )(1 − 2 δ ) = 0 , a 12 b 12 − a 34 b 34 − I 0 C i ¯ γ (1 + δ )(1 + 2 δ ) = 0 , (22) first define δ as a ro ot of the s econd degre e equation γ (1 + δ )(1 + 2 δ ) + ¯ γ (1 − δ )(1 − 2 δ ) = 0 , (23) then put one co nstrain t among a 12 , b 12 , a 34 , b 34 . The equation (23) which defines δ only dep ends on the ar gumen t of γ , (2 cos g ) δ 2 + (3 i s in g ) δ + cos g = 0 , g = ar g γ . (24) When the photorefractive coupling constant γ is pur ely imagina r y , the ex- po nen t δ v anis he s , otherwise it can take tw o pur ely imaginar y v alues δ = i − 3 sin g ± p 9 sin 2 g + 8 co s 2 g 4 cos g . (25) T o conclude, there gener ic ally exist tw o families (13) of mov able singu- larities, defined by the equations (17), (19), (21), (22), i.e.                                      δ = one the tw o ro ots o f Eq. (24) , p 1 = p 4 = P 2 = P 3 = − 1 + δ, P 1 = P 4 = p 2 = p 3 = − 1 − δ, a 1 = N λ p 12 cosh µ, b 2 = − N λ p − 1 12 cosh µ, a 4 = N λ p 34 sinh µ, b 3 = N λ p − 1 34 sinh µ, a 2 = N λ − 1 p 12 cosh µ, b 1 = N λ − 1 p − 1 12 cosh µ, a 3 = − N λ − 1 p 34 sinh µ, b 4 = N λ − 1 p − 1 34 sinh µ, iq 0 = (1 − δ ) λ 2 , ir 0 = ( 1 + δ ) λ − 2 , N 2 = − I 0 C iG (1 − δ 2 )(1 − 4 δ 2 ) , G = (1 + δ )(1 + 2 δ ) γ = − (1 − δ )(1 − 2 δ ) ¯ γ . (26) and they dep end o n four ar bitrary complex functions λ, µ, p 12 , p 34 of ( z , t ). 2.2. F uchs indi c es The lea ding b eha viour (13) is the firs t term of a Laurent series χ → 0 :            A k = χ p k + ∞ X j =0 a k,j χ j , ∆ ε = χ α + ∞ X j =0 q j χ j , A k = χ P k + ∞ X j =0 b k,j χ j , ∆ ¯ ε = χ β + ∞ X j =0 r j χ j , (27) 28 December 2007 16:20 WSPC - Proceedings T rim Size: 9in x 6i n 2008˙Scicli˙arXiv˙fourw ave 6 and the indices j at whic h a r bitrary co efficients enter this expansion are computed as follows. If o ne symbolica lly denotes E ( A k , A k , ∆ ε, ∆ ¯ ε ) = 0 the ten equations (1), one builds the linearized system of (1), lim ζ → 0 E ( A k + ζ C k , A k + ζ C k , ∆ ε + ζ D , ∆ ¯ ε + ζ D ) − E ( A k , A k , ∆ ε, ∆ ¯ ε ) ζ = 0 , i.e.    ∂ z C 1 + i ∆ εC 2 + iA 2 D = 0 , . . .  ∂ t + 1 τ  D − γ I 0  A 2 C 1 + A 1 C 2 + A 3 C 4 + A 4 C 3  = 0 . (28) A t the p oin t (13), this ten-dimensional linear sys tem displa ys near χ = 0 a singularity which has the F uchsian type 7 , therefore it a dmits a solution χ → 0 :  C k ∼ c k χ p k + j , C k ∼ d k χ P k + j , k = 1 , 2 , 3 , 4 , D ∼ s 0 χ α + j , D ∼ t 0 χ β + j . (29) The condition that this solution b e noniden tically zero results in the v an- ishing of a tenth order determinant whose ro ots j , called F uc hs indices, take the ten v alues j = − 1 , 0 , 0 , 0 , 0 , 2 , 2 , 2 , 5 ± √ 1 + 48 δ 2 2 . (30) In the tw o -w av e mixing case, the four r o ots 0 , 0 , 2 , 2 a re to b e remov ed from this list. The necessar y condition, r equired by the Painlev´ e test, tha t all indices b e in teger is viola ted when δ is nonzero , therefore the FWM mo del fails the test in this case. This failure is quite similar to what o ccurs in the complex cubic Ginzburg-La ndau eq uation 8 , and conclusions simila r to those of Ref. 9 can probably b e drawn. Let us remar k that th e F uchs index 0 has m ultiplicity four, in ag reemen t with the n umber of arbitrary functions inv olved in the leading b eha viour , see Eq. (26). 2.3. Conditi ons at the F uchs indi c es The co efficients a k,j , b k,j , q j , r j of the Laurent ser ies (27) are computed for j ≥ 1 by solv ing a linea r system. Whenever j reaches an int eger F uchs index, an obstructio n may o ccur, resulting in the intro duction of a mov able logarithmic branchin g, a nd some necessary conditions need to b e satisfied in order to av oid it. In the generic ca se δ 6 = 0, these obstructions may only arise at j = 2, v alue of a triple F uchs index. In the no ngeneric case δ = 0, in addition to the quadruple index j = 2 , o ne must a lso check the simple index j = 3. The results are as follows. 28 December 2007 16:20 WSPC - Proceedings T rim Size: 9in x 6i n 2008˙Scicli˙arXiv˙fourw ave 7 A t j = 2, wha tev er be δ , a mov able logarithm exists, unless the following necessary condition is satisfied, Q 2 ≡ a ( C t + C C z − 2 aC ) = 0 , a = 1 τ . (31) Since a = 0 is excluded, the second factor defines a precis e depe ndence o f C on ( z , t ), hence a similar dep endence for S, χ, ϕ and ultimately for the five complex amplitudes vi a their Laurent expa ns ion (27). Let us find this depe ndence explicitly . The genera l solution of (31) is pro vided by the metho d of characteristics and is defined implicitly by the rela tio n 2 az = C + F ( e − 2 at C ) , (32) in which F is an arbitra ry function o f one v ar iable. The inv a rian t S is then defined by the first order linear PDE (10), whic h reads S ( x, t ) = Σ( C, t ) , D = 1 + e − 2 at F ′ , e 2 at  Σ t + 2 aC Σ C + 4 a Σ D  + 24 a 3 e − 6 at F ′′ 2 D 5 − 8 a 3 e − 4 at F ′′′ D 4 = 0 , (33) and a dmits the gener al solution S = c 0 e − 4 at D − 2 − 4 a 2 e − 6 at D − 3 F ′′′ + 6 a 2 e − 8 at D − 4 F ′′ 2 , (34) in which c 0 is an arbitrar y constan t. W e have not b een able to prov e whether the function F is a gauge which can be arbitrar ily c ho sen (e.g. F = 0) or whether it is essential. If it is essential, there could exist m uch more intricate so lutions than those which w e now outline. When F is ar bitrary , we hav e not y et succeeded to in tegrate the system (9) for χ ( z , t ), hence to compute the induced dependence o f A j , ∆ ε on ( z , t ). How ever, if o ne re stricts to a constant v alue for F , whic h can then be chosen equal to ze r o, the equatio n (32) for ϕ integrates as F = 0 , ϕ = Φ( ξ ) , ξ = √ z e − at , (35) in which Φ is an arbitrar y function, and it is straightforward to chec k that the L a uren t expa nsion (27) defines the reduction ( z , t ) → ξ = √ z e − at . As opp osed to the stationar y r eduction ∂ t = 0, this reduction is no nc haracter- istic (i.e. it does not lo wer the differential order) it is defined with a reduced 28 December 2007 16:20 WSPC - Proceedings T rim Size: 9in x 6i n 2008˙Scicli˙arXiv˙fourw ave 8 v aria ble of the factorized type ξ = f ( z ) g ( t ),                                ξ = z 1 / 2 e − at , a = 1 /τ , ∆ ε ( z , t ) = e − at − iωt z − 1 / 2 ∆ ε r ( ξ ) , ∆ ¯ ε ( z , t ) = e − at + iωt z − 1 / 2 ∆ ¯ ε r ( ξ ) , A j ( z , t ) = e − at − iωt/ 2 A j,r ( ξ ) , A j ( z , t ) = e − at + iωt/ 2 A j,r ( ξ ) , j = 1 , 4 , A j ( z , t ) = e − at + iωt/ 2 A j,r ( ξ ) , A j ( z , t ) = e − at − iωt/ 2 A j,r ( ξ ) , j = 2 , 3 , d d ξ A 1 ,r = − 2 i ∆ ε r A 2 ,r , d d ξ A 2 ,r = 2 i ∆ ε r A 1 ,r , d d ξ A 3 ,r = − 2 i ∆ ε r A 4 ,r , d d ξ A 4 ,r = 2 i ∆ ε r A 3 ,r , d d ξ ∆ ε r + iω τ ξ − 1 ∆ ε r = − γ I 0  A 1 ,r A 2 ,r + A 3 ,r A 4 ,r  , (36) and it dep ends o n the additional real parameter ω . T o summariz e the information obta ined fro m the F uchs index 2, pro- vided F c a n be arbitrarily chosen, the four-wav e mixing model (1) (as well as the t wo-w av e mixing model) a dmits no sing le v alued dynamical solution other than the p ossible so lutio ns of the reduction ( z , t ) → ξ = √ z e − at . When δ 6 = 0, this ends the P ainlev´ e test. When δ = 0, the F uc hs index 3 is found to b e free of mov a ble logar ithms. Finally , the information provided by the Painlev´ e test is the following. (1) Whatever be γ , and under the mild res triction tha t F can b e arbitrarily chosen, no single v alued dynamical solution of (1) exists other than the po ssible solutio ns of the reduction ( z , t ) → ξ = √ z e − at . (2) F or this reduction, t wo cases m ust b e disting uis hed. When ℜ ( γ ) 6 = 0, the ten-dimens io nal O DE system (36) p ossesses a t mos t a n e ig h t- parameter single v a lue d s o lution. When ℜ ( γ ) = 0 , the system (36) passes the Painlev´ e test therefore it may admit a ten-par ameter single v alued solution, which then would be its g eneral solution. Finding a Lax pair in this case w ould co nsiderably help to per form the e xplicit int egr a tion. Similar conclusions apply to the six-dimensio na l t wo-wa ve mixing: the only p ossibility for a single v alued solution is the reduction (3 6) (with A 3 = A 4 = 0), this s o lution dep ending on four mo v able constants for ℜ ( γ ) 6 = 0, and six mov able consta n ts for ℜ ( γ ) = 0. 3. Conclusion The present study prov es, lik e for the complex Ginzburg-Landa u equation, that ph ysically relev ant analy tic solutions quite certainly ex ist for the four- wa ve mixing model. The present co un ting o f the p ossible a rbitrary constants 28 December 2007 16:20 WSPC - Proceedings T rim Size: 9in x 6i n 2008˙Scicli˙arXiv˙fourw ave 9 in the solutions displays, as expected, the crucial role of the photorefractive complex constant. Explicit solutions based on the present study will b e presented elsewhere. The results can b e used for predicting new phenomena in optical self- diffraction o f wa ves in photo refractive media which use the non-lo cal re- sp onse as well as for optimiza tion of optical dynamic hologr aphic settings. Among these a pplications let us quote: (i) the formation of a lo calized grating to incr ease optical infor mation density; (ii) the metho ds o f all- optical co n tr ol of o utput wa ve characteristics versus input b eam intensi- ties and phases; (iii) the o ptimization of the parameters of optical phase- conjugation; (iv) the use of the new ho lographic top ographic technique to material parameter characterization. Just lik e similar pro cesses of nonlinear self-action of w av es a rise in m o dels of optical net works, optical information pro cessing, quantum inf orma tion pro cessing, interacted neural chains, then v ario us other problems of nonlinear wa ve interaction can become the sub- ject of further independent research. Ac knowledgmen ts W e w armly ackno wledge the financial support of the Max Planck Institut f¨ ur P h ysik komplexer Systeme, and RC thanks the W ASCOM orga nizers for in vitation. References 1. S . Bugayc huk, R .A. R upp, G . Mandula and L. K´ ov acs, Soliton profile of the dynamic grating amplitude and its alteration by p hotorefra ctive wa ve mix- ing, 404–40 9, Phot or ef r active effe cts, m ate rials and devic es , eds. P . Delay er, C. Den z, L. Mager and G. Montemezzani, T r ends in optics and photonics series 87 (Optical so ciet y of America, W ashington D C, 200 3). 2. S . Buga ych uk, L. K ´ ov acs, G. Mandula, K . Polg´ ar and R.A. 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