A steady state solution to four-wave mixing utilising the SU(2,2) group symmetry with mixed gratings in a Kerr type media

Backward four-wave mixing in a Kerr type media with mixed refractive gratings of equal coupling strength are described by a set of nonlinear coupled wave equations. These are formulated into a compact matrix representation which then allows the
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  1 June 1995 Optics ~omtnuni~ations I17 f 1995) 283-289 OPTICS COMMUNICATIONS A steady state solution to four-wave mixing utilising the SU( 2,2) group symmetry with mixed gratings in a Kerr type media P.M. Darbyshire a, T.J. Hall b a Department of Physics, Kings College London. Strand, London WC2R 2L.S. UK h Depurtment of Electronic and Electrical Engineering, Kings College London, Strand, London WC22R .G, UK Received 19 January 1995 Abstract Backward four-wave mixing in a Kerr type media with mixed refractive gratings of equal coupling strength are described by a set of nonlinear coupled wave equations. These are formulated into a compact matrix representation which then allows the identification of the four-dimensional symmetry group SU( 2,2). When considering the two-point boundary value problem it is found that the mixed grating analysis reduces to the solution of a transmission only grating. This formulism is then used to find a solution for steady state phase conjugation and plots of reflectivity show the multiplicity of the solutions. 1. ~Rt~duction Optical phase conjugation has generated a great deal of interest over the past few years. It involves the use of nonlinear optical techniques for real time processing and offers some exciting applications such as aberra- tion correction, holographic interferometry and infor- mation processing in the form of optical associative memories [ 11, Degenerate four-wave mixing (DJ?WM) is an ex- tremely important mechanism for the generation of phase conjugate waves. In this process four beams pass through a nonlinear optical material and interfere. This interference creates a complex interference pat- tern which modulates the refractive index of the ma- terial. The refractive index modulation acts as a thick phase grating causing beams to diffract and exchange energy. The theoretical treatment of this phenomena is based on solutions to nonlinear coupled wave equa- tions. These describe the nonlinear coupling of the Eisevier Science B.V. SSDZOO30-4OlS(95)00210-3 wave interactions through the refractive index grat- ing. First analyses concentrated on simple linearised models using the undepleted pumps approximation [21. Then more realistic models were analysed with pump depletion and the inclusion of linear absorp- tion [3-61. The majority of work focusing on one coupling mechanism, that is only one refractive grat- ing. However, in materials with large nonlinearities many complex processes can occur together and it is therefore possible that a number of different gratings can be written simultaneously within the nonlinear material. A typical example is the serf-pumps phase conjugate mirror [ 71 where a stongly nonlin~ crys- tal is illuminated by a single laser beam. Through the fanning effect [ 8J and total internal reflections in the crystal, the pump beams are generated. Since the beams are coherent and have the same polarisation, different gratings can be written in the crystal during the interaction. Previous studies of mixed grating formation have been concerned with specific cases of intensity and  284 P.M. lkbyshire, T.J. Half Optics Communications 117 (1995) 283-289 coupling strength relationships in photorefractive crys- tals [ 9 J More recently, by direct numerical integra- tion of the coupled wave equations and various an- alytical techniques when pump depletion, linear ab- sorption and arbitrary coupling strengths are included [lO,llJ. A novel approach to the study of dynamic grating formation through the nonlinear coupled wave equa- tions has been based upon the identification of un- derlying group symmetries. The solution to the FWM problem in the single grating regime can be described as a dyn~ical system acting on the special unitary group of two-dimensional transformations. SU( 2) in the case of the transmission only grating and SU( 1,l) for the reflection only grating [ 6,12-141. This paper focuses on Kerr type nonlinearities where no satura- tion of the nonlinearity occurs and all possible refrac- tive gratings are present with equal coupling strength. The group fo~ulation is developed further and a new symmetry group is uncovered for the mixed grating case. The two point boundary vaIue problem is then solved and a solution for steady state phase conjuga- tion is found where the results clearly show the mul- tiplicity of the solutions. 2. Coupled-wave analysis In the standard FWM geometry a slab of nonlin- ear material, in this case a Kerr type media, is placed between the planes z = 0 and z = L. This is illumi- nated by two counterpropagating pump beams and a signal beam. The nonlinear interaction inside the ma- terial creates a phase conjugate replica of the signal beam propagating in the opposite direction, Fig. 1. In general there are four gratings written as a result of the complex coupling mechanisms within the four-wave mixing interaction, see Fig. 2. Assuming that each beam has a plane wave form and self-diffraction terms are ignored, the FWh4 in- teraction in the slowly varying and paraxial approx- :.>TK; ~~~~~~~~ z=o 2s z Fig. 1. The standard FWh4 geometry and beam definitions. (d) ‘A4 Fig. 2. The four possible gratings written simultaneously during the FWh4 intemction: (a) T~s~ssion grating. (b) Reflection grating. (c) urns-pump grating. (d) Signal-conjugate grating. imation without absorption, can be described by the following compact matrix equation, (2.1) Es0 belongs to the set of 4 x 4 Dirac matrices, 1y: s a 4 x 4 matrix containing all the grating terms and A is a four vector of beam amplitudes which are functions of z in the steady state, where * denotes complex conjugate. The complete set of 4 x 4 Dirac matrices may be written, where ci,j are the Pauli spin matrices, and a = The grating terms ~~ = yr(AIA; + AsA;) and K = y,(A,A; + AzA$) are associated with the transmis-  P.M. Darbyshire, T.J. Hall/Optics Communications 117 (199s) 283-289 285 sion and reelection gratings, respectively. The other two gratings K~ = y, AzA;> and K, = ys AIAG) arise due to two-wave mixing between the counterpropa- gating pump beams and the signal-conjugate. The yj are the complex coupling constants which are material dependent parameters. The phase of the yj describes the spatial phase shift between the index grating and the interference pattern. Note that there is no division by la, the total intensity, and we take yj to be real in the final results, because we are dealing with Kerr type media as opposed to photorefractive media which have a saturated nonline~ity. Eq. (2.1) has the solution, A(z) =U(z)A(O), where, dU lE3’ dz -=xxJ. 2.2) Since K: is Hermitian and traceless it follows that, UEsaU+ = E3c that is, the Dirac matrix, E3a is con- served under tr~sfo~ations, U. In addition, the Det (U) = 1 hence U must belong to the special uni- tary group of four-dimensional symmetry SU(2,2). It can be shown that the higher dimensional group for the mixed grating case contains the lower dimen- sional groups for the single grating cases. That is, the SU(2) and SU( 1,l) groups for the obsession and reflection only gratings [ 12-141, respectively are irreducible subgroups of the SU( 2,2) group. The matrix, 6 = AA+, where t denotes the adjoint operation on a matrix, explicitly contains all the rel- evant interference terms that give rise to each of the refractive gratings. It therefore follows that X: will be proportional to projections of 6. From Eq_ (2.1) we have, (2.3) If we introduce an involutiono~rator which utilises the reflection symmetry across the trailing diagonal of 6 given by, M0 = EttM*Ett, where M is any 4 x 4 matrix, Eit is a Dirac matrix and MT is the matrix transpose, then, where we have utilised the properties Ic* = X: and E$, = -E30. If 5, represents the sum of the c and J* matrices then combining E&s. (2.3) and (2.4) gives, dc 2 = iEs&Js - i~,?CE30. dz If we make the approximation that K: = ~(5, + C) (i.e. y, = yr = yP/2 = y,/2 = y with real r), where C represents the diagonal correction terms in our 5, matrix construction, and substitute for X: then, E = @(II&, - j,H) + ir(Df; - SD) , (2.5) where, H = E304’, + j 30 3 and, (2.6) D=E30C. Eq. (2.5) has the solution, (2.7) L(z) =utz)~~(O)U+(z) > where, dU (2.8) - =iy(HfD)U. dz (2.9) Since H and D are Hermitian, traceless and commute with E3e it follows that thou+ = E30, and UU+ = Em. Using these properties and I@. (2.6) we have, H(z) =E3o&‘s(z) +&,(z>E30r that is, H(z) = E3oU~~tO)U+ ~U~~ O)U~E30, therefore, H(z) = UH(O>U+, (2.10) and hence, dU - = iyUH(0) + i?DU dz U(0) = Em, (2.11) which has the formal solution, U(z) = exp(~~~Dd~) exp~i~~H~0)). (2.12)  286 P.M. Darbyshire, T.J. Hall/Optics Cotnnzunications 1 I7 (1995) 283-289 It is now necessary to soive the boundary condition problem in terms of the unitary matrix, U and the matrix, H(0). 3. The boundary value problem Backward four-wave mixing consists of a two-point boundary value problem i.e. only two of the beams at either side of the interaction region are known. These beam amplitudes can be represented by two column vectors f and g, A solution requires the matrix, H(0) to be solved in terms of the two vectors f( 0) and g(L) . Recall Eq. (2.6) at the boundary z = 0, Also, H(O) = lfiO(0)I 83 [-ii(O)1 , where, (3.7) H(0) = E30YJO) + <JO)&o. (3.1) If we substitute g, = y+ l0 into Eq. (3.1) and noting that c(z) = U(z)c(O)U+(z) we have, H(0) = CE30Y03W i- 5°UNE301 ii(O) = * I1 “; 14 > . (3.8) + (U~(~~(E3~~~~) + ~ ~)E3~)U ~)l- (3.2) That is, If we substitute Eqs. (3.6) and (3.7) into Eq. (3.3) noting that P(O) and G(L) are known from the bound~y conditions then, -ii(O) = -P(O) - WVG(L)V+W+ . H(0) =-7@(O) +U+(L)hl(L)U(L), where, (3.3) If we consider the phase conjugate condition AJ( L) = 0 then 6 = gcrs, where g = 13 L) /2. Therefore 6 and G commute at z = L, hence, ?-f(z) = [E@ z)1 633 -G z)1 I B(0) = P(0) + W(L)G(L)W+(L). (3.9) CB enotes the direct sum of two 2 x 2 matrices. i?(z) and G(z) are given by, Q(z) = ff - ~~Trtff~~~~}, Denoting the traceless parts of Bi( 0) , g(O) and G(L) as H, F and G and Ietting W = exp(iyLH) E SU(2) we have the following boundary condition, and G(z) = ggt - ~Tr ggt)rro), where Tr( .) represents the trace of a matrix. We may also write U( z ) and H(0) in terms of the direct sum of two 2 x 2 matrices. Let the unitary operator in Eq. (2.12) be written in terms of two other operators V and W so that, U(z) =V(z)W(z) t (3.4) H=F+WGW+. (3.10) From Eq. (3.8) we can see that the analysis has re- duced the mixed grating problem to that of the trans- mission only grating case. That is, in our matrix repre- sentation of a( 0) only the trans~ssion grating term, K~ s present. Indeed our boundary condition is very similar to that already obtained in Ref. [ 121, where the trans~ssion only grating problem was solved us- ing the SU(2) group. We must also note that in Ref. [12] the 2 x 2 K matrix containing the transmission where V(z) = exp iyJ,“Ddz) and W(z) = ex~(iyzH(0)). It follows that, U(z) = [e z)~~+~z~l~\?tr z>cf,~~~z~l, (3.5) where %‘(z), %(z) E SU(2). Therefore, U(z) = [6(z)] @ rat(z)], (3.6) and so U(z) E SU(2) = [exp(iyp@o) exp(@Ps)] exp(iyz&O)), where, dp I and f ? dz ’ dz -fl+14-(12+13)-  P.M. Darbyshire, T.J. Hull Optics Communications I1 7 I 995) 283-289 287 grating terms, ~~ is traceless, our matrix representation for fi( 0) has a trace equal to the total intensity, IO. The matrices, H, F and G are Hermitian and trace- less and so can be expressed in terms of real three- dimensional vectors. This involves representing the matrices as a dot product of an associated vector with a vector of Pauli spin matrices. The unitary matrix, W represented as an exponential operator with a 2 x 2 Hermitian matrix as its argument can be expanded in a linear form as, w =cos(yLh) + h i sin( yLh) H , (3.11) where h is the positive eigenvalue of the matrix, H. Substituting Fq. (3.11) into (3.10) and making use of the commutator and anticommutator relations given by, [R,S]_=[RS-SR]=2i[rxs].o, (3.12a) and, [R,S]., = [RS + SR] = 2 r. S)CQ,, (3.12b) where CY has components which represent the Pauli spin matrices, ~71, ~72 and trs, and aa is the identity matrix, gives the following three-dimensional vector equation, tan f yLh) J(f-s) x~l~I f+g)--~l h [ f+g) - hl. [ f+g) -hl =T. (3.13) From Eq. (3.13) it is possible to find expressions for the real and imaginary part of the coupling constant, y. If we re-express the tangent function in terms of exponentials then, -i(Z - 1) r= h(Zs-1) ’ where Z = exp( 2iyLh) and hence, (3.14) (3.15) Substituting the complex coupling constant, y = YR + iy into Eq. (3.15) and calculating the argument and modulus of the complex function, 2 gives the follow- ing for the real and imaginary parts of y, YR = arg(Z) + 2F2V 2hL (3.16a) and, -In/Z/ YI=x. (3.16b) From Eq. (3.16a), where n = 0, f I, 32, f3 . . . . it can be seen that multiple solutions for 7~ will be produced as II takes each integer value in the 2nr ambiguity. An expression for h can be found from the dot prod- uct representation for H that is, H=h.c~, (3.17) where h has components hi, h2 and hs. From Eq. (3.17) H can be written as, ’ (3.18) where the eigenvalue h is given by, h=(hf+h;+h$‘2. (3.19) Relations for h,, h: and h3 can be found using the boundary condition and the property that any unitary element of the group W(2) has a representation in terms of the three Euler angles, 5, p and v given by, w= cos( 6) ei(lt+v)/z sin(P) eifp--P)/2 _ sin(S) e--iW--VI/2 cos(e) e-i(fi+v)/2 > ’ (3.20) Thus h is parameterised by 5, p and v and hence from Eq. (3.13) so are A, T, Z and the real and imaginary parts of the coupling constant, YR and ye. In fact under the phase conjugate condition A4( L) = 0 only two variables, the Euler angles, 5 and or. are required for a solution. The problem then involves letting 5 and ,u range through their allowed values to find ail the values of A, yR and y, which are consistent with the boundary conditions, noting that in our case yl = 0. 4. Phase conjugate reflectivity The steady state phase conjugate reflectivity is de- fined by, (4.1)
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