OPTICAL MODES IN PT-SYMMETRIC DOUBLE-CHANNEL WAVEGUIDES

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 3, Number /, pp OPTICAL MODES IN PT-SYMMETRIC DOUBLE-CHANNEL WAVEGUIDES Li CHEN, Rujiang LI, Na YANG, Da CHEN, Lu LI* Institute of Theoretical Physics, Shanxi University, Taiyuan 36, China We investigate the unique properties of various analytical optical moes, incluing the funamental moes an the excite moes, in a ouble-channel waveguie with parity-time (PT) symmetry. Base on these optical moes, the epenence of the threshol values for the gain/loss parameter, i.e., PT symmetry breaking points, on the structure parameters is iscusse. We fin that the threshol value for the excite moes is larger than that of the funamental moe. In aition, the beam ynamics in the ouble-channel waveguie with PT symmetry is also investigate. Key wors: optical moes, PT symmetry, ouble-channel waveguies. INTRODUCTION The theoretical an experimental stuies on quantum-optical analogies have seen a spectacular resurgence, in which the propagation of optical waves in waveguies an optical lattices has become an important platform to investigate these unique phenomena []. For example, in quantum mechanics, one of the funamental axioms is the Hermiticity of the Hamiltonian operator associate with the physical observable, which not only implies real eigenvalues but also guarantees probability conservation. However, replacing the Hermiticity conition, a new concept of parity-time (PT) symmetry has been propose in the framework of quantum mechanics [], an can exhibit entirely real eigenvalue spectra even for the non- Hermitian Hamiltonians [ 5]. Although the impact of PT symmetry in quantum mechanics is still ebate, it has been shown that, in optics, PT-relate notions can be implemente in PT symmetric coupler [6, 7] an PT symmetric optical lattices [8-], an the experimental observations have been emonstrate [ 3]. Thus, optics can provie a fertile groun to investigate PT-relate beam ynamics incluing the nonreciprocal responses, the power oscillations, an the optical transparency. Furthermore, PT-relate beam ynamics in the nonlinear regimes has been stuie extensively, such as, optical solitons in PT-symmetric potentials [8, 4, 5] an uniirectional invisibility inuce by PT-symmetric perioic structures [6, 7]. Among these investigations, the ynamics of light-beam propagation in a coupler compose of a ouble-channel waveguie with PT symmetry gaine particular attention because it can exhibit some universal properties in linear an nonlinear regimes [6, 7,, ]. Inee, the in past years, the oublechannel waveguie structure as an important platform has been stuie extensively, an the results have shown that the coupler can support symmetry-preserving solutions, which have linear counterparts, an symmetry-breaking solutions without any linear counterparts [8 ]. In this paper, we mainly focus on various analytical stationary solutions in a ouble-channel waveguie with PT symmetry, an iscuss the properties of the symmetry-preserving solutions. In aition, we will also iscuss the beam ynamics in the ouble-channel waveguie with PT symmetry. The paper is organize as follows. In Sec. II, the moel equations escribing beam propagation in a ouble-channel couple waveguie with PT symmetry are presente. The properties of symmetry-preserving optical moes an the epenence of the PT symmetry breaking points on the structure parameters are iscusse in Sec. III. Meanwhile, the beam ynamics in the ouble-channel waveguie with PT symmetry is also emonstrate. The conclusions are summarize in Sec. IV.

2 Optical moes in PT-symmetric ouble-channel waveguies 47. MODEL EQUATIONS We consier a planar ouble-channel waveguie with the complex refractive inex istribution n(x), which is of the form nx () = nr ini as L/ D< x< L/ an nx ( ) = nr ini as L/ < x< L/ D, otherwise, nx ( ) = n ( < n r ), where D an L represent the with of channel an the separation between channels, respectively; while n an n r are the substrate inex an channel inex, respectively, an n i is the gain/loss parameter. Without loss of generality, here we assume that n i >. One can easily see that the complex * refractive inex istribution nx ( ) satisfies the necessary conition nx ( ) = n( x) for PT symmetry. Thus the complex refractive inex istribution presents a ouble-channel waveguie structure with PT symmetry [6,7]. Uner slowly varying envelope approximation, the wave equation governing beam propagation in such a waveguie can be written as knx [ ( ) n] i = z k x n, () where (z, x) is the envelope function an k = πn /λ is wave number, with λ being wavelength of the optical source generating the beam. Equation () escribes the propagation of an optical beam in a ouble-channel waveguie with PT symmetry. It shoul be pointe out that, in contrast to conventional optical systems, the actual total power Pz ( ) = xis no longer conserve in PT symmetric structures, but the quasipower Qz ( ) = ( zx, ) ( z, x)xis a constant of motion inepenent of istance z [6 ]. * Introucing the following normalize transformations ( z, x) = P / lϕζξ (, ), ξ = x / l an ζ= z/kl, the imensionless form of Eq. () is of the form ϕ ϕ i V ( ξ) ϕ=, () ξ ξ Here V( ξ ) = U( ξ ) i W( ξ, ) which escribes the imensionless PT symmetric ouble-channel waveguie structure, with an U, / /, L < ξ < L D U() ξ =, others, W, L/ < ξ < L/ D, W() ξ= W, L/ D< ξ <L/,, others, where U = k l (n r n )/n is the moulation epth of the refractive inex, W = k l n i /n is the imensionless gain/loss parameter, an L = L /l an D = D /l correspon to the scale separation an with of channels, respectively. Thus, the moel () presents a imensionless form for the beam propagation in the PT symmetric ouble-channel waveguie, in which the refractive inex profiles of the oublechannel waveguie structure given by Eqs. (3) an (4) are plotte in Fig.. We consier the stationary solution for Eq. () in the form ϕ( ζ, ξ ) = φ( ξ)exp(i βζ) Fig. The real part (re soli) an the imaginary part (blue ashe) of the refractive inex profile of a ouble-channel waveguie structure with PT symmetry. (3) (4) [ u( ξ ) v( ξ)]exp(i βζ ) where u( ξ ) an v( ξ ) are the real an the imaginary parts of ϕζξ (, ),

3 48 Li Chen, Rujiang Li, Na Yang, Da Chen, Lu Li 3 respectively, an β is the propagation constant. Substituting it into Eq. (), we can fin that the real functions u( ξ ) an v( ξ ) satisfy the following orinary ifferential equation u [ U(ξ) β] u = W(ξ) v, (5) v [ U(ξ) β] v = W(ξ) u, (6) Equations (5) an (6) are a couple system of ifferential equations, which is couple by the gain/loss istribute parameter W (ξ). Here we require that the moes must be normalize to one, i. e. φ =, to ensure their uniqueness for a fixe propagation constant. Thus, we can obtain the optical fiel moes in ouble-channel waveguie with PT symmetry by solving Eqs. (5) an (6). In the following Section, we will etermine the optical moes an we will iscuss their salient properties. 3. OPTICAL MODES AND THEIR PROPERTIES In this Section, we first present the exact solutions of Eqs. (5) an (6). Inee, in the region of L/< ξ < L/ D, the solutions for Eqs. (5) an (6) are of the form u( ξ; A, A, δ, δ ) = A exp( λξ) sin( λξδ ) A exp( λξ) sin( λξδ), v( ξ; A, A, δ, δ ) = A exp( λξ) cos( λξδ) A exp( λξ) cos( λξδ), (7) where λ = {[( U β ) W ] ( U β )} / an λ = {[( U β ) W ] ( U β )} /, an here we have use the assumption W > ue to n i >. In the region L/ D <ξ< L/, the solutions for Eqs. (5) an (6) are of the form u( ξ; B, B, σ, σ ) = B exp( λξ) sin( λξσ ) B exp( λξ) sin( λξσ), v( ξ; B, B, σ, σ ) = B exp( λξ) cos( λξσ ) B exp( λξ) cos( λξσ). (8) In the region ξ< L /, the solutions for Eqs. (5) an (6) shoul be of the form u3( ξ ; C, C) = C cosh( βξ ) C sinh( βξ), v3( ξ ; C, C) = C cosh( βξ ) C sinh( βξ). (9) Also, it is easy to show that the solutions of Eqs. (5) an (6) can be written as βξ u4(; ξ D) = De, βξ v4(; ξ D) = De, () in the region ξ> L/ D, an βξ u5(; ξ E) = Ee, βξ v5(; ξ E) = Ee, () in the region ξ<l/ D. Thus one can construct the analytical global solutions of Eqs. (5) an (6) by employing Eqs. (7 )

4 4 Optical moes in PT-symmetric ouble-channel waveguies 49 where the continuity conitions of u, v, u u5( ξ; E), ξ<l/ D, u( ξ; B, B, σ, σ), L/ D <ξ<l/, u( ξ ) = u3( ξ; C, C), ξ < L/, u( ξ; A, A, δ, δ ), L/ <ξ< L/ D, u4( ξ; D), ξ> L/ D, v5( ξ; E), ξ<l/ D, v( ξ; B,B, σ, σ), L/ D <ξ<l/, v( ξ ) = v3( ξ; C,C ), ξ < L/, v( ξ; A,A, δ, δ ), L/ <ξ< L/ D, v4( ξ; D), ξ> L/ D, ξan v ξ at the bounaries require u( L/ D; B, B, σ, σ ) = u5( L/ D; E), u( L/ D; B, B, σ, σ ) = u5( L/ D; E), u( L/ ; B, B, σ, σ ) = u3( L/ ; C, C), u( L/ ; B, B, σ, σ ) = u3( L/ ; C, C), u( L/; A, A, δ, δ ) = u3( L/; C, C), u( L/ ; A, A, δ, δ ) = u3( L/ ; C, C), u( L/ D; A, A, δ, δ ) = u4( L/ D; D), u( L/ D; A, A, δ, δ ) = u4( L/ D; D), () (3) an v ( L/ D; B, B, σ, σ ) = v ( L/ D; E ), 5 v( L/ D; B, B, σ, σ ) = v5( L/ D; E), v ( L/ ; B, B, σ, σ ) = v ( L/ ; C C ), 3, v( L/ ; B, B, σ, σ ) = v3( L/ ; C, C), v ( L/ ; A, A, δ, δ ) = v ( L/ ; C C ), 3, v( L/ ; A, A, δ, δ ) = v3( L/ ; C, C), v ( L/ D; A, A, δ, δ ) = v ( L/ D; D ), 4 v( L/ D; A, A, δ, δ ) = v4( L/ D; D), (4) an the normalization conition requires that ( u v ) ( ξ ) ( ξ) =. (5) In Eq. (), there are seventeen parameters A, A, δ, δ, B, B, σ, σ, C, C, C, C, D, D, E, E an the propagation constant β, which can be calculate by solving numerically Eqs. (3), (4), an (5). Once these parameters are etermine, one can obtain the exact optical moes for the ouble-channel waveguie with PT symmetry.

5 5 Li Chen, Rujiang Li, Na Yang, Da Chen, Lu Li 5 Fig. The istribution plots of the real an the imaginary parts of optical moe with symmetric real part for: a) W =; b) W =.5; c) W =., respectively. Note that the corresponing propagation constants are β=.75,.75, an.747, respectively; ) the corresponing phases. Here the system parameters are D=3.5, L=5 an U =.5. Fig. 3 The istribution plots of the real an imaginary parts of optical moe with antisymmetric real part for: a) W =; b) W =.5; c) W =., respectively, where the corresponing propagation constant is β=.7,.7, an.74, respectively; ) the corresponing phases. Here the system parameters are the same as in Fig.. Figures an 3 present the istribution plots of two funamental optical moes with symmetric an antisymmetric real parts for the ifferent gain/loss parameter W, respectively. It shoul be pointe out that the two moes are reuce from the symmetric an the antisymmetric optical moes in Ref. [], respectively, as shown in Figs. a an 3a. From them, it can be seen that, with the increasing of the gain/loss parameter W, for the moe with symmetric real part, the real part u always remains a symmetric function an the imaginary part v is antisymmetrically increasing, as shown in Figs. b an c, whereas, for the moe with antisymmetric real part, the real part u always remains an antisymmetric function an the imaginary part v is symmetrically increasing, as shown in Figs. 3b an 3c. In aition, we also fin that the phase ifference at the central position ξ = increases for the moe with symmetric real part, while the phase ifference ecreases for the moe with antisymmetric real part, as shown in Figs. an 3, respectively. The propagation constants of the two moes with symmetric an antisymmetric real parts as a function of the gain/loss parameter W are plotte in Fig. 4a. From it, one can see that, with the increasing of the gain/loss parameter, the propagation constants of the two moes close towars each other, an eventually merge together at a threshol value W th =.6 (for our choice of the parameters). Below the threshol value the propagation constants of the two moes are real, however, once the gain/loss parameter excees the threshol value they become complex. Thus, the threshol value correspons to a branch point, i. e., an exceptional point, at which PT symmetry is spontaneously broken. Because the two moes at the branch point have the same propagation Fig. 4 a) The real part of propagation constant β versus the gain/loss parameter W ; b) an c) the istribution plots of optical moes shown in Fig. an Fig. 4 for W =W th, respectively. Here the system parameters are the same as in Fig.. constant, they form a pair of egenerate moes, as shown in Figs. 4b an 4c. By comparing Fig. 4b an Fig. 4c, one fin that the two egenerate moes have only the phase ifference of π/. Note that this egenerate moe completely iffer from the egenerate state in the nonlinear case []. Furthermore, the excite optical moes in the ouble-channel waveguie structure are iscusse. Figures 5 an 6 show the istribution plots of two excite optical moes with symmetric an antisymmetric real parts for ifferent gain/loss parameters W, respectively. Similarly, one can see that the real part u always keeps a symmetric profile an the imaginary part v is an antisymmetric function with the increasing

6 6 Optical moes in PT-symmetric ouble-channel waveguies 5 of the gain/loss parameter W, for the excite moe with symmetric real part, as shown in Figs. 5b an 5c. For the excite moe with antisymmetric real part, the real part u always keeps an antisymmetric profile an the imaginary part v is a symmetric function with the increasing of the gain/loss parameter W, as shown in Figs. 6b an 6c. The corresponing phase istributions exhibit a more complex structure, as shown in Figs. 5 an 6, respectively. From the results presente in Figs. 5 an 6, one can see that the phase ifference at the central position ξ = increases for the moe with symmetric real part, while ecreases for the moe with antisymmetric real part. However, the phase ifferences at the noes keep invariance. Fig. 5 The istributions of excite moe with a symmetric real part for: a) W =; b) W =.; c) W =.4, where the corresponing propagation constant is β=.93,.978, an.957, respectively; ) the corresponing phases. Here the system parameters are the same as in Fig.. Fig. 6 The istributions of excite moe with antisymmetric real part for: a) W =; b) W =.; c) W =.4, where the corresponing propagation constant is β=.89554,.89586, an.8977, respectively; ) the corresponing phases. Here the system parameters are the same as in Fig.. For the two excite optical moes with symmetric an antisymmetric real parts, the epenence of their propagation constants on the gain/loss parameter is also stuie, an the relevant results are summarize in Fig. 7. The results show that there exists a threshol value W th =.5 (for our choice of the parameters). Below the threshol value the propagation constants of the two excite moes are real, but when the gain/loss parameter excees the threshol value they become complex. Thus a branch point appears at the threshol value, as shown in Fig. 7a. The two excite moes at the branch point have the same propagation constant, so they present a pair of egenerate excite moes, as shown in Figs. 7b an 7c. It is similarly foun that the two egenerate excite moes have only the phase Fig. 7 a) The real part of propagation constant β versus the gain/loss parameter W ; b) an c) the istribution plots of optical moes shown in Fig. 7 an in Fig. 9 for the threshol value W =W th, respectively. The system parameters are the same as in Fig.. ifference of π /, which also iffer from the egenerate excite moe in the nonlinear case []. It shoul be pointe out that the threshol value for the excite optical moes is larger than that of the funamental optical moes, which means that in the ouble-channel waveguie with PT symmetry, it is possible that the funamental moe oes not exist, but the excite moe may exist. It shoul be pointe out that all moes shown in Figs. 7 are symmetry-preserving moes because when W = they can be reuce to what was previously shown in Ref. []. In orer to better unerstan the properties of the optical moes in the ouble-channel waveguies with PT symmetry, we also iscuss the influences of the system parameters on the optical moes.

7 5 Li Chen, Rujiang Li, Na Yang, Da Chen, Lu Li 7 Fig. 8 The epenence of the threshol value of: a) the funamental moes; b) the excite moes on the channel-with D, respectively. Here the system parameters are L=5 an U =.5. Fig. 9 The epenence of the threshol value of: a) the funamental moes; b) the excite moes on the channel separation L, respectively. Here the system parameters are D=3.5 an U =.5. Fig. The epenence of the threshol value of: a) the funamental moes; b) the excite moes on the moulation epth U, respectively. Here the system parameters are L=5 an D=3.5. Figures 8 present the epenence of the threshol value for both the funamental an the excite moes, on the channel-with D, the channel separation L an the moulation epth U, respectively. From these figures, one fin that the value of W th varies strongly with the change of D, L, an U. This means that one can exten the region of the gain/loss parameter W by tuning properly the system s parameters. For example, we can enhance the threshol value of the gain/loss parameter by ecreasing the separation between the channels, as shown in Fig. 9. In the following, we iscuss the beam ynamics. For the linear system, the superposition of two eigenmoes with symmetric an antisymmetric real parts is also a solution of the system. We assume that the solution with symmetric real part is of the form ( u i v)exp(i β z), an the solution with antisymmetric real part is of the form ( u i v) exp(i β z). Note that the two moes have the following symmetry properties on the axis ξ =, i. e., u u, v v an u u, v v. We consier both the sum moe an the ifference moe for the above two moes as follows [( i i i )e z ( i )e z u v β β = u v ], (6) [( i i i )e z ( i )e z u v β β = u v ], (7) in which the corresponing power istribution for the sum moe an the ifference moe are ( ) ( )cos( ) ( )sin( ) u v u v u u v v z u v vu z = β β, (8)

8 8 Optical moes in PT-symmetric ouble-channel waveguies 53 ( ) ( )cos( ) ( )sin( ) u v u v u u v v z u v vu z = β β, (9) respectively, where β = β β. The case W = correspons to the conventional Hermitian system, i. e., v = v =. Thus, Eqs. (8) an (9) can be reuce to ( ) cos( ) u u u u z = β, () ( ) cos( ) u u u u z = β, () respectively. From the expressions () an (), one can see that the wave propagation exhibits left-right symmetric oscillations with a beat length of L = π/ β, which means a reciprocal wave propagation, as shown in Figs. a an b for the superpositions of the two funamental state moes, an in Figs. a an b for the superpositions of the two excite state moes. In this case, the total power is conserve. Fig. The evolution plots of the power istributions (top panel) an (bottom panel) of the superposition of two funamental state moes for the ifferent gain/loss parameter W, where W = in a) an b) W =.8, in c) an ), an W =.6 in e) an f). Here the system parameters are L = 5, D = 3.5 an U =.5. Fig. The evolution plots of the power istributions (top panel) an (bottom panel) of the superposition of two excite state moes for the ifferent gain/loss parameter W, where W = in a) an b) W =.5 in c) an ), an W =.5 in e) an f). Here the system parameters are L = 5, D = 3.5 an U =.5. With the increasing of the gain/loss parameter W, as expresse by Eqs. (8) an (9), the beat length becomes larger because the propagation constant ifference β ecreases (see the plots of the propagation constant versus the gain/loss parameter in Figs. 4a an 7a), an the wave propagation exhibits the characteristic feature of the nonreciprocal propagation, as shown in Figs. c an for the superposition of the funamental state moes, an in Figs. c an for the superposition of the excite state moes [7]. In this case, the total power is no longer conserve. However, when W is increase to the threshol value W th, the beat length is infinite because the two propagation constants at the threshol value are the same. In this case, the power in the ouble-channel waveguies shares the same istribution because of the superposition of the two egenerate moes (see Figs. 4b, 4c an Figs. 7b, 7c), in which the corresponing power istributions of an are of the form ( ) ( ), = u u v v = ( ) ( ), u u v v respectively. Thus, boun states are forme in the ouble-channel waveguies, as shown in Figs. e an f for the funamental state moes, an Figs. e an f for the excite state moes.

9 54 Li Chen, Rujiang Li, Na Yang, Da Chen, Lu Li 9 4. CONCLUSIONS In summary, we have analytically iscusse the funamental an the excite optical moes in a oublechannel waveguie with PT symmetry. The results have shown that these moes are symmetry-preserving solutions. The spontaneous PT symmetry breaking point (the occurrence of a threshol value for the gain/loss parameter) has been foun by employing the epenence of the propagation constant on the gain/loss parameter. Also, the epenence of the threshol value on the structure parameters has been stuie, an it was foun that the threshol value for the excite moe is larger than that of the funamental moe. The obtaine results have shown that the PT symmetry conition may result in a large gain/loss effect by properly tuning the waveguie structure parameters. Moreover, the beam ynamics has been investigate in etail. ACKNOWLEDGEMENT This research was supporte by the National Natural Science Founation of China grant 67879, the Shanxi Scholarship Council of China Grant No. -, an the Provincial Program of Unergrauate Innovative Training of Shanxi University. REFERENCES. S. LONGHI, Quantum-optical analogies using photonic structures, Laser an Photon. Rev., 3, pp. 43 6, 9.. C.M. BENDER, S. BOETTCHER, Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry, Phys. Rev. Lett., 8, pp , C.M. BENDER, S. BOETTCHER, P.N. MEISINGER, PT-symmetric quantum mechanics, J. Math. Phys., 4, pp. 9, C.M. BENDER, D.C. BRODY, H.F. JONES, Complex Extension of Quantum Mechanics, Phys. Rev. Lett., 89, p. 74,. 5. A. MOSTAFAZADEH, Spectral Singularities of Complex Scattering Potentials an Reflection an Transmission Coefficients at Real Energies, Phys. Rev. Lett.,, p. 4, R. El-GANAINY, K.G. MAKRIS, D.N. CHRISTODOULIDES, Z.H. MUSSLIMANI, Theory of couple optical PT-symmetric structures, Opt. Lett., 3, pp , S. KLAIMAN, U. GUNTHER, N. MOISEYEV, Visualization of Branch Point in PT-Symmetric Waveguies, Phys. Rev. Lett.,, p. 84, Z.H. MUSSLIMANI, R. El-GANAINY, K.G. MAKRIS, D.N. CHRISTODOULIDES, Optical Solitons in PT Perioic Potentials, Phys. Rev. Lett.,, p. 34, K.G. MAKRIS, R. El-GANAINY, D.N. CHRISTODOULIDES, Z.H. MUSSLIMANI, Beam Dynamics in PT Symmetric Optical Lattices, Phys. Rev. Lett.,, p. 394, 8.. K.G. MAKRIS, R. El- GANAINY, D.N. CHRISTODOULIDES Z.H. MUSSLIMANI, PT-symmetric optical lattices, Phys. Rev. Lett., 8, p. 6387,.. A. GUO, G.J. SALAMO, D. DUCHESNE, R. MORANDOTTI, M. VOLATIER-RAVAT, V. AIMEZ, G.A. SIVILOGLOU, D. N. CHRISTODOULIDES, Observation of PT-Symmetry Breaking in Complex Optical Potentials, Phys. Rev. Lett., 3, p. 939, 9.. C.E. RUTER, K.G. MAKRIS, R. El-GANAINY, D.N. CHRISTODOULIDES, M. SEGEV, D. KIP, Observation of parity-time symmetry in optics, Nat. Phys., 6, pp. 9 95,. 3. T. KOTTOS, Broken symmetry makes light work, Nat. Phys., 6, pp ,. 4. X. ZHU, H. WANG, L.X. ZHENG, H.G. LI, Y.J. HE, Gap solitons in parity-time complex perioic optical lattices with the real part of superlattices, Opt. Lett., 36, pp ,. 5. H.G. LI, Z.W. SHI, X.J. JIANG, X. ZHU, Gray solitons in parity-time symmetric potentials, Opt. Lett., 36, pp ,. 6. H. RAMEZANI, T. KOTTOS, R. El-GANAINY, D.N. CHRISTODOULIDES, Uniirectional nonlinear PT-symmetric optical structures, Phys. Rev., A 8, p. 4383,. 7. Z. LIN, H. RAMEZANI, T. EICHELKRAUT, T. KOTTOS, H. CAO, D.N. CHRISTODOULIDES, Uniirectional Invisibility Inuce by PT-Symmetric Perioic Structures, Phys. Rev. Lett., 6, p. 39,. 8. Y.V. KARTASHOV, B.A. MALOMED, L. TORNER, Solitons in nonlinear lattices, Rev. Mo. Phys., 83, pp ,. 9. P.G. KEVREKIDIS, ZHIGANG CHEN, B.A. MALOMED, D.J. FRANTZESKAKIS, M.I. WEINSTEIN, Spontaneous symmetry breaking in photonic lattices: Theory an experiment, Phys. Lett., A 34, 758, 5.. ZE EV BIRNBAUM, B.A. MALOMED, Families of spatial solitons in a two-channel waveguie with the cubic-quintic nonlinearity, Physics, D 37, pp , 8.. J.F. JIA, Y.P. ZHANG, W.D. LI, L. LI, Optical moes in a nonlinear ouble-channel waveguie, Opt. Commun., 83, pp. 3 37,.. R.J. LI, F. LV, L. Li, Z.Y. XU, Symmetry breaking an manipulation of nonlinear optical moes in an asymmetric oublechannel waveguie, Phys. Rev., A 84, p. 3385,. Receive December 8,