NONLINEAR PARITY-TIME-SYMMETRY BREAKING IN OPTICAL WAVEGUIDES WITH COMPLEX GAUSSIAN-TYPE POTENTIALS

Transcription

1 NONLINEAR PARITY-TIME-SYMMETRY BREAKING IN OPTICAL WAVEGUIDES WITH COMPLEX GAUSSIAN-TYPE POTENTIALS PENGFEI LI 1, BIN LIU 1, LU LI 1,, DUMITRU MIHALACHE 2,3 1 Institute of Theoretical Physics, Shanxi University, Taiyuan , China llz@sxu.edu.cn 2 Academy of Romanian Scientists, 54 Splaiul Independentei, RO Bucharest, Romania 3 Horia Hulubei National Institute of Physics and Nuclear Engineering, Magurele-Bucharest, Romania Received January 15, 2016 In this paper, the effect of the input optical power on the nonlinear parity-time (PT )-symmetry breaking in optical waveguides with complex-valued Gaussian-type potentials is investigated. The results show that in the presence of nonlinearity, the eigenvalue spectra undergo two phase-transition-like behaviors associated with the increasing of gain and loss strength. The first one arises at a PT -symmetry breaking point from which the eigenvalue spectrum of the ground state is bifurcated into two branches, the real-valued and the complex-valued ones, describing the typical feature of the broken PT -symmetry. The other one arises at a coalescence point for two modes, at which they are terminated. These two nonlinear transition points are separated from their linear counterparts. Furthermore, it is found that with the increasing of the input power, the coalescence point as a function of the input power undergoes also a transition from the coalescence of the ground and first excited modes to the coalescence of the first and second excited modes. Finally, the linear stability of the stationary solutions of the nonlinear dynamical system is also systematically analyzed. Key words: Parity-time-symmetric waveguide; phase transition. PACS: Tg, Ge, Er. 1. INTRODUCTION In quantum mechanics, one of fundamental postulates is hermiticity of the Hamiltonian operators associated with physical observables, which not only implies real eigenvalues but also guarantees probability conservation. Bender and Boettcher [1] generalized this axiom in a complex domain by applying the concept of paritytime (PT )-symmetry and showed that non-hermitian Hamiltonians with PT -symmetry can have entirely real spectra [2]. In this case, the complex-valued potential U(x) is requested to satisfy a necessary condition U(x) = U ( x), where asterisk stands for complex conjugation [3]. While the subject of PT -symmetric systems was initially studied in the context of quantum mechanics, the concept of PT -symmetry was also applied in optical systems, such as PT -symmetric couplers [4 6] and PT -symmetric optical lattices [7 9]. Relevant experimental observations were demonstrated by using passive elements [10], by introducing gain or loss via photorefractive two-wave mixing [11], RJP Rom. 61(Nos. Journ. Phys., 3-4), Vol , Nos. 3-4, (2016) P , (c) 2016 Bucharest, - v.1.3a*

2 578 Pengfei Li et al. 2 and in temporal photonic PT -symmetric lattices [12]. Subsequently, the concept of PT -symmetry was extended in other fields, ranging from electronic circuits [13 15] and metamaterials [16 19] to multilevel atomic systems [20], and it provides a fertile ground to implement PT -related beam dynamics including non-reciprocal light propagation, power oscillations, optical transparency and so on. Furthermore, PT -related beam dynamics in nonlinear regimes has been extensively investigated. In Kerr media, optical solitons, including bright solitons, gap solitons, Bragg solitons, gray or dark solitons and vortices, supported by various complex PT -symmetric (or periodic) potentials have been studied during the past years [7, 21 32]. Of much interest is the fact that stable bright solitons can exist in defocusing Kerr media with PT -symmetric potentials [33]. In media with competing nonlinearities, solitons in PT -symmetric potentials have been investigated analytically [34]. Also, optical solitons in mixed linear-nonlinear lattices, optical lattice solitons in media described by the complex Ginzburg-Landau model with PT -symmetric periodic potentials, vector solitons in PT -symmetric coupled waveguides, defect solitons in PT -symmetric lattices, solitons in chains of PT -invariant dimers, solitons in nonlocal media, solitons and breathers in PT -symmetric nonlinear couplers, unidirectional optical transport induced by the balanced gain-loss profiles, the nonlinearly induced PT transition in photonic systems, and asymmetric optical amplifiers based on parity-time symmetry have been reported [35 51]. It is well known that for linear PT -symmetric systems one of the key features is that there exists a threshold value, i.e., a PT -symmetry breaking point, for the strength of the imaginary part of complex potential, beyond which the spectrum is no longer real-valued but instead it becomes complex-valued. Two eigenstates coalesce at the threshold value, which means that a phase-transition-like behavior takes place in the system [6, 52]. However, in the presence of nonlinearity, the impact of the input power on this phenomenon has been rarely studied. In this paper, we will investigate in detail the effects of the input power on nonlinear phase-transition-like points. Our results show that the eigenvalue spectrum exhibits two transition points: one is a bifurcation point from a real eigenvalue to a complex one, at which PT -symmetry is broken, and the other one is a coalescence point for two modes, at which the two modes are terminated. The paper is organized as follows. In the next section, the model and its reductions are introduced. Nonlinear eigenvalue spectrum diagrams for a specific input power and the corresponding eigenstates are presented in Sec. 3. In Sec. 4, the influence of the input power values on the bifurcation and coalescence points is analyzed by employing both a variational method and direct numerical simulations. In Sec. 5, we discuss systematically the problem of linear stability of stationary solutions in both Gaussian and super-gaussian complex potentials. Finally, our conclusions are summarized in Sec. 6.

3 3 Nonlinear parity-time-symmetry breaking in optical waveguides MODEL AND ITS REDUCTIONS We begin our analysis by considering optical wave propagation in a local Kerr nonlinear planar graded-index waveguide, which is governed by the following (1+1)- dimensional paraxial wave equation i ψ z + 1 2k 0 2 ψ x 2 + k 0 [F (x) n 0 ] n 0 ψ + k 0n 2 n 0 ψ 2 ψ = 0, (1) where ψ(z,x) is the optical field envelope function, k 0 = 2πn 0 /λ is the wavenumber with λ and n 0 being the wavelength of the optical source and the background refractive index, respectively, F (x) = F R (x) + if I (x) is a complex function, in which the real part represents the refractive index distribution and the imaginary part stands for the gain/loss, and n 2 is the Kerr nonlinear parameter. Introducing the normalized transformations ψ(z,x) = (k 0 n 2 L D /n 0 ) 1/2 Ψ(ζ,ξ), ξ = x/w 0, and ζ = z/l D with the diffraction length L D = 2k 0 w0 2, respectively, Eq. (1) can be rewritten in a dimensionless form i Ψ ζ + 2 Ψ ξ 2 + U (ξ)ψ + σ Ψ 2 Ψ = 0, (2) where σ = n 2 / n 2 = ±1 corresponds to self-focusing (+) and self-defocusing ( ) Kerr nonlinearity, respectively. Here U(ξ) V (ξ)+iw (ξ), V (ξ) = 2k0 2w2 0 [F R(x) n 0 ]/n 0, and W (ξ) = 2k0 2w2 0 F I(x)/n 0, which are required to be even and odd functions, respectively, for the PT -symmetric system. In the following, we only consider the case of self-focusing nonlinearity, i.e., σ = 1. We search for the solutions of Eq. (2) in the form Ψ(ζ,ξ) = ϕ(ξ)e iβζ, where ϕ(ξ) is a complex-valued function and β = β R +iβ I. Substitution into Eq. (2) yields [ ] d 2 dξ 2 + U (ξ) + σ ϕ(ξ) 2 e 2βIζ β ϕ(ξ) = 0. (3) Note that here we introduced the complex propagation constant or eigenvalue β in order to better explain PT -symmetry breaking in the system. Indeed, in the absence of the nonlinear term, i.e., for σ = 0, Eq. (3) is selfconsistent whether the eigenvalue β is real or complex. Based on this, PT -symmetry breaking in the linear regime has been studied extensively and the results showed that there exists a threshold value for the gain and loss strength, the so-called PT - symmetry breaking point, so that beyond this value the eigenvalue β is no longer real but instead it becomes complex and the two modes merge together at the threshold value [6, 52]. Once the eigenvalue becomes complex, the solution for Eq. (3) without the nonlinear term is no longer the stationary solution for the system. In this case, the imaginary part β I of the eigenvalue would cause an exponential growth or decay depending on the sign of β I.

4 580 Pengfei Li et al. 4 However, considering the nonlinear effect, Eq. (3) is only valid for β I = 0 or ζ = 0. In this case, Eq. (3) can be rewritten as [ ] d 2 dξ 2 + U (ξ) + σ ϕ(ξ) 2 β ϕ(ξ) = 0. (4) This implies that solving Eq. (4), we can obtain the stationary solution for the system (2) only if the eigenvalue β is real, while when β I 0 it can only present the onset of optical field. Unlike the above linear case, such solutions for Eq. (4) lose their physical relevance. Based on Eq. (4), PT -related properties in nonlinear regimes have been extensively investigated in various physical settings both in optics and in Bose-Einstein condensates (BECs). However, the PT -symmetry breaking transition in the linear setting has been investigated in detail during the past years, while in the presence of nonlinearity, the relevant studies in the context of BECs within the mean-field approximation of the Gross-Pitaevskii equation were quite recently reported [53 55]; see also a few recent overview papers in the area of BECs [56 59]. Here, we aim to study the effects of the input power on the PT -symmetry breaking behavior of planar graded-index waveguides in the presence of Kerr-type optical nonlinearities. As a typical example, we take the super Gaussian-type potential in the form )2m ( ) 2m ξξ0 ξξ0 V (ξ) = V 0 e ( ξ, W (ξ) = W 0 e, (5) ξ 0 where the parameters V 0 and W 0 are the normalized modulation strengths of the refractive index and the balanced gain and loss, in which W 0 characterizes the degree of non-hermiticity of the PT -symmetric system, ξ 0 is the potential width, and m is the power index of super-gaussian function. The value m = 1 is for a Gaussian potential, whereas the profile of V (ξ) gradually resembles a rectangular distribution with the increase of m. 3. THE NONLINEAR EIGENVALUE SPECTRUM DIAGRAM In the Section, we will present the eigenvalue spectrum diagram for a special choice of the value of the input power, and we will obtain the corresponding eigenstates. It should be emphasized that when the eigenvalue spectra for Eq. (4) are real, the corresponding eigenstates are the stationary solutions or the modes for the system (2), while for the complex eigenvalues, these eigenstates describe only the onset of the optical field and have no physical relevance though they can explain the broken PT -symmetry [53, 54]. Based on Eq. (4), we calculated numerically the nonlinear eigenvalue spectra for the ground and first excited states, respectively. Fig. 1 presents the dependence of

5 5 Nonlinear parity-time-symmetry breaking in optical waveguides 581 Fig. 1 (Color online) The dependence of β R and β I on the gain and loss strength W 0 for P 0 = 1.5. (a) The relations between β R and W 0, where the thin blue-solid and red-dotted curves are for the linear case (σ = 0), the thick blue-solid and red-dotted curves are for the nonlinear case (σ = 1), and the opencircle curve represents the branch of eigenvalue spectrum emerging from the nonlinear bifurcation point at W b = 7.1. The nonlinear coalescence point of the upper branch (thick blue-solid curve) and the lower branch (thick red-dotted curve) is at W crn = 7.4, and the PT -symmetry breaking point in the linear case is at W crl = (b) The relations between β I and W 0, where the dash-dotted and square curves are for the linear and nonlinear cases, respectively. Here m = 1, σ = 1, ξ 0 = 2, and V 0 = 6. the eigenvalue β on the gain and loss strength W 0 for a given input power P 0 = 1.5, where the thick blue-solid and thick red-dotted curves correspond to the ground and first excited states, respectively. For comparison, the corresponding results for the linear case are also depicted. From Fig. 1, one can see that, with the increasing of the gain and loss strength W 0, the eigenvalue spectra undergo two different transition points. One transition point is a bifurcation point W b from the real eigenvalue spectrum of the ground state, from which the eigenvalue spectrum is bifurcated into two branches, the real and the complex branches, where on the complex branch a pair of complex conjugate numbers occurs [see the open-circle curve in Fig. 1(a) and the square curve in Fig. 1(b)]. The other transition point is the coalescence point W crn for the eigenvalue spectra of the ground and first excited states, at which the two states are terminated; see the thick blue-solid and red-dotted curves in Fig. 1(a). This feature is different from its counterpart in the linear case, in which the bifurcation and the coalescence points coincide; see the thin blue-solid and red-dotted curves in Fig. 1(a) and the dash-dotted curve in Fig. 1(b). Now we turn to the consideration of the corresponding eigenstates. When the eigenvalue is a real number, the corresponding eigenstate is the optical mode or the stationary solution for the system (2), as shown in Fig. 2, which depicts the profiles of the ground and first excited modes and the corresponding phases for different gain

6 582 Pengfei Li et al. 6 Fig. 2 (Color online) The profiles of the modes and the corresponding phases with the increasing of the gain and loss strength W0 until WcrN. (a) and (b) the ground state mode and the corresponding phase; (c) and (d) the first excited state mode and the corresponding phase. Here the parameters are the same as in Fig. 1. and loss strength W0 until WcrN. The results show that the amplitudes of these modes are evenly symmetric. Also, from Fig. 2, one can see that the ground mode has only a peak and its amplitude decreases with the increasing of the gain and loss strength, while the first excited mode possesses two peaks for smaller W0 and evolves gradually into a single peak when the gain and loss strength W0 is close to the coalescence point WcrN, as shown in Figs. 2(a) and (c). This is a natural result because the ground and first excited modes merge together at WcrN. Figures 2(b) and 2(d) present the corresponding phases for the ground and first excited modes. From Figs. 2(b) and 2(d) one can see that for the Hermitian system (W0 = 0), the profiles of the phases for the ground and first excited modes are constant and step function, respectively, and with the increasing of W0, the phase s amplitude becomes gradually large up to the coalescence point. Also, we calculate the phase difference between the ground and first excited modes, and find that it decreases with the increasing of W0. Especially, when reaching at the coalescence point WcrN, it becomes a constant of π/2, which means that the ground and first excited modes coalesce at WcrN except for the phase factor eiπ/2. Thus, we can analyze the transverse power flow S = Sdξ in the PT

7 7 Nonlinear parity-time-symmetry breaking in optical waveguides 583 symmetric system, where S is the transverse component of the Poynting vector [7]: S = i ( ΨΨ ξ Ψ Ψ ξ ) = 2Φξ ϕ 2, (6) which gives the transverse power-flow density, where we have used Ψ = ϕ(ξ) e iφ(ξ)+iβζ for the real eigenvalue β. From Eq. (6), one can find that the phase s gradient and the optical field intensity determine the transverse power-flow density. Obviously, for the Hermitian system (W 0 = 0), the transverse power flow vanishes. However, when W 0 0, the phase distributions are smoothly and monotonously increasing, as shown in Figs. 2(b) and 2(d). In this case, the transverse power flow S is always larger than zero and increases with the increasing of W 0. This means that the transverse power flow reaches a maximum at the coalescence point W crn, at which both the transverse power flows for the ground and first excited modes are equal. Fig. 3 (Color online) The distributions of the eigenstate and the corresponding phase on the broken PT -symmetry branch for different gain and loss strengths W 0 > W b. (a) The eigenstate and (b) the corresponding phase. Here the parameters are the same as in Fig. 1. On the complex branch of eigenvalue spectrum for the ground state, the corresponding eigenstates for Eq. (4) are asymmetric when W 0 > W b, as shown in Fig. 3 (here only the states with β I > 0 are given). This means that the P T -symmetry is broken at the bifurcation point W b. So the complex branch is a broken PT -symmetry branch. In order to understand better the main characteristics of the PT -symmetry breaking phenomenon, we begin with the continuity equation, which can be derived from Eq. (2) and is of the form [45, 60 62] ρ ζ + S = 2W (ξ)ρ, (7) ξ

8 584 Pengfei Li et al. 8 where ρ = ΨΨ is the optical field intensity. Integration of Eq. (7) over all ξ yields d dζ ρ(ξ,ζ)dξ = 2 W (ξ)ρ(ξ,ζ)dξ, (8) where the term on right-hand side can be interpreted as a source or a sink for optical waves. Applying Ψ(ζ,ξ) = ϕ(ξ)e iβζ into Eq. (8) and restricting that the eigenvalue β is real, the optical field intensity ρ = ϕ(ξ) 2 is independent of ζ. Thus from Eq. (8), we can obtain W (ξ) ϕ(ξ) 2 dξ = 0. (9) This implies that the transport of optical power in the transverse direction can make the PT -symmetric system to keep a balanced gain and loss, which is equivalent to a source-free system. However, once the eigenvalue falls into the broken PT - symmetry branch, Eq. (8) leads to W (ξ) ϕ(ξ) 2 dξ = β I ϕ(ξ) 2 ρdξ 0. (10) This is a necessary condition that the eigenvalue falls into the complex branch when W 0 > W b. 4. INFLUENCE OF THE INPUT POWER ON THE NONLINEAR PT -TRANSITION POINTS In the nonlinear case, the input power plays an important role in the optical system because nonlinearity is related to the beam power. In this Section, we focus on the influence of the input power on the nonlinear transition points by employing both the variational method and direct numerical simulations. By rewriting Eq. (2) as [63, 64] i Ψ ζ + 2 Ψ ξ 2 + V (ξ)ψ + σ Ψ 2 Ψ = iw (ξ)ψ, (11) and introducing the Lagrangian density L 0 = i 2 ( Ψ Ψ ζ ΨΨ ζ) Ψξ 2 + V ψ 2 + σ 2 ψ 4, (12) it is easy to verify that Eq. (11) can be recovered from the equation δl 0 /δψ = iw (ξ)ψ, where δ/δψ = n=0 ( 1)n ( n / ξ n )( / Ψ nξ ) ( / ζ)( / Ψ ζ ) with Ψ nξ = n Ψ / ξ n and Ψ ζ = Ψ / ζ. We introduce the test function in the form Ψ(ξ,ζ) = AF (X)e iφ(x)+iµ with X = (ξ q)/a, where A is the amplitude, a represents the width and q is the center

9 9 Nonlinear parity-time-symmetry breaking in optical waveguides 585 position, φ and µ are the phase in the transverse and propagation directions, respectively. Here we assume that these parameters are ζ-dependent and are real except for µ = µ R + iµ I. Substituting the test function into the expression of the Lagrangian we have Here I 0 = I 4 = L 0 = L 0 (Ψ,Ψ )dξ, L 0 = A 2 da dζ I 2 + A 2 dq dζ I 1 A 2 a dµ R dζ I 0 A2 a I 3 A2 a I 4 F 2 dx,i 1 = F 2 φ 2 XdX,I 5 = + A 2 ai σa4 ae 2µ I I 6 e 2µ I. F 2 φ X dx,i 2 = V F 2 dx,i 6 = XF 2 φ X dx,i 3 = F 4 dx, F 2 XdX, where I 0 is related to the input power, I 1 represents the power flow in transverse direction, I 2 represents the net transverse power flow, I 3 and I 4 refer to the diffraction effect of optical field, and I 5 and I 6 are related to the effects of refraction and nonlinearity. By the principle of least action, we have the Euler-Lagrange equation as L 0 p d dζ ( L0 p ζ ) = i W ( Ψ Ψ ) p Ψ Ψ dξ, (13) p where p represents one of the test function parameters A, a, q, µ R, and µ I. Substituting A and µ R for p into Eq. (13), we can obtain the following differential equations for µ R and µ I dµ R dζ = dq dζ I 1 + da I 2 ai 0 dζ dµ I dζ = 1 da A dζ + 1 2a + I 5 + σa 2 e 2µ I I 6 I 3 ai 0 I 0 I 0 a 2 I 4 I 0 a 2, (14) I 0 da dζ + 1 di 0 2I 0 dζ + 1 ai 0 W F 2 dξ. (15) To explain qualitatively the influence of the input power on the nonlinear PT phase transition, we take µ R = β R ζ, µ I = β I ζ, and assume the parameters A, q, and a are constants. Thus Eqs. (14) and (15) can be reduced to β R = I 5 + σa 2 e 2β Iζ I 6 I 3 I 0 I 0 a 2 I 4 I 0 a 2, (16) I 0

10 586 Pengfei Li et al. 10 β I = 1 W F 2 dξ, (17) ai 0 where I 0, I 3, I 4, I 5, and I 6 are constants for a given ansatz function F and the input power P 0 = A 2 ai 0. Note that Eq. (16) is only valid for β I = 0 or ζ = 0. In the case of β I = 0, the eigenvalue has only the real part β R. From Eq. (16), one find that β R increases with the increasing of the input power, and is a linear function of I 5, which is determined by the distribution of the refractive index. To verify our analysis, we performed the corresponding numerical simulations, as shown in Fig. 4, which presents the real eigenvalue spectra for the ground, the first, and the second excited modes in the plane of W 0 -P 0 for the super Gaussian-type potential with different indices m. Note that Fig. 4 has not included the complex-valued branch. From it one can see that for a given W 0, β R is a monotonically increasing function of P 0, which agrees with the result predicted by Eq. (16). From Fig. 4 one can obtain the dependence of the coalescence point W crn on the input power P 0, as shown in Fig. 5. The result shows that there exists a turning point so that W crn is decreasing instead of increasing when the input power P 0 is larger than the corresponding value at the turning point. This is because that the real eigenvalue spectrum as a function of W 0 has a transition at the turning point, beyond which the coalescence point for the ground and first excited modes is transferred to that for the first and second excited modes. For our choice of the parameters, the turning points are at P 0 = 1.7 (for m = 1), 3.6 (for m = 2), and 3.8 (for m = 3), respectively. Because Fig. 4 has not included the broken PT -symmetry branch, it can be used to ascertain the number of modes of the system. From it one can see that for a given P 0, the system has three modes when W 0 < W crn, while when W 0 > W crn the number of modes is reduced to one from three because two of the modes are terminated at the coalescence point W crn. Furthermore, comparing the results for the super Gaussian-type potentials with different indices, it is found that the coalescence points are appearing earlier for the case of super Gaussian-type potentials than for the case of purely Gaussian potentials. In the case ζ = 0, the imaginary part β I can be nonzero. Thus Eq. (17) can be used to acquire some information on the broken PT -symmetry branch. From it, one can see that β I equals to zero when F is an even symmetric distribution, as shown in Figs. 3(a) and 3(c). However, if the state s distribution deviates from the center position (ξ = 0), as shown in Fig. 2(a) with W 0 > W b, β I 0 due to + W F 2 dξ 0. In this case, the imaginary part β I decreases with the increasing of P 0 due to P 0 ai 0. This qualitative result can be verified by direct numerical simulations, as shown in Fig. 6, which presents the contour plots of the real and imaginary parts of the eigenvalue for the ground state as a function of W 0 and P 0.

11 11 Nonlinear parity-time-symmetry breaking in optical waveguides 587 Fig. 4 (Color online) The real eigenvalue spectra βr for the ground, the first, and the second excited modes in the plane W0 -P0. (a), (b), and (c) are for the super Gaussian-type potentials with m = 1, 2, and 3, respectively. The other parameters are the same as in Fig. 1.

12 588 Pengfei Li et al. 12 Fig. 5 (Color online) The dependence of the bifurcation point W b, the coalescence point W crn, and the linear PT -symmetry breaking point W crl on the input power P 0, for m = 1, 2, and 3. Here the short-dotted, short-dashed, and solid curves are for the bifurcation points, the curves with circles, triangles, and rhombuses are for the coalescence points, and the dotted lines are for the corresponding PT -symmetry breaking points W crl in the linear case. Here the parameters are the same as in Fig. 4. Fig. 6 (Color online) The contour plots of the real and imaginary parts of the eigenvalue as a function of W 0 and P 0 for the super Gaussian-type potentials with different indices m, where (a) m = 1, (b) m = 2, and (c) m = 3, and the top and bottom panels are for β R and β I, respectively. The parameters are the same as in Fig. 4.

13 13 Nonlinear parity-time-symmetry breaking in optical waveguides 589 Furthermore, from Figs. 6(a 2 ), 6(b 2 ), and 6(c 2 ), one can easily obtain the dependence of the value of the bifurcation point W b on the input power P 0. W b is an increasing function of the input power P 0, as shown in Fig. 5. Similarly, from Fig. 6 one can see that the eigenvalue spectrum for the super Gaussian-type potential is bifurcated earlier than for the Gaussian potential. The dependence of the bifurcation point W b and the coalescence point W crn on the input power P 0 is summarized in Fig. 5, in which for the sake of comparison, the corresponding PT -symmetry breaking point W crl in the linear case is also presented. From Fig. 5, we can get several conclusions as follows. First, the nonlinear PT phase transition points W b and W crn are always larger than their linear conterpart W crl. Physically, the self-focusing nonlinearity plays the role of an effective potential that enhances the real component of the potential and leads to the delay of the PT transition point. Second, the bifurcation point W b is a monotonically increasing function of the input power P 0, while for the coalescence point W crl, there exists a turning point so that W crl is decreasing instead of increasing when the input power P 0 exceeds the turning point. Indeed, this turning point is the transition point from the coalescence of the ground and first excited modes to the coalescence of the first and second excited modes, as shown in Fig. 4. Third, in the presence of the nonlinearity, the two nonlinear PT transition points, the bifurcation point W b and the coalescence point W crn are separated from the linear counterpart W crl with the increasing of the input power, where the former causes the breaking of PT -symmetry and the latter leads to the coalescence of modes and their termination. 5. LINEAR STABILITY ANALYSIS In this Section, we will discuss the stability of the optical modes by employing both the linear stability analysis and direct numerical simulations. The linear stability analysis can be performed by adding a small perturbation to a known solution ϕ(ξ) Ψ(ξ,ζ) = e iβζ [ ϕ(ξ) + u(ξ)e δζ + v (ξ)e δ ζ ], (18) where ϕ(ξ) is the stationary solution with real propagation constant β, and u(ξ) and v(ξ) are small perturbations with u, v ϕ. Substituting Eq. (18) into Eq. (2) and keeping only the linear terms, we obtain the following linear eigenvalue problem ( )( ) ( ) L11 L i 12 u u = δ, (19) L 21 L 22 v v where L 11 = d 2 /dξ 2 + U β + 2σ ϕ 2, L 12 = σϕ 2, L 21 = L 12 and L 22 = L 11, and δ is the eigenvalue. If δ contains a positive real part, the solution ϕ(ξ) is linearly unstable, otherwise, ϕ(ξ) is linearly stable. In the following, the linear stability of the stationary solution is characterized by the largest real part of δ. Thus, if it is

14 590 Pengfei Li et al. 14 zero, the solution is linearly stable, otherwise, it is linearly unstable. Here, the linear eigenvalue problem (19) can be solved by making use of the Fourier collocation method [65]. Fig. 7 (Color online) The dependence of the eigenvalue for Eq. (19) on the gain and loss strength W 0, where (a 1 ), (b 1 ), and (c 1 ) are the imaginary components of eigenvalues for the ground, the first, and the second excited modes in the super Gaussian-type potential with m = 2 and P 0 = 4, respectively. The corresponding real components are shown in (a 2 ), (b 2 ), and (c 2 ), respectively. The other parameters are the same as in Fig. 1. As a generic example, we calculated the dependence of the imaginary and real components of eigenvalues for Eq. (19) on the gain and loss strength W 0 in the super Gaussian-type potential with m = 2 and the input power P 0 = 4, as shown in Fig. 7. From it, one can see that the imaginary and the real components of the eigenvalue appear in pairs, and so the eigenvalues for Eq. (19) are quartet symmetric, which is different from the Hermitian system. Also, it can be found that the ground and first excited modes are linearly stable in the interval 0 < W 0 < 4.1 [see Fig. 7(a 2 )] and 3.75 < W 0 < 4.6 [see Fig. 7(b 2 )], respectively. The second excited mode is unstable throughout its range of existence, as shown in Figs. 7(c 1 ) and 7(c 2 ). To confirm the results of linear stability analysis, we perturb the three modes in Figs. 7(a 2 ), 7(b 2 ), and 7(c 2 ) by 1% random-noise perturbations. The initial distributions and nonlinear evolutions are summarized in Fig. 8. From it, one can see that the ground and first excited modes with zero real component can propagate robustly, as shown in Figs. 8(a 2 ) and 8(b 2 ), while the second excited mode is unstable due to the presence of the non-zero real component, as shown in Fig. 8(c 2 ).

15 15 Nonlinear parity-time-symmetry breaking in optical waveguides 591 Fig. 8 (Color online) The initial field distributions and the nonlinear evolution plots. (a 1 ), (b 1 ), and (c 1 ) present the initial field profiles for the ground, the first, and the second excited modes (see the open circles in Fig. 7), and (a 2 ), (b 2 ), and (c 2 ) are the corresponding evolution plots, respectively. Here W 0 = 4 and the other parameters are the same as in Fig. 7. Fig. 9 (Color online) The largest real component of the eigenvalue for Eq. (19) with the super Gaussian-type potential for m = 1,2, and 3 versus the input power P 0 and the gain and loss strength W 0. The panels from top to bottom are for the ground, the first, and the second excited modes, respectively, and the panels from left to right correspond to m = 1,2, and 3, respectively. The regions of the broken PT -symmetry are shown by shaded areas. The other parameters are the same as in Fig. 4.

16 592 Pengfei Li et al. 16 Finally, we present the dependence of the largest real part of the eigenvalue on both the input power P 0 and the gain and loss strength W 0 for the super Gaussiantype potential with m = 1, 2, and 3, as shown in Fig. 9. The ground mode is unstable [see Figs. 9(a 1 ), 9(b 1 ), and 9(c 1 )], when the input power and the gain and loss strength are large. The stability domains for the first excited mode is mainly concentrated on two separated regions; see the arrows in Figs. 9(a 2 ), 9(b 2 ), and 9(c 2 ). The region of stability for the second excited mode is smaller than that for the ground mode, and is mainly concentrated on the range of smaller input power and gain and loss strength, as shown in Figs. 9(a 3 ), 9(b 3 ), and 9(c 3 ). Also, it is found that for the super Gaussian-type potential, the stability regions become much smaller with the increasing of the index m. 6. CONCLUSIONS In summary, we have investigated the effects of the input power on the PT - symmetry phase transition in nonlinear PT -symmetric waveguides with the super Gaussian-type potentials. The obtained results have shown that in the presence of Kerr-type nonlinearity, the eigenvalue spectra exhibit two nonlinear transition points with the increasing of the gain and loss strength. The first one is the bifurcation point W b from which the eigenvalue spectrum for the ground state is bifurcated into two branches, the real and complex branches, which means that the PT -symmetry is broken. The second one is the coalescence point W crn, at which two modes merge together and are terminated. Indeed, the two nonlinear PT transition points W b and W crn are separated from the linear PT -symmetry breaking point W crl due to the presence of nonlinearity. Also, it was found that with the increasing of the input power, the coalescence point W crn as a function of the input power undergoes a transition from the coalescence of the ground and first excited modes to the coalescence of the first and second excited modes. Finally, the linear stability of the corresponding nonlinear optical modes has been investigated by numerical methods. Our findings can be applied to other classes of PT -symmetric complex-valued potentials. Acknowledgements. This research was supported by the National Natural Science Foundation of China, through Grants No and , and by the Shanxi Scholarship Council of China, through Grant No

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