nonlinear parity-time-symmetry breaking in optical …solitons, bragg solitons, gray or dark...

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

PENGFEI LI1, BIN LIU1, LU LI1,∗, DUMITRU MIHALACHE2,3

1Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, ChinaE-mail∗: [email protected]

2Academy of Romanian Scientists, 54 Splaiul Independentei, RO-050094 Bucharest, Romania3Horia 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-typepotentials is investigated. The results show that in the presence of nonlinearity, theeigenvalue spectra undergo two phase-transition-like behaviors associated with the in-creasing of gain and loss strength. The first one arises at a PT -symmetry breakingpoint from which the eigenvalue spectrum of the ground state is bifurcated into twobranches, the real-valued and the complex-valued ones, describing the typical featureof 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 fromtheir linear counterparts. Furthermore, it is found that with the increasing of the inputpower, the coalescence point as a function of the input power undergoes also a transitionfrom the coalescence of the ground and first excited modes to the coalescence of thefirst and second excited modes. Finally, the linear stability of the stationary solutionsof the nonlinear dynamical system is also systematically analyzed.

Key words: Parity-time-symmetric waveguide; phase transition.

PACS: 42.65.Tg, 03.65.Ge, 11.30.Er.

1. INTRODUCTION

In quantum mechanics, one of fundamental postulates is hermiticity of theHamiltonian operators associated with physical observables, which not only impliesreal eigenvalues but also guarantees probability conservation. Bender and Boettcher[1] generalized this axiom in a complex domain by applying the concept of parity-time (PT )-symmetry and showed that non-Hermitian Hamiltonians with PT -sym-metry can have entirely real spectra [2]. In this case, the complex-valued potentialU(x) is requested to satisfy a necessary condition U(x) = U∗(−x), where asteriskstands for complex conjugation [3].

While the subject of PT -symmetric systems was initially studied in the con-text of quantum mechanics, the concept of PT -symmetry was also applied in opticalsystems, such as PT -symmetric couplers [4–6] and PT -symmetric optical lattices[7–9]. Relevant experimental observations were demonstrated by using passive el-ements [10], by introducing gain or loss via photorefractive two-wave mixing [11],

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24Rom. Journ. Phys., Vol. 61, Nos. 3-4, P. 577–594, Bucharest, 2016

578 Pengfei Li et al. 2

and in temporal photonic PT -symmetric lattices [12]. Subsequently, the concept ofPT -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 fer-tile ground to implement PT -related beam dynamics including non-reciprocal lightpropagation, power oscillations, optical transparency and so on.

Furthermore, PT -related beam dynamics in nonlinear regimes has been exten-sively investigated. In Kerr media, optical solitons, including bright solitons, gapsolitons, Bragg solitons, gray or dark solitons and vortices, supported by variouscomplex PT -symmetric (or periodic) potentials have been studied during the pastyears [7, 21–32]. Of much interest is the fact that stable bright solitons can existin defocusing Kerr media with PT -symmetric potentials [33]. In media with com-peting nonlinearities, solitons in PT -symmetric potentials have been investigatedanalytically [34]. Also, optical solitons in mixed linear-nonlinear lattices, opticallattice solitons in media described by the complex Ginzburg-Landau model withPT -symmetric periodic potentials, vector solitons in PT -symmetric coupled waveg-uides, defect solitons in PT -symmetric lattices, solitons in chains of PT -invariantdimers, solitons in nonlocal media, solitons and breathers in PT -symmetric non-linear couplers, unidirectional optical transport induced by the balanced gain-lossprofiles, the nonlinearly induced PT transition in photonic systems, and asymmetricoptical 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 featuresis that there exists a threshold value, i.e., a PT -symmetry breaking point, for thestrength of the imaginary part of complex potential, beyond which the spectrum is nolonger real-valued but instead it becomes complex-valued. Two eigenstates coalesceat the threshold value, which means that a phase-transition-like behavior takes placein the system [6, 52]. However, in the presence of nonlinearity, the impact of theinput power on this phenomenon has been rarely studied.

In this paper, we will investigate in detail the effects of the input power onnonlinear phase-transition-like points. Our results show that the eigenvalue spectrumexhibits two transition points: one is a bifurcation point from a real eigenvalue to acomplex one, at which PT -symmetry is broken, and the other one is a coalescencepoint for two modes, at which the two modes are terminated.

The paper is organized as follows. In the next section, the model and its reduc-tions are introduced. Nonlinear eigenvalue spectrum diagrams for a specific inputpower and the corresponding eigenstates are presented in Sec. 3. In Sec. 4, the influ-ence of the input power values on the bifurcation and coalescence points is analyzedby employing both a variational method and direct numerical simulations. In Sec.5, we discuss systematically the problem of linear stability of stationary solutions inboth Gaussian and super-Gaussian complex potentials. Finally, our conclusions aresummarized in Sec. 6.

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24

3 Nonlinear parity-time-symmetry breaking in optical waveguides 579

2. MODEL AND ITS REDUCTIONS

We begin our analysis by considering optical wave propagation in a local Kerrnonlinear planar graded-index waveguide, which is governed by the following (1+1)-dimensional paraxial wave equation

i∂ψ

∂z+

1

2k0

∂2ψ

∂x2+k0 [F (x)−n0]

n0ψ+

k0n2n0

|ψ|2ψ = 0, (1)

where ψ(z,x) is the optical field envelope function, k0 = 2πn0/λ is the wavenumberwith λ and n0 being the wavelength of the optical source and the background refrac-tive index, respectively, F (x) = FR(x)+ iFI(x) is a complex function, in which thereal part represents the refractive index distribution and the imaginary part stands forthe gain/loss, and n2 is the Kerr nonlinear parameter. Introducing the normalizedtransformations ψ(z,x) = (k0 |n2|LD/n0)

−1/2Ψ(ζ,ξ), ξ = x/w0, and ζ = z/LD

with the diffraction length LD = 2k0w20, respectively, Eq. (1) can be rewritten in a

dimensionless form

i∂Ψ

∂ζ+∂2Ψ

∂ξ2+U (ξ)Ψ+σ |Ψ|2Ψ= 0, (2)

where σ = n2/ |n2| = ±1 corresponds to self-focusing (+) and self-defocusing (−)Kerr nonlinearity, respectively. Here U(ξ)≡ V (ξ)+iW (ξ), V (ξ) = 2k20w

20[FR(x)−

n0]/n0, and W (ξ) = 2k20w20FI(x)/n0, which are required to be even and odd func-

tions, respectively, for the PT -symmetric system. In the following, we only considerthe 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)eiβζ , whereϕ(ξ) is a complex-valued function and β = βR+ iβI . Substitution into Eq. (2) yields[

d2

dξ2+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 inorder 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 self-consistent whether the eigenvalue β is real or complex. Based on this, PT -symmetrybreaking in the linear regime has been studied extensively and the results showedthat 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 realbut instead it becomes complex and the two modes merge together at the thresholdvalue [6, 52]. Once the eigenvalue becomes complex, the solution for Eq. (3) withoutthe nonlinear term is no longer the stationary solution for the system. In this case,the imaginary part βI of the eigenvalue would cause an exponential growth or decaydepending on the sign of βI .

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


However, considering the nonlinear effect, Eq. (3) is only valid for βI = 0 orζ = 0. In this case, Eq. (3) can be rewritten as[

d2

dξ2+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 βI = 0 it can only present the onsetof optical field. Unlike the above linear case, such solutions for Eq. (4) lose theirphysical relevance.

Based on Eq. (4), PT -related properties in nonlinear regimes have been ex-tensively investigated in various physical settings both in optics and in Bose-Einsteincondensates (BECs). However, the PT -symmetry breaking transition in the linearsetting has been investigated in detail during the past years, while in the presence ofnonlinearity, the relevant studies in the context of BECs within the mean-field ap-proximation 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 aimto study the effects of the input power on the PT -symmetry breaking behavior ofplanar 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

V (ξ) = V0e−(ξξ0

)2m

, W (ξ) =W0ξ

ξ0e−(ξξ0

)2m

, (5)

where the parameters V0 and W0 are the normalized modulation strengths of therefractive index and the balanced gain and loss, in whichW0 characterizes the degreeof non-hermiticity of the PT -symmetric system, ξ0 is the potential width, and m isthe power index of super-Gaussian function. The value m = 1 is for a Gaussianpotential, whereas the profile of V (ξ) gradually resembles a rectangular distributionwith the increase of m.

3. THE NONLINEAR EIGENVALUE SPECTRUM DIAGRAM

In the Section, we will present the eigenvalue spectrum diagram for a specialchoice of the value of the input power, and we will obtain the corresponding eigen-states. 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 ofthe optical field and have no physical relevance though they can explain the brokenPT -symmetry [53, 54].

Based on Eq. (4), we calculated numerically the nonlinear eigenvalue spectrafor the ground and first excited states, respectively. Fig. 1 presents the dependence of

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


Fig. 1 – (Color online) The dependence of βR and βI on the gain and loss strength W0 for P0 = 1.5.(a) The relations between βR and W0, where the thin blue-solid and red-dotted curves are for the linearcase (σ=0), the thick blue-solid and red-dotted curves are for the nonlinear case (σ=1), and the open-circle curve represents the branch of eigenvalue spectrum emerging from the nonlinear bifurcation pointat Wb =7.1. The nonlinear coalescence point of the upper branch (thick blue-solid curve) and the lowerbranch (thick red-dotted curve) is at WcrN = 7.4, and the PT -symmetry breaking point in the linearcase is at WcrL = 5.56. (b) The relations between βI and W0, where the dash-dotted and square curvesare for the linear and nonlinear cases, respectively. Here m= 1, σ = 1, ξ0 = 2, and V0 = 6.

the eigenvalue β on the gain and loss strength W0 for a given input power P0 = 1.5,where the thick blue-solid and thick red-dotted curves correspond to the ground andfirst excited states, respectively. For comparison, the corresponding results for thelinear case are also depicted. From Fig. 1, one can see that, with the increasing ofthe gain and loss strengthW0, the eigenvalue spectra undergo two different transitionpoints. One transition point is a bifurcation point Wb from the real eigenvalue spec-trum of the ground state, from which the eigenvalue spectrum is bifurcated into twobranches, the real and the complex branches, where on the complex branch a pair ofcomplex conjugate numbers occurs [see the open-circle curve in Fig. 1(a) and thesquare curve in Fig. 1(b)]. The other transition point is the coalescence point WcrNfor the eigenvalue spectra of the ground and first excited states, at which the twostates are terminated; see the thick blue-solid and red-dotted curves in Fig. 1(a). Thisfeature is different from its counterpart in the linear case, in which the bifurcationand the coalescence points coincide; see the thin blue-solid and red-dotted curves inFig. 1(a) and the dash-dotted curve in Fig. 1(b).

Now we turn to the consideration of the corresponding eigenstates. When theeigenvalue is a real number, the corresponding eigenstate is the optical mode or thestationary solution for the system (2), as shown in Fig. 2, which depicts the profilesof the ground and first excited modes and the corresponding phases for different gain

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


Fig. 2 – (Color online) The profiles of the modes and the corresponding phases with the increasing ofthe gain and loss strength W0 until WcrN. (a) and (b) the ground state mode and the correspondingphase; (c) and (d) the first excited state mode and the corresponding phase. Here the parameters are thesame as in Fig. 1.

and loss strengthW0 untilWcrN. The results show that the amplitudes of these modesare evenly symmetric. Also, from Fig. 2, one can see that the ground mode has onlya 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 gradu-ally into a single peak when the gain and loss strength W0 is close to the coalescencepointWcrN, as shown in Figs. 2(a) and (c). This is a natural result because the groundand first excited modes merge together at WcrN.

Figures 2(b) and 2(d) present the corresponding phases for the ground and firstexcited modes. From Figs. 2(b) and 2(d) one can see that for the Hermitian sys-tem (W0 = 0), the profiles of the phases for the ground and first excited modes areconstant and step function, respectively, and with the increasing of W0, the phase’samplitude becomes gradually large up to the coalescence point. Also, we calculatethe phase difference between the ground and first excited modes, and find that it de-creases with the increasing ofW0. Especially, when reaching at the coalescence pointWcrN, it becomes a constant of π/2, which means that the ground and first excitedmodes coalesce at WcrN except for the phase factor eiπ/2.

Thus, we can analyze the transverse power flow S =∫ +∞−∞ Sdξ in the PT -

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


symmetric system, where S is the transverse component of the Poynting vector [7]:

S = i(ΨΨ∗

ξ −Ψ∗Ψξ

)= 2Φξ |ϕ|2 , (6)

which gives the transverse power-flow density, where we have used

Ψ= |ϕ(ξ)|eiΦ(ξ)+iβζ

for the real eigenvalue β. From Eq. (6), one can find that the phase’s gradient and theoptical field intensity determine the transverse power-flow density. Obviously, forthe Hermitian system (W0 = 0), the transverse power flow vanishes. However, whenW0 =0, the phase distributions are smoothly and monotonously increasing, as shownin Figs. 2(b) and 2(d). In this case, the transverse power flow S is always larger thanzero and increases with the increasing of W0. This means that the transverse powerflow reaches a maximum at the coalescence point WcrN, at which both the transversepower 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 brokenPT -symmetry branch for different gain and loss strengths W0 > Wb. (a) The eigenstate and (b) thecorresponding phase. Here the parameters are the same as in Fig. 1.

On the complex branch of eigenvalue spectrum for the ground state, the cor-responding eigenstates for Eq. (4) are asymmetric when W0 >Wb, as shown in Fig.3 (here only the states with βI > 0 are given). This means that the PT -symmetry isbroken at the bifurcation pointWb. So the complex branch is a broken PT -symmetrybranch.

In order to understand better the main characteristics of the PT -symmetrybreaking phenomenon, we begin with the continuity equation, which can be derivedfrom Eq. (2) and is of the form [45, 60–62]

∂ρ

∂ζ+∂S∂ξ

=−2W (ξ)ρ, (7)

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


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 opticalwaves.

Applying Ψ(ζ,ξ) = ϕ(ξ)eiβζ into Eq. (8) and restricting that the eigenvalue βis real, the optical field intensity ρ= |ϕ(ξ)|2 is independent of ζ. Thus from Eq. (8),we can obtain ∫ +∞

−∞W (ξ) |ϕ(ξ)|2dξ = 0. (9)

This implies that the transport of optical power in the transverse direction can makethe PT -symmetric system to keep a balanced gain and loss, which is equivalentto 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 = βI

∫ +∞

−∞|ϕ(ξ)|2 ρdξ = 0. (10)

This is a necessary condition that the eigenvalue falls into the complex branch whenW0 >Wb.

4. INFLUENCE OF THE INPUT POWER ON THE NONLINEAR PT -TRANSITIONPOINTS

In the nonlinear case, the input power plays an important role in the opticalsystem because nonlinearity is related to the beam power. In this Section, we focuson the influence of the input power on the nonlinear transition points by employingboth the variational method and direct numerical simulations.

By rewriting Eq. (2) as [63, 64]

i∂Ψ

∂ζ+∂2Ψ

∂ξ2+V (ξ)Ψ+σ |Ψ|2Ψ=−iW (ξ)Ψ, (11)

and introducing the Lagrangian density

L0 =i

2

(Ψ∗Ψζ −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 δL0/δΨ∗ =

−iW (ξ)Ψ, where δ/δΨ∗=∑∞

n=0(−1)n(∂n/∂ξn)(∂/∂Ψ∗nξ)−(∂/∂ζ)(∂/∂Ψ∗

ζ) withΨ∗

nξ = ∂nΨ∗/∂ξn and Ψ∗ζ = ∂Ψ∗/∂ζ.

We introduce the test function in the form Ψ(ξ,ζ) = AF (X)eiφ(X)+iµ withX = (ξ− q)/a, where A is the amplitude, a represents the width and q is the center

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


position, φ and µ are the phase in the transverse and propagation directions, respec-tively. Here we assume that these parameters are ζ-dependent and are real except forµ= µR+ iµI .

Substituting the test function into the expression of the Lagrangian

L0 =

∫ +∞

−∞L0 (Ψ,Ψ

∗)dξ,

we have

L0 =A2da

dζI2+A

2 dq

dζI1−A2a

dµRdζ

I0−A2

aI3−

A2

aI4

+A2aI5+1

2σA4ae−2µI I6e

−2µI .

Here

I0 =

∫ +∞

−∞F 2dX,I1 =

∫ +∞

−∞F 2φXdX,I2 =

∫ +∞

−∞XF 2φXdX,I3 =

∫ +∞

−∞F 2XdX,

I4 =

∫ +∞

−∞F 2φ2

XdX,I5 =

∫ +∞

−∞V F 2dX,I6 =

∫ +∞

−∞F 4dX,

where I0 is related to the input power, I1 represents the power flow in transversedirection, I2 represents the net transverse power flow, I3 and I4 refer to the diffrac-tion effect of optical field, and I5 and I6 are related to the effects of refraction andnonlinearity. By the principle of least action, we have the Euler-Lagrange equationas

∂L0

∂p− d

dζ

(∂L0

∂pζ

)= i

∫ +∞

−∞W

(Ψ∗∂Ψ

∂p−Ψ

∂Ψ∗

∂p

)dξ, (13)

where p represents one of the test function parameters A, a, q, µR, and µI . Substitut-ing A and µR for p into Eq. (13), we can obtain the following differential equationsfor µR and µI

dµRdζ

=dq

dζ

I1aI0

+da

dζ

I2aI0

+I5I0

+σA2e−2µII6I0

− I3a2I0

− I4a2I0

, (14)

dµIdζ

=1

A

dA

dζ+

1

2a

da

dζ+

1

2I0

dI0dζ

+1

aI0

∫ +∞

−∞WF 2dξ. (15)

To explain qualitatively the influence of the input power on the nonlinear PTphase transition, we take µR = βRζ, µI = βIζ, and assume the parameters A, q, anda are constants. Thus Eqs. (14) and (15) can be reduced to

βR =I5I0

+σA2e−2βIζI6I0

− I3a2I0

− I4a2I0

, (16)

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


βI =1

aI0

∫ +∞

−∞WF 2dξ, (17)

where I0, I3, I4, I5, and I6 are constants for a given ansatz function F and the inputpower P0 =A2aI0. Note that Eq. (16) is only valid for βI = 0 or ζ = 0.

In the case of βI = 0, the eigenvalue has only the real part βR. From Eq. (16),one find that βR increases with the increasing of the input power, and is a linearfunction of I5, which is determined by the distribution of the refractive index. Toverify our analysis, we performed the corresponding numerical simulations, as shownin Fig. 4, which presents the real eigenvalue spectra for the ground, the first, and thesecond excited modes in the plane of W0-P0 for the super Gaussian-type potentialwith different indices m. Note that Fig. 4 has not included the complex-valuedbranch. From it one can see that for a given W0, βR is a monotonically increasingfunction of P0, which agrees with the result predicted by Eq. (16).

From Fig. 4 one can obtain the dependence of the coalescence point WcrN onthe input power P0, as shown in Fig. 5. The result shows that there exists a turningpoint so that WcrN is decreasing instead of increasing when the input power P0 islarger than the corresponding value at the turning point. This is because that the realeigenvalue spectrum as a function of W0 has a transition at the turning point, beyondwhich the coalescence point for the ground and first excited modes is transferred tothat for the first and second excited modes. For our choice of the parameters, theturning points are at P0 = 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 beused to ascertain the number of modes of the system. From it one can see that fora given P0, the system has three modes when W0 <WcrN, while when W0 >WcrNthe number of modes is reduced to one from three because two of the modes areterminated at the coalescence point WcrN.

Furthermore, comparing the results for the super Gaussian-type potentials withdifferent indices, it is found that the coalescence points are appearing earlier for thecase of super Gaussian-type potentials than for the case of purely Gaussian potentials.

In the case ζ = 0, the imaginary part βI can be nonzero. Thus Eq. (17) canbe used to acquire some information on the broken PT -symmetry branch. Fromit, one can see that βI 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 fromthe center position (ξ = 0), as shown in Fig. 2(a) with W0 > Wb, βI = 0 due to∫ +∞−∞ WF 2dξ = 0. In this case, the imaginary part βI decreases with the increasing

of P0 due to P0 ∝ aI0. This qualitative result can be verified by direct numericalsimulations, as shown in Fig. 6, which presents the contour plots of the real andimaginary parts of the eigenvalue for the ground state as a function of W0 and P0.

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


Fig. 4 – (Color online) The real eigenvalue spectra βR for the ground, the first, and the second excitedmodes 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.

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


Fig. 5 – (Color online) The dependence of the bifurcation point Wb, the coalescence point WcrN, andthe linear PT -symmetry breaking point WcrL on the input power P0, for m = 1, 2, and 3. Herethe 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 correspondingPT -symmetry breaking points WcrL 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 functionof W0 and P0 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 βR and βI , respectively. The parametersare the same as in Fig. 4.

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


Furthermore, from Figs. 6(a2), 6(b2), and 6(c2), one can easily obtain the de-pendence of the value of the bifurcation point Wb on the input power P0. Wb is anincreasing function of the input power P0, as shown in Fig. 5. Similarly, from Fig.6 one can see that the eigenvalue spectrum for the super Gaussian-type potential isbifurcated earlier than for the Gaussian potential.

The dependence of the bifurcation pointWb and the coalescence pointWcrN onthe input power P0 is summarized in Fig. 5, in which for the sake of comparison,the corresponding PT -symmetry breaking point WcrL in the linear case is also pre-sented. From Fig. 5, we can get several conclusions as follows. First, the nonlinearPT phase transition pointsWb andWcrN are always larger than their linear conterpartWcrL. Physically, the self-focusing nonlinearity plays the role of an effective poten-tial that enhances the real component of the potential and leads to the delay of thePT transition point. Second, the bifurcation point Wb is a monotonically increasingfunction of the input power P0, while for the coalescence point WcrL, there exists aturning point so that WcrL is decreasing instead of increasing when the input powerP0 exceeds the turning point. Indeed, this turning point is the transition point fromthe coalescence of the ground and first excited modes to the coalescence of the firstand second excited modes, as shown in Fig. 4. Third, in the presence of the non-linearity, the two nonlinear PT transition points, the bifurcation point Wb and thecoalescence point WcrN are separated from the linear counterpart WcrL with the in-creasing of the input power, where the former causes the breaking of PT -symmetryand 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 employingboth the linear stability analysis and direct numerical simulations. The linear stabilityanalysis can be performed by adding a small perturbation to a known solution ϕ(ξ)

Ψ(ξ,ζ) = eiβζ[ϕ(ξ)+u(ξ)eδζ +v∗ (ξ)eδ

∗ζ]

, (18)

where ϕ(ξ) is the stationary solution with real propagation constant β, and u(ξ) andv(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

i

(L11 L12

L21 L22

)(uv

)= δ

(uv

), (19)

where L11 = d2/dξ2+U −β+2σ |ϕ|2, L12 = σϕ2, L21 = −L∗12 and L22 = −L∗

11,and δ is the eigenvalue. If δ contains a positive real part, the solution ϕ(ξ) is linearlyunstable, otherwise, ϕ(ξ) is linearly stable. In the following, the linear stability ofthe stationary solution is characterized by the largest real part of δ. Thus, if it is

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


zero, the solution is linearly stable, otherwise, it is linearly unstable. Here, the lineareigenvalue problem (19) can be solved by making use of the Fourier collocationmethod [65].

Fig. 7 – (Color online) The dependence of the eigenvalue for Eq. (19) on the gain and loss strength W0,where (a1), (b1), and (c1) are the imaginary components of eigenvalues for the ground, the first, and thesecond excited modes in the super Gaussian-type potential with m = 2 and P0 = 4, respectively. Thecorresponding real components are shown in (a2), (b2), and (c2), respectively. The other parametersare the same as in Fig. 1.

As a generic example, we calculated the dependence of the imaginary and realcomponents of eigenvalues for Eq. (19) on the gain and loss strength W0 in the superGaussian-type potential with m= 2 and the input power P0 = 4, as shown in Fig. 7.From it, one can see that the imaginary and the real components of the eigenvalueappear in pairs, and so the eigenvalues for Eq. (19) are quartet symmetric, which isdifferent from the Hermitian system. Also, it can be found that the ground and firstexcited modes are linearly stable in the interval 0 <W0 < 4.1 [see Fig. 7(a2)] and3.75<W0 < 4.6 [see Fig. 7(b2)], respectively. The second excited mode is unstablethroughout its range of existence, as shown in Figs. 7(c1) and 7(c2).

To confirm the results of linear stability analysis, we perturb the three modesin Figs. 7(a2), 7(b2), and 7(c2) by 1% random-noise perturbations. The initial distri-butions and nonlinear evolutions are summarized in Fig. 8. From it, one can see thatthe ground and first excited modes with zero real component can propagate robustly,as shown in Figs. 8(a2) and 8(b2), while the second excited mode is unstable due tothe presence of the non-zero real component, as shown in Fig. 8(c2).

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


Fig. 8 – (Color online) The initial field distributions and the nonlinear evolution plots. (a1), (b1), and(c1) present the initial field profiles for the ground, the first, and the second excited modes (see theopen circles in Fig. 7), and (a2), (b2), and (c2) are the corresponding evolution plots, respectively. HereW0 = 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 superGaussian-type potential for m = 1,2, and 3 versus the input power P0 and the gain and loss strengthW0. The panels from top to bottom are for the ground, the first, and the second excited modes, respec-tively, and the panels from left to right correspond to m= 1,2, and 3, respectively. The regions of thebroken PT -symmetry are shown by shaded areas. The other parameters are the same as in Fig. 4.

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


Finally, we present the dependence of the largest real part of the eigenvalue onboth the input power P0 and the gain and loss strength W0 for the super Gaussian-type potential with m = 1, 2, and 3, as shown in Fig. 9. The ground mode is unsta-ble [see Figs. 9(a1), 9(b1), and 9(c1)], when the input power and the gain and lossstrength are large. The stability domains for the first excited mode is mainly concen-trated on two separated regions; see the arrows in Figs. 9(a2), 9(b2), and 9(c2). Theregion of stability for the second excited mode is smaller than that for the groundmode, and is mainly concentrated on the range of smaller input power and gain andloss strength, as shown in Figs. 9(a3), 9(b3), and 9(c3). Also, it is found that for thesuper Gaussian-type potential, the stability regions become much smaller with theincreasing 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 superGaussian-type potentials. The obtained results have shown that in the presence ofKerr-type nonlinearity, the eigenvalue spectra exhibit two nonlinear transition pointswith the increasing of the gain and loss strength. The first one is the bifurcationpoint Wb from which the eigenvalue spectrum for the ground state is bifurcated intotwo branches, the real and complex branches, which means that the PT -symmetryis broken. The second one is the coalescence point WcrN, at which two modes mergetogether and are terminated. Indeed, the two nonlinear PT transition points Wb andWcrN are separated from the linear PT -symmetry breaking point WcrL due to thepresence of nonlinearity.

Also, it was found that with the increasing of the input power, the coalescencepoint WcrN as a function of the input power undergoes a transition from the coales-cence of the ground and first excited modes to the coalescence of the first and secondexcited modes. Finally, the linear stability of the corresponding nonlinear opticalmodes has been investigated by numerical methods. Our findings can be applied toother classes of PT -symmetric complex-valued potentials.

Acknowledgements. This research was supported by the National Natural Science Foundationof China, through Grants No. 61475198 and 61078079, and by the Shanxi Scholarship Council ofChina, through Grant No. 2011-010.

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


REFERENCES

1. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).2. C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).3. C. M. Bender, S. Boettcher, and P. N. Meisinger, J. Math. Phys. 40, 2201 (1999).4. A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A: Math. Gen. 38, L171 (2005).5. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632

(2007).6. S. Klaiman, U. Gunther, and N. Moiseyev, Phys. Rev. Lett. 101, 080402 (2008).7. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100,

030402 (2008).8. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100,

103904 (2008).9. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81,

063807 (2010).10. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou,

and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).11. C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys.

6, 192 (2010); T. Kottos, ibid. 6, 166 (2010).12. A. Regensburger, C. Bersch, M. -A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel,

Nature 488, 167 (2012).13. J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, Phys. Rev. A 84, 040101(R) (2011).14. Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, Phys. Rev. A 85, 050101(R) (2012).15. H. Ramezani, J. Schindler, F. M. Ellis, U. Gunther, and T. Kottos, Phys. Rev. A 85, 062122 (2012).16. N. Lazarides and G. P. Tsironis, Phys. Rev. Lett. 110, 053901 (2013).17. M. Kang, F. Liu, and J. Li, Phys. Rev. A 87, 053824 (2013).18. G. Castaldi, S. Savoia, V. Galdi, A. Alu, and N. Engheta, Phys. Rev. Lett. 110, 173901 (2013).19. X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, Opt. Lett. 38, 2821 (2013).20. C. Hang, G. Huang, and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).21. F. Kh. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805(R)

(2011).22. X. Zhu, H. Wang, L. Zheng, H. Li, and Y. He, Opt. Lett. 36, 2680 (2011).23. H. Li, Z. Shi, X. Jiang, and X. Zhu, Opt. Lett. 36, 3290 (2011).24. S. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, Phys. Rev. A 84, 043818 (2011).25. S. Hu and W. Hu, J. Phys. B: At. Mol. Opt. Phys. 45, 225401 (2012).26. B. Midya and R. Roychoudhury, Phys. Rev. A 87, 045803 (2013).27. S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85, 023822 (2012).28. D. A. Zezyulin and V. V. Konotop, Phys. Rev. A 85, 043840 (2012).29. V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-Gonzalez, Phys. Rev. A 86,

013808 (2012).30. M.-A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, Phys. Rev. A 86,

033801 (2012).31. H. Xu, P. G. Kevrekidis, Q. Zhou, D. J. Frantzeskakis, V. Achilleos, R. Carretero-Gonzalez, Rom.

J. Phys. 59, 185 (2014).32. J. Yang, Opt. Lett. 39, 5547 (2014).33. Z. Shi, X. Jiang, X. Zhu, and H. Li, Phys. Rev. A 84, 053855 (2011).

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24


34. A. Khare, S. M. Al-Marzoug, and H. Bahlouli, Phys. Lett. A 376, 2880 (2012).35. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, 013831 (2012).36. Y. He and D. Mihalache, Phys. Rev. A 87, 013812 (2013).37. D. Mihalache, Rom. Rep. Phys. 67, 1383 (2015).38. B. Liu, L. Li, and D. Mihalache, Rom. Rep. Phys. 67, 802 (2015).39. S. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. E 84, 046609 (2011).40. K. Zhou, Z. Guo, J. Wang, and S. Liu, Opt. Lett. 35, 2928 (2010).41. H. Wang and J. Wang, Opt. Express 19, 4030 (2011).42. Z. Lu and Z. Zhang, Opt. Express 19, 11457 (2011).43. R. Driben and B. A. Malomed, Opt. Lett. 36, 4323 (2011).44. H. Li, X. Jiang, X. Zhu, and Z. Shi, Phys. Rev. A 86, 023840 (2012).45. C. P. Jisha, A. Alberucci, V. A. Brazhnyi, and G. Assanto, Phys. Rev. A 89, 013812 (2014).46. I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev.

A 86, 053809 (2012).47. D. A. Zezyulin and V. V. Konotop, Phys. Rev. Lett. 108, 213906 (2012).48. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803

(2010).49. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett.

106, 213901 (2011).50. Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, Phys. Rev. Lett. 111, 263901 (2013).51. R. Li, P. Li, and L. Li, Proc. Romanian Acad. A 14, 121 (2013).52. L. Chen, R. Li, N. Yang, D. Chen, and L. Li, Proc. Romanian Acad. A 13, 46 (2012).53. H. Cartarius and G. Wunner, Phys. Rev. A 86, 013612 (2012).54. D. Dast, D. Haag, H. Cartarius, G. Wunner, R. Eichler, and J. Main, Fortschr. Phys. 61, 124 (2013).55. R. Fortanier, D. Dast, D. Haag, H. Cartarius, J. Main, G. Wunner, and R. Gutohrlein, Phys. Rev. A

89, 063608 (2014).56. D. Mihalache, Rom. J. Phys. 59, 295 (2014).57. V. S. Bagnato, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and D. Mihalache, Rom.

Rep. Phys. 67, 5 (2015).58. R. Radha and P. S. Vinayagam, Rom. Rep. Phys. 67, 89 (2015).59. A. I. Nicolin, M. C. Raportaru, and A. Balaz, Rom. Rep. Phys. 67, 143 (2015).60. B. Bagchi, C. Quesne, and M. Znojil, Mod. Phys. Lett. A 16, 2047 (2001).61. C. P. Jisha, L. Devassy, A. Alberucci, and V. C. Kuriakose, Phys. Rev. A 90, 043855 (2014).62. L. Devassy, C. P. Jisha, A. Alberucci, and V. C. Kuriakose, Phys. Rev. E 92, 022914 (2015).63. X. Shi, L. Li, R. Hao, Z. Li, and G. Zhou, Opt. Commun. 241, 185 (2004).64. S. M. Al-Marzoug, Opt. Express 22, 22080 (2014).65. Jianke Yang, Nonlinear waves in integrable and nonintegrable systems, Society for Industrial and

Applied Mathematics, Philadelphia, 2010.

RJP 61(Nos. 3-4), 577–594 (2016) (c) 2016 - v.1.3a*2016.4.24

nonlinear parity-time-symmetry breaking in optical …solitons, bragg solitons, gray or dark...

Documents