PHYSICAL REVIEW A 90, 033804 (2014)
Solitons in PT -symmetric nonlinear dissipative gratings Xiang Li and Xiao-Tao Xie* College of Physics, Northwest University, Xi’an 710069, China (Received 2 June 2014; published 3 September 2014) We investigate the soliton behaviors of laser beams in two types of two coupled PT -symmetric dissipative gratings which contain different coupling forms (constant coupling and periodic coupling). The dynamics of such systems is governed by coupled nonlinear time-dependent Schr¨odinger-like equations and the corresponding soliton solutions are obtained. Long-time soliton oscillations are observed in the case of constant coupling, but absent in the case of periodic coupling. The stable regions of the parameters for gain or loss strength and modulation frequency are estimated by the variational method and numerical simulation. It is interesting that the stable region in the parameter space in the periodic-coupling case is wider than that in the constant coupling case. DOI: 10.1103/PhysRevA.90.033804
PACS number(s): 42.25.Bs, 11.30.Er, 42.82.Et
I. INTRODUCTION
In general, the concept of dissipative solitons is used to describe the localized states in nonlinear dissipative systems, i.e., pulses or spatial wave packets in systems exhibiting gain and loss [1]. Stable dissipative solitons have special amplitudes which are determined by the complex balance between the effects of gain or loss, dispersion, and nonlinearity [2]. Recently, the knowledge of dissipative systems has been renewed. The systems with non-Hermitian but parity-time (PT ) symmetric Hamiltonians were theoretically predicted to have pure real eigenvalues [3,4]. These PT -symmetric effects were experimentally confirmed by coupled optical waveguides with gain and loss [5–7]. It turns out that the amplitudes of solitons in nonlinear optical systems with dissipative and PT -symmetric structures could change continuously over a wide range such as the ones in conservative systems [8–20]. At the same time, the propagation of solitons was investigated in PT -symmetric structures with transverse periodic modulation of the complex refractive index, which are the so-called PT -symmetric optical lattices [7–13,20–24]. Many novel features were revealed, including double refraction, power oscillations, nonreciprocal transport, phase dislocations, superluminal transport dynamics, and so on. The situations of transverse modulation of nonlinear coefficients were also discussed in previous studies [25,26]. Generally speaking, the patterns of laser beams in the above systems are depicted by time-independent Schr¨odinger-like equations. Moreover, the topics of the refractive index modulation along longitudinal direction in PT -symmetric systems attracted considerable attention in recent years [20,27–33]. It is worthwhile to note that the propagation properties of those optical systems are described by time-dependent Schr¨odingerlike equations which are of fundamental importance in physics. Many novel phenomena were discovered, for instance, Rabi oscillations [34], dynamic localization [35], coherent destruction of tunneling [36,37], and Autler-Townes splitting [38]. Here we need to point out that there is a new kind of optical structures with periodic gain or loss distribution instead of periodic modulations of the refractive index, which may be termed as dissipative grating. The studies of linear dissipative gratings have revealed many interesting properties, e.g., the
*
[email protected]
1050-2947/2014/90(3)/033804(6)
self-orthogonal state [29], linear dispersion curve [33], and so on. Our interest is focused on the coupled nonlinear dissipative gratings with PT -symmetric structures. In this paper, the propagation of laser beams in two types of coupled nonlinear PT -symmetric dissipative gratings with different coupled forms (constant coupling and periodic coupling) is to be considered. Such kinds of optical structures could be experimentally achieved by two doped nonlinear waveguides as reported in Refs. [6,30]. The implementation of the periodic distribution of gain or loss coefficients can be realized by a specific design of amplitude mask. In the meanwhile, the coupled coefficients can be modulated by varying the thickness of the gap between waveguides [39,40]. The top view of the configuration of dissipative gratings discussed here is shown in Fig. 1. We find that the dissipative gratings with constant coupling can support oscillatory solitons and another kind of dissipative grating can maintain the standard soliton solutions. The stability analysis indicates that the solitons in the both cases may be stable over wide ranges of gain or loss coefficients and modulation frequency. Furthermore, our results show that the stability parameter region would be extended when the synchronous modulation of coupled coefficients is introduced. The paper is organized as follows. A general mathematical model of nonlinear dissipative gratings is formulated in Sec. II. In the next part, the dynamic evolution of solitons in dissipative gratings with constant coupling is investigated. Then we examine the soliton behavior in the same dissipative gratings but with periodic coupling (Sec. IV). Section V is devoted to our conclusion. II. MODELS
The nonlinear-coupled dissipative gratings shown in Fig. 1 can be described by the following coupled nonlinear Schr¨odinger equations: i
∂u ∂ 2 u + 2 + 2|u|2 u + R(z)v − iγ (z)u = 0, ∂z ∂x
(1a)
i
∂v ∂ 2v + 2 + 2|v|2 v + R(z)u + iγ (z)v = 0. ∂z ∂x
(1b)
Here u(x,z) and v(x,z) are the amplitudes of normalized complex modes. In Eqs. 1(a) and 1(b), a series of dimensionless √ quantities are introduced: z =
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nk |E0 |2 k z and x 4
=
ng nk |E0 |2 k √ x, 2
©2014 American Physical Society
XIANG LI AND XIAO-TAO XIE
PHYSICAL REVIEW A 90, 033804 (2014)
i
∂ 2v ∂v + 2 + 2|v|2 v + Ru + iγ cos(ωz)v = 0, (2b) ∂z ∂x
where ω is the modulation frequency. We restrict ourselves to take R, γ , and ω as positive real constants. Proceeding to solve the system, we assume v = u exp[i(σ z + π/2)] with real constant σ . It casts Eqs. 2(a) and 2(b) in the form i
∂u ∂ 2 u + 2 + 2|u|2 u − R sin(σ z)u ∂z ∂x + i[R cos(σ z) − γ cos(ωz)]u = 0,
(3a)
2
i
∂v ∂ v + 2 + 2|v|2 v − R sin(σ z)v ∂z ∂x − i[R cos(σ z) − γ cos(ωz)]v = 0.
(3b)
In the specific situation R = γ and σ = ω, u and v obeys the same nonlinear Schr¨odinger equation FIG. 1. (Color online) A schematic of modulated nonlinear PT symmetric coupled slab waveguides with gain (red part) and loss (blue part). Two waveguides are on the plane where z and x denote the propagating direction and transversal spatial coordinate, respectively. The coordinate z plays the role of time if the standard Schr¨odinger equations are adopted to describe the propagation of laser beams. The pattern of amplitude mask realizes the modulation of gain or loss coefficient, and the periodic modulation of coupled coefficients are achieved by reversing the S bend of waveguides.
i
∂ 2φ ∂φ + 2 + 2|φ|2 φ − γ sin(ωz)φ = 0, ∂z ∂x
which has soliton solutions [42] γ
III. NONLINEAR PT -SYMMETRIC DISSIPATIVE GRATINGS WITH CONSTANT COUPLING A. Model and solutions
The coupled equations for cosine-driven nonlinear PT symmetric dissipative gratings are i
∂u ∂ 2 u + 2 + 2|u|2 u + Rv − iγ cos(ωz)u = 0, ∂z ∂x
(2a)
1
φ = sech(x)ei[ ω cos(ωz)+z+ 2 ] .
(5)
The solutions of coupled equations (2) then can be written as γ
1
u = sech(x)ei[ ω cos(ωz)+z+ 2 ] , v = sech(x)e
where z is the coordinate in the propagation direction, x is the traversal coordinate, k is the wave number in the vacuum, and |E0 |2 is considered as the characteristic power. ng and nk are the ground refractive index and Kerr nonlinear refractive index, respectively. The apostrophes are dropped in Eqs. 1(a) and 1(b) for convenience. To ensure the existence of bright solitons [14], we take the same sign in front of the diffraction and Kerr nonlinear term. The longitudinal modulation of coupled coefficient R(z) = nk4g(z) , where g(z) means the |E0 |2 k coupled coefficient between the two dissipative gratings, can be performed by the curved waveguides [39,41] as shown 0 (z) in Fig. 1. The periodic imaginary component γ (z) = n4γ 2 k |E0 | k with the gain distribution γ0 (z) describes the distribution of gain or loss. It should noted that the PT symmetry requires γ (z) = γ (−z). It is difficult to reveal the impacts of each modulation term by analyzing Eqs. 1(a) and 1(b) directly. Therefore, we investigate solitons in PT -symmetric dissipative gratings first with constant coupling. For comparison, the case with modulated coupling is considered after that.
(4)
i[ ωγ cos(ωz)+(1+ω)z+ 1+π 2 ]
(6a) .
(6b)
These solutions can be represented as the Jacobi-Anger expansion +∞ γ 1 n sech(x)ei[(nω+1)z+ 2 ] , u= i Jn (7a) ω n=−∞ γ 1+π v= sech(x)ei[(n+1)ωz+z+ 2 ] , i Jn ω n=−∞ +∞
n
(7b)
where Jn (ξ ) is the Bessel function of the first kind of order n. These solutions implicate that many components of solitons with different carrier frequencies are excited. This is a direct outcome of the periodic modulation along the direction of propagation. B. Stability analysis and numerical results
To estimate the stability of soliton solutions, the variational approximation is used. Consider the soliton solutions with perturbation of amplitude and phase difference [43], our ansatz can be written as 1 (8a) u = 1 + a(z)φ(x,z)ei 2 ϕ(z) , 1 v = 1 − a(z)φ(x,z)e−i 2 ϕ(z) , (8b) where φ(x,z) is the solution (5) of system (4). Two variational z-dependent functions a(z) and φ(z) describe the amplitude difference and phase difference of two modes u and v, respectively. It is necessary to explain that the signs in front of the gain or loss parts in two equations are always reversed. So we assume that the effects of perturbation on amplitudes
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SOLITONS IN PT -SYMMETRIC NONLINEAR . . .
PHYSICAL REVIEW A 90, 033804 (2014)
and phase difference is the gain (loss) of u accompanied with the loss (gain) of v. The Lagrangian corresponding to coupled system (2) is ⎡ ∞ ⎣i 1 (ψj ∂z ψj∗ − ψj∗ ∂z ψj ) − |∂x ψj |2 L= 2 ψ =u,v −∞ ψ =u,v
3 2.5
j
2
|ψj |4 + γ (vu∗ + uv ∗ )
ψj =u,v
1.5
⎤
− iγ cos(ωz)(|u|2 − |v|2 )⎦ dx.
1
(9)
0.5
It should be noted that the coupled coefficient R has been set as γ for the localized solution. Substituting the ansatz (8) into Eq. (9) yields the effective Lagrangian and the Euler-Lagrange equations for variational functions a(z) and ϕ(z) as follows: −i6γ cos(ωz)+8a(z) −
6γ cos[ϕ(z)]a(z) dϕ +3 = 0, √ dz 1 − a2
(10a) da = 0. (10b) −2γ sin[ϕ(z)] 1 − a 2 − dz In the limit |a(z)| 1 and |ϕ(z)| 1, we linearize the above equations, which yields two coupled differential equations dϕ = 0, −i6γ cos(ωz) + (8 − 6γ )a(z) + 3 dz da 2γ ϕ(z) + = 0. dz
(11a) (11b)
The solutions of Eqs. 11(a) and 11(b) are
where
4/3
ω/γ
+
j
3.5
ϕ(z) = i [ sinh( z) − ω sin(ωz)],
(12)
a(z) = i2γ [cos(ωz) − cosh( z)],
(13)
√ 2 3 (γ ) = 4γ − 3γ 2 , 3 6γ . (γ ,ω) = 16γ − 12γ 2 + 3ω2
(14) (15)
We can find that the stability of ansatz (8) is represented by the imaginary part of ϕ(z) (12) and the real part of a(z) (13). Since a(z) is a pure imaginary number, we focus our discussion on phase difference ϕ(z). The real part of the hyperbolic sine function in Eq. (12) could be 0 only if (γ ) is a pure imaginary number. Inspecting Eq. (14), it indicates the stable region of γ is γ 4/3. Otherwise, sinh( z) would grow exponentially with the increase of z. On the other hand, the requirement of stability (i.e., |ϕ(z)| 1) leads to a restrictive condition of the coefficient of sin(ωz) in Eq. (12), i.e., ω (γ ,ω) 1. According to the complementary set of this inequality, we can achieve the for√ 6γ −
0 0
4
γ
6
8
10
FIG. 2. The forbidden parameter regions via modulation amplitude γ and frequency ω of nonlinear PT -symmetric dissipative gratings. The gray region is for the gratings √ with constant coupling, 6γ +
180γ 2 −192γ
. The region which consists of γ 4/3 and 0 < ω < 6 covered with black lines corresponds to the gratings with modulated √ coupled coefficient, which consists of 0 < ω < 2+2 6 γ . The vertical coordinate is uniform as ω/γ .
ering the assumption of positive modulation frequency, the forbidden modulation frequency region can be rewritten as √ 6γ +
180γ 2 −192γ
0
