Nonautonomous solitons in parity-time symmetric potentials

Optics Communications 315 (2014) 303–309

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Nonautonomous solitons in parity-time symmetric potentials Chao-Qing Dai a,b,n, Yue-Yue Wang a a b

School of Sciences, Zhejiang Agriculture and Forestry University, Lin'an, Zhejiang 311300, PR China Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia

art ic l e i nf o

a b s t r a c t

Article history: Received 21 October 2013 Received in revised form 18 November 2013 Accepted 19 November 2013 Available online 2 December 2013

Some analytical solutions of a (1 þ1)-dimensional nonlinear Schrödinger equation with inhomogeneous diffraction and nonlinearity in the presence of the parity-time symmetric potential are derived. All characteristic parameters such as amplitudes, speeds and widths of solutions are adapted both to the diffraction and to nonlinearity variations. The dynamical behaviors of two kinds of nonautonomous solitons with different parity-time symmetric potentials are investigated respectively both in the exponential diffraction decreasing waveguide and in the periodic modulated system. Moreover, phase changes for two kinds of nonautonomous solitons are illustrated in detail. These results may provide alternative methods in potential applications of synthetic parity-time symmetric systems. & 2013 Elsevier B.V. All rights reserved.

Keywords: Nonautonomous soliton Nonlinear Schrödinger equation Parity-time symmetry Phase transitions

1. Introduction Solitons solitary waves that maintain their shape as they propagate occur as light pulses in optical fibres [1–3]. For the classical solitons, time has only played the role of the independent variable and has not appeared explicitly in the characteristic of soliton such as the amplitude and velocity and width. This kind of solitons is also called “autonomous solitons”. An uncommon situation is one which could include repeated stress testing of a soliton in nonuniform media with time-dependent density gradients, and it is typical both for experiments with temporal/spatial optical solitons, soliton lasers and ultrafast soliton switches and logic gates. In such situations, solitons are called “nonautonomous soliton” [4–6] because their propagation behaviors possess varying amplitudes, speeds and spectra adapted both to the dispersion and to nonlinearity variations. Nowadays, a large amount of studies [7–10] for nonautonomous solitons was done in various context of optical fiber communication and BECs since nonautonomous soliton was introduced by Serkin et al. [5]. Jiang et al. [7] investigated dynamical behaviors of nonautonomous soliton in an inhomogeneous cubic-quintic nonlinear medium. Yang et al. [8] discussed bright chirp-free and chirped nonautonomous solitons under dispersion and nonlinearity management. Chen and Lu [9] investigated nonlinear tunneling of nonautonomous solitons. Mani Rajan et al. [10] studied dispersion management and cascade compression of femtosecond nonautonomous soliton in birefringent fiber.

n Corresponding author at: School of Sciences, Zhejiang Agriculture and Forestry University, Lin'an, Zhejiang 311300, PR China. Tel.: þ 86 13567179564. E-mail address: [email protected] (C.-Q. Dai).

0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.11.030

However, nonautonomous solitons in inhomogeneous Kerr media with parity-time (PT) symmetric potentials are hardly reported. The PT symmetry first appeared in quantum mechanics since Bender and Boettcher [11] in 1998 pointed that non-Hermitian Hamiltonians exhibit entirely real spectra, provided that they respect both the parity and time-reversal symmetries. In other words, the PT symmetric potential satisfies V ðxÞ ¼ V n ð  xÞ with n denoting complex conjugation [11,12], that is, the real component of a PT complex potential must be a symmetric function of position whereas the imaginary part should be anti-symmetric. Owing to the mathematical similarity and the parallels between the Schrödinger-like equations in quantum mechanics and the nonlinear Schrödinger equation (NLSE) in optics, the role of potential VðxÞ in quantum mechanics can be played as an optical potential by a complex refractive-index distribution nðxÞ ¼ nR ðxÞ þ inI ðxÞ, wherein the real component nR ðxÞ describes the refractive index profile, while the imaginary part nI ðxÞ denotes the gain (þ) or loss ( ) in the system. In optics, the PT-symmetric conditions imply nR ðxÞ ¼ nR ð  xÞ and nI ðxÞ ¼  nI ð xÞ, namely, the index distribution must be an even function of position, whereas the gain and loss must be odd function. Soliton propagation in optical media with PT symmetric potential is presently attracting a great interest both from the theoretical and from the applicative point of view. The related applications were initiated by the key contributions of Christodoulides and co-workers [12]. The proposed PT systems can be realistically implemented through a judicious inclusion of gain/loss regions in optical waveguides [13]. Studies of PT-symmetry have been undertaken from nonlinear lattices [14,15], to double-channel waveguides [16], to double-well potentials [17], and to asymmetric optical amplifier [18]. Two-dimensional (2D) solitons in nonlocal media [19] with PT-symmetric potentials have been exhibited. Moreover, PT symmetry can lead to altogether new

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C.-Q. Dai, Y.-Y. Wang / Optics Communications 315 (2014) 303–309

optical dynamics [12,20,21]. For example, abrupt phase transitions will appear in PT symmetry case along with the occurrence of the so-called exceptional points [20], and the appearance of power oscillations and secondary emissions [13]. In addition, new classes of optical solitons [12,21] have also been suggested. Experimental realizations of such PT-symmetry systems have also been reported [22,23]. Guo et al. [22] demonstrated experimentally passive PT-symmetry breaking within the realm of optics. Rüter et al. [23] reported observation of PT symmetry in an optical coupled system. In a real fibre, the core medium is inhomogeneous [24] because of the variation in the lattice parameters of the fibre medium and the fibre geometry (diameter fluctuations, etc). In these cases, the governing equations are various variable-coefficient NLSEs. In this paper, we discuss dynamical behaviors of nonautonomous soliton in an inhomogeneous Kerr nonlinear medium of Kerr index, n2, whose linear refractive index along the transverse x-direction are perturbed by an inhomogeneous and complex profile with n ¼ n0 ½1 þnR ðz; xÞ þ inI ðz; xÞ. The corresponding governing equation is the following (1 þ1)-dimensional variable-coefficient NLSE iuz þ

βðzÞ 2


uxx þ γ ðzÞ
uj2 u þ ½vðz; xÞ þ iwðz; xÞu ¼ 0;

ð1Þ

where the complex envelope of the electrical field uðz; xÞ is normalized as ðk0 w0 Þ  1 ðn2 =n0 Þ  1=2 , longitudinal z and transverse x coordinates are respectively scaled to the diffraction length LD  k0 w20 and the input beam width unit w0 with the wavenumber k0  2π n0 =λ at the input wavelength λ. Functions βðzÞ and γ ðzÞ represent the diffraction and nonlinearity coefficients, respec2 2 tively. Functions vðz; xÞ  k0 w20 nR ðz; xÞ and wðz; xÞ  k0 w20 nI ðz; xÞ are the real and imaginary components of the complex PT-symmetric potential, respectively. v and w are even and odd functions that correspond to the index guiding and the gain or loss distribution of the optical potential, respectively.

Under the following constraints:

Γ BA20

βðzÞ;

ð2Þ

When the complex potential is of the Scarff II type 2

VðXÞ ¼ V 0 sech ðXÞ;

ð9Þ

and ð10Þ

WðXÞ ¼ W 0 sechðXÞtanhðXÞ;

with two arbitrary constants V0 and W0, Eq. (8) possesses soliton solution as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W 20 þ18B2  9V 0 B UðZ; XÞ ¼ 7 sechðXÞexpðiΦÞ; 9Γ B

ð11Þ

with Φ ¼ BZ þW 0 =3B½arctanðsinhðXÞÞ. Note that when B ¼ Γ ¼ 1, solution (11) is the corresponding solution in [25]. From the expression (4), under the complex PT potential vðz; xÞ ¼

V 0 βðzÞ

2

sech ðXÞ;

ð12Þ

W 0 β ðzÞ sechðXÞtanhðXÞ; Bϖ ðzÞ2

ð13Þ

Bϖ ðzÞ2

and wðz; xÞ ¼

soliton solution of Eq. (1) reads sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W 20 þ 18B2  9V 0 B sech ½Xðz; xÞexp½iψ ðz; xÞ; uðz; xÞ ¼ 7 A0 9Γ Bϖ ðzÞ

ð14Þ

2.2. Solution 2 When the complex potential is in the form

and vðz; xÞ ¼

βðzÞ V ðXÞ; Bϖ ðzÞ2

wðz; xÞ ¼

βðzÞ WðXÞ; Bϖ ðzÞ2

ð3Þ

uðz; xÞ ¼ AðzÞU½Xðz; xÞ; ZðzÞexp½iϕðz; xÞ;

ð4Þ

where the amplitude A(z), formal variable Xðz; xÞ, effective propagation distance Z(z) and phase ϕðz; xÞ are expressed as A0 A ¼ pffiffiffiffiffiffiffiffiffiffiffi; ϖ ðzÞ

X¼

x

ϖ ðzÞ

;

Z¼

ΩðzÞΘðzÞ Bw20

;

ð5Þ

ð6Þ

and the beam width ϖ ðzÞ reads w0

W 20 4 tanh ðXÞ; 16B

ð15Þ

;

ð7Þ Rz

with the accumulated diffraction ΘðzÞ ¼ 0 β ðsÞ ds, chirp function ΩðzÞ ¼ ½1 s0 ΘðzÞ  1 , and constants w0 and s0, Eq. (1) can be reduced to NLSE
B iU Z þ U XX þ Γ
Uj2 U þ ½VðXÞ þ iWðXÞU ¼ 0; 2 with two constants B and Γ .

2

WðXÞ ¼ W 0 sech ðXÞtanhðXÞ;

ð16Þ

with two arbitrary constants V0 and W0, Eq. (8) has the following soliton solution: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16B2 þ W 20 þ 8V 0 B UðZ; XÞ ¼ 7 ð17Þ sechðXÞexpðiΦÞ; 8Γ B W2

s ΩðzÞ 2 x ; ϕ¼  0 2

ΩðzÞ

2

VðXÞ ¼ V 0 tanh ðXÞ þ and

via the transformation

ϖ ðzÞ ¼

2.1. Solution 1

where ψ ðz; xÞ ¼ BZðzÞ þ W 0 =3B½arctanðsinhðXÞÞ þ ϕðz; xÞ and ϖ ðzÞ; Xðz; xÞ; ϕðz; xÞ satisfy Eqs. (5)–(7). Here the phase ψ ðz; xÞ is made up of the phase Φ in solution (11) and the chirped phase ϕ expressed by Eq. (6).

2. Nonautonomous soliton solution

γ ðzÞ ¼

Therefore, if we substitute solutions of Eq. (8) into this transformation (4), nonautonomous solitons of Eq. (1) can be obtained. Solutions of Eq. (8) can be as seeds to generate various solutions of Eq. (1) under conditions (2) and (3).

ð8Þ

with Φ ¼ λZ þW 0 =4BtanhðXÞ and λ ¼ B þ V 0 þ 16B0 . From the expression (4), under the complex PT potential ( ) βðzÞ W 20 2 4 vðz; xÞ ¼ sech ðXÞ ; V 0 sech ðXÞ þ ð18Þ 16B Bϖ ðzÞ2 and wðz; xÞ ¼

W 0 β ðzÞ 2 sech ðXÞtanhðXÞ; Bϖ ðzÞ2

soliton solution of Eq. (1) reads sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16B2 þ W 20 þ 8V 0 B sech½Xðz; xÞexp½iψ ðz; xÞ; uðz; xÞ ¼ 7 A0 8Γ Bϖ ðzÞ

ð19Þ

ð20Þ

C.-Q. Dai, Y.-Y. Wang / Optics Communications 315 (2014) 303–309

where ψ ðz; xÞ ¼ ðB þ V 0 þ W 20 =16BÞZðzÞ þW 0 =4B tanhðXÞ þ ϕðz; xÞ and ϖ ðzÞ; Xðz; xÞ; ϕðz; xÞ satisfy Eqs. (5)–(7). Here the phase ψ ðz; xÞ is composed of the phase Φ in solution (17) and the chirped phase ϕexpressed by Eq. (6).

3. Dynamical behavior of nonautonomous solitons In this section, we discuss dynamical evolution of nonautonomous solitons in two kinds of different waveguides, that is, the exponential diffraction decreasing waveguide (DDW) [26,27]

βðzÞ ¼ β0 expð  szÞ;

ð21Þ

305

and the periodic modulated system (PMS) [28,29]

βðzÞ ¼ β0 cos ðωzÞ;

ð22Þ

where constant β0 describes the initial diffraction in system, and constants s and ω are related to the type of diffraction for different waveguides. Eq. (21) corresponds to a typical case of the exponentially DDW for s 4 0. The periodic modulation in Eq. (22) leads to alternating regions of positive and negative values of β, which are required for the eventual stability of solutions [28]. From these expressions (12) and (18), the complex PT-symmetric potential satisfies vðxÞ ¼ vð xÞ and wðxÞ ¼  wð  xÞ, that is, the index guiding and the gain or loss distribution are even and

1 2.5 0.5

2

20

v 1.5 1

20

5

0 20

10

0 x

–10

–20

15

–0.5

10 z

0.5

0

w

15

10 z 5

–1

0

20

10

3

0 x

–10

–20

0

Amplitude Width

Magnitude

2.5 4 50

3 I

40

2

30 20

1

–10

–5

0 x

1 0.5

0

5

1.5

z

10

0

2

20

10

40

60

z

80

100

Fig. 1. (a) and (b) The profiles of even function v in (12) and odd function w in (13) in the exponential DDW. (c) and (d) Intensity and characteristic parameters of nonautonomous soliton (14) in the exponential DDW. The parameters are β0 ¼ 0:3; s ¼ 0:02; V 0 ¼ 15; W 0 ¼ 10; A0 ¼ 1:5; B ¼ 1; Γ ¼  1; w0 ¼ 2, and s0 ¼ 0:06.

15

0

Φ

ψ

φ

10

–20

0

5

–40

–20

–60

–40

–80

–60

0 –5 50

50

50

40 z

40

40

30

30 z

20 10 0

–40

–20

0

20 x

40

z

20 10 0

–40

–20

0

20 x

40

30 20 10 0

–40

–20

0

20

40

x

Fig. 2. The profiles of (a) Φ in (11), (b) ϕ in (6) and (c) phase ψ of nonautonomous soliton (14) in the exponential DDW. The parameters are chosen as the same as that in Fig. 1.

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C.-Q. Dai, Y.-Y. Wang / Optics Communications 315 (2014) 303–309

60

v

1

40 15

20 0 20

w

–10

–20

15 10 z

–1 20

5 0 x

0 –0.5

10 z 10

20

0.5

20

5 10

0 x

0

–10

0

–20

140 120

50

80 I

40

60

30

40

20 z

20

10

0 –10

–5

0 x

5

Phase

100 100

80

z=0

60

z=30

40 20 0 –20

0 10

–20

–10

0

10 x

20

30

40

Fig. 3. (a) and (b) The profiles of even function v in (18) and odd function w in (19) in the exponential DDW. (c) and (d) Intensity and phase of nonautonomous soliton (20) in the exponential DDW. The parameters are chosen as the same as that in Fig. 1.

odd functions respectively with regard to x. An example for the even function v in (12) and odd function w in (13) in the exponential DDW is shown in Fig. 1(a) and (b). Under the complex PT-symmetric potential expressed by Eqs. (12) and (13), the intensity I ¼ juj2 of nonautonomous soliton (14) in the exponential DDW (21) is shown in Fig. 1(c). Along the propagation distance, the amplitude of nonautonomous soliton (14) increases, and the width of nonautonomous soliton (14) decreases [see Fig. 1(d)]. The nonautonomous soliton (14) is compressed in the exponential DDW (21). Note that an abrupt phase transition can be found in solution (11) for Eq. (8) as shown in Fig. 2(a). This kind of phase transition is the new optical characteristic for PT symmetry. Fig. 2(b) exhibits the parabolic shape for ϕ expressed as (6). As shown in Fig. 2(c), the corresponding phase ψ of nonautonomous soliton (14) is a result of the superposition of the parabolic shape originating from ϕ shown in Fig. 2(b) and the abrupt phase change from Φ in Fig. 2(a). The phase ψ of nonautonomous soliton (14) appears the change from small to big values at x¼0. Under another complex PT-symmetric potential expressed by Eqs. (18) and (19), which also possesses the even index guiding v and the odd gain or loss distribution w shown in Fig. 3(a) and (b), nonautonomous soliton (20) is also compressed along the propagation distance in the exponential DDW [see Fig. 3(c)]. The amplitude of nonautonomous soliton adds, and the width and the center of nonautonomous soliton both attenuate. As shown in Fig. 3(c), the phase ψ of nonautonomous soliton (20) also appears the change from small to big values at x ¼0. In the PMS (22), the complex PT-symmetric potential expressed by Eqs. (12) and (13) demonstrates a periodic structure. As shown in Fig. 4(a) and (b), the index guiding v is even function with

regards to x, i.e. vðxÞ ¼ vð  xÞ, and the gain or loss distribution w is odd function for x, i.e. wðxÞ ¼  wð  xÞ. Under this kind of the complex PT-symmetric potential, nonautonomous soliton (14) exhibits a periodic propagation in the PMS [see Fig. 4(c)]. Moreover, the transition for phase of nonautonomous soliton (14) also displays a periodic behavior in the PMS [see Fig. 4(d)]. Similarly, for nonautonomous soliton (20), it exists under complex PT-symmetric potential in Eqs. (18) and (19). In the PMS, this complex PT-symmetric potential shows another periodic structure. This banding structure for v as shown in Fig. 5(a) is symmetrical for x, and concave and convex structure for w as shown in Fig. 5(b) is antisymmetrical with regard to x. Under this complex PT-symmetric potential, nonautonomous soliton (20) propagates periodically in the PMS as shown in Fig. 5(c). Along the propagation distance, the amplitude and the width of nonautonomous soliton (20) all vary periodically [see Fig. 5(d)], however, the variation of the amplitude has the opposite pace compared with that of the width. The amplitude adds to the maximum value and the width attenuates to the minimum value. Then, the amplitude attenuates from the maximum value to the minimum value, and the width adds from the minimum value to the maximum value. In the PMS, the abrupt phase transition for Φ also exhibits periodic behavior in solution (17) for Eq. (8) [see Fig. 6(a)]. When it superposes the parabolic shape for ϕ expressed as (6) in the phase ψ of nonautonomous soliton (20), the phase ψ of nonautonomous soliton (20) appears the periodic change from small to big values at x¼ 0 shown in Fig. 5(c). At last, we discuss the stability of these solitons. We study analytical solutions evolving along distance when they are disturbed from their analytically given forms. We perform the direct numerical simulation (the split-step Fourier technique) with initial

C.-Q. Dai, Y.-Y. Wang / Optics Communications 315 (2014) 303–309

307

140

140

120

120

100

100 80 v 0

80 z

z

60 8

6

0

w

40 4

2

0 –2 x –4

15

20 –6

–8

140 100

I 1

80 z 60 40

5

10

20

0 x

–5

–10

–15

0

120

–10

80

–20 –30

20 0 x

5

0 Phase

2

–5

40 10

0

120

0 –10

60

–40

0

z

40 –20

0 x

20

0

40

Fig. 4. (a) and (b) The profiles of even function v in (12) and odd function w in (13) in the PMS. (c) and (d) Intensity and phase of nonautonomous soliton (14) in the PMS. The parameters are chosen as the same as that in Fig. 1 except for ω ¼ 0:02.

100

100 80

80 v

0

w 0

60

20

40

10 x

0

60

10

z

20 –10 –20

40

5 x

0

z

20 –5

0

–10

0

7 Amplitude Width

40 I

20 0 –10

–5

0 x

5

10

140 120 100 80 z 60 40 20 0

Magnitude

6 5 4 3 2 0

20

40

60

80

100

z Fig. 5. (a) and (b) The profiles of even function v in (18) and odd function w in (19) in the PMS. (c) and (d) Intensity and characteristic parameters of nonautonomous soliton (20) in the PMS. The parameters are chosen as the same as that in Fig. 4 except for V 0 ¼ 2; W 0 ¼ 18.

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30 Φ 20 10 0 –10 –20 –15 –10

φ

0 80

–20

60 z

–30

40 –5

20 0 x

5

10

15

80

20

–20

0 x

20

40

80 60 z

–40

40

0

100

0 –20

60 z

–40 –40

ψ

20 100

–10

100

40

–40

0

20

–20

0 x

20

40

0

Fig. 6. The profiles of (a) Φ in (17), (b) ϕ in (6) and (c) phase ψ of nonautonomous soliton (20) in the PMS. The parameters are chosen as the same as that in Fig. 5.

10

60 z=50 z=100

9

I

8

50

1 0

7

−5

6 I

z=50 z=100

2

0 x

50 I 0

5

40

−5

0 x

5

I 30

5 4

20

3 2

10

1 0 −6

−4

−2

0 x

2

4

6

0 −6

−4

−2

0 x

2

4

6

Fig. 7. Numerical rerun of solitons in Figs. 1(c) and 5(c). An added 5% white noise are added to the initial values. The parameters are the same as those in the corresponding analytical plots. (Inset) The corresponding initial values given by solutions (14) and (20).

white noise for Eq. (1) with initial fields coming from solutions (14) and (20) in some cases. Two examples of such behaviors are displayed in Fig. 7, which essentially presents a numerical rerun of solitons in Figs. 1(c) and 5(c). From Fig. 7(a), one can find that soliton is compressed along propagation distance, and the white noise hardly influences the propagation of soliton. In Fig. 7(b), the white noise has a weak impact on the peak of soliton. From two examples in Fig. 7, no collapses are found; instead, the stable propagation over tens of diffraction lengths are observed.

Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant no. 11375007) and the Zhejiang Provincial Natural Science Foundation of China (Grant no. LY13F050006). Dr. Chao-Qing Dai is also sponsored by the Foundation of New Century "151 Talent Engineering" of Zhejiang Province of China.

References

4. Conclusions In conclusion, a (1þ 1)-dimensional NLSE with inhomogeneous diffraction and nonlinearity in the presence of the PT-symmetric potential is studied, and some analytical solutions are derived. All characteristic parameters such as amplitudes, speeds and widths of solutions are adapted both to the diffraction and to nonlinearity variations. The dynamical behaviors of two kinds of nonautonomous solitons with different PT-symmetric potentials are investigated respectively both in the exponential DDW and in the PMS. Two kinds of nonautonomous solitons are both compressed in the exponential DDW, and exhibit periodic behaviors in the PMS. Moreover, phase changes for two kinds of nonautonomous solitons are illustrated in detail. The superposition of the abrupt phase transition in the system with constant diffraction and nonlinearity and the parabolic shape from chirped phase leads to phase changes of nonautonomous solitons. These results may provide alternative methods in potential applications of synthetic PT-symmetric systems. More practical application of the theoretical results might be an interesting task.

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