Soliton dynamics in a PT-symmetric optical lattice with a longitudinal potential barrier Keya Zhou,1,2,∗ Tingting Wei,1 Haipeng Sun,1 Yingji He,3 and Shutian Liu1 1 Department

of Physics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China 2 State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences, Xi’an, 710119, China 3 School of Electronics and Information, Guangdong Polytechnic Normal University, Guangzhou, 510665, China ∗ [email protected]

Abstract: We present dynamics of spatial solitons propagating through a PT symmetric optical lattice with a longitudinal potential barrier. We find that a spatial soliton evolves a transverse drift motion after transmitting through the lattice barrier. The gain/loss coefficient of the PT symmetric potential barrier plays an essential role on such soliton dynamics. The bending angle of solitons depends on the lattice parameters including the modulation frequency, incident position, potential depth and the barrier length. Besides, solitons tend to gain a certain amount of energy from the barrier, which can also be tuned by barrier parameters. © 2015 Optical Society of America OCIS codes: (190.6135) Spatial solitons; (190.4420) Nonlinear optics, transverse effects in.

References and links 1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998). 2. C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Amer. J. Phys. 71, 1095 (2003). 3. Z. Musslimani, K. Makris, R. El-Ganainy, and D. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008). 4. K. G. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008). 5. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010). 6. K. Y. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35, 2928 (2010). 7. H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express 19, 4030 (2011). 8. Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84, 053855 (2011). 9. S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, “Binary parity-time-symmetric nonlinear lattices with balanced gain and loss,” Opt. Lett. 35, 2976 (2010). 10. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012). 11. Y. He and D. Mihalache, “Spatial solitons in parity-time-symmetric mixed linear-nonlinear optical lattices: recent theoretical results,” Rom. Rep. Phys. 64, 1243 (2012). 12. H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).

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Received 28 Apr 2015; revised 13 Jun 2015; accepted 15 Jun 2015; published 18 Jun 2015 29 Jun 2015 | Vol. 23, No. 13 | DOI:10.1364/OE.23.016903 | OPTICS EXPRESS 16903

13. S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85, 043826 (2012). 14. R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Opt. Lett. 36, 4323 (2011). 15. I. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, “Breathers in PT-symmetric optical couplers,” Phys. Rev. A 86, 053809 (2012). 16. H. Eisenberg, Y. Silberberg, R. Morandotti, and J. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863 (2000). 17. M. J. Ablowitz and Z. H. Musslimani, “Discrete diffraction managed spatial solitons,” Phys. Rev. Lett. 87, 254102 (2001). 18. L. Torner and V. A. Vysloukh, “Parametric amplification of soliton steering in optical lattices,” Opt. Lett. 29, 1102 (2004). 19. I. Garanovich, A. Sukhorukov, and Y. Kivshar, “Soliton control in modulated optically-induced photonic lattices,” Opt. Express 13, 5704 (2005). 20. Y. V. Kartashov, L. Torner, and D. N. Christodoulides, “Soliton dragging by dynamic optical lattices,” Opt. Lett. 30, 1378 (2005). 21. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton control in fading optical lattices,” Opt. Lett. 31, 2181 (2006). 22. Y. He, D. Mihalache, X. Zhu, L. Guo, and Y. V. Kartashov, “Stable surface solitons in truncated complex potentials,” Opt. Lett. 37, 2526 (2012). 23. R. Yang and X. Wu, “Spatial soliton tunneling, compression and splitting,” Opt. Express 16, 17759 (2008). 24. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003). 25. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski,“Spatial solitons in optically induced gratings,” Opt. Lett. 28, 710-712(2003). 26. C. E. R¨uter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of paritytime symmetry in optics,” Nat. Phys. 6, 192-195 (2010). 27. K. Y. Zhou, Z. Y. Guo, and S. T. Liu, “Position dependent splitting of bound states in periodic photonic lattices,” J. Opt. Soc. Am. B 27, 1099 (2010).

1.

Introduction

In 1998, Bender and Boettcher firstly introduced the concept of non-Hermitian parity-time (PT) symmetric complex potentials, whose real part must be an even function of position whereas the imaginary component is odd. There exists a critical threshold below which the eigenvalues are entirely real, which is referred to a phase transition point [1]. Since then, there has been a lot activity in this area especially in the field of optics and photonics [2–5]. Beam dynamics in PT-symmetric periodic lattices exhibit unique properties. PT-symmetric periodic potentials supporting defect modes [6,7], bright spatial solitons in 1D and 2D defocusing Kerr media with PT-symmetric potentials [8], binary PT-symmetric nonlinear lattices with balanced gain and loss [9] have been studied. Besides, solitons in PT symmetric mixed linear-nonlinear Optical lattices [10, 11] and PT-symmetric potentials with nonlocal nonlinearity were also investigated [12, 13]. In [14] it was investigated the stability of solitons in PT-symmetric nonlinear optical couplers. Also, it was shown recently that PT-symmetric coupled optical waveguides with gain and loss support localized oscillatory structures similar to breathers of a classical model [15]. Optical lattices with refractive index modulation have also intrigued the interest of researchers [16–21]. It was revealed that truncated periodic complex potentials with homogeneous losses can support stable surface solitons in both focusing and defocusing media [22]. Optical solitons in optical lattices with periodic modulation of the refractive index offer more opportunities for the applications in all-optical devices based on spatial solitons. In this Letter, we investigate the propagation of spatial solitons in PT-symmetric optical lattices with a longitudinal barrier. We demonstrate the transmission drift of a spatial soliton when it propagates through a special lattice barrier with different parameters modulation. We explore the relations of the transmission deflection angle to various parameters of the potential barrier. The imaginary part coefficient of the PT symmetric potential barrier induces an anti-symmetric

#239892 (C) 2015 OSA

Received 28 Apr 2015; revised 13 Jun 2015; accepted 15 Jun 2015; published 18 Jun 2015 29 Jun 2015 | Vol. 23, No. 13 | DOI:10.1364/OE.23.016903 | OPTICS EXPRESS 16904

force upon soliton propagating and pushes it to a certain direction accordingly. Soliton dynamics in such complex potential barrier could be tuned through adjusting the parameters of the barrier, which may have the possibility for controllable deflection of spatial solitons. 2.

The model

Light propagation in a focusing Kerr nonlinear medium with linear refractive index modulation in both transverse and longitudinal directions is described by the dimensionless nonlinear Schrodinger equation [6, 7]

∂q 1 ∂ 2q =− − q|q|2 − pR(x, z)q, (1) ∂z 2 ∂ 2x The normalized transverse x and longitudinal z coordinates are scaled to the input beam width a and diffraction length Ldiff = n0 k0 a2 /2, respectively, where k0 = 2π /λ0 and n0 is the back−1/2 ground refractive index. q(x, z) = (Ldiff /Lnl )1/2 A(x, z)I0 , where A(x, z) is the slowly varying envelope, I0 is the input peak intensity, Lnl = 2/k0 n2 I0 is the nonlinear length, n2 is the Kerr nonlinear coefficient. p = Ldiff /Lref , Lref = 1/δ nk0 is the linear refraction length and δ n is the refraction index modulation depth. In this way, R(x, z) describes the complex refractive-index distribution along the transverse and longitudinal axes. We consider the following PT symmetric optical potential barrier similar to that in [23]:  V (x) exp[δ (z − zb )] 0 ≤ z ≤ zb , (2) R(x, z) = V (x) exp[−δ (z − zb )] zb ≤ z ≤ 2zb . i

in which V (x) = [cos2 (Ωx x) + iω0 sin(2Ωx x)] denotes a conventional PT-symmetric lattice. Ωx is the modulation frequency, ω0 is the gain/loss coefficient of the PT symmetric lattice, zb is the dimensionless propagation distance which is set as 5 in this Letter. The barrier length is determined by δ , which determines the slope rate in the longitudinal direction. Generally, we set δ > 0, and the barrier given by Eq. (2) can be described as Fig. 1 where its real and imaginary parts are schematically demonstrated. Such kind of lattice can be realized by an exponential rising lattice followed by a decaying lattice with periodic gain and losses, which can be technologically fabricated or induced optically in photorefractive crystals [24–26].

Fig. 1. Schematic Profile of a PT symmetric lattice potential. barrier described by Eq. (2)

3.

Numerical results and analysis

In order to investigate the influence of the PT symmetric potential barrier on the propagation of spatial solitons, we adopt the split-step Fourier method to simulate Eq. (1) which has been used in our previous work [27].We assume the profile of the input beam to be a fundermental soliton solution with an amplitude A, q(x, 0) = Asech [A (x − x0 )] ,

#239892 (C) 2015 OSA

(3)

Received 28 Apr 2015; revised 13 Jun 2015; accepted 15 Jun 2015; published 18 Jun 2015 29 Jun 2015 | Vol. 23, No. 13 | DOI:10.1364/OE.23.016903 | OPTICS EXPRESS 16905

where x0 is the incident location, and the initial energy of soliton is resulted to be P0 =

 +∞ −∞

|q (x, 0)|2 dx = 2A.

(4)

For simplicity, we consider the A = 1 soliton, whose FWHM is about 1.76, with an initial energy of 2, incident normally into the PT symmetric potential barrier at the site x0 = 0. The slope rate , gain/loss coefficient and the potential depth are firstly set as δ = 1, ω0 = 1 and p = 6, respectively. In Figs. 2 (a)–2(c), we demonstrate three cases with typical modulation frequencies Ωx = 2.4, 2.8 and 4. The period of the PT symmetric optical potential barrier T = π /Ωx corresponds to 1.31, 1.12 and 0.79, respectively. Our simulation results demonstrate that solitons are more likely to penetrate through the potential barrier with a smaller Ωx , accompanied by a deflection towards the right side of the propagation direction. The reason for this is because we takes ω0 > 1 here, which makes the imaginary part of the lattice Im{V } an increasing function of x around the point x0 . Otherwise, if a negative ω0 is used, solitons could be expected to bend to the opposite side. However, for larger Ωx , no such deflection is observed (see Fig. 2(c)). We also drew the beam profile of the solitons at z = 0, zb , and 2zb shown in Figs. 2(d)–2(f). Interestingly, after passing through the barrier, the soliton not only endures a deflection, but also its FWHM is narrowed to match the lattice period, a certain amount of energy is also obtained from the barrier meanwhile. (a) |q|2

5 0 −5

0

5

(b)

0 −5 10

0

5

(c)

5 x

10

5

10 z=0 z=5 z=10

1

1

0

0 (e)

0 −5

10

5 0 −5

2

2

5

z=0 z=5 z=10

4

0 −5

10

|q|2

z

10

z

(d)

|q|2

z

10

0

5

(f)

10 z=0 z=5 z=10

0.5 0 −5

0

5

10

x

Fig. 2. Propagation of spatial soliton through lattice barriers with typical transverse modulation frequencies (a) Ωx = 2.4, (b) 2.8 and (c) 4; (d-f) Beam profiles of solitons at z = 0, zb , and 2zb accordingly. Other parameters of the barrier are set as ω0 = 1, p = 6 and δ = 1.

To explore the inherent physical dynamics of solitons in such barrier, we define the bending angle φ as in Fig. 3(a) to show the deflection of the beam transmission, which satisfies tan φ = Δx/Δz. We set Δx and Δz to be the beam traverse and longitudinal displacement respect to its initial location as shown in Fig. 3(a). As a result, the dependence of bending angle on the modulation frequency Ωx is depicted in Fig. 3(b) with blue line. It should be noted here that the bending angles are not the real angles in unscaled units. In a real spatial scheme, Δx and Δz should be multiplied by the input beam width a and diffraction length Ldiff , respectively. The emergent beam energy can be integrated by a similar equation to Eq. (4), and its dependence on Ωx is shown in the same figure in red line as well. Obviously, there exists a critical point Ωcr ≈ 3 for such transversely drift motion. If the modulation frequency is larger than Ωcr , the bending angle is close to zero, as demonstrated in Fig. 2(c). The reason for this apparent by #239892 (C) 2015 OSA

Received 28 Apr 2015; revised 13 Jun 2015; accepted 15 Jun 2015; published 18 Jun 2015 29 Jun 2015 | Vol. 23, No. 13 | DOI:10.1364/OE.23.016903 | OPTICS EXPRESS 16906

comparing the beam profiles of solitons at z = 0, zb , and 2zb shown in Figs. 2(d)–2(f). The beam waist covers several lattice sites of V (x) at larger values of Ωx . As a result, the variation of Im{V } is averaged and its influence on the dynamic motion of solitons becomes negligibly small. According to the red line in Fig. 3(b), the emergent beam energy decreases with the modulation frequency, a similar trend with the bending angle. Such a beam dynamics is valid in a modulation frequency ranging from 2.5 to 3. 10

Δx

50

φ

40

(a) Δz

(b)

Bending angle φ Beam power P

4 3

z

30 5

20

2

10 1 0 0 −10

0 x

10

−10 2.5

3

3.5

Ω

4

4.5

0 5

x

Fig. 3. (a) Definition of bending angle; (b) Dependence of soliton bending angle (blue line) and emergent beam energy (red line) on modulation frequency.

Intuitively, the gain/loss coefficient of the PT symmetric potential barrier ω0 could be the main cause of such beam deflection. Upon this purpose, we still consider the A = 1 fundamental soliton, which is incident on site at the point x0 = 0. The barrier length, the modulation frequency and the potential depth are fixed at Ωx = 2 and p = 6, respectively. We swept ω0 from -1 to 1, and drew the dependence of bending angle on ω0 in Fig. 4 (a). Apparently, the motion of negative ω0 demonstrates an anti-symmetric way due to the lattice property. Furthermore, the larger ω0 is, the bending angle becomes larger. It is well known that ω0 = 0.5 is a phase transition point of a PT symmetric lattice with a form of V (x) = [cos2 (Ωx x) + iω0 sin(2Ωx x)]. When ω0 < 0.5, the eigenvalues are entirely real, which makes the band-gap structure of such PT symmetric lattice resembles that of a traditional lattice. While for ω0 > 0.5, the first two bands start to merge together and form an oval-like structure with a related complex spectrum [6]. Actually, it is clearly seen from the curve in Fig. 4(a) that for ω0 below such transition point, solitons experience only a small deflection and the bending angle increases very slowly. Once ω0 reached above 0.5, the soltion could feel an increasing force caused by the barrier, and bend to the right side of the propagation direction. To verify the fact that the above-mentioned soliton transversely drift motion results from the spatially inhomogeneous loss, we investigate the soliton propagation with the optical lattice barrier without such PT symmetry, i.e., ω0 = 0 in Eq. 2. It is discovered that the soliton bending angle equals to zero and the energy remains to be 2 invariably whatever the modulation frequency Ωx changes. Actually in this case, the imaginary part of V (x) becomes 0, and the system turns to non-dispersive. As the lattice potential is symmetric about the beam center, and solitons are incident normally into the lattice, no transverse forces could be expected. We drew such differences in Figs. 4(b)–4(e) for typical barriers both with and without gain/losses. Though the modulation frequency Ωx = 2 is far below the critical modulation frequency Ωcr , no transverse drift is observed in lattices without such PT symmetry, as shown in Figs. 4(d) and 4(e). In addition, the emergent beam energy in Fig. 4(b) is stronger than that of Fig. 4(c), which is proved to be a unique feature of PT symmetric lattice barriers. Physically, when ω0 > 0, the imaginary part of V (x) is positive on the right half side of the soliton, while negative on the other side. The soliton would obtain energy from the lattice on the right side while lose energy on the other side. Such an asymmetric effect generally increases upon propagation, and interacts severely with the lattice barrier at z = 5. As for ω0 < 0, the bending angle would be #239892 (C) 2015 OSA

Received 28 Apr 2015; revised 13 Jun 2015; accepted 15 Jun 2015; published 18 Jun 2015 29 Jun 2015 | Vol. 23, No. 13 | DOI:10.1364/OE.23.016903 | OPTICS EXPRESS 16907

30 (a) 20

4

10

3

3.5

0

2.5

−10

2 Bending angle φ Beam power P 1.5 1 0 0.5 1 ω

−20 −30 −1

−0.5

0

3 (d) 2

(b)

5

z=0 z=5 z=10

2

|q|

z

10

1 0

5

0 −5

10

0

5

1.5 (e) 2

(c)

5

|q|

z

0 −5 10

10 z=0 z=5 z=10

1 0.5

0 −5

0

5 x

10

0 −5

0

5

10

x

Fig. 4. (a) Dependence of soliton bending angle(blue line) and emergent beam energy (red line) on the gain/loss coefficients; (b) and (c) are solitons propagating through lattice barriers with ω0 = 1 and 0, respectively. other lattice parameters are Ωx = 2 and p = 3, and δ = 1. (d) and (e) are beam profiles corresponding to the cases in (b) and (c).

reflected back towards left side. Another essential factor that causes a soliton self-bending is its incident position relative to optical lattices, which has been reported in our previous work [27]. Thus in the following study, we fixed Ωx = 2.4, ω0 = 1, A = 1, p = 6, δ = 1, and let the incident position varies in a lattice period, ranging from −0.5T to 0.5T . Typical examples are shown in Fig. 5. It is interesting to find that the bending angle still has a similar variation trend with the beam energy, as shown in Fig. 5(a). Notably, there exists a range near 0.5T where the beam dynamics change dramatically, which is different from other cases. Typically we drew the case of x0 = 0.4T and 0.3T in Figs. 5(b) and 5(c) for comparison in the same plot. Although the soliton is incident at two points very closely, their propagation dynamics are totally different. This phenomenon reflects the instability of solitons in a PT symmetric lattice barrier. Physically, this result can be understood by the modulation instability of the beam itself. When the beam is incident near the position of 0.5T , where the local refractive index (RI) reaches its minimum, the beam tends to bend towards regions with higher RI. Upon this fact, the soliton dynamics is much more sensitive to its incident positions compared with other cases, especially at a position around x0 /T ∼ 0.3, where the real part of V (x) is asymmetric across the beam profile. As the splitting of beams occurs at the same time, the variation in beam power could be expected. In a word, the bending of solitons could be a combined effect of the RI modulation and the loss/gain anti-symmetric properties of the lattice barriers, which can be tuned sharply. There might have some other factors that could influence the transmission deflection of solitons, for example the lattice modulation depth , which determines the relative refractive index variation in structures. Here we fix a set of parameters as Ωx = 2, ω0 = 1, A = 1, x0 = 0 and δ = 1. The dependence of bending angle on the potential depth is shown in Fig. 6(a). When the potential depth p is below 1, the soliton propagation direction remains almost unchanged due to the fact that the lattice barrier is too weak to affect the soliton propagation. However, as the lattice modulation depth becomes larger, the bending angle of the solitons would be increased accordingly. Especially when p is larger than 2, the bending angle grows very sharply with the

#239892 (C) 2015 OSA

Received 28 Apr 2015; revised 13 Jun 2015; accepted 15 Jun 2015; published 18 Jun 2015 29 Jun 2015 | Vol. 23, No. 13 | DOI:10.1364/OE.23.016903 | OPTICS EXPRESS 16908

50 (a) 40

5

30

4

4.5

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3.5

10 0 −10 −0.5 10

3

Bending angle φ Beam power P −0.25

0 x0/T

(b)

2.5 2 0.5

0.25 (d)

z=0 z=5 z=10

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|q|

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|q|

z

0 −5 10

10 z=0 z=5 z=10

2 1

0 −5

0

5

0 −5

10

0

x

5

10

x

Fig. 5. Dependence of soliton bending angle (blue line) and emergent beam energy (red line) on the incident position; (b-c) are soliton propagation through the lattice barrier at x0 = 0.4T and 0.3T respectively, other parameters are Ωx = 2.4, ω0 = 1, A = 1, p = 6, δ = 1; (d) and (e) are the beam profiles corresponds to the cases of (b) and (c), respectively.

50 40

(a)

30

5 Bending angle φ Beam power P

4.5 4

20

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−10 2 0 0.5 1 1.5 2 2.5 3 3.5 4 p 10

(b) 2

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z=0 z=5 z=10

1

|q|

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1.5

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0 −5

2

|q|

z

4 5

0

5 x

10

10 z=0 z=5 z=10

2 0 −5

0

5

10

x

Fig. 6. Dependence of soliton bending angle (blue line) and emergent beam energy (red line) on the potential depth; (b-c) are soliton propagation through the lattice barrier at p = 2 and 4 respectively, other parameters are Ωx = 2.4, ω0 = 1, A = 1, x0 = 0, δ = 1; (d) and (e) are the beam profiles corresponds to the cases of (b) and (c), respectively.

#239892 (C) 2015 OSA

Received 28 Apr 2015; revised 13 Jun 2015; accepted 15 Jun 2015; published 18 Jun 2015 29 Jun 2015 | Vol. 23, No. 13 | DOI:10.1364/OE.23.016903 | OPTICS EXPRESS 16909

increase of the lattice modulation depth. In the Figs. 6(b)–6(e), we give two typical examples of the soliton propagation at p = 2 and 4, respectively. Apparently, in the latter case, the soliton not only bend to its right side, but also gained a certain amount of energy from the PT-symmetric lattice barrier, the corresponding beam energy reaches 3.78 in the latter case, which is almost twice of the initial soltion energy. 50 (a) 40

5

Bending angle φ 4.5 Beam power P

30

4

20

3.5

10

3

0

2.5

−10 2 1 1.5 2 2.5 3 3.5 4 4.5 5 δ 3 (d) 2

(b)

5

|q|

z

10

z=0 z=5 z=10

2 1

0

5

0

5

2 (e)

0

5 x

10

10 z=0 z=5 z=10

2

5

0 −5

0 −5

10

(c)

|q|

z

0 −5 10

1 0 −5

0

5

10

x

Fig. 7. Dependence of soliton bending angle (blue line) and emergent beam energy (red line) on slope rate δ of the barrier; (b-c) are soliton propagation through the lattice barrier at δ = 2 and δ = 4, respectively, other parameters are Ωx = 2.4, ω0 = 1, p = 6, A = 1, x0 = 0.;(d) and (e) are the beam profiles corresponds to the cases of (b) and (c), respectively.

Lastly, we investigate the influence of the barrier length, which is determined by the slope rate of the lattice δ . Optically, when the barrier length approaches zero, the resulted barrier is close to a uniform PT-symmetric lattice. At this time, the modulation of the PT-symmetric lattice is neglectable and solitons will not deflect when passing through the lattice. As the barrier length shrinks with the slope rate, we only consider the slope rate to be a range of 1 < δ < 5. Other parameters are set as x0 = 0, ω0 = 1,A = 1,Ωx = 2.4, and p = 6. Figure 7(a) demonstrates a decay of bending angle with the barrier length. Actually, as the barrier length increases, the height of the barrier will damp much faster. The interaction time between the incident soliton and the barrier would be shortened. In Figs. 7(b)–7(e), we give two typical examples of the soliton propagation with δ = 2 and δ = 4. The difference in emergent beam energy and the bending angle could be clearly seen in Figs. 7(d) and 7(e), which can be tuned by the barrier length. 4.

conclusion

In conclusion, we have investigated the soliton dynamics in PT-symmetric optical lattices with a longitudinal barrier. The results proved that solitons can penetrate through the barrier and exhibit a transverse drift together with a gain of energy from the PT symmetric lattice barrier. Such a transverse drift motion of solitons arises from an external force resulting from the inhomogeneous gain and loss of the potential. The bending angle of the soliton propagation can be efficiently tuned by adjusting barrier parameters such as the modulation frequency, potential depth, gain/loss coefficient, incident position, slope rate and so on. These results suggest new

#239892 (C) 2015 OSA

Received 28 Apr 2015; revised 13 Jun 2015; accepted 15 Jun 2015; published 18 Jun 2015 29 Jun 2015 | Vol. 23, No. 13 | DOI:10.1364/OE.23.016903 | OPTICS EXPRESS 16910

possibilities for experimental and theoretical studies of the solitons dynamics in media with a PT symmetric barrier. It may be interesting to extend the analysis for similar two-dimensional settings. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant nos. 61308017, 61377016 and 11174061), the Fundamental Research Funds for the Central Universities (Grant no. HIT.NSRIF.2011015), and the Guangdong Province Natural Science Foundation of China (Grant no. S2013010015795).

#239892 (C) 2015 OSA

Received 28 Apr 2015; revised 13 Jun 2015; accepted 15 Jun 2015; published 18 Jun 2015 29 Jun 2015 | Vol. 23, No. 13 | DOI:10.1364/OE.23.016903 | OPTICS EXPRESS 16911

Soliton dynamics in a PT-symmetric optical lattice with a longitudinal potential barrier.

We present dynamics of spatial solitons propagating through a PT symmetric optical lattice with a longitudinal potential barrier. We find that a spati...
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