FAMILIES OF DIPOLE SOLITONS IN SELF-DEFOCUSING KERR MEDIA AND PARTIAL PARITY-TIME-SYMMETRIC OPTICAL POTENTIALS

Transcription

1 FAMILIES OF DIPOLE SOLITONS IN SELF-DEFOCUSING KERR MEDIA AND PARTIAL PARITY-TIME-SYMMETRIC OPTICAL POTENTIALS HONG WANG 1,*, JING HUANG 1,2, XIAOPING REN 1, YUANGHANG WENG 1, DUMITRU MIHALACHE 3, YINGJI HE 4 1 South China University of Technology, Engineering Research Center for Optoelectronics of Guangdong Province, School of Electronic and Information Engineering, Guangzhou , China 2 Guizhou Mnizu University, School of Materials Science and Engineerings, Guiyang, , China 3 Horia Hulubei National Institute for Physics and Nuclear Engineering, P.O. Box MG-6, RO , Bucharest-Magurele, Romania 4 Guangdong Polytechnic Normal University, School of Photoelectric Engineering, Guangzhou , China * Corresponding author: phhwang@scut.edu.cn Received January 29, 2018 Abstract. We report the evolution dynamics of continuous families of dipole solitons in media with self-defocusing Kerr nonlinearity and partial parity-timesymmetric optical potentials. It is found that this type of complex-valued potentials can support one-parameter families of diagonal and adjacent dipole solitons with two humps that are either out-of-phase or in-phase. Both diagonal out-of-phase and diagonal in-phase dipole solitons can propagate stably in the moderate power regime. However, only adjacent in-phase dipole solitons can be stable in a certain region of their existence domain. The adjacent out-of-phase dipole solitons cannot maintain their initial shapes during propagation. Moreover, the magnitude of the gain/loss component of the complex-valued optical potential has a great effect on the field structure and stability of dipole solitons. Key words: PT-symmetry, partially PT-symmetric optical potentials, dipole solitons. 1. INTRODUCTION It is a common fact that light propagating in free space tends to broaden due to diffraction. However, when light propagates in nonlinear media, the exact balance between nonlinearity and diffraction (or dispersion) will lead to the selftrapping of light and the formation of spatial (or temporal) optical solitons [1]. Over the past decades, the studies of optical solitons have aroused a great attention due to the potential applications of optical solitons to all-optical switching and routing of optical signals [2]. Romanian Journal of Physics 63, 205 (2018)

2 Article no. 205 Hong Wang et al. 2 In particular, the existence, stability, and evolution dynamics of various nonlinear localized structures in the form of fundamental, dipole, quadrupole, and vortex solitons have been extensively investigated in optics and photonics and in the area of matter waves (Bose-Einstein condensates) [3 11], see also the review papers [12 15]. In 1998, Bender and Boettcher pointed out that even non-hermitian Hamiltonians can exhibit entirely real spectra if they respect the so-called paritytime-(pt)-symmetry [16]. The PT-symmetry was soon introduced in optics and photonics [17 20]. In one-dimensional settings, the PT-symmetry means that the complex-valued external potential obeys the condition V*(x) = V( x), where the symbol * stands for complex conjugation. Thus the complex-valued potential is invariant under complex conjugation and reflection of the spatial coordinate x. This means that the real part of the PT-symmetric optical potential is an even function, while its imaginary part is an odd one. In the context of PT-symmetry, the existence, stability, and propagation dynamics of both one and two-dimensional spatial optical solitons in linear [21 25] and nonlinear [26, 27] optical lattices have been widely investigated. Moreover, spatial optical solitons have been also found to propagate stably in mixed linear-nonlinear PT-symmetric optical lattices [28 30]. Furthermore, stable vortex solitons have been also reported, see, for example, Ref. [31]. We also refer to other relevant recent papers in the area of linear and nonlinear structures that form in PT-symmetric complex-valued potentials [32 42], see the comprehensive review papers [43, 44]. Multidimensional partially PT-symmetric complex-valued optical potentials possessing all-real spectra and continuous, one-parameter families of solitons were also proposed [45]. The complex-valued potential is invariant under complex conjugation and reflection in a single spatial coordinate [45]. In this paper, we investigate the existence, stability, and evolution dynamics of dipole solitons in two-dimensional partial PT-symmetric optical potentials with self-defocusing Kerr nonlinearity. These solitons can exist in the first gap with their two lobes located in either two nearest diagonal lattice sites or in two nearest adjacent lattice sites. We call them diagonal and adjacent dipole solitons, respectively. These two types of dipole solitons can exist either in-phase or out-ofphase between their two humps. The influence of gain/loss component of the complex-valued optical potential on the formation and stability of dipole solitons is also analyzed in detail. In Sec. 2 we discuss the basic physical model. Then in Sec. 3 we analyze in detail the existence, stability, and dynamical features of various one-parameter families of dipole solitons. In Sec. 4 we give our conclusions. 2. THE BASIC MODEL The evolution of a beam propagating in complex-valued optical potentials and self-defocusing Kerr media is governed by the following normalized nonlinear Schrödinger equation [18]

3 3 Families of dipole solitons in self-defocusing Kerr media Article no i ( ) V ( x, y) 0, 2 2 z x y (1) where is the slowly varying envelope of the optical field, z is the normalized propagation distance, and ( xy, ) are the normalized transverse coordinates. The complex-valued potential V ( x, y) is described as follows: V( x, y) V ( R iw), R sin x sin y, W W sin(2 x), (2) where V 0 and W 0 are the lattice depth and the amplitude of gain-loss component, respectively. Obviously, V * ( x, y) V( x, y), thus the complex-valued potential V ( x, y) is partially PT-symmetric. In this paper, we set V 9, W 0.1. The stationary solutions of Eq. (1) are 0 0 thought in the form of ( x, y) u( x, y)exp( i z), where is the propagation constant and u is a complex-valued function satisfying: 2 u u u Vu u u 0. (3) xx yy By using the plane-wave expansion method, we calculate numerically the linear counterpart of Eq. (3) and we get the band structure of the partially PT-symmetric potential given by Eq. (2). The band structure is displayed in Fig. 1. In Fig. 1a, we fix W 0 = 0.1 and vary V 0 from V 0 = 6 to V 0 = 12, while in Fig. 1b, we fix V 0 = 9 and vary W 0 from W 0 = 0.1 to W 0 = 0.5. From Fig. 1b, we see that the band gaps shrink with the increase of W 0, and at the phase transition point W 0 = 0.5 (i.e., at the corresponding critical point), all Bloch bands merge. Fig. 1 The band structure of Eq. (2) for (a) W 0 = 0.1 and varying V 0 from V 0 = 6 to V 0 = 12; (b) V 0 = 9 and varying W 0 from W 0 = 0.1 to W 0 = 0.5. The blue regions correspond to Bloch bands and the white ones represent the associated band gaps. By using the modified squared operator iteration method (MSOM) [46], we can obtain the stationary solutions of these dipole solitons from Eq. (3). The power

4 Article no. 205 Hong Wang et al. 4 of dipole solitons is defined as the double integral of the intensity of the optical field u 2 over the two transverse coordinates x and y; see e.g. Ref. [22]. In order to analyze the linear stability of these dipole solitons, we set * * ( x, y, z) { u( x, y) [ p( x, y) q( x, y)]exp( z) [ p( x, y) q( x, y)] exp( z)}exp( i z), (4) where p( x, y) and q( x, y) are small perturbations satisfying p q u and u, and the asterisk * stands for complex conjugation. Substituting Eq. (4) into Eq. (1) and linearizing it, we obtain the following coupled equations: (5) where The eigenvalues can be obtained numerically by using the Fourier collocation method [47]. If the eigenvalue has a positive real part Re( ) 0, the dipole soliton is linearly unstable, otherwise it is linearly stable. 3. NUMERICAL SIMULATIONS First, we investigate the propagation dynamics of diagonal dipole solitons that have the corresponding two humps located in the two nearest diagonal lattice sites. These solitons can exist in the form of either out-of-phase or in-phase optical fields associated to their two humps. We call them diagonal out-of-phase dipole solitons and diagonal in-phase dipole solitons, respectively. The corresponding field profiles are displayed in Fig. 2. Diagonal out-of-phase dipole soliton having its two out-of-phase humps located in the two nearest diagonal lattices has the power P versus the propagation constant shown in Fig. 3a. From Fig. 2a f and Fig. 3a, we see that the dipole soliton with the propagation constant near the first Bloch band (e.g. for μ = 12.30) has its field profile less localized and its imaginary part consists of four out-ofphase humps. However, by decreasing the propagation constant, the power of the dipole soliton increases monotonously, the soliton becomes more localized, and its imaginary part exhibits two in-phase humps.

5 5 Families of dipole solitons in self-defocusing Kerr media Article no. 205 Fig. 2 The field profiles of diagonal dipole solitons. The first row is for diagonal out-of phase dipole soliton at μ =12.30: (a) u, (b) real part, (c) imaginary part. The middle row is for diagonal out-ofphase dipole soliton at μ = 9.65: (d) u, (e) real part, (f) imaginary part. The bottom row is for diagonal in-phase dipole soliton at μ = 12.30: (h) u, (i) real part, (j) imaginary part. Fig. 3 The power of diagonal dipole solitons versus the propagation constant. (a) out-of-phase dipole soliton, (b) in-phase dipole soliton. The solid line stands for stable solitons whereas the red dashed lines indicate unstable ones. The black shaded regions indicate the Bloch bands. In order to investigate the propagation dynamics of diagonal out-of-phase dipole soliton, we perturb its amplitude by a 5% random noise. From Fig. 3a, we see that this type of dipole soliton exists in the first gap and can be stable in a large range of propagation constant μ, while it cannot be stable in both lower and higher

6 Article no. 205 Hong Wang et al. 6 power regions. The evolution of these stable and unstable diagonal out-of-phase dipole solitons are displayed in Fig. 4. From the upper row of Fig. 4, we see that when this type of dipole soliton is near the first Bloch band (the lower power region), e.g. at μ = 12.35, it cannot maintain its initial shape during propagation. Moreover, by solving Eq. (5), we obtain Fig. 4c and find that its linear-stability spectrum contains a pair of purely real eigenvalues, thus it will suffer an exponential-instability propagation regime. Furthermore, we also find that at the high power regime, taking μ = 9.30 as a typical example (the bottom row of Fig. 4), the diagonal out-of-phase dipole soliton cannot maintain its initial shape either, and its linear-stability spectrum contains two quadruples of complex eigenvalues with nonzero real parts, thus it will suffer an oscillatory instability during propagation. However, in the moderate power regime, the diagonal out-of-phase dipole soliton propagates stably without distortion even at z = 500; this result is in accordance with the associated linear-stability spectrum, which consists of only purely imaginary eigenvalues. Fig. 4 Evolution dynamics and linear-stability spectrum of diagonal dipole solitons. The first row is for μ = (unstable soliton) at (a) z = 0, (b) z = 400, (c) the corresponding linear-stability spectrum. The middle row is for μ = (stable soliton) at (d) z = 0, (e) z = 500, (f) the corresponding linear-stability spectrum. The bottom row is for μ = 9.30 (unstable soliton) at (h) z = 0, (i) z = 470, (j) the corresponding linear stability spectrum.

7 7 Families of dipole solitons in self-defocusing Kerr media Article no. 205 Fig. 5 Field profiles of adjacent dipole solitons. The first row is for adjacent in-phase dipole soliton in vertical direction; the second row is for adjacent out-of-phase dipole soliton in vertical direction; the third and bottom row are for adjacent in-phase and out-of-phase dipole solitons in horizontal direction, respectively. The first, middle, and last column are the corresponding profiles of u, real parts, and imaginary parts of these dipole solitons, respectively. The diagonal in-phase dipole soliton has its two humps located in two nearest diagonal lattice sites, but with its two humps in-phase. Its field profile is

8 Article no. 205 Hong Wang et al. 8 displayed in the bottom row of Fig. 2. The corresponding power diagram versus propagation constant is shown in Fig. 3b. We see that the diagonal in-phase dipole soliton exists in the first gap, can be stable in a certain region of its existence domain, and the stability characteristics are similar to those of diagonal out-ofphase dipole soliton. However, by comparing Fig. 3a and Fig. 3b we see that the diagonal in-phase dipole soliton is less robust than the diagonal out-of-phase dipole soliton. Second, we discuss the adjacent dipole soliton that has its two humps located in two nearest lattice sites. The two humps can be either in horizontal or vertical direction. Moreover, it can also have its humps either in-phase or out-of-phase. The corresponding field profile is shown in Fig. 5. However, the dipole soliton has the same key properties no matter it is located in horizontal or in vertical direction. Thus, we only discuss the adjacent dipole soliton located in vertical direction hereafter. Fig. 6 The power of adjacent dipole solitons versus the propagation constant. (a) in-phase dipole soliton (b) out-of-phase dipole soliton. The blue solid line stands for stable solitons whereas the red dashed lines indicate unstable ones. The black shaded regions indicate the Bloch bands. We find that adjacent in-phase dipole soliton exists in the first gap and can be stable in the moderate power regime while is unstable in both the low and the high power existence domains. Its power P versus propagation constant is displayed in Fig. 6a. It is found that its power increases monotonously with decreasing the propagation constant. By analyzing the evolution and stability of this type of dipole soliton, we found that it has the same generic properties as the diagonal out-of-phase dipole soliton. That is, it will suffer exponential and oscillatory instabilities in both low and high power regimes, while it propagates stably in the moderate power regime. These features can be clearly seen in Fig. 7.

9 9 Families of dipole solitons in self-defocusing Kerr media Article no. 205 Fig. 7 Evolution and linear-stability spectrum of adjacent in-phase dipole solitons. The first row is for μ = (unstable) at (a) z = 0, (b) z = 400, (c) the corresponding linear-stability spectrum. The middle row is for μ = 12 (stable) at (d) z = 0, (e) z = 500, (f) the corresponding linear-stability spectrum. The bottom row is for μ = 11 (unstable) at (h) z = 0, (i) z = 380, (j) the corresponding linear-stability spectrum. Fig. 8 The evolution of adjacent out-of phase dipole solitons at μ = 12. (a) z = 0, (b) z = 380, (c) the corresponding linear-stability spectrum.

10 Article no. 205 Hong Wang et al. 10 Fig. 9 The field profile of diagonal out-of-phase dipole solitons for V 0 = 9, W 0 = 0.4, and μ = 9.3. (a) the field u, (b) the real part, (c) the imaginary part, (d) the stable region of diagonal out-of-phase dipole solitons as function of gain/loss parameter W 0. The blue shaded region corresponds to the domain of stability of diagonal out-of-phase dipole solitons. The adjacent out-of-phase dipole soliton also exists in the first gap, but it cannot be stable. Its power diagram is displayed in Fig. 6b, and its propagation dynamics is shown in Fig. 8. We see that one out of the two humps of the soliton becomes very weak at z = 380, and that the corresponding linear-stability spectrum of the soliton contains a pair of real eigenvalues. Third, we discuss the influence of gain/loss component of the complexvalued potential on the key features of dipole solitons. As a typical example we consider the diagonal out-of-phase dipole soliton. The field profile of diagonal out-of-phase dipole soliton for V 0 = 9, W 0 = 0.4, and μ = 9.3 and the diagram of the domain of stability corresponding to diagonal out-of-phase dipole solitons (the propagation constant μ as function of the gain/loss component W 0 ) is displayed in Fig. 9d. From Fig. 9a c, we see that the tails of the soliton tend to appear along the x-axis with increasing the gain/loss component of the complex-valued optical potential. In addition, from Fig. 9d, we see that the region of stability of diagonal out-of-phase dipole soliton shrinks sharply with the increase of gain/loss component W 0, and before the phase-transition point the stable diagonal out-ofphase dipole soliton no longer exists.

11 11 Families of dipole solitons in self-defocusing Kerr media Article no CONCLUSIONS We have investigated the continuous families of dipole solitons and their unique propagation dynamics in partial parity-time-symmetric optical potentials and self-defocusing Kerr media. It is found that these dipole solitons can have their two humps located in the two nearest diagonal lattice sites or in the two nearest adjacent lattice sites. They can have their two humps either in-phase or out-ofphase. All these dipole solitons can exist in the first gap. The diagonal in-phase and out-of-phase dipole solitons and adjacent in-phase dipole solitons can be stable in the moderate power regime, whereas in the low and high power regimes the solitons will suffer exponential and oscillatory instability, respectively. However, the adjacent out-of-phase dipole solitons cannot be stable. In addition, we found that the diagonal out-of-phase dipole soliton is more robust than the diagonal inphase one. Moreover, the gain/loss component of the complex-valued potential has a great influence on the key features of these one-parameter families of dipole solitons. With increasing the gain/loss component, the tails of dipole solitons appear along the x-axis and their domains of stability shrink sharply. Acknowledgments. This work was supported by the Key technologies R&D Program of Guangdong Province (Nos. 2014B , 2017B ), the Applied technologies R&D Major Programs of Guangdong Province (Nos. 2015B , 2016B ), the Key Technologies R&D Major Programs of Guangzhou City (Nos , ), the National Natural Science Foundations of China (No ), the Guangdong Province Nature Foundation of China (No. 2017A ), and the Guangdong Province Education Department Foundation of China (No. 2014KZDXM059). REFERENCES 1. R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964). 2. D. K. Campbell, S. Flach, and Y. S. Kivshar, Phys. Today 57, 43 (2004). 3. Z. H. Musslimani and J. Yang, J. Opt. Soc. Am. B. 21, 5 (2004). 4. J. Yang, New J. Phys. 6, 47 (2004). 5. J. Yang and Z. H. Musslimani, Opt. Lett. 28, 2094 (2003). 6. J. Zeng and B. A. Malomed, Phys. Scr. T149, (2012). 7. J. Zheng and L. Dong, J. Opt. Soc. Am. B. 28, 780 (2011). 8. N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, (2003). 9. Y. J. He and H. Z. Wang, Opt. Express 14, 9832 (2006). 10. H. Susanto, K. J. H. Law, P. G. Kevrekidis, L. Tang, C. Lou, X. Wang, and Z. Chen, Physica D 237, 3123 (2008). 11. J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, Stud. Appl. Math. 113, 389 (2004). 12. B. A. Malomed, D. Mihalache, F. Wise, L. Torner, J. Opt. B 7, R53 (2005). 13. V. S. Bagnato, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, D. Mihalache, Rom. Rep. Phys. 67, 5 (2015). 14. B. Malomed, L. Torner, F. Wise, D. Mihalache, J. Phys. B 49, (2016). 15. D. Mihalache, Rom. Rep. Phys. 69, 403 (2017).

12 Article no. 205 Hong Wang et al C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998). 17. A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A: Math. Gen. 38, L171 (2005). 18. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007). 19. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, (2010). 20. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010). 21. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, (2008). 22. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, (2008). 23. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, J. Phys. A 41, (2008). 24. Z. Shi, X. Jiang, X. Zhu, and H. Li, Phys. Rev. A 84, (2011). 25. S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85, (2012). 26. D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, EPL 96, (2011). 27. F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, (2011). 28. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Opt. Commun. 285, 3320 (2012). 29. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, (2012). 30. C. Huang, C. Li, and L. Dong, Opt. Express 21, 3917 (2013). 31. X. Ren, H. Wang, H. Wang, and Y. He, Opt. Express 22, (2014). 32. P. Li, L. Li, and D. Mihalache, Rom. Rep. Phys. 70, 408 (2018). 33. P. Li and D. Mihalache, Proc. Romanian Acad. A 19, 61 (2018). 34. Y. Chen, Z. Yan, D. Mihalache, and B. A. Malomed, Sci. Rep. 7, 1257 (2017). 35. Q. Zhou, Proc. Romanian Acad. A 18, 223 (2017). 36. Y. Zhang, D. Qiu, Y. Cheng, and J. He, Rom. J. Phys. 62, 108 (2017). 37. R. J. Lombard and R. Mezhoud, Rom. J. Phys. 62, 112 (2017). 38. P. Li, D. Mihalache, and L. Li, Rom. J. Phys. 61, 1028 (2016). 39. Y. He, X. Zhu, and D. Mihalache, Rom. J. Phys. 61, 595 (2016). 40. P. Li, B. Liu, L. Li, and D. Mihalache, Rom. J. Phys. 61, 577 (2016). 41. D. Mihalache, Rom. Rep. Phys. 67, 1383 (2015). 42. B. Liu, L. Li, and D. Mihalache, Rom. Rep. Phys. 67, 802 (2015). 43. V. V. Konotop, J. Yang, and D. A. Zezyulin, Rev. Mod. Phys. 88, (2016). 44. S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, Laser and Photonics Reviews 10, 177 (2016). 45. J. Yang, Opt. Lett. 39, 1133 (2014). 46. J. Yang and T. I. Lakoba, Stud. Appl. Math. 118, 153 (2007). 47. J. Yang, Nonlinear Waves in Integrable Systems, SIAM, 2010.