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Beat patterns of vector solitons in nonlocal nonlinear lattices with PT symmetry
Optics Communications 451 (2019) 276–280
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Beat patterns of vector solitons in nonlocal nonlinear lattices with PT symmetry Yuanhang Weng a , Jing Huang a,c , Hong Wang a,b ,∗ a
Engineering Research Center for Optoelectronics of Guangdong Province, School of Electronic and Information Engineering, South China University of Technology, Guangzhou, 510640, China b Zhongshan Institute of Modern Industrial Technology, South China University of Technology, Zhongshan 528437, China c School of Materials Science and Engineering, Guizhou Minzu University, Guiyang 550025, China
ARTICLE Keywords: Vector soliton Beat pattern Nonlocal nonlinearity PT-symmetry
INFO
ABSTRACT We investigate the propagation of two-component vector solitons in the nonlocal nonlinear lattices with PT symmetry. We find and analyze the beat patterns of power evolution and gravity center evolution. The beat patterns are the result of the combined effect of PT symmetric potential and nonlocal nonlinearity. In particular, PT symmetry has been demonstrated to be necessary for beat patterns. The nonlocal nonlinearity and the propagation constants of vector solitons greatly affect the stability of propagation and the beat pattern. The most obvious beat patterns can be observed in the propagation of vector solitons with appropriate propagation constants in the intermediate nonlocal nonlinearity.
1. Introduction Optical spatial solitons, the result of the balance of linear diffraction and nonlinear self-focusing, are found to exist in many types of nonlinear media. Generally, these nonlinearities can be divided into local type and nonlocal type. For nonlocal nonlinear media, the nonlinear response domain contains the light incidence point and its neighborhood. Various materials exhibit the nonlocal nonlinear response, such as nematic liquid crystal [1–3], thermal nonlinear liquid [4], lead glass [5,6], and so on. In these nonlocal media, the existence and evolution of scalar solitons were studied, including bright solitons [2,3,7], dark solitons [8,9], surface solitons [10], breather solitons [11] and multipole solitons [12,13]. Besides, vector solitons are also demonstrated to be stabilized in nonlocal media [14,15]. In particular, the (1+1) dimensional vector soliton solutions and the interaction patterns of their components can be derived by the Darboux transformation method [16]. Different to scalar solitons, vector solitons are multicomponent self-trapped states that result from the nonlinear interaction between several optical beams. Take two-component vector solitons as an example, they can generate from two orthogonally polarized beams, or two beams with different wavelengths, or two mutually incoherent beams but with same wavelength and polarization. The linear potentials imprinted in nonlinear media affect solitons as well. One type of complex potentials is the Parity-Time (PT) symmetric potential. The concept of PT symmetry was first proposed by Bender and Boettcher, which a non-Hermitian operator has completely real
spectra if the system is PT symmetric [17]. In recent decades, PT symmetry then extends from quantum mechanics to optics. Optical PT symmetry is realized by manufacturing complex refractive index that meets the condition 𝑛 (𝑟) = 𝑛′ (−𝑟). That is, the real part of the complex refractive index should be symmetric. And the imaginary part, which corresponds to optical gain and loss, should be antisymmetric [18,19]. PT symmetric and nonlinear optical media support various types of solitons, such as vector solitons [20–22], scalar solitons in nonlocal nonlinear media [23–25], fundamental solitons or vortex solitons in competing nonlinear media [26–29], and multipole solitons in modulated optical lattices [30–32]. Recently, the theoretical studies and the relevant experiments of PT symmetric systems have been reviewed in Refs. [33–36], revealing the potential applications in the field of optics, photonic and matter waves. Most studies of vector spatial solitons focus on the case when the two components of vector soliton are overlapped [14,15,20–22]. As a result of the coupling effect provided by the common nonlinear waveguide, mixed-gap vector solitons, of which two components locate in different bandgaps, can be found in PT symmetric lattices [15,20,22], and some unstable scalar solitons may be stabilized in their vector form [14,15]. The nonlinear waveguides can be partially overlapped due to the nonlocal nonlinearity, which is demonstrated in scalar multipole solitons [13]. Following this line of interest, we investigate the vector solitons propagating in nonlocal nonlinear optical lattices with PT symmetry. Their two components are separated but close
Abbreviations: PT, parity and time; 2D, two dimensional; CNLSE, coupled nonlinear Schrödinger equation; MSOM, modified squared-operator method ∗ Corresponding author at: Zhongshan Institute of Modern Industrial Technology, South China University of Technology, Zhongshan 528437, China. E-mail address: [email protected] (H. Wang). https://doi.org/10.1016/j.optcom.2019.06.070 Received 6 May 2019; Received in revised form 26 June 2019; Accepted 27 June 2019 Available online 1 July 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
Y. Weng, J. Huang and H. Wang
Optics Communications 451 (2019) 276–280
enough, instead of exactly overlapped. In this case, we obtain the novel beat patterns during the propagation of these vector solitons, which are similar to optical beats when two lights with different frequencies co-propagate in the same direction. In this article, we investigate the influences of PT symmetric potential, the degree of nonlocality and the propagation constant on these beat patterns. We demonstrate that the PT symmetric potential is necessary for beat patterns, and the degrees of nonlocality and the propagation constants greatly affect the stability of vector solitons and their beat patterns. 2. Theory and models To investigate the propagation of vector solitons in nonlocal nonlinear lattices with PT symmetry, we consider the normalized twocomponent coupled nonlinear Schrödinger equation (CNLSE): ⎧ 𝜕𝑈1 ⎪𝑖 𝜕𝑧 + ∇⊥ 𝑈1 + (𝑉 + 𝑖𝑊 ) 𝑈1 + 𝜎𝑛𝑈1 = 0, ⎪ 𝜕𝑈 ⎨𝑖 2 + ∇⊥ 𝑈2 + (𝑉 + 𝑖𝑊 ) 𝑈2 + 𝜎𝑛𝑈2 = 0 ⎪ 𝜕𝑧 ⎪ 2 2 ⎩𝑑∇⊥ 𝑛 − 𝑛 + ||𝑈1 || + ||𝑈2 || = 0
(1)
where ∇⊥ = 𝜕𝑥𝑥 + 𝜕𝑦𝑦 indicates 2D Laplace operator, 𝑈1 and 𝑈2 are the complex slowly varying amplitudes of electric field, and 𝜎 = −1 represents the defocusing nonlinearity, respectively. Eq. (1) is derived from the normalized two dimensional nonlocal NLSE for one soliton [37,38]. The optical PT symmetric potential is composed of the spatially modulated refractive index 𝑉 and spatially distributed loss and gain 𝑊 . Without loss of generality, we consider a periodic PT symmetric potential, which is set as ( ) 𝑉 = 𝑉0 cos2 𝑥 + cos2 𝑦 , 𝑊 = 𝑉0 𝑊0 (sin 2𝑥 + sin 2𝑦) . (2)
Fig. 1. (a) The profiles of the vector soliton at 𝑧 = 0. (b) The phase structures of S1 and S2 at 𝑧 = 0. The black circles correspond to the location of S1 and S2. (c) The profile of refractive index at 𝑧 = 0. (d) The profiles of S1 and S2 at 𝑧 = 530. The propagation constants of S1 and S2 is 𝜇1 = 𝜇2 = 8.6. The other parameters are 𝑊0 = 0.2 and 𝑑 = 0.5 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where 𝑉0 is the depth of potential and is preset as 𝑉0 = 8, 𝑊0 is the gain–loss coefficient. Before solving Eq. (1), we consider the linear spectral problem 𝐋𝜓 (𝑥, 𝑦) = 𝜆𝜓 (𝑥, 𝑦) with eigenvalues 𝜆 and eigenfunctions 𝜓 (𝑥, 𝑦), where the operator 𝐋 = 𝑖𝑑∕𝑑𝑧 + 𝑑 2 ∕𝑑𝑥2 + 𝑑 2 ∕𝑑𝑦2 + 𝑉 + 𝑖𝑊 . By solving this linear function, we obtain the threshold 𝑊0𝑡ℎ = 0.5, below which all 𝜆 are real. Otherwise 𝜆 becomes complex, which indicates the PT symmetry is broken [39]. Interaction between solitons can be coherent, partially coherent or mutually incoherent, depending on their relative phases. In practice, phase control of mutual solitons may be difficult to achieve, while the mutually incoherent interaction can be realized by utilizing a piezoelectric transducer with the AC signal [40,41]. To describe the nonlocal nonlinear and mutually incoherent interaction of the two components of vector soliton, the nonlocal refractive index 𝑛 is codetermined by |𝑈1 |2 + |𝑈2 |2 and the degree of nonlocality 𝑑 in Eq. (1). | | | | The solution of Eq. (1) can be sought in the form ( ) 𝑈1 (𝑥, 𝑦; 𝑧) = 𝑢1 (𝑥, 𝑦) exp 𝑖𝜇1 𝑧 , ( ) (3) 𝑈2 (𝑥, 𝑦; 𝑧) = 𝑢2 (𝑥, 𝑦) exp 𝑖𝜇2 𝑧 .
This gravity center that involve the power distribution of light in space is proper to represent the location of two components of vector soliton. At the beginning of the propagation (𝑧 = 0), the gravity centers of the two components and the whole optical field are (−𝜋, −𝜋), (𝜋, 𝜋) and (0, 0), respectively. Comparing to the gravity center itself, the change of gravity center is more important because it reveals the shift of√light energy. The change of gravity center can be defined as 𝛥 = ( ( ) )2 ( )2 ± 𝑥′ − 𝑥0 + 𝑦′ − 𝑦0 , where ± represents 𝑥′ , 𝑦′ is on the positive ( ) direction (+) or negative direction (−) of 𝑥0 , 𝑦0 . 3. Numerical results 3.1. The beat patterns We consider vector solitons of which the two components are close enough for example. As shown in Fig. 1(a), one component (S1) is located on (−𝜋, −𝜋), and another (S2) is located on (𝜋, 𝜋). They have the same propagation constants, i.e. 𝜇1 = 𝜇2 = 8.6. The parameters of nonlinear medium are set as 𝑊0 = 0.2 and 𝑑 = 0.5. S1 and S2 are in-phase, as shown in Fig. 1(b). Because of nonlocal nonlinearity, the distribution of refractive index is wider than the profile of solitons, as shown in Fig. 1(c). A stable propagation requires the soliton to maintain its profile and power. S1 and S2 can stably propagate before 𝑧 ≈ 530. At 𝑧 ≈ 530, part of the power of S1 flows to the adjacent lattice point, causing the uncontrollable growth of the power and the break of stable propagation after that, as shown in Fig. 1(d). To investigate the interaction between S1 and S2 in detail, we show the evolution of the power and the gravity center of the whole optical field in Fig. 2. From Fig. 2(a), a beat pattern is observed. At the beginning of propagation, the power undergoes a periodic oscillation, and there is no sign of beats. When the distance of propagation reaches 𝑧 ≈ 205, the oscillation is suppressed. After that, the power oscillation is enhanced and suppressed again. The beats occur until one of these solitons (S1) becomes unstable. It is worth noting that the persistent distance of each beat changes in the course of propagation. For the first
where 𝜇1 and 𝜇2 are the corresponding propagation constants. Substituting Eq. (3) into Eq. (1), we obtain the following eigen-equation: ⎧𝜇𝑢 = (∇ + 𝑉 + 𝑖𝑊 + 𝜎𝑛) 𝑢 , ) 1 ⎪ 1 ( ⊥ ⎨𝜇𝑢2 = ∇⊥ + 𝑉 + 𝑖𝑊 + 𝜎𝑛 𝑢2 , ⎪𝑑∇ 𝑛 − 𝑛 + |𝑢 |2 + |𝑢 |2 = 0. | 1| | 2| ⎩ ⊥
(4)
Eq. (4) then can be numerically solved by the developed modified squared-operator method (MSOM) [42]. Actually, the MSOM calculates the ‘‘squared-operator’’ evolution equation deduced from Eq. (4), and it introduces a speed-up operator. The propagation of 𝑈1 and 𝑈2 is simulated by the split-step method [42], which calculates linear and nonlinear parts of the evolution equation separately. To investigate the interaction between the two components of vector soliton in detail, we consider their powers and the gravity centers. The total power of a vec+∞ 2 tor soliton is defined as 𝑃 = ∬−∞ ||𝑢1 + 𝑢2 || 𝑑𝑥𝑑𝑦. We defined the grav( ) +∞ 2 ity center of an optical field as 𝑥0 , 𝑦0 = 𝑃1 ∬−∞ (𝑥, 𝑦) ||𝑢1 + 𝑢2 || 𝑑𝑥𝑑𝑦. 277
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Optics Communications 451 (2019) 276–280
Fig. 3. The evolution of the power with different gain–loss coefficients 𝑊0 . The propagation constants of S1 and S2 is 𝜇1 = 𝜇2 = 8.6. 𝑑 = 0.5.
nonlocal nonlinearity affect the beat patterns, we investigate the power evolution during the propagation of S1 and S2 in the media with different gain–loss coefficients 𝑊0 in Fig. 3 and different degrees of nonlocality 𝑑 in Fig. 4. As for the gravity center evolution, its beat pattern can be derived from the beat patterns of the power evolution. Thus the gravity center evolution is self-evident and is unnecessary to be shown in our following work. Fig. 3 shows the evolution of the power of the whole optical field with different 𝑊0 . The other parameters are set as 𝜇1 = 𝜇2 = 8.6 and 𝑑 = 0.5. The beat pattern of 𝑊0 = 0.2 is the most obvious. As 𝑊0 becomes smaller, the power oscillation is weaker and the persistent distances of beats are longer. This is because the gain–loss coefficient 𝑊0 of PT symmetric potential indicates the intensity of energy exchange between solitons and media. When 𝑊0 = 0, the potential becomes real, which means no energy exchanges (if the radiation and other losses are ignored). Thus the power curve of 𝑊0 = 0 is a straight line, which means the PT symmetric potential is necessary for beat patterns. When 𝑊0 is close to the threshold 0.5, the energy exchange between S1 and S2 becomes stronger. Thus the power oscillation is unexpected and no beat can be found on the power curve, such as the power curve of 𝑊0 = 0.3. Moreover, the larger 𝑊0 leads to lower power and faster power growth.
Fig. 2. The evolution of the power and the gravity center during propagation of the vector soliton in Fig. 1. (a) is the power of the whole light field. (b) is the corresponding power of S1 (red solid line) and S2 (green solid line). (c) is the change of gravity center of the whole light field. The blue dash lines are auxiliary lines that mark the location of the bulges or the nodes of beats. And the blue numbers in circles correspond to those auxiliary lines . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
beat, the persistent distance is about 𝛥𝑧 ≈ 115. For the next three beats, the persistent distance is 𝛥𝑧 ≈ 80, 55, 50. To study the beats in more detail, the power evolutions of S1 and S2 are drawn in Fig. 2(b). Both of S1 and S2 undergo periodic power oscillations, and their frequencies are close, which results the beats. Before 𝑧 ≈ 100, the frequencies of power oscillation for S1 and S2 are equal, thus no beat can be found for the oscillation of total power. After that, the frequencies for S1 and S2 become slightly different, and beats occur on the evolution of total power. The differences between the S1 and S2 frequencies become larger during the propagation, which causes the persistent distances of beats being smaller. Due to the nonlinear effect, the power of S1 and S2 increases during propagation, which leads to the increase of the total power. With the help of auxiliary lines in Fig. 2, it is clear that the bulges of the beats of power oscillation (location ①, ③, ⑤, ⑦ and ⑨) correspond to the nodes of the beats of gravity center oscillation, and vice versa. That can be explained as follows. Beat bulges or beat nodes of total power oscillation depend on the phase different of power oscillation for S1 and S2. When their power oscillation are in-phase, beat bulges of total power oscillation occur. When their power oscillation are out-of-phase, beat nodes occur. If the power oscillation of S1 and S2 is in-phase (i.e. there is not phase difference between them), the energies of S1 and S2 will not exchange, and thus the gravity center of the whole light field holds, which corresponds to the nodes in Fig. 2(c). Otherwise, the energies of S1 and S2 will exchange, which leads to the gravity center oscillation. As the phase difference becomes larger, the gravity center oscillation is stronger, which corresponds to the bulges in Fig. 2(c).
3.3. The effect of different degrees of nonlocality To investigate the influence of the degree of nonlocality on the beat patterns of the power evolution, the power curves with different 𝑑 are drawn in Fig. 4. The other parameters are set as 𝜇1 = 𝜇2 = 8.6 and 𝑊0 = 0.2. The range of 𝑑 that we investigate is from 𝑑 = 0.005 (nearly local case) to 𝑑 = 2 (strongly nonlocal case). Five power curves are chosen to show in Fig. 4. When the nonlinearity is local or nearly local, we find that the solitons is very hard to propagate stably, which indicates that nonlinear nonlocality is necessary for stable propagation. When 𝑑 = 0.05 (weakly nonlocal case), S1 and S2 can propagate stably before 𝑧 = 690. The power oscillation becomes slower and weaker during the propagation, but no beat can be found. When 𝑑 = 0.2 (intermediately nonlocal case), the beat pattern occurs but it is not obvious. A much more obvious beat pattern is observed when 𝑑 = 0.5, as shown in Fig. 4. S1 and S2 can propagate stably until 𝑧 = 530. The dash line on its power curve indicates unstable propagation, which is valueless to our analysis. As 𝑑 becomes larger (the nonlinearity becomes more nonlocal), the distance of stable propagation is shorter (𝑧 = 100 for 𝑑 = 1, 𝑧 = 93 for 𝑑 = 1.5 and 𝑧 = 76 for 𝑑 = 2), the power oscillation becomes stronger and the beat pattern is more compact. This is because the two solitons that are close enough can affect each other via the nonlocal refractive index. The insets in Fig. 4 are the corresponding profiles of refractive index. It is obvious that the refractive index is more nonlocal with larger 𝑑, which means one soliton is easier to affect another one.
3.2. The effect of different gain–loss coefficients The PT symmetric potential and the nonlocal nonlinearity of media are the causes of the beat patterns of the power and the gravity center evolution. To understand how the PT symmetric potential and the 278
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Optics Communications 451 (2019) 276–280
Fig. 6. The evolution of the power when 𝜇1 ≠ 𝜇2 . 𝜇2 is fixed as 𝜇2 = 8.6. The solid lines or the dash line indicate the propagation is stable or not. From top to bottom, the propagation constants of S1 are 7.8, 8.2, 8.6, 9.0 and 9.4. 𝑑 = 0.5, 𝑊0 = 0.2.
beats becomes longer. For the vector solitons with 𝜇0 > 9.4, their power curves are almost smooth lines, which indicates the components of vector solitons with large propagation constant have no interaction between them. That is because when the power of vector solitons with large propagation are too small that their behaviors are similar to linear modes. In other words, the nonlocal nonlinear effect becomes weak for these vector solitons with small power. When propagation constants decrease, the power oscillation becomes stronger. We can see that the oscillation for 𝜇0 = 7.8 is much more violent than for 𝜇0 = 8.6, while the distance of stable propagation for 𝜇0 = 7.8 is much short (𝑧 = 200). This is due to the strong nonlocal nonlinear effect for vector solitons with large power. When 𝜇1 ≠ 𝜇2 , the difference between the two components of vector solitons should be considered. We fix the propagation constant of S2 as 𝜇2 = 8.6, and investigate the beat patterns with different 𝜇1 . Four cases (𝜇1 = 7.8, 8.2, 9.0 and 9.4) are chosen to compare the beat pattern for 𝜇1 = 𝜇2 = 8.6, as shown in Fig. 6. As the difference ||𝜇1 − 𝜇2 || becomes larger, the beat patterns are harder to be recognized. The power curves of the vector solitons with 𝜇1 = 7.7 and 𝜇1 = 9.5 are equalamplitude oscillation curves with no beat pattern. Thus for 𝜇2 = 8.6, |𝜇1 − 𝜇2 | < 0.8 is a necessary condition to obtain a beat pattern for a | | vector soliton propagating in nonlocal nonlinear PT-symmetric lattices. When 𝜇1 decreases, the power oscillation becomes stronger and the distance of stable propagation for vector solitons becomes shorter, of which the cause is the same as the case of 𝜇1 = 𝜇2 .
Fig. 4. The evolution of the power with different degrees of nonlocality d. The solid lines or the dash line indicate the propagation is stable or not. The insets are the profiles of refractive index under different d. The propagation constants of S1 and S2 is 𝜇1 = 𝜇2 = 8.6. 𝑊0 = 0.2.
4. Conclusions and discusses In conclusion, we found and investigated the beat patterns during the propagation of vector solitons in the nonlocal nonlinear lattices with PT symmetry. Because of the combined effect of PT symmetric potential and nonlocal nonlinearity, the power and the gravity center of the whole light field undergo oscillations, which results the beat patterns in the power and the gravity center evolution curves. We demonstrated that the PT symmetric potential is necessary for beat patterns. But if the gain–loss coefficient is too large and close to the threshold of PT symmetry breaking, the beat patterns would disappear. The degree of nonlocality also affect the stability of propagation and the beat patterns. The intermediate nonlocal nonlinearity (𝑑 ∼ 10−1 ) is most suitable for stable propagation and distinct beat patterns. We also considered the influence of different propagation constants of the two components of vector solitons. When the propagation constants are equal, the vector solitons with intermediate power have the most distinct beat patterns. When the difference between the two propagation constants are large, the beat patterns degenerate into equal-amplitude oscillations. In practical, optical beats can be used to measure the wavelength of the tested lasers by measuring the frequency difference between the reference and tested lasers. This method is more accurate than the other
Fig. 5. The evolution of the power when 𝜇1 = 𝜇2 . The solid lines or the dash line indicate the propagation is stable or not. From top to bottom, the propagation constants are 7.8, 8.2, 8.6, 9.0 and 9.4. 𝑑 = 0.5, 𝑊0 = 0.2.
3.4. The effect of different propagation constants We also investigate the influence of the propagation constants 𝜇1 and 𝜇2 of vector solitons on beat patterns. In this section, the degree of nonlocality and the gain–loss coefficient are fixed as 𝑑 = 0.5 and 𝑊0 = 0.2. Two cases are considered: 𝜇1 = 𝜇2 and 𝜇1 ≠ 𝜇2 . When 𝜇1 = 𝜇2 = 𝜇0 , the characters and the situation of the two components of vector solitons are exactly the same. The evolution of the power for these vector solitons with different 𝜇0 are shown in Fig. 5. Comparing the power curves in Fig. 5, As propagation constants increase, the power of vector solitons decreases. Meanwhile, the power oscillation becomes weaker and the persistent distance of 279
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methods, and does not need an interferometer [43]. Inspired by the optical beat technique, we believe the beat pattern of optical solitons is a potential method to detect the differences between solitons or to calibrate optical solitons.
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Acknowledgments This work was supported by Science and Technologies plan Projects of Guangdong Province, China (Nos. 2017B010112003, 2017A050506013); Applied Technologies R & D Major Programs of Guangdong Province, China (Nos. 2015B010127013, 2016B01012300); Science and Technologies Projects of Guangzhou City, China (Nos. 201604046021, 201704030139, 201905010001); Science and Technology Development Special Fund Projects of Zhongshan City, China (Nos. 2017F2FC0002, 2017A1009, 2019AG014). References [1] M. Peccianti, C. Conti, G. Assanto, Interplay between nonlocality and nonlinearity in nematic liquid crystals, Opt. Lett. 30 (2005) 41, http://dx.doi.org/10.1364/ OL.30.000415. [2] A. Piccardi, S. Residori, G. Assanto, Nonlocal soliton scattering in random potentials, J. Opt. 18 (2016) 07LT01, http://dx.doi.org/10.1088/2040-8978/18/ 7/07lt01. [3] G. Assanto, M. Peccianti, C. Conti, One-dimensional transverse modulational instability in nonlocal media with a reorientational nonlinearity, IEEE J. Sel. Topics Quantum Electron 10 (2004) 862, http://dx.doi.org/10.1109/JSTQE. 2004.835999. [4] A. Dreischuh, D. Neshev, D.E. Petersen, O. Bang, W. Królikowski, Observation of attraction between dark solitons, Phys. Rev. Lett. 96 (2006) 043901, http: //dx.doi.org/10.1103/Physrevlett.96.043901. [5] C. Rotschild, O. Cohen, O. Manela, M. Segev, T. Carmon, Solitons in nonlinear media with an infinite range of nonlocality: first observation of conherent elliptic solitons and vortex-ring solitons, Phys. Rev. Lett. 95 (2005) 213904, http://dx.doi.org/10.1103/Physrevlett.95.213904. [6] C. Rotschild, B. Alfassi, O. Cohen, M. Segev, Long-range interactions between optical solitons, Nat. Phys. 2 (2006) 769, http://dx.doi.org/10.1038/nphys445. [7] Z. Shi, H. Li, X. Zhu, X. Jiang, Nonlocal bright spatial solitons in defocusing Kerr media supported by PT symmetric potentials, Europhys. Lett. 98 (2012) 64006, http://dx.doi.org/10.1209/0295-5075/98/64006. [8] L. Chen, Q. Wang, M. Shen, H. Zhao, Y.Y. Lin, C.C. Jeng, R.K. Lee, W. Królikowski, Nonlocal dark solitons under competing cubic-quintic nonlinearities, Opt. Lett. 38 (2013) 13, http://dx.doi.org/10.1364/OL.38.000013. [9] W. Chen, M. Shen, Q. Kong, J. Shi, Q. Wang, W. Królikowski, Interactions of nonlocal dark solitons under competing cubic-quintic nonlinearities, Opt. Lett. 39 (2014) 1764, http://dx.doi.org/10.1364/Ol.39.001764. [10] Y.V. Kartashov, L. Torner, V.A. Vysloukh, Lattice-supported surface solitons in nonlocal nonlinear media, Opt. Lett. 31 (2006) 2595, http://dx.doi.org/10.1364/ Ol.31.002595. [11] A. Alberucci, C.P. Jisha, G. Assanto, Breather solitons in highly nonlocal media, J. Opt. 18 (2016) 125501, http://dx.doi.org/10.1088/2040-8978/18/12/125501. [12] C. Rotschild, M. Segev, Z. Xu, Y.V. Kartashov, L. Torner, O. Cohen, Twodimensional multipole solitons in nonlocal nonlinear media, Opt. Lett. 31 (2006) 3312, http://dx.doi.org/10.1364/Ol.31.003312. [13] J. Huang, Y. Weng, H. Wang, Stability and internal interaction of multipole solitons in nonlocal PT-symmetric lattices, Opt. Express 26 (2018) 11667, http: //dx.doi.org/10.1364/Oe.26.011667. [14] Y.V. Izdebskaya, G. Assanto, W. Królikowski, Observation of stable-vector vortex solitons, Opt. Lett. 40 (2015) 4182, http://dx.doi.org/10.1364/Ol.40.004182. [15] L. Li, X. Zhu, H. Li, T. Lai, Vector solitons in parity-time symmetric lattices with nonlocal nonlinearity, J. Opt. 18 (2016) 095501, http://dx.doi.org/10.1088/ 2040-8978/18/9/095501. [16] Q. Zhang, M. Gao. Rational, Rational soliton solutions in the parity-timesymmetric nonlocal coupled nonlinear Schrödinger equations, Commun. Nonlinear Sci. Numer. Simul. 63 (2018) 253, http://dx.doi.org/10.1016/j.cnsns.2018. 02.029. [17] C.M. Bender, S. Boettcher, Real spectra in non-hermitian hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998) 5243, http://dx.doi.org/10.1103/ PhysRevLett.80.5243. [18] A. Guo, G.J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G.A. Siviloglou, D.N. Christodoulides, Observation of PT-symmetry breaking in complex optical potentials, Phys. Rev. Lett. 103 (2009) 093902, http://dx.doi. org/10.1103/Physrevlett.103.093902.
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Beat patterns of vector solitons in nonlocal nonlinear lattices with PT symmetry Yuanhang Weng a , Jing Huang a,c , Hong Wang a,b ,∗ a
Engineering Research Center for Optoelectronics of Guangdong Province, School of Electronic and Information Engineering, South China University of Technology, Guangzhou, 510640, China b Zhongshan Institute of Modern Industrial Technology, South China University of Technology, Zhongshan 528437, China c School of Materials Science and Engineering, Guizhou Minzu University, Guiyang 550025, China
ARTICLE Keywords: Vector soliton Beat pattern Nonlocal nonlinearity PT-symmetry
INFO
ABSTRACT We investigate the propagation of two-component vector solitons in the nonlocal nonlinear lattices with PT symmetry. We find and analyze the beat patterns of power evolution and gravity center evolution. The beat patterns are the result of the combined effect of PT symmetric potential and nonlocal nonlinearity. In particular, PT symmetry has been demonstrated to be necessary for beat patterns. The nonlocal nonlinearity and the propagation constants of vector solitons greatly affect the stability of propagation and the beat pattern. The most obvious beat patterns can be observed in the propagation of vector solitons with appropriate propagation constants in the intermediate nonlocal nonlinearity.
1. Introduction Optical spatial solitons, the result of the balance of linear diffraction and nonlinear self-focusing, are found to exist in many types of nonlinear media. Generally, these nonlinearities can be divided into local type and nonlocal type. For nonlocal nonlinear media, the nonlinear response domain contains the light incidence point and its neighborhood. Various materials exhibit the nonlocal nonlinear response, such as nematic liquid crystal [1–3], thermal nonlinear liquid [4], lead glass [5,6], and so on. In these nonlocal media, the existence and evolution of scalar solitons were studied, including bright solitons [2,3,7], dark solitons [8,9], surface solitons [10], breather solitons [11] and multipole solitons [12,13]. Besides, vector solitons are also demonstrated to be stabilized in nonlocal media [14,15]. In particular, the (1+1) dimensional vector soliton solutions and the interaction patterns of their components can be derived by the Darboux transformation method [16]. Different to scalar solitons, vector solitons are multicomponent self-trapped states that result from the nonlinear interaction between several optical beams. Take two-component vector solitons as an example, they can generate from two orthogonally polarized beams, or two beams with different wavelengths, or two mutually incoherent beams but with same wavelength and polarization. The linear potentials imprinted in nonlinear media affect solitons as well. One type of complex potentials is the Parity-Time (PT) symmetric potential. The concept of PT symmetry was first proposed by Bender and Boettcher, which a non-Hermitian operator has completely real
spectra if the system is PT symmetric [17]. In recent decades, PT symmetry then extends from quantum mechanics to optics. Optical PT symmetry is realized by manufacturing complex refractive index that meets the condition 𝑛 (𝑟) = 𝑛′ (−𝑟). That is, the real part of the complex refractive index should be symmetric. And the imaginary part, which corresponds to optical gain and loss, should be antisymmetric [18,19]. PT symmetric and nonlinear optical media support various types of solitons, such as vector solitons [20–22], scalar solitons in nonlocal nonlinear media [23–25], fundamental solitons or vortex solitons in competing nonlinear media [26–29], and multipole solitons in modulated optical lattices [30–32]. Recently, the theoretical studies and the relevant experiments of PT symmetric systems have been reviewed in Refs. [33–36], revealing the potential applications in the field of optics, photonic and matter waves. Most studies of vector spatial solitons focus on the case when the two components of vector soliton are overlapped [14,15,20–22]. As a result of the coupling effect provided by the common nonlinear waveguide, mixed-gap vector solitons, of which two components locate in different bandgaps, can be found in PT symmetric lattices [15,20,22], and some unstable scalar solitons may be stabilized in their vector form [14,15]. The nonlinear waveguides can be partially overlapped due to the nonlocal nonlinearity, which is demonstrated in scalar multipole solitons [13]. Following this line of interest, we investigate the vector solitons propagating in nonlocal nonlinear optical lattices with PT symmetry. Their two components are separated but close
Abbreviations: PT, parity and time; 2D, two dimensional; CNLSE, coupled nonlinear Schrödinger equation; MSOM, modified squared-operator method ∗ Corresponding author at: Zhongshan Institute of Modern Industrial Technology, South China University of Technology, Zhongshan 528437, China. E-mail address: [email protected] (H. Wang). https://doi.org/10.1016/j.optcom.2019.06.070 Received 6 May 2019; Received in revised form 26 June 2019; Accepted 27 June 2019 Available online 1 July 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
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Optics Communications 451 (2019) 276–280
enough, instead of exactly overlapped. In this case, we obtain the novel beat patterns during the propagation of these vector solitons, which are similar to optical beats when two lights with different frequencies co-propagate in the same direction. In this article, we investigate the influences of PT symmetric potential, the degree of nonlocality and the propagation constant on these beat patterns. We demonstrate that the PT symmetric potential is necessary for beat patterns, and the degrees of nonlocality and the propagation constants greatly affect the stability of vector solitons and their beat patterns. 2. Theory and models To investigate the propagation of vector solitons in nonlocal nonlinear lattices with PT symmetry, we consider the normalized twocomponent coupled nonlinear Schrödinger equation (CNLSE): ⎧ 𝜕𝑈1 ⎪𝑖 𝜕𝑧 + ∇⊥ 𝑈1 + (𝑉 + 𝑖𝑊 ) 𝑈1 + 𝜎𝑛𝑈1 = 0, ⎪ 𝜕𝑈 ⎨𝑖 2 + ∇⊥ 𝑈2 + (𝑉 + 𝑖𝑊 ) 𝑈2 + 𝜎𝑛𝑈2 = 0 ⎪ 𝜕𝑧 ⎪ 2 2 ⎩𝑑∇⊥ 𝑛 − 𝑛 + ||𝑈1 || + ||𝑈2 || = 0
(1)
where ∇⊥ = 𝜕𝑥𝑥 + 𝜕𝑦𝑦 indicates 2D Laplace operator, 𝑈1 and 𝑈2 are the complex slowly varying amplitudes of electric field, and 𝜎 = −1 represents the defocusing nonlinearity, respectively. Eq. (1) is derived from the normalized two dimensional nonlocal NLSE for one soliton [37,38]. The optical PT symmetric potential is composed of the spatially modulated refractive index 𝑉 and spatially distributed loss and gain 𝑊 . Without loss of generality, we consider a periodic PT symmetric potential, which is set as ( ) 𝑉 = 𝑉0 cos2 𝑥 + cos2 𝑦 , 𝑊 = 𝑉0 𝑊0 (sin 2𝑥 + sin 2𝑦) . (2)
Fig. 1. (a) The profiles of the vector soliton at 𝑧 = 0. (b) The phase structures of S1 and S2 at 𝑧 = 0. The black circles correspond to the location of S1 and S2. (c) The profile of refractive index at 𝑧 = 0. (d) The profiles of S1 and S2 at 𝑧 = 530. The propagation constants of S1 and S2 is 𝜇1 = 𝜇2 = 8.6. The other parameters are 𝑊0 = 0.2 and 𝑑 = 0.5 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where 𝑉0 is the depth of potential and is preset as 𝑉0 = 8, 𝑊0 is the gain–loss coefficient. Before solving Eq. (1), we consider the linear spectral problem 𝐋𝜓 (𝑥, 𝑦) = 𝜆𝜓 (𝑥, 𝑦) with eigenvalues 𝜆 and eigenfunctions 𝜓 (𝑥, 𝑦), where the operator 𝐋 = 𝑖𝑑∕𝑑𝑧 + 𝑑 2 ∕𝑑𝑥2 + 𝑑 2 ∕𝑑𝑦2 + 𝑉 + 𝑖𝑊 . By solving this linear function, we obtain the threshold 𝑊0𝑡ℎ = 0.5, below which all 𝜆 are real. Otherwise 𝜆 becomes complex, which indicates the PT symmetry is broken [39]. Interaction between solitons can be coherent, partially coherent or mutually incoherent, depending on their relative phases. In practice, phase control of mutual solitons may be difficult to achieve, while the mutually incoherent interaction can be realized by utilizing a piezoelectric transducer with the AC signal [40,41]. To describe the nonlocal nonlinear and mutually incoherent interaction of the two components of vector soliton, the nonlocal refractive index 𝑛 is codetermined by |𝑈1 |2 + |𝑈2 |2 and the degree of nonlocality 𝑑 in Eq. (1). | | | | The solution of Eq. (1) can be sought in the form ( ) 𝑈1 (𝑥, 𝑦; 𝑧) = 𝑢1 (𝑥, 𝑦) exp 𝑖𝜇1 𝑧 , ( ) (3) 𝑈2 (𝑥, 𝑦; 𝑧) = 𝑢2 (𝑥, 𝑦) exp 𝑖𝜇2 𝑧 .
This gravity center that involve the power distribution of light in space is proper to represent the location of two components of vector soliton. At the beginning of the propagation (𝑧 = 0), the gravity centers of the two components and the whole optical field are (−𝜋, −𝜋), (𝜋, 𝜋) and (0, 0), respectively. Comparing to the gravity center itself, the change of gravity center is more important because it reveals the shift of√light energy. The change of gravity center can be defined as 𝛥 = ( ( ) )2 ( )2 ± 𝑥′ − 𝑥0 + 𝑦′ − 𝑦0 , where ± represents 𝑥′ , 𝑦′ is on the positive ( ) direction (+) or negative direction (−) of 𝑥0 , 𝑦0 . 3. Numerical results 3.1. The beat patterns We consider vector solitons of which the two components are close enough for example. As shown in Fig. 1(a), one component (S1) is located on (−𝜋, −𝜋), and another (S2) is located on (𝜋, 𝜋). They have the same propagation constants, i.e. 𝜇1 = 𝜇2 = 8.6. The parameters of nonlinear medium are set as 𝑊0 = 0.2 and 𝑑 = 0.5. S1 and S2 are in-phase, as shown in Fig. 1(b). Because of nonlocal nonlinearity, the distribution of refractive index is wider than the profile of solitons, as shown in Fig. 1(c). A stable propagation requires the soliton to maintain its profile and power. S1 and S2 can stably propagate before 𝑧 ≈ 530. At 𝑧 ≈ 530, part of the power of S1 flows to the adjacent lattice point, causing the uncontrollable growth of the power and the break of stable propagation after that, as shown in Fig. 1(d). To investigate the interaction between S1 and S2 in detail, we show the evolution of the power and the gravity center of the whole optical field in Fig. 2. From Fig. 2(a), a beat pattern is observed. At the beginning of propagation, the power undergoes a periodic oscillation, and there is no sign of beats. When the distance of propagation reaches 𝑧 ≈ 205, the oscillation is suppressed. After that, the power oscillation is enhanced and suppressed again. The beats occur until one of these solitons (S1) becomes unstable. It is worth noting that the persistent distance of each beat changes in the course of propagation. For the first
where 𝜇1 and 𝜇2 are the corresponding propagation constants. Substituting Eq. (3) into Eq. (1), we obtain the following eigen-equation: ⎧𝜇𝑢 = (∇ + 𝑉 + 𝑖𝑊 + 𝜎𝑛) 𝑢 , ) 1 ⎪ 1 ( ⊥ ⎨𝜇𝑢2 = ∇⊥ + 𝑉 + 𝑖𝑊 + 𝜎𝑛 𝑢2 , ⎪𝑑∇ 𝑛 − 𝑛 + |𝑢 |2 + |𝑢 |2 = 0. | 1| | 2| ⎩ ⊥
(4)
Eq. (4) then can be numerically solved by the developed modified squared-operator method (MSOM) [42]. Actually, the MSOM calculates the ‘‘squared-operator’’ evolution equation deduced from Eq. (4), and it introduces a speed-up operator. The propagation of 𝑈1 and 𝑈2 is simulated by the split-step method [42], which calculates linear and nonlinear parts of the evolution equation separately. To investigate the interaction between the two components of vector soliton in detail, we consider their powers and the gravity centers. The total power of a vec+∞ 2 tor soliton is defined as 𝑃 = ∬−∞ ||𝑢1 + 𝑢2 || 𝑑𝑥𝑑𝑦. We defined the grav( ) +∞ 2 ity center of an optical field as 𝑥0 , 𝑦0 = 𝑃1 ∬−∞ (𝑥, 𝑦) ||𝑢1 + 𝑢2 || 𝑑𝑥𝑑𝑦. 277
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Optics Communications 451 (2019) 276–280
Fig. 3. The evolution of the power with different gain–loss coefficients 𝑊0 . The propagation constants of S1 and S2 is 𝜇1 = 𝜇2 = 8.6. 𝑑 = 0.5.
nonlocal nonlinearity affect the beat patterns, we investigate the power evolution during the propagation of S1 and S2 in the media with different gain–loss coefficients 𝑊0 in Fig. 3 and different degrees of nonlocality 𝑑 in Fig. 4. As for the gravity center evolution, its beat pattern can be derived from the beat patterns of the power evolution. Thus the gravity center evolution is self-evident and is unnecessary to be shown in our following work. Fig. 3 shows the evolution of the power of the whole optical field with different 𝑊0 . The other parameters are set as 𝜇1 = 𝜇2 = 8.6 and 𝑑 = 0.5. The beat pattern of 𝑊0 = 0.2 is the most obvious. As 𝑊0 becomes smaller, the power oscillation is weaker and the persistent distances of beats are longer. This is because the gain–loss coefficient 𝑊0 of PT symmetric potential indicates the intensity of energy exchange between solitons and media. When 𝑊0 = 0, the potential becomes real, which means no energy exchanges (if the radiation and other losses are ignored). Thus the power curve of 𝑊0 = 0 is a straight line, which means the PT symmetric potential is necessary for beat patterns. When 𝑊0 is close to the threshold 0.5, the energy exchange between S1 and S2 becomes stronger. Thus the power oscillation is unexpected and no beat can be found on the power curve, such as the power curve of 𝑊0 = 0.3. Moreover, the larger 𝑊0 leads to lower power and faster power growth.
Fig. 2. The evolution of the power and the gravity center during propagation of the vector soliton in Fig. 1. (a) is the power of the whole light field. (b) is the corresponding power of S1 (red solid line) and S2 (green solid line). (c) is the change of gravity center of the whole light field. The blue dash lines are auxiliary lines that mark the location of the bulges or the nodes of beats. And the blue numbers in circles correspond to those auxiliary lines . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
beat, the persistent distance is about 𝛥𝑧 ≈ 115. For the next three beats, the persistent distance is 𝛥𝑧 ≈ 80, 55, 50. To study the beats in more detail, the power evolutions of S1 and S2 are drawn in Fig. 2(b). Both of S1 and S2 undergo periodic power oscillations, and their frequencies are close, which results the beats. Before 𝑧 ≈ 100, the frequencies of power oscillation for S1 and S2 are equal, thus no beat can be found for the oscillation of total power. After that, the frequencies for S1 and S2 become slightly different, and beats occur on the evolution of total power. The differences between the S1 and S2 frequencies become larger during the propagation, which causes the persistent distances of beats being smaller. Due to the nonlinear effect, the power of S1 and S2 increases during propagation, which leads to the increase of the total power. With the help of auxiliary lines in Fig. 2, it is clear that the bulges of the beats of power oscillation (location ①, ③, ⑤, ⑦ and ⑨) correspond to the nodes of the beats of gravity center oscillation, and vice versa. That can be explained as follows. Beat bulges or beat nodes of total power oscillation depend on the phase different of power oscillation for S1 and S2. When their power oscillation are in-phase, beat bulges of total power oscillation occur. When their power oscillation are out-of-phase, beat nodes occur. If the power oscillation of S1 and S2 is in-phase (i.e. there is not phase difference between them), the energies of S1 and S2 will not exchange, and thus the gravity center of the whole light field holds, which corresponds to the nodes in Fig. 2(c). Otherwise, the energies of S1 and S2 will exchange, which leads to the gravity center oscillation. As the phase difference becomes larger, the gravity center oscillation is stronger, which corresponds to the bulges in Fig. 2(c).
3.3. The effect of different degrees of nonlocality To investigate the influence of the degree of nonlocality on the beat patterns of the power evolution, the power curves with different 𝑑 are drawn in Fig. 4. The other parameters are set as 𝜇1 = 𝜇2 = 8.6 and 𝑊0 = 0.2. The range of 𝑑 that we investigate is from 𝑑 = 0.005 (nearly local case) to 𝑑 = 2 (strongly nonlocal case). Five power curves are chosen to show in Fig. 4. When the nonlinearity is local or nearly local, we find that the solitons is very hard to propagate stably, which indicates that nonlinear nonlocality is necessary for stable propagation. When 𝑑 = 0.05 (weakly nonlocal case), S1 and S2 can propagate stably before 𝑧 = 690. The power oscillation becomes slower and weaker during the propagation, but no beat can be found. When 𝑑 = 0.2 (intermediately nonlocal case), the beat pattern occurs but it is not obvious. A much more obvious beat pattern is observed when 𝑑 = 0.5, as shown in Fig. 4. S1 and S2 can propagate stably until 𝑧 = 530. The dash line on its power curve indicates unstable propagation, which is valueless to our analysis. As 𝑑 becomes larger (the nonlinearity becomes more nonlocal), the distance of stable propagation is shorter (𝑧 = 100 for 𝑑 = 1, 𝑧 = 93 for 𝑑 = 1.5 and 𝑧 = 76 for 𝑑 = 2), the power oscillation becomes stronger and the beat pattern is more compact. This is because the two solitons that are close enough can affect each other via the nonlocal refractive index. The insets in Fig. 4 are the corresponding profiles of refractive index. It is obvious that the refractive index is more nonlocal with larger 𝑑, which means one soliton is easier to affect another one.
3.2. The effect of different gain–loss coefficients The PT symmetric potential and the nonlocal nonlinearity of media are the causes of the beat patterns of the power and the gravity center evolution. To understand how the PT symmetric potential and the 278
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Fig. 6. The evolution of the power when 𝜇1 ≠ 𝜇2 . 𝜇2 is fixed as 𝜇2 = 8.6. The solid lines or the dash line indicate the propagation is stable or not. From top to bottom, the propagation constants of S1 are 7.8, 8.2, 8.6, 9.0 and 9.4. 𝑑 = 0.5, 𝑊0 = 0.2.
beats becomes longer. For the vector solitons with 𝜇0 > 9.4, their power curves are almost smooth lines, which indicates the components of vector solitons with large propagation constant have no interaction between them. That is because when the power of vector solitons with large propagation are too small that their behaviors are similar to linear modes. In other words, the nonlocal nonlinear effect becomes weak for these vector solitons with small power. When propagation constants decrease, the power oscillation becomes stronger. We can see that the oscillation for 𝜇0 = 7.8 is much more violent than for 𝜇0 = 8.6, while the distance of stable propagation for 𝜇0 = 7.8 is much short (𝑧 = 200). This is due to the strong nonlocal nonlinear effect for vector solitons with large power. When 𝜇1 ≠ 𝜇2 , the difference between the two components of vector solitons should be considered. We fix the propagation constant of S2 as 𝜇2 = 8.6, and investigate the beat patterns with different 𝜇1 . Four cases (𝜇1 = 7.8, 8.2, 9.0 and 9.4) are chosen to compare the beat pattern for 𝜇1 = 𝜇2 = 8.6, as shown in Fig. 6. As the difference ||𝜇1 − 𝜇2 || becomes larger, the beat patterns are harder to be recognized. The power curves of the vector solitons with 𝜇1 = 7.7 and 𝜇1 = 9.5 are equalamplitude oscillation curves with no beat pattern. Thus for 𝜇2 = 8.6, |𝜇1 − 𝜇2 | < 0.8 is a necessary condition to obtain a beat pattern for a | | vector soliton propagating in nonlocal nonlinear PT-symmetric lattices. When 𝜇1 decreases, the power oscillation becomes stronger and the distance of stable propagation for vector solitons becomes shorter, of which the cause is the same as the case of 𝜇1 = 𝜇2 .
Fig. 4. The evolution of the power with different degrees of nonlocality d. The solid lines or the dash line indicate the propagation is stable or not. The insets are the profiles of refractive index under different d. The propagation constants of S1 and S2 is 𝜇1 = 𝜇2 = 8.6. 𝑊0 = 0.2.
4. Conclusions and discusses In conclusion, we found and investigated the beat patterns during the propagation of vector solitons in the nonlocal nonlinear lattices with PT symmetry. Because of the combined effect of PT symmetric potential and nonlocal nonlinearity, the power and the gravity center of the whole light field undergo oscillations, which results the beat patterns in the power and the gravity center evolution curves. We demonstrated that the PT symmetric potential is necessary for beat patterns. But if the gain–loss coefficient is too large and close to the threshold of PT symmetry breaking, the beat patterns would disappear. The degree of nonlocality also affect the stability of propagation and the beat patterns. The intermediate nonlocal nonlinearity (𝑑 ∼ 10−1 ) is most suitable for stable propagation and distinct beat patterns. We also considered the influence of different propagation constants of the two components of vector solitons. When the propagation constants are equal, the vector solitons with intermediate power have the most distinct beat patterns. When the difference between the two propagation constants are large, the beat patterns degenerate into equal-amplitude oscillations. In practical, optical beats can be used to measure the wavelength of the tested lasers by measuring the frequency difference between the reference and tested lasers. This method is more accurate than the other
Fig. 5. The evolution of the power when 𝜇1 = 𝜇2 . The solid lines or the dash line indicate the propagation is stable or not. From top to bottom, the propagation constants are 7.8, 8.2, 8.6, 9.0 and 9.4. 𝑑 = 0.5, 𝑊0 = 0.2.
3.4. The effect of different propagation constants We also investigate the influence of the propagation constants 𝜇1 and 𝜇2 of vector solitons on beat patterns. In this section, the degree of nonlocality and the gain–loss coefficient are fixed as 𝑑 = 0.5 and 𝑊0 = 0.2. Two cases are considered: 𝜇1 = 𝜇2 and 𝜇1 ≠ 𝜇2 . When 𝜇1 = 𝜇2 = 𝜇0 , the characters and the situation of the two components of vector solitons are exactly the same. The evolution of the power for these vector solitons with different 𝜇0 are shown in Fig. 5. Comparing the power curves in Fig. 5, As propagation constants increase, the power of vector solitons decreases. Meanwhile, the power oscillation becomes weaker and the persistent distance of 279
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methods, and does not need an interferometer [43]. Inspired by the optical beat technique, we believe the beat pattern of optical solitons is a potential method to detect the differences between solitons or to calibrate optical solitons.
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