Gap solitons in PT-symmetric optical lattices with higher-order diffraction Lijuan Ge,1,∗ Ming Shen,2 Chunlan Ma,1 Taocheng Zang,1 and Lu Dai1 1
School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China 2 Department of Physics, Shanghai University, 99 Shangda Road, Shanghai 200444, China ∗
[email protected]
Abstract: The existence and stability of gap solitons are investigated in the semi-infinite gap of a parity-time (PT)-symmetric periodic potential (optical lattice) with a higher-order diffraction. The Bloch bands and band gaps of this PT-symmetric optical lattice depend crucially on the coupling constant of the fourth-order diffraction, whereas the phase transition point of this PT optical lattice remains unchangeable. The fourth-order diffraction plays a significant role in destabilizing the propagation of dipole solitons. Specifically, when the fourth-order diffraction coupling constant increases, the stable region of the dipole solitons shrinks as new regions of instability appear. However, fundamental solitons are found to be always linearly stable with arbitrary positive value of the coupling constant. We also investigate nonlinear evolution of the PT solitons under perturbation. © 2014 Optical Society of America OCIS codes: (190.0190) Nonlinear optics; (190.6135) Spatial solitons; (160.5293) Photonic bandgap materials.
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#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29435
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#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29436
1.
Introduction
In 1998, Bender and Boettcher [1] found the quite remarkable phenomenon that even nonHermitian Hamiltonians can still have completely real eigenvalue spectra if they respect paritytime (PT) symmetry. From then on, the unique properties of parity-time symmetric systems have drawn considerable attention in both quantum mechanism [2] and optics [3, 4]. For the Hamiltonian of a system to be PT-symmetric, the real part of the PT potential should be even function of position, while the imaginary part should be odd. In the context of optics, PT-symmetric systems have attracted much attention both theoretically and experimentally [5–7]. With PT-symmetric optical lattices, optical solitons has been studied for the first time in 2008 [8]. Ever since, PT-symmetric solitons [9–14] have been widely studied, which shown that the PT symmetric systems can support a series of novel solitons states [15–32]. Stability properties of these PT solitons have been carefully examined in our previous work [33]. Stable families of solitons can exist in PT lattices, increasing the gain-loss component has an overall destabilizing effect on soliton propagation. During the past decades, when studying the solitons results through the interplay between diffraction/dispersion and nonlinearity, the high-order diffraction/dispersion effect can not be neglected [34, 35]. In particular, the fourth-order dispersion/diffraction affects the dynamics of both bright [36] and dark solitons [37] in fiber devices. Furthermore, it has also been shown that the fourth-order dispersion/diffraction effect determine the modulation instability in photonic crystal fiber cavity [38]. Most recently, band gaps and lattice solitons were investigated in a periodic potential with the fourth-order dispersion/diffraction [39]. Thus far, nobody pays attention to the solitons in PT-symmetric structures with higher (fourth)-order diffraction/dispersion. In this paper, we perform a systematic study of the existence and stability of optical solitons in PT-symmetric optical lattices, and the propagation of such solitons is governed by the following nonlinear Schr¨odinger (NLS) equation with a PT lattice potential and a fourth-order diffraction (1) iUz + Uxx − β Uxxxx + V (x)U + σ |U|2U = 0, where the complex-valued function U(x, z) corresponds to the slowly varying amplitude of the electric field, z is a scaled propagation distance, β is a fourth-order diffraction coupling constant taken to be either positive (normal diffraction) or negative (anomalous diffraction), and when β = 0, the system returns to the nonlinear Schr¨odinger equation with a PT lattice potential [33], the potential V (x) is periodic in x and satisfies the PT symmetry V (x) = V ∗ (−x), and sign(σ ) = ±1 denotes the focusing and defocusing nonlinearity. For convenience [33], we take this PT lattice potential to be V (x) = V0 cos2 (x) + iW0 sin(2x) . (2) Here V0 is the depth of the real component of the potential, W0 is the relative magnitude of the imaginary component, and the period of this PT lattice is π . It should be emphasized that the physical model of Eq. (1) with fourth-order diffraction was previously used to study the subdiffractive band-edge solitons in periodic potentials (e.g. optical lattices) [40]. For this system, we first study the band-gap structure of this PT-symmetric lattice with higher-order diffraction by using the Floquet-Bloch theory. As increasing the fourth-order diffraction coupling constant, the Bloch bands and band gaps are found to shift up. The phase transition point of the PT lattice with higher-order diffraction remains as W0 = 0.5, above which all the linear Bloch bands turn to be complex simultaneously. Second, we focus on studying the effect of the fourthorder diffraction on the propagation of fundamental and dipole solitons in the semi-infinite gap. Fundamental solitons are determined to be all linearly stable with any value of the coupling constant, whereas, when the fourth-order coupling constant increases, the stable region of the
#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29437
dipole solitons shrinks as new regions of instability appear. Nonlinear evolution of linearly stable and unstable PT solitons under perturbation is also investigated to confirm the stability of PT solitons. 2.
Band-gap structure
To determine the properties of the band-gap structure, we first consider the linear Schr¨odinger equation (3) iUz + Uxx − β Uxxxx + V (x)U = 0, where the PT lattice potential V (x) is given in Eq. (2). Using Floquet-Bloch theory, we seek the band-gap structure and Bloch modes in the following form U(x, z) = p(x; k)eikx−iµ z ,
(4)
where p(x; k) is periodic with the same period π as the potential function V (x), k is the wave number in the first Brillouin zone −1 ≤ k ≤ 1, and µ is the propagation constant. The relation µ = µ (k) is called the diffraction relation. β =0.25
β =0
β =0.5
10
10
5
5
5
0
0
0
µ
10
−1
0 k
1
−1
W 0 = 0.5
0 k
1
−1
µ
5 0 1
1 Im(µ)
Re(µ)
10
0 k
1
W 0 = 0.6
10
−1
0 k
5 0 −1
0 k
1
0 −1 −1
0 k
1
Fig. 1. Diffraction relations of PT lattices with a fourth-order diffraction (top) for three β values 0 (top left), 0.25 (top middle) and 0.5 (top right) at V0 = 6 and W0 = 0.45, and (bottom) for W0 = 0.5 (bottom left) and W0 = 0.6 (bottom middle and right) at V0 = 6 and β = 0.25.
β =0.25
W 0 =0.45 0.6
1 0.8
0.4
β
W0
0.6 0.4
0.2
0.2 0 −4 −2
0
2 µ
4
6
8
0 −4 −2
0
2 µ
4
6
8
Fig. 2. Band-gap structure of PT lattices with a fourth-order diffraction (left) as increasing the β value from zero to one at V0 = 6 and W0 = 0.45, (right) as W0 crosses the phase transition point with β = 0.25 and V0 = 6.
#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29438
According to the Floquet-Bloch theory, diffraction relations of PT lattices with a fourth-order diffraction are obtained, shown in Fig. 1, for three different β values 0 (left), 0.25 (middle) and 0.5 (right) at V0 = 6 and W0 = 0.45. The band-gap structure is also shown in Fig. 2. In Fig. 2, the shaded regions represent spectral bands, whereas the unshaded areas correspond to band gaps; the largest, which contains everything to the left of the continuous spectrum, is the semi-infinite gap and further gaps are numbered (in our case from left to right). It is clearly shown that as increasing the value of the positive coupling constant β , the Bloch bands and band gaps of this PT lattice with fourth-order diffraction shift up, and the width of Bloch bands is getting bigger. The width of the second gap is getting smaller, but the width of the first gap changes very little. And we fix β = 0.25 and let W0 increase from zero. It is found that for the PT lattice (2) with a fourth-order diffraction, the phase transition point remains W0 = 0.5, shown in Fig. 1 (bottom left) and Fig. 2 (right), similar to what happens for β = 0 [8, 33]. Below this phase transition point (W0 < 0.5), the continuous spectrum is all real and comprises an infinite number of Bloch bands. When the strength of the gain-loss component in the PT lattice rises above the certain threshold, the band structure becomes complex, starting from the first two bands, shown in Fig. 1 (bottom middle and right). Above the phase transition point (W0 > 0.5), linear waves amplify exponentially during propagation. Thus any solitons would also be unstable to perturbations. We also study the band-gap structures for the negative β case numerically and we find that it is highly nontrivial. For simplicity, we only show the first two bands in Fig. 3. When W0 = 0.5, the first two bands merge, not only in the edges of the first Brillouin zone, but also in the center. Once W0 is bigger than 0.5, the complex eigenvalues appears. Also it is worthy to mention that for negative β case, no localized non-radiating solitons exist under self-focusing nonlinearity with PT optical lattices (even with real potentials [39]). So we will only consider the normal diffraction β > 0 case below. β = −0.1
β = −0.25
β = −0.5
−1
0
−2
−2
µ
0 −1
−4
−3 −2
−6
−1
0 k
1
−4 −1
W 0 = 0.5
0 k
1
−1
0 k
1
W 0 = 0.51 0.5
−2.5 −3 −1
−2
Im(µ)
Re(µ)
µ
−2
−2.5
0
−3 0 k
1
−1
0 k
1
−0.5 −1
0 k
1
Fig. 3. Diffraction relations of PT lattices with a fourth-order diffraction (top) for three negative β values −0.1 (top left), −0.25 (top middle) and −0.5 (top right) at V0 = 6 and W0 = 0.45, and (bottom) for W0 = 0.5 (bottom left) and W0 = 0.51 (bottom middle and right) at V0 = 6 and β = −0.25.
3.
Stability of PT solitons
The mathematical model becomes the NLS equation (1) with a PT lattice potential when we consider light waves propagation in the cubic nonlinear media. Next, we will study the PT solitons and their linear-stability. Solitons in the Eq. (1) are sought in the form U(x, z) = u(x)e−iµ z ,
(5)
#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29439
where u(x) is a localized function, and µ is a real propagation constant. These solitons can be computed by either the squared operator iteration method or the Newton-conjugate-gradient method (the latter is faster) [41, 42]. In our calculation, the PT lattice V (x) is taken as (2) with V0 = 6 and W0 = 0.45 [33]. To determine the linear stability of these PT solitons, we perturb them as h i ∗ U = e−iµ t u(x) + f (x) eλ z + g∗ (x) eλ z , (6) where | f |, |g| ≪ |u|. After substitution into equation (1) and linearizing, we arrive at the eigenvalue problem f f , (7) iL =λ g g where
L11 L21
L12 L22
L
=
,
(8)
L11
= µ + ∂xx − β ∂xxxx + V (x) + 2σ |u|2,
(9)
L12
= σu ,
L21 L22
2
(10) 2 ∗
= −σ u , = − µ + ∂xx − β ∂xxxx + V ∗ (x) + 2σ |u|2 .
(11) (12)
The eigenvalue problem (7) can be computed by the Fourier collocation method (for the full spectrum) or the Newton-conjugate-gradient method (for individual discrete eigenvalues) [41]. If eigenvalues are entirely imaginary, the soliton is linearly stable; otherwise it is linearly unstable. Now we focus on fundamental and dipole PT solitons in the semi-infinite gap under focusing cubic nonlinearity (σ = 1). In the semi-infinite gap, real parts of the fundamental solitons possess a single dominant peak, while real parts of the dipole solitons possess two dominant peaks with opposite phase. The solutions of these two families of PT solitons and their power curves are displayed in Fig. 4 for three β values: 0 (left), 0.25 (middle) and 0.5 (right). Here the power of a soliton is defined as Z ∞ P(µ ) = |u(x; µ )|2 dx. (13) −∞
In this figure, the lower power curves are for fundamental solitons and the upper power curves are for dipole solitons. Similar to the phenomenon occurs in purely real lattices [41] and PT lattice in cubic nonlinearity [33], the fundamental soliton family bifurcates out of the first Bloch band, and the solitons near this first band are low-amplitude Bloch-wave packets. The power curve for dipole solitons also features double branches which terminate before reaching the first Bloch band, and dipole solitons on the upper one with larger power are always unstable, even in purely real lattices [41], so we only need to consider the dipole solitons on the lower one below. We can see that the soliton power increases with the increase of the coupling constant β . Similar to the case without the fourth-order diffraction (β = 0), shown in Fig. 4 (left), the fundamental solitons are always linearly stable for arbitrary positive β values, which is indicated by solid lines of their power curves. We show the profile of the fundamental soliton and its entirely imaginary spectra at µ = −3.5 and β = 0.5, in Fig. 5 (left). However, the unstable region of dipole solitons increase as β increases [see Fig. 4 (middle and right)], and the instability is due to a quadruple of complex eigenvalues [see Fig. 5 (bottom middle)]. Also when µ approaches to be near the left side of the power minimum, similar to the case of PT lattice with the secondorder diffraction [33], there is another pair of real eigenvalues bifurcating from zero eigenvalue [see in Fig. 5 (bottom right)]. #222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29440
β =0
β =0.25
β =0.5
10
10
5
5
5
P
10
0
−4
−3
µ
0
−4
µ
−3
0
−4
µ
−3
Re[U ] and Im[U ]
Fig. 4. Power curves of PT solitons in the semi-infinite gap under focusing cubic nonlinearity (σ = 1) for three β values 0 (left), 0.25(middle) and 0.5(right) at V0 = 6 and W0 = 0.45. The lower curves are for fundamental solitons and the upper curves for dipole solitons. Solid blue and dashed red lines represent stable and unstable solitons, respectively (the same holds for all other figures). µ = −3.5 1 0.5
µ = −3.9
µ = −3.3
1
1
0
0
0 −1 −0.5
−10
0 x
10
Im(λ)
5
−1 −10
0 x
10
5
−10
0
0
0
−5
−5
0 0.1 Re(λ)
−0.1
10
5
−5 −0.1
0 x
0 0.1 Re(λ)
−0.2 0 0.2 Re(λ)
Fig. 5. Fundamental solitons (top left) and dipole solitons (top middle and right) and their linear-stability spectra (bottom) for three different µ values −3.5 (left), −3.9 (middle) and −3.3 (right) in the semi-infinite gap at V0 = 6, W0 = 0.45 and β = 0.5. The solid blue lines are for the real part and dashed pink lines for the imaginary part. The power curves of these solitons are shown in Fig. 4 (right), and the locations of these solitons are marked by dots on that power curves.
In order to state the problem about the instability caused by the fourth-order diffraction clearly, we set the propagation constant µ value, and let β change from zero to one. We find that the fundamental solitons are still linearly stable over the whole β interval, but the stable region of dipole solitons shrinks as µ increasing in the semi-infinite gap. As an example, at three µ values −4.5, −4 and −3.5, the power curves of fundamental and dipole soliton families versus the parameter β , are displayed in Fig. 6. Specifically, the dipole solitons become linearly unstable when β exceeds about 0.74 at µ = −4.5 [see Fig. 6 (left)], and the dipole soliton switches its linear-stability at the point β ≈ 0.4 when µ = −4 [see Fig. 6 (middle)], with a quadruple of complex eigenvalues [see Fig. 7 (bottom middle)]. The dipole solitons is always linearly unstable at any β values when µ = −3.5 [see Fig. 6 (right)], and the instability is due to a quadruple of complex eigenvalues when β < 0.9. At the point of β = 0.9, another pair of real eigenvalues bifurcates from the zero eigenvalue [see Fig. 7 (bottom right)].
#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29441
µ = −4.5
µ = −4
µ = −3.5
10
10
5
5
5
P
10
0 0
0.5 β
1
0 0
0.5 β
1
0 0
0.5 β
1
Re[U ] and Im[U ]
Fig. 6. Power curves of fundamental and dipole solitons for three µ values −4.5 (left), −4 (middle) and −3.5 (right) as increasing the fourth-order coupling constant from zero to 1 at V0 = 6 and W0 = 0.45. The lower curves are for fundamental solitons and the upper curves for dipole solitons. µ = −4.5
µ = −3.5
1
1
1
0
0
0
−1
−1 −10
Im(λ)
µ = −4
0 x
10
−1 −10
0 x
10
−10
5
5
5
0
0
0
−5 −0.1
−5 0 0.1 Re(λ)
−0.1
0 x
10
−5 0 0.1 Re(λ)
−0.1
0 0.1 Re(λ)
Fig. 7. Dipole solitons (top) and their linear-stability spectra (bottom) for three µ values −4.5 (left), −4 (middle) and −3.5 (right) in the semi-infinite gap when β = 0.5 (left and middle) and β = 0.95 (right). The power curves of these dipole solitons are shown in Fig. 6, and the locations of these solitons are marked by dots on that power curve.
4.
Nonlinear evolution of PT solitons under perturbations
In this section, we examine the nonlinear evolution of PT solitons with higher-order diffraction under weak perturbations. It is found that if a PT soliton is linearly stable, then it is also nonlinearly stable and propagates robustly against perturbations. When the soliton is linearly unstable, then it breaks up under perturbations, and its amplitude and energy can grow unbounded over distance. First we consider the linearly stable fundamental solitons [shown in Fig. 5 (left)] and the linearly stable dipole solitons [shown in Fig. 7 (left)] in the semi-infinite gap under focusing cubic nonlinearity (σ = 1). We perturb them by 10% random noise perturbations and then simulate their evolution in Eq. (1). The simulation results are shown in Fig. 8. It is seen that even after z = 200 units of propagation, these two solitons remain robust and do not break up. Thus both linearly stable solitons are also nonlinearly stable. Lastly we consider the linearly unstable dipole solitons shown in Fig. 7 (middle and right), which reside in the semi-infinite gap under focusing cubic nonlinearity (σ = 1). When µ = −4 and β = 0.5, this dipole soliton is linearly unstable with a quadruple of complex eigenvalues. Its evolution is shown in Fig. 9 (left) when perturbed by 10% random noise perturbations. We can see that this soliton grows oscillatorily without bound, thus this soliton is nonlinearly unstable.
#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29442
β = 0.5, µ = −3.5
β = 0.5, µ = −4.5 200
150
150
100
100
50
50
z
200
0
−10
0 x
10
0
−10
0 x
10
Fig. 8. Nonlinear evolution of the linearly stable fundamental solitons with µ = −3.5 (left) shown in Fig. 5 (left) and the linearly stable dipole solitons with µ = −4.5 (right) shown in Fig. 7 (left) at β = 0.5 under 10% random noise perturbations. Shown is the field |U(x, z)| in the (x, z) plane. β = 0.95, µ = −3.5
β = 0.5, µ = −4 45
40
30
z
60
15
20
0
−10
0 x
10
0
−10
0 x
10
Fig. 9. Nonlinear evolution of the linearly unstable dipole solitons shown in Fig. 7 (middle and right) under 10% random noise perturbations. Shown is the field |U(x, z)| in the (x, z) plane.
This oscillatory growth occurs since the unstable eigenvalues of this soliton are complex [see Fig. 7 (middle)]. When µ = −3.5 and β = 0.95, the instability of the dipole soliton is due to the pair of real eigenvalues. We perturb this soliton by 10% random noise perturbations, its evolution is shown in Fig. 9 (right). It is seen that this dipole soliton quick spread up, and it is also nonlinearly unstable. 5.
Conclusion
To conclude, we have analyzed the existence and stability of optical solitons in PT-symmetric optical lattices with the fourth-order diffraction. We have studied the band-gap structure of this PT-symmetric lattice with higher-order diffraction. It has been found that the Bloch bands and band gaps rises up with increasing the coupling constant of higher-order diffraction, and the phase transition point of the PT lattice remains as W0 = 0.5, above which all the linear Bloch bands turn to be complex simultaneously. We have also shown that the fourth-order diffraction plays a significant role in destabilizing the propagation of dipole solitons in the cubic nonlinear systems with higher-order diffraction. Specifically, when the fourth-order diffraction coupling
#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29443
constant increases, the stable region of the dipole solitons shrinks as new regions of instability appear. However, fundamental solitons have been found to be all linearly stable with any value of the coupling constant. In addition, we have also investigated nonlinear evolution of linearly stable and unstable PT solitons under perturbation. Acknowledgments This work was supported in part by the National Natural Science Foundation of China (nos. 11347180,11347136 and 61405135), the Natural Science Foundation of Jiangsu Province (no. BK20130265), the Research Fund of Suzhou University of Science and Technology (no. XKQ201210), and the Innovation Program of Shanghai Municipal Education Commission.
#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29444
School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China 2 Department of Physics, Shanghai University, 99 Shangda Road, Shanghai 200444, China ∗
[email protected]
Abstract: The existence and stability of gap solitons are investigated in the semi-infinite gap of a parity-time (PT)-symmetric periodic potential (optical lattice) with a higher-order diffraction. The Bloch bands and band gaps of this PT-symmetric optical lattice depend crucially on the coupling constant of the fourth-order diffraction, whereas the phase transition point of this PT optical lattice remains unchangeable. The fourth-order diffraction plays a significant role in destabilizing the propagation of dipole solitons. Specifically, when the fourth-order diffraction coupling constant increases, the stable region of the dipole solitons shrinks as new regions of instability appear. However, fundamental solitons are found to be always linearly stable with arbitrary positive value of the coupling constant. We also investigate nonlinear evolution of the PT solitons under perturbation. © 2014 Optical Society of America OCIS codes: (190.0190) Nonlinear optics; (190.6135) Spatial solitons; (160.5293) Photonic bandgap materials.
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#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29435
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#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29436
1.
Introduction
In 1998, Bender and Boettcher [1] found the quite remarkable phenomenon that even nonHermitian Hamiltonians can still have completely real eigenvalue spectra if they respect paritytime (PT) symmetry. From then on, the unique properties of parity-time symmetric systems have drawn considerable attention in both quantum mechanism [2] and optics [3, 4]. For the Hamiltonian of a system to be PT-symmetric, the real part of the PT potential should be even function of position, while the imaginary part should be odd. In the context of optics, PT-symmetric systems have attracted much attention both theoretically and experimentally [5–7]. With PT-symmetric optical lattices, optical solitons has been studied for the first time in 2008 [8]. Ever since, PT-symmetric solitons [9–14] have been widely studied, which shown that the PT symmetric systems can support a series of novel solitons states [15–32]. Stability properties of these PT solitons have been carefully examined in our previous work [33]. Stable families of solitons can exist in PT lattices, increasing the gain-loss component has an overall destabilizing effect on soliton propagation. During the past decades, when studying the solitons results through the interplay between diffraction/dispersion and nonlinearity, the high-order diffraction/dispersion effect can not be neglected [34, 35]. In particular, the fourth-order dispersion/diffraction affects the dynamics of both bright [36] and dark solitons [37] in fiber devices. Furthermore, it has also been shown that the fourth-order dispersion/diffraction effect determine the modulation instability in photonic crystal fiber cavity [38]. Most recently, band gaps and lattice solitons were investigated in a periodic potential with the fourth-order dispersion/diffraction [39]. Thus far, nobody pays attention to the solitons in PT-symmetric structures with higher (fourth)-order diffraction/dispersion. In this paper, we perform a systematic study of the existence and stability of optical solitons in PT-symmetric optical lattices, and the propagation of such solitons is governed by the following nonlinear Schr¨odinger (NLS) equation with a PT lattice potential and a fourth-order diffraction (1) iUz + Uxx − β Uxxxx + V (x)U + σ |U|2U = 0, where the complex-valued function U(x, z) corresponds to the slowly varying amplitude of the electric field, z is a scaled propagation distance, β is a fourth-order diffraction coupling constant taken to be either positive (normal diffraction) or negative (anomalous diffraction), and when β = 0, the system returns to the nonlinear Schr¨odinger equation with a PT lattice potential [33], the potential V (x) is periodic in x and satisfies the PT symmetry V (x) = V ∗ (−x), and sign(σ ) = ±1 denotes the focusing and defocusing nonlinearity. For convenience [33], we take this PT lattice potential to be V (x) = V0 cos2 (x) + iW0 sin(2x) . (2) Here V0 is the depth of the real component of the potential, W0 is the relative magnitude of the imaginary component, and the period of this PT lattice is π . It should be emphasized that the physical model of Eq. (1) with fourth-order diffraction was previously used to study the subdiffractive band-edge solitons in periodic potentials (e.g. optical lattices) [40]. For this system, we first study the band-gap structure of this PT-symmetric lattice with higher-order diffraction by using the Floquet-Bloch theory. As increasing the fourth-order diffraction coupling constant, the Bloch bands and band gaps are found to shift up. The phase transition point of the PT lattice with higher-order diffraction remains as W0 = 0.5, above which all the linear Bloch bands turn to be complex simultaneously. Second, we focus on studying the effect of the fourthorder diffraction on the propagation of fundamental and dipole solitons in the semi-infinite gap. Fundamental solitons are determined to be all linearly stable with any value of the coupling constant, whereas, when the fourth-order coupling constant increases, the stable region of the
#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29437
dipole solitons shrinks as new regions of instability appear. Nonlinear evolution of linearly stable and unstable PT solitons under perturbation is also investigated to confirm the stability of PT solitons. 2.
Band-gap structure
To determine the properties of the band-gap structure, we first consider the linear Schr¨odinger equation (3) iUz + Uxx − β Uxxxx + V (x)U = 0, where the PT lattice potential V (x) is given in Eq. (2). Using Floquet-Bloch theory, we seek the band-gap structure and Bloch modes in the following form U(x, z) = p(x; k)eikx−iµ z ,
(4)
where p(x; k) is periodic with the same period π as the potential function V (x), k is the wave number in the first Brillouin zone −1 ≤ k ≤ 1, and µ is the propagation constant. The relation µ = µ (k) is called the diffraction relation. β =0.25
β =0
β =0.5
10
10
5
5
5
0
0
0
µ
10
−1
0 k
1
−1
W 0 = 0.5
0 k
1
−1
µ
5 0 1
1 Im(µ)
Re(µ)
10
0 k
1
W 0 = 0.6
10
−1
0 k
5 0 −1
0 k
1
0 −1 −1
0 k
1
Fig. 1. Diffraction relations of PT lattices with a fourth-order diffraction (top) for three β values 0 (top left), 0.25 (top middle) and 0.5 (top right) at V0 = 6 and W0 = 0.45, and (bottom) for W0 = 0.5 (bottom left) and W0 = 0.6 (bottom middle and right) at V0 = 6 and β = 0.25.
β =0.25
W 0 =0.45 0.6
1 0.8
0.4
β
W0
0.6 0.4
0.2
0.2 0 −4 −2
0
2 µ
4
6
8
0 −4 −2
0
2 µ
4
6
8
Fig. 2. Band-gap structure of PT lattices with a fourth-order diffraction (left) as increasing the β value from zero to one at V0 = 6 and W0 = 0.45, (right) as W0 crosses the phase transition point with β = 0.25 and V0 = 6.
#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29438
According to the Floquet-Bloch theory, diffraction relations of PT lattices with a fourth-order diffraction are obtained, shown in Fig. 1, for three different β values 0 (left), 0.25 (middle) and 0.5 (right) at V0 = 6 and W0 = 0.45. The band-gap structure is also shown in Fig. 2. In Fig. 2, the shaded regions represent spectral bands, whereas the unshaded areas correspond to band gaps; the largest, which contains everything to the left of the continuous spectrum, is the semi-infinite gap and further gaps are numbered (in our case from left to right). It is clearly shown that as increasing the value of the positive coupling constant β , the Bloch bands and band gaps of this PT lattice with fourth-order diffraction shift up, and the width of Bloch bands is getting bigger. The width of the second gap is getting smaller, but the width of the first gap changes very little. And we fix β = 0.25 and let W0 increase from zero. It is found that for the PT lattice (2) with a fourth-order diffraction, the phase transition point remains W0 = 0.5, shown in Fig. 1 (bottom left) and Fig. 2 (right), similar to what happens for β = 0 [8, 33]. Below this phase transition point (W0 < 0.5), the continuous spectrum is all real and comprises an infinite number of Bloch bands. When the strength of the gain-loss component in the PT lattice rises above the certain threshold, the band structure becomes complex, starting from the first two bands, shown in Fig. 1 (bottom middle and right). Above the phase transition point (W0 > 0.5), linear waves amplify exponentially during propagation. Thus any solitons would also be unstable to perturbations. We also study the band-gap structures for the negative β case numerically and we find that it is highly nontrivial. For simplicity, we only show the first two bands in Fig. 3. When W0 = 0.5, the first two bands merge, not only in the edges of the first Brillouin zone, but also in the center. Once W0 is bigger than 0.5, the complex eigenvalues appears. Also it is worthy to mention that for negative β case, no localized non-radiating solitons exist under self-focusing nonlinearity with PT optical lattices (even with real potentials [39]). So we will only consider the normal diffraction β > 0 case below. β = −0.1
β = −0.25
β = −0.5
−1
0
−2
−2
µ
0 −1
−4
−3 −2
−6
−1
0 k
1
−4 −1
W 0 = 0.5
0 k
1
−1
0 k
1
W 0 = 0.51 0.5
−2.5 −3 −1
−2
Im(µ)
Re(µ)
µ
−2
−2.5
0
−3 0 k
1
−1
0 k
1
−0.5 −1
0 k
1
Fig. 3. Diffraction relations of PT lattices with a fourth-order diffraction (top) for three negative β values −0.1 (top left), −0.25 (top middle) and −0.5 (top right) at V0 = 6 and W0 = 0.45, and (bottom) for W0 = 0.5 (bottom left) and W0 = 0.51 (bottom middle and right) at V0 = 6 and β = −0.25.
3.
Stability of PT solitons
The mathematical model becomes the NLS equation (1) with a PT lattice potential when we consider light waves propagation in the cubic nonlinear media. Next, we will study the PT solitons and their linear-stability. Solitons in the Eq. (1) are sought in the form U(x, z) = u(x)e−iµ z ,
(5)
#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29439
where u(x) is a localized function, and µ is a real propagation constant. These solitons can be computed by either the squared operator iteration method or the Newton-conjugate-gradient method (the latter is faster) [41, 42]. In our calculation, the PT lattice V (x) is taken as (2) with V0 = 6 and W0 = 0.45 [33]. To determine the linear stability of these PT solitons, we perturb them as h i ∗ U = e−iµ t u(x) + f (x) eλ z + g∗ (x) eλ z , (6) where | f |, |g| ≪ |u|. After substitution into equation (1) and linearizing, we arrive at the eigenvalue problem f f , (7) iL =λ g g where
L11 L21
L12 L22
L
=
,
(8)
L11
= µ + ∂xx − β ∂xxxx + V (x) + 2σ |u|2,
(9)
L12
= σu ,
L21 L22
2
(10) 2 ∗
= −σ u , = − µ + ∂xx − β ∂xxxx + V ∗ (x) + 2σ |u|2 .
(11) (12)
The eigenvalue problem (7) can be computed by the Fourier collocation method (for the full spectrum) or the Newton-conjugate-gradient method (for individual discrete eigenvalues) [41]. If eigenvalues are entirely imaginary, the soliton is linearly stable; otherwise it is linearly unstable. Now we focus on fundamental and dipole PT solitons in the semi-infinite gap under focusing cubic nonlinearity (σ = 1). In the semi-infinite gap, real parts of the fundamental solitons possess a single dominant peak, while real parts of the dipole solitons possess two dominant peaks with opposite phase. The solutions of these two families of PT solitons and their power curves are displayed in Fig. 4 for three β values: 0 (left), 0.25 (middle) and 0.5 (right). Here the power of a soliton is defined as Z ∞ P(µ ) = |u(x; µ )|2 dx. (13) −∞
In this figure, the lower power curves are for fundamental solitons and the upper power curves are for dipole solitons. Similar to the phenomenon occurs in purely real lattices [41] and PT lattice in cubic nonlinearity [33], the fundamental soliton family bifurcates out of the first Bloch band, and the solitons near this first band are low-amplitude Bloch-wave packets. The power curve for dipole solitons also features double branches which terminate before reaching the first Bloch band, and dipole solitons on the upper one with larger power are always unstable, even in purely real lattices [41], so we only need to consider the dipole solitons on the lower one below. We can see that the soliton power increases with the increase of the coupling constant β . Similar to the case without the fourth-order diffraction (β = 0), shown in Fig. 4 (left), the fundamental solitons are always linearly stable for arbitrary positive β values, which is indicated by solid lines of their power curves. We show the profile of the fundamental soliton and its entirely imaginary spectra at µ = −3.5 and β = 0.5, in Fig. 5 (left). However, the unstable region of dipole solitons increase as β increases [see Fig. 4 (middle and right)], and the instability is due to a quadruple of complex eigenvalues [see Fig. 5 (bottom middle)]. Also when µ approaches to be near the left side of the power minimum, similar to the case of PT lattice with the secondorder diffraction [33], there is another pair of real eigenvalues bifurcating from zero eigenvalue [see in Fig. 5 (bottom right)]. #222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29440
β =0
β =0.25
β =0.5
10
10
5
5
5
P
10
0
−4
−3
µ
0
−4
µ
−3
0
−4
µ
−3
Re[U ] and Im[U ]
Fig. 4. Power curves of PT solitons in the semi-infinite gap under focusing cubic nonlinearity (σ = 1) for three β values 0 (left), 0.25(middle) and 0.5(right) at V0 = 6 and W0 = 0.45. The lower curves are for fundamental solitons and the upper curves for dipole solitons. Solid blue and dashed red lines represent stable and unstable solitons, respectively (the same holds for all other figures). µ = −3.5 1 0.5
µ = −3.9
µ = −3.3
1
1
0
0
0 −1 −0.5
−10
0 x
10
Im(λ)
5
−1 −10
0 x
10
5
−10
0
0
0
−5
−5
0 0.1 Re(λ)
−0.1
10
5
−5 −0.1
0 x
0 0.1 Re(λ)
−0.2 0 0.2 Re(λ)
Fig. 5. Fundamental solitons (top left) and dipole solitons (top middle and right) and their linear-stability spectra (bottom) for three different µ values −3.5 (left), −3.9 (middle) and −3.3 (right) in the semi-infinite gap at V0 = 6, W0 = 0.45 and β = 0.5. The solid blue lines are for the real part and dashed pink lines for the imaginary part. The power curves of these solitons are shown in Fig. 4 (right), and the locations of these solitons are marked by dots on that power curves.
In order to state the problem about the instability caused by the fourth-order diffraction clearly, we set the propagation constant µ value, and let β change from zero to one. We find that the fundamental solitons are still linearly stable over the whole β interval, but the stable region of dipole solitons shrinks as µ increasing in the semi-infinite gap. As an example, at three µ values −4.5, −4 and −3.5, the power curves of fundamental and dipole soliton families versus the parameter β , are displayed in Fig. 6. Specifically, the dipole solitons become linearly unstable when β exceeds about 0.74 at µ = −4.5 [see Fig. 6 (left)], and the dipole soliton switches its linear-stability at the point β ≈ 0.4 when µ = −4 [see Fig. 6 (middle)], with a quadruple of complex eigenvalues [see Fig. 7 (bottom middle)]. The dipole solitons is always linearly unstable at any β values when µ = −3.5 [see Fig. 6 (right)], and the instability is due to a quadruple of complex eigenvalues when β < 0.9. At the point of β = 0.9, another pair of real eigenvalues bifurcates from the zero eigenvalue [see Fig. 7 (bottom right)].
#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29441
µ = −4.5
µ = −4
µ = −3.5
10
10
5
5
5
P
10
0 0
0.5 β
1
0 0
0.5 β
1
0 0
0.5 β
1
Re[U ] and Im[U ]
Fig. 6. Power curves of fundamental and dipole solitons for three µ values −4.5 (left), −4 (middle) and −3.5 (right) as increasing the fourth-order coupling constant from zero to 1 at V0 = 6 and W0 = 0.45. The lower curves are for fundamental solitons and the upper curves for dipole solitons. µ = −4.5
µ = −3.5
1
1
1
0
0
0
−1
−1 −10
Im(λ)
µ = −4
0 x
10
−1 −10
0 x
10
−10
5
5
5
0
0
0
−5 −0.1
−5 0 0.1 Re(λ)
−0.1
0 x
10
−5 0 0.1 Re(λ)
−0.1
0 0.1 Re(λ)
Fig. 7. Dipole solitons (top) and their linear-stability spectra (bottom) for three µ values −4.5 (left), −4 (middle) and −3.5 (right) in the semi-infinite gap when β = 0.5 (left and middle) and β = 0.95 (right). The power curves of these dipole solitons are shown in Fig. 6, and the locations of these solitons are marked by dots on that power curve.
4.
Nonlinear evolution of PT solitons under perturbations
In this section, we examine the nonlinear evolution of PT solitons with higher-order diffraction under weak perturbations. It is found that if a PT soliton is linearly stable, then it is also nonlinearly stable and propagates robustly against perturbations. When the soliton is linearly unstable, then it breaks up under perturbations, and its amplitude and energy can grow unbounded over distance. First we consider the linearly stable fundamental solitons [shown in Fig. 5 (left)] and the linearly stable dipole solitons [shown in Fig. 7 (left)] in the semi-infinite gap under focusing cubic nonlinearity (σ = 1). We perturb them by 10% random noise perturbations and then simulate their evolution in Eq. (1). The simulation results are shown in Fig. 8. It is seen that even after z = 200 units of propagation, these two solitons remain robust and do not break up. Thus both linearly stable solitons are also nonlinearly stable. Lastly we consider the linearly unstable dipole solitons shown in Fig. 7 (middle and right), which reside in the semi-infinite gap under focusing cubic nonlinearity (σ = 1). When µ = −4 and β = 0.5, this dipole soliton is linearly unstable with a quadruple of complex eigenvalues. Its evolution is shown in Fig. 9 (left) when perturbed by 10% random noise perturbations. We can see that this soliton grows oscillatorily without bound, thus this soliton is nonlinearly unstable.
#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29442
β = 0.5, µ = −3.5
β = 0.5, µ = −4.5 200
150
150
100
100
50
50
z
200
0
−10
0 x
10
0
−10
0 x
10
Fig. 8. Nonlinear evolution of the linearly stable fundamental solitons with µ = −3.5 (left) shown in Fig. 5 (left) and the linearly stable dipole solitons with µ = −4.5 (right) shown in Fig. 7 (left) at β = 0.5 under 10% random noise perturbations. Shown is the field |U(x, z)| in the (x, z) plane. β = 0.95, µ = −3.5
β = 0.5, µ = −4 45
40
30
z
60
15
20
0
−10
0 x
10
0
−10
0 x
10
Fig. 9. Nonlinear evolution of the linearly unstable dipole solitons shown in Fig. 7 (middle and right) under 10% random noise perturbations. Shown is the field |U(x, z)| in the (x, z) plane.
This oscillatory growth occurs since the unstable eigenvalues of this soliton are complex [see Fig. 7 (middle)]. When µ = −3.5 and β = 0.95, the instability of the dipole soliton is due to the pair of real eigenvalues. We perturb this soliton by 10% random noise perturbations, its evolution is shown in Fig. 9 (right). It is seen that this dipole soliton quick spread up, and it is also nonlinearly unstable. 5.
Conclusion
To conclude, we have analyzed the existence and stability of optical solitons in PT-symmetric optical lattices with the fourth-order diffraction. We have studied the band-gap structure of this PT-symmetric lattice with higher-order diffraction. It has been found that the Bloch bands and band gaps rises up with increasing the coupling constant of higher-order diffraction, and the phase transition point of the PT lattice remains as W0 = 0.5, above which all the linear Bloch bands turn to be complex simultaneously. We have also shown that the fourth-order diffraction plays a significant role in destabilizing the propagation of dipole solitons in the cubic nonlinear systems with higher-order diffraction. Specifically, when the fourth-order diffraction coupling
#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29443
constant increases, the stable region of the dipole solitons shrinks as new regions of instability appear. However, fundamental solitons have been found to be all linearly stable with any value of the coupling constant. In addition, we have also investigated nonlinear evolution of linearly stable and unstable PT solitons under perturbation. Acknowledgments This work was supported in part by the National Natural Science Foundation of China (nos. 11347180,11347136 and 61405135), the Natural Science Foundation of Jiangsu Province (no. BK20130265), the Research Fund of Suzhou University of Science and Technology (no. XKQ201210), and the Innovation Program of Shanghai Municipal Education Commission.
#222647 - $15.00 USD Received 8 Sep 2014; revised 25 Oct 2014; accepted 12 Nov 2014; published 18 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029435 | OPTICS EXPRESS 29444
