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Defect solitons in parity-time symmetric superlattices with focusing saturable nonlinearity
Optics Communications 349 (2015) 171–179
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Defect solitons in parity-time symmetric superlattices with focusing saturable nonlinearity Lei Li a, Huagang Li a,b, Tianshu Lai a,n a b
State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China Department of Physics, Guangdong University of Education, Guangzhou 510303, China
art ic l e i nf o
a b s t r a c t
Article history: Received 14 February 2015 Received in revised form 21 March 2015 Accepted 25 March 2015 Available online 28 March 2015
In this paper, we find that defect superlattice solitons (DSSs) exist at the defect site in one-dimensional optical superlattices with a parity-time (PT) symmetric potential and focusing saturable nonlinearity. We discuss the existence and stability of DSSs numerically and find that both the saturation parameter s and the defect strength ε can affect the power, the existence and stable regions of solitons significantly. For given saturation parameter s and defect strength ε (propagation constant μ), the power of DSSs increases with the increase (decrease) of μ (ε). For positive and zero defects, DSSs only exist in the semi-infinite band gap, whereas for negative defects, DSSs exist not only in the semi-infinite band gap but also in the first finite band gap. Besides, our numerical calculations show that when the saturation nonlinearity increases up to some level, the increasing rate of soliton power with μ is very fast and the stable regions of DSSs will become obviously narrow and eventually disappear with a decrease of ε for negative defects. & 2015 Elsevier B.V. All rights reserved.
Keywords: Defect solitons Saturable nonlinearity PT-symmetric Superlattices
1. Introduction Gap solitons have been extensively studied in the past decades, and found to able to exist and stabilize in a periodic potential (lattice) with Kerr nonlinearity [1,2]. It was even found that gap solitons could be trapped at the defect site in a lattice, that is the so-called defect soliton [3,4]. Such defect solitons might have potential applications in optical switching [5]. In addition, it was also reported that stable gap solitons can exist in saturable nonlinear medium [6–9]. Huang et al. [9] found that saturable nonlinearity can suppress the existence domain of kink solitons. It is possible to realize “higher power” gap solitons using saturable nonlinearity. On the other hand, it was found that non-Hermitian Hamiltonians might have real eigenvalue spectra [10] if their complex potentials possess the parity-time (PT) symmetry. In optics, PTsymmetric potentials mean the balance of gain and loss. Musslimani et al. [11] first studied the beam propagation in a PT symmetric optical lattice with Kerr nonlinearity and found that gap solitons can exist and propagate robustly as the loss modulation parameter is below breaking point. Meanwhile, Makris et al. [12] revealed some unusual features of the gap solitons, such as energy oscillations, double refraction, and secondary emissions. Since then, gap solitons in PT-symmetric media have drawn much attention. Li et al. [13,14] even found stable PT surface gap solitons at n
Corresponding author. E-mail address: [email protected] (T. Lai).
http://dx.doi.org/10.1016/j.optcom.2015.03.067 0030-4018/& 2015 Elsevier B.V. All rights reserved.
the interface between a linear medium and a Kerr nonlinear medium. Cao et al. [15] also reported stable PT gap solitons in a saturable self-defocusing nonlinear medium. In addition, it was reported as well that gap and defect solitons could exist and stabilize in PT-symmetric optical superlattices with Kerr nonlinearity [16–18], where the real part of superlattice potential was a superposed structure of two periodic potentials. Superlattices provide an additional freedom of modulation over usual lattices, the relative strength of the two potentials, and thus may more effectively control the soliton behaviors, while saturable nonlinearity may be useful in controlling the power of solitons. Therefore, the beam propagation in a PT-symmetric superlattice with a saturable nonlinearity should be interesting and deserves to be studied. However, no reports are available yet on the beam propagation in such a new system. It is unknown whether solitons can exist, and how the defect and saturable nonlinearity can affect the properties of gap solitons if solitons exist. It is necessary to explore their influences for the integrality of the knowledge system of optical solitons. In this article, we have studied the effects of defect and focusing saturable nonlinearity on the existence and stability of solitons in one-dimensional PT-symmetric superlattices, and found some new phenomena. First, defect superlattice solitons (DSSs) may exist and stabilize in one-dimensional PT-symmetric superlattices with a focusing saturable nonlinearity. Second, the power of solitons increases rapidly with the increasing of the propagation constant when the saturable nonlinearity increases up to some level. DSSs become unstable as their power is too high.
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Fig. 1. The PT-symmetric superlattice potentials with a zero defect (ε ¼0) (a), positive defect (ε ¼0.5) (b), negative defect (ε¼ 0.5) (c) and without defect (d). Blue solid and red dotted lines represent the real and imaginary parts, respectively. (e) The Bloch band versus different relative strength of dual frequency components in superlattices in (d). Black filled regions are Bloch bands. (f) The band structure of the superlattices with α ¼0.5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Third, defect strength ε significantly influences the existence and stability of DSSs. Negative defect (ε o 0) leads to narrowing of the stable regions. For positive and zero defects, the DSSs are stable in most of their existence regions.
2. Modeling existence and stability of DSSs The evolution of complex normalized amplitude U of the light field in a PT-symmetric potential with focusing saturable nonlinearity is governed by the following normalized nonlinear Schrödinger equation [19]:
iUz +
1 U 2U Uxx + [V (x) + iW (x)]U + =0 2 1 + s U2
(1)
where V(x) and W(x) is the real and imaginary parts of the complex potential. The s is saturation parameter of saturable nonlinearity. In this paper, we consider the following one-dimensional PTsymmetric superlattice potential [16],
⎧⎧ ⎡⎛ ⎡ ⎛ π ⎞⎤ π ⎞⎤⎫ π ⎪ V0⎨αsin2⎢⎜x + ⎟⎥ + (1 − α)sin2⎢2⎜x + ⎟⎥⎬ , x > ⎣⎝ ⎣ ⎝ 2 2 ⎠⎦ 2 ⎠⎦⎭ ⎪⎩ V (x) = ⎨ ⎛ ⎪ π ⎞⎤⎡ π x 8 ⎞⎤⎥ 2⎡⎛ ⎟ , x ≤ ⎪ Msin ⎢⎣⎜⎝x + ⎟⎠⎥⎦⎢⎢1 + εexp⎜ − ⎥ 2 128 2 ⎝ ⎠ ⎣ ⎦ ⎩ ⎛ ⎧ ⎡⎛ ⎡ ⎛ π ⎞⎤ π ⎞⎤⎫⎞ M = max⎜V0⎨αsin2⎢⎜x + ⎟⎥ + (1 − α)sin2⎢2⎜x + ⎟⎥⎬⎟ ⎣⎝ ⎣ ⎝ 2 ⎠⎦ 2 ⎠⎦⎭⎠ ⎝ ⎩
(2)
and W (x) = W0 sin(2x) Here, V0 and W0 represent the modulation depth of the real and
imaginary parts of the PT symmetric superlattice, respectively. α denotes the modulation parameter of superlattice. We will later consider a set of reasonable parameters V0 ¼4; W0 ¼0.4; ε ¼ 0, 0.5, 0.5; α ¼0.5. Function εnexp( x8/128) stands for the defect of the lattice, where ε is the strength of the defect. As ε 40 (ε o0), the potential field at defect site is stronger (weaker) than that of normal superlattice in amplitude, such a defect is defined as positive (negative) defect. As ε ¼0, the potential field at defect site and in normal superlattice is identical in amplitude, which is referred as zero defect. It is worth reminding that zero defect does not mean no defect (without defect), but only the amplitude of potential field at defect site and in normal superlattice is identical. The intensity distributions of the PT superlattice potential mentioned above are plotted in Fig. 1(a)–(c), respectively for various defects ε ¼0, 0.5 and 0.5, while the intensity distribution without defect is displayed in Fig. 1(d). One can note the difference between panels (a) and (d). We search the stationary soliton solutions of Eq. (1) with the form of,
U (x, z) = f (x)exp(iμz)
(3)
where μ is a real propagation constant and f(x) is a complex function. Here the physical essence of μ deserves to be paid attention for the convenience of discussions later. It is actually a phase shift (with respect to the phase in linear homogenous media) induced by all effects, including nonlinear effect, the modulation of lattice field and defect etc as beam propagates in media. Substituting Eq. (3) into Eq. (1) leads to,
1 f 2f fxx + [V (x) + iW (x)]f + 2 1+sf
2
− μf = 0
(4)
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Fig. 2. (a) The power and (b) the perturbation growth rate versus propagation constant for different defect strengths in the semi-infinite gap (the black region is Bloch band). (c)–(e) The profiles of DSSs (the blue solid and red dotted curves are the real and imaginary parts respectively). Other parameter s¼ 0.1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Soliton solutions f can be explored by solving Eq. (4) numerically [20]. The power of a soliton can be calculated by, +∞
2
P=∫ f dx . −∞ The linear version of nonlinear differential Eq. (4) is,
1 f + [V (x) + iW (x)]f − μf = 0 2 xx
(5)
According to Bloch theorem, the eigenfunctions of Eq. (5) may be in the form of f ¼Fkexp(ikx), where k is the Bloch wave number, and Fk is a periodic function of x with the same period as the function V(x) and W(x). Substituting the Bloch solution to Eq. (5), we can get the eigenvalue equation,
⎞ 1 ⎛ d2 d ⎜ + 2ik − k 2⎟Fk + [V (x) + iW (x)]Fk = μFk 2 ⎝ dx2 dx ⎠
(6)
For the no defect superlattice potential described in Fig. 1(d), V
(x) and W(x), we can calculate the band structure numerically using the plane wave expansion method. For V0 ¼ 4 and W0 ¼ 0.4, the calculated Bloch band structure is shown in Fig. 1(e) as a function of α, while Fig. 1(f) shows the band structure as a function of Bloch wave vector k at a given α ¼0.5. One can see from Fig. 1 (f) that the semi-infinite gap region is in the range of μ 42.18 and the first gap one is confined in the range of 1.36 o μ o1.92. To examine the stability of defect solitons, it is necessary to explore the perturbed solutions for Eq. (1) with the following form,
q(x, z) = exp(iμz){f (x) + [F (x)exp(δz) + G⁎(x)exp(δ ⁎z)]}
(7)
where the superscript ‘∗’ represents complex conjugation. F and G are the given perturbations with F⪡1 and G⪡1. δ is a growth rate of perburbations. Substituting Eq. (7) into Eq. (1) and then linearing it, a set of equations on the perturbation is given as follows,
⎧ ⎤ ⎡ ⎪ ⎥ ⎢ 1 ∂ 2F 2f2 sf4 f2 s f 2f 2 − μ F + ( V + iW ) F + F − F + G − G ⎪ δF = i⎢ ⎥ 2 2 2 ∂x2 2 2 2 2 1 + s f 1 + s f ⎪ 1 + s f 1 + s f ⎢⎣ ( ) ( ) ⎥⎦ ⎪ ⎨ ⁎ ⁎ ⎤ ⎡ ⎪ (f 2 ) F + s f 2(f 2 ) F ⎥ ⎢ 1 ∂ 2G 2f2 sf4 ⎪ δ G = i − + μ G − ( V − iW ) G − G + G − ⎢ 2 2 2 2 ⎥ ⎪ ∂x 1+sf2 1+sf2 ⎪ ⎢⎣ (1 + s f 2) (1 + s f 2) ⎥⎦ ⎩
(8)
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Fig. 3. Propagation of DSSs with s¼ 0.1 and μ¼4.5.for (a) ε¼ 0.5, (b) ε ¼0, (c) ε¼ 0.5 and (d) ε ¼ 0.5, μ¼ 2.3.
These equations can be numerically solved to get the perturbation growth rate Re(δ). Solitons are considered linearly stable only if max{Re(δ)}¼ 0 [21]. On the other hand, the robustness of the spatial soliton propagation can also be tested directly by numerical simulations of Eq. (1) with the initial soliton profile added a random noise (10% of the soliton amplitude).
3. Numerical studies of DSSs In this section, we study numerically DSSs in the superlattice potential with a given α ¼ 0.5 and two different strengths (s ¼0.1 and 0.4) of saturable nonlinearity. The evolution and properties of DSSs with the variation of defect strength ε are discussed in details for semi-infinite and first band gaps. 3.1. DSSs in weak saturable nonlinearity. Here, we first study DSSs in the superlattice with a weak saturable nonlinearity of s¼0.1. The power of DSSs in the semi-infinite band gap is shown in Fig. 2(a) as a function of μ for different ε values. One can see that the power of DSSs increases with μ for a given defect strength ε, but decreases with increasing defect strength ε for a given μ. These phenomena can be easily explained if one bears in mind the physical meaning of μ (an additional phase shift). In our case, μ is contributed by three effects, selffocusing, defect and superlattice potential, and hence can be expressed by a phenomenological summation for the simplicity of discussions later, μ ¼ μD þ μL þ μF, where μD is the contribution from the defect, μL from the lattice field and μF from the self-
focusing effect. For a given defect ε, the increase of soliton power leads to the enhancement of self-focusing effect, that is the increase of μF which leads to the increase of μ. That is why the power of DSSs increases with μ in Fig. 2(a) for each given defect ε. On the other hand, for a given μ, the phenomenon that the power of DSSs can be adjusted by the strength of defect ε, can be explained by the constraint, μ ¼ μD þ μL þ μF ¼ constant. Because μD increases with ε, μF must decrease to meet the constraint, μ ¼ μD þ μL þ μF ¼constant, which demands lowering the power of DSSs. The linear-stability analysis method is used to analyze the stability of the solitons mentioned in Fig. 2(a). The maximum of perturbation growth rate δ, max{Re(δ)}, in the semi-infinite band gap is calculated and plotted in Fig. 2(b) as a function of μ for the three defect cases corresponding to (a). As mentioned before, the solitons are stable only if max{Re(δ)} ¼ 0. According to such a criterion, one can discern from (b) that the stable region of negative defect solitons of ε ¼ 0.5 is in the range of μ 42.75, one of zero defect (ε ¼0) solitions in the range of μ 4 2.21, and one of positive defect solitons of ε ¼0.5 in the range of μ 43.7. From Fig. 2 (b), one can see that solitons are very unstable in the region close to the Bloch band edge. The stable and unstable ranges are also indicated in (a) by solid and dashed curves respectively. The stable soliton profiles (f(x)) at μ ¼4.5 are displayed in Fig. 2(c), (d) and (e), respectively for ε ¼ 0.5, 0 and 0.5. As seen from them, the real and imaginary parts of these soliton profiles have even and odd symmetries respectively. The robustness of the propagation of some solitons is directly tested as well by numerical simulation of Eq. (1) for the initial input soliton with 10% random noise. The simulating propagation is shown in Fig. 3 for some typical solitons in the semi-infinite gap.
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Fig. 4. (a) The power and (b) the perturbation growth rate versus propagation constant for ε ¼ 0.5 and s ¼ 0.1 in the first gap (the black regions are Bloch bands). (c) The profile of DSS (the blue solid and red dotted curves are the real and imaginary parts respectively) at μ¼1.8. (d) Stable propagation of DSS in (c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The propagations of three kind of stable defect solitons with μ ¼ 4.5 are shown in Fig. 3(a), (b) and (c), respectively for ε ¼ 0.5, 0 and 0.5. The propagation of unstable negative defect soliton with
μ ¼2.3 is shown in Fig. 3(d) for ε ¼ 0.5, which corresponds to the filled circle in the inset of Fig. 2(a). The simulation results in Fig. 3 are in good agreement with the linear stability analyses shown in
Fig. 5. (a) The power and (b) the perturbation growth rate versus defect strength for different propagation constants. (c)–(e) The profiles of DSSs. The blue solid and red dotted curves are the real and imaginary parts respectively. (f)–(h) Propagation of DSSs, respectively corresponding to the solitons in (c)–(e). Other parameter s¼ 0.1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 6. (a) The power and (b) the perturbation growth rate versus propagation constant for different defect strengths at s ¼ 0.4 in the semi-infinite gap (the black region is Bloch band). (c)–(h) The profiles of DSSs corresponding to filled circles in (a). The blue solid and red dotted curves are the real and imaginary parts respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 2(b). Similarly, we search for possible DSSs in the first gap, and find that only negative defect solitons can exist. The power of negative defect solitons at ε ¼ 0.5 is plotted in Fig. 4(a) as a function of μ. The maximum of perturbation growth rate, max{Re(δ)}, is also calculated and plotted in Fig. 4(b). Obviously, negative defect solitons are stable only if propagation constant μ locates between 1.77 and 1.91. The profile of a stable soliton at μ ¼ 1.8 is plotted in Fig. 4(c). Comparing Fig. 2(a) with Fig. 4(a), one can find that the power of negative defect DSSs is larger in semi-infinite gap than in the first gap, which is because μ is reduced in the first gap so that smaller μF or lower power is needed. Contrarily, the occupied region of the DSSs is much larger in the first gap than in semi-infinite gap if one compares Fig. 2(c) with Fig. 4(c). The negative defect soliton in the first gap obviously extends outside defect site in space and presents side lobes in the two sides of central main peak, whereas one in semi-infinite gap is mainly confined in the defect site and only presents a central main peak.
The obvious difference of profile size may have different applications. Fig. 4(d) shows the robust propagation of the DSSs, being very stable. In previous discussions, all the results are shown only for particular fixed ε values. Here we analyze the dependences of the existence and stability of DSS on ε to know how the defect affects properties of the solitons for fixed μ values. The power of DSSs is shown in Fig. 5(a) as a function of ε for different μ values. In the first gap, we choose μ ¼ 1.44, while μ ¼ 2.3 and μ ¼4.5 are chosen in the semi-infinite gap. One can see that the power of DSSs decreases with increasing ε for a given propagation constant μ because the change of μF and μD must compensate for each other to keep μ constant. To know about the stability of solitons, we plot the change of max{Re(δ)} as a function of ε in Fig. 5(b). One can discern that the stable region of solitons at μ ¼1.44 is in the range of 0.41 r ε r 0.23, while one for μ ¼2.3 in the range of 0.34 r ε r0.12 and at μ ¼ 4.5 in the full range of 0.5 r ε r0.5. Then we study the propagation stability of solitons for three states
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Fig. 7. Propagation of DSSs with s ¼0.4. (a) ε¼ 0.5, μ¼3, (b) ε ¼ 0.4,μ¼2.5 (c) ε¼ 0.4, μ¼ 3.3 (d) ε¼ 0.4, μ¼3 (e) ε ¼0, μ¼4.3 (f) ε¼ 0.5, μ¼ 5.
Fig. 8. (a) The curves of power (b) the unstable growth rate versus propagation constant at different negative defect strengths for s¼ 0.4 in the first gap (the black regions are Bloch bands) (c) DSSs profile (the blue solid curve is the real parts and the red dotted curve represents the corresponding imaginary parts), here s ¼0.4,ε¼ 0.4,μ¼ 1.6. (d) Propagation of (c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 9. (a) The power and (b) the perturbation growth rate versus defect strength for different propagation constants. (c)–(e) The profiles of DSSs, where the blue solid and red dotted curves are the real and imaginary parts respectively. (f)–(h) Propagation of DSSs, respectively corresponding to the solitons in (c)–(e). Other parameter s ¼ 0.4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
corresponding to filled circles in Fig. 5(a). When μ is fixed at 1.44, we select an unstable example, ε ¼ 0.48 and plot its soliton profile in Fig. 5(c). The corresponding soliton propagation is shown in Fig. 5(f). The profiles of two stable solitons (μ ¼2.3, ε ¼ 0.1; μ ¼4.5, ε ¼ 0.3) are plotted in Fig. 5(d) and (e). The corresponding soliton propagations are shown in Fig. 5(g) and (h), respectively. 3.2. DSSs in deeper saturable nonlinearity We first calculate the power of DSSs in PT-symmetric superlattice with a deeper saturable nonlinearity of s¼ 0.4 for different defect strengths ε. The calculated power versus μ is plotted in Fig. 6(a) for semi-infinite gap. One can see that the power of solitons still increases with μ for each given ε, but is reduced with increasing ε for a given μ. These phenomena are very similar to those shown in Fig. 2(a), and can be explained as similarly as Fig. 2 (a) is done before. However, one prominent feature is that
increasing rate of soliton power with μ is much faster in Fig. 6 (a) than in Fig. 2(a) for each given ε. It is the deeper saturable nonlinearity (s¼ 0.4) that suppresses increasing rate of self-focusing nonlinearity. The maximum of the perturbation growth rate, max{Re(δ)}, is also calculated based on the linear-stability analysis method, and is plotted in Fig. 6(b). The stable and unstable ranges are also indicated in Fig. 6(a) by solid and dashed curves respectively. One can discern that positive and zero defect solitons are stable in most of their existence regions. The stable region of positive defect solitons of ε ¼ 0.5 is in the range of 3.7 r μ r5.72, while for ε ¼ 0, the stable region is in the range of 2.22 r μ r4.4. But for negative defect solitons of ε ¼ 0.4, the stable region is narrow and only confined in a small range of 2.7 r μ r3.1. Besides, solitons are unstable in the whole existence region for ε ¼ 0.5. These results show that the stable region of the solitons becomes narrow with a decrease of ε, and eventually disappear, as the blue curve shows
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ε ¼ 0.5 in Fig. 6(a). Next, we study the propagation stability of solitons for six states corresponding to filled circles in Fig. 6(a). The profiles of those solitons are plotted in Fig. 6(c)–(h) for different μ and ε, where the real and imaginary parts of profiles are shown by blue solid and red dotted lines respectively. One can see that those profiles have PT-symmetry. Their propagations are shown in Fig. 7 (a)–(f), respectively. One can also see that the soliton propagation is unstable in Fig. 7(a)–(c), but stable in Fig. 7(d)–(f), which are in good agreement with the linear stability analyses shown in Fig. 6 (b). In the first gap, possible solitons are also searched. It is found that only negative defect solitons exist. The power of negative defect (ε ¼-0.5 and ε ¼ 0.4) solitons is plotted in Fig. 8(a) as a function of μ. The maximum of perturbation growth rate, max{Re (δ)}, is also calculated to verify the stability and plotted in Fig. 8(b). It shows that solitons are unstable throughout their existence region for ε ¼ 0.5, whereas the solitons at ε ¼ 0.4 are stable only in the range of 1.45 o μ o 1.7. The stable and unstable ranges are also indicated in Fig. 8(a) by solid and dashed curves respectively. The profile of a stable soliton at μ ¼ 1.6 and ε ¼ 0.4 is plotted in Fig. 8(c). Comparing Fig. 6(f) with Fig. 8(c), one can find that the negative defect soliton in the first gap obviously extends outside defect site in space, whereas in semi-infinite gap it is mainly confined in the defect site. The main reason may be the reduction of self-focusing effect due to low power of solitons in the first gap. Fig. 8(d) shows the robust propagation of the DSSs, being very stable. The numerical results show that the analysis of the propagation properties is in good agreement with the linear stability analysis. Finally, we analyze the dependences of the existence and stability properties on ε for s ¼0.4. As before, ε continuously changes from 0.5 to 0.5, and the change of P and max{Re(δ)} are plotted as a function of ε for three fixed μ values in Fig. 9(a) and (b), respectively. In the first gap, we choose μ ¼1.46, while μ ¼3 and μ ¼ 4 are chosen in the semi-infinite gap. As one can see from (a), the power of DSSs also decreases with the increase of ε for a given propagation constant μ. Moreover, one can discern from (b) that the stable region of solitons for μ ¼ 1.46 is in the range of 0.4 r ε r 0.22, while one of solitons at μ ¼3 in the range of 0.42 r ε r0.39, and for μ ¼4 in the range of 0.175r ε r0.5. Next, we study the propagation stability of solitons for three states corresponding to filled circles in Fig. 9(a). When μ is fixed at 1.46, we select an unstable example, ε ¼ 0.48 and plot its soliton profile in Fig. 9(c). The propagation of the soliton is shown in Fig. 9 (f), and obviously unstable. The profiles of two stable solitons (μ ¼3, ε ¼ 0.2; μ ¼ 4, ε ¼0.2) are plotted in Fig. 9(d) and (e). Their corresponding propagations are shown in Fig. 9(g) and (h), respectively. The simulation results of propagation are in accordance with the linear stability analyses shown in Fig. 9(b). for
4. Conclusions In conclusion, we have studied the existence and stability of defect solitons supported by PT symmetric superlattices with focusing saturation nonlinearity. The negative, zero and positive defects are considered. By numerical calculation, it is found that for positive and zero defects, DSSs only exist in the semi-infinite gap, whereas for negative defects, DSSs exist not only in the semiinfinite gap but also in the first finite gap. Due to the reduction of self-focusing effect with power, solitons in the first gap obviously extend outside defect site in space. Besides, the calculation results show that both the saturation parameter s and the defect strength ε can affect the power, the existence and stable regions of DSSs.
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For a given s, the power of DSSs increases with propagation constant μ (“phase shift” induced by all effects) when ε is kept at a constant, and when μ remains unchanged, the power must vary inversely with ε to meet the constraint μF (“phase shift” induced by self-focusing) þ μD (“phase shift” induced by defect) ¼ constant. Besides, when s increases to some value, the power of DSSs can grow dramatically with μ for a given ε. It is the deeper saturable nonlinearity that suppresses increasing rate of μF with power. On the other hand, linear stability analyses indicate that high-power DSSs are unstable, especially for negative defects, and that the stable regions will become narrow rapidly and eventually disappear with a decrease of ε. While for positive and zero defects, the unstable high-power solitons only focus on a very small-scale region of propagation constant.
Acknowledgments We are grateful to Dr. X. Zhu for his helpful discussions with us. This work was supported by National Basic Research Program of China under grant no. 2013CB922403, National Natural Science Foundation of China under grant nos. 11274399 and 61475195 as well as Guangdong Natural Science Foundation, China under grant no. 2014A030311029.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Defect solitons in parity-time symmetric superlattices with focusing saturable nonlinearity Lei Li a, Huagang Li a,b, Tianshu Lai a,n a b
State Key Laboratory of Optoelectronic Materials and Technologies, School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China Department of Physics, Guangdong University of Education, Guangzhou 510303, China
art ic l e i nf o
a b s t r a c t
Article history: Received 14 February 2015 Received in revised form 21 March 2015 Accepted 25 March 2015 Available online 28 March 2015
In this paper, we find that defect superlattice solitons (DSSs) exist at the defect site in one-dimensional optical superlattices with a parity-time (PT) symmetric potential and focusing saturable nonlinearity. We discuss the existence and stability of DSSs numerically and find that both the saturation parameter s and the defect strength ε can affect the power, the existence and stable regions of solitons significantly. For given saturation parameter s and defect strength ε (propagation constant μ), the power of DSSs increases with the increase (decrease) of μ (ε). For positive and zero defects, DSSs only exist in the semi-infinite band gap, whereas for negative defects, DSSs exist not only in the semi-infinite band gap but also in the first finite band gap. Besides, our numerical calculations show that when the saturation nonlinearity increases up to some level, the increasing rate of soliton power with μ is very fast and the stable regions of DSSs will become obviously narrow and eventually disappear with a decrease of ε for negative defects. & 2015 Elsevier B.V. All rights reserved.
Keywords: Defect solitons Saturable nonlinearity PT-symmetric Superlattices
1. Introduction Gap solitons have been extensively studied in the past decades, and found to able to exist and stabilize in a periodic potential (lattice) with Kerr nonlinearity [1,2]. It was even found that gap solitons could be trapped at the defect site in a lattice, that is the so-called defect soliton [3,4]. Such defect solitons might have potential applications in optical switching [5]. In addition, it was also reported that stable gap solitons can exist in saturable nonlinear medium [6–9]. Huang et al. [9] found that saturable nonlinearity can suppress the existence domain of kink solitons. It is possible to realize “higher power” gap solitons using saturable nonlinearity. On the other hand, it was found that non-Hermitian Hamiltonians might have real eigenvalue spectra [10] if their complex potentials possess the parity-time (PT) symmetry. In optics, PTsymmetric potentials mean the balance of gain and loss. Musslimani et al. [11] first studied the beam propagation in a PT symmetric optical lattice with Kerr nonlinearity and found that gap solitons can exist and propagate robustly as the loss modulation parameter is below breaking point. Meanwhile, Makris et al. [12] revealed some unusual features of the gap solitons, such as energy oscillations, double refraction, and secondary emissions. Since then, gap solitons in PT-symmetric media have drawn much attention. Li et al. [13,14] even found stable PT surface gap solitons at n
Corresponding author. E-mail address: [email protected] (T. Lai).
http://dx.doi.org/10.1016/j.optcom.2015.03.067 0030-4018/& 2015 Elsevier B.V. All rights reserved.
the interface between a linear medium and a Kerr nonlinear medium. Cao et al. [15] also reported stable PT gap solitons in a saturable self-defocusing nonlinear medium. In addition, it was reported as well that gap and defect solitons could exist and stabilize in PT-symmetric optical superlattices with Kerr nonlinearity [16–18], where the real part of superlattice potential was a superposed structure of two periodic potentials. Superlattices provide an additional freedom of modulation over usual lattices, the relative strength of the two potentials, and thus may more effectively control the soliton behaviors, while saturable nonlinearity may be useful in controlling the power of solitons. Therefore, the beam propagation in a PT-symmetric superlattice with a saturable nonlinearity should be interesting and deserves to be studied. However, no reports are available yet on the beam propagation in such a new system. It is unknown whether solitons can exist, and how the defect and saturable nonlinearity can affect the properties of gap solitons if solitons exist. It is necessary to explore their influences for the integrality of the knowledge system of optical solitons. In this article, we have studied the effects of defect and focusing saturable nonlinearity on the existence and stability of solitons in one-dimensional PT-symmetric superlattices, and found some new phenomena. First, defect superlattice solitons (DSSs) may exist and stabilize in one-dimensional PT-symmetric superlattices with a focusing saturable nonlinearity. Second, the power of solitons increases rapidly with the increasing of the propagation constant when the saturable nonlinearity increases up to some level. DSSs become unstable as their power is too high.
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Fig. 1. The PT-symmetric superlattice potentials with a zero defect (ε ¼0) (a), positive defect (ε ¼0.5) (b), negative defect (ε¼ 0.5) (c) and without defect (d). Blue solid and red dotted lines represent the real and imaginary parts, respectively. (e) The Bloch band versus different relative strength of dual frequency components in superlattices in (d). Black filled regions are Bloch bands. (f) The band structure of the superlattices with α ¼0.5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Third, defect strength ε significantly influences the existence and stability of DSSs. Negative defect (ε o 0) leads to narrowing of the stable regions. For positive and zero defects, the DSSs are stable in most of their existence regions.
2. Modeling existence and stability of DSSs The evolution of complex normalized amplitude U of the light field in a PT-symmetric potential with focusing saturable nonlinearity is governed by the following normalized nonlinear Schrödinger equation [19]:
iUz +
1 U 2U Uxx + [V (x) + iW (x)]U + =0 2 1 + s U2
(1)
where V(x) and W(x) is the real and imaginary parts of the complex potential. The s is saturation parameter of saturable nonlinearity. In this paper, we consider the following one-dimensional PTsymmetric superlattice potential [16],
⎧⎧ ⎡⎛ ⎡ ⎛ π ⎞⎤ π ⎞⎤⎫ π ⎪ V0⎨αsin2⎢⎜x + ⎟⎥ + (1 − α)sin2⎢2⎜x + ⎟⎥⎬ , x > ⎣⎝ ⎣ ⎝ 2 2 ⎠⎦ 2 ⎠⎦⎭ ⎪⎩ V (x) = ⎨ ⎛ ⎪ π ⎞⎤⎡ π x 8 ⎞⎤⎥ 2⎡⎛ ⎟ , x ≤ ⎪ Msin ⎢⎣⎜⎝x + ⎟⎠⎥⎦⎢⎢1 + εexp⎜ − ⎥ 2 128 2 ⎝ ⎠ ⎣ ⎦ ⎩ ⎛ ⎧ ⎡⎛ ⎡ ⎛ π ⎞⎤ π ⎞⎤⎫⎞ M = max⎜V0⎨αsin2⎢⎜x + ⎟⎥ + (1 − α)sin2⎢2⎜x + ⎟⎥⎬⎟ ⎣⎝ ⎣ ⎝ 2 ⎠⎦ 2 ⎠⎦⎭⎠ ⎝ ⎩
(2)
and W (x) = W0 sin(2x) Here, V0 and W0 represent the modulation depth of the real and
imaginary parts of the PT symmetric superlattice, respectively. α denotes the modulation parameter of superlattice. We will later consider a set of reasonable parameters V0 ¼4; W0 ¼0.4; ε ¼ 0, 0.5, 0.5; α ¼0.5. Function εnexp( x8/128) stands for the defect of the lattice, where ε is the strength of the defect. As ε 40 (ε o0), the potential field at defect site is stronger (weaker) than that of normal superlattice in amplitude, such a defect is defined as positive (negative) defect. As ε ¼0, the potential field at defect site and in normal superlattice is identical in amplitude, which is referred as zero defect. It is worth reminding that zero defect does not mean no defect (without defect), but only the amplitude of potential field at defect site and in normal superlattice is identical. The intensity distributions of the PT superlattice potential mentioned above are plotted in Fig. 1(a)–(c), respectively for various defects ε ¼0, 0.5 and 0.5, while the intensity distribution without defect is displayed in Fig. 1(d). One can note the difference between panels (a) and (d). We search the stationary soliton solutions of Eq. (1) with the form of,
U (x, z) = f (x)exp(iμz)
(3)
where μ is a real propagation constant and f(x) is a complex function. Here the physical essence of μ deserves to be paid attention for the convenience of discussions later. It is actually a phase shift (with respect to the phase in linear homogenous media) induced by all effects, including nonlinear effect, the modulation of lattice field and defect etc as beam propagates in media. Substituting Eq. (3) into Eq. (1) leads to,
1 f 2f fxx + [V (x) + iW (x)]f + 2 1+sf
2
− μf = 0
(4)
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Fig. 2. (a) The power and (b) the perturbation growth rate versus propagation constant for different defect strengths in the semi-infinite gap (the black region is Bloch band). (c)–(e) The profiles of DSSs (the blue solid and red dotted curves are the real and imaginary parts respectively). Other parameter s¼ 0.1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Soliton solutions f can be explored by solving Eq. (4) numerically [20]. The power of a soliton can be calculated by, +∞
2
P=∫ f dx . −∞ The linear version of nonlinear differential Eq. (4) is,
1 f + [V (x) + iW (x)]f − μf = 0 2 xx
(5)
According to Bloch theorem, the eigenfunctions of Eq. (5) may be in the form of f ¼Fkexp(ikx), where k is the Bloch wave number, and Fk is a periodic function of x with the same period as the function V(x) and W(x). Substituting the Bloch solution to Eq. (5), we can get the eigenvalue equation,
⎞ 1 ⎛ d2 d ⎜ + 2ik − k 2⎟Fk + [V (x) + iW (x)]Fk = μFk 2 ⎝ dx2 dx ⎠
(6)
For the no defect superlattice potential described in Fig. 1(d), V
(x) and W(x), we can calculate the band structure numerically using the plane wave expansion method. For V0 ¼ 4 and W0 ¼ 0.4, the calculated Bloch band structure is shown in Fig. 1(e) as a function of α, while Fig. 1(f) shows the band structure as a function of Bloch wave vector k at a given α ¼0.5. One can see from Fig. 1 (f) that the semi-infinite gap region is in the range of μ 42.18 and the first gap one is confined in the range of 1.36 o μ o1.92. To examine the stability of defect solitons, it is necessary to explore the perturbed solutions for Eq. (1) with the following form,
q(x, z) = exp(iμz){f (x) + [F (x)exp(δz) + G⁎(x)exp(δ ⁎z)]}
(7)
where the superscript ‘∗’ represents complex conjugation. F and G are the given perturbations with F⪡1 and G⪡1. δ is a growth rate of perburbations. Substituting Eq. (7) into Eq. (1) and then linearing it, a set of equations on the perturbation is given as follows,
⎧ ⎤ ⎡ ⎪ ⎥ ⎢ 1 ∂ 2F 2f2 sf4 f2 s f 2f 2 − μ F + ( V + iW ) F + F − F + G − G ⎪ δF = i⎢ ⎥ 2 2 2 ∂x2 2 2 2 2 1 + s f 1 + s f ⎪ 1 + s f 1 + s f ⎢⎣ ( ) ( ) ⎥⎦ ⎪ ⎨ ⁎ ⁎ ⎤ ⎡ ⎪ (f 2 ) F + s f 2(f 2 ) F ⎥ ⎢ 1 ∂ 2G 2f2 sf4 ⎪ δ G = i − + μ G − ( V − iW ) G − G + G − ⎢ 2 2 2 2 ⎥ ⎪ ∂x 1+sf2 1+sf2 ⎪ ⎢⎣ (1 + s f 2) (1 + s f 2) ⎥⎦ ⎩
(8)
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Fig. 3. Propagation of DSSs with s¼ 0.1 and μ¼4.5.for (a) ε¼ 0.5, (b) ε ¼0, (c) ε¼ 0.5 and (d) ε ¼ 0.5, μ¼ 2.3.
These equations can be numerically solved to get the perturbation growth rate Re(δ). Solitons are considered linearly stable only if max{Re(δ)}¼ 0 [21]. On the other hand, the robustness of the spatial soliton propagation can also be tested directly by numerical simulations of Eq. (1) with the initial soliton profile added a random noise (10% of the soliton amplitude).
3. Numerical studies of DSSs In this section, we study numerically DSSs in the superlattice potential with a given α ¼ 0.5 and two different strengths (s ¼0.1 and 0.4) of saturable nonlinearity. The evolution and properties of DSSs with the variation of defect strength ε are discussed in details for semi-infinite and first band gaps. 3.1. DSSs in weak saturable nonlinearity. Here, we first study DSSs in the superlattice with a weak saturable nonlinearity of s¼0.1. The power of DSSs in the semi-infinite band gap is shown in Fig. 2(a) as a function of μ for different ε values. One can see that the power of DSSs increases with μ for a given defect strength ε, but decreases with increasing defect strength ε for a given μ. These phenomena can be easily explained if one bears in mind the physical meaning of μ (an additional phase shift). In our case, μ is contributed by three effects, selffocusing, defect and superlattice potential, and hence can be expressed by a phenomenological summation for the simplicity of discussions later, μ ¼ μD þ μL þ μF, where μD is the contribution from the defect, μL from the lattice field and μF from the self-
focusing effect. For a given defect ε, the increase of soliton power leads to the enhancement of self-focusing effect, that is the increase of μF which leads to the increase of μ. That is why the power of DSSs increases with μ in Fig. 2(a) for each given defect ε. On the other hand, for a given μ, the phenomenon that the power of DSSs can be adjusted by the strength of defect ε, can be explained by the constraint, μ ¼ μD þ μL þ μF ¼ constant. Because μD increases with ε, μF must decrease to meet the constraint, μ ¼ μD þ μL þ μF ¼constant, which demands lowering the power of DSSs. The linear-stability analysis method is used to analyze the stability of the solitons mentioned in Fig. 2(a). The maximum of perturbation growth rate δ, max{Re(δ)}, in the semi-infinite band gap is calculated and plotted in Fig. 2(b) as a function of μ for the three defect cases corresponding to (a). As mentioned before, the solitons are stable only if max{Re(δ)} ¼ 0. According to such a criterion, one can discern from (b) that the stable region of negative defect solitons of ε ¼ 0.5 is in the range of μ 42.75, one of zero defect (ε ¼0) solitions in the range of μ 4 2.21, and one of positive defect solitons of ε ¼0.5 in the range of μ 43.7. From Fig. 2 (b), one can see that solitons are very unstable in the region close to the Bloch band edge. The stable and unstable ranges are also indicated in (a) by solid and dashed curves respectively. The stable soliton profiles (f(x)) at μ ¼4.5 are displayed in Fig. 2(c), (d) and (e), respectively for ε ¼ 0.5, 0 and 0.5. As seen from them, the real and imaginary parts of these soliton profiles have even and odd symmetries respectively. The robustness of the propagation of some solitons is directly tested as well by numerical simulation of Eq. (1) for the initial input soliton with 10% random noise. The simulating propagation is shown in Fig. 3 for some typical solitons in the semi-infinite gap.
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Fig. 4. (a) The power and (b) the perturbation growth rate versus propagation constant for ε ¼ 0.5 and s ¼ 0.1 in the first gap (the black regions are Bloch bands). (c) The profile of DSS (the blue solid and red dotted curves are the real and imaginary parts respectively) at μ¼1.8. (d) Stable propagation of DSS in (c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The propagations of three kind of stable defect solitons with μ ¼ 4.5 are shown in Fig. 3(a), (b) and (c), respectively for ε ¼ 0.5, 0 and 0.5. The propagation of unstable negative defect soliton with
μ ¼2.3 is shown in Fig. 3(d) for ε ¼ 0.5, which corresponds to the filled circle in the inset of Fig. 2(a). The simulation results in Fig. 3 are in good agreement with the linear stability analyses shown in
Fig. 5. (a) The power and (b) the perturbation growth rate versus defect strength for different propagation constants. (c)–(e) The profiles of DSSs. The blue solid and red dotted curves are the real and imaginary parts respectively. (f)–(h) Propagation of DSSs, respectively corresponding to the solitons in (c)–(e). Other parameter s¼ 0.1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 6. (a) The power and (b) the perturbation growth rate versus propagation constant for different defect strengths at s ¼ 0.4 in the semi-infinite gap (the black region is Bloch band). (c)–(h) The profiles of DSSs corresponding to filled circles in (a). The blue solid and red dotted curves are the real and imaginary parts respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 2(b). Similarly, we search for possible DSSs in the first gap, and find that only negative defect solitons can exist. The power of negative defect solitons at ε ¼ 0.5 is plotted in Fig. 4(a) as a function of μ. The maximum of perturbation growth rate, max{Re(δ)}, is also calculated and plotted in Fig. 4(b). Obviously, negative defect solitons are stable only if propagation constant μ locates between 1.77 and 1.91. The profile of a stable soliton at μ ¼ 1.8 is plotted in Fig. 4(c). Comparing Fig. 2(a) with Fig. 4(a), one can find that the power of negative defect DSSs is larger in semi-infinite gap than in the first gap, which is because μ is reduced in the first gap so that smaller μF or lower power is needed. Contrarily, the occupied region of the DSSs is much larger in the first gap than in semi-infinite gap if one compares Fig. 2(c) with Fig. 4(c). The negative defect soliton in the first gap obviously extends outside defect site in space and presents side lobes in the two sides of central main peak, whereas one in semi-infinite gap is mainly confined in the defect site and only presents a central main peak.
The obvious difference of profile size may have different applications. Fig. 4(d) shows the robust propagation of the DSSs, being very stable. In previous discussions, all the results are shown only for particular fixed ε values. Here we analyze the dependences of the existence and stability of DSS on ε to know how the defect affects properties of the solitons for fixed μ values. The power of DSSs is shown in Fig. 5(a) as a function of ε for different μ values. In the first gap, we choose μ ¼ 1.44, while μ ¼ 2.3 and μ ¼4.5 are chosen in the semi-infinite gap. One can see that the power of DSSs decreases with increasing ε for a given propagation constant μ because the change of μF and μD must compensate for each other to keep μ constant. To know about the stability of solitons, we plot the change of max{Re(δ)} as a function of ε in Fig. 5(b). One can discern that the stable region of solitons at μ ¼1.44 is in the range of 0.41 r ε r 0.23, while one for μ ¼2.3 in the range of 0.34 r ε r0.12 and at μ ¼ 4.5 in the full range of 0.5 r ε r0.5. Then we study the propagation stability of solitons for three states
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Fig. 7. Propagation of DSSs with s ¼0.4. (a) ε¼ 0.5, μ¼3, (b) ε ¼ 0.4,μ¼2.5 (c) ε¼ 0.4, μ¼ 3.3 (d) ε¼ 0.4, μ¼3 (e) ε ¼0, μ¼4.3 (f) ε¼ 0.5, μ¼ 5.
Fig. 8. (a) The curves of power (b) the unstable growth rate versus propagation constant at different negative defect strengths for s¼ 0.4 in the first gap (the black regions are Bloch bands) (c) DSSs profile (the blue solid curve is the real parts and the red dotted curve represents the corresponding imaginary parts), here s ¼0.4,ε¼ 0.4,μ¼ 1.6. (d) Propagation of (c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 9. (a) The power and (b) the perturbation growth rate versus defect strength for different propagation constants. (c)–(e) The profiles of DSSs, where the blue solid and red dotted curves are the real and imaginary parts respectively. (f)–(h) Propagation of DSSs, respectively corresponding to the solitons in (c)–(e). Other parameter s ¼ 0.4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
corresponding to filled circles in Fig. 5(a). When μ is fixed at 1.44, we select an unstable example, ε ¼ 0.48 and plot its soliton profile in Fig. 5(c). The corresponding soliton propagation is shown in Fig. 5(f). The profiles of two stable solitons (μ ¼2.3, ε ¼ 0.1; μ ¼4.5, ε ¼ 0.3) are plotted in Fig. 5(d) and (e). The corresponding soliton propagations are shown in Fig. 5(g) and (h), respectively. 3.2. DSSs in deeper saturable nonlinearity We first calculate the power of DSSs in PT-symmetric superlattice with a deeper saturable nonlinearity of s¼ 0.4 for different defect strengths ε. The calculated power versus μ is plotted in Fig. 6(a) for semi-infinite gap. One can see that the power of solitons still increases with μ for each given ε, but is reduced with increasing ε for a given μ. These phenomena are very similar to those shown in Fig. 2(a), and can be explained as similarly as Fig. 2 (a) is done before. However, one prominent feature is that
increasing rate of soliton power with μ is much faster in Fig. 6 (a) than in Fig. 2(a) for each given ε. It is the deeper saturable nonlinearity (s¼ 0.4) that suppresses increasing rate of self-focusing nonlinearity. The maximum of the perturbation growth rate, max{Re(δ)}, is also calculated based on the linear-stability analysis method, and is plotted in Fig. 6(b). The stable and unstable ranges are also indicated in Fig. 6(a) by solid and dashed curves respectively. One can discern that positive and zero defect solitons are stable in most of their existence regions. The stable region of positive defect solitons of ε ¼ 0.5 is in the range of 3.7 r μ r5.72, while for ε ¼ 0, the stable region is in the range of 2.22 r μ r4.4. But for negative defect solitons of ε ¼ 0.4, the stable region is narrow and only confined in a small range of 2.7 r μ r3.1. Besides, solitons are unstable in the whole existence region for ε ¼ 0.5. These results show that the stable region of the solitons becomes narrow with a decrease of ε, and eventually disappear, as the blue curve shows
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ε ¼ 0.5 in Fig. 6(a). Next, we study the propagation stability of solitons for six states corresponding to filled circles in Fig. 6(a). The profiles of those solitons are plotted in Fig. 6(c)–(h) for different μ and ε, where the real and imaginary parts of profiles are shown by blue solid and red dotted lines respectively. One can see that those profiles have PT-symmetry. Their propagations are shown in Fig. 7 (a)–(f), respectively. One can also see that the soliton propagation is unstable in Fig. 7(a)–(c), but stable in Fig. 7(d)–(f), which are in good agreement with the linear stability analyses shown in Fig. 6 (b). In the first gap, possible solitons are also searched. It is found that only negative defect solitons exist. The power of negative defect (ε ¼-0.5 and ε ¼ 0.4) solitons is plotted in Fig. 8(a) as a function of μ. The maximum of perturbation growth rate, max{Re (δ)}, is also calculated to verify the stability and plotted in Fig. 8(b). It shows that solitons are unstable throughout their existence region for ε ¼ 0.5, whereas the solitons at ε ¼ 0.4 are stable only in the range of 1.45 o μ o 1.7. The stable and unstable ranges are also indicated in Fig. 8(a) by solid and dashed curves respectively. The profile of a stable soliton at μ ¼ 1.6 and ε ¼ 0.4 is plotted in Fig. 8(c). Comparing Fig. 6(f) with Fig. 8(c), one can find that the negative defect soliton in the first gap obviously extends outside defect site in space, whereas in semi-infinite gap it is mainly confined in the defect site. The main reason may be the reduction of self-focusing effect due to low power of solitons in the first gap. Fig. 8(d) shows the robust propagation of the DSSs, being very stable. The numerical results show that the analysis of the propagation properties is in good agreement with the linear stability analysis. Finally, we analyze the dependences of the existence and stability properties on ε for s ¼0.4. As before, ε continuously changes from 0.5 to 0.5, and the change of P and max{Re(δ)} are plotted as a function of ε for three fixed μ values in Fig. 9(a) and (b), respectively. In the first gap, we choose μ ¼1.46, while μ ¼3 and μ ¼ 4 are chosen in the semi-infinite gap. As one can see from (a), the power of DSSs also decreases with the increase of ε for a given propagation constant μ. Moreover, one can discern from (b) that the stable region of solitons for μ ¼ 1.46 is in the range of 0.4 r ε r 0.22, while one of solitons at μ ¼3 in the range of 0.42 r ε r0.39, and for μ ¼4 in the range of 0.175r ε r0.5. Next, we study the propagation stability of solitons for three states corresponding to filled circles in Fig. 9(a). When μ is fixed at 1.46, we select an unstable example, ε ¼ 0.48 and plot its soliton profile in Fig. 9(c). The propagation of the soliton is shown in Fig. 9 (f), and obviously unstable. The profiles of two stable solitons (μ ¼3, ε ¼ 0.2; μ ¼ 4, ε ¼0.2) are plotted in Fig. 9(d) and (e). Their corresponding propagations are shown in Fig. 9(g) and (h), respectively. The simulation results of propagation are in accordance with the linear stability analyses shown in Fig. 9(b). for
4. Conclusions In conclusion, we have studied the existence and stability of defect solitons supported by PT symmetric superlattices with focusing saturation nonlinearity. The negative, zero and positive defects are considered. By numerical calculation, it is found that for positive and zero defects, DSSs only exist in the semi-infinite gap, whereas for negative defects, DSSs exist not only in the semiinfinite gap but also in the first finite gap. Due to the reduction of self-focusing effect with power, solitons in the first gap obviously extend outside defect site in space. Besides, the calculation results show that both the saturation parameter s and the defect strength ε can affect the power, the existence and stable regions of DSSs.
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For a given s, the power of DSSs increases with propagation constant μ (“phase shift” induced by all effects) when ε is kept at a constant, and when μ remains unchanged, the power must vary inversely with ε to meet the constraint μF (“phase shift” induced by self-focusing) þ μD (“phase shift” induced by defect) ¼ constant. Besides, when s increases to some value, the power of DSSs can grow dramatically with μ for a given ε. It is the deeper saturable nonlinearity that suppresses increasing rate of μF with power. On the other hand, linear stability analyses indicate that high-power DSSs are unstable, especially for negative defects, and that the stable regions will become narrow rapidly and eventually disappear with a decrease of ε. While for positive and zero defects, the unstable high-power solitons only focus on a very small-scale region of propagation constant.
Acknowledgments We are grateful to Dr. X. Zhu for his helpful discussions with us. This work was supported by National Basic Research Program of China under grant no. 2013CB922403, National Natural Science Foundation of China under grant nos. 11274399 and 61475195 as well as Guangdong Natural Science Foundation, China under grant no. 2014A030311029.
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