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We report the existence and stability of solitons in kagome optical lattices with a defect in photorefractive crystal under focusing saturable nonlinearity. For different types of defects, solitons will exist in different gaps. For a positive defect, the solitons only exist in the semi-infinite gap and only stably exist in the low power region. For a negative defect, the solitons exist both in the semi-infinite gap and the first gap. With an increasing of the negative defect depth, the stable region in the semi-infinite will be narrowed, while solitons will be firstly unstable in the high power region of the first gap, and finally solitons will be not stable in the whole first gap.
©2010 Optical Society of America
1. Introduction
Recently, solitons in two-dimensional (2D) optical periodic lattices have been widely studied both theoretically and experimentally [1–5]. We can see solitons in different 2D optical lattices with different nonlinearity [6,7]. Light propagation in periodic nonlinear media take on unique phenomena that cannot be seen in homogeneous media [8], Periodic 2D optical lattices offer the unique ways for controlling light propagation [9], and the interplay between the periodicity and nonlinearity is important not only in the physics of optical lattices in photorefractive crystals, but also in a variety of other contexts in optical and atomic physics [10]. Gap solitons can exist in different gaps and different nonlinear media [11,12], and gap solitons can be utilized for different applications [13]. Gap solitons have many types in different gaps, including dipole solitons, vortex solitons, quadrupole solitons, and fundamental solitons [14–16].
The existence and stability of defect solitons in two-dimensional square lattices with focusing saturable nonlinearity have been theoretically studied [17], and the two-dimensional defect surface solitons were observed experimentally in a hexagonal waveguide array under Kerr nonlinearity [18]. Some kinds of solitons were studied in kagome lattices [11]. In this paper, the existence and stability of defect solitons in kagome lattices are studied carefully, and the similarities and differences of defect solitons properties between kagome lattices and square lattices are also reported.
2. The theoretical model
We consider an ordinary beam launched through a mask to form kagome optical lattices with a defect in photorefractive crystal under focusing saturable nonlinearity. Meanwhile, an extraordinarily polarized probe beam is launched into the defect site. The probe beam propagates along Z for the varying amplitude U is described by the normalized 2D nonlinearity Schrödinger equation [1,15]
The intensity profile of kagome lattices is described by [11] In Eq. (1), IL is the intensity profile of kagome lattices with a defect that described by Here, p = 3/2, k = 4π/d, h = (1 + 4p/3) [11]. Z is the propagation distance, its real units is 2k1D2/π2, D is the lattice spacing, (x, y) are the transverse distance, its real units is D/π, k1 = k0ne, k0 is the wavenumber in vacuum, ne is the refractive index along the extraordinary axis, γ33 is the electrooptic coefficient of the crystal, and E0 is the applied DC field voltage, its real unit is π2/(k02ne4D2γ33) [15]. We choose E0 = 15 [19], d =3π, V0 = 0.375, D = 20 μm, ne = 2.3, γ33 = 280 pm/V, and λ0 = 0.5 μm. Thus, one x or y unit corresponds to 6.4 μm, one z unit corresponds 2.3 mm, one E0 unit corresponds to 20 V/mm, and the uniform lattice peak intensity is 6.We use plane wave expansion method to get the band structure, which is shown in Fig. 1(a) . Using the above parameters, we obtain the region of the semi-infinite gap as μ≤6.33, and the first gap as 6.86≤μ≤8.96. The bandgap diagram is shown in Fig. 1(a). Figure 1(b) is the intensity distribution of kagome lattices with a negative defect (ε = −0.5), while Fig. 1(c) is the intensity distribution of kagome lattices with a positive defect (ε = 0.5).
Fig. 1 (Color online) (a) Band structure of kagome lattices (blank region corresponds to Bloch band). (b) The kagome lattices with a negative defect (ε = −0.5). (c) The kagome lattices with a positive defect (ε = 0.5).
We search the stationary solitons’ numerical solution of Eq. (1) in form of , where μ is the propagation constant and is the real function. By substituting into Eq. (1), we find that satisfies the equation
The power of solitons is defined as and we obtain the numerical solution of Eq. (2) by the modified square-operator method [16].
To numerically analyze the linear stability of solitons in kagome lattices, we perturb them as , Here the superscript “*” represents complex conjugation, and , <<1. By substituting the perturbation into Eq. (1) and linearizing, we get the eigenvalue equation
If there exists Re(δ)>0, the defect solitons are linearly unstable. Otherwise they are linearly stable. We get the Re(δ) by a numerical method that is called OOM [20].3. Numerical results
We add the solitons by 10% random-noise perturbations into the initial input to simulate the soliton propagation.
At first, we study the gap solitons in uniform kagome lattices. In such a case (ε = 0), we find solitons only exist in the semi-infinite gap. Figure 2(a) shows the power diagram of gap solitons versus the propagation constant μ. In the region of 3.24≤μ≤6.25, where the power of solitons is moderate and the solitons are stable, Fig. 2(c) shows the profile (|u|) of soliton for μ = 4.6 [point A in Fig. 2(a)]. Figures 2(d) and 2(e) show the profile (|u|) at z = 100 and z = 200, respectively. Obviously the soliton can stably transmit. In the region of 3.24≤μ≤6.25, the slope of the power curve is negative, that is dp/dμ<0. According to the VK criterion, solitons can stably exist. So the stability of gap solitons in the region is in accordance with the VK criterion. In the region of 6.26≤μ≤6.32, where the power of solitons is low and the slope of power diagram changes, dp/dμ>0. In the region, we find that the solitons are unstable. Figure 2(b) shows the Re(δ), Re(δ)>0 indicate the solitons are unstable. Figure 2(f) shows the profile (|u|) of gap soliton for μ = 6.30 [point B in Fig. 2(a)]. Figures 2(g) and 2(h) show the profile (|u|) at z = 100 and z = 200, respectively. The soliton is not stable.
Fig. 2 (Color online) ε = 0. (a) Power diagram of gap solitons (blue regions correspond to Bloch bands). (b) Re(δ) versus constant μ for gap solitons. (c) Profile (|u|) of gap soliton for μ = 4.6 (point A). Its profile (|u|) at (d) z = 100 and (e) z = 200. (f) Profile (|u|) of gap soliton for μ = 6.3 (point B). Its profile (|u|) at (g) z = 100 and (h) z = 200.
When μ = 6.33 (very close to the band), soliton can stably propagate. Figure 3(d) shows the profile (|u|) of the soliton. Figures 3(e) and 3(f) show the profile (|u|) at z = 100 and z = 200, respectively.
Fig. 3 (Color online) ε = 0. (a) Profile (|u|) of DS for μ = 2.5 (point C). Its profile (|u|) at (b) z = 100 and (c) z = 200. (d) Profile (|u|) of DS for μ = 6.33. Its profile (|u|) at (e) z = 100 and (f) z = 200.
When μ<3.24, in the region, the power of solitons is high and exponentially grows with the decreasing of the propagation constant μ. Solitons can’t stably propagate. This kind of instability is different from the VK instability caused by the slope of power curve is positive [21]. Figure 2(b) shows the Re(δ) in the region, obviously Re(δ)>0. Figure 3(a) shows the profile (|u|) of soliton for μ = 2.5 [point C in Fig. 2(a)]. Figures 3(b) and 3(c) show the profile (|u|) at z = 100 and z = 200, respectively. The soliton changes the place along the propagation.
Next, we choose ε = −0.5 as a case for the negative defect. It is found that defect solitons (DSs) exist in both the semi-infinite gap and the first gap. Figure 4(a) shows the power diagram of DSs versus the propagation constant μ. In the semi-infinite gap, the stable region is 4.16≤μ≤6.05, where the power of DSs is moderate. In the region, dp/dμ<0, according to the VK criterion, DSs in the region are stable. Figure 4(c) shows the profile (|u|) of DS for μ = 5.0 [point A in Fig. 4(a)]. Figures 4(d) and 4(e) show the profile (|u|) at z = 100 and z = 200, respectively. The DS can stably propagate.
Fig. 4 (Color online) ε = −0.5. (a) Power diagram of DSs (blue regions correspond to Bloch bands). (b) Re(δ) versus constant μ for negative DSs. (c) Profile (|u|) of DS for μ = 5.0 (point A). Its profile (|u|) at (d) z = 100 and (e) z = 200. (f) Profile (|u|) of DS for μ = 6.16 (point B). Its profile (|u|) at (g) z = 100 and (h) z = 200.
When 6.06≤μ≤6.33, the power curve is not smooth. That means DSs in the region aren’t stable, Fig. 4(b) shows the Re(δ) in the region, where we can see Re(δ)>0 . Figure 4(f) shows the profile (|u|) of defect soliton for μ = 6.16 [point B in Fig. 4(a)]. Figures 4(g) and 4(h) show the profile (|u|) at z = 100 and z = 200, respectively. DS can’t stably propagate.
While μ<4.16, the power of DSs is high and exponentially grows with the decreasing of the propagation constant μ, DSs cannot propagate stably. Figure 4(b) shows the Re(δ)>0 obviously in the region. Figure 5(a) shows the profile (|u|) of DS for μ = 3.5 [point C in Fig. 4(a)]. Figures 5(b) and 5(c) show the profile (|u|) at z = 100 and z = 200, respectively. The DS can’t stay at the same site along the propagation.
Fig. 5 (Color online) ε = −0.5. (a) Profile (|u|) of DS for μ = 3.5. Its profile (|u|) at (b) z = 100, and (c) z = 200. (d) Profile (|u|) of DS for μ = 7.0. Its profile (|u|) at (e) z = 100 and (f) z = 200. (g) Profile (|u|) of DS for μ = 7.5. Its profile (|u|) at (h) z = 100 and (i) z = 200.
In the first gap, DSs are stable. Figure 5(d) shows the profile (|u|) of DS for μ = 7.0 [point D in Fig. 4(a)]. Figures 5(e) and 5(f) show the profile (|u|) at z = 100 and z = 200, respectively. We can see the DS can stably transmit. Figure 5(g) shows the profile (|u|) of DS for μ = 7.5 [point E in Fig. 4(a)]. Figures 5(h) and 5(i) show the profile (|u|) at z = 100 and z = 200, respectively. The DS is stable. Figure 4(a) show the power curve of DSs in the first gap, the power of DSs is gradually decreasing with the increasing of the propagation constant μ, obviously dp/dμ<0, so the stability of DSs in the first gap also is in accordance with the VK criterion.
By increasing the negative defect depth, the stable region in the semi-infinite gap will be narrowed. When the negative defect depth increases to −0.6, the stable region in the semi-infinite gap is 4.60≤μ≤6.01, but when ε = −0.7, the DSs in the whole semi-infinite gap are unstable. In the first gap, with an increasing of the negative defect depth, the DSs will not all stably exist, when ε = −0.8, DSs in the high power region are not stable, when ε = −1, DSs in the whole first gap are all unstable.
Finally, we choose ε = 0.5 as a case for the positive defect. In this case, the DSs only exist in the semi-infinite gap. Figure 6(a) shows the power diagram of DSs versus the propagation constant μ. For the positive defects, solitons can stably exist in the low power region, but cannot stably exist in the high power region, which is similar to the DSs in 2D square optical lattices [17]. In the case μ>2.81, DSs are stable. When μ>2.81, Re(δ) = 0, that also means solitons are stable. Figure 6(c) shows the profile (|u|) of DS for μ = 3[point A in Fig. 6(a)]. Figures 6(d) and 6(e) show the profile (|u|) at z = 100 and z = 200, respectively. The DS can stably propagate. In the region of μ≤2.81, the power of DSs exponentially grows with the decreasing of the propagation constant μ, DSs are not stable, Fig. 6(b) show the Re(δ) versus the propagation constant μ. In the region of μ≤2.81, Re(δ)>0, it means solitons can’t stably exist. Figure 6(f) shows the profile (|u|) of DS for μ = 2.2 [point B in Fig. 6(a)]. Figures 6(g) and 6(h) show the profile (|u|) at z = 100 and z = 200, respectively. DS can’t maintain its original shape at z = 100 and z = 200.
Fig. 6 (Color online) ε = 0.5. (a) Power diagram of DSs (blue regions correspond to Bloch bands, the dashed line represents the unstable regions and the solid line represents the stable regions). (b) Re(δ) versus constant μ for positive DSs. (c) Profile (|u|) of DS for μ = 3.0 (point A). Its profile (|u|) at (d) z = 100 and (e) z = 200. (f) Profile (|u|) of DS for μ = 2.2 (point B). Its profile (|u|) at (g) z = 100 and (h) z = 200.
The lattice peak intensity and the applied DC field voltage (E0 in this paper) can affect the properties of defect solitons in optical lattices. To compare the properties of defect solitons in kagome optical lattices with that in square optical lattices which has been reported recently [17], we change the lattice peak intensity I0 to 6 and E0 to 15 in [17] to correspond that in this paper, thus the band structure of square lattices is different from [17]. In such a case, there are three gaps of square lattices, the semi-infinite gap: μ≤5.53, the first gap: 5.76≤μ≤9.21, and the second gap: 10.13≤μ≤12.68. For positive defects, DSs in the two types of optical lattices all only exist in the semi-infinite gap, and are only stable in the low power region while not stable in the high power region. The stable region of DSs in kagome lattices is wider than that in square lattices. For negative defects, DSs in kagome lattices and square lattices can exist in the semi-infinite gap and the first gap at the same time. In the semi-infinite gap, DSs in the two types of lattices both stably exist in the moderate power region, the stable region of DSs in square lattices is wider than that in kagome lattices. In the first gap, with an increasing of the negative defect depth, DSs will not always be stable in the gap in the two kinds of lattices, with an increasing of the negative defect depth, for kagome lattices, at first DSs in the high power region will not be stable, and then DSs in the whole first gap are unstable, while for square lattices, the finding varies a little: Firstly, DSs in the low power region are unstable, and later DSs also are unstable in the whole first gap. In the first gap, the stable region of DSs in square lattices is wider than that in kagome lattices. For the uniform lattices, solitons in kagome lattices and square lattices both only exist in the semi-infinite gap. Solitons in kagome lattices and square lattices stably exist in the moderate power region, the stable region of solitons in square lattices is wider than that in kagome lattices. When propagation constant µ is very close to the band, solitons in square lattices are not stable, which is a little different from kagome lattices.
By the way, we find that for different applied DC field voltage (E0) and lattice peak intensity (the peak intensity of kagome lattices and the square lattices are the same, and E0 in the two types of lattices are also the same), the similarities and differences between kagome lattices and square lattices will be different
4. Conclusions
We have analytically and numerically studied the existence and stability of defect solitons in kagome lattices with focusing saturable nonlinearity. For the negative defect, DSs can exist in the semi-infinite gap and the first gap at the same time. In the semi-infinite gap, the DSs are only stable in the moderate power region. With an increasing of the negative defect depth, the stable region in the semi-infinite will be narrowed, finally DSs will be unstable in the whole semi-infinite gap. In the first gap, with an increasing of the negative defect depth, at first, DSs are unstable in the high power region, and then they are unstable in the whole first gap. For the positive gap, DSs only exist in the semi-infinite gap and only stably exist in the low power region. In the uniform lattices, solitons only exist in the semi-infinite gap, and can stably transmit in the moderate power region and the low power region very near the band. In the high power region and the low power region a little away from the band, solitons can’t stably transmit.
Acknowledgment
This work was supported by The Key Technologies R&D Program of Guangdong Province, Major Research Plan (No.2009A080301013), and The Key Technologies R&D Program of Guangzhou City, Major Plan Program (No.2009A1-D081), and partially by the National Natural Science Foundation of China (No.60835001).
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