Open Access
Add to BibSonomyAbstract
We study the surface line defect gap solitons (SLDGSs) in an interface between a line defect of two-dimensional (2D) square optical lattice and the uniform media with focusing saturable nonlinearity. Some unique properties are revealed that the surface line defect of square optical lattice can profoundly affect the shape and stability of soliton. Stable soltion for the case of negative defect can exist in both of the semi-infinite gap and the first gap; unlike in the square lattice without defect, soliton can only exist in the semi-infinite gap. For the case of the positive defect, the solitons exist in the semi-inifite gap and stably exist in the low power region.
© 2011 Optical Society of America
1. Introduction
When forward- and backward-propagating waves at the interface between uniform media and an optical lattice experience scattering, their nonlinear coupling can form surface gap solitons [1]. Surface gap solitons (SGSs) are a type of solitons that exist at the interface between two media with different refractive index and possess specific features of both nonlinear surface wave and gap soliton. Because of their unique properties, it makes lots of theoretical and experimental work devoted to study SGSs in various optical lattices [1–8] and thus these research results have been widely applied in diverse fields. For example, He et al. [9] have theoretically studied surface superlattice gap solitons at an interface between one-dimensional superlattice and an uniform medium. Its main result is that the solitons are only stable in the semi-infinite gap. In experiment, surface gap solitons in 2D optical lattices have been observed by Wang et al. [10]. It has been pointed out that the experimental results are in good agreement with theoretical predications.
The nonlinear defect modes, defect solitons (DSs), have been extensively studied in some system [11–16] and have many potential applications such as the all-optical switch and the routing of optical signal. Recently, the existence and stability of defect gap soliton (DGS) in 2D square and kagome optical lattices have been studied theoretically [17,18]. In the experimental field, Szameit et al. [19] have observed surface defect solitons in 2D hexagonal wave guide array. However, the research on surface line defect gap soliton is comparatively lacked. To our knowledge, it has only been mentioned in Ref.[16] and then an in-depth study on surface line defect gap solitons is very interesting. In this paper, we will study DGSs at the surface of 2D square optical lattices with defect line. We reveal that stable solitons for the case of negative defect can exist in both of the semi-infinite gap and the first gap, while for the case of the positive defect, the solitons exist in the semi-infinite gap and can stably propagate in the low power region. The shape and stability of the 2D SLDGSs have also been discussed analytically and numerically.
2. Theory
Considering a light beam propagating along z axis and illuminating at the middle site of the line defect at the surface of square optical lattice under focusing saturable nonlinearity, we can get the expression of the light field amplitude q which obeys the normalized 2D nonlinearity schrödinger equation [20,21,22]:
In Eq.(1), coordinate z [in units of 2klD2/π2] is propagation distance, where kl = k0ne, k0 = 2π/λ0 is the wave number (λ0 is the wavelength), ne is the unperturbed refractive index, and D is the lattice spacing. Coordinate (x,y) [in units of D/π] are the transverse distances. E0 [in units of ] is the applied DC field [22], where γ33 is the eletro-optical coefficient of the crystal. IL is the intensity profile of optical lattice that described by and IL = 0, for . In Eq.(2), I0 is the lattice peak intensity and ɛ is modulation parameter of defect depth. Some reasonable parameters are chosen in our paper: E0 = 6, D = 20μm, ne = 2.3, γ33 = 280pm/v, and λ0 = 0.5μm [22]. We search the stationary solitons numerical solution of Eq.(1) in the form of q = u(x,y) exp(−iμz), where μ is the propagation constant and u is the real function which satisfies the following equation We can use the modified square-operator method [23] to obtain the soliton solutions u(x,y) and the power of solitons is defined as .Figure 1(a) shows band structure of the lattice at I0 = 3, which is obtained by plane wave expansion method. For E0 = 6, we find that the region of the semi-infinite gap is μ ≤ 3.58 and the first gap is 4.41 < μ ≤ 5.55. The intensity distribution of square lattice with a positive line defect: ɛ = 0.5 and with a negative line defect: ɛ = −0.5 are displayed in figure 1(b) and figure 1(c), respectively.
Fig. 1 (Color online) (a) Band structure of square optical lattices (blank region corresponding to Bloch band);Semi-infinite gap (SI gap) and the first gap (1st gap) have been noted. (b) The square optical lattices with a positive line defect(ɛ = 0.5). (c) The square optical lattices with a negative line defect(ɛ = −0.5). In (b) and (c), a line defect exists at the surface between the uniform media (the left side) and square lattice (the right side).
3. Numerical results and discussion
To examine the stability of surface gap solitons at the surface of 2D square optical lattice, we search the perturbed solutions to Eq.(1) in the form
where v(x,y) and w(x,y) are the real and imaginary part of the eigenfunction, respectively, and δ is the corresponding eigenvalue and also represents the complex growth rate. The superscript “*” means complex conjunction, and v(x,y), w(x,y) << 1. Substituting Eq. (4) into Eq. (1) and then linearing Eq.(1), we arrive at the coupled eigenvalue equationsThese equations are solved by a number method called OOM [24] to get the perturbation growth rate Re(δ). For Re(δ) = 0, the SGSs are linearly stable; otherwise, they are linearly unstable. Random-noise perturbation whose amplitude is set at 10% is added to the initial input light to simulate the soliton propagation.
Figure 2(a) plots power P versus propagation constant μ for the case of a zero defect (ɛ = 0) at the surface of square optical lattice. This figure indicates that surface gap soliton only exists in the semi-infinite gap. In the high power region: μ < 1.63, SGSs cannot stably propagate. As an example, the profile (|u|) of soliton for μ = 1.55 [ point A in figure 2(a)] at z = 0, 100, and 200 are shown in figure 2(c), 2(d), and 2(e), respectively. When the propagation distance z increases, the SGS will obviously jump away from the original site of the initial soliton. In the moderate power region: 1.63 ≤ μ < 2.0 and 2.03 < μ ≤ 3.49, the SGS can stably transmit. A stable example [μ = 2.6 corresponds to point B in figure 2(a)] is shown in figure 2(f), 2(g), and 2(h). We can find in these figures that the shape of surface gap soliton is not centrosymmetric for the asymmetric spatial distribution of media; unlike the case of Ref.[17], the soliton shape is centrosymmetric. In the little region: 2.0 ≤ μ ≤ 2.03, the SGS cannot stably transmit because of the change of slope in power diagram [see figure 2(a)]. In the low power region: μ > 3.49, the surface gap soliton cannot stably propagate. We choose μ = 3.52 [point C in figure 2(a)] as an unstable example which is shown in figure 3(a), 3(b), and 3(c). The soliton will be gradually scattered in the course of propagation. The stability of soliton propagation is mainly determined by light diffraction and self-focusing resulting from nonlinearity. When the nonlinearity can contract with light diffraction, soliton pulses can stably propagate. Otherwise, diffraction cannot be suppressed by nonlinearity and soliton pulses finally decay into linear diffractive waves [25]. To further verify the instability, we numerically calculate Eq.(5) to obtain the perturbation growth rate Re(δ) as shown in figure 2(b). The Re(δ) is obviously larger than zero in the region: μ < 1.63 and μ > 3.49, and the soliton cannot stably transmit. In the region μ > 3.49, the positive slope of power diagram (dP/dμ > 0) and Re(δ) > 0 both can indicate the solitons are unstable in this region. However, in the region: μ < 1.63, the slope of power diagram is negative, but Re(δ) > 0. So we can conclude that this instability is different from the VK instability caused by the positive slope of power curve [11]. It should be pointed out that a consequence of the instability in this region is soliton pulses will not decay and will be shift away from the original place. For the negative slope of power curve (dP/dμ < 0) and Re(δ) = 0 in the region: 1.63 ≤ μ < 2.0 and 2.03 < μ ≤ 3.49, the stability of SGS is in accordance with the VK criterion. The instability of SGS in the little region: 2.0 ≤ μ ≤ 2.03 is also in accordance with the VK criterion.
Fig. 2 (Color online)(ɛ = 0). (a) Power diagram of surface gap solitons (blue regions corresponding to Bloch bands. (b) Re(δ) versus constant μ for surface gap solitons. Profile (|u|) of surface gap soliton for μ = 1.55 (point A) at (c) z=0, (d) z=100, and (e) z=200. Profile (|u|) of surface gap soliton for μ = 2.6 (point B) at (f) z=0, (g) z=100, and (h) z=200.
Fig. 3 (Color online)(ɛ = 0). Profile (|u|) of surface gap soliton for μ = 3.52 (point C) at (a) z=0, (b) z=100, and (c) z=200.
Power P versus propagation constant μ for a negative defect (ɛ = −0.5) at the surface of square optical lattice is presented in figure 3(a). It can be found in this figure that surface line defect soliton can exist in both the semi-infinite gap and in the first gap. In the semi-infinite gap, the stable region is 2.18 ≤ μ ≤ 3.13, where the power of SLDGSs is moderate. In the high power region: μ < 2.18 and low power region: μ > 3.13, the SLDGSs cannot stably propagate along the propagation. In the high power region, we choose an unstable example [μ = 1.85 corresponds to point A in figure 4(a)] as shown in figure 4(c), 4(d), and 4(e); while in the low power region, we choose an unstable example [μ = 3.54 corresponds to point C in figure 4(a)] as shown in figure 5(a), 5(b), and 5(c). In the stable region, μ = 2.95 [point B in figure 4(a)] is chosen to show the stability of soliton in this region. At this point, the profile (|u|) of SLDGS at z = 0, 100, and 200 are plotted in figure 4(f), 4(g), and 4(h), respectively. Figure 4(b) shows the change of the growth rate Re(δ) with propagation constant μ. For Re(δ ) > 0 in the region: μ < 2.18 and μ > 3.13, the SLDGSs cannot stably propagate. In the moderate power region: 2.18 ≤ μ ≤ 3.13, the slope of power curve is negative and Re(δ) = 0, then the stability of SLDGSs in this region is in accordance with the VK criterion. In the first gap, SLDGSs are stable. Choosing μ = 4.45 [point D in figure 4(a)] as an example in the first gap, we will show the stable SLDGS propagation in this region in figure 5(d), 5(e), and 5(f). We also can find in these figures that the shape of stable SLDGS in the first gap is very different from that in the semi-infinite gap. For dP/dμ < 0 in the first gap that is obtained form the gradually decreasing power of DSs with the increasing of the propagation constant μ, we can conclude that the stability of DSs in the first gap also is in accordance with the VK criterion. From the numerical results in figures (4) and (5), we can see that the interplay of surface and defect modes can give rise to a new state. The negative defect site performs as a repulsion because the light intensity at the defect site is lower than the normal ones. The repulsion from the defect will increase the light diffraction and then will change the stability of surface gap soliton. As a result, the stable region of surface gap soliton will be obviously changed. With the introduction of negative defect, the surface gap soliton can not only stably propagate in the semi-infinite gap but also in the first gap.
Fig. 4 (Color online)(ɛ = −0.5). (a) Power diagram of SLDGSs (blue regions corresponding to Bloch bands. (b) Re(δ) versus constant μfor SLDGSs Profile (|u|) of SLDGS for μ = 1.85 (point A) at (c) z=0, (d) z=100, and (e) z=200. Profile (|u|) of SLDGS for μ = 2.95 (point B) at (f) z=0, (g) z=100, and (h) z=200.
Fig. 5 (Color online)(ɛ = −0.5). Profile (|u|) of SLDGSs for μ = 3.54 (point C) at (c) z=0, (d) z=100, and (e) z=200. Profile (|u|) of SLDGS for μ = 4.45 (point D) at (f) z=0, (g) z=100, and (h) z=200.
Finally, we choose ɛ = 0.5 as a case for the positive line defect at the surface of square optical lattice and the power of SLDGSs versus the propagation constant μ for this case is shown in figure 6(a). In this figure, we can see that the SLDGSs only exist in the semi-infinite gap. In addition, in the semi-infinite gap, the SLDGSs can stably exist in low power region: 2.01 ≤ μ ≤ 3.28, but cannot stably exist in high power region: μ < 2.01. This result is very close to the DGSs in 2D square optical and kagome optical lattices [17,18]. The stable example [ μ = 2.35 corresponds to point A in figure 6(a)] in low power region is shown in figure 6(c), 6(d), and 6(e); while the unstable example [μ = 1.5 corresponds to point B in figure 6(a)] in high power region is shown in figure 6(f), 6(g), and 6(h). Figure 6(b) shows the change of the Re(δ) with propagation constant μ. For the low power region, Re(δ ) = 0 means that soliton can stably exist; and for the high power region, Re(δ) > 0 means that solitons cannot stably exist. In the case of positive defect, the defect site has a higher light intensity and is attractive to the light field. The attraction from the positive defect will decrease the light diffraction and then the stable region of surface gap soliton will also be altered, just like the case of negative defect. With the effect of positive defect mode, the surface gap soliton can stably propagate in the low power region of the semi-infinite gap.
Fig. 6 (Color online)(ɛ = 0.5). (a) Power diagram of SLDGSs (blue regions corresponding to Bloch bands. (b) Re(δ) versus propagate constant μ for SLDGSs. Profile (|u|) of SLDGS for μ = 2.35 (point A) at (c) z=0, (d) z=100, and (e) z=200. Profile (|u|) of SLDGS for μ = 1.5 (point B) at (f) z=0, (g) z=100, and (h) z=200.
4. Conclusion
To summarize, we have revealed the existence of surface line defect gap soliton. Such new type of solitons are supported by an interface between the line defect of two-dimensional square optical lattice and the uniform media with focusing saturable nonlinearity. The surface line defect of two-dimensional optical lattice offers new properties of solitons. For the negative defect, DGS can exist in both the semi-infinite gap and the first gap. In the semi-infinite gap, the DGS are stable in the moderate power region. For the positive gap, DGS exist in the semi-infinite gap and stably exist in the low power region but not in high power region.
Acknowledgments
This work was supported by the National Natural Science Foundation of China(No. 10774101, and No. 60978009), the National Minister of Education Program for Training Ph.D., and the State Key Development Program for Basic Research of China (No. 2007CB925204 and No. 2009CB929604).
References and links
1. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. 96, 073901 (2006). [CrossRef] [PubMed]
2. M. Cronin-Golomb, “Photorefractive surface waves,” Opt. Lett. 20, 2075–2077 (1995). [CrossRef] [PubMed]
3. K. G. Makris, S. Suntsov, D. N. Christodoulides, and G. I. Stegeman, “Discrete surface solitons,” Opt. Lett. 30, 2466–2468 (2005). [CrossRef] [PubMed]
4. M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, “Discrete surface solitons in semi-inifinite binary waveguide arrays,” Opt. Lett. 31, 2332–2334 (2006). [CrossRef] [PubMed]
5. Y. V. Kartashov, V. A Vysloukh, D. Mihalache, and L. Torner, “Generation of surface soliton arrays,” Opt. Lett. 31, 2329–2331 (2006). [CrossRef] [PubMed]
6. E. Smironov, M. Stepić, C. E. Rüter, D. Kip, and V. Shandarov, “Observation of staggered surface solitary waves in one-dimensional waveguide arrays,” Opt. Lett. 31, 2338–2340 (2006). [CrossRef]
7. C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, Y. S. Kivshar, and A. Hache, “Observation of surface gap solitons in semi-infinite wave guide arrays,” Phys. Rev, Lett. 97, 083901 (2006). [CrossRef]
8. Y. J. He and B. A. Malomed, “Surface waves and boundary effects in DNLS equations,”, in The Discrete Nonlinear Schrodinger Equation: Mathematical Analysis, Numerical Computations, and Physcal Perspectives, P. G. Kevrekidis, ed. (Springer, 2009), pp. 259–276. [PubMed]
9. Y. J. He, W. H. Chen, H. Z. Wang, and B. A. Malomed, “Surface superlattice gap solitons,” Opt. Lett. 32, 1390–1392 (2007). [CrossRef] [PubMed]
10. X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett. 98, 123903 (2007). [CrossRef] [PubMed]
11. J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E 73, 026609 (2006). [CrossRef]
12. W. H. Chen, Y. J. He, and H. Z. Wang, “Surface defect gap solitons,” Opt.Express 14, 11271–11276 (2006). [CrossRef] [PubMed]
13. J. Xie, Y. He, and H. Wang, “Surface defect gap solitons in one-dimensional chirped optical lattices,” J. Opt. Soc. Am. B 27(3), 484–487 (2010). [CrossRef]
14. W. H. Chen, Y. J. He, and H. Z. Wang, “Defect superlattice solitons,” Opt.Express 15(22), 14498–14503 (2007). [CrossRef] [PubMed]
15. M. J. Ablowitz, B. Ilan, E. Schonbrun, and R. Piestun, “Solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures,” Phys. Rev. E 74, 035601 (2006). [CrossRef]
16. Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009). [CrossRef]
17. W. H. Chen, X. Zhu, T. W. Wu, and R. H. Li, “Defect solitons in two-dimensional optical lattices,” Opt. Express 18(11), 10956–10962 (2010). [CrossRef] [PubMed]
18. X. Zhu, H. Wang, and L. X. Zheng, “Defect solitons in kagome optical lattices,” Opt. Express 18(20), 20786–20792 (2010). [CrossRef] [PubMed]
19. A. Szameit, Y. V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional defect surface solitons,” Opt. Lett. 34(6), 797–799 (2009). [CrossRef] [PubMed]
20. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in opticallly indued real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003). [CrossRef] [PubMed]
21. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in opticallly indued nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003). [CrossRef] [PubMed]
22. J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole and quadrupole solitons in optically induced two-dimensional photonic lattices: theory and experiment,” Stud. Appl. Math. 113(4), 389–412 (2004). [CrossRef]
23. J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary wave in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153–197 (2007). [CrossRef]
24. J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 277(14), 6862–6876 (2008). [CrossRef]
25. D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, “Insability of solitons governed by quadratic nonlinearity,” Phys. Rev. Lett. 75, 591–595 (1995). [CrossRef] [PubMed]
