Surface defect gap solitons in two-dimensional optical lattices

Transcription

1 Chin. hs. B Vol. 21, No. 7 (212) 7426 Surface defect gap solitons in two-dimensional optical lattices Meng Yun-Ji( 孟云吉 ), Liu You-Wen( 刘友文 ), and Tang Yu-Huang( 唐宇煌 ) Department of Applied hsics, Nanjing Universit of Aeronautics and Astronautics, Nanjing 2116, China (Received 28 November 211; revised manuscript received 7 December 211) We investigate the eistence and stabilit of surface defect gap solitons at an interface between a defect in a two-dimensional optical lattice and a uniform saturable Kerr nonlinear medium. The surface defect embedded in the two-dimensional optical lattice gives rise to some unique properties. It is interestingl found that for the negative defect, stable surface defect gap solitons can eist both in the semi-infinite gap and in the first gap. The deeper the negative defect, the narrower the stable region in the semi-infinite gap will be. For a positive defect, the surface defect gap solitons eist onl in the semi-infinite gap and the stable region localizes in a low power region. Kewords: surface defect gap soliton, optical lattice, saturable Kerr nonlinear media ACS: Tg, J, Wi DOI: 1.188/1674-6/21/7/ Introduction Surface waves, localized waves propagating at the interface separated b two media with different properties, are ubiquitous wave phenomena in optics. Owing to their unique features of having no analogues in bulk, surface waves have been found to possess potential applications in the eploration of the surface characteristics of optical materials. In the linear optical domain, the surface modes at the interface between uniform and periodicall laered media began to be studied long time ago. [1] Recentl, the research interest in this domain was renewed after the metamaterial had been used to construct multilaered media. [2] In the nonlinear optical domain, the studies of the surface waves pinned to interfaces between different nonlinear optical media, at least one of which carries a lattice structure, have also attracted intense attention. [3 1] Etensive theoretical and eperimental investigations have been carried out in the contet of obtaining stable surface solitons in diverse settings. In one-dimensional (1D) arras, the Kerr surface solitons near the edge of a semi-infinite waveguide with self-focusing nonlinearit were theoreticall predicted and eperimentall observed b Makris et al. [3] and Suntsov et al. [4] for the first time. The also found that surface solitons can eist at the heterointerface between two dissimilar semi-infinite 1D waveguide arras, as well as at the boundaries of a two-dimensional (2D) nonlinear lattice. [5] In addition, Kartashov et al. [6] suggested the eistence of surface gap solitons at the interface separated b uniform media and an optical lattice (OL) with selfdefocusing nonlinearit, which was observed in semiinfinite waveguide arras. [7] In 2D arras, the theoretical prediction of the edge and corner solitons inspired two eperiments, one in asmmetric waveguide arras of Kerr tpe (fused silica) [8] and the other in a photorefractive medium. [9] Apart from the above two eperiments, one should note that Szameit et al. [1] reported on the observation of 2D surface solitons at the interface between square and heagonal waveguides. In addition, the nonlocal surface-wave solitons occurring at the interface between a dielectric medium (air) and a nonlinear material with a long-range nonlocal response were demonstrated theoreticall and eperimentall b Alfassi et al. [11] Surface solitons at the interface between a 1D photonic superlattice and a uniform medium with weak nonlocal nonlinearit were also proposed. [12] Defect modes (DMs) in photonic lattices have also been considered in Refs. [13] [22]. Fedele et al. [13] investigated the linear DMs in 1D photonic lattices and found that a variet of DMs in the negative (repulsive) defect case. The nonlinear 1D defect DMs (defect solitons) and their stabilit were studied b Yang roject supported b the National Natural Science Foundation of China (Grant No ) and the Natural Science Foundation of Jiangsu rovince, China (Grant No. BK29366). Corresponding author. wliu@nuaa.edu.cn 212 Chinese hsical Societ and IO ublishing Ltd

2 Chin. hs. B Vol. 21, No. 7 (212) 7426 and Chen [14] in detail. One of the striking results in their work is that for a repulsive defect, defect solitons in the first gap are linearl stable, while the reverse scenario is found in the attractive case. In the 2D geometr, linear and nonlinear localized DMs were sstematicall studied b Wang et al. [] and Chen et al. [16] respectivel. The defect solitons in Kagome OLs were also reported. [17] Apart from the above work concentrating on the single-site defect, defect solitons, and their stabilit in a defect that covers several lattice sites have also been studied. [18,19] Motivated b the above theoretical predictions, some tpical eperiments have been performed to observe defect modes in 1D negative defects [2] and 2D negative defects in opticall-induced OLs, [21] as well as 2D negative and positive defects in a heagonal waveguide arra. [22] Surface defect gap solitons (SDGSs) have been analsed in 1D simple lattices [23] and dual-frequenc lattices. [24] Despite the above progress, SDGSs in 2D geometr are still poorl understood. In the present work, we report the eistence and stabilit of the surface defect gap solitons at an interface between a defect of 2D OLs and uniform saturable Kerr nonlinear media. For the repulsive defect, SDGSs with moderate power could stabl eist in the semi-infinite (SI) gap (corresponding to the first gap in Ref. [23]). What is worth emphasizing is that for some specific repulsive defect, SDGSs can stabl eist in the whole first gap (corresponding to the second gap in Ref. [23]). For the attractive defect, SDGSs eist onl in SI gaps and are stable in the low power region. 2. Theoretical modeling To elucidate the dnamics of the beam propagating at the interface between uniform and twodimensional OLs with a single defect embedded in the self-focusing saturable nonlinear medium, we consider the normalized 2D nonlinear Schrödinger equation for the light field q [25] i q z + 2 q q 2 E 2 q =, (1) 1 + I L + q where I L is the intensit profile of the OLs with a defect located in the centre of the OLs, I L = at < π/2; I L = I cos 2 () cos 2 (){1 + ε ep[ ( ) 4 /128]}, (2) at π/2, with I being the total strength of the OLs; ε the modulation depth of the defect intensit; and are transverse coordinates measured in units of D/π, with D being the lattice spacing; z is the propagation distance in units of 2k 1 D 2 /π 2, where k 1 = k n e, with k = 2π/λ being the wave number, λ the wavelength in vacuum, n e the unperturbed refractive inde along the etraordinar ais; E is the applied bias field in units of π 2 /(kn 2 4 e D2 γ 33 ), with γ 33 being the electro-optic coefficient of the crstal. We set the parameters to be D = 2 m, λ =.5 m, n e = 2.3, and γ 33 = 28 pm/v. [13] As a consequence, one or unit corresponds to 6.4 m, one z unit corresponds to 2.3 m, and one E unit corresponds to 2 V/mm. The intensit distribution of the defect lattices are illustrated in Figs. 1 1(d) for zero, negative (ε =.5), and positive (ε =.5) defects, respectivel, from which we note that if ε < (repulsive), the light intensit I L at the defect site is lower than those at the surrounding sites. In contrast, when ε > (attractive), the light intensit I L at the defect site is higher than those at the surrounding sites. Other parameters are I = 3 and E = 6 (corresponding to a maimal refractive change without defect, and a minimal refractive change without defect, with defect ε =.5, with defect ε =.5) which are fied unless otherwise stated. B illuminating the sample with a broad laser beam passing through a properl designed amplitude mask, the potential given b Eq. (2) can be realized opticall in a photorefractive crstal. [25] First, we utilize the plane-wave epansion method based on the Bloch theorem to obtain the bandgap structure of the sstem. We substitute q(,, z) = u(, ) ep[i(k + k ) iz] into the linearized epression of Eq. (1), where u(, ) is the periodic function with the same period as the OLs, k and k are the Bloch wave-numbers, and is the propagation constant. The obtained bandgap spectrum is shown Fig. 1, from which we can find that the boundaries and range of the semi-infinite gap and the first gap are < and 4.41 < < 5.552, respectivel. Net, we search for the stationar soliton solutions in the form of q(,, z) = u(, ) ep( iz), where u(, ) is the real function obeing the following nonlinear equation: 2 u u 2 E u = u. (3) 1 + I L + u 2 The power of the soliton is defined as = u 2 dd. Solving Eq. (3) b dint of the modified square-operator iteration method (MSOM) proposed b Yang and Lakoba, [26] we can acquire the soliton profiles

3 Chin. hs. B Vol. 21, No. 7 (212) (c) E (d) Fig. 1. (colour online) Bandgap spectrum of the square optical lattices. The intensit distribution without a defect. (c) The intensit distribution with a repulsive defect ε =.5. (d) The intensit distribution with an attractive defect ε =.5. Lastl, we eamine the linear stabilit of SDGSs b considering the perturbed stationar solution form as q(,, z) = {u(, ) + [v(, ) w(, )] ep(δz) + [v(, ) + w(, )] ep(δ z)} ep( iz), where δ is the associated growth rate, the superscript represents comple conjugation, and the perturbed components v, w 1. Linearization of Eq. (1) around u ields the eigenvalue problem i L L 1 v = δ v, (4) w w L = E 1 + I L + u 2, (5) L 1 = E 1 + I L + u 2 2E u 2 + (1 + I L + u 2 ) 2. (6) We solve the above equations b adopting the original operator method (OOM) [27] to find perturbation profiles and associated growth rates. The stabilit criterion of the sstem is that if >, the SDGSs are linearl unstable, otherwise the are linearl stable. [27] 3. Numerical simulations and discussion We first stud the repulsive defect case and take ε =.5 as a tpical eample. In such a case, we find that the SDGSs can eist stabl in both SI gap and the first gap. The dependence of the total power on the propagation constant is displaed in Fig. 2, from which one can draw a conclusion that the soliton power is a monotonicall decreasing function of propagation constant with eception of the propagation constant approaching the gap edge. This means that the slope of the curve is negative when < According to the Vakhitov Kolokolov (VK) criterion, i.e., d /d < is known as the necessar condition for the stabilit of solitons supported b the selffocusing nonlinearit, [28] the SDGSs ma be stable in a region of < Moreover, we calculate the dependence of the real part of the growth rate δ on propagation constant as shown in Fig. 2, and ascertain that the stable SDGSs emerge onl in a narrow region i.e., 2.77 < < 3.193, which accords with the VK criterion. The eistence region 2.77 < < of the surface defect gap solitons in this case is narrower than the region 2.42 < < 3.43 of the defect solitons in the corresponding bulk case. [16] To confirm the predictions of the linear analsis, we solve Eq. (1) with the input condition q(,, z = ) = u(, )[1+ρ(, )] b emploing the split-step Fourier method, where ρ(, ) is the random function with Gaussian distribution with its relative amplitude set at 1% level. First, we demonstrate the stable region 2.77 < < For instance, the profile of SDGSs for = 2.85 is shown in Fig. 2(c), and the SDGSs are trapped at the defect site and maintain their profiles at z = and z = 3, as obviousl shown in Figs. 2(d) and 2(e), respectivel. Net, we turn to one of the unstable regions ( 2.77), and choose = 1.75 as a representative eample. The profiles of the SDGSs corresponding to ecitations at,, and 3 unit distances in the nonlinear medium are illustrated in Figs. 2(f) 2(h), respectivel. The results reveal that the SDGSs cannot be trapped at the defect site and the VK instabilit leads to the snakelike position oscillations of the SDGSs. It is worth emphasizing that in this unstable region, the power is even higher and grows eponentiall with the decreasing of propagation constant, as shown in Fig. 2. Apparentl, such instabilit of the SDGSs in the high power region is different from the usual VK instabilit caused b a sign change in the slope of the power diagram. [14] Then we stud the other unstable region of < < 3.58, in which the curve is not smooth, as shown in Fig. 2. During propagating in the medium, the SDGSs ( = 3.45) deca as shown in Figs. 3 3(c), which demonstrates great consistence with the analtic prediction shown in the Fig. 2. In the first gap, the SDGSs are alwas stable under perturbations

4 Chin. hs. B Vol. 21, No. 7 (212) 7426 in their eisting region of 4.41 < < 4.83, and the slope of the curve is negative, which is in accordance with the VK criterion. The results mentioned above are confirmed b Figs. 3(d) 3(f) ( = 4.45) and Figs. 3(g) 3(i) ( = 4.75) (c) (d) (e) (f) (g) (h) - As a controllable parameter, the influence of the applied electric field E on the stabilit properties of the SDGSs is investigated. We increase the E value from 6 to 6.5 (while keeping I = 3), and obtain the dependence of the real part of the growth rate on propagation constant, which is plotted in Fig. 4. At the higher E value of 6.5, one should note that the upper boundar value of SI gap is = 3.836, and accordingl, the stabilit region of SDGSs becomes broader (i.e., 2.62 < < 3.548), namel, increasing the applied bias electric field could significantl stabilize the SDGSs. This ma be eplained phsicall as follows: with I fied, the depth of the lattice induced potential E /(1 + I L ) increases with E increasing, and the SDGSs become more stable. A similar finding has been reported for vorte solitons in a saturable medium. [29] Fig. 2. (colour online) ower diagram of SDGSs (dark regions correspond to Bloch bands). versus. (c) (e) rofiles of SDGS for = 2.85 at z =, and 3 unit distances respectivel. (f) (h) rofiles of SDGS for = 1.75 at z =, and 3 unit distances, respectivel. ε = (c) (d) (e) (f) (g) (h) (i) Fig. 3. (colour online) rofiles of SDGS for = 3.45, z = ; = 3.45, z = ; (c) = 3.45, z = 3; (d) = 4.45, z = ; (e) = 4.45, z = ; (f) = 4.45, z = 3; (g) = 4.75, z = ; (h) = 4.75, z = ; (i) = 4.75, z = 3. ε = Fig. 4. Dependence of on for ε =.5 and E = 6.5. We increase the depth of the repulsive defect and choose ε = 1, and plot the diagram and curve in Figs. 5 and 5, respectivel. The relationship in the SI gap is not shown in the figure, since the SDGSs are alwas not stable in this gap. As shown in Fig. 5, the soliton power first decreases eponentiall and then increases at = 4.946, after having reached a peak value (not apparent in the figure), the power ehibits a similar decrease behaviour. According to the VK criterion, there might be two stable regions, but surprisingl, onl one stable region (4.479 < < 4.944) eists under moderate soliton power in the first gap, because < in this region as shown in Fig. 5. One ma note that the stable region is narrower than that in the bulk case. [16] As we have done previousl, in order to verif the analtic prediction, the ecitations of SGDSs for different propagation constants are illustrated in Figs. 5(c) 5(e) (stable for = 4.75), Figs. 5(f) 5(h) (unstable for = 4.45), and Figs. 5(i) 5(k) (unstable for = 5.35), respectivel

5 Chin. hs. B Vol. 21, No. 7 (212) (c) (d) (e) (f) (g) (h) - (i) - - (j) (k) - Fig. 5. (colour online) ower diagram of SDGSs (dark regions correspond to Bloch bands). versus. (c) (k) rofiles of SDGS for (c) = 4.75, z = ; (d) = 4.75, z = ; (e) = 4.75, z = 3; (f) = 4.45, z = ; (g) = 4.45, z = ; (h) = 4.45, z = 3; (i) = 5.35, z = ; (j) = 5.35, z = ; (k) = 5.35, z = 3. ε = (c) (d) (e) (f) (g) (h) Fig. 6. (colour online) ower diagram of SDGSs (dark regions correspond to Bloch bands). versus. (c) (h) rofiles of SDGS for (c) = 2.35, z = ; (d) = 2.35, z = ; (e) = 2.35, z = 3; (f) = 1.25, z = ; (g) = 1.25, z = ; (h) = 1.25, z = 3. ε =.5. We turn to the attractive defect case and set ε =.5. B numerical evaluation, we find that there are no SGDSs eisting in the first gap, so we shed light on the SGDSs eisting in the SI gap. We plot the dependences of the soliton power and the real part of the growth rate on propagation constant in Figs. 6 and 6, respectivel. As shown in Fig. 6, the power decreases with propagation constant increasing, i.e., d /d < in the SI gap, and there eists a stable region of < < The above prediction is justified b the direct simulation of Eq. (1), and the results are shown in Figs. 6(c) 6(e) (stable for = 2.35) and Fig. 6(f) (h) (unstable for = 1.25). Comparing the case with the bulk case, [16] we notice that the stable region is even broader. One of the interesting results is that the SDGSs do not require the threshold power for their ecitation provided that the modulation depth of the defect intensit ε eceeds a critical value of.25 as shown in Fig. 7. When ε.25, the dependence for SDGSs is non-monotonic and the slope of curve tends to infinit as the propagation constant approaches to the first band as shown in Fig. 7. Therefore, the eistence of SDGSs requires a threshold power th. In contrast, for ε >.25, th vanishes as shown in Fig. 7. The above phenomena are analogous to the results proposed b Szameit et al. [22] in optical lattices of Kerr tpe. However, what differs from Ref. [22] is that for ε >.25 the stabilit domain of SDGSs decreases as ε increases (Fig. 8) band ε=.5 ε=.2 ε=.1 ε=.25 ε= th ε Fig. 7. (colour online) ower diagrams of SDGSs versus for different values of ε in the SI band, and threshold power th versus ε in the SI band ε The SI gap 4.8 The first gap ε Fig. 8. Stabilit (shaded) and instabilit (white) regions of SDGSs for dependences of the propagation constant on defect parameter ε in the SI gap and the first gap

6 Chin. hs. B Vol. 21, No. 7 (212) 7426 Finall, after a huge of numerical work, we obtain the stabilit and instabilit regions of SDGSs for dependences of the propagation constant on defect parameter ε in the SI gap and the first gap as shown in Figs. 8 and 8, respectivel. We find that the stable region of SDGSs eisting in the SI gap for repulsive defect is narrower than that for the attractive defect. And when ε is less than a critical value (i.e., ε =.55), all the SDGSs are unstable in the SI gap. Similarl, there is a threshold value of ε (i.e., ε =.24), above which the powers of SDGSs in the first gap are nearl zero, i.e., the SDGSs are noneistent. However it is worthwhile to remark that the SDGSs are stable in the whole first gap for the specific region of.79 < ε < Conclusion In this paper, we stud SDGSs at an interface between a defect of 2D OLs and uniform saturable Kerr nonlinear medium. For repulsive defect, the SDGSs can eist in both SI gap and first gap. But in the SI gap, if ε is lower than a critical value, there will be no stable SDGSs. In the first gap, the power of SDGSs is almost zero provided that the defect parameter ε eceeds a threshold value. For the attractive defect, SDGSs eist onl in SI gap and are stable in the low power region. References [1] Yeh, Yariv A and Cho A Y 1978 Appl. hs. Lett [2] Shadrivov I V, Sukhorukov A A and Kivshar Y S 23 hs. Rev. E [3] Makris K G, Suntsov S, Christodoulides D N, Stegeman G I and Hache A 25 Opt. Lett [4] Suntsov S, Makris K G, Christodoulides D N, Stegeman G I, Hache A, Morandotti R, Yang H, Salamo G and Sorel M 26 hs. Rev. Lett [5] Makris K G, Hudock J, Christodoulides D N, Stegeman G I, Manela O and Segev M 25 Opt. Lett [6] Kartashov Y V, Vsloukh V A and Torner L 26 hs. Rev. Lett [7] Rosberg C R, Neshev D N, Krolikowski W, Mitchell A, Vicencio R A, Molina M I and Kivshar Y S 26 hs. Rev. Lett [8] Szameit A, Kartashov Y V, Dreisow F, ersch T, Nolte S, Tünnermann A and Torner L 27 hs. Rev. Lett [9] Wang X, Bezradina A, Chen Z, Makris K G, Christodoulides D N and Stegeman G I 27 hs. Rev. Lett [1] Szameit A, Kartashov Y V, Dreisow F, Heinrich M, Vsloukh V A, ersch T, Nolte S, Tünnermann A, Lederer F and Torner L 28 Opt. Lett [11] Alfassi B, Rotschild C, Manela O, Segev M and Christodoulides D N 27 hs. Rev. Lett [12] Huang H C, He Y J and Wang H Z 29 Chin. hs. B [13] Fedele F, Yang J and Chen Z 25 Opt. Lett. 3 6 [14] Yang J and Chen Z 26 hs. Rev. E [] Wang J, Yang J and Chen Z 27 hs. Rev. A [16] Chen W, Zhu X, Wu T and Li R 21 Opt. Epress [17] Zhu X, Wang H and Zheng L 21 Opt. Epress [18] Dong L and Ye F 21 hs. Rev. A [19] Yang X Y, Zheng J B and Dong L W 211 Chin. hs. B [2] Wang X, Young J, Chen Z, Weinstein D and Yang J 26 Opt. Epress [21] Makasuk I, Chen Z and Yang J 26 hs. Rev. Lett [22] Szameit A, Kartashov Y V, Heinrich M, Dreisow F, ersch T, Nolte T S, Tünnermann A, Lederer F, Vsloukh V A and Torner L 29 Opt. Lett [23] Chen W, He Y, and Wang H 26 Opt. Epress [24] Zhu W, Luo L, He Y and Wang H 29 Chin. hs. B [25] Fleischer J W, Segev M, Efremidis N K and Christodoulides D N 23 Nature [26] Yang J and Lakoba T I 27 Stud. Appl. Math [27] Yang J 28 J. Comput. hs [28] Vakhitov M G and Kolokolov A A 1973 Sov. J. Radiophs. Quantum. Electron [29] Yang J 24 New. J. hs