- Email: [email protected]
Defect solitons in optically induced kagome photonic lattices in photovoltaic–photorefractive crystals
Optics Communications 312 (2014) 258–262
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Defect solitons in optically induced kagome photonic lattices in photovoltaic–photorefractive crystals Shuqin Liu a, Keqing Lu a,n, Yiqi Zhang b, Weijun Chen a, Tianrun Feng a, Niu Pingjuan a a
School of Electronics and Information Engineering, Tianjin Polytechnic University, Tianjin 300387, China Key Laboratory for Physical Electronics and Devices of the Ministry of Education & Shaanxi Key Lab of Information Photonic Technique, Xi'an Jiaotong University, Xi'an 710049, China
b
art ic l e i nf o
a b s t r a c t
Article history: Received 12 July 2013 Received in revised form 2 September 2013 Accepted 3 September 2013 Available online 27 September 2013
We study defect solitons (DSs) and their stability in optically induced kagome photonic lattices with a defect in photovoltaic–photorefractive crystals. We show that these DSs exist only in the semi-infinite gap when the defect is positive and both in the semi-infinite gap and the first gap when the defect is negative. For a positive defect, DSs are stable in the low power region and unstable in the high power region. For a negative defect, DSs in the semi-infinite gap are stable in the moderate power region and unstable in the high and low power regions. In the first gap, DSs are stable in the all power region for the low negative defect depth and in the high power region for the high negative defect depth. We find that the stable region of DSs increases with the defect depth when the defect is positive and decreases with an increase in the defect depth when the defect is negative. Crown Copyright & 2013 Published by Elsevier B.V. All rights reserved.
Keywords: Defect solitons Photonic lattices Photorefractive effect Nonlinear optics
1. Introduction Light propagation in periodic optical systems such as waveguide arrays, optically-induced photonic lattices, and photonic crystals has attracted a great deal of attention because of its physics and lightrouting applications. In such periodic systems, linear light propagation exhibits Bloch bands and forbidden bandgaps. Gap solitons can exist in different bandgaps and form by the nonlinear coupling between forward- and backward-propagating waves when both experience Bragg scattering from the periodic structures. At present, a wide variety of gap solitons in different gaps are known: fundamental solitons [1–5], dipole solitons [6,7], vortex solitons [8–10], quadrupole solitons [11], and so on. On the other hand, a defect in the periodic medium can support defect modes (DMs) and defect solitons (DSs) in bandgaps of the periodic medium. DMs in photonic lattices in biased non-photovoltaic–photorefractive (non-PP) [12–14], PP [15], and biased PP [16] crystals have been proposed. DSs in one-dimensional (1D) photonic lattices in biased non-PP crystals have been theoretically analyzed [17]. In 2D square photonic lattices and kagome photonic lattices in biased non-PP crystals, DSs have also been predicted [18,19]. Therefore, it is interesting to know whether optically induced kagome photonic lattices in PP crystals can support DSs. In this letter, we study DSs and their stability in optically induced kagome photonic lattices with a defect in PP crystals.
n
Corresponding author. Tel.: þ 86 2283955164. E-mail address: [email protected] (K. Lu).
These DSs exist in different bandgaps due to the change of defect intensity. For a positive defect, DSs exist only in the semi-infinite gap and are stable in the low power region but unstable in the high power region. When the positive defect depth is increased, the stable region of DSs is extended. For a negative defect, DSs exist in the semi-infinite and first gaps. In the semi-infinite gap, DSs are stable in the moderate power region but unstable in the high and low power regions. In the first gap, DSs are stable in the all power region for the low negative defect depth and in the high power region for the high negative defect depth, but unstable in the low power region for the high negative defect depth. When the negative defect depth is increased, the stable region of DSs is narrowed.
2. Theoretical model Let us consider an ordinarily polarized two-dimensional kagome lattice beam with a single-site defect that propagates in a PP crystal along the z axis. This defected lattice beam is assumed to be uniform along the direction of propagation. Meanwhile, an extraordinarily polarized probe beam is launched into the defect site, propagating collinearly with the lattice beam. In this situation, the nondimensionalized model equation for the probe beam is [15,19] ∂U ∂2 U ∂2 U I L þ jUj2 i þ 2 þ 2 þ E0 U ¼ 0; ∂Z ∂X I L þjUj2 þ 1 ∂Y
0030-4018/$ - see front matter Crown Copyright & 2013 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.09.007
ð1Þ
S. Liu et al. / Optics Communications 312 (2014) 258–262
obtain eigenfunction equation as follow
where I L ¼ I f1 þ εexp½ ð4X 2 þ 3Y 2 Þ4 =128g;
259
ð2Þ
is the intensity profile of kagome lattices with a defect, I is the intensity profile of kagome lattices as described by [20] I ¼ V 0 2expðik1 pY=hÞ cos ðk1 pY=hÞexpðik1 Y=hÞ pffiffiffi pffiffiffi þ exp½ ik1 Y=ð2hÞ ið 3=2Þk1 X þ exp½ ik1 Y=ð2hÞ þ ið 3=2Þk1 Xj2 ;
ð3Þ p ¼ 3=2; V 0 ¼ 0:375; k1 ¼ 4π =d; h ¼ 1 þ4p=3; ε controls the strength of the defect, U is the slowly varying amplitude of the probe 2 beam, E0 ¼ Ep =½π 2 =ðT 2 k n2e r 33 Þ, Ep is the photovoltaic field constant, T is the lattice spacing, k ¼ 2π ne =λ is the optical wave number in the PP crystal, λ is the wavelength, ne is the unperturbed extraordinary index of refraction, and r 33 is the electrooptic coefficient. In obtaining Eq. (1), we have used normalized coordinates, i.e., Z ¼ z=ð2kT 2 =π 2 Þ, X ¼ x= T=π , and Y ¼ y=ðT=π Þ. For a positive defect (ε 4 0), the lattice light intensity I L at the defect site is higher than that at the surrounding sites. For a negative defect (ε o 0), the lattice intensity I L at the defect site is lower than that at the surrounding sites. The intensity distributions of optical lattices with ε ¼ 0:5 and 0:5 are shown in Fig. 1(c) andp(d), ffiffiffi respectively. In this paper, let us choose T ¼ 20 μm, d ¼ 3π , 12 ne ¼ 2:365, r 33 ¼ 80 10 m=V, Ep ¼ 16 KV=cm, and λ ¼ 0:5 μm [17,21]. Thus E0 25, one X or Y unit corresponds to 6:4 μm, and one Z unit corresponds to 2:4 mm. In order to obtain the soliton solutions in the bandgaps, we look for Floquet-Bloch spectrum by substituting UðX; Y; ZÞ ¼ uðX; YÞexp½iðkX X þ kY Y μZÞ into the linear version of Eq. (1), and
∂2 u ∂X
2
þ
∂2 u ∂Y 2
þ2ikX
∂u ∂u 2 2 þ 2ikY ðkX þ kY Þu þ VðX; YÞu ¼ μu; ∂X ∂Y
ð4Þ
where uðX; YÞ is a periodic function with the same periodicity as the lattices, V ðX; YÞ ¼ E0 I=ð1 þ IÞ is the uniform periodic potential, kX and kY are wave numbers in the first Brillouin zone, and μ is the Bloch-wave propagation constant. We calculate Eq. (4) by the plane wave expansion method to obtain the bandgap diagram, which is shown in Fig. 1(a). Using the above parameters, we obtain the region of the semi-infinite, first, and second gaps as μ o 15:61, 15:42 o μ o 11:44, and 6:78 o μ o 5:01, respectively. Fig. 1(b) is the intensity distribution of kagome lattices. The stationary soliton solutions in Eq. (1) are sought in the form of UðX; Y; ZÞ ¼ uðX; YÞexpð iμZÞ;
ð5Þ
where uðX; YÞ is a localized function in X and Y. By substituting Eq. (5) into Eq. (1), we find that uðX; YÞ satisfies the nonlinear equation ∂2 u ∂X
2
þ
∂2 u ∂Y
2
þE0
ðI L þ u2 Þu ¼ μu; I L þ u2 þ 1
ð6Þ
from which DSs uðX; YÞ can be determined by the modified squared-operator iteration method [22]. The power of DSs is R1 R1 defined as P ¼ 1 1 juj2 dXdY. To investigate the stability of DSs, we search for the solution of Eq. (1), describing the propagation of a DS with a perturbed initial profile of the following form: UðX; Y; ZÞ ¼ uðX; YÞ þ ½vðX; YÞ wðX; YÞexpðδZÞ n ð7Þ þ½vðX; YÞ þ wðX; YÞn expðδ ZÞ expð iμZÞ;
Fig. 1. (a) Band structure of kagome photonic lattices. (b) The kagome photonic lattices. (c) The kagome photonic lattices with a positive defect ε ¼ 0:5. (d) The kagome photonic lattices with a negative defect ε ¼ 0:5.
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S. Liu et al. / Optics Communications 312 (2014) 258–262
n where the superscript “ ” represents complex conjugation, and vðX; YÞ, wðX; YÞ≪1 are the small perturbations. Substituting Eq. (7) into Eq. (1) and linearizing the resulting equation, we obtain the following system of linear equations:
δ v ¼ i μw þ
∂2 w
∂2 w
∂X
∂Y
þ 2
þ E0 2
ðI L þ u2 Þw ; I L þ u2 þ1
h i 9 8 < ðI L þ u2 þ 1Þ2 I L þ u2 1 v= ∂2 v ∂ 2 v δw ¼ i μv þ 2 þ 2 þ E0 ; : ; ðI L þ u2 þ 1Þ2 ∂X ∂Y
ð8Þ
ð9Þ
from which the perturbation growth rate ReðδÞ can be obtained by a numerical method [23]. When ReðδÞ 4 0, DSs are linearly unstable, otherwise they are linearly stable.
3. Numerical results We begin our analysis by considering first DSs for the positive defect lattice. In this case (ε 4 0), DSs only exist in the semiinfinite gap. Fig. 2(a) depicts the power diagrams of DSs for three different values of the defect strength parameter ε. When μ o 21:32 with ε ¼ 0:2, μ o 21:52 with ε ¼ 0:5, and μ o 21:68 with ε ¼ 0:8 in Fig. 2(a), where the power of DSs is high and the slope of the power curves is negative (dP=dμ o0), DSs cannot stably exist, as shown in Fig. 2(b). As an example, Fig. 3 (a) shows the profile of a DS for μ ¼ 21:52 with ε ¼ 0:2 [point A in Fig. 2(a)]. Fig. 3(b) shows the profile of the DS at Z ¼ 300, where the DS changes its original shape. In this study, we add a noise to inputted DSs by multiplying them with ½1 þ φðXÞ and let them propagate along the Z direction, where φðXÞ is a Gaussian random
ε=0.2
0.6
ε=0.5
Re(δ)
A
ε=0.8
0.4 0.2 0
B C
−22
−20 μ
−18
Fig. 2. (Color online) (a) Power versus propagation constant (blue regions are Bloch band) when ε ¼ 0:2 (solid curve), 0:5 (dashed curve), and 0:8 (dash–dot curve). (b) Perturbation growth rates ReðδÞ for unstable DSs when ε ¼ 0:2, 0:5, and 0:8. In (a), the solid, dashed, and dash–dot curves indicate the stable DSs, and the dot curves indicate the unstable DSs, see (b). Profiles of DSs at the circled points in (a) and (b) are displayed in Fig. 3.
Fig. 3. (a) Profile of DS at the circled point A in Fig. 2(a) with ðε; μÞ ¼ ð0:2; 21:52Þ and (b) its profile at Z ¼ 300. (c) Profile of DS at the circled point B in Fig. 2(a) with ðε; μÞ ¼ ð0:5; 21:52Þ and (d) its profile at Z ¼ 300. (e) Profile of DS at the circled point C in Fig. 2(a) with ðε; μÞ ¼ ð0:8; 21:52Þ and (f) its profile at Z ¼ 300. The DS in (a) is unstable, while the DSs in (c) and (e) are stable.
S. Liu et al. / Optics Communications 312 (2014) 258–262
function with o φ 4 ¼ 0 and o φ2 4 ¼ s2 , and s is equal to 20% of the input soliton amplitude. When 21:32 r μ r 16:14 with ε ¼ 0:2, 21:52 r μ r 16:89 with ε ¼ 0:5, and 21:68 r μ r 17:44 with ε ¼ 0:8 in Fig. 2(a), where the power of DSs is low and dP=dμ o 0, DSs can stably exist because of ReðδÞ ¼ 0 [see Fig. 2(b)]. When μ ¼ 21:52 with ε ¼ 0:5 and 0:8 [points B and C in Fig. 2(a)], the profiles of DSs are displayed in Fig. 3(c) and (e), whereas their profiles at Z ¼ 300 are displayed in Fig. 3(d) and (f). Evidently, DSs are stable since they propagate unchanged. In the semi-infinite gap, the shape of DSs is very similar to that displayed in Fig. 3(a), (c), and (e). This phenomenon is similar to the intraband solitons in the first band in lattice potentials with the single defect and self-focusing nonlinearity [24]. Moreover, Fig. 2(b) indicates that the stable region of DSs increases with the positive defect depth.
261
Next, we consider the case of negative defects. It is found that DSs exist in the semi-infinite and first gaps. Fig. 4(a) shows the power diagrams of DSs for ε ¼ 0:2, 0:5, and 0:8. In the semi-infinite gap, when 15:71 o μ o 15:61 with ε ¼ 0:2, 15:76 o μ o 15:61 with ε ¼ 0:5, and 15:81 o μ o 15:61 with ε ¼ 0:8 in Fig. 4(a), where the power of DSs is low and dP=dμ 4 0 in a limited region [see Fig. 4(a)], DSs are not stable because of ReðδÞ 40 [see Fig. 4(b)]. For example, taking μ ¼ 15:69 with ε ¼ 0:2 [point A in Fig. 4(a)], the profile of the DS is displayed in Fig. 5(a), where the slope of the power curve is positive (dP=dμ 40). The DS breaks up at Z ¼ 300, which is shown in Fig. 5(b). When 20:84 r μ r 15:71 with ε ¼ 0:2, 20:44 r μ r 15:76 with ε ¼ 0:5, and 17:46 r μ r 15:81 with ε ¼ 0:8 in Fig. 4(a), where the power of DSs is moderate and dP=dμ o 0, DSs are stable because of ReðδÞ ¼ 0 [see Fig. 4(b)].
Fig. 4. (Color online) (a) Power versus propagation constant (blue regions are Bloch band) when ε ¼ 0:2 (solid curve), 0:5 (dashed curve), and 0:8 (dash–dot curve). (b) Perturbation growth rates ReðδÞ for DSs when ε ¼ 0:2, 0:5, and 0:8. In (a), the solid, dashed, and dash–dot curves indicate the stable DSs, and the dot curves indicate the unstable DSs, see (b). Profiles of DSs at the circled points in (a) and (b) are displayed in Fig. 5.
Fig. 5. (a) Profile of DS at the circled point A in Fig. 4(a) with ðε; μÞ ¼ ð 0:2; 15:69Þ and (b) its profile at Z ¼ 300. (c)–(f) Profiles of DSs at the circled point B–E in Fig. 4(a), with ðε; μÞ ¼ ð 0:5; 16:43Þ, ð 0:5; 13:73Þ, ð 0:8; 16:81Þ, and ð 0:8; 13:09Þ respectively. The DS in (a) is unstable, while the DSs in (c)–(f) are stable.
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S. Liu et al. / Optics Communications 312 (2014) 258–262
For example, for μ ¼ 16:43 with ε ¼ 0:5 [point B in Fig. 4(a)] and μ ¼ 16:81 with ε ¼ 0:8 [point D in Fig. 4(a)], Fig. 5(c) and (e) show the profiles of DSs, respectively. When μ o 20:84 with ε ¼ 0:2, μ o 20:44 with 0:5, and μ o 17:46 with 0:8 in Fig. 4(a), the power of DSs is high and dP=dμ o 0. In high power regions, DSs cannot propagate stably because of ReðδÞ 40 [see Fig. 4(b)]. In the first gap, DSs with ε ¼ 0:2 and 0:5 can stably exist in the all power regions [see Fig. 4(b)]. For ε ¼ 0:8, DSs are stable in the region of 15:42 o μ o 12:56 [see Fig. 4(b)] but unstable in the region of 12:56 o μ o 11:44 [see Fig. 4(b)]. Taking μ ¼ 13:73 with ε ¼ 0:5 [point C in Fig. 4(a)] and μ ¼ 13:09 with ε ¼ 0:8 [point E in Fig. 4(a)], the profiles of the stable DSs are shown in Fig. 5(d) and (f). On the other hand, Fig. 4(b) demonstrates that the stable region of DSs decreases with an increase in the negative defect depth. It is very interesting to apply Vakhitov–Kolokolov (VK) stability criterion to DSs in optically induced kagome photonic lattices with a defect in PP crystals. The usual VK criterion is that the solitons are stable if dP=dμ o 0 and unstable if dP=dμ 4 0. In previous studies by Pelinovsky et al. [25], the authors found that the usual VK instability leads to either soliton decay or amplitude oscillations, depending on the sign of perturbations. For our physical system, when the slope of the power curve is negative (dP=dμ o 0), the VK instability leads to the DS of change shape with no relation to the sign of perturbations [see Fig. 3(a) and (b)]. When the slope of the power curve is positive (dP=dμ 40), the VK instability leads to the breakup DS regardless of the sign of perturbations [see Figs. 5(a) and (b)]. In the present case, the VK instability is different from the usual VK instability based on the sign of dP=dμ. On the other hand, it has been reported that when the slope of the power curve is positive, the VK instability leads to the stable intra-band solitons in the first band in lattice potentials with the single defect and self-focusing nonlinearity [24]. 4. Conclusion In conclusion, we have investigated DSs and their stability in optically induced kagome photonic lattices with a defect in PP crystals. In our numerical studies, we have shown that these DSs exist only in the semi-infinite gap for a positive defect and both in the semi-infinite gap and the first gap for a negative defect. In the semi-infinite gap, DSs are stable in the low power region for a positive defect and in the moderate power region for a negative defect, but unstable in the high power region for a positive defect and in the high and low power regions for a negative defect. In the
first gap, DSs are stable in the all power regions for the low negative defect depth. For the high negative defect depth, the high-power DSs are stable, whereas low-power DSs are unstable. We have demonstrated that the stable region of DSs increases with the positive defect depth and decreases with an increase in the negative defect depth.
Acknowledgment The work was supported by the Natural Science Foundation of Chinese Tianjin (No. 13JCYBJC16400). References [1] H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, J.S. Aitchison, Physical Review Letters 81 (1998) 3383. [2] J. Yang, Z.H. Musslimani, Optics Letters 28 (2003) 2094. [3] N.K. Efremidis, J. Hudock, D.N. Christodoulides, J.W. Fleischer, O. Cohen, M. Segev, Physical Review Letters 91 (2003) 213906. [4] D.E. Pelinovsky, A.A. Sukhorukov, Y.S. Kivshar, Physical Review E 70 (2004) 036618. [5] R. Iwanow, R. Schiek, G.I. Stegeman, T. Pertsch, F. Lederer, Y. Min, W. Sohler, Physical Review Letters 93 (2004) 113902. [6] D. Neshev, E. Ostrovskaya, Yu.S. Kivshar, W. Krolikowski, Optics Letters 28 (2003) 710. [7] J. Yang, I. Makasyuk, A. Bezryadina, Z. Chen, Studies in Applied Mathematics 113 (2004) 389. [8] B.B. Baizakov, B.A. Malomed, M. Salerno, Europhysics Letters 63 (2003) 642. [9] D.N. Neshev, T.J. Alexander, E.A. Ostrovskaya, Y.S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, Physical Review Letters 92 (2004) 123903. [10] J.W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D.N. Christodoulides, Physical Review Letters 92 (2004) 123904. [11] T.J. Alexander, A.A. Sukhorukov, Yu.S. Kivshar, Physical Review Letters 93 (2004) 063901. [12] F. Fedele, J. Yang, Z. Chen, Optics Letters 30 (2005) 1506. [13] X. Wang, J. Young, Z. Chen, D. Weinstein, J. Yang, Optics Express 14 (2006) 7362. [14] J. Wang, J. Yang, Z. Chen, Physical Review A 76 (2007) 013828. [15] Kehao Li, Keqing Lu, Jianbang Guo, Weijun Chen, Tongtong Sun, Fengxue Yao, Jingjun Xu, Optics Communications 285 (2012) 3187. [16] Keqing Lu, Jianbang Guo, Kehao Li, Weijun Chen, Tongtong Sun, Fengxue Yao, Pingjuan Niu, Jingjun Xu, Optical Materials 34 (2012) 1277. [17] J. Yang, Z. Chen, Physical Review E 73 (2006) 026609. [18] W.H. Chen, X. Zhu, T.W. Wu, R.H. Li, Optics Express 18 (2010) 10956. [19] Xing Zhu, Hong Wang, Li-Xian Zheng, Optics Express 18 (2010) 20786. [20] K.J.H. Law, A. Saxena, P.G. Kevrekidis, A.R. Bishop, Physical Review A 79 (2009) 053818. [21] B.I. Sturman, V.M. Fridkin, The Photovoltaic and Photorefractive Effect in Noncentrosymmetric Materias, Gordon and Breach, Philadelphia, 1992. [22] J. Yang, T.I. Lakoba, Studies in Applied Mathematics 118 (2007) 153. [23] J. Yang, Journal Computational Physics 227 (2008) 6862. [24] JianhuaZeng, Boris A. Malomed, Journal of the Optical Society of America B 30 (2013) 1786. [25] D.E. Pelinovsky, V.V. Afanasjev., Yu.S. Kivshar, Physical Review E 53 (1996) 1940.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Defect solitons in optically induced kagome photonic lattices in photovoltaic–photorefractive crystals Shuqin Liu a, Keqing Lu a,n, Yiqi Zhang b, Weijun Chen a, Tianrun Feng a, Niu Pingjuan a a
School of Electronics and Information Engineering, Tianjin Polytechnic University, Tianjin 300387, China Key Laboratory for Physical Electronics and Devices of the Ministry of Education & Shaanxi Key Lab of Information Photonic Technique, Xi'an Jiaotong University, Xi'an 710049, China
b
art ic l e i nf o
a b s t r a c t
Article history: Received 12 July 2013 Received in revised form 2 September 2013 Accepted 3 September 2013 Available online 27 September 2013
We study defect solitons (DSs) and their stability in optically induced kagome photonic lattices with a defect in photovoltaic–photorefractive crystals. We show that these DSs exist only in the semi-infinite gap when the defect is positive and both in the semi-infinite gap and the first gap when the defect is negative. For a positive defect, DSs are stable in the low power region and unstable in the high power region. For a negative defect, DSs in the semi-infinite gap are stable in the moderate power region and unstable in the high and low power regions. In the first gap, DSs are stable in the all power region for the low negative defect depth and in the high power region for the high negative defect depth. We find that the stable region of DSs increases with the defect depth when the defect is positive and decreases with an increase in the defect depth when the defect is negative. Crown Copyright & 2013 Published by Elsevier B.V. All rights reserved.
Keywords: Defect solitons Photonic lattices Photorefractive effect Nonlinear optics
1. Introduction Light propagation in periodic optical systems such as waveguide arrays, optically-induced photonic lattices, and photonic crystals has attracted a great deal of attention because of its physics and lightrouting applications. In such periodic systems, linear light propagation exhibits Bloch bands and forbidden bandgaps. Gap solitons can exist in different bandgaps and form by the nonlinear coupling between forward- and backward-propagating waves when both experience Bragg scattering from the periodic structures. At present, a wide variety of gap solitons in different gaps are known: fundamental solitons [1–5], dipole solitons [6,7], vortex solitons [8–10], quadrupole solitons [11], and so on. On the other hand, a defect in the periodic medium can support defect modes (DMs) and defect solitons (DSs) in bandgaps of the periodic medium. DMs in photonic lattices in biased non-photovoltaic–photorefractive (non-PP) [12–14], PP [15], and biased PP [16] crystals have been proposed. DSs in one-dimensional (1D) photonic lattices in biased non-PP crystals have been theoretically analyzed [17]. In 2D square photonic lattices and kagome photonic lattices in biased non-PP crystals, DSs have also been predicted [18,19]. Therefore, it is interesting to know whether optically induced kagome photonic lattices in PP crystals can support DSs. In this letter, we study DSs and their stability in optically induced kagome photonic lattices with a defect in PP crystals.
n
Corresponding author. Tel.: þ 86 2283955164. E-mail address: [email protected] (K. Lu).
These DSs exist in different bandgaps due to the change of defect intensity. For a positive defect, DSs exist only in the semi-infinite gap and are stable in the low power region but unstable in the high power region. When the positive defect depth is increased, the stable region of DSs is extended. For a negative defect, DSs exist in the semi-infinite and first gaps. In the semi-infinite gap, DSs are stable in the moderate power region but unstable in the high and low power regions. In the first gap, DSs are stable in the all power region for the low negative defect depth and in the high power region for the high negative defect depth, but unstable in the low power region for the high negative defect depth. When the negative defect depth is increased, the stable region of DSs is narrowed.
2. Theoretical model Let us consider an ordinarily polarized two-dimensional kagome lattice beam with a single-site defect that propagates in a PP crystal along the z axis. This defected lattice beam is assumed to be uniform along the direction of propagation. Meanwhile, an extraordinarily polarized probe beam is launched into the defect site, propagating collinearly with the lattice beam. In this situation, the nondimensionalized model equation for the probe beam is [15,19] ∂U ∂2 U ∂2 U I L þ jUj2 i þ 2 þ 2 þ E0 U ¼ 0; ∂Z ∂X I L þjUj2 þ 1 ∂Y
0030-4018/$ - see front matter Crown Copyright & 2013 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.09.007
ð1Þ
S. Liu et al. / Optics Communications 312 (2014) 258–262
obtain eigenfunction equation as follow
where I L ¼ I f1 þ εexp½ ð4X 2 þ 3Y 2 Þ4 =128g;
259
ð2Þ
is the intensity profile of kagome lattices with a defect, I is the intensity profile of kagome lattices as described by [20] I ¼ V 0 2expðik1 pY=hÞ cos ðk1 pY=hÞexpðik1 Y=hÞ pffiffiffi pffiffiffi þ exp½ ik1 Y=ð2hÞ ið 3=2Þk1 X þ exp½ ik1 Y=ð2hÞ þ ið 3=2Þk1 Xj2 ;
ð3Þ p ¼ 3=2; V 0 ¼ 0:375; k1 ¼ 4π =d; h ¼ 1 þ4p=3; ε controls the strength of the defect, U is the slowly varying amplitude of the probe 2 beam, E0 ¼ Ep =½π 2 =ðT 2 k n2e r 33 Þ, Ep is the photovoltaic field constant, T is the lattice spacing, k ¼ 2π ne =λ is the optical wave number in the PP crystal, λ is the wavelength, ne is the unperturbed extraordinary index of refraction, and r 33 is the electrooptic coefficient. In obtaining Eq. (1), we have used normalized coordinates, i.e., Z ¼ z=ð2kT 2 =π 2 Þ, X ¼ x= T=π , and Y ¼ y=ðT=π Þ. For a positive defect (ε 4 0), the lattice light intensity I L at the defect site is higher than that at the surrounding sites. For a negative defect (ε o 0), the lattice intensity I L at the defect site is lower than that at the surrounding sites. The intensity distributions of optical lattices with ε ¼ 0:5 and 0:5 are shown in Fig. 1(c) andp(d), ffiffiffi respectively. In this paper, let us choose T ¼ 20 μm, d ¼ 3π , 12 ne ¼ 2:365, r 33 ¼ 80 10 m=V, Ep ¼ 16 KV=cm, and λ ¼ 0:5 μm [17,21]. Thus E0 25, one X or Y unit corresponds to 6:4 μm, and one Z unit corresponds to 2:4 mm. In order to obtain the soliton solutions in the bandgaps, we look for Floquet-Bloch spectrum by substituting UðX; Y; ZÞ ¼ uðX; YÞexp½iðkX X þ kY Y μZÞ into the linear version of Eq. (1), and
∂2 u ∂X
2
þ
∂2 u ∂Y 2
þ2ikX
∂u ∂u 2 2 þ 2ikY ðkX þ kY Þu þ VðX; YÞu ¼ μu; ∂X ∂Y
ð4Þ
where uðX; YÞ is a periodic function with the same periodicity as the lattices, V ðX; YÞ ¼ E0 I=ð1 þ IÞ is the uniform periodic potential, kX and kY are wave numbers in the first Brillouin zone, and μ is the Bloch-wave propagation constant. We calculate Eq. (4) by the plane wave expansion method to obtain the bandgap diagram, which is shown in Fig. 1(a). Using the above parameters, we obtain the region of the semi-infinite, first, and second gaps as μ o 15:61, 15:42 o μ o 11:44, and 6:78 o μ o 5:01, respectively. Fig. 1(b) is the intensity distribution of kagome lattices. The stationary soliton solutions in Eq. (1) are sought in the form of UðX; Y; ZÞ ¼ uðX; YÞexpð iμZÞ;
ð5Þ
where uðX; YÞ is a localized function in X and Y. By substituting Eq. (5) into Eq. (1), we find that uðX; YÞ satisfies the nonlinear equation ∂2 u ∂X
2
þ
∂2 u ∂Y
2
þE0
ðI L þ u2 Þu ¼ μu; I L þ u2 þ 1
ð6Þ
from which DSs uðX; YÞ can be determined by the modified squared-operator iteration method [22]. The power of DSs is R1 R1 defined as P ¼ 1 1 juj2 dXdY. To investigate the stability of DSs, we search for the solution of Eq. (1), describing the propagation of a DS with a perturbed initial profile of the following form: UðX; Y; ZÞ ¼ uðX; YÞ þ ½vðX; YÞ wðX; YÞexpðδZÞ n ð7Þ þ½vðX; YÞ þ wðX; YÞn expðδ ZÞ expð iμZÞ;
Fig. 1. (a) Band structure of kagome photonic lattices. (b) The kagome photonic lattices. (c) The kagome photonic lattices with a positive defect ε ¼ 0:5. (d) The kagome photonic lattices with a negative defect ε ¼ 0:5.
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n where the superscript “ ” represents complex conjugation, and vðX; YÞ, wðX; YÞ≪1 are the small perturbations. Substituting Eq. (7) into Eq. (1) and linearizing the resulting equation, we obtain the following system of linear equations:
δ v ¼ i μw þ
∂2 w
∂2 w
∂X
∂Y
þ 2
þ E0 2
ðI L þ u2 Þw ; I L þ u2 þ1
h i 9 8 < ðI L þ u2 þ 1Þ2 I L þ u2 1 v= ∂2 v ∂ 2 v δw ¼ i μv þ 2 þ 2 þ E0 ; : ; ðI L þ u2 þ 1Þ2 ∂X ∂Y
ð8Þ
ð9Þ
from which the perturbation growth rate ReðδÞ can be obtained by a numerical method [23]. When ReðδÞ 4 0, DSs are linearly unstable, otherwise they are linearly stable.
3. Numerical results We begin our analysis by considering first DSs for the positive defect lattice. In this case (ε 4 0), DSs only exist in the semiinfinite gap. Fig. 2(a) depicts the power diagrams of DSs for three different values of the defect strength parameter ε. When μ o 21:32 with ε ¼ 0:2, μ o 21:52 with ε ¼ 0:5, and μ o 21:68 with ε ¼ 0:8 in Fig. 2(a), where the power of DSs is high and the slope of the power curves is negative (dP=dμ o0), DSs cannot stably exist, as shown in Fig. 2(b). As an example, Fig. 3 (a) shows the profile of a DS for μ ¼ 21:52 with ε ¼ 0:2 [point A in Fig. 2(a)]. Fig. 3(b) shows the profile of the DS at Z ¼ 300, where the DS changes its original shape. In this study, we add a noise to inputted DSs by multiplying them with ½1 þ φðXÞ and let them propagate along the Z direction, where φðXÞ is a Gaussian random
ε=0.2
0.6
ε=0.5
Re(δ)
A
ε=0.8
0.4 0.2 0
B C
−22
−20 μ
−18
Fig. 2. (Color online) (a) Power versus propagation constant (blue regions are Bloch band) when ε ¼ 0:2 (solid curve), 0:5 (dashed curve), and 0:8 (dash–dot curve). (b) Perturbation growth rates ReðδÞ for unstable DSs when ε ¼ 0:2, 0:5, and 0:8. In (a), the solid, dashed, and dash–dot curves indicate the stable DSs, and the dot curves indicate the unstable DSs, see (b). Profiles of DSs at the circled points in (a) and (b) are displayed in Fig. 3.
Fig. 3. (a) Profile of DS at the circled point A in Fig. 2(a) with ðε; μÞ ¼ ð0:2; 21:52Þ and (b) its profile at Z ¼ 300. (c) Profile of DS at the circled point B in Fig. 2(a) with ðε; μÞ ¼ ð0:5; 21:52Þ and (d) its profile at Z ¼ 300. (e) Profile of DS at the circled point C in Fig. 2(a) with ðε; μÞ ¼ ð0:8; 21:52Þ and (f) its profile at Z ¼ 300. The DS in (a) is unstable, while the DSs in (c) and (e) are stable.
S. Liu et al. / Optics Communications 312 (2014) 258–262
function with o φ 4 ¼ 0 and o φ2 4 ¼ s2 , and s is equal to 20% of the input soliton amplitude. When 21:32 r μ r 16:14 with ε ¼ 0:2, 21:52 r μ r 16:89 with ε ¼ 0:5, and 21:68 r μ r 17:44 with ε ¼ 0:8 in Fig. 2(a), where the power of DSs is low and dP=dμ o 0, DSs can stably exist because of ReðδÞ ¼ 0 [see Fig. 2(b)]. When μ ¼ 21:52 with ε ¼ 0:5 and 0:8 [points B and C in Fig. 2(a)], the profiles of DSs are displayed in Fig. 3(c) and (e), whereas their profiles at Z ¼ 300 are displayed in Fig. 3(d) and (f). Evidently, DSs are stable since they propagate unchanged. In the semi-infinite gap, the shape of DSs is very similar to that displayed in Fig. 3(a), (c), and (e). This phenomenon is similar to the intraband solitons in the first band in lattice potentials with the single defect and self-focusing nonlinearity [24]. Moreover, Fig. 2(b) indicates that the stable region of DSs increases with the positive defect depth.
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Next, we consider the case of negative defects. It is found that DSs exist in the semi-infinite and first gaps. Fig. 4(a) shows the power diagrams of DSs for ε ¼ 0:2, 0:5, and 0:8. In the semi-infinite gap, when 15:71 o μ o 15:61 with ε ¼ 0:2, 15:76 o μ o 15:61 with ε ¼ 0:5, and 15:81 o μ o 15:61 with ε ¼ 0:8 in Fig. 4(a), where the power of DSs is low and dP=dμ 4 0 in a limited region [see Fig. 4(a)], DSs are not stable because of ReðδÞ 40 [see Fig. 4(b)]. For example, taking μ ¼ 15:69 with ε ¼ 0:2 [point A in Fig. 4(a)], the profile of the DS is displayed in Fig. 5(a), where the slope of the power curve is positive (dP=dμ 40). The DS breaks up at Z ¼ 300, which is shown in Fig. 5(b). When 20:84 r μ r 15:71 with ε ¼ 0:2, 20:44 r μ r 15:76 with ε ¼ 0:5, and 17:46 r μ r 15:81 with ε ¼ 0:8 in Fig. 4(a), where the power of DSs is moderate and dP=dμ o 0, DSs are stable because of ReðδÞ ¼ 0 [see Fig. 4(b)].
Fig. 4. (Color online) (a) Power versus propagation constant (blue regions are Bloch band) when ε ¼ 0:2 (solid curve), 0:5 (dashed curve), and 0:8 (dash–dot curve). (b) Perturbation growth rates ReðδÞ for DSs when ε ¼ 0:2, 0:5, and 0:8. In (a), the solid, dashed, and dash–dot curves indicate the stable DSs, and the dot curves indicate the unstable DSs, see (b). Profiles of DSs at the circled points in (a) and (b) are displayed in Fig. 5.
Fig. 5. (a) Profile of DS at the circled point A in Fig. 4(a) with ðε; μÞ ¼ ð 0:2; 15:69Þ and (b) its profile at Z ¼ 300. (c)–(f) Profiles of DSs at the circled point B–E in Fig. 4(a), with ðε; μÞ ¼ ð 0:5; 16:43Þ, ð 0:5; 13:73Þ, ð 0:8; 16:81Þ, and ð 0:8; 13:09Þ respectively. The DS in (a) is unstable, while the DSs in (c)–(f) are stable.
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For example, for μ ¼ 16:43 with ε ¼ 0:5 [point B in Fig. 4(a)] and μ ¼ 16:81 with ε ¼ 0:8 [point D in Fig. 4(a)], Fig. 5(c) and (e) show the profiles of DSs, respectively. When μ o 20:84 with ε ¼ 0:2, μ o 20:44 with 0:5, and μ o 17:46 with 0:8 in Fig. 4(a), the power of DSs is high and dP=dμ o 0. In high power regions, DSs cannot propagate stably because of ReðδÞ 40 [see Fig. 4(b)]. In the first gap, DSs with ε ¼ 0:2 and 0:5 can stably exist in the all power regions [see Fig. 4(b)]. For ε ¼ 0:8, DSs are stable in the region of 15:42 o μ o 12:56 [see Fig. 4(b)] but unstable in the region of 12:56 o μ o 11:44 [see Fig. 4(b)]. Taking μ ¼ 13:73 with ε ¼ 0:5 [point C in Fig. 4(a)] and μ ¼ 13:09 with ε ¼ 0:8 [point E in Fig. 4(a)], the profiles of the stable DSs are shown in Fig. 5(d) and (f). On the other hand, Fig. 4(b) demonstrates that the stable region of DSs decreases with an increase in the negative defect depth. It is very interesting to apply Vakhitov–Kolokolov (VK) stability criterion to DSs in optically induced kagome photonic lattices with a defect in PP crystals. The usual VK criterion is that the solitons are stable if dP=dμ o 0 and unstable if dP=dμ 4 0. In previous studies by Pelinovsky et al. [25], the authors found that the usual VK instability leads to either soliton decay or amplitude oscillations, depending on the sign of perturbations. For our physical system, when the slope of the power curve is negative (dP=dμ o 0), the VK instability leads to the DS of change shape with no relation to the sign of perturbations [see Fig. 3(a) and (b)]. When the slope of the power curve is positive (dP=dμ 40), the VK instability leads to the breakup DS regardless of the sign of perturbations [see Figs. 5(a) and (b)]. In the present case, the VK instability is different from the usual VK instability based on the sign of dP=dμ. On the other hand, it has been reported that when the slope of the power curve is positive, the VK instability leads to the stable intra-band solitons in the first band in lattice potentials with the single defect and self-focusing nonlinearity [24]. 4. Conclusion In conclusion, we have investigated DSs and their stability in optically induced kagome photonic lattices with a defect in PP crystals. In our numerical studies, we have shown that these DSs exist only in the semi-infinite gap for a positive defect and both in the semi-infinite gap and the first gap for a negative defect. In the semi-infinite gap, DSs are stable in the low power region for a positive defect and in the moderate power region for a negative defect, but unstable in the high power region for a positive defect and in the high and low power regions for a negative defect. In the
first gap, DSs are stable in the all power regions for the low negative defect depth. For the high negative defect depth, the high-power DSs are stable, whereas low-power DSs are unstable. We have demonstrated that the stable region of DSs increases with the positive defect depth and decreases with an increase in the negative defect depth.
Acknowledgment The work was supported by the Natural Science Foundation of Chinese Tianjin (No. 13JCYBJC16400). References [1] H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, J.S. Aitchison, Physical Review Letters 81 (1998) 3383. [2] J. Yang, Z.H. Musslimani, Optics Letters 28 (2003) 2094. [3] N.K. Efremidis, J. Hudock, D.N. Christodoulides, J.W. Fleischer, O. Cohen, M. Segev, Physical Review Letters 91 (2003) 213906. [4] D.E. Pelinovsky, A.A. Sukhorukov, Y.S. Kivshar, Physical Review E 70 (2004) 036618. [5] R. Iwanow, R. Schiek, G.I. Stegeman, T. Pertsch, F. Lederer, Y. Min, W. Sohler, Physical Review Letters 93 (2004) 113902. [6] D. Neshev, E. Ostrovskaya, Yu.S. Kivshar, W. Krolikowski, Optics Letters 28 (2003) 710. [7] J. Yang, I. Makasyuk, A. Bezryadina, Z. Chen, Studies in Applied Mathematics 113 (2004) 389. [8] B.B. Baizakov, B.A. Malomed, M. Salerno, Europhysics Letters 63 (2003) 642. [9] D.N. Neshev, T.J. Alexander, E.A. Ostrovskaya, Y.S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, Physical Review Letters 92 (2004) 123903. [10] J.W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D.N. Christodoulides, Physical Review Letters 92 (2004) 123904. [11] T.J. Alexander, A.A. Sukhorukov, Yu.S. Kivshar, Physical Review Letters 93 (2004) 063901. [12] F. Fedele, J. Yang, Z. Chen, Optics Letters 30 (2005) 1506. [13] X. Wang, J. Young, Z. Chen, D. Weinstein, J. Yang, Optics Express 14 (2006) 7362. [14] J. Wang, J. Yang, Z. Chen, Physical Review A 76 (2007) 013828. [15] Kehao Li, Keqing Lu, Jianbang Guo, Weijun Chen, Tongtong Sun, Fengxue Yao, Jingjun Xu, Optics Communications 285 (2012) 3187. [16] Keqing Lu, Jianbang Guo, Kehao Li, Weijun Chen, Tongtong Sun, Fengxue Yao, Pingjuan Niu, Jingjun Xu, Optical Materials 34 (2012) 1277. [17] J. Yang, Z. Chen, Physical Review E 73 (2006) 026609. [18] W.H. Chen, X. Zhu, T.W. Wu, R.H. Li, Optics Express 18 (2010) 10956. [19] Xing Zhu, Hong Wang, Li-Xian Zheng, Optics Express 18 (2010) 20786. [20] K.J.H. Law, A. Saxena, P.G. Kevrekidis, A.R. Bishop, Physical Review A 79 (2009) 053818. [21] B.I. Sturman, V.M. Fridkin, The Photovoltaic and Photorefractive Effect in Noncentrosymmetric Materias, Gordon and Breach, Philadelphia, 1992. [22] J. Yang, T.I. Lakoba, Studies in Applied Mathematics 118 (2007) 153. [23] J. Yang, Journal Computational Physics 227 (2008) 6862. [24] JianhuaZeng, Boris A. Malomed, Journal of the Optical Society of America B 30 (2013) 1786. [25] D.E. Pelinovsky, V.V. Afanasjev., Yu.S. Kivshar, Physical Review E 53 (1996) 1940.