We investigate linear and nonlinear light propagation at the interface of two one-dimensional homogeneous waveguide arrays containing a single defect of different strength. For the linear case and in a limited region of the defect size, we find trapped staggered and unstaggered modes. In the nonlinear case, we study the dependence of power thresholds for discrete soliton formation in different channels as a function of defect strength. All experimental results are confirmed theoretically using an adequate discrete model.
©2011 Optical Society of America
It is well known that the translational symmetry of periodic photonic lattices allows for the existence of band-gap structures for light, similar to the propagation of electron waves in crystalline solids [1–3]. In such optical media, light propagation is governed by the dispersion relation. Intensive research has been triggered because these systems exhibit numerous phenomena that do not exist in continuous media, such as linear discrete diffraction [4], dispersion management [5], or Rabi-like band-band transitions [6]. In the nonlinear regime, the generation of different types of discrete solitons [7–11], as well as the experimental investigation of nonlinear interactions [12–14], has put this direction at the forefront of optical soliton research.
Within the scope of applications, future optical devices may use different types and/or combinations of photonic lattices, including interfaces of two different media. It is well known that breaking the translational symmetry leads to the occurrence of various types of localized surface structures at such interfaces. Beside soliton formation at the boundary of a homogeneous array [15–18], energy localization can be achieved by introducing a single defect inside the lattice, either by locally changing refractive index, or by altering local geometrical properties of the array [19]. On the one hand, for proper conditions such defect states are known to allow for diffraction-less light propagation in the linear power regime [20,21]. On the other hand, by choosing adequate parameters of photonic lattices in combination with suitable defects, it is possible to achieve lower threshold powers required for nonlinear soliton formation [22].
Recent theoretical and experimental publications have shown the existence of self-localized structures that occur at the interface between two equivalent [23] as well as between two dissimilar one-dimensional (1D) arrays [24,25]. In the latter case, the two arrays are connected by a defect channel that has a different width when compared to all of its neighbors. This leads to a mismatch in the related propagation constants, and thus the linear discrete diffraction (coupling coefficients) differs between the left and right semi-infinite arrays. The influence of the defect on light localization has been theoretically investigated for different types of lattices, such as quadratically nonlinear photonic lattices [26], 1D photonic waveguide arrays consisting of two discrete meta-materials [27], and 2D systems made of hexagonal and square lattices [28]. Experimentally, the existence of localized states has been analyzed in [29] for a 1D lattice fabricated in AlGaAs possessing a self-focusing Kerr-type nonlinearity.
In this work, the existence of linear and nonlinear optical modes at and close to the interface between two 1D dissimilar photonic lattices, exhibiting a self-defocusing saturable nonlinearity, is investigated both theoretically and experimentally. It is shown that, due to the presence of the defect and in a limited range of defect strength, both staggered and unstaggered linear modes can propagate, extending into the two arrays. In the nonlinear regime, we find an increase of the power threshold to form discrete solitons in the neighborhood of the defect. To investigate this behavior experimentally, 1D waveguide arrays are fabricated using lithium niobate (LiNbO3) as the nonlinear medium. First, we prove the theoretical predictions on existence and phase profile of localized linear modes. Furthermore, we systematically determine the power thresholds to form discrete solitons at and close to the interface. Depending on the width of the created defect channel, we observe a rather large increase of necessary power for light trapping, which is in good agreement with our theoretical predictions.
In our investigated system two semi-infinite 1D photonic lattices, with identical channel width w but different separations dL and dR, are coupled by a single defect of width d, which is varied in a certain range. A geometrical representation is given in Fig. 1 . According to the coupled-mode theory, the interaction of nearest neighbors can be described by a coupling constant C. This parameter depends exponentially on the separation between channels and is a direct result of the individual fields’ overlap. In the case of two identical semi-infinite photonic lattices separated by a defect, due to the symmetry of the system, the field overlap does not depend on which boundary channel, left or right, is excited. Hence the inter-lattice coupling constants between two boundary channels are the same (CL→R=CR→L).
Fig. 1 Geometrical representation of our waveguide array fabricated in LiNbO3. Here n is the waveguide number, and the defect is located between waveguides 0 and 1. Array parameters are w = 5 μm, dL = 4 μm, dR = 3 μm, and 2 μm < d < 4.5 μm.
Download Full Size | PDF
Generally, for two dissimilar photonic lattices breaking mirror symmetry, the inter-lattice coupling constants differ, since the field overlap depends on the parameters of the system, such as channel and gap widths, refractive indices, nonlinearity of the material, etc. However, in the investigated model system (Fig. 1) differences between inter-lattice coupling constants are sufficiently small and thus can be neglected. This approximation is valid due to identical individual channel widths and relatively small differences in gap widths between the two lattices.
Presuming a 1D photonic lattice with saturable type of nonlinearity and light propagation along z direction, the field evolution of our system can be described by the discrete nonlinear Schrödinger equation (DNLS) [30]:
(1)idEndz+Cn,n−1En−1+Cn,n+1En+1+α|En|21+κ|En|2En=0 ,n is the channel index of a mode with amplitude En, α = −1 denotes the defocusing type of the nonlinearity, and κ describes saturation strength. Considering that the defect is placed between channels n = 0 and n = 1, the following notations are used: Cn = CL for n < 0, Cn = CR for n > 1, and C 0,1 = C 1,0 = C at the defect.To study light propagation at the interface of two periodic systems including a single defect, we fabricated a set of 1D waveguide arrays using non-doped LiNbO3 crystals. This material possesses a defocusing nonlinearity due to the photorefractive effect at moderate light intensities. Sample dimensions are 1 × 20 × 7.8 mm3 with the ferroelectric c-axis pointing along the 7.8 mm-long direction. Arrays of parallel-aligned channel waveguides, each being 5 μm wide, are fabricated by patterning a 10 nm-thick Ti layer formed by sputtering on the sample surface, using standard photolithographic techniques. In-diffusion of the Ti stripes takes place for 2 hours at a temperature of 1000 °C in wet Ar atmosphere. Finally, input and output facets are polished to optical quality to allow for direct coupling of light into the 20 mm-long channels. In order to ensure equal (nonlinear) waveguide properties, on a single substrate 11 different waveguide arrays are formed at the same time. Each array consists of two homogeneous parts with separations (gaps) of 4 μm (left array, grating period ΛL = 9 μm) and 3 μm (right array, grating period ΛR = 8 μm), respectively, separated by a defect. The defect width d is varied between 2 µm and 4.5µm in steps of 0.25 µm.
To investigate light propagation in the fabricated samples, we use a standard endfacet-coupling setup [31] and a frequency-doubled Nd:YVO4 laser with wavelength 532 nm as the light source. By using a chrome-glass mask with sets of adjacent pinholes, either single or multiple neighbored channels of the waveguide array can be excited. A small optional tilt angle of the input light distribution allows for either unstaggered or staggered input conditions. The output intensity distribution is imaged onto a CCD camera with a microscope objective. With the help of a Mach-Zehnder interferometer, which interferes the output amplitude with a plane reference wave, we are able to monitor the phase distribution of the out-coupled light.
As a first step, we investigate the coupling constants relevant to our periodic system. In section 2 we have assumed identical coefficients C 0,1 = C 1,0 = C for light coupling from channel 0 to channel 1 and vice versa. This assumption is confirmed in the example given in Fig. 2 when comparing experimentally obtained linear discrete diffraction patterns in the fabricated lattices with corresponding numerical results.
Fig. 2 Linear discrete diffraction when (a) channel 0 and (b) channel 1 is excited. Numerical results are shown on the left panels, while experimental output distributions are depicted on the right panels. Defect width is d = 2.75 μm, CL = 66.5 m−1, CR = 160.5 m−1, and CL→R=CR→L=202.6m−1.
Download Full Size | PDF
The dependence of the coupling constant C on the defect width d is calculated from Eq. (1) for the linear case (α = 0), using the experimentally obtained diffraction patterns, and the results are depicted in Fig. 3 . As can be seen, the results are well described by an exponential function of the form C ~exp(−d/d 0) with d 0 being a constant.
Fig. 3 Calculated and experimentally determined coupling coefficient C of the defect as a function of defect width d.
Download Full Size | PDF
The stationary solutions of Eq. (1) can be written in the form En(z)=Enexp(iβz), with β being the nonlinear propagation constant. In the linear low-power regime (α = 0), we find localized solutions of the form En=ELξL|n| for n≤0, and En=ERξR|n−1| forn≥1, where |ξL|,|ξR|≤1. Inserting these solutions into the set of linearly coupled stationary equations gives the following relations: β=CL(ξL+1/ξL)=CR(ξR+1/ξR), C2=CLCR/(ξLξR), EL/ER=CξL/CL, ξL=±CL/C⋅(CR2−C2)/(CL2−C2), and ξR=±CR/C⋅(CL2−C2)/(CR2−C2). When analyzing the previous expressions, we find that in the presented model staggered as well as unstaggered linear modes can exist. The first case may occur when the condition ξL,ξR<0 is satisfied, while unstaggered solitons require ξL,ξR>0. Examples of both solutions are given in Fig. 4 .
Fig. 4 (a) Unstaggered and (b) staggered linear stationary solutions of DNLS for d = 2.5 μm.
Download Full Size | PDF
The existence of linear localized modes can be sketched by the phase diagram in coupling space as shown in Fig. 5 . The dark turquoise region represents the area in which both, unstaggered and staggered modes can exist and its boundary is determined by two curves: CR/C=(2−(CL/C)−2)1/2 (lower one) and CR/C=(2−(CL/C)2)−1/2 (upper one), obtained from the conditions ξL2=1 and ξR2=1, respectively. Out of this region no linear trapped states can be found. Dots represent waveguide arrays with various defect widths d in the range from 2 μm to 3 μm. As can be seen, linear localized states exist only for d ≤ 2.5 μm.
Fig. 5 Phase diagram in coupling space of linear localized modes for various defect widths.
Download Full Size | PDF
Experimentally, linear trapped states can be observed for defect widths up to 2.5 µm, i.e. in the region where a local increase of average refractive index arises. In Fig. 6 we present the output intensity distribution of linear modes (b, f, h) and the corresponding input intensity distributions (a, e, g). For smaller d a higher average refractive index occurs, resulting in a stronger localization of the corresponding modes. By slightly tilting the incident angle we find both, staggered and unstaggered light distributions, for each guided mode. As an example, Fig. 6c and 6d monitor the out-of-phase and in-phase output patterns obtained by interference for the case d = 2 μm given in Fig. 6b.
Fig. 6 Excitation of linear trapped modes at the interface. (a), (b) input and output intensity distribution (unstaggered modes) for d = 2µm; (e), (f) d = 2.25µm; (g), (h) for d = 2.5 µm. The dashed vertical lines depict the defect location. For the case d = 2 μm panels (c, d) show the interferograms of the output distribution for staggered (c) and unstaggered (d, corresponding to the output profile monitored in panel b) modes, respectively.
Download Full Size | PDF
The existence curves of nonlinear localized states related to LiNbO3 (defocusing nonlinearity, α = −1) are shown in Fig. 7 . This figure depicts the power (dimensionless units) of each nonlinear localized state versus the nonlinear propagation constant β for two different defect widths. For each width, only staggered solutions are analyzed, while the existence of unstaggered solutions has been confirmed neither experimentally nor numerically. A linear stability analysis of the obtained solutions was performed, and the results show that solutions are stable when dP/dβ < 0, and vice versa.
Fig. 7 Power versus nonlinear propagation constant for (a) d = 2.5 μm and (b) d = 4.25 μm. The insets in the left panel show the magnitude of the field amplitude of staggered surface solitons.
Download Full Size | PDF
Existence curves for defect widths for which also linear localized states exist are depicted in Fig. 7a. Obviously, surface modes located at the waveguide next to the defect (n = 0) have higher power threshold compared to the homogeneous arrays, which represent waveguides placed far from the defect. Also, lower power is necessary to achieve light localization in the homogeneous array with 4 μm gaps than in an array with 3 μm gaps. In these regions no localized linear modes can occur. The waveguide n = 1 exhibits a different behavior, since there is no power threshold for achieving surface modes, i.e. there is continuous transition from linear to nonlinear localized states. The three insets show the formation of highly localized states by increasing the input power. The bottom one represents the linear mode, the middle one monitors a solution where 60% of total soliton power is localized in the central waveguide n = 1, while the top inset shows a solution with 95% of total power concentrated in the excited waveguide. On the other hand, Fig. 7b depicts power versus nonlinear propagation constant for a defect too wide to achieve localization in the linear regime. Now, a rather high power threshold appears for (nonlinear) modes localized at n = 1. It is also evident that the increase of the defect width reduces the repulsive potential for channel n = 0 (and all other neighboured ones).
The power threshold as a function of waveguide number n in which the localized state is excited is shown in Fig. 8 . For defect widths d < 3 μm (Fig. 8a, where d is smaller than the gaps of the array with ΛR = 8 μm) strong coupling between lattices causes high thresholds for nonlinear localized states for all waveguides closer to the defect, reaching maximum values for boundary channels (n = 0 and n = 1, solid lines). Here, for waveguide n = 1, we adopted the criterion of 90% of total soliton power in the central waveguide element for nonlinear localization. This was done because of the existence of linear modes in this channel, which means that its real power threshold equals zero (dashed lines). These linear localized solutions can be obtained for structures with defect widths less or equal to 2.5 μm. Also, the increase of the defect width causes a decrease of the threshold for light localization in every channel. While the highest threshold for d = 3 μm is obtained for n = 0, in the range of 3 μm < d < 4 μm this maximum transits to waveguide n = 1 (Fig. 8b). Further increase of the defect width does not affect the power threshold, since the lattices act as two nearly-independent semi-infinite arrays (Fig. 8c). The shape of the power threshold curves for d > 4 μm is dictated by the interplay between a repulsive edge potential and Bragg reflection inside the arrays causing Tamm-like oscillations [32].
Fig. 8 Numerically (left panels: a, b, c) and experimentally (right panels: d, e, f) obtained power thresholds for narrow soliton formation: (a, d) 2 μm ≤ d ≤ 2.75 μm; (b, e) 3 μm ≤ d ≤ 3.75 μm; and (c, f) 4 μm ≤ d ≤ 4.50 μm. The vertical solid lines mark the position of the defect, while the two horizontal dashed lines indicate the power thresholds in the two homogeneous arrays.
Download Full Size | PDF
Experimentally, for the nonlinear case we study the temporal evolution of the output intensity as a function of the power coupled into a single channel n. Coupling light into only one element provides both, simple and stable input conditions, and has proven to be an effective method to excite discrete (surface) solitons in waveguide arrays fabricated in LiNbO3 [16,17]. The power threshold for the formation of a (narrow) discrete soliton is then defined by two criteria: (i) the output light is strongly concentrated on the excited channel n, and (ii) this situation is achieved by a rather small increase of necessary input power. In fact, using these criteria, at and above threshold more than 80% of the total output power is located in the excited element.
Results for the measured power thresholds are given in the right panels of Fig. 8. The necessary power to form narrow discrete solitons in the homogeneous parts of the sample are exemplarily given by exciting either channel n = − 6 (left array, ΛL = 9 μm) and n = 8 (right array, ΛL = 8 μm). Around the border between channels n = 0 and n = 1, the repulsive character of the defect causes increased power thresholds for all investigated defect widths, as predicted by our numerical modeling. With decreasing width d, the measured power thresholds increase. For small defect sizes d < 2.75 μm, we are not able to trap light in the excited channel anymore, which is due to limited input power available with our setup (low transmission of the pinhole mask).
The repulsive potential of the defect can also be observed by continuously monitoring the temporal evolution during soliton formation. An example using the sample with d = 2.75 μm is given in Fig. 9 , showing the two cases of excitation of channel n = 0 (Fig. 9a) and n = 1 (Fig. 9b). After the input light is switched on, the growth of the (defocusing) nonlinearity Δn follows a well-known exponential law, Δn(t) = Δnsat (1−exp(−t/τ)), where τ is the Maxwell time constant and Δnsat is the nonlinear index change in saturation, i.e. for t → ∞.
Fig. 9 Temporal dynamics of soliton formation in the sample with d = 2.75 µm when channel n = 0 (a) or channel n = 1 (b) is excited. The white arrows points to the excited channel while the dashed line marks the position of the defect. The interferograms given in the right panels show the staggered phase of the output light distribution in steady-state (t = 80 min).
Download Full Size | PDF
For excitation of channel n = 0, the output light distribution for smaller nonlinearities is dominated by the strongly asymmetric discrete diffraction towards channels with positive index. In the intermediate regime for t ~(10 – 16) min, the temporal trapping of the light mostly in channel n = 1 may be related to a breather-like solution that may exist in this parameter range. When the nonlinearity reaches its saturation value the output power is concentrated on channel n = 0 and a narrow staggered soliton is formed. This solution is stable for an observation time of at least 80 min. In the second case (n = 1), for small nonlinearity broader light distributions similar to the linear mode localized on that channel are obtained. For intermediate time t ~13 min (where nonlinearity Δn is only slightly smaller compared to Δnsat), a narrow soliton on channel n = 0 is formed, which might be due to the lower power threshold for this case. Finally, in saturation light is trapped on channel n = 1 and a stable staggered soliton occurs.
In summary, we have investigated linear and nonlinear optical wave propagation at the interface between two different one-dimensional photonic lattices connected to each other by a defect (gap) of different width (i.e., different coupling constant). Such interfaces may play an important role in tailoring the flow of light in future devices using combinations of different photonic lattices. Theoretical modeling of our system predicts the existence of both, staggered and unstaggered linear trapped states provided that the defect width is below some threshold value. Both types of localized solutions are found experimentally using a waveguide sample fabricated in lithium niobate. For higher input powers the defocusing (saturable) nonlinearity allows for the formation of staggered interface lattice solitons bounded to the defect. We experimentally find a power threshold for narrow interface solitons which increases with decreasing gap width. Again, this behavior, which is a direct consequence of the repulsive potential induced by the defect, is in full agreement with the corresponding numerical results.
We gratefully acknowledge support from the Deutsche Forschungsgemeinschaft (grant (KI482/11-2) and from the German-Serbian Academic Exchange Programme (DAAD grant 500243908 and 451-03-00245/2009-01-8). Authors also acknowledge support provided by the Ministry of Science of Republic of Serbia (P141034).
1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003). [CrossRef] [PubMed]
2. P. St, J. Russell, T. A. Birks, and F. D. Lloyd Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein and C. Weisbuch, Eds., (Plenum Press, 1995), pp. 585–633.
3. Yu. S. Kivshar, and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).
4. R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86(15), 3296–3299 (2001). [CrossRef] [PubMed]
5. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000). [CrossRef] [PubMed]
6. K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102(12), 123905 (2009). [CrossRef] [PubMed]
7. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998). [CrossRef]
8. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003). [CrossRef] [PubMed]
9. J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005). [CrossRef] [PubMed]
10. F. Chen, M. Stepić, C. E. Rüter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, “Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays,” Opt. Express 13(11), 4314–4324 (2005). [CrossRef] [PubMed]
11. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. G. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004). [CrossRef] [PubMed]
12. M. Stepić, C. Wirth, C. E. Rüter, and D. Kip, “Observation of modulational instability in discrete media with self-defocusing nonlinearity,” Opt. Lett. 31(2), 247–249 (2006). [CrossRef] [PubMed]
13. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003). [CrossRef] [PubMed]
14. R. Dong, C. E. Rüter, D. Kip, O. Manela, M. Segev, C. Yang, and J. Xu, “Spatial frequency combs and supercontinuum generation in one-dimensional photonic lattices,” Phys. Rev. Lett. 101(18), 183903 (2008). [CrossRef] [PubMed]
15. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Haché, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, “Observation of discrete surface solitons,” Phys. Rev. Lett. 96(6), 063901 (2006). [CrossRef] [PubMed]
16. E. Smirnov, M. Stepić, C. E. Rüter, D. Kip, and V. Shandarov, “Observation of staggered surface solitary waves in one-dimensional waveguide arrays,” Opt. Lett. 31(15), 2338–2340 (2006). [CrossRef] [PubMed]
17. C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Observation of surface gap solitons in semi-infinite waveguide arrays,” Phys. Rev. Lett. 97(8), 083901 (2006). [CrossRef] [PubMed]
18. N. Malkova, I. Hromada, X. S. Wang, G. Bryant, and Z. G. Chen, “Observation of optical Shockley-like surface states in photonic superlattices,” Opt. Lett. 34(11), 1633–1635 (2009). [CrossRef] [PubMed]
19. H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, and A. Bräuer, “Tailoring guided modes in waveguide arrays,” Opt. Express 11(25), 3404–3411 (2003). [CrossRef] [PubMed]
20. F. Fedele, J. K. Yang, and Z. G. Chen, “Defect modes in one-dimensional photonic lattices,” Opt. Lett. 30(12), 1506–1508 (2005). [CrossRef] [PubMed]
21. A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Observation of defect-free surface modes in optical waveguide arrays,” Phys. Rev. Lett. 101(20), 203902 (2008). [CrossRef] [PubMed]
22. K. G. Makris, J. Hudock, D. N. Christodoulides, G. I. Stegeman, O. Manela, and M. Segev, “Surface lattice solitons,” Opt. Lett. 31(18), 2774–2776 (2006). [CrossRef] [PubMed]
23. M. I. Molina and Y. S. Kivshar, “Nonlinear localized modes at phase-slip defects in waveguide arrays,” Opt. Lett. 33(9), 917–919 (2008). [CrossRef] [PubMed]
24. M. I. Molina and Yu. S. Kivshar, “Interface localized modes and hybrid lattice solitons in waveguide arrays,” Phys. Lett. A 362(4), 280–282 (2007). [CrossRef]
25. Y. Hu, R. Egger, P. Zhang, X. S. Wang, and Z. G. Chen, “Interface solitons excited between a simple lattice and a superlattice,” Opt. Express 18(14), 14679–14684 (2010). [CrossRef] [PubMed]
26. Z. Y. Xu, M. I. Molina, and Yu. S. Kivshar, “Interface solitons in quadratic nonlinear photonic lattices,” Phys. Rev. A 80(1), 013817 (2009). [CrossRef]
27. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, R. Morandotti, M. Volatier, V. Aimez, R. Arès, C. E. Rüter, and D. Kip, “Optical modes at the interface between two dissimilar discrete meta-materials,” Opt. Express 15(8), 4663–4670 (2007). [CrossRef] [PubMed]
28. A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, “Observation of two-dimensional surface solitons in asymmetric waveguide arrays,” Phys. Rev. Lett. 98(17), 173903 (2007). [CrossRef]
29. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, R. Morandotti, M. Volatier, V. Aimez, R. Arès, E. H. Yang, and G. Salamo, “Optical spatial solitons at the interface between two dissimilar periodic media: theory and experiment,” Opt. Express 16(14), 10480–10492 (2008). [CrossRef] [PubMed]
30. M. Stepić, D. Kip, Lj. Hadžievski, and A. Maluckov, “One-dimensional bright discrete solitons in media with saturable nonlinearity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6 Pt 2), 066618 (2004). [CrossRef] [PubMed]
31. E. Smirnov, C. E. Rüter, M. Stepić, V. Shandarov, and D. Kip, “Dark and bright blocker soliton interaction in defocusing waveguide arrays,” Opt. Express 14(23), 11248–11255 (2006). [CrossRef] [PubMed]
32. M. Stepi, E. Smirnov, C. E. Rüter, D. Kip, A. Maluckov, and L. O. Hadžievski, “Tamm oscillations in semi-infinite nonlinear waveguide arrays,” Opt. Lett. 32(7), 823–825 (2007). [CrossRef] [PubMed]
If you have any questions on Linear Downlight. We will give the professional answers to your questions.