## Abstract

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

## 1. Introduction

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 (LiNbO_{3}) 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.

## 2. Model system

In our investigated system two semi-infinite 1D photonic lattices, with identical channel width *w* but different separations *d _{L}* and

*d*, are coupled by a single defect of width

_{R}*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 (${C}_{L\to R}={C}_{R\to L}$).

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]:

*n*is the channel index of a mode with amplitude

*E*,

_{n}*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:

*C*=

_{n}*C*for

_{L}*n*< 0,

*C*=

_{n}*C*for

_{R}*n*> 1, and

*C*

_{0,1}=

*C*

_{1,0}=

*C*at the defect.

## 3. Sample fabrication and experimental methods

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 LiNbO_{3} crystals. This material possesses a defocusing nonlinearity due to the photorefractive effect at moderate light intensities. Sample dimensions are 1 × 20 × 7.8 mm^{3} 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 Λ* _{L}* = 9 μm) and 3 μm (right array, grating period Λ

*= 8 μm), respectively, separated by a defect. The defect width*

_{R}*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:YVO_{4} 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.

## 4. Linear propagation

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.

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.

The stationary solutions of Eq. (1) can be written in the form ${E}_{n}(z)={E}_{n}\mathrm{exp}(i\beta z)$, with *β* being the nonlinear propagation constant. In the linear low-power regime (*α* = 0), we find localized solutions of the form ${E}_{n}={E}_{L}{\xi}_{L}^{\left|n\right|}$ for $n\le 0$, and ${E}_{n}={E}_{R}{\xi}_{R}^{\left|n-1\right|}$ for$n\ge 1$, where $\left|{\xi}_{L}\right|,\left|{\xi}_{R}\right|\le 1$. Inserting these solutions into the set of linearly coupled stationary equations gives the following relations: $\beta ={C}_{L}({\xi}_{L}+1/{\xi}_{L})={C}_{R}({\xi}_{R}+1/{\xi}_{R})$, ${C}^{2}={C}_{L}{C}_{R}/({\xi}_{L}{\xi}_{R})$, ${E}_{L}/{E}_{R}=C{\xi}_{L}/{C}_{L}$, ${\xi}_{L}=\pm {C}_{L}/C\cdot \sqrt{({C}_{R}^{2}-{C}^{2})/({C}_{L}^{2}-{C}^{2})}$, and ${\xi}_{R}=\pm {C}_{R}/C\cdot \sqrt{({C}_{L}^{2}-{C}^{2})/({C}_{R}^{2}-{C}^{2})}$. 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 ${\xi}_{L},{\xi}_{R}<0$ is satisfied, while unstaggered solitons require ${\xi}_{L},{\xi}_{R}>0$. Examples of both solutions are given in Fig. 4
.

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: ${C}_{R}/C={(2-{({C}_{L}/C)}^{-2})}^{1/2}$ (lower one) and ${C}_{R}/C={(2-{({C}_{L}/C)}^{2})}^{-1/2}$ (upper one), obtained from the conditions ${\xi}_{L}{}^{2}=1$ and ${\xi}_{R}{}^{2}=1$, 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.

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.

## 5. Nonlinear propagation

The existence curves of nonlinear localized states related to LiNbO_{3} (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.

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 Λ* _{R}* = 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].

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 LiNbO_{3} [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, Λ* _{L}* = 9 μm) and

*n*= 8 (right array, Λ

*= 8 μm). Around the border between channels*

_{L}*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 Δ*n* follows a well-known exponential law, Δ*n*(*t*) = Δ*n _{sat}* (1−exp(−

*t*/

*τ*)), where

*τ*is the Maxwell time constant and Δ

*n*is the nonlinear index change in saturation, i.e. for

_{sat}*t*→ ∞.

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 Δ*n* is only slightly smaller compared to Δ*n _{sat}*), 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.

## 6. Summary

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.

## Acknowledgments

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).

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