1. Introduction

The field of surface science is one of the richest in physics simply due to that surface wave phenomena are ubiquitous in nature, from classical electronic Tamm and Shockley surface states to acoustic and plasmonic surface waves. In the field of optics, there has been a great deal of interest recently in generation of linear surface modes and nonlinear low-threshold surface solitons in discrete optical systems [1–3]. One example is nonlinear Tamm-like optical surface states at the interface with small controlled index contrast between homogenous media and semi-infinite waveguide arrays, as have been predicted and demonstrated in a number of experiments [4–13]. With a self-focusing or -defocusing nonlinearity, both one- and twodimensional nonlinear surface bound states as discrete in-phase or “staggered” out-of-phase surface solitons were observed at the edge of a semi-infinite photonic lattice with a single periodicity. Linear and nonlinear surface states excited at the interface between two different semi-infinite photonic lattices have also been studied [14–19].

When a waveguide array or photonic lattice is made with two or more periodicities, it is called a superlattice. Binary superlattices are usually made from alternating layers of two different semiconductor materials with advanced fabrication techniques [20], but recent experimental realizations of photonic superlattices with alternating strong and weak couplings are made possible with the optical induction technique [21,22] as well as the femtosecond laser writing technology [23–25]. While superlattices are widely used for tuning the photonic bandgap structures, more exotic wave phenomena can exist in photonic superlattices such as recently observed transitions between optical Shockley-like and Tamm-like surface states [21] and Bloch-Zener oscillations [23], in addition to nonlinear superlattice solitons [24–26].

In this paper, we study numerically and demonstrate experimentally nonlinear surface states at the interface between an optically induced simple (periodic) and super (bi-periodic) lattice. Depending on the excitation condition at the interface, we indentify a few typical interface solitons with different phase structures under a self-focusing nonlinearity. The first type is in-phase interface solitons since their propagation constants are located in the semiinfinite gaps of both lattices and the surface modes have uniform phase among lattice sites. The second type is staggered interface gap solitons since their propagation constants are located in the first photonic gap of the superlattice, and the surface modes have staggered (out-of-phase) phase structures between adjacent lattice sites. These interface gap solitons have similar phase structures, but the main soliton power can be located at different sides of the interface with a dipole-like double humps. In fact, they represent a novel type of interface solitons whose field profiles have uniform phase in the simple lattice but staggered phase in the superlattice, with characteristic mode properties from both 1st-band of the simple lattice and 2nd-band of the superlattice. Our numerical simulations show that, when the main soliton power is located at the superlattice side of the interface, these “mixed-band” interface solitons are stable. However, when the main soliton power is located at the simple-lattice side of the interface, they are not stable upon long distance propagation, although they remain invariant within short distances as observed in our experiments with a 10-mm-long nonlinear crystal.

2. Numerical solutions

Let us start with finding the numerical solutions of solitons at an interface between a simple lattice and a superlattice. The nonlinear evolution of a probe beam in a lattice potential optically induced in a photorefractive crystal can be expressed by the following scalar equation:

where φ is the slowly varying amplitude of the optical field of the probe beam (or solitonforming beam), and E ₀ is the normalized bias electric field across the crystal. In this paper, we only consider the case of self-focusing nonlinearity, so E ₀ is taken to be positive. The normalized intensity of the lattice-inducing beam [as illustrated in Fig. 1(a)] is expressed as V = I_gcos²[π(x-Λ/4)/(Λ/2)] when (x≥Λ/2) and V = I_g{cos²[π(x-Λ/4)/(Λ/2)] + cos²(πx/Λ)} when (x<Λ/2), where I_g is the normalized peak intensity of the lattice beam, and Λ is the period of the superlattice. The interface established by this way can be viewed as a superposition of an infinite lattice (period is Λ/2) and a semi-infinite lattice (period is Λ) [21,22]. Since the induced index change depends on the intensity of the lattice beam, the resulting average refractive index of the superlattice is higher than that of the simple lattice. Therefore, the first photonic band of the superlattice is located higher than that of the simple lattice according to the propagation constant β [see Fig. 1(b)].

 

Fig. 1. (a) Optically induced interface between a simple lattice (x>Λ/2) and a binary superlattice (x<Λ/2), where i, j, k mark the waveguides near the interface (dashed white line) to be excited in experiment. (b) Soliton existence curves obtained numerically, where the blue and yellow shaded areas illustrate the transmission bands of the simple lattice and of the superlattice, respectively. (c)–(e) depict three different interface soliton solutions corresponding to the marked points in (b).

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To find the solitons at the interface of the two different lattices, we look for solutions of Eq. (1) in the form of φ(x, z) = u(x)exp(iβz), where u is the field profile of the soliton solution and β is the propagation constant. Three families of soliton solutions are found at the interface, which are marked by the letters C, D, E in their existence curves plotted in Fig. 1(b). Typical solutions obtained at marked points C, D, E are shown in Figs. 1(c)–1(e). From these results along with in-depth numerical simulation and analysis, we can identify the difference between these interface solitons. The solitons along curve C have their propagation constants located in the semi-infinite gaps of both lattices, and thus the soliton profiles have uniform phase in both the simple (right) and super (left) lattice sides. These are in-phase interface solitons. The solitons along curves D and E, have their propagation constants located in the semi-infinite gap of the simple lattice but in the first photonic gap of the superlattice, thus the phase of soliton profiles is uniform at right but staggered at left side. These are staggered interface gap solitons. The difference between solitons in curve D and those in curve E is that solitons in curve D have their intensity located mostly in the simple-lattice side while those along curve E have their intensity located mostly in the superlattice side. In addition, the existence curve E does not “touch” the 1st band of the superlattice, and this family of solitons always has double-humps corresponding to the double-peak of the superlattice. These interface solitons have all been observed in our experiment.

3. Experimental and simulation results

Our experiment setup is similar to that used in Refs [21,22]. The superlattice structures are created by using the optical induction technique with two sets of ordinarily-polarized latticeinducing beams. Superimposition of the two periodic intensity patterns (one is an infinite-long lattice with a spatial period of 20 µm and the other is a semi-infinite-long lattice with 40 µm period) leads to an interface between binary superlattice and homogeneous simple lattice as shown in Fig. 1(a). By suppressing the Talbot effect, such a lattice interface remains stationary while propagating through a photorefractive SBN crystal. With a positive bias field, the intensity pattern induces a refractive index pattern so that two sets of waveguide lattices with an interface are established. To excite nonlinear surface states at the interface discussed above, an extraordinarily-polarized stripe beam (soliton-forming beam) is sent into different surface waveguides near the interface as illustrated in Fig. 1(a). The input/output intensity patterns and phase structures of the output surface solitons are monitored by a CCD camera.

 

Fig. 2. Experimental (a) and simulation (b) results of in-phase interface soliton. (1)–(3) show linear output, nonlinear output and the interferogram of nonlinear output after 1cm of propagation, respectively. (c) and (d) show numerical beam propagation results for 3cm of linear and nonlinear propagation of the probe beam, where the dashed line marks the position at z = 1cm. At input, the probe beam is launched between two lattice sites i and j near the interface.

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First, we excite the in-phase interface solitons by launching a quasi-1D stripe beam centered between the sites i and j shown in Fig. 1(a). In the linear regime, we observe that the probe beam diffracts asymmetrically and most of its intensity couples towards the superlattice side after 1cm of propagation through the crystal [Fig. 2(a1)]. This asymmetric discrete diffraction is due to different waveguide couplings at two sides of the interface, which is shown much more clearly in our numerical simulation of linear propagation up to 3cm [Fig. 2(c)]. When a proper self-focusing nonlinearity is applied, the probe beam self-traps into an interface soliton [Fig. 2(a2)] with the peak intensity returning to its input location (between sites i and j) and uniform phase structure as monitored by its interferogram with a titled planewave reference beam [Fig. 2(a3)]. This soliton has the characteristics of the soliton solution shown in Fig. 1(c). Numerical simulations to 1cm propagation distance [Fig. 2(b1)–2(b3)] using experiment parameters show good agreement with the experimental results. Difference between linear discrete diffraction and nonlinear self-trapping is more evident in the plots of beam evolution up to 3cm in Figs. 2(c), 2(d).

Next, we excite the staggered interface gap solitons in the simple-lattice side [solitons shown in Fig. 1(d)] by launching the quasi-1D stripe beam into the boundary waveguide of the simple lattice [site k, as shown in Fig. 1(a) and Fig. 3]. In this case, the probe beam again exhibits asymmetric discrete diffraction without nonlinearity, but its intensity flows more to the simple-lattice side [Fig. 3(a1)], which can be seen clearly from the simulation of 3cm linear propagation in Fig. 3(c). When the self-focusing nonlinearity is present, self-trapping of the probe beam is realized, with most of its intensity going back to the input waveguide channel [Fig. 3(a2)]. From the intensity pattern and interferogram [Fig. 3(a3)] of the soliton beam, we can see that the soliton has two uneven humps in two lattice sites next to the interface which are out of phase with each other, in consistence with the soliton solution obtained in Fig. 1(d). Simulation results at 1cm [Fig. 3(b1)–3(b3)] show qualitatively the same behavior, while longer propagation [Fig. 3(c), 3(d)] reveals again the distinct difference between linear and nonlinear propagation. We point out that this interface gap soliton is not stable after sufficient long distance propagation.

 

Fig. 3. Experimental (a) and simulation (b) results of staggered interface gap soliton with main power located in the simple-lattice side. At input, the probe beam is a single-hump Gaussian beam launched at surface waveguide k in simple-lattice side. Other description is the same as for Fig. 2.

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Finally, we excite the staggered interface gap solitons in the superlattice side [solitons shown in Fig. 1(e)] by launching a dipole-like probe beam into two boundary waveguides of the superlattice [sites i and j, as shown in Fig. 1(a) and Fig. 4]. Different from that in Fig.3(a1), the linear diffraction of the probe beam goes to the superlattice side [Figs. 4(a1), 4(b1, 4(c)]. Upon nonlinear evolution under appropriate bias condition, a dipole-like interface soliton is generated, with most of its intensity restored to the input sites. The intensity [Fig. 4(a2)] and phase [Fig. 4(a3)] of the soliton show two major peaks at sites i and j with a dipole-like π phase shift, in consistence with the soliton solution obtained in Fig. 1(e). Again, simulation results obtained at 1cm of crystal length [Fig. 4(b1)–4(b3)] show qualitatively the same behavior, while longer propagation [Fig. 4(c), 4(d)] reveals the distinct difference between linear and nonlinear propagation. However, comparing with the soliton in Fig. 3, we found by numerical simulations that these dipole-like interface gap solitons are more stable and robust even after 10-cm of propagation.

 

Fig. 4. Experimental (a) and simulation (b) results of staggered interface gap soliton with main power located in the superlattice side. At input, the probe beam is a double-hump dipole-like beam launched at sites i and j, the two surface waveguides in superlattice side. Other description is the same as for Fig. 2.

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4. Discussion and summary

In our lattice structure, we found that the interface solitons tend to be localized at the surface waveguide in the superlattice site, as shown in Figs. 1(c), 1(e), perhaps due to the higher average index change in the superlattice side. If the nonlinear interface states are localized more on the simple lattice side, as shown in Fig. 1(d), they cannot maintain their intensity profiles after sufficient long distance propagation beyond 3cm, although they can be observed in our 1cm nonlinear crystal. Another issue is about the constituting modes of the interface solitons in Figs. 1(d), 1(e). As we mentioned before, the profiles of these solitons have the features of the Bloch modes from both the 1st band of the simple lattice and the 2nd band of the superlattice. In fact, careful numerical analysis found that solitons belong to the family of curve D can bifurcate either from the 1st or the 2nd Bloch band of the superlattice. However, the solitons belong to the family of curve E always have dipole-like double humps with respect to the binary lattice sites since their existence curve does not touch the superlattice 1st band. Since these soliton solutions have a dipole-like double-hump profile in superlattice side, our experiment with a dipole-like beam excitation makes their observation clear by making the gap between the 1st bands of the two lattices large enough through, for example, increasing the bias field or the modulation depth of the lattice beam. To our knowledge, these “mixed-band” superlattice interface solitons have not been studied before. In summary, we have demonstrated the nonlinear surface states at the interface between optically-induced simple and super photonic lattices. Depending on the excitation condition at the interface, we indentify numerically and observe experimentally different types of interface solitons with different phase structures under a self-focusing nonlinearity, including higher band dipole-like interface gap solitons.