Nontrivial coupling of light into a defect: the interplay of nonlinearity and topology

ByAUTHOR

Aug 19, 2020

We study the propagation of light in photonic lattices with a refractive-index variation given by (n_0 + delta n_Lleft( {mathbf{x}} right) + delta n_{NL}left( {left| psi right|^2} right)), where n0 is the constant part of the material’s index of refraction, δnL(x) describes the linear photonic lattice, which is uniform along the propagation axis z, and (delta n_{NL}left( {left| psi right|^2} right)) is the nonlinear index change, which depends on the intensity of the light (with ψ(x, z) being the complex amplitude of the electric field). In the paraxial approximation, the propagation of the light is modeled by the following Schrödinger-type equation with a nonlinear term:

$$ifrac{{partial psi }}{{partial z}} = – frac{1}{{2k_0}}nabla ^2psi – frac{{k_0delta n_Lleft( {mathbf{x}} right)}}{{n_0}}psi – frac{{k_0delta n_{NL}left( {left| psi right|^2} right)}}{{n_0}}psi left( {{mathbf{x}},z} right) = (K + V_L + V_{NL})psi$$

(1)

which includes the kinetic term K, the linear index potential VL from δnL(x), and the nonlinear index potential VNL due to (delta n_{NL}left( {left| psi right|^2} right)); k0 is the wavenumber of light in the medium. The above equation holds for both 1D and 2D photonic lattices. In 1D systems, the spatial coordinate is a scalar x, and in 2D systems, it is a vector ({mathbf{x}} = xhat x + yhat y). Here, we consider a 1D topological system; that is, we assume that the photonic lattice VL can have nontrivial topological invariants. In our experiments and numerical simulations, we use the SSH lattice for VL(x). The photonic lattice and excitation scheme are illustrated in Fig. 1, where Fig. 1(a1) corresponds to a nontrivial lattice (Zak phase π) with two topological edge modes in the gap, and Fig. 1(c1) corresponds to a trivial lattice (Zak phase 0) without an edge state. In our theory, we use the above continuum model to describe the wave dynamics rather than its discrete version to obtain better correspondence with the experiments.

In our experiment, a 1D SSH photonic lattice, as illustrated in Fig. 1, is established by the continuous-wave (CW) laser-writing technique, which writes the waveguide lattice site-to-site in the bulk of a 20-mm-long nonlinear photorefractive crystal60. This technique allows a topological defect to be induced not only at the edge (Fig. 1(a1)) but also at the center, forming an interface (Fig. 2(a1)). Unlike femtosecond-laser writing in fused silica61, the lattice written in the nonlinear crystal is reconfigurable, so it can be readily changed from a trivial to a nontrivial structure in the same crystal. Once a chosen structure is written, it remains invariant during the period of experimental measurements (see “Methods”). In fact, since the SSH lattice is established here in a nonlinear crystal, it provides a convenient platform to investigate nonlinear wave dynamics in such a topological system, where the photorefractive nonlinear index potential VNL is easily controlled by a bias field and the beam intensity19,62. Below, we demonstrate nonlinearity-induced coupling of light into topologically protected states in two different cases.

In the first case, the topological defect is located at the SSH lattice edge (Fig. 1, left panels). When a narrow stripe beam (FWHM 12 μm; input power 2.5 μW) is launched straight into the edge waveguide under linear conditions (the beam itself does not exhibit nonlinear self-action when the bias field is turned off), it evolves into a topological edge state (Fig. 1(a2)). Such an edge state, with a characteristic amplitude and phase populating only the odd-numbered waveguides counting from the edge, is topologically protected by the chiral symmetry of the SSH lattice2, as previously observed in the 1D photonic superlattice43. On the other hand, when the excitation is shifted away from the edge with a tilted broad beam to pump the defect ((k_x = 1.4pi /a,) where a = 38 μm is the lattice constant), we observe that the beam does not couple into the edge channel under linear conditions (Fig. 1b1). However, when the beam experiences a self-focusing nonlinearity (at a bias field of 160kV/m), a significant portion of the beam is coupled into the edge channel (Fig. 1b2), indicating that the nonlinearity somehow enables the energy to flow from the bulk modes into the topological edge mode of the SSH lattice. According to Eq. (1), we perform numerical simulation to examine the nonlinear beam dynamics using the parameters from the experiments, and the results are shown in Fig. 1(b3). We clearly see nonlinear coupling of the beam to the topological edge state of the SSH lattice, in agreement with the experiment.

For direct comparison, in the right panels of Fig. 1, we present the corresponding results obtained with the trivial SSH lattice. A dramatic difference is observed: (1) Under straight excitation, the input beam transports to quite a few waveguides close to the edge, but there is no dominant coupling to the first waveguide to form an edge state under linear conditions (Fig. 1(c2)). (2) For tilted excitation, however, the beam can easily enter the edge waveguide under linear conditions (Fig. 1(d1)), while it does not efficiently excite the edge waveguide within 20 mm of nonlinear propagation (Fig. 1(d2) and (d3)). Simulations for much longer distances beyond the crystal length indicate that the energy of the initial beam will eventually dissipate into the bulk under linear propagation. There is a key difference between trivial and nontrivial lattices under nonlinear propagation for tilted excitation: a distinct edge state persists in the nontrivial lattice, but no edge state exists in the trivial lattice. The underlying mechanism is analysed below in detail on the basis of nonlinear wave theory.

In the second case, the topological defect is located inside the SSH lattice (Fig. 2). To validate the nontrivial lattice established by laser writing, as shown in Fig. 2(a1), a single probe beam is launched straight into the defect channel, which leads to a topological interface state (Fig. 2(a2)). Then, two tilted beams are launched from opposite directions ((k_x = pm 1.4pi /a)) to pump the interface defect simultaneously, as illustrated in the left panel of Fig. 2. When the two beams are in-phase, light cannot couple into the defect channel in the linear condition (Fig. 2b1), but significantly enhanced coupling into the channel occurs in the nonlinear condition (Fig. 2(b2)). For comparison, similar experiments were performed on the same lattice under the same conditions except for two out-of-phase beams, which cannot couple into the defect channel under either linear or nonlinear excitation conditions (Fig. 2(c1) and (c2)). For linear excitation, topological protection prevents energy from flowing into the defect. For the nonlinear excitation, the nonlinear interaction of the two out-of-phase beams leads them to repel each other. This remarkable difference can be seen more clearly in the numerical simulation, where the nonlinearity-induced coupling (Fig. 2(b3)) and “repulsion” (Fig. 2(c3)) are evident. These results clearly show that optical beams from different directions can be pumped into a nontrivial defect channel due to optical nonlinearity under proper excitation conditions.

Now that we have presented our experiment and simulation results, which demonstrate nonlinear coupling into topologically protected states, we develop a general theoretical protocol for interpreting dynamics in nonlinear topological systems and employ it for our experiments. Let us assume that the linear component of the index of refraction VL(x) in Eq. (1) represents a topological photonic lattice, which is characterized by a topological invariant such as the Chern number for 2D lattices or the Zak phase for 1D lattices. The initial excitation is given by (psi ({mathbf{x}},z = 0)). The subsequent propagation, governed by Eq. (1), gives us the complex amplitude of the electric field ψ(x, z) along the propagation direction, which in turn modulates the total index potential (linear and nonlinear) for any z: (Vleft( {{mathbf{x}},z} right) = V_Lleft( {mathbf{x}} right) + V_{NL}({mathbf{x}},z)). To determine and interpret the topological properties of the dynamically evolving nonlinear system, we use the total index potential V(x, z). The corresponding nonlinear eigenmodes (varphi _{NL,n}left( {{mathbf{x}},z} right)) and nonlinear eigenvalues (beta _{NL,n}(z)) are defined by the equation:

$$(K + V_L + V_{NL})varphi _{NL,n} = – beta _{NL,n}varphi _{NL,n}$$

(2)

We note that nonlinear eigenmodes and their eigenvalues are a function of the propagation distance z because nonlinear beam dynamics are generally not stationary. In contrast, the topological invariants of a linear system are drawn from the linear eigenmodes (varphi _{L,n}({mathbf{x}})) with propagation constants βL,n, obtained from

$$(K + V_L)varphi _{L,n} = – beta _{L,n}varphi _{L,n}$$

(3)

which are obviously not z-dependent. In both cases, n denotes the “quantum” numbers associated with the eigenmode, which can be associated with the Bloch wavevector and the band index for periodic photonic structures.

We emphasize several consequences of this approach: (i) The topological properties depend on the state of the system ψ(x, z) (this is natural because the system is nonlinear). These properties can be inherited from the underlying linear topological system, or they can emerge due to nonlinearity (see, e.g., Ref. 24). The inherited and emergent topological properties should be distinguished, as explained in the “Discussion” section below. (ii) The topological properties can change along the propagation direction. For example, we envision that for some initial conditions, the gap in the nonlinear spectrum (beta _{NL,n}(z)) could dynamically close and re-open, leading to a topological phase transition driven by nonlinearity. (iii) The evolution of the topological properties depends on the initial condition (psi ({mathbf{x}},z = 0)). For a given initial condition, the subsequent dynamics yielding V(x, z) are unique.

Let us apply the protocol to interpret the dynamics observed in the experiment of Fig. 1 (left panel). The linear SSH lattice with VL(x) is in the topologically nontrivial regime, which has two degenerate edge states, as illustrated in Fig. 3a. The propagation constants of the linear eigenmodes βL,n are illustrated in Fig. 3b, c (they are plotted in the region z < 0 for clarity, although they are z-independent); there are two bands corresponding to extended states, while the propagation constants of the localized edge states are in the middle of the gap as expected41. However, they are not at “zero energy” because we employ a continuous model with the experimental parameters. One can obtain the zero-energy states by adjusting the bottom of the linear potential through a transformation (V_Lleft( x right) to V_Lleft( x right) + {mathrm{constant}}), but shifting the zero energy by a constant does not change the physics.

First, we analyse the initial excitation, which has the shape of the left edge state (colored red in Fig. 3a): (psi left( {{mathbf{x}},z = 0} right) = sqrt {I_0} varphi _{L,edge}). This corresponds to the observation of Fig. 1a2. This linear edge state (varphi _{L,edge}) has a typical mode profile of topological characteristics: populating only odd-numbered waveguides with alternating opposite phases along the SSH lattice44. It is convenient to introduce the following quantities: (i) the edge state of the nonlinear system, (varphi _{NL,edge}({mathrm{x}},z)), as the eigenmode of the potential K + V, which has the largest overlap with the linear edge state (varphi _{L,edge}), defined as (F_{edge}left( z right) = left| {leftlangle {varphi _{NL,edge}{mathrm{|}}varphi _{L,edge}} rightrangle } right|^2); (ii) the overlap of the overall complex amplitude (psi left( {{mathrm{x}},z} right)) with the linear edge state, defined as (F_{all}left( z right) = left| {leftlangle {psi left( {{mathrm{x}},z} right){mathrm{|}}varphi _{L,edge}} rightrangle } right|^2/left| {leftlangle {psi {mathrm{|}}psi } rightrangle } right|^2). The values of the overlaps Fedge(z) and Fall(z) are always between 0 and 1 by definition; the former tells us how similar the nonlinear and linear edge states are, and the latter tells us how much of the power of the beam populates the linear topological edge state.

In Fig. 3b, we show the eigenvalue evolution of the nonlinear system (beta _{NL,n}(z)) for a low nonlinearity (see the Supplementary Material for the calculation details and parameter values). The bands and the nonlinear eigenvalue of the right edge state (plotted for z > 0) are essentially identical to those of the linear spectrum (beta _{L,n}) (for comparison, (beta _{L,n}) is also plotted for z < 0, even though it is independent of z). However, the nonlinear eigenvalue (beta _{NL,edge}) of (varphi _{NL,edge}) (for the left edge state) is pushed towards the higher band, although it is still in the gap34. The nonlinear spectrum (beta_{NL,n}(Z)) is almost z-independent for this initial excitation. Our calculation shows that in this case, Fedge(z) ≈ 0.99, while most of the power populates the left edge state, as Fall(z) ≈ 0.99. The inset in Fig. 3b shows the profile of the topological linear edge state (varphi_{L, edge},), along with that of the nonlinear edge state (varphi _{NL,edge}) (at z = 15 mm). We see that the profile of the nonlinear edge state has the proper oscillations pertaining to the topological edge state, with the amplitude in odd waveguides (starting from the edge waveguide as the first one) and opposite phases in neighboring peaks. The edge state has an amplitude mainly in the first (edge) waveguide and then the third waveguide. If the nonlinearity is increased above some threshold value, the nonlinear eigenvalue (beta _{NL,edge}) moves across the band to appear above the first band, as illustrated in Fig. 3c. From the mode profiles shown in the inset of Fig. 3c, we find that the nonlinear edge state is essentially identical to the linear one in the edge waveguide, but it lacks an amplitude in the third waveguide. This difference is more easily seen when we use a larger lattice coupling than the one obtained from the experimental parameters.

We conclude that for the initial excitation (psi left( {{mathbf{x}},z = 0} right) = sqrt {I_0} varphi _{L,edge}), when (beta _{NL,edge}) is in the gap, the localization is induced by the topology, and the nonlinear edge state can be regarded as a topological edge state. When (beta _{NL,edge}) is above the upper band (in the semi-infinite gap), the localization is induced by nonlinearity. Even though the mode profile in the edge channel is inherited from the linear topological system (see the inset in Fig. 3c), due to the lack of mode features in the third waveguide, the nonlinear edge mode should not be characterized as topological when (beta_{NL,edge}) is in the semi-infinite gap. A related analysis of similar scenarios can be found in Refs. 30,59.

A theoretical analysis of the experiments corresponding to tilted excitation in Fig. 1(b2) is more involved because in this case, the dynamics are far from stationary, yet this case captures the essence of the theoretical protocol. The beam is launched at x = 1.2a at an angle (k_{x}=-1.4 pi/a) towards the edge located at x = 0 (see Fig. 3a). For this initial excitation, Fall(z = 0) ≈ 0; i.e., at the input of the medium, the beam does not excite the linear edge state. Figure 1(b1) is easily understood, as Fall(z) is z-independent in the linear dynamics. The evolution of the nonlinear spectrum (beta _{NL,n}(z)) is depicted in Fig. 3d. First, we note that the band structure (thick blue lines) corresponding to the bulk states is essentially z-invariant and is equivalent to that of the linear system. Due to the self-focusing nonlinearity, the dynamics are manifested in the localized modes of (Vleft( {x,z} right) = V_Lleft( x right) + V_{NL}(x,z)); there are quite a few evolving localized modes of V(x,z), with eigenvalues (beta _{NL,n}(z)) indicated by the dotted blue lines in Fig. 3d. We focus only on the nonlinear edge state (varphi _{NL,edge}) and its eigenvalue (beta _{NL,edge}), plotted in Fig. 3d with a solid red line. From Fig. 3e, f, which illustrate Fall(z) and Fedge(z), respectively, we see that the dynamics can be divided into three stages. More specifically, the sudden drop of Fedge(z) at z = 5 mm indicates the end of the first stage, while the sudden increase at z = 11 mm indicates the end of the second stage of the dynamics (see Fig. 3f). In the first stage (shaded magenta in Fig. 3d, f), the launched beam travels towards the edge, and the edge state is not populated, as (F_{all}(z) approx 0); consequently, (beta _{NL,edge}) is in the gap (see the left red line in Fig. 3d), and Fedge(z) is close to unity. In the second stage (shaded gray), when the beam is at the edge, the linear edge state becomes populated, and Fall(z) increases. In this stage, the beam strongly perturbs the local structure of the lattice at the edge, as seen from the drop in Fedge(z) in Fig. 3f, which means that none of the nonlinear localized states are similar to (varphi _{L,edge}) (thus, none of the nonlinear eigenvalues is colored red in Fig. 3d in the second stage). In the third stage (shaded green), a large portion of the beam is reflected, but ~30% of the beam becomes trapped in a localized edge state: Fall(z) ≈ 0.3, as shown in Fig. 3e. There is a well-defined nonlinear edge state with eigenvalue (beta _{NL,edge}) above the first band, not in the gap, as indicated by the right red line in Fig. 3d. The profile of this nonlinear edge state is mostly inherited from the topology of the linear structure, as seen in the inset of Fig. 3e and the overlapping Fedge(z) ≈ 0.98 shown in Fig. 3f; however, it lacks the topological mode feature in the third waveguide. We conclude that the localization is dominantly induced by nonlinearity. We should emphasize that the linear edge state does not continuously transform into the nonlinear edge state during propagation, because of the strong deformation of the lattice in the second stage of the dynamics. After this distortion, one of the localized states from stage two re-emerges as the new nonlinear localized edge state in stage three, as can be traced by following the nonlinear eigenmodes alongside the F-functions plotted in Fig. 3d, e.

The details of the theoretical analysis corresponding to the right panel of Fig. 1 (for the SSH lattice in the topologically trivial regime) and Fig. 2 (for excitation of the topological defect with two beams) are shown in the Supplementary Material and summarized here. The results in the right panels of Fig. 1 can be interpreted as follows: all the linear modes are extended, as the SSH lattice is in the topologically trivial regime. The beam initially excites many of these states. In the linear regime illustrated in Fig. 1(d1), the beam approaches the waveguide at the edge and then travels along the edge for the length of the crystal. In other words, for short propagation distances (smaller than the length of the crystal), the phases of all linearly excited (extended) modes add together such that the intensity of the beam populates the waveguides close to the edge in Fig. 1(d1). However, for a very long propagation distance, due to the de-phasing of the excited bulk modes, the beam will spread into the lattice. In the nonlinear case corresponding to Fig. 1(d2), (d3), the nonlinearity creates evolving localized states, which are not related to the topological origin, as none of the nonlinear modes resemble the linear topological edge state. In fact, in this trivial lattice structure, the localized modes arise purely due to the nonlinear index change, as is typically the case with optical solitons. A light beam forms a few self-trapped filaments around these states and evolves in this fashion for the propagation distance of the crystal length in the experiment. As the initial excitation is not at the edge, the location of the self-trapped filaments is also not at the edge, as illustrated in Fig. 1(d2), (d3).

Regarding the excitation of the defect mode with two beams at opposite angles, when the beams are in phase, there are again three stages of the dynamics, which are equivalent to those shown in Fig. 3d–f. In the first stage, the beams travel towards the linear defect channel in the center of the lattice; the defect state is not yet populated, and its eigenvalue is in the gap. Many evolving nonlinear localized states arise due to nonlinearity but not to topology. In the second stage, the linear defect state starts to become populated, but the lattice is distorted locally due to nonlinear action, so none of the nonlinear states are similar to the linear defect state. In the third stage, some of the incident light (~20–30% for the parameters used here) is trapped in the defect state, while the rest is repelled. There is a well-defined nonlinear defect state with a profile in the defect channel inherited from the linear defect state and a nonlinear eigenvalue emerging above the first band. Thus, conceptually, an identical scenario to that shown in Fig. 3d–f occurs. The difference is that the defect state can now be coupled from both sides, and this could be extended to coupling light from all directions in a 2D SSH-type system, leading to a nonlinear “tapered” topological waveguide. Such potential applications certainly merit further research.

When the two incident input beams at opposite angles are out of phase, again there are three stages of the dynamics analogous to those presented above (see the Supplementary Material). However, the linear defect state is not populated by any of them. The eigenvalue of the nonlinear defect state is within the gap in the first and third stages of the dynamics. In the second stage, when the light is close to the defect state, the lattice structure is distorted, and none of the nonlinear localized states are very similar to the linear defect state. In fact, the two beams stay away from each other, and the defect in this case is related to the nonlinear interaction of out-of-phase soliton-like beams rather than to topology.