Optical soliton formation controlled by angle twisting in photonic moiré lattices

Optical soliton formation controlled by angle-twisting in photonic moir´e lattices Qidong Fu,1Peng Wang,1Changming Huang,2Yaroslav V. Kartashov,3,4Lluis Torner,3,5Vladimir V. Konotop,6,7Fangwei Ye1∗ 1School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China 2Department of Electronic Information and Physics, Changzhi University, Shanxi 046011, China 3ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 4Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 108840, Russia 5Universitat Politecnica de Catalunya, 08034 Barcelona, Spain 6Departamento de F´ısica, Faculdade de Ciˆencias, Universidade de Lisboa, Campo Grande, Ed. C8, Lisboa 1749-016, Portugal 7Centro de F´ısica Te´orica e Computacional, Universidade de Lisboa, Campo Grande, Ed. C8, Lisboa 1749-016, Portugal ∗Corresponding author: [email protected]du.cn (Dated: June 2, 2020) 1 Exploration of the impact of synthetic material landscapes featuring tunable geometrical properties on physical processes is a research direction of the highest current interest because of the outstanding phenomena that are continuously uncovered. Twistronics and the properties of wave excitations in moir´e lattices are salient examples. Moir´e patterns bridge the gap between aperiodic structures and perfect crystals, thus opening the door to the exploration of effects accompanying the transition from commensurate to incommensurate phases. Moir´e patterns have revealed profound effects in graphene-based systems [1–5], they are used to manipulate ultracold atoms [6,7] and to create gauge potentials [8], and are observed in colloidal clusters [9]. Recently, it was shown that photonic moir´e lattices enable the observation of the two-dimensional localization-to-delocalization transition of light in purely linear systems [10,11]. Here we employ moir´e lattices optically-induced in photorefractive nonlinear media [12–14] to elucidate the formation of optical solitons under different geometrical conditions controlled by the twisting angle between the constitutive sublattices. We observe the formation of solitons in lattices that smoothly transit from fully periodic geometries to aperiodic ones, with threshold properties that are a pristine direct manifestation of flat-band physics [11]. By and large, the linear transport and localization properties of excitations in a material are intimately determined by its inner symmetry and geometrical properties, including a periodic or aperiodic nature, as it occurs in electronic [15], atomic [16,17], optical [18,19], or two-dimensional material [20] systems. This feature is directly related to the nature of the eigenstates of the system, which can be extended or localized. When an underlying material exhibits nonlinear response, the formation of self-sustained excitation, alias solitons, becomes possible [21–23]. Nevertheless, the properties of solitons are still strongly impacted by the linear spectrum of the system. Optical media offer a unique laboratory for the investigation of solitons in different environments. Thus, two-dimensional self-trapping and soliton formation have been investigated in fully periodic optical lattices [13,24–26], as well as in quasicrystals, which are characterized by broken translational invariance [14,27–30]. A 2 common feature of all these results is the occurrence of a power threshold for two-dimensional soliton existence whose value can be tuned by varying the lattice configuration [27]; however, the threshold never vanishes. At the same time, the formation of solitons at the transition from aperiodic to periodic systems has been never explored experimentally because of lacking of a suitable setting. Optical moir´e lattices [11] offer a powerful platform enabling such study. Created with incommensurate geometries, moir´e patterns may enable localization of light even in the linear limit due to the existence of extremely flat bands in their spectra, a property that therefore strongly impacts the diffraction of beams in such media. Thus, since solitons can emerge due to a balance between diffraction and self-phase-modulation induced by nonlinearity, moir´e lattices allow the investigation of the formation of solitons controlled by the geometry of the induced optical potential. Here we provide the experimental evidence of such possibility, by reporting qualitative differences in soliton excitation dynamics in commensurate and incommensurate moir´e lattices. We present the first ever observation consistent with the formation of thresholdless two-dimensional optical solitons in aperiodic system, which is enabled by the creation of extremely flat bands in the incommensurate geometry. In addition, importantly, we demonstrate that in the commensurate moir´e lattices the threshold for soliton formation is dictated by the area of the primitive lattice cell, which determines the width of the allowed bands. Therefore, our observations show that the properties of the entire soliton families are determined not only by the characteristics of the respective gaps, but that they also fundamentally depend on the properties of the allowed bands. A photonic moir´e lattice can be created in a photorefractive crystal by a shallow modulation of the refractive index in the (x, y) plane induced by two mutually rotated, or twisted, periodic square sublattices generated by light interference. In our SBN:61 crystal, where the linear electro-optic effect determines the refractive index modulation experienced by light with different polarizations, the electro-optic coefficient r13 = 45 pm/V is much smaller than r33 = 250 pm/V . Accordingly, we used ordinary polarized light for the lattice induction, so that the corresponding beams do not experience any noticeable self-action in the crystal and thus propagate undistorted as in a linear medium. In contrast, light with extraordinary 3 polarization experiences a strong nonlinear response. Propagation along the z-direction of a signal beam in such polarization is governed by the nonlinear Schr¨odinger equation for the dimensionless amplitude Ψ(r⊥, z) [12,13]: i∂Ψ ∂z =−1 2∇2 ⊥Ψ + V0 1 + I(r⊥) + |Ψ|2Ψ.(1) Here ∇⊥= (∂/∂x, ∂/∂y); r⊥= (x, y) is the radius-vector in the transverse plane; zis the longitudinal coordinate scaled to the characteristic length 2πneλ, where λis the wavelength (in our experiments λ= 632.8 nm); neis the unperturbed refractive index of the crystal experienced by the extraordinary-polarized light; and V0>0 is the dimensionless applied dc field. Here we set V0= 5, which corresponds to 5.7×104V/m dc electric field applied to the crystal; I(r⊥) = |p1V(r⊥) + p2V(Sr⊥)|2is the moir´e pattern composed by two ordinarypolarized periodic sublattices V(r⊥) and V(Sr⊥) interfering in the (x, y) plane [here S=S(θ) is the matrix of rotation in the (x, y) plane by the angle θ]; p1and p2are the amplitudes of the first and second square sublattices, respectively. Each square sublattice V(r⊥) is formed by the interference of four plane waves [11]. In the following we set the amplitude of the first sublattice to p1= 0.5, which corresponds to an average intensity Iav ≈1.9 mW/cm2, and tune the amplitude p2of the second sublattice. For such parameters, the actual refractive index modulation depth in the moir´e pattern illustrated in Fig. 1(a-c) is of the order of δn ∼10−4. Two-dimensional Pythagorean moir´e lattices composed of two square Bravais sublattices p1V(r⊥) and p2V(Sr⊥) (the point group D4) rotated with respect to each other around common lattice site, are periodic (commensurate) structures only when the rotation angle θsatisfies cos θ=a/c, sin θ=b/c, where the positive integers (a, b, c) having no common divisors except 1, constitute a primitive Pythagorean triple, i.e., a2+b2=c2(Fig. 1a,c). We call such angles Pythagorean. For other rotation angles the pattern is aperiodic (incommensurate, or almost periodic in mathematical terms), as illustrated in Fig. 1b. The linear spectrum of the lattices [10,11] can be obtained by omitting the nonlinear term |Ψ|2in Eq. (1) and searching for the corresponding linear eigenmodes in the form Ψ(r⊥, z) = ψ(r⊥)eibz, where bis the linear propagation constant and ψ(r⊥) is the transverse field distribution. To characterize the mode localization we use the integral form-factor χ= (RR |Ψ|4d2r⊥)1/2/U, 4 FIG. 1: Moir´e patterns and properties of their linear eigenmodes. Example of periodic (a,c) and aperiodic (b) moir´e lattices I(r⊥) produced by two superimposed square sublattices with p1= 0.5 and p2= 0.3 rotated by the angles θ= arctan(5/12) (a), θ≈0.167π(b), θ= arctan(3/4) (c). Blue arrows in (a),(c) indicate the primitive lattice vectors. (d) Form-factor (inverse width) of the most localized linear eigenmode of the lattice versus rotation angle θand depth p2of the second sublattice at p1= 0.5. Green arrows indicate angles corresponding to the (3,4,5) and (5,12,13) Pythagorean triples. The top lattice bands for two Pythagorean angles θ= arctan(3/4) and θ= arctan(5/12) superimposed in one plot for p2= 0.1 (e) and p2= 0.3 (f). Bloch momenta kx,y are normalized to the width of the Brillouin zone equal to K≈1.265 for θ= arctan(3/4) and to K≈0.785 for θ= arctan(5/12). with Ubeing the mode power U=RR |Ψ|2d2r⊥.Note that the form-factor χis inversely proportional to the mode width, thus the larger its value the stronger the mode localization. The dependence of the form-factor of the mode with the largest b(this is the most localized mode) on θand p2is shown in Fig. 1d. For Pythagorean angles the mode is delocalized for any depth p2of the second sublattice because in this case the moir´e pattern is periodic, but for non-Pythagorean angles the mode becomes localized if p2exceeds some critical value, pcr 2≈0.18 corresponding to the linear localization-delocalization threshold. The physical origin of this phenomenon is the suppressed diffraction due to flatness of the allowed bands of the effective Pythagorean lattice approximating real incommensurate moir´e pattern at p2> pcr 2[11]. This observation comes from a general rule: the higher the order of the primitive Pythagorean triple (determined by the integer c) the larger the area of the respective primitive cell of the lattice (see the blue arrows in Fig. 1a,c) and the smaller the width (in b) of the allowed bands, thus indicating a reduced diffraction strength. This behavior is illus- 5 trated in Fig. 1e,fthat compares top bands of the Floquet-Bloch spectra calculated for two different Pythagorean angles and two different sublattice amplitudes p2. As discussed below, the angular-dependent band flattening has direct implications for soliton formation in the lattices, because it exposes that the diffraction strength experienced by narrow linear inputs in the lattices notably decreases with increase of the order of the primitive Pythagorean triple. This is particularly clear at p2> pcr 2, where patterns akin to discrete diffraction are observed in the linear limit and is less pronounced at p2< pcr 2, when such patterns co-exist with a rapidly expanding broader background. Turning now to the nonlinear regime, we look for soliton solutions of Eq. (1) in the form Ψ(r⊥, z) = ψ(r⊥)eibz, where bis the nonlinear propagation constant, which for V0>0 (focusing nonlinearity) exceeds the propagation constants of the linear eigenmodes. Solitons form families characterized by the dependencies of the peak amplitude ψmax = max|ψ|and power Uon b, as shown in Fig. 2. Since incommensurate moir´e lattices have either delocalized (at p2< pcr 2) or localized (at p2> pcr 2) linear modes, solitons in such lattices show a completely different behaviour in the low-amplitude limit depending on the p2value. When linear localized modes do not exist (p2< pcr 2), then consistent with the behavior in homogeneous media [31], solitons in incommensurate moir´e lattices can exist only if they carry a power Uthat exceeds a certain threshold value Uth (Fig. 2a) below which they quickly diffract. However, when p2> pcr 2and linear localized states exist, the soliton family bifurcates from the respective linear localized mode, remaining well-localized at any power U. Under such conditions, solitons do not feature a power existence threshold (Fig. 2b). In contrast, in commensurate moir´e lattices corresponding to the Pythagorean angles, solitons always exhibit a nonzero existence power threshold for any depth of the second sublattice p2 (Fig. 2c), and strongly expand at low amplitudes. In all the cases shown in Fig. 2, the peak soliton amplitude vanishes in the linear limit (see red curves in Fig. 2a-c), while far from it the solitons become strongly localized. All shown soliton families are either completely stable, when power threshold is zero, or unstable at low amplitudes and get stabilized at high amplitudes, when the power threshold does not vanish. Here we report the experimental observation of all such scenarios using lattices with read- 6 FIG. 2: Families of two-dimensional solitons in moir´e lattices. Soliton power Uand peak amplitude ψmax versus propagation constant b(left column a,b,c) and representative soliton profiles (right column) are shown for incommensurate moir´e lattice with the rotation angle θ≈ 0.139πbelow (p2= 0.1, the upper panels) and above (p2= 0.3, the middle panels) the critical value pcr 2, as well as for a lattice with the Pythagorean rotation angle θ= arctan(3/4) with p2= 0.3 (bottom panels). The profiles shown in the right column correspond to red dots on the ψmax(b) curves. The horizontal dashed lines in (a),(c) indicate the corresponding power thresholds for soliton existence. ily tunable twisting angle. More specifically, the transition between commensurate and incommensurate moir´e lattices or between the regimes, where delocalization and localization takes place in the linear limit, can be explored by adjusting the phase mask used for the creation of the sublattices or by tuning their relative amplitudes. The comparison between 7 FIG. 3: Thresholds for soliton formation in moir´e lattices. Comparison of theoretically calculated (a,c) and experimentally measured (b,d) dependencies of threshold for soliton formation on rotation angle θfor the depth of the second sublattice above localization-delocalization threshold p2= 0.3 (a,b) and below localization-delocalization threshold p2= 0.1 (c,d). the theoretical predictions and the experimental observations of the threshold power Uth as a function of the rotation angle θbetween the sublattices is shown in Fig. 3. For p2above the critical value (Fig. 3a), Uth vanishes for non-Pythagorean angles while it exhibits narrow peaks around the Pythagorean angles. As visible in Fig. 1e,f, the band curvature decreases with the increase of the order of the Pythagorean triple. Thus, the nonlinearity required to 8 balance the curvature-induced diffraction decreases too, in agreement with the observations depicted in Fig. 3a,b. The first, second, and third highest peaks occur near the Pythagorean angles corresponding to the primitive triples (3,4,5), (5,12,13), and (8,15,17), respectively. The important conclusion to be highlighted is that the observed power thresholds for soliton formation are directly related to the areas of the respective primitive cells, and thus to the structure of both, the gaps and the allowed bands of the lattice spectrum. The area of the primitive cell of a commensurate Pythagorean lattice, defined by the third integer c, non-monotonically depends on the twist angle indeed. Hence the observed non-monotonic variation in the heights of the peaks in Fig. 3a,b, which is in agreement with the variation of primitive cell areas for the corresponding Pythagorean triples. For instance, the smallest area of the primitive (square) cell of a Pythagorean moir´e pattern corresponds to the primitive triple (3,4,5), that explains the locations of the absolute maxima in all panels of Fig. 3. Because the bands become flatter when the areas of the primitive cells increase, other Uth maxima are almost undetectable. This is consistent with the approximation of incommensurate Pythagorean moir´e patterns by commensurate ones established in [11]. In all experimental results reported, as well as in all numerical simulations showing dynamical soliton excitation, we observed stability of two-dimensional beams. Remarkably, the above relation between the order of the Pythagorean triple associated with the lattice and the threshold for soliton formation, even though less pronounced, was still observed in the regime p2< pcr 2, where localization in the linear system is impossible even in incommensurate lattices (Fig. 3c,d). Under these conditions the soliton formation threshold Uth does not vanish for any rotation angles, although it remains sensitive to the diffraction properties of the lattice. Our observations show that it achieves the maximal value around θ= arctan(3/4) angle that corresponds to the commensurate Pythagorean lattice associated with the (3,4,5) triple and having the smallest possible primitive cell. In all cases we observed an excellent agreement between the theoretical predictions (Fig. 3a,c) and the experimental (Fig. 3b,d) results. The threshold power Uth for soliton formation was measured by evaluating the soliton content C=Uout/Uin, where Uin is the power of the input beam measured at z= 0 and 9 Author contributions Q. F and P. W contribute equally to this work. All authors contribute significantly to the work. Competing interests The authors declare no competing interests. 16