Floquet topological insulator laser
Sergey K. Ivanov, Yiqi Zhang, Yaroslav V. Kartashov, Dmitry V. Skryabin
11 Floquet topological insulator laser
Sergey K. Ivanov,
Yiqi Zhang,
Yaroslav V. Kartashov, and Dmitry V. Skryabin Department of Applied Physics, School of Science, Xi’an Jiaotong University, Xi’an 710049, China Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, 141700, Russia Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 108840, Russia Department of Physics, University of Bath, BA2 7AY Bath, UK *Corresponding author: [email protected]
Abstract:
We introduce a class of topological lasers based on the photonic Floquet topological insulator concept. The proposed system is realized as a truncated array of the lasing helical waveguides, where the pseudo-magnetic field arises due to twisting of the waveguides along the propagation direction that breaks the time-reversal symmetry and opens up a topological gap. When sufficient gain is provided in the edge channels of the array then the system lases into the topological edge states. Topological lasing is stable only in certain intervals of the Bloch momenta, that ensure a dynamic, but stable balance between the linear amplification and nonlinear absorption leading to the formation of the breathing edge states. We also illus-trate topological robustness of the edge currents by simulating lattice defects and triangular arrangements of the waveguides. © An important property of topological insulators is the existence of the topologically pro-tected states at its edges with energies inside a topological gap and connecting two bands with different topological invariants. In real space such edge states may demonstrate unidirectional propagation and topological robustness to the lattice and edge distortions . Originated in sol-id-state physics, the concept of topological insulators is now interdisciplinary: It has been in-troduced also in mechanical , acoustic , atomic , photonic , optoelectronic and many other systems, where diverse potential applications of topologically protected transport are envisioned. Recent progress in the sub-area of photonic topological insulators is described in, e.g., reviews . Floquet topological insulators is a special case in a family of photonic realizations of topo-logical systems, where a system is also periodic in an evolution variable, which can be either time or a longitudinal coordinate. Following the first proposal of such a system in semiconduc-tor quantum wells , Floquet topological insulators have been realized with honeycomb arrays of helical waveguides . In the latter case, the waveguide helicity gives rise to the pseudo-magnetic field breaking the time-reversal symmetry and leading to the appearance of the uni-directional edge states. Helical waveguide array is a photonic analogue of the Haldane sys-tem in high-frequency driving limit, and it can be used to verify the anomalous quantum Hall effect. A variety of new phenomena were theoretical predicted or experimentally ob-served with helical waveguide arrays, including anomalous topological insulators , topolog-ically protected path entanglement , unpaired Dirac cones , topological edge states in quasi-crystals , solitons , topological Anderson insulator , topological phases in synthetic di-mensions , guiding light by artificial gauge fields , and others. Notice that driven topological systems, such as helical arrays, may be characterized by special topological invariants . Topological phases of matter are nowadays under active investigation not only in con-servative, but also in dissipative settings, see for example . Among the most exciting oppor-tunities in this direction is the realization of lasing in topological edge states in active systems that promise remarkable stability of topological lasers, inherited from robustness and re-sistance to disorder of conservative topological systems. Theoretically topological lasers were proposed in photonic crystals . Later they were realized in one-dimensional polaritonic and photonic structures employing Su-Schrieffer-Heeger model , which, however, did not allow to demonstrate topological currents due to the reduced dimensionality. Two-dimensional top-ological lasing was very recently observed in photonic crystals , lattices of coupled-ring reso-nators , and proposed theoretically in polaritonic arrays . In these static systems, edge states appear either due to the external magnetic , or due to judicious engineering of coupling be-tween elements leading to Haldane model . At the same time, our proposal offers ad-vantages of not using external magnetic fields, operating at optical frequencies, and relying on conventional nonlinear transparent materials. The aim of this work is to show that such Floquet topological lasers exhibiting stable dis-order- and defect-immune lasing in topologically protected edge states can be implemented using truncated honeycomb array of helical waveguides written or fabricated in the nonlinear optical material with gain saturation. Broken time-reversal symmetry guarantees the existence of unidirectional edge states that can lase, when spatially-inhomogeneous gain is provided for them. Nonlinear losses result in stabilization of the nonlinear edge states at certain Bloch mo-menta determining their group velocity. We describe dynamics of light in Floquet topological lasers using the nonlinear Schrö-dinger equation for field amplitude that in dimensionless units takes the form: i x y z i x y z i iz (1) where is the scaled field amplitude; , x y are the transverse coordinates normalized to the character-istic transverse scale w ; z is the propagation distance scaled to diffraction length w ; r n is the wavenumber; r n is the real part of the unperturbed refractive index of the material; / w n n is the scaled linear loss that is assumed uniform; i r n n is the imaginary part of the refractive index; is the nonlinear loss parameter. Further we consider focusing cubic (Kerr) nonline-arity, typical for many solid materials, including optical fibres, but Floquet laser can be realized in the defocusing case too. We assume that the Floquet laser is composed from honeycomb array of helical waveguides that modulates the linear refractive index re re , ( , , ) ( , ) n mn m x y z p x x y y , where / p w n n is the modulation depth, , n m x y are the nodes of the honeycomb grid; sin( ) x x r z and cos( ) y y r r z , where r is the helix radius, Z 2 / is the helix period, and exp[ ( ) / ] x y d is the function describing profile of individual waveguides of width d [see Fig. 1(a) with schematic array representation and Fig. 4(c)]. The separation between the waveguides in the array is a . We assume that the array is truncated along the x -axis to form two zig-zag edges and that gain is provided only on its left edge im im , ( , , ) ( , ) q lq l x y z p x x y y , where , q l x y are the coordinates of edge waveguides, see green waveguides in Fig. 1(a), while im re p p is the gain amplitude. The array is periodic in y with period Y 3 a . Results do not change qualitatively for z -independent gain acting only inside the edge waveguides. It should be not-ed that by moving into the coordinate frame co-rotating with the waveguides x x , y y Eq. (1) can be rewritten in the form: i i z x y i x y r i iz A (2) where [ cos( ), sin( )] r z z A is the gauge potential, re, im do not depend on z z . Further we select parameters of helical waveguide array in accordance with recent experiments , see caption to Fig. 1. Dissipative helical arrays exhibiting gain in certain waveguides can be fabricated in different ways. Most tried approach relies on the direct laser writing with femtosecond pulses available in a broad range of transparent materials , including those containing amplifying dopants. Thus, various wave- guides were already realized in Er-doped active phosphate , silicate , tellurite , and Baccarat glass-es, and also in lithium niobate allowing realization of inhomogeneous parametric gain used for ob-servation of parity-time symmetry . Another viable alternative is the infiltration of hollow photonic crystal fibres with helical channels with active index-matching liquids . First, we consider topological properties of conservative linear helical waveguide array by setting im , , 0 p and neglecting nonlinearity in Eq. (1). The eigenstates ( , , ) ( , , )exp( ) x y z u x y z i z iky of Eq. (1) are Bloch waves, where u is localized in x : x u and periodic both in y and z directions: Z z z u u and Y y y u u , k is the Bloch momentum along the y -axis, is the quasienergy. The latter is periodic function of k with period K 2 / Y and is defined modulo due to longitudinal periodicity of the array. Typical quasienergy spectrum for helical array is presented in Fig. 1(b). Since we consider real-world continu-ous system, quasienergy spectrum was calculated using following approach. First, Bloch modes st st ( , )exp( ) i i u x y i z iky from two top bands of the static truncated array with straight channels were obtained using a plane-wave expansion method. The number of such modes is n , where n is the number of waveguides in one y -period of the array (unit cell). Each such mode st i , normalized as st st ( , ) i j ij , where Hermitian product involves integral over one unit cell of the array, was propa-gated in helical array for one period Z . Rotation couples modes from the first two bands (coupling to the lower bands can be neglected, since they remain well-separated). The output distributions out j corresponding to input st j were then projected on the initial basis of modes st i that yields n n projection matrix st out ( , ) ij i j , whose eigenvalues are Floquet exponents exp( ) j i . Quasienergies are found as / Z j j ; their imaginary part is negligible as long as radiative losses are small that is the case for parameters used below. Fig. 1. (a) Schematic illustration of truncated helical waveguide array. Amplifying edge wave-guides are shown green, waveguides with losses in the bulk are shown brown. (b) Quasiener-gies of linear modes supported by conservative truncated helical array versus Bloch mo-mentum k . (c) Velocity (solid circles) and dispersion (open circles) of the edge states versus Bloch momentum k . Red (green) circles correspond to the edge states from the left (right) edges of the array, black circles correspond to bulk modes. Here and below helix radius r , period Z 6 , waveguide width d and separation a , array depth re p . Waveguide rotation opens topological gap with the edge states existing for
K/ 3 2 K/ 3 k and the zigzag-zigzag interface [Fig. 1(b)]. The width of the gap increases with increase of helix radius r or decrease of rotation period Z , but so do also radiative losses, so for each Z there is certain optimal r . There are two topological edge states in the gap – red curve corresponds to the states on the left edge that moves in positive y -direction, green curve corresponds to the right edge states moving in the negative y -direction. Black circles correspond to bulk modes. First / k and second / k derivatives of the quasienergy that quantify group velocity and dispersion of the edge states are shown in Fig. 1(c). Inversion of the waveguide rotation direction also inverts the direc-tion of edge currents. The sign and magnitude of determines domains of Bloch momentum, where modulational instability of the edge state can develop in the presence of nonlinearity in a conservative case (thus, for focusing nonlinearity this is possible when ), but in a dissipative system such instabilities may be suppressed by linear and nonlinear losses, as shown below. An array with the bearded edges can be analysed similarly and the edge states were found for K/ 3 k and k . Our system retains its topological properties in the presence of the spatially uniform linear losses and gain im p concentrated in the edge channels. The edge states are well localized, and hence they have largest overlap with gain area relative to the other modes and therefore they experience preferen-tial amplification. We have found that there exists a sharp threshold in im p above which lasing in edge states occurs. Most efficient amplification occurs for the edge states with Bloch momentum, which most localised around the edge. The interval of k values, where the edge states get amplified increas-es with im p until lasing becomes possible in the entire topological gap. Because gain was provided on the left edge only, the right edge states were attenuated. Fig. 2. (a) Typical evolution of peak amplitude in stable nonlinear edge state in Floquet laser at im p , k and (b),(c) representative field modulus distributions at Z z n and ( 1 / 2)Z z n . Examples of the nonlinear edge states at k , im p (d) and im p (e); k , im p (f); and k , im p (g). The edge states in (b)-(d) are stable, while edge states in (e)-(g) are unstable. In all cases . To achieve stable lasing we now add focusing nonlinearity and nonlinear absorption into system. This leads to appearance of attractors – nonlinear edge states performing periodic stable breathing in the course of propagation. A typical breathing dynamics of the edge state is illustrated in Figs. 2(a)-2(c). Being stable attractor, this state was excited using a linear conservative state with K/ 2 k as an ini-tial condition. After some transient stage the state has evolved into completely stable nonlinear dissi-pative mode existing due to balance between nonlinearity and diffraction, gain and losses, that exactly replicates its transverse profile after each period Z . The amplitude of this nonlinear edge state shows 0 complex, but regular periodic oscillations without damping or growth and the period of these oscilla-tions coincides with helix period Z 6 . Figure 2(a) shows thirty of these periods to stress that the state is practically stable. Amplitude oscillations notably increase with the increasing gain im p . Comparison of wave profiles at Z z n [Fig. 2(b)] and ( 1 / 2)Z z n [Fig. 2(c)] shows slightly larger penetra-tion of the latter state into the depth of array. Increasing gain amplitude leads to gradual expansion of the dissipative state into the depth of array [Figs. 2(d) and 2(e)] and may finally cause its destabiliza-tion [Fig. 2(d) shows a state from the boundary of the stability domain, while the state in Fig. 2(e) is unstable]. The extend of the edge states away from the edge and into the crystal strongly depends on Bloch momentum k and increases for the quasienergies approaching the gap boundaries [Figs. 2(f) and 2(g)]. Fig. 3. Average amplitude (a) and quasienergy (b) of the nonlinear edge state versus gain amplitude im p at k for the different values of nonlinear absorption, , , and (the direction of increase of is shown by arrows). Lower and upper dashed lines in (b) indicate quasienergy of lin-1 ear edge state and the border of topological gap for k . (c) Quasienergy of the nonlinear edge state versus gain amplitude im p for k and k , at . Upper dashed line in-dicates the border of the gap, identical for k and k , while two lower dashed lines indicate energies of linear edge states, which are different for k and k . Stable branches are shown black, unstable branches are shown red. To prove that nonlinear dissipative edge states reported here are topological, we traced families of such states by gradually increasing gain amplitude im p . Since amplitude of the lasing state exhibits complex behaviour over one helix period [see Fig. 2(a)], we introduce a new quantity – averaged am-plitude ( 1)Z1av maxZ Z nn a dz . This amplitude is depicted in Fig. 3(a) as a function of im p for K/ 2 k and different values of the nonlinear absorption coefficient . Moreover, we introduced quasienergy of the nonlinear dissipative edge states by analogy with quasienergy of linear con-servative states. It can be determined from phase accumulated by the edge state over one helix pe-riod / Z , where phase is calculated numerically from the product Z ( 1)Z ( , ) exp( ) z n z n
U i , where U is the norm of the state per one unit cell ( y -period). The dependencies im ( ) p , calculated for the same values of nonlinear absorption as in av im ( ) a p curves, are shown in Fig. 3(b). The presence of lasing threshold in im p is obvious in Fig. 3(a) – it corresponds to the point where averaged amplitude of the edge state becomes nonzero. Lasing threshold is mini-mal for Bloch momentum K/ 2 k (in this case im p ) and it slightly increases for other momen-tum values reaching maximal values at K/ 3 k or k , the property connected with de-creasing overlap of the edge states for latter momentum values with gain landscape leading to less ef-2 ficient amplification. The quasienergy of the nonlinear edge state at its generation threshold coin-cides with that of the conservative linear state, as indicated by the bottom dashed line in Fig. 3(b), and it increases almost linearly with increasing gain until it reaches the upper edge of the topological gap, as indicated by the top dashed line in Fig. 3(b). Thus nonlinear edge states bifurcate from linear ones, once gain im p exceeds corresponding k -dependent threshold. This is illustrated in Fig. 3(c), where dependencies im ( ) p for different momentum values k and k clearly start at different levels coinciding with quasienergies of corresponding linear edge states from red branch of Fig. 1(b). If quasienergy of the nonlinear edge state moves out of the topological gap for a given k , this state acquires nonzero background inside the array due to coupling to bulk modes, thus we truncate the av im ( ) a p and im ( ) p dependences in Figs. 3 accordingly. Notice that in Fig. 3(c) the upper edge of the gap is the same for two presented k values. Fig. 4. Peak amplitude versus distance illustrating (a) stable propagation of the edge state at im p in regular Floquet laser (c),(d) [corresponding | | distributions are shown in (e),(f)]; and (b) stable propagation in the presence of edge defect in the form of missing channel (h) at im p [corresponding | | distributions at different times are shown in (i),(j)]. In-3 stability development is shown in (g) for edge state with im p . In all cases k , and input states were perturbed by amplitude noise We also tested stability of all obtained dissipative edge states by perturbing them with ampli-tude noise and modelling their long-distance propagation on huge transverse windows (100 y -periods) to accommodate for all possible perturbations that could lead to instability of these states. The outcome is that for K/ 2 k considerable portions of branches of nonlinear states close to lasing threshold are stable. In Fig. 3 stable families are marked with black dots, while the unstable ones are marked with red dots. Two observations can be made: Increasing gain eventually leads to destabilisa-tion of the nonlinear states, but higher nonlinear absorption extends stability intervals, Fig. 3(a). The interval of stability in gain amplitudes im p quickly decreases away from momentum K/ 2 k , so that all states with momenta k and k were found formally unstable. However, corresponding instabilities are very weak, so in practical experimental conditions with finite samples and close to lasing threshold such states will appear as stable ones too. Stable propagation of the perturbed dissipative edge state in Floquet laser is illustrated in Figs. 4(a),(e),(f) for K/ 2 k . In Fig. 4(a), the peak amplitude max | | of the launched state is shown during propagation that clearly performs regular periodic oscillations reflecting helical structure of the wave-guide array. Note that the curve in Fig. 2(a) is a portion (in the range z ) of dependence in Fig. 4(a). Comparison of initial and output field modulus distributions in Figs. 4(e) and 4(f) showing only small fraction of actual integration window in y , reveals complete stability of the wave. In con-trast, development of instability is shown in Fig. 4(g) for large gain amplitude im p . Even in this 4 case, despite the appearance of weak irregular modulations travelling along array interface and weak radiation into the bulk, the state remains confined near the interface at any propagation distance. The striking advantage of dissipative topological edge states in Floquet laser is that they inherit topological protection of conservative edge states. To illustrate this we remove one waveguide from the left edge of helical waveguide array. The real part of corresponding array is depicted in Fig. 4(h); there is similar defect in gain profile too (not shown). Dissipative edge state launched into such a heli-cal waveguide array at im p experience some reshaping and amplitude oscillations due to presence of defect [see Fig. 4(b)], but finally reaches new stationary state shown in Figs. 4(i) and 4(j) for different distances. The representative feature of these distributions is that state is perturbed only local-ly around the defect and no radiation into bulk is visible. Finally, we note that practical Floquet lasers should be spatially compact, hence we considered also triangular geometry of the array, in which gain is again provided only in edge channels, see Figs. 5(c) and 5(d). Such a geometry may be beneficial for formation of stable edge currents, because it allows to effectively eliminate instabilities to low-frequency perturbations. To simultaneously illustrate edge currents and formation of stable attractor in this system, we start with localized excitation on the left edge of the triangle with broad Gaussian envelope [Fig. 5(e)] and let it evolve at im p and . Figures 5(f)-5(h) reveal clockwise circulation of the state accompanied by its gradual expan-sion only along the edge of array. As mentioned above, inversion of the rotation direction of wave-guides inverts also the direction of edge current in this system. Already at distances ~ 500 z the entire edge of the array becomes excited. After some transient stage the wave reaches steady-state profile depicted in Fig. 5(h), while peak amplitude of the wave stops changing [Fig. 5(a)]. Notice excellent lo-5 calization near the edge of the structure. To verify the topological protection of states in this finite sys-tem we introduced two defects to the triangular insulator by removing two channels from top and bottom edges; the corresponding real part of the array is shown in Fig. 5(i). As for the gain landscape, we removed only one channel on the bottom edge, but kept corresponding channel on the top one (not shown here). Using the same initial excitation as in Fig. 5(e) we arrived to the final steady-state profile shown in Fig. 5(j) that exhibits local deformations only around defect channels. In the presence of defects steady-state regime is reached at somewhat larger propagation distances [see Fig. 5(b) with corresponding dependence of peak amplitude on z ]. Interestingly, deformed patterns on the top and bottom edges look practically the same, that indicates that the particular type of the defect (purely con-servative or dissipative) is not important due to topological protection in Floquet laser Fig. 5. Peak amplitude versus distance illustrating stable circulation in triangular Floquet laser without edge defects (a) and with edge defects (b) at im p , . Refractive index (c) and gain (d) distributions in Floquet laser without defects and | | snapshots (e)-(h) illustrat-ing circulation in this structure. Refractive index (i) and | | distribution at large distance (j) in the Floquet laser with two edge defects 6 Summarizing, we have investigated the topological lasing in photonic Floquet topological insula-tors. We demonstrated that the edge states in this system are topologically protected against large structural perturbations and can be either dynamically stable or unstable depending on the system pa-rameters and, in particular, on the gain amplitude. We demonstrated lasing not only for an idealized infinite edge but also for a more practical triangular geometry. This work provides a practically feasi-ble scheme to obtain topological lasing without the external magnetic fields. This work was supported by Natural Science Foundation of Guangdong province (2018A0303130057), Fundamental Research Funds for the Central Universities (xzy012019038, xzy022019076), and RFBR and DFG according to the research project № 18-502-12080.
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