Self-organized vortex and antivortex patterns in laser arrays
Mostafa Honari-Latifpour, Jiajie Ding, So Takei, Mohammad-Ali Miri
SSelf-organized vortex and antivortex patterns in laser arrays
M. Honari-Latifpour, J. Ding, S. Takei, and M.-A. Miri a) Department of Physics, Queens College of the City University of New York, Queens, New York 11367,USAPhysics Program, The Graduate Center, City University of New York, New York, New York 10016,USA (Dated: 23 February 2021)
Recently it is shown that dissipatively coupled laser arrays simulate the classical XY model. We show that phase-locking of laser arrays can give rise to the spontaneous formation of vortex and antivortex phase patterns that areanalogous to topological defects of the XY model. These patterns are stable although their formation is less likely incomparison to the ground state lasing mode. In addition, we show that small ratios of photon to gain lifetime destabilizesvortex and antivortex phase patterns. These findings are important for studying topological effects in optics as well asfor designing laser array devices.
I. INTRODUCTION
The two-dimensional XY model consists of a lattice of in-teracting fixed-length spins that are constrained to rotate in aplane . The classical XY model in two spatial dimensions isgoverned by the Hamiltonian H ( φ , φ , · · · , φ n ) : H = ∑ (cid:104) i , j (cid:105) κ i j cos ( φ i − φ j ) (1)where, φ , · · · , φ n represent the orientation of the n spins, and κ i j is the interaction for the pair i , j . This model supports non-trivial equilibrium spin configurations — a class of topologi-cal defects — known as vortices, which are characterized bythe phase of the spins going through a multiple of 2 π as onetraces a loop enclosing the vortex, e.g., (cid:90) d φ = ± π (2)for a single vortex/antivortex. In the 2D XY model, vorticeshave found applications in various areas of condensed mat-ter physics including superfluid helium-4 , superconductiv-ity in thin films , liquid crystals , and the melting of 2Dcrystals . (a) (b) FIG. 1. A schematic of the equilibrium phase patterns of a dissipa-tively coupled laser array. (a) The ground state. (b) A vortex state.Here, the arrows represent the phase of each element. a) Electronic mail: [email protected]
Recently, it has been realized that the classical XY modelcan be optically simulated with an array of coupled opticaloscillators . What makes this possible is the random phaseof a laser above oscillation thresholds, which emulates a clas-sical spin confined to a two-dimensional plane. In addition,dissipative interaction facilitates synchronization of an arrayof lasers to a globally-phase-locked state . In this case, onecan show that the laser array reaches an equilibrium phasepattern that locally minimizes a cost function that is asymp-totically equivalent with the classical XY Hamiltonian . Ac-cordingly, laser arrays have been utilized for simulating inter-esting phenomena related to spin systems such as geometricfrustration .In this work, we investigate the formation of vortex and an-tivortex singularities as self-organized phase patterns in laserarrays. As shown schematically in Fig. 1, these patterns areequilibrium states of laser arrays when reaching a globally-phase-locked state. From a nonlinear dynamics point of view,these topological defects are fixed-point solutions of nonlineardynamical equations governing laser arrays. However, in gen-eral, they can be considered as metastable states with finitebasins of attraction which restricts their formation to properinitial conditions. In addition, we find that the stability ofthese states depends critically on the gain level and lifetime.In the following, after introducing a dynamical model govern-ing laser arrays, first we numerically investigate the formationof vortices. Next, we draw a connection between the govern-ing dynamical model and a class of Ginzburg-Landau systemsthat are known to support vortex patterns. Finally, we inves-tigate the stability of the vortex patterns with respect to thecompeting time scales of optical cavity and gain decay ratesof the lasers. II. MODEL
To build a dynamical model governing laser arrays, wefirst consider an array of passive and single-mode optical res-onators that dissipatively interact . For the sake of sim-plicity, we assume all resonators being identical in resonancefrequency ω and linewidth 1 / τ p . Thus, in the framework ofthe temporal coupled mode theory , the complex modal am- a r X i v : . [ phy s i c s . op ti c s ] F e b a b c d H = -480 H = -469 H = -462H = -469
FIG. 2. Equilibrium phase patterns of an array of dissipatively coupled lasers arranged on a 16 ×
16 square lattice. (a-d) The ground state,vortex, antivortex and bound vortex-antivortex states for the case of attractive coupling ( κ > κ < g = κ = − κ = plitude of the electric field in the i th resonator is governed by:˙ a i ( t ) = − a i − γ i a i − ∑ j (cid:54) = i κ i j a j , (3)where, the equations are written within the gauge a i → a i e − i ω t and time is normalized to the photon lifetime τ p . Inthis relation, κ i j represents the rate of dissipative coupling be-tween the i th and j th resonators, γ i = ∑ j | κ i j | is the externalloss of the i th resonator due to its coupling with other res-onators as demanded by conservation relations . Here, allcoupling coefficients κ i j are normalized to the photon decayrate 1 / τ p , while the choice of normalization for the complexamplitudes depends on the gain as discussed next.By incorporating a saturable gain mechanism, equations (3)can be modified to support self-sustained finite stationary so-lutions of the field amplitudes. Here, we consider the gainbeing a dynamical variable as in the so-called class-B lasermodel . In this model, the gain of a laser oscillator is drivenat a finite pump rate, while it decays linearly for small field in-tensities and nonlinearly when the field intensity grows. Thenormalized rate equations for the i th oscillator can then bewritten as: ˙ a i ( t ) = [ g i ( t ) − ] a i − γ i a i − ∑ j (cid:54) = i κ i j a j , (4a)˙ g i ( t ) = ( τ p / τ g )[ g − ( + | a i | ) g i ] , (4b)where g i represents the gain of the i th oscillator, g is thepump parameter, and 1 / τ g is the gain decay rate. In these rela-tions both the field amplitude and gain are dimensionless and the time is normalized to the photon lifetime τ p . This modelhas been applied to solid-state lasers . In addition, it can begeneralized to model semiconductor lasers by incorporatingthe linewidth enhancement factor which plays an importantrole in the dynamics .Equations (4) can be greatly simplified when the gain decayrate is much larger than the photon decay rate, i.e., 1 / τ g (cid:29) / τ p . In this case, the gain almost instantaneously follows -3 -3 -3 % -485-480-475-470-465-460-455 H XY -3 -3 -3 % -485-480-475-470-465-460-455 H XY FIG. 3. The XY energy distribution associated with the equilibriumphase patterns of a 16 ×
16 rectangular lattice laser array for (a) fer-romagnetic, and (b) anti-ferromagnetic coupling. The histograms areproduced based on 1000 simulations with initial phases drawn ran-domly with uniform distribution from the range [ , π ] . All parame-ters are the same as in Fig. 2. a b c FIG. 4. The steady state phase pattern of a laser array arranged on a triangular lattice. Here, κ = − g = the dynamics of the field. Thus, one can adiabatically elim-inate the dynamics of the gain, i.e., ˙ g i ( t ) ≈
0, to reach at aninstantaneous nonlinear gain g i ( | a i | ) = g / ( + | a i | ) . In thismanner, one reaches at a reduced model that is suitable for aso-called class-A laser . Here, we use a polynomial gainterm g i ( a i ) = g ( − | a i | ) , which is enough to guarantee thebounded oscillations of the laser. Thus, we reach at the fol-lowing reduced model,˙ a i ( t ) = [ g ( − | a i | ) − ] a i − γ i a i − ∑ j (cid:54) = i κ i j a j , (5)which is accurate as long as the photon decay rate is smallerthan the decay rates of the atomic degrees of freedom that giveresult to the gain. In this work, first we focus on simulatingequations (5). Next, we return to equations (4) and discuss theeffect of the gain dynamics.The dynamical model of Eq. (5) admits a Lyapunov func-tion F such that ˙ a i = − ∂ F / ∂ a ∗ i , where F = ∑ i [ − ( g − − γ i ) | a i | + g | a i | ] + ∑ i , j a ∗ i κ i j a j . (6)The fixed point solutions of the dynamical system of Eq. (5)are local minima of this Lyapunov function. In a recent work -3 -3 -3 % -755-750-745-740 H XY -3 -3 -3 % -380-375-370-365-360-355-350 H XY FIG. 5. The XY energy distribution associated with the equilibriumphase patterns of a triangular lattice arrangement of lasers with (a)ferromagnetic ( κ < κ >
0) coupling. Thelattice size and array parameters are the same as in Fig. 4 we showed that the diagonal term in this cost function be-haves like a penalty term that tends to force all oscillators toa constant amplitude in the large gain limit, g (cid:29) . Thus,considering the stationary state solution of the oscillators as a i = √ I i e i φ i by enforcing the condition of I i = I , the cost func-tion of Eq. (6) reduces to the XY Hamiltonian of Eq. (1). III. FORMATION OF TOPOLOGICAL DEFECTS
In the following, we investigate self-organization of topo-logical defects by numerically simulating the dynamicalmodel of Eqs. (5). We first consider an array of lasers arrangedon a square lattice with uniform nearest-neighbor coupling ofstrength κ . Here, the gain is assumed to be large ( g (cid:29) ×
16 oscillators for both sce-narios of attractive ( κ <
0) and repulsive ( κ >
0) coupling,which are respectively associated with ferromagnetic and an-tiferromagnetic cases. In both cases, different stable patternsare observed, including the ground state, isolated vortex andantivortex states, and paired vortex-antivortex states. The XYenergy levels associated with these equilibrium phase patternsare listed in Fig. 2, which shows higher energy levels for thetopological defects. The difference between the XY energy ofthe vortex (Fig. 2(b)) and the ground state (Fig. 2(a)) is com-parable with the approximate formula ∆ E = πκ ln L , where L is the lattice length.The stability of these fixed point solutions is directly eval-uated through the Jacobian matrix of the dynamical system ofEqs. (5): J = g (cid:18) diag ( − a (cid:12) ¯ a ∗ ) − I − Q − diag ( ¯ a (cid:12) ¯ a ) − diag ( ¯ a ∗ (cid:12) ¯ a ∗ ) diag ( − a (cid:12) ¯ a ∗ ) − I − Q (cid:19) . (7) In this relation, ¯ a = ( ¯ a , · · · , ¯ a n ) t is the stationary state, Irepresents the identity matrix, Q is the coupling matrix, where q i j = κ i j and q ii = ∑ j | κ i j | , (cid:12) represents the entry-wise prod-uct, and diag creates a diagonal matrix of a given vector. TheJacobian matrix turns out to be a negative semi-definite matrixfor all cases shown in Fig. 2. i ii iiiiv v vi a b i ii iii iv v vi FIG. 6. The transient dynamics of a 16 ×
16 laser array. (a) Snapshots of the phase pattern at intermediate times. (b) Amplitudes and andphases of the array elements. In part (b), the dashed lines show the time instants associated with the snapshots in part (a). All parameters arethe same as in Fig. 2.
Given the finite attraction basins of the phase patternsshown in Fig. 2, formation of these patterns depends on theinitial conditions. In a realistic scenario, one can assume thatthe initial conditions are uniformly sampled from the high-dimensional phase space of the laser array. Thus, as localminima of the Lyapunov function, topological defects are ex-pected to have lower chances of formation. To explore this as-pect, we simulated the dynamical system for a large ensembleof random initial conditions. In these simulations, the initialphases were drawn from a uniform random distribution withinthe range [ , π ] , while the initial amplitudes were taken to besmall values. The resulting equilibrium energy distribution isshown in Fig. 3 for both cases of ferromagnetic and antiferro-magnetic systems. As this figure clearly indicates, topologi-cal defects form at much lower probabilities compared to theground state.The vortex and antivortex phase patterns can also form inthe triangular lattice geometry as shown in the simulations ofFig. 4. In this lattice geometry, the antiferromagnetic case ismore complex due to the geometric frustration. The XY en-ergy distributions associated with the equilibrium phase pat-terns of the ferromagnetic and antiferromagnetic cases are de-picted in Fig. 5. This figure indicates the complexity of theantiferromagnetic system due to its large number of higherenergy states.It is worth noting that the transient dynamics of the laserarray reveals unstable topological defects that disappear byreaching the lattice boundaries or by collapsing of vortex-antivortex pairs. An exemplary dynamics of a laser network isshown in Fig. 6. To visualize the transient dynamics, the phasepattern in depicted at intermediate time scales. It is worthnoting that similar behavior was reported in a reduced modelbased on Kuramoto phase oscillators, which showed rapid de-cay of a large number of transient topological defects . IV. DISCRETE GINZBURG LANDAU EQUATION
In order to provide insight to the formation vortices in laserarrays, we draw a connection between the lattice model ofEq. (5) with its continuum counterpart, which turns out tobe the well-known Ginzburg-Landau Equation (GLE). To un-derstand this analogy, we focus our attention to the case ofa 2D square lattice arrangement of lasers with uniform near-est neighbor coupling κ , and consider the ferromagnetic case( κ < ( m , n ) fordescribing the horizontal and vertical coordinates of a squarelattice, one can rewrite Eqs. (5) as:˙ a m , n ( t ) = [ g ( − | a m , n | ) − ] a m , n − κ L a m , n (8)where, L is the discrete Laplacian operator on a square latticegraph that acts as L a m , n = a m − , n + a m + , n + a m , n − + a m , n + − a m , n . This relation is clearly in a finite difference form. Toconstruct the continuum counterpart of this relation, we use ( m , n ) → ( x , y ) , a m , n ( t ) → ψ ( x , y , t ) , and L → ∇ , which re-sults in the Ginzburg-Landau Equation (GLE):˙ ψ ( x , y , t ) = [ g ( − | ψ | ) − ] ψ − κ ∇ ψ . (9)Given that all coefficients are real-valued, despite the fact that ψ is complex, this equation is often referred to as the realGinzburg-Landau equation .The dynamical equation (9) can be written in terms of avariation of a functional F [ ψ ] , as ˙ ψ = − δ F / δ ψ ∗ . The func-tional F is found to be: F = (cid:90) dxdy (cid:104) − ( g − ) | ψ | + g | ψ | + | ∇ ψ | (cid:105) , (10)which, can be considered as a counterpart of the Lyapunovfunction of Eq. (6). In an amplitude and phase representation a m p li t ude
05 05100 5 10 15 20 t ga i n t t
05t = 0 t = 5 t =10 t = 15 t = 20 t = 0 t = 5 t =10 t = 15 t = 20 t = 0 t = 5 t =10 t = 15 t = 20 a b c
FIG. 7. The dynamics of a 6 × τ p / τ g =
10 (b) 1, and (c) 0 .
1. In all cases, κ = − g = ψ ( r , φ , t ) = R ( r , φ , t ) exp [ i Φ ( φ ( r , φ , t ))] , Eq. (9) can be rewrit-ten as:˙ R ( r , φ , t ) = [ g ( − R ) − ] R − κ ( ∇ − | ∇Φ | ) R , (11a) R ˙ Φ ( r , φ , t ) = − κ ( ∇ R · ∇Φ + R ∇ Φ ) . (11b)These equations are a special class of a reaction-diffusion sys-tem, named λ − ω systems . It is shown that these equationssupport stable single-arm spiral wave solutions of the form R ( r , φ , t ) = R ( r ) and Φ ( r , φ , t ) = φ , while multi-arm spiralwaves are unstable . Driven from this intuition, one wouldexpect discrete counterparts of the spiral patterns in lattices ofcoupled lasers. An obvious deviation of the lattice model fromthe continuum case is the finite amplitude of the field at thecenter of the vortex. In the continuum scenario, the vanishingamplitude at the phase singularity guarantees a well-definedfield. In the case of the lattice model, however, there is nosuch requirement. This is clearly irrelevant in the case of thelattice model, although the amplitudes still tend to be lower atthe lattice sites that are located near the vortex center. Never-theless, the amplitudes become uniform in the large gain limit( g → ∞ ), where the laser system approaches the XY model. V. VORTICES IN THE DYNAMIC GAIN REGIME
The results presented in Figs. 2-6 are based on the complexamplitude model with an instantaneous nonlinear saturablegain described by Eqs. (5). Here, we consider the more real-istic scenario that involves gain as a dynamical variable as de-scribed through Eqs. (4). The gain lifetime τ g , which in realityis dictated by the laser gain material, is found to play a criticalrole in the dynamics and in formation of stationary topologi-cal defects. As mentioned earlier, when the ratio of the photonlifetime over the gain lifetime is large ( τ p / τ g (cid:29) τ p / τ g ∼ τ p / τ g (cid:28) τ p / τ g while usingidentical initial conditions. The results are depicted in Figs. 7.As these exemplary simulations indicate, by decreasing theratio of τ p / τ g , the transient dynamics becomes more com-plicated, involving effects like self-pulsations that are knownprocesses in class-B lasers. On the other hand, these figuresshow that the vortex state does not form for smaller ratios of τ p / τ g (Figs. 7(b,c)).This can be explained through the dynamics of the gain.According to Figs. 7(b,c), by increasing τ g , the gain increasesslowly and during this process, the laser array finds the time toescape from patterns associated with higher XY energy levelsand to stabilize into the ground state phase pattern. This is inagreement with our recent study based on the reduced model,while the gain was adiabatically increased to avoid trappinginto local minima . Here, the dynamics allows the gain to au-tomatically increase over a time scale governed by τ g . To sys-tematically explore this aspect, the laser array was simulatedwith an ensemble of random initial conditions and for threedifferent time scale ratios. In addition, both cases of rectan-gular and triangular lattices were considered. The XY energydistributions of the associated steady-state phase patterns areshown in Fig. 8. As expected, for τ p / τ g (cid:29)
1, the equilibriumenergy distribution is similar to the case of instantaneous gain(Figs. 3(b) and 5(b)). As this ratio increases, however, theoccurrence of higher energy patterns drops significantly.These results clearly suggest that smaller ratios of τ p / τ g destabilize the vortex and antivortex patterns. However, it isquite interesting to note that these patterns pass the linear sta-bility test in the both regimes of small and large time scaleratios. To show this, first we consider the fixed point solutionsof Eqs. (4), i.e., ( a i ( t ) , g i ( t )) = ( ¯ a i , ¯ g i ) . The fixed points are -3 -2 -1 % -375-370-365-360-355-350 H XY -485-480-475-470-465-460-455 H XY -3 -2 -1 % -375-370-365-360-355-350 H XY -3 -2 -1 % -375-370-365-360-355-350 H XY -485-480-475-470-465-460-455 H XY -485-480-475-470-465-460-455 H XY FIG. 8. The XY energy distribution associated with the equilibriumphase pattern of laser arrays governed by the dynamical equations (4)for different photon to gain lifetime ratios. Top: The square latticefor (a) τ p / τ g =
10, (b) 1 .
0, and (c) 0 .
01. Bottom: The triangularlattice for (a) τ p / τ g =
10, (b) 1 .
0, and (c) 0 .
01. Here, κ = − g =
30, while the lattice size is the same as in Figs. 2 and 4 for thetop and bottom rows, respectively. governed by a set of nonlinear algebraic equations: [ ¯ g i − ] ¯ a i − γ i ¯ a i − ∑ j (cid:54) = i κ i j ¯ a j = , (12a) [ g − ( + | ¯ a i | ) ¯ g i ] = , (12b)which, are independent from the time scale ratio τ p / τ g . Thus,as an example, the vortex state found in Fig. 7(a) for τ p / τ g =
10 is also a valid solution of the system explored in Fig. 7(c)for τ p / τ g = .
1. However, while numerical simulations in-dicate that the vortex state is stable in the former case, it re-mains to investigate its stability in the latter scenario. In thiscase again, it is straightforward to perform a linear stabilityanalysis through the Jacobian matrix: J = diag ( ¯ g ) − I − Q diag ( ¯ a ) − τ diag ( ¯ a (cid:12) ¯ g ) ∗ − τ ( I + diag ( ¯ a (cid:12) ¯ a ∗ )) − τ diag ( ¯ a (cid:12) ¯ g ) diag ( ¯ a ) ∗ diag ( ¯ g ) − I − Q (13) where, ¯ a = ( ¯ a , · · · , ¯ a n ) t , ¯ g = ( ¯ g , · · · , ¯ g n ) t , and τ = τ p / τ g . Byconsidering the vortex pattern of Fig. 7(a) as the fixed pointsolution, the eigenvalues of the Jacobian matrix were numer-ically calculated for a range of values of τ p / τ g . Figure 9 de-picts the real part of all eigenvalues versus τ p / τ g , while thelead eigenvalue, i.e., the one with the largest real part, is high-lighted in red. Clearly, all eigenvalues exhibit negative realparts in the entire range of the time scale ratio, which showslinear stability of the vortex state in all regimes. However, ac-cording to this figure, the eigenvalues undergo a square-roottransition and increase toward zero for small values of τ p / τ g .This clearly suggests the evolution of the vortex from a stableto a metastable state that can disappear with a small perturba-tion, which is in agreement with the simulations. FIG. 9. The real part of the Jacobian matrix eigenvalues versus thephoton to gain lifetime ratio τ p / τ g . These eigenvalues represent thelinear stability test for a vortex pattern (shown as inset) in a 6 × VI. CONCLUSION
In summary, we investigated the formation of vortex andantivortex phase patterns in laser arrays. We showed that forlarge gains these patterns exist in direct analogy with topolog-ical defects of the XY model. However, their stability dependscritically on the ratio of photon to gain lifetime τ p / τ g . An im-portant finding was that large gain lifetimes ( τ g ) destabilizetopological defects and force the system into the ground state.The presence of self-organized vortex and antivortex phasepatterns in laser arrays could have practical applications giventhat these patterns can emit optical vortex beams. It is worthnoting that both solid-state and semiconductor lasers are con-sidered class-B lasers given that in these systems the fluores-cence lifetime is by several orders of magnitudes larger thanthe photon lifetime ( τ p / τ g (cid:28) . Nevertheless, certain 2D systems can exhibitsigns of a BKT transition from a high-temperature disorderedstate, with short-range correlations, to a quasi-ordered state,with algebraic correlations, below a critical temperature .When viewed in terms of the vortices, the BKT transition is avortex-unbinding transition. These topological objects, whiletightly bound in pairs in the low-temperature state, unbind andfreely proliferate above the critical temperature, causing thesystem to melt its quasi-order. While the present work sug-gests such a transition, a rigorous investigation of this aspectrequires a non-equilibrium thermodynamic treatment of thelaser array which could be the subject of future studies. ACKNOWLEDGMENTS
We thank Dr. Morrel H. Cohen for bringing the subject ofthe BKT phase transition to our attention.
DATA AVAILABILITY
The data that support the findings of this study are availablefrom the corresponding author upon reasonable request.
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