Theory and simulations employing coupled-cavity rate equations in an active photonic lattice show excitation of coherent low frequency collective oscillations in the photon and carrier densities, analogous to photonic sound. For parameters just below the lattice stability threshold, long range wave propagation results from external excitation of a few cavities. Above threshold long range coherent oscillations are self-excited without external stirring.
©2004 Optical Society of America
Closely packed microlaser arrays, such as guided mode VCSEL cavities and active defect superlattices, interact through their evanescent fields, Fig. 1. Fringe field interference during stimulated emission introduces active cavity coupling, whereby photons confined in one cavity induce emission in neighboring cavities. Thusly coupled cavities lead to active photonic lattice behavior that is similar in certain aspects to solid crystal, whereby the radiation envelopes play the role of atomic wavefunctions.
Specifically a weak interaction among eigenmodes that are laterally confined at each site, implying either paraxial k ⊥≡k x,y≪kz or laterally evanescent k x,y→iκ x.y modes, generates propagating pass-bands ω=ωo(kz)+Ω(K ⊥) where ωo is the unperturbed eigenfrequency and Ω is the slow frequency of the modulation envelope with lattice wavevector K ⊥. (By contrast, interference among radiating sites k ⊥~kz creates photonic bandgaps inside a continuum.) The situation may arise by the clustering of microlaser cavities characterized by index- or gain-guided stand-alone modes. Similar mechanism applies to fringe interactions among active defect-localized modes in a photonic lattice; in that case the pass-band lies inside the lattice band-gap. One may also consider a passive dielectric photonic crystal and load each site with active material emitting at a bang-gap frequency; an evanescent mode is excited in each site and the lattice turns into a coupled defect lattice.
Active coupling differs from passive dielectric interference in that it involves stimulated cross-cavity emission and absorption via photon “tunneling”. It is therefore accompanied by a periodic modulation of the carrier density over the cavity lattice. The slow coupled oscillations in the photon and carrier number constitute a “photonic sound” excited in the lattice. The role of the traditional collision-driven pressure is taken over here by stimulated photon absorption and emission. Radiation pressure is negligible because the electron recoil during emission is very small. The observed group velocity, of the order µm/ps, is comparable to the sound velocity. Near the stability boundaries of the dispersion relation the decay constant of these waves is small and long range propagation effects appear.
The coupled equations for an active photonic lattice, the spontaneous lattice relaxation into Bloch steady-states, and the stability analysis for the lattice oscillations around steady-state has been presented earlier. In this paper we focus on the numerical observation of photonic sound waves over active photonic lattices. In the stable parameter regime these waves are excited by a periodic external driving of selected array sites around the steady-state values. Sizable 1-D and 2-D arrays (up to 1×73 and 21×21) are simulated with either periodic or finite boundaries. For parameters well inside the stable regime the excited waves are decaying over few sites; they become long range waves reaching the system boundaries for parameters near the stability boundary. Above the stability boundary we observe self-excitation of non-linear waves without any external stirring and under uniform in space and time applied laser biasing. They appear as photonic “convection cells”, superficially resembling convection patterns in uniformly heated layers. Chaotic lattice patterns emerge without stirring at even higher bias.
We consider 1-D or 2-D slab array, Fig. 1, with a periodic xy (lateral)plane arrangement of laser cavities centered at lattice points R ij=i bx+j b y. Each stand-alone cavity is characterized by profiles of complex gain coefficient g, mirror losses µ, dielectric constant ∊ and passive absorption α of the general form ν(r)=νo+δνχν(r) where ν stands for g, µ, ∊, α respectively and νo denotes a uniform background. The eigenmodes of the uncoupled cavities are U eikz-i ω t where U(r) is the transverse mode profile and z the axial direction. The material property variations over the lattice have the general form ν(r)=νo+δν∑ij χν(r-Rij); the ij-th site experiences the other sites as a perturbation of the form Δνij(r)=δν∑i′≠i, j′≠j χν(r-R i′j′). For weakly interacting cavities the collective eigenmodes are written in the tight-binding approximation as
εij being the ij-cavity complex amplitude. We will focus on the evolution of the discrete lattice envelope εij(t), factoring out the fact dependence eikz-iωt. By substituting the form (1) inside the Maxwell-Bloch equations, averaging over axial length z, and taking the integral projections with U around each lattice site, the coupled lattice equations retaining near neighbor interactions i ′=i±1, j ′=j±1 for the carrier density N, the photon density ℵ and the phase difference Φij;i′j′≡φ ij-φ i ′ j ′ among cavities are
Here gr, gi are the real and imaginary parts of the gain coefficient, ℵ is given in terms of the original electric field amplitude ε=Eeiφ as ℵ=∊εε*/8π.ħω and N̂ij≡Nij/Ntr with Ntr the transparency density. The cavity coupling strength enters via and the combined phase Ψij;i′j′=Φij;i′j′+ϑij where ϑ=tan -1Ξ/Z and Z and Ξ given by
involve, respectively, the lump-summed contributions from the imaginary dielectric (real gain gr, absorption δα, mirror losses δµ) and the real dielectric (imaginary gain gi and refractive δ∊) response. The coefficients ϒg, ϒµ, ϒα and ϒ∊ express thintercavity overlap strengths among cavity eigenmode profiles U(r-Rij). These modes are lattice orthonormal ∫dr2 U(r-Rij)U(r-R i′j′)=∫dr2 U(r)U(r±ΔR i-i′,j-j′)=δ i-i′,j-j′ thus the near-neighbor ΔR i-i′,j-j′=±b coupling strengths Ya≡∫dr U(r)χa(r)U(r±b) stem entirely from the variations in the gain, mirror reflectivity δµ, absorption δα, and dielectric constant δ∊ from their respective “floor” values. The different strengths of Yν, ν=g, µ, ∊, α reflect the differences in the spatial profiles χν(r); for identical profiles χ all coupling terms assume the same value ϒ. Finally the term S ij;i′j′=Λg ζgrlnN̂i′j′+Λµ µ-Λ∊ δα̂ lump-sums the phase-independent contributions Λν≡∫dr2 U(r±b)χν(r)U(r±b) from inter-cavity interactions. Notice that the carrier density evolution involves only real gain-coupling terms. For small coupling strengths ϒν≪1, and for around steady-state variations, the carrier and photon densities scale as N≃No+𝒪(ϒ), ℵ≃ℵo+𝒪(ϒ), where the subscript o means the steady-state values for an isolated-cavity; for linear oscillations we may approximate Z ij;i′j′≃Zo and Ξij;i′j′Ξo with ℵo inside Eqs. (5)–(6).
The collective array behavior has been investigated via numerical integration of the coupled lattice Eqs. (2)–(4), starting with random initial conditions. Under lattice-periodic J ij(r)=JoχJ(r), constant in time drive currents we observe a spontaneous relaxation to steady-states dN/dt=dℵ/dt=0. For periodic arrays the lattice envelope assumes a Bloch wave form, where all amplitudes are constant Nij=No, ℵij=ℵo and the phase-difference among all neighbor sites is fixed Φx,y ij;i′j′≡φij-φ i′j′=K mn·b x,y and proportional to an inverse lattice vector Kmn=(m, n)2π/L. In general
Any value Φ yields a steady state for the rate Eqs. (2)–(4) but configurations maximizing the collective gain are favored; thus Φ is either zero or π for Z positive or negative. The steady-state values No, ℵo are obtained analytically from the zeros of Eqs. (2)–(4).
We proceed to examine the collective evolution of small perturbations about given steady state values Φij=Φo, εij=εo, Nij=No. Collective modes over a periodic lattice conform with the discretized version of a Bloch wave sampled over lattice sites
δA being the lattice envelope of either carrier or photon density. Similar variations in the intercavity phase shift, of amplitude δΦo, are caused by out-of-phase electric field changes δε relative to the steady state field εo. Replacing independent phase variations by a lattice-periodic envelope phase manifests the transition to collective behavior with long range coherence (in general situations far from equilibrium, Φij in Eqs. (2)–(4) are independent variables). The description resembles the dynamic behavior of a periodic, spring-coupled mass system. Due to the periodicity, linearization of Eqs. (2)–(4) about any site leads to identical stability equations. Expanding around equilibrium and using Eq. (8) yields the following stability matrix for perturbations about steady-state
The elements DXY≡∂(XẊ)/∂Y with X, Y being either of N, ε, Φ are found from the rhs of Eqs. (2)–(4) at steady-state - the linearization is performed using complex E notation and the results are then recast in terms of ℵ∝|ε|2. The roots of the cubic characteristic equation
yield the dispersion relation for lattice perturbation wavenumber κ, affected by the perturbed steady state period K. It has been shown that the real (non-oscillating) root is negative (stable) λ3<0 over the entire parameter space of interest. Lattice stability is thus determined by the Γ1,2 sign of the complex root pair λ1,2. Linear stability depends primarily on the ratio gi/gr between the imaginary and real gain coefficients. Unconditional stability exists for gi/gr≤1, yielding a dispersion of stable, time decaying collective oscillations. For gi/gr>1 the stability is conditional and depends on the coupling strength Ƶ. At larger coupling strengths we observe transition to limit cycles, and eventually chaotic oscillations. Finally, for gi/gr≪1, the complex roots are approximated by the eigenvalues of the upper 2×2 submatrix.
Dispersion Eq. (10) represents complex frequency ω for a real wavenumber κ. It applies to initial value problems, regarding the time evolution of periodic lattice perturbations. Equivalently, complex wavenumber solutions κ→κ+iξ exist for real excited frequencies ω. That applies to boundary value problems, related to the excitation of selected lattice sites with real frequency ωo, and subsequent wave propagation. Near the boundary of the stability regime, where Γ≪Ω, imposing ℑω=ℑ(λ)=0 to the analytic continuation of Eq. (10) with complex κ yields the decay constant ξ,
where Vgr is the lattice group velocity. Since ξ is proportional to the decay rate Γ, when the lattice parameters are well inside the stability region the decay constant is fairly large, ξb x,y~1, and driven waves decay over few lattice sites. Figure 2 shows various snapshots of the photon density envelope ℵij obtained during a simulation of 21×21 finite VCSEL arrays. Three central sites i=10, j=10-12 are driven by a sinusoidal drive current of amplitude Idr/Ith=1.1 superimposed on the steady-state drive Io/Ith=3.1 for the entire array. The coupling strength in all cases is Z=0.0066 and yields in-phase K x,y b x,y=0 steady-state. In Fig. 2a (46ns) the ratio gi/gr=0.5 lies well inside the stability regime and yields evanescent waves within two lattice sites, regardless of the drive frequency ωo. Increasing the ratio gi/gr=1.5 near the stability boundary excites small ξb x,y≪1, long range waves propagating over the entire array, Figs. 2b (44ns) and 2c (37ns). The wave periods κ x,y b x,y=π/2 and κ x,y b x,y=π respectively match the drive current frequencies ωo=Ω(κ x,y;K=0) from Eq. (10). The wave decay is in large part due to the 1/r dependence of the “point”-source radiation, due to energy conservation.
Figure 3 plots the dynamic in time evolution of a 1×49 array excited at j=25, gi/gr=1.5 and same drive currents as before. The coupling strength Z=0.0133 corresponds to a site separation-to-radiation waist ratio b/ω≃2. The excited wave pattern of κb=π/2 (four sites per wavelength) matches the applied drive frequency ωo=Ω(κ). The amplitude decay rate in 1D is due solely to the decay constant γ value. For typical site separation b=4µm, the group velocity estimated from the Mach cone is υgr≃2.4×104 cm/s, of the order of the sound velocity in solids.
When a lattice parameter crosses the stability boundary there is no steady-state relaxation. Instead a limit cycle emerges where the lattice undergoes a periodic sequence of coherent non-uniform patterns, without external stirring. Figure 4 shows two snap-shots of the electric field envelope for a finite 21×21 array with gi/gr=4 and Ƶ=0.00155. Surface of section plots Nij(tn) vs. ℵij(tn) at a given site, recorded at time increments tn, exhibit periodic non-linear cycles. The trajectories for all lattice sites fall into the same curve, each site separated by a phase shift around the cycle relative to its neighbors. The transition from a fixed point Nij=No to a limit cycle is an example of a Hopf bifurcation. For even higher coupling strengths Ƶ we observe transitions to intermittent and finally fully spatio/temporary chaotic states, under constant and uniform external biasing.
References and links
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