Abstract

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

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References

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  1. S. Riyopoulos, “Coherent phase locking, collective oscillations and stability in coupled VCSEL arrays,” Phys. Rev. A 66, 53820 (2002).
    [Crossref]
  2. J. D. Joannopoulos, R. D. Meade, and J. N. Winn in Photonic Crystals (Princeton University, Princeton N.J., 1995).
  3. O. Painter, J. Vuckovic, and A. Scherer. “Defect Modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B 16, 275 (1999).
    [Crossref]
  4. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
    [Crossref]

2002 (1)

S. Riyopoulos, “Coherent phase locking, collective oscillations and stability in coupled VCSEL arrays,” Phys. Rev. A 66, 53820 (2002).
[Crossref]

1999 (2)

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn in Photonic Crystals (Princeton University, Princeton N.J., 1995).

Lee, R. K.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn in Photonic Crystals (Princeton University, Princeton N.J., 1995).

Painter, O.

Riyopoulos, S.

S. Riyopoulos, “Coherent phase locking, collective oscillations and stability in coupled VCSEL arrays,” Phys. Rev. A 66, 53820 (2002).
[Crossref]

Scherer, A.

Vuckovic, J.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn in Photonic Crystals (Princeton University, Princeton N.J., 1995).

Xu, Y.

Yariv, A.

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Phys. Rev. A (1)

S. Riyopoulos, “Coherent phase locking, collective oscillations and stability in coupled VCSEL arrays,” Phys. Rev. A 66, 53820 (2002).
[Crossref]

Other (1)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn in Photonic Crystals (Princeton University, Princeton N.J., 1995).

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Figures (4)

Fig. 1.
Fig. 1. Schematic, top view of 2-D slab array configuration. a active region radius, am mirror radius, b center separation, w the 1/e 2 mode waist
Fig. 2.
Fig. 2. Time snap-shots of photon density in each lattice site (cavity) over 21×21 array of coupling strength Ƶ=0.0066. Sites i=10, j=10-12 driven by a sinusoidal current of amplitude Idr /Ith =1.1 on top of the steady-state Io /Ith =3.1 over entire array. (a) Stable array with gi /gr =0.5 driven by ωo =Ω(κyby =π/2), after t=46ns (b) Marginally stable array gi /gr =1.5 driven by ωo =Ω(κyby =π/2), after t=44ns (c) Marginally stable array gi /gr =1.5 driven by ωo =Ω(κyby =π), after t=37ns.
Fig. 3.
Fig. 3. Sound wave propagation over a 1×49 array of coupling strength Ƶ=0.0133 and gi /gr =1.5. Site j=25 driven by a sinusoidal current of amplitude Idr /Ith =1.1 and ωo =Ω(κyby =π/2). (a) Time evolution of photon density over the array sites for 400 ns (b) Contour plot over 600ns.
Fig. 4.
Fig. 4. Time snap-shots of photon density in each lattice site (cavity) over 21×21 array. For gi /gr =4 the coupling strength Z=0.0017 lies deeply inside the unstable regime, yielding cyclic self excitation of non-linear periodic patterns, shown at (a) 60 ns (b) 80 ns.

Equations (14)

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ε ( r , t ) = e ikz i ω t i , j ε ij ( t ) e i φ ij ( t ) U ( r R ij )
N ij t = J ij e d w γ N ij B N ij 2 υ g g r ln N ̂ ij ij υ g g r ln N ̂ ij Λ g i , j i j
υ g g r ln N ̂ ij 2 ϒ g i ' , j ' ij i j cos Φ ij ; i j
ij t = υ g ζ g r ln N ̂ ij ij υ g μ ij + i ' , j ' υ g S ij ; i j ij
+ i , j υ g 2 Z ij ; i j ij i j cos Ψ ij ; i j
Φ ij ; i j t = υ g ζ g i 2 ln ( N ij N i , j 1 ) + Λ g i , j υ g ζ g i 2 ln ( N i j N i j 1 )
+ i , j υ g 2 2 { Z i j ; i j i j ij sin Ψ ij ; i j Z ij 1 ; i j 1 i j 1 i , j 1 sin Ψ i , j 1 ; i , j 1 }
Z ij ; i j = ϒ g ζ g r ln N ̂ ij N ̂ i j ϒ α δ α ̂ + ϒ μ μ
Ξ ij ; i j = ϒ g ζ g i ln N ̂ ij N ̂ i j + ϒ ω υ g 1 δ ̂ .
mn ( r ) = o e i δ ω t i , t e i K mn · R ij U ( r R ij ) + cc .
δ A ( r ) = δ A o e λ t i , j e κ · R ij ,
( D N N λ D N F D N Φ D F N D F F λ D N Φ D Φ N D Φ F D Φ Φ λ ) = 0
λ q Γ q ( κ ; K ) + i Ω q ( κ ; K ) , q = 1 , 2 , 3
ξ ( ω ; K ) = Γ ( κ ; K ) ( Ω κ ) Ω = ω 1 = Γ ( κ ; K ) V gr 1

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