Abstract

Deviations in the cold cavity parameters, random or systematic, produce incoherently phased-locked laser arrays with unevenly distributed phase difference and intensity. The collective radiation fields constitute “fuzzy” eigenmodes; the phasing among cavities is constant in time but changes randomly from site-to-site. The existence and structure of such eigenmodes is demonstrated numerically and analyzed theoretically using the rate equations for coupled semiconductor laser cavities. Active coupling, whereby one cavity’s radiation field modulates the complex gain of nearby cavities (cross-cavity hole burning), is essential for the frequency pulling allowing synchronization of the laser operating frequencies.

© 2006 Optical Society of America

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References

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  1. J. K. Butler, D. E. Ackley and M. Ettenberg, "Coupled-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays," IEEE J. Quantum Electron. 21, 458-464 (1990).
    [CrossRef]
  2. S. Riyopoulos, "Tolerance of phase-locked VCSEL arrays to random and systematic parameter deviations among cavities," IEEE JQE 41, 1450-1460 (2005).
    [CrossRef]
  3. S. Riyopoulos,"Effects of non-linear frequency-pulling on the cavity phasing and the collective mode structure in phase-locked VCSEL arrays," J. Opt. Soc. Am. B 23, 250-256 (2006).
    [CrossRef]
  4. J. D. Joannopoulos, R. D. Meade and J. N. Winn in "Photonic Crystals," (Princeton University, Princeton N.J., 1995).
  5. S. Riyopoulos, "Simulations of boundary layers and point defects in coupled VCSEL arrays," IEEE J. Sel. Top. Quantum Electron. 11, 945-957 (2005).
    [CrossRef]
  6. S. Riyopoulos, "Phase stability theory of Bloch eigenstates in active photonic lattices with coupled microlaser arrays," Eur. Phys. J. D 36, 295-317 (2005).
    [CrossRef]
  7. S. S. Wang and H. G. Winful, "Dynamics of phase-locked semiconductor arrays", Appl. Phys. Lett. 53, 1894- 1896 (1988).
  8. K. Otsuka, "Self-induced chaotic turbulence and phase itinerancy in coupled lasers," Phys. Rev. Lett. 65, 329-332 (1990).
    [CrossRef] [PubMed]

2006 (1)

2005 (3)

S. Riyopoulos, "Tolerance of phase-locked VCSEL arrays to random and systematic parameter deviations among cavities," IEEE JQE 41, 1450-1460 (2005).
[CrossRef]

S. Riyopoulos, "Simulations of boundary layers and point defects in coupled VCSEL arrays," IEEE J. Sel. Top. Quantum Electron. 11, 945-957 (2005).
[CrossRef]

S. Riyopoulos, "Phase stability theory of Bloch eigenstates in active photonic lattices with coupled microlaser arrays," Eur. Phys. J. D 36, 295-317 (2005).
[CrossRef]

1990 (2)

K. Otsuka, "Self-induced chaotic turbulence and phase itinerancy in coupled lasers," Phys. Rev. Lett. 65, 329-332 (1990).
[CrossRef] [PubMed]

J. K. Butler, D. E. Ackley and M. Ettenberg, "Coupled-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays," IEEE J. Quantum Electron. 21, 458-464 (1990).
[CrossRef]

1988 (1)

S. S. Wang and H. G. Winful, "Dynamics of phase-locked semiconductor arrays", Appl. Phys. Lett. 53, 1894- 1896 (1988).

Ackley, D. E.

J. K. Butler, D. E. Ackley and M. Ettenberg, "Coupled-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays," IEEE J. Quantum Electron. 21, 458-464 (1990).
[CrossRef]

Butler, J. K.

J. K. Butler, D. E. Ackley and M. Ettenberg, "Coupled-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays," IEEE J. Quantum Electron. 21, 458-464 (1990).
[CrossRef]

Ettenberg, M.

J. K. Butler, D. E. Ackley and M. Ettenberg, "Coupled-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays," IEEE J. Quantum Electron. 21, 458-464 (1990).
[CrossRef]

Otsuka, K.

K. Otsuka, "Self-induced chaotic turbulence and phase itinerancy in coupled lasers," Phys. Rev. Lett. 65, 329-332 (1990).
[CrossRef] [PubMed]

Riyopoulos, S.

S. Riyopoulos,"Effects of non-linear frequency-pulling on the cavity phasing and the collective mode structure in phase-locked VCSEL arrays," J. Opt. Soc. Am. B 23, 250-256 (2006).
[CrossRef]

S. Riyopoulos, "Simulations of boundary layers and point defects in coupled VCSEL arrays," IEEE J. Sel. Top. Quantum Electron. 11, 945-957 (2005).
[CrossRef]

S. Riyopoulos, "Tolerance of phase-locked VCSEL arrays to random and systematic parameter deviations among cavities," IEEE JQE 41, 1450-1460 (2005).
[CrossRef]

S. Riyopoulos, "Phase stability theory of Bloch eigenstates in active photonic lattices with coupled microlaser arrays," Eur. Phys. J. D 36, 295-317 (2005).
[CrossRef]

Wang, S. S.

S. S. Wang and H. G. Winful, "Dynamics of phase-locked semiconductor arrays", Appl. Phys. Lett. 53, 1894- 1896 (1988).

Winful, H. G.

S. S. Wang and H. G. Winful, "Dynamics of phase-locked semiconductor arrays", Appl. Phys. Lett. 53, 1894- 1896 (1988).

Appl. Phys. Lett. (1)

S. S. Wang and H. G. Winful, "Dynamics of phase-locked semiconductor arrays", Appl. Phys. Lett. 53, 1894- 1896 (1988).

Eur. Phys. J. D (1)

S. Riyopoulos, "Phase stability theory of Bloch eigenstates in active photonic lattices with coupled microlaser arrays," Eur. Phys. J. D 36, 295-317 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. K. Butler, D. E. Ackley and M. Ettenberg, "Coupled-mode analysis of gain and wavelength oscillation characteristics of diode laser phased arrays," IEEE J. Quantum Electron. 21, 458-464 (1990).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

S. Riyopoulos, "Simulations of boundary layers and point defects in coupled VCSEL arrays," IEEE J. Sel. Top. Quantum Electron. 11, 945-957 (2005).
[CrossRef]

IEEE JQE (1)

S. Riyopoulos, "Tolerance of phase-locked VCSEL arrays to random and systematic parameter deviations among cavities," IEEE JQE 41, 1450-1460 (2005).
[CrossRef]

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

Phys. Rev. Lett. (1)

K. Otsuka, "Self-induced chaotic turbulence and phase itinerancy in coupled lasers," Phys. Rev. Lett. 65, 329-332 (1990).
[CrossRef] [PubMed]

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 (6)

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 (b) Inter-cavity radiation overlap responsible for cross cavity gain modulation and the gain coupling coefficient

Fig. 2.
Fig. 2.

Schematic, top view of (a) 2-D perfect lattice array with varying cold-cavity parameters and lasing frequencies, symbolized by different gray shades (b) 2-D array of identical cavities on a disorganized pattern with cavity centers shifted from lattice locations - some shifts marked by arrows.

Fig. 3.
Fig. 3.

Randomly phase-locked steady-state from a simulation of a 32 × 32 array with uniform current bias Io /Ith = 3.1 and gi /gr = 0.5. The coupling strength is ϒ = 0.250, V = 0 and the rms deviation for the cold cavity frequencies 〈〈δ ω ˜ ij 〉〉/ωo = 1 × 10-5. (a) Histogram of the circulating cavity power at each array site at t =48ns (b) Neighbor cavity phasing showing the departures from zero-phasing that would result for an ideal array with 〈〈δ ω ˜ ij 〉〉 = 0 (c) Snap-shot of the collective electric field over the central array segment according to Eq. (14) (d) The change in the phasing between t =48ns and t =24ns is nearly zero and of the order of the spontaneous noise.

Fig. 4.
Fig. 4.

Dynamic evolution of the phase differences Φijx (t), Φijy (t) among neighbor sites vs. time, starting from randomly distibuted initial phases between (-π,π) (as well as random subthreshold cavity densities and intensities), for the array parameters of Fig. 3. Values for 11 out of the 32 × 32 array cavites are shown.

Fig. 5.
Fig. 5.

Phase-locked steady-state from a simulation of a 32 × 32 array with a systematic distribution of cold cavity frequencies according to the quadratic law δ ω ˜ ij /ωo = ∆Ω[(i - N/2)2 + (j - N/2)2] /(N 2) with ∆Ω = 5 × 10-6 with ϒ = 0.250, V = 0. Same other parameters as in Fig. 3 (a) Histogram of the circulating power at ij-th cavity at t =48ns (b) Snap-shot of the collective electric field over the central array segment. (c) Histogram of cavity phasing along x-direction, showing the departure from zero-phasing for an ideal array. (d) Same for the cavity phasing along y.

Fig. 6.
Fig. 6.

Phase-locked steady-state for a 21 × 21 “amorphous” array of identical-cavities with a random distribution in cavity center off-sets, corresponding to uniformly distributed coupling strengths between 0.125 ≤ ϒ ˜ ij ≤ 0.375, and same other parameters as in Fig. 3 (a) Histogram of the circulating cavity power at each array site at t =48ns (b) Histogram of the neighbor cavity phasing in the y-direction. The same quantities are shown in Figs. (c) and (d) respectively, but for fixed gain coupling ϒ = 0.021 combined with a uniform random distribution in feedback coupling strengths 0.051 ≤ ij ≤ 0.151.

Equations (56)

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( r , t ) = e ikz iωt i , j ij ( t ) e i φ ij ( t ) U ( r R ij )
N ( r , t ) = N ij ( t ) i , j χ ( r R ij )
F ij = ε v g E ij 2 8 π h ̅ ω .
N ij t = J ij e d w γ N ij B N ij 2 ξ g ̂ r ln N ̂ ij F ij ξ g ̂ r ln N ̂ ij Λ g i ' , j ' F i ' j '
ξ g ̂ r ln N ̂ i j 2 ϒ i ' j ' F ij F i ' j ' cos Φ ij ; i ' j '
F i j t = v g ζ g ̂ i ln N ̂ i j F ij v g μ F ij v g ζ g ̂ r Λ F ij i ' , j ' l n N ̂ i ' j ' + v g μ Π F ij i ' , j ' 1
+ v g ζ g ̂ r 2 + g ̂ i 2 2 ϒ i',j' l n N ̂ ij N ̂ i ' j ' F ij F i ' j ' cos ( Φ i j ; i ' j ' + ϑ )
+ v g σ r 2 + σ i 2 2 V i ' j ' F ij F i ' j ' cos ( Φ i j ; i ' j ' + ψ )
Φ ij ; i , j 1 t = v g 2 ζ g ̂ i ln ( N ij N i , j 1 ) + v g 2 ζ g ̂ i Λ i ' , j ' ln ( N i ' j ' N i ' , j ' 1 )
+ v g 2 ζ g ̂ r 2 + g ̂ i 2 2 ϒ i ' , j ' { ln N ̂ ij N ̂ i ' , j ' F i ' j ' F ij sin ( Φ ij ; i ' j ' + ϑ )
ln N ̂ ij 1 N ̂ i ' j ' 1 F i ' j ' 1 F ij 1 sin ( Φ ij 1 ; i ' j ' 1 + ϑ ) }
+ v g 2 σ r 2 + σ i 2 2 V i ' j ' { F i ' j ' F ij sin ( Φ ij ; i ' j ' + ψ )
F i ' j ' 1 F ij 1 sin ( Φ ij 1 ; i ' j ' 1 + ψ ) } .
ϒ dr 2 U ( r ) χ ( r ) U ( r ± b ) , Λ dr 2 U ( r ± b ) χ ( r ) U ( r ± b )
( σ r + i σ i ) V , ( σ r + i σ i )
V dr 2 U ( r ) χ m ( r ) U ( r ± b ) , dr 2 U ( r ± b ) χ m ( r ) U ( r ± b ) ,
φ ij t v ij = Δ ω ij + δ ω ˜ ij
Δ ω ij v g 2 ζ g ̂ i ln N ̂ ij + v g 2 ζ g ̂ i Λ i ' , j ' ln N ̂ i ' j '
+ v g 2 ζ g ̂ r 2 + g ̂ i 2 2 ϒ i ' , j ' ln N ̂ ij N ̂ i ' j ' F i ' j ' F ij sin ( Φ ij ; i ' j ' + ϑ )
+ v g 2 σ r 2 + σ i 2 2 V , i ' , j ' F i ' j ' F ij sin ( Φ ij ; i ' j ' + ψ )
Φ ij x , y t = v ij v i ± 1 , j ± 1 = 0 for all i , j v ij = Δ ω ij + δ ω ˜ ij = v o
( r , t ) = e i Δ ω o t i , j E o e i ( i + j ) Φ o U ( r R ij )
( r , t ) = e i v o t i , j E ij o e i ( i + j ) Φ o + Δ φ ij U ( r R ij )
N mn o = N tr exp [ μ ζ g r Ξ ( K mn ) ]
F mn o = ζ γ μ { J o e d w γ γ ˜ γ N tr exp [ μ ζ g r Ξ ( K mn ) ] }
Ξ mn = 1 4 Π 4 V ( cos Φ m o + cos Φ n o ) 1 + 4 Λ + 4 ϒ cos Φ m o + cos Φ n o )
1 2 [ g r C i ( 1 + α 2 ) + σ r D i ( 1 + αβ ) + σ r D r ( β α ) ] j + 1
1 2 [ g r C i ( 1 + α 2 ) + σ r D i ( 1 + αβ ) + σ r D r ( β α ) ] j = δ ω ˜ j δ ω ˜ j 1 v g
C r ζ [ ln N j + Λ ± ln N j ± 1 + 2 ϒ ± ln N j N j ± 1 F j ± 1 F j cos Φ j ± 1 ; j ]
C i ζ 2 ϒ ± ln N j N j ± 1 F j ± 1 F j sin Φ j ± 1 ; j
D r 2 V ± F j ± 1 F j cos Φ j ± 1 ; j
D r 2 V ± F j ± 1 F j sin Φ j ± 1 ; j
A 2 1 + B A 0 0 0 1 + B A 2 1 + B A 0 0 0 1 + B A 2 1 + B A 0 0 0 0 0 0 1 + B A 2 1 + B A 0 0 0 1 + B A 2 δ Φ ˜ 1 δ Φ ˜ 2 δ Φ ˜ 3 δ Φ ˜ N 1 δ Φ ˜ N = 1 v g Δδ ω ˜ 1 , N Δδ ω ˜ 2,1 Δδ ω ˜ 3,2 Δδ ω ˜ N 1 , N 2 Δδ ω ˜ N , N 1
A cos Φ o [ ζ g r ( 1 + α 2 ) ϒ ln N o + σ r ( 1 + αβ ) V ]
B sin Φ o σ r ( β α ) V
δ Φ ˜ j δ ω j ω o c k o v g 1 ζ g r ( 1 + α 2 ) ln N o ϒ + σ r ( 1 + αβ ) V
δ ω j ω o < μ ( 1 α 2 ) ϒ + σ r ( 1 + αβ ) V n k o ,
δ ω j ω o < ζ g r ( 1 + α 2 ) ln N o ϒ n k o 1 M Q
ϒ ϒ + δ ϒ ˜ ij ; i ' j ' , V V + δ V ˜ ij ; i ' j '
ϒ ( b ) = 4 e ( b ̅ x 2 + b ̅ y 2 ) 0 a ̅ d ρ ̅ ρ ̅ e 2 ρ ̅ 2 I o ( 2 ρ ̅ b ̅ x ) I o ( 2 ρ ̅ b ̅ y )
δ ϒ ˜ ij ; i ' j ' = 2 b . δ b w 2 ϒ + 4 e ( b ̅ x 2 + b ̅ y 2 ) 0 a ̅ d ρ ̅ 2 ρ ̅ 2 e 2 ρ ̅ 2 { I o ( 2 ρ ̅ b ̅ x ) I o ( 2 ρ ̅ b ̅ y )
× [ b x δ b x w I 1 ( 2 ρ ̅ b ̅ x ) I o ( 2 ρ ̅ b ̅ x ) + b y δ b y w I 1 ( 2 ρ ̅ b ̅ y ) I o ( 2 ρ ̅ b ̅ y ) ] }
A 2 1 + B A 0 0 0 1 + B A 2 1 + B A 0 0 0 1 + B A 2 1 + B A 0 0 0 0 0 0 1 + B A 2 1 + B A 0 0 0 1 + B A 2 δ Φ ˜ 1 δ Φ ˜ 2 δ Φ ˜ 3 δ Φ ˜ N 1 δ Φ ˜ N = ν 2 ν N ν 3 ν 1 ν 4 ν 2 ν N ν N 2 ν 1 ν N 1
ν j [ g r ( 1 + α 2 ) ln N o sin Φ o ] δ ϒ j [ σ r ( 1 + αβ ) sin Φ o σ r ( β α ) cos Φ o ] δ V j
F ij t = [ g r C r + σ r D r g i C i + σ i D i ] F ij μ F ij
φ ij t = δ ω ˜ ij + 1 2 [ g i C i + σ r D i + g i C r + σ i D r ] = 0
d Φ ij ; ij 1 dt = ( δ ω ˜ ij δ ω ˜ ij 1 )
+ 1 2 [ g r C i + σ r D i + g i C r + σ i D r ] ij 1 2 [ g r C i + σ r D i + g i C r + σ i D r ] i j 1
± sin Φ j ± 1 ; j ± sin Φ ( j 1 ) ± 1 ; j 1 = cos Φ o ( δ Φ ˜ j + 1 2 δ Φ ˜ j + δ Φ ˜ j 1 )
± cos Φ j ± 1 ; j ± cos Φ ( j 1 ) ± 1 ; j 1 = sin Φ o ( δ Φ ˜ j + 1 δ Φ ˜ j 1 )
1 2 [ g r ( 1 + α 2 ) 2 ϒ ln N o + σ r ( 1 + αβ ) 2 V ] ( cos Φ o ) ( δ Φ ˜ j + 1 2 δ Φ ˜ j + δ Φ ˜ j 1 ) +
1 2 [ σ r ( β α ) 2 V ] ( sin Φ o ) ( δ Φ ˜ j + 1 δ Φ ˜ j 1 ) + ( ε 2 ) = δ ω ˜ j δ ω ˜ j 1 v g
1 2 [ g r ( 1 + α 2 ) 2 ϒ ln N o + σ r ( 1 + αβ ) 2 V ] ( cos Φ o ) ( δ Φ ˜ j + 1 2 δ Φ ˜ j + δ Φ ˜ j 1 ) +
1 2 [ σ r ( β α ) 2 V ] ( sin Φ o ) ( δ Φ ˜ j + 1 δ Φ ˜ j 1 ) + ( ε 2 ) = δ ω ˜ j δ ω ˜ j 1 v g
1 2 [ g r ( 1 + α 2 ) ln N o sin Φ o ] 2 ( δ ϒ j + 1 δ ϒ j 1 ) + 1 2 [ σ r ( 1 + αβ ) sin Φ o
σ r ( β α ) cos Φ o ] 2 ( δ V j + 1 δ V j 1 )

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