Abstract

We develop a generalized coupled-mode model for multistripe index-guided laser arrays that includes explicitly the influence of carrier-induced antiguiding, gain guiding, and carrier diffusion in the gain stripe. As an illustration of an application of the model, stability criteria for two-element laser arrays are derived that show that the phase-locked solution is intrinsically unstable. We find that the phase-locked solution can be stabilized at a low external injection-locking power at the suitably chosen injection-locking frequency. We have tested our model in the large carrier-diffusion case and still find good qualitative agreement between the coupled-mode model and a full coupled partial differential equation model.

© 1993 Optical Society of America

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References

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  1. S. S. Wang and H. G. Winful, Appl. Phys. Lett. 52, 1774 (1988).
    [Crossref]
  2. H. G. Winful and S. S. Wang, Appl. Phys. Lett. 53, 1894 (1988).
    [Crossref]
  3. J. K. Butler, D. E. Ackley, and M. Ettenberg, IEEE J. Quantum Electron. QE-21, 458 (1985).
    [Crossref]
  4. G. A. Wilson, K. Defreez, and H. G. Winful, IEEE J. Quantum. Electron. QE-27, 1696 (1991); Opt. Commun. 82, 293 (1991).
    [Crossref]
  5. E. Doedel, “auto: software for continuation and bifurcation problems in ordinary differential equations” (California Institute of Technology, Pasadena, Calif., May1986).
  6. P. K. Jakobsen, R. A. Indik, J. V. Moloney, A. C. Newell, H. G. Winful, and L. Rahman, J. Opt. Soc. Am. B 8, 1674 (1991).
    [Crossref]
  7. G. R. Hadley, J. P. Hohimer, and A. Owyoung, IEEE J. Quantum Electron. QE-23, 765 (1987).
    [Crossref]
  8. H. Adachihara, O. Hess, R. Indik, and J. V. Moloney, “Semiconductor laser array dynamics: numerical simulations on multistripe index-guided lasers,” J. Opt. Soc. Am. B 10, 496 (1993).
    [Crossref]
  9. C. Henry, IEEE J. Quantum Electron. QE-18, 259 (1982).
    [Crossref]
  10. G. Thompson, Opto-electronics 4, 257 (1972).
    [Crossref]
  11. G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
    [Crossref]

1993 (1)

1991 (2)

G. A. Wilson, K. Defreez, and H. G. Winful, IEEE J. Quantum. Electron. QE-27, 1696 (1991); Opt. Commun. 82, 293 (1991).
[Crossref]

P. K. Jakobsen, R. A. Indik, J. V. Moloney, A. C. Newell, H. G. Winful, and L. Rahman, J. Opt. Soc. Am. B 8, 1674 (1991).
[Crossref]

1988 (2)

S. S. Wang and H. G. Winful, Appl. Phys. Lett. 52, 1774 (1988).
[Crossref]

H. G. Winful and S. S. Wang, Appl. Phys. Lett. 53, 1894 (1988).
[Crossref]

1987 (1)

G. R. Hadley, J. P. Hohimer, and A. Owyoung, IEEE J. Quantum Electron. QE-23, 765 (1987).
[Crossref]

1985 (1)

J. K. Butler, D. E. Ackley, and M. Ettenberg, IEEE J. Quantum Electron. QE-21, 458 (1985).
[Crossref]

1982 (1)

C. Henry, IEEE J. Quantum Electron. QE-18, 259 (1982).
[Crossref]

1972 (1)

G. Thompson, Opto-electronics 4, 257 (1972).
[Crossref]

Ackley, D. E.

J. K. Butler, D. E. Ackley, and M. Ettenberg, IEEE J. Quantum Electron. QE-21, 458 (1985).
[Crossref]

Adachihara, H.

Agrawal, G. P.

G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
[Crossref]

Butler, J. K.

J. K. Butler, D. E. Ackley, and M. Ettenberg, IEEE J. Quantum Electron. QE-21, 458 (1985).
[Crossref]

Defreez, K.

G. A. Wilson, K. Defreez, and H. G. Winful, IEEE J. Quantum. Electron. QE-27, 1696 (1991); Opt. Commun. 82, 293 (1991).
[Crossref]

Doedel, E.

E. Doedel, “auto: software for continuation and bifurcation problems in ordinary differential equations” (California Institute of Technology, Pasadena, Calif., May1986).

Dutta, N. K.

G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
[Crossref]

Ettenberg, M.

J. K. Butler, D. E. Ackley, and M. Ettenberg, IEEE J. Quantum Electron. QE-21, 458 (1985).
[Crossref]

Hadley, G. R.

G. R. Hadley, J. P. Hohimer, and A. Owyoung, IEEE J. Quantum Electron. QE-23, 765 (1987).
[Crossref]

Henry, C.

C. Henry, IEEE J. Quantum Electron. QE-18, 259 (1982).
[Crossref]

Hess, O.

Hohimer, J. P.

G. R. Hadley, J. P. Hohimer, and A. Owyoung, IEEE J. Quantum Electron. QE-23, 765 (1987).
[Crossref]

Indik, R.

Indik, R. A.

Jakobsen, P. K.

Moloney, J. V.

Newell, A. C.

Owyoung, A.

G. R. Hadley, J. P. Hohimer, and A. Owyoung, IEEE J. Quantum Electron. QE-23, 765 (1987).
[Crossref]

Rahman, L.

Thompson, G.

G. Thompson, Opto-electronics 4, 257 (1972).
[Crossref]

Wang, S. S.

H. G. Winful and S. S. Wang, Appl. Phys. Lett. 53, 1894 (1988).
[Crossref]

S. S. Wang and H. G. Winful, Appl. Phys. Lett. 52, 1774 (1988).
[Crossref]

Wilson, G. A.

G. A. Wilson, K. Defreez, and H. G. Winful, IEEE J. Quantum. Electron. QE-27, 1696 (1991); Opt. Commun. 82, 293 (1991).
[Crossref]

Winful, H. G.

G. A. Wilson, K. Defreez, and H. G. Winful, IEEE J. Quantum. Electron. QE-27, 1696 (1991); Opt. Commun. 82, 293 (1991).
[Crossref]

P. K. Jakobsen, R. A. Indik, J. V. Moloney, A. C. Newell, H. G. Winful, and L. Rahman, J. Opt. Soc. Am. B 8, 1674 (1991).
[Crossref]

S. S. Wang and H. G. Winful, Appl. Phys. Lett. 52, 1774 (1988).
[Crossref]

H. G. Winful and S. S. Wang, Appl. Phys. Lett. 53, 1894 (1988).
[Crossref]

Appl. Phys. Lett. (2)

S. S. Wang and H. G. Winful, Appl. Phys. Lett. 52, 1774 (1988).
[Crossref]

H. G. Winful and S. S. Wang, Appl. Phys. Lett. 53, 1894 (1988).
[Crossref]

IEEE J. Quantum Electron. (3)

J. K. Butler, D. E. Ackley, and M. Ettenberg, IEEE J. Quantum Electron. QE-21, 458 (1985).
[Crossref]

G. R. Hadley, J. P. Hohimer, and A. Owyoung, IEEE J. Quantum Electron. QE-23, 765 (1987).
[Crossref]

C. Henry, IEEE J. Quantum Electron. QE-18, 259 (1982).
[Crossref]

IEEE J. Quantum. Electron. (1)

G. A. Wilson, K. Defreez, and H. G. Winful, IEEE J. Quantum. Electron. QE-27, 1696 (1991); Opt. Commun. 82, 293 (1991).
[Crossref]

J. Opt. Soc. Am. B (2)

Opto-electronics (1)

G. Thompson, Opto-electronics 4, 257 (1972).
[Crossref]

Other (2)

G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
[Crossref]

E. Doedel, “auto: software for continuation and bifurcation problems in ordinary differential equations” (California Institute of Technology, Pasadena, Calif., May1986).

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Figures (7)

Fig. 1
Fig. 1

Stability diagram for the phase-locked solutions. The stable regime lies below the curve.

Fig. 2
Fig. 2

Stability diagram for the phase-antilocked solutions. Both shaded regions correspond to stable output.

Fig. 3
Fig. 3

Bifurcation diagram with the injection amplitude A as a control parameter detuned at the supermode frequency. The solid line is the stable solution, and the dashed lines are the unstable solution. The solid squares indicate the Hopf bifurcations, and the open square is the branch point where the symmetric and the asymmetric solutions cross.

Fig. 4
Fig. 4

Bifurcation diagram with the injection amplitude A as a control parameter detuned at the frequency at which the minimum injection locking is achieved. The minimum injection locking power is 0.1 mW. The stable branch extends to the branch point.

Fig. 5
Fig. 5

Bifurcation diagram with the injection frequency detuning δ as a control parameter at the injection power of 1 mW. The solid curve is the stable branch of equilibrium solutions, and the dashed curve corresponds to the unstable branch. The unstable branch has several Hopf bifurcation points, and the symmetric and asymmetric solutions cross at the branch point.

Fig. 6
Fig. 6

Bifurcation diagram with the stripe separation d as a control parameter at a pumping current level of 30 mA. The solid curve is the stable branch of phase-antilocked solutions, and the dashed line is the unstable branch. The filled square indicates the Hopf bifurcation point, which is at d = 4.8 μm. The open square indicates the data from the full numerical simulations. The filled circles indicate the stable periodic solutions after they pass through the Hopf bifurcation point, and the open circles indicate the unstable periodic solutions.

Fig. 7
Fig. 7

Stability boundary curve for the phase-antilocked solutions. The curve is obtained from the coupled-mode analysis by using auto, and the open squares are the data from the full numerical simulations. The shadowed region is the stable region of the solutions

Tables (1)

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Table 1 Device Description

Equations (75)

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E ˙ j = ( 1 - i R ) z j E j + i η ( E j + 1 + E j - 1 ) ,
T z ˙ j = p j - z j - ( z j + 1 ) E j 2 ,
s ˙ 1 = z 1 s 1 - ( η r sin θ + η i cos θ ) s 2 ,
s ˙ 2 = z 2 s 2 + ( η r sin θ - η i cos θ ) s 1 ,
θ ˙ = R ( z 1 - z 2 ) + η r cos θ ( s 1 s 2 - s 2 s 1 ) + η i sin θ ( s 1 s 2 + s 2 s 1 ) ,
T z ˙ 1 = p 1 - z 1 - ( z 1 + 1 ) s 1 2 ,
T z ˙ 2 = p 2 - z 2 - ( z 2 + 1 ) s 2 2 ,
s 1 = s 2 = s = ( p - η i 1 + η i ) 1 / 2 ,             z 1 = z 2 = z = η i , θ = 0.
s 1 = s 2 = s = ( p + η i 1 - η i ) 1 / 2 ,             z 1 = z 2 = z = - η i , θ = π .
R < - γ + η r ( 1 + s 2 ) ( 1 + γ 2 ) p - z ,
R > γ - 1 + s 2 2 η r T + 4 γ η r 2 T p - z [ ( 1 + s 2 2 η r T - γ ) 2 + 1 ] ,
R < γ + 1 + s 2 2 η r T + 4 γ η r 2 T p - z [ ( 1 + s 2 2 η r T + γ ) 2 + 1 ] .
E ˙ j = ( 1 - i R ) z j E j + i η ( E j + 1 + E j - 1 ) + A exp ( - i δ τ ) ,
T z ˙ j = p j - z j - ( z j + 1 ) E j 2 .
s ˙ 1 = [ z 1 + ξ ( 1 + z 2 ) ] s 1 - ( η r sin θ + η i cos θ ) s 2 ,
s ˙ 2 = [ z 2 + ξ ( 1 + z 1 ) ] s 2 + ( η r sin θ - η i cos θ ) s 1 ,
θ ˙ = R ( 1 - ξ ) ( z 1 - z 2 ) + η r cos θ ( s 1 s 2 - s 2 s 1 ) + η i sin θ ( s 1 s 2 + s 2 s 1 ) ,
T z ˙ 1 = p 1 - z 1 - ( z 1 + 1 ) s 1 2 ,
T z ˙ 2 = p 2 - z 2 - ( z 2 + 1 ) s 2 2 .
s 1 = s 2 = s = ( p - z 1 + z ) 1 / 2 ,             z 1 = z 2 = z = η i - ξ 1 + ξ , θ = 0 ,
s 1 = s 2 = s = ( p - z 1 + z ) 1 / 2 ,             z 1 = z 2 = z = - η i + ξ 1 + ξ , θ = π ,
R < - γ + η r ( 1 + s 2 ) ( 1 + γ 2 ) ( p - z ) ( 1 - ξ ) ,
R > γ - 1 + s 2 2 η r T + 4 γ η r 2 T ( p - z ) ( 1 - ξ ) [ ( 1 + s 2 2 η r T - γ ) 2 + 1 ] .
R < γ + 1 + s 2 2 η r T + 4 γ η r 2 T ( p - z ) ( 1 - ξ ) [ ( 1 + s 2 2 η r T + γ ) 2 + 1 ] .
F t + F z = i D p 2 F x 2 - i Δ ( x ) F + κ Γ ( x ) [ g ( N ) - i R N ] F ,
B t - B z = i D p 2 B x 2 - i Δ ( x ) B + κ Γ ( x ) [ g ( N ) - i R N ] B ,
N t = D f 2 N x 2 + J ( x ) - γ N - Γ ( x ) g ( N ) × ( F 2 + B 2 ) ,
F ( t , x , z = 0 ) = - ( R 1 ) 1 / 2 B ( t , x , z = 0 ) + ( T 1 ) 1 / 2 F inj ( x ) exp ( - i δ ω t ) ,
B ( t , x , z = 1 ) = - ( R 2 ) 1 / 2 F ( t , x , z = 1 ) ,
F ¯ ( t , z , x ) = exp ( - i ω ¯ t ) F s t ( z , x ) , B ¯ ( t , z , x ) = exp ( - i ω ¯ t ) B s t ( z , x ) , N ¯ ( t , z , x ) = N s t ( z , x ) ,
F ( t , z , x ) = i = 1 M f i ( t , z ) F ¯ i ( t , z , x ) = i = 1 M f i ( t , z ) F ¯ ( t , z , x - i d ) ,
B ( t , z , x ) = i = 1 M b i ( t , z ) B ¯ i ( t , z , x ) = i = 1 M n i ( t , z ) B ¯ ( t , z , x - i d ) ,
N ( t , z , x ) = i = 1 M n i ( t , z ) N ¯ i ( t , z , x ) = i = 1 M n i ( t , z ) N ¯ ( t , z , x - i d ) ,
f j t + f j z = - c 1 ( z ) ( 1 - i R ) f j + c 1 ( z ) ( 1 - i R ) n j f j + c 2 ( F ) ( z ) ( f j + 1 + f j - 1 ) + ( 1 - i R ) [ c 3 ( z ) ( n j + 1 + n j - 1 ) f j + c 4 ( F ) ( z ) n j ( f j + 1 + f j - 1 ) + c 5 ( F ) ( z ) ( n j + 1 f j + 1 + n j - 1 f j - 1 ) ] ,
c 1 ( z ) = κ - + d x Γ j ( x ) N ¯ j F ¯ j B ¯ j - + d x F ¯ j B ¯ j , c 2 ( F ) = ( j ) d x [ - i δ Δ j ( x ) - κ δ Γ j ( x ) ] F ¯ j + 1 B ¯ j - + d x F ¯ j B ¯ j - ( 1 - i R ) c 5 ( F ) ( z ) , c 3 ( z ) = κ - + d x Γ ( x ) N ¯ j + 1 F ¯ j B ¯ j - + d x F ¯ j B ¯ j , c 4 ( F ) ( z ) = κ - + d x Γ ( x ) N ¯ j F ¯ j + 1 B ¯ j - + d x F ¯ j B ¯ j , c 5 ( F ) ( z ) = κ - + d x Γ ( x ) N ¯ j + 1 F ¯ j + 1 B ¯ j - + d x F ¯ j B ¯ j ,
b j t + b j z = - c 1 ( z ) ( 1 - i R ) b j + c 1 ( z ) ( 1 - i R ) n j b j + c 2 ( B ) ( z ) ( b j + 1 + b j - 1 ) + ( 1 - i R ) [ c 3 ( z ) ( n j + 1 + n j - 1 ) b j + c 4 ( B ) ( z ) n j ( b j + 1 + b j - 1 ) + c 5 ( B ) ( z ) ( n j + 1 b j + 1 + n j - 1 b j - 1 ) ] ,
c 2 ( B ) ( z ) = ( j ) d x [ - i δ Δ j ( x ) - κ δ Γ j ( x ) ] B ¯ j + 1 F ¯ j - + d x F ¯ j B ¯ j - ( 1 - i R ) c 5 ( B ) ( z ) , c 4 ( B ) ( z ) = κ - + d x Γ ( x ) N ¯ j B ¯ j + 1 F ¯ j - + d x F ¯ j B ¯ j , c 5 ( B ) ( z ) = κ - + d x Γ ( x ) N ¯ j + 1 B ¯ j + 1 F ¯ j - + d x F ¯ j B ¯ j .
n j t = J j ( z ) - c 6 ( z ) n j - n j [ c 7 ( F ) ( z ) f j 2 + c 7 ( B ) ( z ) b j 2 ] + c 8 ( F ) ( z ) f j 2 + c 8 ( B ) ( z ) b j 2 ,
J j ( z ) = ( j ) d x J ( x ) ( j ) d x N ¯ j , c 6 ( z ) = J 0 - ( j ) d x Γ j ( x ) ( N ¯ j - 1 ) ( F ¯ j 2 + B ¯ j 2 ) ( j ) d x N ¯ j , c 7 ( F ) ( z ) = ( j ) d x Γ ( x ) N ¯ j F ¯ j 2 ( j ) d x N ¯ j , c 7 ( B ) ( z ) = ( j ) d x Γ ( x ) N ¯ j B ¯ j 2 ( j ) d x N ¯ j , c 8 ( F ) ( z ) = ( j ) d x Γ ( x ) F ¯ j 2 ( j ) d x N ¯ j , c 8 ( B ) ( z ) = ( j ) d x Γ ( x ) B ¯ j 2 ( j ) d x N ¯ j .
f j ( z = 0 ) = b j ( z = 0 ) + a j ( t ) ,
f j ( z = 1 ) = b j ( z = 1 ) ,
a j ( t ) = ( T 1 ) 1 / 2 exp ( - i δ ω t ) - + d x F inj B ¯ j ( z = 0 ) - + d x F ¯ j ( z = 0 ) B ¯ j ( z = 0 ) .
f ˜ j = f j + z - 1 2 a j ( t ) ,
b ˜ j = b j - z - 1 2 a j ( t ) .
f ˜ j ( z = 0 ) = b ˜ j ( z = 0 ) ,
f ˜ j ( z = 1 ) = b ˜ j ( z = 1 ) .
e j ( t ) = 1 2 0 1 d z ( f ˜ j + b ˜ j ) ;
e ˙ j = - c 1 ( 1 - i R ) e j + c 1 ( 1 - i R ) n j e j + c 2 ( e j + 1 + e j - 1 ) + ( 1 - i R ) [ c 3 ( n j + 1 + n j - 1 ) e j + c 4 n j ( e j + 1 + e j - 1 ) + c 5 ( n j + 1 e j + 1 + n j - 1 e j - 1 ) ] + a j ( t ) ,
n ˙ j = J j - c 6 n j - c 7 n j e j 2 + c 8 e j 2 ,
c 1 = 0 1 d z c 1 ( z ) , c 2 = 1 2 0 1 d z [ c 2 ( F ) ( z ) + c 2 ( B ) ( z ) ] , c 3 = 0 1 d z c 3 ( z ) , c 4 = 1 2 0 1 d z [ c 4 ( F ) ( z ) + c 4 ( B ) ( z ) ] , c 5 = 1 2 0 1 d z [ c 5 ( F ) ( z ) + c 5 ( B ) ( z ) ] , J j = 0 1 d z J j ( z ) , c 6 = 0 1 d z c 6 ( z ) , c 7 = 0 1 d z [ c 7 ( F ) ( z ) + c 7 ( B ) ( z ) ] , c 8 = 0 1 d z [ c 8 ( F ) ( z ) + c 8 ( B ) ( z ) ] .
z j = n j - 1 1 - ( c 8 / c 7 ) ,
τ = t × c 1 ( 1 - c 8 c 7 ) .
E j = ( c 7 c 6 ) 1 / 2 e j ;
E ˙ j = ( 1 - i R ) z j E j + i η ( E j + 1 + E j - 1 ) + A j ( t ) + ( 1 - i R ) ξ ( 2 - δ j 1 - δ j M + z j + 1 + z j - 1 ) E j ,
T z ˙ j = p j - z j - ( z j + 1 ) e j 2 ,
η = - i c 2 + ( 1 - i R ) ( c 4 + c 5 ) c 1 [ 1 - ( c 8 / c 7 ) ] , ξ = c 3 c 1 [ 1 - ( c 8 / c 7 ) ] A j ( t ) = c 7 c 6 a j ( t ) c 1 [ 1 - ( c 8 / c 7 ) ] , p j = ( J j / c 6 ) - 1 1 - ( c 8 / c 7 ) ,
T = c 1 [ 1 - ( c 8 / c 7 ) ] c 6 .
s ˙ 1 = z 1 s 1 - ( η r sin θ + η i cos θ ) s 2 ,
s ˙ 2 = z 2 s 2 + ( η r sin θ - η i cos θ ) s 1 ,
θ ˙ = R ( z 1 - z 2 ) + η r cos θ ( s 1 s 2 - s 2 s 1 ) + η i sin θ ( s 1 s 2 + s 2 s 1 ) ,
T z ˙ 1 = p 1 - z 1 - ( z 1 + 1 ) s 1 2 ,
T Z ˙ 2 = p 2 - z 2 - ( z 2 + 1 ) s 2 2 .
s 1 = s 2 = s = ( p - z 1 + z ) 1 / 2 ,             z 1 = z 2 = z = η i cos θ ,
λ 2 + 1 + s 2 T λ + 2 ( p - z ) T = 0 ,
( λ + 1 + s 2 T ) [ ( λ - 2 z ) 2 + 4 η r 2 ] - 2 ( p - z ) T ( 2 R η r cos θ - λ + 2 z ) = 0.
P = 1 + s 2 2 η r T , q = p - z 2 η r 2 T , S = λ 2 η r , γ = η i η r .
( S + P ) [ ( S - γ cos θ ) 2 + 1 ] - q [ ( R + γ ) cos θ - S ] = 0.
q R + γ P < 1 + γ 2
R < - γ + η r ( 1 + γ 2 ) ( 1 + s 2 ) p - z .
y 2 = ( x - γ cos θ ) 2 + 1 + 2 ( x + P ) ( x - γ cos θ ) + q ,
( x - γ cos θ ) 2 + 1 + ( x + P ) 2 = - 2 ( x + P ) ( x - γ cos θ ) - q - q [ P + ( R + γ ) cos θ ] 2 ( x - γ cos θ ) .
( P - γ ) 2 + 1 < q 2 γ ( P + R - γ ) ;
R > γ - 1 + s 2 2 η r T + 4 γ η r 2 T p - z [ ( 1 + s 2 2 η r T - γ ) 2 + 1 ] .
( P + γ ) 2 + 1 > - q 2 γ ( P - R + γ ) ;
R < γ + 1 + s 2 2 η r T + 4 γ η r 2 T p - z [ ( 1 + s 2 2 η r T + γ ) 2 + 1 ] .

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