Abstract

A longitudinal-mode analysis of a system of laterally coupled waveguided resonators is presented in the coupled-mode approximation. It is shown that variations in the mirror reflectivity of the individual channels result in coupling between the supermodes of the structure. This may lead to mode suppression by modulation of the threshold gain of different Fabry–Perot modes.

© 1985 Optical Society of America

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References

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  1. J. E. Ripper, T. L. Paoli, Appl. Phys. Lett. 17, 371 (1970).
    [CrossRef]
  2. D. R. Scifres, W. Streifer, R. D. Burnham, IEEE J. Quantum Electron. QE-15, 917 (1979).
    [CrossRef]
  3. D. Botez, J. C. Connolly, Appl. Phys. Lett. 43, 1096 (1983).
    [CrossRef]
  4. D. E. Ackley, Appl. Phys. Lett. 42, 152 (1983).
    [CrossRef]
  5. E. Kapon, J. Katz, A. Yariv. Opt. Lett. 10, 125 (1984).
    [CrossRef]
  6. E. Kapon, C. Lindsey, J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 45, 200 (1984).
    [CrossRef]
  7. T. W. Kumar, R. F. Ormondroyd, T. E. Rozzi, IEEE J. Quantum Electron. QE-20, 364 (1984).
    [CrossRef]
  8. S. R. Chinn, R. J. Spiers, IEEE J. Quantum Electron. QE-20, 358 (1984).
    [CrossRef]
  9. J. Salzman, T. Venkatesan, S. Margalit, A. Yariv, “Double heterostructure lasers with facets formed by a hybrid wet and reactive ion etching techniques,” J. Appl. Phys. 57, 2948 (1985).
    [CrossRef]
  10. A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
    [CrossRef]
  11. The reflectivities in r1 and r2 are complex numbers. A constant phase in both can be absorbed as a small additional length in the propagation matrix P(L). Therefore the relative phase between them will affect the final result.
  12. E. Kapon, J. Katz, S. Margalit, A. Yariv. Appl. Phys. Lett. 44, 157 (1984).
    [CrossRef]
  13. Y. Suematsu, M. Yamada, K. Hayashi, IEEE J. Quantum Electron. QE-11, 457 (1975).
    [CrossRef]

1985 (1)

J. Salzman, T. Venkatesan, S. Margalit, A. Yariv, “Double heterostructure lasers with facets formed by a hybrid wet and reactive ion etching techniques,” J. Appl. Phys. 57, 2948 (1985).
[CrossRef]

1984 (5)

E. Kapon, J. Katz, S. Margalit, A. Yariv. Appl. Phys. Lett. 44, 157 (1984).
[CrossRef]

E. Kapon, J. Katz, A. Yariv. Opt. Lett. 10, 125 (1984).
[CrossRef]

E. Kapon, C. Lindsey, J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 45, 200 (1984).
[CrossRef]

T. W. Kumar, R. F. Ormondroyd, T. E. Rozzi, IEEE J. Quantum Electron. QE-20, 364 (1984).
[CrossRef]

S. R. Chinn, R. J. Spiers, IEEE J. Quantum Electron. QE-20, 358 (1984).
[CrossRef]

1983 (2)

D. Botez, J. C. Connolly, Appl. Phys. Lett. 43, 1096 (1983).
[CrossRef]

D. E. Ackley, Appl. Phys. Lett. 42, 152 (1983).
[CrossRef]

1979 (1)

D. R. Scifres, W. Streifer, R. D. Burnham, IEEE J. Quantum Electron. QE-15, 917 (1979).
[CrossRef]

1975 (1)

Y. Suematsu, M. Yamada, K. Hayashi, IEEE J. Quantum Electron. QE-11, 457 (1975).
[CrossRef]

1973 (1)

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

1970 (1)

J. E. Ripper, T. L. Paoli, Appl. Phys. Lett. 17, 371 (1970).
[CrossRef]

Ackley, D. E.

D. E. Ackley, Appl. Phys. Lett. 42, 152 (1983).
[CrossRef]

Botez, D.

D. Botez, J. C. Connolly, Appl. Phys. Lett. 43, 1096 (1983).
[CrossRef]

Burnham, R. D.

D. R. Scifres, W. Streifer, R. D. Burnham, IEEE J. Quantum Electron. QE-15, 917 (1979).
[CrossRef]

Chinn, S. R.

S. R. Chinn, R. J. Spiers, IEEE J. Quantum Electron. QE-20, 358 (1984).
[CrossRef]

Connolly, J. C.

D. Botez, J. C. Connolly, Appl. Phys. Lett. 43, 1096 (1983).
[CrossRef]

Hayashi, K.

Y. Suematsu, M. Yamada, K. Hayashi, IEEE J. Quantum Electron. QE-11, 457 (1975).
[CrossRef]

Kapon, E.

E. Kapon, J. Katz, S. Margalit, A. Yariv. Appl. Phys. Lett. 44, 157 (1984).
[CrossRef]

E. Kapon, C. Lindsey, J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 45, 200 (1984).
[CrossRef]

E. Kapon, J. Katz, A. Yariv. Opt. Lett. 10, 125 (1984).
[CrossRef]

Katz, J.

E. Kapon, J. Katz, A. Yariv. Opt. Lett. 10, 125 (1984).
[CrossRef]

E. Kapon, C. Lindsey, J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 45, 200 (1984).
[CrossRef]

E. Kapon, J. Katz, S. Margalit, A. Yariv. Appl. Phys. Lett. 44, 157 (1984).
[CrossRef]

Kumar, T. W.

T. W. Kumar, R. F. Ormondroyd, T. E. Rozzi, IEEE J. Quantum Electron. QE-20, 364 (1984).
[CrossRef]

Lindsey, C.

E. Kapon, C. Lindsey, J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 45, 200 (1984).
[CrossRef]

Margalit, S.

J. Salzman, T. Venkatesan, S. Margalit, A. Yariv, “Double heterostructure lasers with facets formed by a hybrid wet and reactive ion etching techniques,” J. Appl. Phys. 57, 2948 (1985).
[CrossRef]

E. Kapon, J. Katz, S. Margalit, A. Yariv. Appl. Phys. Lett. 44, 157 (1984).
[CrossRef]

E. Kapon, C. Lindsey, J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 45, 200 (1984).
[CrossRef]

Ormondroyd, R. F.

T. W. Kumar, R. F. Ormondroyd, T. E. Rozzi, IEEE J. Quantum Electron. QE-20, 364 (1984).
[CrossRef]

Paoli, T. L.

J. E. Ripper, T. L. Paoli, Appl. Phys. Lett. 17, 371 (1970).
[CrossRef]

Ripper, J. E.

J. E. Ripper, T. L. Paoli, Appl. Phys. Lett. 17, 371 (1970).
[CrossRef]

Rozzi, T. E.

T. W. Kumar, R. F. Ormondroyd, T. E. Rozzi, IEEE J. Quantum Electron. QE-20, 364 (1984).
[CrossRef]

Salzman, J.

J. Salzman, T. Venkatesan, S. Margalit, A. Yariv, “Double heterostructure lasers with facets formed by a hybrid wet and reactive ion etching techniques,” J. Appl. Phys. 57, 2948 (1985).
[CrossRef]

Scifres, D. R.

D. R. Scifres, W. Streifer, R. D. Burnham, IEEE J. Quantum Electron. QE-15, 917 (1979).
[CrossRef]

Spiers, R. J.

S. R. Chinn, R. J. Spiers, IEEE J. Quantum Electron. QE-20, 358 (1984).
[CrossRef]

Streifer, W.

D. R. Scifres, W. Streifer, R. D. Burnham, IEEE J. Quantum Electron. QE-15, 917 (1979).
[CrossRef]

Suematsu, Y.

Y. Suematsu, M. Yamada, K. Hayashi, IEEE J. Quantum Electron. QE-11, 457 (1975).
[CrossRef]

Venkatesan, T.

J. Salzman, T. Venkatesan, S. Margalit, A. Yariv, “Double heterostructure lasers with facets formed by a hybrid wet and reactive ion etching techniques,” J. Appl. Phys. 57, 2948 (1985).
[CrossRef]

Yamada, M.

Y. Suematsu, M. Yamada, K. Hayashi, IEEE J. Quantum Electron. QE-11, 457 (1975).
[CrossRef]

Yariv, A.

J. Salzman, T. Venkatesan, S. Margalit, A. Yariv, “Double heterostructure lasers with facets formed by a hybrid wet and reactive ion etching techniques,” J. Appl. Phys. 57, 2948 (1985).
[CrossRef]

E. Kapon, J. Katz, S. Margalit, A. Yariv. Appl. Phys. Lett. 44, 157 (1984).
[CrossRef]

E. Kapon, J. Katz, A. Yariv. Opt. Lett. 10, 125 (1984).
[CrossRef]

E. Kapon, C. Lindsey, J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 45, 200 (1984).
[CrossRef]

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

Appl. Phys. Lett. (5)

J. E. Ripper, T. L. Paoli, Appl. Phys. Lett. 17, 371 (1970).
[CrossRef]

D. Botez, J. C. Connolly, Appl. Phys. Lett. 43, 1096 (1983).
[CrossRef]

D. E. Ackley, Appl. Phys. Lett. 42, 152 (1983).
[CrossRef]

E. Kapon, C. Lindsey, J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 45, 200 (1984).
[CrossRef]

E. Kapon, J. Katz, S. Margalit, A. Yariv. Appl. Phys. Lett. 44, 157 (1984).
[CrossRef]

IEEE J. Quantum Electron. (5)

Y. Suematsu, M. Yamada, K. Hayashi, IEEE J. Quantum Electron. QE-11, 457 (1975).
[CrossRef]

A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

T. W. Kumar, R. F. Ormondroyd, T. E. Rozzi, IEEE J. Quantum Electron. QE-20, 364 (1984).
[CrossRef]

S. R. Chinn, R. J. Spiers, IEEE J. Quantum Electron. QE-20, 358 (1984).
[CrossRef]

D. R. Scifres, W. Streifer, R. D. Burnham, IEEE J. Quantum Electron. QE-15, 917 (1979).
[CrossRef]

J. Appl. Phys. (1)

J. Salzman, T. Venkatesan, S. Margalit, A. Yariv, “Double heterostructure lasers with facets formed by a hybrid wet and reactive ion etching techniques,” J. Appl. Phys. 57, 2948 (1985).
[CrossRef]

Opt. Lett. (1)

Other (1)

The reflectivities in r1 and r2 are complex numbers. A constant phase in both can be absorbed as a small additional length in the propagation matrix P(L). Therefore the relative phase between them will affect the final result.

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Figures (2)

Fig. 1
Fig. 1

Schematic description of the laterally coupled waveguided lasers.

Fig. 2
Fig. 2

Cavity resonances for the coupled supermodes with normalized modal gain x1 = x2, coupling factor Ks = 1, L = 300 μm, s = 25 cm−1, (Δs)FP = 1 cm−1, and β ¯ = 2.5 × 105 cm−1.

Equations (19)

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E ( x , z ) = a 1 ( z ) E 1 ( x ) + a 2 ( z ) E 2 ( x ) ,
E ( x , z ) = b 1 ( z ) w 1 ( x ) + b 2 ( z ) w 2 ( x ) .
σ 1 , 2 = β ¯ s ,
β ¯ = β 1 + β 2 2 , s = ( Δ 2 + k 2 ) 1 / 2 , Δ = β 2 - β 1 2 .
A = [ a 1 a 2 ] ,             B = [ b 1 b 2 ] ,
A = V B ,
V = [ cos θ - sin θ sin θ cos θ ]
P ( z ) = [ e - i σ 1 z 0 0 e - i σ 2 z ] .
R = [ r 1 0 0 r 2 ] ,
L × B = B ,
det L - I = 0.
[ r exp ( - i 2 σ 1 L ) - 1 ] [ r exp ( - i 2 σ 2 L ) - 1 ] = 0.
Re ( β ¯ s ) L = 2 N π ,
r exp [ Im ( β s ) L ] = 1.
{ exp ( 2 i σ 1 L ) r [ 1 - δ ( Δ s ) ] - 1 } { exp ( 2 i σ 2 L ) r [ 1 + δ ( Δ s ) ] - 1 } = K s ,
r = r 1 + r 2 2 ,             δ = r 1 - r 2 2 r ,             K s = δ 2 ( k s ) 2 1 - δ 2 ( Δ s ) 2 .
x i = exp [ - Im ( σ i ) L ] r [ 1 δ ( Δ s ) ] ,
φ i = 2 π N i - 2 × Re { σ i } × L ,
[ x 1 exp ( φ 1 ) - 1 ] [ x 2 exp ( φ 2 ) - 1 ] = K s .

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