Abstract

The optical characteristics of phase-locked semiconductor laser arrays are formulated in terms of the array supermodes, which are the eigenmodes of the composite-array waveguide, by using coupled-mode theory. These supermodes are employed to calculate the near fields, the far fields, and the difference in the longitudinal-mode oscillation wavelengths of the array. It is shown that the broadening in the far-field beam divergence, as well as the broadening of each of the longitudinal modes that were observed in phase-locked arrays, may arise from the excitation of an increasing number of supermodes at increasing pumping levels.

© 1984 Optical Society of America

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Corrections

E. Kapon, J. Katz, and A. Yariv, "Supermode analysis of phase-locked arrays of semiconductor lasers: erratum," Opt. Lett. 9, 318-318 (1984)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-9-7-318

References

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  1. D. R. Scifres, R. D. Burnham, W. Streifer, Appl. Phys. Lett. 33, 1015 (1978).
    [CrossRef]
  2. D. R. Scifres, W. Streifer, R. D. Burnham, IEEE J. Quantum Electron. QE-15, 917 (1979).
    [CrossRef]
  3. D. R. Scifres, C. Lindstrom, R. D. Burnham, W. Streifer, T. L. Paoli, Electron. Lett. 19, 169 (1983).
    [CrossRef]
  4. D. E. Ackley, R. W. H. Engleman, Appl. Phys. Lett. 39, 27 (1981).
    [CrossRef]
  5. W. T. Tsang, R. A. Logan, R. P. Salathe, Appl. Phys. Lett. 34, 162 (1979).
    [CrossRef]
  6. D. Botez, J. C. Connolly, Appl. Phys. Lett. 43, 1096 (1983).
    [CrossRef]
  7. J. Katz, E. Kapon, C. Lindsey, S. Margalit, U. Shreter, A. Yariv, Appl. Phys. Lett. 43, 521 (1983).
    [CrossRef]
  8. E. Kapon, J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 44, 157 (1984).
    [CrossRef]
  9. K. Otsuka, Electron. Lett. 19, 723 (1983).
    [CrossRef]
  10. A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
    [CrossRef]
  11. A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1976), Chap. 13.
  12. J. Katz, E. Kapon, S. Margalit, A. Yariv (unpublished).
  13. E. Kapon, J. Katz, C. Lindsey, S. Margalit, A. Yariv, Appl. Phys. Lett. 43, 421 (1983).
    [CrossRef]

1984 (1)

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

1983 (5)

K. Otsuka, Electron. Lett. 19, 723 (1983).
[CrossRef]

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

J. Katz, E. Kapon, C. Lindsey, S. Margalit, U. Shreter, A. Yariv, Appl. Phys. Lett. 43, 521 (1983).
[CrossRef]

D. R. Scifres, C. Lindstrom, R. D. Burnham, W. Streifer, T. L. Paoli, Electron. Lett. 19, 169 (1983).
[CrossRef]

E. Kapon, J. Katz, C. Lindsey, S. Margalit, A. Yariv, Appl. Phys. Lett. 43, 421 (1983).
[CrossRef]

1981 (1)

D. E. Ackley, R. W. H. Engleman, Appl. Phys. Lett. 39, 27 (1981).
[CrossRef]

1979 (2)

W. T. Tsang, R. A. Logan, R. P. Salathe, Appl. Phys. Lett. 34, 162 (1979).
[CrossRef]

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

1978 (1)

D. R. Scifres, R. D. Burnham, W. Streifer, Appl. Phys. Lett. 33, 1015 (1978).
[CrossRef]

1973 (1)

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

Ackley, D. E.

D. E. Ackley, R. W. H. Engleman, Appl. Phys. Lett. 39, 27 (1981).
[CrossRef]

Botez, D.

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

Burnham, R. D.

D. R. Scifres, C. Lindstrom, R. D. Burnham, W. Streifer, T. L. Paoli, Electron. Lett. 19, 169 (1983).
[CrossRef]

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

D. R. Scifres, R. D. Burnham, W. Streifer, Appl. Phys. Lett. 33, 1015 (1978).
[CrossRef]

Connolly, J. C.

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

Engleman, R. W. H.

D. E. Ackley, R. W. H. Engleman, Appl. Phys. Lett. 39, 27 (1981).
[CrossRef]

Kapon, E.

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

J. Katz, E. Kapon, C. Lindsey, S. Margalit, U. Shreter, A. Yariv, Appl. Phys. Lett. 43, 521 (1983).
[CrossRef]

E. Kapon, J. Katz, C. Lindsey, S. Margalit, A. Yariv, Appl. Phys. Lett. 43, 421 (1983).
[CrossRef]

J. Katz, E. Kapon, S. Margalit, A. Yariv (unpublished).

Katz, J.

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

E. Kapon, J. Katz, C. Lindsey, S. Margalit, A. Yariv, Appl. Phys. Lett. 43, 421 (1983).
[CrossRef]

J. Katz, E. Kapon, C. Lindsey, S. Margalit, U. Shreter, A. Yariv, Appl. Phys. Lett. 43, 521 (1983).
[CrossRef]

J. Katz, E. Kapon, S. Margalit, A. Yariv (unpublished).

Lindsey, C.

E. Kapon, J. Katz, C. Lindsey, S. Margalit, A. Yariv, Appl. Phys. Lett. 43, 421 (1983).
[CrossRef]

J. Katz, E. Kapon, C. Lindsey, S. Margalit, U. Shreter, A. Yariv, Appl. Phys. Lett. 43, 521 (1983).
[CrossRef]

Lindstrom, C.

D. R. Scifres, C. Lindstrom, R. D. Burnham, W. Streifer, T. L. Paoli, Electron. Lett. 19, 169 (1983).
[CrossRef]

Logan, R. A.

W. T. Tsang, R. A. Logan, R. P. Salathe, Appl. Phys. Lett. 34, 162 (1979).
[CrossRef]

Margalit, S.

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

J. Katz, E. Kapon, C. Lindsey, S. Margalit, U. Shreter, A. Yariv, Appl. Phys. Lett. 43, 521 (1983).
[CrossRef]

E. Kapon, J. Katz, C. Lindsey, S. Margalit, A. Yariv, Appl. Phys. Lett. 43, 421 (1983).
[CrossRef]

J. Katz, E. Kapon, S. Margalit, A. Yariv (unpublished).

Otsuka, K.

K. Otsuka, Electron. Lett. 19, 723 (1983).
[CrossRef]

Paoli, T. L.

D. R. Scifres, C. Lindstrom, R. D. Burnham, W. Streifer, T. L. Paoli, Electron. Lett. 19, 169 (1983).
[CrossRef]

Salathe, R. P.

W. T. Tsang, R. A. Logan, R. P. Salathe, Appl. Phys. Lett. 34, 162 (1979).
[CrossRef]

Scifres, D. R.

D. R. Scifres, C. Lindstrom, R. D. Burnham, W. Streifer, T. L. Paoli, Electron. Lett. 19, 169 (1983).
[CrossRef]

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

D. R. Scifres, R. D. Burnham, W. Streifer, Appl. Phys. Lett. 33, 1015 (1978).
[CrossRef]

Shreter, U.

J. Katz, E. Kapon, C. Lindsey, S. Margalit, U. Shreter, A. Yariv, Appl. Phys. Lett. 43, 521 (1983).
[CrossRef]

Streifer, W.

D. R. Scifres, C. Lindstrom, R. D. Burnham, W. Streifer, T. L. Paoli, Electron. Lett. 19, 169 (1983).
[CrossRef]

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

D. R. Scifres, R. D. Burnham, W. Streifer, Appl. Phys. Lett. 33, 1015 (1978).
[CrossRef]

Tsang, W. T.

W. T. Tsang, R. A. Logan, R. P. Salathe, Appl. Phys. Lett. 34, 162 (1979).
[CrossRef]

Yariv, A.

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

J. Katz, E. Kapon, C. Lindsey, S. Margalit, U. Shreter, A. Yariv, Appl. Phys. Lett. 43, 521 (1983).
[CrossRef]

E. Kapon, J. Katz, C. Lindsey, S. Margalit, A. Yariv, Appl. Phys. Lett. 43, 421 (1983).
[CrossRef]

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

A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1976), Chap. 13.

J. Katz, E. Kapon, S. Margalit, A. Yariv (unpublished).

Appl. Phys. Lett. (7)

D. E. Ackley, R. W. H. Engleman, Appl. Phys. Lett. 39, 27 (1981).
[CrossRef]

W. T. Tsang, R. A. Logan, R. P. Salathe, Appl. Phys. Lett. 34, 162 (1979).
[CrossRef]

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

J. Katz, E. Kapon, C. Lindsey, S. Margalit, U. Shreter, A. Yariv, Appl. Phys. Lett. 43, 521 (1983).
[CrossRef]

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

D. R. Scifres, R. D. Burnham, W. Streifer, Appl. Phys. Lett. 33, 1015 (1978).
[CrossRef]

E. Kapon, J. Katz, C. Lindsey, S. Margalit, A. Yariv, Appl. Phys. Lett. 43, 421 (1983).
[CrossRef]

Electron. Lett. (2)

D. R. Scifres, C. Lindstrom, R. D. Burnham, W. Streifer, T. L. Paoli, Electron. Lett. 19, 169 (1983).
[CrossRef]

K. Otsuka, Electron. Lett. 19, 723 (1983).
[CrossRef]

IEEE J. Quantum Electron. (2)

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

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

Other (2)

A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart & Winston, New York, 1976), Chap. 13.

J. Katz, E. Kapon, S. Margalit, A. Yariv (unpublished).

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Figures (4)

Fig. 1
Fig. 1

Schematic illustration of an N-channel laser array. The y axis is in the pn-junction plane.

Fig. 2
Fig. 2

Schematic illustration of the supermodes in a five-channel array of equal waveguides ( β ˆ j = β ˆ ) and uniform coupling (κij = κ). The numbers beside the near-field lobes indicate the relative magnitude of the field amplitude. The expressions beside each field pattern are the corresponding supermode eigenvalues.

Fig. 3
Fig. 3

The grating function G for the supermodes of Fig. 2. The dashed curve corresponds to an array of five identical radiators.

Fig. 4
Fig. 4

Effect of variation of the coupling coefficients on the near field of the (+++++) supermode for κ23 = κ34

Tables (1)

Tables Icon

Table 1 Propagation Constants of the Supermodes of Identical-Channel Arrays

Equations (10)

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E y ( x , y , z ) = l = 1 N E l ( x , y ) A l ( z ) exp ( i β ˆ l z ) ,
d E / d z = i M ˜ E ,
E ν ( z ) = E ν ( 0 ) exp ( i σ ν z ) ,
( M ˜ σ ν I ˜ ) E ν = 0 ,
Δ λ 0 S = ( Δ β L / π ) Δ λ 0 FP ,
F ν ( θ ) = E ( θ ) G ν ( θ ) ,
G ν ( θ ) | l = 1 N ρ ˆ 11 exp ( i k 0 S sin θ ) | 2
E l v = sin ( l π ν N + 1 ) ,
σ ν = β ˆ + 2 κ cos ( π ν N + 1 ) , ν = 1 , 2 , N .
Δ λ 0 S = ( 4 / π ) κ L Δ λ 0 FP .

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