Abstract

Array modes of cyclic vertical cavity laser arrays with odd numbers of elements were explored experimentally and theoretically. The odd antiphase array mode is prohibited by the circular symmetry, and the lowest-loss modes are degenerated. Slight inhomogenities drive the array mode structure to experience spontaneous symmetry breaking, leading by means of mode degeneracy removal to stable operation characterized by an antiphase lasing mode with an even number of lobes. Circular laser arrays of 3–19 elements were measured; all were lasing in a stable even-array mode, exhibiting 2–18 intensity lobes for both the near and the far fields.

© 1996 Optical Society of America

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References

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  1. M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, Appl. Phys. Lett. 60, 1535 (1992).
    [CrossRef]
  2. R. A. Morgan, K. Kojiama, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, M. Asom, Appl. Phys. Lett. 61, 1160 (1992).
    [CrossRef]
  3. R. J. Black, L. Gagnon, R. C. Youngquist, R. H. Wentworth, Electron. Lett. 22, 1311 (1986).
    [CrossRef]
  4. H. J. Yoo, R. J. Hayes, E. G. Paek, A. Scherer, Y. S. Kwon, IEEE J. Quantum Electron. 26, 1093 (1990).
    [CrossRef]
  5. T. Fishman, M. Oreinstein, presented at the 13th IEEE International Semiconductor Laser Conference, Takamatsu, Japan, 1992.
  6. E. Kapon, J. Katz, A. Yariv, Opt. Lett. 9, 125 (1984).
    [CrossRef] [PubMed]
  7. G. Lentz, J. Salzman, J. Lightwave Technol. 8, 1803 (1990).

1992 (2)

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

R. A. Morgan, K. Kojiama, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, M. Asom, Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

1990 (2)

H. J. Yoo, R. J. Hayes, E. G. Paek, A. Scherer, Y. S. Kwon, IEEE J. Quantum Electron. 26, 1093 (1990).
[CrossRef]

G. Lentz, J. Salzman, J. Lightwave Technol. 8, 1803 (1990).

1986 (1)

R. J. Black, L. Gagnon, R. C. Youngquist, R. H. Wentworth, Electron. Lett. 22, 1311 (1986).
[CrossRef]

1984 (1)

Asom, M.

R. A. Morgan, K. Kojiama, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, M. Asom, Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Black, R. J.

R. J. Black, L. Gagnon, R. C. Youngquist, R. H. Wentworth, Electron. Lett. 22, 1311 (1986).
[CrossRef]

Fishman, T.

T. Fishman, M. Oreinstein, presented at the 13th IEEE International Semiconductor Laser Conference, Takamatsu, Japan, 1992.

Florez, L. T.

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

Focht, M. W.

R. A. Morgan, K. Kojiama, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, M. Asom, Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Gagnon, L.

R. J. Black, L. Gagnon, R. C. Youngquist, R. H. Wentworth, Electron. Lett. 22, 1311 (1986).
[CrossRef]

Guth, G. D.

R. A. Morgan, K. Kojiama, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, M. Asom, Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Harbison, J. P.

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

Hayes, R. J.

H. J. Yoo, R. J. Hayes, E. G. Paek, A. Scherer, Y. S. Kwon, IEEE J. Quantum Electron. 26, 1093 (1990).
[CrossRef]

Kapon, E.

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

E. Kapon, J. Katz, A. Yariv, Opt. Lett. 9, 125 (1984).
[CrossRef] [PubMed]

Katz, J.

Kojiama, K.

R. A. Morgan, K. Kojiama, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, M. Asom, Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Kwon, Y. S.

H. J. Yoo, R. J. Hayes, E. G. Paek, A. Scherer, Y. S. Kwon, IEEE J. Quantum Electron. 26, 1093 (1990).
[CrossRef]

Leibenguth, R. E.

R. A. Morgan, K. Kojiama, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, M. Asom, Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Lentz, G.

G. Lentz, J. Salzman, J. Lightwave Technol. 8, 1803 (1990).

Morgan, R. A.

R. A. Morgan, K. Kojiama, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, M. Asom, Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Mullally, T.

R. A. Morgan, K. Kojiama, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, M. Asom, Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Oreinstein, M.

T. Fishman, M. Oreinstein, presented at the 13th IEEE International Semiconductor Laser Conference, Takamatsu, Japan, 1992.

Orenstein, M.

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

Paek, E. G.

H. J. Yoo, R. J. Hayes, E. G. Paek, A. Scherer, Y. S. Kwon, IEEE J. Quantum Electron. 26, 1093 (1990).
[CrossRef]

Salzman, J.

G. Lentz, J. Salzman, J. Lightwave Technol. 8, 1803 (1990).

Scherer, A.

H. J. Yoo, R. J. Hayes, E. G. Paek, A. Scherer, Y. S. Kwon, IEEE J. Quantum Electron. 26, 1093 (1990).
[CrossRef]

Stoffel, N. G.

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

Wentworth, R. H.

R. J. Black, L. Gagnon, R. C. Youngquist, R. H. Wentworth, Electron. Lett. 22, 1311 (1986).
[CrossRef]

Yariv, A.

Yoo, H. J.

H. J. Yoo, R. J. Hayes, E. G. Paek, A. Scherer, Y. S. Kwon, IEEE J. Quantum Electron. 26, 1093 (1990).
[CrossRef]

Youngquist, R. C.

R. J. Black, L. Gagnon, R. C. Youngquist, R. H. Wentworth, Electron. Lett. 22, 1311 (1986).
[CrossRef]

Appl. Phys. Lett. (2)

M. Orenstein, E. Kapon, N. G. Stoffel, J. P. Harbison, L. T. Florez, Appl. Phys. Lett. 60, 1535 (1992).
[CrossRef]

R. A. Morgan, K. Kojiama, T. Mullally, G. D. Guth, M. W. Focht, R. E. Leibenguth, M. Asom, Appl. Phys. Lett. 61, 1160 (1992).
[CrossRef]

Electron. Lett. (1)

R. J. Black, L. Gagnon, R. C. Youngquist, R. H. Wentworth, Electron. Lett. 22, 1311 (1986).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. J. Yoo, R. J. Hayes, E. G. Paek, A. Scherer, Y. S. Kwon, IEEE J. Quantum Electron. 26, 1093 (1990).
[CrossRef]

J. Lightwave Technol. (1)

G. Lentz, J. Salzman, J. Lightwave Technol. 8, 1803 (1990).

Opt. Lett. (1)

Other (1)

T. Fishman, M. Oreinstein, presented at the 13th IEEE International Semiconductor Laser Conference, Takamatsu, Japan, 1992.

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Figures (5)

Fig. 1
Fig. 1

a, Mirror-reflectivity-modulated VCSEL transformed into a gain-guided laser array. b, The cyclic laser array employed in the model. Shaded areas, regions where the 2D model is not applicable.

Fig. 2
Fig. 2

Two degenerate dominant constant-phase modes of a seven-element circular array with an effective index neff = 3.47 − i5.84 × 10−4.

Fig. 3
Fig. 3

a, Dominant mode of a seven-element array. Continuous curve, model results for a slight perturbation (r = 0.99, rd = 0.9899); dashed curve, simulation of an unperturbed array. b, Experimental results for an 11-element array compared with model results for a slightly perturbed array. Arrows indicate the location of the defective laser.

Fig. 4
Fig. 4

Measurements of an 11-element array: a, top contact and spontaneous emission patterns (I = 30 mA); b, 10-lobed NF and FF patterns (I = 50 mA, Ith = 40 mA); c, 12-lobed NF and FF patterns of a different array (I = 70 mA, Ith = 60 mA).

Fig. 5
Fig. 5

Two rotated modes of a five-element cyclic array.

Equations (10)

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[ 2 r 2 + 1 r r + 1 r 2 2 ϕ 2 + ( k 2 - β 2 ) ] E t = 0 ,
ɛ ( r , ϕ ) n r 2 ( r ) + n ϕ 2 ( ϕ )
{ 2 r 2 + 1 r r + k 0 2 [ n r 2 ( r ) - n er 2 ] } R ( r ) = 0 ,
{ 2 x 2 + k 0 2 [ n ϕ 2 ( x ) - n e ϕ 2 ] } Φ ( x ) = 0 ,
Φ m ( x ) = { A m cos ( k 2 x m ) + B m sin ( k 2 x m ) x m < d 2 / 2 C m cos ( k 1 x m ) + D m sin ( k 1 x m ) d 2 / 2 < x m < d 1 + d 2 / 2 E m cos ( k 2 x m ) + F m sin ( k 2 x m ) d 1 + d 2 / 2 < x m < d 2 + d 1 ,
[ A N B N ] = M N - N d M d M N d - 1 [ A 0 B 0 ] ,
exp ( ± i N K Λ ) [ A B ] ± = [ cos ( N K Λ ) ) i / η ) sin ( N K Λ ) i η sin ( N K Λ ) cos ( N K Λ ) ] [ A B ] ± ,
E p l = exp ( ± i 2 π l p / N ) ,             p = 0 , 1 , , N - 1 ,
E CP , p l , 1 = E p l + E p * l 2 ,             E CP , p l , 2 = E p l + i E p * l 2 ( E p l and E p * l degenerate ) ,
E CP , p l = { cos ( 2 π p l / N ) sin ( 2 π p l / N ) , p = 0 , 1 , , N - 1 .

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