Abstract

We present numerical simulations of the propagation of the field emitted by laser arrays consisting of a small number of elements. Our computations show how propagation transforms part of an amplitude modulation into a phase modulation. This fact can be used to enhance the central lobe energy in the far field of high-power laser arrays by means of a simple corrective optical element. We discuss optimal filtering conditions and prove that the quarter-Talbot distance generally used in previous studies is not the best choice.

© 1993 Optical Society of America

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  1. V. V. Antyukhov, A. I. Bondarenko, A. F. Glova, V. S. Golubev, O. R. Kachurin, L. L. Kolesov, A. E. Lebedev, F. V. Lebedev, Y. F. Suslov, V. A. Timofeev, Sov. J. Quantum Electron. 11, 1363 (1981).
    [CrossRef]
  2. L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. Demaria, W. B. Bridges, Appl. Phys. Lett. 48, 1701 (1986).
    [CrossRef]
  3. R. A. Hart, L. A. Newman, A. J. Cantor, A. J. Demaria, J. T. Kennedy, Appl. Phys. Lett. 51, 1057 (1987).
    [CrossRef]
  4. K. M. Abramsky, A. D. Colley, H. J. Baker, D. R. Hall, Appl. Phys. Lett. 60, 530 (1992).
    [CrossRef]
  5. A. Lapucci, G. Cangioli, Appl. Phys. Lett. 62, 7 (1993).
    [CrossRef]
  6. G. J. Swanson, J. R. Leger, M. Holtz, Opt. Lett. 12, 245 (1987).
    [CrossRef] [PubMed]
  7. J. R. Leger, M. L. Scott, W. B. Veldkamp, Appl. Phys. Lett. 52, 1771 (1988).
    [CrossRef]
  8. See, for example, K. Patorski, in Progress in Optics XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Chap. 1 and references therein.
  9. V. V. Antyukhov, A. F. Glova, O. R. Kachurin, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pismennyi, JETP Lett. 44, 78 (1986).
  10. J. R. Leger, M. Griswold, P. Bundman, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL1.
  11. M. Jansen, J. J. Yang, S. S. Ou, J. Wilcox, D. Botez, L. Mawst, W. W. Simmons, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL2.
  12. F. X. D’Amato, E. T. Siebert, C. Roychoudhuri, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL3.
  13. A. W. Lohmann, Optik 79, 41 (1988).
  14. J. R. Leger, G. J. Swanson, Opt. Lett. 15, 288 (1990).
    [CrossRef] [PubMed]
  15. V. V. Likhanskii, A. P. Napartovich, Sov. Phys. Usp. 33, 228 (1990), and references therein.
    [CrossRef]
  16. J. T. Winthorp, C. R. Worthington, J. Opt. Soc. Am. 55, 373 (1965).
    [CrossRef]

1993 (1)

A. Lapucci, G. Cangioli, Appl. Phys. Lett. 62, 7 (1993).
[CrossRef]

1992 (1)

K. M. Abramsky, A. D. Colley, H. J. Baker, D. R. Hall, Appl. Phys. Lett. 60, 530 (1992).
[CrossRef]

1990 (2)

J. R. Leger, G. J. Swanson, Opt. Lett. 15, 288 (1990).
[CrossRef] [PubMed]

V. V. Likhanskii, A. P. Napartovich, Sov. Phys. Usp. 33, 228 (1990), and references therein.
[CrossRef]

1988 (2)

A. W. Lohmann, Optik 79, 41 (1988).

J. R. Leger, M. L. Scott, W. B. Veldkamp, Appl. Phys. Lett. 52, 1771 (1988).
[CrossRef]

1987 (2)

G. J. Swanson, J. R. Leger, M. Holtz, Opt. Lett. 12, 245 (1987).
[CrossRef] [PubMed]

R. A. Hart, L. A. Newman, A. J. Cantor, A. J. Demaria, J. T. Kennedy, Appl. Phys. Lett. 51, 1057 (1987).
[CrossRef]

1986 (2)

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. Demaria, W. B. Bridges, Appl. Phys. Lett. 48, 1701 (1986).
[CrossRef]

V. V. Antyukhov, A. F. Glova, O. R. Kachurin, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pismennyi, JETP Lett. 44, 78 (1986).

1981 (1)

V. V. Antyukhov, A. I. Bondarenko, A. F. Glova, V. S. Golubev, O. R. Kachurin, L. L. Kolesov, A. E. Lebedev, F. V. Lebedev, Y. F. Suslov, V. A. Timofeev, Sov. J. Quantum Electron. 11, 1363 (1981).
[CrossRef]

1965 (1)

Abramsky, K. M.

K. M. Abramsky, A. D. Colley, H. J. Baker, D. R. Hall, Appl. Phys. Lett. 60, 530 (1992).
[CrossRef]

Antyukhov, V. V.

V. V. Antyukhov, A. F. Glova, O. R. Kachurin, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pismennyi, JETP Lett. 44, 78 (1986).

V. V. Antyukhov, A. I. Bondarenko, A. F. Glova, V. S. Golubev, O. R. Kachurin, L. L. Kolesov, A. E. Lebedev, F. V. Lebedev, Y. F. Suslov, V. A. Timofeev, Sov. J. Quantum Electron. 11, 1363 (1981).
[CrossRef]

Baker, H. J.

K. M. Abramsky, A. D. Colley, H. J. Baker, D. R. Hall, Appl. Phys. Lett. 60, 530 (1992).
[CrossRef]

Bondarenko, A. I.

V. V. Antyukhov, A. I. Bondarenko, A. F. Glova, V. S. Golubev, O. R. Kachurin, L. L. Kolesov, A. E. Lebedev, F. V. Lebedev, Y. F. Suslov, V. A. Timofeev, Sov. J. Quantum Electron. 11, 1363 (1981).
[CrossRef]

Botez, D.

M. Jansen, J. J. Yang, S. S. Ou, J. Wilcox, D. Botez, L. Mawst, W. W. Simmons, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL2.

Bridges, W. B.

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. Demaria, W. B. Bridges, Appl. Phys. Lett. 48, 1701 (1986).
[CrossRef]

Bundman, P.

J. R. Leger, M. Griswold, P. Bundman, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL1.

Cangioli, G.

A. Lapucci, G. Cangioli, Appl. Phys. Lett. 62, 7 (1993).
[CrossRef]

Cantor, A. J.

R. A. Hart, L. A. Newman, A. J. Cantor, A. J. Demaria, J. T. Kennedy, Appl. Phys. Lett. 51, 1057 (1987).
[CrossRef]

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. Demaria, W. B. Bridges, Appl. Phys. Lett. 48, 1701 (1986).
[CrossRef]

Colley, A. D.

K. M. Abramsky, A. D. Colley, H. J. Baker, D. R. Hall, Appl. Phys. Lett. 60, 530 (1992).
[CrossRef]

D’Amato, F. X.

F. X. D’Amato, E. T. Siebert, C. Roychoudhuri, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL3.

Demaria, A. J.

R. A. Hart, L. A. Newman, A. J. Cantor, A. J. Demaria, J. T. Kennedy, Appl. Phys. Lett. 51, 1057 (1987).
[CrossRef]

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. Demaria, W. B. Bridges, Appl. Phys. Lett. 48, 1701 (1986).
[CrossRef]

Glova, A. F.

V. V. Antyukhov, A. F. Glova, O. R. Kachurin, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pismennyi, JETP Lett. 44, 78 (1986).

V. V. Antyukhov, A. I. Bondarenko, A. F. Glova, V. S. Golubev, O. R. Kachurin, L. L. Kolesov, A. E. Lebedev, F. V. Lebedev, Y. F. Suslov, V. A. Timofeev, Sov. J. Quantum Electron. 11, 1363 (1981).
[CrossRef]

Golubev, V. S.

V. V. Antyukhov, A. I. Bondarenko, A. F. Glova, V. S. Golubev, O. R. Kachurin, L. L. Kolesov, A. E. Lebedev, F. V. Lebedev, Y. F. Suslov, V. A. Timofeev, Sov. J. Quantum Electron. 11, 1363 (1981).
[CrossRef]

Griswold, M.

J. R. Leger, M. Griswold, P. Bundman, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL1.

Hall, D. R.

K. M. Abramsky, A. D. Colley, H. J. Baker, D. R. Hall, Appl. Phys. Lett. 60, 530 (1992).
[CrossRef]

Hart, R. A.

R. A. Hart, L. A. Newman, A. J. Cantor, A. J. Demaria, J. T. Kennedy, Appl. Phys. Lett. 51, 1057 (1987).
[CrossRef]

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. Demaria, W. B. Bridges, Appl. Phys. Lett. 48, 1701 (1986).
[CrossRef]

Holtz, M.

Jansen, M.

M. Jansen, J. J. Yang, S. S. Ou, J. Wilcox, D. Botez, L. Mawst, W. W. Simmons, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL2.

Kachurin, O. R.

V. V. Antyukhov, A. F. Glova, O. R. Kachurin, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pismennyi, JETP Lett. 44, 78 (1986).

V. V. Antyukhov, A. I. Bondarenko, A. F. Glova, V. S. Golubev, O. R. Kachurin, L. L. Kolesov, A. E. Lebedev, F. V. Lebedev, Y. F. Suslov, V. A. Timofeev, Sov. J. Quantum Electron. 11, 1363 (1981).
[CrossRef]

Kennedy, J. T.

R. A. Hart, L. A. Newman, A. J. Cantor, A. J. Demaria, J. T. Kennedy, Appl. Phys. Lett. 51, 1057 (1987).
[CrossRef]

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. Demaria, W. B. Bridges, Appl. Phys. Lett. 48, 1701 (1986).
[CrossRef]

Kolesov, L. L.

V. V. Antyukhov, A. I. Bondarenko, A. F. Glova, V. S. Golubev, O. R. Kachurin, L. L. Kolesov, A. E. Lebedev, F. V. Lebedev, Y. F. Suslov, V. A. Timofeev, Sov. J. Quantum Electron. 11, 1363 (1981).
[CrossRef]

Lapucci, A.

A. Lapucci, G. Cangioli, Appl. Phys. Lett. 62, 7 (1993).
[CrossRef]

Lebedev, A. E.

V. V. Antyukhov, A. I. Bondarenko, A. F. Glova, V. S. Golubev, O. R. Kachurin, L. L. Kolesov, A. E. Lebedev, F. V. Lebedev, Y. F. Suslov, V. A. Timofeev, Sov. J. Quantum Electron. 11, 1363 (1981).
[CrossRef]

Lebedev, F. V.

V. V. Antyukhov, A. F. Glova, O. R. Kachurin, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pismennyi, JETP Lett. 44, 78 (1986).

V. V. Antyukhov, A. I. Bondarenko, A. F. Glova, V. S. Golubev, O. R. Kachurin, L. L. Kolesov, A. E. Lebedev, F. V. Lebedev, Y. F. Suslov, V. A. Timofeev, Sov. J. Quantum Electron. 11, 1363 (1981).
[CrossRef]

Leger, J. R.

J. R. Leger, G. J. Swanson, Opt. Lett. 15, 288 (1990).
[CrossRef] [PubMed]

J. R. Leger, M. L. Scott, W. B. Veldkamp, Appl. Phys. Lett. 52, 1771 (1988).
[CrossRef]

G. J. Swanson, J. R. Leger, M. Holtz, Opt. Lett. 12, 245 (1987).
[CrossRef] [PubMed]

J. R. Leger, M. Griswold, P. Bundman, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL1.

Likhanskii, V. V.

V. V. Likhanskii, A. P. Napartovich, Sov. Phys. Usp. 33, 228 (1990), and references therein.
[CrossRef]

V. V. Antyukhov, A. F. Glova, O. R. Kachurin, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pismennyi, JETP Lett. 44, 78 (1986).

Lohmann, A. W.

A. W. Lohmann, Optik 79, 41 (1988).

Mawst, L.

M. Jansen, J. J. Yang, S. S. Ou, J. Wilcox, D. Botez, L. Mawst, W. W. Simmons, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL2.

Napartovich, A. P.

V. V. Likhanskii, A. P. Napartovich, Sov. Phys. Usp. 33, 228 (1990), and references therein.
[CrossRef]

V. V. Antyukhov, A. F. Glova, O. R. Kachurin, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pismennyi, JETP Lett. 44, 78 (1986).

Newman, L. A.

R. A. Hart, L. A. Newman, A. J. Cantor, A. J. Demaria, J. T. Kennedy, Appl. Phys. Lett. 51, 1057 (1987).
[CrossRef]

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. Demaria, W. B. Bridges, Appl. Phys. Lett. 48, 1701 (1986).
[CrossRef]

Ou, S. S.

M. Jansen, J. J. Yang, S. S. Ou, J. Wilcox, D. Botez, L. Mawst, W. W. Simmons, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL2.

Patorski, K.

See, for example, K. Patorski, in Progress in Optics XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Chap. 1 and references therein.

Pismennyi, V. D.

V. V. Antyukhov, A. F. Glova, O. R. Kachurin, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pismennyi, JETP Lett. 44, 78 (1986).

Roychoudhuri, C.

F. X. D’Amato, E. T. Siebert, C. Roychoudhuri, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL3.

Scott, M. L.

J. R. Leger, M. L. Scott, W. B. Veldkamp, Appl. Phys. Lett. 52, 1771 (1988).
[CrossRef]

Siebert, E. T.

F. X. D’Amato, E. T. Siebert, C. Roychoudhuri, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL3.

Simmons, W. W.

M. Jansen, J. J. Yang, S. S. Ou, J. Wilcox, D. Botez, L. Mawst, W. W. Simmons, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL2.

Suslov, Y. F.

V. V. Antyukhov, A. I. Bondarenko, A. F. Glova, V. S. Golubev, O. R. Kachurin, L. L. Kolesov, A. E. Lebedev, F. V. Lebedev, Y. F. Suslov, V. A. Timofeev, Sov. J. Quantum Electron. 11, 1363 (1981).
[CrossRef]

Swanson, G. J.

Timofeev, V. A.

V. V. Antyukhov, A. I. Bondarenko, A. F. Glova, V. S. Golubev, O. R. Kachurin, L. L. Kolesov, A. E. Lebedev, F. V. Lebedev, Y. F. Suslov, V. A. Timofeev, Sov. J. Quantum Electron. 11, 1363 (1981).
[CrossRef]

Veldkamp, W. B.

J. R. Leger, M. L. Scott, W. B. Veldkamp, Appl. Phys. Lett. 52, 1771 (1988).
[CrossRef]

Wilcox, J.

M. Jansen, J. J. Yang, S. S. Ou, J. Wilcox, D. Botez, L. Mawst, W. W. Simmons, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL2.

Winthorp, J. T.

Worthington, C. R.

Yang, J. J.

M. Jansen, J. J. Yang, S. S. Ou, J. Wilcox, D. Botez, L. Mawst, W. W. Simmons, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL2.

Appl. Phys. Lett. (5)

L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, A. J. Demaria, W. B. Bridges, Appl. Phys. Lett. 48, 1701 (1986).
[CrossRef]

R. A. Hart, L. A. Newman, A. J. Cantor, A. J. Demaria, J. T. Kennedy, Appl. Phys. Lett. 51, 1057 (1987).
[CrossRef]

K. M. Abramsky, A. D. Colley, H. J. Baker, D. R. Hall, Appl. Phys. Lett. 60, 530 (1992).
[CrossRef]

A. Lapucci, G. Cangioli, Appl. Phys. Lett. 62, 7 (1993).
[CrossRef]

J. R. Leger, M. L. Scott, W. B. Veldkamp, Appl. Phys. Lett. 52, 1771 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

JETP Lett. (1)

V. V. Antyukhov, A. F. Glova, O. R. Kachurin, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pismennyi, JETP Lett. 44, 78 (1986).

Opt. Lett. (2)

Optik (1)

A. W. Lohmann, Optik 79, 41 (1988).

Sov. J. Quantum Electron. (1)

V. V. Antyukhov, A. I. Bondarenko, A. F. Glova, V. S. Golubev, O. R. Kachurin, L. L. Kolesov, A. E. Lebedev, F. V. Lebedev, Y. F. Suslov, V. A. Timofeev, Sov. J. Quantum Electron. 11, 1363 (1981).
[CrossRef]

Sov. Phys. Usp. (1)

V. V. Likhanskii, A. P. Napartovich, Sov. Phys. Usp. 33, 228 (1990), and references therein.
[CrossRef]

Other (4)

See, for example, K. Patorski, in Progress in Optics XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Chap. 1 and references therein.

J. R. Leger, M. Griswold, P. Bundman, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL1.

M. Jansen, J. J. Yang, S. S. Ou, J. Wilcox, D. Botez, L. Mawst, W. W. Simmons, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL2.

F. X. D’Amato, E. T. Siebert, C. Roychoudhuri, in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper FL3.

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Figures (4)

Fig. 1
Fig. 1

Intensity and phase distributions of a field generated by an ideal periodic array at different propagation distances.

Fig. 2
Fig. 2

Same as in Fig. 1 for a three-element array.

Fig. 3
Fig. 3

Effect of a phase filtering on the central lobe energy. The solid curve is obtained by zeroing the phase modulation after a variable propagation distance. The stars refer to removal of the fundamental-frequency component only, (a) A periodic array, (b) a six-element array, (c) a three-element array.

Fig. 4
Fig. 4

Far-field intensity distribution for a three-element array: (a) unfiltered, (b) sinusoidally filtered, (c) phase-matched filtered.

Equations (2)

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u ( x , z ) = h ( x , z ) * u ( x , 0 ) ,
h ( x , z ) = exp ( j k z ) j λ z exp ( j k x 2 2 z )

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