Abstract

The Lau cavity is the self-imaging cavity with a phase corrector under the Lau reimaging condition. I propose the use of the Lau cavity to utilize both the Talbot and the Lau effects for phase locking one-dimensional and two-dimensional diode-laser arrays into a single-lobe coherent beam. Analyses on the self-reproducing of a coherent lasing field and the reimaging of initial incoherent radiation are given.

© 1989 Optical Society of America

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  1. R. H. Rediker, R. P. Schloss, L. J. Van Ruyven, Appl. Phys. Lett. 46, 133 (1985).
    [CrossRef]
  2. J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 42, 554 (1983).
    [CrossRef]
  3. J. J. Yang, M. Jansen, Electron. Lett. 22, 2 (1986).
    [CrossRef]
  4. J. R. Leger, G. J. Swanson, W. B. Veldkamp, Appl. Opt. 26, 4391 (1987).
    [CrossRef] [PubMed]
  5. M. Jansen, J. J. Yang, S. S. Ou, J. Wilcox, D. Botez, L. Mawst, W. W. Simmons, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989), paper FL2.
  6. F. X. D’Amato, E. T. Siebert, C. Roychoudhuri, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989), paper FL3.
  7. J. Z. Wilcox, W. W. Simmons, D. Botez, M. Jansen, L. J. Mawst, G. Peterson, T. J. Wilcox, J. J. Yang, Appl. Phys. Lett. 5, 1848 (1989).
    [CrossRef]
  8. J. R. Leger, M. L. Scott, W. B. Veldkamp, Appl. Phys. Lett. 52, 1771 (1988).
    [CrossRef]
  9. A. A. Golubentsev, V. V. Likhanskii, A. P. Napartovich, Sov. Phys. JETP 66, 676 (1987).
  10. L. Liu, L. Zhao, Chin. J. Lasers 16, 37 (1989).
  11. G. J. Swanson, E. N. Leith, J. Opt. Soc. Am. A 2, 789 (1985).
    [CrossRef]
  12. L. Liu, Opt. Commun. 70, 267 (1989).
    [CrossRef]
  13. R. Sudol, B. J. Thompson, Appl. Opt. 20, 1107 (1981).
    [CrossRef] [PubMed]
  14. J. K. Butler, D. E. Ackley, D. Botez, Appl. Phys. Lett. 44, 293 (1984); E. Kapon, J. Katz, A. Yariv, Opt. Lett. 10, 125 (1984).
    [CrossRef]
  15. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).
  16. L. Liu, Acta Phys. Sin. 35, 1556 (1986).
  17. L. Liu, J. Opt. Soc. Am. A 5, 1709 (1988).
    [CrossRef]
  18. J. T. Winthrop, C. R. Worthington, J. Opt. Soc. Am. 55, 373 (1965).
    [CrossRef]
  19. L. Liu, Opt. Acta Sin. 6, 807 (1986).

1989 (3)

J. Z. Wilcox, W. W. Simmons, D. Botez, M. Jansen, L. J. Mawst, G. Peterson, T. J. Wilcox, J. J. Yang, Appl. Phys. Lett. 5, 1848 (1989).
[CrossRef]

L. Liu, L. Zhao, Chin. J. Lasers 16, 37 (1989).

L. Liu, Opt. Commun. 70, 267 (1989).
[CrossRef]

1988 (2)

L. Liu, J. Opt. Soc. Am. A 5, 1709 (1988).
[CrossRef]

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

1987 (2)

A. A. Golubentsev, V. V. Likhanskii, A. P. Napartovich, Sov. Phys. JETP 66, 676 (1987).

J. R. Leger, G. J. Swanson, W. B. Veldkamp, Appl. Opt. 26, 4391 (1987).
[CrossRef] [PubMed]

1986 (3)

J. J. Yang, M. Jansen, Electron. Lett. 22, 2 (1986).
[CrossRef]

L. Liu, Acta Phys. Sin. 35, 1556 (1986).

L. Liu, Opt. Acta Sin. 6, 807 (1986).

1985 (2)

R. H. Rediker, R. P. Schloss, L. J. Van Ruyven, Appl. Phys. Lett. 46, 133 (1985).
[CrossRef]

G. J. Swanson, E. N. Leith, J. Opt. Soc. Am. A 2, 789 (1985).
[CrossRef]

1984 (1)

J. K. Butler, D. E. Ackley, D. Botez, Appl. Phys. Lett. 44, 293 (1984); E. Kapon, J. Katz, A. Yariv, Opt. Lett. 10, 125 (1984).
[CrossRef]

1983 (1)

J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 42, 554 (1983).
[CrossRef]

1981 (1)

1965 (1)

1961 (1)

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Ackley, D. E.

J. K. Butler, D. E. Ackley, D. Botez, Appl. Phys. Lett. 44, 293 (1984); E. Kapon, J. Katz, A. Yariv, Opt. Lett. 10, 125 (1984).
[CrossRef]

Botez, D.

J. Z. Wilcox, W. W. Simmons, D. Botez, M. Jansen, L. J. Mawst, G. Peterson, T. J. Wilcox, J. J. Yang, Appl. Phys. Lett. 5, 1848 (1989).
[CrossRef]

J. K. Butler, D. E. Ackley, D. Botez, Appl. Phys. Lett. 44, 293 (1984); E. Kapon, J. Katz, A. Yariv, Opt. Lett. 10, 125 (1984).
[CrossRef]

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

Butler, J. K.

J. K. Butler, D. E. Ackley, D. Botez, Appl. Phys. Lett. 44, 293 (1984); E. Kapon, J. Katz, A. Yariv, Opt. Lett. 10, 125 (1984).
[CrossRef]

D’Amato, F. X.

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

Fox, A. G.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Golubentsev, A. A.

A. A. Golubentsev, V. V. Likhanskii, A. P. Napartovich, Sov. Phys. JETP 66, 676 (1987).

Jansen, M.

J. Z. Wilcox, W. W. Simmons, D. Botez, M. Jansen, L. J. Mawst, G. Peterson, T. J. Wilcox, J. J. Yang, Appl. Phys. Lett. 5, 1848 (1989).
[CrossRef]

J. J. Yang, M. Jansen, Electron. Lett. 22, 2 (1986).
[CrossRef]

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

Katz, J.

J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 42, 554 (1983).
[CrossRef]

Leger, J. R.

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

J. R. Leger, G. J. Swanson, W. B. Veldkamp, Appl. Opt. 26, 4391 (1987).
[CrossRef] [PubMed]

Leith, E. N.

Li, T.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Likhanskii, V. V.

A. A. Golubentsev, V. V. Likhanskii, A. P. Napartovich, Sov. Phys. JETP 66, 676 (1987).

Liu, L.

L. Liu, Opt. Commun. 70, 267 (1989).
[CrossRef]

L. Liu, L. Zhao, Chin. J. Lasers 16, 37 (1989).

L. Liu, J. Opt. Soc. Am. A 5, 1709 (1988).
[CrossRef]

L. Liu, Opt. Acta Sin. 6, 807 (1986).

L. Liu, Acta Phys. Sin. 35, 1556 (1986).

Margalit, S.

J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 42, 554 (1983).
[CrossRef]

Mawst, L.

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

Mawst, L. J.

J. Z. Wilcox, W. W. Simmons, D. Botez, M. Jansen, L. J. Mawst, G. Peterson, T. J. Wilcox, J. J. Yang, Appl. Phys. Lett. 5, 1848 (1989).
[CrossRef]

Napartovich, A. P.

A. A. Golubentsev, V. V. Likhanskii, A. P. Napartovich, Sov. Phys. JETP 66, 676 (1987).

Ou, S. S.

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

Peterson, G.

J. Z. Wilcox, W. W. Simmons, D. Botez, M. Jansen, L. J. Mawst, G. Peterson, T. J. Wilcox, J. J. Yang, Appl. Phys. Lett. 5, 1848 (1989).
[CrossRef]

Rediker, R. H.

R. H. Rediker, R. P. Schloss, L. J. Van Ruyven, Appl. Phys. Lett. 46, 133 (1985).
[CrossRef]

Roychoudhuri, C.

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

Schloss, R. P.

R. H. Rediker, R. P. Schloss, L. J. Van Ruyven, Appl. Phys. Lett. 46, 133 (1985).
[CrossRef]

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 Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989), paper FL3.

Simmons, W. W.

J. Z. Wilcox, W. W. Simmons, D. Botez, M. Jansen, L. J. Mawst, G. Peterson, T. J. Wilcox, J. J. Yang, Appl. Phys. Lett. 5, 1848 (1989).
[CrossRef]

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

Sudol, R.

Swanson, G. J.

Thompson, B. J.

Van Ruyven, L. J.

R. H. Rediker, R. P. Schloss, L. J. Van Ruyven, Appl. Phys. Lett. 46, 133 (1985).
[CrossRef]

Veldkamp, W. B.

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

J. R. Leger, G. J. Swanson, W. B. Veldkamp, Appl. Opt. 26, 4391 (1987).
[CrossRef] [PubMed]

Wilcox, J.

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

Wilcox, J. Z.

J. Z. Wilcox, W. W. Simmons, D. Botez, M. Jansen, L. J. Mawst, G. Peterson, T. J. Wilcox, J. J. Yang, Appl. Phys. Lett. 5, 1848 (1989).
[CrossRef]

Wilcox, T. J.

J. Z. Wilcox, W. W. Simmons, D. Botez, M. Jansen, L. J. Mawst, G. Peterson, T. J. Wilcox, J. J. Yang, Appl. Phys. Lett. 5, 1848 (1989).
[CrossRef]

Winthrop, J. T.

Worthington, C. R.

Yang, J. J.

J. Z. Wilcox, W. W. Simmons, D. Botez, M. Jansen, L. J. Mawst, G. Peterson, T. J. Wilcox, J. J. Yang, Appl. Phys. Lett. 5, 1848 (1989).
[CrossRef]

J. J. Yang, M. Jansen, Electron. Lett. 22, 2 (1986).
[CrossRef]

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

Yariv, A.

J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 42, 554 (1983).
[CrossRef]

Zhao, L.

L. Liu, L. Zhao, Chin. J. Lasers 16, 37 (1989).

Acta Phys. Sin. (1)

L. Liu, Acta Phys. Sin. 35, 1556 (1986).

Appl. Opt. (2)

Appl. Phys. Lett. (5)

R. H. Rediker, R. P. Schloss, L. J. Van Ruyven, Appl. Phys. Lett. 46, 133 (1985).
[CrossRef]

J. Katz, S. Margalit, A. Yariv, Appl. Phys. Lett. 42, 554 (1983).
[CrossRef]

J. Z. Wilcox, W. W. Simmons, D. Botez, M. Jansen, L. J. Mawst, G. Peterson, T. J. Wilcox, J. J. Yang, Appl. Phys. Lett. 5, 1848 (1989).
[CrossRef]

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

J. K. Butler, D. E. Ackley, D. Botez, Appl. Phys. Lett. 44, 293 (1984); E. Kapon, J. Katz, A. Yariv, Opt. Lett. 10, 125 (1984).
[CrossRef]

Bell Syst. Tech. J. (1)

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Chin. J. Lasers (1)

L. Liu, L. Zhao, Chin. J. Lasers 16, 37 (1989).

Electron. Lett. (1)

J. J. Yang, M. Jansen, Electron. Lett. 22, 2 (1986).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta Sin. (1)

L. Liu, Opt. Acta Sin. 6, 807 (1986).

Opt. Commun. (1)

L. Liu, Opt. Commun. 70, 267 (1989).
[CrossRef]

Sov. Phys. JETP (1)

A. A. Golubentsev, V. V. Likhanskii, A. P. Napartovich, Sov. Phys. JETP 66, 676 (1987).

Other (2)

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

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

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the Lau cavity. Z1 and Z2 are the corresponding distances.

Fig. 2
Fig. 2

(1) Laser array with Tx/hx = Ty/hyy = 2 and Tx = Ty. (b) The phase correction plate located at z1 = 3/2 Tx2/λ, where +, −, and 0 represent exp[j(π/2)], exp[−j(π/2)], and exp(j0), respectively.

Fig. 3
Fig. 3

(a) Intensity function of the individual emitter radiation and (b) the corresponding reimaged Lau intensity function.

Equations (15)

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E 1 ( x , y ) = E s ( x , y ) [ n δ ( x n T x ) n δ ( y n T y ) ] ,
E 2 ( x , y ) = exp ( j k z 1 ) j λ z 1 E 1 ( x , y ) ⊗⊗ exp [ j π λ z 1 ( x 2 + y 2 ) ] .
E 1 ( x , y ) K = 1 j λ z 1 T ( x , y ) exp [ j π λ z 1 ( x 2 + y 2 ) ] .
I 1 ( x , y ) = I s ( x , y ) [ n δ ( x n T x ) n δ ( y n T y ) ] ,
I 1 ( x , y ) = r 2 2 K 4 I 1 ( ξ , η ) d ξ d η × | T i { 1 p [ ξ + s ( ξ x ) , η + s ( η y ) ] } | 2 × | T i [ 1 p ( ξ , η ) ] | 2 d ξ d η ,
z 2 z 1 .
I 1 ( x , y ) = r 2 2 I s ( x , y ) [ n δ ( x n T x ) n δ ( y n T y ) ] ,
I s ( x , y ) = I s ( ξ , η ) d ξ d η × I s ( ξ x , η ) y ) I s ( ξ , η ) d ξ d η .
z = α β T 2 λ ,
S ( x : T β ) = { 1 β k R k δ ( x k T β ) α and / or β are even 1 β k R k δ ( x k T β T 2 β ) both α and β are odd ,
R k = 1 β n = 0 ( β 1 ) exp [ j ( 2 π k n β + π α β n 2 ) ] , R k = 1 2 β n = 0 ( 2 β 1 ) exp [ j ( π n ( 2 k 1 ) β + π α β n 2 ) ] .
E s ( x , y ) = rect ( x h x ) rect ( y h y ) ,
T x h x = β x and T y h y = β y ,
z 1 = α x β x T x 2 λ = α y β y T y 2 λ .
T ( x , y ) = β x β y rect ( x h x ) S * ( x : T x β x ) × rect ( y h y ) S * ( y : T y β y ) .

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