Abstract

Implementation of a beam-shaping system, whereby an arbitrary array of spots is generated, is proposed. The suggested beam-shaping generator, which is based on the Gerchberg–Saxton algorithm, contains two phase-only filters and thus yields a very-high-power throughput. The flexibility of the suggested approach is demonstrated with some computer simulations.

© 1996 Optical Society of America

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References

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  1. H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [Crossref]
  2. H. Dammann, E. Klotz, “Coherent-optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
    [Crossref]
  3. U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
    [Crossref]
  4. W. B. Veldkamp, J. R. Leger, G. J. Swanson, “Coherent summation of laser beams using binary phase gratings,” Opt. Lett. 11, 303–305 (1986).
    [Crossref] [PubMed]
  5. D. Mendlovic, I. Ouzieli, I. Kiryuschev, E. Marom “Two-dimensional wavelet transform achieved by computer generated multireference matched filter and Dammann grating,” Appl. Opt. 34, 8213–8219 (1995).
    [Crossref] [PubMed]
  6. L. P. Bolvin, “Multiple imaging using various types of simple phase gratings,” Appl. Opt. 11, 1782–1792 (1972).
    [Crossref]
  7. M. T. Gale, M. Rossi, H. Schütz, P. Ehbets, H. P. Herzig, D. Prongué, “Continuous-relief diffractive optical elements for two-dimensional array-generation,” Appl. Opt. 32, 2526–2533 (1993).
    [Crossref] [PubMed]
  8. J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
  9. Z. Zalevsky, D. Mendlovic, A. W. Lohmann, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
    [Crossref] [PubMed]
  10. E. Noel, R. R. Khan, H. S. Dhadwal, “Optical implementation of a regularized Gerchberg iterative algorithm for super resolution,” Opt. Eng. 32, 2866–2870 (1993).
    [Crossref]
  11. R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [Crossref]
  12. G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
    [Crossref] [PubMed]
  13. M. Hedley, H. Yan, D. Rosenfeld, “A modified Gerchberg–Saxton algorithm for one-dimensional motion artifact correction in MRI,” IEEE Trans. Signal Process. 39, 1428–1432 (1991).
    [Crossref]
  14. R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Stuttgart) 34, 275–284 (1971).
  15. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 227–246 (1972).
  16. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

1996 (1)

1995 (1)

1994 (1)

1993 (2)

E. Noel, R. R. Khan, H. S. Dhadwal, “Optical implementation of a regularized Gerchberg iterative algorithm for super resolution,” Opt. Eng. 32, 2866–2870 (1993).
[Crossref]

M. T. Gale, M. Rossi, H. Schütz, P. Ehbets, H. P. Herzig, D. Prongué, “Continuous-relief diffractive optical elements for two-dimensional array-generation,” Appl. Opt. 32, 2526–2533 (1993).
[Crossref] [PubMed]

1991 (1)

M. Hedley, H. Yan, D. Rosenfeld, “A modified Gerchberg–Saxton algorithm for one-dimensional motion artifact correction in MRI,” IEEE Trans. Signal Process. 39, 1428–1432 (1991).
[Crossref]

1989 (1)

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

1986 (1)

1982 (1)

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[Crossref]

1977 (1)

H. Dammann, E. Klotz, “Coherent-optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[Crossref]

1974 (1)

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

1972 (2)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 227–246 (1972).

L. P. Bolvin, “Multiple imaging using various types of simple phase gratings,” Appl. Opt. 11, 1782–1792 (1972).
[Crossref]

1971 (2)

H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[Crossref]

R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Stuttgart) 34, 275–284 (1971).

Bolvin, L. P.

Dammann, H.

H. Dammann, E. Klotz, “Coherent-optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[Crossref]

H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[Crossref]

Dhadwal, H. S.

E. Noel, R. R. Khan, H. S. Dhadwal, “Optical implementation of a regularized Gerchberg iterative algorithm for super resolution,” Opt. Eng. 32, 2866–2870 (1993).
[Crossref]

Dong, B. Z.

Downs, M. M.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Ehbets, P.

Ersoy, O. K.

Gale, M. T.

Gerchberg, R. W.

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 227–246 (1972).

R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Stuttgart) 34, 275–284 (1971).

Gortler, K.

H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[Crossref]

Gu, B. Y.

Hedley, M.

M. Hedley, H. Yan, D. Rosenfeld, “A modified Gerchberg–Saxton algorithm for one-dimensional motion artifact correction in MRI,” IEEE Trans. Signal Process. 39, 1428–1432 (1991).
[Crossref]

Herzig, H. P.

Jahns, J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Khan, R. R.

E. Noel, R. R. Khan, H. S. Dhadwal, “Optical implementation of a regularized Gerchberg iterative algorithm for super resolution,” Opt. Eng. 32, 2866–2870 (1993).
[Crossref]

Killat, U.

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[Crossref]

Kiryuschev, I.

Klotz, E.

H. Dammann, E. Klotz, “Coherent-optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[Crossref]

Leger, J. R.

Lohmann, A. W.

Marom, E.

Mendlovic, D.

Noel, E.

E. Noel, R. R. Khan, H. S. Dhadwal, “Optical implementation of a regularized Gerchberg iterative algorithm for super resolution,” Opt. Eng. 32, 2866–2870 (1993).
[Crossref]

Ouzieli, I.

Prise, M. E.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Prongué, D.

Rabe, G.

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[Crossref]

Rave, W.

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[Crossref]

Rosenfeld, D.

M. Hedley, H. Yan, D. Rosenfeld, “A modified Gerchberg–Saxton algorithm for one-dimensional motion artifact correction in MRI,” IEEE Trans. Signal Process. 39, 1428–1432 (1991).
[Crossref]

Rossi, M.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 227–246 (1972).

R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Stuttgart) 34, 275–284 (1971).

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

Schütz, H.

Streibl, N.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Swanson, G. J.

Veldkamp, W. B.

Walker, S. J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Yan, H.

M. Hedley, H. Yan, D. Rosenfeld, “A modified Gerchberg–Saxton algorithm for one-dimensional motion artifact correction in MRI,” IEEE Trans. Signal Process. 39, 1428–1432 (1991).
[Crossref]

Yang, G. Z.

Zalevsky, Z.

Zhuang, J. Y.

Appl. Opt. (4)

Fiber Integr. Opt. (1)

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[Crossref]

IEEE Trans. Signal Process. (1)

M. Hedley, H. Yan, D. Rosenfeld, “A modified Gerchberg–Saxton algorithm for one-dimensional motion artifact correction in MRI,” IEEE Trans. Signal Process. 39, 1428–1432 (1991).
[Crossref]

Opt. Acta (2)

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

H. Dammann, E. Klotz, “Coherent-optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[Crossref]

Opt. Commun. (1)

H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[Crossref]

Opt. Eng. (2)

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

E. Noel, R. R. Khan, H. S. Dhadwal, “Optical implementation of a regularized Gerchberg iterative algorithm for super resolution,” Opt. Eng. 32, 2866–2870 (1993).
[Crossref]

Opt. Lett. (2)

Optik (Stuttgart) (2)

R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Stuttgart) 34, 275–284 (1971).

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 227–246 (1972).

Other (1)

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

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Figures (10)

Fig. 1
Fig. 1

Common optical setup for implementing a beam-shaping generator by the use of the G–S algorithm.

Fig. 2
Fig. 2

G–S iterative algorithm for calculating the phase of the input phase-only filter.

Fig. 3
Fig. 3

Optical setup for implementing a beam-shaping generator.

Fig. 4
Fig. 4

Optical setup for implementing image replication.

Fig. 5
Fig. 5

Desired beam shape: four impulses equally spaced.

Fig. 6
Fig. 6

Generated array of four impulses.

Fig. 7
Fig. 7

Desired beam shape: an asymmetric array of seven impulses.

Fig. 8
Fig. 8

Beam-shaping result achieved with the proposed method.

Fig. 9
Fig. 9

Generation of an array of four impulses in the presence of noise.

Fig. 10
Fig. 10

Generation of an asymmetrical array of seven impulses in the presence of noise.

Equations (20)

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t ( x , y ) = t 1 ( x ) t 2 ( y ) .
f ( x , y ) = A in ( x , y ) exp [ i ϕ in ( x , y ) ] ,
F ( u , v ) = F T { f ( x , y ) } = F out ( u , v ) exp [ i ϕ out ( u , v ) ] .
G ( u , v ) = A out ( u , v ) exp [ i ϕ out ( u , v ) ] .
t out ( u , v ) = exp { i [ - ϕ out ( u , v ) + ϕ d ( u , v ) ] } ,
f ( x , y ) = A in ( x , y ) exp [ i ϕ 1 ( x , y ) ] ,
g ( x , y ) = exp ( i k Z 0 ) i λ Z 0 - - f ( x , y ) × exp { i k 2 Z 0 [ ( x - x ) 2 + ( y - y ) 2 ] } d x d y ,
s ( x , y ) = g ( x , y ) exp [ i ϕ 2 ( x , y ) ] .
S ( u , v ) = - - s ( x , y ) exp [ - 2 π i λ f ( x u + y v ) ] d x d y ,
D ( u , v ) = A out ( u , v ) exp [ i ϕ out ( u , v ) ] .
d ( x , y ) = C - - D ( u , v ) exp [ 2 π i λ f ( x u + y v ) ] d u d v ,
d ( x , y ) = A d ( x , y ) exp [ i ϕ d ( x , y ) ] .
g ( x , y ) = A 0 ( x , y ) exp [ i ϕ 0 ( x , y ) ] ,
A d ( x , y ) = A 0 ( x , y ) ,
ϕ 2 ( x , y ) = ϕ d ( x , y ) - ϕ 0 ( x , y ) .
f ( x , y ) = A in ( x , y ) exp [ i ϕ 1 ( x , y ) ] ,
g ( x , y ) = exp ( i k Z 0 ) i λ Z 0 - - f ( x , y ) × exp { i k 2 Z 0 [ ( x - x ) 2 + ( y - y ) 2 ] } d x d y .
g ˜ ( x , y ) = A d ( x , y ) exp [ i ϕ 0 ( x , y ) ] ,
D ( u , v ) = m n A m , n · δ ( u - u m , v - v n ) ,
L v = λ f N L x ,

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