Abstract

We present an exact mathematical description of beam shaping and indicate that a rigorous solution does not exist: only an optimal solution can be found. An optimization method is proposed to search for the solution. The simulation results for an example are given in detail.

© 1998 Optical Society of America

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References

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  1. M. T. Eismann, A. M. Tai, J. N. Cederquist, “Iterative design of a holographic beam former,” Appl. Opt. 28, 2641–2650 (1989).
    [CrossRef] [PubMed]
  2. C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
    [CrossRef]
  3. M. Quintanilla, A. M. Frutos, “Holographic filter that transforms a Gaussian beam into a uniform beam,” Appl. Opt. 20, 879–880 (1981).
    [CrossRef] [PubMed]
  4. F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 30, 529–536 (1991).
    [CrossRef]
  5. Y. H. Chang, I. Yukihiro, M. Kazumi, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
    [CrossRef]
  6. W. B. Veldkamp, “Laser beam profile shaping with interlaced binary diffractive gratings,” Appl. Opt. 21, 3209–3212 (1982).
    [CrossRef] [PubMed]
  7. W. B. Veldkamp, C. J. Kastner, “Beam profile shaping for laser radars that use detector arrays,” Appl. Opt. 21, 345–356 (1982).
    [CrossRef] [PubMed]
  8. J. Cordingley, “Application of a binary diffractive optic for beam shaping in semiconductor processing by lasers,” Appl. Opt. 32, 2538–2542 (1993).
    [CrossRef] [PubMed]
  9. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
    [CrossRef]
  10. O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–166 (1974).
    [CrossRef]
  11. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  12. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef]
  13. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
    [CrossRef] [PubMed]
  14. X. Tan, B. Y. Gu, G. Z. Yang, B. Z. Dong, “Diffractive phase elements for beam shaping: a new design method,” Appl. Opt. 34, 1314–1320 (1995).
    [CrossRef] [PubMed]
  15. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  16. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  17. R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
    [CrossRef] [PubMed]
  18. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]
  19. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]
  20. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  21. Z. Zalevsky, R. G. Dorsch, “Gerchberg–Saxton algorithms applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
    [CrossRef] [PubMed]
  22. X. G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Proc. Lett. 3, 72–74 (1996).
    [CrossRef]
  23. R. Fletcher, Practical Methods of Optimization (Wiley, New York, 1980), pp. 33–62.

1996 (2)

1995 (2)

1994 (3)

1993 (4)

1991 (2)

F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 30, 529–536 (1991).
[CrossRef]

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

1989 (1)

1983 (1)

1982 (2)

1981 (1)

1980 (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

1974 (2)

O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
[CrossRef]

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–166 (1974).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Aleksoff, C. C.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Barshan, B.

Bitran, Y.

Bryngdahl, O.

O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
[CrossRef]

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–166 (1974).
[CrossRef]

Cederquist, J. N.

Chang, Y. H.

Cordingley, J.

Dong, B. Z.

Dorsch, R. G.

Eismann, M. T.

Ellis, K. K.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Fienup, J. R.

J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
[CrossRef] [PubMed]

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

Fletcher, R.

R. Fletcher, Practical Methods of Optimization (Wiley, New York, 1980), pp. 33–62.

Frutos, A. M.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Gu, B. Y.

Kastner, C. J.

Kazumi, M.

Lohmann, A. W.

Mendlovic, D.

Neagle, B. D.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Onural, L.

Ozaktas, H. M.

Quintanilla, M.

Roux, F. S.

F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 30, 529–536 (1991).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Tai, A. M.

Tan, X.

Veldkamp, W. B.

Xia, X. G.

X. G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Proc. Lett. 3, 72–74 (1996).
[CrossRef]

Yang, G. Z.

Yukihiro, I.

Zalevsky, Z.

Appl. Opt. (10)

M. T. Eismann, A. M. Tai, J. N. Cederquist, “Iterative design of a holographic beam former,” Appl. Opt. 28, 2641–2650 (1989).
[CrossRef] [PubMed]

M. Quintanilla, A. M. Frutos, “Holographic filter that transforms a Gaussian beam into a uniform beam,” Appl. Opt. 20, 879–880 (1981).
[CrossRef] [PubMed]

Y. H. Chang, I. Yukihiro, M. Kazumi, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
[CrossRef]

W. B. Veldkamp, “Laser beam profile shaping with interlaced binary diffractive gratings,” Appl. Opt. 21, 3209–3212 (1982).
[CrossRef] [PubMed]

W. B. Veldkamp, C. J. Kastner, “Beam profile shaping for laser radars that use detector arrays,” Appl. Opt. 21, 345–356 (1982).
[CrossRef] [PubMed]

J. Cordingley, “Application of a binary diffractive optic for beam shaping in semiconductor processing by lasers,” Appl. Opt. 32, 2538–2542 (1993).
[CrossRef] [PubMed]

J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
[CrossRef] [PubMed]

X. Tan, B. Y. Gu, G. Z. Yang, B. Z. Dong, “Diffractive phase elements for beam shaping: a new design method,” Appl. Opt. 34, 1314–1320 (1995).
[CrossRef] [PubMed]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[CrossRef] [PubMed]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
[CrossRef] [PubMed]

IEEE Signal Proc. Lett. (1)

X. G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Proc. Lett. 3, 72–74 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–166 (1974).
[CrossRef]

Opt. Eng. (3)

F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 30, 529–536 (1991).
[CrossRef]

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Other (1)

R. Fletcher, Practical Methods of Optimization (Wiley, New York, 1980), pp. 33–62.

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Figures (2)

Fig. 1
Fig. 1

Optical setup for the fractional Fourier transform.

Fig. 2
Fig. 2

Gaussian beam shaping with the optimization technique. The solid curve corresponds to the output amplitude with the optimization method. The dashed curve corresponds to the ideal amplitude of the uniform beam. The dotted curve corresponds to the Gaussian amplitude.

Equations (17)

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p g x = -   B p t ,   x g t d t ,
B p t ,   x = exp - i π / 4 + i α / 2 | sin   α | 1 / 2 exp i π x 2 cot   α - 2 xt   csc   α + t 2 cot   α .
-   B p t ,   x g in t d t = v ˜ x exp i ψ out x ,
- +   g * t - y g t + y exp i 4 π yt   cot   α d t = - +   | p g x | 2 exp i 4 π xy   csc   α d x ,
- +   g * t - y g t + y exp i 4 π yt   cot   α d t = - + g t - y exp i π t - y 2 cot   α * g t + y × exp i π t + y 2 cot   α d t = - +   g t - y exp i π t - y 2 cot   α * x × g t + y exp i π t + y 2 cot   α x d x = - +   g t exp i π t 2 cot   α * x × g t exp i π t 2 cot   α x exp i 4 π xy d x = - + p g * x p g x exp i 4 π xy   csc   α d x ,
- +   v ˜ 2 x exp i 4 π xy   csc   α d x - +   | g * t - y g t + y | d t .
V 2 sin 4 π ay   csc   α 2 π y   csc   α σ π exp - y 2 σ 2 .
1 4 n + 1 c   exp - s 4 n + 1 2 ,     n = 1 ,   2 ,   3 , ,
min - ( | p u in t exp i ϕ in t x | - v ˜ x ) 2 d x .
v n exp i ψ n = m = 1 M   G mn u m exp i ϕ m , n = 1 ,   2 , ,   N .
min   D Φ in = n = 1 N m = 1 M   G mn u m exp i ϕ m - v ˜ n 2 ,
D Φ in ϕ m = iu m exp - i ϕ m n = 1 N   G mn * v ˜ n - u n × exp i ψ n + cc , m = 1 ,   2 ,     ,   M ,
grad k = D Φ in k ϕ 1 k ,   D Φ in k ϕ 2 k , , D Φ in k ϕ M k .
D Φ in k + 1 = min   D Φ in k + λ P k .
H k + 1 = H k + 1 + W T · H k · W Δ q T · W Δ q · Δ q T Δ q T · W - Δ q · W T · H k Δ q T · W - H k · W · Δ q T Δ q T · W ,
W = w 1 ,   w 2 ,     ,   w M T , w m = D Φ in k + 1 ϕ m k + 1 - D Φ in k ϕ m k ,     m = 1 ,   2 , ,   M , Δ q = Φ in k + 1 - Φ in k .
MSE = D Φ in k + 1 m   u m 2 < ,

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