Abstract

A design method based on the Yang–Gu algorithm [Appl. Opt. 33, 209 (1994)] is proposed for computing the phase distributions of an optical system composed of diffractive phase elements that achieve beam shaping with a high transfer efficiency in energy. Simulation computations are detailed for rotationally symmetric beam shaping in which a laser beam with a radially symmetric Gaussian intensity distribution is converted into a uniform beam with a circular region of support. To present a comparison of the efficiency and the performance of the designed diffractive phase elements by use of the geometrical transformation technique, the Gerchberg–Saxton algorithm and the Yang–Gu algorithm for beam shaping, we carry out in detail simulation calculations for a specific one-dimensional beam-shaping example.

© 1995 Optical Society of America

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    [CrossRef]
  10. O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164 (1974).
    [CrossRef]
  11. R. W. Gerchburg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
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    [CrossRef] [PubMed]
  14. J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
  15. G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude–phase retrieval in any linear transform system and its applications,” Intl. J. Mod. Phys. B 7, 3152–3224 (1993).
  16. 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]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968) Chap. 4, pp. 57–61.
  18. W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transformation with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
    [CrossRef]

1994 (1)

1993 (3)

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)

1987 (1)

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transformation with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[CrossRef]

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).

1974 (2)

1972 (1)

R. W. Gerchburg, W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik 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]

Bryngdahl, O.

Cederquist, J. N.

Chang, Y. H.

Cordingley, J.

Dahdouh, A.

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transformation with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[CrossRef]

Darling, A. M.

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transformation with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[CrossRef]

de Frutos, A. M.

Dong, B. Z.

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]

G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude–phase retrieval in any linear transform system and its applications,” Intl. J. Mod. Phys. B 7, 3152–3224 (1993).

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]

Ersoy, O. K.

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).

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).

Gerchburg, R. W.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968) Chap. 4, pp. 57–61.

Gu, B. Y.

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]

G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude–phase retrieval in any linear transform system and its applications,” Intl. J. Mod. Phys. B 7, 3152–3224 (1993).

Hossack, W. J.

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transformation with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[CrossRef]

Kastner, C. J.

Kazumi, M.

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]

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

Tai, A. M.

Veldkamp, W. B.

Yang, G. Z.

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]

G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude–phase retrieval in any linear transform system and its applications,” Intl. J. Mod. Phys. B 7, 3152–3224 (1993).

Yukihiro, I.

Zhuang, J. Y.

Appl. Opt. (8)

Intl. J. Mod. Phys. B (1)

G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude–phase retrieval in any linear transform system and its applications,” Intl. J. Mod. Phys. B 7, 3152–3224 (1993).

J. Mod. Opt. (1)

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transformation with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164 (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).

Optik (1)

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

Other (2)

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968) Chap. 4, pp. 57–61.

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Figures (8)

Fig. 1
Fig. 1

Configuration of a beam-shaping system composed of diffractive phase elements.

Fig. 2
Fig. 2

Results of 2-D rotationally symmetric beam shaping with the YG algorithm with a constant initial phase estimate: (a) phase profile of DPE#1; (b) relative amplitude distributions of the beams, where curve i is the Gaussian profile of the incident laser beam in reference to abscissa ρ1, curve ii is the relative amplitude distribution of the wave front generated by the designed DPE, and curve iii is the ideal uniform beam with a steplike function provided as a reference. Both curves ii and iii are in reference to abscissas ρ2. We repeat this notation for Figs. 3 5.

Fig. 3
Fig. 3

Results of 2-D rotationally symmetric beam shaping with the YG algorithm with an initial phase distribution derived from geometrical transformation: (a) phase profile of DPE #1, (b) relative amplitude distributions. All symbols have the same meaning as in Fig. 2.

Fig. 4
Fig. 4

Results of 2-D rotationally symmetric beam shaping with the geometrical transformation technique: (a) phase profile, (b) relative amplitude distributions.

Fig. 5
Fig. 5

Beam-shaping results after an eight-level phase quantization of the phase given in Fig. 3(a): (a) profile of the quantized phase of order 2 k = 8, (b) corresponding relative amplitude distributions. For clarity the inset of (b) shows large fluctuations of the amplitude around the small central region on the output plane.

Fig. 6
Fig. 6

One-dimensional Gaussian beam-shaping with the geometrical transformation technique: (a) phase of DPE#1, (b) relative amplitude distributions. Curve i is the 1-D Gaussian profile of the incident beam in reference to the abscissa x 1, curve ii is the relative amplitude of the wave front generated by the designed DPE, and curve iii is the ideal uniform beam with a top-hat shape provided as a reference. Both curves ii and iii are in reference to the abscissa x 2. We retain this notation for Figs. 7 and 8.

Fig. 7
Fig. 7

One-dimensional Gaussian beam shaping with the GS algorithm with an initial phase distribution derived from the geometrical transformation of Fig. 6: (a) phase of DPE#1, (b) relative amplitude distributions.

Fig. 8
Fig. 8

One-dimensional Gaussian beam shaping with the YG algorithm with an initial phase distribution derived from the GS algorithm of Fig. 7: (a) phase of DPE#1, (b) relative amplitude distributions.

Equations (17)

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U 1 ( X 1 ) = u 1 ( X 1 ) exp [ i ϕ 1 ( X 1 ) ] ,
U 2 ( X 2 ) = u 2 ( X 2 ) exp [ i ϕ 2 ( X 2 ) ] .
U 2 ( X 2 ) = G ( X 2 , X 1 ) U 1 ( X 1 ) d X 1 ,
U 2 ( X 2 ) = Ĝ U 1 ( X 1 ) ,
Ĝ + Ĝ = Â Î ,
U 1 n = u 1 n exp ( i ϕ 1 n ) for n = 1 , 2 , 3 , , N 1 ,
U 2 m = u 2 m exp ( i ϕ 2 m ) for m = 1 , 2 , 3 , , N 2 ,
U 2 m = n = 1 N 1 G m n U 1 n .
D 2 ( u 1 , ϕ 1 ; u 2 , ϕ 2 ) = U 2 Ĝ U 1 = 1 N 2 m = 1 N 2 | U 2 m ( Ĝ U 1 ) m | 2 .
ϕ 1 k = 1 A k k arg [ G j k * u 2 j exp ( i ϕ 2 j ) j k A k j u 1 j exp ( i ϕ 1 j ) ] ,
ϕ 2 k = arg [ j G k j u 1 j exp ( i ϕ 1 j ) ] ,
U 2 ( ρ 2 ) = G ( ρ 2 , ρ 1 ) U 1 ( ρ 1 ) d ρ 1 , G ( ρ 2 , ρ 1 ) = 2 π i λ l exp ( i 2 π l / λ ) exp [ i π ( ρ 1 2 + ρ 2 2 ) λ l ] × J 0 ( 2 π ρ 2 ρ 1 λ l ) ρ 1 ,
SSE = k = 1 N 2 | u 2 k exp ( i ϕ 2 k ) j G k j u 1 j exp ( i ϕ 1 j ) | 2 k u 2 k 2 .
ϕ 1 ( ρ 1 ) = 2 π λ l 0 ρ 1 [ μ ( ξ ) ξ ] d ξ , μ ( ξ ) = w 2 2 [ 1 exp ( 8 ξ 2 / w 1 2 ) 1 exp ( 2 x 1max 2 / w 1 2 ) ] 1 / 2 .
G ( x 2 , x 1 ) = ( 1 / i λ l ) 1 / 2 exp ( i π l / λ ) exp [ i π ( x 2 x 1 ) 2 / λ l ] ,
U 2 ( x 2 ) = G ( x 2 , x 1 ) U 1 ( x 1 ) d x 1 .
ϕ 1 ( x 1 ) = ( 2 π λ l ) 0.5 x 1 max x 1 [ μ ( ξ ) ξ ] d ξ , μ ( ξ ) = w 2 [ S ( ξ ) / S 0 0.5 ] , S ( ξ ) = 0.5 x 1 max ξ exp ( 8 ξ 2 / w 1 2 ) d ξ , S 0 ( ξ ) = 0.5 x 1 max 0.5 x 1 max exp ( 8 ξ 2 / w 1 2 ) d ξ .

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