Abstract

A new design approach for the diffractive phase elements (DPE’s) that implement beam shaping in the fractional Fourier transform (FRFT) domain is presented. The new algorithm can successfully achieve the design of DPE’s for beam shaping in both unitary and nonunitary transform systems. The unitarity transform condition of the FRFT domain is discussed. Modeling designs of the DPE’s are carried out for several fractional orders and different parameters of the beam for converting a Gaussian profile into a uniform beam. Our approach can realize beam shaping well for a nonunitary transform system in the FRFT domain.

© 1998 Optical Society of America

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  1. M. Quintanilla, A. M. de Frutos, “Holographic filter that transforms a Gaussian into a uniform beam,” Appl. Opt. 20, 879–880 (1981).
    [CrossRef] [PubMed]
  2. Y. H. Chang, I. Yukihire, M. Kazumi, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
    [CrossRef]
  3. M. T. Eismann, A. M. Tai, J. N. Cederquist, “Iterative design of a holographic beam former,” Appl. Opt. 28, 2641–2650 (1989).
    [CrossRef] [PubMed]
  4. J. Cordingley, “Application of a binary diffractive optic for beam shaping in semiconductor processing by lasers,” Appl. Opt. 32, 2538–2549 (1993).
    [CrossRef] [PubMed]
  5. F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. (Bellingham) 30, 529–536 (1991).
    [CrossRef]
  6. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their implementations: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  7. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their implementations: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  8. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  9. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  10. 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]
  11. S. Liu, J. Xu, Y. Zhang, L. Chen, C. Li, “General optical implementations of fractional Fourier transforms,” Opt. Lett. 20, 1053–1055 (1995).
    [CrossRef] [PubMed]
  12. B. Dong, Y. Zhang, B. Gu, G. Yang, “Numerical investigation of phase retrieval in a fractional Fourier transform,” J. Opt. Soc. Am. A 14, 2709–2714 (1997).
    [CrossRef]
  13. R. W. Gerchberg, W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik (Stuttgart) 34, 275–284 (1971).
  14. 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).
  15. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
    [CrossRef] [PubMed]
  16. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. (Bellingham) 19, 297–305 (1980).
    [CrossRef]
  17. Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  20. G. Yang, B. Dong, B. Gu, J. 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]
  21. X. Tan, B. Gu, G. Yang, B. Dong, “Diffractive phase elements for beam shaping: a new design method,” Appl. Opt. 34, 1314–1320 (1995).
    [CrossRef] [PubMed]
  22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 2, p. 21–25.
  23. E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1986).
  24. Xiang-Gen Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Process. Lett. 3, 72–74 (1996).
    [CrossRef]
  25. R. Rollestion, N. George, “Image reconstruction from partial Fresnel zone information,” Appl. Opt. 25, 178–183 (1986).
    [CrossRef]
  26. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  27. D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
    [CrossRef]
  28. D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D 6, 2200–2225 (1973).
    [CrossRef]
  29. R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” in International Optical Computing Conference, W. T. Rhodes, ed., Proc. SPIE231, 130–141 (1980).
    [CrossRef]

1997

1996

1995

1994

1993

1991

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

1989

1986

1983

1982

1981

1980

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

1973

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D 6, 2200–2225 (1973).
[CrossRef]

1972

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).

1971

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

Barshan, B.

Boucher, R. H.

R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” in International Optical Computing Conference, W. T. Rhodes, ed., Proc. SPIE231, 130–141 (1980).
[CrossRef]

Cederquist, J. N.

Chang, Y. H.

Chen, L.

Cordingley, J.

de Frutos, A. M.

Dong, B.

Dorsch, R. G.

Eismann, M. T.

Ersoy, O. K.

Fienup, J. R.

George, N.

Gerchberg, R. W.

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).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 2, p. 21–25.

Gu, B.

Kazumi, M.

Kreyszig, E.

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1986).

Li, C.

Liu, S.

Lohmann, A. W.

Mendlovic, D.

Misell, D. L.

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D 6, 2200–2225 (1973).
[CrossRef]

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

Onural, L.

Ozaktas, H. M.

Quintanilla, M.

Rollestion, R.

Roux, F. S.

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

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).

Tai, A. M.

Tan, X.

Xia, Xiang-Gen

Xiang-Gen Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Process. Lett. 3, 72–74 (1996).
[CrossRef]

Xu, J.

Yang, G.

Yukihire, I.

Zalevsky, Z.

Zhang, Y.

Zhuang, J.

Appl. Opt.

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

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

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

R. Rollestion, N. George, “Image reconstruction from partial Fresnel zone information,” Appl. Opt. 25, 178–183 (1986).
[CrossRef]

B. Gu, G. Yang, B. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206 (1986).
[CrossRef] [PubMed]

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

J. Cordingley, “Application of a binary diffractive optic for beam shaping in semiconductor processing by lasers,” Appl. Opt. 32, 2538–2549 (1993).
[CrossRef] [PubMed]

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

G. Yang, B. Dong, B. Gu, J. 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]

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

IEEE Signal Process. Lett.

Xiang-Gen Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Process. Lett. 3, 72–74 (1996).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. D

D. L. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D 6, 2200–2225 (1973).
[CrossRef]

Opt. Commun.

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

Opt. Eng. (Bellingham)

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

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

Opt. Lett.

Optik (Stuttgart)

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

G. Yang, B. Gu, B. Dong, “Theory of the amplitude-phase retrieval in an any linear transform system and its applications,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 457–479 (1992).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 2, p. 21–25.

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1986).

R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” in International Optical Computing Conference, W. T. Rhodes, ed., Proc. SPIE231, 130–141 (1980).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Variation of the ratio B/C with fractional order P of the FRFT.

Fig. 2
Fig. 2

Relative amplitude distributions for Gaussian-profile beam shaping in the FRFT domain with order P=0.14: (a) with the GS algorithm, (b) with our algorithm. Curves i, amplitude distributions of the input Gaussian-profile beam; curves ii, amplitude distributions of the ideal uniform beam; curves iii, amplitude distributions of the output beam generated by the designed DPE’s.

Fig. 3
Fig. 3

Same as Fig. 2, except for P=0.6.

Fig. 4
Fig. 4

Same as Fig. 2, except for P=0.8.

Fig. 5
Fig. 5

Relative amplitude distributions for beam shaping in the FRFT domain with order P=0.8: (a) with the GS algorithm, (b) with our algorithm. The waist size of the Gaussian beam is ω1=0.5a1, and the width of the desired uniform beam is ω2=0.4a2.

Fig. 6
Fig. 6

Same as Fig. 5, except for ω1=0.875a1 and ω2=0.7a2.

Fig. 7
Fig. 7

Relative amplitude distributions for beam shaping in the FRFT domain: (a) with the GS algorithm, (b) with the input–output algorithm, (c) with the steepest-decent method, (d) with the conjugate-gradient method, and (e) with our algorithm. The order of the FRFT is P=0.8, the waist size of the Gaussian beam is ω1=0.875a1, and the width of the desired uniform beam is ω2=0.7a2.

Tables (1)

Tables Icon

Table 1 Comparison of Values of the SSE for Several Algorithms in Beam Shaping

Equations (21)

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U1l=ρ1l exp(iϕ1l),l=1, 2, 3, , N1,
U2m=ρ2m exp(iϕ2m),
U2m=l=1N1GmlU1l,m=1, 2,3, , N2.
D(ρ1, ϕ1; ρ2, ϕ2)=U2-GˆU1=dx2U2(x2)-GˆU121/2=m=1N2U2m-(GˆU1)m21/2.
δϕ1D2=0,δϕ2D2=0,
ϕ1k=argj=1N2Gjk*ρ2j exp(iϕ2j)-jkAkjρ1j exp(iϕ1j),
ϕ2k=argj=1N1Gkjρ1j exp(iϕ1j),
ϕ2=arg(GˆU1),
ϕ1=arg(Gˆ+U2),
SSE=[ρ2(x2)-|Gˆρ1 exp(iϕ˜1(n))|]2dx2/ρ22(x2)dx2,
U2(x2)=Fα{U1(x1)}=B(x2, x1, α)U1(x1)dx1,
B(x2, x1, α)=1-i cot αλFI  ×expiπλFI[(x12+x22) ×cot α-2x1x2 csc α],
Δ1=a1N1λFI sin αa2.
Akl=s=1N2Gsk*Gsl=a1a2|cα|2N1N2 ×exp-iπa12λFIN12(k2-l2)cot α ×expiπN2T(N2+1)(k-l) ×sin[(π/T)(k-l)]sinπN2T(k-lü),
cα=1-i cot αλFI,
l=1, 2, 3, .N1,k=1, 2, 3, ..N2,
T=N1λFI sin αa1a2.
T=N1λFI sin αa1a2=1
Akl=δkl,
C(α)=1N1i=1N1|Aii|,
B(α)=1N1(N1-1)i=1N1jiN1|Aij|.

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