Abstract

An improved iterative algorithm for designing diffractive phase elements for laser beam shaping in free space is presented. The algorithm begins with the Gerchberg–Saxton approach to obtain a stable solution. This is followed by several new iterations, in which modified constraining functions are imposed in the Fourier domain while the phase distribution of each iteration remains unchanged. For super-Gaussian beam shaping suitable for inertial confinement fusion applications the mean-square errors of the amplitude and the intensity profile of the entire beam fitted to the corresponding parameters of the 12th-power super-Gaussian beam are approximately 0.035 and 9.75×10-3, respectively. Approximately 97.4% of the incident energy is converged into the desired region.

© 2002 Optical Society of America

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References

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  1. R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).
  2. J. R. Fienup, Appl. Opt. 21, 2758 (1982).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  6. J. D. Moody, B. J. MacGowan, J. E. Rothenberg, R. L. Berger, L. Divol, S. H. Glenzer, R. K. Kirkwood, E. A. Williams, and P. E. Young, Phys. Rev. Lett. 86, 2810 (2001).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [PubMed]
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    [CrossRef] [PubMed]
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2001 (2)

Q. F. Tan, Y. B. Yan, G. F. Jin, and M. X. Wu, Opt. Laser. Eng. 35, 165 (2001).
[CrossRef]

J. D. Moody, B. J. MacGowan, J. E. Rothenberg, R. L. Berger, L. Divol, S. H. Glenzer, R. K. Kirkwood, E. A. Williams, and P. E. Young, Phys. Rev. Lett. 86, 2810 (2001).
[CrossRef] [PubMed]

1996 (3)

1995 (1)

1994 (1)

1989 (1)

1983 (1)

S. Kirkpartrick, C. D. Gelatt, and M. P. Vecchi, Science 220, 671 (1983).
[CrossRef]

1982 (1)

1974 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Berger, R. L.

J. D. Moody, B. J. MacGowan, J. E. Rothenberg, R. L. Berger, L. Divol, S. H. Glenzer, R. K. Kirkwood, E. A. Williams, and P. E. Young, Phys. Rev. Lett. 86, 2810 (2001).
[CrossRef] [PubMed]

Bryndahl, O.

Deng, X.

Divol, L.

J. D. Moody, B. J. MacGowan, J. E. Rothenberg, R. L. Berger, L. Divol, S. H. Glenzer, R. K. Kirkwood, E. A. Williams, and P. E. Young, Phys. Rev. Lett. 86, 2810 (2001).
[CrossRef] [PubMed]

Dixit, S. N.

Fan, D.

Feit, M. D.

Fienup, J. R.

Gelatt, C. D.

S. Kirkpartrick, C. D. Gelatt, and M. P. Vecchi, Science 220, 671 (1983).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Glenzer, S. H.

J. D. Moody, B. J. MacGowan, J. E. Rothenberg, R. L. Berger, L. Divol, S. H. Glenzer, R. K. Kirkwood, E. A. Williams, and P. E. Young, Phys. Rev. Lett. 86, 2810 (2001).
[CrossRef] [PubMed]

Huntley, J. M.

Jin, G. F.

Q. F. Tan, Y. B. Yan, G. F. Jin, and M. X. Wu, Opt. Laser. Eng. 35, 165 (2001).
[CrossRef]

Kessler, T. J.

Kirkpartrick, S.

S. Kirkpartrick, C. D. Gelatt, and M. P. Vecchi, Science 220, 671 (1983).
[CrossRef]

Kirkwood, R. K.

J. D. Moody, B. J. MacGowan, J. E. Rothenberg, R. L. Berger, L. Divol, S. H. Glenzer, R. K. Kirkwood, E. A. Williams, and P. E. Young, Phys. Rev. Lett. 86, 2810 (2001).
[CrossRef] [PubMed]

Lawrence, G. N.

Lawson, J. K.

Li, Y.

Lin, Y.

MacGowan, B. J.

J. D. Moody, B. J. MacGowan, J. E. Rothenberg, R. L. Berger, L. Divol, S. H. Glenzer, R. K. Kirkwood, E. A. Williams, and P. E. Young, Phys. Rev. Lett. 86, 2810 (2001).
[CrossRef] [PubMed]

Manes, K. R.

Moody, J. D.

J. D. Moody, B. J. MacGowan, J. E. Rothenberg, R. L. Berger, L. Divol, S. H. Glenzer, R. K. Kirkwood, E. A. Williams, and P. E. Young, Phys. Rev. Lett. 86, 2810 (2001).
[CrossRef] [PubMed]

Nugent, K. A.

Perry, M. D.

Powell, H. T.

Qiu, Y.

Rothenberg, J. E.

J. D. Moody, B. J. MacGowan, J. E. Rothenberg, R. L. Berger, L. Divol, S. H. Glenzer, R. K. Kirkwood, E. A. Williams, and P. E. Young, Phys. Rev. Lett. 86, 2810 (2001).
[CrossRef] [PubMed]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Tan, Q. F.

Q. F. Tan, Y. B. Yan, G. F. Jin, and M. X. Wu, Opt. Laser. Eng. 35, 165 (2001).
[CrossRef]

Vecchi, M. P.

S. Kirkpartrick, C. D. Gelatt, and M. P. Vecchi, Science 220, 671 (1983).
[CrossRef]

Williams, E. A.

J. D. Moody, B. J. MacGowan, J. E. Rothenberg, R. L. Berger, L. Divol, S. H. Glenzer, R. K. Kirkwood, E. A. Williams, and P. E. Young, Phys. Rev. Lett. 86, 2810 (2001).
[CrossRef] [PubMed]

Wu, M. X.

Q. F. Tan, Y. B. Yan, G. F. Jin, and M. X. Wu, Opt. Laser. Eng. 35, 165 (2001).
[CrossRef]

Yan, Y. B.

Q. F. Tan, Y. B. Yan, G. F. Jin, and M. X. Wu, Opt. Laser. Eng. 35, 165 (2001).
[CrossRef]

Young, P. E.

J. D. Moody, B. J. MacGowan, J. E. Rothenberg, R. L. Berger, L. Divol, S. H. Glenzer, R. K. Kirkwood, E. A. Williams, and P. E. Young, Phys. Rev. Lett. 86, 2810 (2001).
[CrossRef] [PubMed]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Opt. Laser. Eng. (1)

Q. F. Tan, Y. B. Yan, G. F. Jin, and M. X. Wu, Opt. Laser. Eng. 35, 165 (2001).
[CrossRef]

Opt. Lett. (5)

Optik (1)

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Phys. Rev. Lett. (1)

J. D. Moody, B. J. MacGowan, J. E. Rothenberg, R. L. Berger, L. Divol, S. H. Glenzer, R. K. Kirkwood, E. A. Williams, and P. E. Young, Phys. Rev. Lett. 86, 2810 (2001).
[CrossRef] [PubMed]

Science (1)

S. Kirkpartrick, C. D. Gelatt, and M. P. Vecchi, Science 220, 671 (1983).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Far-field intensity distribution of the designed super-Gaussian beam after 50 GS iterations. The beam top is far from flat, and there is an intensity decrease in the beam center.

Fig. 2
Fig. 2

Modified Fourier-domain constraint function. Basically the intensity distribution of the modified Fourier-domain constraint function is symmetrical with that of Fig. 1 about the ideal super-Gaussian beam; i.e., the position where a high intensity exists in Fig. 1 corresponds to a lower intensity in the modified Fourier-domain constraint function.

Fig. 3
Fig. 3

Final designed far-field output-beam profile after use of four new iterations. The beam intensity is uniform across the top and decreases to zero abruptly at the edges of the target region.

Fig. 4
Fig. 4

Converged energy fraction and the MSE of the intensity contained within different dimensions. For the designed super-Gaussian beam, 97.4% of the energy was converged into the target region of 500 µm×500 µm, whereas the value is 1-5.52×10-5 for an ideal 12th-power super-Gaussian beam. The MSE of the intensity of the beam top fitted to the 12th-power super-Gaussian beam is of the order of 10-4, and the MSE of the entire beam is 9.75×10-3 because of the imperfectly steep beam edges.

Equations (14)

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Iinput=exp-x2+y2/r02,
Ioutput=exp-x2/rf2200+y2/rf2200,
Gku=Fgkx=FIinputx1/2×expiϕgk-1x,
Gku=FuexpiϕGku=Ioutputu1/2exp{iϕ[Gku},
gkx=F-1Gku,
gk+1x=fexpiϕgkx=Iinputx1/2expiϕgkx,
MSEk=Eku2ΣuFu2=N-2ΣuGku-Gku2ΣuFu2.
ϕK0=ϕgK0-1x.
GK0u=FIinputx1/2expiϕgK0-1x.
MSE=ΣuGK0_Nu-Fu2ΣuFu2,
MSE=ΣuGK0_Nu2-Ioutputu2ΣuIoutputu2,
FK0+kmodifiedu=FuGK0+k_Nuc1if uγ-δFuif uγ-δ,
FK0+kmodifiedu=c2Fu-GK0+k_Nuif uγ-δFuif uγ-δ,
GK0+ku=FK0+kmodifieduexpiϕGK0u=IK0+kmodified_output1/2expiϕGK0u.

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