Abstract

To design a fully continuous wave-front distribution suitable for focused beam shaping by a deformable mirror, we modify the phase-retrieval algorithm by employing a uniformly distributed phase as a starting phase screen and spatial filtering for the near-field phase retrieved during the iteration process. A special phase unwrapping algorithm is not required to obtain a continuous phase distribution from the retrieved phase since the boundary of the 2π-phase-jumped region in the designed phase distribution is perfectly closed. From the computational result producing a uniform square beam transformation from a circular defocused beam, this algorithm has provided a fully continuous wave-front distribution with a lower spatial frequency for a deformable mirror. The transformed square beam has a normalized intensity nonuniformity of σrms = 0.14 with respect to a desired flat-topped square beam pattern. This beam-shaping method also provides a high energy-concentration rate of more than 98%.

© 1998 Optical Society of America

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  1. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
    [CrossRef]
  2. S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
    [CrossRef]
  3. R. H. Lehmberg, A. J. Schmitt, S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680–2701 (1987).
    [CrossRef]
  4. S. N. Dixit, J. K. Lawson, K. R. Manes, H. T. Powell, K. A. Nugent, “Kinoform phase plates for focal plane irradiance profile control,” Opt. Lett. 19, 417–419 (1994).
    [PubMed]
  5. G. Y. Yoon, T. Jitsuno, M. Nakatsuka, Y. Kato, S. Nakai, “Arbitrary profile control of laser beam spot by deformable mirror from fusion technology,” in Conference on Lasers and Electro-Optics, Vol. 15 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), Postdeadline paper PD1.12.
  6. G. Y. Yoon, T. Jitsuno, Y. Kato, M. Nakatsuka, “High-aspect-ratio line focus for an x-ray laser by a deformable mirror,” Appl. Opt. 36, 847–852 (1997).
    [CrossRef] [PubMed]
  7. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 237–246 (1972).
  8. N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [CrossRef] [PubMed]
  9. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef]
  10. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  11. M. T. Eismann, A. M. Tai, J. N. Cederquist, “Iterative design of a holographic beam former,” Appl. Opt. 28, 2641–2650 (1989).
    [CrossRef] [PubMed]
  12. C. Yamanaka, Y. Kato, Y. Izawa, K. Yoshida, T. Yamanaka, T. Sasaki, M. Nakatsuka, T. Mochizuki, J. Kuroda, S. Nakai, “Nd-doped phosphate glass laser systems for laser-fusion research,” IEEE J. Quantum Electron. QE-17, 1639–1649 (1981).
    [CrossRef]

1997 (1)

1994 (1)

1989 (2)

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

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

1987 (1)

R. H. Lehmberg, A. J. Schmitt, S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680–2701 (1987).
[CrossRef]

1984 (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

1982 (1)

1981 (1)

C. Yamanaka, Y. Kato, Y. Izawa, K. Yoshida, T. Yamanaka, T. Sasaki, M. Nakatsuka, T. Mochizuki, J. Kuroda, S. Nakai, “Nd-doped phosphate glass laser systems for laser-fusion research,” IEEE J. Quantum Electron. QE-17, 1639–1649 (1981).
[CrossRef]

1980 (1)

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

1973 (1)

1972 (1)

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

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Bodner, S. E.

R. H. Lehmberg, A. J. Schmitt, S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680–2701 (1987).
[CrossRef]

Cederquist, J. N.

Craxton, R. S.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

Dixit, S. N.

Eismann, M. T.

Fienup, J. R.

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

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

Gallagher, N. C.

Gerchberg, R. W.

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

Izawa, Y.

C. Yamanaka, Y. Kato, Y. Izawa, K. Yoshida, T. Yamanaka, T. Sasaki, M. Nakatsuka, T. Mochizuki, J. Kuroda, S. Nakai, “Nd-doped phosphate glass laser systems for laser-fusion research,” IEEE J. Quantum Electron. QE-17, 1639–1649 (1981).
[CrossRef]

Jitsuno, T.

G. Y. Yoon, T. Jitsuno, Y. Kato, M. Nakatsuka, “High-aspect-ratio line focus for an x-ray laser by a deformable mirror,” Appl. Opt. 36, 847–852 (1997).
[CrossRef] [PubMed]

G. Y. Yoon, T. Jitsuno, M. Nakatsuka, Y. Kato, S. Nakai, “Arbitrary profile control of laser beam spot by deformable mirror from fusion technology,” in Conference on Lasers and Electro-Optics, Vol. 15 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), Postdeadline paper PD1.12.

Kato, Y.

G. Y. Yoon, T. Jitsuno, Y. Kato, M. Nakatsuka, “High-aspect-ratio line focus for an x-ray laser by a deformable mirror,” Appl. Opt. 36, 847–852 (1997).
[CrossRef] [PubMed]

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

C. Yamanaka, Y. Kato, Y. Izawa, K. Yoshida, T. Yamanaka, T. Sasaki, M. Nakatsuka, T. Mochizuki, J. Kuroda, S. Nakai, “Nd-doped phosphate glass laser systems for laser-fusion research,” IEEE J. Quantum Electron. QE-17, 1639–1649 (1981).
[CrossRef]

G. Y. Yoon, T. Jitsuno, M. Nakatsuka, Y. Kato, S. Nakai, “Arbitrary profile control of laser beam spot by deformable mirror from fusion technology,” in Conference on Lasers and Electro-Optics, Vol. 15 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), Postdeadline paper PD1.12.

Kessler, T.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

Kitagawa, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Kuroda, J.

C. Yamanaka, Y. Kato, Y. Izawa, K. Yoshida, T. Yamanaka, T. Sasaki, M. Nakatsuka, T. Mochizuki, J. Kuroda, S. Nakai, “Nd-doped phosphate glass laser systems for laser-fusion research,” IEEE J. Quantum Electron. QE-17, 1639–1649 (1981).
[CrossRef]

Lawson, J. K.

Lehmberg, R. H.

R. H. Lehmberg, A. J. Schmitt, S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680–2701 (1987).
[CrossRef]

Letzring, S.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

Liu, B.

Manes, K. R.

Mima, K.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Miyanaga, N.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Mochizuki, T.

C. Yamanaka, Y. Kato, Y. Izawa, K. Yoshida, T. Yamanaka, T. Sasaki, M. Nakatsuka, T. Mochizuki, J. Kuroda, S. Nakai, “Nd-doped phosphate glass laser systems for laser-fusion research,” IEEE J. Quantum Electron. QE-17, 1639–1649 (1981).
[CrossRef]

Nakai, S.

C. Yamanaka, Y. Kato, Y. Izawa, K. Yoshida, T. Yamanaka, T. Sasaki, M. Nakatsuka, T. Mochizuki, J. Kuroda, S. Nakai, “Nd-doped phosphate glass laser systems for laser-fusion research,” IEEE J. Quantum Electron. QE-17, 1639–1649 (1981).
[CrossRef]

G. Y. Yoon, T. Jitsuno, M. Nakatsuka, Y. Kato, S. Nakai, “Arbitrary profile control of laser beam spot by deformable mirror from fusion technology,” in Conference on Lasers and Electro-Optics, Vol. 15 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), Postdeadline paper PD1.12.

Nakatsuka, M.

G. Y. Yoon, T. Jitsuno, Y. Kato, M. Nakatsuka, “High-aspect-ratio line focus for an x-ray laser by a deformable mirror,” Appl. Opt. 36, 847–852 (1997).
[CrossRef] [PubMed]

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

C. Yamanaka, Y. Kato, Y. Izawa, K. Yoshida, T. Yamanaka, T. Sasaki, M. Nakatsuka, T. Mochizuki, J. Kuroda, S. Nakai, “Nd-doped phosphate glass laser systems for laser-fusion research,” IEEE J. Quantum Electron. QE-17, 1639–1649 (1981).
[CrossRef]

G. Y. Yoon, T. Jitsuno, M. Nakatsuka, Y. Kato, S. Nakai, “Arbitrary profile control of laser beam spot by deformable mirror from fusion technology,” in Conference on Lasers and Electro-Optics, Vol. 15 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), Postdeadline paper PD1.12.

Nugent, K. A.

Powell, H. T.

Sasaki, T.

C. Yamanaka, Y. Kato, Y. Izawa, K. Yoshida, T. Yamanaka, T. Sasaki, M. Nakatsuka, T. Mochizuki, J. Kuroda, S. Nakai, “Nd-doped phosphate glass laser systems for laser-fusion research,” IEEE J. Quantum Electron. QE-17, 1639–1649 (1981).
[CrossRef]

Saxton, W. O.

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

Schmitt, A. J.

R. H. Lehmberg, A. J. Schmitt, S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680–2701 (1987).
[CrossRef]

Short, R. W.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

Skupsky, S.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

Soures, J. M.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

Tai, A. M.

Yamanaka, C.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

C. Yamanaka, Y. Kato, Y. Izawa, K. Yoshida, T. Yamanaka, T. Sasaki, M. Nakatsuka, T. Mochizuki, J. Kuroda, S. Nakai, “Nd-doped phosphate glass laser systems for laser-fusion research,” IEEE J. Quantum Electron. QE-17, 1639–1649 (1981).
[CrossRef]

Yamanaka, T.

C. Yamanaka, Y. Kato, Y. Izawa, K. Yoshida, T. Yamanaka, T. Sasaki, M. Nakatsuka, T. Mochizuki, J. Kuroda, S. Nakai, “Nd-doped phosphate glass laser systems for laser-fusion research,” IEEE J. Quantum Electron. QE-17, 1639–1649 (1981).
[CrossRef]

Yoon, G. Y.

G. Y. Yoon, T. Jitsuno, Y. Kato, M. Nakatsuka, “High-aspect-ratio line focus for an x-ray laser by a deformable mirror,” Appl. Opt. 36, 847–852 (1997).
[CrossRef] [PubMed]

G. Y. Yoon, T. Jitsuno, M. Nakatsuka, Y. Kato, S. Nakai, “Arbitrary profile control of laser beam spot by deformable mirror from fusion technology,” in Conference on Lasers and Electro-Optics, Vol. 15 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), Postdeadline paper PD1.12.

Yoshida, K.

C. Yamanaka, Y. Kato, Y. Izawa, K. Yoshida, T. Yamanaka, T. Sasaki, M. Nakatsuka, T. Mochizuki, J. Kuroda, S. Nakai, “Nd-doped phosphate glass laser systems for laser-fusion research,” IEEE J. Quantum Electron. QE-17, 1639–1649 (1981).
[CrossRef]

Appl. Opt. (4)

IEEE J. Quantum Electron. (1)

C. Yamanaka, Y. Kato, Y. Izawa, K. Yoshida, T. Yamanaka, T. Sasaki, M. Nakatsuka, T. Mochizuki, J. Kuroda, S. Nakai, “Nd-doped phosphate glass laser systems for laser-fusion research,” IEEE J. Quantum Electron. QE-17, 1639–1649 (1981).
[CrossRef]

J. Appl. Phys. (2)

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

R. H. Lehmberg, A. J. Schmitt, S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680–2701 (1987).
[CrossRef]

Opt. Eng. (1)

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 (1)

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

Phys. Rev. Lett. (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Other (1)

G. Y. Yoon, T. Jitsuno, M. Nakatsuka, Y. Kato, S. Nakai, “Arbitrary profile control of laser beam spot by deformable mirror from fusion technology,” in Conference on Lasers and Electro-Optics, Vol. 15 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), Postdeadline paper PD1.12.

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Figures (8)

Fig. 1
Fig. 1

Retrieved phase distribution by use of the G–S algorithm started with a randomly distributed phase. The phase difference between two points P and Q changes with respect to path A with a 2π-phase jump and with respect to path B without a 2π-phase jump because a 2π-phase jump occurs.

Fig. 2
Fig. 2

Schematic diagram of the algorithm for designing a fully continuous wave-front distribution limited to a lower spatial frequency for beam shaping: fft, fast Fourier transformation; ifft, inverse fast Fourier transformation; A 0(x, y), input near-field amplitude; A(x, y), desired far-field amplitude; ϕlens(x, y), lens phase to defocus; ϕ0(x, y), initial uniform phase distribution; ϕ F (ξ, η), phase distribution in the Fourier domain; ϕ r (x, y), retrieved phase distribution; ϕsf(x, y), spatially filtered phase distribution.

Fig. 3
Fig. 3

Schematic of numerical defocused beam shaping.

Fig. 4
Fig. 4

Retrieved phase distribution for rectangular beam shaping after 429 iterations and a phase unwrapping process: (a) Retrieved phase distribution with a closed 2π phase jump. (b) One-dimensional phase unwrapping in the horizontal direction from (a). (c) One-dimensional phase unwrapping in the vertical direction from (b).

Fig. 5
Fig. 5

(a) Defocused circular beam pattern with a diameter of 166 μm before wave-front control. (b) Rectangular beam pattern of 166 μm × 166 μm shaped by the continuous wave-front distribution shown in Fig. 4(c).

Fig. 6
Fig. 6

Characteristics of (a) a convergent speed and (b) an energy-concentration rate of our algorithm in comparison with the G–S algorithm.

Fig. 7
Fig. 7

Variation of the intensity uniformity indicated as the rms error with respect to the defocused distance (the optimized defocused position is at 500 μm).

Fig. 8
Fig. 8

Rectangular spot pattern when the intensity modulation included in the input beam was almost the same as that of the spot pattern [see Fig. 5(b)] without input-beam modulation.

Equations (6)

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ϕ lens x ,   y = - k   x 2 + y 2 2 f a ,
f a = - f m D spot   D beam ,
D = f m 2 f a - f m + d ,
J i = - 2 π + 2 π 0 P i - P i - 1 + 2 π P i - P i - 1 - 2 π otherwise ,
C i = J i - 1 - 2 π J i - 1 + 2 π J i - 1 J i = - 2 π J i = + 2 π J i = 0 .
U i = J i + C i .

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