Abstract

We discuss the design and fabrication of 80-cm-diameter random phase plates for target-plane beam smoothing on the Nova laser. Random phase plates have been used in a variety of inertial confinement fusion target experiments, such as studying direct-drive hydrodynamic stability and producing spatially smooth x-ray backlighting sources. These phase plates were produced by using a novel sol-gel dip-coating technique developed by us. The sol-gel phase plates have a high optical damage threshold at the second- and third-harmonic wavelengths of the Nd:glass laser and have excellent optical performance.

© 1993 Optical Society of America

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  1. J. Nuckolls, L. Wood, A. Thiessen, G. Zimmerman, “Laser compression of matter to super-high densities: thermonuclear (CTR) applications,” Nature (London) 239, 139–142 (1972).
    [CrossRef]
  2. W. W. Simmons, R. O. Goodwin, “Nova laser fusion facility: design, engineering, and assembly overview,” J. Nucl. Technol. Fusion 4, 8–24 (1983).
  3. 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]
  4. X. Deng, X. Liang, Z. Chen, W. Yu, R. Ma, “Uniform illumination of large targets using a lens array,” Appl. Opt. 25, 377–381 (1986).
    [CrossRef] [PubMed]
  5. R. H. Lehmberg, S. P. Obenschain, “Use of induced spatial incoherence for uniform illumination of laser fusion targets,” Opt. Commun. 46, 27–31 (1983).
    [CrossRef]
  6. 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]
  7. D. Veron, H. Ayral, C. Geouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaong, C. Sauteret, “Optical spatial smoothing of Nd:glass laser beam,” Opt. Commun. 65, 42–46 (1988).
    [CrossRef]
  8. H. Nakano, K. Tsubakimoto, K. Yagi, T. Jitsuno, M. Nakatuska, S. Nakai, T. Kanabe, C. Yamanaka, “Irradiation uniformity improvement by incoherence of amplified spontaneous emission in glass laser fusion systems,” in Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 280–282.
  9. C. N. Danson, R. Bann, D. Pepler, N. Rizvi, I. Ross, T. Afsharrad, S. Coe, M. Dusselberger, O. Willi, “Focal spot smoothing on the vulcan glass laser system,” 1989 OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) p. 86.
  10. H. T. Powell, S. N. Dixit, M. A. Henesian, “Beam smoothing capability on the Nova laser,” in Lawrence Liver-more National Laboratory ICF Quarterly Report, Rep. UCRL-LR-105821-91-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1990), Vol. 1, pp. 28–38.
  11. B. W. Woods, I. M. Thomas, M. A. Henesian, S. N. Dixit, H. T. Powell, “Large aperture (80-cm diameter) phase plates for beam smoothing on Nova,” in Solid-State Lasers II, G. Duber, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1410, 47–54 (1991).
  12. Laboratory for Laser Energetics, “OMEGA phase conversion with distributed phase plates,” in Laboratory for Laser Energetics, Annual Report, Vol. 33, Q. Rep. DOE/DP40200-65 (University of Rochester, Rochester, N.Y., 1987), pp. 1–10.
  13. C. N. Danson, R. Bann, D. Pepler, I. Ross, J. Exley, D. Hardie, S. Sails, “Development of random phase plate smoothing technology,” in Rutherford Appleton Laboratory Annual Report RAL-91-025 (Rutherford Appleton Laboratory, Abingdon, UK, 1991), pp. 62–65; M. Desselberger, L. Gizzi, V. Barrow, J. Edwards, F. Khattak, S. Viana, O. Willi, C. N. Danson, “High-aspect-ratio line focus and plasma production using a random phase plate,” Appl. Opt. 31, 3759–3766 (1992).
    [CrossRef] [PubMed]
  14. See the papers in Diffractive Optics: Design, Fabrication, and Applications,” in Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992).
  15. B. W. Woods, I. M. Thomas, S. N. Dixit, H. T. Powell, M. A. Henesian, “Large aperture sol-gel phase plates for beam smoothing on Nova,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 456.
  16. I. M. Thomas, “High laser damage threshold porous silica antireflective coating,” Appl. Opt. 25, 1481–1483 (1986).
    [CrossRef] [PubMed]
  17. J.-L. Nogues, “Pure silica diffraction grating and microlens arrays made by the sol-gel process,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 15–17.
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, pp. 85–90.
  19. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8, pp. 393–401.
  20. R. E. English, “Diffraction theory for polygonal apertures,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1988).
  21. J. W. Goodman, “Statistical properties of speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer, New York, 1984), Chap. 2, pp. 9–75.
  22. C. J. Brinker, C. W. Scherer, Sol-Gel Science (Academic, New York, 1990), Chap. 13.
  23. F. Abeles, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1963), Vol. 2, Chap. 6, pp. 251–288.
    [CrossRef]
  24. R. Epstein, S. Skupsky, “Anticipated improvement in laser beam uniformity using distributed phase plates with quasi-random patterns,” J. Appl. Phys. 68, 924–931 (1990).
    [CrossRef]
  25. P. J. Wegner, M. A. Henesian, “Precision high-power solid-state laser diagnostics for target-irradiation studies and target plane irradiation modeling,” in Laser Beam Diagnostics,” R. N. Hindy, Y. Kohanzadeh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1414, 162–174 (1991).
  26. L. G. Seppala, R. B. Ehrlich, S. N. Dixit, M. A. Henesian, H. T. Powell, P. J. Wegner, B. W. Woods, “Intense peaks in near-field diffraction from random phase plates,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991) p. 274.

1990 (1)

R. Epstein, S. Skupsky, “Anticipated improvement in laser beam uniformity using distributed phase plates with quasi-random patterns,” J. Appl. Phys. 68, 924–931 (1990).
[CrossRef]

1989 (1)

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]

1988 (1)

D. Veron, H. Ayral, C. Geouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaong, C. Sauteret, “Optical spatial smoothing of Nd:glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[CrossRef]

1986 (2)

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]

1983 (2)

R. H. Lehmberg, S. P. Obenschain, “Use of induced spatial incoherence for uniform illumination of laser fusion targets,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

W. W. Simmons, R. O. Goodwin, “Nova laser fusion facility: design, engineering, and assembly overview,” J. Nucl. Technol. Fusion 4, 8–24 (1983).

1972 (1)

J. Nuckolls, L. Wood, A. Thiessen, G. Zimmerman, “Laser compression of matter to super-high densities: thermonuclear (CTR) applications,” Nature (London) 239, 139–142 (1972).
[CrossRef]

Abeles, F.

F. Abeles, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1963), Vol. 2, Chap. 6, pp. 251–288.
[CrossRef]

Afsharrad, T.

C. N. Danson, R. Bann, D. Pepler, N. Rizvi, I. Ross, T. Afsharrad, S. Coe, M. Dusselberger, O. Willi, “Focal spot smoothing on the vulcan glass laser system,” 1989 OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) p. 86.

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]

Ayral, H.

D. Veron, H. Ayral, C. Geouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaong, C. Sauteret, “Optical spatial smoothing of Nd:glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[CrossRef]

Bann, R.

C. N. Danson, R. Bann, D. Pepler, N. Rizvi, I. Ross, T. Afsharrad, S. Coe, M. Dusselberger, O. Willi, “Focal spot smoothing on the vulcan glass laser system,” 1989 OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) p. 86.

C. N. Danson, R. Bann, D. Pepler, I. Ross, J. Exley, D. Hardie, S. Sails, “Development of random phase plate smoothing technology,” in Rutherford Appleton Laboratory Annual Report RAL-91-025 (Rutherford Appleton Laboratory, Abingdon, UK, 1991), pp. 62–65; M. Desselberger, L. Gizzi, V. Barrow, J. Edwards, F. Khattak, S. Viana, O. Willi, C. N. Danson, “High-aspect-ratio line focus and plasma production using a random phase plate,” Appl. Opt. 31, 3759–3766 (1992).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8, pp. 393–401.

Brinker, C. J.

C. J. Brinker, C. W. Scherer, Sol-Gel Science (Academic, New York, 1990), Chap. 13.

Chen, Z.

Coe, S.

C. N. Danson, R. Bann, D. Pepler, N. Rizvi, I. Ross, T. Afsharrad, S. Coe, M. Dusselberger, O. Willi, “Focal spot smoothing on the vulcan glass laser system,” 1989 OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) p. 86.

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]

Danson, C. N.

C. N. Danson, R. Bann, D. Pepler, I. Ross, J. Exley, D. Hardie, S. Sails, “Development of random phase plate smoothing technology,” in Rutherford Appleton Laboratory Annual Report RAL-91-025 (Rutherford Appleton Laboratory, Abingdon, UK, 1991), pp. 62–65; M. Desselberger, L. Gizzi, V. Barrow, J. Edwards, F. Khattak, S. Viana, O. Willi, C. N. Danson, “High-aspect-ratio line focus and plasma production using a random phase plate,” Appl. Opt. 31, 3759–3766 (1992).
[CrossRef] [PubMed]

C. N. Danson, R. Bann, D. Pepler, N. Rizvi, I. Ross, T. Afsharrad, S. Coe, M. Dusselberger, O. Willi, “Focal spot smoothing on the vulcan glass laser system,” 1989 OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) p. 86.

Deng, X.

Dixit, S. N.

B. W. Woods, I. M. Thomas, S. N. Dixit, H. T. Powell, M. A. Henesian, “Large aperture sol-gel phase plates for beam smoothing on Nova,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 456.

L. G. Seppala, R. B. Ehrlich, S. N. Dixit, M. A. Henesian, H. T. Powell, P. J. Wegner, B. W. Woods, “Intense peaks in near-field diffraction from random phase plates,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991) p. 274.

B. W. Woods, I. M. Thomas, M. A. Henesian, S. N. Dixit, H. T. Powell, “Large aperture (80-cm diameter) phase plates for beam smoothing on Nova,” in Solid-State Lasers II, G. Duber, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1410, 47–54 (1991).

H. T. Powell, S. N. Dixit, M. A. Henesian, “Beam smoothing capability on the Nova laser,” in Lawrence Liver-more National Laboratory ICF Quarterly Report, Rep. UCRL-LR-105821-91-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1990), Vol. 1, pp. 28–38.

Dusselberger, M.

C. N. Danson, R. Bann, D. Pepler, N. Rizvi, I. Ross, T. Afsharrad, S. Coe, M. Dusselberger, O. Willi, “Focal spot smoothing on the vulcan glass laser system,” 1989 OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) p. 86.

Ehrlich, R. B.

L. G. Seppala, R. B. Ehrlich, S. N. Dixit, M. A. Henesian, H. T. Powell, P. J. Wegner, B. W. Woods, “Intense peaks in near-field diffraction from random phase plates,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991) p. 274.

English, R. E.

R. E. English, “Diffraction theory for polygonal apertures,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1988).

Epstein, R.

R. Epstein, S. Skupsky, “Anticipated improvement in laser beam uniformity using distributed phase plates with quasi-random patterns,” J. Appl. Phys. 68, 924–931 (1990).
[CrossRef]

Exley, J.

C. N. Danson, R. Bann, D. Pepler, I. Ross, J. Exley, D. Hardie, S. Sails, “Development of random phase plate smoothing technology,” in Rutherford Appleton Laboratory Annual Report RAL-91-025 (Rutherford Appleton Laboratory, Abingdon, UK, 1991), pp. 62–65; M. Desselberger, L. Gizzi, V. Barrow, J. Edwards, F. Khattak, S. Viana, O. Willi, C. N. Danson, “High-aspect-ratio line focus and plasma production using a random phase plate,” Appl. Opt. 31, 3759–3766 (1992).
[CrossRef] [PubMed]

Geouedard, C.

D. Veron, H. Ayral, C. Geouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaong, C. Sauteret, “Optical spatial smoothing of Nd:glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, pp. 85–90.

J. W. Goodman, “Statistical properties of speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer, New York, 1984), Chap. 2, pp. 9–75.

Goodwin, R. O.

W. W. Simmons, R. O. Goodwin, “Nova laser fusion facility: design, engineering, and assembly overview,” J. Nucl. Technol. Fusion 4, 8–24 (1983).

Hardie, D.

C. N. Danson, R. Bann, D. Pepler, I. Ross, J. Exley, D. Hardie, S. Sails, “Development of random phase plate smoothing technology,” in Rutherford Appleton Laboratory Annual Report RAL-91-025 (Rutherford Appleton Laboratory, Abingdon, UK, 1991), pp. 62–65; M. Desselberger, L. Gizzi, V. Barrow, J. Edwards, F. Khattak, S. Viana, O. Willi, C. N. Danson, “High-aspect-ratio line focus and plasma production using a random phase plate,” Appl. Opt. 31, 3759–3766 (1992).
[CrossRef] [PubMed]

Henesian, M. A.

H. T. Powell, S. N. Dixit, M. A. Henesian, “Beam smoothing capability on the Nova laser,” in Lawrence Liver-more National Laboratory ICF Quarterly Report, Rep. UCRL-LR-105821-91-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1990), Vol. 1, pp. 28–38.

L. G. Seppala, R. B. Ehrlich, S. N. Dixit, M. A. Henesian, H. T. Powell, P. J. Wegner, B. W. Woods, “Intense peaks in near-field diffraction from random phase plates,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991) p. 274.

B. W. Woods, I. M. Thomas, M. A. Henesian, S. N. Dixit, H. T. Powell, “Large aperture (80-cm diameter) phase plates for beam smoothing on Nova,” in Solid-State Lasers II, G. Duber, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1410, 47–54 (1991).

P. J. Wegner, M. A. Henesian, “Precision high-power solid-state laser diagnostics for target-irradiation studies and target plane irradiation modeling,” in Laser Beam Diagnostics,” R. N. Hindy, Y. Kohanzadeh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1414, 162–174 (1991).

B. W. Woods, I. M. Thomas, S. N. Dixit, H. T. Powell, M. A. Henesian, “Large aperture sol-gel phase plates for beam smoothing on Nova,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 456.

Husson, D.

D. Veron, H. Ayral, C. Geouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaong, C. Sauteret, “Optical spatial smoothing of Nd:glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[CrossRef]

Jitsuno, T.

H. Nakano, K. Tsubakimoto, K. Yagi, T. Jitsuno, M. Nakatuska, S. Nakai, T. Kanabe, C. Yamanaka, “Irradiation uniformity improvement by incoherence of amplified spontaneous emission in glass laser fusion systems,” in Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 280–282.

Kanabe, T.

H. Nakano, K. Tsubakimoto, K. Yagi, T. Jitsuno, M. Nakatuska, S. Nakai, T. Kanabe, C. Yamanaka, “Irradiation uniformity improvement by incoherence of amplified spontaneous emission in glass laser fusion systems,” in Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 280–282.

Kato, 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]

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]

Lauriou, J.

D. Veron, H. Ayral, C. Geouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaong, C. Sauteret, “Optical spatial smoothing of Nd:glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[CrossRef]

Lehmberg, R. H.

R. H. Lehmberg, S. P. Obenschain, “Use of induced spatial incoherence for uniform illumination of laser fusion targets,” Opt. Commun. 46, 27–31 (1983).
[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]

Liang, X.

Ma, R.

Martin, O.

D. Veron, H. Ayral, C. Geouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaong, C. Sauteret, “Optical spatial smoothing of Nd:glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[CrossRef]

Meyer, B.

D. Veron, H. Ayral, C. Geouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaong, C. Sauteret, “Optical spatial smoothing of Nd:glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[CrossRef]

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]

Nakai, S.

H. Nakano, K. Tsubakimoto, K. Yagi, T. Jitsuno, M. Nakatuska, S. Nakai, T. Kanabe, C. Yamanaka, “Irradiation uniformity improvement by incoherence of amplified spontaneous emission in glass laser fusion systems,” in Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 280–282.

Nakano, H.

H. Nakano, K. Tsubakimoto, K. Yagi, T. Jitsuno, M. Nakatuska, S. Nakai, T. Kanabe, C. Yamanaka, “Irradiation uniformity improvement by incoherence of amplified spontaneous emission in glass laser fusion systems,” in Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 280–282.

Nakatsuka, M.

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]

Nakatuska, M.

H. Nakano, K. Tsubakimoto, K. Yagi, T. Jitsuno, M. Nakatuska, S. Nakai, T. Kanabe, C. Yamanaka, “Irradiation uniformity improvement by incoherence of amplified spontaneous emission in glass laser fusion systems,” in Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 280–282.

Nogues, J.-L.

J.-L. Nogues, “Pure silica diffraction grating and microlens arrays made by the sol-gel process,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 15–17.

Nuckolls, J.

J. Nuckolls, L. Wood, A. Thiessen, G. Zimmerman, “Laser compression of matter to super-high densities: thermonuclear (CTR) applications,” Nature (London) 239, 139–142 (1972).
[CrossRef]

Obenschain, S. P.

R. H. Lehmberg, S. P. Obenschain, “Use of induced spatial incoherence for uniform illumination of laser fusion targets,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

Pepler, D.

C. N. Danson, R. Bann, D. Pepler, I. Ross, J. Exley, D. Hardie, S. Sails, “Development of random phase plate smoothing technology,” in Rutherford Appleton Laboratory Annual Report RAL-91-025 (Rutherford Appleton Laboratory, Abingdon, UK, 1991), pp. 62–65; M. Desselberger, L. Gizzi, V. Barrow, J. Edwards, F. Khattak, S. Viana, O. Willi, C. N. Danson, “High-aspect-ratio line focus and plasma production using a random phase plate,” Appl. Opt. 31, 3759–3766 (1992).
[CrossRef] [PubMed]

C. N. Danson, R. Bann, D. Pepler, N. Rizvi, I. Ross, T. Afsharrad, S. Coe, M. Dusselberger, O. Willi, “Focal spot smoothing on the vulcan glass laser system,” 1989 OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) p. 86.

Powell, H. T.

H. T. Powell, S. N. Dixit, M. A. Henesian, “Beam smoothing capability on the Nova laser,” in Lawrence Liver-more National Laboratory ICF Quarterly Report, Rep. UCRL-LR-105821-91-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1990), Vol. 1, pp. 28–38.

B. W. Woods, I. M. Thomas, M. A. Henesian, S. N. Dixit, H. T. Powell, “Large aperture (80-cm diameter) phase plates for beam smoothing on Nova,” in Solid-State Lasers II, G. Duber, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1410, 47–54 (1991).

L. G. Seppala, R. B. Ehrlich, S. N. Dixit, M. A. Henesian, H. T. Powell, P. J. Wegner, B. W. Woods, “Intense peaks in near-field diffraction from random phase plates,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991) p. 274.

B. W. Woods, I. M. Thomas, S. N. Dixit, H. T. Powell, M. A. Henesian, “Large aperture sol-gel phase plates for beam smoothing on Nova,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 456.

Rizvi, N.

C. N. Danson, R. Bann, D. Pepler, N. Rizvi, I. Ross, T. Afsharrad, S. Coe, M. Dusselberger, O. Willi, “Focal spot smoothing on the vulcan glass laser system,” 1989 OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) p. 86.

Ross, I.

C. N. Danson, R. Bann, D. Pepler, N. Rizvi, I. Ross, T. Afsharrad, S. Coe, M. Dusselberger, O. Willi, “Focal spot smoothing on the vulcan glass laser system,” 1989 OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) p. 86.

C. N. Danson, R. Bann, D. Pepler, I. Ross, J. Exley, D. Hardie, S. Sails, “Development of random phase plate smoothing technology,” in Rutherford Appleton Laboratory Annual Report RAL-91-025 (Rutherford Appleton Laboratory, Abingdon, UK, 1991), pp. 62–65; M. Desselberger, L. Gizzi, V. Barrow, J. Edwards, F. Khattak, S. Viana, O. Willi, C. N. Danson, “High-aspect-ratio line focus and plasma production using a random phase plate,” Appl. Opt. 31, 3759–3766 (1992).
[CrossRef] [PubMed]

Rostaong, M.

D. Veron, H. Ayral, C. Geouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaong, C. Sauteret, “Optical spatial smoothing of Nd:glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[CrossRef]

Sails, S.

C. N. Danson, R. Bann, D. Pepler, I. Ross, J. Exley, D. Hardie, S. Sails, “Development of random phase plate smoothing technology,” in Rutherford Appleton Laboratory Annual Report RAL-91-025 (Rutherford Appleton Laboratory, Abingdon, UK, 1991), pp. 62–65; M. Desselberger, L. Gizzi, V. Barrow, J. Edwards, F. Khattak, S. Viana, O. Willi, C. N. Danson, “High-aspect-ratio line focus and plasma production using a random phase plate,” Appl. Opt. 31, 3759–3766 (1992).
[CrossRef] [PubMed]

Sauteret, C.

D. Veron, H. Ayral, C. Geouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaong, C. Sauteret, “Optical spatial smoothing of Nd:glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[CrossRef]

Scherer, C. W.

C. J. Brinker, C. W. Scherer, Sol-Gel Science (Academic, New York, 1990), Chap. 13.

Seppala, L. G.

L. G. Seppala, R. B. Ehrlich, S. N. Dixit, M. A. Henesian, H. T. Powell, P. J. Wegner, B. W. Woods, “Intense peaks in near-field diffraction from random phase plates,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991) p. 274.

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]

Simmons, W. W.

W. W. Simmons, R. O. Goodwin, “Nova laser fusion facility: design, engineering, and assembly overview,” J. Nucl. Technol. Fusion 4, 8–24 (1983).

Skupsky, S.

R. Epstein, S. Skupsky, “Anticipated improvement in laser beam uniformity using distributed phase plates with quasi-random patterns,” J. Appl. Phys. 68, 924–931 (1990).
[CrossRef]

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]

Thiessen, A.

J. Nuckolls, L. Wood, A. Thiessen, G. Zimmerman, “Laser compression of matter to super-high densities: thermonuclear (CTR) applications,” Nature (London) 239, 139–142 (1972).
[CrossRef]

Thomas, I. M.

I. M. Thomas, “High laser damage threshold porous silica antireflective coating,” Appl. Opt. 25, 1481–1483 (1986).
[CrossRef] [PubMed]

B. W. Woods, I. M. Thomas, M. A. Henesian, S. N. Dixit, H. T. Powell, “Large aperture (80-cm diameter) phase plates for beam smoothing on Nova,” in Solid-State Lasers II, G. Duber, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1410, 47–54 (1991).

B. W. Woods, I. M. Thomas, S. N. Dixit, H. T. Powell, M. A. Henesian, “Large aperture sol-gel phase plates for beam smoothing on Nova,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 456.

Tsubakimoto, K.

H. Nakano, K. Tsubakimoto, K. Yagi, T. Jitsuno, M. Nakatuska, S. Nakai, T. Kanabe, C. Yamanaka, “Irradiation uniformity improvement by incoherence of amplified spontaneous emission in glass laser fusion systems,” in Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 280–282.

Veron, D.

D. Veron, H. Ayral, C. Geouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaong, C. Sauteret, “Optical spatial smoothing of Nd:glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[CrossRef]

Wegner, P. J.

L. G. Seppala, R. B. Ehrlich, S. N. Dixit, M. A. Henesian, H. T. Powell, P. J. Wegner, B. W. Woods, “Intense peaks in near-field diffraction from random phase plates,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991) p. 274.

P. J. Wegner, M. A. Henesian, “Precision high-power solid-state laser diagnostics for target-irradiation studies and target plane irradiation modeling,” in Laser Beam Diagnostics,” R. N. Hindy, Y. Kohanzadeh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1414, 162–174 (1991).

Willi, O.

C. N. Danson, R. Bann, D. Pepler, N. Rizvi, I. Ross, T. Afsharrad, S. Coe, M. Dusselberger, O. Willi, “Focal spot smoothing on the vulcan glass laser system,” 1989 OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) p. 86.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8, pp. 393–401.

Wood, L.

J. Nuckolls, L. Wood, A. Thiessen, G. Zimmerman, “Laser compression of matter to super-high densities: thermonuclear (CTR) applications,” Nature (London) 239, 139–142 (1972).
[CrossRef]

Woods, B. W.

B. W. Woods, I. M. Thomas, M. A. Henesian, S. N. Dixit, H. T. Powell, “Large aperture (80-cm diameter) phase plates for beam smoothing on Nova,” in Solid-State Lasers II, G. Duber, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1410, 47–54 (1991).

L. G. Seppala, R. B. Ehrlich, S. N. Dixit, M. A. Henesian, H. T. Powell, P. J. Wegner, B. W. Woods, “Intense peaks in near-field diffraction from random phase plates,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991) p. 274.

B. W. Woods, I. M. Thomas, S. N. Dixit, H. T. Powell, M. A. Henesian, “Large aperture sol-gel phase plates for beam smoothing on Nova,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 456.

Yagi, K.

H. Nakano, K. Tsubakimoto, K. Yagi, T. Jitsuno, M. Nakatuska, S. Nakai, T. Kanabe, C. Yamanaka, “Irradiation uniformity improvement by incoherence of amplified spontaneous emission in glass laser fusion systems,” in Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 280–282.

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]

H. Nakano, K. Tsubakimoto, K. Yagi, T. Jitsuno, M. Nakatuska, S. Nakai, T. Kanabe, C. Yamanaka, “Irradiation uniformity improvement by incoherence of amplified spontaneous emission in glass laser fusion systems,” in Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 280–282.

Yu, W.

Zimmerman, G.

J. Nuckolls, L. Wood, A. Thiessen, G. Zimmerman, “Laser compression of matter to super-high densities: thermonuclear (CTR) applications,” Nature (London) 239, 139–142 (1972).
[CrossRef]

Appl. Opt. (2)

J. Appl. Phys. (2)

R. Epstein, S. Skupsky, “Anticipated improvement in laser beam uniformity using distributed phase plates with quasi-random patterns,” J. Appl. Phys. 68, 924–931 (1990).
[CrossRef]

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]

J. Nucl. Technol. Fusion (1)

W. W. Simmons, R. O. Goodwin, “Nova laser fusion facility: design, engineering, and assembly overview,” J. Nucl. Technol. Fusion 4, 8–24 (1983).

Nature (London) (1)

J. Nuckolls, L. Wood, A. Thiessen, G. Zimmerman, “Laser compression of matter to super-high densities: thermonuclear (CTR) applications,” Nature (London) 239, 139–142 (1972).
[CrossRef]

Opt. Commun. (2)

R. H. Lehmberg, S. P. Obenschain, “Use of induced spatial incoherence for uniform illumination of laser fusion targets,” Opt. Commun. 46, 27–31 (1983).
[CrossRef]

D. Veron, H. Ayral, C. Geouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaong, C. Sauteret, “Optical spatial smoothing of Nd:glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[CrossRef]

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 (17)

P. J. Wegner, M. A. Henesian, “Precision high-power solid-state laser diagnostics for target-irradiation studies and target plane irradiation modeling,” in Laser Beam Diagnostics,” R. N. Hindy, Y. Kohanzadeh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1414, 162–174 (1991).

L. G. Seppala, R. B. Ehrlich, S. N. Dixit, M. A. Henesian, H. T. Powell, P. J. Wegner, B. W. Woods, “Intense peaks in near-field diffraction from random phase plates,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991) p. 274.

H. Nakano, K. Tsubakimoto, K. Yagi, T. Jitsuno, M. Nakatuska, S. Nakai, T. Kanabe, C. Yamanaka, “Irradiation uniformity improvement by incoherence of amplified spontaneous emission in glass laser fusion systems,” in Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), pp. 280–282.

C. N. Danson, R. Bann, D. Pepler, N. Rizvi, I. Ross, T. Afsharrad, S. Coe, M. Dusselberger, O. Willi, “Focal spot smoothing on the vulcan glass laser system,” 1989 OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) p. 86.

H. T. Powell, S. N. Dixit, M. A. Henesian, “Beam smoothing capability on the Nova laser,” in Lawrence Liver-more National Laboratory ICF Quarterly Report, Rep. UCRL-LR-105821-91-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1990), Vol. 1, pp. 28–38.

B. W. Woods, I. M. Thomas, M. A. Henesian, S. N. Dixit, H. T. Powell, “Large aperture (80-cm diameter) phase plates for beam smoothing on Nova,” in Solid-State Lasers II, G. Duber, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1410, 47–54 (1991).

Laboratory for Laser Energetics, “OMEGA phase conversion with distributed phase plates,” in Laboratory for Laser Energetics, Annual Report, Vol. 33, Q. Rep. DOE/DP40200-65 (University of Rochester, Rochester, N.Y., 1987), pp. 1–10.

C. N. Danson, R. Bann, D. Pepler, I. Ross, J. Exley, D. Hardie, S. Sails, “Development of random phase plate smoothing technology,” in Rutherford Appleton Laboratory Annual Report RAL-91-025 (Rutherford Appleton Laboratory, Abingdon, UK, 1991), pp. 62–65; M. Desselberger, L. Gizzi, V. Barrow, J. Edwards, F. Khattak, S. Viana, O. Willi, C. N. Danson, “High-aspect-ratio line focus and plasma production using a random phase plate,” Appl. Opt. 31, 3759–3766 (1992).
[CrossRef] [PubMed]

See the papers in Diffractive Optics: Design, Fabrication, and Applications,” in Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992).

B. W. Woods, I. M. Thomas, S. N. Dixit, H. T. Powell, M. A. Henesian, “Large aperture sol-gel phase plates for beam smoothing on Nova,” in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 456.

J.-L. Nogues, “Pure silica diffraction grating and microlens arrays made by the sol-gel process,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 15–17.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, pp. 85–90.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8, pp. 393–401.

R. E. English, “Diffraction theory for polygonal apertures,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1988).

J. W. Goodman, “Statistical properties of speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer, New York, 1984), Chap. 2, pp. 9–75.

C. J. Brinker, C. W. Scherer, Sol-Gel Science (Academic, New York, 1990), Chap. 13.

F. Abeles, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1963), Vol. 2, Chap. 6, pp. 251–288.
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Schematic layout of a random phase plate consisting of randomly distributed on–off phase-plate elements. A relative optical phase delay of π beween the on and the off elements is achieved by selectively depositing a thin SiO2 layer, as shown on the right.

Fig. 2
Fig. 2

Parametrization of a stretched hexagonal aperture. On the right, a rectangular aperture is shown as a special case of the stretched hexagonal aperture.

Fig. 3
Fig. 3

Far-field diffraction patterns |S(x, y, f)|2 [from Eq. (5)] for (a) a regular hexagonal aperture ( h 3 = 2 h 2 = 2 h 1 / 3 ) and (b) for a stretched hexagonal aperture with h2 = 2h1 and h3 = 3h1. The transverse distances for both cases are in units of λf/h1, and the peak intensity is normalized to 1.

Fig. 4
Fig. 4

Dependence of the position of the first minimum in the x direction on the ratio h3/h2. The arrow at h3 = 2h2 corresponds to the regular hexagonal case.

Fig. 5
Fig. 5

(a) Line scan through a calculated speckle pattern generated by the random phase plate. See the text for the parameters used in the calculation, (b) Histogram of the intensity distribution for the calculated speckle pattern over a 200 μm × 20 μm area is compared with the negative exponential probability distribution. The maximum on the abscissa is truncated at 4.0 for clarity. Intensity peaks much higher than 4 times the average are commonly observed.

Fig. 6
Fig. 6

Dependence of the coherence spike on the number of coherently illuminated phase-plate elements N and on the error in the relative optical phase difference δ. Desirable phase plates would have a coherent-spike-to-background ratio near or below unity.

Fig. 7
Fig. 7

Schematic illustration of the sol-gel phase-plate fabrication procedure.

Fig. 8
Fig. 8

Measurement of the transmission through the sol-gel coating as a function of the wavelength. From the positions of the extrema in transmission and the amount of transmission at these positions, the coating thickness and its refractive index can be calculated as discussed in the text.

Fig. 9
Fig. 9

(a) Far-field film image of a coherently illuminated 30-cm-diameter random phase plate composed of 1.3-mm square elements. The light wavelength is 351 nm, and the focal length of the lens is 3 m. (b) Line scan through a central portion of an unsaturated image similar to that shown in (a).

Fig. 10
Fig. 10

Photographs of the intensity distributions for the Nova laser operating at 351 nm at (a) best focus and (b) in a plane 4 mm past best focus. The spatial scales are the same for both pictures. Horizontal line scans through the figures of (a) and (b) are shown in (c) and (d), respectively. The intensity values correspond to a 1-TW incident beam power.

Fig. 11
Fig. 11

Schematic illustration of the placement of the RPP on the Nova beam: 1 ω, the fundamental 1.053-μm beam; 2 ω, its second harmonic at 0.527 μm following the conversion through the potassium dihydrogen phosphate (KDP) array.

Fig. 12
Fig. 12

Photographs of the far-field distributions on Nova with the random phase plates inserted in the beam. For (a), (b), and (c) the sizes of the phase-plate elements and the corresponding operating wavelengths are as follows: (a) 7-mm hexagons (527 nm), (b) 1.3-mm squares (351 nm), (c) stretched hexagons (351 nm). The size of the stretched hexagonal phase-plate element is h1 = 3 mm, h2 = 3.4 mm, and h3 = 6.4 mm. The spatial scale is the same for all three pictures. Horizontal line scans through (a), (b), and (c) are shown in (d), (e), and (f), respectively. The intensity values correspond to a 1-TW incident beam power.

Equations (20)

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E ( ρ , f ) = i λ f exp ( π i ρ 2 λ f ) × aperture d ρ E 0 ( ρ , 0 ) exp ( 2 π i ρ · ρ λ f ) ,
E ( ρ , f ) = exp ( π i ρ 2 λ f ) S ( x , y , f ) × elements j E 0 ( ρ j , 0 ) exp ( 2 π i ρ · ρ j λ f )
S ( x , y , f ) = i λ f phase-plate element d ρ exp ( 2 π i ρ · ρ λ f )
S ( x , y , f ) = i λ f a b sinc ( π a x λ f ) sinc ( π b y λ f ) ,
sinc ( x ) = sin ( x ) / x .
S ( x , y , f ) = i A λ f ( 2 h 2 h 2 + h 3 sinc ( π x h 1 λ f ) sinc ( π y h 2 λ f ) + h 3 h 2 h 3 + h 2 λ f π x h 1 { sinc [ π x h 1 2 λ f + π y ( h 3 h 2 ) 2 λ f ] × sin [ π x h 1 2 λ f π y ( h 3 + h 2 ) 2 λ f ] + sinc [ π x h 1 2 λ f π y ( h 3 h 2 ) 2 λ f ] × sin [ π x h 1 2 λ f + π y ( h 3 + h 2 ) 2 λ f ] } ) ,
A = h 1 ( h 2 + h 3 ) 2
S ( x , 0 , f ) = i A λ f [ 2 h 2 h 2 + h 3 sinc ( π x h 1 λ f ) + ( h 3 h 2 ) h 2 + h 3 sinc 2 ( π x h 1 2 λ f ) ] ,
S ( 0 , y , f ) = i A λ f { sinc [ π y ( h 3 h 2 ) 2 λ f ] sinc [ π y ( h 3 + h 2 ) 2 λ f ] } .
y min = ± 2 λ f ( h 3 + h 2 ) .
π 4 d ˜ 2 = 3 2 h 1 2 .
r min = 1 . 22 λ f d ˜ = 1 . 22 ( π 2 3 ) 1 / 2 λ f h 1 1 . 16 λ f h 1 .
E ( ρ , f ) = exp ( π i ρ 2 λ f ) E 0 S ( x , y , f ) × elements j exp [ 2 π i ( x x j + y y j ) λ f + ϕ j ] ,
I ( x , y , f ) = I 0 | S ( x , y , f ) | 2 × | elements j exp [ 2 π i ( x x j + y y j ) λ f + ϕ j ] | 2 ,
spike background = 1 N | N + + N exp ( i δ ) | 2 ,
spike background = N cos 2 ( δ 2 ) .
n c = n s ( 1 + R ) ( 1 R ) ,
n c t = λ p 4 ( 2 K + 1 ) ,
OPD = ( n c 1 ) t .
OPD = n c t ( 1 sin 2 θ n c 2 ) 1 / 2 t cos θ ( n c 1 ) t ( 1 + 1 2 n c θ 2 ) ,

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