Abstract

We present a novel moiré diagnostic that allows the time-averaged spatial coherence properties of soft-x-ray lasers to be measured. The technique is an extension of the theoretical work in a recent paper [J. Opt. Soc. Am. A 16, 323 (1999)] and is based on the concept of decompositions of the far-field divergence for partially coherent beams. An appealing feature of the approach is that the spatial coherence width at other planes can be evaluated from the measured one at a given plane on the assumption of shape-invariant beams.

© 2000 Optical Society of America

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  1. G. Hazak, A. Bar-Shalom, “Mode-selecting effects and coherence in hot-plasma x-ray lasers,” Phys. Rev. A 40, 7055–7064 (1989).
    [CrossRef] [PubMed]
  2. R. A. London, M. Strauss, M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
    [CrossRef] [PubMed]
  3. M. D. Feit, J. A. Fleck, “Wave-optics description of laboratory soft-x-ray lasers,” J. Opt. Soc. Am. B 7, 2048–2060 (1990).
    [CrossRef]
  4. R. P. Ratowsky, R. A. London, “Propagation of mutual coherence in refractive x-ray lasers using a WKB method,” Phys. Rev. A 51, 2361–2370 (1995).
    [CrossRef] [PubMed]
  5. J. Yang, D. Fan, S. Wang, Y. Gu, “Generalized matrix-optics description of soft-x-ray lasers,” J. Opt. Soc. Am. B 16, 1016–1024 (1999).
    [CrossRef]
  6. P. Amendt, M. Strauss, R. A. London, “Plasma fluctuations and x-ray laser transverse coherence,” Phys. Rev. A 53, R23–R26 (1996).
    [CrossRef] [PubMed]
  7. J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
    [CrossRef] [PubMed]
  8. K. A. Nugent, J. E. Trebes, “Coherence measurement technique for short-wavelength light sources,” Rev. Sci. Instrum. 63, 2146–2151 (1992).
    [CrossRef]
  9. Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.
  10. R. E. Burge, G. E. Slark, M. T. Browne, X.-C. Yuan, P. Charalambous, X.-H. Cheng, C. L. S. Lewis, A. MacPhee, D. Neely, “Spatial coherence of x-ray laser emission from neonlike germanium after prepulse,” J. Opt. Soc. Am. B 14, 2742–2751 (1997).
    [CrossRef]
  11. P. Lu, E. Fill, Y. Li, J. Maruhn, G. Pretzler, “Spatial coherence of prepulse-induced neonlike x-ray lasers,” Phys. Rev. A 58, 628–635 (1998).
    [CrossRef]
  12. J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.
  13. F. Albert, B. Rus, Ph. Zeitoun, A. Carillon, P. Jaeglé, G. Jamelot, A. Klisnick, D. Ros, S. Sebban, “New approach for measurement of the x-ray laser transverse coherence,” in X-Ray Lasers 1996, S. Svanverg, C.-G. Wahlström, eds., Vol. 151 of Institute of Physics Conference Proceedings (Institute of Physics, Bristol, UK, 1996), pp. 427–429.
  14. M. C. Marconi, J. L. A. Chilla, C. H. Moreno, B. R. Benware, J. J. Rocca, “Measurement of the spatial coherence buildup in a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 79, 2799–2802 (1997).
    [CrossRef]
  15. P. Celliers, F. Weber, L. B. Da Silva, T. W. Barbee, R. Cauble, A. S. Wan, J. C. Moreno, “Fringe formation and coherence of a soft-x-ray laser beam illuminating a Mach–Zehnder interferometer,” Opt. Lett. 20, 1907–1909 (1995).
    [CrossRef] [PubMed]
  16. J. Yang, D. Fan, S. Wang, Y. Gu, “Technique for diagnosing x-ray laser beam quality by use of the moiré signal,” J. Opt. Soc. Am. A 16, 323–330 (1999).
    [CrossRef]
  17. F. Gori, S. Vicalvi, M. Santarsiero, R. Borghi, “Shape-invariance range of a light beam,” Opt. Lett. 21, 1205–1207 (1996).
    [CrossRef] [PubMed]
  18. F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
    [CrossRef]
  19. K. A. Nugent, “Partially coherent diffraction patterns and coherence measurement,” J. Opt. Soc. Am. A 8, 1574–1579 (1991).
    [CrossRef]
  20. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
    [CrossRef] [PubMed]

1999

1998

P. Lu, E. Fill, Y. Li, J. Maruhn, G. Pretzler, “Spatial coherence of prepulse-induced neonlike x-ray lasers,” Phys. Rev. A 58, 628–635 (1998).
[CrossRef]

1997

M. C. Marconi, J. L. A. Chilla, C. H. Moreno, B. R. Benware, J. J. Rocca, “Measurement of the spatial coherence buildup in a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 79, 2799–2802 (1997).
[CrossRef]

R. E. Burge, G. E. Slark, M. T. Browne, X.-C. Yuan, P. Charalambous, X.-H. Cheng, C. L. S. Lewis, A. MacPhee, D. Neely, “Spatial coherence of x-ray laser emission from neonlike germanium after prepulse,” J. Opt. Soc. Am. B 14, 2742–2751 (1997).
[CrossRef]

1996

F. Gori, S. Vicalvi, M. Santarsiero, R. Borghi, “Shape-invariance range of a light beam,” Opt. Lett. 21, 1205–1207 (1996).
[CrossRef] [PubMed]

P. Amendt, M. Strauss, R. A. London, “Plasma fluctuations and x-ray laser transverse coherence,” Phys. Rev. A 53, R23–R26 (1996).
[CrossRef] [PubMed]

1995

1992

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

K. A. Nugent, J. E. Trebes, “Coherence measurement technique for short-wavelength light sources,” Rev. Sci. Instrum. 63, 2146–2151 (1992).
[CrossRef]

1991

1990

R. A. London, M. Strauss, M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[CrossRef] [PubMed]

M. D. Feit, J. A. Fleck, “Wave-optics description of laboratory soft-x-ray lasers,” J. Opt. Soc. Am. B 7, 2048–2060 (1990).
[CrossRef]

1989

G. Hazak, A. Bar-Shalom, “Mode-selecting effects and coherence in hot-plasma x-ray lasers,” Phys. Rev. A 40, 7055–7064 (1989).
[CrossRef] [PubMed]

1983

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

Albert, F.

F. Albert, B. Rus, Ph. Zeitoun, A. Carillon, P. Jaeglé, G. Jamelot, A. Klisnick, D. Ros, S. Sebban, “New approach for measurement of the x-ray laser transverse coherence,” in X-Ray Lasers 1996, S. Svanverg, C.-G. Wahlström, eds., Vol. 151 of Institute of Physics Conference Proceedings (Institute of Physics, Bristol, UK, 1996), pp. 427–429.

Amendt, P.

P. Amendt, M. Strauss, R. A. London, “Plasma fluctuations and x-ray laser transverse coherence,” Phys. Rev. A 53, R23–R26 (1996).
[CrossRef] [PubMed]

Azuma, H.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Barbee, T. W.

P. Celliers, F. Weber, L. B. Da Silva, T. W. Barbee, R. Cauble, A. S. Wan, J. C. Moreno, “Fringe formation and coherence of a soft-x-ray laser beam illuminating a Mach–Zehnder interferometer,” Opt. Lett. 20, 1907–1909 (1995).
[CrossRef] [PubMed]

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Bar-Shalom, A.

G. Hazak, A. Bar-Shalom, “Mode-selecting effects and coherence in hot-plasma x-ray lasers,” Phys. Rev. A 40, 7055–7064 (1989).
[CrossRef] [PubMed]

Bélanger, P. A.

Benware, B. R.

M. C. Marconi, J. L. A. Chilla, C. H. Moreno, B. R. Benware, J. J. Rocca, “Measurement of the spatial coherence buildup in a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 79, 2799–2802 (1997).
[CrossRef]

Borghi, R.

Browne, M. T.

Burge, R. E.

Cairns, C.

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

Carillon, A.

F. Albert, B. Rus, Ph. Zeitoun, A. Carillon, P. Jaeglé, G. Jamelot, A. Klisnick, D. Ros, S. Sebban, “New approach for measurement of the x-ray laser transverse coherence,” in X-Ray Lasers 1996, S. Svanverg, C.-G. Wahlström, eds., Vol. 151 of Institute of Physics Conference Proceedings (Institute of Physics, Bristol, UK, 1996), pp. 427–429.

Carter, M. R.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Cauble, R.

Celliers, P.

Charalambous, P.

Cheng, X.-H.

Chilla, J. L. A.

M. C. Marconi, J. L. A. Chilla, C. H. Moreno, B. R. Benware, J. J. Rocca, “Measurement of the spatial coherence buildup in a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 79, 2799–2802 (1997).
[CrossRef]

Da Silva, L. B.

P. Celliers, F. Weber, L. B. Da Silva, T. W. Barbee, R. Cauble, A. S. Wan, J. C. Moreno, “Fringe formation and coherence of a soft-x-ray laser beam illuminating a Mach–Zehnder interferometer,” Opt. Lett. 20, 1907–1909 (1995).
[CrossRef] [PubMed]

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Daido, H.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Dwivedi, L.

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

Fan, D.

Feit, M. D.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

M. D. Feit, J. A. Fleck, “Wave-optics description of laboratory soft-x-ray lasers,” J. Opt. Soc. Am. B 7, 2048–2060 (1990).
[CrossRef]

Fill, E.

P. Lu, E. Fill, Y. Li, J. Maruhn, G. Pretzler, “Spatial coherence of prepulse-induced neonlike x-ray lasers,” Phys. Rev. A 58, 628–635 (1998).
[CrossRef]

Fleck, J. A.

Fukuda, Y.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Gori, F.

F. Gori, S. Vicalvi, M. Santarsiero, R. Borghi, “Shape-invariance range of a light beam,” Opt. Lett. 21, 1205–1207 (1996).
[CrossRef] [PubMed]

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

Gu, Y.

Hazak, G.

G. Hazak, A. Bar-Shalom, “Mode-selecting effects and coherence in hot-plasma x-ray lasers,” Phys. Rev. A 40, 7055–7064 (1989).
[CrossRef] [PubMed]

Holden, M.

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

Jaeglé, P.

F. Albert, B. Rus, Ph. Zeitoun, A. Carillon, P. Jaeglé, G. Jamelot, A. Klisnick, D. Ros, S. Sebban, “New approach for measurement of the x-ray laser transverse coherence,” in X-Ray Lasers 1996, S. Svanverg, C.-G. Wahlström, eds., Vol. 151 of Institute of Physics Conference Proceedings (Institute of Physics, Bristol, UK, 1996), pp. 427–429.

Jamelot, G.

F. Albert, B. Rus, Ph. Zeitoun, A. Carillon, P. Jaeglé, G. Jamelot, A. Klisnick, D. Ros, S. Sebban, “New approach for measurement of the x-ray laser transverse coherence,” in X-Ray Lasers 1996, S. Svanverg, C.-G. Wahlström, eds., Vol. 151 of Institute of Physics Conference Proceedings (Institute of Physics, Bristol, UK, 1996), pp. 427–429.

Kanabe, T.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Kato, Y.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Key, M. H.

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

Klisnick, A.

F. Albert, B. Rus, Ph. Zeitoun, A. Carillon, P. Jaeglé, G. Jamelot, A. Klisnick, D. Ros, S. Sebban, “New approach for measurement of the x-ray laser transverse coherence,” in X-Ray Lasers 1996, S. Svanverg, C.-G. Wahlström, eds., Vol. 151 of Institute of Physics Conference Proceedings (Institute of Physics, Bristol, UK, 1996), pp. 427–429.

Koch, J. A.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Krishnan, J.

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

Lewis, C. L. S.

R. E. Burge, G. E. Slark, M. T. Browne, X.-C. Yuan, P. Charalambous, X.-H. Cheng, C. L. S. Lewis, A. MacPhee, D. Neely, “Spatial coherence of x-ray laser emission from neonlike germanium after prepulse,” J. Opt. Soc. Am. B 14, 2742–2751 (1997).
[CrossRef]

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

Li, Y.

P. Lu, E. Fill, Y. Li, J. Maruhn, G. Pretzler, “Spatial coherence of prepulse-induced neonlike x-ray lasers,” Phys. Rev. A 58, 628–635 (1998).
[CrossRef]

London, R. A.

P. Amendt, M. Strauss, R. A. London, “Plasma fluctuations and x-ray laser transverse coherence,” Phys. Rev. A 53, R23–R26 (1996).
[CrossRef] [PubMed]

R. P. Ratowsky, R. A. London, “Propagation of mutual coherence in refractive x-ray lasers using a WKB method,” Phys. Rev. A 51, 2361–2370 (1995).
[CrossRef] [PubMed]

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

R. A. London, M. Strauss, M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[CrossRef] [PubMed]

Lu, P.

P. Lu, E. Fill, Y. Li, J. Maruhn, G. Pretzler, “Spatial coherence of prepulse-induced neonlike x-ray lasers,” Phys. Rev. A 58, 628–635 (1998).
[CrossRef]

MacGowan, B. J.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

MacPhee, A.

R. E. Burge, G. E. Slark, M. T. Browne, X.-C. Yuan, P. Charalambous, X.-H. Cheng, C. L. S. Lewis, A. MacPhee, D. Neely, “Spatial coherence of x-ray laser emission from neonlike germanium after prepulse,” J. Opt. Soc. Am. B 14, 2742–2751 (1997).
[CrossRef]

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

Marconi, M. C.

M. C. Marconi, J. L. A. Chilla, C. H. Moreno, B. R. Benware, J. J. Rocca, “Measurement of the spatial coherence buildup in a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 79, 2799–2802 (1997).
[CrossRef]

Maruhn, J.

P. Lu, E. Fill, Y. Li, J. Maruhn, G. Pretzler, “Spatial coherence of prepulse-induced neonlike x-ray lasers,” Phys. Rev. A 58, 628–635 (1998).
[CrossRef]

Matthews, D. L.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Miura, E.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Moreno, C. H.

M. C. Marconi, J. L. A. Chilla, C. H. Moreno, B. R. Benware, J. J. Rocca, “Measurement of the spatial coherence buildup in a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 79, 2799–2802 (1997).
[CrossRef]

Moreno, J. C.

Mrowka, S.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Murai, K.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Nakai, S.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Neely, D.

R. E. Burge, G. E. Slark, M. T. Browne, X.-C. Yuan, P. Charalambous, X.-H. Cheng, C. L. S. Lewis, A. MacPhee, D. Neely, “Spatial coherence of x-ray laser emission from neonlike germanium after prepulse,” J. Opt. Soc. Am. B 14, 2742–2751 (1997).
[CrossRef]

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

Niibe, M.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Norreys, P. A.

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

Nugent, K. A.

K. A. Nugent, J. E. Trebes, “Coherence measurement technique for short-wavelength light sources,” Rev. Sci. Instrum. 63, 2146–2151 (1992).
[CrossRef]

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

K. A. Nugent, “Partially coherent diffraction patterns and coherence measurement,” J. Opt. Soc. Am. A 8, 1574–1579 (1991).
[CrossRef]

O’Neill, D. M.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Pert, G. J.

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

Pretzler, G.

P. Lu, E. Fill, Y. Li, J. Maruhn, G. Pretzler, “Spatial coherence of prepulse-induced neonlike x-ray lasers,” Phys. Rev. A 58, 628–635 (1998).
[CrossRef]

Ramsden, S. A.

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

Ratowsky, R. P.

R. P. Ratowsky, R. A. London, “Propagation of mutual coherence in refractive x-ray lasers using a WKB method,” Phys. Rev. A 51, 2361–2370 (1995).
[CrossRef] [PubMed]

Rocca, J. J.

M. C. Marconi, J. L. A. Chilla, C. H. Moreno, B. R. Benware, J. J. Rocca, “Measurement of the spatial coherence buildup in a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 79, 2799–2802 (1997).
[CrossRef]

Ros, D.

F. Albert, B. Rus, Ph. Zeitoun, A. Carillon, P. Jaeglé, G. Jamelot, A. Klisnick, D. Ros, S. Sebban, “New approach for measurement of the x-ray laser transverse coherence,” in X-Ray Lasers 1996, S. Svanverg, C.-G. Wahlström, eds., Vol. 151 of Institute of Physics Conference Proceedings (Institute of Physics, Bristol, UK, 1996), pp. 427–429.

Rosen, M. D.

R. A. London, M. Strauss, M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[CrossRef] [PubMed]

Rus, B.

F. Albert, B. Rus, Ph. Zeitoun, A. Carillon, P. Jaeglé, G. Jamelot, A. Klisnick, D. Ros, S. Sebban, “New approach for measurement of the x-ray laser transverse coherence,” in X-Ray Lasers 1996, S. Svanverg, C.-G. Wahlström, eds., Vol. 151 of Institute of Physics Conference Proceedings (Institute of Physics, Bristol, UK, 1996), pp. 427–429.

Santarsiero, M.

Sebban, S.

F. Albert, B. Rus, Ph. Zeitoun, A. Carillon, P. Jaeglé, G. Jamelot, A. Klisnick, D. Ros, S. Sebban, “New approach for measurement of the x-ray laser transverse coherence,” in X-Ray Lasers 1996, S. Svanverg, C.-G. Wahlström, eds., Vol. 151 of Institute of Physics Conference Proceedings (Institute of Physics, Bristol, UK, 1996), pp. 427–429.

Shiraga, H.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Slark, G. E.

Smith, C. G.

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

Stone, G. F.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

Strauss, M.

P. Amendt, M. Strauss, R. A. London, “Plasma fluctuations and x-ray laser transverse coherence,” Phys. Rev. A 53, R23–R26 (1996).
[CrossRef] [PubMed]

R. A. London, M. Strauss, M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[CrossRef] [PubMed]

Takagi, M.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Tallents, G. J.

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

Trebes, J. E.

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

K. A. Nugent, J. E. Trebes, “Coherence measurement technique for short-wavelength light sources,” Rev. Sci. Instrum. 63, 2146–2151 (1992).
[CrossRef]

Tsukamoto, N.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Vicalvi, S.

Wan, A. S.

Wang, S.

Weber, F.

Yamanaka, M.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Yang, J.

Yuan, G.

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

Yuan, X.-C.

Zeitoun, Ph.

F. Albert, B. Rus, Ph. Zeitoun, A. Carillon, P. Jaeglé, G. Jamelot, A. Klisnick, D. Ros, S. Sebban, “New approach for measurement of the x-ray laser transverse coherence,” in X-Ray Lasers 1996, S. Svanverg, C.-G. Wahlström, eds., Vol. 151 of Institute of Physics Conference Proceedings (Institute of Physics, Bristol, UK, 1996), pp. 427–429.

Zhang, J.

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

Opt. Lett.

Phys. Rev. A

P. Lu, E. Fill, Y. Li, J. Maruhn, G. Pretzler, “Spatial coherence of prepulse-induced neonlike x-ray lasers,” Phys. Rev. A 58, 628–635 (1998).
[CrossRef]

P. Amendt, M. Strauss, R. A. London, “Plasma fluctuations and x-ray laser transverse coherence,” Phys. Rev. A 53, R23–R26 (1996).
[CrossRef] [PubMed]

R. P. Ratowsky, R. A. London, “Propagation of mutual coherence in refractive x-ray lasers using a WKB method,” Phys. Rev. A 51, 2361–2370 (1995).
[CrossRef] [PubMed]

G. Hazak, A. Bar-Shalom, “Mode-selecting effects and coherence in hot-plasma x-ray lasers,” Phys. Rev. A 40, 7055–7064 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett.

R. A. London, M. Strauss, M. D. Rosen, “Modal analysis of x-ray laser coherence,” Phys. Rev. Lett. 65, 563–566 (1990).
[CrossRef] [PubMed]

J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, M. D. Feit, “Measurement of the spatial coherence of a soft-x-ray laser,” Phys. Rev. Lett. 68, 588–591 (1992).
[CrossRef] [PubMed]

M. C. Marconi, J. L. A. Chilla, C. H. Moreno, B. R. Benware, J. J. Rocca, “Measurement of the spatial coherence buildup in a discharge pumped table-top soft x-ray laser,” Phys. Rev. Lett. 79, 2799–2802 (1997).
[CrossRef]

Rev. Sci. Instrum.

K. A. Nugent, J. E. Trebes, “Coherence measurement technique for short-wavelength light sources,” Rev. Sci. Instrum. 63, 2146–2151 (1992).
[CrossRef]

Other

Y. Kato, M. Yamanaka, H. Daido, H. Shiraga, H. Azuma, K. Murai, G. Yuan, E. Miura, T. Kanabe, M. Takagi, S. Nakai, C. L. S. Lewis, D. M. O’Neill, D. Neely, M. Niibe, N. Tsukamoto, Y. Fukuda, “Coherence and double-pass amplification in Ge soft x-ray laser,” in X-Ray Lasers 1992, E. E. Fill, ed., Vol. 125 of Institute of Physics Conference Series (Institute of Physics, Bristol, UK, 1992), pp. 9–16.

J. Krishnan, C. Cairns, L. Dwivedi, M. Holden, M. H. Key, C. L. S. Lewis, A. MacPhee, D. Neely, P. A. Norreys, G. J. Pert, S. A. Ramsden, C. G. Smith, G. J. Tallents, J. Zhang, “Gain measurements and spatial coherence in neon-like x-ray lasers,” in X-Ray Lasers 1994, D. C. Eder, D. L. Matthews, eds., Vol. 332 of American Institute of Physics (AIP) Conference Proceedings (AIP, New York, 1994), pp. 483–487.

F. Albert, B. Rus, Ph. Zeitoun, A. Carillon, P. Jaeglé, G. Jamelot, A. Klisnick, D. Ros, S. Sebban, “New approach for measurement of the x-ray laser transverse coherence,” in X-Ray Lasers 1996, S. Svanverg, C.-G. Wahlström, eds., Vol. 151 of Institute of Physics Conference Proceedings (Institute of Physics, Bristol, UK, 1996), pp. 427–429.

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Figures (1)

Fig. 1
Fig. 1

Schematic of an arrangement for observing the moiré pattern of an XRL.

Tables (2)

Tables Icon

Table 1 Calculated Values of Beam Coherence Width and Quality Parameters at the Plane of the First Grating (z=0.8 m) and Waist Location (z=z0)

Tables Icon

Table 2 Actual Values of Beam Coherence Width and Quality Parameters

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

J(x,Δ)=ψ(x+Δ/2)ψ*(x-Δ/2)μ(Δ),
Re=-d/θ tan γ.
M2=4.346 pWdλ11+c¯,
W2αf2-(W2/Re)2=(λM2/π)2,
αf=M2λ/πW0,
W2(z)=W02(z0)1+z-z0zR2,
G(α,δ)=λ-2J(x, Δ)exp[-j2π(αΔ+δx)/λ]dx dΔ.
U(α)=μ˜(α)*|ψ˜(α)|2,
μ˜(α)=λ-1μ(Δ)exp(-j2παΔ/λ)dΔ
ψ˜(α)=λ-1ψ(Δ)exp(-j2παΔ/λ)dΔ
αf2=αincoh2+αcoh2,
αcoh2=(λMcoh2/πW)2+W2/Re2,
αincoh2=αf2-(λMcoh2/πW)2-W2/Re2.
σc=λ/παincoh.
Imin=-XsXs(2|X|/P)exp(-2X2/Xs2)dX
Imax=-XsXs(1-2|X|/P)exp(-2X2/Xs2)dX.
M20.867 8πpWdλ11+c4.346 pWdλ11+c,
J(x, Δ, z)ms-2J0(x/ms, Δ/ms, z0)×exp{j[φ(x+Δ/2, z-z0)-φ(x-Δ/2, z-z0)]}.
J(x, Δ, t)=n=1N|an(t)|2un(x+Δ/2)un*(x-Δ/2)+n=1NlnNan(t)al*(t)un(x+Δ/2)×ul*(x-Δ/2)exp [j(ωn-ωl)t],

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