Abstract

We derive a novel expression for the mutual-coherence function for plasma x-ray lasers using generalized matrix optics. The expression is valid when optical systems can be described by complex ABCDGH beam matrices and when gain saturation can be ignored. The model is used to analyze the output-beam characteristics and the spatial-coherence properties of a general class of x-ray lasers that contain exploding-foil, slab, and curved targets. Comparison of our results with numerical solutions of the paraxial wave equation and previous experimental data shows good agreement.

© 1999 Optical Society of America

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    [CrossRef] [PubMed]

1997

J. Zhang, A. G. MacPhee, J. Lin, E. Wolfrum, R. Smith, C. Danson, M. H. Key, C. L. S. Lewis, D. Neely, J. Nilsen, G. J. Pert, G. J. Tallents, and J. S. Wark, “A saturated x-ray laser beam at 7 nanometers,” Science 276, 1097–1100 (1997), and references therein.
[CrossRef]

1995

J. Nilsen and J. C. Moreno, “Monochromatic lasing at 182A° in neonlike selenium,” Phys. Rev. Lett. 74, 3376–3379 (1995).
[CrossRef] [PubMed]

R. P. Ratowsky and 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]

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

1994

S. Wang, Y. Gu, G. Zhou, S. Yu, Y. Ni, Y. Xiong, S. Fu, C. Mao, Z. Zhou, B. Wang, Y. Zeng, G. Han, G. Huang, D. Fan, Z. Lin, and X. Deng, “Achievement of soft-x-ray laser with nearly saturated intensity and minimum divergence angle ~1 mrad,” Chin. J. Lasers B 3, 507–510 (1994), and references therein.

R. Kodama, D. Neely, Y. Kato, H. Daido, K. Murai, G. Yuan, A. MacPhee, and C. L. S. Lewis, “Generation of small-divergence soft x-ray laser by plasma waveguiding with a curved target,” Phys. Rev. Lett. 73, 3215–3218 (1994).
[CrossRef] [PubMed]

1993

1992

S. Wang, Y. Gu, G. Zhou, S. Yu, S. Fu, Y. Ni, J. Wu, Z. Zhou, G. Han, Z. Tao, Z. Lin, S. Wang, W. Chen, D. Fan, G. Zhang, J. Sheng, H. Peng, T. Zhang, and Y. Shao, “Experimental investigation of high-gain Ne-like Ge soft x-ray laser by double-massive-target coupling,” J. Opt. Soc. Am. B 9, 360–368 (1992).
[CrossRef]

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

1990

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

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

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

1989

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

1988

1987

1986

1984

R. Simon, E. C. G. Sudarsdhan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

1983

D. Fan, “The Fresnel number in terms of ray matrix elements,” Acta Opt. Sin. 3, 43–49 (1983), in Chinese.

1982

1980

D. Fan, “Diffraction integral of optical systems and its application,” Chin. J. Lasers 7, 23–33 (1980), in Chinese.

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

1973

1970

1969

P. Baues, “Huygens’ principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-Electron. 1, 37–44 (1969).
[CrossRef]

Amendt, P.

P. Amendt and R. A. London, “Optimization of single-stage x-ray laser coherence,” Phys. Rev. A 47, 4348–4363 (1993).
[CrossRef] [PubMed]

Barbee Jr., T. W.

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

Bar-Shalom, A.

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

Bastiaans, M. J.

Baues, P.

P. Baues, “Huygens’ principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-Electron. 1, 37–44 (1969).
[CrossRef]

Brown, S.

Carillon, A.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Casperson, L. W.

Cauble, R.

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

Ceglio, N.

Ceglio, N. M.

Celliers, P.

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

Chen, H. Z.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Chen, W.

Ciarlo, D.

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

Collins, S. A.

Da Silva, L. B.

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

Daido, H.

R. Kodama, D. Neely, Y. Kato, H. Daido, K. Murai, G. Yuan, A. MacPhee, and C. L. S. Lewis, “Generation of small-divergence soft x-ray laser by plasma waveguiding with a curved target,” Phys. Rev. Lett. 73, 3215–3218 (1994).
[CrossRef] [PubMed]

Danson, C.

J. Zhang, A. G. MacPhee, J. Lin, E. Wolfrum, R. Smith, C. Danson, M. H. Key, C. L. S. Lewis, D. Neely, J. Nilsen, G. J. Pert, G. J. Tallents, and J. S. Wark, “A saturated x-ray laser beam at 7 nanometers,” Science 276, 1097–1100 (1997), and references therein.
[CrossRef]

Deng, X.

S. Wang, Y. Gu, G. Zhou, S. Yu, Y. Ni, Y. Xiong, S. Fu, C. Mao, Z. Zhou, B. Wang, Y. Zeng, G. Han, G. Huang, D. Fan, Z. Lin, and X. Deng, “Achievement of soft-x-ray laser with nearly saturated intensity and minimum divergence angle ~1 mrad,” Chin. J. Lasers B 3, 507–510 (1994), and references therein.

Dhez, P.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Dwivedi, L.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Eder, D.

Fan, D.

S. Wang, Y. Gu, G. Zhou, S. Yu, Y. Ni, Y. Xiong, S. Fu, C. Mao, Z. Zhou, B. Wang, Y. Zeng, G. Han, G. Huang, D. Fan, Z. Lin, and X. Deng, “Achievement of soft-x-ray laser with nearly saturated intensity and minimum divergence angle ~1 mrad,” Chin. J. Lasers B 3, 507–510 (1994), and references therein.

S. Wang, Y. Gu, G. Zhou, S. Yu, S. Fu, Y. Ni, J. Wu, Z. Zhou, G. Han, Z. Tao, Z. Lin, S. Wang, W. Chen, D. Fan, G. Zhang, J. Sheng, H. Peng, T. Zhang, and Y. Shao, “Experimental investigation of high-gain Ne-like Ge soft x-ray laser by double-massive-target coupling,” J. Opt. Soc. Am. B 9, 360–368 (1992).
[CrossRef]

D. Fan, “The Fresnel number in terms of ray matrix elements,” Acta Opt. Sin. 3, 43–49 (1983), in Chinese.

D. Fan, “Diffraction integral of optical systems and its application,” Chin. J. Lasers 7, 23–33 (1980), in Chinese.

Feit, M. D.

Fill, E. E.

E. E. Fill, “Gain guiding of x-ray laser beams,” Opt. Commun. 67, 441–445 (1988).
[CrossRef]

Fleck Jr., J. A.

Friberg, A. T.

Fu, S.

S. Wang, Y. Gu, G. Zhou, S. Yu, Y. Ni, Y. Xiong, S. Fu, C. Mao, Z. Zhou, B. Wang, Y. Zeng, G. Han, G. Huang, D. Fan, Z. Lin, and X. Deng, “Achievement of soft-x-ray laser with nearly saturated intensity and minimum divergence angle ~1 mrad,” Chin. J. Lasers B 3, 507–510 (1994), and references therein.

S. Wang, Y. Gu, G. Zhou, S. Yu, S. Fu, Y. Ni, J. Wu, Z. Zhou, G. Han, Z. Tao, Z. Lin, S. Wang, W. Chen, D. Fan, G. Zhang, J. Sheng, H. Peng, T. Zhang, and Y. Shao, “Experimental investigation of high-gain Ne-like Ge soft x-ray laser by double-massive-target coupling,” J. Opt. Soc. Am. B 9, 360–368 (1992).
[CrossRef]

Gaines, D. P.

Gori, F.

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Gu, Y.

S. Wang, Y. Gu, G. Zhou, S. Yu, Y. Ni, Y. Xiong, S. Fu, C. Mao, Z. Zhou, B. Wang, Y. Zeng, G. Han, G. Huang, D. Fan, Z. Lin, and X. Deng, “Achievement of soft-x-ray laser with nearly saturated intensity and minimum divergence angle ~1 mrad,” Chin. J. Lasers B 3, 507–510 (1994), and references therein.

S. Wang, Y. Gu, G. Zhou, S. Yu, S. Fu, Y. Ni, J. Wu, Z. Zhou, G. Han, Z. Tao, Z. Lin, S. Wang, W. Chen, D. Fan, G. Zhang, J. Sheng, H. Peng, T. Zhang, and Y. Shao, “Experimental investigation of high-gain Ne-like Ge soft x-ray laser by double-massive-target coupling,” J. Opt. Soc. Am. B 9, 360–368 (1992).
[CrossRef]

Han, G.

S. Wang, Y. Gu, G. Zhou, S. Yu, Y. Ni, Y. Xiong, S. Fu, C. Mao, Z. Zhou, B. Wang, Y. Zeng, G. Han, G. Huang, D. Fan, Z. Lin, and X. Deng, “Achievement of soft-x-ray laser with nearly saturated intensity and minimum divergence angle ~1 mrad,” Chin. J. Lasers B 3, 507–510 (1994), and references therein.

S. Wang, Y. Gu, G. Zhou, S. Yu, S. Fu, Y. Ni, J. Wu, Z. Zhou, G. Han, Z. Tao, Z. Lin, S. Wang, W. Chen, D. Fan, G. Zhang, J. Sheng, H. Peng, T. Zhang, and Y. Shao, “Experimental investigation of high-gain Ne-like Ge soft x-ray laser by double-massive-target coupling,” J. Opt. Soc. Am. B 9, 360–368 (1992).
[CrossRef]

Hawryluk, A.

Hawryluk, A. M.

Hazak, G.

O. Zahavi, G. Hazak, and Z. Zinamon, “Study of amplified spontaneous emission systems by the ray-tracing technique,” J. Opt. Soc. Am. B 10, 271–278 (1993).
[CrossRef]

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

Huang, G.

S. Wang, Y. Gu, G. Zhou, S. Yu, Y. Ni, Y. Xiong, S. Fu, C. Mao, Z. Zhou, B. Wang, Y. Zeng, G. Han, G. Huang, D. Fan, Z. Lin, and X. Deng, “Achievement of soft-x-ray laser with nearly saturated intensity and minimum divergence angle ~1 mrad,” Chin. J. Lasers B 3, 507–510 (1994), and references therein.

Jacoby, J.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Jaegle, P.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Jamelot, G.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Kato, Y.

R. Kodama, D. Neely, Y. Kato, H. Daido, K. Murai, G. Yuan, A. MacPhee, and C. L. S. Lewis, “Generation of small-divergence soft x-ray laser by plasma waveguiding with a curved target,” Phys. Rev. Lett. 73, 3215–3218 (1994).
[CrossRef] [PubMed]

Kauderer, M.

Keane, C.

Key, M. H.

J. Zhang, A. G. MacPhee, J. Lin, E. Wolfrum, R. Smith, C. Danson, M. H. Key, C. L. S. Lewis, D. Neely, J. Nilsen, G. J. Pert, G. J. Tallents, and J. S. Wark, “A saturated x-ray laser beam at 7 nanometers,” Science 276, 1097–1100 (1997), and references therein.
[CrossRef]

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Kidd, A.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Klisnick, A.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Kodama, R.

R. Kodama, D. Neely, Y. Kato, H. Daido, K. Murai, G. Yuan, A. MacPhee, and C. L. S. Lewis, “Generation of small-divergence soft x-ray laser by plasma waveguiding with a curved target,” Phys. Rev. Lett. 73, 3215–3218 (1994).
[CrossRef] [PubMed]

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Krishnan, J.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Lewis, C. L. S.

J. Zhang, A. G. MacPhee, J. Lin, E. Wolfrum, R. Smith, C. Danson, M. H. Key, C. L. S. Lewis, D. Neely, J. Nilsen, G. J. Pert, G. J. Tallents, and J. S. Wark, “A saturated x-ray laser beam at 7 nanometers,” Science 276, 1097–1100 (1997), and references therein.
[CrossRef]

R. Kodama, D. Neely, Y. Kato, H. Daido, K. Murai, G. Yuan, A. MacPhee, and C. L. S. Lewis, “Generation of small-divergence soft x-ray laser by plasma waveguiding with a curved target,” Phys. Rev. Lett. 73, 3215–3218 (1994).
[CrossRef] [PubMed]

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Libby, S.

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

Lin, J.

J. Zhang, A. G. MacPhee, J. Lin, E. Wolfrum, R. Smith, C. Danson, M. H. Key, C. L. S. Lewis, D. Neely, J. Nilsen, G. J. Pert, G. J. Tallents, and J. S. Wark, “A saturated x-ray laser beam at 7 nanometers,” Science 276, 1097–1100 (1997), and references therein.
[CrossRef]

Lin, Z.

S. Wang, Y. Gu, G. Zhou, S. Yu, Y. Ni, Y. Xiong, S. Fu, C. Mao, Z. Zhou, B. Wang, Y. Zeng, G. Han, G. Huang, D. Fan, Z. Lin, and X. Deng, “Achievement of soft-x-ray laser with nearly saturated intensity and minimum divergence angle ~1 mrad,” Chin. J. Lasers B 3, 507–510 (1994), and references therein.

S. Wang, Y. Gu, G. Zhou, S. Yu, S. Fu, Y. Ni, J. Wu, Z. Zhou, G. Han, Z. Tao, Z. Lin, S. Wang, W. Chen, D. Fan, G. Zhang, J. Sheng, H. Peng, T. Zhang, and Y. Shao, “Experimental investigation of high-gain Ne-like Ge soft x-ray laser by double-massive-target coupling,” J. Opt. Soc. Am. B 9, 360–368 (1992).
[CrossRef]

London, R.

London, R. A.

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

R. P. Ratowsky and 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]

R. A. London, “Development of coherent x-ray lasers,” Phys. Fluids B 5, 2707–2713 (1993).
[CrossRef]

P. Amendt and R. A. London, “Optimization of single-stage x-ray laser coherence,” Phys. Rev. A 47, 4348–4363 (1993).
[CrossRef] [PubMed]

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

R. A. London, “Beam optics of exploding foil plasma x-ray lasers,” Phys. Fluids 31, 184–192 (1988).
[CrossRef]

MacGowan, B.

MacPhee, A.

R. Kodama, D. Neely, Y. Kato, H. Daido, K. Murai, G. Yuan, A. MacPhee, and C. L. S. Lewis, “Generation of small-divergence soft x-ray laser by plasma waveguiding with a curved target,” Phys. Rev. Lett. 73, 3215–3218 (1994).
[CrossRef] [PubMed]

MacPhee, A. G.

J. Zhang, A. G. MacPhee, J. Lin, E. Wolfrum, R. Smith, C. Danson, M. H. Key, C. L. S. Lewis, D. Neely, J. Nilsen, G. J. Pert, G. J. Tallents, and J. S. Wark, “A saturated x-ray laser beam at 7 nanometers,” Science 276, 1097–1100 (1997), and references therein.
[CrossRef]

Mao, C.

S. Wang, Y. Gu, G. Zhou, S. Yu, Y. Ni, Y. Xiong, S. Fu, C. Mao, Z. Zhou, B. Wang, Y. Zeng, G. Han, G. Huang, D. Fan, Z. Lin, and X. Deng, “Achievement of soft-x-ray laser with nearly saturated intensity and minimum divergence angle ~1 mrad,” Chin. J. Lasers B 3, 507–510 (1994), and references therein.

Matthews, D.

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

D. Matthews, M. Rosen, S. Brown, N. Ceglio, D. Eder, A. Hawryluk, C. Keane, R. London, B. MacGowan, S. Maxon, D. Nilson, J. Scofield, and J. Trebes, “X-ray laser research at the Lawrence Livermore National Laboratory Nova laser facility,” J. Opt. Soc. Am. B 4, 575–587 (1987).
[CrossRef]

Maxon, S.

Moreno, J. C.

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

J. Nilsen and J. C. Moreno, “Monochromatic lasing at 182A° in neonlike selenium,” Phys. Rev. Lett. 74, 3376–3379 (1995).
[CrossRef] [PubMed]

Mrowka, S.

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

Mukunda, N.

R. Simon, E. C. G. Sudarsdhan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Murai, K.

R. Kodama, D. Neely, Y. Kato, H. Daido, K. Murai, G. Yuan, A. MacPhee, and C. L. S. Lewis, “Generation of small-divergence soft x-ray laser by plasma waveguiding with a curved target,” Phys. Rev. Lett. 73, 3215–3218 (1994).
[CrossRef] [PubMed]

Nazarathy, M.

Neely, D.

J. Zhang, A. G. MacPhee, J. Lin, E. Wolfrum, R. Smith, C. Danson, M. H. Key, C. L. S. Lewis, D. Neely, J. Nilsen, G. J. Pert, G. J. Tallents, and J. S. Wark, “A saturated x-ray laser beam at 7 nanometers,” Science 276, 1097–1100 (1997), and references therein.
[CrossRef]

R. Kodama, D. Neely, Y. Kato, H. Daido, K. Murai, G. Yuan, A. MacPhee, and C. L. S. Lewis, “Generation of small-divergence soft x-ray laser by plasma waveguiding with a curved target,” Phys. Rev. Lett. 73, 3215–3218 (1994).
[CrossRef] [PubMed]

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Ni, Y.

S. Wang, Y. Gu, G. Zhou, S. Yu, Y. Ni, Y. Xiong, S. Fu, C. Mao, Z. Zhou, B. Wang, Y. Zeng, G. Han, G. Huang, D. Fan, Z. Lin, and X. Deng, “Achievement of soft-x-ray laser with nearly saturated intensity and minimum divergence angle ~1 mrad,” Chin. J. Lasers B 3, 507–510 (1994), and references therein.

S. Wang, Y. Gu, G. Zhou, S. Yu, S. Fu, Y. Ni, J. Wu, Z. Zhou, G. Han, Z. Tao, Z. Lin, S. Wang, W. Chen, D. Fan, G. Zhang, J. Sheng, H. Peng, T. Zhang, and Y. Shao, “Experimental investigation of high-gain Ne-like Ge soft x-ray laser by double-massive-target coupling,” J. Opt. Soc. Am. B 9, 360–368 (1992).
[CrossRef]

Nilsen, J.

J. Zhang, A. G. MacPhee, J. Lin, E. Wolfrum, R. Smith, C. Danson, M. H. Key, C. L. S. Lewis, D. Neely, J. Nilsen, G. J. Pert, G. J. Tallents, and J. S. Wark, “A saturated x-ray laser beam at 7 nanometers,” Science 276, 1097–1100 (1997), and references therein.
[CrossRef]

J. Nilsen and J. C. Moreno, “Monochromatic lasing at 182A° in neonlike selenium,” Phys. Rev. Lett. 74, 3376–3379 (1995).
[CrossRef] [PubMed]

Nilson, D.

Norreys, P.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

O’Neill, D.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Peng, H.

Pert, G. J.

J. Zhang, A. G. MacPhee, J. Lin, E. Wolfrum, R. Smith, C. Danson, M. H. Key, C. L. S. Lewis, D. Neely, J. Nilsen, G. J. Pert, G. J. Tallents, and J. S. Wark, “A saturated x-ray laser beam at 7 nanometers,” Science 276, 1097–1100 (1997), and references therein.
[CrossRef]

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Ramsden, S. A.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Ratowsky, R. P.

R. P. Ratowsky and 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]

Raucourt, J. P.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Ress, D.

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

Rosen, M.

Rosen, M. D.

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

Scofield, J.

Shamir, J.

Shao, Y.

Sheng, J.

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

Simon, R.

R. Simon, E. C. G. Sudarsdhan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Smith, R.

J. Zhang, A. G. MacPhee, J. Lin, E. Wolfrum, R. Smith, C. Danson, M. H. Key, C. L. S. Lewis, D. Neely, J. Nilsen, G. J. Pert, G. J. Tallents, and J. S. Wark, “A saturated x-ray laser beam at 7 nanometers,” Science 276, 1097–1100 (1997), and references therein.
[CrossRef]

Starikov, A.

Stearns, D. G.

Strauss, M.

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

Sudarsdhan, E. C. G.

R. Simon, E. C. G. Sudarsdhan, and N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[CrossRef]

Tallents, G. J.

J. Zhang, A. G. MacPhee, J. Lin, E. Wolfrum, R. Smith, C. Danson, M. H. Key, C. L. S. Lewis, D. Neely, J. Nilsen, G. J. Pert, G. J. Tallents, and J. S. Wark, “A saturated x-ray laser beam at 7 nanometers,” Science 276, 1097–1100 (1997), and references therein.
[CrossRef]

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Tao, Z.

Trebes, J.

Trebes, J. E.

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

N. M. Ceglio, D. G. Stearns, D. P. Gaines, A. M. Hawryluk, and J. E. Trebes, “Mutipass amplification of soft x-rays in a laser cavity,” Opt. Lett. 13, 108–110 (1988).
[CrossRef]

Turunen, J.

Uhomoibhi, J.

A. Carillon, H. Z. Chen, P. Dhez, L. Dwivedi, J. Jacoby, P. Jaegle, G. Jamelot, J. Zhang, M. H. Key, A. Kidd, A. Klisnick, R. Kodama, J. Krishnan, C. L. S. Lewis, D. Neely, P. Norreys, D. O’Neill, G. J. Pert, S. A. Ramsden, J. P. Raucourt, G. J. Tallents, and J. Uhomoibhi, “Saturated and near-diffraction-limited operation of an XUV laser at 23.6 nm,” Phys. Rev. Lett. 68, 2917–2920 (1992).
[CrossRef] [PubMed]

Wan, A. S.

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

Wang, B.

S. Wang, Y. Gu, G. Zhou, S. Yu, Y. Ni, Y. Xiong, S. Fu, C. Mao, Z. Zhou, B. Wang, Y. Zeng, G. Han, G. Huang, D. Fan, Z. Lin, and X. Deng, “Achievement of soft-x-ray laser with nearly saturated intensity and minimum divergence angle ~1 mrad,” Chin. J. Lasers B 3, 507–510 (1994), and references therein.

Wang, S.

Wark, J. S.

J. Zhang, A. G. MacPhee, J. Lin, E. Wolfrum, R. Smith, C. Danson, M. H. Key, C. L. S. Lewis, D. Neely, J. Nilsen, G. J. Pert, G. J. Tallents, and J. S. Wark, “A saturated x-ray laser beam at 7 nanometers,” Science 276, 1097–1100 (1997), and references therein.
[CrossRef]

Weber, F.

L. B. Da Silva, T. W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo, S. Libby, R. A. London, D. Matthews, S. Mrowka, J. C. Moreno, D. Ress, J. E. Trebes, A. S. Wan, and F. Weber, “Electron density measurements of high density plasma using soft x-ray laser interferometry,” Phys. Rev. Lett. 74, 3991–3994 (1995).
[CrossRef] [PubMed]

Wolf, E.

Wolfrum, E.

J. Zhang, A. G. MacPhee, J. Lin, E. Wolfrum, R. Smith, C. Danson, M. H. Key, C. L. S. Lewis, D. Neely, J. Nilsen, G. J. Pert, G. J. Tallents, and J. S. Wark, “A saturated x-ray laser beam at 7 nanometers,” Science 276, 1097–1100 (1997), and references therein.
[CrossRef]

Wu, J.

Xiong, Y.

S. Wang, Y. Gu, G. Zhou, S. Yu, Y. Ni, Y. Xiong, S. Fu, C. Mao, Z. Zhou, B. Wang, Y. Zeng, G. Han, G. Huang, D. Fan, Z. Lin, and X. Deng, “Achievement of soft-x-ray laser with nearly saturated intensity and minimum divergence angle ~1 mrad,” Chin. J. Lasers B 3, 507–510 (1994), and references therein.

Yu, S.

S. Wang, Y. Gu, G. Zhou, S. Yu, Y. Ni, Y. Xiong, S. Fu, C. Mao, Z. Zhou, B. Wang, Y. Zeng, G. Han, G. Huang, D. Fan, Z. Lin, and X. Deng, “Achievement of soft-x-ray laser with nearly saturated intensity and minimum divergence angle ~1 mrad,” Chin. J. Lasers B 3, 507–510 (1994), and references therein.

S. Wang, Y. Gu, G. Zhou, S. Yu, S. Fu, Y. Ni, J. Wu, Z. Zhou, G. Han, Z. Tao, Z. Lin, S. Wang, W. Chen, D. Fan, G. Zhang, J. Sheng, H. Peng, T. Zhang, and Y. Shao, “Experimental investigation of high-gain Ne-like Ge soft x-ray laser by double-massive-target coupling,” J. Opt. Soc. Am. B 9, 360–368 (1992).
[CrossRef]

Yuan, G.

R. Kodama, D. Neely, Y. Kato, H. Daido, K. Murai, G. Yuan, A. MacPhee, and C. L. S. Lewis, “Generation of small-divergence soft x-ray laser by plasma waveguiding with a curved target,” Phys. Rev. Lett. 73, 3215–3218 (1994).
[CrossRef] [PubMed]

Zahavi, O.

Zeng, Y.

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Figures (8)

Fig. 1
Fig. 1

Illustration of a Gaussian beam through a complex lens-like medium with a curved z axis.

Fig. 2
Fig. 2

Deflection angle versus the length of a slab plasma with Lx=100 µm (curves 1 and 2) and 70 µm (curves 3 and 4). Both cases have Δ=1 µm. Curves 1 and 3 refer to the GSM, and curves 2 and 4 refer to the diffraction integral.

Fig. 3
Fig. 3

Beam radius versus plasma length for the exploding-foil target calculated by the GSM (curve 1) and the diffraction integral (curve 2) for a curved target (Lx=100 µm) calculated by the GSM (curve 3) and the diffraction integral (curve 4). Also, the fundamental-mode radius for exploding foils and curved targets are displayed as curves 5 and 6, respectively.

Fig. 4
Fig. 4

Coherence length at the exit plane versus plasma length. The notes are the same as those for Fig. 3.

Fig. 5
Fig. 5

Far-field divergence angle versus plasma length. The notes are the same as those for Fig. 3.

Fig. 6
Fig. 6

Schematic of double-pass amplification x-ray laser geometry: (a) exploding-foil target and (b) double-slab target.

Fig. 7
Fig. 7

Far-field intensity distribution: (a) single-exploding-foil target and (b) double-slab-target coupling (Lx=100 µm). In both cases the DPA is calculated by the diffraction integral (curve 1) and the wave equation (curve 2); the SPA is calculated by the diffraction integral (curve 3) and the wave equation (curve 4).

Fig. 8
Fig. 8

Same as Fig. 7, but for the spatial degree of coherence at the exit plane.

Equations (59)

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2E(x, z)x2-2ik0E(x, z)z+k2(x, z)-k02[1-2C(z)x]-idk0dzE(x, z)=0,
k2(x, z)=k02(z)-k0(z)[k1(z)x+k2(z)x2].
E(x, z)=E0 exp-iQ(z)2x2+S(z)x+P(z),
Q2+k0(z)dQdz+k0(z)k2(z)=0,
QS+k0(z)dSdz+k0(z)k1(z)2-k02(z)C(z)=0,
dPdz=-iQ2k0(z)-S22k0(z)-i2k0(z)dk0(z)dz.
G(z)=0zk0(z)C(z)-k1(z)2A(z)dz,
H(z)=0zk0(z)C(z)-k1(z)2B(z)dz.
E(x, z)=iλB exp(-iφ) Ein(ζ, z0)×exp-iπλB(Aζ2-2ζx+Dx2+eζ+fx),
e=λπ(BG-AH),f=λπH,
φ=-λH4πBG-AHB+λ4πz0zGdHdz-HdGdzdz+k0(z-z0).
Γ(x1, x2; z)=E*(x1, z)E(x2, z),
Γ(x1, x2; z)=exp[2 Im(φ)]λ|B| dζ1ζ2Γin(ζ1, ζ2; z0)×expiπλ(A*ζ12-2ζ1x1+D*x12+e*ζ1+f*x1)B*-(Aζ22-2ζ2x2+Dx22+eζ2+fx2)B,
x=x1+x22,s=x1-x2,ζ=ζ1+ζ22,
η=ζ1-ζ2.
Γin(ζ, η; z0)=Iin(ζ)δ(η),
Γ(x, s; z)=exp[2 Im(φ)]λ|B|exp2πλx2+s24ImDB+x ImfB+i2πλxs ReDB+s2RefB×-+ Iin(ζ, x)exp-i2πλζs Re1Bdζ,
Iin(ζ, x)=Iin(ζ)exp2πλζ2 ImAB-2ζx Im1B+ζ ImeB.
I(x)=Γ(x, 0),
μ(x1, 0)=Γ(x1/2, x1)I(0)I(x1).
W(x, θ)=1λ-+ exp(-i2πθs/λ)Γ(x, s)ds,
U(θ)=-+ W(x, θ)dx.
W2=4-+ x2I(x)dx-+ I(x)dx,
Lc2=-+ s2|μ(s)|ds-+|μ(s)|ds,
Θ2=16-+ θ2U(θ)dθ-+ U(θ)dθ.
Γin(ζ1, ζ2; z0)=A0 exp-ζ12+ζ224σI02-(ζ1-ζ2)22σc02,
Γin(ζ1, ζ2; z0)=nαnϕn*(ζ1, z0)ϕn(ζ2, z0).
αn=α0pn,
α0=8πA0σI0M2+1,p=M2-1M2+1,M2=1+4σI02σc02.
Γ(x1, x2; z)=nαnϕn*(x1, z)ϕn(x2, z).
ϕn(x, z)=2π2nn!ω0u vun/2Hn2x-e2ω0uv×exp-iπλx2τu+xBeu+f×exp(-iφ),
ω0=2σI0M,1q0=-iλπω02,
u=A+B/q0,v=A+B/q0*,
τ=C+D/q0,andφ=φ-πe2/4λBu.
n Hn(x1)Hn(x2)tnn!2n=exp2x1x2t-(x12+x22)t21-t21-t2,
Γ(x1, x2; z)=A(z)exp-x12+x224σI2(z)-(x1-x2)22σc2(z)+iπλR(z)(x12-x22)×exp[ΔI(x1+x2)+iΔc(x1-x2)+Δconst],
A(z)=2α0 exp[2 Im(φ)]πω02(|u|2-p2|v|2),
σI2(z)=(|u|2-p2|v|2)|u|2ω0242p2 Re(uv)+πω02λ(|u|2-p2|v|2)Im(τ*u)-2p|u|2,
σc2(z)=(|u|2-p2|v|2)ω024p,
R(z)=(|u|2-p2|v|2)|u|2ω02-2λπp2 Im(uv)+ω02(|u|2-p2|v|2)Re(τ*u),
ΔI=2ω02(|u|2-p2|v|2)Re(-pe*+p2ev*/u)+2πλIm(e/u+f)2B,
Δc=-2ω02(|u|2-p2|v|2)Im(-pe*+p2ev*/u)+2πλRe(e/u+f)2B,
Δconst=p|e|2-p2 Re(e2v*/u)ω02(|u|2-p2|v|2).
W(z)=2σI(z),
Lc(z)=σc(z),
Θ=λπ21σI2+4σc2+16σI2R2,
θdef=2σI2ΔIR+λΔc2π.
ne(x)=a1-[x-(Lx-b)]2b2,
g(x)=g01-x2Lx2(-LxxLx).
k=2πλn(x)+ig(x)2=k0-k12x-k22x2,
k0=2πλ1-a2ncr+a(Lx-b)22ncrb2+ig02,
k1=4πa(Lx-b)λncrb2,k2=-2πaλncrb2+ig0Lx2.
cos(γz)γ-1 sin(γz)0-γ sin(γz)cos(γz)00z[k0C(z)-k1/2]cos(γz)dzγ-10z[k0C(z)-k1/2]sin(γz)dz1.
Γ(x, s)=1Bλ-+ rectζLx×expi2πsλBDx-ζ+f2dζ.
W(x, θ)=rectDx-Bθ+f/2Lx.
Ws=Lx/|D|,
Ldis=|B/D|.
θdiv=Ws/Ldis+θD(|D|<1)Lx/Ldis+θD(|D|1),
θdef=f/2B.

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