Abstract

The problem of radiation propagating along the axial direction of an elongated laser plasma is treated analytically and numerically. With the paraxial-ray equation, simple expressions for ray trajectories in such plasmas are derived that allow one to calculate beam trajectories in an expanding plasma. Linear, quadratic, exponential, and Gaussian electron-density distributions are considered for plane and curved targets. Several of the trajectories are exact solutions of the paraxial-ray equation; others are useful approximations that are compared with numerical solutions. This theory finds its main application in the field of x-ray lasers, in which it may be helpful to design targets for optimum beam propagation and to understand the effect of beam deflection on the effective gain and the beam divergence. Furthermore, with this theory, the maximum electron density at which there is gain can be estimated from the far-field deflection angle of an x-ray laser beam.

© 1997 Optical Society of America

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    [CrossRef]
  2. M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
    [CrossRef] [PubMed]
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1996 (2)

Y. Li, G. Pretzler, P. Lu, and E. E. Fill, Phys. Rev. A 53, R652–R654 (1996).
[CrossRef]

P. Lu, Y. Li, and E. E. Fill, Phys. Rev. A 54, 5193–5200 (1996).
[CrossRef] [PubMed]

1995 (5)

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

H. Milchberg, C. G. Durfee III, and J. Lynch, J. Opt. Soc. Am. B 12, 731–737 (1995).
[CrossRef]

H. Daido, Y. Kato, K. Murai, S. Ninomiya, R. Kodama, G. Yuan, Y. Oshikane, M. Takagi, H. Takabe, and F. Koike, Phys. Rev. Lett. 75, 1074–1077 (1995).
[CrossRef] [PubMed]

J. Nilsen and J. C. Moreno, Opt. Lett. 20, 1386–1388 (1995).
[CrossRef] [PubMed]

E. E. Fill, Y. Li, D. Schlögl, J. Steingruber, and J. Nilsen, Opt. Lett. 20, 374–376 (1995).
[CrossRef]

1994 (2)

B. Rus, A. Carillon, B. Gauthé, P. Goedtkindt, P. Jaeglé, G. Jamelot, A. Klisnick, A. Sureau, and P. Zeitoun, J. Opt. Soc. Am. B 11, 564–573 (1994).
[CrossRef]

H. Daido, Y. Kato, R. Kodama, K. Murai, G. Yuan, M. Takagi, H. Takabe, S. Nakai, C. H. Nam, I. W. Choi, D. Neely, and A. MacPhee, AIP Conf. Proc. 332, 1 (1994).
[CrossRef]

1993 (2)

J. Nilsen, B. J. MacGowan, L. B. Da Silva, and J. C. Moreno, Phys. Rev. A 48, 4682–4685 (1993).
[CrossRef] [PubMed]

J. Nilsen, J. C. Moreno, B. J. MacGowan, and J. A. Koch, Appl. Phys. B 57, 309–311 (1993).
[CrossRef]

1988 (1)

R. A. London, Phys. Fluids 31, 184–192 (1988).
[CrossRef]

1987 (1)

1986 (2)

J. G. Lunney, Appl. Phys. Lett. 48, 891–893 (1986).
[CrossRef]

R. A. London and M. D. Rosen, Phys. Fluids 29, 3813–3822 (1986).
[CrossRef]

1985 (1)

M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
[CrossRef] [PubMed]

1984 (1)

V. A. Chirkov, Sov. J. Quantum Electron. 14, 1497–1505 (1984).
[CrossRef]

1982 (1)

W. M. Manheimer and D. G. Colombant, Phys. Fluids 25, 1644–1652 (1982).
[CrossRef]

1965 (1)

P. K. Tien, J. P. Gordon, and J. R. Whinnery, Proc. IEEE 53, 129–136 (1965).
[CrossRef]

Brown, S.

Busch, Gar. E.

M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
[CrossRef] [PubMed]

Campbell, E. M.

M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
[CrossRef] [PubMed]

Carillon, A.

Ceglio, N.

Charatis, G.

M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
[CrossRef] [PubMed]

Chilla, J. L. A.

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

Chirkov, V. A.

V. A. Chirkov, Sov. J. Quantum Electron. 14, 1497–1505 (1984).
[CrossRef]

Choi, I. W.

H. Daido, Y. Kato, R. Kodama, K. Murai, G. Yuan, M. Takagi, H. Takabe, S. Nakai, C. H. Nam, I. W. Choi, D. Neely, and A. MacPhee, AIP Conf. Proc. 332, 1 (1994).
[CrossRef]

Clark, D. P.

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

Colombant, D. G.

W. M. Manheimer and D. G. Colombant, Phys. Fluids 25, 1644–1652 (1982).
[CrossRef]

Da Silva, L. B.

J. Nilsen, B. J. MacGowan, L. B. Da Silva, and J. C. Moreno, Phys. Rev. A 48, 4682–4685 (1993).
[CrossRef] [PubMed]

Daido, H.

H. Daido, Y. Kato, K. Murai, S. Ninomiya, R. Kodama, G. Yuan, Y. Oshikane, M. Takagi, H. Takabe, and F. Koike, Phys. Rev. Lett. 75, 1074–1077 (1995).
[CrossRef] [PubMed]

H. Daido, Y. Kato, R. Kodama, K. Murai, G. Yuan, M. Takagi, H. Takabe, S. Nakai, C. H. Nam, I. W. Choi, D. Neely, and A. MacPhee, AIP Conf. Proc. 332, 1 (1994).
[CrossRef]

Durfee III, C. G.

Eder, D.

Fill, E. E.

Y. Li, G. Pretzler, P. Lu, and E. E. Fill, Phys. Rev. A 53, R652–R654 (1996).
[CrossRef]

P. Lu, Y. Li, and E. E. Fill, Phys. Rev. A 54, 5193–5200 (1996).
[CrossRef] [PubMed]

E. E. Fill, Y. Li, D. Schlögl, J. Steingruber, and J. Nilsen, Opt. Lett. 20, 374–376 (1995).
[CrossRef]

Gauthé, B.

Goedtkindt, P.

Gordon, J. P.

P. K. Tien, J. P. Gordon, and J. R. Whinnery, Proc. IEEE 53, 129–136 (1965).
[CrossRef]

Hagelstein, P. L.

M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
[CrossRef] [PubMed]

Hawryluk, A.

Hazi, A. U.

M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
[CrossRef] [PubMed]

Jaeglé, P.

Jamelot, G.

Kato, Y.

H. Daido, Y. Kato, K. Murai, S. Ninomiya, R. Kodama, G. Yuan, Y. Oshikane, M. Takagi, H. Takabe, and F. Koike, Phys. Rev. Lett. 75, 1074–1077 (1995).
[CrossRef] [PubMed]

H. Daido, Y. Kato, R. Kodama, K. Murai, G. Yuan, M. Takagi, H. Takabe, S. Nakai, C. H. Nam, I. W. Choi, D. Neely, and A. MacPhee, AIP Conf. Proc. 332, 1 (1994).
[CrossRef]

Keane, C.

Klisnick, A.

Koch, J. A.

J. Nilsen, J. C. Moreno, B. J. MacGowan, and J. A. Koch, Appl. Phys. B 57, 309–311 (1993).
[CrossRef]

Kodama, R.

H. Daido, Y. Kato, K. Murai, S. Ninomiya, R. Kodama, G. Yuan, Y. Oshikane, M. Takagi, H. Takabe, and F. Koike, Phys. Rev. Lett. 75, 1074–1077 (1995).
[CrossRef] [PubMed]

H. Daido, Y. Kato, R. Kodama, K. Murai, G. Yuan, M. Takagi, H. Takabe, S. Nakai, C. H. Nam, I. W. Choi, D. Neely, and A. MacPhee, AIP Conf. Proc. 332, 1 (1994).
[CrossRef]

Koike, F.

H. Daido, Y. Kato, K. Murai, S. Ninomiya, R. Kodama, G. Yuan, Y. Oshikane, M. Takagi, H. Takabe, and F. Koike, Phys. Rev. Lett. 75, 1074–1077 (1995).
[CrossRef] [PubMed]

Lee, R. W.

M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
[CrossRef] [PubMed]

Li, Y.

Y. Li, G. Pretzler, P. Lu, and E. E. Fill, Phys. Rev. A 53, R652–R654 (1996).
[CrossRef]

P. Lu, Y. Li, and E. E. Fill, Phys. Rev. A 54, 5193–5200 (1996).
[CrossRef] [PubMed]

E. E. Fill, Y. Li, D. Schlögl, J. Steingruber, and J. Nilsen, Opt. Lett. 20, 374–376 (1995).
[CrossRef]

London, R.

London, R. A.

R. A. London, Phys. Fluids 31, 184–192 (1988).
[CrossRef]

R. A. London and M. D. Rosen, Phys. Fluids 29, 3813–3822 (1986).
[CrossRef]

Lu, P.

Y. Li, G. Pretzler, P. Lu, and E. E. Fill, Phys. Rev. A 53, R652–R654 (1996).
[CrossRef]

P. Lu, Y. Li, and E. E. Fill, Phys. Rev. A 54, 5193–5200 (1996).
[CrossRef] [PubMed]

Lunney, J. G.

J. G. Lunney, Appl. Phys. Lett. 48, 891–893 (1986).
[CrossRef]

Lynch, J.

MacGowan, B.

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, J. Opt. Soc. Am. B 4, 575–587 (1987).
[CrossRef]

M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
[CrossRef] [PubMed]

MacGowan, B. J.

J. Nilsen, B. J. MacGowan, L. B. Da Silva, and J. C. Moreno, Phys. Rev. A 48, 4682–4685 (1993).
[CrossRef] [PubMed]

J. Nilsen, J. C. Moreno, B. J. MacGowan, and J. A. Koch, Appl. Phys. B 57, 309–311 (1993).
[CrossRef]

MacPhee, A.

H. Daido, Y. Kato, R. Kodama, K. Murai, G. Yuan, M. Takagi, H. Takabe, S. Nakai, C. H. Nam, I. W. Choi, D. Neely, and A. MacPhee, AIP Conf. Proc. 332, 1 (1994).
[CrossRef]

Manheimer, W. M.

W. M. Manheimer and D. G. Colombant, Phys. Fluids 25, 1644–1652 (1982).
[CrossRef]

Marconi, M. C.

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

Matthews, D.

Matthews, D. L.

M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
[CrossRef] [PubMed]

Maxon, S.

Milchberg, H.

Moreno, J. C.

J. Nilsen and J. C. Moreno, Opt. Lett. 20, 1386–1388 (1995).
[CrossRef] [PubMed]

J. Nilsen, J. C. Moreno, B. J. MacGowan, and J. A. Koch, Appl. Phys. B 57, 309–311 (1993).
[CrossRef]

J. Nilsen, B. J. MacGowan, L. B. Da Silva, and J. C. Moreno, Phys. Rev. A 48, 4682–4685 (1993).
[CrossRef] [PubMed]

Murai, K.

H. Daido, Y. Kato, K. Murai, S. Ninomiya, R. Kodama, G. Yuan, Y. Oshikane, M. Takagi, H. Takabe, and F. Koike, Phys. Rev. Lett. 75, 1074–1077 (1995).
[CrossRef] [PubMed]

H. Daido, Y. Kato, R. Kodama, K. Murai, G. Yuan, M. Takagi, H. Takabe, S. Nakai, C. H. Nam, I. W. Choi, D. Neely, and A. MacPhee, AIP Conf. Proc. 332, 1 (1994).
[CrossRef]

Nakai, S.

H. Daido, Y. Kato, R. Kodama, K. Murai, G. Yuan, M. Takagi, H. Takabe, S. Nakai, C. H. Nam, I. W. Choi, D. Neely, and A. MacPhee, AIP Conf. Proc. 332, 1 (1994).
[CrossRef]

Nam, C. H.

H. Daido, Y. Kato, R. Kodama, K. Murai, G. Yuan, M. Takagi, H. Takabe, S. Nakai, C. H. Nam, I. W. Choi, D. Neely, and A. MacPhee, AIP Conf. Proc. 332, 1 (1994).
[CrossRef]

Neely, D.

H. Daido, Y. Kato, R. Kodama, K. Murai, G. Yuan, M. Takagi, H. Takabe, S. Nakai, C. H. Nam, I. W. Choi, D. Neely, and A. MacPhee, AIP Conf. Proc. 332, 1 (1994).
[CrossRef]

Nilsen, J.

E. E. Fill, Y. Li, D. Schlögl, J. Steingruber, and J. Nilsen, Opt. Lett. 20, 374–376 (1995).
[CrossRef]

J. Nilsen and J. C. Moreno, Opt. Lett. 20, 1386–1388 (1995).
[CrossRef] [PubMed]

J. Nilsen, J. C. Moreno, B. J. MacGowan, and J. A. Koch, Appl. Phys. B 57, 309–311 (1993).
[CrossRef]

J. Nilsen, B. J. MacGowan, L. B. Da Silva, and J. C. Moreno, Phys. Rev. A 48, 4682–4685 (1993).
[CrossRef] [PubMed]

Nilson, D.

Ninomiya, S.

H. Daido, Y. Kato, K. Murai, S. Ninomiya, R. Kodama, G. Yuan, Y. Oshikane, M. Takagi, H. Takabe, and F. Koike, Phys. Rev. Lett. 75, 1074–1077 (1995).
[CrossRef] [PubMed]

Oshikane, Y.

H. Daido, Y. Kato, K. Murai, S. Ninomiya, R. Kodama, G. Yuan, Y. Oshikane, M. Takagi, H. Takabe, and F. Koike, Phys. Rev. Lett. 75, 1074–1077 (1995).
[CrossRef] [PubMed]

Pretzler, G.

Y. Li, G. Pretzler, P. Lu, and E. E. Fill, Phys. Rev. A 53, R652–R654 (1996).
[CrossRef]

Rocca, J. J.

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

Rockett, P. D.

M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
[CrossRef] [PubMed]

Rosen, M.

Rosen, M. D.

R. A. London and M. D. Rosen, Phys. Fluids 29, 3813–3822 (1986).
[CrossRef]

M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
[CrossRef] [PubMed]

Rus, B.

Schlögl, D.

Scofield, J.

Shepard, C. L.

M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
[CrossRef] [PubMed]

Shlyaptsev, V. N.

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

Steingruber, J.

Sureau, A.

Takabe, H.

H. Daido, Y. Kato, K. Murai, S. Ninomiya, R. Kodama, G. Yuan, Y. Oshikane, M. Takagi, H. Takabe, and F. Koike, Phys. Rev. Lett. 75, 1074–1077 (1995).
[CrossRef] [PubMed]

H. Daido, Y. Kato, R. Kodama, K. Murai, G. Yuan, M. Takagi, H. Takabe, S. Nakai, C. H. Nam, I. W. Choi, D. Neely, and A. MacPhee, AIP Conf. Proc. 332, 1 (1994).
[CrossRef]

Takagi, M.

H. Daido, Y. Kato, K. Murai, S. Ninomiya, R. Kodama, G. Yuan, Y. Oshikane, M. Takagi, H. Takabe, and F. Koike, Phys. Rev. Lett. 75, 1074–1077 (1995).
[CrossRef] [PubMed]

H. Daido, Y. Kato, R. Kodama, K. Murai, G. Yuan, M. Takagi, H. Takabe, S. Nakai, C. H. Nam, I. W. Choi, D. Neely, and A. MacPhee, AIP Conf. Proc. 332, 1 (1994).
[CrossRef]

Tien, P. K.

P. K. Tien, J. P. Gordon, and J. R. Whinnery, Proc. IEEE 53, 129–136 (1965).
[CrossRef]

Tomasel, F. G.

J. J. Rocca, M. C. Marconi, J. L. A. Chilla, D. P. Clark, F. G. Tomasel, and V. N. Shlyaptsev, IEEE J. Sel. Topics Quantum Electron. 1, 945–948 (1995).
[CrossRef]

Trebes, J.

Turner, R. E.

M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
[CrossRef] [PubMed]

Whinnery, J. R.

P. K. Tien, J. P. Gordon, and J. R. Whinnery, Proc. IEEE 53, 129–136 (1965).
[CrossRef]

Whitten, B. L.

M. D. Rosen, P. L. Hagelstein, D. L. Matthews, E. M. Campbell, A. U. Hazi, B. L. Whitten, B. MacGowan, R. E. Turner, R. W. Lee, G. Charatis, Gar. E. Busch, C. L. Shepard, and P. D. Rockett, Phys. Rev. Lett. 54, 106–109 (1985).
[CrossRef] [PubMed]

Yuan, G.

H. Daido, Y. Kato, K. Murai, S. Ninomiya, R. Kodama, G. Yuan, Y. Oshikane, M. Takagi, H. Takabe, and F. Koike, Phys. Rev. Lett. 75, 1074–1077 (1995).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Ray trajectories for a flat target with exponential and Gaussian electron-density distributions. The electron-density parameters for both distributions are N0/Nc=10-3 and x0 =50 µm. The rays shown start parallel to the target surface with initial distances of r0=50 µm and r0=100 µm. The ray trajectories for the exponential electron-density profile are exact solutions of the paraxial-ray equation. For the Gaussian density profile, both the analytical approximations and numerical solutions of the ray equation are plotted. Note that the far-field angle for exponential distribution for a ray starting at r0 =100 µm is different from the one for Gaussian distribution, since the electron densities at that distance are different.

Fig. 2
Fig. 2

Ray trajectories for a curved target and an exponential electron-density profile. The electron-density parameters are N0/Nc=10-3 and x0=50 µm (same as in Fig. 1), and the radius of curvature is R=50 cm, resulting in an equilibrium distance req=80.5 µm. Analytical and numerical ray paths are shown for angles of φeq=1 mrad and φeq=5 mrad. For 1 mrad the analytical approximation is almost indistinguishable from the exact numerical ray path. For 5 mrad the analytical solution is still useful.

Fig. 3
Fig. 3

Ray trajectories for a curved target and a Gaussian electron-density profile with N0/Nc=10-3 and x0=50 µm. The radius of curvature is 50 cm. The resulting equilibrium distance is 84 µm. Analytical and numerical trajectories for φeq =1 mrad and φeq=5 mrad are compared. Note that the spatial period of oscillation is smaller than for the exponential case, since req>x0, resulting in q=1.66 [see Eq. (25)].

Fig. 4
Fig. 4

Electron density at closest approach to the target for neonlike x-ray lasers of various materials, calculated from the far-field deflection angle (triangles). The error bars reflect the uncertainty in target alignment of ±1 mrad plus 10% of the beam divergence. For details of the experiment, see text. The drawn curve represents the maximum electron density for gain as derived by Elton (Ref. 4, p. 106).

Equations (27)

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ddsndrds=n,
n=(1-N/2Nc),
d2rdz2=nx,
dr/dz=±[2n(r)+C]1/2,
dr/dz=[2n(r)-2n(r0)]1/2.
dr/dz=[N(r0)/Nc]1/2[1-N(r)/N(r0)]1/2.
φ=[N(r0)/Nc]1/2,
z=[Nc/N(r0)]1/2r0rdx[1-N(x)/N(r0)]1/2.
r=r0+(N0/Nc)z2/(4x0),
r=r0 cosh[(z/x0)(N0/Nc)1/2].
r=r0 sin[(N0/Nc)1/2z/x0],
r=r0+x0 ln(cosh2{(z/2x0)[N(r0)/Nc]1/2}).
zf=2x0[Nc/N(r0)]1/2 ln[f1/2+(f-1)].
r=r0{1+[N(r0)/Nc]z2/2x02}.
r=r0{1+[N(r0)/Nc]z2/2x02}forzz1,
r=r0+[N(r0)/Nc]1/2(z-z1)forzz1,
d2rdz2=nx-1R.
dr/dz=[2n(r)-2r/R+C]1/2,
r=r0+[N0/(4Ncx0)-1/(2R)]z2.
req=x0 lnN02NcRx0,
dr/dz=[2n(r)-2n(req)-(2r-2req)/R+φeq2]1/2.
z=reqrdx{[Neq-N(x)]/Nc+(2req-2x)/R+φeq2}1/2,
r=req+φeq(Rx0)1/2 sin[z/(Rx0)1/2].
req=x0lnN0Ncreqx02R1/2,
req(0)=x0;req(i)=x0lnN0Ncreq(i-1)x02R1/2.
r=req+(φeq/q)(Rx0)1/2 sin[qz/(Rx0)1/2],
q=(2req/x0-x0/req)1/2.

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