Abstract

Propagation of x rays generated by a small-diameter incoherent source through the capillary waveguide that satisfies the multimode condition is studied with the Fresnel–Kirchhoff diffraction theory. The strong influence of diffraction on the propagation is demonstrated. The diffraction phenomenon is manifested by the appearance of diffraction fringes in both the guide channel and the far-field zone of the capillary output. Experimental data that confirm such behavior of the x-ray radiation is also presented. The results confirm the interference effects recently observed in some experiments on the grazing reflections of x rays in single- and multiple-capillary optics.

© 2000 Optical Society of America

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    [CrossRef]
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    [CrossRef]

1999 (1)

T. A. Bobrova, L. Ognev, “On the ‘supercollimation’ of x-ray beams in rough interfacial channels,” JETP Lett. 69, 734–738 (1999).
[CrossRef]

1998 (4)

Yu. M. Alexandrov, S. B. Dabagov, M. A. Kumakhov, V. A. Murashova, D. A. Fedin, R. V. Fedorchuk, M. N. Yakimenko, “Peculiarities of photon transmission through capillary systems,” Nucl. Instrum. Methods Phys. Res. B 134, 174–180 (1998).
[CrossRef]

S. V. Kukhlevsky, L. Kozma, “Guiding of light by short-length multimode waveguides. I,” Nuovo Cimento D 20, 783–789 (1998).
[CrossRef]

S. V. Kukhlevsky, L. Kozma, “Modal and focusing properties of the plasma based waveguides for x-ray beams,” Contrib. Plasma Phys. 38, 583–597 (1998).
[CrossRef]

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

1997 (2)

L. I. Ognev, “On the structure of x-ray glass capillary lens focal spots,” Nucl. Instrum. Methods Phys. Res. B 124, 624–626 (1997).
[CrossRef]

V. L. Kantsyrev, R. Bruch, R. Phaneuf, N. G. Publicover, “New concepts for x-ray, soft x-ray, and EUV optical instrumentation including applications in spectroscopy, plasma diagnostics, and biomedical microscopy: a status report,” J. X-Ray Sci. Technol. 7, 139–158 (1997).
[CrossRef]

1996 (2)

S. V. Kukhlevsky, L. Kozma, K. Negrea, “Pulsed-mode analysis of soft-x-ray radiation passing through capillary waveguides,” J. Mod. Opt. 43, 2595–2604 (1996).
[CrossRef]

S. B. Dabagov, R. V. Fedorchuk, V. A. Murashova, S. V. Nikitina, M. N. Yakimenko, “Interference phenomenon under focusing of synchrotron radiation by a Kumakhov lens,” Nucl. Instrum. Methods Phys. Res. B 108, 213–218 (1996).
[CrossRef]

1995 (6)

L. Vincze, K. Janssens, F. Adams, A. Rindby, “Detailed ray-tracing code for capillary optics,” X-Ray Spectrom. 24, 27–37 (1995).
[CrossRef]

S. B. Dabagov, S. V. Nikitina, M. A. Kumakhov, N. S. Ibraimov, G. A. Vartaniants, A. N. Nikitin, L. Spielberger, “Focusing of x rays by capillary systems,” Nucl. Instrum. Methods Phys. Res. B 103, 99–105 (1995).
[CrossRef]

Yu. M. Aleksandrov, M. N. Yakimenko, “Investigations on the S-60 SR source of the Lebedev Physical Institute,” Nucl. Instrum. Methods Phys. Res. A 359, 12–14 (1995).
[CrossRef]

S. B. Dabagov, M. A. Kumakhov, S. V. Nikitina, “On the interference of x rays in multiple reflection optics,” Phys. Lett. A 203, 279–282 (1995).
[CrossRef]

S. P. Pogossian, H. Le Gall, “Neutron and x-ray propagation laws in thin film waveguides,” Opt. Commun. 114, 235–241 (1995).
[CrossRef]

J. B. Hastings, S. L. Hulbert, G. P. Williams, eds., Proceedings of the 5th International Conference on Synchrotron Radiation Instrumentation, Rev. Sci. Instrum. 66, 1271–2385 (1995).

1994 (1)

1993 (1)

B. L. Henke, E. M. Gullikson, J. C. Davis, “Low-energy x-ray interaction coefficients: photoabsorption, scattering and reflection, E = 30 - 3000 eV, Z = 1 - 92,” At. Data Nucl. Data Tables 53, 340–388 (1993).

1990 (1)

M. A. Kumakhov, F. A. Komarov, “Multiple reflections from surface x-ray optics,” Phys. Rep. 191, 289–350 (1990).
[CrossRef]

1988 (1)

1986 (1)

A. Rindby, “Application of fiber technique in the x-ray region,” Nucl. Instrum. Methods Phys. Res. A 249, 536–540 (1986).
[CrossRef]

1982 (1)

T. Hidaka, “Extremely low-loss hollow core waveguide for VUV light,” Opt. Commun. 44, 90–93 (1982).
[CrossRef]

1979 (1)

1976 (1)

D. Mosher, S. J. Stephanakis, “X-ray light pipes,” Appl. Phys. Lett. 29, 105–107 (1976).
[CrossRef]

1974 (2)

P. Mallozzi, H. M. Epstein, R. G. Jung, D. C. Applebaum, B. P. Fairand, W. T. Gallagher, R. L. Uecker, M. C. Muckerheide, “Laser-generated plasmas as a source of x rays for medical applications,” J. Appl. Phys. 45, 1891–1895 (1974).
[CrossRef]

E. E. Spiller, A. Segmuller, “Propagation of x rays in waveguides,” Appl. Phys. Lett. 24, 60–61 (1974).
[CrossRef]

1951 (1)

P. B. Hirsch, J. N. Kellar, “An x-ray micro-beam technique: I. Collimation,” Proc. Phys. Soc. London 64, 369–374 (1951).

1949 (1)

J. A. Lely, T. W. Van Rijssel, “X-ray collimator producing a beam of very small divergence and large intensity,” Acta Crystallogr. 2, 337–338 (1949).
[CrossRef]

1931 (1)

F. Jentzsch, E. Nähring, “Die Fortleitung von Licht- und Röntgen durch Röhren,” Z. Tech. Phys. 12, 185–189 (1931).

1923 (1)

A. H. Compton, “The total reflection of x rays,” Philos. Mag. 45, 1121–1131 (1923).
[CrossRef]

Adams, F.

L. Vincze, K. Janssens, F. Adams, A. Rindby, “Detailed ray-tracing code for capillary optics,” X-Ray Spectrom. 24, 27–37 (1995).
[CrossRef]

Albertano, P.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Aleksandrov, Yu. M.

Yu. M. Aleksandrov, M. N. Yakimenko, “Investigations on the S-60 SR source of the Lebedev Physical Institute,” Nucl. Instrum. Methods Phys. Res. A 359, 12–14 (1995).
[CrossRef]

Alexandrov, Yu. M.

Yu. M. Alexandrov, S. B. Dabagov, M. A. Kumakhov, V. A. Murashova, D. A. Fedin, R. V. Fedorchuk, M. N. Yakimenko, “Peculiarities of photon transmission through capillary systems,” Nucl. Instrum. Methods Phys. Res. B 134, 174–180 (1998).
[CrossRef]

Applebaum, D. C.

P. Mallozzi, H. M. Epstein, R. G. Jung, D. C. Applebaum, B. P. Fairand, W. T. Gallagher, R. L. Uecker, M. C. Muckerheide, “Laser-generated plasmas as a source of x rays for medical applications,” J. Appl. Phys. 45, 1891–1895 (1974).
[CrossRef]

Belli, M.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Bobrova, T. A.

T. A. Bobrova, L. Ognev, “On the ‘supercollimation’ of x-ray beams in rough interfacial channels,” JETP Lett. 69, 734–738 (1999).
[CrossRef]

Bollanti, S.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980)

Bruch, R.

V. L. Kantsyrev, R. Bruch, R. Phaneuf, N. G. Publicover, “New concepts for x-ray, soft x-ray, and EUV optical instrumentation including applications in spectroscopy, plasma diagnostics, and biomedical microscopy: a status report,” J. X-Ray Sci. Technol. 7, 139–158 (1997).
[CrossRef]

Chung, P. S.

Compton, A. H.

A. H. Compton, “The total reflection of x rays,” Philos. Mag. 45, 1121–1131 (1923).
[CrossRef]

Dabagov, S. B.

Yu. M. Alexandrov, S. B. Dabagov, M. A. Kumakhov, V. A. Murashova, D. A. Fedin, R. V. Fedorchuk, M. N. Yakimenko, “Peculiarities of photon transmission through capillary systems,” Nucl. Instrum. Methods Phys. Res. B 134, 174–180 (1998).
[CrossRef]

S. B. Dabagov, R. V. Fedorchuk, V. A. Murashova, S. V. Nikitina, M. N. Yakimenko, “Interference phenomenon under focusing of synchrotron radiation by a Kumakhov lens,” Nucl. Instrum. Methods Phys. Res. B 108, 213–218 (1996).
[CrossRef]

S. B. Dabagov, M. A. Kumakhov, S. V. Nikitina, “On the interference of x rays in multiple reflection optics,” Phys. Lett. A 203, 279–282 (1995).
[CrossRef]

S. B. Dabagov, S. V. Nikitina, M. A. Kumakhov, N. S. Ibraimov, G. A. Vartaniants, A. N. Nikitin, L. Spielberger, “Focusing of x rays by capillary systems,” Nucl. Instrum. Methods Phys. Res. B 103, 99–105 (1995).
[CrossRef]

Davis, J. C.

B. L. Henke, E. M. Gullikson, J. C. Davis, “Low-energy x-ray interaction coefficients: photoabsorption, scattering and reflection, E = 30 - 3000 eV, Z = 1 - 92,” At. Data Nucl. Data Tables 53, 340–388 (1993).

Di Lazzaro, P.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Dozier, C. M.

C. M. Dozier, D. A. Newman, J. P. Gilfrich, R. K. Freitag, J. P. Kirkland, “Capillary optics for x-ray analysis,” in Vol. 37 of Advances in X-Ray Analysis, J. P. Gilfrich, ed. (Plenum, New York, 1994), pp. 507–514.

Edmond Turcu, I. C.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Epstein, H. M.

P. Mallozzi, H. M. Epstein, R. G. Jung, D. C. Applebaum, B. P. Fairand, W. T. Gallagher, R. L. Uecker, M. C. Muckerheide, “Laser-generated plasmas as a source of x rays for medical applications,” J. Appl. Phys. 45, 1891–1895 (1974).
[CrossRef]

Faenov, A. Yu.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Fairand, B. P.

P. Mallozzi, H. M. Epstein, R. G. Jung, D. C. Applebaum, B. P. Fairand, W. T. Gallagher, R. L. Uecker, M. C. Muckerheide, “Laser-generated plasmas as a source of x rays for medical applications,” J. Appl. Phys. 45, 1891–1895 (1974).
[CrossRef]

Fedin, D. A.

Yu. M. Alexandrov, S. B. Dabagov, M. A. Kumakhov, V. A. Murashova, D. A. Fedin, R. V. Fedorchuk, M. N. Yakimenko, “Peculiarities of photon transmission through capillary systems,” Nucl. Instrum. Methods Phys. Res. B 134, 174–180 (1998).
[CrossRef]

Fedorchuk, R. V.

Yu. M. Alexandrov, S. B. Dabagov, M. A. Kumakhov, V. A. Murashova, D. A. Fedin, R. V. Fedorchuk, M. N. Yakimenko, “Peculiarities of photon transmission through capillary systems,” Nucl. Instrum. Methods Phys. Res. B 134, 174–180 (1998).
[CrossRef]

S. B. Dabagov, R. V. Fedorchuk, V. A. Murashova, S. V. Nikitina, M. N. Yakimenko, “Interference phenomenon under focusing of synchrotron radiation by a Kumakhov lens,” Nucl. Instrum. Methods Phys. Res. B 108, 213–218 (1996).
[CrossRef]

Flora, F.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

S. V. Kukhlevsky, L. Kozma, L. Palladino, A. Reale, F. Flora, L. Mezi, G. Giordano, “Double-pulse excitation of x-ray capillary lasers,” Soft X-ray Lasers and Applications II, J. J. Rocca, L. B. Da Silva, eds., Proc. SPIE.3156, 180–187 (1997).

Freitag, R. K.

C. M. Dozier, D. A. Newman, J. P. Gilfrich, R. K. Freitag, J. P. Kirkland, “Capillary optics for x-ray analysis,” in Vol. 37 of Advances in X-Ray Analysis, J. P. Gilfrich, ed. (Plenum, New York, 1994), pp. 507–514.

Gallagher, W. T.

P. Mallozzi, H. M. Epstein, R. G. Jung, D. C. Applebaum, B. P. Fairand, W. T. Gallagher, R. L. Uecker, M. C. Muckerheide, “Laser-generated plasmas as a source of x rays for medical applications,” J. Appl. Phys. 45, 1891–1895 (1974).
[CrossRef]

Gilfrich, J. P.

C. M. Dozier, D. A. Newman, J. P. Gilfrich, R. K. Freitag, J. P. Kirkland, “Capillary optics for x-ray analysis,” in Vol. 37 of Advances in X-Ray Analysis, J. P. Gilfrich, ed. (Plenum, New York, 1994), pp. 507–514.

Giordano, G.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

S. V. Kukhlevsky, L. Kozma, L. Palladino, A. Reale, F. Flora, L. Mezi, G. Giordano, “Double-pulse excitation of x-ray capillary lasers,” Soft X-ray Lasers and Applications II, J. J. Rocca, L. B. Da Silva, eds., Proc. SPIE.3156, 180–187 (1997).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Grilli, A.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Gullikson, E. M.

B. L. Henke, E. M. Gullikson, J. C. Davis, “Low-energy x-ray interaction coefficients: photoabsorption, scattering and reflection, E = 30 - 3000 eV, Z = 1 - 92,” At. Data Nucl. Data Tables 53, 340–388 (1993).

Henke, B. L.

B. L. Henke, E. M. Gullikson, J. C. Davis, “Low-energy x-ray interaction coefficients: photoabsorption, scattering and reflection, E = 30 - 3000 eV, Z = 1 - 92,” At. Data Nucl. Data Tables 53, 340–388 (1993).

Hidaka, T.

T. Hidaka, “Extremely low-loss hollow core waveguide for VUV light,” Opt. Commun. 44, 90–93 (1982).
[CrossRef]

Hirsch, P. B.

P. B. Hirsch, J. N. Kellar, “An x-ray micro-beam technique: I. Collimation,” Proc. Phys. Soc. London 64, 369–374 (1951).

Ianzini, F.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Ibraimov, N. S.

S. B. Dabagov, S. V. Nikitina, M. A. Kumakhov, N. S. Ibraimov, G. A. Vartaniants, A. N. Nikitin, L. Spielberger, “Focusing of x rays by capillary systems,” Nucl. Instrum. Methods Phys. Res. B 103, 99–105 (1995).
[CrossRef]

Jans, J. C.

J. C. Jans, Philips Research Laboratories Eindhoven, Netherlandse Philips Bedrijven B. V., Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands (personal communication, 1997).

Janssens, K.

L. Vincze, K. Janssens, F. Adams, A. Rindby, “Detailed ray-tracing code for capillary optics,” X-Ray Spectrom. 24, 27–37 (1995).
[CrossRef]

Jentzsch, F.

F. Jentzsch, E. Nähring, “Die Fortleitung von Licht- und Röntgen durch Röhren,” Z. Tech. Phys. 12, 185–189 (1931).

Jung, R. G.

P. Mallozzi, H. M. Epstein, R. G. Jung, D. C. Applebaum, B. P. Fairand, W. T. Gallagher, R. L. Uecker, M. C. Muckerheide, “Laser-generated plasmas as a source of x rays for medical applications,” J. Appl. Phys. 45, 1891–1895 (1974).
[CrossRef]

Kalman, Z.

Kantsyrev, V. L.

V. L. Kantsyrev, R. Bruch, R. Phaneuf, N. G. Publicover, “New concepts for x-ray, soft x-ray, and EUV optical instrumentation including applications in spectroscopy, plasma diagnostics, and biomedical microscopy: a status report,” J. X-Ray Sci. Technol. 7, 139–158 (1997).
[CrossRef]

Kellar, J. N.

P. B. Hirsch, J. N. Kellar, “An x-ray micro-beam technique: I. Collimation,” Proc. Phys. Soc. London 64, 369–374 (1951).

Kirkland, J. P.

C. M. Dozier, D. A. Newman, J. P. Gilfrich, R. K. Freitag, J. P. Kirkland, “Capillary optics for x-ray analysis,” in Vol. 37 of Advances in X-Ray Analysis, J. P. Gilfrich, ed. (Plenum, New York, 1994), pp. 507–514.

Komarov, F. A.

M. A. Kumakhov, F. A. Komarov, “Multiple reflections from surface x-ray optics,” Phys. Rep. 191, 289–350 (1990).
[CrossRef]

Kozma, L.

S. V. Kukhlevsky, L. Kozma, “Guiding of light by short-length multimode waveguides. I,” Nuovo Cimento D 20, 783–789 (1998).
[CrossRef]

S. V. Kukhlevsky, L. Kozma, “Modal and focusing properties of the plasma based waveguides for x-ray beams,” Contrib. Plasma Phys. 38, 583–597 (1998).
[CrossRef]

S. V. Kukhlevsky, L. Kozma, K. Negrea, “Pulsed-mode analysis of soft-x-ray radiation passing through capillary waveguides,” J. Mod. Opt. 43, 2595–2604 (1996).
[CrossRef]

S. V. Kukhlevsky, L. Kozma, L. Palladino, A. Reale, F. Flora, L. Mezi, G. Giordano, “Double-pulse excitation of x-ray capillary lasers,” Soft X-ray Lasers and Applications II, J. J. Rocca, L. B. Da Silva, eds., Proc. SPIE.3156, 180–187 (1997).

Kuczumow, A.

Kukhlevsky, S. V.

S. V. Kukhlevsky, L. Kozma, “Guiding of light by short-length multimode waveguides. I,” Nuovo Cimento D 20, 783–789 (1998).
[CrossRef]

S. V. Kukhlevsky, L. Kozma, “Modal and focusing properties of the plasma based waveguides for x-ray beams,” Contrib. Plasma Phys. 38, 583–597 (1998).
[CrossRef]

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

S. V. Kukhlevsky, L. Kozma, K. Negrea, “Pulsed-mode analysis of soft-x-ray radiation passing through capillary waveguides,” J. Mod. Opt. 43, 2595–2604 (1996).
[CrossRef]

S. V. Kukhlevsky, L. Kozma, L. Palladino, A. Reale, F. Flora, L. Mezi, G. Giordano, “Double-pulse excitation of x-ray capillary lasers,” Soft X-ray Lasers and Applications II, J. J. Rocca, L. B. Da Silva, eds., Proc. SPIE.3156, 180–187 (1997).

Kumakhov, M. A.

Yu. M. Alexandrov, S. B. Dabagov, M. A. Kumakhov, V. A. Murashova, D. A. Fedin, R. V. Fedorchuk, M. N. Yakimenko, “Peculiarities of photon transmission through capillary systems,” Nucl. Instrum. Methods Phys. Res. B 134, 174–180 (1998).
[CrossRef]

S. B. Dabagov, M. A. Kumakhov, S. V. Nikitina, “On the interference of x rays in multiple reflection optics,” Phys. Lett. A 203, 279–282 (1995).
[CrossRef]

S. B. Dabagov, S. V. Nikitina, M. A. Kumakhov, N. S. Ibraimov, G. A. Vartaniants, A. N. Nikitin, L. Spielberger, “Focusing of x rays by capillary systems,” Nucl. Instrum. Methods Phys. Res. B 103, 99–105 (1995).
[CrossRef]

M. A. Kumakhov, F. A. Komarov, “Multiple reflections from surface x-ray optics,” Phys. Rep. 191, 289–350 (1990).
[CrossRef]

Larsson, S.

Le Gall, H.

S. P. Pogossian, H. Le Gall, “Neutron and x-ray propagation laws in thin film waveguides,” Opt. Commun. 114, 235–241 (1995).
[CrossRef]

Lely, J. A.

J. A. Lely, T. W. Van Rijssel, “X-ray collimator producing a beam of very small divergence and large intensity,” Acta Crystallogr. 2, 337–338 (1949).
[CrossRef]

Letardi, T.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Lewis, A.

Lieberman, K.

Mallozzi, P.

P. Mallozzi, H. M. Epstein, R. G. Jung, D. C. Applebaum, B. P. Fairand, W. T. Gallagher, R. L. Uecker, M. C. Muckerheide, “Laser-generated plasmas as a source of x rays for medical applications,” J. Appl. Phys. 45, 1891–1895 (1974).
[CrossRef]

Mezi, L.

S. V. Kukhlevsky, L. Kozma, L. Palladino, A. Reale, F. Flora, L. Mezi, G. Giordano, “Double-pulse excitation of x-ray capillary lasers,” Soft X-ray Lasers and Applications II, J. J. Rocca, L. B. Da Silva, eds., Proc. SPIE.3156, 180–187 (1997).

Mosher, D.

D. Mosher, S. J. Stephanakis, “X-ray light pipes,” Appl. Phys. Lett. 29, 105–107 (1976).
[CrossRef]

Muckerheide, M. C.

P. Mallozzi, H. M. Epstein, R. G. Jung, D. C. Applebaum, B. P. Fairand, W. T. Gallagher, R. L. Uecker, M. C. Muckerheide, “Laser-generated plasmas as a source of x rays for medical applications,” J. Appl. Phys. 45, 1891–1895 (1974).
[CrossRef]

Murashova, V. A.

Yu. M. Alexandrov, S. B. Dabagov, M. A. Kumakhov, V. A. Murashova, D. A. Fedin, R. V. Fedorchuk, M. N. Yakimenko, “Peculiarities of photon transmission through capillary systems,” Nucl. Instrum. Methods Phys. Res. B 134, 174–180 (1998).
[CrossRef]

S. B. Dabagov, R. V. Fedorchuk, V. A. Murashova, S. V. Nikitina, M. N. Yakimenko, “Interference phenomenon under focusing of synchrotron radiation by a Kumakhov lens,” Nucl. Instrum. Methods Phys. Res. B 108, 213–218 (1996).
[CrossRef]

Nähring, E.

F. Jentzsch, E. Nähring, “Die Fortleitung von Licht- und Röntgen durch Röhren,” Z. Tech. Phys. 12, 185–189 (1931).

Negrea, K.

S. V. Kukhlevsky, L. Kozma, K. Negrea, “Pulsed-mode analysis of soft-x-ray radiation passing through capillary waveguides,” J. Mod. Opt. 43, 2595–2604 (1996).
[CrossRef]

Newman, D. A.

C. M. Dozier, D. A. Newman, J. P. Gilfrich, R. K. Freitag, J. P. Kirkland, “Capillary optics for x-ray analysis,” in Vol. 37 of Advances in X-Ray Analysis, J. P. Gilfrich, ed. (Plenum, New York, 1994), pp. 507–514.

Nikitin, A. N.

S. B. Dabagov, S. V. Nikitina, M. A. Kumakhov, N. S. Ibraimov, G. A. Vartaniants, A. N. Nikitin, L. Spielberger, “Focusing of x rays by capillary systems,” Nucl. Instrum. Methods Phys. Res. B 103, 99–105 (1995).
[CrossRef]

Nikitina, S. V.

S. B. Dabagov, R. V. Fedorchuk, V. A. Murashova, S. V. Nikitina, M. N. Yakimenko, “Interference phenomenon under focusing of synchrotron radiation by a Kumakhov lens,” Nucl. Instrum. Methods Phys. Res. B 108, 213–218 (1996).
[CrossRef]

S. B. Dabagov, M. A. Kumakhov, S. V. Nikitina, “On the interference of x rays in multiple reflection optics,” Phys. Lett. A 203, 279–282 (1995).
[CrossRef]

S. B. Dabagov, S. V. Nikitina, M. A. Kumakhov, N. S. Ibraimov, G. A. Vartaniants, A. N. Nikitin, L. Spielberger, “Focusing of x rays by capillary systems,” Nucl. Instrum. Methods Phys. Res. B 103, 99–105 (1995).
[CrossRef]

Nottola, A.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Ognev, L.

T. A. Bobrova, L. Ognev, “On the ‘supercollimation’ of x-ray beams in rough interfacial channels,” JETP Lett. 69, 734–738 (1999).
[CrossRef]

Ognev, L. I.

L. I. Ognev, “On the structure of x-ray glass capillary lens focal spots,” Nucl. Instrum. Methods Phys. Res. B 124, 624–626 (1997).
[CrossRef]

Palladino, L.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

S. V. Kukhlevsky, L. Kozma, L. Palladino, A. Reale, F. Flora, L. Mezi, G. Giordano, “Double-pulse excitation of x-ray capillary lasers,” Soft X-ray Lasers and Applications II, J. J. Rocca, L. B. Da Silva, eds., Proc. SPIE.3156, 180–187 (1997).

Pantell, R. H.

Phaneuf, R.

V. L. Kantsyrev, R. Bruch, R. Phaneuf, N. G. Publicover, “New concepts for x-ray, soft x-ray, and EUV optical instrumentation including applications in spectroscopy, plasma diagnostics, and biomedical microscopy: a status report,” J. X-Ray Sci. Technol. 7, 139–158 (1997).
[CrossRef]

Pikuz, T.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Pogossian, S. P.

S. P. Pogossian, H. Le Gall, “Neutron and x-ray propagation laws in thin film waveguides,” Opt. Commun. 114, 235–241 (1995).
[CrossRef]

Publicover, N. G.

V. L. Kantsyrev, R. Bruch, R. Phaneuf, N. G. Publicover, “New concepts for x-ray, soft x-ray, and EUV optical instrumentation including applications in spectroscopy, plasma diagnostics, and biomedical microscopy: a status report,” J. X-Ray Sci. Technol. 7, 139–158 (1997).
[CrossRef]

Reale, A.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

S. V. Kukhlevsky, L. Kozma, L. Palladino, A. Reale, F. Flora, L. Mezi, G. Giordano, “Double-pulse excitation of x-ray capillary lasers,” Soft X-ray Lasers and Applications II, J. J. Rocca, L. B. Da Silva, eds., Proc. SPIE.3156, 180–187 (1997).

Reale, L.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Rindby, A.

L. Vincze, K. Janssens, F. Adams, A. Rindby, “Detailed ray-tracing code for capillary optics,” X-Ray Spectrom. 24, 27–37 (1995).
[CrossRef]

A. Rindby, “Application of fiber technique in the x-ray region,” Nucl. Instrum. Methods Phys. Res. A 249, 536–540 (1986).
[CrossRef]

Scafati, A.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Schina, G.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Segmuller, A.

E. E. Spiller, A. Segmuller, “Propagation of x rays in waveguides,” Appl. Phys. Lett. 24, 60–61 (1974).
[CrossRef]

Spielberger, L.

S. B. Dabagov, S. V. Nikitina, M. A. Kumakhov, N. S. Ibraimov, G. A. Vartaniants, A. N. Nikitin, L. Spielberger, “Focusing of x rays by capillary systems,” Nucl. Instrum. Methods Phys. Res. B 103, 99–105 (1995).
[CrossRef]

Spiller, E. E.

E. E. Spiller, A. Segmuller, “Propagation of x rays in waveguides,” Appl. Phys. Lett. 24, 60–61 (1974).
[CrossRef]

Stephanakis, S. J.

D. Mosher, S. J. Stephanakis, “X-ray light pipes,” Appl. Phys. Lett. 29, 105–107 (1976).
[CrossRef]

Stern, E. A.

Tabocchini, M. A.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Uecker, R. L.

P. Mallozzi, H. M. Epstein, R. G. Jung, D. C. Applebaum, B. P. Fairand, W. T. Gallagher, R. L. Uecker, M. C. Muckerheide, “Laser-generated plasmas as a source of x rays for medical applications,” J. Appl. Phys. 45, 1891–1895 (1974).
[CrossRef]

Van Rijssel, T. W.

J. A. Lely, T. W. Van Rijssel, “X-ray collimator producing a beam of very small divergence and large intensity,” Acta Crystallogr. 2, 337–338 (1949).
[CrossRef]

Vartaniants, G. A.

S. B. Dabagov, S. V. Nikitina, M. A. Kumakhov, N. S. Ibraimov, G. A. Vartaniants, A. N. Nikitin, L. Spielberger, “Focusing of x rays by capillary systems,” Nucl. Instrum. Methods Phys. Res. B 103, 99–105 (1995).
[CrossRef]

Vigli-Papadaki, K.

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

Vincze, L.

L. Vincze, K. Janssens, F. Adams, A. Rindby, “Detailed ray-tracing code for capillary optics,” X-Ray Spectrom. 24, 27–37 (1995).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980)

Yakimenko, M. N.

Yu. M. Alexandrov, S. B. Dabagov, M. A. Kumakhov, V. A. Murashova, D. A. Fedin, R. V. Fedorchuk, M. N. Yakimenko, “Peculiarities of photon transmission through capillary systems,” Nucl. Instrum. Methods Phys. Res. B 134, 174–180 (1998).
[CrossRef]

S. B. Dabagov, R. V. Fedorchuk, V. A. Murashova, S. V. Nikitina, M. N. Yakimenko, “Interference phenomenon under focusing of synchrotron radiation by a Kumakhov lens,” Nucl. Instrum. Methods Phys. Res. B 108, 213–218 (1996).
[CrossRef]

Yu. M. Aleksandrov, M. N. Yakimenko, “Investigations on the S-60 SR source of the Lebedev Physical Institute,” Nucl. Instrum. Methods Phys. Res. A 359, 12–14 (1995).
[CrossRef]

Acta Crystallogr. (1)

J. A. Lely, T. W. Van Rijssel, “X-ray collimator producing a beam of very small divergence and large intensity,” Acta Crystallogr. 2, 337–338 (1949).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (2)

D. Mosher, S. J. Stephanakis, “X-ray light pipes,” Appl. Phys. Lett. 29, 105–107 (1976).
[CrossRef]

E. E. Spiller, A. Segmuller, “Propagation of x rays in waveguides,” Appl. Phys. Lett. 24, 60–61 (1974).
[CrossRef]

At. Data Nucl. Data Tables (1)

B. L. Henke, E. M. Gullikson, J. C. Davis, “Low-energy x-ray interaction coefficients: photoabsorption, scattering and reflection, E = 30 - 3000 eV, Z = 1 - 92,” At. Data Nucl. Data Tables 53, 340–388 (1993).

Contrib. Plasma Phys. (1)

S. V. Kukhlevsky, L. Kozma, “Modal and focusing properties of the plasma based waveguides for x-ray beams,” Contrib. Plasma Phys. 38, 583–597 (1998).
[CrossRef]

J. Appl. Phys. (1)

P. Mallozzi, H. M. Epstein, R. G. Jung, D. C. Applebaum, B. P. Fairand, W. T. Gallagher, R. L. Uecker, M. C. Muckerheide, “Laser-generated plasmas as a source of x rays for medical applications,” J. Appl. Phys. 45, 1891–1895 (1974).
[CrossRef]

J. Mod. Opt. (1)

S. V. Kukhlevsky, L. Kozma, K. Negrea, “Pulsed-mode analysis of soft-x-ray radiation passing through capillary waveguides,” J. Mod. Opt. 43, 2595–2604 (1996).
[CrossRef]

J. X-Ray Sci. Technol. (1)

V. L. Kantsyrev, R. Bruch, R. Phaneuf, N. G. Publicover, “New concepts for x-ray, soft x-ray, and EUV optical instrumentation including applications in spectroscopy, plasma diagnostics, and biomedical microscopy: a status report,” J. X-Ray Sci. Technol. 7, 139–158 (1997).
[CrossRef]

JETP Lett. (1)

T. A. Bobrova, L. Ognev, “On the ‘supercollimation’ of x-ray beams in rough interfacial channels,” JETP Lett. 69, 734–738 (1999).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A (2)

A. Rindby, “Application of fiber technique in the x-ray region,” Nucl. Instrum. Methods Phys. Res. A 249, 536–540 (1986).
[CrossRef]

Yu. M. Aleksandrov, M. N. Yakimenko, “Investigations on the S-60 SR source of the Lebedev Physical Institute,” Nucl. Instrum. Methods Phys. Res. A 359, 12–14 (1995).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. B (4)

S. B. Dabagov, R. V. Fedorchuk, V. A. Murashova, S. V. Nikitina, M. N. Yakimenko, “Interference phenomenon under focusing of synchrotron radiation by a Kumakhov lens,” Nucl. Instrum. Methods Phys. Res. B 108, 213–218 (1996).
[CrossRef]

Yu. M. Alexandrov, S. B. Dabagov, M. A. Kumakhov, V. A. Murashova, D. A. Fedin, R. V. Fedorchuk, M. N. Yakimenko, “Peculiarities of photon transmission through capillary systems,” Nucl. Instrum. Methods Phys. Res. B 134, 174–180 (1998).
[CrossRef]

S. B. Dabagov, S. V. Nikitina, M. A. Kumakhov, N. S. Ibraimov, G. A. Vartaniants, A. N. Nikitin, L. Spielberger, “Focusing of x rays by capillary systems,” Nucl. Instrum. Methods Phys. Res. B 103, 99–105 (1995).
[CrossRef]

L. I. Ognev, “On the structure of x-ray glass capillary lens focal spots,” Nucl. Instrum. Methods Phys. Res. B 124, 624–626 (1997).
[CrossRef]

Nuovo Cimento D (2)

S. Bollanti, P. Albertano, M. Belli, P. Di Lazzaro, A. Yu. Faenov, F. Flora, G. Giordano, A. Grilli, F. Ianzini, S. V. Kukhlevsky, T. Letardi, A. Nottola, L. Palladino, T. Pikuz, A. Reale, L. Reale, A. Scafati, M. A. Tabocchini, I. C. Edmond Turcu, K. Vigli-Papadaki, G. Schina, “Soft x-ray plasma source for atmospheric pressure microscopy, radiobiology and other applications,” Nuovo Cimento D 20, 1685–1696 (1998).

S. V. Kukhlevsky, L. Kozma, “Guiding of light by short-length multimode waveguides. I,” Nuovo Cimento D 20, 783–789 (1998).
[CrossRef]

Opt. Commun. (2)

T. Hidaka, “Extremely low-loss hollow core waveguide for VUV light,” Opt. Commun. 44, 90–93 (1982).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

(a) Propagation of the x-ray radiation through the slab hollow waveguide. (b) Schematic diagram of the free-space equivalent source.

Fig. 2
Fig. 2

Intensity distribution of x-ray radiation inside the guide (z ∊ [10 cm, 10.5 cm]), calculated for the first source-guide configuration.

Fig. 3
Fig. 3

Intensity distribution of x-ray radiation inside the guide at z = 10.5 cm calculated for the first (C, D) and the second (A, B) source-guide configurations. Curves (A) and (C) were computed taking into account the absorption (γ ≠ 0). Curves (B) and (D) correspond to the guided propagation without absorption (γ = 0).

Fig. 4
Fig. 4

Intensity distribution of x-ray radiation in the far-field zone computed for the first (A) and the second (B) source-guide configurations. The intensity distribution of x-ray radiation in the far-field zone calculated with the geometric optics for the first (C) and the second (D) configurations are shown for comparison.

Fig. 5
Fig. 5

Observed intensity distribution of the soft-x-ray radiation in the far-field zone (A) for the on-axis configuration (α = 0). The intensity distribution of x-ray radiation in the far-field zone calculated with the new approach (B) is shown for comparison.

Fig. 6
Fig. 6

Observed intensity distribution of the soft-x-ray radiation in the far-field zone (A) for the off-axis configuration (α = 1.1 mrad). The intensity distribution of x-ray radiation in the far-field zone calculated with the new approach (B) is shown for the comparison.

Equations (6)

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Em(x, t)=12iλxmminxmmaxexpi 2πλ rr×(1+cos Θ)Em(x, t)dx,
Emx, t=RmΘ, n2E0x2am-1m×expi-ωt+ϕ0+Δϕm,
Rm=12tan(Θ-Ψ)tan(Θ+Ψ)m+cos(Θ+Ψ)cos(Θ-Ψ)m,
Ψ=arcsincos(Θ)n2,
E(x, t)=m=-MMEm(x, t),
E(x, t)=12iλm=-MM-aaexpi 2πλ rr×(1+cos Θ)Em(x, t)dx,

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