Abstract

The reflectivities of most surfaces are higher for grazing or near-90-deg angles of incidence than for more perpendicular or near-zero-deg angles. Grazing-incidence configurations are especially important in the development of lasers and optical systems that operate in the far-ultraviolet and soft-x-ray regions of the spectrum, where transparent or highly reflecting media are almost unknown. Analytical solutions of the paraxial wave equation are obtained for the grazing reflection and complex interference effects that take place when a Gaussian beam interacts at shallow angles with a reflecting surface.

© 1999 Optical Society of America

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  7. W. S. Bacsa, A. Kulik, “Interference scanning optical probe microscopy,” Appl. Phys. Lett. 70, 3507–3509 (1997).
    [CrossRef]
  8. See, for example, J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley, New York, 1967), Section 2.8 and references therein.
  9. See, for example, A. N. Zaidel, E. Y. Shreider, Vacuum Ultraviolet Spectroscopy (Ann Arbor-Humphrey Science Publishers, Inc., Ann Arbor, Mich., 1970), pp. 119–126.
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    [CrossRef]
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    [CrossRef]
  17. K. C. Sun, “Grazing incidence ring resonator for visible wavelength free electron laser,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 409–422 (1990).
    [CrossRef]
  18. D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
    [CrossRef]
  19. D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
    [CrossRef]
  20. D. R. Gabardi, D. L. Shealy, “Optical analysis of grazing incidence ring resonators for free-electron lasers,” Opt. Eng. 29, 641–648 (1990).
    [CrossRef]
  21. M. C. Wang, Y. Yuan, Z. Wang, “The grazing incidence ring resonator for a free-electron laser,” Nucl. Instrum. Methods A 318, 874–876 (1992).
    [CrossRef]
  22. J. P. Braud, “Laser cavities and polarization optics for soft X-rays and the extreme ultraviolet,” Appl. Phys. B 50, 205–212 (1990).
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  23. J. F. Young, J. J. Macklin, S. E. Harris, “Grazing-incidence ellipsoidal reflector for longitudinally pumping short-wavelength lasers,” Opt. Lett. 12, 90–92 (1987).
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  25. A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995), Eq. (24).

1997

W. S. Bacsa, A. Kulik, “Interference scanning optical probe microscopy,” Appl. Phys. Lett. 70, 3507–3509 (1997).
[CrossRef]

1995

A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. 12, 1522–1533 (1995).
[CrossRef]

A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995), Eq. (24).

1992

M. C. Wang, Y. Yuan, Z. Wang, “The grazing incidence ring resonator for a free-electron laser,” Nucl. Instrum. Methods A 318, 874–876 (1992).
[CrossRef]

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

1991

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

1990

D. R. Gabardi, D. L. Shealy, “Optical analysis of grazing incidence ring resonators for free-electron lasers,” Opt. Eng. 29, 641–648 (1990).
[CrossRef]

J. P. Braud, “Laser cavities and polarization optics for soft X-rays and the extreme ultraviolet,” Appl. Phys. B 50, 205–212 (1990).
[CrossRef]

1989

1987

E. Sklar, “The tilt sensitivity of a grazing incidence confocal unstable resonator with applications to free-electron lasers,” IEEE J. Quantum Electron. QE-23, 229–233 (1987).
[CrossRef]

J. F. Young, J. J. Macklin, S. E. Harris, “Grazing-incidence ellipsoidal reflector for longitudinally pumping short-wavelength lasers,” Opt. Lett. 12, 90–92 (1987).
[CrossRef] [PubMed]

1983

1979

L. W. Casperson, T. S. Garfield, “Guided beams in concave metallic waveguides,” IEEE J. Quantum Electron. QE-15, 491–496 (1979), and references therein.
[CrossRef]

M. E. Marhic, L. I. Kwan, M. Epstein, “Whispering-gallery CO2 laser,” IEEE J. Quantum Electron. QE-15, 487–490 (1979).
[CrossRef]

1978

1976

1975

1973

1965

1961

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Sys. Tech. J. 40, 489–508 (1961).
[CrossRef]

Adamski, J. L.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

Al-Mashaabi, F. S.

Bacsa, W. S.

W. S. Bacsa, A. Kulik, “Interference scanning optical probe microscopy,” Appl. Phys. Lett. 70, 3507–3509 (1997).
[CrossRef]

Bender, S.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

Bender, S. C.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

Bentz, M.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

Bentz, M. P.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

Boyd, G. D.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Sys. Tech. J. 40, 489–508 (1961).
[CrossRef]

Braud, J. P.

J. P. Braud, “Laser cavities and polarization optics for soft X-rays and the extreme ultraviolet,” Appl. Phys. B 50, 205–212 (1990).
[CrossRef]

Burns, R.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

Burns, R. H.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

Byrd, D.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

Casperson, L. W.

Dowell, D. H.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

Eggleston, J. M.

J. M. Eggleston, “Angularly stable ring resonators for high power FELs,” in Proceedings of the International Conference on Lasers 1983, R. Powell, ed. (STS, McLean, Va., 1985), p. 305.

Epstein, M.

M. E. Marhic, L. I. Kwan, M. Epstein, “Whispering-gallery CO2 laser,” IEEE J. Quantum Electron. QE-15, 487–490 (1979).
[CrossRef]

Gabardi, D. R.

D. R. Gabardi, D. L. Shealy, “Optical analysis of grazing incidence ring resonators for free-electron lasers,” Opt. Eng. 29, 641–648 (1990).
[CrossRef]

Garfield, T. S.

L. W. Casperson, T. S. Garfield, “Guided beams in concave metallic waveguides,” IEEE J. Quantum Electron. QE-15, 491–496 (1979), and references therein.
[CrossRef]

Gordon, J. P.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Sys. Tech. J. 40, 489–508 (1961).
[CrossRef]

Grossman, J. G.

Guha, J.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

Harris, S. E.

Kogelnik, H.

Krammer, H.

Kulik, A.

W. S. Bacsa, A. Kulik, “Interference scanning optical probe microscopy,” Appl. Phys. Lett. 70, 3507–3509 (1997).
[CrossRef]

Kwan, L. I.

M. E. Marhic, L. I. Kwan, M. Epstein, “Whispering-gallery CO2 laser,” IEEE J. Quantum Electron. QE-15, 487–490 (1979).
[CrossRef]

Laucks, M. L.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

Lowrey, A. R.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

Lumpkin, A. H.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

Lunnam, S. D.

Macklin, J. J.

Marhic, M. E.

M. E. Marhic, L. I. Kwan, M. Epstein, “Whispering-gallery CO2 laser,” IEEE J. Quantum Electron. QE-15, 487–490 (1979).
[CrossRef]

Pistoresi, D. J.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

Samson, J. A. R.

See, for example, J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley, New York, 1967), Section 2.8 and references therein.

Shealy, D. L.

D. R. Gabardi, D. L. Shealy, “Optical analysis of grazing incidence ring resonators for free-electron lasers,” Opt. Eng. 29, 641–648 (1990).
[CrossRef]

Shoffstall, D. R.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

Shreider, E. Y.

See, for example, A. N. Zaidel, E. Y. Shreider, Vacuum Ultraviolet Spectroscopy (Ann Arbor-Humphrey Science Publishers, Inc., Ann Arbor, Mich., 1970), pp. 119–126.

Sklar, E.

E. Sklar, “The tilt sensitivity of a grazing incidence confocal unstable resonator with applications to free-electron lasers,” IEEE J. Quantum Electron. QE-23, 229–233 (1987).
[CrossRef]

Stafsudd, O. M.

Sun, K.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

Sun, K. C.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

K. C. Sun, “Grazing incidence ring resonator for visible wavelength free electron laser,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 409–422 (1990).
[CrossRef]

Tokar, R. L.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

Tomita, W.

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

Tovar, A. A.

A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. 12, 1522–1533 (1995).
[CrossRef]

A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995), Eq. (24).

Wang, M. C.

M. C. Wang, Y. Yuan, Z. Wang, “The grazing incidence ring resonator for a free-electron laser,” Nucl. Instrum. Methods A 318, 874–876 (1992).
[CrossRef]

Wang, Z.

M. C. Wang, Y. Yuan, Z. Wang, “The grazing incidence ring resonator for a free-electron laser,” Nucl. Instrum. Methods A 318, 874–876 (1992).
[CrossRef]

Young, J. F.

Yuan, Y.

M. C. Wang, Y. Yuan, Z. Wang, “The grazing incidence ring resonator for a free-electron laser,” Nucl. Instrum. Methods A 318, 874–876 (1992).
[CrossRef]

Zaidel, A. N.

See, for example, A. N. Zaidel, E. Y. Shreider, Vacuum Ultraviolet Spectroscopy (Ann Arbor-Humphrey Science Publishers, Inc., Ann Arbor, Mich., 1970), pp. 119–126.

Appl. Opt.

Appl. Phys. B

J. P. Braud, “Laser cavities and polarization optics for soft X-rays and the extreme ultraviolet,” Appl. Phys. B 50, 205–212 (1990).
[CrossRef]

Appl. Phys. Lett.

W. S. Bacsa, A. Kulik, “Interference scanning optical probe microscopy,” Appl. Phys. Lett. 70, 3507–3509 (1997).
[CrossRef]

Bell Sys. Tech. J.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Sys. Tech. J. 40, 489–508 (1961).
[CrossRef]

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Sys. Tech. J. 44, 455–494 (1965).
[CrossRef]

IEEE J. Quantum Electron.

L. W. Casperson, T. S. Garfield, “Guided beams in concave metallic waveguides,” IEEE J. Quantum Electron. QE-15, 491–496 (1979), and references therein.
[CrossRef]

M. E. Marhic, L. I. Kwan, M. Epstein, “Whispering-gallery CO2 laser,” IEEE J. Quantum Electron. QE-15, 487–490 (1979).
[CrossRef]

E. Sklar, “The tilt sensitivity of a grazing incidence confocal unstable resonator with applications to free-electron lasers,” IEEE J. Quantum Electron. QE-23, 229–233 (1987).
[CrossRef]

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, M. P. Bentz, R. H. Burns, J. Guha, K. C. Sun, W. Tomita, A. H. Lumpkin, S. C. Bender, D. Byrd, R. L. Tokar, “Tests of a grazing-incidence ring resonator free-electron laser,” IEEE J. Quantum Electron. 27, 2613–2625 (1991).
[CrossRef]

J. Opt. Soc. Am.

L. W. Casperson, “Beam modes in complex lenslike media and resonators,” J. Opt. Soc. Am. 66, 1373–1379 (1976).
[CrossRef]

A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. 12, 1522–1533 (1995).
[CrossRef]

J. Opt. Soc. Am. A

Nucl. Instrum. Methods A

M. C. Wang, Y. Yuan, Z. Wang, “The grazing incidence ring resonator for a free-electron laser,” Nucl. Instrum. Methods A 318, 874–876 (1992).
[CrossRef]

D. H. Dowell, M. L. Laucks, A. R. Lowrey, J. L. Adamski, D. J. Pistoresi, D. R. Shoffstall, A. H. Lumpkin, S. Bender, D. Byrd, R. L. Tokar, K. Sun, M. Bentz, R. Burns, J. Guha, W. Tomita, “Final results of the Boeing and Los Alamos grazing incidence ring-resonator free electron laser experiment,” Nucl. Instrum. Methods A 318, 74–80 (1992).
[CrossRef]

Opt. Eng.

D. R. Gabardi, D. L. Shealy, “Optical analysis of grazing incidence ring resonators for free-electron lasers,” Opt. Eng. 29, 641–648 (1990).
[CrossRef]

Opt. Lett.

Other

K. C. Sun, “Grazing incidence ring resonator for visible wavelength free electron laser,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 409–422 (1990).
[CrossRef]

J. M. Eggleston, “Angularly stable ring resonators for high power FELs,” in Proceedings of the International Conference on Lasers 1983, R. Powell, ed. (STS, McLean, Va., 1985), p. 305.

See, for example, J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley, New York, 1967), Section 2.8 and references therein.

See, for example, A. N. Zaidel, E. Y. Shreider, Vacuum Ultraviolet Spectroscopy (Ann Arbor-Humphrey Science Publishers, Inc., Ann Arbor, Mich., 1970), pp. 119–126.

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of a Gaussian beam undergoing grazing reflection from a flat surface. The coordinate system used in the analysis is also shown.

Fig. 2
Fig. 2

Series of transverse intensity profiles of a normalized Gaussian beam undergoing grazing reflection from a flat surface located at the position x′ = 0.5 on the right-hand side of the figure. The arrows to the left-hand side of the figure distinguish between the propagation directions of the more localized incident beam profiles (right arrow) and the less localized reflected beam profiles (left arrow). The beam is polarized parallel to the surface, the waist spot size is w0 = 0.2, the velocity of the input beam toward the surface is v 1 = 0.5, and the propagation distance between successive profiles is z′ = 0.2 (five plots per Rayleigh length). In (a) the beam waist (maximum beam-axis intensity) occurs just as the beam axis crosses the location x′ = -0.5. Because of increasing diffraction, the reflected beam in this example has a lower intensity and greater width than the incident beam. In (b) the waist occurs at x′ = 0.0.

Fig. 3
Fig. 3

Transverse intensity profiles of a reflecting Gaussian beam whose waist occurs at the reflecting surface. The two-headed arrows to the left-hand sides of the figures indicate that in these cases the incident and reflected beam profiles are identical. The beam is polarized parallel to the surface, the waist spot size is w0 = 0.2, and the propagation distance between successive profiles is z′ = 0.1. In (a) the incident beam is moving toward the surface at a velocity of v 1 = 0.5, and in (b) the velocity of the beam is v 1 = 1.0.

Fig. 4
Fig. 4

Transverse intensity profiles of a reflecting Gaussian beam. The beam is polarized perpendicular to the surface, the waist spot size is w0 = 0.2, the velocity of the input beam toward the surface is v 1 = 0.5, and the propagation distance between successive profiles is z′ = 0.2. In (a) the beam waist occurs as the beam axis crosses the location x′ = -0.5, and in (b) the waist is at x′ = 0.0.

Fig. 5
Fig. 5

Transverse intensity profiles of a reflecting Gaussian beam whose waist occurs at the reflecting surface. The beam is polarized perpendicular to the surface, the waist spot size is w0 = 0.2, and the propagation distance between successive profiles is z′ = 0.1. In (a) the incident beam is moving toward the surface at a velocity of v 1 = 0.5, and in (b) the velocity of the beam is v 1 = 1.0.

Equations (36)

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2Ex, y, z+k2x, y, zEx, y, z=0,
k2x, y, z=k0zk0z-k1xzx-k2xzx2.
Ex, y, z=Ax, y, zexp-i  k0zdz,
2Ax2+2Ay2-2ik0Az-i dk0dz A-k0k1xx+k2xx2A=0,
Ax, y, z=A0 exp-iQxzx22+Sxzx+Pz,
Qx2+k0dQxdz+k0k2x=0,
QxSx+k0dSxdz+k0k1x2=0,
dPdz=-i Qx2k0-Sx22k0-i2k0dk0dz.
Qxz=β0zRxz-i 2wx2z,
Sxz=-Qxzdxaz+β0zdxaz-α2xzdxaz+α1xz/2/Qxiz,
Qx2+β0dQxdz=0,
QxSx+β0dSxdz=0,
dPdz=-i Qx2β0-Sx22β0.   
Sxz=-Qxzdxaz+β0dxa0=-Qxzdxa0+β0-Qxzzdxa0,
1qxz=1/qx11+z/qx1,
Qxzβ0=1qxz=1Rxz-iλnπwx2z.
1SxzdSxzdz=-1/qx11+z/qx1.
Sxz=Sx11+z/qx1,
dPzdz=-i21/qx11+z/qx1-Sx12/2β01+z/qx12.
Pz=P1-i2 ln1+zqx1-Sx12qx12β0z/qx11+z/qx1,
1qxz=-i/z01-iz/z0.
Sxz=Sx11-iz/z0,
Pz=P1-i2 ln1-izz0-Sx12z02β0z/z01-iz/z0.
Sxz=-β0z0z/z0-i1+z/z02dxa0+zz0ddxadz/z0+β0z0ddxadz/z0=- 2w02z/z0-i1+z/z02dxa0+zz0ddxadz/z0+2w02ddxadz/z0,
Sxz=-2w02z-i1+z2dxa0+z ddxadz+2w02ddxadz.
Sxz=-2 z-i1+z2p1+zv1w02d+2 v1w02d.
Sx1=2 v1+ip1w02d.
Pz=P1-i2 ln1-iz-zw02v1+ip121-iz=-i2ln1+z21/2-i tan-1z]-zw02×v12-p12-2v1p1z+iv12-p12z+2v1p11+z2,
Ax, z=A0 exp-1w021+z2-i zw021+z2x2+2p1+zv1w021+z2+i 2zp1+zv1w021+z2-i 2v1w02x+-ln1+z21/4-z2v12-p12+2zv1p1w021+z2+i zv12-p12-2z2v1p1w021+z2+i2 tan-1z,
Ix, z=A*x, zAx, z.
- Ix, zdx=1.
Ax, z=2/π1/2w01+z21/21/2 exp-1w021+z2 x2+2p1+zv1w021+z2 x-p12+2zv1p1+z2v12w021+z2×exp-izw021+z2 x2+2v1w02-2zp1+zv1w021+z2x-zv12-p12-2z2v1p1w021+z2-12 tan-1z=2/π1/2w01+z21/21/2×exp-x-p1+zv1w01+z21/22×exp-izx-p1+zv1w01+z21/22+2v1x-zv12w02-12 tan-1z.
Ax, z=Ax, z-A1-x, z.
Ix, z=A*x, zAx, z,
Ax, z=Ax, z+A1-x, z.
Ix, z=A*x, zAx, z,

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