Abstract

We present a theoretical study of the emission from a superluminal polarization current whose distribution pattern rotates (with an angular frequency ω) and oscillates (with a frequency Ω) at the same time and that comprises both poloidal and toroidal components. This type of polarization current is found in recent practical machines designed to investigate superluminal emission. We find that the superluminal motion of the distribution pattern of the emitting current generates localized electromagnetic waves that do not decay spherically, i.e., that do not have an intensity diminishing as RP-2 with the distance RP from their source. The nonspherical decay of the focused wave packets that are emitted by the polarization currents does not contravene conservation of energy: The constructive interference of the constituent waves of such propagating caustics takes place within different solid angles on spheres of different radii (RP) centered on the source. For a polarization current whose longitudinal distribution (over an azimuthal interval of length 2π) consists of m cycles of a sinusoidal wave train, the nonspherically decaying part of the emitted radiation contains the frequencies Ω±mω; i.e., it contains only the frequencies involved in the creation and implementation of the source. This is in contrast to recent studies of the spherically decaying emission, which was shown to contain much higher frequencies. The polarization of the emitted radiation is found to be linear for most configurations of the source.

© 2004 Optical Society of America

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  1. A. Sommerfeld , “Zur Elektronentheorie (3 Teile),” Nach. Kgl. Ges. Wiss. Göttingen Math. Naturwiss. Klasse, 99–130, 363–439 (1904); 201–235 (1905).
  2. B. M. Bolotovskii, V. L. Ginzburg, “The Vavilov–Cerenkov effect and the Doppler effect in the motion of sources with superluminal velocity in vacuum,” Sov. Phys. Usp. 15, 184–192 (1972).
    [CrossRef]
  3. V. L. Ginzburg, Theoretical Physics and Astrophysics (Pergamon, Oxford, UK, 1979), Chap. VIII.
  4. B. M. Bolotovskii, V. P. Bykov, “Radiation by charges moving faster than light,” Sov. Phys. Usp. 33, 477–487 (1990).
    [CrossRef]
  5. H. Ardavan, “Generation of focused, nonspherically decaying pulses of electromagnetic radiation,” Phys. Rev. E 58, 6659–6684 (1998).
    [CrossRef]
  6. A. Hewish, “Comment I on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3007 (2000).
    [CrossRef]
  7. J. H. Hannay, “Comment II on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3008–3009 (2000).
    [CrossRef]
  8. H. Ardavan, “Reply to Comments on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3010–3013 (2000).
    [CrossRef]
  9. A. Ardavan, H. Ardavan, “Apparatus for generating focused electromagnetic radiation,” International patent application, PCT-GB99-02943 (September6, 1999).
  10. M. Durrani, “Revolutionary device polarizes opinions,” Phys. World, August2000, p. 9.
  11. N. Appleyard, B. Appleby, “Warp speed,” New Sci.April 2001, 28–31 (2001).
  12. E. Cartlidge, “Money spinner or loopy idea?” Science, September 2003, 1463 (2003).
  13. J. Fopma, A. Ardavan, D. Halliday, J. Singleton, “Phase control electronics for a polarization synchrotron,” preprint available from the author (contact email: jsingle@lanl.gov).
  14. A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, David Halliday, A. Narduzzo, P. Goddard, “Experimental observation of non-spherically decaying emission from a rotating superlumnal source,” preprint available from the author (contact email: jsingle@lanl.gov).
  15. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  16. P. Saari, K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
    [CrossRef]
  17. K. Reivelt, P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 56611 (2002).
    [CrossRef]
  18. E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
    [CrossRef]
  19. A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, S. M. Sedky, “Generation of approximate focus wave mode pulses from wide-band dynamic apertures,” J. Opt. Soc. Am. A 12, 1954–1964 (1995).
    [CrossRef]
  20. A. M. Shaarawi, “Comparison of two localized wave fields generated from dynamic apertures,” J. Opt. Soc. Am. A 14, 1804–1816 (1997).
    [CrossRef]
  21. L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, UK, 1975).
  22. J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Yale U. Press, New Haven, Conn., 1923), Dover reprint (1952).
  23. F. J. Bureau, “Divergent integrals and partial differential equations,” Commun. Pure Appl. Math. 8, 143–202 (1955).
    [CrossRef]
  24. R. F. Hoskins, Generalised Functions (Harwood, London, 1979), Chap. 7.
  25. H. Ardavan, A. Ardavan, J. Singleton, “Frequency spec-trum of focused broadband pulses of electromagnetic radiation generated by polarization currents with superlumi-nally rotating distribution patterns,” J. Opt. Soc. Am. A 20, 2137–2155 (2003).
    [CrossRef]
  26. C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descent,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
    [CrossRef]
  27. R. Burridge, “Asymptotic evaluation of integrals related to time-dependent fields near caustics,” SIAM J. Appl. Math. 55, 390–409 (1995).
    [CrossRef]
  28. J. J. Stamnes, Waves in Focal Regions (Hilger, Boston, Mass., 1986).
  29. N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986).
  30. R. Wong, Asymptotic Approximations of Integrals (Academic, Boston, Mass., 1989).
  31. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1995).

2003

E. Cartlidge, “Money spinner or loopy idea?” Science, September 2003, 1463 (2003).

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

H. Ardavan, A. Ardavan, J. Singleton, “Frequency spec-trum of focused broadband pulses of electromagnetic radiation generated by polarization currents with superlumi-nally rotating distribution patterns,” J. Opt. Soc. Am. A 20, 2137–2155 (2003).
[CrossRef]

2002

K. Reivelt, P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 56611 (2002).
[CrossRef]

2001

N. Appleyard, B. Appleby, “Warp speed,” New Sci.April 2001, 28–31 (2001).

2000

A. Hewish, “Comment I on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3007 (2000).
[CrossRef]

J. H. Hannay, “Comment II on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3008–3009 (2000).
[CrossRef]

H. Ardavan, “Reply to Comments on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3010–3013 (2000).
[CrossRef]

M. Durrani, “Revolutionary device polarizes opinions,” Phys. World, August2000, p. 9.

1998

H. Ardavan, “Generation of focused, nonspherically decaying pulses of electromagnetic radiation,” Phys. Rev. E 58, 6659–6684 (1998).
[CrossRef]

1997

P. Saari, K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

A. M. Shaarawi, “Comparison of two localized wave fields generated from dynamic apertures,” J. Opt. Soc. Am. A 14, 1804–1816 (1997).
[CrossRef]

1995

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, S. M. Sedky, “Generation of approximate focus wave mode pulses from wide-band dynamic apertures,” J. Opt. Soc. Am. A 12, 1954–1964 (1995).
[CrossRef]

R. Burridge, “Asymptotic evaluation of integrals related to time-dependent fields near caustics,” SIAM J. Appl. Math. 55, 390–409 (1995).
[CrossRef]

1990

B. M. Bolotovskii, V. P. Bykov, “Radiation by charges moving faster than light,” Sov. Phys. Usp. 33, 477–487 (1990).
[CrossRef]

1987

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

1972

B. M. Bolotovskii, V. L. Ginzburg, “The Vavilov–Cerenkov effect and the Doppler effect in the motion of sources with superluminal velocity in vacuum,” Sov. Phys. Usp. 15, 184–192 (1972).
[CrossRef]

1957

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descent,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

1955

F. J. Bureau, “Divergent integrals and partial differential equations,” Commun. Pure Appl. Math. 8, 143–202 (1955).
[CrossRef]

Appleby, B.

N. Appleyard, B. Appleby, “Warp speed,” New Sci.April 2001, 28–31 (2001).

Appleyard, N.

N. Appleyard, B. Appleby, “Warp speed,” New Sci.April 2001, 28–31 (2001).

Ardavan, A.

H. Ardavan, A. Ardavan, J. Singleton, “Frequency spec-trum of focused broadband pulses of electromagnetic radiation generated by polarization currents with superlumi-nally rotating distribution patterns,” J. Opt. Soc. Am. A 20, 2137–2155 (2003).
[CrossRef]

A. Ardavan, H. Ardavan, “Apparatus for generating focused electromagnetic radiation,” International patent application, PCT-GB99-02943 (September6, 1999).

J. Fopma, A. Ardavan, D. Halliday, J. Singleton, “Phase control electronics for a polarization synchrotron,” preprint available from the author (contact email: jsingle@lanl.gov).

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, David Halliday, A. Narduzzo, P. Goddard, “Experimental observation of non-spherically decaying emission from a rotating superlumnal source,” preprint available from the author (contact email: jsingle@lanl.gov).

Ardavan, H.

H. Ardavan, A. Ardavan, J. Singleton, “Frequency spec-trum of focused broadband pulses of electromagnetic radiation generated by polarization currents with superlumi-nally rotating distribution patterns,” J. Opt. Soc. Am. A 20, 2137–2155 (2003).
[CrossRef]

H. Ardavan, “Reply to Comments on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3010–3013 (2000).
[CrossRef]

H. Ardavan, “Generation of focused, nonspherically decaying pulses of electromagnetic radiation,” Phys. Rev. E 58, 6659–6684 (1998).
[CrossRef]

A. Ardavan, H. Ardavan, “Apparatus for generating focused electromagnetic radiation,” International patent application, PCT-GB99-02943 (September6, 1999).

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, David Halliday, A. Narduzzo, P. Goddard, “Experimental observation of non-spherically decaying emission from a rotating superlumnal source,” preprint available from the author (contact email: jsingle@lanl.gov).

Besieris, I. M.

Bleistein, N.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986).

Bolotovskii, B. M.

B. M. Bolotovskii, V. P. Bykov, “Radiation by charges moving faster than light,” Sov. Phys. Usp. 33, 477–487 (1990).
[CrossRef]

B. M. Bolotovskii, V. L. Ginzburg, “The Vavilov–Cerenkov effect and the Doppler effect in the motion of sources with superluminal velocity in vacuum,” Sov. Phys. Usp. 15, 184–192 (1972).
[CrossRef]

Bureau, F. J.

F. J. Bureau, “Divergent integrals and partial differential equations,” Commun. Pure Appl. Math. 8, 143–202 (1955).
[CrossRef]

Burridge, R.

R. Burridge, “Asymptotic evaluation of integrals related to time-dependent fields near caustics,” SIAM J. Appl. Math. 55, 390–409 (1995).
[CrossRef]

Bykov, V. P.

B. M. Bolotovskii, V. P. Bykov, “Radiation by charges moving faster than light,” Sov. Phys. Usp. 33, 477–487 (1990).
[CrossRef]

Cartlidge, E.

E. Cartlidge, “Money spinner or loopy idea?” Science, September 2003, 1463 (2003).

Chester, C.

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descent,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

Dartora, C. A.

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Durrani, M.

M. Durrani, “Revolutionary device polarizes opinions,” Phys. World, August2000, p. 9.

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Fopma, J.

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, David Halliday, A. Narduzzo, P. Goddard, “Experimental observation of non-spherically decaying emission from a rotating superlumnal source,” preprint available from the author (contact email: jsingle@lanl.gov).

J. Fopma, A. Ardavan, D. Halliday, J. Singleton, “Phase control electronics for a polarization synchrotron,” preprint available from the author (contact email: jsingle@lanl.gov).

Friedman, B.

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descent,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

Ginzburg, V. L.

B. M. Bolotovskii, V. L. Ginzburg, “The Vavilov–Cerenkov effect and the Doppler effect in the motion of sources with superluminal velocity in vacuum,” Sov. Phys. Usp. 15, 184–192 (1972).
[CrossRef]

V. L. Ginzburg, Theoretical Physics and Astrophysics (Pergamon, Oxford, UK, 1979), Chap. VIII.

Goddard, P.

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, David Halliday, A. Narduzzo, P. Goddard, “Experimental observation of non-spherically decaying emission from a rotating superlumnal source,” preprint available from the author (contact email: jsingle@lanl.gov).

Hadamard, J.

J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Yale U. Press, New Haven, Conn., 1923), Dover reprint (1952).

Halliday, D.

J. Fopma, A. Ardavan, D. Halliday, J. Singleton, “Phase control electronics for a polarization synchrotron,” preprint available from the author (contact email: jsingle@lanl.gov).

Halliday, David

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, David Halliday, A. Narduzzo, P. Goddard, “Experimental observation of non-spherically decaying emission from a rotating superlumnal source,” preprint available from the author (contact email: jsingle@lanl.gov).

Handelsman, R. A.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986).

Hannay, J. H.

J. H. Hannay, “Comment II on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3008–3009 (2000).
[CrossRef]

Hewish, A.

A. Hewish, “Comment I on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3007 (2000).
[CrossRef]

Hoskins, R. F.

R. F. Hoskins, Generalised Functions (Harwood, London, 1979), Chap. 7.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, UK, 1975).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, UK, 1975).

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Narduzzo, A.

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, David Halliday, A. Narduzzo, P. Goddard, “Experimental observation of non-spherically decaying emission from a rotating superlumnal source,” preprint available from the author (contact email: jsingle@lanl.gov).

Nobrega, K. Z.

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

Recami, E.

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

Reivelt, K.

K. Reivelt, P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 56611 (2002).
[CrossRef]

P. Saari, K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

Saari, P.

K. Reivelt, P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 56611 (2002).
[CrossRef]

P. Saari, K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

Sedky, S. M.

Shaarawi, A. M.

Singleton, J.

H. Ardavan, A. Ardavan, J. Singleton, “Frequency spec-trum of focused broadband pulses of electromagnetic radiation generated by polarization currents with superlumi-nally rotating distribution patterns,” J. Opt. Soc. Am. A 20, 2137–2155 (2003).
[CrossRef]

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, David Halliday, A. Narduzzo, P. Goddard, “Experimental observation of non-spherically decaying emission from a rotating superlumnal source,” preprint available from the author (contact email: jsingle@lanl.gov).

J. Fopma, A. Ardavan, D. Halliday, J. Singleton, “Phase control electronics for a polarization synchrotron,” preprint available from the author (contact email: jsingle@lanl.gov).

Sommerfeld, A.

A. Sommerfeld , “Zur Elektronentheorie (3 Teile),” Nach. Kgl. Ges. Wiss. Göttingen Math. Naturwiss. Klasse, 99–130, 363–439 (1904); 201–235 (1905).

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Boston, Mass., 1986).

Ursell, F.

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descent,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1995).

Wong, R.

R. Wong, Asymptotic Approximations of Integrals (Academic, Boston, Mass., 1989).

Zamboni-Rached, M.

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

Ziolkowski, R. W.

Commun. Pure Appl. Math.

F. J. Bureau, “Divergent integrals and partial differential equations,” Commun. Pure Appl. Math. 8, 143–202 (1955).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[CrossRef]

J. Opt. Soc. Am. A

Nach. Kgl. Ges. Wiss. Göttingen Math. Naturwiss. Klasse

A. Sommerfeld , “Zur Elektronentheorie (3 Teile),” Nach. Kgl. Ges. Wiss. Göttingen Math. Naturwiss. Klasse, 99–130, 363–439 (1904); 201–235 (1905).

New Sci.

N. Appleyard, B. Appleby, “Warp speed,” New Sci.April 2001, 28–31 (2001).

Phys. Rev. E

H. Ardavan, “Generation of focused, nonspherically decaying pulses of electromagnetic radiation,” Phys. Rev. E 58, 6659–6684 (1998).
[CrossRef]

A. Hewish, “Comment I on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3007 (2000).
[CrossRef]

J. H. Hannay, “Comment II on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3008–3009 (2000).
[CrossRef]

H. Ardavan, “Reply to Comments on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3010–3013 (2000).
[CrossRef]

K. Reivelt, P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 56611 (2002).
[CrossRef]

Phys. Rev. Lett.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

P. Saari, K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

Phys. World

M. Durrani, “Revolutionary device polarizes opinions,” Phys. World, August2000, p. 9.

Proc. Cambridge Philos. Soc.

C. Chester, B. Friedman, F. Ursell, “An extension of the method of steepest descent,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[CrossRef]

Science

E. Cartlidge, “Money spinner or loopy idea?” Science, September 2003, 1463 (2003).

SIAM J. Appl. Math.

R. Burridge, “Asymptotic evaluation of integrals related to time-dependent fields near caustics,” SIAM J. Appl. Math. 55, 390–409 (1995).
[CrossRef]

Sov. Phys. Usp.

B. M. Bolotovskii, V. P. Bykov, “Radiation by charges moving faster than light,” Sov. Phys. Usp. 33, 477–487 (1990).
[CrossRef]

B. M. Bolotovskii, V. L. Ginzburg, “The Vavilov–Cerenkov effect and the Doppler effect in the motion of sources with superluminal velocity in vacuum,” Sov. Phys. Usp. 15, 184–192 (1972).
[CrossRef]

Other

V. L. Ginzburg, Theoretical Physics and Astrophysics (Pergamon, Oxford, UK, 1979), Chap. VIII.

A. Ardavan, H. Ardavan, “Apparatus for generating focused electromagnetic radiation,” International patent application, PCT-GB99-02943 (September6, 1999).

J. Fopma, A. Ardavan, D. Halliday, J. Singleton, “Phase control electronics for a polarization synchrotron,” preprint available from the author (contact email: jsingle@lanl.gov).

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, David Halliday, A. Narduzzo, P. Goddard, “Experimental observation of non-spherically decaying emission from a rotating superlumnal source,” preprint available from the author (contact email: jsingle@lanl.gov).

L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, UK, 1975).

J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Yale U. Press, New Haven, Conn., 1923), Dover reprint (1952).

J. J. Stamnes, Waves in Focal Regions (Hilger, Boston, Mass., 1986).

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986).

R. Wong, Asymptotic Approximations of Integrals (Academic, Boston, Mass., 1989).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1995).

R. F. Hoskins, Generalised Functions (Harwood, London, 1979), Chap. 7.

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Figures (6)

Fig. 1
Fig. 1

Relationship between observation time t P and emission time t for an observation point that lies (a) inside or on, (b) on the cusp of, and (c) outside the envelope of the wave fronts or the bifurcation surface shown in Figs. 2 and 3. This relationship is given by t P = t + R ( t ) / c h ( t ;   r ,   φ ,   z ;   r P ,   φ P ,   z P ) , which applies to the envelope when the position ( r ,   φ ,   z ) of the source point is fixed and to the bifurcation surface when the location ( r P ,   φ P ,   z P ) of the observer is fixed. The maxima and minima of curve (a), at which d R / d t = - c , occur on the sheets ϕ + and ϕ - of the envelope or the bifurcation surface, respectively (Figs. 2 and 3). The inflection points of curve (b), at which d 2 R / d t 2 = 0 , occur on the cusp curve of the envelope or the bifurcation surface (Fig. 4).

Fig. 2
Fig. 2

(a) Envelope of the spherical wave fronts emanating from a source point S that moves with a constant angular velocity ω on a circle of radius r = 2.5 c / ω   ( r ˆ r ω / c = 2.5 ) . The dashed circles designate the orbit of S and the light cylinder r P = c / ω   ( r^P = 1 ) . The curves to which the emitted wave fronts are tangent are the cross sections of the two sheets ϕ ± of the envelope with the plane of source’s orbit. (b) Three-dimensional view of the light cylinder and the envelope of wave fronts for the same source point S. The tube-like surface constituting the envelope is symmetric with respect to the plane of the orbit. The cusp along which the two sheets of this envelope meet touches, and is tangential to, the light cylinder at a point on the plane of the source’s orbit and spirals around the rotation axis out into the radiation zone. It approaches the cone θ P = arcsin ( 1 / r ˆ ) as R P tends to infinity ( R P , θ P , and φ P are the spherical coordinates of the observation point P).

Fig. 3
Fig. 3

Bifurcation surface (i.e., locus of source points that approach the observer along the radiation direction with the speed of light at the retarded time) associated with the observation point P at the observation time t P (the motion of the source is clockwise). The cusp C b , along which the two sheets of the bifurcation surface meet, touches and is tangent to the light cylinder ( r ˆ = 1 ) at a point on the plane passing through P normal to the rotation axis. This cusp curve is the locus of source points that approach the observer not only with the speed of light ( d R / d t = - c ) but also with zero acceleration ( d 2 R / d t 2 = 0 ) along the radiation direction. For an observation point in the radiation zone, the spiraling surface that issues from P undergoes a large number of turns, in which its two sheets intersect one another, before reaching the light cylinder.

Fig. 4
Fig. 4

Close-up of a segment of the cusp curve in Fig. 3. The figure shows the section 0 < z ˆ - z ^ P < 5 of the light cylinder ( r ˆ = 1 ) and the two sheets ϕ ± of the bifurcation surface (the locus of source points that approach the observer along the radiation direction with the speed of light at the retarded time) in the vicinity of its cusp curve C b (the locus of source points that approach the observer with the speed of light and zero acceleration) for an observer who is located at r^P = 3 , φ^P = 0 . The cusp curve C b is symmetrical with respect to the plane z = z P passing through the observation point P. The value G j in of the Green’s function G j inside the bifurcation surface diverges on the inner sides of the two sheets ϕ + and ϕ - . The value G j out of G j outside the bifurcation surface undergoes a jump across the strip bordering on the cusp curve onto which these two sheets coalescence (in the limit R^P 1 ) .

Fig. 5
Fig. 5

Projection Δ = 0 of the cusp curve of the bifurcation surface onto ( r ,   z ) space. The two sheets ϕ ± ( r ,   z ) of the bifurcation surface exist only for the values of ( r ,   z ) in Δ 0 . For a given ( r ,   z ) in Δ 0 , the source point ( r ,   φ ˆ ,   z ) lies within the bifurcation surface for ϕ - < φ ˆ - φ^P < ϕ + and outside this surface for other values of φ ˆ . The source elements whose ( r ,   z ) coordinates fall in Δ < 0 approach the observer with a speed d R / d t < c at the retarded time and so make contributions toward the field that are no different from those made in the subluminal regime. The projection of the cusp curve of the envelope of wave fronts onto the ( r P ,   z P ) space is also given by Δ = 0 and has the same shape: The function Δ is invariant under the transformation ( r ,   z ;   r P ,   z P ) ( r P ,   z P ;   r ,   z ) . That the cusp curve of the envelope approaches the cone θ P = arcsin ( 1 / r ˆ ) as R P tends to infinity can be seen here from the slopes of the asymptotes of the curve Δ = 0 : These asymptotes, and so the segments z ˆ > z ^ P and z ˆ < z ^ P of the cusp curve, lie in the plane z ˆ = z ^ P for r ˆ = 1 + but open up and tend toward the vertical as r ˆ becomes increasingly greater than 1.

Fig. 6
Fig. 6

The χ dependence of the Green’s function G 1 in the far-field limit where p 1 = 0 . The values of this function inside and outside the interval - 1 < χ < 1 represent G 1 in and G 1 out , respectively: The space inside the bifurcation surface [the interval ϕ - < ϕ < ϕ + at a given ( r ,   z ) ] is mapped into | χ | < 1 and that outside the bifurcation surface into | χ | > 1 . The function G 1 in diverges on the inner sides, χ = 1 - and χ = - 1 + , of the two sheets ϕ + and ϕ - of the bifurcation surface. The function G 1 out is discontinuous across a two-dimensional strip bordering on the cusp curve of the bifurcation surface: Even at points close to this cusp curve where the separation ϕ + - ϕ - of the two sheets of the bifurcation surface tends to zero (Figs. 3 and 4), the vanishingly small interval ϕ - < ϕ < ϕ + in ϕ is mapped into the finite interval - 1 < χ < 1 in χ, so that the difference G 1 out | χ = 1 + - G 1 out | χ = - 1 - between the values of the function G 1 out on opposite sides of the strip in question remains nonzero.

Tables (2)

Tables Icon

Table 1 Definition of Frequencies and Numbers Used To Describe the Source and the Emitted Radiation

Tables Icon

Table 2 State of Polarization of the Nonspherically Decaying Component of the Emitted Radiation for Different Ranges of Ω/ω and Different Orientations of the Emitting Polarization Current

Equations (90)

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r = const . , z = const . , φ = φ ˆ + ω t ,
E ( x P ,   t P ) = q t ret ( 1 - | x ˙ | 2 / c 2 ) ( n ˆ - x ˙ / c ) | 1 - n ˆ     x ˙ / c | 3 R 2 ( t ) + n ˆ × { ( n ˆ - x ˙ / c ) × x ¨ } c 2 | 1 - n ˆ     x ˙ / c | 3 R ( t )
R ( t ) = [ ( z P - z ) 2 + r P 2 + r 2 - 2 r P r   cos ( φ P - φ ˆ - ω t ) ] 1 / 2
1 - n ˆ     x ˙ / c = h ˙ ( t ) = 1 - r P ( r ω / c ) sin ( φ P - φ + ω t ) / R ( t ) = 0 ,
t P = t P ± + 1 2   h ¨ ( t ± ) ( t - t ± ) 2 + ,
| 1 - n ˆ     x ˙ / c |  =  | h ˙ ( t ) | [ 2 h ¨ ( t ± ) ( t P - t P ± ) ] 1 / 2
J ( a ) 0 a d x f / a = - ( a - x ) - 1 / 2 | x = a + a - 1 / 2 ,
0 a d x ( a - x ) - 3 / 2 F ( x )
= 2 ( a - x ) - 1 / 2 F ( x ) | a - 2 a - 1 / 2 F ( 0 ) - 2 0 a d x ( a - x ) - 1 / 2 F ( x ) / x ,
P r , φ , z ( r ,   φ ,   z ,   t ) = s r , φ , z ( r ,   z ) cos ( m φ ˆ ) cos ( Ω t ) ,
- π < φ ˆ π ,
φ ˆ φ - ω t ,
A μ ( x P ,   t P ) = c - 1 d 3 x d tj μ ( x ,   t ) δ ( t P - t - R / c ) / R ,
μ = 0 , , 3 ,
E = - P A 0 - A / ( ct P ) , B = P × A
j = 1 2   i ω μ = μ ± μ   exp [ - i ( μ φ ˆ - Ω φ / ω ) ] s ,
- π < φ ˆ π ,
A = 1 2   i ( ω / c ) μ = μ ± V d V μ   exp ( - i μ φ ˆ ) s× Δ φ d φ   exp ( i Ω φ / ω ) δ ( g - ϕ ) / R ( φ ) ,
R ( φ ) = [ ( z P - z ) 2 + r P 2 + r 2 - 2 r P r   cos ( φ P - φ ) ] 1 / 2 ;
g φ - φ P + R ˆ ( φ ) ,
B 1 2   i ( ω / c ) 2 μ = μ ± V d V μ   exp ( - i μ φ ˆ ) Δ φ d φ   exp ( i Ω φ / ω ) n ˆ × s δ ( g - ϕ ) / R ( φ ) ,
e^r e^φ e^z = cos ( φ - φ P ) sin ( φ - φ P ) 0 - sin ( φ - φ P ) cos ( φ - φ P ) 0 0 0 1 e^rP e^φP e^zP .
lim R   n ˆ = sin   θ P e^rP + cos   θ P e^zP , θ P arctan ( r P / z P ) ,
n ˆ × s = [ s r   cos   θ P   cos ( φ - φ P ) - s φ   cos   θ P   sin ( φ - φ P ) - s z   sin   θ P ] e^ + [ s φ   cos ( φ - φ P ) + s r   sin ( φ - φ P ) ] e^ ,
B - 1 2   i ( ω / c ) 2 μ = μ ± V d V μ   exp ( - i μ φ ˆ ) j = 1 3 u j G j / φ ˆ ,
u 1 s r   cos   θ P e^ + s φ e^ , u 2 - s φ   cos   θ P e^ + s r e^ ,
u 3 - s z   sin   θ P e^ ,
G 1 G 2 G 3 = Δ φ d φ   δ ( g - ϕ ) R   exp ( i Ω φ / ω ) cos ( φ - φ P ) sin ( φ - φ P ) 1 .
G 3 | Ω = 0 = φ = φ j   1 R | g / φ | ,
φ ± = φ P + 2 π - arccos [ ( 1 Δ 1 / 2 ) / ( r ˆ r^P ) ] ,
Δ ( r ^ P 2 - 1 ) ( r ^ 2 - 1 ) - ( z ˆ - z ^ P ) 2 ,
ϕ = ϕ ± g ( φ ± ) = 2 π - arccos [ ( 1 Δ 1 / 2 ) / ( r ˆ r^P ) ] + R ^ ± ,
R ^ ± R ˆ ( φ ± ) = [ ( z ˆ - z ^ P ) 2 + r ^ 2 + r ^ P 2 - 2 ( 1 Δ 1 / 2 ) ] 1 / 2 .
g ( φ ) = 1 3 ν 3 - c 1 2 ν + c 2 ,
c 1 ( 3 4 ) 1 / 3 ( ϕ + - ϕ - ) 1 / 3 , c 2 1 2 ( ϕ + + ϕ - ) .
G j = Δ ν d ν f j ( ν ) δ ( 1 3   ν 3 - c 1 2 ν + c 2 - ϕ ) ,
f 1 f 2 f 3 = R - 1 ( d φ / d ν ) exp ( i Ω φ / ω ) cos ( φ - φ P ) sin ( φ - φ P ) 1 ,
G j - d ν ( p j + q j ν ) δ ( 1 3   ν 3 - c 1 2 ν + c 2 - ϕ ) ,
p j = 1 2   ( f j | ν = c 1 + f j | ν = - c 1 ) ,
q j = 1 2   c 1 - 1 ( f j | ν = c 1 - f j | ν = - c 1 )
G j = G j in | χ | < 1 G j out | χ | > 1 ,
G j in 2 c 1 - 2 ( 1 - χ 2 ) - 1 / 2 [ p j   cos ( 1 3   arcsin   χ ) - c 1 q j   sin ( 2 3   arcsin   χ ) ] ,
G j out c 1 - 2 ( χ 2 - 1 ) - 1 / 2 [ p j   sinh ( 1 3   arccosh | χ | ) + c 1 q j   sgn ( χ ) sinh ( 2 3   arccosh | χ | ) ] ,
G j out | ϕ = ϕ ± = G j out | χ = ± 1 ( p j ± 2 c 1 q j ) / ( 3 c 1 2 ) ,
B in , out = - 1 2 i ( ω / c ) 2 j = 1 3 Δ 0 r d r d z u j K j in , out ,
K j in μ = μ ± ϕ - ϕ + d ϕ μ   exp ( - i μ φ ˆ ) G j in / φ ˆ ,
K j out μ = μ ± - π - φ^P ϕ - + ϕ + π - φ^P d ϕ μ   exp ( - i μ φ ˆ ) G j out / φ ˆ .
K j in = μ = μ ± μ   exp ( - i μ φ ˆ ) G j in ϕ - ϕ + + i ϕ - ϕ + d ϕ μ 2 exp ( - i μ φ ˆ ) G j in ,
F { K j in } = i μ = μ ± ϕ - ϕ + d ϕ μ 2   exp ( - i μ φ ˆ ) G j in .
K j out = μ = μ ± [ μ   exp ( - i μ φ ˆ ) G j out ] ϕ - ϕ + + i - π - φ^P ϕ - + ϕ + π - φ^P d ϕ μ 2 exp ( - i μ φ ˆ ) G j out ,
B s = 1 2   ( ω / c ) 2 μ = μ ± μ 2 Δ 0 r d r d z - π π d φ ˆ   exp ( - i μ φ ˆ ) j = 1 3 u j G j
B ns - 1 2   i ( ω / c ) 2 j = 1 3 Δ 0 r d r d z u j K j edge
K j edge μ = μ ± [ μ   exp ( - i μ φ ˆ ) G j out ] ϕ - ϕ +
K j edge = 2 3   μ = μ ± μ   exp [ - i μ ( φ^P + c 2 ) ]× [ 2 c 1 - 1 q j   cos ( 2 3   μ c 1 3 ) - i c 1 - 2 p j   sin ( 2 3   μ c 1 3 ) ] ,
K j edge = 4 3   c 1 - 1 q j μ = μ ± μ   exp [ - i μ ( φ^P + ϕ - ) ] + ,
c 1 = 2 - 1 / 3 ( r ^ 2 r ^ P 2 - 1 ) - 1 / 2 Δ 1 / 2 + O ( Δ ) .
q j 2 2 / 3 ( ω / c ) R ^ P - 1   exp [ i ( Ω / ω ) ( φ P + 3 π / 2 ) ] q ¯ j ,
q ¯ j ( 1 - i Ω / ω i Ω / ω )
B ns - 4 3   i   exp [ i ( Ω / ω ) ( φ P + 3 π / 2 ) ]× μ = μ ± μ   exp ( - i μ φ^P ) j = 1 3 q ¯ j× Δ 0 r ˆ d r ˆ d z ˆ Δ - 1 / 2 u j   exp ( - i μ ϕ - ) .
r ˆ = r ^ C ( z ˆ ) { 1 2 ( r ^ P 2 + 1 ) - [ 1 4 ( r ^ P 2 - 1 ) 2 - ( z ˆ - z ^ P ) 2 ] 1 / 2 } 1 / 2 ,
φ = φ C ( z ˆ ) φ P + 2 π - arccos ( r ^ C / r^P ) .
φ = φ P + 2 π - arccos [ 1 / ( r ˆ r^P ) ] ,
ϕ - = ϕ C - [ 1 2 ( r ^ P 2 - 1 ) ( r ^ C 2 - 1 ) - 1 - 1 ] R ^ C - 1 ( r ˆ - r ^ C ) 2 + ,
R ^ C [ ( z ^ P - z ˆ ) 2 + r ^ P 2 - r ^ C 2 ] 1 / 2 ,
Δ 0 r ˆ d r ˆ Δ - 1 / 2 u j   exp ( - i μ ϕ - )
exp ( - i μ ϕ C ) r ^ C ( r ^ C 2 - 1 ) - 1 u j | C 0 r ^ > - r ^ C d η× exp { i μ [ 1 2 ( r ^ P 2 - 1 ) ( r ^ C 2 - 1 ) - 1 - 1 ] R ^ C - 1 η 2 } ,
r ^ C cosec   θ P , ϕ C 3 π / 2 + R ^ C ,
R ^ C R^P - z ˆ   cos   θ P ,
Δ 0 r ˆ d r ˆ Δ - 1 / 2 u j   exp ( - i μ ϕ - )
( π / 2 ) 1 / 2 R ^ P - 1 / 2 | sin   θ P   cos   θ P | - 1 | μ | - 1 / 2 × exp [ - i ( μ ϕ C - π 4   sgn   μ ) ] u j | C
B ns - 4 3   i ( 2 π ) 1 / 2 R ^ P - 1 / 2 | sin   2 θ P | - 1   exp ( i Ω φ C / ω )× μ = μ ± | μ | 1 / 2   sgn ( μ ) exp ( i   π 4   sgn   μ ) × exp [ - i μ ( R^P - ω t P + φ C ) ] j = 1 3 q ¯ j - d z ˆ u j | C   exp ( i μ z ˆ   cos   θ P ) ,
s ¯ r , φ , z - d z ˆ s r , φ , z | C   exp ( i μ z ˆ   cos   θ P )
E ns 4 3   ( 2 π ) 1 / 2 R ^ P - 1 / 2 | sin   2 θ P | - 1   exp ( i Ω φ C / ω )× μ = μ ± | μ | 1 / 2   sgn ( μ ) exp ( i   π 4   sgn   μ ) × exp [ - i μ ( R^P - ω t P + φ C ) ] { ( i s ¯ φ + Ω s ¯ r / ω ) e^ - [ ( i s ¯ r - Ω s ¯ φ / ω ) cos   θ P + Ω s ¯ z   sin   θ P / ω ] e^ }
E s = E ˜ 0 s + 2 n = 1 E ˜ n s exp ( - i n φ^P ) ,
E ˜ n s 1 2 r ^ P - 1 exp { - i [ n ( R^P + 3 2 π ) - ( Ω / ω ) ( φ P + 3 2 π ) ] } ( r ^ > - r ^ < ) Q φ ˆ Q ¯ z + { m - m ,   Ω - Ω } ,
Q φ ˆ = - μ = μ ± μ 2 sin [ π ( n - μ ) ] / ( n - μ ) ,
Q ¯ z [ s ¯ r J n - Ω / ω ( n ) + i s ¯ φ J n - Ω / ω ( n ) ] e^ + [ ( s ¯ φ cos   θ P - s ¯ z sin   θ P ) J n - Ω / ω ( n ) - i s ¯ r cos   θ P J n - Ω / ω ( n ) ] e^
| E μ + ns |  4 3 ( 2 π ) 1 / 2 R ^ P - 1 / 2 | sin   2 θ P | - 1 μ + 1 / 2 | ( i s ¯ φ + Ω s ¯ r / ω ) e^ - [ ( i s ¯ r - Ω s ¯ φ / ω ) cos   θ P + Ω s ¯ z sin   θ P / ω ] e^ |
| E ˜ μ + s |  1 2 π r ^ P - 1 ( r ^ > - r ^ < ) μ + 2 | [ s ¯ r J m ( μ + ) + i s ¯ φ J m ( μ + ) ] e^ + [ ( s ¯ φ cos   θ P - s ¯ z sin   θ P ) J m ( μ + ) - i s ¯ r cos   θ P J m ( μ + ) ] e^ |
| E μ + ns | / | E ˜ μ + s | 4 3 ( 2 π ) 1 / 2 | sec   θ P | μ + - 3 / 2 ( Ω / ω ) | J m ( μ + ) | - 1 R ^ P 1 / 2× ( r ^ > - r ^ < ) - 1 ,
φ ± = φ c ( r ^ 2 r ^ P 2 - 1 ) - 1 / 2 Δ 1 / 2 + O ( Δ ) ,
R ^ ± = ( r ^ 2 r ^ P 2 - 1 ) 1 / 2 ± ( r ^ 2 r ^ P 2 - 1 ) - 1 / 2 Δ 1 / 2 + O ( Δ ) ,
( d g / d φ ) ( d φ / d ν ) = ν 2 - c 1 2 ,
( d 2 g / d φ 2 ) ( d φ / d ν ) 2 + ( d g / d φ ) ( d 2 φ / d ν 2 ) = 2 ν ,
d φ / d ν | ν = ± c 1 = ( 2 c 1 R ^ ) 1 / 2 / Δ 1 / 4
d 2 φ / d ν 2 | ν = ± c 1 = ± 1 3 Δ - 1 / 4 ( 2 R ^ / c 1 ) 1 / 2 [ 1 - ( 2 1 / 3 c 1 R ^ / Δ 1 / 2 ) 3 / 2 ( 1 ± 3 Δ 1 / 2 / R ^ 2 ) ]
p 1 2 1 / 3 ( ω / c ) R ^ P - 2 exp ( i Ω φ c / ω ) ,
p 2 - 2 1 / 3 ( ω / c ) R ^ P - 1 exp ( i Ω φ c / ω ) , p 3 - p 2 ,
q 1 2 2 / 3 ( ω / c ) R ^ P - 1 exp ( i Ω φ c / ω ) ,
q 2 - q 3 - i ( Ω / ω ) q 1 ,

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