Abstract

We consider the nonspherically decaying radiation field that is generated by a polarization current with a superluminally rotating distribution pattern in vacuum, a field that decays with the distance RP from its source as RP12, instead of RP1. It is shown (i) that the nonspherical decay of this emission remains in force at all distances from its source independently of the frequency of the radiation, (ii) that the part of the source that makes the main contribution toward the value of the nonspherically decaying field has a filamentary structure whose radial and azimuthal widths become narrower (as RP2 and RP3, respectively) the farther the observer is from the source, (iii) that the loci on which the waves emanating from this filament interfere constructively delineate a radiation subbeam that is nondiffracting in the polar direction, (iv) that the cross-sectional area of each nondiffracting subbeam increases as RP, instead of RP2, so that the requirements of conservation of energy are met by the nonspherically decaying radiation automatically, and (v) that the overall radiation beam within which the field decays nonspherically consists, in general, of the incoherent superposition of such coherent nondiffracting subbeams. These findings are related to the recent construction and use of superluminal sources in the laboratory and numerical models of the emission from them. We also briefly discuss the relevance of these results to the giant pulses received from pulsars.

© 2007 Optical Society of America

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    [CrossRef]
  5. A. V. Bessarab, A. A. Gorbunov, S. P. Martynenko, and N. A. Prudkoy, "Faster-than-light EMP source initiated by short X-ray pulse of laser plasma," IEEE Trans. Plasma Sci. 32, 1400-1403 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. H. Ardavan, "Generation of focused, nonspherically decaying pulses of electromagnetic radiation," Phys. Rev. E 58, 6659-6684 (1998).
    [CrossRef]
  10. A. Ardavan and H. Ardavan, "Apparatus for generating focused electromagnetic radiation," International patent application PCT-GB99-02943 (September 6, 1999).
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  12. A. Schmidt, H. Ardavan, J. Fasel, J. Singleton, and A. Ardavan, "Occurrence of concurrent 'orthogonal' polarization modes in the Liénard-Wiechert field of a rotating superluminal source," in Proceedings of the 363rd WE-Heraeus Seminar on Neutron Stars and Pulsars, W.Becker and H.H.Huang, eds. (2007), pp. 124-127. ArXiv:astro-ph/0701257.
  13. H. Ardavan, A. Ardavan, and J. Singleton, "Spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns," J. Opt. Soc. Am. A 21, 858-872 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. M. V. Popov, V. A. Soglasnov, V. I. Kondrat'ev, S. V. Kostyuk, and Y. P. Ilyasov, "Giant pulses--the main component of the radio emission of the Crab pulsar," Astron. Rep. 50, 55-61 (2006).
    [CrossRef]
  21. G. M. Lilley, R. Westley, A. H. Yates, and Y. R. Busing, "Some aspects of noise from supersonic aircraft," J. R. Aeronaut. Soc. 57, 396-414 (1953).
  22. M. V. Lowson, "Focusing of helicopter BVI noise," J. Sound Vib. 190, 477-494 (1996).
    [CrossRef]
  23. J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations (Yale U. Press, 1923; Dover reprint, 1952).
  24. R. F. Hoskins, Generalized Functions (Harwood, 1979), Chap. 7.
  25. V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).
  26. N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, 1986).
  27. A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, and S. M. Sedky, "Generation of approximate focus-wave-mode pulses from wide-band dynamic Gaussian apertures," J. Opt. Soc. Am. A 12, 1954-1964 (1995).
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  28. K. Reivelt and P. Saari, "Experimental demonstration of realizability of optical focus wave modes," Phys. Rev. E 66, 056611 (2002).
    [CrossRef]
  29. E. Recami, M. Zamboni-Rached, K. Z. Nobrega, C. A. Dartora, and H. E. Hernandez, "On the localized superluminal solutions to the Maxwell equations," IEEE J. Sel. Top. Quantum Electron. 9, 59-73 (2003).
    [CrossRef]
  30. H. Ardavan, A. Ardavan, and J. Singleton, "Frequency spectrum of focused broadband pulses of electromagnetic radiation generated by polarization currents with superluminally rotating distribution patterns," J. Opt. Soc. Am. A 20, 2137-2155 (2003).
    [CrossRef]

2006 (3)

2005 (1)

B. M. Bolotovskii and A. V. Serov, "Radiation of superluminal sources in empty space," Phys. Usp. 48, 903-915 (2005).
[CrossRef]

2004 (4)

A. V. Bessarab, A. A. Gorbunov, S. P. Martynenko, and N. A. Prudkoy, "Faster-than-light EMP source initiated by short X-ray pulse of laser plasma," IEEE Trans. Plasma Sci. 32, 1400-1403 (2004).
[CrossRef]

A. Ardavan, W. Hayes, J. Singleton, H. Ardavan, J. Fopma, and D. Halliday, "Experimental observation of nonspherically-decaying radiation from a rotating superluminal source," J. Appl. Phys. 96, 7760-7777(E) (2004) (corrected version of 96, 4614-4613).
[CrossRef]

V. A. Soglasnov, M. V. Popov, N. Bartel, W. Cannon, A. Y. Novikov, V. I. Kondratiev, and V. I. Altunin, "Giant pulses from PSR B1937+21 with widths ⩽15 nanoseconds and tb=5×1039 K, the highest brightness temperature observed in the universe," Astrophys. J. 616, 439-451 (2004).
[CrossRef]

H. Ardavan, A. Ardavan, and J. Singleton, "Spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns," J. Opt. Soc. Am. A 21, 858-872 (2004).
[CrossRef]

2003 (3)

H. Ardavan, A. Ardavan, and J. Singleton, "Frequency spectrum of focused broadband pulses of electromagnetic radiation generated by polarization currents with superluminally rotating distribution patterns," J. Opt. Soc. Am. A 20, 2137-2155 (2003).
[CrossRef]

T. H. Hankins, J. S. Kern, J. C. Weatherall, and J. A. Eilek, "Nanosecond radio bursts from strong plasma turbulence in the Crab pulsar," Nature 422, 141-143 (2003).
[CrossRef] [PubMed]

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

2002 (1)

K. Reivelt and P. Saari, "Experimental demonstration of realizability of optical focus wave modes," Phys. Rev. E 66, 056611 (2002).
[CrossRef]

1999 (1)

S. Sallmen, D. C. Backer, T. H. Hankins, D. Moffett, and S. Lundgren, "Simultaneous dual-frequency observations of giant pulses from the Crab pulsar," Astrophys. J. 517, 460-471 (1999).
[CrossRef]

1998 (1)

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

1996 (1)

M. V. Lowson, "Focusing of helicopter BVI noise," J. Sound Vib. 190, 477-494 (1996).
[CrossRef]

1995 (1)

1990 (1)

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

1972 (2)

B. M. Bolotovskii and 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, "Vavilov-Cerenkov effect and anomalous Doppler effect in a medium in which wave phase velocity exceeds velocity of light in vacuum," Sov. Phys. JETP 35, 92-93 (1972).

1953 (1)

G. M. Lilley, R. Westley, A. H. Yates, and Y. R. Busing, "Some aspects of noise from supersonic aircraft," J. R. Aeronaut. Soc. 57, 396-414 (1953).

Altunin, V. I.

V. A. Soglasnov, M. V. Popov, N. Bartel, W. Cannon, A. Y. Novikov, V. I. Kondratiev, and V. I. Altunin, "Giant pulses from PSR B1937+21 with widths ⩽15 nanoseconds and tb=5×1039 K, the highest brightness temperature observed in the universe," Astrophys. J. 616, 439-451 (2004).
[CrossRef]

Ardavan, A.

H. Ardavan, A. Ardavan, and J. Singleton, "Spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns: reply to comment," J. Opt. Soc. Am. A 23, 1535-1539 (2006).
[CrossRef]

H. Ardavan, A. Ardavan, and J. Singleton, "Spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns," J. Opt. Soc. Am. A 21, 858-872 (2004).
[CrossRef]

A. Ardavan, W. Hayes, J. Singleton, H. Ardavan, J. Fopma, and D. Halliday, "Experimental observation of nonspherically-decaying radiation from a rotating superluminal source," J. Appl. Phys. 96, 7760-7777(E) (2004) (corrected version of 96, 4614-4613).
[CrossRef]

H. Ardavan, A. Ardavan, and J. Singleton, "Frequency spectrum of focused broadband pulses of electromagnetic radiation generated by polarization currents with superluminally rotating distribution patterns," J. Opt. Soc. Am. A 20, 2137-2155 (2003).
[CrossRef]

A. Ardavan and H. Ardavan, "Apparatus for generating focused electromagnetic radiation," International patent application PCT-GB99-02943 (September 6, 1999).

A. Schmidt, H. Ardavan, J. Fasel, J. Singleton, and A. Ardavan, "Occurrence of concurrent 'orthogonal' polarization modes in the Liénard-Wiechert field of a rotating superluminal source," in Proceedings of the 363rd WE-Heraeus Seminar on Neutron Stars and Pulsars, W.Becker and H.H.Huang, eds. (2007), pp. 124-127. ArXiv:astro-ph/0701257.

J. Singleton, A. Ardavan, H. Ardavan, J. Fopma, D. Halliday, and W. Hayes, "Experimental demonstration of emission from a superluminal polarization current--a new class of solid-state source for MHz-THz and beyond," in Digest of the 2004 Joint 29th International Conference on Infrared and Millimeter Waves and 12th International Conference on Terahertz Electronics (IEEE, 2004), pp. 591-592.
[CrossRef]

Ardavan, H.

H. Ardavan, A. Ardavan, and J. Singleton, "Spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns: reply to comment," J. Opt. Soc. Am. A 23, 1535-1539 (2006).
[CrossRef]

H. Ardavan, A. Ardavan, and J. Singleton, "Spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns," J. Opt. Soc. Am. A 21, 858-872 (2004).
[CrossRef]

A. Ardavan, W. Hayes, J. Singleton, H. Ardavan, J. Fopma, and D. Halliday, "Experimental observation of nonspherically-decaying radiation from a rotating superluminal source," J. Appl. Phys. 96, 7760-7777(E) (2004) (corrected version of 96, 4614-4613).
[CrossRef]

H. Ardavan, A. Ardavan, and J. Singleton, "Frequency spectrum of focused broadband pulses of electromagnetic radiation generated by polarization currents with superluminally rotating distribution patterns," J. Opt. Soc. Am. A 20, 2137-2155 (2003).
[CrossRef]

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

A. Schmidt, H. Ardavan, J. Fasel, J. Singleton, and A. Ardavan, "Occurrence of concurrent 'orthogonal' polarization modes in the Liénard-Wiechert field of a rotating superluminal source," in Proceedings of the 363rd WE-Heraeus Seminar on Neutron Stars and Pulsars, W.Becker and H.H.Huang, eds. (2007), pp. 124-127. ArXiv:astro-ph/0701257.

A. Ardavan and H. Ardavan, "Apparatus for generating focused electromagnetic radiation," International patent application PCT-GB99-02943 (September 6, 1999).

H. Ardavan, "The superluminal model of pulsars," in Pulsar Astronomy--2000 and Beyond, Vol. 202 of Astronomical Society of the Pacific Conference Series, M.Kramer, N.Wex, and Wielebinski, eds. (Astronomical Society of the Pacific, 2000), pp. 365-366.

J. Singleton, A. Ardavan, H. Ardavan, J. Fopma, D. Halliday, and W. Hayes, "Experimental demonstration of emission from a superluminal polarization current--a new class of solid-state source for MHz-THz and beyond," in Digest of the 2004 Joint 29th International Conference on Infrared and Millimeter Waves and 12th International Conference on Terahertz Electronics (IEEE, 2004), pp. 591-592.
[CrossRef]

Backer, D. C.

S. Sallmen, D. C. Backer, T. H. Hankins, D. Moffett, and S. Lundgren, "Simultaneous dual-frequency observations of giant pulses from the Crab pulsar," Astrophys. J. 517, 460-471 (1999).
[CrossRef]

Bartel, N.

V. A. Soglasnov, M. V. Popov, N. Bartel, W. Cannon, A. Y. Novikov, V. I. Kondratiev, and V. I. Altunin, "Giant pulses from PSR B1937+21 with widths ⩽15 nanoseconds and tb=5×1039 K, the highest brightness temperature observed in the universe," Astrophys. J. 616, 439-451 (2004).
[CrossRef]

Besieris, I. M.

Bessarab, A. V.

A. V. Bessarab, A. A. Gorbunov, S. P. Martynenko, and N. A. Prudkoy, "Faster-than-light EMP source initiated by short X-ray pulse of laser plasma," IEEE Trans. Plasma Sci. 32, 1400-1403 (2004).
[CrossRef]

Bleistein, N.

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

Bolotovskii, B. M.

B. M. Bolotovskii and A. V. Serov, "Radiation of superluminal sources in empty space," Phys. Usp. 48, 903-915 (2005).
[CrossRef]

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

B. M. Bolotovskii and 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]

Borovikov, V. A.

V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).

Busing, Y. R.

G. M. Lilley, R. Westley, A. H. Yates, and Y. R. Busing, "Some aspects of noise from supersonic aircraft," J. R. Aeronaut. Soc. 57, 396-414 (1953).

Bykov, V. P.

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

Cannon, W.

V. A. Soglasnov, M. V. Popov, N. Bartel, W. Cannon, A. Y. Novikov, V. I. Kondratiev, and V. I. Altunin, "Giant pulses from PSR B1937+21 with widths ⩽15 nanoseconds and tb=5×1039 K, the highest brightness temperature observed in the universe," Astrophys. J. 616, 439-451 (2004).
[CrossRef]

Dartora, C. A.

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

Eilek, J. A.

T. H. Hankins, J. S. Kern, J. C. Weatherall, and J. A. Eilek, "Nanosecond radio bursts from strong plasma turbulence in the Crab pulsar," Nature 422, 141-143 (2003).
[CrossRef] [PubMed]

Fasel, J.

A. Schmidt, H. Ardavan, J. Fasel, J. Singleton, and A. Ardavan, "Occurrence of concurrent 'orthogonal' polarization modes in the Liénard-Wiechert field of a rotating superluminal source," in Proceedings of the 363rd WE-Heraeus Seminar on Neutron Stars and Pulsars, W.Becker and H.H.Huang, eds. (2007), pp. 124-127. ArXiv:astro-ph/0701257.

Fopma, J.

A. Ardavan, W. Hayes, J. Singleton, H. Ardavan, J. Fopma, and D. Halliday, "Experimental observation of nonspherically-decaying radiation from a rotating superluminal source," J. Appl. Phys. 96, 7760-7777(E) (2004) (corrected version of 96, 4614-4613).
[CrossRef]

J. Singleton, A. Ardavan, H. Ardavan, J. Fopma, D. Halliday, and W. Hayes, "Experimental demonstration of emission from a superluminal polarization current--a new class of solid-state source for MHz-THz and beyond," in Digest of the 2004 Joint 29th International Conference on Infrared and Millimeter Waves and 12th International Conference on Terahertz Electronics (IEEE, 2004), pp. 591-592.
[CrossRef]

Ginzburg, V. L.

V. L. Ginzburg, "Vavilov-Cerenkov effect and anomalous Doppler effect in a medium in which wave phase velocity exceeds velocity of light in vacuum," Sov. Phys. JETP 35, 92-93 (1972).

B. M. Bolotovskii and 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]

Gorbunov, A. A.

A. V. Bessarab, A. A. Gorbunov, S. P. Martynenko, and N. A. Prudkoy, "Faster-than-light EMP source initiated by short X-ray pulse of laser plasma," IEEE Trans. Plasma Sci. 32, 1400-1403 (2004).
[CrossRef]

Hadamard, J.

J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations (Yale U. Press, 1923; Dover reprint, 1952).

Halliday, D.

A. Ardavan, W. Hayes, J. Singleton, H. Ardavan, J. Fopma, and D. Halliday, "Experimental observation of nonspherically-decaying radiation from a rotating superluminal source," J. Appl. Phys. 96, 7760-7777(E) (2004) (corrected version of 96, 4614-4613).
[CrossRef]

J. Singleton, A. Ardavan, H. Ardavan, J. Fopma, D. Halliday, and W. Hayes, "Experimental demonstration of emission from a superluminal polarization current--a new class of solid-state source for MHz-THz and beyond," in Digest of the 2004 Joint 29th International Conference on Infrared and Millimeter Waves and 12th International Conference on Terahertz Electronics (IEEE, 2004), pp. 591-592.
[CrossRef]

Handelsman, R. A.

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

Hankins, T. H.

T. H. Hankins, J. S. Kern, J. C. Weatherall, and J. A. Eilek, "Nanosecond radio bursts from strong plasma turbulence in the Crab pulsar," Nature 422, 141-143 (2003).
[CrossRef] [PubMed]

S. Sallmen, D. C. Backer, T. H. Hankins, D. Moffett, and S. Lundgren, "Simultaneous dual-frequency observations of giant pulses from the Crab pulsar," Astrophys. J. 517, 460-471 (1999).
[CrossRef]

Hannay, J. H.

Hayes, W.

A. Ardavan, W. Hayes, J. Singleton, H. Ardavan, J. Fopma, and D. Halliday, "Experimental observation of nonspherically-decaying radiation from a rotating superluminal source," J. Appl. Phys. 96, 7760-7777(E) (2004) (corrected version of 96, 4614-4613).
[CrossRef]

J. Singleton, A. Ardavan, H. Ardavan, J. Fopma, D. Halliday, and W. Hayes, "Experimental demonstration of emission from a superluminal polarization current--a new class of solid-state source for MHz-THz and beyond," in Digest of the 2004 Joint 29th International Conference on Infrared and Millimeter Waves and 12th International Conference on Terahertz Electronics (IEEE, 2004), pp. 591-592.
[CrossRef]

Hernandez, H. E.

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

Hoskins, R. F.

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

Ilyasov, Y. P.

M. V. Popov, V. A. Soglasnov, V. I. Kondrat'ev, S. V. Kostyuk, and Y. P. Ilyasov, "Giant pulses--the main component of the radio emission of the Crab pulsar," Astron. Rep. 50, 55-61 (2006).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Kern, J. S.

T. H. Hankins, J. S. Kern, J. C. Weatherall, and J. A. Eilek, "Nanosecond radio bursts from strong plasma turbulence in the Crab pulsar," Nature 422, 141-143 (2003).
[CrossRef] [PubMed]

Kondrat'ev, V. I.

M. V. Popov, V. A. Soglasnov, V. I. Kondrat'ev, S. V. Kostyuk, and Y. P. Ilyasov, "Giant pulses--the main component of the radio emission of the Crab pulsar," Astron. Rep. 50, 55-61 (2006).
[CrossRef]

Kondratiev, V. I.

V. A. Soglasnov, M. V. Popov, N. Bartel, W. Cannon, A. Y. Novikov, V. I. Kondratiev, and V. I. Altunin, "Giant pulses from PSR B1937+21 with widths ⩽15 nanoseconds and tb=5×1039 K, the highest brightness temperature observed in the universe," Astrophys. J. 616, 439-451 (2004).
[CrossRef]

Kostyuk, S. V.

M. V. Popov, V. A. Soglasnov, V. I. Kondrat'ev, S. V. Kostyuk, and Y. P. Ilyasov, "Giant pulses--the main component of the radio emission of the Crab pulsar," Astron. Rep. 50, 55-61 (2006).
[CrossRef]

Kramer, M.

D. Lorimer and M. Kramer, Handbook of Pulsar Astronomy (Cambridge U. Press, 2005).

Lilley, G. M.

G. M. Lilley, R. Westley, A. H. Yates, and Y. R. Busing, "Some aspects of noise from supersonic aircraft," J. R. Aeronaut. Soc. 57, 396-414 (1953).

Lorimer, D.

D. Lorimer and M. Kramer, Handbook of Pulsar Astronomy (Cambridge U. Press, 2005).

Lowson, M. V.

M. V. Lowson, "Focusing of helicopter BVI noise," J. Sound Vib. 190, 477-494 (1996).
[CrossRef]

Lundgren, S.

S. Sallmen, D. C. Backer, T. H. Hankins, D. Moffett, and S. Lundgren, "Simultaneous dual-frequency observations of giant pulses from the Crab pulsar," Astrophys. J. 517, 460-471 (1999).
[CrossRef]

Martynenko, S. P.

A. V. Bessarab, A. A. Gorbunov, S. P. Martynenko, and N. A. Prudkoy, "Faster-than-light EMP source initiated by short X-ray pulse of laser plasma," IEEE Trans. Plasma Sci. 32, 1400-1403 (2004).
[CrossRef]

Moffett, D.

S. Sallmen, D. C. Backer, T. H. Hankins, D. Moffett, and S. Lundgren, "Simultaneous dual-frequency observations of giant pulses from the Crab pulsar," Astrophys. J. 517, 460-471 (1999).
[CrossRef]

Nobrega, K. Z.

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

Novikov, A. Y.

V. A. Soglasnov, M. V. Popov, N. Bartel, W. Cannon, A. Y. Novikov, V. I. Kondratiev, and V. I. Altunin, "Giant pulses from PSR B1937+21 with widths ⩽15 nanoseconds and tb=5×1039 K, the highest brightness temperature observed in the universe," Astrophys. J. 616, 439-451 (2004).
[CrossRef]

Popov, M. V.

M. V. Popov, V. A. Soglasnov, V. I. Kondrat'ev, S. V. Kostyuk, and Y. P. Ilyasov, "Giant pulses--the main component of the radio emission of the Crab pulsar," Astron. Rep. 50, 55-61 (2006).
[CrossRef]

V. A. Soglasnov, M. V. Popov, N. Bartel, W. Cannon, A. Y. Novikov, V. I. Kondratiev, and V. I. Altunin, "Giant pulses from PSR B1937+21 with widths ⩽15 nanoseconds and tb=5×1039 K, the highest brightness temperature observed in the universe," Astrophys. J. 616, 439-451 (2004).
[CrossRef]

Prudkoy, N. A.

A. V. Bessarab, A. A. Gorbunov, S. P. Martynenko, and N. A. Prudkoy, "Faster-than-light EMP source initiated by short X-ray pulse of laser plasma," IEEE Trans. Plasma Sci. 32, 1400-1403 (2004).
[CrossRef]

Recami, E.

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

Reivelt, K.

K. Reivelt and P. Saari, "Experimental demonstration of realizability of optical focus wave modes," Phys. Rev. E 66, 056611 (2002).
[CrossRef]

Saari, P.

K. Reivelt and P. Saari, "Experimental demonstration of realizability of optical focus wave modes," Phys. Rev. E 66, 056611 (2002).
[CrossRef]

Sallmen, S.

S. Sallmen, D. C. Backer, T. H. Hankins, D. Moffett, and S. Lundgren, "Simultaneous dual-frequency observations of giant pulses from the Crab pulsar," Astrophys. J. 517, 460-471 (1999).
[CrossRef]

Schmidt, A.

A. Schmidt, H. Ardavan, J. Fasel, J. Singleton, and A. Ardavan, "Occurrence of concurrent 'orthogonal' polarization modes in the Liénard-Wiechert field of a rotating superluminal source," in Proceedings of the 363rd WE-Heraeus Seminar on Neutron Stars and Pulsars, W.Becker and H.H.Huang, eds. (2007), pp. 124-127. ArXiv:astro-ph/0701257.

Sedky, S. M.

Serov, A. V.

B. M. Bolotovskii and A. V. Serov, "Radiation of superluminal sources in empty space," Phys. Usp. 48, 903-915 (2005).
[CrossRef]

Shaarawi, A. M.

Singleton, J.

H. Ardavan, A. Ardavan, and J. Singleton, "Spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns: reply to comment," J. Opt. Soc. Am. A 23, 1535-1539 (2006).
[CrossRef]

H. Ardavan, A. Ardavan, and J. Singleton, "Spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns," J. Opt. Soc. Am. A 21, 858-872 (2004).
[CrossRef]

A. Ardavan, W. Hayes, J. Singleton, H. Ardavan, J. Fopma, and D. Halliday, "Experimental observation of nonspherically-decaying radiation from a rotating superluminal source," J. Appl. Phys. 96, 7760-7777(E) (2004) (corrected version of 96, 4614-4613).
[CrossRef]

H. Ardavan, A. Ardavan, and J. Singleton, "Frequency spectrum of focused broadband pulses of electromagnetic radiation generated by polarization currents with superluminally rotating distribution patterns," J. Opt. Soc. Am. A 20, 2137-2155 (2003).
[CrossRef]

A. Schmidt, H. Ardavan, J. Fasel, J. Singleton, and A. Ardavan, "Occurrence of concurrent 'orthogonal' polarization modes in the Liénard-Wiechert field of a rotating superluminal source," in Proceedings of the 363rd WE-Heraeus Seminar on Neutron Stars and Pulsars, W.Becker and H.H.Huang, eds. (2007), pp. 124-127. ArXiv:astro-ph/0701257.

J. Singleton, A. Ardavan, H. Ardavan, J. Fopma, D. Halliday, and W. Hayes, "Experimental demonstration of emission from a superluminal polarization current--a new class of solid-state source for MHz-THz and beyond," in Digest of the 2004 Joint 29th International Conference on Infrared and Millimeter Waves and 12th International Conference on Terahertz Electronics (IEEE, 2004), pp. 591-592.
[CrossRef]

Soglasnov, V. A.

M. V. Popov, V. A. Soglasnov, V. I. Kondrat'ev, S. V. Kostyuk, and Y. P. Ilyasov, "Giant pulses--the main component of the radio emission of the Crab pulsar," Astron. Rep. 50, 55-61 (2006).
[CrossRef]

V. A. Soglasnov, M. V. Popov, N. Bartel, W. Cannon, A. Y. Novikov, V. I. Kondratiev, and V. I. Altunin, "Giant pulses from PSR B1937+21 with widths ⩽15 nanoseconds and tb=5×1039 K, the highest brightness temperature observed in the universe," Astrophys. J. 616, 439-451 (2004).
[CrossRef]

Weatherall, J. C.

T. H. Hankins, J. S. Kern, J. C. Weatherall, and J. A. Eilek, "Nanosecond radio bursts from strong plasma turbulence in the Crab pulsar," Nature 422, 141-143 (2003).
[CrossRef] [PubMed]

Westley, R.

G. M. Lilley, R. Westley, A. H. Yates, and Y. R. Busing, "Some aspects of noise from supersonic aircraft," J. R. Aeronaut. Soc. 57, 396-414 (1953).

Yates, A. H.

G. M. Lilley, R. Westley, A. H. Yates, and Y. R. Busing, "Some aspects of noise from supersonic aircraft," J. R. Aeronaut. Soc. 57, 396-414 (1953).

Zamboni-Rached, M.

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

Ziolkowski, R. W.

Astron. Rep. (1)

M. V. Popov, V. A. Soglasnov, V. I. Kondrat'ev, S. V. Kostyuk, and Y. P. Ilyasov, "Giant pulses--the main component of the radio emission of the Crab pulsar," Astron. Rep. 50, 55-61 (2006).
[CrossRef]

Astrophys. J. (2)

S. Sallmen, D. C. Backer, T. H. Hankins, D. Moffett, and S. Lundgren, "Simultaneous dual-frequency observations of giant pulses from the Crab pulsar," Astrophys. J. 517, 460-471 (1999).
[CrossRef]

V. A. Soglasnov, M. V. Popov, N. Bartel, W. Cannon, A. Y. Novikov, V. I. Kondratiev, and V. I. Altunin, "Giant pulses from PSR B1937+21 with widths ⩽15 nanoseconds and tb=5×1039 K, the highest brightness temperature observed in the universe," Astrophys. J. 616, 439-451 (2004).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

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

IEEE Trans. Plasma Sci. (1)

A. V. Bessarab, A. A. Gorbunov, S. P. Martynenko, and N. A. Prudkoy, "Faster-than-light EMP source initiated by short X-ray pulse of laser plasma," IEEE Trans. Plasma Sci. 32, 1400-1403 (2004).
[CrossRef]

J. Appl. Phys. (1)

A. Ardavan, W. Hayes, J. Singleton, H. Ardavan, J. Fopma, and D. Halliday, "Experimental observation of nonspherically-decaying radiation from a rotating superluminal source," J. Appl. Phys. 96, 7760-7777(E) (2004) (corrected version of 96, 4614-4613).
[CrossRef]

J. Opt. Soc. Am. A (5)

J. R. Aeronaut. Soc. (1)

G. M. Lilley, R. Westley, A. H. Yates, and Y. R. Busing, "Some aspects of noise from supersonic aircraft," J. R. Aeronaut. Soc. 57, 396-414 (1953).

J. Sound Vib. (1)

M. V. Lowson, "Focusing of helicopter BVI noise," J. Sound Vib. 190, 477-494 (1996).
[CrossRef]

Nature (1)

T. H. Hankins, J. S. Kern, J. C. Weatherall, and J. A. Eilek, "Nanosecond radio bursts from strong plasma turbulence in the Crab pulsar," Nature 422, 141-143 (2003).
[CrossRef] [PubMed]

Phys. Rev. E (2)

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

K. Reivelt and P. Saari, "Experimental demonstration of realizability of optical focus wave modes," Phys. Rev. E 66, 056611 (2002).
[CrossRef]

Phys. Usp. (1)

B. M. Bolotovskii and A. V. Serov, "Radiation of superluminal sources in empty space," Phys. Usp. 48, 903-915 (2005).
[CrossRef]

Sov. Phys. JETP (1)

V. L. Ginzburg, "Vavilov-Cerenkov effect and anomalous Doppler effect in a medium in which wave phase velocity exceeds velocity of light in vacuum," Sov. Phys. JETP 35, 92-93 (1972).

Sov. Phys. Usp. (2)

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

B. M. Bolotovskii and 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 (10)

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

A. Ardavan and H. Ardavan, "Apparatus for generating focused electromagnetic radiation," International patent application PCT-GB99-02943 (September 6, 1999).

H. Ardavan, "The superluminal model of pulsars," in Pulsar Astronomy--2000 and Beyond, Vol. 202 of Astronomical Society of the Pacific Conference Series, M.Kramer, N.Wex, and Wielebinski, eds. (Astronomical Society of the Pacific, 2000), pp. 365-366.

A. Schmidt, H. Ardavan, J. Fasel, J. Singleton, and A. Ardavan, "Occurrence of concurrent 'orthogonal' polarization modes in the Liénard-Wiechert field of a rotating superluminal source," in Proceedings of the 363rd WE-Heraeus Seminar on Neutron Stars and Pulsars, W.Becker and H.H.Huang, eds. (2007), pp. 124-127. ArXiv:astro-ph/0701257.

D. Lorimer and M. Kramer, Handbook of Pulsar Astronomy (Cambridge U. Press, 2005).

J. Singleton, A. Ardavan, H. Ardavan, J. Fopma, D. Halliday, and W. Hayes, "Experimental demonstration of emission from a superluminal polarization current--a new class of solid-state source for MHz-THz and beyond," in Digest of the 2004 Joint 29th International Conference on Infrared and Millimeter Waves and 12th International Conference on Terahertz Electronics (IEEE, 2004), pp. 591-592.
[CrossRef]

J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations (Yale U. Press, 1923; Dover reprint, 1952).

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

V. A. Borovikov, Uniform Stationary Phase Method (Institution of Electrical Engineers, 1994).

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

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Figures (5)

Fig. 1
Fig. 1

(a) Cross section of the Čerenkov-like envelope (bold curves) of the spherical Huygens wavefronts (fine circles) emitted by a small element S within an extended, rotating superluminal source of angular velocity ω. S is on a circle of radius r = 2.5 c ω , or, in our dimensionless units, r ̂ r ω c = 2.5 ; i.e., its instantaneous linear velocity is r ω = 2.5 c . The cross section is in the plane of S’s rotation; dashed circles designate the light cylinder r P = c ω ( r ̂ P = 1 ) and the orbit of S. (b) Three-dimensional view of the light cylinder, the envelope of wavefronts emanating from S, and the cusp along which the two sheets ϕ ± of this envelope meet tangentially. (c) The relationship between reception time t P and source (retarded) time t [Eq. (4)] plotted for r ̂ = 2.5 and three different observation points. The maxima and minima of curve (i) occur on the sheets ϕ ± of the envelope, respectively. Curve (ii) corresponds to an observation point that is located on the cusp. Note that the waves emitted during an interval of retarded time centered at t c are received over a much shorter interval of observation time at t P c . Curve (iii) is for an observation point that is never crossed by the rotating sheets of the envelope (after [13].)

Fig. 2
Fig. 2

Schematic illustration of the light cylinder r = c ω , the filamentary part of the source that approaches the observeration point with the speed of light and zero acceleration at the retarded time, the orbit of this filamentary source, and the subbeam formed by the bundle of cusps that emanate from the constituent volume elements of this filament. The subbeam is diffractionless in the direction of θ P . The figure represents a snapshot corresponding to a fixed value of the observation time t P . The polar width δ θ P of this subbeam decreases with the distance R ̂ P in such a way that the thickness R ̂ P δ θ P of the subbeam in the polar direction remains constant: it equals the projection, δ z ̂ sin θ P , of the z ̂ extent, δ z ̂ , of the contributing filamentary source onto a direction normal to the line of sight. The azimuthal width of the subbeam, on the other hand, is subject to diffraction as in any other radiation beam: δ φ P is independent of R ̂ P .

Fig. 3
Fig. 3

Bifurcation surface of the observation point P for a source whose rotational motion is counterclockwise. The source points that lie inside this surface influence the field at P at three distinct values of the retarded time, while those that lie outside this surface influence the field at only a single value of the retarded time. The source elements on the filamentary locus at which the cusp curve of this surface intersects the source distribution approach P with the speed of light and zero acceleration at the retarded time and so generate a nonspherically decaying field at P.

Fig. 4
Fig. 4

(a) Segment of the cusp of the envelope of wavefronts emitted by a rotating point source with the speed r ω = 3 c . This curve is tangent to the light cylinder at the point ( r ̂ P = 1 , φ P = φ 3 π 2 , z ̂ P = z ̂ ) on the plane of the orbit and spirals outward into the far zone. Note that this figure represents a snapshot at a fixed value of the observation time t P . The cusp curve of the bifurcation surface of an observer P shown in Fig. 3 has precisely the same shape, except that it resides in the space of source points, instead of the space of observation points, and spirals in the counterclockwise direction: it is tangent to the light cylinder at the point ( r ̂ = 1 , φ = φ P + 3 π 2 , z ̂ = z ̂ P ). (b) The projections of the cusp curve of the bifurcation surface and a localized source distribution onto the ( r ̂ , z ̂ ) plane. Only the part of the source that lies close to the cusp in Δ > 0 contributes to the nonspherically decaying radiation. 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 asymptotes of the hyperbola Δ = 0 make the angles arcsin ( 1 r ̂ P ) and π arcsin ( 1 r ̂ P ) with the z axis, so that for an observation point in the far zone ( r ̂ P 1 ) the projection of the cusp onto the ( r ̂ , z ̂ ) plane is (as depicted in Fig. 2) effectively parallel to the rotation axis.

Fig. 5
Fig. 5

Integration contours in the complex plane ξ = u + i v . The critical point C lies at the origin, and u S and u > are the images under transformation (31) of the radial boundaries r ̂ = r ̂ S ( z ̂ ) and r ̂ = r ̂ > ( z ̂ ) of the part of the source that lies within Δ > 0 (see Fig. 4). The contours C 1 , C 2 , and C 3 are the paths of steepest descent through the stationary point C and the lower and upper boundaries of the integration domain, respectively.

Equations (70)

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× H = 4 π c j + 1 c D t = 4 π c ( j + P t ) + 1 c E t ;
r = const. , φ ( t ) = φ ̂ + ω t , z = const. ,
R ( t ) = [ ( z P z ) 2 + r P 2 + r 2 2 r r P cos ( φ P φ ̂ ω t ) ] 1 2 .
t P = t + R ( t ) c ,
d R d t = c , d 2 R d t 2 = 0
d t P d t = 0 , d 2 t P d t 2 = 0 .
φ = φ ± φ P + 2 π arccos ( 1 Δ 1 2 r ̂ r ̂ P ) ,
Δ ( r ̂ P 2 1 ) ( r ̂ 2 1 ) ( z ̂ z ̂ P ) 2 .
φ P = ω t P + φ ̂ ϕ ± ( r P , z P ) ,
ϕ ± R ̂ ± + 2 π arccos ( 1 Δ 1 2 r ̂ r ̂ P ) ,
R ̂ ± [ ( z ̂ z ̂ P ) 2 + r ̂ 2 + r ̂ P 2 2 ( 1 Δ 1 2 ) ] 1 2 .
Δ = 0 , φ P = ω t P + φ ̂ ϕ ± ( r P , z P ) Δ = 0 ,
θ P = arcsin ( r ̂ 1 ) + , φ P = φ 3 2 π + ,
arcsin ( 1 r ̂ > ) θ P arcsin ( 1 r ̂ < ) .
P r , φ , z ( r , φ , z , t ) = s r , φ , z ( r , z ) cos ( m φ ̂ ) cos ( Ω t ) , π < φ ̂ π ,
φ ̂ φ ω t ,
E = P A 0 A ( c t P ) , B = P × A
A μ ( x P , t P ) = c 1 d 3 x d t j μ ( x , t ) δ ( t P t R c ) R , μ = 0 , , 3 .
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 ̂ d r ̂ d z ̂ Δ 1 2 u j exp ( i μ ϕ ) ,
μ ± ( Ω ω ) ± m ,
φ ̂ P φ P ω t P ,
q ¯ j ( 1 i Ω ω i Ω ω ) ,
u j { s r cos θ P e ̂ + s φ e ̂ s φ cos θ P e ̂ + s r e ̂ s z sin θ P e ̂ } ,
r ̂ = r ̂ C ( z ̂ ) { 1 2 ( r ̂ P 2 + 1 ) [ 1 4 ( r ̂ P 2 1 ) 2 ( z ̂ z ̂ P ) 2 ] 1 2 } 1 2 .
r ̂ = r ̂ S [ 1 + ( z ̂ z ̂ P ) 2 ( r ̂ P 2 1 ) ] 1 2 ,
ϕ r ̂ = r ̂ C ϕ C = R ̂ C + φ C φ P ,
2 ϕ r ̂ 2 r ̂ = r ̂ C a = R ̂ C 1 [ ( r ̂ P 2 1 ) ( r ̂ C 2 1 ) 1 2 ] ,
φ C = φ P + 2 π arccos ( r ̂ C r ̂ P ) ,
R ̂ C = r ̂ C ( r ̂ P 2 r ̂ C 2 ) 1 2 .
a R ̂ P sin 4 θ P sec 2 θ P
ϕ ( r ̂ , z ̂ ) = ϕ C ( z ̂ ) + 1 2 a ( z ̂ ) ξ 2 ,
Δ 0 r ̂ d r ̂ d z ̂ Δ 1 2 u j exp ( i μ ϕ ) = ξ ξ S d z ̂ d ξ A ( ξ , z ̂ ) exp ( i α ξ 2 ) ,
A ( ξ , z ̂ ) r ̂ Δ 1 2 u j r ̂ ξ exp ( i μ ϕ C ) ,
r ̂ ξ = a ξ r ̂ R ̂ ( r ̂ 2 1 Δ 1 2 ) 1 ,
ξ = ξ S [ 2 a 1 ( ϕ S ϕ C ) ] 1 2 ,
ϕ S ϕ r ̂ = r ̂ S = 2 π arccos [ 1 ( r ̂ S r ̂ P ) ] + ( r ̂ S 2 r ̂ P 2 1 ) 1 2 .
ϕ r ̂ r ̂ ξ = a ξ ,
2 ϕ r ̂ 2 ( r ̂ ξ ) 2 + ϕ r ̂ 2 r ̂ ξ 2 = a ,
ξ S 3 1 2 cos 4 θ P csc 5 θ P R ̂ P 2 ,
I ( z ̂ ) ξ S ξ > d ξ A ( ξ , z ̂ ) exp ( i α ξ 2 )
i ξ 2 = 2 u v + i ( u 2 v 2 ) ,
C 1 d ξ A ( ξ , z ̂ ) exp ( i α ξ 2 ) = ( 1 + i ) d v A ξ = ( 1 + i ) v exp ( 2 α v 2 ) ( 2 π μ ) 1 2 exp [ i ( μ ϕ C π 4 ) ] u j C csc θ P sec θ P R ̂ P 1 2 ,
i ξ 2 C 2 = 2 v ( v 2 + u S 2 ) 1 2 + i u S 2
ξ C 2 = ( u S 2 + i σ ) 1 2
C 2 d ξ A ( ξ , z ̂ ) exp ( i α ξ 2 ) = 1 2 exp [ i ( α u S 2 π 2 ) ] 0 d σ ( u S 2 + i σ ) 1 2 A ξ = ( u S 2 + i σ ) 1 2 exp ( α σ ) .
ϕ ± = ϕ S + r ̂ S 1 ( r ̂ S 2 1 ) ( r ̂ S 2 r ̂ P 2 1 ) 1 2 ( r ̂ r ̂ S ) ± 1 3 ( 2 r ̂ S ) 3 2 ( r ̂ P 2 1 ) 3 2 ( r ̂ S 2 r ̂ P 2 1 ) 3 2 ( r ̂ r ̂ S ) 3 2 + .
ϕ ϕ S = 1 2 a ( ξ 2 ξ S 2 ) .
r ̂ r ̂ S + 1 2 sin 5 θ P sec 4 θ P R ̂ P 2 ( ξ S 2 ξ 2 ) ,
Δ 1 2 ( 2 sin θ P ) 1 2 R ̂ P ( r ̂ r ̂ S ) 1 2 , r ̂ r ̂ S 1 , R ̂ P 1 .
A C 2 exp [ i ( μ ϕ C π 4 ) ] u j S sin θ P sec 2 θ P ( u S 2 + i σ ) 1 2 σ 1 2
C 2 d ξ A ( ξ , z ̂ ) exp ( i α ξ 2 ) sin θ P sec 2 θ P exp [ i ( μ ϕ C + π 4 ) ] u j S 0 d τ exp ( α τ 2 ) 1 2 ( 2 π μ ) 1 2 csc θ P sec θ P exp [ i ( μ ϕ C + π 4 ) ] u j S R ̂ P 1 2 ,
i ξ 2 C 3 = 2 v ( v 2 + u > 2 ) 1 2 + i u > 2
ξ C 3 = ( u > 2 + i χ ) 1 2
C 3 d ξ A ( ξ , z ̂ ) exp ( i α ξ 2 ) = 1 2 exp [ i ( α u > 2 π 2 ) ] 0 d χ ( u > 2 + i χ ) 1 2 A ξ = ( u > 2 + i χ ) 1 2 exp ( α χ ) .
A C 3 , χ = 0 r ̂ > 2 sin 4 θ P sec 2 θ P ( r ̂ > 2 sin 2 θ P 1 ) 1 u j r ̂ = r ̂ > exp ( i μ ϕ C ) u >
C 3 d ξ A ( ξ , z ̂ ) exp ( i α ξ 2 ) r ̂ > 2 ( r ̂ > 2 sin 2 θ P 1 ) 1 u j r ̂ > exp [ i ( μ ϕ r ̂ > + π 2 ) ] μ 1 R ̂ P 1 ,
B ns 2 3 ( 1 + 2 i ) ( 2 π ) 1 2 R ̂ P 1 2 sec θ P csc θ P exp [ i ( Ω ω ) ( φ P + 3 π 2 ) ] μ = μ ± μ 1 2 sgn ( μ ) exp ( i π 4 sgn μ ) j = 1 3 q ¯ j d z ̂ u j C exp [ i μ ( ϕ C + φ ̂ P ) ] ,
δ r ̂ r ̂ C r ̂ S 1 2 cos 4 θ P csc 5 θ P R ̂ P 2
δ φ ( ϕ + ϕ ) r ̂ = r ̂ C 2 3 cot 6 θ P R ̂ P 3 .
ϕ C R ̂ P z ̂ cos θ P + 3 π 2
E ns 4 3 ( 2 π ) 1 2 R ̂ P 1 2 sec θ P csc θ P exp [ i ( Ω ω ) ( φ P + 3 π 2 ) ] μ = μ ± μ 1 2 sgn ( μ ) exp ( i π 4 sgn μ ) exp [ i μ ( R ̂ P + φ ̂ P + 3 π 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 ̂ } ,
s ¯ r , φ , z d z ̂ s r , φ , z ( r ̂ , z ̂ ) r ̂ = csc θ P exp ( i μ z ̂ cos θ P ) .
z ̂ P = z ̂ ± ( r ̂ P 2 1 ) 1 2 ( r ̂ 2 1 ) 1 2 ,
φ P = φ 2 π + arccos [ 1 ( r ̂ r ̂ P ) ] .
θ P = arccos { 1 r ̂ R ̂ P [ z ̂ r ̂ ± ( r ̂ 2 1 ) 1 2 ( R ̂ P 2 1 z ̂ 2 r ̂ 2 ) 1 2 ] } ,
φ P = φ 2 π + arccos [ 1 ( R ̂ P r ̂ sin θ P ) ] ,
θ P = arcsin ( 1 r ̂ ) ( z ̂ r ̂ ) R ̂ P 1 ± 1 2 ( r ̂ 2 1 ) 1 2 R ̂ P 2 + ,
φ P = φ 3 π 2 R ̂ P 1 + .
δ θ P = r ̂ 1 ( r ̂ 2 1 ) 1 2 δ r ̂ r ̂ 1 δ z ̂ R ̂ P 1 +
δ θ P δ z ̂ sin θ P R ̂ P 1 , δ φ P δ φ

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