Abstract

We investigate the spectral features of the emission from a superluminal polarization current whose distribution pattern rotates (with an angular frequency ω) and oscillates (with a frequency Ω>ω differing from an integral multiple of ω) at the same time. This type of polarization current is found in recent practical machines designed to investigate superluminal emission. Although all of the processes involved are linear, we find that the broadband emission contains frequencies that are higher than Ω by a factor of the order of (Ω/ω)2. This generation of frequencies not required for the creation of the source stems from mathematically rigorous consequences of the familiar classical expression for the retarded potential. The results suggest practical applications for superluminal polarization currents as broadband radio-frequency and infrared sources.

© 2003 Optical Society of America

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References

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  1. A. Ardavan, H. Ardavan, “Apparatus for generating focused electromagnetic radiation,” International patent applicationPCT-GB99-02943 (September6, 1999).
  2. M. Durrani, “Revolutionary device polarizes opinions,” Phys. World, August2001, p. 9.
  3. N. Appleyard, B. Appleby, “Warp speed,” New Sci. 170 (No. 2288), 28–31 (2001).
  4. J. Fopma, A. Ardavan, D. Halliday, J. Singleton, “Phase control electronics for a polarization synchrotron,” manuscript available from J. Singleton (jsingle@lanl.gov).
  5. A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, D. Halliday, A. Narduzzo, P. Goddard, “Experimental observation of nonspherically decaying emission from a rotating superluminal source,” manuscript available from J. Singleton (jsingle@lanl.gov).
  6. H. Ardavan, “Generation of focused, nonspherically decaying pulses of electromagnetic radiation,” Phys. Rev. E 58, 6659–6684 (1998).
    [CrossRef]
  7. A. Hewish, “Comment I on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3007 (2000).
    [CrossRef]
  8. J. H. Hannay, “Comment II on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3008–3009 (2000).
    [CrossRef]
  9. H. Ardavan, “Reply to Comments on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’,” Phys. Rev. E 62, 3010–3013 (2000).
    [CrossRef]
  10. 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]
  11. V. L. Ginzburg, Theoretical Physics and Astrophysics (Pergamon, Oxford, UK, 1979), Chap. VIII.
  12. B. M. Bolotovskii, V. P. Bykov, “Radiation by charges moving faster than light,” Sov. Phys. Usp. 33, 477–487 (1990).
    [CrossRef]
  13. T. T. Wu, H. Lehmann, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
    [CrossRef]
  14. A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
    [CrossRef]
  15. E. Recami, “On localized ‘X-shaped’ superluminal solutions to Maxwell equations,” Physica A 252, 586–610 (1998).
    [CrossRef]
  16. A. M. Shaarawi, “Comparison of two localized wavefields generated from dynamic apertures,” J. Opt. Soc. Am. A 14, 1804–1816 (1997).
    [CrossRef]
  17. 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]
  18. C. H. Walter, Traveling Wave Antennas (McGraw-Hill, New York, 1965), p. 349.
  19. T. Tamir, “Leaky-wave antennas,” in Antenna Theory: Part 2, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, New York, 1969), pp. 253–297.
  20. G. M. Lilley, R. Westley, A. H. Yates, J. R. Busing, “Some aspects of noise from supersonic aircraft,” J. Aeronaut. Sci. 57, 396–414 (1953).
  21. L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, UK, 1975).
  22. H. Ardavan, A. Ardavan, J. Singleton, “The spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns,” http://xxx.lanl.gov/abs/physics/0301083 .
  23. H. Ardavan, “Method of handling the divergences in the radiation theory of sources that move faster than their waves,” J. Math. Phys. 40, 4331–4336 (1999).
    [CrossRef]
  24. N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1986).
  25. R. Wong, Asymptotic Approximations of Integrals (Academic, Boston, 1989).
  26. J. J. Stamnes, Waves in Focal Regions (Hilgar, Boston, 1986).
  27. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  28. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1995).
  29. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

2001 (2)

M. Durrani, “Revolutionary device polarizes opinions,” Phys. World, August2001, p. 9.

N. Appleyard, B. Appleby, “Warp speed,” New Sci. 170 (No. 2288), 28–31 (2001).

2000 (3)

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]

1999 (1)

H. Ardavan, “Method of handling the divergences in the radiation theory of sources that move faster than their waves,” J. Math. Phys. 40, 4331–4336 (1999).
[CrossRef]

1998 (2)

E. Recami, “On localized ‘X-shaped’ superluminal solutions to Maxwell equations,” Physica A 252, 586–610 (1998).
[CrossRef]

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

1997 (1)

1995 (1)

1990 (1)

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

1985 (2)

T. T. Wu, H. Lehmann, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
[CrossRef]

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[CrossRef]

1972 (1)

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]

1953 (1)

G. M. Lilley, R. Westley, A. H. Yates, J. R. Busing, “Some aspects of noise from supersonic aircraft,” J. Aeronaut. Sci. 57, 396–414 (1953).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Appleby, B.

N. Appleyard, B. Appleby, “Warp speed,” New Sci. 170 (No. 2288), 28–31 (2001).

Appleyard, N.

N. Appleyard, B. Appleby, “Warp speed,” New Sci. 170 (No. 2288), 28–31 (2001).

Ardavan, A.

J. Fopma, A. Ardavan, D. Halliday, J. Singleton, “Phase control electronics for a polarization synchrotron,” manuscript available from J. Singleton (jsingle@lanl.gov).

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

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, D. Halliday, A. Narduzzo, P. Goddard, “Experimental observation of nonspherically decaying emission from a rotating superluminal source,” manuscript available from J. Singleton (jsingle@lanl.gov).

Ardavan, H.

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, “Method of handling the divergences in the radiation theory of sources that move faster than their waves,” J. Math. Phys. 40, 4331–4336 (1999).
[CrossRef]

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

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, D. Halliday, A. Narduzzo, P. Goddard, “Experimental observation of nonspherically decaying emission from a rotating superluminal source,” manuscript available from J. Singleton (jsingle@lanl.gov).

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

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]

Busing, J. R.

G. M. Lilley, R. Westley, A. H. Yates, J. R. Busing, “Some aspects of noise from supersonic aircraft,” J. Aeronaut. Sci. 57, 396–414 (1953).

Bykov, V. P.

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

Durrani, M.

M. Durrani, “Revolutionary device polarizes opinions,” Phys. World, August2001, p. 9.

Fopma, J.

J. Fopma, A. Ardavan, D. Halliday, J. Singleton, “Phase control electronics for a polarization synchrotron,” manuscript available from J. Singleton (jsingle@lanl.gov).

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, D. Halliday, A. Narduzzo, P. Goddard, “Experimental observation of nonspherically decaying emission from a rotating superluminal source,” manuscript available from J. Singleton (jsingle@lanl.gov).

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, D. Halliday, A. Narduzzo, P. Goddard, “Experimental observation of nonspherically decaying emission from a rotating superluminal source,” manuscript available from J. Singleton (jsingle@lanl.gov).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Halliday, D.

J. Fopma, A. Ardavan, D. Halliday, J. Singleton, “Phase control electronics for a polarization synchrotron,” manuscript available from J. Singleton (jsingle@lanl.gov).

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, D. Halliday, A. Narduzzo, P. Goddard, “Experimental observation of nonspherically decaying emission from a rotating superluminal source,” manuscript available from J. Singleton (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]

Landau, L. D.

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

Lehmann, H.

T. T. Wu, H. Lehmann, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
[CrossRef]

Lifshitz, E. M.

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

Lilley, G. M.

G. M. Lilley, R. Westley, A. H. Yates, J. R. Busing, “Some aspects of noise from supersonic aircraft,” J. Aeronaut. Sci. 57, 396–414 (1953).

Narduzzo, A.

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, D. Halliday, A. Narduzzo, P. Goddard, “Experimental observation of nonspherically decaying emission from a rotating superluminal source,” manuscript available from J. Singleton (jsingle@lanl.gov).

Recami, E.

E. Recami, “On localized ‘X-shaped’ superluminal solutions to Maxwell equations,” Physica A 252, 586–610 (1998).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Sedky, S. M.

Sezginer, A.

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[CrossRef]

Shaarawi, A. M.

Singleton, J.

J. Fopma, A. Ardavan, D. Halliday, J. Singleton, “Phase control electronics for a polarization synchrotron,” manuscript available from J. Singleton (jsingle@lanl.gov).

J. Fopma, A. Ardavan, D. Halliday, J. Singleton, “Phase control electronics for a polarization synchrotron,” manuscript available from J. Singleton (jsingle@lanl.gov).

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, D. Halliday, A. Narduzzo, P. Goddard, “Experimental observation of nonspherically decaying emission from a rotating superluminal source,” manuscript available from J. Singleton (jsingle@lanl.gov).

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, D. Halliday, A. Narduzzo, P. Goddard, “Experimental observation of nonspherically decaying emission from a rotating superluminal source,” manuscript available from J. Singleton (jsingle@lanl.gov).

Stamnes, J. J.

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

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Tamir, T.

T. Tamir, “Leaky-wave antennas,” in Antenna Theory: Part 2, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, New York, 1969), pp. 253–297.

Walter, C. H.

C. H. Walter, Traveling Wave Antennas (McGraw-Hill, New York, 1965), p. 349.

Watson, G. N.

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

Westley, R.

G. M. Lilley, R. Westley, A. H. Yates, J. R. Busing, “Some aspects of noise from supersonic aircraft,” J. Aeronaut. Sci. 57, 396–414 (1953).

Wong, R.

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

Wu, T. T.

T. T. Wu, H. Lehmann, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
[CrossRef]

Yates, A. H.

G. M. Lilley, R. Westley, A. H. Yates, J. R. Busing, “Some aspects of noise from supersonic aircraft,” J. Aeronaut. Sci. 57, 396–414 (1953).

Ziolkowski, R. W.

J. Aeronaut. Sci. (1)

G. M. Lilley, R. Westley, A. H. Yates, J. R. Busing, “Some aspects of noise from supersonic aircraft,” J. Aeronaut. Sci. 57, 396–414 (1953).

J. Appl. Phys. (2)

T. T. Wu, H. Lehmann, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
[CrossRef]

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[CrossRef]

J. Math. Phys. (1)

H. Ardavan, “Method of handling the divergences in the radiation theory of sources that move faster than their waves,” J. Math. Phys. 40, 4331–4336 (1999).
[CrossRef]

J. Opt. Soc. Am. A (2)

New Sci. (1)

N. Appleyard, B. Appleby, “Warp speed,” New Sci. 170 (No. 2288), 28–31 (2001).

Phys. Rev. E (4)

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]

Phys. World (1)

M. Durrani, “Revolutionary device polarizes opinions,” Phys. World, August2001, p. 9.

Physica A (1)

E. Recami, “On localized ‘X-shaped’ superluminal solutions to Maxwell equations,” Physica A 252, 586–610 (1998).
[CrossRef]

Sov. Phys. Usp. (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]

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

Other (14)

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

H. Ardavan, A. Ardavan, J. Singleton, “The spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns,” http://xxx.lanl.gov/abs/physics/0301083 .

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 applicationPCT-GB99-02943 (September6, 1999).

J. Fopma, A. Ardavan, D. Halliday, J. Singleton, “Phase control electronics for a polarization synchrotron,” manuscript available from J. Singleton (jsingle@lanl.gov).

A. Ardavan, J. Singleton, H. Ardavan, J. Fopma, D. Halliday, A. Narduzzo, P. Goddard, “Experimental observation of nonspherically decaying emission from a rotating superluminal source,” manuscript available from J. Singleton (jsingle@lanl.gov).

C. H. Walter, Traveling Wave Antennas (McGraw-Hill, New York, 1965), p. 349.

T. Tamir, “Leaky-wave antennas,” in Antenna Theory: Part 2, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, New York, 1969), pp. 253–297.

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

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

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

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

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

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

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Figures (10)

Fig. 1
Fig. 1

(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=2.5c/ω (rˆrω/c=2.5). The dashed circles designate the orbit of S and the light cylinder rP=c/ω (rˆP=1). The curves to which the emitted wave fronts are tangential are the cross sections of the two sheets ϕ± of the envelope with the plane of the source’s orbit. (b) Three-dimensional view of the light cylinder and the envelope of wave fronts for the same source point S. The tubelike 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 (based on Ref. 6).

Fig. 2
Fig. 2

Bifurcation surface (i.e., the 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 for a clockwise source motion. The cusp Cb, along which the two sheets of the bifurcation surface meet tangentially, touches the light cylinder (rˆ=1) at point O. This cusp curve is the locus of source points that approach the observer not only with the speed of light but also with zero acceleration 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 each other, before reaching the light cylinder. Note that the bifurcation surface issues from the observation point P and resides in the space (r, φˆ, z) of source points, while the envelope of wave fronts issues from a source point and resides in the space (rP, φˆP, zP) of observation points; the similarity between these two surfaces reflects the reciprocity properties of the Green’s function for the problem (based on Ref. 6).

Fig. 3
Fig. 3

Relationship between the observation time tP and the 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. 1 and 2. This relationship is given by g(r, φ, z; rP, φP, zP)=φˆ-φˆP, an equation that applies to the envelope when the position (r, φˆ, z) of the source point is fixed and to the bifurcation surface when the location (rP, φˆP, zP) of the observer is fixed. (Note that, by virtue of the linear relation φ=φˆ+ωt, the motion of the source may be parameterized either by t or by φ.) The maxima and the minima of curve (a), at which dR/dt=-c, occur on the sheets ϕ+ and ϕ- of the envelope (or the bifurcation surface), respectively (see Figs. 1 and 2). The inflection points of curve (b), at which d2R/dt2=0, occur on the cusp curve of the envelope (or the bifurcation surface).

Fig. 4
Fig. 4

Projection of the trajectories (world lines) of the volume elements of a uniformly rotating extended source onto the (φ, t) space. The Lagrangian coordinate φˆφ-ωt designating the initial (t=0) position of each source element lies in (-π, π), while both φ and t range over (-∞, ∞). The space–time trajectory (world tube) of the extended source itself in (φ, t) space consists of the array of trajectories of its constituent volume elements (dashed lines) encompassed by the lines φˆ=-π and φˆ=π.

Fig. 5
Fig. 5

φˆ dependence cos(mφˆ)cos[Ω(φˆ-φ)/ω] of the source density described by Eq. (7) at a fixed (r, φ, z) for m=1, Ω/ω=3.5, and φ=π/5. This φˆ dependence is, by virtue of the relationship t=(φˆ-φ)/ω, equivalent to a time dependence: The space–time of source elements may be marked either by the coordinates (r, φ, z, t), as in Eq. (7), or by the coordinates (r, φ, z, φˆ), as in relation (19). The plotted function is physically meaningful only within the interval -π<φˆ<π. However, once this function is expanded into a Fourier series over the interval (-π, π), a new periodic function results, represented by the series, which is coincident with the original function within (-π, π) and periodically reproduces the original function outside this interval. The periodic function represented by the series outside (-π, π) is designated by the dashed curves.

Fig. 6
Fig. 6

Spectral distribution of the Green’s function G˜2(n) for Ω/ω=15.5, normalized by the value G˜2(16) of this function at a harmonic number (n=16) close to Ω/ω. (Frequency f and harmonic number n are related through 2πf=nω.) The inset highlights the highest-frequency peak of the spectrum. Note that the ranges of frequencies shown in the figure and its inset are complementary. This function represents the spectral distribution of the emission arising from the single comoving point O of a polarization along the r or the z direction [see relations (24) and (40)].

Fig. 7
Fig. 7

Spectral distribution of the Green’s function G˜1(n) for Ω/ω=15.5, normalized by the value G˜1(16) of this function at a harmonic number (n=16) close to Ω/ω. (Frequency f and harmonic number n are related through 2πf=nω.) The inset highlights the highest-frequency peak of the spectrum. Note that the ranges of frequencies shown in the figure and its inset are complementary. This function represents the spectral distribution of the emission arising from the single comoving point O of a polarization current that flows in the φ direction [see relation (40) and relation (24) for θP=π/2].

Fig. 8
Fig. 8

Spectral distribution of the radiation that is generated by a poloidal polarization current for Ω/ω=15.5 and m=72. The normalization factor is the magnitude E˜87 of the radiation field at a harmonic number (n=87) close to the peak emission at n=(m+Ω/ω) or f=(mω+Ω)/(2π) (see Fig. 9); frequency f and harmonic number n are related through 2πf=nω. The plotted quantity has the value of unity at n=87 and decreases monotonically over the range 87<n<400 of harmonic numbers not shown here (see Fig. 9). The inset highlights the highest-frequency peak of the spectrum.

Fig. 9
Fig. 9

Logarithm of the normalized intensity, log10|E˜n/E˜87|2, shown in Fig. 8 over the lower range (30<n<400) of harmonic numbers (recall that Ω/ω=15.5 and m=72). Note that the ranges of frequencies in this figure and in Fig. 8 are complementary. The contribution from |μ-ω|=mω-Ω (see Table 1) to the radiation field is in this case too small to make a difference to the figure, even at these lower frequencies.

Fig. 10
Fig. 10

View of the experimental device (a) from the side and (b) from above, showing an arc of the dielectric medium (lightly shaded regions), its polarized part (darkly shaded region), and the electrode pairs (designated by ± where on and by 00 where off).

Tables (2)

Tables Icon

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

Tables Icon

Table 2 State of Polarization of the Radiation In and Out of the Plane of Rotation for Different Orientations of the Emitting Polarization Current

Equations (106)

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r=const.,z=const.,φ=φˆ+ωt,
|xP-x(t)|=c(tP-t),
R=[(zP-z)2+rP2+r2-2rPr cos(φP-φ)]1/2,
tP=t+[(zP-z)2+rP2+r2-2rPr×cos(φP-φˆ-ωt)]1/2/c.
tP=tPc+16ω2(t-tc)3+,
ρ(r, φ, z, t)=qδ(r-r0)δ(φ-ωt-φˆ)δ(z)/r,
Pr,φ,z(r, φ, z, t)=sr,φ,z(r, z)cos(mφˆ)cos(Ωt),
-π<φˆπ,
Pr,φ,z(r, φ, z, t)=sr,φ,z(r, z)n=-Cn exp(inωt),
Cn=(2π/ω)-1(φ-π)/ω(φ+π)/ωdt cos[m(φ-ωt)]×cos(Ωt)exp(-inωt)=(-1)m+n+1(2π)-1(n-Ω/ω)[(n-Ω/ω)2-m2]-1×sin(πΩ/ω)exp[-i(n-Ω/ω)φ]+{Ω-Ω}.
Aμ(xP, tP)=c-1d3xdt jμ(x, t)δ(tP-t-R/c)/R,
μ=0,,3,
E=-PA0-A/(ctP),B=P×A
E=-1cd3xdtδ(tP-t-R/c)R×1cjt-1cj0t+j0RRR,
B=-1cd3xdtδ(tP-t-R/c)RRR×1cjt+jR,
E1c2d3xdtδ(tP-t-R/c)Rnˆ×nˆ×jt
j=14iωμ=μ±μ exp[-i(μφˆ-Ωφ/ω)]×(sreˆr+sφeˆφ+szeˆz)+{m-m, Ω-Ω},-π<φˆπ,
eˆreˆφeˆz=cos(φ-φP)sin(φ-φP)0-sin(φ-φP)cos(φ-φP)0001eˆrPeˆφPeˆzP.
limR nˆ=(sin θP)eˆrP+(cos θP)eˆzP,
θParctan(rP/zP),
nˆ×(nˆ×j/t)=14ω2μ=μ±μ2 exp[-i(μφˆ-Ωφ/ω)]×{[sφ cos θP sin(φ-φP)-sr cos θP cos(φ-φP)+sz sin θP]eˆ+[sr sin(φ-φP)+sφ cos(φ-φP)]eˆ}+{m-m, Ω-Ω},
E14(ω/c)2μ=μ±0rdr-+dz-π+πdφˆ μ2×exp(-iμφˆ){(srG2+sφG1)eˆ+[(cos θP)(sφG2-srG1)+(sin θP)szG3]eˆ}+{m-m, Ω-Ω},
G1G2G3=Δφdφδ(g-ϕ)Rexp(iΩφ/ω)cos(φ-φP)sin(φ-φP)1.
gφ-φP+Rˆ
δ(g-ϕ)=(2π)-1n=- exp[-in(g-ϕ)],
E=Re{E˜0+2n=1E˜n exp(-inφˆP)},
E˜n14(ω/c)2μ=μ±0rdr-+dz-π+πdφˆ μ2 exp(-iμφˆ)×{(srG˜2+sφG˜1)eˆ+[(cos θP)(sφG˜2-srG˜1)+(sin θP)szG˜3]eˆ}+{m-m, Ω-Ω}
G˜i=(2π)-1 exp(inφˆ)Δφdφ fi exp[-i(ng-Ωφ/ω)],
f1f2f3R-1cos(φ-φP)sin(φ-φP)1,
O:rˆ=1,φ=φP+2π-arccos(1/rˆP)φO,
z=zP,
C:rˆ=rˆC(zˆ){12(rˆP2+1)-[14(rˆP2-1)2-(zˆ-zˆP)2]1/2}1/2,
φ=φC(zˆ)φP+2π-arccos(rˆC/rˆP).
Cb:rˆ=[1+(zˆ-zˆP)2/(rˆP2-1)]1/2,
φ=φP+2π-arccos[1/(rˆrˆP)]
gRˆP-zˆ cos θP+φ-φP-rˆ sin θP cos(φ-φP),
RˆP1,
ξφ-φO,ηrˆ-1,ζzˆ-zˆP
g=ϕO+ξ-(1+η)sin ξ+[4(1+η)sin2(ξ/2)-(1+η)2 sin2 ξ+η2+ζ2]/(2rˆP)+,
g=ϕO+ξ-(1+η)sin ξ+(η2+ζ2)/(2rˆP)+.
ρrˆ-rˆC,μφ-φC.
g=φC-φP+μ+[RˆC2-2(1+ρ/rˆC)RˆC sin μ+4rˆC(rˆC+ρ)sin2(μ/2)+ρ2]1/2,
RˆC[(zˆP-zˆ)2+rˆP2-rˆC2]1/2
g=ϕC+μ-(1+ρ/rˆC)sin μ+[ρ2+(rˆC2-1)μ2]/(2RˆC)+,
fi|O=rP-1(sin ξ-11)
G˜i(2π)-1 exp{in[φˆ-ϕO-(η2+ζ2)/(2rˆP)]+iΩφO/ω}-ππdξ fi|O×exp{-i[(n-Ω/ω)ξ-nrˆ sin ξ]},
G˜i-rP-1exp{in[φˆ-φO-(η2+ζ2)/(2rˆP)]+iΩφO/ω}×[iJn-Ω/ω(nrˆ)Jn-Ω/ω(nrˆ)-Jn-Ω/ω(nrˆ)],
Jν(χ)(2π)-1-ππdξ exp[-i(νξ-χ sin ξ)]=Jν(χ)+π-1 sin(νπ)0dτ exp(-ντ-χ sinh τ),
0dτ exp(-ντ-χ sinh τ)
(1+ν/χ)-1χ-1-(1+ν/χ)-4χ-3+(9-ν/χ)(1+ν/χ)-7χ-5+
Jν(ν sech α)(2πν tanh α)-1/2 exp[ν(tanh α-α)],
Jν(ν sech α)(4πν/sinh 2α)-1/2 exp[ν(tanh α-α)]
Jν(ν sec β)(12πν tan β)-1/2 cos[ν(tan β-β)-π4],
Jν(ν sec β)(πν/sin 2β)-1/2 sin[ν(tan β-β)-π4].
E˜n12rˆP-1 exp[-i(nϕO-ΩφO/ω)]Qφˆ-dzˆ×exp-12inrˆP-1(zˆ-zˆP)2×0rˆdrˆ exp-12inrˆP-1(rˆ-1)2×V+{m-m, Ω-Ω},
Qφˆ-12μ=μ±-π+πdφˆ μ2 exp[i(n-μ)φˆ]=(-1)n+m sin(πΩ/ω)[μ+2(n-μ+)-1+μ-2(n-μ-)-1]
V(srW2+sφW1)eˆ+[(cos θP)(sφW2-srW1)+(sin θP)szW3]eˆ,
Wj[iJn-Ω/ω(nrˆ)Jn-Ω/ω(nrˆ)-Jn-Ω/ω(nrˆ)].
0rˆdrˆ exp-12inrˆP-1(rˆ-1)2VV|rˆ=1Qr,
Qrrˆ<rˆ>drˆ exp-12inrˆP-1(rˆ-1)2=(πrˆP/n)1/2{C(η>)-C(η<)-i[S(η>)-S(η<)]},
η>[n/(πrˆP)]1/2(rˆ>-1),
η<[n/(πrˆP)]1/2(rˆ<-1),
E˜n12rˆP-1exp[-i(nϕO-ΩφO/ω)]QφˆQrQzV|rˆ=1,zˆ=zˆP+{m-m, Ω-Ω},
Qzzˆ<zˆ>dzˆ exp-12inrˆP-1(zˆ-zˆP)2=(πrˆP/n)1/2{C(ζ>)-C(ζ<)-i[S(ζ>)-S(ζ<)]},
ζ>[n/(πrˆP)]1/2(zˆ>-zˆP),
ζ<[n/(πrˆP)]1/2(zˆ<-zˆP).
V|rˆ=1,zˆ=zˆP=Jn-Ω/ω(n)(srOeˆ-szOeˆ)+iJn-Ω/ω(n)sφOeˆ,
E˜n(-1)n+mrˆP-1exp[-i(nϕO-ΩφO/ω)](m2+Ω2/ω2)×sin(πΩ/ω)QrQzn-1[Jn-Ω/ω(n)(srOeˆ-szOeˆ)+iJn-Ω/ω(n)sφOeˆ],
Qr(2πrˆP/n)1/2 exp(-iπ/4),rˆP(n/π)(rˆ>-rˆ<)2,
Qz(2πrˆP/n)1/2 exp(-iπ/4),rˆP(n/π)(zˆ>-zˆ<)2.
Qrrˆ>-rˆ<,rˆP(n/π)(rˆ>-rˆ<)2,
Qzzˆ>-zˆ<,rˆP(n/π)(zˆ>-zˆ<)2,
dPn/dΩP(2πc)-1ω2(m2+Ω2/ω2)2×sin2(πΩ/ω)(r>-r<)2×(z>-z<)2n-2|Jn-Ω/ω(n)(srOeˆ-szOeˆ)+iJn-Ω/ω(n)sφOeˆ|2.
G˜i(2π)-1 exp{i[n(φˆ-φC-12RˆC-1ρ2)+ΩφC/ω]}×-ππdμ fi|C exp{-i[(n-Ω/ω)μ-n(rˆ/rˆC)sin μ]},
rˆCcsc θP,φCφP+3π/2,
RˆCRˆP-zˆ cos θP
fi|C=RP-1(sin μ-11).
G˜i-RP-1 exp{i[n(φˆ-32π-RˆP+zˆ cos θP-12RˆP-1ρ2)+ΩφC/ω]}W¯i,
W¯j[iJn-Ω/ω(nrˆ sin θP)Jn-Ω/ω(nrˆ sin θP)
-Jn-Ω/ω(nrˆ sin θP)].
E˜n12RˆP-1 exp{-i[n(RˆP+32π)-ΩφC/ω]}Qφˆ×-dzˆ exp(inzˆ cos θP)×0rˆdrˆ exp-12inRˆP-1(rˆ-csc θP)2V¯+{m-m, Ω-Ω},
0rˆdrˆ exp[-12inRˆP-1(rˆ-csc θP)2]V¯
(csc θP)V¯|rˆ=csc θPQ¯r,
Q¯r=(πRˆP/n)1/2{C(η¯>)-C(η¯<)-i[S(η¯>)-S(η¯<)]},
η¯>[n/(πRˆP)]1/2(rˆ>-csc θP),
η¯<[n/(πRˆP)]1/2(rˆ<-csc θP),
E˜n12rˆP-1exp{-i[n(RˆP+32π)-ΩφC/ω]}QφˆQ¯rQ¯z+{m-m, Ω-Ω},
Q¯z[s¯rJn-Ω/ω(n)+is¯φJn-Ω/ω(n)]eˆ+[(s¯φ cos θP-s¯z sin θP)Jn-Ω/ω(n)-is¯r(cos θP)Jn-Ω/ω(n)]eˆ,
s¯r,φ,z-dzˆ exp(inzˆ cos θP)sr,φ,z|C.
dPn/dΩP(2π)-1c csc2 θP sin2(πΩ/ω)×(m2+Ω2/ω2)2(r>-r<)2n-2|Q¯z|2
j(r, φ, z, t)=qωδ(r-r0)δ(φˆ-φˆ0)δ(z-z0)eˆφ,
Eq(ω/c)20rdr-+dz-π+πdφˆ δ(r-r0)×δ(φˆ-φˆ0)δ(z-z0)(G1eˆ+(cos θP)G2eˆ),
E˜n-in(ω/c)2qr0[G˜1eˆ+(cos θP)G˜2eˆ].
E˜ninq(ω/c)2(r0/RP)×exp[in(φˆ0-32π-RˆP+zˆ0 cos θP)]×[iJn(nrˆ0 sin θP)eˆ+rˆ0-1Jn(nrˆ0 sin θP)×(cot θP)eˆ],
|E˜n|πrˆP-1|s|(rˆ>-rˆ<)(zˆ>-zˆ<)n2|Jn-Ω/ω(n)|
|E˜n|12rP-1(nω/c)2|nJn-Ω/ω(n)p|.
PinLrLφˆLz|j|2/(χeΩ)
Poutc-1Lr2Lz2|j|2n-2ΔnΔΩP,
Pout/Pinχe(Lr/Lφˆ)(Lzω/c)n-5/3ΔnΔΩP,
δtP=12(a/c)2[(v/c)2-1]-1(δt)3+,
P(φ, t)=cos(Ωt)k=0N-1Π(k-Nφ/2π)×cos[m(ωt-2πk/N)],
Π(k-Nφ/2π)=N-1+n=1(nπ/2)-1 sin(nπ/N)cos[n(φ-2πk/N)].
P(φ, t)=cos(Ωt)n=-(nπ)-1 sin(nπ/N)×k=0N-1 cos[mωt-nφ+2π(n-m)k/N],
k=0N-1 cos[mωt-nφ+2π(n-m)k/N]
=cos[mωt-nφ+π(n-m)(N-1)/N]×sin[(n-m)π]csc[(n-m)π/N].
P(φ, t)=(mπ/N)-1 sin(mπ/N)cos(Ωt)×cos[m(φ-ωt)]+l0(-1)l(1+Nl/m)-1×cos[(Nl+m)φ-mωt],

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