Abstract

We calculate the gradient of the radiation field generated by a polarization current with a superluminally rotating distribution pattern and show that the absolute value of this gradient increases as R72 with distance R, within the sharply focused subbeams that constitute the overall radiation beam from such a source. In addition to supporting the earlier finding that the azimuthal and polar widths of these subbeams become narrower (as R3 and R1, respectively) with distance from the source, this result implies that the boundary contribution to the solution of the wave equation governing the radiation field does not always vanish in the limit where the boundary tends to infinity (as is commonly assumed in textbooks and the published literature). While the boundary contribution to the retarded solution for the potential can always be rendered equal to zero by means of a gauge transformation that preserves the Lorenz condition, the boundary contribution to the retarded solution of the wave equation for the field may be neglected only if it diminishes with distance faster than the contribution of the source density. In the case of a rotating superluminal source, however, the boundary term in the retarded solution for the field is by a factor of the order of R12 larger than the source term of this solution, in the limit where the boundary tends to infinity. This result explains why an argument based on the solution of the wave equation governing the field in which the boundary term is neglected [such as that presented by Hannay, J. Opt. Soc. A 23, 1530 (2006)] misses the nonspherical decay of the field that is generated by a rotating superluminal source. The only way one can calculate the free-space radiation field of an accelerated superluminal source is via the retarded solution for the potential. Our findings have implications also for the observations of the pulsar emission: The more distant a pulsar, the narrower and brighter its giant pulses should be.

© 2008 Optical Society of America

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References

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  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]
  2. 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, 4614-4631.
    [CrossRef]
  3. A. V. Bessarab, S. P. Martynenko, N. A. Prudkoi, A. V. Soldatov, and V. A. Terekhin, "Experimental study of electromagnetic radiation from a faster-than-light vacuum macroscopic source," Radiat. Phys. Chem. 75, 825-831 (2006).
    [CrossRef]
  4. B. M. Bolotovskii and A. V. Serov, "Radiation of superluminal sources in vacuum," Radiat. Phys. Chem. 75, 813-824 (2006).
    [CrossRef]
  5. 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]
  6. 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).
  7. B. M. Bolotovskii and V. P. Bykov, "Radiation by charges moving faster than light," Sov. Phys. Usp. 33, 477-487 (1990).
    [CrossRef]
  8. H. Ardavan, "Generation of focused, nonspherically decaying pulses of electromagnetic radiation," Phys. Rev. E 58, 6659-6684 (1998).
    [CrossRef]
  9. 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]
  10. H. Ardavan, A. Ardavan, J. Singleton, J. Fasel, and A. Schmidt, "Morphology of the nonspherically decaying radiation beam generated by a rotating superluminal source," J. Opt. Soc. Am. A 24, 2443-2456 (2007).
    [CrossRef]
  11. The superposition of the subbeams is necessarily incoherent because the subbeams that are detected at two neighboring points within the overall beam arise from two distinct filamentary parts of the source with essentially no common elements. The incoherence of this superposition would ensure that, though the field amplitude within a subbeam, which narrows with distance, decays nonspherically, the field amplitude associated with the overall radiation beam, which occupies a constant solid angle, does not.
  12. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
  13. J. H. Hannay, "Bounds on fields from fast rotating sources, and others," Proc. R. Soc. London, Ser. A 452, 2351-2354 (1996).
    [CrossRef]
  14. J. H. Hannay, "Comment II on 'Generation of focused, nonspherically decaying pulses of electromagnetic radiation'," Phys. Rev. E 62, 3008-3009 (2000).
    [CrossRef]
  15. J. H. Hannay, "Comment on 'Method of handling the divergences in the radiation theory of sources that move faster than their waves'," J. Math. Phys. 42, 3973-3974 (2001).
    [CrossRef]
  16. J. H. Hannay, "Spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns: comment," J. Opt. Soc. Am. A 23, 1530-1534 (2006).
    [CrossRef]
  17. 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]
  18. E. Recami, M. Zamboni-Rached, and H. Hernández-Fiueroa, "Superluminal x-shaped waves and localized waves: a historical and scientific introduction," in Localized Waves, H.Hernández-Fiueroa, M.Zamboni-Rached, and E.Recami, eds. (Wiley, 2008), arXiv:0708.1655v2 [physics.gen-ph], 16 August 2007, http://arxiv.org/abs/0708.1655v2.
  19. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, 1953).
  20. R. F. Hoskins, Delta Functions: An Introduction to Generalised Functions (Horwood, 1999), Chap. 7.
  21. C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descent," Proc. Cambridge Philos. Soc. 53, 599-611 (1957).
    [CrossRef]
  22. R. Burridge, "Asymptotic evaluation of integrals related to time-dependent fields near caustics," SIAM J. Appl. Math. 55, 390-409 (1995).
    [CrossRef]
  23. N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, 1986).
  24. That these components of the gradient are of the same order of magnitude is a consequence of the fact that the spiraling cusps that emanate from this source propagate to infinity along a conical surface centered at the origin and so have nonzero pitch angles.
  25. 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]
  26. 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 (London) 422, 141-143 (2003).
    [CrossRef]
  27. 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]
  28. 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. (Max-Plânck Institute für extraterrestrische Physik, 2007), pp. 124-127, arXiv:astro-ph/0701257, 9 January 2007, http://arxiv.org/abs/astro-ph/0701257v1.
  29. H. Ardavan, A. Ardavan, J. Singleton, J. Fasel, and A. Schmidt, "Spectral properties of the nonspherically decaying radiation generated by a rotating superluminal source," arXiv:0710.3364 [astro-ph], 17 October 2007, http://arxiv.org/abs/0710.3364.

2007 (1)

2006 (5)

J. H. Hannay, "Spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns: comment," J. Opt. Soc. Am. A 23, 1530-1534 (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: reply to comment," J. Opt. Soc. Am. A 23, 1535-1539 (2006).
[CrossRef]

A. V. Bessarab, S. P. Martynenko, N. A. Prudkoi, A. V. Soldatov, and V. A. Terekhin, "Experimental study of electromagnetic radiation from a faster-than-light vacuum macroscopic source," Radiat. Phys. Chem. 75, 825-831 (2006).
[CrossRef]

B. M. Bolotovskii and A. V. Serov, "Radiation of superluminal sources in vacuum," Radiat. Phys. Chem. 75, 813-824 (2006).
[CrossRef]

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]

2004 (4)

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]

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, 4614-4631.
[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 (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 (London) 422, 141-143 (2003).
[CrossRef]

2001 (1)

J. H. Hannay, "Comment on 'Method of handling the divergences in the radiation theory of sources that move faster than their waves'," J. Math. Phys. 42, 3973-3974 (2001).
[CrossRef]

2000 (1)

J. H. Hannay, "Comment II on 'Generation of focused, nonspherically decaying pulses of electromagnetic radiation'," Phys. Rev. E 62, 3008-3009 (2000).
[CrossRef]

1998 (1)

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

1996 (1)

J. H. Hannay, "Bounds on fields from fast rotating sources, and others," Proc. R. Soc. London, Ser. A 452, 2351-2354 (1996).
[CrossRef]

1995 (1)

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

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).

1957 (1)

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

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. (1)

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 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, 4614-4631.
[CrossRef]

J. Math. Phys. (1)

J. H. Hannay, "Comment on 'Method of handling the divergences in the radiation theory of sources that move faster than their waves'," J. Math. Phys. 42, 3973-3974 (2001).
[CrossRef]

J. Opt. Soc. Am. A (4)

Nature (London) (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 (London) 422, 141-143 (2003).
[CrossRef]

Phys. Rev. E (2)

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

Proc. Cambridge Philos. Soc. (1)

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

Proc. R. Soc. London, Ser. A (1)

J. H. Hannay, "Bounds on fields from fast rotating sources, and others," Proc. R. Soc. London, Ser. A 452, 2351-2354 (1996).
[CrossRef]

Radiat. Phys. Chem. (2)

A. V. Bessarab, S. P. Martynenko, N. A. Prudkoi, A. V. Soldatov, and V. A. Terekhin, "Experimental study of electromagnetic radiation from a faster-than-light vacuum macroscopic source," Radiat. Phys. Chem. 75, 825-831 (2006).
[CrossRef]

B. M. Bolotovskii and A. V. Serov, "Radiation of superluminal sources in vacuum," Radiat. Phys. Chem. 75, 813-824 (2006).
[CrossRef]

SIAM J. Appl. Math. (1)

R. Burridge, "Asymptotic evaluation of integrals related to time-dependent fields near caustics," SIAM J. Appl. Math. 55, 390-409 (1995).
[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 (9)

The superposition of the subbeams is necessarily incoherent because the subbeams that are detected at two neighboring points within the overall beam arise from two distinct filamentary parts of the source with essentially no common elements. The incoherence of this superposition would ensure that, though the field amplitude within a subbeam, which narrows with distance, decays nonspherically, the field amplitude associated with the overall radiation beam, which occupies a constant solid angle, does not.

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

E. Recami, M. Zamboni-Rached, and H. Hernández-Fiueroa, "Superluminal x-shaped waves and localized waves: a historical and scientific introduction," in Localized Waves, H.Hernández-Fiueroa, M.Zamboni-Rached, and E.Recami, eds. (Wiley, 2008), arXiv:0708.1655v2 [physics.gen-ph], 16 August 2007, http://arxiv.org/abs/0708.1655v2.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw-Hill, 1953).

R. F. Hoskins, Delta Functions: An Introduction to Generalised Functions (Horwood, 1999), Chap. 7.

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

That these components of the gradient are of the same order of magnitude is a consequence of the fact that the spiraling cusps that emanate from this source propagate to infinity along a conical surface centered at the origin and so have nonzero pitch angles.

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. (Max-Plânck Institute für extraterrestrische Physik, 2007), pp. 124-127, arXiv:astro-ph/0701257, 9 January 2007, http://arxiv.org/abs/astro-ph/0701257v1.

H. Ardavan, A. Ardavan, J. Singleton, J. Fasel, and A. Schmidt, "Spectral properties of the nonspherically decaying radiation generated by a rotating superluminal source," arXiv:0710.3364 [astro-ph], 17 October 2007, http://arxiv.org/abs/0710.3364.

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Figures (1)

Fig. 1
Fig. 1

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 (61) of the radial boundaries r ̂ = r ̂ S ( z ̂ ) and r ̂ = r ̂ > ( z ̂ ) of the part of the source that lies within Δ > 0 . The contours C 1 , C 2 , and C 3 are the paths of steepest descent through the stationary point C and through the lower and upper boundaries of the integration domain, respectively.

Equations (135)

Equations on this page are rendered with MathJax. Learn more.

A ( R P , φ P , t P ) = A 0 R ̂ P 1 2 exp [ ( R ̂ P 3 φ ̂ P ) 2 ] ,
A 2 R P 2 sin θ P d θ P d φ P = ( 2 π ) 1 2 ( c ω ) 2 A 0 2 ,
A φ ̂ P = 2 A 0 R ̂ P 7 2 ( R ̂ P 3 φ ̂ P ) exp [ ( R ̂ P 3 φ ̂ P ) 2 ] ,
E = P A 0 A ( c t P ) , B = P × A ,
2 A μ 1 c 2 2 A μ t 2 = 4 π c j μ , μ = 0 , , 3 ,
A μ ( x P , t P ) = 1 c 0 t P d t V d 3 x j μ G + 1 4 π 0 t P d t Σ d S ( G A μ A μ G ) 1 4 π c 2 V d 3 x ( A μ G t G A μ t ) t = 0 ,
A A + Λ , A 0 A 0 Λ t
G ( x , t ; x P , t P ) = δ ( t P t R c ) R ,
A μ ( x P , t P ) = c 1 d 3 x d t j μ ( x , t ) δ ( t P t R c ) R ,
2 B 1 c 2 2 B t 2 = 4 π c × j .
B k ( x P , t P ) = 1 c 0 t P d t V d 3 x ( × j ) k G + 1 4 π 0 t P d t Σ d S ( G B k B k G ) 1 4 π c 2 V d 3 x ( B k G t G B k t ) t = 0 ,
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 ,
B = 1 2 i ( ω c ) 2 μ = μ ± V r d r d φ ̂ d z μ exp ( i μ φ ̂ ) j = 1 3 u j G j φ ̂ ,
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 ] .
R = [ ( z P z ) 2 + r P 2 + r 2 2 r P r cos ( φ P φ ) ] 1 2 ,
g φ φ P + R ̂ ,
Δ 0 , ϕ ϕ ϕ + ,
Δ = ( r ̂ P 2 1 ) ( r ̂ 2 1 ) ( z ̂ z ̂ P ) 2 ,
ϕ ± = 2 π arccos [ ( 1 Δ 1 2 ) ( r ̂ r ̂ P ) ] + R ̂ ± ,
R ̂ ± = [ ( z ̂ z ̂ P ) 2 + r ̂ 2 + r ̂ P 2 2 ( 1 Δ 1 2 ) ] 1 2 .
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 ) ] ,
χ 3 ( ϕ c 2 ) ( 2 c 1 3 ) ,
c 1 ( 3 4 ) 1 3 ( ϕ + ϕ ) 1 3 , c 2 1 2 ( ϕ + + ϕ ) ,
p 1 2 1 3 ( ω c ) R ̂ P 2 exp ( i Ω φ c ω ) ,
p 2 R ̂ P p 1 , 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 ,
G j out ϕ = ϕ ± = G j out χ = ± 1 ( p j ± 2 c 1 q j ) ( 3 c 1 2 ) ;
( B φ ̂ P ) Δ 0 = ( B φ ̂ P ) in + ( B φ ̂ P ) out
( B φ ̂ P ) in , out = 1 2 i ( ω c ) 2 j = 1 3 Δ 0 r d r d z u j L j in , out ,
L j in = μ = μ ± ϕ ϕ + d ϕ μ exp ( i μ φ ̂ ) ( 2 G j φ ̂ 2 ) in ,
L j out = μ = μ ± ( π φ ̂ P ϕ + ϕ + π φ ̂ P ) d ϕ μ exp ( i μ φ ̂ ) ( 2 G j φ ̂ 2 ) out .
L j in = μ = μ ± { μ exp ( i μ φ ̂ ) [ ( G j φ ̂ ) in + i μ G j in ] ϕ ϕ + μ 3 ϕ ϕ + d ϕ exp ( i μ φ ̂ ) G j in } ,
F { L j in } = μ 3 μ = μ ± ϕ ϕ + d ϕ exp ( i μ φ ̂ ) G j in .
L j out = μ = μ ± { μ exp ( i μ φ ̂ ) [ ( G j φ ̂ ) out + i μ G j out ] ϕ ϕ + + ( π φ ̂ P ϕ + ϕ + π φ ̂ P ) d ϕ μ 3 exp ( i μ φ ̂ ) G j out } ,
F { ( B φ ̂ P ) Δ 0 } = ( B φ ̂ P ) s + ( B φ ̂ P ) ns .
( B φ ̂ P ) s = 1 2 i ( ω c ) 2 μ = μ ± μ 3 Δ 0 r d r d z π π d φ ̂ exp ( i μ φ ̂ ) × j = i 3 u j G j
( B φ ̂ P ) ns 1 2 i ( ω c ) 2 j = 1 3 Δ 0 r d r d z u j L j edge
L j edge μ = μ ± μ exp ( i μ φ ̂ ) [ ( G j φ ̂ ) out + i μ G j out ] ϕ ϕ +
G j φ ̂ = Δ φ d φ h j ( φ ) δ ( g ϕ ) ,
[ h 1 h 2 h 3 ] = exp ( i Ω φ ω ) R [ cos ( φ φ P ) sin ( φ φ P ) 1 ] .
G j φ ̂ = Δ φ d φ h j ( φ ) g φ d d φ δ ( g ϕ ) = Δ φ d φ d d φ [ h j ( φ ) g φ ] δ ( g ϕ ) ,
g ( φ ) = 1 3 ν 3 c 1 2 ν + c 2 ,
g φ = ν 2 c 1 2 d φ d ν
G j φ ̂ = Δ ν d ν [ F j ( ν 2 c 1 2 ) 2 + F j ν 2 c 1 2 ] δ ( 1 3 ν 3 c 1 2 ν + c 2 ϕ ) ,
F j ( d φ d ν ) 3 2 g φ 2 h j ,
F j ( d φ d ν ) 2 h j φ ,
G j φ ̂ d ν [ P j + Q j ν ( ν 2 c 1 2 ) 2 + P j + Q j ν ν 2 c 1 2 ] 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 ) ,
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 ) .
φ ± = 2 π arccos [ ( 1 Δ 1 2 ) ( r ̂ r ̂ P ) ]
( 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 .
F j ν = ± c 1 = ± 2 c 1 f j ν = ± c 1 ,
ν = ν * = 2 c 1 sgn ( χ ) cosh ( 1 3 arccosh χ ) , χ 1 ,
( G j φ ̂ ) out 1 ν 2 c 1 2 [ P j + Q j ν ( ν 2 c 1 2 ) 2 + P j + Q j ν ν 2 c 1 2 ] ν * .
( G j φ ̂ ) out 2 sinh 3 ( 1 3 arccosh χ ) c 1 5 ( χ 2 1 ) 3 2 [ c 1 q j + 2 p j sgn ( χ ) cosh ( 1 3 arccosh χ ) ] ,
exp ( i μ φ ̂ ) ( G j φ ̂ ) out ϕ ϕ + ( 2 3 ) 3 c 1 5 exp [ i μ ( φ ̂ P + c 2 ) ] × [ p j cos ( 2 3 μ c 1 3 ) 1 2 i c 1 q j sin ( 2 3 μ c 1 3 ) ] .
L j edge 2 1 3 ( 2 3 ) 3 R P 1 p ¯ j c 1 5 exp ( i Ω φ c ω ) μ = μ ± μ exp [ i μ ( φ ̂ P + ϕ ) ] ,
c 1 2 1 3 R ̂ P 1 Δ 1 2
( B φ ̂ P ) ns 2 ( 2 3 ) 3 R ̂ P 4 exp [ i ( Ω φ c ω π 2 ) ] μ = μ ± μ exp ( i μ φ ̂ P ) × j = 1 3 p ¯ j Δ 0 r ̂ d r ̂ d z ̂ Δ 5 2 u j exp ( i μ ϕ ) .
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 5 2 u j exp ( i μ ϕ ) = ξ ξ S d z ̂ d ξ F ( ξ , z ̂ ) exp ( i α ξ 2 ) ,
F ( ξ , z ̂ ) r ̂ Δ 5 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 ,
J ( z ̂ ) ξ S ξ > d ξ F ( ξ , z ̂ ) exp ( i α ξ 2 )
i ξ 2 = 2 u v + i ( u 2 v 2 ) ,
C 1 d ξ F ( ξ , z ̂ ) exp ( i α ξ 2 ) = ( 1 + i ) d v F ξ = ( 1 + i ) v exp ( 2 α v 2 ) ( 2 π μ ) 1 2 exp [ i ( μ ϕ C π 4 ) ] u j C sin 7 θ P sec θ P 9 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 τ 2 ) 1 2
C 2 d ξ F ( ξ , z ̂ ) exp ( i α ξ 2 ) = exp [ i ( α u S 2 π 2 ) ] × 0 d τ τ ( u S 2 + i τ 2 ) 1 2 F ξ = ( u S 2 + i τ 2 ) 1 2 exp ( α τ 2 ) .
ϕ ( r ̂ , z ̂ ) ϕ S ( z ̂ ) = 1 2 i a τ 2 .
F { C 2 d ξ F ( ξ , z ̂ ) exp ( i α ξ 2 ) } = ( 35 6 4 ) ( 2 π μ ) 1 2 exp [ i ( μ ϕ S 3 π 4 ) ] × u j S sin 7 θ P sec θ P 9 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 F ( 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 F ξ = ( u > 2 + i χ ) 1 2 exp ( α χ ) .
F C 3 , χ = 0 r ̂ > 2 R ̂ P 4 sin 4 θ P sec 2 θ P ( r ̂ > 2 sin 2 θ P 1 ) 3 u j r ̂ = r ̂ > exp ( i μ ϕ C ) u >
C 3 d ξ F ( ξ , z ̂ ) exp ( i α ξ 2 ) r ̂ > 2 ( r ̂ > 2 sin 2 θ P 1 ) 3 u j r ̂ = r ̂ > × exp [ i ( μ ϕ r ̂ = r ̂ > + π 2 ) ] μ 1 R ̂ P 5 ,
( B φ ̂ P ) ns 1 3 7 ( 35 6 4 i ) R ̂ P 7 2 sin 7 θ P sec θ P 9 exp ( i Ω φ C ω ) × μ = μ ± ( 2 π μ ) 1 2 sgn ( μ ) exp ( i π 4 sgn μ ) × j = 1 3 p ¯ j d z ̂ exp [ i μ ( ϕ C + φ ̂ P ) ] u j C ,
ϕ C R ̂ P z ̂ cos θ P + 3 π 2 ,
( B φ ̂ P ) ns 1 3 7 ( 35 6 4 i ) R ̂ P 7 2 sin 7 θ P sec θ P 9 × exp [ i ( Ω ω ) ( φ P + 3 π 2 ) ] × μ = μ ± ( 2 π μ ) 1 2 sgn ( μ ) exp { i [ π 4 sgn ( μ ) μ ( R ̂ P + φ ̂ P + 3 π 2 ) ] } × [ ( s ¯ φ cos θ P s ¯ z sin θ P ) e ̂ s ¯ r e ̂ ] ,
s ¯ r , φ , z d z ̂ s r , φ , z ( r ̂ , z ̂ ) r ̂ = csc θ P exp ( i μ z ̂ cos θ P ) .
ρ ( r 2 + z 2 ) 1 2 , θ arctan ( r z ) ,
B boundary = ρ 2 d t Σ d φ d θ sin θ ( G B R P R P = ρ B G ρ ) ,
B boundary = ρ ̂ 2 Σ d φ ̂ d θ sin θ ( G b B R ̂ P R P = ρ B G b ρ ̂ ) ,
G b d φ R ̂ 1 δ ( g ϕ ) ,
G b = φ = φ j 1 R ̂ g φ = φ = φ j R ̂ + ρ ̂ R ̂ P sin θ sin θ P sin ( φ j φ P ) 1 ,
B R ̂ P = 1 2 i ( ω c ) 2 μ = μ ± V r d r d φ ̂ d z μ exp ( i μ φ ̂ ) × j = 1 3 u j 2 G j R ̂ P φ ̂ ,
2 G j R ̂ P φ ̂ = ( ω c ) Δ φ d φ h j ( φ ) R ̂ 1 [ δ ( g ϕ ) R ̂ 1 δ ( g ϕ ) ] R ̂ R ̂ P ,
R ̂ R ̂ P = R ̂ 1 [ R ̂ P z ̂ cos θ P r ̂ sin θ P cos ( φ φ P ) ]
G b Σ φ = φ j [ ρ ̂ R ̂ P sin θ sin θ P sin ( φ j φ P ) ] 1 ;
B boundary ρ ̂ 2 × ρ ̂ 3 × ρ ̂ 1 × ( ρ ̂ R ̂ P ) 1 × ρ ̂ 7 2 ρ ̂ 1 2 R ̂ P 1 .
Δ 1 2 = ( r ̂ P 2 1 ) 1 2 ( r ̂ 2 r ̂ S 2 ) 1 2 ,
η ( r ̂ 2 r ̂ S 2 ) 1 2 ,
ϕ ( η , z ̂ ) = { r ̂ P 2 ( r ̂ S 2 + η 2 ) [ 1 + ( r ̂ P 2 1 ) 1 2 η ] 2 } 1 2 + 2 π arccos { r ̂ P 1 ( r ̂ S 2 + η 2 ) 1 2 [ 1 + ( r ̂ P 2 1 ) 1 2 η ] }
ϕ = ϕ S + 1 2 R ̂ P 1 cos 2 θ P η 2 1 3 sin 3 θ P η 3 + 1 8 R ̂ P 3 cos 2 θ P ( 5 sin 2 θ P 1 ) η 4 + 1 5 sin 5 θ P η 5 + .
( ϕ η ) ( η τ ) = i a τ ,
( ϕ η ) ( 2 η τ 2 ) + ( 2 ϕ η 2 ) ( η τ ) 2 = i a ,
η ̂ = τ ̂ + τ ̂ 2 + 5 2 τ ̂ 3 + 8 τ ̂ 4 + 231 8 τ ̂ 5 + ,
η ̂ 1 3 R ̂ P sin 3 θ P sec 2 θ P η ,
τ ̂ 1 3 exp ( i π 4 ) R ̂ P 2 sin 5 θ P sec 4 θ P τ .
F ξ = ( u S 2 + i τ 2 ) 1 2 1 3 R ̂ P 5 csc 5 θ P u j τ 1 ( ξ S 2 + i τ 2 ) 1 2 × exp [ i ( μ ϕ C π 2 ) ] ( η 3 ) τ C 2
C 2 d ξ F ( ξ , z ̂ ) exp ( i α ξ 2 ) = 1 3 R ̂ P 8 cot 6 θ P csc 5 θ P exp [ i ( μ ϕ S + π 4 ) ] × u j ( 3 I 1 I 2 ) C 2 ,
I 1 0 d τ τ 4 exp ( α τ 2 ) ψ ( τ ) ,
I 2 0 d τ τ 3 exp ( α τ 2 ) ( d ψ d τ ) ,
ψ ( τ ) = ψ ( 0 ) + ψ ( 0 ) τ + 1 2 ψ ( 0 ) τ 2 + 1 3 ! ψ ( 0 ) τ 3 + 1 4 ! ψ ( κ τ ) τ 4 ,
I 1 = lim ϵ 0 [ ψ ( 0 ) ϵ d τ τ 4 exp ( α τ 2 ) + ψ ( 0 ) ϵ d τ τ 3 exp ( α τ 2 ) + 1 2 ψ ( 0 ) ϵ d τ τ 2 exp ( α τ 2 ) + 1 3 ! ψ ( 0 ) ϵ d τ τ 1 exp ( α τ 2 ) + 1 4 ! ϵ d τ ψ ( κ τ ) exp ( α τ 2 ) ] .
ϵ d τ τ 4 exp ( α τ 2 ) = 1 3 ϵ 3 ( 1 2 α ϵ 2 ) exp ( α ϵ 2 ) + 2 3 π 1 2 α 3 2 erfc ( α 1 2 ϵ ) ,
ϵ d τ ψ ( κ τ ) exp ( α τ 2 ) = 4 ! ϵ d τ τ 4 [ ψ ( τ ) ψ ( 0 ) ψ ( 0 ) τ 1 2 ψ ( 0 ) τ 2 1 3 ! ψ ( 0 ) τ 3 ] exp ( α τ 2 )
0 d τ ψ ( κ τ ) exp ( α τ 2 ) ψ ( 0 ) 0 d τ exp ( α τ 2 ) = 1 2 ( π α ) 1 2 ψ ( 0 ) .
F { 0 d τ τ 4 exp ( α τ 2 ) } = 2 3 π 1 2 α 3 2
F { I 1 } 2 3 π 1 2 ψ ( 0 ) α 3 2 + 1 2 ψ ( 0 ) ( ln α + γ ) α 1 2 π 1 2 ψ ( 0 ) α 1 2 1 12 ψ ( 0 ) ( ln α + γ ) + 1 48 π 1 2 ψ ( 0 ) α 1 2 , α 1 ,
F { I 2 } 1 2 ψ ( 0 ) ( ln α + γ ) α π 1 2 ψ ( 0 ) α 1 2 1 4 ψ ( 0 ) ( ln α + γ ) + 1 12 π 1 2 ψ ( 0 ) α 1 2 , α 1 .
ψ = 1 3 τ ̂ 3 2 τ ̂ 2 4 τ ̂ 3 105 8 τ ̂ 4 + ,
F { 3 I 1 I 2 } = ( 105 6 4 ) ( 2 π μ ) 1 2 sin 18 θ P sec θ P 15 R ̂ P 15 2 ,
R ̂ P 1 ,

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