Abstract

Neither Eq. (6.52) of Jackson [Classical Electrodynamics, 3rd ed. (Wiley, 1999)] nor Hannay’s derivation of that equation in the preceding Comment [J. Opt. Soc. Am. A 26, 2107 (2009) ] is applicable to a source whose distribution pattern moves faster than light in vacuo with nonzero acceleration. It is assumed in Hannay’s derivation that the retarded distribution of the density of any moving source will be smooth and differentiable if its rest-frame distribution is. By working out an explicit example of a rotating superluminal source with a bounded and smooth density profile, we show that this assumption is erroneous. The retarded distribution of a rotating source with a moderate superluminal speed is, in general, spread over three disjoint volumes (differing in shape from one another and from the volume occupied by the source in its rest frame) whose boundaries depend on the space–time position of the observer. Hannay overlooks the fact that the limits of integration in his expression for the retarded potential are not differentiable, as functions of the coordinates of the observer, when the distribution pattern of the source moves faster than light. These limits, which delineate the boundaries of the retarded distribution of the source, have divergent gradients at those points on the source boundary that approach the observer, along the radiation direction, with the speed of light at the retarded time. In the superluminal regime, derivatives of the integral representing the retarded potential are well defined only as generalized functions.

© 2009 Optical Society of America

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References

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  1. J. H. Hannay, “Fundamental role of the retarded potential in the electrodynamics of superluminal sources: comment,” J. Opt. Soc. Am. A 26, 2107-2108 (2009).
    [CrossRef]
  2. H. Ardavan, A. Ardavan, J. Singleton, J. Fasel, and A. Schmidt, “Fundamental role of the retarded potential in the electrodynamics of superluminal sources,” J. Opt. Soc. Am. A 25, 543-557 (2008).
    [CrossRef]
  3. R. Courant, Differential and Integral Calculus (Blackie, 1967), Vol. 2, Chap. 4.
  4. J. Hadamard, Lectures on Cauchy's problem in Linear Partial Differential Equations (Yale Univ. Press, 1923). Dover reprint, 1952.
  5. R. F. Hoskins, Delta Functions: An Introduction to Generalised Functions (Harwood, 1999), Chap. 7.
  6. A. H. Zemanian, Distribution Theory and Transform Analysis (McGraw-Hill, 1965).
  7. 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]
  8. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
  9. 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]

2009 (1)

2008 (1)

2007 (1)

2004 (1)

Ardavan, A.

Ardavan, H.

Courant, R.

R. Courant, Differential and Integral Calculus (Blackie, 1967), Vol. 2, Chap. 4.

Fasel, J.

Hadamard, J.

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

Hannay, J. H.

Hoskins, R. F.

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

Jackson, J. D.

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

Schmidt, A.

Singleton, J.

Zemanian, A. H.

A. H. Zemanian, Distribution Theory and Transform Analysis (McGraw-Hill, 1965).

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Figures (3)

Fig. 1
Fig. 1

The function g ( φ ) , defined in Eq. (10), for φ P = 0 , r P = 3 c ω , r = 2 c ω , and curve (a) Δ > 0 , curve (b) Δ = 0 , and curve (c) Δ < 0 . The marked adjacent turning points of curve (a) have the coordinates ( φ ± , ϕ ± ) .

Fig. 2
Fig. 2

The smallest circle (in dahsed lines) designates the boundary of the source distribution described by Eq. (1), in its rest frame, for r c = 2 c ω and a = 1 4 c ω , where c is the speed of light in vacuo. (The axes are marked in units of c ω .) The two circles (in dots and dashes) with the radii 1 and 2 (in units of c ω ) represent the light cylinder and the orbit of the center of the source, respectively. The closed curves (in solid lines) show the cross-section of the retarded distribution of the source with the plane of the orbit (i.e., the multiple images of the circle in dashed lines) for an observer who is located at r P = 3 c ω , φ P = arccos ( 1 3 ) 8 1 2 , z P = 0 , at the observation time t P = 0 . The intersections of these boundaries of the retarded source distribution with r = const. , z = 0 (the dotted circle) specify the limits ( φ l ( n ) , φ u ( n ) ) of the φ integration in the expression for the retarded potential.

Fig. 3
Fig. 3

The smallest circle (in dashed lines) designates the boundary of the source distiribution described by Eq. (1) in its rest frame for r c = 3 2 c ω and a = 1 2 c ω . (The axes are marked in units of c ω .) The two dotted circles with radii 1 and 3 2 (in units of c ω ) represent the light cylinder and the orbit of the center of the source, respectively. The closed solid curve shows the cross-section of the retarded distribution of the the source with the plane of the orbit for an observer who is located at r P = 5 2 c ω , φ P = arccos ( 2 5 ) ( 21 ) 1 2 2 , z P = 0 , at the observation time t P = 0 .

Equations (24)

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ρ ( r , φ ̂ , z ) = { ρ 0 cos 2 [ π R s ( 2 a ) ] if R s a , π < φ ̂ π , 0 otherwise , }
φ ̂ φ ω t ,
R s ( z 2 + r 2 + r c 2 2 r r c cos φ ̂ ) 1 2
A ( x P , t P ) = 1 c d 3 x [ j ( x , t ) ] | x x P | ,
[ j ( x , t ) ] j ( x , t P | x x P | c ) = r ω ρ ( r , φ ̂ ret , z ) e ̂ φ ,
φ ̂ ret φ ω ( t P R c )
R = [ ( z z P ) 2 + r 2 + r P 2 2 r r P cos ( φ φ P ) ] 1 2
( r ̂ , z ̂ ; r ̂ P , z ̂ P ) ( r ω c , z ω c ; r P ω c , z P ω c ) ,
φ ̂ ret = φ ̂ P + g ( r , r P , φ φ P , z z P ) ,
g φ φ P + [ ( z ̂ z ̂ P ) 2 + r ̂ 2 + r ̂ P 2 2 r ̂ r ̂ P cos ( φ φ P ) ] 1 2 ,
| R s | φ ̂ = φ ̂ ret = [ z 2 + r 2 + r c 2 2 r r c cos ( φ ̂ P + g ) ] 1 2 = a
g = ϕ b ± ( r , z , φ ̂ P ) ,
ϕ b ± ± 2 arcsin { [ a 2 z 2 ( r r c ) 2 4 r r c ] 1 2 } φ ̂ P .
Δ ( r ̂ P 2 1 ) ( r ̂ 2 1 ) ( z ̂ P z ̂ ) 2
φ ± φ P + 2 π arccos ( 1 Δ 1 2 r ̂ P r ̂ )
Δ = 0 , φ = φ P + 2 π arccos [ 1 ( r ̂ r ̂ P ) ]
A = ω c ρ 0 n = 1 3 a a d z r c ( a 2 z 2 ) 1 2 r c + ( a 2 z 2 ) 1 2 d r φ l ( n ) φ u ( n ) d φ r 2 R cos 2 ( π 2 a R s | φ ̂ = φ ̂ ret ) e ̂ φ .
A ( r P , φ ̂ P , z P ) = ω c ρ 0 n = 1 3 a a d z r c ( a 2 z 2 ) 1 2 r c + ( a 2 z 2 ) 1 2 d r r 2 σ l ( n ) σ u ( n ) d σ cos 2 { π [ z 2 + r 2 + r c 2 2 r r c cos ( φ ̂ P + g ) ] 1 2 ( 2 a ) } [ ( z z P ) 2 + r 2 + r P 2 2 r r P cos σ ] 1 2 e ̂ φ ,
g = σ + [ ( z ̂ z ̂ P ) 2 + r ̂ 2 + r ̂ P 2 2 r ̂ r ̂ P cos σ ] 1 2
d d x α ( x ) β ( x ) f ( x , ξ ) d ξ = α ( x ) β ( x ) f x d ξ f ( x , α ) d α d x + f ( x , β ) d β d x
g = σ + [ ( z ̂ z ̂ P ) 2 + r ̂ 2 + r ̂ P 2 2 r ̂ r ̂ P cos σ ] 1 2 = ϕ b ± ( r ̂ , z ̂ , φ ̂ P ) ,
P σ = 1 g σ ( g r P e ̂ r P + g z P e ̂ z P 1 r P ϕ b ± φ P e ̂ φ P ) = ω c [ r P r cos ( φ φ P ) ] e ̂ r P + ( z P z ) e ̂ z P + ( R r ̂ P ) e ̂ φ P R + r r ̂ P sin ( φ φ P ) ,
A ( x P , t P ) = 1 c d 3 x d t j ( x , t ) δ ( t P t | x P x | c ) | x P x | ,
P × A = 1 c d 3 x d t j × x P x | x P x | 2 [ 1 c δ ( t P t | x P x | c ) + δ ( t P t | x P x | c ) | x P x | ]

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