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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. A26, 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.
Journal of the Optical Society of America A 10/2009; 26(10):2109-13. · 1.56 Impact Factor
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ABSTRACT: There is a fundamental difference between the classical expression for the retarded electromagnetic potential and the corresponding retarded solution of the wave equation that governs the electromagnetic field. While the boundary contribution to the retarded solution for the {\em 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 governing the {\em field} may be neglected only if it diminishes with distance faster than the contribution of the source density in the far zone. In the case of a source whose distribution pattern both rotates and travels faster than light {\em in vacuo}, as realized in recent experiments, the boundary term in the retarded solution governing the field is by a factor of the order of $R^{1/2}$ {\em larger} than the source term of this solution in the limit that the distance $R$ of the boundary from the source tends to infinity. This result is consistent with the prediction of the retarded potential that part of the radiation field generated by a rotating superluminal source decays as $R^{-1/2}$, instead of $R^{-1}$, a prediction that is confirmed experimentally. More importantly, it pinpoints the reason why an argument based on a solution of the wave equation governing the field in which the boundary term is neglected (such as appears in the published literature) misses the nonspherical decay of the field.
08/2009;
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ABSTRACT: Observational data imply the presence of superluminal electric currents in pulsar magnetospheres. Such sources are not inconsistent with special relativity; they have already been created in the laboratory. Here we describe the distinctive features of the radiation beam that is generated by a rotating superluminal source and show that (i) it consists of subbeams that are narrower the farther the observer is from the source: subbeams whose intensities decay as 1/R instead of 1/R^2 with distance (R), (ii) the fields of its subbeams are characterized by three concurrent polarization modes: two modes that are 'orthogonal' and a third mode whose position angle swings across the subbeam bridging those of the other two, (iii) its overall beam consists of an incoherent superposition of such coherent subbeams and has an intensity profile that reflects the azimuthal distribution of the contributing part of the source (the part of the source that approaches the observer with the speed of light and zero acceleration), (iv) its spectrum (the superluminal counterpart of synchrotron spectrum) is broader than that of any other known emission and entails oscillations whose spacings and amplitudes respectively increase and decrease algebraically with increasing frequency, and (v) the degree of its mean polarization and the fraction of its linear polarization both increase with frequency beyond the frequency for which the observer falls within the Fresnel zone. We also compare these features with those of the radiation received from the Crab pulsar.
04/2009;
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ABSTRACT: The fact that the formula used by Hannay in the preceding Comment [J. Opt. Soc. Am. A25, 2165 (2008)] is "from a standard text on electrodynamics" neither warrants that it is universally applicable nor that it is unequivocally correct. We have explicitly shown [J. Opt. Soc. Am. A25, 543 (2008)] that, since it does not include the boundary contribution toward the value of the field, the formula in question is not applicable when the source is extended and has a distribution pattern that rotates faster than light in vacuo. The neglected boundary term in the retarded solution to the wave equation governing the electromagnetic field forms the basis of diffraction theory. If this term were identically zero, for the reasons given by Hannay, the diffraction of electromagnetic waves through apertures on a surface enclosing a source would have been impossible.
Journal of the Optical Society of America A 10/2008; 25(9):2167-9. · 1.56 Impact Factor
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ABSTRACT: There is a fundamental difference between the classical expression for the retarded electromagnetic potential and the corresponding retarded solution of the wave equation that governs the electromagnetic field. 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 governing the field may be neglected only if it diminishes with distance faster than the contribution of the source density in the far zone. In the case of a source whose distribution pattern rotates superluminally (i.e., faster than the speed of light in vacuo), the boundary term in the retarded solution governing the field is by a factor of the order of R^(1/2) larger than the source term of this solution in the limit where the distance R of the boundary from the source tends to infinity. This result is consistent with the prediction of the retarded potential that the radiation field generated by a rotating superluminal source decays as 1/R^(1/2), instead of 1/R. It also explains why an argument based on the solution of the wave equation governing the field in which the boundary term is neglected, such as Hannay presents in his Comment, misses the nonspherical decay of the field.
06/2008;
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ABSTRACT: The focusing of the radiation generated by a polarization current with a superluminally rotating distribution pattern is of a higher order in the plane of rotation than in other directions. Consequently, our previously published [J. Opt. Soc. Am. A24, 2443 (2007)] asymptotic approximation to the value of this field outside the equatorial plane breaks down as the line of sight approaches a direction normal to the rotation axis, i.e., is nonuniform with respect to the polar angle. Here we employ an alternative asymptotic expansion to show that, though having a rate of decay with frequency (mu) that is by a factor of order mu(2/3) slower, the equatorial radiation field has the same dependence on distance as the nonspherically decaying component of the generated field in other directions: It, too, diminishes as the inverse square root of the distance from its source. We also briefly discuss the relevance of these results to the giant pulses received from pulsars: The focused, nonspherically decaying pulses that arise from a superluminal polarization current in a highly magnetized plasma have a power-law spectrum (i.e., a flux density S infinity mu(alpha)) whose index (alpha) is given by one of the values -2/3, -2, -8/3, or -4.
Journal of the Optical Society of America A 04/2008; 25(3):780-4. · 1.56 Impact Factor
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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 R(7/2) 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 R(-3) and R(-1), 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 R(1/2)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.
Journal of the Optical Society of America A 04/2008; 25(3):543-57. · 1.56 Impact Factor
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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 R(P) from its source as R(P)(-1/2), instead of R(P)(-1). 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 R(P)(-2) and R(P)(-3), 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 R(P), instead of R(P)(2), 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.
Journal of the Optical Society of America A 09/2007; 24(8):2443-56. · 1.56 Impact Factor
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ABSTRACT: There is a fundamental difference between the classical expression for the
retarded electromagnetic potential and the corresponding retarded solution of
the wave equation that governs the electromagnetic field. 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 governing the field may be neglected only if it diminishes with
distance faster than the contribution of the source density in the far zone. In
the case of a source whose distribution pattern rotates superluminally (i.e.,
faster than the speed of light in vacuo), the boundary term in the retarded
solution governing the field is by a factor of the order of R^(1/2) larger than
the source term of this solution in the limit where the distance R of the
boundary from the source tends to infinity. This result is consistent with the
prediction of the retarded potential that the radiation field generated by a
rotating superluminal source decays as 1/R^(1/2), instead of 1/R. It also
explains why an argument based on the solution of the wave equation governing
the field in which the boundary term is neglected, such as Hannay presents in
his Comment, misses the nonspherical decay of the field.
