Morphology of the nonspherically decaying radiation generated by a rotating superluminal source: reply to comment

The fact that the formula used by Hannay in his Comment 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. A 25, 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 iffraction of electromagnetic waves through apertures on a surface enclosing a source would have been impossible. 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

Mechanism of generation of the emission bands in the dynamic spectrum of the Crab pulsar

We show that the proportionately spaced emission bands in the dynamic spectrum of the Crab pulsar (Hankins T. H. & Eilek J. A., 2007, ApJ, 670, 693) fit the oscillations of the square of a Bessel function whose argument exceeds its order. This function has already been encountered in the analysis of the emission from a polarization current with a superluminal distribution pattern: a current whose distribution pattern rotates (with an angular frequency $\omega$) and oscillates (with a frequency $\Omega>\omega$ differing from an integral multiple of $\omega$) at the same time (Ardavan H., Ardavan A. & Singleton J., 2003, J Opt Soc Am A, 20, 2137). Using the results of our earlier analysis, we find that the dependence on frequency of the spacing and width of the observed emission bands can be quantitatively accounted for by an appropriate choice of the value of the single free parameter $\Omega/\omega$. In addition, the value of this parameter, thus implied by Hankins & Eilek's data, places the last peak in the amplitude of the oscillating Bessel function in question at a frequency ($\sim\Omega^3/\omega^2$) that agrees with the position of the observed ultraviolet peak in the spectrum of the Crab pulsar. We also show how the suppression of the emission bands by the interference of the contributions from differring polarizations can account for the differences in the time and frequency signatures of the interpulse and the main pulse in the Crab pulsar. Finally, we put the emission bands in the context of the observed continuum spectrum of the Crab pulsar by fitting this broadband spectrum (over 16 orders of magnitude of frequency) with that generated by an electric current with a superluminally rotating distribution pattern

The spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns

We present a theoretical study of the emission from a superluminal polarization current whose distribution pattern rotates (with an angular frequency $\omega$) and oscillates (with a frequency $\Omega$) at the same time, and which comprises both poloidal and toroidal components. This type of polarization current is found in recent practical machines designed to investigate superluminal emission. We find that the superluminal motion of the distribution pattern of the emitting current generates localized electromagnetic waves that do not decay spherically, i.e. that do not have an intensity diminishing like ${R_P}^{-2}$ with the distance $R_P$ from their source. The nonspherical decay of the focused wave packets that are emitted by the polarization currents does not contravene conservation of energy: the constructive interference of the constituent waves of such propagating caustics takes place within different solid angles on spheres of different radii ($R_P$) centred on the source. For a polarization current whose longitudinal distribution (over an azimuthal interval of length $2\pi$) consists of $m$ cycles of a sinusoidal wave train, the nonspherically decaying part of the emitted radiation contains the frequencies $\Omega \pm m\omega$; i.e. it contains {\it only} the frequencies involved in the creation and implementation of the source. This is in contrast to recent studies of the spherically decaying emission, which was shown to contain much higher frequencies. The polarization of the emitted radiation is found to be linear for most configurations of the source.Comment: 19 pages, six figure

Spectral properties of the nonspherically decaying radiation generated by a rotating superluminal source

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 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 proportional to mu^alpha) whose index (alpha) is given by one of the values -2/3, -2, -8/3, or -4

The fundamental role of the retarded potential in the electrodynamics of superluminal sources

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 constituting the overall radiation beam. This result not only supports the earlier finding that the azimuthal and polar widths of these subbeams narrow with distance (as R^(-3) and R^(-1), respectively), but also 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. There is a fundamental difference between the classical expressions for the retarded potential and field: while the boundary contribution for the potential can always be made zero via a gauge transformation preserving the Lorenz condition, that for 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 rotating superluminal source, however, the boundary term in the retarded solution for the field is by a factor of order R^(1/2) larger than the source term of this solution in the limit, which explains why an argument based on the solution of the wave equation governing the field that neglects the boundary term (such as that presented by J. H. Hannay) misses the nonspherical decay of the field. Given that the distribution of the radiation field of an accelerated superluminal source in the far zone is not known a priori, the only way to calculate the free-space radiation field of such sources is via the retarded solution for the potential. Finally, we apply these findings to pulsar observational data: the more distant a pulsar, the narrower and brighter its giant pulses should be