Skip to main content Accessibility help
×
Home

The electromagnetic radiation whose decay violates the inverse-square law: detailed mathematical treatment of an experimentally realized example

  • Houshang Ardavan (a1)

Abstract

I analyse and numerically evaluate the radiation field generated by an experimentally realized embodiment of an electric polarization current whose rotating distribution pattern moves with linear speeds exceeding the speed of light in vacuum. I find that the flux density of the resulting emission (i) has a dominant value and is linearly polarized within a sharply delineated radiation beam whose orientation and polar width are determined by the range of values of the linear speeds of the rotating source distribution, and (ii) decays with the distance $d$ from the source as $d^{-\unicode[STIX]{x1D6FC}}$ in which the value of $\unicode[STIX]{x1D6FC}$ lies between $1$ and $2$ (instead of being equal to $2$ as in a conventional radiation) across the beam. In that the rate at which boundaries of the retarded distribution of such a source change with time depends on its duration monotonically, this is an intrinsically transient emission process: temporal rate of change of the energy density of the radiation generated by it has a time-averaged value that is negative (instead of being zero as in a conventional radiation) at points where the envelopes of the wave fronts emanating from the constituent volume elements of the source distribution are cusped. The difference in the fluxes of power across any two spheres centred on the source is in this case balanced by the change with time of the energy contained inside the shell bounded by those spheres. These results are relevant not only to long-range transmitters in communications technology but also to astrophysical objects containing rapidly rotating neutron stars (such as pulsars) and to the interpretation of the energetics of the multi-wavelength emissions from sources that lie at cosmological distances (such as radio and gamma-ray bursts). The analysis presented in this paper is self-contained and supersedes my earlier works on this problem.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      The electromagnetic radiation whose decay violates the inverse-square law: detailed mathematical treatment of an experimentally realized example
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      The electromagnetic radiation whose decay violates the inverse-square law: detailed mathematical treatment of an experimentally realized example
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      The electromagnetic radiation whose decay violates the inverse-square law: detailed mathematical treatment of an experimentally realized example
      Available formats
      ×

Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

Corresponding author

Email address for correspondence: ardavan@ast.cam.ac.uk

References

Hide All
Ardavan, A. & Ardavan, H.2010 Apparatus for generating focused electromagnetic radiation. European patent EP1112578.
Ardavan, A., Ardavan, H. & Singleton, J. 2004a Synchrotron-Čerenkov radiation. Science 303 (5656), 311.
Ardavan, A., Hayes, W., Singleton, J., Ardavan, H., Fopma, J. & Halliday, D. 2004b Experimental observation of nonspherically-decaying radiation from a rotating superluminal source. J. Appl. Phys. 96, 77607777(E).
Ardavan, H. 1981 Is the light cylinder the site of emission in pulsars? Nature 289, 4445.
Ardavan, H. 1998 Generation of focused, nonspherically decaying pulses of electromagnetic radiation. Phys. Rev. E 58, 66596684.
Ardavan, H. 1999 Method of handling the divergences in the radiation theory of sources that move faster than their waves. J. Math. Phys. 40, 43314336.
Ardavan, H. 2000 Reply to Comments on Generation of focused, nonspherically decaying pulses of electromagnetic radiation. Phys. Rev. E 62 (2), 30103013.
Ardavan, H., Ardavan, A. & Singleton, J. 2004c Spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns. J. Opt. Soc. Am. A 21, 858872.
Ardavan, H., Ardavan, A. & Singleton, J. 2006 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 (6), 15351539.
Ardavan, H., Ardavan, A., Singleton, J., Fasel, J. & Schmidt, A. 2007 Morphology of the nonspherically decaying radiation beam generated by a rotating superluminal source. J. Opt. Soc. Am. A 24, 24432456.
Ardavan, H., Ardavan, A., Singleton, J., Fasel, J. & Schmidt, A. 2008a Fundamental role of the retarded potential in the electrodynamics of superluminal sources. J. Opt. Soc. Am. A 25, 543557.
Ardavan, H., Ardavan, A., Singleton, J., Fasel, J. & Schmidt, A. 2008b Morphology of the nonspherically decaying radiation beam generated by a rotating superluminal source: reply to comment. J. Opt. Soc. Am. A 25 (9), 21672169.
Ardavan, H., Ardavan, A., Singleton, J., Fasel, J. & Schmidt, A. 2009a Fundamental role of the retarded potential in the electrodynamics of superluminal sources: reply to comment. J. Opt. Soc. Am. A 26 (10), 21092113.
Ardavan, H., Ardavan, A., Singleton, J., Fasel, J. & Schmidt, A. 2009b Inadequacies in the conventional treatment of the radiation field of moving sources. J. Math. Phys. 50, 103510(1)–103510(12).
Ardavan, H., Ardavan, A., Singleton, J. & Perez, M. R. 2008c Mechanism of generation of the emission bands in the dynamic spectrum of the Crab pulsar. Mon. Not. R. Astron. Soc. 388, 873883.
Bender, C. M. & Orszag, S. A. 1999 Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer.
Bolotovskii, B. M. & Bykov, V. P. 1990 Radiation by charges moving faster than light. Sov. Phys. Uspekhi 33, 477487.
Bolotovskii, B. M. & Ginzburg, V. L. 1972 The Vavilov-Čerenkov effect and the doppler effect in the motion of sources with superluminal velocity in vacuum. Sov. Phys. Uspekhi 15, 184192.
Bolotovskii, B. M. & Serov, A. V. 2005 Radiation of superhumanal sources in empty space. Sov. Phys. Uspekhi 48, 903915.
Burridge, R. 1995 Asymptotic evaluation of integrals related to time-dependent fields near caustics. SIAM J. Appl. Maths 55, 390409.
Chatterjee, S., Law, C. J., Wharton, R. S., Burke-Spolaor, M., Hessels, J. W. T., Bower, G. C., Cordes, J. M., Tendulkar, S. P., Bassa, C. G., Demorest, P. et al. 2017 A direct localization of a fast radio burst and its host. Nature 541, 5861.
Chester, C., Friedman, B. & Ursell, F. 1957 An extension of the method of steepest descent. Proc. Camb. Phil. Soc. 53, 599611.
Courant, R. 1967 Differential and Integral Calculus, vol. 2, chap. 4. Blackie.
Ginzburg, V. L. 1972 Vavilov-Čerenkov effect and anomalous doppler effect in a medium in which the wave phase velocity exceeds the velocity of light in vacuum. Sov. Phys. JETP 35, 9293.
Gradshteyn, I. S. & Ryzhik, I. M. 1980 Table of Integrals, Series and Products. Academic.
Hadamard, J. 2003 Lectures on Cauchy’s Problem in Linear Partial Differental Equations. Dover.
Hannay, J. H. 2000 Comment II on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’. Phys. Rev. E 62 (2), 30083009.
Hannay, J. H. 2001 Comment on ‘Method of handling the divergences in the radiation theory of sources that move faster than their waves’. J. Math. Phys. 42, 39733974.
Hannay, J. H. 2006 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 (6), 10847529.
Hannay, J. H. 2008 Morphology of the nonspherically decaying radiation generated by a rotating superluminal source: comment. J. Opt. Soc. Am. A 25, 21652166.
Hannay, J. H. 2009 Fundamental role of the retarded potential in the electrodynamics of superluminal sources: comment. J. Opt. Soc. Am. A 26, 21072109.
Hewish, A. 2000 Comment I on ‘Generation of focused, nonspherically decaying pulses of electromagnetic radiation’. Phys. Rev. E 62, 3007.
Hoskins, R. F. 2009 Delta Functions: an Introduction to Generalised Functions, 2nd edn. Oxford.
Jackson, J. D. 1999 Classical Electrodynamics, 3rd edn. Wiley.
Jay, F.(Ed.) 1984 IEEE Standard Dictionary of Electrical and Electronics Terms, 3rd edn. Institute of Electrical and Electronics Engineers.
Kalapotharakos, C., Contopoulos, I. & Kazanas, D. 2012 The extended pulsar magnetosphere. Mon. Not. Astron. Soc. 420, 27932798.
McDonald, K. T. 2004 Synchrotron-Čerenkov radiation. Science 303 (5656), 310.
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics, vol. 1. McGraw-Hill.
Olver, F. W. J., Lozier, D. W., Boisvert, R. F. & Clark, C. W.(Eds) 2010 NIST Handbook of Mathematical Functions. Cambridge University Press.
Philippov, A., Uzdensky, D. A., Spitkovsky, A. & Cerutti, B. 2019 Pulsar radio emission mechanism: radio nanoshots as a low frequency afterglow of relativistic magnetic reconnection. Astrophys. J. 876, L6L12.
Piron, F. 2016 Gamma-ray bursts at high and very high energies. C. R. Phys. 17, 617631.
Spitkovsky, A. 2006 Time-dependent force-free pulsar magnetospheres: axisymmetric and oblique rotators. Astrophys. J. 648, L51L54.
Tchekhovskoy, A., Philippov, A. & Spitkovsky, A. 2016 Three-dimensional analytical description of magnetized winds from oblique pulsars. Mon. Not. R. Astron. Soc. 457, 33843395.
Uzdensky, D. A. & Spitkovsky, A. 2014 Physical conditions in the reconnection layer in pulsar magnetospheres. Astrophys. J. 780, 3(1)–3(7).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Related content

Powered by UNSILO

The electromagnetic radiation whose decay violates the inverse-square law: detailed mathematical treatment of an experimentally realized example

  • Houshang Ardavan (a1)

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.