Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The time integrated far field for Maxwell's and D'Alembert's equations


Authors: Jeffrey Rauch and Gérard Mourou
Journal: Proc. Amer. Math. Soc. 134 (2006), 851-858
MSC (2000): Primary 78M35, 78A40, 78A60, 35L05
Published electronically: July 18, 2005
MathSciNet review: 2180903
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For $x$ large consider the electric field, $E(t,x)$, and its temporal Fourier Transform, $\hat E(\omega,x)$. The D.C. component $\hat E(0,x)$ is equal to the time integral of the electric field. Experimentally, one observes that the D.C. component is negligible compared to the field. In this paper we show that this is true in the far field for all solutions of Maxwell's equations. It is not true for typical solutions of the scalar wave equation. The difference is explained by the fact that though each component of the field satisfies the scalar wave equation, the spatial integral of $\partial_t E(t,x)$vanishes identically. For the scalar wave equation the spatial integral of $\partial_t u(t,x)$ need not vanish. This conserved quantity gives the leading contribution to the time integrated far field. We also give explicit formulas for the far field behavior of the time integrals of the intensity.


References [Enhancements On Off] (What's this?)

  • 1. J.F. Whitaker, F. Gao, and Y. Liu, in Nonlinear Optics for High-Speed Electronics and Optical Frequency Conversion, N. Peygambarian, H. Everitt, R.C. Eckardt, D.D. Lowenthal, eds., Proc. SPIE, vol. 2145, pp. 168-177 (1994).
  • 2. R. Courant, Methods of Mathematical Physics Volume II, Interscience Publ. 1962.
  • 3. F. G. Friedlander, On the radiation field of pulse solutions of the wave equation. II, Proc. Roy. Soc. Ser. A 279 (1964), 386–394. MR 0164132 (29 #1431)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 78M35, 78A40, 78A60, 35L05

Retrieve articles in all journals with MSC (2000): 78M35, 78A40, 78A60, 35L05


Additional Information

Jeffrey Rauch
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109
Email: rauch@umich.edu

Gérard Mourou
Affiliation: Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan 48109-2122
Email: mourou@umich.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08140-2
PII: S 0002-9939(05)08140-2
Keywords: Far field, Maxwell's equations, wave equation, D.C. component
Received by editor(s): October 1, 2004
Published electronically: July 18, 2005
Additional Notes: The first author was partially supported by the US National Science Foundation grant NSF-DMS-0104096.
The second author was partially supported by the US National Science Foundation grant NSF-PHY-0114336.
Communicated by: M. Gregory Forest
Article copyright: © Copyright 2005 American Mathematical Society