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

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Abstract: For large consider the electric field, , and its temporal Fourier Transform, . The D.C. component 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 vanishes identically. For the scalar wave equation the spatial integral of 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.

**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**

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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:
https://doi.org/10.1090/S0002-9939-05-08140-2

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