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

Author:

Rauch Jeffrey,Mourou Gérard

Abstract

For x x large consider the electric field, E ( t , x ) E(t,x) , and its temporal Fourier Transform, E ^ ( ω , x ) \hat E(\omega ,x) . The D.C. component E ^ ( 0 , x ) \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 t E ( t , x ) \partial _t E(t,x) vanishes identically. For the scalar wave equation the spatial integral of t u ( t , x ) \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.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference3 articles.

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. On the radiation field of pulse solutions of the wave equation. II;Friedlander, F. G.;Proc. Roy. Soc. London Ser. A,1964

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