Ultra-Wideband, Short-Pulse Electromagnetics 3 pp 165-176 | Cite as

# Theorems on Time-Domain Far Fields

## Abstract

The far-field characteristics of classical electromagnetic fields satisfying Maxwell’s equations have been investigated quite thoroughly for sources radiating at a single frequency, that is, for frequency-domain or time-harmonic fields [1]. For example, frequency-domain far fields radiated by integrable sources in a volume of finite extent decay as 1/*r* or faster as the distance *r* to the far field approaches infinity. In addition, these far fields are entire analytic functions of their angular variables *θ* and *ø*. Using a plane-wave decomposition, frequency-domain near fields can be expressed as an integral of the far-field pattern and its analytic continuation to complex angles of observation [2],[3].

## Keywords

Point Charge Finite Time Interval Finite Region Finite Extent Entire Analytic Function## Preview

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## References

- [1]C. Müller,
*Foundations of the Mathematical Theory of Electromagnetic Waves*( Springer-Verlag, New York, 1969 ).zbMATHGoogle Scholar - [2]D.M. Kerns,
*Plane-Wave Scattering-Matrix Theory of Antennas and Antenna-Antenna Interactions*(NBS Monograph 162, U.S. Gov. Printing Off., Washington, D.C., 1982 ).Google Scholar - [3]T.B.Hansen and A.D. Yaghjian,
*Formulation of time-domain planar near-field measurements without probe correction*(Rome Laboratory Technical Report RL-TR-93–210, Hanscom AFB, MA 01731, 1993 ).Google Scholar - [4]T.B.Hansen and A.D. Yaghjian, IEEE Trans. Antennas Propagat.,
*42*, 1280–1300 (1994).MathSciNetADSzbMATHCrossRefGoogle Scholar - [5]J.A. Stratton,
*Electromagnetic Theory*( McGraw-Hill, New York, 1941 ).zbMATHGoogle Scholar - [6]The interchange of the time integration with the curl operator on the right side of (2) can be proven valid by interchanging the space and time integrations in the integral form of Maxwell’s second equation to get \( \in \int_S E .dS = \int_C {(\int_{t0}^t {Hdt'} )} .d1\) where no point of the surface
*S*or the curve*C*is in the source region V. Then Stokes’ theorem is applied to obtain (2). This interchange of space and time integrations is permitted by standard theorems of integration [7, sec.237] under the very weak condition that**H**(r, t)1is integrable over any finite space-time domain (*C*,*t*), where no point of the curve*C*is in the source region*V*. Alternatively, one could simply take the above integral equation as the fundamental form of Maxwell’s second equation.Google Scholar - [7]E.W. Hobson,
*The Theory of Functions of a Real Variable*(Dover, New York, 1957 ), Vol. II.Google Scholar - [8]J. van Bladel,
*Electromagnetic Fields*( McGraw-Hill, New York, 1964 ).Google Scholar - [9]H.E. Moses, R.J. Nagem and G.V.H. Sandri, J. Math. Phys.,
**33**, 86–101 (1992).MathSciNetADSzbMATHCrossRefGoogle Scholar - [10]J.W. Dettman,
*Applied Complex Variables*( Dover, New York, 1984 ).zbMATHGoogle Scholar - [11]H.E. Moses and R.T. Prosser, SIAM J. Appl. Math.,
**50**, 1325–1340 (1990).MathSciNetADSzbMATHCrossRefGoogle Scholar - [12]E.T. Whittaker and G.N. Watson, A
*Course of Modern Analysis*(Cambridge University Press, London, 1952), sec. 5. 32.Google Scholar - [13]A.D. Yaghjian and T.B. Hansen, J. Appl. Phys.,
**79**, 2822–2830 (1996).ADSCrossRefGoogle Scholar - [14]T.T. Wu, J. Appl. Phys.,
**57**, 2370–2373 (1985).ADSCrossRefGoogle Scholar - [15]A.D. Yaghjian,
*Relativistic Dynamics of a Charged Sphere: Updating the Lorentz-Abraham Model*( Springer-Verlag, New York, 1992 ).Google Scholar - [16]If the first time derivative does not exist, bringing the curl operator under the integral sign to obtain (8) and (9) is no longer a valid interchange. In that case, the derivatives operating on the vector potential integral can be expressed in terms of their defining limits to show that the far fields decay as 1/
*r*unless the secant slope, [**J**(**r**,t + Δt) −**J**(**r**,*t*)]Δ*/t*,becomes infinite for some*t*and Δ*t*[17],[7, sec.246, p.355]. Of course, if the limit of the secant slope exists as Δt → 0, the limit equals the time derivative. These concepts are illustrated in Blejer*et al*. [17] for the specific case of the fields radiated by the current on a disk.Google Scholar - [17]D.J. Blejer, R.C. Wittmann and A.D. Yaghjian, in
*Ultra-Wideband*,*Short-Pulse Electromagnetics*( Plenum Press, New York, 1993 ), pp. 285–292.CrossRefGoogle Scholar - [18]Note that the first time derivative of the current \(\frac{\partial }{{\partial t}}J(r,t)\) must become infinite as a function of time
*t*(not just position r)to generate an electromagnetic missile. For example, the current parallel to a perfectly conducting sharp edge is infinite right at the edge. Yet this singularity is a function of the spatial coordinates and will not generate an electromagnetic missile because the singularity is integrable with respect to the spatial coordinates.Google Scholar - [19]J.M. Myers,
*Developmental Study of Electromagnetic Missiles*(Gordon McKay Laboratory Annual Progress Report No. 3, Harvard University, Cambridge, MA 02138, January 1989), sec.5.Google Scholar