Thorkild B. Hansen

Time‐domain far fields

The behavior of classical time‐domain electromagnetic far fields is investigated with emphasis placed on those characteristics that are unique to the time domain. Necessary conditions are derived for the time‐domain far fields to decay slower than 1/r and for the energy in a far‐field pulse to decay slower than 1/r2. Time‐domain near fields are expressed in terms of integrals over the time‐domain far‐field pattern. Analyticity with respect to angular variation of the time‐domain far‐field pattern is determined, and restrictions are derived on the time dependence of the electromagnetic far fields.

Electromagnetic and Acoustic Field Equations

This chapter contains sections titled: Time-Domain Electromagnetic Field Equations Time-Domain Acoustic Field Equations Frequency-Domain Electromagnetic Field Equations Frequency-Domain Acoustic Field Equations Equations for Lossy Media ]]>

Appendix D: Validation of the PlaneWave Spectrum Representation

This chapter contains sections titled: D.1 Evaluation of the Spectrum for ( k x , k y ) ?????? (0,0) D.2 Evaluation of the Spectrum for ( k x , k y ) = (0,0)

Appendix B: Proofs of Theorems 2I and 2II

This chapter contains sections titled: B.1 Proof of Theorem 2-I B.2 Proof of Theorem 2-II

Wave equations end transmission formulas for the output of a receiving antenna

The output of a linear receiving antenna satisfies the homogeneous wave equation and the outgoing wave condition in both the frequency and time domains. The homogeneous wave equations are satisfied throughout all space except where the minimum convex surface enclosing the receiving antenna (probe) intersects the minimum convex surface enclosing the transmitting antenna. It is assumed that the probe translates without rotation with respect to the transmitting antenna, and that multiple interactions between the probe and transmitting antenna are negligible (or gated out in the time domain). The frequency-domain wave equation (Helmholtz equation) has been used in the past to efficiently compute the coupling between co-sited antennas, and to suggest a simplified probe-corrected spherical near-field measurement method. In this paper we first give a rigorous derivation of these wave equations and their region of validity. We then use the wave equations to derive both the frequency-domain and time-domain probe-corrected plane-wave transmission formulas that determine the plane-wave spectrum (and thus the near and far fields) of the transmitting antenna in terms of the measured output and the plane-wave receiving characteristic of the probe.

Theorems on Time-Domain Far Fields

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 o. 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].