Plane-wave expansions have proven useful for solving numerous problems involving the radiation, reception, propagation, and scattering of electromagnetic and acoustic fields. Several textbooks and monographs have dealt with the subject of expanding fields in a spectrum of plane waves [1] [10]. However, most of the previous work with plane-wave representations has been confined to the frequency domain, and relatively little work has been devoted to deriving general plane-wave representations for time-domain fields. In geophysics, Schultz and Claerbout [11], Chapman [12], [13], Phinney et al. [14], and Stoffa et al. [15] have used time-domain Radon transforms and "slant stacks" to analyze seismic reflection and refraction data. Tygel and Hubral [16] developed a plane-wave theory for transient acoustic waves in layered media. Hey man et al. [17] [19] presented plane-wave expressions for scalar transient fields, and Marengo and Devaney [20] derived time-domain angular spectrum expressions for electromagnetic fields. Our own previous contributions [21]-[26] to the theory and application of time-domain plane-wave representations form the nucleus of this book in which we develop a general plane-wave theory of electromagnetic and acoustic fields produced by sources that have arbitrary time dependence and are located in homogeneous, isotropic media. The resulting time-domain plane-wave spectrum representations have a variety of applications, ranging from the general decomposition and analysis of transient fields to the formulation of time-domain near-field measurement techniques. A large body of literature exists that documents the development of near-field measurements in the frequency domain. The discovery by Kerns [27] in the early 1960s of a simple, rigorous method to correct for the effects of the measurement probe used in planar near-field scanning initiated the development of practical, accurate near-field measurement techniques. Since that time, the theory for probe-corrected near-field measurements has been successfully formulated and applied to the measurement of electromagnetic and acoustic radiators on planar [28] [31], cylindrical [32], [33], and spherical [34]-[37] surfaces. Accurate experimental and computational techniques have been developed and implemented for near-field scanning [32], [38]-[51], and a number of books [8], [40], [52] and overviews [42], [53]-[55] have been published that document the modern development of near-field measurements and attest to its status as a mature science. 1
2 Chapter 1 Introduction Because all this development of probe-corrected near-field measurement techniques during the past 35 years has been limited to the frequency domain, an important objective of this book is to formulate planar near-field measurement techniques for radiators that are excited by wide-band short pulses and must be measured in the time domain. However, the time-domain formulation of planar near-field scanning may also be applied to radiators that are designed to operate at many discrete frequencies. If such a radiator is excited by a pulse with a frequency bandwidth that covers all the discrete operating frequencies, the radiator can be measured in the time domain, and by means of the Fourier transform, its near and far fields can be determined in the frequency domain at each of the discrete operating frequencies. Alternatively, the radiator could be measured in the frequency domain at each of the discrete operating frequencies. Since the frequency response of a pulsed radiator can be determined from a single scan in the time-domain near field, time-domain near-field measurements can be used to determine the wide-band or out-of-band frequency response of radiators at a reduced measurement time [56], [57]. We find in Chapter 8 that time gating can be used with time-domain measurements to remove the errors caused by the finite size of the scan plane. These finite-scan errors can be prohibitively large for the frequency-domain near-field measurement of broadbeam radiators. Therefore, for broadbeam radiators it may be advantageous to excite the radiator by a relatively wide-band, short pulse and take near-field measurements in the time domain, even if the broadbeam radiator is to be used at a single frequency. Although this book does not treat time-domain measurements on cylindrical and spherical surfaces, we mention the recent work in these areas. The theory, sampling theorems, and computation schemes for probe-corrected spherical near-field scanning of time-domain acoustic and electromagnetic fields are developed in references [58] [61]. Preliminary results on the formulation of cylindrical scanning for scalar time-domain fields were presented in [62]. An outline of the material contained in each of the succeeding chapters of the book is given as follows. Chapter 2 presents the basic electromagnetic and acoustic field equations that are used in Chapters 3 through 8 of the book. We also include as background material in Chapter 2, a number of derivations, theorems, and expressions that are new or not found in the classic textbooks on electromagnetics and acoustics. The time-domain and frequency-domain fields are treated in separate sections. For each domain, the electromagnetic field equations are given first, and the acoustic field equations are then developed along parallel lines. The basic time-independent (static) field equations are included as part of the time-domain sections. In Chapter 2 we prove, with the help of causality and far-field conditions, the equivalence of the first- and second-order differential equations for the electromagnetic and acoustic fields. Rigorous uniqueness and existence theorems are also proven, and sufficient conditions on the fields and sources are found for the validity of surface and volume integral representations of the frequency-domain and time-domain fields. Expressions are derived for cavity-defined and mathematically defined fields in the source regions. The electromagnetic volume integral representations are applied to obtain the Lienard-Wiechert fields of accelerating point charges used in Chapter 5 for analyzing "electromagnetic missiles." Care is taken to derive unambiguous expressions for the force, power, momentum, and energy supplied by the electromagnetic and acoustic fields. In the last section of Chapter 2, the basic electromagnetic and acoustic field equations are extended to lossy dispersive media, and a direct method is used to derive the dispersion relations. In Chapter 3, we derive the frequency-domain Green's function and plane-wave spectrum representations for thefieldsin the half space in front of an electromagnetic or acoustic radiator in terms of the fields in a scan plane. (In Chapter 3, the measurement probe is assumed to be ideal in the sense that it can be used to directly measure the required near fields in the scan plane.) The equations are given in a form that is convenient for their translation into the time
3 domain in Chapter 5. The far fields and plane-wave spectra are expressed in terms of volume sources, Huygens' sources on a closed surface, and the fields in the scan plane. Relationships between the low-frequency behavior of the far fields and their sources are found from the volume-source expressions. The frequency-domain plane-wave spectra of electromagnetic and acoustic sources are first derived in Chapter 3 without regard to mathematical rigor and are then verified rigorously in Appendix D. The plane-wave spectrum divides into propagating and evanescent parts, and explicit expressions are found for the propagating and evanescent contributions to the fields of an acoustic point source and to the far fields of arbitrary electromagnetic and acoustic sources. We determine the singularities and asymptotic behavior of the spectra, and we find that the spectra, the far-field patterns, and the fields outside the source region are analytic functions in certain complex domains of their arguments. Many of the plane-wave representations are rewritten in terms of the far-field patterns with complex angles of observation. Homogeneous fields and inverse source formulas are found that can be expressed throughout all space in terms of far-field patterns evaluated at real angles of observation only. The asymptotic behavior of the plane-wave spectrum is used to derive simple expressions for the smallest possible convex region that can contain the sources of a given far-field pattern. The power and radiation force transmitted through an infinite plane in front of the electromagnetic or acoustic radiator are expressed in terms of the plane-wave spectrum and far-field pattern. In the last section of Chapter 3, we determine the changes required in the preceding equations if the electromagnetic and acoustic sources radiate into lossy regions. A major difference between the equations in the lossless and lossy media is that the entire spectrum of plane waves, not just the evanescent part of the spectrum, is attenuated in lossy media. In Chapters 3 and 6, which derive the frequency-domain formulas for planar near-field scanning with and without probe correction, respectively, the frequency is chosen positive (w > 0) because the fields, sources, and equations for negative frequencies can be found from those for positive frequencies simply through use of the reality condition (f- 0J = f*). However, many of the formulas and relationships in Chapters 3 and 6 do not hold for co = 0, that is, for time-independent fields. Moreover, many of the time-domain formulas and relationships in Chapters 5 and 7 dealing with time-domain planar near-field scanning with and without probe correction, respectively, do not hold for time-independent fields. Therefore, Chapter 4 is devoted to deriving Green's function and plane-wave spectrum representations for static electric and magnetic fields. Emphasis is placed on formulas that determine the total charge and dipole moments of the static sources, as well as on formulas that determine the static fields in a half space in front of the sources in terms of the static fields measured in a scan plane. Many of the electrostatic results in Chapter 4 can also be applied to the measurement of Newtonian gravitational fields. In Chapter 5 the frequency-domain representations of Chapter 3 are translated into the time domain. The time-domain plane-wave representations are expressed in terms of real time-domain plane-wave spectra and fields, and, alternatively, in terms of complex analyticsignal spectra and fields. The analytic-signal formulas are simpler in form than the real timedomain formulas, and thus they prove useful for analysis. However, they involve complex functions of complex times, and, at least for planar near-field measurements, are ultimately more cumbersome to use for computations than the real time-domain formulas. Both the analytic-signal and real time-domain spectra are written as space-time Radon transforms of the fields in the scan plane. The time-domain Green's function and plane-wave spectrum representations are derived in Chapter 5 for the acoustic and electromagnetic fields in the half space in front of the sources in terms of the time-domain fields measured in a scan plane. The time-domain far fields and plane-wave spectra are expressed in terms of the volume sources, the Huygens' sources on a
4 Chapter 1 Introduction closed surface, and the fields in the scan plane. Closed-form expressions are found for the separate contributions of the propagating and evanescent parts of the time-domain spectrum to the fields of an acoustic point source and a rectangular-pulse point source. Time-domain far fields are written in terms of their spectra, and the analytic properties of the far-field patterns and the spectra are determined. At times after all sources have been turned off, the timedomain fields everywhere can be expressed in terms of their far-field patterns integrated over the far-field sphere. Expressions are obtained in terms of the time-domain plane-wave spectra for the total acoustic and electromagnetic energy and momentum radiated into a half space. In addition to presenting the time-domain field expansions, Chapter 5 also derives a number of general theorems on the behavior of time-domain fields and investigates the phenomena of electromagnetic "bullets" and "missiles." In Chapter 6, we derive the frequency-domain probe-corrected formulas that determine the acoustic and electromagnetic fields in the half space in front of the sources (test antenna) in terms of the output on a scan plane of an arbitrary measurement probe (with known receiving characteristic). A straightforward approach based on linearity is taken in the frequencydomain derivations that proves useful in the corresponding derivations in Chapter 7 of the time-domain probe-corrected formulas. With this approach it is not necessary to introduce the full scattering-matrix description of antennas and electroacoustic transducers [8], [29], or the Lorentz reciprocity theorem [30], to obtain the probe-corrected transmission formula first derived by Kerns. The frequency-domain probe-corrected formulas for planar near-field measurements are also derived in Chapter 6 by a novel method based on the probe output satisfying the homogeneous Helmholtz equation. This latter result and its region of validity are rigorously proven for both acoustic and electromagnetic probes. The receiving characteristic for a probe is defined, and reciprocity relations are given for the plane-wave transmitting spectra and receiving characteristics of reciprocal electromagnetic antennas and electroacoustic transducers. General "system 2-port" reciprocity relations between the outputs of two reciprocal antennas are also found. For a reciprocal acoustic monopole or a reciprocal electric dipole, it is proven that the output of the probe becomes proportional to the field, and the probe-corrected formulas of Chapter 6 reduce to the non-probe-corrected formulas of Chapter 3. The probe-corrected formulas for planar near-field scanning in the time-domain are obtained in Chapter 7 byfirsttaking the inverse Fourier transform of the corresponding frequencydomain probe-corrected formulas in Chapter 6, then by deriving the formulas directly in the time domain, and finally by using the result that the time-dependent output of the measurement probe satisfies the time-dependent homogeneous wave equation. These general probecorrected formulas are then shown to greatly simplify through the use of "time-derivative" probes. Reciprocity relations in the time domain are derived between the plane-wave transmitting spectrum and the receiving characteristic of a single reciprocal antenna, as well as between the outputs of any two reciprocal antennas excited by identical input pulses. As in the frequency domain, the time-domain probe-corrected formulas are shown to reduce to the non-probe-corrected formulas if the probe is a reciprocal acoustic monopole or a reciprocal electric dipole. Lastly, in Chapter 7, we consider the time-domain near-field scanning of antennas with a reciprocal probe whose far field is frequency independent over the measurement bandwidth. To determine the time-domain near and far fields of an antenna from the time-domain output data of the near-field measurement probe, one can compute the required integrals over the scan plane directly in the time domain using the formulas of Chapters 5 and 7. Alternatively, one can compute the near and far fields of the antenna by Fourier transforming the time-domain near-field data to the frequency domain, evaluating the frequency-domain integrals over the scan plane found in Chapters 3 and 6, and finally Fourier transforming back into the time
5 domain. Chapter 8 derives the sampling theorems and reconstruction formulas needed to adequately sample the near field and accurately evaluate the time-domain far fields using either of the two computation schemes. The advantages and disadvantages of both computation schemes are illustrated by applying the two schemes to compute the fields of simple timedomain acoustic and electromagnetic radiators. We compare the number of operations required by each computation scheme and the errors in the time-domain caused by the finite size of the scan plane. With either computation scheme, it is found that one of the advantages of time-domain near-field measurements is that time gating the output of the probe can be used to remove the errors caused by the finite size of the scan plane as well as by the multiple interactions between the probe and test antenna. Throughout the book, we have used the International System of Units (mksa). We have consistently defined the Fourier transform of a time-domain function f(t) as +00 fa> = ~ I fit)e iu "dt (1.1) 00 so that f(t) is given in terms of its frequency-domain function f (O by the inverse Fourier transform +00 f(t) = I fa>e- iaa dt. (1.2) 00 The choice of e lcot and the 1 / (2n) factor in the Fourier transform of / (t) given by (1.1) produces a time-harmonic function f w e~ i(i)t in (1.2) that conforms to the time-harmonic functions used in previous frequency-domain, plane-wave formulations of near-field measurement techniques [8]. However, f 0J has the dimensions of f(t) multiplied by time f, and this fact must be taken into account when checking and comparing the dimensions of the frequency-domain and time-domain expressions.