Ehud Heyman

Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics.

Paraxial Gaussian beams (GBs) are collimated wave objects that have found wide application in optical system analysis and design. A GB propagates in physical space according to well-established quasi-geometric-optical rules that can accommodate weakly inhomogeneous media as well as reflection from and transmission through curved interfaces and thin-lens configurations. We examine the GB concept from a broad perspective in the frequency domain (FD) and the short-pulse time domain (TD) and within as well as arbitrarily beyond the paraxial constraint. For the formal analysis, which is followed by physics-matched high-frequency asymptotics, we use a (space-time)-(wavenumber-frequency) phase-space format to discuss the exact complex-source-point method and the associated asymptotic beam tracking by means of complex rays, the TD pulsed-beam (PB) ultrawideband wave-packet counterpart of the FD GB, GBs and PBs as basis functions for representing arbitrary fields, GB and PB diffraction, and FD-TD radiation from extended continuous aperture distributions in which the GB and the PB bases, installed through windowed transforms, yield numerically compact physics-matched a priori localization in the plane-wave-based nonwindowed spectral representations.

Complex-source pulsed-beam fields

Pulsed beams (PB’s) are time-dependent wave fields that are confined in beamlike fashion in transverse planes perpendicular to the propagation axis, whereas confinement along the axis is due to temporal windowing. Because they have these properties, pulsed beams are useful wave objects for generating and synthesizing highly focused transient fields. The PB problem is addressed here within the context of fundamental Green’s-function propagators for the time-dependent field equations. In a departure from known results in the frequency domain, by which beam solutions can be generated from point-source solutions by displacing the source coordinate location into a complex coordinate space, the complex extension is applied here as well to the source initiation time. This procedure converts the conventional causal impulsive-source Green’s-function propagator into a noncausal PB propagator, which must be defined as an analytic signal because, owing to causality, the analytic continuation into the complex domain cannot be performed by direct substitution. This being done, PB’s can be manipulated as conventional Green’s functions. Some previous results obtained by similar methods are viewed here from a sharper perspective, and new results, both analytical and numerical, are presented that grant basic insight into the PB behavior, including the ability to excite these fields by finite-causal-aperture-source distributions. Besides the basic (analytic Green’s-function) PB, examples include PB’s with frequency spectra of special interest. Particular attention is paid to the PB synthesis of focus-wave modes, which are source-free solutions of the time-dependent wave equation, and to the compact PB formulation of wave fields synthesized by focus-wave-mode spectral superposition.

Weakly dispersive spectral theory of transients, part I: Formulation and interpretation

Dispersive effects in transient propagation and scattering are usually negligible over the high frequency portion of the signal spectrum, and for certain configurations, they may be neglected altogether. The source-excited field may then be expressed as a continuous spatial spectrum of nondispersive time-harmonic local plane waves, which can be inverted in closed form into the time domain to yield a fundamental field representation in terms of a spatial spectrum of transient local plane waves. By exploiting its analytic properties, one may evaluate the basic spectral integral in terms of its singularities-real and complex, time dependent and time independent-in the complex spectral plane. These singularities describe distinct features of the propagation and scattering process appropriate to a given environment. The theory is developed in detail for the generic local plane wave spectra representative of a broad class of two-dimensional propagation and diffraction problems, with emphasis on physical interpretation of the various spectral contributions. Moreover, the theory is compared with a similar approach that restricts all spectra to be real, thereby forcing certain wave processes into a spectral mold less natural than that admitting complex spectra. Finally, application of the theory is illustrated by specific examples. The presentation is divided into three parts. Part I, in this paper, deals with the formulation of the theory and the classification of the singularities. Parts II and III, to appear subsequently, contain the evaluation and interpretation of the spectral integral and the applications, respectively.

Pulsed beam propagation in inhomogeneous medium

Pulsed beams (PB) are localized space-time wavepackets that propagate along ray trajectories. This paper deals with general PB solutions in inhomogeneous medium. We derive an approximate form of the time-dependent wave equation (termed the wavepacket equation), valid within a moving space-time window that brackets the wavepacket, and then construct its exact PB solutions. This is done first in a free-space and latter on in a general smoothly varying medium where the propagation trajectories are curved. We also determine the reflection and transmission laws at curved interfaces. These new PBs are related to the so called complex source pulsed beams which are exact solutions in free-space, but they have more general form that admits wavepacket astigmatism and medium inhomogeneity. Since they maintain their wavepacket structure throughout the propagation process they are identified as eigen-wavepacket solutions of the time dependent wave equation. >

Spectral analysis of focus wave modes

The focus wave mode (FWM), which is a time-dependent beam field that propagates at the speed of light without dispersion and retains its shape in space, is an interesting wave object with possible implications for synthesizing focused fields under transient conditions. To explore this potential, it is necessary to understand fully the properties of this wave field. It is already known that the FWM is a homogeneous solution of the wave equation, which is related in a special way to the field of a source moving on a complex trajectory parallel to the real axis of propagation. This suggests that there may be a connection between the FWM and the conventional free-space Green’s function. It is shown here that the FWM is related, in fact, to a source-free combination of causal and anticausal free-space Green’s functions and that one can formulate a bilateral transform pair relating these solutions. This new representation is then analyzed by using the spectral theory of transients to establish the properties of the FWM in terms of a distribution of transient plane waves. The spectral decomposition in the spatial wave-number domain reveals that the FWM is synthesized by both forward- and backward-propagating plane waves that are restricted to the visible spectrum. Asymptotic considerations show that the dominant mechanism is constructive interference of the backward-propagating waves. Taken together, the Green’s-function and spectral approaches grant further insight into the physical and spectral properties of the FWM. The conclusions cast doubt on the possibility of embedding the FWM within a causal excitation scheme.

Wave analysis of Airy beams

The Airy beams are analyzed in order to provide a cogent physical explanation to their intriguing features which include weak diffraction, curved propagation trajectories in free-space, and self healing. The asymptotically exact analysis utilizes the method of uniform geometrical optics (UGO), and it is also verified via a uniform asymptotic evaluation of the Kirchhoff-Huygens integral. Both formulations are shown to fully agree with the exact Airy beam solution in the paraxial zone where the latter is valid, but they are also valid outside this zone. Specifically it is shown that the beam along the curved propagation trajectory is not generated by contributions from the main lobe in the aperture, i.e., it is not described by a local wave-dynamics along this trajectory. Actually, this beam is identified as a caustic of rays that emerge sideways from points in the initial aperture that are located far away from the main lobe. The field of these focusing rays, described here by the UGO, fully agrees with the Airy beam solution. These observations explain that the “weak-diffraction” and the “self healing” properties are generated, in fact, by a continuum of sideways contributions to the field. The uniform ray representation provides a systematic framework to synthesize aperture sources for other beam solutions with similar properties in uniform or in non-uniform media.

Phase-space beam summation for time-dependent radiation from large apertures : continuous parameterization

We extend the study of alternative phase-space formulations of time-harmonic radiation from extended but truncated aperture source distributions to the time domain. Included are nonwindowed continuous forms spanning the space–time (configuration) domains, wave-number-frequency (spectrum) domains, and windowed (local beam-type) continuous forms. Synthesized in the frequency domain by nonwindowed or windowed Fourier transforms, field synthesis in the time domain involves nonwindowed or windowed radon transforms combined with the theory of analytic signals. Because the properties of suitable wave objects used in the analysis and synthesis of the field are strongly tied to relevant configurational and spectral parameters, the incorporation of these aspects into the various formats is referred to as phase-space parameterization. In the continuous parameterization the resulting time-dependent field radiated from the aperture is expressed as a superposition of pulsed beams whose phase-space parameters are their initiation time, initiation location, and initial direction. The properties of these formulations are discussed in detail, within a rigorous format and also with more physically transparent asymptotic approximations. As in the time-harmonic case, major stress is placed on localization in the phase space, which is achievable with various alternatives, and on the corresponding implications. Specific examples include analytic δ windows that yield as propagators complex-source pulsed beams, and numerical implementation of field synthesis for nonfocused and focused pulsed aperture distributions.

Phase-space beam summation for time-harmonic radiation from large apertures

Analytical modeling of high-frequency time-harmonic and transient radiation from extended aperture sources and of propagation of the resulting fields through perturbing environments is facilitated by simultaneous use of configurational (space-time) and spectral (wave number–frequency) information for suitably defined synthesizing wave objects. Such a bilateral approach can be embodied within a configuration-spectrum phase space. The present investigation deals with radiation from extended aperture sources, with emphasis on alternative uses of the phase space at high frequencies, on promising wave objects as basis elements for field synthesis, and on extraction of physical information from exact wave solutions by asymptotic methods. Of special interest are beam-type wave objects that exhibit localization in the phase space because localized wave fields have favorable propagation characteristics in complex external environments. In this paper, alternative phase-space parameterizations are applied to time-harmonic plane aperture distributions and to the corresponding fields radiated into a homogeneous half-space. The parameterizations include nonwindowed continuum versions, in which localization occurs asymptotically through constructive interference; windowed continuum versions, in which localization is embedded inherently; and windowed discretized versions, in which the basis elements are situated on a self-consistent configuration–wave number lattice. By analysis and illustrative examples, it is shown how these alternative formulations are interrelated, how the localization around well-defined regions in the phase space takes place in each formulation, and how these localization properties, through the beam propagators, influence the synthesis of the radiation field. Transient phenomena will be addressed in separate publications.

Time-domain near-field analysis of short-pulse antennas .I. Spherical wave (multipole) expansion

The radiation from a time-dependent source distribution in free-space is analyzed using time-domain (TD) spherical wave (multipole) expansion. The multipole moment functions are calculated from the time-dependent source distribution. The series convergence rate in the near and far zone and the bounds on the near-zone reactive field are determined as functions of the source support and of the pulse length. The formulation involves a spherical transmission line representation that can be extended to more general spherical configurations. This formulation also describes the field and energy transmission mechanisms in a physically transparent fashion that will be used in a companion paper to define and explore fundamental concepts such as TD reactive energy and Q and to derive bounds on the antenna properties. Finally, the concepts discussed above are demonstrated numerically for pulsed radiation by a circular current disk.

Weakly dispersive spectral theory of transients, part II: Evaluation of the spectral integral

In the spectral theory of transients formulated in Part I of this paper, the transient response for weakly dispersive wave processes has been expressed in terms of canonical integrals in the complex spatial wavenumber domain. The real and complex singularities in the integrands, which dominate the behavior of the spectral integrals, have been categorized and associated with generic physical wave processes. The integrals are now evaluated by Contour deformation around the singularities. This yields general expressions for the transient Greens function that are applicable to a broad class of propagation and diffraction problems. The generic results, which can be grouped into contributions from real or complex singularities; express the transient field in terms of compact (and therefore physically incisive) wave spectra, in contrast to alternative procedures that always constrain the spectra to be real. These aspects, together with simplifying explicit wavefront approximations, are explored in the present paper, with the application to specific problems relegated to Part III.