Techniques based on the method of modal expansions, the Rayleigh-Stevenson expansion in inverse powers of the wavelength, and also the method of moments solution of integral equations are essentially restricted to the analysis of electromagnetic radiating structures which are small in terms of the wavelength. It therefore becomes necessary to employ approximations based on “high-frequency techniques” for performing an efficient analysis of electromagnetic radiating systems that are large in terms of the wavelength.

Techniques for High-Frequency Problems

Paraxial Gaussian beams (GB’s) 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, GB’s and PB’s 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.

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

Summary. Recently, a method using superposition of Gaussian beams has been proposed for the solution of high-frequency wave problems. The method is a potentially useful approach when the more usual techniques of ray theory fail: it gives answers which are finite at caustics, computes a nonzero field in shadow zones, and exhibits critical angle phenomena, including head waves. Subsequent tests by several authors have been encouraging, although some reported solutions show an unexplained dependence on the ‘free’ complex parameter e which specifies the initial widths and phases of the Gaussian beams.



We use methods of uniform asymptotic expansions to explain the behaviour of the Gaussian beam method. We show how it computes correctly the entire caustic boundary layer of a caustic of arbitrary complexity, and computes correctly in a region of critical reflection. However, the beam solution for head waves and in edge-diffracted shadow zones are shown to have the correct asymptotic form, but with governing parameters that are explicitly e-dependent. We also explain the mechanism by which the beam solution degrades when there are strong lateral inhomogeneities. We compare numerically our predictions for some representative, model problems, with exact solutions obtained by other means.

Some remarks on the Gaussian beam summation method

This paper determines the ray structure of a spinning acoustic mode propagating inside a semi-infinite circular cylindrical duct, and thereby determines the ray structure of the field radiated from the end of the duct. Inside the duct, but outside of a caustic cylindrical surface, the rays are piecewise linear helices; on striking the rim of the end-face of the duct, these rays produce ‘Keller cones’ of diffracted rays. The cones determine the structure of the radiated field: for example, no rays penetrate two cone-shaped far-field quiet zones centred on the duct axis; two rays pass through each point in a forward loud zone; and one ray passes through each point in a rearward loud zone. The two rays through each point in the forward loud zone interfere to produce an oscillatory directivity pattern. One quarter of the rays on each cone point back inside the duct and produce the reflected field. Thus the rim of the end-face of the duct acts as a ‘ring source’, in which the radiated and reflected fields have their origin. Every propagating duct mode determines a polar angle and an azimuthal angle; these are taken as parameters specifying the mode and are used to calculate the positions and angles of all the rays. The mathematical method on which the paper is based is Debye's approximation for the Bessel function which appears in the expression for the duct modes; the approximation shows also that the duct contains a region of smooth helical rays on which the field consists of inhomogeneous waves: this region is the inner cylinder, lying inside the annulus of piecewise linear helical rays. The results of the paper are very promising for the application of Keller's geometrical theory of diffraction to detailed calculations of the sound radiated from aeroengine ducts. An alternative description of the field, using Cargill's meridional rays, is summarized.

Sound radiation from a cylindrical duct. Part 1. Ray structure of the duct modes and of the external field

We present numerical results concerning the properties of the
electromagnetic field scattered by an infinite circular cylinder
illuminated by a circular Gaussian beam. The cylinder is
arbitrarily located and arbitrarily oriented with respect to the
illuminating Gaussian beam. Numerical evaluations are provided
within the framework of a rigorous electromagnetic theory, the
generalized Lorenz–Mie theory, for infinite cylinders. This theory
provides new insights that could not be obtained from older
formulations, i.e., geometrical optics and plane-wave
scattering. In particular, some emphasis is laid on the waveguiding
effect and on the rainbow phenomenon whose fine structure is hardly
predictable by use of geometrical optics.

Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: numerical results

Inhomogeneous waves behave locally like A(r) exp[ikS(r)], where A and S are spatially dependent complex amplitude and phase functions, and k is the (large) free-space wavenumber. A previously developed asymptotic theory for high-frequency propagation and scattering of such waves is here applied to the propagation and scattering of paraxial Gaussian beams. Attention is given to Gaussian beams in free space, to beams in a lens-like medium with parabolic variation of the refractive index, and to beam reflection by a cylindrical obstacle. In the latter instance, the obstacle size may be comparable to the incident beamwidth, thereby introducing substantial distortion into the reflected beam. The results obtained from the asymptotic theory are verified by comparison with rigorously derived solutions, thereby confirming the validity of the theory, which can also be applied to more general medium and obstacle configurations.

S. Choudhary

Papers

Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking