Manipulation of curved beams using beam-domain optimization.

An efficient scheme for the design of aperture fields (distributed sources) that radiate arbitrary trajectory curved (accelerating) beams, with enhanced controllability of various beam features, is presented. The scheme utilizes a frame-based phase-space representation of aperture fields to overcome the main hurdles in the design for large apertures: First, it uses the a-priory localization of caustic beams to significantly reduce the optimization problem's variable space, to that of few Gaussian window coefficients accurately capturing those beams. Then, the optimization problem is solved in the reduced (local) spectral domain. We adopt a linearization approach that enables the solution by sequential application of conventional convex optimization tools, which are naturally compatible with the proposed phase-space representation. The localized nature of the Gaussian windows' radiation is used also for fast field evaluation at a greatly reduced number of optimization constraint points. The significant enhancement in the controllability over the various beam parameters is demonstrated through a range of examples.

Manipulation and control of 3-D caustic beams over an arbitrary trajectory.

We present an algorithm for manipulating and controlling 3-D field patterns, with energy confined to the narrow vicinity of predefined 3-D trajectories in free-space, which are of arbitrary curvature and torsion. This is done by setting the aperture field's phase to form smooth caustic surfaces that include the desired trajectory. The aperture amplitude distribution is constructed to manipulate both the on-axis intensity profile and the off-axis beam-width, and is updated iteratively. Once the aperture distribution is calculated, the radiation from a finite sampled aperture is computed numerically using a Fast Fourier Transform-based scheme. This allows for both verification of the design and examination of its sensitivity to parameters of realistic discrete implementation. The algorithm is demonstrated for the cases of an Airy beam of a planar trajectory, as well as for helical and conical-helical trajectory beams.

Practical algorithm for custom-made caustic beams.

We present a practical algorithm for designing an aperture field (source) that propagates along a predefined generic beam trajectory that consists of both convex and concave sections. We employ here the mechanism that forms the well-known Airy beam in which the beam trajectory follows a smooth convex caustic of the geometric optics rays and generalize it for a class of beams that are referred to as "caustic beams" (CBs). The implementation is based on "back-tracing" rays from the predefined beam trajectory to the source's aperture to form its phase distribution. The amplitude is set in order to form a uniform smooth amplitude of the CBs along their trajectories. Several numerical examples are included.

Transverse electric and transverse magnetic pulsed-beam decomposition of time-dependent aperture fields.

The present contribution is concerned with applying beam-type expansion to a planar aperture time-dependent (TD) electromagnetic field in which the propagating elements, the electromagnetic pulsed-beams, are a priori decomposed into transverse electric (TE) and transverse magnetic (TM) field polarizations. The propagating field is described as a discrete superposition of tilted, shifted, and delayed TE and TM electromagnetic pulsed-beam propagators over the frame spectral lattice. These waveobjects are evaluated by using TD plane-wave spectral representations. Explicit asymptotic expressions for electromagnetic isodiffracting pulsed-quadratic beam propagators are presented, as well as a numerical example.

Two-dimensional relativistic longitudinal Green's function in the presence of a moving planar dielectric-magnetic discontinuity.

The current contribution is concerned with obtaining the relativistic two-dimensional (three-dimensional in relativity jargon) Green's function of a time-harmonic line current that is embedded in a moving dielectric-magnetic medium with a planar discontinuity. By applying a plane-wave (PW) spectral representation for the relativistic electromagnetic Green's function of a dielectric-magnetic medium that is moving in a uniform velocity, the exact reflected and transmitted (refracted) fields are obtained in the form of a spectral integral over PWs in the so-called laboratory and comoving frames. We investigate these spectral representations, as well as their asymptotic evaluations, and discuss the associated relativistic wave phenomena of direct reflected/transmitted rays and relativistic head waves (lateral waves).

TE and TM beam decomposition of time-harmonic electromagnetic waves.

The present contribution is concerned with applying beam-type expansion to planar aperture time-harmonic electromagnetic field distribution in which the propagating elements, the electromagnetic beam-type wave objects, are decomposed into transverse electric (TE) and transverse magnetic (TM) field constituents. This procedure is essential for applying Maxwell's boundary conditions for solving different scattering problems. The propagating field is described as a discrete superposition of tilted and shifted TE and TM electromagnetic beams over the frame-based spatial-directional expansion lattice. These vector wave objects are evaluated either by applying differential operators to scalar beam propagators, or by using plane-wave spectral representations. Explicit asymptotic expressions for scalar, as well as for electromagnetic, Gaussian beam propagators are presented as well.

Tilted Gaussian beam propagation in inhomogeneous media.

The present work is concerned with applying a ray-centered non-orthogonal coordinate system which is a priori matched to linearly-phased localized aperture field distributions. The resulting beam-waveobjects serve as the building blocks for beam-type spectral expansions of aperture fields in 2D inhomogeneous media that are characterized by a generic wave-velocity profile. By applying a rigorous paraxial-asymptotic analysis, a novel parabolic wave equation is obtained and termed "Non-orthogonal domain parabolic equation"--NoDope. Tilted Gaussian beams, which are exact solutions to this equation, match Gaussian aperture distributions over a plane that is tilted with respect to the beam-axes initial directions. A numerical example, which demonstrates the enhanced accuracy of the tilted Gaussian beams over the conventional ones, is presented as well.

Local spectrum analysis of field propagation in an anisotropic medium. Part I. Time-harmonic fields.

The phase-space beam summation is a general analytical framework for local analysis and modeling of radiation from extended source distributions. In this formulation, the field is expressed as a superposition of beam propagators that emanate from all points in the source domain and in all directions. In this Part I of a two-part investigation, the theory is extended to include propagation in anisotropic medium characterized by a generic wave-number profile for time-harmonic fields; in a companion paper [J. Opt. Soc. Am. A 22, 1208 (2005)], the theory is extended to time-dependent fields. The propagation characteristics of the beam propagators in a homogeneous anisotropic medium are considered. With use of Gaussian windows for the local processing of either ordinary or extraordinary electromagnetic field distributions, the field is represented by a phase-space spectral distribution in which the propagating elements are Gaussian beams that are formulated by using Gaussian plane-wave spectral distributions over the extended source plane. By applying saddle-point asymptotics, we extract the Gaussian beam phenomenology in the anisotropic environment. The resulting field is parameterized in terms of the spatial evolution of the beam curvature, beam width, etc., which are mapped to local geometrical properties of the generic wave-number profile. The general results are applied to the special case of uniaxial crystal, and it is found that the asymptotics for the Gaussian beam propagators, as well as the physical phenomenology attached, perform remarkably well.

Local spectrum analysis of field propagation in an anisotropic medium. Part II. Time-dependent fields.

In Part I of this two-part investigation [J. Opt. Soc. Am. A 22, 1200 (2005)], we presented a theory for phase-space propagation of time-harmonic electromagnetic fields in an anisotropic medium characterized by a generic wave-number profile. In this Part II, these investigations are extended to transient fields, setting a general analytical framework for local analysis and modeling of radiation from time-dependent extended-source distributions. In this formulation the field is expressed as a superposition of pulsed-beam propagators that emanate from all space-time points in the source domain and in all directions. Using time-dependent quadratic-Lorentzian windows, we represent the field by a phase-space spectral distribution in which the propagating elements are pulsed beams, which are formulated by a transient plane-wave spectrum over the extended-source plane. By applying saddle-point asymptotics, we extract the beam phenomenology in the anisotropic environment resulting from short-pulsed processing. Finally, the general results are applied to the special case of uniaxial crystal and compared with a reference solution.