Source-free Field Research Articles

Spin raising and lowering operators for Rarita-Schwinger fields

Spin raising and lowering operators for massless field equations constructed from twistor spinors are considered. Solutions of the spin-$\frac{3}{2}$ massless Rarita-Schwinger equation from source-free Maxwell fields and twistor spinors are constructed. It is shown that this construction requires Ricci-flat backgrounds due to the gauge invariance of the massless Rarita-Schwinger equation. Constraints to construct spin raising and lowering operators for Rarita-Schwinger fields are found. Symmetry operators for Rarita-Schwinger fields via twistor spinors are obtained.

ELECTRIC QUADRUPOLARIZABILITY OF A SOURCE-DRIVEN DIELECTRIC SPHERE

Since both metamaterials composed of artificial molecules (inclusions in a host material) and natural molecular materials at optical and greater frequencies can exhibit significant electric quadrupolarization as well as electric and magnetic dipolarization, we determine the passive, causal electric quadrupolarizability for a spherically symmetric molecule, namely a dielectric sphere subject to source-driven applied fields. For source-driven excitations, it is found that two electric quadrupolarizability constants are generally required to characterize the electric quadrupolar response of the sphere, with one of the constants multiplying the divergence of the applied electric field. For source-free fields, such as the fields of the eigenmodes of an electric quadrupolar array, the local electric field illuminating each inclusion is solenoidal, the constitutive relation is characterized by just one quadrupolarizability constant, and the electric quadrupolarization becomes traceless. It is also found that the electric quadrupolarization becomes very large and effectively traceless near the resonant frequencies of electrically small plasmonic spheres with negative permittivity and for somewhat larger spheres with positive permittivity.

Some uniqueness results for stationary solutions to the Maxwell–Born–Infeld field equations and their physical consequences

Uniqueness is established for time-independent finite-energy electromagnetic fields which solve the nonlinear Maxwell–Born–Infeld equations in boundary-free space under the condition that either the charge or current density vanishes. In addition, it is also shown that the simpler Maxwell–Born equations admit at most one stationary finite-energy electromagnetic field solution, without the above condition. In these theories of electromagnetism the following physical consequences emerge: source-free field solitons moving at speeds less than the vacuum speed of light c do not exist; any electrostatic (resp. magnetostatic) field is the unique stationary electromagnetic field for the same current-density-free (resp. charge-density-free) sources. These results put to rest some interesting speculations in the physics literature.

Volume-preserving maps, source-free systems and their local structures

In this paper, we study local structures of volume-preserving maps and source-free vector fields, which are defined in the Euclidean n-space Rn with n ≥ 3. First, we prove that any volume-preserving map, defined in some neighbourhood of the origin, can be represented as a composition of n − 1 essentially two-dimensional area-preserving maps. This result can be viewed as an analogue of the following known fact (Feng and Shang 1995 Volume-preserving algorithms for source-free dynamical systems Numer. Math. 71 451–63): any source-free vector field on Rn can be represented as a sum of n − 1 essentially two-dimensional Hamiltonian vector fields. Then, we present a local representation of source-free vector fields under volume-preserving coordinate changes. Finally, we construct a Lie algebra of skew-symmetric tensor potentials of second order associated with source-free vector fields. The Lie algebra turns out to be isomorphic to the Lie algebra of source-free vector fields.

ASYMPTOTIC ANALYSIS OF A LINE SOURCE DIFFRACTION BY A PERFECTLY CONDUCTING HALF-PLANE IN A BI-ISOTROPIC MEDIUM

This paper is concerned with the diffraction of an electromagnetic wave by a perfectly conducting half-plane in a homogeneous bi-isotropic medium (asymptotically).Similar analysis in a source-free field is done in S.Asghar and A.Lakhtakia (1994), Plane- wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium. Int.J.Appl.Electromagnetics in materials, 5, (1994), 181-188. In this paper attention is focused on the wave coming from a line source. The objective is to study the scattering of an electromagnetic wave from the boundary of a half-plane and thereby to provide a theoretical framework for the line source diffraction asymptotically. In far field approximation it is shown that an incident wave coming from a line source behaves like a plane wave.The scattered field is calculated by using the Fourier transform and the Wiener-Hopf techniques.The scattered field in the far zone is determined by using contour integration.

Algebraic and differential Rainich conditions for symmetric trace-free tensors of higher rank

We present a study of Rainich-like conditions for symmetric and trace-free tensors T . For arbitrary even rank we find a necessary and sufficient differential condition for a tensor to satisfy the source-free field equation. For rank 4, in a generic case, we combine these conditions with previously obtained algebraic conditions to gain a complete set of algebraic and differential conditions on T for it to be a superenergy tensor of a Weyl candidate tensor, satisfying the Bianchi vacuum equations. By a result of Bell and Szekeres, this implies that in vacuum, generically, T must be the Bel–Robinson tensor of the spacetime. For the rank 3 case, we derive a complete set of necessary algebraic and differential conditions for T to be the superenergy tensor of a massless spin-3/2 field, satisfying the source-free field equation.

Gravitation and Electrodynamics over SO(3,3)

In a series of papers, an approach to field theory is developed in which matter appears by interpreting source-free (homogeneous) fields over a 6-dimensional space of signature (3,3), as interacting (inhomogeneous) fields in space-time. The extra dimensions are given a physical meaning as “coordinatized matter.” The inhomogeneous energy-momentum relations for the interacting fields in space-time are automatically generated by the simple homogeneous relations in 6-d. We then develop a Weyl geometry over SO(3,3) as base, under which gravity and electromagnetism are essentially unified via an irreducible 6-calibration invariant Lagrange density and corresponding variational principle. The Einstein–Maxwell equations are shown to represent a low-order approximation, and the cosmological constant must vanish in order that this limit exist.

Generalized Forms, Connections, and Gauge Theories

Generalized differential forms of type N = 2, and flat generalized connections are used to describe the SO(p, q) form of Cartan's structure equations for metric geometries, source-free Yang-Mills fields, and the Einstein–Yang-Mills equations in four dimensions. Maxwell's equations for type N = 2 forms are also constructed.

RADIATION REACTION AND THE PRINCIPLE OF EQUIVALENCE

The principle of equivalence is shown to extend to situations involving radiation reaction. For example, the Lorentz force law governs the motion of an isolated charge undergoing radiation reaction. In the case of an isolated uncharged particle of small mass, it is the geodesic equation which governs the motion, even when radiation reaction is included. For a local observer to understand such motion he must subtract the singular field of the particle from the actual electromagnetic and gravitational fields he measures. The remaining source-free fields are then used in computing the motion of the particle. With only local measurements, the observer has no knowledge of the existence of radiation and sees no effect which he would be compelled to describe as radiation reaction.

Resonance and Bound States in Photonic Crystal Slabs

Using boundary-integral projections for time-harmonic electromagnetic (EM) fields, and their numerical implementation, we analyze EM resonance in slabs of two-phase dielectric photonic crystal materials. We characterize resonant frequencies by a complex Floquet--Bloch dispersion relation $\omega = W(\beta)$ defined by the existence of a nontrivial nullspace of a pair of boundary-integral projections parameterized by the wave number $\beta$ and the time-frequency $\omega$. At resonant frequencies, the crystal slab supports a source-free EM field. We link complex resonant frequencies, where the imaginary part is small, to resonant scattering behavior of incident source fields at nearby real frequencies and anomalous transmission of energy through the slab. At a real resonant frequency, the source-free field supported by the slab is a bound state. We present numerical examples which demonstrate the effects of structural defects on the resonant properties of a crystal slab and surface waves supported by a die...

On the uniqueness of the problem of acoustic diffraction by an infinite plate with local irregularities

The question concerning the uniqueness of the solution to the problem of the acoustic diffraction by an immersed and isolated thin infinite plate with a finite scatterer is studied. It is shown that, to provide the uniqueness of the solution, the conditions at the scatterer must lead to an energy inequality for a source-free field, which determines the absence of the energy-carrying field components at infinity. A formula that generalizes the Sommerfeld formula is obtained and is used to prove the uniqueness of the solution to the problem of diffraction by a plate immersed in an acoustic medium. For the problem of diffraction of a flexural wave by an irregularity of the plate, the uniqueness theorem is proved only for the case of a fixed or hinged edge. When boundary conditions of a general form are imposed on the scatterer in an isolated plate, the uniqueness of the solution is generally lost, which is also corroborated by an example.

A Maxwell Field Equations Class of Exact Solutions on Matter-Dominated Spatially Closed Friedman-Robertson-Walker Universe

Working in a spatially closed Friedman-Robertson-Walker universe with cosmological dust, we investigate a particular source-free Maxwell field generated by a rotationally-symmetric potential A with one of the components in the direction of θ and the other along β. Using the (α, β, θ)-Euler coordinates on S3 and a compact timelike coordinate f, we obtain a class of parametric solutions that allows us to write down the essential components of the Maxwell tensor as well as the induction and the electric field intensity pointing out, besides the non-propagating fundamental electric field, a “burst” of electromagnetic radiation.

The angular spectrum representation and the Sherman expansion of pulsed electromagnetic beam fields in dispersive, attenuative media

The angular spectrum of plane-waves representation of a pulsed electromagnetic beam field propagating in a dispersive, attenuative medium occupying the half-space expresses that wave field as a superposition of both homogeneous and inhomogeneous plane waves. A generalization of the Sherman expansion for source-free wave fields in lossless media to the lossy, dispersive medium case is used to derive a spatial series representation of a pulsed, source-free electromagnetic beam field from its angular spectrum representation. This spatial series representation displays the temporal evolution of the pulsed beam field explicitly through a single-contour integral that is of the same form as that obtained in the Fourier-Laplace description of a pulsed plane-wave field that is propagating in the positive z-direction in the lossy, dispersive medium. The temporal pulse evolution is found to depend upon the transverse spatial position in the beam field through the derivatives of the field boundary values.

Optimum field squeezing from atomic sources: Three-level atoms.

Light emitted from laser-driven atoms often has squeezed quadrature fluctuations reflecting the phase dependence of the atomic source excited into a coherent superposition by the driving field. In this paper, we study the maximum squeezing in the emitted light fields from three-level atoms, paying particular attention to the role of atomic coherences in governing the optimum squeezing which is possible. Some phase dependent nonlinear optical processes have the potential for substantial quantum noise reduction at useful intensities: for example, four-wave mixing involving atomic systems has already been used to generate quadrature-squeezed light. The threoretical description of such a source is generally complicated with damping processes, coherent and incoherent pumping, and so on, all needing to be taken into account. Fully quantum treatments are available, but the basic physics of the process is often difficult to grasp in the necessarily complicated analysis. Although less important from the point of view of efficient practical sources of squeezed light, studying the generation of squeezed light in simpler systems such as two- or three-level atoms interacting with one or two modes (as in Jaynes-Cummings models) or in spontaneous emission or resonance fluorescence gives greater insight into the fundamentals of the squeezing process. In particular, the relationship between the atomic coherences and the degree of squeezing in the generated fields can be explored. We study the maximum (optimized) squeezing that can be obtained in the emitted fields from the irreversible decay from three-level atoms for any choice of initial conditions, excitations and decay process, for all quadrature components and for all possible configurations (V, , and ladder systems). We show that for V and systems, optimum squeezing is actually associated with a two-level state (in a suitable basis) involving a single one-photon coherence, but for ladder systems the state for maximum noise reduction is not equivalent to a two-level system and involves a single two-photon coherence and no intermediate state population. We pay particular attention to whether the total field is in a minimum uncertainty state, and the nature of the atomic state associated with maximum squeezing, especially whether it is a mixed or pure state. In the case where the initial free field has a zero amplitude at the detector and the source atoms are confined to a region small compared to the wavelength (Dicke source) we show that the squeezing in the total field is given in terms of the squeezing in the source field, and hence related to atomic populations and atomic one-photon and two-photon coherences for the case of three level atom. Previously unconsidered source-free field interferences terms are shown to be zero. The choice of quadrature phase and frequency is optimized to minimize the source field normally ordered variance. We then minimize further with respect to the choice of atomic density matrix elements subject to the constraints that the density matrix is Hermitean, positive, and has a trace equal to unity. In all cases we find that the optimum squeezing is produced when the source field is in a minimum uncertainty state.

Analysis of natural frequencies of cavities and scatterers using an impulsive current model

A recently suggested (see Y. Leviatan et al., ibid., vol.36, p.1722-1734, Dec. 1988) method of moments solution to generalized formulations of electromagnetic scattering from perfectly conducting and material bodies of smooth shape is used to compute the natural frequencies of such bodies. The field inside the body and the field scattered from the body are expanded by means of the field of impulsive current sources (elemental dipoles) situated a distance away from (rather than on) the boundary. The key feature of this approach is the fields anywhere in space can be evaluated analytically, yielding a substantial saving of computation time in comparison with standard on-surface formulations. Impulsive expansion functions are used for the fictitious currents that simulate the source-free fields in the relevant regions, and a point matching procedure is adopted for the enforcement of the boundary conditions. The result is a generalized impedance matrix whose singularities in the complex frequency plane represent the natural frequencies of the body. The numerical solution is simple to execute, computationally efficient, and can be applied to a broad class of bodies of smooth shape. Results demonstrating the excellent accuracy of the procedure are given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Incoming and outgoing components of source-free wavefields

Angular spectrum representations for source-free fields are compared with those for incoming and outgoing fields. It is shown that a source-free field is the difference between an outgoing and an incoming field, each of them generated from the corresponding boundary values of the analytic continuation of their angular spectrum.

Intrinsic modes in tapered optical waveguides

Intrinsic modes are recently developed source-free field solutions in homogeneous wedge-shaped layers with penetrable boundaries. Expressed as spectral integrals over a continuum of plane waves, they can be computed to chart the continuous progress of a propagating wave field as it passes from its well-trapped regime in the wider section to the radiating regime through local cutoff in the narrower part. Intrinsic modes are explored here for application to tapered optical waveguides embedded within different exterior media. Two-dimensional electromagnetic TGE (transverse electric) and TM (transverse magnetic) field solutions are presented and evaluated numerically for small as well as large taper angles and various material parameters. The results reveal the effect of asymmetry in the exterior media on the intrinsic mode shapes, and the control that can be exercised thereby to transfer the optical energy from the taper to the surrounding environment.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Generalized Wigner functions in curved spaces: A new approach.

It is well known that, given a quantum field in Minkowski space, one can define Wigner functions ${f}_{W}^{N}$(${x}_{1}$,${p}_{1}$,...,${x}_{N}$,${p}_{N}$) which (a) are convenient to analyze since, unlike the field itself, they are c-number quantities and (b) can be interpreted in a limited sense as ``quantum distribution functions.'' Recently, Winter and Calzetta, Habib and Hu have shown one way in which these flat-space Wigner functions can be generalized to a curved-space setting, deriving thereby approximate kinetic equations which make sense ``quasilocally'' for ``short-wavelength modes.'' This paper suggests a completely orthogonal approach for defining curved-space Wigner functions which generalizes instead an object such as the Fourier-transformed ${f}_{W}^{1}$(k,p), which is effectively a two-point function viewed in terms of the ``natural'' creation and annihilation operators ${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$(p-(1/2k) and a(p+(1/2k). The approach suggested here lacks the precise phase-space interpretation implicit in the approach of Winter or Calzetta, Habib, and Hu, but it is useful in that (a) it is geared to handle any ``natural'' mode decomposition, so that (b) it can facilitate exact calculations at least in certain limits, such as for a source-free linear field in a static spacetime.

Separability of the Killing–Maxwell system underlying the generalized angular momentum constant in the Kerr–Newman black hole metrics

The concept of a Killing–Maxwell system may be defined by the relation Â[μ;v];ρ =(4π/3)ĵ[μgν]. In such a system the one-form Âμ is interpretable as the four-potential of an electromagnetic field F̂μv, whose source current ĵ μ is an ordinary Killing vector. Such a system determines a canonically associated duality class of source-free electromagnetic fields, its own dual being a Killing–Yano tensor, such as was found by Penrose [Ann. N.Y. Acad. Sci. 224, 125 (1973)] (with Floyd) to underlie the generalized angular momentum conservation law in the Kerr black hole metrics, the existence of the Killing–Yano tensor being also a sufficient condition for that of the Killing–Maxwell system. In the Kerr pure vacuum metric and more generally in the Kerr–Newman metrics for which a member of the associated family of source-free fields is coupled in gravitationally, it is shown that the gauge of the Killing–Maxwell one-form may be chosen so that it is expressible (in the standard Boyer–Lindquist coordinates) by 1/2 (a2 cos 2 θ−r2)dt+ 1/2 a(r2−a2)sin2 θ dφ, the corresponding source current being just (4π/3)(∂/∂t). It is found that this one-form (like that of the standard four-potential for the associated source-free field) satisfies the special requirement for separability of the corresponding coupled charged (scalar or Dirac spinor) wave equations.

Axially symmetric Brans-Dicke-Maxwell solutions

Following a method of John and Goswami new solutions of coupled Brans-Dicke-Maxwell theory are generated from Zipoy's solutions in oblate and prolate spheroidal coordinates for source-free gravitational field. All these solutions become Euclidean at infinity. The asymptotic behavior and the singularity of the solutions are discussed and a comparative study made with the corresponding Einstein-Maxwell solutions. The possibility of a very large red shift from the boundary of the spheroids is also discussed.