Linear Homogeneous Media Research Articles

A new matrix representation of the Maxwell equations based on the Riemann–Silberstein–Weber vector for a linear inhomogeneous medium

We derive a new eight dimensional matrix representation of the Maxwell equations for a linear homogeneous medium and extend it to the case of a linear inhomogneous medium. This derivation starts ab initio with the Maxwell equations and uses arguments based on the algebra of the Pauli matrices. This process leads automatically to the matrix representation based on the Riemann–Silberstein–Weber (RSW) vector. The new representation for the homogeneous medium is a direct sum of four Pauli matrix blocks. This aspect of the new representation should make it suitable for studying the propagation of electromagnetic waves in a linear inhomogeneous medium adopting the techniques of quantum mechanics treating the inhomogeneity as a perturbation. The new representation is used to rederive the Mukunda–Simon–Sudarshan matrix substitution rule for transition from the Helmholtz scalar wave optics to the Maxwell vector wave optics.

Instability of High-Order Optical Phase and Polarization Singularities in a Homogeneous Linear Medium

Instability of High-Order Optical Phase and Polarization Singularities in a Homogeneous Linear Medium

Homogeneous microwave near-field power focusing using a cylindrical antenna array

This paper presents an investigation of a microwave near-field power focusing (NFPF) analytical approach utilizing a cylindrical array of electric line sources along the z-axis, applicable to any arbitrary homogeneous linear medium. A novel parameter, termed the “focus ability” (FA), is introduced to quantitatively assess a method's capabilities in achieving power focusing. In the following, the phases and amplitudes of the excitation signals of the array elements are obtained with optimization to maximize FA over a certain studied area. Each stage of the theoretical analysis is supported through COMSOL full-wave simulations, ensuring the validity of the results. Theoretical results relevant to focused power are then compared to outcomes obtained from CST full-wave simulations with the same array configuration while employing half-wave dipole antenna elements with excitations obtained via optimization. These comparisons indicate that the excitation signals obtained by theoretical analysis can be used for power focusing when the array of half-wave dipole antenna sources is utilized. Furthermore, the proposed method leads to a significant reduction in the optimization time. Also, in the provided examples, the FA parameter and NFPF are investigated for the different conditions consisting of the type of the homogeneous medium and the number of antennas.

New principle for scattering inside a Huygens bianisotropic medium

A Huygens bianisotropic medium is a linear homogeneous medium for which the Huygens principle can be formulated. When a bounded 3D scattering object composed of a linear bianisotropic medium, whether homogeneous or not, is embedded in a Huygens bianisotropic medium, the excess field phasors inside that object act as volume current densities, and the tangential components of the internal field phasors on the surface of the same object act as surface current densities, to radiate identical field phasors in the external region.

Propagation dynamics of off-axis noncanonical vortices in a collimated Gaussian beam**Project supported by the Fundamental Research Funds for the Central Universities, China (Grant No. 2017B14914), China Postdoctoral Science Foundation (Grant No. 2016M601586), and the National Natural Science Foundation of China (Grant Nos. 11874140 and 11404092).

It is widely accepted that an off-axis noncanonical vortex moves across the free-space diffracting Gaussian beam without rotation. But our analysis shows that the vortex swirls a while before it approaches infinite. By neglecting the divergence of the host beam, we focus on this rotation characteristics of the vortices in linear homogeneous media. For the symmetrical host beam, it is found that the vortex moves along an elliptical trajectory, while the topological charge and the angular momentum of the vortex core relative to the beam axis are conserved. For the asymmetrical host beam, the vortex trajectory is rather complicated, since the noncanonical parameter varies as the light propagates, resulting in topological charge inversion. But we find that the vortices are always confined in a rectangular area, and the rotation direction is determined by the topological charge.

Coupling finite elements and auxiliary sources for Maxwell's equations

AbstractThe Multiple Multipole Program is a Trefftz method approximating the electromagnetic field in a domain filled with a homogeneous linear medium. MMP can easily handle unbounded domains; yet, it cannot accommodate inhomogeneous or nonlinear materials, situations well within the scope of the standard finite element method. We propose to couple FEM and MMP to model Maxwell's equations for materials with spatially varying properties in an unbounded domain. In some bounded parts of the domain, we use Nédélec's first family of curl‐conforming elements; in the unbounded complement, multipole expansions. Several approaches are developed to couple both discretizations across the common interface: Least‐squares–based coupling using techniques from PDE‐constrained optimization. Multifield variational formulation in the spirit of mortar finite element methods. Discontinuous Galerkin coupling between the FEM mesh and the single‐entity MMP subdomain. Coupling by tangential components traces. We study the convergence of these approaches in a series of numerical experiments.

On the Huygens principle for bianisotropic mediums with symmetric permittivity and permeability dyadics

Mathematical statements of the Huygens principle relate the electric and magnetic field phasors at an arbitrary location in a source-free region enclosed by a surface to the tangential components of the electric and magnetic field phasors over that surface, via the dyadic Green functions applicable to the linear homogeneous medium occupying that region. We have mathematically formulated the Huygens principle for the electric and magnetic field phasors when the permittivity and permeability dyadics of the medium are symmetric, the symmetric parts of the two magnetoelectric dyadics of the medium are negative of each other, and both magnetoelectric dyadics also contain anti-symmetric terms. We have also formulated the Huygens principle for the electric (resp. magnetic) field phasor in a medium whose permittivity (resp. permeability) is scalar, the permeability (resp. permittivity) is symmetric, the symmetric parts of the two magnetoelectric dyadics reduce to dissimilar scalars, and anti-symmetric parts of the two magnetoelectric dyadics are identical.

Derivation of the state matrix for dynamic analysis of linear homogeneous media.

A method to obtain the state matrix of an arbitrary linear homogeneous medium excited by a plane wave is proposed. The approach is based on projections on the eigenspace of the governing equations matrix. It is an alternative to manually obtaining a linearly independent set of equations by combining the governing equations. The resulting matrix has been validated against previously published derivations for an anisotropic poroelastic medium.

Метод последовательных приближений для расчета вихревых токов в протяженных проводниках, движущихся вблизи частично насыщенных ферромагнетиков

A system of extended conductors located in an external variable magnetic field near the partially saturated ferromagnets is considered. Conductors are moved in a plane with the given velocities. The density of the eddy currents induced in conductors satisfies the system of integral-differential equations. The calculation of nonlinearity of ferromagnetic materials is carried out by replacing the homogeneous nonlinear medium to piecewise homogeneous linear medium. It is assumed that the permeability in each region of the homogeneity depends of time and currents of magnetization induced on boundaries of piecewise homogeneous medium. To find the density of secondary sources is proposed a double iterative process on the inner iterations are determined by the instantaneous values of the magnetic permeability and magnetization density currents and external iterations to solve a system of integral-differential equations. A linear induction motor is given as an example of applying of the method of calculation.

Digital reverse propagation in focusing Kerr media

Lenses allow the formation of clear images in homogeneous linear media. Holography is an alternative imaging method, but its use is limited to cases in which it provides an advantage, such as three-dimensional imaging. In nonlinear media, lenses no longer work. The light produces intensity-dependent aberrations. The reverse propagation method used in digital holography to form images from recorded holograms works even in Kerr media [M. Tsang, D. Psaltis, and F. G. Omenetto, Opt. Lett. 28, 1873 (2003).]. The principle has been experimentally demonstrated recently in defocusing media [C. Barsi, W.Wan, and J.W. Fleischer, Nat. Photonics 3, 211 (2009).]. Here, we report experimental results in focusing media.

Surface-wave solitons between linear media and nonlocal nonlinear media

We address the properties of surface-wave solitons on the interface between a semi-infinite homogeneous linear medium and a semi-infinite homogeneous nonlinear nonlocal medium. The stability, energy flow and FWHM of the surface wave solitons can be affected by the degree of nonlocality of the nonlinear medium. We find that the refractive index difference affects the power distribution of the surface solitons in two media. We show that the different boundary values at the interface can lead to the different peak position of the surface solitons, but it can not influence the solitons stability with a certain degree of nonlocality.

Relativistic invariance of dispersion‐relations and their associated wave‐operators and Green‐functions

AbstractIdentifying invariance properties helps in simplifying calculations and consolidating concepts. Presently the Special Relativistic invariance of dispersion relations and their associated scalar wave operators is investigated for general dispersive homogeneous linear media. Invariance properties of the four‐dimensional Fourier‐transform integrals is demonstrated, from which the invariance of the scalar Green‐function is inferred. Dispersion relations and the associated group velocities feature in Hamiltonian ray tracing theory. The derivation of group velocities for moving media from the dispersion relation for these media at rest is discussed. It is verified that the group velocity concept satisfies the relativistic velocity‐addition formula. In this respect it is considered to be ‘real’, i.e., substantial, physically measurable, and not merely a mathematical artifact. Conversely, if we assume the group velocity to be substantial, it follows that the dispersion relation must be a relativistic invariant.

Subluminal wave bullets: Exact localized subluminal solutions to the wave equations

In this work it is shown how to obtain, in a simple way, localized (nondiffractive) subluminal pulses as exact analytic solutions to the wave equations. These ideal subluminal solutions, which propagate without distortion in any homogeneous linear media, are herein obtained for arbitrarily chosen frequencies and bandwidths, avoiding in particular any recourse to the noncausal (backward moving) components that so frequently plague the previously known localized waves. Such solutions are suitable superpositions of---zeroth order, in general---Bessel beams, which can be performed either by integrating with respect to (w.r.t.) the angular frequency $\ensuremath{\omega}$, or by integrating w.r.t. the longitudinal wave number ${k}_{z}$: Both methods are expounded in this paper. The first one appears to be powerful enough; we study the second method as well, however, since it allows us to deal even with the limiting case of zero-speed solutions (and furnishes a way, in terms of continuous spectra, for obtaining the so-called ``frozen waves,'' so promising also from the point of view of applications). We briefly treat the case, moreover, of nonaxially symmetric solutions, in terms of higher-order Bessel beams. Finally, some attention is paid to the known role of special relativity, and to the fact that the localized waves are expected to be transformed one into the other by suitable Lorentz transformations. In this work we fix our attention especially on acoustics and optics: However, results of the present kind are valid whenever an essential role is played by a wave equation (such as electromagnetism, seismology, geophysics, gravitation, elementary particle physics, etc.).

Finite element computation of Green's functions

Green's functions are important mathematical tools in mechanics and in other parts of physics. For instance, the boundary element method needs to know the Green's function of the problem to compute its numerical solution. However, Green's functions are only known in a limited number of cases, often under the form of complex analytical expressions. In this article, a new method is proposed to calculate Green's functions for any linear homogeneous medium from a simple finite element model. The method relies on the theory of wave propagation in periodic media and requires the knowledge of the finite element dynamic stiffness matrix of only one period. Several examples are given to check the accuracy and the efficiency of the proposed numerical Green's function.

A discussion of the properties of the Rayleigh perturbative solution in diffraction theory

The problem of two-dimensional plane-wave scattering from an infinite periodic profile with a Dirichlet boundary condition is considered. It is shown that the Rayleigh theory yields the same perturbation series for the reflection coefficients as the extinction-theorem formalism. This identity holds up to any order if the boundary profile is infinitely continuously differentiable, and at least up to some finite order if the boundary profile is finitely continuously differentiable. The nature of the derivation proposed allows the extension of these results to all linear homogeneous media encountered in electromagnetism and elastodynamics, to all usual linear boundary conditions, and to the case of scattering from finite bodies. It is indicated that, for interfaces which are periodic finite linear combinations of sinusoids, the Rayleigh perturbative solution is identical to the perturbative solutions obtained with any other approach, and verifies the reciprocity relationships.

Influence of off-waist incidence on evolution of the Gaussian beam in the nonlocal nonlinear media

In this paper, we discuss the evolution of the offwaist incident Gaussian beam in the nonlocal nonlinear media by use of the (1+1)dimensional strongly nonlocal model. An exact analytical solution for the evolution is obtained. We find that no matter how high the power of the incident beam is, the beam width will oscillate periodically and no spatial soliton will exist, which is substantially different from the onwaist incident case. The influence of the input power and incident position on beam evolution is also discussed. The higher the input power, the narrower the beam width is. The incident position determines the initial evolution trend. The definition of a “spatial chirp” is given, to our knowledge, for the first time. The physical origin of the above phenomenon is that the different incident position corresponds to the different spatial chirp. The concept of the spatial chirp provides a new physical qualitative insight into the evolution of optical beams. The comparisons of beam evolutions in focusing media, in defocusing media and in linear homogeneous media are also given.

Optical Eigenmodes in General Linear Optical Media

We consider analytic solutions to Maxwell's equations in general homogeneous linear media. In general, the dielectric tensor is not Hermitian, and it is a function of the wavevector. We present our formalism for obtaining closed form analytic solutions for the eigenmodes in the general case. The dispersion relation is described by a quartic equation using covariant representation. Explicit expressions are obtained for the eigenmodes in terms of the propagation direction, the permittivity and the permeability. The results may be useful to describe the optical fields as well as energy and momentum propagation in a wide variety of optical elements, such as magneto-optical films, dye doped liquid crystals, gyrotropic, dichroic and magneto-chiral materials. A simple example of a birefringent dichroic crystal is considered.

Spectrum-induced changes in diffraction of pulsed optical beams

Propagation of ultrashort optical pulses in a linear homogeneous medium is studied using the standard diffraction theory of electromagnetic fields. Diffraction pattern of a Gaussian beam consisting of such short pulses depends on the pulse shape. In general, the pulsed beam does not remain Gaussian on propagation and its spot size is enhanced. In the visible region, changes in the diffracted intensity become noticeable only for pulse widths below 10 fs and are larger for chirped pulses and for pulses whose spectrum has long tails. We also show that the diffraction characteristics of coherent but pulsed and CW but partially coherent Gaussian beams are identical if the two have identical optical spectra.

Optical pulse propagation

We analyse in the domain t>0, z>0, the propagation of an optical pulse launched at t=0, z=0 in an isotropic, homogeneous linear medium. This corresponds for Maxwell's equations to a boundary-initial value problem that we solve by using the 2D-Laplace transform. We show that the refractive index must satisfy a particular condition to make sure that the second order partial differential equation obtained from Maxwell's equations is hyperbolic. Then, we discuss the inverse Laplace transform of the solution for a boundary-initial Dirac pulse in three different material media so that the solution for another type of boundary-initial pulse is given by a simple convolution product. We consider in particular a harmonic pulse and a sequence of digital pulses.

On a medium constraint arising directly from Maxwell's equations

The mathematical structure of Maxwell's equations provides a constraint on the constitutive parameters of linear homogeneous media. This is exemplified by showing that the nonreciprocity parameter of bi-isotropic media is superfluous, confirming independent results (based on covariance and uniqueness arguments) that nonreciprocal bi-isotropic media do not exist.