Macroscopic Maxwell Equations Research Articles

Electromagnetic forces in photonic crystals

We develop a general methodology for numerical computations of electromagnetic (EM) fields and forces in matter, based on solving the macroscopic Maxwell's equations in real space and adopting the Maxwell Stress Tensor formalism. Our approach can be applied to both dielectric and metallic systems of frequency-dependent dielectric function; as well as to objects of any size and geometrical properties in principle. We are particularly interested in calculating forces on nanostructures. We find that a body reacts to the EM field by minimising its energy, i.e. it is attracted (repelled) by regions of lower (higher) EM energy. When travelling waves are involved, forces can be additionally understood in terms of momentum exchange between the body and its environment. However when evanescent waves dominate, the forces are complicated, often become attractive and cannot be explained by means of real momentum being exchanged. We study the EM forces induced by a laser beam on a crysta! l of dielectric GaP spheres. We ob serve effects due to the lattice structure, as well as due to single scattering from each sphere. In the former case the two main features are a maximum momentum exchange when the frequency lies within a band gap; and a multitude of force orientations when the Bragg conditions for multiple outgoing waves are met. In the latter case the radiation couples to the EM eigenmodes of isolated spheres (Mie resonances) and very sharp attractive and repulsive forces occur. Depending on the intensity of the incident radiation these forces can overcome all other interactions present (gravitational, thermal and Van der Waals) and may provide the main mechanism for formation of stable structures in colloidal systems.

An idea for electromagnetic "feedforward-feedbackward" media

An idea for a new class of complex media that we name feedforward-feedbackward (FFFB) media is presented and some of the results of our theoretical work in analyzing plane wave propagation in the axial direction through these media are described. The concept of FFFB media, as introduced here, was inspired by the theoretical research of Saadoun and Engheta (1992, 1994) on a variation of artificial chiral media. Like chiral media, to our knowledge there are no naturally occurring FFFB media for the microwave frequency band; for this reason we introduce an idea for artificial FFFB media. The focus of this paper is on one conceptualization of such media, namely dipole-dipole FFFB media. First, we present the calculation of the necessary constitutive parameters for studying axial plane wave propagation. Then we solve the macroscopic Maxwell equations in the k domain for axial plane wave propagation in an unbounded source-free crossed-dipole FFFB medium. Finally, we present the dispersion equation for this medium in this case, discuss some of the physical properties of its roots and certain features of the polarization eigenstates, and speculate some of the potential applications of this medium.

Quasistatic principles in the macroscopic electrodynamics of bianisotropic media

Only very few theoretical results in macroscopic electrodynamics of bianisotropic composites have necessary experimental justifications. This fact, it seems, is not accidental. Artificial bianisotropic materials based on a composition of helices and \ensuremath{\Omega} particles do not meet the requirements of macroscopic electrodynamics of condensed media. Our standpoint is based on the principle that in the description of the electromagnetic properties of a bianisotropic medium, one has to be able to separate the microscopic and macroscopic levels of consideration. In other words, specific properties of a bianisotropic medium should be defined separately from macroscopic Maxwell's equations. In this paper we formulate some principles that should underlie the main laws of macroscopic electrodynamics of bianisotropic media. Our consideration is based on the notion of two types of dual quasistatic (quasimagnetostatic and quasielectrostatic) bianisotropic particles.

The incorporation of microscopic material models into the FDTD approach for ultrafast optical pulse simulations

We are developing full-wave vector Maxwell equation solvers for use in studying the physics and engineering of linear and nonlinear integrated photonics systems. Particular emphasis has been given to the interaction of ultrafast optical pulses with nonresonant and resonant optical materials and structures. Results are reviewed that simulate the interaction of ultrafast optical pulses with structures (e.g., gratings of finite length) filled with materials exhibiting resonant loss or gain. In particular, we consider structures loaded with atomic media resonant at or near the frequency of the incident optical radiation. Interest in these problems follows from our desire to design micron-sized linear and nonlinear guided-wave couplers, modulators, and switches. These resonant problems pose interesting FDTD modeling issues because of the many time and length scales involved. To understand the physics underlying the small-distance scale and short-time scale interactions, particularly in the resonance regime of the materials and the associated device structures, a first principles approach is desirable. Thus, the results presented are based upon a quantum mechanical two-level atom model for the materials. The resulting Maxwell-Bloch model requires a careful marriage between microscopic (quantum mechanical) material models of the resonant material systems and the multidimensional, macroscopic Maxwell's equations solver. The FDTD numerical issues are discussed. Examples are given to illustrate the design and control of these resonant large-scale optical structures. An optical triode is designed and characterized with the FDTD Maxwell-Bloch simulator.

Quantized fields in a nonlinear dielectric medium: A microscopic approach

Theories which have been used to describe the quantized electromagnetic field interacting with a nonlinear dielectric medium are either phenomenological or derived by quantizing the macroscopic Maxwell equations. Here we take a different approach and derive a Hamiltonian describing interacting fields from one which contains both field and matter degrees of freedom. The medium is modelled as a collection of two-level atoms, and these interact with the electromagnetic field. The atoms are grouped into effective spins and the Holstein- Primakoff representation of the spin operators is used to expand them in one over the total spin. When the lowest-order term is combined with the free atomic and field Hamiltonians, a theory of noninteracting polaritons results. When higher-order terms are expressed in terms of polariton operators, standard nonlinear optical interactions emerge.

Dynamics of Dispersive and Nonlinear Media.

The macroscopic Maxwell equations are closed, meaningful, and account for the dynamics of the fields E, B, D, and H, if the latter two are given in terms of the former, such as in Dstd ­ R est 2 t0dEst0 d dt0. The information about the medium is contained in the coefficient e, where the real and imaginary parts of its Fourier-transformed ev account for dispersion and dissipation, respectively. While these statements are elementary, further questions may signal a steep ascent in rocky terrain: One example is about the electromagnetic energy, another the force exerted by these fields on a volume element of a dielectric liquid. Few results have been obtained in this direction, since Maxwell proposed his equations a century ago. Two stand out as gems. The first is by Brillouin [1], who established in 1921 that the average field energy in a dispersive, transparent, linear, and stationary system is kE2y 2ldsvedydv 1 kH2y2ldsvmdydv. (Transparency implies one may neglect dissipation, or the imaginary part of ev , for the given frequency.) So the difference to the average thermodynamic expression is ,vdeydv and vdmydv. Four decades later, Pitaevskiĭ [1,2] calculated the average stress tensor for the same system by examining an ingenious setup, containing a resonance circuit and a capacitor filled with dielectrics. (Given the stress tensor, one can easily calculate the electromagnetic force.) Remarkably, he did not find any analogous term , vdeydv. In addition, he also obtained an expression for the dynamic correction to the dielectric function: estd ­ estats% stdd 1 edyn, where estats% d is the static dielectric function depending on a parameter % , e.g., the density, while edyn is the correction due to the time dependency of % . Finally, he calculated the average magnetization of an otherwise unmagnetizable system, in the presence of both a strong, static magnetic field and a weak electric one of higher frequency. (This sounds cryptic, but should become quite clear later.) Because of his special setup, however, Pitaevskiĭ was forced to confine his considerations to a single frequency for the electromagnetic field. Any loosening of this restriction, to a slowly time-dependent amplitude [3] (i.e., a “quasimonochromatic” field), proved at first difficult—as the stress tensor was then no longer symmetric [4]—but was finally accomplished by the heroic efforts of Barash and Karpman [5]. A nice review is provided by Kentwell and Jones [6].

Two-photon spectroscopy of ZnSe under uniaxial stress.

Resonances on the upper polariton branch of ZnSe are studied under uniaxial stress by two-photon excitation spectroscopy. The spectra are measured for uniaxial stress up to 1.2 kbar in configurations with stress directions [001], [111] and [11 \(\overline{2}\)]. The polariton resonances are calculated by diagonalizing the Pikus-Bir-Hamiltonian and solving the macroscopic Maxwell equations. The stress-induced shifts and splittings of the resonances on the polariton dispersion curves are described by the hydrostatic, tetragonal and trigonal deformation potentials a, b, and d, respectively. The values for the deformation potentials resulting from a least square fit are a=-4.65 eV, b=3.72 eV and d=7.36 eV. Due to the much smaller linewidths (here: < 0.25 meV) of nonlinear resonances as compared to linear optical absorption or reflection lines, the deformation potentials are determined with high accuracy in this work.

Can the macroscopic Maxwell equations be obtained from the microscopic Maxwell-Lorentz equations by performing averages?

It is shown that the usual procedures of obtaining the macroscopic Maxwell equations from the microscopic Maxwell-Lorentz equations by performing averages contain an arbitrary choice of gauge. By a suitable different choice of the gauge the so-obtained Maxwell equations can be cast back to the form of the starting Maxwell-Lorentz equations. Therefore one cannot consider the Maxwell equations to be obtainable from the Maxwell-Lorentz equations by simply performing averages. The implication of this result is that besides the electromagnetic fields produced by the moving electric charges, as given by the Maxwell-Lorentz equations, there may be some other agents that cannot be identified as some kind of motion of the electric charges and that participate in the production of the electromagnetic fields.

How multipole electric moments enter into macroscopic Maxwell equations

This paper is concerned with the correspondence between the macroscopic electric displacementD and the analogous quantityD m of microscopic derivation. Instead of identifying these two fields one with the other, we will equate them, while the identification of the macroscopic polarization with that of microscopic derivation is retained. The result is a new field equation along with the Maxwell equations.

Hydrodynamics of Polarizable Liquids

The complete hydrodynamic theory of polarizable and magnetizable isotropic liquids is presented. This is a century-old problem that we claim to have solved. Starting from an extensive discussion of the results and controversies of the past 90 years, and of the present literature, the obvious difficulties are first presented and then cleansed of their notational content. Using only input of universal acceptance, the Lorentz transformation, the macroscopic Maxwell equations, and the thermodynamics, a set of hydrodynamic equations of motion is derived rigorously, without any ad-hoc assumptions or residual ambiguities. Definite expressions are especially provided for the total momentum density, its stress tensor, and the acceleration of a polarized body. Other controversial concepts, such as the field momentum density, the pressure, and the force density in the polarized body, are shown to be arbitrary, the extent of which, however, is clarified.

Ground-state quantum fluctuations in a scalar Hopfield dielectric

In quantum optics dielectric materials may be dealt with in two ways. In a fundamental approach there are separate terms in the Lagrangian density for matter and radiation. Alternatively, the macroscopic Maxwell equations may be developed from a Lagrangian scheme. In the former method the electric field is conjugate to the vector potential, but in the latter the dielectric displacement is the conjugate momentum. There is therefore reason to question the compatibility of quantized versions of these theories. Using an exactly solvable model for the matter-radiation system, which is a scalar version of Hopfield's classical dielectric model, it is shown that the predictions of the ground-state quantum fluctuations in the two frameworks disagree. It is suggested that the contributions to the fluctuations which arise from the matter in a system should not be neglected in a proper treatment of quantum noise and squeezing in dielectrics.

Model for scanning tunneling optical microscopy: A microscopic self-consistent approach.

A general method based on a microscopic picture of matter is developed in order to describe the optical interaction between a thin dielectric tip and a corrugated sample lighted in total reflection. Such a model is expected to interpret recent images obtained from scanning tunneling optical microscopy (steps, infinite tracks, glass plate with local scratches). The conversion of evanescent waves into homogeneous propagating ones is studied from a molecular-physics perspective (multipolar interactions between each atom of the tip and of the object). Our approach is concerned with the study of subwavelength details lying at the surface of a transparent medium. So, instead of solving the macroscopic Maxwell equations and applying the corresponding boundary conditions at the surface of the tip and of the object, we prefer a microscopic treatment in which the dielectric surrounding is taken into account from a set of dynamical matrices introducing all correlations between each elementary volume inside the object. Relations with experiments are discussed.

Theory for statistical continuum magnetic fluid

This paper deals with the statistical theory of the flow of suspensions of rigid spherical magnetic particles embedded in an incompressible Newtonian liquid. The basic postulates are in the theory of electrons, the principle of statistical mechanics and the point particle method, the magnetic particles being modelled by magneto-hydrodynamic multipoles. This theory gives the macroscopic Maxwell equations for moving matter from statistical considerations of microscopic processes. It clearly separates the forces that are generated by the external electromagnetic fields from the interatomic forces. It also identifies the nature of the long-range force and that of the short-range force.

Electrodynamics at a metal surface with applications to the spectroscopy of adsorbed molecules. I. General theory

Recently, a large number of surface-spectroscopic phenomena have been discussed in terms of models which provide phenomenological equations for the response of the adsorbed molecule to the external fields and compute these fields by assuming that the macroscopic Maxwell equations are valid. Since the macroscopic equations neglect the spatial dispersion of the dielectric constant of the metal and its variation with the distance to the surface, one expects that the computation of the field near or inside the interface may have substantial errors. In order to study the extent of such errors, we use a jellium-electron-gas model to develop a microscopic theory of the electromagnetic fields at the interface. The electron-gas properties needed in such calculations are obtained by using the random-phase approximation. Numerical results are planned to be presented in future papers.

Bulk-selvedge coupling theory for the optical properties of surfaces

We derive expressions for the Fresnel coefficients using a theory which takes into account the coupling between bulk and selvedge; the selvedge is the region near the surface of thickness $l$, $\frac{l}{\ensuremath{\lambda}}\ensuremath{\ll}1$, where $\ensuremath{\lambda}$ is the wavelength of light, in which the macroscopic Maxwell equations with a bulk dielectric constant cannot be used. The expressions involve selvedge response coefficients, and we show how to evaluate them for a given microscopic model of the surface. Since our expressions describe the coupling of bulk and selvedge to all orders, unlike earlier simple perturbation expressions, the dispersion relations of surface excitations can be determined from their poles. Using jellium metal with random-phase-approximation electron dynamics as an example, we derive a formula for the surface-plasmon dispersion relation, and compare it with earlier approximate formulas.

Ionization losses of fast charged particles in an anisotropic medium

The energy losses of fast charged particles in anisotropic media are investigated. The macroscopic Maxwell equations are used to find the electromagnetic field of particles moving according to a given law in an anisotropic medium. A solution in quadratures is obtained for the energy loss of a charge moving at an angle to the optical axis of a weakly anisotropic uniaxial crystal; the result is in the form of a correction to the ionization losses in an isotropic medium. In the case of a medium consisting of anisotropic oscillators, an analytic formula is obtained for the correction: It is inversely proportional to the square of the velocity at particle velocities much less than the velocity of light and tends to zero for ultrarelativistic particles.

Susceptibilities, polarization fluctuations, and the dielectric constant in polar media

A procedure which avoids the necessity of the artificial introduction of cavities has been used in a previous paper to relate the dielectric constant ε of polar media to polarization and orientational fluctuations calculated in the absence of the transverse electromagnetic field. Here the arguments are extended to relate ε to the polarization fluctuations calculated with the inclusion of the transverse fields. This modification leads to the appearance of a propagating mode in the expression for the fluctuations. For a system of nonpolarizable molecules, the polarization fluctuations are directly related to the orientational fluctuations. The expression for the propagating mode of these latter fluctuations agrees with that given by Titulaer and Deutch when the media is nonmagnetic. The expressions for the short and long range fluctuations are the same as previously obtained. It is pointed out that it is quite common to ignore the transverse fields in a microscopic calculation of the orientational fluctuations. Such calculations should not give a propagating mode if the conventional macroscopic Maxwell equations are to be obeyed.

Macroscopic electromagnetic theory of resonant dielectrics

The application of the macroscopic Maxwell equations to the resonant interaction of electromagnetic radiation and dielectric crystals consisting of molecules coupled via retarded dipole fields is investigated. The macroscopic fields are defined by space averaging over volume elements of linear dimensions $\ensuremath{\Delta}$ satisfying $a\ensuremath{\ll}\ensuremath{\Delta}\ensuremath{\ll}\ensuremath{\lambda}$, where $a$ is the intermolecular separation and $\ensuremath{\lambda}$ the wavelength in vacuo. The essential point of our method is the direct derivation from the microscopic equations of a constitutive relation for a finite dielectric, taking proper account of the radiation reaction terms and avoiding the use of an expansion in powers of the molecular polarizability. The results lead to a simple interpretation of the expressions for the internal fields derived by Vlieger along similar lines, and verify the validity of the standard equations of dispersion theory to all points inside the medium a distance $\ensuremath{\Delta}$ away from the surface. The transmission and scattering of radiation at or near a molecular resonance by media of over-all extent small and large compared to the wavelength are discussed for sphere and slab geometries, and the effective natural linewidths are calculated. The limits of validity of the constitutive relation are discussed, and the existence of frequency regions where the macroscopic description breaks down due to the appearance of large spatial variations in the dipole moments is pointed out. As an example of such antiresonant behavior and of the role played by the higher-order radiation-damping terms, the scattering of light from two interacting molecules is treated in detail. In the absence of sufficiently strong dissipative damping, the occurrence of antiresonance is predicted to be a general phenomenon giving rise to large oscillations in the scattering cross section even in the presence of spatial dispersion in a narrow region around the frequency where the expression for the macroscopic index of refraction diverges.

Three distribution identities and Maxwell's atomic field equations including the Röntgen current

Maxwell's equations within a dielectric and/or a magnetic medium were first developed macroscopically and must be complemented by constitutive relations to obtain solutions. These relations connect D with E (and with B in optically active material) and H with B (and again with E in optically active material). The atomic and molecular theories of quantum mechanics allow a microscopic approach to derive these constitutive relations where the macroscopic electric and magnetic fields are averages of the microscopic fields e and b. In classical electromagnetic theory Lorentz [1] originally showed how to derive the Maxwell's macroscopic equations from electron theory using microscopic fields obeying the Maxwell's equations in vacuo but coupled to electronic and ionic sources. There are two distinct steps in this procedure. The first introduces microscopic polarization fields, both electric amd magnetic, p and m from which microscopic electric displacement vector field d and auxiliary magnetic field h are simply constructed. The resulting equations for the microscopic fields e, b, d, and h are called the atomic field equations. The second step is the statistical one where the macroscopic fields E, B, D, and H are defined as averages of the microscopic fields and these macroscopic fields are then shown to obey the phenomenological macroscopic Maxwell's equations. A historical appraisal may be found in the recent book by de Groot [2].

The microscopic and macroscopic equations of the electromagnetic field

The connection between the microscopic field equations of Lorentz, in which all electromagnetic phenomena are attributed to charged particles, and the macroscopic Maxwell equations in which electricity is treated as a continuous fluid and the properties of material media are described by multipole moment densities P, M, Q etc., has been discussed at intervals ever since Lorentz's time. In this paper we discuss a formal mathematical procedure which links the two sets of equations and is free from the objections which can be raised to either the volume averages introduced by Lorentz or the ensemble averages used by more recent authors. The method is based on a procedure for eliminating unwanted high spatial frequency components of the fields, which, though mathematically similar to forming an ensemble average has a different physical interpretation which is much closer to Lorentz's original ideas.