Electromagnetic Energy Density Research Articles

Continuity equations for bound electromagnetic field and the electromagnetic energy–momentum tensor

We analyze the application of the Poynting theorem to the bound (velocity-dependent) electromagnetic (EM) field and show that an often-used arbitrary elimination of the term of self-interaction in the product j·E (where j is the current density and E the electric field) represents, in general, an illegitimate operation, which leads to incorrect physical consequences. We propose correct ways of eliminating the terms of self-interaction from the Poynting theorem to transform it into the form that is convenient for problems with bound EM field, which yield the continuity equations for the proper EM energy density, the interaction part of EM energy density and the total EM energy density of bound fields, respectively. These equations indicate the incompleteness of the common EM energy–momentum tensor, and in our analysis, we find a missed term in its structure, which makes its trace non-vanished. Some implications of these results are discussed, in particular, in view of the notion of EM mass of charged particles.

Photon energy dependence of the light pressure exerted onto a thin silicon slab

We review the theory of ponderomotive forces of classical nonionizing electromagnetic (EM) radiation exerted on dispersive matter. Minkowski's EM energy and momentum density lack any dispersion term in contrast to Nelson's theory, where they are included naturally. By considering force experiments on a dielectric mirror immersed in weakly dispersive liquids [R. V. Jones and B. Leslie, Proc. R. Soc. London, Ser. A 360, 347 (1978)], we found that the appearance of the dispersive term should depend on the phase of the mirror reflectivity. It thus matters if the electric or the magnetic field is dominant at the interface. Accordingly, the force measurements depend on the boundary condition and do not permit to uniquely determine the EM momentum in the liquids. Force measurements as a function of the reflectivity phase would permit to experimentally verify the expressions for the EM energy density in a dispersive medium. In our experiments, we chop light beams of different photon energies to excite the motion of a very thin and long Si slab near its mechanical resonance under UHV conditions. This permits to study the force response, where the power reflectivity of the sample varies from 0.7 to smaller than 10${}^{\ensuremath{-}3}$. The determination of the velocity of the slab with a Doppler interferometer yields the effective force exerted by the light beam. The measurements also confirm our theoretical considerations that the observed forces due to EM radiation cannot be traced to the EM momentum in matter, as the observed forces primarily depend on the boundary conditions. Minkowski's stress tensor remains applicable in our case thanks to the embedding of the Si slab in vacuum. Our quantitative analysis of the experimental data reveals an extra force of thermal origin most likely associated with the difference of the native oxide thickness on the surfaces of the slab. The estimated difference in oxide layer thickness amounts to \ensuremath{\sim}4 nm.

Electromagnetic energy density in a single-resonance chiral metamaterial

We derive the electromagnetic energy density in a chiral metamaterial consisting of uncoupled single-resonance helical resonators. Both the lossless and absorptive cases are studied, and the energy density is shown to be positively definite. The key relation making the derivation successful is the proportionality between the magnetization and the rate of change of the electric polarization of the medium. The same time-domain formulation of energy density also applies to the bianisotropic medium proposed by Zhang et al. [Phys. Rev. Lett.102, 023901 (2009)PRLTAO0031-900710.1103/PhysRevLett.102.023901]. This work may provide insights for studying time-dependent phenomena in metamaterials.

CONSISTENT FORMALISM FOR THE MOMENTUM OF ELECTROMAGNETIC WAVES IN LOSSLESS DISPERSIVE METAMATERIALS AND THE CONSERVATION OF MOMENTUM

A new formalism for electromagnetic and mechanical momenta in a metamaterial is developed by means of the technique of wave-packet integrals. The medium has huge mass density and can therefore be regarded as almost stationary upon incident electromagnetic waves. A clear identification of momentum density and momentum flow, including their electromagnetic and mechanical parts, is obtained by employing this formalism in a lossless dispersive metamaterial (including the cases of impedance matching and mismatching with vacuum). It is found that the ratio of the electromagnetic momentum density to the mechanical momentum density depends on the impedance and group velocity of the electromagnetic wave inside the metamaterial. One of the definite results is that both the electromagnetic momentum and the mechanical momentum in the metamaterial are in the same direction as the energy flow, instead of in the direction of the wave vector. The conservation of total momentum is verified. In addition, the law of energy conservation in the process of normal incidence is also verified by using the wave-packet integral of both the electromagnetic energy density and the electromagnetic power density, of which the latter is caused by the interaction between the induced (polarized) currents and the electromagnetic wave.

DIFFRACTION FIELD BEHAVIOR NEAR THE EDGES OF A SLOT AND STRIP

The difiraction fleld asymptotics on the edges of a slot in the plane conducting screen and of a complementary strip is considered using the exact solutions of corresponding stationary difiraction problems, which have been derived before on the bases of the slot (strip) fleld expansions into discrete Fourier series. It is shown that as nearing the slot (strip) edges, the flelds decrease or increase indeflnitely in magnitude by the power law with an exponent of modulus less than unity, so the given exact difiraction solutions yield flnite value of electromagnetic energy density in any point of space.

On energy and momentum conservation laws for an electromagnetic field in a medium or at diffraction on a conducting plate

For a field-matter system, general nonstationary balance equations for energy and momentum densities and their transport velocities are obtained based on a rigorous nonstationary definition of these densities depending on the creation history of the field. We analyze the simplest dispersion law determined by conductivity dissipation; we find the electromagnetic energy density, phase velocity, group velocity, and energy and momentum transport velocities of a plane monochromatic wave. The low-frequency energy density is shown to be given by the electrostatic density in which the dielectric constant is replaced with its real part and the energy transport velocity is equal to the phase velocity. The group velocity can exceed the speed of light. We prove that the Minkowski form of momentum density must be used in the medium, and find the corresponding transport velocity, which in the case under consideration also coincides with the phase velocity. Energy and momentum conservation laws are shown to hold for a plane electromagnetic wave propagating in a medium or diffracted by a conducting plate.

Electromagnetic field energy in dispersive materials

A general expression for the electromagnetic energy density in a lossy dispersive medium, applicable for a field having a narrow temporal frequency bandwidth, is derived and compared with exact results for an example dielectric constant. Consequently, the possibility of negative time-averaged stored field energy is shown to have physical meaning. This observation is of interest in the study of dispersive metamaterials, such as those which can exhibit a negative refractive index.

On the electromagnetic energy density and energy transfer rate in a medium with dispersion due to conduction

The simplest dispersion relation determined by dissipation due to conduction is considered; the electromagnetic energy density in a plane monochromatic wave and its (phase and group) velocity are determined, as well as the energy and momentum transfer rates. It is shown that the energy density at low frequencies in this case has the form of the electrostatic density, in which the permittivity is replaced by its real part, and the energy transfer rate in a plane electromagnetic wave is equal to the phase velocity. The group velocity may exceed the speed of light.

Understanding interference experiments with polarized light through photon trajectories

Bohmian mechanics allows to visualize and understand the quantum-mechanical behavior of massive particles in terms of trajectories. As shown by Bialynicki-Birula, Electromagnetism also admits a hydrodynamical formulation when the existence of a wave function for photons (properly defined) is assumed. This formulation thus provides an alternative interpretation of optical phenomena in terms of photon trajectories, whose flow yields a pictorial view of the evolution of the electromagnetic energy density in configuration space. This trajectory-based theoretical framework is considered here to study and analyze the outcome from Young-type diffraction experiments within the context of the Arago-Fresnel laws. More specifically, photon trajectories in the region behind the two slits are obtained in the case where the slits are illuminated by a polarized monochromatic plane wave. Expressions to determine electromagnetic energy flow lines and photon trajectories within this scenario are provided, as well as a procedure to compute them in the particular case of gratings totally transparent inside the slits and completely absorbing outside them. As is shown, the electromagnetic energy flow lines obtained allow to monitor at each point of space the behavior of the electromagnetic energy flow and, therefore, to evaluate the effects caused on it by the presence (right behind each slit) of polarizers with the same or different polarization axes. This leads to a trajectory-based picture of the Arago-Fresnel laws for the interference of polarized light.

Dyadic Green's functions and electromagnetic local density of states

A formal proof to relate the concept of electromagnetic local density of states (LDOS) to the electric and magnetic dyadic Green's functions (DGF) is provided. The expression for LDOS is obtained by relating the electromagnetic energy density at any location in a medium at uniform temperature T to the electric and magnetic DGFs. The appropriate boundary conditions governing the DGFs are obtained and it is seen that the two types of DGFs are electromagnetic duals of each other. With this the concept of LDOS is also extended to material media. The LDOS is split into two terms—one that originates from the energy density in an infinite, homogeneous medium and the other that takes into account scattering from inhomogeneities. The second part can always be defined unambiguously, even in lossy materials. For lossy materials, the first part is finite only if spatial dispersion is taken into account.

Response of the high-latitude polar ionosphere to changes in the orientation of the Poynting vector near the Earth orbit relative to the geomagnetic moment orientation

The effect of the mutual orientation of the Poynting vector P of the electromagnetic energy density in the solar wind and the vector M of the Earth’s magnetic moment (taking into account its orbital and diurnal motions) on the geomagnetic activity has been examined for the first time using the measurements of the solar wind parameters on the Earth orbit in 1963–2005. The component Pm of the vector P along the vector M is shown to have a pronounced annual variation with the extrema in November and May and a diurnal variation with the extrema at ∼6 and 18 UT. The phases of the variations are shown to be determined only by the geometric parameters and are independent of the sign of the sector structure of the interplanetary magnetic field. The experimental data on the planetary and high-latitude geomagnetic activity, which is a response to changes in the orientation of P relative to M, are presented. The power of the sources of the electromagnetic energy of the solar wind during strong geomagnetic disturbances is also estimated.

Power loss and electromagnetic energy density in a dispersive metamaterial medium

The power loss and electromagnetic energy density of a metamaterial consisting of arrays of wires and split-ring resonators are investigated. We show that a field energy density formula can be derived consistently from both the electrodynamic (ED) approach and the equivalent circuit (EC) approach. The derivations are based on the knowledge of the dynamical equations of the electric and magnetic dipoles in the medium and the correct form of the power loss. We discuss the role of power loss in determining the form of energy density and explain why the power loss should be identified first in the ED derivation. When the power loss is negligible and the field is harmonic, our energy density formula reduces to the result of Landau's classical formula. For the general case with finite power loss, our investigation resolves the apparent contradiction between the previous results derived by the EC and ED approaches.

A back-to-front derivation: the equal spacing of quantum levels is a proof of simple harmonic oscillator physics

The dynamical behaviour of simple harmonic motion can be found in numerous natural phenomena. Within the quantum realm of atomic, molecular and optical systems, two main features are associated with harmonic oscillations: a finite ground-state energy and equally spaced quantum energy levels. Here it is shown that there is in fact a one-to-one mapping between the provision of equally spaced quantum energy levels and simple harmonic motion. The analysis establishes that the Hamiltonian of any system featuring quantized energy levels in an evenly spaced, infinite set must have a quadratic dependence on a pair of canonically conjugate variables. Moreover, specific physical inferences can be drawn. For example, exploiting this ‘back-to-front’ derivation, and based on the known existence of photons, it can be proved that an electromagnetic energy density is quadratic in both the electric and magnetic fields.

Electrically charged fluids with pressure in Newtonian gravitation and general relativity indspacetime dimensions: Theorems and results for Weyl type systems

Previous theorems concerning Weyl type systems, including Majumdar-Papapetrou systems, are generalized in two ways, namely, we take these theorems into $d$ spacetime dimensions ($d\ensuremath{\ge}4$), and we also consider the very interesting Weyl-Guilfoyle systems, i.e., general relativistic charged fluids with nonzero pressure. In particular within the Newton-Coulomb theory of charged gravitating fluids, a theorem by Bonnor (1980) in three-dimensional space is generalized to arbitrary $(d\ensuremath{-}1)>3$ space dimensions. Then, we prove a new theorem for charged gravitating fluid systems in which we find the condition that the charge density and the matter density should obey. Within general relativity coupled to charged dust fluids, a theorem by De and Raychaudhuri (1968) in four-dimensional spacetime is rendered into arbitrary $d>4$ dimensions. Then a theorem, new in $d=4$ and $d>4$ dimensions, for Weyl-Guilfoyle systems, is stated and proved, in which we find the condition that the charge density, the matter density, the pressure, and the electromagnetic energy density should obey. This theorem comprises, in particular cases, a theorem by Gautreau and Hoffman (1973) and results in four dimensions by Guilfoyle (1999). Upon connection of an interior charged solution to an exterior Tangherlini solution (i.e., a Reissner-Nordstr\"om solution in $d$ dimensions), one is able to give a general definition for gravitational mass for this kind of relativistic systems and find a mass relation with several quantities of the interior solution. It is also shown that for sources of finite extent the mass is identical to the Tolman mass.

Non-linear electromagnetism and special relativity

We continue the study of nonlinear Maxwell equations for electromagnetism in the formalism of B. D. Coleman & E. H. Dill. We exploit here the assumption of Lorentz invariance, following I. Białinicki-Barula. In particular, we show that nonlinearity forbids the convexity of the electromagnetic energy density. This justifies the study of rank-one convex and of polyconvex densities, begun in [8, 16]. We also show the alternative that either electrodynamics is linear, or dispersion is lost as the electromagnetic field becomes intense.

Dynamical laws of superenergy in general relativity

The Bel and Bel–Robinson tensors were introduced nearly 50 years ago in an attempt to generalize to gravitation the energy–momentum tensor of electromagnetism. This generalization was successful from the mathematical point of view because these tensors share mathematical properties which are remarkably similar to those of the energy–momentum tensor of electromagnetism. However, the physical role of these tensors in general relativity has remained obscure and no interpretation has achieved wide acceptance. In principle, they cannot represent energy and the term superenergy has been coined for the hypothetical physical magnitude lying behind them. In this work, we try to shed light on the true physical meaning of superenergy by following the same procedure which enables us to give an interpretation of the electromagnetic energy. This procedure consists in performing an orthogonal splitting of the Bel and Bel–Robinson tensors and analyzing the different parts resulting from the splitting. In the electromagnetic case such splitting gives rise to the electromagnetic energy density, the Poynting vector and the electromagnetic stress tensor, each of them having a precise physical interpretation which is deduced from the dynamical laws of electromagnetism (Poynting theorem). The full orthogonal splitting of the Bel and Bel–Robinson tensors is more complex but, as expected, similarities with electromagnetism are present. Also the covariant divergence of the Bel tensor is analogous to the covariant divergence of the electromagnetic energy–momentum tensor and the orthogonal splitting of the former is found. The ensuing equations are to the superenergy what the Poynting theorem is to electromagnetism. Some consequences of these dynamical laws of superenergy are explored, among them the possibility of defining superenergy radiative states for the gravitational field.

Energy and momentum flux in a high-numerical-aperture beam using the extended Nijboer-Zernike diffraction formalism

We describe the energy and momentum flux in the case of an aberrated optical imaging system with a high numerical aperture (NA). The approach is based on the extended Nijboer-Zernike diffraction theory, that, in its high-NA version, yields an accurate analytic representation of the electromagnetic field vectors in the focal region of imaging systems that suffer from aberrations and/or transmission defects. In an earlier publication, we have derived the electromagnetic energy density from the field vectors. In this paper, we expand our analysis to the energy flow (Poynting vector) and to the quantities related to the linear and angular momentum of the radiation. Several examples of the energy and momentum flow are presented. In particular, we show how the linear and angular momentum distribution in the focal region depend on the initial polarization state and on the parameters describing the wavefront shape of the converging beam. For the angular momentum flow, we show how the separation between spin and orbital momentum is modified when going from the paraxial case to a high-NA focused beam.

Internal Energy, Q-Energy, Poynting's Theorem, and the Stress Dyadic in Dispersive Material

General expressions are derived for time-domain energy density and the time integral of the Poynting vector that are related to the kinetic, potential, and heat energy densities of the bound charge-polarization carriers and the stored electromagnetic field energy density in passive, nonlinear or linear, lossy or lossless, temporally and/or spatially dispersive polarized media. In the most general linear, lossless, spatially nondispersive media, the energy expressions reveal non-negative quadratic forms for the frequency-domain internal energy densities that are used in expressions for the quality factor (Q) of antennas containing lossless dispersive material. The analysis also reveals useful inequalities that imply that the magnitude of the group velocity in lossless material is always less than or equal to the speed of light. The energy expressions do not, however, predict an internal energy in highly lossy temporally dispersive media that can be used to improve upon the expressions for the Q of antennas in such media. To improve upon the accuracy of the Q of antennas with highly lossy dispersive media, a non-negative Q-energy density is found that maintains the accuracy of the inverse relationship between Q and matched VSWR half-power fractional bandwidth for antennas containing highly lossy dispersive material. Lower bounds for this improved Q are found in terms of previously determined lower bounds. The paper also confirms the result that, for general linear lossy or lossless dispersive material, the steady state time averages of the electromagnetic power density, force density, and stress dyadic with sinusoidal time dependence that turns on at some finite time in the past, unlike the internal energy, does not contain derivatives with respect to frequency.

On negative refraction in classical vacuum

Propagation of light in a classical gravitational field can be successfully modelled as electromagnetic waves propagating in a bianisotropic medium. However, some care must be exercised when applying the conventional expressions for the electromagnetic energy density and the power flow in a bianisotropic medium. These expressions are shown to be components of a four-vector that is dual to the wave's 4-wavevector, the two 4-vectors themselves being parallel. The implications and status of claims for negative refraction in vacuum are discussed.

Electromagnetic energy density and velocity in a medium with anomalous positive dispersion

The simple laws of dispersion, the electromagnetic energy density, and the phase, group, and energy transfer velocities in dissipative media are considered. In polar dielectrics with an anomalous positive dispersion described by the Debye formula, the energy transfer velocity in a plane monochromatic wave coincides with the phase velocity, while the group velocity can exceed the speed of light.