本文基于经典偶极子源电磁表述建立能量与动量连续性方程，提出微观质量极化激元模型，并阐明 Abraham 光学力密度源于诱导偶极子与机械应力的相互作用。
摘要：
我们报告了与最近提出的基于经典偶极子源的电磁学表述相关的线动量和能量连续性方程[Eur. Phys. J. Plus 138, 1034 (2023)]。当与质量极化激元（mass-polariton）准粒子动力学相联系时，这些方程为电介质内部的局域光学能量和动量传递提供了一致的微观描述，称为微观质量极化激元表述。该过程还揭示了久为人知的 Abraham 光学力密度的真正微观起源，即诱导偶极子与材料内部产生的机械应力之间的相互作用。
英文摘要：
We report on the continuity equations for linear momentum and energy associated to a recently introduced electromagnetic formulation based on classical dipolar sources [Eur. Phys. J. Plus 138, 1034 (2023)]. When connected to the mass-polariton quasi-particle dynamics, these equations provide a consistent microscopic description of the local optical energy and momentum transfer inside dielectric media, called microscopic mass-polariton formulation. This procedure also unveils the true microscopic origin of the long-known Abraham optical force density as an interplay between induced dipoles and mechanical stresses generated within the material.
When the effects of dispersion are included, neither the Abraham nor the Minkowski expression for electromagnetic momentum in a dielectric medium gives the correct recoil momentum for absorbers or emitters of radiation. The total momentum density associated with a field in a dielectric medium has three contributions: (i) the Abraham momentum density of the field, (ii) the momentum density associated with the Abraham force, and (iii) a momentum density arising from the dispersive part of the response of the medium to the field, the latter having a form evidently first derived by D.F. Nelson [Ph…
We analyse the "Einstein box" thought experiment and the definition of the momentum of light inside matter. We stress the importance of the total energy-momentum tensor of the closed system (electromagnetic field plus material medium) and derive in detail the relativistic expressions for the Abraham and Minkowski momenta, together with the corresponding balance equations for an isotropic and homogeneous medium. We identify some assumptions hidden in the Einstein box argument, which make it weaker than it is usually recognized. In particular, we show that the Abraham momentum is not uniquely se…
After more than a century of debate, there remains continuing discomfort over what is the correct expression for the electromagnetic momentum in a dielectric medium. This is the so-called the Minkowski-Abraham controversy. We show that there is indeed a consistent picture for the electromagnetic momentum associated with waves in a dielectric, but one must start with the fields E and B, not D and H as fundamental objects. The momentum flow is described by a tensor T^EB, where the expression is identical to that in vacuum. The Minkowski tensor and the Abraham force result from a fractured decomp…
Interconnection of alternative equations of conservation of the momentum related with different interpretations of equations of the electromagnetic field is an essential aspect of the Abraham-Minkowski controversy, which is associated in the first place with rivalry of alternative definitions of optical momentum of the electromagnetic wave in a medium. Assuming significance of alternative equations of momentum conservation in accord with their interconnection, potential functional equivalence of alternative descriptions of optical momentum is examined in the context of the radiation pressure. …
According to the scientific literature, the momentum of a photon in a simple linear dielectric is either $\hbarω/(nc)$ or $n\hbarω/c$ with a unit vector ${\bf \hat e}_k$ in the direction of propagation. These momentums are typically used to argue the century-old Abraham--Minkowski controversy in which the momentum density of the electromagnetic field in a dielectric is either the Abraham momentum density, ${\bf g}_A={\bf E}\times{\bf H}/c$, or the Minkowski momentum density, ${\bf g}_M={\bf D}\times{\bf B}/c$. The elementary optical excitations, photons, are typically known as polaritions in t…
In this paper I compare the Abraham and the Minkowski forms for the momentum pertaining to an electromagnetic wave inside a dielectric or a magnetic material. The discussion is based on a careful treatment of the surface charges and currents and of the forces acting on them. While in the dielectric case the Abraham momentum is certainly more appealing from the physical point of view, for a magnetic material it suggests an interpretation in terms of magnetic charges and re- lated magnetic currents. The Minkowski momentum for magnetic non conducting materials, on the contrary, has a natural inte…
The Abraham--Minkowski controversy refers to a long-standing inability to adequately address certain issues involving the conservation of the momentum of an electromagnetic field in a linear dielectric medium. We apply the usual assumption of a material subsystem that couples to the electromagnetic subsystem such that the total energy and total momentum are conserved. We then construct the total energy--momentum tensor from the total energy density and the total momentum density. Applying conservation principles to the total energy--momentum tensor, we construct the tensor energy--momentum con…
A conventional general electromagnetic force definition has been widely used to analyze radiation forces in dielectric media in published research works. However in this paper, we would like to indicate that this conventional force definition is flawed.
Continuum electrodynamics is an axiomatic formal theory based on the macroscopic Maxwell equations and the constitutive relations. We apply the formal theory to a thermodynamically closed system consisting of an antireflection coated block of dielectric situated in free-space and illuminated by a quasimonochromatic field. We show that valid theorems of the formal theory are proven false by relativity and by conservation laws. Then the axioms of the formal theory are proven false at a fundamental level of mathematical logic. We derive a new formal theory of continuum electrodynamics for macrosc…
The problem of assigning a momentum to an electromagnetic wave packet propagating inside an insulator has become known under the name of the Abraham-Minkowski controversy. In the present paper we re-examine the question, first through a power expansion in the polarizability of the medium and assuming the simplest and most natural choice for the force exerted on a dielectric material by an electromagnetic field. It is shown that the Abraham expression is highly favoured. We then show the complete generality of these results.
We treat continuum electrodynamics as an axiomatic formal theory based on the macroscopic Maxwell--Minkowski equations applied to a thermodynamically closed system consisting of an antireflection-coated block of a simple linear dielectric material situated in free-space that is illuminated by a quasimonochromatic field. We prove that valid theorems of the formal theory of Maxwellian continuum electrodynamics are inconsistent with conservation laws for the inviscid incoherent flow of non-interacting particles (photons) in the continuum limit (light field) in the absence of external forces, pres…
In the more than 100 years-old Abraham-Minkowski problem in macroscopic electrodynamics, the issue of how to observe the so-called Abraham term ${\bf f}^{\rm Aterm} = [(\varepsilonμ-1)/c^2] \partial/\partial t ({\bf E}\times {\bf H})$ has been a main point. Recent years have seen a number of beautiful experiments in radiation optics, but these experiments usually give no information about the Abraham term as this term simply fluctuates out. So one is left with somewhat indirect verifications of this force, as in the radiation pressure of Jones {et al.} in the 1950's, testing the radiation pres…
We examine the momentum and angular-momentum (AM) properties of monochromatic optical fields in dispersive and inhomogeneous isotropic media, using the Abraham- and Minkowski-type approaches, as well as the kinetic (Poynting-like) and canonical (with separate spin and orbital degrees of freedom) pictures. While the kinetic Abraham-Poynting momentum describes the energy flux and the group velocity of the wave, the Minkowski-type quantities, with proper dispersion corrections, describe the actual momentum and angular momentum carried by the wave. The kinetic Minkowski-type momentum and AM densit…
Radiation pressure measurements on mirrors submerged in dielectric liquids have consistently shown an effective Minkowski momentum for the photons within the liquid. Using an exact theoretical calculation based on Maxwell's equations and the Lorentz law of force, we demonstrate that this result is a consequence of the fact that conventional mirrors impart, upon reflection, a 180 degree phase-shift to the incident beam of light. If the mirror is designed to impart a different phase, then the effective momentum will turn out to be anywhere between the two extremes of the Minkowski and Abraham mo…
The long-standing resolution of the Abraham--Minkowski electromagnetic momentum controversy is predicated on a decomposition of the total momentum of a closed continuum electrodynamic system into separate field and matter components. Using a microscopic model of a simple linear dielectric, we derive Lagrangian equations of motion for the electric dipoles and show that the dielectric can be treated as a collection of stationary simple harmonic oscillators that are driven by the electric field and produce a polarization field in response. The macroscopic energy and momentum are defined in terms …
This memo contains a collection of formulas describing the electromagnetic energy, momentum and spin distribution of an optical field formed in dielectric media separated by a plane interface when an incident monochromatic plane wave is totally reflected. The formulas are based on the Abraham momentum definition and include the momentum decomposition into the orbital (canonical) and spin parts as well as explicit dual-symmetric separation of the electric and magnetic contributions. This material was prepared in February 2013 but it had not been finalized and published because of the difficulti…
In a continuum setting, the energy-momentum tensor embodies the relations between conservation of energy, conservation of linear momentum, and conservation of angular momentum. The well-defined total energy and the well-defined total momentum in a thermodynamically closed system with complete equations of motion are used to construct the total energy-momentum tensor for a stationary simple linear material with both magnetic and dielectric properties illuminated by a quasimonochromatic pulse of light through a gradient-index antireflection coating. The perplexing issues surrounding the Abraham …
We use a relativistic argument to define an effective photon that travels through a transparent (non-absorbing) nondispersive dielectric medium of index of refraction $n$. If $p$ is the momentum of the photon in a vacuum, then the momentum of an effective photon inside the medium may be of the form $p_{\rm eff}=pn^α$ and still reproduce the observed transverse relativistic drift when the medium is in motion. We employ an energy argument to determine the value of the exponent to be $α=-1$, so that the effective photon momentum is $p_{\rm eff}=p/n$, which is the Abraham momentum.
The dielectric microsphere can be directly embedded into the traditional microscope, which can significantly improve the resolution of the microscope and provide a simple and feasible way to break through the diffraction limit of optical imaging system. However, due to the lack of effective theory and formula system, the resolution limit, magnification and optimal imaging position of the microsphere microsystem cannot be determined, which hinders its application. In this paper, the microscopic theory of dielectric microspheres is studied systematically, and the ray optics is extended to the im…
A transparent linear magneto-dielectric material in free space that is illuminated by a finite quasimonochromatic field is a thermodynamically closed system, definitively, regardless of what field and material subsystems that one defines. The energy--momentum tensor that is formally derived from the Maxwell--Minkowski field equations is inconsistent with both angular and linear momentum conservation in this closed system; this very solid fact is the foundational and continuing issue of the Abraham--Minkowski controversy. The extant resolution of the Abraham--Minkowski dilemma is to treat Maxwe…
We derive an exact expression for the radiation pressure of a quasi- monochromatic plane wave incident from the free space onto the flat surface of a semi-infinite dielectric medium. In order to account for the total optical momentum (incident plus reflected) that is transferred to the dielectric, the mechanical momentum acquired by the medium must be added to the rate of flow of the electromagnetic momentum (the so-called Abraham momentum) inside the dielectric. We confirm that the electromagnetic momentum travels with the group velocity of light inside the medium. The photon drag effect, in …
The force of electromagnetic radiation on a dielectric medium may be derived by a direct application of the Lorentz law of classical electrodynamics. While the light's electric field acts upon the (induced) bound charges in the medium, its magnetic field exerts a force on the bound currents. We use the example of a wedge-shaped solid dielectric, immersed in a transparent liquid and illuminated at Brewster's angle, to demonstrate that the linear momentum of the electromagnetic field within dielectrics has neither the Minkowski nor the Abraham form; rather, the correct expression for momentum de…
We present arguments in favor of the proposition that the momentum of light inside a transparent dielectric medium is the arithmetic average of the Minkowski and Abraham momenta. Using the Lorentz transformation of the fields (and of the coordinates) from a stationary to a moving reference frame, we show the consistent transformation of electromagnetic energy and momentum between the two frames. We also examine the momentum of static (i.e., time-independent) electromagnetic fields, and show that the close connection that exists between the Poynting vector and the momentum density extends all t…
We present an exactly solvable model of a classical dielectric medium that gives an unambiguous local decomposition of field and charge motion and their contribution to the conserved quantities. The result is a set of four branches to the dispersion law that gives full independent freedom in the selection of initial data of the fields and charge motion, in contrast with constitutive laws. This is done with special care to the forces that exist at surfaces, coatings and the ends of packets. As a result the utility of a stress-tensor as a function of field strengths and dielectric response for d…
Electromagnetic waves carry the Abraham momentum, whose density is given by p_EM = S(r,t)/c^2. Here S(r,t) = E(r,t)\timesH(r,t) is the Poynting vector at point r in space and instant t in time, E and H are the local electromagnetic fields, and c is the speed of light in vacuum. The above statement is true irrespective of whether the waves reside in vacuum or within a ponderable medium, which medium may or may not be homogeneous, isotropic, transparent, linear, magnetic, etc. When a light pulse enters an absorbing medium, the force experienced by the medium is only partly due to the absorbed Ab…
Though the interfacial tractor beam experiment supports the increase of photon momentum (i.e., Minkowski momentum), it is still a matter of investigation whether, inside matter, the photon momentum always increases. Considering the inhomogeneous or heterogeneous background, we have demonstrated that if the background and the half-immersed object are both non-absorbing, the transferred photon momentum to the object can be considered as the one of Minkowski exactly at the interface. In contrast, the presence of loss inside matter, either in the half-immersed object (i.e., plasmonic or lossy diel…
The total momentum of a thermodynamically closed system is unique, as is the total energy. Nevertheless, there is continuing confusion concerning the correct form of the momentum and the energy-momentum tensor for an electromagnetic field interacting with a linear dielectric medium. Here we investigate the energy and momentum in a closed system composed of a propagating electromagnetic field and a negligibly reflecting dielectric. The Gordon momentum is easily identified as the total momentum by the fact that it is, by virtue of being invariant in time, conserved. We construct continuity equat…
The macroscopic equations of Maxwell combined with a generalized form of the Lorentz law are a complete and consistent set; not only are these five equations fully compatible with the special theory of relativity, they also conform with the conservation laws of energy, momentum, and angular momentum. The linear momentum density associated with the electromagnetic field is p_EM(r,t)=E(r,t)\timesH(r,t)/c^2, whether the field is in vacuum or in a ponderable medium. [Homogeneous, linear, isotropic media are typically specified by their electric and magnetic permeabilities epsilon_0*epsilon(w) and …
We investigate one of the consequences of the three competing models describing the mechanical interaction of light with a dielectric medium. According to both the Abraham and Minkowski models the time-averaged force density is zero inside a homogeneous dielectric, whereas the induced-current Lorentz force model predicts a non-zero force density. We argue that the latter force, if exists, could drive a hydrodynamic flow inside a homogeneous fluid. Our numerical experiments show that such flows have distinct spatial patterns and may influence the dynamics of particles in a water-based single-be…
We present a classical linear response theory for a magneto-dielectric material and determine the polariton dispersion relations. The electromagnetic field fluctuation spectra are obtained and polariton sum rules for their optical parameters are presented. The electromagnetic field for systems with multiple polariton branches is quantised in 3 dimensions and field operators are converted to 1-dimensional forms appropriate for parallel light beams. We show that the field-operator commutation relations agree with previous calculations that ignored polariton effects. The Abraham (kinetic) and Min…
The Abraham--Minkowski momentum controversy is the outwardly visible symptom of an inconsistency in the use of the energy-momentum tensor in the case of a plane quasimonochromatic field in a simple linear dielectric. We show that the Gordon form of the electromagnetic momentum is conserved in a thermodynamically closed system. We regard conservation of the components of the four-momentum in a thermodynamically closed system as a fundamental property of the energy--momentum tensor. Then the first row and column of the energy--momentum tensor is populated by the electromagnetic energy density an…
In a recent paper, W. She, J. Yu and R. Feng reported the slight deformations observed upon transmission of a light pulse through a fairly short length of a silica glass nano-fiber. Relating the shape and magnitude of these deformations to the momentum of the light pulse both inside and outside the fiber, these authors concluded that, within the fiber, the photons carry the Abraham momentum. In my view, the authors' claim that they have resolved the Abraham-Minkowski controversy surrounding the momentum of photons inside dielectric media is premature. A correct interpretation of the experiment…
That static electric and magnetic fields can store momentum may be perplexing, but is necessary to ensure total conservation of momentum. Simple situations in which such field momentum is transferred to nearby bodies and point charges have often been considered for pedagogical purposes, normally assuming vacuum surroundings. If dielectric media are involved, however, the analysis becomes more delicate, not least since one encounters the electromagnetic energy-momentum problem in matter, the `Abraham-Minkowski enigma', of what the momentum is of a photon in matter. We analyze the momentum balan…
We derive an expression for the macroscopic force density that a narrow-band electromagnetic field imposes on a dissipative isotropic medium. The result is obtained by averaging the microscopic form for Lorentz force density. The derived expression allows us to calculate realistic electromagnetic forces in a wide range of materials that are described by complex-valued electric permittivity and magnetic permeability. The three-dimensional energy-momentum tensor in our expression reduces for lossless media to the so-called Helmholtz tensor that has not been contradicted in any experiment so far.…
We propose to use femtosecond direct laser writing technique to realize dielectric optical elements from photo-resist materials for the generation of structured light from purely geometrical phase transformations. This is illustrated by the fabrication and characterization of spin-to-orbital optical angular momentum couplers generating optical vortices of topological charge from 1 to 20. In addition, the technique is scalable and allows obtaining microscopic to macroscopic flat optics. These results thus demonstrate that direct 3D photopolymerization technology qualifies for the realization of…
We discuss the calculation of the refractive index by means of the ab initio scalar dielectric function and point out its inherent limitations. To overcome these, we start from the recently proposed fundamental, microscopic wave equation in materials in terms of the frequency- and wavevector-dependent dielectric tensor, and investigate under which conditions the standard treatment can be justified. Thereby, we address the question of neglecting the wavelength dependence of microscopic response functions. Furthermore, we analyze in how far the fundamental, microscopic wave equation is equivalen…
A discussion is given on the interpretation and physical importance of the Minkowski momentum in macroscopic electrodynamics (essential for the Abraham-Minkowski problem). We focus on the following two facets: (1) Adopting a simple dielectric model where the refractive index $n$ is constant, we demonstrate by means of a mapping procedure how the electromagnetic field in a medium can be mapped into a corresponding field in vacuum. This mapping was presented many years ago [I. Brevik and B. Lautrup, Mat. Fys. Medd. Dan. Vid. Selsk {\bf 38}(1), 1 (1970)], but is apparently not well known. A chara…
We report the first observation of mechanical momentum transferred to atoms and molecules upon application of crossed electric and magnetic fields. We identify this momentum as the microscopic analogue of the classical Abraham force. Several predictions of additional magneto-electrically induced mechanical momentum are addressed. One of them, proposed to result from the interaction with the quantum vacuum, is experimentally refuted, others are found to be currently below experimental detection.
Treatment of Linear and Nonlinear Dielectric Property of Molecular Monolayer and Submonolayer with Microscopic Dipole Lattice Model: I. Second Harmonic Generation and Sum-Frequency Generation
A simple optical lens plays an important role for exploring the microscopic world in science and technology by refracting light with tailored spatially varying refractive index. Recent advancements in nanotechnology enable novel lenses, such as, superlens, hyperlens, Luneburg lens, with sub-wavelength resolution capabilities by specially designing materials' refractive indices with meta-materials and transformation optics. However, these artificially nano/micro engineered lenses usually suffer high losses from metals and are highly demanding in fabrication. Here we experimentally demonstrate f…
A theory is presented for calculating the effect of the electromagnetic field on the centre of mass of a macroscopic dielectric body that is valid in both quantum and classical regimes. We apply the theory to find the classical equation of motion for the centre of mass of a macroscopic object in a classical field, and the spreading of an initially localized wave-packet representing the centre of mass of a small object, in a quantum field. The classical force is found to be consistent with the identification of the Abraham momentum with the mechanical momentum of light, and the motion of the wa…
Maxwell's macroscopic equations combined with a generalized form of the Lorentz law of force are a complete and consistent set of equations. Not only are these five equations fully compatible with special relativity, they also conform with conservation laws of energy, momentum, and angular momentum. We demonstrate consistency with the conservation laws by showing that, when a beam of light enters a magnetic dielectric, a fraction of the incident linear (or angular) momentum pours into the medium at a rate determined by the Abraham momentum density, ExH/c^2, and the group velocity V_g of the el…
We derive the force of the electromagnetic radiation on material objects by a direct application of the Lorentz law of classical electro-dynamics. The derivation is straightforward in the case of solid metals and solid dielectrics, where the mass density and the optical constants of the media are assumed to remain unchanged under internal and external pressures, and where material flow and deformation can be ignored. For metallic mirrors, we separate the contribution to the radiation pressure of the electrical charge density from that of the current density of the conduction electrons. In the …
A synoptic view on the long-established theory of light propagation in crystalline dielectrics is presented, providing a new exact solution for the microscopic local electromagnetic field thus disclosing the role of the divergence-free (transversal) and curl-free (longitudinal) parts of the electromagnetic field inside a material as a function of the density of polarizable atoms. Our results enable fast and efficient calculation of the photonic bandstructure and also the (non-local) dielectric tensor, solely with the crystalline symmetry and atom-individual polarizabilities as input.
This paper is a continuation of a recent one in which, apparently for the first time, the existence of polaritons in ionic crystals was proven in a microscopic electrodynamic theory. This was obtained through an explicit computation of the dispersion curves. Here the main further contribution consists in studying electric susceptibility, from which the spectrum can be inferred. We show how susceptibility is obtained by the Green--Kubo methods of Hamiltonian statistical mechanics, and give for it a concrete expression in terms of time--correlation functions. As in the previous paper, here too w…
We develop a recursive computational procedure to efficiently calculate the macroscopic dielectric function of multi-component metamaterials of arbitrary geometry and composition within the long wavelength approximation. Although the microscopic response of the system might correspond to non-Hermitian operators, we develop a representation of the microscopic fields and of the response, and we introduce an appropriate metric that makes all operators symmetric. This allows us to use a modified Haydock recursion, introducing complex Haydock coefficients that allow an efficient computation of the …
