Invoking Maxwell's classical equations in conjunction with expressions for the electromagnetic (EM) energy, momentum, force, and torque, we use a few simple examples to demonstrate the nature of the EM angular momentum. The energy and the angular momentum of an EM field will be shown to have an intimate relationship; a source radiating EM angular momentum will, of necessity, pick up an equal but opposite amount of mechanical angular momentum; and the spin and orbital angular momenta of the EM field, when absorbed by a small particle, will be seen to elicit different responses from the particle.
The classical theory of electromagnetism is based on Maxwell's macroscopic equations, an energy postulate, a momentum postulate, and a generalized form of the Lorentz law of force. These seven postulates constitute the foundation of a complete and consistent theory, thus eliminating the need for physical models of polarization P and magnetization M - these being the distinguishing features of Maxwell's macroscopic equations. In the proposed formulation, P(r,t) and M(r,t) are arbitrary functions of space and time, their physical properties being embedded in the seven postulates of the theory. T…
The classical theory of electrodynamics is built upon Maxwell's equations and the concepts of electromagnetic (EM) field, force, energy, and momentum, which are intimately tied together by Poynting's theorem and by the Lorentz force law. Whereas Maxwell's equations relate the fields to their material sources, Poynting's theorem governs the flow of EM energy and its exchange between fields and material media, while the Lorentz law regulates the back-and-forth transfer of momentum between the media and the fields. An alternative force law, first proposed by Einstein and Laub, exists that is cons…
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 energy--momentum tensor and the tensor continuity equation serve as the conservation laws of energy, linear momentum, and angular momentum for a continuous flow. Previously, we derived equations of motion for macroscopic electromagnetic fields in a homogeneous linear dielectric medium that is draped with a gradient-index antireflection coating (J. Math Phys. 55, 042901 (2014) ). These results are consistent with the electromagnetic tensor continuity equation in the limit that reflections and the accompanying surface forces are negligible thereby satisfying the condition of an unimpeded flo…
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…
It is shown that the momentum density of free electromagnetic field splits into two parts. One has no contribution to the net momentum due to the transversality condition. The other yields all the momentum. The angular momentum that is associated with the former part is spin, and the angular momentum that is associated with the latter part is orbital angular momentum. Expressions for the spin and orbital angular momentum are given in terms of the electric vector in reciprocal space. The spin and orbital angular momentum defined this way are used to investigate the angular momentum of nonparaxi…
The classical theory of electrodynamics is built upon Maxwell's equations and the concepts of electromagnetic field, force, energy and momentum, which are intimately tied together by Poynting's theorem and the Lorentz force law. Whereas Maxwell's macroscopic equations relate the electric and magnetic fields to their material sources (i.e., charge, current, polarization and magnetization), Poynting's theorem governs the flow of electromagnetic energy and its exchange between fields and material media, while the Lorentz law regulates the back-and-forth transfer of momentum between the media and …
The expression for the total angular momentum carried by a laser optical vortex beam, splits, in the paraxial approximation, into two terms which seem to represent orbital and spin angular momentum respectively. There are, however, two very different competing versions of the formula for the spin angular momentum, one based on the use of the Poynting vector, as in classical electrodynamics, the other related to the canonical expression for the angular momentum which occurs in Quantum Electrodynamic. I analyze the possibility that a sufficiently sensitive optical measurement could decide which …
A necessary and sufficient condition for energy-momentum conservation is proved within a topological, pre-metric approach to classical electrodynamics including magnetic as well as electric charges. The extended Lorentz force, consisting of mutual actions by F=(E, B) on the electric current and G=(H, D) on the magnetic current, can be derived from an energy-momentum "potential" if and only if the constitutive relation G=G(F) satisfies a certain vanishing condition. The electric-magnetic reciprocity introduced by Hehl and Obukhov is seen to define a complex structure on the tensor product of 2-…
Some consequences of a fully classical unified theory of gravity and electromagnetism are worked out for the electromagnetic sector such as the occurrence of classical light beams with spin and orbital angular momenta that are topologically quantized in units of $q_e q_m=σ$, independent of the beam size. Empirical fits require $σ= \hbar$. The theory also predicts a generalized coherency matrix whose consequences are testable.
From Newtons third law, the principle of actio et reactio, we expect the forces between interacting particles to be equal and opposite. However, non-reciprocal forces can arise. Specifically, this has recently been shown theoretically in the interaction between dissimilar optically trapped particles mediated by an external field. As a result, despite the incident external field not having a transverse component of momentum, the particle pair experiences a force in a direction transverse to the light propagation direction. In this letter, we directly measure the net non-reciprocal force in elec…
Invoking tools and techniques from elementary theories of classical electrodynamics and special relativity, we analyze some of the thought experiments that have contributed substantively to the conceptual development and understanding of the linear and angular momenta of light.
A circularly polarized rotationally symmetric paraxial laser beams carries hbar angular momentum per photon as spin. Focussing the beam with a rotationally symmetric lens cannot change this angular momentum flux, yet the focussed beam must have spin less than hbar per photon. The remainder of the original spin is converted to orbital angular momentum, manifesting itself as a longitudinal optical vortex at the focus. This demonstrates that optical orbital angular momentum can be generated by a rotationally symmetric optical system which preserves the total angular momentum of the beam.
Starting from covariant expressions, a gauge independent separation of orbital and spin angular momentum for electrodynamics is presented. This results from the non-symmetric canonical energy momentum tensor of the electromagnetic field. The origin of the difficulty is discussed and a covariant gauge invariant spin vector is derived. The paradox concerning the spin angular momentum of a plane wave finds a natural solution.
We describe the coupling between orbital angular momentum (OAM) and temporal degrees of freedom in pulsed Laguerre-Gauss beams. The effects of this coupling are to increase significantly the duration of few-cycle pulses when the OAM carried by the pulse is large, and to enhance spatiotemporal couplings. Contrary to other detrimental but retrievable effects such as spatial, group velocity and topological charge dispersions in these pulses, OAM-temporal coupling effects are unavoidable, and have therefore an impact in applications such as OAM-based optical communications, or in the generation of…
The interplay between spin and orbital angular momentum in the up-conversion process allows us to control the macroscopic wave front of high harmonics by manipulating the microscopic polarizations of the driving field. We demonstrate control of orbital angular momentum in high harmonic generation from both solid and gas phase targets using the selection rules of spin angular momentum. The gas phase harmonics extend the control of angular momentum to extreme-ultraviolet wavelength. We also propose a bi-color scheme to produce spectrally separated extreme-ultraviolet radiation carrying orbital a…
It is shown that the momentum density of free electromagnetic field splits into two parts. One has no contribution to the net momentum due to the transversality condition. The other yields all the momentum. The angular momentum that originates from the former part is spin, and the angular momentum that originates from the latter part is orbital angular momentum. Expressions for the spin and orbital angular momentum are given in terms of the electric vector in reciprocal space. The spin and orbital angular momentum defined this way are used to investigate the angular momentum of nonparaxial bea…
We theoretically analyze and experimentally measure the extrinsic angular momentum contribution of topologically structured darkness found within fractional vortex beams, and show that this structured darkness can be explained by evanescent waves at phase discontinuities in the generating optic. We also demonstrate the first direct measurement of the intrinsic orbital angular momentum of light with both intrinsic and extrinsic angular momentum, and explain why the total orbital angular momenta of fractional vortices do not match the winding number of their generating phases.
We demonstrate the Fourier relationship between angular position and angular momentum for a light mode. In particular we measure the distribution of orbital angular momentum states of light that has passed through an aperture and verify that the orbital angular momentum distribution is given by the complex Fourier-transform of the aperture function. We use spatial light modulators, configured as diffractive optical components, to define the initial orbital angular momentum state of the beam, set the defining aperture, and measure the angular momentum spread of the resulting beam. These measure…
The angular dependence of the differential cross section of unpolarized light-by-light scattering summed over final polarizations is the same in any low-energy effective theory of quantum electrodynamics and also in Born-Infeld electrodynamics. In this paper we derive general expressions for polarization-dependent low-energy scattering amplitudes, including a hypothetical parity-violating situation. These are evaluated for quantum electrodynamics with charged scalar or spinor particles, which give strikingly different polarization effects. Ordinary quantum electrodynamics is found to exhibit r…
Many methods now exist to prepare free electrons into orbital angular momentum states, and the predicted applications of these electron states as probes of materials and scattering processes are numerous. The development of electron orbital angular momentum measurement techniques has lagged behind. We show that coupling between electron orbital angular momentum and a spatially varying magnetic field produces an angular momentum-dependent focusing effect. We propose a design for an orbital angular momentum measurement device built on this principle. As the method of measurement is non-interfero…
Photon carries linear momentum and angular momentum simultaneously. Within the light-matter interaction process, exchange of linear momentum results in optical forces, whereas exchange of angular momentum leads to optical torques. Use of optical forces (light pressure or damping) have been long and wide in quantum optomechanics, however, those of optical torque and optical angular momentum are not. Here we propose a theoretical framework based on optical angular momentum and optical torques to derive the Hamiltonians of cavity orbital and spin angular momentum optomechanical systems, respectiv…
The angular momentum density and flux of light in a dispersive, rotationally symmetric medium are derived from Noether's theorem. Optical angular momentum in a dispersive medium has no simple relation to optical linear momentum, even if the medium is homogeneous. A circularly polarized monochromatic beam in a homogeneous, dispersive medium carries a spin angular momentum of $\pm\hbar$ per energy $\hbarω$, as in vacuum. This result demonstrates the non-trivial interplay of dispersive contributions to optical angular momentum and energy.
Spin and orbital angular momentum of light plays a central role in quantum nanophotonics as well as topological electrodynamics. Here, we show that the thermal radiation from finite-sized bodies comprising of nonreciprocal magneto-optical materials can exert a spin torque even in global thermal equilibrium. Moving beyond the paradigm of near-field heat transfer, we calculate near-field radiative angular momentum transfer between finite-sized nonreciprocal objects by combining Rytov's fluctuational electrodynamics with the theory of optical angular momentum. We prove that a single magneto-optic…
