Alberto Favaro

The non‐birefringent limit of all linear, skewonless media and its unique light‐cone structure

Based on a recent work by Schuller et al., a geometric representation of all skewonless, non-birefringent linear media is obtained. The derived constitutive law is based on a “core”, encoding the optical metric up to a constant. All further corrections are provided by two (anti-)selfdual bivectors, and an “axion”. The bivectors are found to vanish if the optical metric has signature (3,1) – that is, if the Fresnel equation is hyperbolic. We propose applications of this result in the context of transformation optics and premetric electrodynamics.

Optimal Frames for Polarization State Reconstruction.

Complete determination of the polarization state of light requires at least four distinct projective measurements of the associated Stokes vector. Stability of state reconstruction, however, hinges on the condition number κ of the corresponding instrument matrix. Optimization of redundant measurement frames with an arbitrary number of analysis states, m, is considered in this Letter in the sense of minimization of κ. The minimum achievable κ is analytically found and shown to be independent of m, except for m=5 where this minimum is unachievable. Distribution of the optimal analysis states over the Poincaré sphere is found to be described by spherical 2 designs, including the Platonic solids as special cases. Higher order polarization properties also play a key role in nonlinear, stochastic, and quantum processes. Optimal measurement schemes for nonlinear measurands of degree t are hence also considered and found to correspond to spherical 2t designs, thereby constituting a generalization of the concept of mutually unbiased bases.

The Kummer tensor density in electrodynamics and in gravity

Abstract Guided by results in the premetric electrodynamics of local and linear media, we introduce on 4-dimensional spacetime the new abstract notion of a Kummer tensor density of rank four, K i j k l . This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four T i j k l , which is antisymmetric in its first two and its last two indices: T i j k l = − T j i k l = − T i j l k . Thus, K ∼ T 3 , see Eq. (46) . (i) If T is identified with the electromagnetic response tensor of local and linear media, the Kummer tensor density encompasses the generalized Fresnel wave surfaces for propagating light. In the reversible case, the wave surfaces turn out to be Kummer surfaces as defined in algebraic geometry (Bateman 1910). (ii) If T is identified with the curvature tensor R i j k l of a Riemann–Cartan spacetime, then K ∼ R 3 and, in the special case of general relativity, K reduces to the Kummer tensor of Zund (1969). This K is related to the principal null directions of the curvature. We discuss the properties of the general Kummer tensor density. In particular, we decompose K irreducibly under the 4-dimensional linear group G L ( 4 , R ) and, subsequently, under the Lorentz group S O ( 1 , 3 ) .

Four Poynting theorems

The Poynting vector is an invaluable tool for analysing electromagnetic problems. However, even a rigorous stress–energy tensor approach can still leave us with the question: is it best defined as E × H or as D × B? Typical electromagnetic treatments provide yet another perspective: they regard E × B as the appropriate definition, because E and B are taken to be the fundamental electromagnetic fields. The astute reader will even notice the fourth possible combination of fields, i.e. D × H. Faced with this diverse selection, we have decided to treat each possible flux vector on its merits, deriving its associated energy continuity equation but applying minimal restrictions to the allowed host media. We then discuss each form, and how it represents the response of the medium. Finally, we derive a propagation equation for each flux vector using a directional fields approach, a useful result which enables further interpretation of each flux and its interaction with the medium.

Light propagation in local and linear media: Fresnel-Kummer wave surfaces with 16 singular points

It is known that the Fresnel wave surfaces of transparent biaxial media have 4 singular points, located on two special directions. We show that, in more general media, the number of singularities can exceed 4. In fact, a highly symmetric linear material is proposed whose Fresnel surface exhibits 16 singular points. Because, for every linear material, the dispersion equation is quartic, we conclude that 16 is the maximum number of isolated singularities. The identity of Fresnel and Kummer surfaces, which holds true for media with a certain symmetry (zero skewon piece), provides an elegant interpretation of the results. We describe a metamaterial realization for our linear medium with 16 singular points. It is found that an appropriate combination of metal bars, split-ring resonators, and magnetized particles can generate the correct permittivity, permeability, and magnetoelectric moduli. Lastly, we discuss the arrangement of the singularities in terms of Kummers (16,6)-configuration of points and planes. An investigation parallel to ours, but in linear elasticity, is suggested for future research.

Decomposable Medium Conditions in Four-Dimensional Representation

The well-known TE/TM decomposition of time-harmonic electromagnetic fields in uniaxial anisotropic media is generalized in terms of four-dimensional differential-form formalism by requiring that the field two-form satisfies an orthogonality condition with respect to two given bivectors. Conditions for the electromagnetic medium in which such a decomposition is possible are derived and found to define three subclasses of media. It is shown that the previously known classes of generalized Q-media and generalized P-media are particular cases of the proposed decomposable media (DCM) associated to a quadratic equation for the medium dyadic. As a novel solution, another class of special decomposable media (SDCM) is defined by a linear dyadic equation. The paper further discusses the properties of medium dyadics and plane-wave propagation in all the identified cases of DCM and SDCM.

Comment on "Correct definition of the Poynting vector in electrically and magnetically polarizable medium reveals that negative refraction is impossible.".

Physically valid electromagnetic continuity equations can be generated from either the usual form of the Poynting vector E x H or the alternative E x B form. However, the continuity equations are not identical, which means that quantities following from E x H cannot always be compared directly to those from E x B. In particular, the work done on the bound current densities are attributed differently in the two representations.We also comment on the negative refraction condition used.

Decomposition of Electromagnetic Q and P Media

Two previously studied classes of electromagnetic media, labeled as those of Q media and P media, are decomposed according to the natural decomposition introduced by Hehl and Obukhov. Six special cases based on either non-existence or sole existence of the three Hehl-Obukhov components, are defined for both medium classes.

Electromagnetic wave propagation in metamaterials: A visual guide to Fresnel-Kummer surfaces and their singular points

The propagation of light through bianisotropic materials is studied in the geometrical optics approximation. For that purpose, we use the quartic general dispersion equation specified by the Tamm-Rubilar tensor, which is cubic in the electromagnetic response tensor of the medium. A collection of different and remarkable Fresnel (wave) surfaces is gathered, and unified via the projective geometry of Kummer surfaces.

Plane wave propagation in extreme magnetoelectric (EME) medium

EME medium is defined as one with zero permittivity and inverse permeability. Dispersion equations for the plane wave are shown to be cubic and homogeneous or identically satisfied by the wave vector. Various special cases and reflection from a uniaxial EME medium interface are considered.