Black polarization sandwiches are square roots of zero

The singularities of light: intensity, phase, polarisation

In modern optics, light can be described at different levels: as rays, as scalar waves, as vector fields, and as quantum fields. In the first three levels, there are singularities-characteristic features, useful in interpreting phenomena at that level. In geometrical optics, the singularities are ray caustics; in scalar wave optics, they are phase singularities (=wave dislocations= wave vortices = nodal manifolds); in vector waves, they are singularities where the polarisation of light is purely linear or purely circular. The singularities at each level are dissolved at the next level. Similar singularities occur in all waves, not just light.

How inhomogeneous can an inhomogeneous Jones matrix be?

We determine the interval of the inhomogeneity parameter of a Jones matrix to get physically realizable optical systems satisfying the passivity condition. It is found that the inhomogeneity parameter depends on the inner product of the eigenvectors of the Jones matrix, but its maximum value depends exclusively on its eigenvalues.

Geometric phase morphology of Jones matrices

We demonstrate an innovative technique based on the Pancharatnam-Berry phase that can be used to determine whether an optical system characterized by a Jones matrix is homogeneous or inhomogeneous, containing orthogonal or nonorthogonal eigenpolarizations, respectively. Homogeneous systems have a symmetric geometric phase morphology showing line dislocations and sets of polarization states with an equal geometric phase. In contrast, the morphology of inhomogeneous systems exhibits phase singularities, where the Pancharatnam-Berry phase is undetermined. The results show an alternative to extract polarization properties such as diattenuation and retardance, and can be used to study the transformation of space-variant polarized beams.

Nonnormal operators in physics, a singular-vectors approach: illustration in polarization optics

The singular-vectors analysis of a general nonnormal operator defined on a finite-dimensional complex vector space is given in the frame of a pure operatorial ("nonmatrix," "coordinate-free") approach, performed in a Dirac language. The general results are applied in the field of polarization optics, where the nonnormal operators are widespread as operators of various polarization devices. Two nonnormal polarization devices representative for the class of nonnormal and even pathological operators-the standard two-layer elliptical ideal polarizer (singular operator) and the three-layer ambidextrous ideal polarizer (singular and defective operator)-are analyzed in detail. It is pointed out that the unitary polar component of the operator exists and preserves, in such pathological case too, its role of converting the input singular basis of the operator in its output singular basis. It is shown that for any nonnormal ideal polarizer a complementary one exists, so that the tandem of their operators uniquely determines their (common) unitary polar component.

Partial isometries, unitary operators, and complementary operators in polarization optics

We show that for nonnormal singular operators corresponding to the nonorthogonal polarizers, the unitary polar component is constituted by two partial isometries, one of them "active" and the other "hidden" or "mute." For each such operator there exists a complementary one, corresponding also to a nonorthogonal polarizer, which has the same unitary polar component and whose partial isometries reverse their roles.

Nonorthogonal polarizers: a polar analysis

An analysis of the operators of some widespread nonorthogonal polarizers is performed on the basis of the polar factorization theorem, in pure operatorial (nonmatrix) Dirac algebraic language. The role of the unitary polar component as a converter of the two sets of singular eigenvectors of the operator, one in the other, is emphasized in each case; this role is maintained for the singular operators corresponding to these special nonorthogonal polarizers.

Disorientated optical polarization devices

By using a pure operatorial (nonmatrix) approach to some of the variable composite polarization devices, we show that for some values of the variable internal parameters, these devices lose their anisotropy. Their principal axes, or more generally their eigenvectors, become undetermined, i.e., in this sense, the anisotropic devices get "disorientated."

Non-Hermitian polarizers: a biorthogonal analysis

The non-Hermitian operators of the ideal nonorthogonal multilayer optical polarizers are spectrally analyzed in the framework of skew-angular biorthonormal vector bases. It is shown that these polarizers correspond to skew projectors and their operators are generated by skew projectors, exactly as the canonical ideal polarizers correspond to Hermitian projectors. Thus the common feature of all the polarizers (Hermitian and non-Hermitian) is that their "nuclei" are (orthogonal or skew) projectors--the generating projectors. It is shown that if these nonorthogonal polarizers are looked upon as variable devices, two kinds of degeneracy may occur for suitable values of the inner parameter of the device: The corresponding operators may become normal (more precisely, Hermitian) or, on the contrary, very pathological--defective and singular. In the first case their eigenvectors and biorthogonal conjugate eigenvectors collapse into a unique pair of eigenvectors; in the second case their eigenvectors (as well as their biorthogonal conjugates) collapse into a single vector.

The optical singularities of bianisotropic crystals

The electric and magnetic polarization states for plane waves in arbitrary linear crystals, in which each of D and B is coupled to both of E and H , can be characterized by their typical singularities in direction space: degeneracies, where two refractive index eigenvalues coincide; C e and C m points, where the electric or magnetic field is circularly polarized; and L e and L m lines, where either field is linearly polarized. The well-known 4×4 matrix formalism, expressed in terms of the stereographic projection of directions, enables extensive numerical and visual exploration of the singularities in the general case (which involves 65 crystal parameters), incorporating bianisotropy, natural and Faraday optical activity, and absorption, as well as special cases where one or more effect is absent. For crystals whose anisotropy is weak but which are otherwise general, an unusual perturbation theory leads to a powerful 2×2 formalism capturing all the essential singularity phenomena, including the principal feature of the general case, namely the separation between the electric and magnetic singularities.