Eigenanalysis of dichroic, birefringent, and degenerate polarization elements: a Jones-calculus study

2015

In a pure operatorial (nonmatrix) Pauli algebraic approach, this Letter shows that the Poincaré vector of the light transmitted by a dichroic device can be expressed as function of the Poincaré vectors of the incoming light and of the device by a composition law of the same kind as the composition law of the noncolinear relativistic velocities. This is, in fact, a general law of composition for three-dimensional (3D) vectors remaining in the Poincaré ball (where they have a group-like structure). The differences between this problem and that of the composition law of two dichroic devices are pointed out and justified.The quasi-relativistic approach in polarization theory has a long history since, in the 1970s, Richard Barakat [1] noticed that the polarization Stokes vector is a Minkowskian quadrivector and Hiroshi Takenaka [2] noticed that the transformations of the polarization states given by the deterministic [6,22] polarization devices are Lorentz transformations.A special line of these researches was opened two decades after the pioneering works of Barakat and Takenaka, by J.M. Vigoureux, who showed that the reflection coefficient of any stratified planar structure can be obtained using a complex generalization of Einstein's composition law for parallel relativistic velocities [8,10,11]. Recently, Lages et al. [21] extended this quasi-relativistic result, showing that the composition law for polarizers is analogous with the composition law for noncolinear relativistic velocities. They also have determined the Wigner angle that corresponds to a sequence of two polarizers, as a natural consequence of the noncomutativity of this composition law. This Letter presents, using a Pauli algebraic approach, a similar result, referring this time to the action of a dichroic device (e.g., partial polarizer) on the partially polarized light: The Poincaré vector of the light transmitted by a dichroic device can be expressed as a function of the Poincaré vector of the incoming light and the Poincaré vector of the device by a composition rule analogous with Einstein's composition rule for nonparallel relativistic velocities. No Wigner rotation occurs here.In obtaining this result, I adopted a pure operatorial (nonmatrix) Pauli algebraic approach [23], because it is the most compact, straightforwardly connected with the intuitive geometric representation of the Poincaré sphere, and best adapted to the internal symmetry of the problem.In such an approach, the density operator of a partially polarized (mixed) state of light takes the form [23,24]:where I and p are the intensity and the degree of polarization of the light SOP (state of polarization); σσ 1 ; σ 2 ; σ 3 is the Pauli vectorial operator (σ i -the Pauli matrices); σ 0 -the unit operator in the C 2 space (the unit 2 × 2 matrix); and n is the Pauli axis of the operator, the unit vector defining the direction of the respective SOP in the Poincaré sphere Σ 1 2 representation of the SOPs. Then pn is the vector defining the position of this mixed SOP in the Poincaré ball, ...

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confidence: 99%

On a quasi-relativistic formula in polarization theory

2015

“…Since then the problem of nonorthogonality in polarization theory has grown in interest [7][8][9][10], and even some strange nonorthogonal polarization devices (black polarization sandwiches-square roots of zero [7]; white polarization sandwiches-nontrivial square roots of unity [10]) have received attention and have been rigorously analyzed.…”

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confidence: 99%

“…Gil and Bernabeu [13] introduced the polar decomposition theorem in handling the Mueller matrices. Since then, the polar as well as the singular value decomposition of the Jones and Mueller matrices has become a common method of analyzing the properties of various polarization devices and anisotropic media in general [8,9,[14][15][16][17][18][19].…”

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confidence: 99%

Nonorthogonal polarizers: a polar analysis

2014

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.

“…This situation is general in polarization optics. The two eigenstates corresponding to an arbitrary direction of propagation in some crystals are generally nonorthogonal [2][3][4].…”

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confidence: 99%

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

2014

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.