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Physical Review A

Degree of polarization of type-II unpolarized light

Alfredo Luis

Published May 7, 2007• 20 citations
Publisher

Abstract

We address a quantitative determination of the degree of polarization of type-II unpolarized light via the computation of the distance between the polarization distribution and the uniform distribution associated with fully unpolarized light (i.e., type-I unpolarized light or natural light). We determine the maximum degree of polarization for type-II unpolarized light and the states reaching it. We show that the degree of polarization can be arbitrarily large, approaching complete polarization for increasing mean photon numbers.

DOI
10.1103/physreva.75.053806

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We address a quantitative determination of the degree of polarization of type-II unpolarized light via the computation of the distance between the polarization distribution and the uniform distribution associated with fully unpolarized light (i.e., type-I unpolarized light or natural light). We determine the maximum degree of polarization for type-II unpolarized light and the states reaching it. We show that the degree of polarization can be arbitrarily large, approaching complete polarization for increasing mean photon numbers.
Quantum optics entails polarization properties that cannot be fully described by the classical Stokes parameters. In this work, we characterize the polarization of classical as well as quantum fields by means of a probability distribution on the Poincar\'e sphere. This serves to define the degree of polarization of a field state as the distance between the corresponding polarization distribution and the uniform distribution representing unpolarized light. We apply this definition to relevant quantum field states such as SU(2) coherent, squeezed, number, and phase states.
We introduce a probability distribution for polarization of three-dimensional quantum light fields as a marginal of the quadrature QQQ function for a three-mode field by removing the variables irrelevant for polarization (total intensity and global phase). The probability distribution turns out to be determined by projection on SU(3) coherent states. We introduce a degree of polarization as the distance between the polarization distribution and the uniform distribution associated with completely unpolarized light. We study the relation between two- and three-dimensional polarization by considering field states with a component in the vacuum state. We apply this formalism to some relevant field states.
We give a general definition of unpolarized light that rests on invariance properties rather than specific values of second-order field correlations or, equivalently, the Stokes parameters. It turns out that to require invariance with respect to rotation around the propagation axis and symmetry with respect to left- and right-handed circular polarization is not sufficient to fully characterize completely unpolarized light, such as natural light. Actually, a third requirement demanding invariance with respect to phase retardation is needed. Depending on whether it is satisfied or not, we speak of type I or type II unpolarized light, and we specify the general form of both classical distribution functions and density operators consistent with the respective conditions. \textcopyright{} 1996 The American Physical Society.
The degree of polarization of a quantum field can be defined as its distance to an appropriate set of states. When we take unpolarized states as this reference set, the states optimizing this degree for a fixed average number of photons N‾\overline{N}N present a fairly symmetric, parabolic photon statistic, with a variance scaling as N‾2{\overline{N}}^{2}N2. Although no standard optical process yields such a statistic, we show that, to an excellent approximation, a highly squeezed vacuum can be taken as maximally polarized. We also consider the distance of a field to the set of its SU(2) transformed, finding that certain linear superpositions of SU(2) coherent states make this degree to be unity.
It is well established that unpolarized light is invariant with respect to any SU(2) polarization transformation. This requirement fully characterizes the set of density matrices representing unpolarized states. We introduce the degree of polarization of a quantum state as its distance to the set of unpolarized states. We use two different candidates of distance, namely the Hilbert-Schmidt and the Bures metric, showing that they induce fundamentally different degrees of polarization. We apply these notions to relevant field states and we demonstrate that they avoid some of the problems arising with the classical definition.