Stokes parameter

A complete description of the polarization state can be given by the set of Stokes parameters... 

In situ polarimetry could be performed using an eight axis SXR polarimeter [87] with multilayer optics mounted immediately behind the experimental chamber. Full determinations of the four Stokes parameters are time consuming. 

While a proper solution of the radiative transfer problem in media at B Bqed — 4.4 x 1013 G has necessarily to wait for a description in terms of the Stokes parameters, the search for the proton cyclotron feature in the spectra of AXPs and SGRs begun. Up to now no evidence for the proton line has been found in the thermal components of SGRs and AXPs, although these... 

We have now done enough thought experiments to determine the Stokes parameters I, Q,U,V ... 

The Stokes parameters are related to the ellipsometric parameters as follows ... 

Thus, the Stokes parameters are equivalent to the ellipsometric parameters although less easily visualized, they are operationally defined in terms of measurable quantities (irradiances). Additional advantages of the Stokes parameters will become evident as we proceed. Note that Q and U depend on the choice of horizontal and vertical directions. If the basis vectors 0 ( and , ... 

It is fairly obvious from either (2.81) or (2.83) that there are three quantities associated with the Stokes parameters that are invariant under rotation of the reference directions I, Q2 + U2, and V. In addition, the Stokes parameters are not all independent ... 

A few representatives sets of Stokes parameters, written as column vectors, are shown in Table 2.2 the irradiance / is normalized to unity. 

The Stokes parameters of a quasi-monochromatic beam are given by... 

If two or more quasi-monochromatic beams propagating in the same direction are superposed incoherently, that is to say, there is no fixed relationship among the phases of the separate beams, the total irradiance is merely the sum of the individual beam irradiances. Because the definition of the Stokes parameters involves only irradiances, it follows that the Stokes parameters of a collection of incoherent sources are additive. 

In the derivations above of the Stokes parameters we began with monochromatic light and then extended our results to the more general case of quasi-monochromatic light. However, the operational definition of the Stokes parameters in terms of a set of elementary experiments involving a detector and various polarizers, as opposed to the formal mathematical definitions (2.80) and (2.84), is independent of any assumed properties of the beam. Unless otherwise stated, we shall assume that all beams of interest are quasi-monochromatic, which includes as a special case monochromatic light. 

We may represent a beam of arbitrary polarization, including partially polarized light, by a column vector, the Stokes vector, the four elements of which are the Stokes parameters. In general, the state of polarization of a beam is changed on interaction with an optical element (e.g., polarizer, retarder, reflector, scatterer). Thus, it is possible to represent such optical elements by a 4 X 4 matrix (Mueller, 1948). The Mueller matrix describes the relation between incident and transmitted Stokes vectors by incident is meant before interaction with the optical element, and by transmitted is meant after interaction. As an example, consider the Mueller matrix for an ideal linear polarizer. Such a polarizer transmits, without change of amplitude, only electric field components parallel to a particular axis called the transmission axis. Electric field components in other directions are completely removed from the transmitted beam by some means which we need not explicitly consider. The relation between incident field components (E, E i) and field components ( l, E () transmitted by the polarizer is... 

McMaster (1954) takes a quantum-mechanical approach to the Stokes parameters and polarized light. Two books are devoted entirely to polarization Shurcliff (1962) and Clarke and Grainger (1971). And a splendid collection of papers on many aspects of polarized light has been edited by Gehrels (1974a). Another collection worth consulting is that compiled by Swindell (1975) it contains several of the classical papers on polarization. 

By interposing various polarizers between particle and detector and recording the resulting irradiances in a manner identical to that discussed for a plane wave in Section 2.11, we obtain the Stokes parameters of the light scattered by... 

We again omit the multiplicative factor k/2cojii. The relation between incident and scattered Stokes parameters follows from the amplitude scattering matrix (3.12) ... 

If unpolarized light of irradiance /, is incident on one or more particles, the Stokes parameters of the scattered light are... 

Circular dichroism and optical rotation for particulate media may be operationally defined in terms of the Stokes parameters (2.80), which in the circular polarization representation are written... 

The azimuth y and ellipticity tan rj of the vibration ellipse for an arbitrary beam can be determined from the Stokes parameters by (2.82). 

Although we have chosen to emphasize the more common Stokes parameters (/, Q,U,V), the system (/((, I , 0, V) is more suited to measurements in which linear polarizers are interposed in the incident and scattered beams. In the latter system, the Stokes parameters are defined by... 

The simplest, and probably most obvious, way to measure scattering matrix elements is with a conventional nephelometer (Fig. 13.5) and various optical elements fore and aft of the scattering medium. Recall that we introduced Stokes parameters in Section 2.11 by way of a series of conceptual measurements of differences between irradiances with different polarizers in the beam. Although we did not specify the origin of the beam, it could be light scattered in any direction. Combinations of scattering matrix elements can therefore be extracted from these kinds of measurements. There are now, however, two beams—incident and scattered—and many possible pairs of optical elements these are discussed below. 

The incident beam, which we may take to be unpolarized, encounters three optical elements, each of which is represented by a matrix we recall from Section 2.11 that matrices for polarizers and retarders depend on their orientation. The Stokes parameters of the light emerging successively from the polarizer, the modulator, and the scattering sample are (from right to left)... 

The detector response is proportional to the first Stokes parameter fC(Sn + Sncos8 +, S14sinS),... 

Walker, M. J., 1954. Matrix calculus and the Stokes parameters of polarized radiation, Am. J. Phys., 22, 170-174. 

U. Eano. A stokes-parameter technique for the treatment of polarization in quantum mechanics. Phys. Rev., 93(1) 121-123 (1954). 

From the foregoing, U(l) electrodynamics was never a complete theory, although it is rigidly adhered to in the received view. It has been argued already that the Maxwell-Heaviside theory is a U(l) Yang-Mills gauge theory that discards the basic commutator A(1) x A(2). However, this commutator appears in the fundamental definition of circular polarity in the Maxwell-Heaviside theory through the third Stokes parameter... 

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