Stokes parameters polarization measurement

Figure 4. Average values of (a) the percentage polarization p and (b) the Stokes parameter Q measured at the center of the Jovian disk in the 727 nm methane absorption band. Dots correspond to measurements on 26/27 June and 10/11 July 1986 (phase angle a 11 ) circles are for data on October 8/9, 13/14, 14/15 (a 6. 9) crosses are for October 1/2, 2/3 (a 4. 6). The wavelength resolution is 2.5nm. The error of measurements remains constant within the band, and is indicated by a vertical bar for each phase angle [27].

The polarization characteristics of the incident field can also be described by the coherency and Stokes vectors. Although the ellipsometric parameters completely specify the polarization state of a monochromatic wave, they are difEcult to measure directly (with the exception of the intensity Eq). In contrast, the Stokes parameters are measurable quantities and are of greater usefulness in scattering problems. The coherency vector is defined as... 

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. 

Figure 10. Time-resolved measurements of a very large polarization fluctuation, where the size of the fluctuation is about half (in fact, 45%) of that of a complete polarization switch [84],, V, , v2 and Si are the normalized Stokes parameters representing the polarization state on the Poincare sphere [84],...

The Q values in these tensors represent the Stokes parameters for the spin sensitivity of the detector in its x", y", z" frame. For example, Qx- describes the detector efficiency for measuring spin projections along +x" and —x", respectively (for the definition of the spin polarization vector see Section 9.2.1). 

These parameters have very simple physical meanings. The parameter. vq measures the contribution of all three components into the energy density. The parameters 51,2,5,6,7,8 give the phase information about the phase differences between the circular and linear polarized components. The parameter s3 gives the preponderance of negative helicity over positive helicity and the parameter 54 gives the preponderance of transversal circular polarization over linear (radial) polarization. Unlike (131), the multipole Stokes parameters (132) describe the local properties of polarization due to the position dependence of the mode functions (18). It is clear that, at far distances (kr 1) when 0 is negligibly small in comparison with E , the set (132) formally coincides with (131) because... 

Most detectors of electromagnetic radiation, especially those in the visible and infrared spectral range, are insensitive to the polarization state of the beam impinging on the detector surface and can measure only the first Stokes parameter of the beam, viz., the intensity. Therefore, to measure the entire Stokes vector of the beam, one has to insert between the source of light and the detector one or several optical elements that modify the first Stokes parameter of the radiation reaching the detector in such a way that it contains information about the second, third, and fourth Stokes parameters of the original beam. This is usually done with so called polarizers and retarders. 

Figure 11.4 Measurement of the Stokes parameters with a retarder and an ideal perfect linear polarizer rotated with respect to the laboratory reference frame.

When a quarter-wave rotating phase plate is used in a polarization modulator, it is possible to measure parameters of linear and eireular polarization simultaneously, but the aeeuraey of determining Q and U parameters is half that of parameter V. When a PSP with a phase shift of A = 126.52° is used, the accuracy of measurements of all three Stokes parameters is idenrieal. When 92= 0, the formula for right modulation beeomes... 

The above relations show that the Stokes parameters carry information about the amplitudes and the phase difference, and are operationally defined in terms of measurable quantities (intensities). For a linearly polarized plane wave, X = 0 and 14 = 0, while for a circularly polarized plane wave, x = 7t/4 and Qe = Ue = 0. Thus, the Stokes vector of a linearly polarized wave of unit amplitude is given by Jg = [l,cos2apoi, -sin2o poi,0], while the Stokes vector of a circularly polarized wave of unit amplitude is /g = [1,0,0, Tl]" -The Stokes parameters of a monochromatic plane wave are not independent since... 

Reviews of concentration polarization have been reported (14,38,39). Because solute wall concentration may not be experimentally measurable, models relating solute and solvent fluxes to hydrodynamic parameters are needed for system design. The Navier-Stokes diffusion—convection equation has been numerically solved to calculate wall concentration, and thus the water flux and permeate quaUty (40). 

The second and third of these measurements can be made with a precisely oriented linear polarizer, while the fourth assumes the additional use of a wave plate. Stokes vector components can be expressed by means of the parameters of the polarization ellipse ... 

In a QELS experiment, a monochromatic beam of light from a laser is focused on to a dilute suspension of particles and the scattering intensity is measured at some angle 0 by a detector. The phase and the polarization of the scattered light depend on the position and orientation of each scatterer. Because molecules or particles in solution are in constant Brownian motion, scattered light will result that is spectrally broadened by the Doppler effect. The key parameter determined by QELS is the diffusion coefficient, D, or particle di sivity which can be related to particle diameter, d, via the Stokes-Einstein equation ... 

Another way to identify the dominant contributions of the order parameter for a particular line, is to experimentally determine the reflecting Mueller matrix of the Bragg-reflecting blue phase. The polarizations of light incident and scattered from a surface can be described by four-component Stokes vectors. The matrix which transforms one to the other (and describes the blue phase) is a 4 x 4 Mueller matrix. By analyzing the reflected Stokes vectors for a range of incident Stokes vectors, the complete Mueller matrix can be determined. The result of such measurements [61], [62] was that the e 2 coefficient completely dominates the other coefficients, which rules out certain space groups. 

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