Stokes parameters, measurement

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 , ... 

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. 

At Pennsylvania s Flower and Cook Observatory, instrumentation has been developed to measure simultaneously the four Stokes parameters of the filtered radiation field from a celestial source. The instrumental Q/U/V-parameters have been found to be very small and well-behaved. 

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],...

Figure 1.18 Example of the determination of the Stokes parameter S, and tilt angle X measured at 40 eV photon energy. It is based on an angular distribution measurement of ls-photoelectrons in helium for which equ. (1.55a) holds with P = 2. The arrows point to the measured values (with error bars, some of which are too small to be visible) the solid curve is a fit of the experimental data to this equation from which = 0.819(3) and...

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). 

Here x is a phenomenological parameter measuring the chirality and / is a size scale factor. Since here the Reynolds number is small ( 10 s), the Stokes equation can be used to get r = DS2. where D is the hydrodynamic drag coefficient and 2 is the rotational speed. The drag coefficient for a cylindrical object rotating about its axis with cross-sectional radius r and length L is D = 4ztT)r2L, where tj is the viscosity of the medium [19]. Therefore, D /3 and the rotational speed 2 of the rotor will scale as... 

For P — transitions, however, the excited atoms are completely described by three parameters as discussed in subsection 8.1.4. Thus in this case the four Stokes parameters are not independent quantities and only three measurements are required for a complete determination of the excited state, e.g. a, I and lt]2- The intensity of light radiated in the transition i) /) and emitted into a solid angle dQ at (0,) measured by an ideal detector sensitive only to polarisation in the time interval t, t -I- dt, is given by... 

A measurement of circular polarisation also gives the sign of x and (Lj ), whereas the measurement of the angular correlation alone (expression (8.35a)) does not. Similarly at 0 = 0 = 90° the Stokes parameters for linear polarisation are given by... 

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... 

Figure 19. The experimental setup for measurement of the Stokes parameters.

Where /, denotes the intensity measured by the detector d,. From these relations, the classical Stokes parameters (130) can be found as... 

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.

This formula suggests a simple way to determine the Stokes parameters of the original beam by measuring the intensity of the resulting beam using four... 

Other methods for measuring the Stokes parameters and practical aspects of polarimetry are discussed in detail in [14-17]. 

Measurements of four Stokes parameters over the coma in comet Halley by Dollfiis and Suchail [6]. There is evolution of dust physical properties with distance from the nucleus. 

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].

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 same idea was used in the development of a spectropolarimeter for simultaneous measurement of all four Stokes parameters [3-5]. A 127-degree SPSP is rotated with frequency 61 Hz and modulated intensity (4) was divided into 8 equal intervals with the use of special electronic keys ... 

There are few spectropolarimeters for simultaneous measurement of all four Stokes parameters of solar system bodies. One such device was developed at the Main Astronomical Observatory of National Academy of Sciences of Ukraine. The scheme of its optical-mechanical constraction is shown in Fig. 1. 

Stokes, C.L., Lauffenhurger, D.A. and Williams, S.K. (1991). Migration of individual microvessel endothelial cells stochastic model and parameter measurement. J. Cell Sci. 99, 419- 0. 

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