Mueller vector

The equation (4.5) is the most general equation for Mueller matrix measurement. This equation can be rewritten in a vector-vector product form. For that, the Mueller matrix to be measured is rewritten as a 16x1 Mueller vector M Then, the... 

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... 

DDT, discovered by Dr. Mueller in Switzerland, and used for insect vector control during World War II, quickly found a place in forestry, as well as agriculture. The material proved highly effective in the control of such insects as the spruce budworm, tussock moth, hemlock looper, and many others. It was widely used in the Northeast for control of the introduced Gypsy moth during these early years. The low toxicity of DDT to mammals made it to appear to be an excellent insecticide for forestry use. It was only after subsequent studies revealed the impact on other species that reservations about its use was raised. 

The connection between the Stokes and Jones vectors, given by equation (1.59) can be used to relate the sixteen-component Mueller matrix to the four-component Jones matrix. Combining equations (2.1), (2.2), and (1.59), we have, using a notation similar to that developed in Azzam and Bashara [5],... 

Most optical experiments consist of a cascade of optical elements, and each will be represented by a Jones or Mueller matrix. Such a series is shown in Figure 2.2. The polarization vector generated by a train of n elements is... 

Since the Maxwell equations involve the components of the Jones vector, it is normally easier to derive the Jones matrix, J, for complex, anisotropic media. Once J is obtained, it is generally convenient to transform it to the Mueller matrix representation for the purpose of analyzing the quantities measured in specific optical trains. This is because the components of the Stokes vector are observable, whereas the Jones vector components are not. Since it is the intensity of light that is normally required, only the first element of Sn,... 

Once a Jones or Mueller matrix of an optical element is obtained for one orthonormal basis set (ep e2, for example), the corresponding matrices for the element relative to other basis sets can be obtained using standard rotation transformation rules. The action of rotating an optical element through an angle 0 and onto a new basis set ej, e2 is pictured in Figure 2.3. In the nonrotated frame, the exiting polarization vector is ... 

The intensity of the light generated in this experiment is easily calculated using equations (2.5) and (2.6) combined with the appropriate Jones and Mueller matrices selected from Appendix I. The Jones and Stokes vectors, Aj and, exiting this cascade are... 

A rotary polarization modulator simply consists of an optical element that rotates uniformly at a frequency Q about the transmission axis of light. In practice, retardation plates and polarizers are used. In either case, the Mueller matrix of such a device is found by simply replacing the angle 6 by Q.t in the equations listed in Appendix I. Typical PSGs based on rotary modulators and the associated Stokes vectors, Sp G, that are produced are listed in table 8.2. 

Using these vectors, the intensity signal can be readily calculated for a sample with an arbitrary Mueller matrix and the result is of the same form as equation (8.46). The coefficients a- and bi will again be linear combinations of the sample s Mueller matrix... 

The appearance of Stokes publication, in which the concept of a vector representation of the beam description parameters was introduced, renders back to 1852. It, however, has remained for many years unnoticed (Shurcliff, 1962). The need for a general systematic and effective procedure for solving the exponentially growing quantity of optical problems has led to the rediscovery of Stokes approach to polarized radiation. Mueller s contribution consists of developing a matrix calculus for evaluation of the four elements of the Stokes vector which are a set of quantities describing the intensity and polarization of a light beam. 

The basic concept of the Stokes-Mueller calculus is that the transformation of the state of a beam under the action of an optical element could unequivocally be described by multiplying its Stokes vector So by a matrix M (Mueller matrix) from the left. The resulting new Stokes vector Sr represents the altered state of the beam. 

The purpose of such a device consists in changing the orientation of the polarization plane of a beam by 90°. That means the initial Stokes vector 1,1,0,0 of a horizontally polarized beam becomes 1,-1,0,0 after passing through the retarder. Retarders are most often birefringent crystals of definite thickness. If the fast and slow axes of such a crystal orthogonal to each other are crossed at 45° with respect to the polarization plane, the retarder rotates the latter by 90°. The Stokes-Mueller transformation corresponding to this experiment should be ... 

F. Mueller-Plathe and D. Brown, Comput. Phys. Commun., 64, 7 (1991). Multi-Color Algorithms m Molecular Simulation Vectorization and Parallelization of Internal Forces and Constraints. 

The first organochlorine synthesized was DDT. Although it was first synthesized by Zeidler in 1874, it was not produced or used for many years. Mueller rediscovered DDT in 1939 and won the Nobel Prize for his efforts in 1948. The first major uses for DDT were vector control of typhus and malaria and control of lice and other pests during World War II. 

If the transmission axes of a polarizer are rotated relative to the laboratory coordinate system (Fig. 11.2) then its Mueller matrix with respect to the laboratory coordinate system also changes. To obtain the resulting Stokes vector with respect to the laboratory coordinate system, we need to... 

When light with arbitrary polarization propagates through or interacts with an object, its polarization changes. Thus, the effect of an object on input light is to transform the Stokes vectors describing the light. Therefore, the object can be represented by a transformation matrix t he Mueller matrix ... 

In this section we present a general theory for the measurement of polarization characteristics of light, Stokes vector, Mueller matrix, and optimization procedures for minimizing time and errors of measurement. Stokes vector for a beam of light is determined by carrying out a series of measurements for the intensity of light transmitted through a set of polarization elements. Fig. 1. shows the polarization state analyzer (PSA). 

Eqn. (4.7) describes any existence scheme for Mueller polarimeters. Below we present examples to illustrate the application of equation (4.6) in complete and incomplete Mueller polarimeters. Consider a prevailing scheme of a Mueller polarimeter in which PSG forms fom independent polarizations of input light, and PSA measures the complete Stokes vector. For this polarimeter it is convenient to rewrite (4.5) in the form ... 

The quantity 5 exp is the measured Stokes vector, exaot is the corresponding exact Stokes vector, Mjxp and Mexact are the measured and exact Mueller matrices of the object, respectively. The value of >S always can be obtained... 

When the polarization state of a light beam is represented by the Stokes vector, the effect of an optical element can be represented by the Mueller matrix M which operates on the Stokes vector, Si, of the incident light to generate the Stokes vector, So, of the outgoing light ... 

We now consider the Mueller matrix of a rotator. In the xy frame, the electric field vector is E = EyX + Ey-y. In another frame x y, which is in the same plane but the x axis makes the angle... 

We consider how the three-component Stokes vector S evolves on the Poincare sphere under the action of retardation films. The Mueller matrix of a uniform uniaxial retarder with the retardation angle F and the slow axis making the angle (p with the x axis is given by (from Equations (3.77) and (3.78))... 

We now consider the Mueller matrix of a uniformly twisted nematic (or cholesteric) liquid crystal. The problem can be simplified if we consider the Stokes vector and the Mueller matrix in the local frame x y, in which the liquid crystal director lies along the x axis. Divide the liquid crystal film into N thin slabs. The thickness of each slab is dz = h/N, where h is the thickness of the liquid crystal film. The angle between the liquid crystal director of two neighboring slabs is dxff = qdz, where q is the twisting rate. The retardation angle of a slab is liT = k Andz. If the... 

Using the definition of Stokes vector, derive the Mueller matrix given by Equation (3.70) of an optical element whose Jones matrix given by Equation (3.69). 

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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