The Jones and Mueller Matrices

The polarization properties of light can be represented by the Jones or Stokes vectors, A or S, respectively. The latter prescription has the advantages of describing partial polarization and contains directly observable quantities. When light is transmitted through a polarizing element with an incident electric vector A0 or S0, the light will exit with 

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], 

For perfectly polarized light, the elements are constrained by the equality of equation (1.60). For that reason, only seven of the elements of the Mueller matrix will be linearly independent. In these circumstances, a sample can be properly characterized by determining a limited set of the components of M. For more complex, depolarizing systems, it may be necessary to determine all sixteen Mueller matrix components. 

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Therefore, the relationship between the Jones and Mueller matrices in the rotated frame, J and M, are related to their unrotated forms by... 

The majority of Jones matrices for transmission polarizing elements have been presented in this chapter. As mentioned earlier, it is generally more convenient to use Mueller matrices and not Jones matrices when analyzing cascades of optical elements making up a particular experiment. For the purpose of such calculations, a list of Jones and Mueller matrices for most of the elements encountered in practice can be found compiled in Appendix I. 

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... 

The following list of Jones and Mueller matrices has been compiled for most optical elements encountered in optical instruments where polarization effects must be taken into account. In writing these matrices, the following notation has been used a - 2nn d/X a" = 2nn"d/ k, where n = n -iti" is the isotropic refractive index, d is the sample thickness ... 

Matrix representations are commonly used to provide a mathematical description of optical measurements in a compact and convenient manner. The Jones and the Mueller matrix representations are often used to describe the change in polarization state of the light upon interaction with a sample. In these representations, Jones and Stokes vectors represent the different polarization states of light, and matrices are used to represent elements which change those states, as optical elements or the sample being analyzed. Figure 2.14 illustrates a simple ellipsometer and shows the Jones matrix representations for each component in a general case. 

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 coherency matrix method permits the so-called dominant type of deterministic polarization transformation i.e., the corresponding deterministic part, Mueller-Jones matrix, of the initial Mueller matrix. A Jones matrix J is a 2x2 complex valued matrix containing generally eight independent parameters from the real and imaginary parts for each the four matrix elements, or seven parameters if the absolute (isotropic) phase which is not of interest for polarizations is excluded. Every Jones matrix can be transformed into an equivalent Mueller matrix but the converse assertion is not necessarily true. Between Jones J and Mueller Mj matrices that describe deterministic objects there exist a one-to-one correspondence ... 

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