Mueller matrix

The usefulness of the Mueller formulation becomes apparent when we realize that Mueller matrices give us a simple means of determining the state of polarization of a beam transmitted by an optical element for an arbitrarily polarized incident beam. Moreover, if a series of optical elements is interposed in a beam, the combined effect of all these elements may be determined by merely multif ying their associated Mueller matrices. As an example, let us consider how a circular polarizer can be constructed by superposing a linear polarizer and a hnear retarder. The beam is first incident on a linear polarizer with horizontal transmission axis ( = 0°), the Mueller matrix for which is obtained from (2.87) ... 

Thus, if unpolarized light or, indeed, light of arbitrary polarization is incident on the optical system described by the Mueller matrix (2.92), the transmitted light will be 100% right-circularly polarized. Note that matrix multiplication is not commutative the order of elements in a train must be properly taken into account. Further details about Mueller matrices and experimental means for realizing polarizers, retarders, and other optical elements are found in the excellent book by Shurcliff (1962). 

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 ... 

The effect of several elements through which a beam may pass is described by the matrix product of the Mueller matrices M corresponding to each of the optical elements E. The matrix product is build up from right to left in the sequence in which the beam enters each of these elements, 3, 2, . ... 

Similar approaches can be applied to each type of polarization devices. A generalised representation of the Mueller matrices can be created separately for polarizers and retarders which acquires then a very compact form from which Mueller matrices for all possible types of polarizers and retarders can be derived. Since the matter exceeds the scope of the present consideration only a few Mueller matrices will be listed while for more details the reader could again refer to Shurcliff (1962). 

Table 3.2-2 Classification of ideal optical devices and their corresponding Mueller matrices...

The same matrix product is obtained if the multiplication MpMs is performed instead of MsMp. That means the matrices Mp and Ms commute, and since one-to-one correspondence exists between the Mueller matrices and the optical elements which they represent, the mathematical commutation signifies that the places of the polarizer and the sample could be interchanged in the optical train. In other words the optical schemes in Fig. 3.2-3 turn out to be equivalent. 

Table 3.2-3 The Mueller matrices for samples possessing only a single polarization property...

These two examples give some idea about the use of the Stokes-Mueller method and show its effectiveness. All Mueller matrices for optical polarization components used in the present chapter have been taken at exact positions such as 0°, 90°, 45°, etc. in order to avoid complicated expressions which might obscure the principle of the method. Of course, any arbitrary position of the optical elements could be considered by the Stokes-Mueller method. In such cases the rotated Mueller matrix M 6) is obtained as a result of the transformation. 

For external incidence, the Mueller matrices of the reflected and refracted rays (denoted by subscripts r and t) can be obtained from... 

Since the Mueller matrices in Eq. (7) interrelate flux densities that are not conserved, in practical ray tracing, energy conservation is established by renormalizing the refraction coefficients in T so that... 

Abstract. We describe how Mueller matrices from polarization experiments are obtained in practice. A generalized method to determine Mueller matrices and anisotropy parameters from polarimetiy apparatus is given. We then devote a section to leaf light scatter. 

The Mueller matrix is a 4x4 matrix with real elements. Given the directions of input and output light and wavelength, the Mueller matrix contains all polarization properties of an object parameters of depolarization, and amplitude and phase anisotropy. Additional information about the Mueller matrices can be found elsewhere [7-11]. The main problem, which will be addressed in this section, is the following suppose we have a real 4x4 matrix. What conditions must be satisfied in order for it to be a Mueller matrix A... 

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... 

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