The Mueller matrix

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 first consider the Mueller matrix of an absorber. The transmission coefficients along thex and y axes are and py, respectively  

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 

For a retarder with retardation angle F = InAnh/X and the slow axis along x axis 

The Mueller matrix of a retarder, whose slow axis makes the angle with the x axis is 

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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... 

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). 

Azzam, R. M. A., 1978. Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal, Opt. Lett., 2, 148-150. 

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. 

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,... 

The Mueller matrix can then be calculated using equation (2.3). Because the Mueller matrix represents observable intensities, these elements must be averaged over probability distributions functions that represent the system. This matrix is... 

It is required to calculate the Mueller elements by first calculating the differential elements for the slab using equation (2.3) and integrating along the path of light propagation as indicated in equation (5.51). The Mueller matrix elements are... 

The selection of a particular modulation scheme starts by determining which of the sample s Mueller matrix components need to be measured. This decision can be guided by examining the form of the Mueller matrix given in equation (1.18), which contains all... 

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. 

When the material is noncoaxial, the optical train in Figure 8.9 can still be used and provides sufficient information to solve for all four unknowns, An, An", 9, and 9". To demonstrate this capability, the analysis shall be worked out for the (P/RH)psG design. The Mueller matrix for a noncoaxial birefringent and dichroic material was developed in section 2.4.7 and is given by equation (1.18) with 8 = 8"c<- = 0 The intensities... 

Case 1 8 1, S" 1. If both the retardation and extinction are small, the Mueller matrix takes on the simple form given in equation (1.18). In this limit, the signal measured by detector D2 is... 

The constants, ai and fi., are the Fourier coefficients of the intensity and are the following combinations of the Mueller matrix components of the sample ... 

A variety of choices are available for the PSG section of this experiment. As before, the selection is based on the Mueller matrix components of the sample that are sought. A convenient arrangement that has been used for samples subject to uniaxial deformation is described in detail in reference 22 and collects the Raman scattered light in the forward direction. The PSG section of the instrument consists of a polarizer oriented at zero degrees, and a photoelastic modulator at 45°. Following the sample, the PSA section consists of a polarizer oriented at 45°. The signal measured at the photomultiplier tube was shown to have the form ... 

G3 is the apparatus constant related to the sensitivity of the spectrometer with the analyzer inserted, and A is the Mueller matrix for the analyzer. The above equation can be approximated as... 

Only reasoning of technical nature could lead to a decision whether scheme a or b should be preferred in an experiment. Exactly the same result is obtained if one takes from Table 3.2-2 the Mueller matrix for the perpendicular polarizer. 

R(6) is an orthogonal rotation matrix. The symbol tilde ( ) means here transposed and M(0) is the Mueller matrix of the given optical element corresponding to 0°. If e.g. a linear polarizer is rotated at an angle 6 the above transformation should look like as follows ... 

Hence the Mueller matrix of the rotated polarizer computed with respect to the laboratory coordinate system is given by... 

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... 

Thus, we have the generalized Jones matrix. Its four elements are a function of six parameters P, R, S, a, which have physical meanings. In our applications, the values for the anisotropy parameters P, S, a, (j) were calculated numerically. In our applications we calculated the anisotropy parameters numerically from the deterministic Mueller matrix. In section 4 measurement of the Mueller matrix is described. 

Thus, the PSG forms either a sequence of definite polarizations or periodically changing polarization of input light. The output polarizations are analyzed by N configurations of polarization elements in the PSA. From the set of such measurements, a system of linear equations is developed that can be solved for the Mueller matrix elements. This yields ... 

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... 

As in (4.3), the existence, rank, and nniqneness of inverse matrix plays a key role in the solntion of eqnation (4.6). If contains sixteen independent colnnms, then all sixteen elements of the Mueller matrix can be determined. When N = 16, then is unique. If A > 16, then is overdetermined. The optimal least squares estimation for m j is given by the pseudo-inverse of ... 

If A < 16, then the optimal matrix inverse is the pseudo-inverse as well. However, in this case only fifteen or less of the Mueller matrix elements can be measured. The corresponding measurement apparatus is called an incom plete Mneller polarimeter. It is worth pointing ont the following advantages of Mneller polarimetry ... 

In practice, all 16 elements very often may not be independent. Some may be zero and some may be identical to others depending on symmetry and other physical properties of the studied object [7,8], A typical example is the deterministic class of objects which have less than seven independent elements. Hence, in measuring all 16 elements of the deterministic Mueller matrix, more than 50% are uni nformative measurements. This problem is relevant to imaging polarimetry, since time and storage requirements are important considerations. The number of independent elements of the Mueller matrix can be taken into account in developing a polarimeter. If one takes only three input polarizations, (4.9) can be stractured in the following way ... 

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