6Mueller Matrices

6.1Introduction

The Mueller matrix provides a systematic way of representing all of the polarization properties of a sample: the diattenuation, retardance, and depolarization. The Mueller matrix has a simple definition; it is the 4 × 4 matrix that relates a set of incident Stokes parameters to the exiting Stokes parameters, for any type of sample. Despite this simple definition, the properties of Muller matrices are quite complex.

With the introduction of the Mueller matrix, the range of polarization phenomena under description is now greatly expanded. With the Jones calculus, calculations were limited to non-depolarizing interactions; all fully polarized incident states emerge as fully polarized. Indeed, for the majority of optical systems, lenses, telescopes, microscopes, and fiber optic systems, these non-depolarizing interactions are the desired condition. Most optical systems operate extremely close to the non-depolarizing condition.

In a depolarizing interaction, polarized light becomes partially polarized; the degree of polarization is reduced. In a microscope or telescope, such depolarization is usually associated with defects such as dirt, fingerprints, scratches, or a bad optical coating. For such defects, the Mueller matrix, not the Jones matrix, is appropriate to characterize the polarization effects. But more generally when polarized light scatters from everything, paint, paper, dirt, rocks, plastics, the degree of polarization decreases and some depolarization is present. Thus, the Mueller matrix expands the variety of light–matter interactions that can be described.

6.2The Mueller Matrix

The Mueller matrix, M, is a 4 × 4 matrix that transforms the incident Stokes parameters, S, into the exiting Stokes parameters, S′, by matrix vector multiplication,1,2

$$\mathbf{M}\cdot \mathbf{S}={\mathbf{S}}^{\prime }=\left(\begin{array}{llll}{M}_{00}& M_{01}& M_{02}& M_{03}\\ M_{10}& M_{11}& M_{12}& M_{13}\\ M_{20}& M_{21}& M_{22}& M_{23}\\ M_{30}& M_{31}& M_{32}& M_{33}\end{array}\right)\left(\begin{array}{l}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{l}{S^{\prime }}_0\\ {S^{\prime }}_1\\ {S^{\prime }}_2\\ {S^{\prime }}_3\end{array}\right).$$6.1

6.1

From the definition of matrix–vector multiplication, the exiting Stokes parameters S′ are

$${\mathbf{S}}^{\prime }=\left(\begin{array}{l}{S^{\prime }}_0\\ {S^{\prime }}_1\\ {S^{\prime }}_2\\ S^{\prime }\end{array}\right)=\left(\begin{array}{l}{S}_0{M}_{00}+S_1{M}_{01}+S_2{M}_{02}+S_3{M}_{03}\\ S_0{M}_{10}+S_1{M}_{11}+S_2{M}_{12}+S_3{M}_{13}\\ S_0{M}_{20}+S_1{M}_{21}+S_2{M}_{22}+S_3{M}_{23}\\ S_0{M}_{30}+S_1{M}_{31}+S_2{M}_{32}+S_3{M}_{33}\end{array}\right).$$6.2

6.2

Each element of the incident S is related to the four elements of S′ by the elements of M. Since the elements of S and S′ are irradiances, the elements of M are dimensionless ratios of irradiances. Since irradiances are real, the elements of M are real valued, not complex numbers. Our convention numbers the subscripts from 0 to 3 to match the corresponding Stokes parameter subscripts.

Example 6.1Meaning of Mueller Matrix Columns

When unpolarized light is incident, the 0th column of the Mueller matrix (counting from 0 to 3) describes the exiting polarization state,

$$\mathbf{M}\cdot \mathbf{S}={\mathbf{S}}^{\prime }=\left(\begin{array}{llll}{M}_{00}& M_{01}& M_{02}& M_{03}\\ M_{10}& M_{11}& M_{12}& M_{13}\\ M_{20}& M_{21}& M_{22}& M_{23}\\ M_{30}& M_{31}& M_{32}& M_{33}\end{array}\right)\left(\begin{array}{l}1\\ 0\\ 0\\ 0\end{array}\right)=\left(\begin{array}{l}{M}_{00}\\ M_{10}\\ M_{20}\\ M_{30}\end{array}\right).$$6.3

6.3

This is also the average exiting polarization state, since when 0° or 90° polarized light is incident, the exiting state is the 0th column plus or minus the 1st column,

$$\mathbf{M}\cdot \mathbf{S}={\mathbf{S}}^{\prime }=\left(\begin{array}{llll}{M}_{00}& M_{01}& M_{02}& M_{03}\\ M_{10}& M_{11}& M_{12}& M_{13}\\ M_{20}& M_{21}& M_{22}& M_{23}\\ M_{30}& M_{31}& M_{32}& M_{33}\end{array}\right)\left(\begin{array}{l}1\\ 1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{l}{M}_{00}\\ M_{10}\\ M_{20}\\ M_{30}\end{array}\right)+\left(\begin{array}{l}{M}_{01}\\ M_{11}\\ M_{21}\\ M_{31}\end{array}\right),$$6.4

6.4

$$\mathbf{M}\cdot \mathbf{S}={\mathbf{S}}^{\prime }=\left(\begin{array}{llll}{M}_{00}& M_{01}& M_{02}& M_{03}\\ M_{10}& M_{11}& M_{12}& M_{13}\\ M_{20}& M_{21}& M_{22}& M_{23}\\ M_{30}& M_{31}& M_{32}& M_{33}\end{array}\right)\left(\begin{array}{r}\hfill 1\\ \hfill -1\\ \hfill 0\\ \hfill 0\end{array}\right)=\left(\begin{array}{l}{M}_{00}\\ M_{10}\\ M_{20}\\ M_{30}\end{array}\right)-\left(\begin{array}{l}{M}_{01}\\ M_{11}\\ M_{21}\\ M_{31}\end{array}\right).$$6.5

6.5

Thus, the 0th column is the average of these two output states. When 45° or 135° is incident, the exiting state is the 0th column plus or minus the 2nd column. Similarly, when right and left circularly polarized light are incident, the exiting state is the 0th column plus or minus the 3rd column. Thus, the 0th column is seen as the average output.

6.3Sequences of Polarization Elements

The effect of a series of polarization elements or interactions is described by matrix multiplication of their individual Mueller matrices Mq, where q is the index describing the order the elements are encountered. The first element encountered is on the right side of the sequence of matrices being multiplied. The final element MQ is the leftmost matrix in the matrix product,

$$\mathbf{M}={\mathbf{M}}_Q\cdot {\mathbf{M}}_{Q-1}\dots {\mathbf{M}}_2\cdot {\mathbf{M}}_1={\displaystyle \prod _{q=1}^Q{\mathbf{M}}_{Q-q+1}}.$$6.6

6.6

Thus, the properties of sequences of polarization effects are readily calculated using the simple operation of matrix multiplication. In evaluating cascades of Mueller matrices, the associative rule for matrix multiplication can be used,

$$({\mathbf{M}}_3\cdot {\mathbf{M}}_2)\cdot {\mathbf{M}}_1={\mathbf{M}}_3\cdot ({\mathbf{M}}_2\cdot {\mathbf{M}}_1),$$6.7

6.7

and adjacent matrices can be grouped in any order for multiplication. Example 6.2 contains a sequence of polarization elements example.

Example 6.2Mueller Matrix Multiplication Example

A half wave linear retarder with a 0° fast axis followed by a half wave linear retarder with a 45° fast axis becomes a half wave circular retarder,

$$\begin{array}{l}HWLR(45°)\cdot HWLR(0)=\left(\begin{array}{rrrr}\hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill -1\end{array}\right)\cdot \left(\begin{array}{rrrr}\hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill -1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill -1\end{array}\right)\\ \phantom{HWLR(45°)\cdot HWLR(0)}=\left(\begin{array}{rrrr}\hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill -1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill -1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1\end{array}\right)=LHWCR=RHWCR.\end{array}$$6.33

6.33

The Mueller matrices of both left and right half wave circular retarders are equal, so the helicity is not determined.

6.4Non-Polarizing Mueller Matrices

A non-polarizing optical element does not change the polarization state of any incident polarization state, only the amplitude and/or phase change. The Mueller matrix for a non-absorbing, non-polarizing sample is the 4 × 4 identity matrix, I,

$$\mathbf{I}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right).$$6.8

6.8

I is the Mueller matrix for vacuum and the approximate Mueller matrix for air. For a neutral density filter or an element with polarization-independent absorption or loss, the Mueller matrix has Tmax = Tmin = T, and the resulting Mueller matrix is proportional to the identity matrix and can be expressed in terms of our notation for linear diattenuators, LD(Tmax, Tmin,θ) (Equation 6.53), as

$$\mathbf{L}\mathbf{D}(T,T,0)=T\left(\begin{array}{llll}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right).$$6.9

6.9

The Stokes parameters do not contain an absolute phase term, unlike Jones vectors. The Jones matrix for absolute phase change ϕ and optical path length is

$$e^{i\varphi }\left(\begin{array}{ll}1& 0\\ 0& 1\end{array}\right).$$6.10

6.10

There is no corresponding Mueller matrix for phase change. If phase change or optical path length needs to be calculated, such calculations need to be performed in addition to the Mueller calculations. Changes in relative phase between the polarization components of the light, the retardance, is calculated within the Mueller calculus.

6.5Rotating Polarization Elements about the Light Direction

When a polarization element with Mueller matrix M is rotated about the incident beam of light by an angle θ, the angle of incidence is unchanged. For example, consider a normal-incidence beam passing through an element rotating about its normal, the resulting Mueller matrix M(θ) is

$$\begin{array}{l}\mathbf{M}(\theta )={\mathbf{R}}_M(\theta )\cdot \mathbf{M}\cdot {\mathbf{R}}_M(-\theta )\\ \phantom{M(\theta )}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \text{cos}2\theta & -\text{sin}2\theta & 0\\ 0& \text{sin}2\theta & \text{cos}2\theta & 0\\ 0& 0& 0& 1\end{array}\right)\cdot \left(\begin{array}{llll}{M}_{00}& M_{01}& M_{02}& M_{03}\\ M_{10}& M_{11}& M_{12}& M_{13}\\ M_{20}& M_{21}& M_{22}& M_{23}\\ M_{30}& M_{31}& M_{32}& M_{33}\end{array}\right)\cdot \left(\begin{array}{llll}1& 0& 0& 0\\ 0& \text{cos}2\theta & \text{sin}2\theta & 0\\ 0& -\text{sin}2\theta & \text{cos}2\theta & 0\\ 0& 0& 0& 1\end{array}\right).\end{array}$$6.11

6.11

RM(θ), the Mueller rotation matrix,

$${\mathbf{R}}_M(\theta )=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \text{cos}2\theta & -\text{sin}2\theta & 0\\ 0& \text{sin}2\theta & \text{cos}2\theta & 0\\ 0& 0& 0& 1\end{array}\right),$$6.12

6.12

is the matrix for rotations about the optical axis. RM(θ) was introduced in Chapter 3 along with a discussion of the non-orthogonal coordinates for the Stokes parameters.

For vectors, a rotational change of basis is accomplished by left multiplication with a rotation matrix. Hence, Stokes parameters are rotated with respect to their coordinate system by the transformation

$$\mathbf{S}(\theta )={\mathbf{R}}_M(\theta )\cdot \mathbf{S}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \text{cos}2\theta & -\text{sin}2\theta & 0\\ 0& \text{sin}2\theta & \text{cos}2\theta & 0\\ 0& 0& 0& 1\end{array}\right)\cdot \left(\begin{array}{l}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{l}{S}_0\\ S_1\text{cos}2\theta -S_2\text{sin}2\theta \\ S_1\text{sin}2\theta +S_2\text{cos}2\theta \\ S_3\end{array}\right).$$6.13

6.13

The 2θ occurs because the S1 and S2 axes are only 45° apart. A rotation of 180° returns the Stokes parameters to their original state so RM(180°) must equal the identity matrix. By definition, θ is positive for rotations that initially move an x-component toward the +y axis, rotations that are counterclockwise looking into the beam.

For matrices, a rotational change of basis requires two rotation matrices of opposite sign, one multiplied from the left and one from the right. This is an example of a unitary transformation of a matrix. Note the +θ in the first rotation matrix and −θ in the second rotation matrix in Equation 6.11. These signs are understood by comparing a matrix–vector multiplication in un-rotated coordinates

$$\mathbf{M}\cdot \mathbf{S}={\mathbf{S}}^{\prime },$$6.14

6.14

with the multiplication of the rotated matrix and rotated Stokes parameters, each shown initially in brackets,

$$\begin{array}{l}\left[{\mathbf{R}}_M(\theta )\cdot \mathbf{M}\cdot {\mathbf{R}}_M(-\theta )\right]\cdot \left[{\mathbf{R}}_M(\theta )\cdot \mathbf{S}\right]\hfill \\ \phantom{}={\mathbf{R}}_M(\theta )\cdot \mathbf{M}\cdot \left[{\mathbf{R}}_M(-\theta )\cdot {\mathbf{R}}_M(\theta )\right]\cdot \mathbf{S}\hfill \\ \phantom{}={\mathbf{R}}_M(\theta )\cdot \mathbf{M}\cdot \mathbf{S}\hfill \\ \phantom{}={\mathbf{R}}_M(\theta )\cdot {\mathbf{S}}^{\prime }.\hfill \end{array}$$6.15

6.15

Since matrix multiplication is associative, (A·B)·C = A·(B·C), the parenthesis can be rearranged into the second expression. RM(θ) is a unitary rotation matrix and an opposite rotation must undo the effect of a rotation,

$$\begin{array}{l}{R}_M(\theta )\cdot R_M(-\theta )\hfill \\ \phantom{}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \text{cos}2\theta & -\text{sin}2\theta & 0\\ 0& \text{sin}2\theta & \text{cos}2\theta & 0\\ 0& 0& 0& 1\end{array}\right)\cdot \left(\begin{array}{llll}1& 0& 0& 0\\ 0& \text{cos}2\theta & \text{sin}2\theta & 0\\ 0& -\text{sin}2\theta & \text{cos}2\theta & 0\\ 0& 0& 0& 1\end{array}\right)\hfill \\ \phantom{}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right),\hfill \end{array}$$6.16

6.16

yielding the identity matrix. Hence,

$${\mathbf{R}}_M(-\theta )={\mathbf{R}}_M^{-1}(\theta ).$$6.17

6.17

6.6Retarder Mueller Matrices

Retarders are polarization elements that have two polarization states that are transmitted in the incident polarization state (eigenpolarizations) but with different optical path lengths (phases). This section presents Mueller matrix formulas for ideal retarders, where it is assumed that the two eigenpolarizations are transmitted without loss and the element has no diattenuation.

Birefringent retarders operate by dividing the incident light into two modes with orthogonal polarization states and different refractive indices. Propagation delays one mode with respect to the other, resulting in two optical path lengths, and an optical path difference, the retardance, as was seen in Figure 5.3. Other retardance mechanisms include reflections from metals, reflection and transmission through multilayer thin films, stress birefringence, and interactions with diffraction gratings. These interactions also are often diattenuating.

A retarder is specified by the optical path difference between the eigenpolarizations (the retardance δ) and the eigenpolarization states, the state with the smaller optical path length (the fast axis) and the larger optical path length (the slow axis). Retardance is generally expressed in radians, so δ = 2π indicates one wavelength of optical path difference. Note that axis implies a linear polarization state, but the fast eigenpolarization may also be elliptical or circular and the term “axis” can still be applied.

The action of retarders on Stokes parameters is visualized as a rotation of the Poincaré sphere. The line passing through the two retarder eigenpolarizations forms the rotation axis on the Poincaré sphere. The magnitude of the rotation is the retardance. This description of retarders is elaborated at length in Section 6.8.

In the Mueller calculus, retarders are represented by real unitary matrices of the form

$${\mathbf{M}}_{\text{Retarder}}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& & & \\ 0& & \begin{array}{l}3\times 3\text{ Rotation}\\ \text{Matrix}\end{array}& \\ 0& & & \end{array}\right),$$6.18

6.18

where, except for the M00 element, the first row and column are zeros. Real unitary matrices are called orthogonal matrices; the rows of an orthogonal matrix form a set of orthogonal unit vectors, as do the set of columns. The definition of a unitary matrixU is a matrix whose adjoint equals its matrix inverse,

$${\mathbf{U}}^{†}=({\mathbf{U}}^{\text{T}})*={\text{U}}^{-1}.$$6.19

6.19

For a real matrix, the complex conjugate of a matrix equals the matrix, so the transpose of an orthogonal matrix O equals its inverse,

$${\mathbf{O}}^{\text{T}}={\mathbf{O}}^{-1}.$$6.20

6.20

Thus, MT = M−1 tests if a Mueller matrix is a pure retarder. Orthogonal matrices such as retarder Mueller matrices are rotation matrices. The lower right 3 × 3 elements form a rotation matrix in (S1, S2, S3) space, where (S1, S2, S3) are the three Stokes parameters. This is how retarders mathematically rotate the Poincaré sphere. The retardanceδ of a pure retarder Mueller matrix is

$$\delta ={\text{cos}}^{-1}\left(\frac{M_{00}+M_{11}+M_{22}+M_{33}}{2}-1\right)={\text{cos}}^{-1}\left(\frac{\text{Tr}(\mathbf{M})}{2}-1\right).$$6.21

6.21

Tr indicates the trace of a matrix, the sum of the diagonal elements. A horizontal linear retarder with retardance, δ, has the Mueller matrix,

$$\mathbf{L}\mathbf{R}(\delta ,0)=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \text{cos}\delta & \text{sin}\delta \\ 0& 0& -\text{sin}\delta & \text{cos}\delta \end{array}\right).$$6.22

6.22

When LR(δ, 0) operates on a set of Stokes parameters, the first two elements, S0 and S1, are left unchanged; they are the non-zero elements in the two eigenpolarizations, horizontal and vertical linearly polarized light. A linear retarder LR(δ, 0) with its fast axis oriented at an angle θ can be derived using the Mueller rotation operation, as shown in Equation 6.11, and has the Mueller matrix

$$\mathbf{L}\mathbf{R}(\delta ,\theta )=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& {\text{cos}}^{2}2\theta +\text{cos}\delta {\text{sin}}^{2}2\theta & (1-\text{cos}\delta )\text{cos}2\theta \text{sin}2\theta & -\text{sin}\delta \text{sin}2\theta \\ 0& (1-\text{cos}\delta )\text{cos}2\theta \text{sin}2\theta & \text{cos}\delta {\text{cos}}^{2}2\theta +{\text{sin}}^{2}2\theta & \text{cos}2\theta \text{sin}\delta \\ 0& \text{sin}\delta \text{sin}2\theta & -\text{cos}2\theta \text{sin}\delta & \text{cos}\delta \end{array}\right).$$6.23

6.23

A circular retarder, CR(δ), with retardance, δ, has the Mueller matrix

$$\mathbf{C}\mathbf{R}(\delta )=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \text{cos}\delta & \text{sin}\delta & 0\\ 0& -\text{sin}\delta & \text{cos}\delta & 0\\ 0& 0& 0& 1\end{array}\right).$$6.24

6.24

This Mueller matrix can describe optically active materials and Faraday rotation.

The Mueller matrices for quarter wave retarders, δ = π/2, and half wave retarders, δ = π, with fast axes corresponding to the basis polarization states are given in Table 6.1.

Table 6.1Quarter Wave and Half Wave Retarder Mueller Matrices for the Basis Polarization States

Type of Retarder	Symbol	Mueller Matrix
Horizontal quarter wave linear retarder	HQWLR	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right)$
Vertical quarter wave linear retarder	VQWLR	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right)$
45° Quarter wave linear retarder	QWLR(45°)	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\end{array}\right)$
135° Quarter wave linear retarder	QWLR(135°)	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& -1& 0& 0\end{array}\right)$
Quarter wave right circular retarder	QWRCR	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 1\end{array}\right)$
Quarter wave left circular retarder	QWLCR	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& -1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right)$
Horizontal or vertical half wave linear retarder (same matrix)	HHWLR	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)$
45° or 135° Half wave linear retarder	HWLR(45°)	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right)$
Right or left half wave circular retarder	RHWCR	$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right)$

A half wave linear retarder with fast axis at angle θ, HWLR(θ), has the Mueller matrix

$$\mathbf{HWLR}(\theta )=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \text{cos}4\theta & \text{sin}4\theta & 0\\ 0& \text{sin}4\theta & -\text{cos}4\theta & 0\\ 0& 0& 0& -1\end{array}\right).$$6.25

6.25

The Mueller matrix elements for half wave linear retarders are plotted in Figure 6.1 (right). The Mueller matrix is the same for a horizontal half wave linear retarder and a vertical half wave linear retarder because both half wave retarders transform all incident Stokes parameters equally. The Mueller matrix for a quarter wave linear retarder with fast axis at angle θ, QWLR(θ), is

Figure 6.1(Left) The Mueller matrix elements of the family of quarter wave linear retarders. The abscissa is the retarder fast axis orientation, from 0° to 360°. (Right) The Mueller matrix elements of a half wave linear retarder as a function of the fast axis orientation.

$$\begin{array}{l}\mathbf{QWLR}(\theta )=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& {\text{cos}}^{2}2\theta & \text{cos}2\theta \text{sin}2\theta & -\text{sin}2\theta \\ 0& \text{cos}2\theta \text{sin}2\theta & {\text{sin}}^{2}2\theta & \text{cos}2\theta \\ 0& \text{sin}2\theta & -\text{cos}2\theta & 0\end{array}\right)\\ \phantom{QWLR(\theta )}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \frac{1}{2}(1+\text{cos}4\theta )& \frac{1}{2}\text{sin}4\theta & -\text{sin}2\theta \\ 0& \frac{1}{2}\text{sin}4\theta & \frac{1}{2}(1-\text{cos}4\theta )& \text{cos}2\theta \\ 0& \text{sin}2\theta & -\text{cos}2\theta & 0\end{array}\right).\end{array}$$6.26

6.26

The Mueller matrix elements are plotted in Figure 6.1 (left). The Mueller matrix elements for all linear retarders LR(δ, θ) are plotted in two dimensions in Figure 6.2.

Figure 6.2Density plots of the Mueller matrix elements of the linear retarders. Retardance in radians is along the y-axis. Fast axis orientation in degrees is along the x-axis. Coloring represents the value of the Mueller matrix element. The bottom (0 waves) and top (1 wave) sides of each figure represents the identity matrix, δ = (0,2π). Similarly, a horizontal line across the center contains all the half wave linear retarder matrix elements, δ = π.

The general elliptical retarder Mueller matrix can be expressed in terms of retardance components, (δH, δ45, δR). The magnitude of the retardance δ and the associated Stokes eigenpolarization are

$$\delta =\sqrt{{\delta }_{\text{H}}^2+{\delta }_{45}^2+{\delta }_{\text{R}}^2},{\mathbf{S}}_{\text{fast}}=\left(\begin{array}{l}1\\ {\delta }_{\text{H}}/\delta \\ {\delta }_{45}/\delta \\ {\delta }_{\text{R}}/\delta \end{array}\right).$$6.27

6.27

The elliptical retarder’s Mueller matrix is

$$\mathbf{ER}({\delta }_{\text{H}},{\delta }_{45},{\delta }_{\text{R}})=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \frac{{\delta }_{\text{H}}^2+\left({\delta }_{45}^2+{\delta }_{\text{R}}^2\right)C}{{\delta }^2}& \frac{{\delta }_{45}{\delta }_{\text{H}}T}{{\delta }^2}+\frac{{\delta }_{\text{R}}S}{\delta }& \frac{{\delta }_{\text{H}}{\delta }_{\text{R}}T}{{\delta }^2}-\frac{{\delta }_{45}S}{\delta }\\ 0& \frac{{\delta }_{45}{\delta }_{\text{H}}T}{{\delta }^2}-\frac{{\delta }_{\text{R}}\text{S}}{\delta }& \frac{{\delta }_{45}^2+\left({\delta }_{\text{R}}^2+{\delta }_{\text{H}}^2\right)C}{{\delta }^2}& \frac{{\delta }_{\text{R}}{\delta }_{45}T}{{\delta }^2}+\frac{{\delta }_{\text{H}}S}{\delta }\\ 0& \frac{{\delta }_{\text{H}}{\delta }_{\text{R}}T}{{\delta }^2}+\frac{{\delta }_{45}S}{\delta }& \frac{{\delta }_{\text{R}}{\delta }_{45}T}{{\delta }^2}-\frac{{\delta }_{\text{H}}S}{\delta }& \frac{{\delta }_{\text{R}}^2+\left({\delta }_{45}^2+{\delta }_{\text{H}}^2\right)C}{{\delta }^2}\end{array}\right),$$6.28

6.28

where C = cos δ, S = sin δ, T = 1 − cos δ. The Mueller matrix for half wave elliptical retarders, HWR, simplifies to the following form,

$$\mathbf{HWR}(r_1,r_2,r_3)=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& -1+2r_1^2& 2r_2{r}_1& 2r_1{r}_3\\ 0& 2r_2{r}_1& -1+2r_2^2& 2r_2{r}_3\\ 0& 2r_3{r}_1& 2r_2{r}_3& -1+2r_3^2\end{array}\right),$$6.29

6.29

where $\sqrt{r_1^2+r_2^2+r_3^2}=1,r_1=\frac{{\delta }_{\text{H}}}{\pi },r_2=\frac{{\delta }_{45}}{\pi },r_3=\frac{{\delta }_{\text{R}}}{\pi }.$Figure 6.3 shows density plots of the Mueller matrix elements for all quarter wave and half wave elliptical retarders.

Figure 6.3Density plots of quarter wave (left) and half wave (right) elliptical retarder Mueller matrix elements. The x-axis is orientation. The y-axis is Poincaré sphere latitude for the fast axis, varying from left circular across the bottom, to linear across the center, to right circular at the top. Coloring represents the value of the Mueller matrix element. Each square represents a flattened Poincaré sphere. Linear retarders fall along the line across the center.

The retardance parameters (δH, δ45, δR) of a retarder Mueller matrix can be calculated from Equations 6.27 and 6.28 by adding the off-diagonal elements,

$$({\delta }_{\text{H}},{\delta }_{45},{\delta }_{\text{R}})=\frac{\delta }{2\text{sin}\delta }(M_{23}-M_{32},M_{31}-M_{13},M_{12}-M_{21}).$$6.30

6.30

For retardances very near zero, Equation 6.30 can be replaced by

$$({\delta }_{\text{H}},{\delta }_{45},{\delta }_{\text{R}})=\frac{1}{2}(M_{23}-M_{32},M_{31}-M_{13},M_{12}-M_{21}).$$6.31

6.31

Equation 6.30 does not work for half wave retarders due to a zero denominator; for this special case, the following equation, which is derived from the diagonal elements, works well,

$$({\delta }_{\text{H}},{\delta }_{45},{\delta }_{\text{R}})=\pi \left(\sqrt{\frac{M_{11}+1}{2}},\text{sign}(M_{12})\sqrt{\frac{M_{22}+1}{2}},\text{sign}(M_{13})\sqrt{\frac{M_{33}+1}{2}}\right).$$6.32

6.32

Since the square root always returns a positive value, the sign of the off-diagonal elements is used to determine the octant of the retarder axis. Equation 6.32 always returns an axis with a positive δH. For half wave retarders, the retarder with an orthogonal axis also has the same Mueller matrix.

6.7Polarizer and Diattenuator Mueller Matrices

6.7.1Basic Polarizers

A polarizer is specified by the unique exiting polarization state; the orthogonal polarization state is blocked. An ideal polarizer transmits 100% of the transmitted polarization state and completely blocks the orthogonal polarization state. These two states are the two eigenpolarizations of the polarizer.

Example 6.3Mueller Matrix for a Horizontal Linear Polarizer

The Mueller matrix for an ideal horizontal linear polarizer HLP operates on an arbitrary S yielding Stokes parameters where ${S^{\prime }}_0={S^{\prime }}_1$, which must be horizontal linearly polarized light,

$$\mathbf{HLP}\cdot \mathbf{S}=\frac{1}{2}\left(\begin{array}{llll}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\cdot \left(\begin{array}{l}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{l}{S^{\prime }}_0\\ {S^{\prime }}_1\\ {S^{\prime }}_2\\ {S^{\prime }}_3\end{array}\right)=\frac{1}{2}\left(\begin{array}{l}{S}_0+S_1\\ S_0+S_1\\ 0\\ 0\end{array}\right)=\frac{S_0+S_1}{2}\left(\begin{array}{l}1\\ 1\\ 0\\ 0\end{array}\right).$$6.34

6.34

Since the first two rows of HLP are equal, the first two elements of S′ are equal, the S2 and S3 characteristics of the incident light are lost, and the exiting light is always horizontal linearly polarized. Horizontal and vertical linearly polarized light are the two eigenpolarizations of this matrix,

$$\begin{array}{l}\mathbf{HLP}\cdot \mathbf{H}=\frac{1}{2}\left(\begin{array}{llll}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\cdot \left(\begin{array}{l}1\\ 1\\ 0\\ 0\end{array}\right)=1\left(\begin{array}{l}1\\ 1\\ 0\\ 0\end{array}\right),\\ \mathbf{HLP}\cdot \mathbf{V}=\frac{1}{2}\left(\begin{array}{llll}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\cdot \left(\begin{array}{l}1\\ -1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{l}0\\ 0\\ 0\\ 0\end{array}\right)=0\left(\begin{array}{l}1\\ -1\\ 0\\ 0\end{array}\right),\end{array}$$6.35

6.35

with eigenvalues 1 and 0, the intensity transmittances for these two states. Any 4 × 4 matrix has 4 eigenvectors and eigenvalues. The other two eigenvalues and eigenvectors of HLP are as follows:

$$\begin{array}{l}\frac{1}{2}\left(\begin{array}{llll}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\cdot \left(\begin{array}{l}0\\ 0\\ 1\\ 0\end{array}\right)=0\left(\begin{array}{l}0\\ 0\\ 1\\ 0\end{array}\right),\text{and}\\ \frac{1}{2}\left(\begin{array}{llll}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\cdot \left(\begin{array}{l}0\\ 0\\ 0\\ 1\end{array}\right)=0\left(\begin{array}{l}0\\ 0\\ 0\\ 1\end{array}\right).\end{array}$$6.36

6.36

These two eigenvectors are not valid Stokes parameters since they violate the condition on Stokes parameters that $S_0<\sqrt{S_1^2+S_2^2+S_3^2}$; hence, these two eigenvectors do not count as eigenpolarizations.

Using MR(θ), the Mueller matrix for a linear polarizer with transmission axis at θ is readily calculated from HLP, as

$$\begin{array}{l}LP(\theta )=R_M(\theta )\cdot HLP\cdot R_M(-\theta )\\ \phantom{LP(\theta )}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \text{cos}2\theta & -\text{sin}2\theta & 0\\ 0& \text{sin}2\theta & \text{cos}2\theta & 0\\ 0& 0& 0& 1\end{array}\right)\cdot \frac{1}{2}\left(\begin{array}{llll}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\cdot \left(\begin{array}{llll}1& 0& 0& 0\\ 0& \text{cos}2\theta & \text{sin}2\theta & 0\\ 0& -\text{sin}2\theta & \text{cos}2\theta & 0\\ 0& 0& 0& 1\end{array}\right)\\ \phantom{LP(\theta )}=\left(\begin{array}{llll}1& \text{cos}2\theta & \text{sin}2\theta & 0\\ \text{cos}2\theta & {\text{cos}}^{2}2\theta & \text{sin}2\theta \text{cos}2\theta & 0\\ \text{sin}2\theta & \text{sin}2\theta \text{cos}2\theta & {\text{sin}}^{2}2\theta & 0\\ 0& 0& 0& 0\end{array}\right).\end{array}$$6.37

6.37

LP(θ) transmits the linear polarization state oriented at θ and blocks the state at θ + 90°.

Table 6.2 contains the Mueller matrices for the six ideal polarizers that transmit the six basis sets of Stokes parameters. Figure 6.4 graphs the 16 Mueller matrix elements for a linear polarizer as a function of transmission axis angle θ.

Table 6.2Polarizer Mueller Matrices for the Basis Polarization States

Type of Polarizer	Symbol	Mueller Matrix
Horizontal linear polarizer	HLP	$\frac{1}{2}\left(\begin{array}{cccc}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$
Vertical linear polarizer	VLP	$\frac{1}{2}\left(\begin{array}{cccc}1& -1& 0& 0\\ -1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$
45° Linear polarizer	LP(45°)	$\frac{1}{2}\left(\begin{array}{cccc}1& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right)$
135° Linear polarizer	LP(135°)	$\frac{1}{2}\left(\begin{array}{cccc}1& 0& -1& 0\\ 0& 0& 0& 0\\ -1& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right)$
Right circular polarizer	RCP	$\frac{1}{2}\left(\begin{array}{cccc}1& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 1\end{array}\right)$
Left circular polarizer	LCP	$\frac{1}{2}\left(\begin{array}{cccc}1& 0& 0& -1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -1& 0& 0& 1\end{array}\right)$

Figure 6.4The 16 Mueller matrix elements for the set of linear polarizers as a function of transmission axis angle θ.

Consider an elliptical polarizer that transmits the polarization state with the major axis of the ellipse oriented at θ and with the transmitting eigenpolarization located at latitude η on the Poincaré sphere, where −π/2 ≤ η ≤ π/2. The elliptical polarizer Mueller matrixEP(θ, η) is

$$\mathbf{EP}(\theta ,\eta )=\frac{1}{2}\left(\begin{array}{llll}1& \text{cos}2\theta \text{cos}\eta & \text{sin}2\theta \text{cos}\eta & \text{sin}\eta \\ \text{cos}2\theta \text{cos}\eta & {\text{cos}}^{2}2\theta {\text{cos}}^2\eta & \frac{1}{2}\text{sin}4\theta {\text{cos}}^2\eta & \text{cos}2\theta \text{cos}\eta \text{sin}\eta \\ \text{sin}2\theta \text{cos}\eta & \frac{1}{2}\text{sin}4\theta {\text{cos}}^2\eta & {\text{sin}}^{2}2\theta {\text{cos}}^2\eta & 2\text{cos}\theta \text{sin}\theta \text{cos}\eta \text{sin}\eta \\ \text{sin}\eta & \text{cos}2\theta \text{sin}\eta \text{cos}\eta & \text{sin}2\theta \text{cos}\eta \text{sin}\eta & {\text{sin}}^2\eta \end{array}\right).$$6.38

6.38

Problem 6.7 in Section 6.15 describes the derivation. Alternatively, consider an elliptical polarizer that transmits the state with Poincaré sphere coordinates (dH, d45, dR) and blocks the orthogonal state. The transmitted eigenpolarization is

$${\mathbf{S}}_1=(1,d_{\text{H}},d_{45},d_{\text{R}}),d_{\text{H}}^2+d_{45}^2+d_{\text{R}}^2=1.$$6.39

6.39

This elliptical polarizer Mueller matrix is

$$\mathbf{EP}(d_{\text{H}},d_{45},d_{\text{R}})=\frac{1}{2}\left(\begin{array}{llll}1& d_{\text{H}}& d_{45}& d_{\text{R}}\\ d_{\text{H}}& d_{\text{H}}^2& d_{\text{H}}{d}_{45}& d_{\text{H}}{d}_{\text{R}}\\ d_{45}& d_{\text{H}}{d}_{45}& d_{45}^2& d_{45}{d}_{\text{R}}\\ d_{\text{R}}& d_{\text{H}}{d}_{\text{R}}& d_{\text{R}}{d}_{45}& d_{\text{R}}^2\end{array}\right).$$6.40

6.40

6.7.2Transmittance and Diattenuation

Polarizers and partial polarizers are characterized by the property diattenuation, which describes the magnitude of the variation of the transmitted irradiance as a function of the incident polarization state. The diattenuation magnitude, D, usually referred to as the diattenuation, is a function of the maximum, Tmax, and minimum, Tmin, transmittances of a polarization element or optical interaction,

$$D=\frac{T_{\text{max}}-T_{\text{min}}}{T_{\text{max}}+T_{\text{min}}}=\frac{\sqrt{M_{01}^2+M_{02}^2+M_{03}^2}}{M_{00}},0\le D\le 1.$$6.41

6.41

The diattenuation has the useful property that D varies from 1 for a polarizer to 0 for an element that transmits all polarization states equally, such as a retarder or a non-polarizing interaction.

The transmitted irradiance of a Mueller matrix and its diattenuation depends only on the first row, M0 = (M00, M01, M02, M03), because these are the only elements that affect ${S^{\prime }}_0$. The diattenuation is not linear in the extinction ratio Tmin/Tmax as shown in Figure 6.5.

Figure 6.5Relationship between the diattenuation and the extinction ratio, Tmin/Tmax.

To find Tmax and Tmin, first the incident Stokes parameters are normalized, so s0 = 1. The normalized Stokes parameters are defined by a three-element vector s that represents the coordinate of the state on the unit Poincaré sphere,

$$\widehat{\mathbf{S}}=\frac{\mathbf{S}}{S_0}=\left(\begin{array}{l}{s}_0\\ s_1\\ s_2\\ s_3\end{array}\right)=\left(\begin{array}{l}1\\ \mathbf{s}\end{array}\right),\mathbf{s}=(s_1,s_2,s_3).$$6.42

6.42

For unpolarized light, the magnitude of s is zero, |s| = 0. For a completely polarized state, |s| = 1. The introduction of s allows the polarization state to be specified irrespective of the flux.

The transmittance, T(s), of a device with Mueller matrix M is the ratio of the exiting flux to the incident flux,

$$T(\mathbf{M},\mathbf{s})=\frac{{S^{\prime }}_0}{S_0}=\mathbf{M}\cdot \widehat{\mathbf{S}}=M_{00}+M_{01}{s}_1+M_{02}{s}_2+M_{03}{s}_3,$$6.43

6.43

which depends on the dot product of the first row of the Mueller matrix with the incident Stokes parameters. The dependence of the transmission on incident polarization state is characterized by a three-element diattenuation parameter, d, defined as

$$\mathbf{d}=\frac{(M_{01},M_{02},M_{03})}{M_{00}}=(d_{\text{H}},d_{45},d_{\text{R}}).$$6.44

6.44

The diattenuation parameters have three components corresponding to the three components of the Stokes parameters, x/y, 45°/135°, right/left, each of which characterize how the transmission varies with each of the Stokes parameter component. The diattenuation parameter set, d, is often called the diattenuation vector or analyzer vector, but similar to the Stokes parameters, d is not a true vector. Diattenuation parameters do not add. For a Stokes three-element vector, s, the transmission function, T, is

$$T(\mathbf{M},s)=\frac{{S^{\prime }}_0}{S_0}=\mathbf{M}\cdot \widehat{\mathbf{S}}=M_{00}+M_{01}{s}_1+M_{02}{s}_2+M_{03}{s}_3=M_{00}(1+\mathbf{d}\cdot \mathbf{s}).$$6.45

6.45

The average transmission, calculated by averaging over all sets of polarized Stokes parameters, is M00. The average transmission is also the transmission for unpolarized incident light, sU = (0, 0, 0). The polarization-dependent variation of the transmission is contained in the dot product term between the incident Stokes three-element vector and the diattenuation vector, s·d. The maximum transmission, Tmax, occurs when the dot product is maximized, which occurs when s and d are parallel, and the magnitude of ${S^{\prime }}_0$ is also maximized. The incident Stokes parameters with maximum transmittance, Smax, and minimum transmittance, Smin, are

$${\mathbf{S}}_{\text{max}}=\frac{\mathbf{d}}{\left|\mathbf{d}\right|}=\frac{1}{D}\left(\begin{array}{l}D\\ d_{\text{H}}\\ d_{45}\\ d_{\text{R}}\end{array}\right)\text{ and }{\mathbf{S}}_{\text{min}}=\frac{-\mathbf{d}}{\left|\mathbf{d}\right|}=\frac{1}{D}\left(\begin{array}{l}D\\ -d_{\text{H}}\\ -d_{45}\\ -d_{\text{R}}\end{array}\right),$$6.46

6.46

yielding

$$T_{\text{max}}=M_{00}(1+D)\text{and}{T}_{\text{min}}=M_{00}(1-D).$$6.47

6.47

Therefore, the diattenuation of any Mueller matrix is

$$D(\mathbf{M})=\frac{T_{\text{max}}-T_{\text{min}}}{T_{\text{max}}+T_{\text{min}}}=\frac{\sqrt{M_{01}^2+M_{02}^2+M_{03}^2}}{M_{00}}.$$6.48

6.48

For an ideal polarizer, the minimum transmission is zero, D = 1, Tmin = M00(1 − D) = 0. Linear polarization sensitivity or linear diattenuation LD(M) characterizes the variation of intensity transmittance with incident linear polarization states:

$$LD(\mathbf{M})=\frac{\sqrt{M_{01}^2+M_{02}^2}}{M_{00}}.$$6.49

6.49

Linear polarization sensitivity is frequently specified as a performance parameter in remote sensing systems designed to measure incident power independently of any linearly polarized component present in scattered earth-light.3LD(M) = 1 identifies M as a linear analyzer; M is not necessarily a linear polarizer, but may represent a linear polarizer followed by some other polarization element.

Diattenuation in fiber optic components and systems is often characterized by the polarization dependent loss, PDL, specified in decibels:

$$PDL(\mathbf{M})=10{\text{Log}}_{10}\frac{T_{\text{max}}}{T_{\text{min}}}.$$6.50

6.50

Glan Thompson polarizers routinely achieve a PDL of 60 dB. Many different formulations of sheet polarizer are also fabricated and sold at a lower cost. Sheet polarizers are dichroic polarizers that depend on the difference in absorption for light polarized along and perpendicular to a molecular absorption axis, and thus have a strong wavelength dependence for their PDL. The PDL varies from over 50 dB for the best sheet polarizers to 20 dB for some low-cost sheet polarizers. Polarcor1 dichroic polarizers consist of nanometer-size silver crystals aligned in glass, and can achieve a PDL of 60 dB in the near infrared.

When sequences of polarizers have parallel transmission axes, the net polarization-dependent loss is the sum of the individual polarization-dependent losses. The polarization-dependent loss of two diattenuators with orthogonal transmission axes is calculated by subtracting the polarization-dependent loss of one from the other.

6.7.3Polarizance

The polarizance, P(M), is the degree of polarization DoP of the exiting light for unpolarized incident light, U,4

$$P(\mathbf{M})=DoP(\mathbf{M}\cdot \mathbf{U})=\frac{\sqrt{M_{10}^2+M_{20}^2+M_{30}^2}}{M_{00}}.$$6.51

6.51

The exiting polarization state, Sp(M), is the first column of M,

$${\mathbf{S}}_p(\mathbf{M})=\mathbf{M}\cdot \mathbf{U}=\left(\begin{array}{llll}{M}_{00}& M_{01}& M_{02}& M_{03}\\ M_{10}& M_{11}& M_{12}& M\\ M_{20}& M_{21}& M_{22}& M_{23}\\ M_{30}& M_{31}& M_{32}& M_{33}\end{array}\right)\cdot \left(\begin{array}{l}1\\ 0\\ 0\\ 0\end{array}\right)=\left(\begin{array}{l}{M}_{00}\\ M_{10}\\ M_{20}\\ M_{30}\end{array}\right).$$6.52

6.52

The polarizance does not necessarily equal the diattenuation. Nor does Sp necessarily equal Smax, the incident state of maximum transmittance. Polarizance parameters or the polarizance vector is defined as (pH, p45, pR) = (M10, M20, M30)/M00, the normalized Stokes parameters when unpolarized light is incident.

6.7.4Diattenuators

The Mueller matrix for a partial polarizer or homogeneous diattenuator with intensity transmittances Tx and Ty along the x and y axes is LD(Tx, Ty, 0), where

$$\mathbf{LD}(T_x,T_y,0)\cdot \left(\begin{array}{l}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\frac{1}{2}\left(\begin{array}{llll}{T}_x+T_y& T_x-T_y& 0& 0\\ T_x-T_y& T_x+T_y& 0& 0\\ 0& 0& 2\sqrt{T_x{T}_y}& 0\\ 0& 0& 0& 2\sqrt{T_x{T}_y}\end{array}\right).\left(\begin{array}{l}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{l}{S^{\prime }}_0\\ {S^{\prime }}_1\\ {S^{\prime }}_2\\ {S^{\prime }}_3\end{array}\right).$$6.53

6.53

Ideal diattenuators have two different intensity transmittances, Tmax and Tmin, for two orthogonal linear eigenpolarizations, thus the name “di” “attenuator.” A linear diattenuatorLD(Tx, Ty, θ) oriented at angle θ has the Mueller matrix

$$\begin{array}{l}\mathbf{L}\mathbf{D}(T_{\max },T_{\min },\theta )\hfill \\ \phantom{}={\mathbf{R}}_M(\theta )\cdot \mathbf{L}\mathbf{D}(T_{\text{max}},T_{\text{min}},0)\cdot {\mathbf{R}}_M(-\theta )\hfill \\ \phantom{}=\frac{1}{2}\left(\begin{array}{llll}A& B\text{cos}2\theta & B\text{sin}2\theta & 0\\ B\text{cos}2\theta & A{\text{cos}}^{2}2\theta +C{\text{sin}}^{2}2\theta & (A-C)\text{cos}2\theta \text{sin}2\theta & 0\\ B\text{sin}2\theta & (A-C)\text{cos}2\theta \text{sin}2\theta & C{\text{cos}}^{2}2\theta +A{\text{sin}}^{2}2\theta & 0\\ 0& 0& 0& C\end{array}\right),\hfill \end{array}$$6.54

6.54

where

$$A=T_{\text{max}}+T_{\text{min}},B=T_{\text{max}}-T_{\text{min}},C=2\sqrt{T_{\text{max}}{T}_{\text{min}}}.$$6.55

6.55

This is an example of a unitary transformation by the unitary matrix RM(θ).

Figure 6.6 is a density plot of all linear diattenuator Mueller matrices LD(1, Tmin, θ). Ideal diattenuators have no retardance, although, in practice, most diattenuators have some retardance. An example of a pure linear diattenuator without retardance is transmission into a transparent dielectric; Tmax and Tmin are then given by intensity Fresnel coefficients. Reflection at metal surfaces acts as a diattenuator with retardance.

Figure 6.6Mueller matrices for all linear diattenuators with Tmax = 1. The x-axis is the transmission axis orientation θ, which varies from 0° to 180°, and the y-axis is Tmin, which varies from 0 (polarizers) along the bottom to 1 (non-polarizers with identity matrix) along the top.

Ideal diattenuator Mueller matrices are Hermitian matrices; they have real eigenvalues. A Hermitian matrix equals the complex conjugate of its matrix transpose, its Hermitian adjoint, H = H† = (HT)*. Since Mueller matrices are real, H* = H, ideal diattenuator Mueller matrices equal their transpose,

$$\mathbf{H}={\mathbf{H}}^T$$6.56

6.56

and are symmetric about the diagonal. An ideal horizontal linear polarizer has zero transmission along one axis; Tx = 1, Ty = 0, and has the following Mueller matrix,

$$\mathbf{LD}(1,0,0)=\frac{1}{2}\left(\begin{array}{llll}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$6.57

6.57

A real polarizer will have Tx < 1; thus,

$$\mathbf{LD}(T_x,0,0)=\frac{T_x}{2}\left(\begin{array}{llll}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$6.58

6.58

The general equation for a diattenuator Mueller matrixD, either linear, elliptical, or circular, expressed in terms of the first row of the Mueller matrix is

$$\begin{array}{l}\mathbf{D}(d_{\text{H}},d_{45},d_{\text{R}},T_{\text{Avg}})\hfill \\ \phantom{}=T_{\text{Avg}}\left(\begin{array}{llll}1& d_{\text{H}}& d_{45}& d_{\text{R}}\\ d_{\text{H}}& A& 0& 0\\ d_{45}& 0& A& 0\\ d_{\text{R}}& 0& 0& A\end{array}\right)+\frac{T_{\text{Avg}}(1-A)}{D^2}\left(\begin{array}{llll}0& 0& 0& 0\\ 0& d_{\text{H}}^2& d_{45}{d}_{\text{H}}& d_{\text{H}}{d}_{\text{R}}\\ 0& d_{45}{d}_{\text{H}}& d_{45}^2& d_{45}{d}_{\text{R}}\\ 0& d_{\text{H}}{d}_{\text{R}}& d_{45}{d}_{\text{R}}& d_{\text{R}}^2\end{array}\right),\hfill \end{array}$$6.59

6.59

where $D=\sqrt{d_{\text{H}}^2+d_{45}^2+d_{\text{R}}^2},A=\sqrt{1-d_{\text{H}}^2-d_{45}^2-d_{\text{R}}^2},T_{\text{Avg}}=\frac{T_{\text{max}}+T_{\text{min}}}{2}.$

6.8Poincaré Sphere Operations

6.8.1The Operation of Retarders on the Poincaré Sphere

The principal reason for the widespread use of the Poincaré sphere, introduced in Section 3.13, is the simple geometric description it provides for the operation of retarders. The basic model of a retarder is an element that splits the light into two orthogonal polarization states (modes) and applies a different phase shift to each mode (see Figure 5.3). This optical path difference is the retardance, δ.

The Mueller matrix for a retarder contains a 3 × 3 rotation matrix as was shown in Equation 6.18. The Mueller matrix for an elliptical retarder ER, shown in Equation 6.28, is expressed in terms of three retardance components, (δH, δ45, δR), which define the Stokes parameters of the retarder fast axis. This Mueller matrix operates on the incident Stokes parameters by rotating their initial location on the Poincaré sphere about an axis through the retarder fast and slow states by an angle equal to δ. For an incident polarization state propagating through a birefringent retarder, as the retarder’s retardance steadily increases from 0 to δ, the state evolves along a circular arc about the retarder axis.

Figure 6.7 shows the trajectory when 45° linearly polarized light is incident on a vertical quarter wave linear retarder. The polarization evolves along an arc from (0, 1, 0), through (0, −1, 1)/$\sqrt{2}$, ending at (0, 0, 1). In this case, the rotation is about the S1-axis; the direction the polarization evolves is given by applying the left-hand rule about the fast axis. This assumes that the Poincaré sphere is drawn with right-handed coordinates such that2

Figure 6.7Vertical quarter wave linear retarder operating on 45° linearly polarized light, which brings the polarization to right circularly polarized. The trajectory is drawn thicker toward the exiting state.

$$S_1\times S_2=S_3.$$6.60

6.60

By placing your left hand on the sphere above the fast axis with your thumb pointing out from the fast axis, make a fist and the direction your fingers curl indicates the direction the polarization state evolves around the axis of rotation for δ > 0.

Figure 6.8 shows the action of a quarter wave linear retarder with a vertical fast axis operating on different linearly polarized incident states. Each incident state moves 90° from the equator along an arc about the retarder axis around the sphere.

Figure 6.8A vertical quarter wave linear retarder operating on a set of linearly polarized states rotates the polarization 90° about the S1 axis.

Figure 6.9 shows the action of a half wave linear retarder with a 0° fast axis operating on several linear polarized states. Each linear state moves along a 180° trajectory and ends back on the equator, as another linear state. Reasoning with the Poincaré sphere, it is easily seen that all half wave linear retarders bring a state on the equator back onto another state on the equator, leaving the fast and slow axes fixed, as shown in Figure 6.10. Thus, half wave linear retarders are used to rotate the orientation of linearly polarized light.

Figure 6.9A vertical half wave linear retarder operating on a set of linearly polarized states. It rotates the polarization 180° about the S1 axis, mapping linearly polarized input states into linearly polarized output states.

Figure 6.10A 180° rotation of a state on the equator about an axis passing through the equator brings the state back to another state on the equator.

Figure 6.11 shows the action of a one-wave circular retarder (δ = 2π). The rotation axis now passes through the poles. For circular retarders, the Poincaré sphere rotates like the earth with an axis through the poles.

Figure 6.11Action of a one-wave circular retarder rotates the Poincaré sphere through 360° about the poles, so all states exit as the incident polarization state.

Sequences of retarders are analyzed as a series of rotations of the Poincaré sphere. Several important examples will be presented. Figure 6.12 shows the action of a half wave linear retarder with a horizontal fast axis followed by another half wave linear retarder with a 45° fast axis on several polarization states. When right circularly polarized light is incident, the first retarder moves it to the left circular pole, and then the second retarder returns it back to the right circular pole. Thus, right circularly polarized light is an eigenpolarization of this retarder combination. 45° polarized light is rotated to 135° and then is unchanged by the half wave retarder with a 45° fast axis, as shown by the black line in Figure 6.12. The overall rotation is 180°; hence, this retarder combination acts as a half wave circular retarder.

Figure 6.12A sequence of two ½ wave linear retarders at 0° and 45° is a half wave circular retarder. This sequence corresponds to the Pauli matrices multiplication σ1 × σ2 = iσ3 (see Equation 6.108).

For any sequence of rotations of a sphere, two points on the surface must end up back where they started. Thus, a sequence of rotations has the overall result of a single rotation about a rotation axis. For propagation through a sequence of retarders, the overall polarization transformations are equivalent to the operation of a single retarder, usually with an elliptical fast axis. The two states that end up where they started are the eigenpolarizations of the compound retarder. In fact, elliptical retarders are often most easily constructed from a set of two or more linear retarders.

Circular retarders can also be constructed from linear retarders. Consider a linear retarder with retardance δ oriented at 45°. When this retarder is placed between a horizontal quarter wave retarder and a vertical quarter wave linear retarder, the fast and slow axes of the δ retarder rotate to the circularly polarized poles; the three-element compound retarder forms a circular retarder with retardance δ. If the retardance of the horizontal and vertical linear retarders is adjusted to an angle μ, the fast and slow axes move to a latitude of ±μ on the Poincaré sphere.

Example 6.4Sequence of Three Quarter Wave Retarders

Consider a sequence of three quarter wave retarders with fast axes of 0°, 45°, and then left circular state. The trajectories on the Poincaré sphere are shown in Figure 6.13 for two incident elliptical states, one for an eigenstate, $\left(1,1/\sqrt{2},0,1/\sqrt{2}\right)$, and another state (1, 0.383, 0, 0.924) with a horizontally oriented major axis. The eigenstate follows a spherical triangle returning to the initial location; the first arc of the trajectory is the thinnest; the final arc is the thickest. For the other state, light becomes linearly polarized after the horizontal quarter wave retarder and then moves to a slightly elliptical state with a 45° orientation. Finally, the state moves along a circle of latitude due to the circular retarder and ends up elliptically polarized with a horizontally oriented major axis. This output state is the input state rotated 180° about the eigenpolarization; hence, the sequence is a half wave elliptical retarder.

Figure 6.13Trajectories are shown on the Poincaré sphere for a sequence of three quarter wave retarders, (1) horizontal, (2) 45°, and (3) left circular retarders. The two sequences of arcs correspond to two different incident polarization states. The horizontal retarder’s trajectory is the thinnest and the circular retarder’s trajectory is the thickest. One of the trajectories’ sequences returns to its starting point; this is an eigenvector or eigenpolarization state for the compound retarder. Because the other trajectory begins and ends symmetrically with respect to the eigenstate, the assembly of three retarders acts as a half wave retarder with elliptical eigenstates.

6.8.2The Operation of a Rotating Linear Retarder

One important polarization modulator is the rotating linear retarder. Consider horizontal linearly polarized light incident on a spinning quarter wave linear retarder. Figure 6.14 (center) shows the evolution of polarization state as the retarder’s fast axis rotates. When the fast axis is oriented horizontally, the incident polarization state is an eigenpolarization and thus the state is unchanged. When the fast axis moves to 10° from horizontal, the polarization evolves through the retarder along a small 90° arc, moving away from the equator along a 45° angle. As the retarder angle increases, the trajectory approaches the 45° longitude circle. As the angle reaches 45°, the state moves through the right circularly polarized pole. When the fast axis is vertical, the light has returned to horizontal state, and the state has moved around the top half of a figure eight in the northern hemisphere. During the next 90° rotation of the fast axis, the polarization trajectory moves through the southern hemisphere through the left circularly polarized light, completing the figure eight trajectory as shown in Figure 6.14 (center). Figure 6.14 (left) shows the smaller, narrower figure eight trajectory for a rotating 1/8 wave linear retarder. Figure 6.14 (right) shows the trajectory for a rotating 3/8 wave linear retarder that starts in front and then circles around the back side of the sphere. The trajectory for a rotating 1/2 wave linear retarder circles the equator twice in 180° of retarder rotation.

Figure 6.14The action of a rotating linear retarder on horizontal linearly polarized light is shown for three values of retardance, 1/8 wave (left), 1/4 wave (center), and 3/8 wave (right). The resulting polarization states trace a vertical figure eight on the surface of the Poincaré sphere. The 3/8 wave retarder has a smaller average distance from all the sphere states than the 1/4 wave retarder; the 3/8 wave is close to the optimum retardance value for a rotating retarder polarimeter.

6.8.3The Operation of Polarizers and Diattenuators

The transmittance, or the fraction of flux transmitted through dichroic polarizers and diattenuators, can be depicted on and inside the Poincaré sphere as a series of planes perpendicular to the diattenuator’s transmission axis. For example, the action of an ideal horizontal linear polarizer on an arbitrary set of normalized Stokes parameters has the Mueller matrix equation,

$$\mathbf{H}\mathbf{L}\mathbf{P}\cdot \widehat{\mathbf{S}}=\frac{1}{2}\left(\begin{array}{llll}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\cdot \left(\begin{array}{l}1\\ s_1\\ s_2\\ s_3\end{array}\right)=\frac{1}{2}\left(\begin{array}{l}1+s_1\\ 1+s_1\\ 0\\ 0\end{array}\right).$$6.61

6.61

The transmittance, (1 + s1)/2, corresponds to a series of planes through the Poincaré sphere perpendicular to the S1 axis as shown in Figure 6.15. Each plane indicates the set of polarization states with the same transmittance. The plane through the middle of the sphere indicates Stokes parameters with a transmittance of ½, including 45° polarized, 135° polarized, left and right circularly polarized, and unpolarized. The plane tangent to the sphere at (1, 0, 0) represents the transmittance of one at the polarizer’s transmission axis. The plane tangent to the sphere at (1, 0, 0) represents the transmittance of zero at its extinction axis. The same construction can be performed for any polarizer. For a diattenuator, the transmittances are scaled to vary linearly between Tmax and Tmin.

Figure 6.15For diattenuators, lines perpendicular to the transmission axis also indicate a set of states with equal transmission. The transmission varies linearly from Tmin to Tmax for planes perpendicular to the axis.

Next, consider the evolution of the Stokes parameters during propagation through a dichroic material, such as a sheet polarizer. The incident polarization state will move toward the transmission axis on the Poincaré sphere. Figure 6.16 shows some of the corresponding trajectories through a dichroic diattenuator for several different incident polarization states. The diattenuation is zero at the entrance and increases with the propagation distance. For a polarized input beam, the polarization state evolves along the great circles from the absorption axis toward the transmission axis. The arc lengths are shorter when the initial state is closer to the transmitted state or the attenuated state. The trajectories for the two incident states on the axis, the transmitted state and the attenuated state, have zero length; thus, those states do not move.

Figure 6.16Evolution of polarization states through a dichroic medium with a diattenuation of 0.6 where the transmitted state (transmission axis) is vertically polarized and the attenuated state is horizontally polarized is plotted on the Poincaré sphere for several initial polarization states. At the beginning of each trajectory, the diattenuation is zero. The diattenuation steadily increases along the arrow. At the midpoint of each arrow, the diattenuation is 0.3; at the end or tip of the arrow, the diattenuation is 0.6. The trajectories would continue to the transmitted state for a diattenuation of one.

6.8.4Indicating Polarization Properties

Many commercial polarimeters use the Poincaré sphere to represent the eigenstates associated with diattenuation properties, (dH, d45, dR), polarizance properties (pH, p45, pR), and retardance properties (δH, δ45, δR), where indices H, 45, and R indicate the corresponding Stokes parameter components. Two examples are shown in this section.

Example 6.5Sequence of Polarizers

A horizontal linear polarizer followed by a 45° linear polarizer has a Mueller matrix

$$\mathbf{LP}(45°)\cdot \mathbf{LP}(0°)=\frac{1}{2}\left(\begin{array}{llll}1& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right)\cdot \frac{1}{2}\left(\begin{array}{llll}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)=\frac{1}{4}\left(\begin{array}{llll}1& 1& 0& 0\\ 0& 0& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$6.62

6.62

The eigenpolarizations are vertically polarized light and 135° polarized light; hence, this is an inhomogeneous polarizer Mueller matrix. The diattenuation parameters from the top row are

$$\text{(}{M}_{01},M_{02},M_{03}\text{)}=(1,0,0).$$6.63

6.63

The polarizance parameters from the left column are

$$(M_{10},M_{20},M_{30})=(0,1,0).$$6.64

6.64

These properties are represented with the Poincaré sphere in Figure 6.17. The retardance is undefined for a polarizer and so is not shown.

Figure 6.17Poincaré sphere representation of the polarization properties of a particular inhomogeneous polarizer formed from a horizontal polarizer followed by a 90° polarizer. The brown line indicates the diattenuation axis and the blue line represents the polarizance axis.

Example 6.6A Linear Diattenuator Followed by a Retarder

A horizontal linear diattenuator, Tmax = 1, Tmin = 0.5, with diattenuation = 1/3, followed by a 45° quarter wave linear retarder has the Mueller matrix

$$\begin{array}{l}\mathbf{LR}(\pi /2,45°)\cdot \mathbf{LD}(1,1/2,0°)\\ \phantom{}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\end{array}\right)\cdot \frac{1}{4}\left(\begin{array}{llll}3& 1& 0& 0\\ 1& 3& 0& 0\\ 0& 0& \frac{4}{\sqrt{2}}& 0\\ 0& 0& 0& \frac{4}{\sqrt{2}}\end{array}\right)\\ \phantom{}=\frac{1}{4}\left(\begin{array}{llll}3& 1& 0& 0\\ 0& 0& 0& \frac{-4}{\sqrt{2}}\\ 0& 0& \frac{4}{\sqrt{2}}& 0\\ 1& 3& 0& 0\end{array}\right)\end{array}$$6.65

6.65

with diattenuation parameters (1/3, 0, 0), polarizance parameters (0, 0, 1/3), and retardance parameters (0, π/2, 0). These properties are represented on the Poincaré sphere as in Figure 6.18.

Figure 6.18Poincaré sphere representation of the polarization properties of the Mueller matrix of Equation 6.65. The brown line indicates the diattenuation axis, the blue line represents the polarizance axis, and the green line indicates the retardance fast axis.

6.9Weak Polarization Elements

Weak polarization elements cause only small changes to the polarization state. Weak polarization elements have Mueller matrices close to the identity matrix times a constant (to account for absorption or transmission losses). The properties of weak Mueller matrices are much simpler than those of general Mueller matrices because the retardance, diattenuation, and depolarization are close to zero. Some important examples of such weakly polarizing elements would be the lens surfaces and mirror surfaces in lenses, microscopes, and telescopes, where the polarization properties are not zero due to Fresnel equations, antireflection coatings, or mirrored surfaces, but the diattenuation and retardance effects are usually well below 0.05.

The structure of the Mueller calculus and the properties of these weak elements can be explored by performing Taylor series on the polarization property Mueller matrix expressions with respect to diattenuation or retardance. Weak retarders have a retardance near zero. Performing a Taylor series expansion on the general equation for an elliptical retarder (Equation 6.28) and keeping the first-order terms yield the following simple weak retarder Mueller matrix,

$$\underset{{\delta }_{\text{H}},{\delta }_{45},{\delta }_{\text{R}}\to 0}{\lim }\mathbf{E}\mathbf{R}({\delta }_{\text{H}},{\delta }_{45},{\delta }_{\text{R}})=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& 1& {\delta }_{\text{R}}& -{\delta }_{45}\\ 0& -{\delta }_{\text{R}}& 1& {\delta }_H\\ 0& {\delta }_{45}& -{\delta }_{\text{H}}& 1\end{array}\right).$$6.66

6.66

Similarly, a first-order Taylor series expansion on the general diattenuator expression yields the weak diattenuator Mueller matrix,

$$\underset{d_{\text{H}},d_{45},d_{\text{R}}\to 0}{\lim }\mathbf{D}(d_{\text{H}},d_{45},d_{\text{R}},T_{\text{Avg}})=T_{\text{Avg}}\left(\begin{array}{llll}1& d_{\text{H}}& d_{45}& d_{\text{R}}\\ d_{\text{H}}& 1& 0& 0\\ d_{45}& 0& 1& 0\\ d_{\text{R}}& 0& 0& 1\end{array}\right).$$6.67

6.67

Combining these two expressions yields the weak diattenuator and retarder Mueller matrix

$$\mathbf{WDR}(d_{\text{H}},d_{45},d_{\text{R}},{\delta }_{\text{H}},{\delta }_{45},{\delta }_{\text{R}},T_{\text{Avg}})=T_{\text{Avg}}\left(\begin{array}{llll}1& d_{\text{H}}& d_{45}& d_{\text{R}}\\ d_{\text{H}}& 1& {\delta }_{\text{R}}& -{\delta }_{45}\\ d_{45}& -{\delta }_{\text{R}}& 1& {\delta }_{\text{H}}\\ d_{\text{R}}& {\delta }_{45}& -{\delta }_{\text{H}}& 1\end{array}\right).$$6.68

6.68

These three equations are only correct to first order. Higher-order terms, which are present when these parameters are not infinitesimal, are calculated from the exact equations presented earlier.

Weak diattenuators are symmetric in the top row and first column. Weak retarders are anti-symmetric in the off-diagonal lower right 3 × 3 elements. The presence of anti-symmetric components in the top row and column and symmetric components in the lower right 3 × 3 elements of weak polarization element Mueller matrices indicates the presence of depolarization, as discussed in Section 6.11.4.

Example 6.7Weakly Polarized Mueller Matrix

Consider a weakly polarized Mueller matrix

$$\mathbf{M}=\left(\begin{array}{llll}1& 0.02& 0& 0.01\\ 0.02& 1& -0.005& 0\\ 0& 0.005& 1& -0.01\\ 0.01& 0& 0.01& 1\end{array}\right).$$6.69

6.69

The diattenuation has a horizontal component dH = 0.02 and a right circular component dR = 0.01. The diattenuation magnitude is

$$d=\sqrt{{0.02}^2+{0.01}^2}\approx 0.022.$$6.70

6.70

The Stokes parameters for the state with maximum transmission are any vector proportional to

$${\mathbf{S}}_{\text{max}}\propto \left(\begin{array}{l}0.022\\ 0.02\\ 0\\ 0.01\end{array}\right).$$6.71

6.71

The retardance has a horizontal component δH = 0.01 and a right circular component δR = 0.005. Since these parameters are proportional to the diattenuation parameters, the matrix is homogeneous and has orthogonal eigenpolarizations shared by the diattenuation part and the retardance part. The retardance magnitude is

$$\delta =\sqrt{{0.01}^2+{0.005}^2}\approx \mathrm{0.011.}$$6.72

6.72

6.10Non-Depolarizing Mueller Matrices

Non-depolarizing Mueller matrices are the set of Mueller matrices for which completely polarized incident light with DoP(S) = 1 transmits as completely polarized light for all incident polarization states. Non-depolarizing Mueller matrices have a depolarization index of one, which will be described in Equation 6.79. Non-depolarizing Mueller matrices are a subset of the Mueller matrices. Jones matrices can only represent non-depolarizing interactions. The non-depolarizing Mueller matrices are those Mueller matrices with corresponding Jones matrices; thus, non-depolarizing Mueller matrices are also called Mueller–Jones matrices.5

An ideal polarizer is non-depolarizing; when the incident beam is polarized, the exiting beam is polarized. Similarly, an ideal retarder is non-depolarizing. The non-depolarizing Mueller matrices comprise the Mueller matrices for the matrix product of all arbitrary sequences of diattenuation and retardance. A Mueller–Jones matrix must satisfy the following condition for all θ and η,

$$DoP(\mathbf{M}\cdot \mathbf{S})=DoP\left(\left(\begin{array}{llll}{M}_{00}& M_{01}& M_{02}& M_{03}\\ M_{10}& M_{11}& M_{12}& M_{13}\\ M_{20}& M_{21}& M_{22}& M_{23}\\ M_{30}& M_{31}& M_{32}& M_{33}\end{array}\right)\cdot \left(\begin{array}{l}1\\ \cos 2\theta \cos \eta \\ \sin 2\theta \cos \eta \\ \sin \eta \end{array}\right)\right)=1.$$6.73

6.73

One necessary, but not sufficient, condition for non-depolarizing Mueller matrices is6

$$\text{Tr}(\mathbf{M}\cdot {\mathbf{M}}^{\text{T}})=4M_{00}^2,$$6.74

6.74

where Tr is the trace of a matrix, the sum of the diagonal elements. Tr (M · MT) equals the sum of the squares of all the matrix elements,

$$\begin{array}{l}\text{Tr}(\mathbf{M}\cdot {\mathbf{M}}^T)=M_{00}^2+M_{01}^2+M_{02}^2+M_{03}^2+M_{10}^2+M_{11}^2+M_{12}^2+M_{13}^2\\ +M_{20}^2+M_{21}^2+M_{22}^2+M_{23}^2+M_{30}^2+M_{31}^2+M_{32}^2+M_{33}^2.\end{array}$$6.75

6.75

In a typical imaging optical system, depolarization is an undesirable characteristic for lens and mirror surfaces, filters, and polarization elements. Depolarization is associated with scattering, and optical surfaces are carefully fabricated and coated to minimize scattering. Depolarization is generally very small in high-quality optical surfaces. Thus, the majority of optical surfaces are well described by non-depolarizing Mueller matrices.

6.11Depolarization

Depolarization is the reduction of the degree of polarization (DoP) of light. Depolarization was first described by David Brewster in 1815.7 In the Mueller calculus, depolarization can be pictured as a coupling of polarized into unpolarized light. For polarized incident light, the exiting Stokes parameters for a depolarized beam have a DoP < 1 and can be mathematically separated into a fully polarized and an unpolarized set of Stokes parameters.

Optical elements with considerable scattering depolarize light to some extent. Similarly rough metal surfaces, painted surfaces, and natural surfaces such as rock, grass, and sand partially depolarize light. Integrating spheres (Figure 6.19) are frequently used in the laboratory to depolarize light into nearly unpolarized light as are thin plates of opal. Milk and other turbid fluids do a fine job depolarizing. When an integrating sphere is illuminated with laser light, the exiting light is a speckle pattern and the exiting polarization state is scrambled and resembles a depolarized beam. Figure 6.20 shows an ellipse map of a simulated polarized speckle pattern. The x-polarized light forms one speckle pattern and the y-polarized light forms a second pattern. These combine and are described by a Jones vector at each point of a randomly polarized field. For monochromatic illumination, the pattern is polarized at each point, but averaging the Stokes parameters over an area would yield to a depolarized measurement.

Figure 6.19An integrating sphere is a good depolarizer. The interior is coated with a highly reflective diffuse reflector. Most light rays scatter multiple times before reaching the output port, ensuring that each exiting ray contains light from a large variety of paths and scattering angles.

Figure 6.20A simulation of a speckle pattern from a laser-illuminated integrating sphere or another depolarizer. Laser speckle patterns are polarized at each point because the beam is monochromatic. However, the amplitude and polarization state vary from speckle to speckle. Here, the polarization ellipses are mapped over a contour plot of the intensity distribution where brown is darkest and white is brightest. Speckle patterns have large regions of low flux and much smaller bright regions, the speckles.8,9

The Mueller matrix has 16 independent degrees of freedom (DoF), as shown in Table 6.3. Of the 16 degrees of freedom, one corresponds to loss, three to diattenuation, and three to retardance. The remaining nine degrees of freedom describe depolarization.

Table 6.3Degrees of Freedom in Polarization Matrices

	Jones Matrix	Mueller Matrix
Transmission	1	1
Absolute phase	1	0
Diattenuation	3	3
Retardance	3	3
Depolarization	0	9

The depolarization associated with the three diagonal elements of the Mueller matrix, M11, M22, and M33, tend to be the most significant of the nine degrees of freedom. The ideal depolarizer Mueller matrix, ID, transforms all incident beams into unpolarized light,

$$\mathbf{ID}\cdot \mathbf{S}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\cdot \left(\begin{array}{l}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{l}{S}_0\\ 0\\ 0\\ 0\end{array}\right).$$6.76

6.76

Only unpolarized light exits such a device. Although this matrix is an idealization, some devices such as integrating spheres approach this limit of nearly complete depolarization.

The partial depolarizer Mueller matrix depolarizes all incident states equally,

$$\mathbf{PD}\cdot \mathbf{S}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& d& 0& 0\\ 0& 0& d& 0\\ 0& 0& 0& d\end{array}\right)\cdot \left(\begin{array}{l}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=(1-d)\left(\begin{array}{l}{S}_0\\ 0\\ 0\\ 0\end{array}\right)+d\left(\begin{array}{l}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right).$$6.77

6.77

All fully polarized incident states exit partially polarized with DoP(PD·S) = d. The diagonal depolarizer Mueller matrixDD represents a variable partial depolarizer; the degree of polarization of the exiting light is a function of the incident state, with an exiting DoP of a for S1, b for S2, and c for S3,

$$\mathbf{DD}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& a& 0& 0\\ 0& 0& b& 0\\ 0& 0& 0& c\end{array}\right).$$6.78

6.78

Physically, depolarization is closely related to scattering and usually has its origin in retardance or diattenuation, which are rapidly varying in time, space, or wavelength.

6.11.1The Depolarization Index and the Average Degree of Polarization

Two depolarization metrics, the depolarization index and the average degree of polarization, have been introduced to describe the degree to which a Mueller matrix depolarizes incident states.10–12

The depolarization index DI(M) is the Euclidian distance, indicated by $\Vert \Vert$, of the normalized Mueller matrix M/M00 from the ideal depolarizer:

$$DI(\mathbf{M})=\Vert \frac{\mathbf{M}}{M_{00}}-\mathbf{ID}\Vert =\frac{\sqrt{\left({\displaystyle \sum _{i,j}{M}_{ij}^2}\right)-M_{00}^2}}{\sqrt{3}{M}_{00}}.$$6.79

6.79

DI(M) varies from 0 for the ideal depolarizer to 1 for all non-depolarizing Mueller matrices, including all pure diattenuators, pure retarders, and any sequences composed from them. The form of the depolarization index equation is similar to the equation for degree of polarization for Stokes parameters.

The average degree of polarization, or AverageDoP, is the arithmetic mean of the degree of polarization of the exiting light for polarized incident light averaged over the Poincaré sphere,

$$AverageDoP(\mathbf{M})=\frac{{\displaystyle \underset{0}{\overset{\pi }{\int }}{\displaystyle \underset{-\pi /2}{\overset{\pi /2}{\int }}DoP(\mathbf{M}\cdot \mathbf{S}(\theta ,\eta ))\text{cos }(\eta )d\eta d\theta }}}{2\pi },$$6.80

6.80

where, for Stokes parameters S(θ, η), θ is the orientation of the major axis, and η is the latitude in radians on the Poincare sphere,

$$\mathbf{S}(\theta ,\eta )=\left(\begin{array}{l}1\\ \text{cos }(2\theta )\text{cos }(\eta )\\ \text{sin }(2\theta )\text{ cos }(\eta )\\ \text{sin}(\eta )\end{array}\right).$$6.81

6.81

The AverageDoP varies from 0 to 1, summarizing the depolarizing properties in a single number. When AverageDoP is equal to 1, the exiting light is always completely polarized, indicating a non-depolarizing Mueller matrix. Values near 1 indicate a small amount of depolarization. When AverageDoP equals 0, the exiting light is completely depolarized; only unpolarized light exits the interaction. The DI and AverageDoP of an aperture with different polarizers are shown in Figure 6.21; a detailed calculation is shown in Example 6.9.

Figure 6.21Depolarization index (red) and average depolarization (green) of a family of Mueller matrices formed from an aperture covered with a fraction α of horizontal polarizer and a fraction 1 − α of vertical polarizer.

The DI and the AverageDoP are usually close in value. The AverageDoP is the easier metric to understand; it provides the mean DoP of the exiting light averaged over the Poincaré sphere, the expected value. The DI has a clear geometric meaning in the Mueller matrix configuration space, being the fractional distance of a Mueller matrix along a line segment from the ideal depolarizer to the hypersphere of non-depolarizing Mueller matrices.

6.11.2Degree of Polarization Surfaces and Maps

Insight into the nine degrees of freedom of depolarization can be found by examining the variations of DoP with incident state using degree of polarization surfaces and degree of polarization maps.13

The DoP surface for a Mueller matrix, M, is formed by moving each set of normalized Stokes parameters, S, formed from the three Stokes parameters (S1, S2, S3) on the surface of the Poincaré sphere radially inward to a distance DoP(S′ = M⋅S) from the origin. It is plotted for all incident S on the surface of the Poincaré sphere,

$$DoP\text{Surface}(\mathbf{M},\mathbf{S})=\frac{\sqrt{{S^{\prime }}_1{(\mathbf{M},\mathbf{S})}^2+{S^{\prime }}_2{(\mathbf{M},\mathbf{S})}^2+{S^{\prime }}_3{(\mathbf{M},\mathbf{S})}^2}}{{S^{\prime }}_0(\mathbf{M},\mathbf{S})}(S_1,S_2,S_3).$$6.82

6.82

The DoP surface results from the product of a scalar, the DoP, and a vector, (S1, S2, S3), forming a three-dimensional surface. For a non-depolarizing Mueller matrix, the exiting DoP is 1 for all incident states; hence, the DoP surface is the unit sphere; the Poincaré sphere does not shrink in this case.

The DoP map represents DoP as a contour plot on a flat map of the Poincaré sphere. To create the DoP map, the surface of the Poincaré sphere is parameterized in terms of the orientation of the major axis θ and the degree of circular polarization DoCP as

$$\mathbf{S}(\theta ,DoCP)=\left(\begin{array}{l}1\\ \text{cos}(2\theta )\text{cos}({\text{sin}}^{-1}DoCP)\\ \text{sin}(2\theta )\text{cos}({\text{sin}}^{-1}DoCP)\\ DoCP\end{array}\right).$$6.83

6.83

The DoP of the light exiting the Mueller matrix is plotted as a contour plot as a function of θ and DoCP. The DoP surface and the DoP map represent the same information, the exiting DoP as a function of incident polarization state.

A depolarizing example is the Mueller matrix M1 formed from an aperture covered one-half with a horizontal polarizer and the other half with a vertical polarizer,

$${\mathbf{M}}_1=\frac{1}{2}\left(\begin{array}{llll}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)+\frac{1}{2}\left(\begin{array}{rrrr}\hfill 1& \hfill -1& \hfill 0& \hfill 0\\ \hfill -1& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right)=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$6.84

6.84

The DoP surface and DoP map for M1 are plotted in Figure 6.22. The two maxima of DoP occur when (1) vertically polarized light is incident and only vertically polarized light exits and (2) when horizontally polarized light is incident and only horizontally polarized light exits. For incident states residing on a circle of the Poincaré sphere midway between horizontal and vertical, the exiting light is completely depolarized and is the incoherent sum of half horizontally polarized and half vertically polarized light.

Figure 6.22(Left) Degree of polarization surface and (right) degree of polarization map for M1, a Mueller matrix formed from the sum of horizontal and 90° linear polarizers.

As another example, consider an element with a Mueller matrix M2 where one-half of the aperture is covered with a horizontal linear polarizer and the other half is covered by a 45° linear polarizer,

$${\mathbf{M}}_2=\frac{1}{2}\left(\begin{array}{llll}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)+\frac{1}{2}\left(\begin{array}{llll}1& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right)=\left(\begin{array}{llll}1& 0.5& 0.5& 0\\ 0.5& 0.5& 0& 0\\ 0.5& 0& 0.5& 0\\ 0& 0& 0& 0\end{array}\right).$$6.85

6.85

The DoP surface and DoP map for M2 are plotted in Figure 6.23. The two states of maximum DoP, (1,−1,0,0) and (1,0,−1,0), are the states that are blocked by one of the two polarizers. It is seen that the maxima of the DoP map do not need to be orthogonal.

Figure 6.23(Left) Degree of polarization map and (right) degree of polarization surface for M2, a Mueller matrix formed from the sum of horizontal and 45° linear polarizers.

DoP maps and surfaces may exhibit one or two maxima and one or two minima. These maxima or minima may also be degenerate for entire circles of incident states around the Poincaré sphere.

6.11.3Testing for Physically Realizable Mueller Matrices

To be physically realizable,14,15 a Mueller matrix needs to operate on all possible sets of Stokes parameters producing valid exiting Stokes parameters; otherwise, the Mueller matrix is not physically realizable. Only a subset of 4 × 4 matrices are physically realizable Mueller matrices. Mueller matrices that are not physically realizable can be an issue in measured Mueller matrices. Polarimeters always have noise. Retarders, diattenuators, and their combinations lie right on the boundary between physically realizable and not physically realizable Mueller matrices. Thus, a small amount of noise can shift a Mueller matrix with a depolarization index of one into the non-physically realizable region. Goldstein provides algorithms and examples of moving non-physically realizable Mueller matrices to the closest physically realizable matrices.16

For a Mueller matrix to be physically realizable, several conditions must apply. First, the output flux must be non-negative, ${S^{\prime }}_0\ge 0$. Second, for all input polarization states, the output degree of polarization must lie between zero and one,

$$0\le DoP(\mathbf{M}\cdot \mathbf{S})\le 1.$$6.86

6.86

If the degree of polarization is greater than one, the output state is non-physical. One test for physicality is to test that the depolarization index lies between zero and one,

$$0\le DI(\mathbf{M})\le 1.$$6.87

6.87

Several comprehensive and systematic tests have been developed. Givens and Kostinski17 evaluate the physicality of Mueller matrices as follows. Defining the Lorentz metric matrix G as

$$\mathbf{G}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right),$$6.88

6.88

then the Mueller coherence matrix,

$$\mathbf{D}=\mathbf{G}{\mathbf{M}}^{\text{T}}\mathbf{G}\mathbf{M}=\left(\begin{array}{llll}{a}_{00}& a_{01}& a_{02}& a_{03}\\ a_{10}& a_{11}& a_{12}& a_{13}\\ a_{20}& a_{21}& a_{22}& a_{23}\\ a_{30}& a_{31}& a_{32}& a_{33}\end{array}\right),$$6.89

6.89

where

$$\begin{array}{l}{a}_{00}=M_{00}^2-M_{10}^2-M_{20}^2-M_{30}^2\hfill \\ a_{01}=M_{00}{M}_{01}-M_{10}{M}_{11}-M_{20}{M}_{21}-M_{30}{M}_{31}\hfill \\ a_{02}=M_{00}{M}_{02}-M_{10}{M}_{12}-M_{20}{M}_{22}-M_{30}{M}_{32}\hfill \\ a_{03}=M_{00}{M}_{03}-M_{10}{M}_{13}-M_{20}{M}_{23}-M_{30}{M}_{33}\hfill \end{array}$$

$$\begin{array}{l}{a}_{10}=-M_{00}{M}_{01}+M_{10}{M}_{11}+M_{20}{M}_{21}+M_{30}{M}_{31}\hfill \\ a_{11}=-M_{01}^2+M_{11}^2+M_{21}^2+M_{31}^2\hfill \\ a_{12}=-M_{01}{M}_{02}+M_{11}{M}_{12}+M_{21}{M}_{22}+M_{31}{M}_{32}\hfill \\ a_{13}=-M_{01}{M}_{03}+M_{11}{M}_{13}+M_{21}{M}_{23}+M_{31}{M}_{33}\hfill \end{array}$$

$$\begin{array}{l}{a}_{20}=-M_{00}{M}_{02}+M_{10}{M}_{12}+M_{20}{M}_{22}+M_{30}{M}_{32}\hfill \\ a_{21}=-M_{01}{M}_{02}+M_{11}{M}_{12}+M_{21}{M}_{22}+M_{31}{M}_{32}\hfill \\ a_{22}=-M_{02}^2+M_{12}^2+M_{22}^2+M_{32}^2\hfill \\ a_{23}=-M_{02}{M}_{03}+M_{12}{M}_{13}+M_{22}{M}_{23}+M_{32}{M}_{33}\hfill \end{array}$$

$$\begin{array}{l}{a}_{30}=-M_{00}{M}_{03}+M_{10}{M}_{13}+M_{20}{M}_{23}+M_{30}{M}_{33}\hfill \\ a_{31}=-M_{01}{M}_{03}+M_{11}{M}_{13}+M_{21}{M}_{23}+M_{31}{M}_{33}\hfill \\ a_{32}=-M_{02}{M}_{03}+M_{12}{M}_{13}+M_{22}{M}_{23}+M_{32}{M}_{33}\hfill \\ a_{33}=-M_{03}^2+M_{13}^2+M_{23}^2+M_{33}^2\hfill \end{array}$$

is calculated. In order to be a physically realizable Mueller matrix, (1) all the eigenvalues of D must be real, and (2) the eigenvalue associated with the largest eigenvector must be a valid set of Stokes parameters.

Example 6.8Testing Mueller Matrices for Physicality

Consider an example of a family of depolarizing matrices M3 with a single degree of freedom β,

$${\mathbf{M}}_3=\left(\begin{array}{llll}1& 0.1& 0& 0\\ \beta & 0.9& 0& 0\\ 0& 0& 0.9& 0\\ 0& 0& 0& 0.8\end{array}\right).$$6.90

6.90

Over what range of β, the M10 element, is M3 a valid physically realizable Mueller matrix? The associated Mueller coherence matrix

$${\mathbf{D}}_3=\left(\begin{array}{llll}1-{\beta }^2& 0.1-0.9\beta & 0& 0\\ -0.1+0.9\beta & 0.8& 0& 0\\ 0& 0& 0.81& 0\\ 0& 0& 0& 0.64\end{array}\right)$$6.91

6.91

has eigenvalues

$$\lambda =\{\begin{array}{l}0.81,\\ 0.64,\\ 0.9-0.5{\beta }^2-0.5\sqrt{0.72\beta -3.64{\beta }^2+{\beta }^4},\\ 0.9-0.5{\beta }^2+0.5\sqrt{0.72\beta -3.64{\beta }^2+{\beta }^4.}\end{array}$$6.92

6.92

The discriminant 0.72β − 3.64β2 + β4 ≥ 0 for the range 0 ≤ β ≤ 0.2; hence, all four eigenvalues are only real over this range of β. It is readily verified that the associated eigenvector for the fourth eigenvalues has physical Stokes parameters over this range. Thus, M3 is a valid Mueller matrix only when 0 ≤ β ≤ 0.2.

6.11.4Weak Depolarizing Elements

Section 6.9 described the Mueller matrices for weak non-depolarizing elements in the vicinity of the identity matrix in terms of three diattenuation and three retarding degrees of freedom. This formalism readily extends to the nine remaining degrees of freedom and thus the nine forms of depolarization.18 One simple way to view the depolarizing degrees of freedom is as follows. For weak Mueller matrices, the depolarizing degrees of freedom are associated with the following matrix element combinations

$$\mathbf{WDepol}=\left(\begin{array}{llll}1& e_1& e_2& e_3\\ -e_1& 1-g_1& f_3& -f_2\\ -e_2& f_3& 1-g_2& f_1\\ -e_3& -f_2& f_1& 1-g_3\end{array}\right),$$6.93

6.93

where the nine forms have been grouped into three families. g1, g2, and g3 are associated with the diagonal and labeled diagonal depolarization. e1, e2, and e3 are associated with antisymmetric values in the diattenuation elements and are labeled amplitude depolarization. f1, f2, and f3 are associated with symmetric values in the retardance elements and are labeled phase depolarization. Each of the three forms, amplitude depolarization, phase depolarization, and diagonal depolarization, has a term associated with S1, a term associated with S2, and a term associated with S3. When the e, f, and g components are close to 0, the matrix WDepol is close to the identity matrix and the depolarization can be regarded as weak. Because of the symmetry of these elements, WDepol has no diattenuation and no retardance. As the strength of the depolarization increases, the terms do not remain purely linear. Depolarizing matrices outside the weak limit with particular depolarizing terms can be generated by raising WDepol to arbitrary powers,

$$\mathbf{WDepo}{l}^N={\left(\begin{array}{llll}1& e_1& e_2& e_3\\ -e_1& 1-g_1& f_3& f_2\\ -e_2& f_3& 1-g_2& f_1\\ -e_3& f_2& f_1& 1-g_3\end{array}\right)}^N.$$6.94

6.94

Care must be taken, because the matrices of Equation 6.93 are only physically realizable in the limit as the coefficients equal to zeros; thus, Equation 6.93 is slightly non-physical, and the resulting strong matrices need small adjustments for physicality.

6.11.5The Addition of Mueller Matrices

The matrix product of Mueller matrices shown in Equation 6.6 represents sequences of polarization elements. On the other hand, the addition of Mueller matrices represents polarization elements side by side sharing an aperture. Whenever two different non-depolarizing Mueller matrices are added, depolarization must be introduced. In general, across an aperture, different Stokes parameters exit and are combined, reducing the degree of polarization.

Mueller matrix functions can be integrated over time or space to simulate time- or space-varying polarization processes, as in the following two examples.

Example 6.9Two Polarizers over an Aperture

Consider an aperture where one fraction, α, of it is covered by a horizontal linear polarizer and the remainder (1 − α) of it is filled with a vertical linear polarizer. The resulting Mueller matrix is

$$\alpha \mathbf{HLP}+(1-\alpha )\mathbf{VLP}=\frac{\alpha }{2}\left(\begin{array}{llll}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)+\frac{(1-\alpha )}{2}\left(\begin{array}{llll}1& -1& 0& 0\\ -1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)=\frac{1}{2}\left(\begin{array}{llll}1& 2\alpha -1& 0& 0\\ 2\alpha -1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$6.95

6.95

When α = ½, the aperture is half horizontally polarized and half vertically polarized; the combination acts as a partial depolarizer with a depolarization index of $1\text{/}\sqrt{3}$. S2 and S3 are depolarized but S1 exits unchanged.

Example 6.10A Spinning Quarter Wave Linear Retarder

A rapidly spinning quarter wave linear retarder, similar to the case described in Figure 6.14, has a time-averaged Mueller matrix

$${\displaystyle \underset{0}{\overset{2\pi }{\int }}\mathbf{QWLR}(\theta )}\text{d}\theta ={\displaystyle \underset{0}{\overset{2\pi }{\int }}\left(\begin{array}{llll}1& 0& 0& 0\\ 0& {\text{cos}}^{2}2\theta & \text{cos}2\theta \text{sin}2\theta & -\text{sin}2\theta \\ 0& \text{cos}2\theta \text{sin}2\theta & {\text{sin}}^{2}2\theta & \text{cos}2\theta \\ 0& \text{sin}2\theta & -\text{cos}2\theta & 0\end{array}\right)}\text{d}\theta =\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \frac{1}{2}& 0& 0\\ 0& 0& \frac{1}{2}& 0\\ 0& 0& 0& 0\end{array}\right)$$6.96

6.96

with a depolarization index of $1\text{/}\sqrt{6}$ and an average DoP of π/8, which differ by ≈0.016. This Mueller matrix completely depolarizes circularly polarized light and reduces the degree of linear polarization by half.

6.12Relating Jones and Mueller Matrices

6.12.1Transforming Jones Matrices into Mueller Matrices Using Tensor Product

Along with Mueller matrices, Jones matrices (Chapter 5) form a very useful representation of sample polarization, particularly because Jones matrices have simpler properties and are more easily manipulated and interpreted. The complication in mapping Mueller matrices onto Jones matrices and vice versa is that Mueller matrices cannot represent absolute phase and Jones matrices cannot represent depolarization. Only non-depolarizing Mueller matrices or the Mueller–Jones matrices have corresponding Jones matrices. All Jones matrices have a corresponding Mueller matrix. However, since the absolute phase is not represented in Mueller matrices, many Jones matrices with different absolute phase can be mapped to the same Mueller matrix.

Both Jones matrices and Mueller matrices can calculate the polarization properties of sequences of non-depolarizing interactions, the effect of cascading a series of diattenuators and retarders. When this same polarization element sequence is calculated by Jones matrices and alternatively by Mueller matrices, the answer contains the same diattenuating and retarding properties. Either method is suitable.

Math Tip 6.1The Tensor Product

The tensor product of two 2 × 2 matrices, A⊗ B, is the matrix5,19–21

$$\begin{array}{l}\mathbf{A}\otimes \mathbf{B}=\left(\begin{array}{ll}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)\otimes \left(\begin{array}{ll}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right)\\ \phantom{A\otimes B}=\left(\begin{array}{ll}{a}_{11}\left(\begin{array}{ll}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right)& a_{12}\left(\begin{array}{ll}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right)\\ a_{21}\left(\begin{array}{ll}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right)& a_{22}\left(\begin{array}{ll}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right)\end{array}\right)\\ \phantom{A\otimes B}=\left(\begin{array}{llll}{a}_{11}{b}_{11}& a_{12}{b}_{12}& a_{12}{b}_{11}& a_{12}{b}_{12}\\ a_{11}{b}_{21}& a_{11}{b}_{22}& a_{12}{b}_{21}& a_{12}{b}_{22}\\ a_{21}{b}_{11}& a_{21}{b}_{12}& a_{22}{b}_{11}& a_{22}{b}_{12}\\ a_{21}{b}_{21}& a_{21}{b}_{22}& a_{22}{b}_{21}& a_{22}{b}_{22}\end{array}\right).\end{array}$$6.97

6.97

A Jones matrix J is transformed into the equivalent Mueller matrix M by a unitary transformation of the tensor productJT ⊗ J† and the unitary matrix U,

$$\mathbf{U}=\frac{1}{\sqrt{2}}\left(\begin{array}{llll}1& 0& 0& 1\\ 1& 0& 0& -1\\ 0& 1& 1& 0\\ 0& i& -i& 0\end{array}\right)={({\mathbf{U}}^{-1})}^{†}.$$6.98

6.98

Note that each of the four rows are each flattened versions of the Pauli matrices, but the fourth row is the negative of σ3.3 The Mueller matrix corresponding to Jones matrix J is4

$$\mathbf{M}=\mathbf{U}\cdot ({\mathbf{J}}^{*}\otimes \mathbf{J})\cdot {\mathbf{U}}^{-1}.$$6.99

6.99

All Jones matrices of the form J′ = e−iϕJ transform to the same Mueller matrix. Consider the Jones matrix with its complex elements expressed in polar coordinate form,

$$\mathbf{J}=\left(\begin{array}{ll}{j}_{xx}\hfill & j_{xy}\hfill \\ j_{yx}\hfill & j_{yy}\hfill \end{array}\right)=\left(\begin{array}{ll}{\rho }_{xx}{e}^{-i{\varphi }_{xx}}& {\rho }_{xy}{e}^{-i{\varphi }_{xy}}\\ {\rho }_{yx}{e}^{-i{\varphi }_{yx}}& {\rho }_{yy}{e}^{-i{\varphi }_{yy}}\end{array}\right).$$6.100

6.100

The tensor product J* ⊗ J is calculated as Equation 6.97,

$$\begin{array}{l}(\mathbf{J}*\otimes \mathbf{J})=\left(\begin{array}{ll}{\rho }_{xx}{e}^{i{\varphi }_{xx}}\left(\begin{array}{ll}{\rho }_{xx}{e}^{-i{\varphi }_{xx}}& {\rho }_{xy}{e}^{-i{\varphi }_{xy}}\\ {\rho }_{yx}{e}^{-i{\varphi }_{yx}}& {\rho }_{yy}{e}^{-i{\varphi }_{yy}}\end{array}\right)& {\rho }_{xy}{e}^{i{\varphi }_{xy}}\left(\begin{array}{ll}{\rho }_{xx}{e}^{-i{\varphi }_{xx}}& {\rho }_{xy}{e}^{-i{\varphi }_{xy}}\\ {\rho }_{yx}{e}^{-i{\varphi }_{yx}}& {\rho }_{yy}{e}^{-i{\varphi }_{yy}}\end{array}\right)\\ {\rho }_{yx}{e}^{i{\varphi }_{yx}}\left(\begin{array}{ll}{\rho }_{xx}{e}^{-i{\varphi }_{xx}}& {\rho }_{xy}{e}^{-i{\varphi }_{xy}}\\ {\rho }_{yx}{e}^{-i{\varphi }_{yx}}& {\rho }_{yy}{e}^{-i{\varphi }_{yy}}\end{array}\right)& {\rho }_{yy}{e}^{i{\varphi }_{yy}}\left(\begin{array}{ll}{\rho }_{xx}{e}^{-i{\varphi }_{xx}}& {\rho }_{xy}{e}^{-i{\varphi }_{xy}}\\ {\rho }_{yx}{e}^{-i{\varphi }_{yx}}& {\rho }_{yy}{e}^{-i{\varphi }_{yy}}\end{array}\right)\end{array}\right)\\ \phantom{(J*\otimes J)}=\left(\begin{array}{llll}{\rho }_{xx}^2& {\rho }_{xx}{\rho }_{xy}{e}^{i\left({\varphi }_{xx}-{\varphi }_{xy}\right)}& {\rho }_{xy}{\rho }_{xx}{e}^{i\left({\varphi }_{xy}-{\varphi }_{xx}\right)}& {\rho }_{xy}^2\\ {\rho }_{xx}{\rho }_{yx}{e}^{i\left({\varphi }_{xx}-{\varphi }_{yx}\right)}& {\rho }_{xx}{\rho }_{yy}{e}^{i\left({\varphi }_{xx}-{\varphi }_{yy}\right)}& {\rho }_{xy}{\rho }_{yx}{e}^{i\left({\varphi }_{xy}-{\varphi }_{yz}\right)}& {\rho }_{xy}{\rho }_{yy}{e}^{i\left({\varphi }_{xy}-{\varphi }_{yy}\right)}\\ {\rho }_{yx}{\rho }_{xx}{e}^{i\left({\varphi }_{yx}-{\varphi }_{xx}\right)}& {\rho }_{yx}{\rho }_{xy}{e}^{i\left({\varphi }_{yx}-{\varphi }_{xy}\right)}& {\rho }_{yy}{\rho }_{xx}{e}^{i\left({\varphi }_{yy}-{\varphi }_{xx}\right)}& {\rho }_{yy}{\rho }_{xy}{e}^{i\left({\varphi }_{yy}-{\varphi }_{xy}\right)}\\ {\rho }_{yx}^2& {\rho }_{yx}{\rho }_{yy}{e}^{i\left({\varphi }_{yx}-{\varphi }_{yy}\right)}& {\rho }_{yy}{\rho }_{yx}{e}^{i\left({\varphi }_{yy}-{\varphi }_{yx}\right)}& {\rho }_{yy}^2\end{array}\right).\end{array}$$6.101

6.101

When J* ⊗ J is transformed by U and U−1, it gives the Mueller matrix elements as in Equation 6.102.

$$\begin{array}{l}\mathbf{U}\cdot (\mathbf{J}*\otimes \mathbf{J})\cdot {\mathbf{U}}^{-1}=\\ \left(\begin{array}{l}{M}_{00}\\ M_{01}\\ M_{02}\\ M_{03}\\ M_{10}\\ M_{11}\\ M_{12}\\ M_{13}\\ M_{20}\\ M_{21}\\ M_{22}\\ M_{23}\\ M_{30}\\ M_{31}\\ M_{32}\\ M_{33}\end{array}\right)=\left(\begin{array}{l}\frac{1}{2}\left({\rho }_{xx}^2+{\rho }_{xy}^2+{\rho }_{yx}^2+{\rho }_{yy}^2\right)\\ \frac{1}{2}\left({\rho }_{xx}^2-{\rho }_{xy}^2+{\rho }_{yx}^2-{\rho }_{yy}^2\right)\\ {\rho }_{xx}{\rho }_{xy}\text{cos}({\varphi }_{xx}-{\varphi }_{xy})+{\rho }_{yx}{\rho }_{yy}\text{cos}({\varphi }_{yx}-{\varphi }_{yy})\\ {\rho }_{xx}{\rho }_{xy}\text{sin}({\varphi }_{xx}-{\varphi }_{xy})+{\rho }_{yx}{\rho }_{yy}\text{sin}({\varphi }_{yx}-{\varphi }_{yy})\\ \frac{1}{2}\left({\rho }_{xx}^2+{\rho }_{xy}^2-{\rho }_{yx}^2-{\rho }_{yy}^2\right)\\ \frac{1}{2}\left({\rho }_{xx}^2-{\rho }_{xy}^2-{\rho }_{yx}^2+{\rho }_{yy}^2\right)\\ {\rho }_{xx}{\rho }_{xy}\text{cos}({\varphi }_{xx}-{\varphi }_{xy})-{\rho }_{yx}{\rho }_{yy}\text{cos}({\varphi }_{yx}-{\varphi }_{yy})\\ {\rho }_{xx}{\rho }_{xy}\text{sin}({\varphi }_{xx}-{\varphi }_{xy})-{\rho }_{yx}{\rho }_{yy}\text{sin}({\varphi }_{yx}-{\varphi }_{yy})\\ {\rho }_{xx}{\rho }_{yx}\text{cos}({\varphi }_{xx}-{\varphi }_{yx})+{\rho }_{xy}{\rho }_{yy}\text{cos}({\varphi }_{xy}-{\varphi }_{yy})\\ {\rho }_{xx}{\rho }_{yx}\text{cos}({\varphi }_{xx}-{\varphi }_{yx})-{\rho }_{xy}{\rho }_{yy}\text{cos}({\varphi }_{xy}-{\varphi }_{yy})\\ {\rho }_{xy}{\rho }_{yx}\text{cos}({\varphi }_{xy}-{\varphi }_{yx})+{\rho }_{xx}{\rho }_{yy}\text{cos}({\varphi }_{xx}-{\varphi }_{yy})\\ -{\rho }_{xy}{\rho }_{yx}\text{sin}({\varphi }_{xy}-{\varphi }_{yx})+{\rho }_{xx}{\rho }_{yy}\text{sin}({\varphi }_{xx}-{\varphi }_{yy})\\ -{\rho }_{xx}{\rho }_{yx}\text{sin}({\varphi }_{xx}-{\varphi }_{yx})-{\rho }_{xy}{\rho }_{yy}\text{sin}({\varphi }_{xy}-{\varphi }_{yy})\\ -{\rho }_{xx}{\rho }_{yx}\text{sin}({\varphi }_{xx}-{\varphi }_{yx})+{\rho }_{xy}{\rho }_{yy}\text{sin}({\varphi }_{xy}-{\varphi }_{yy})\\ -{\rho }_{xy}{\rho }_{yx}\text{sin}({\varphi }_{xy}-{\varphi }_{yx})-{\rho }_{xx}{\rho }_{yy}\text{sin}({\varphi }_{xx}-{\varphi }_{yy})\\ -{\rho }_{xy}{\rho }_{yx}\text{cos}({\varphi }_{xy}-{\varphi }_{yx})+{\rho }_{xx}{\rho }_{yy}\text{cos}({\varphi }_{xx}-{\varphi }_{yy})\end{array}\right).\end{array}$$6.102

6.102

Example 6.11Numerical Example of Converting a Jones Matrix to a Mueller Matrix

Consider a Jones matrix

$$\mathbf{J}=\left(\begin{array}{ll}\frac{1}{4}& \frac{1}{4}\\ \frac{e^{i\frac{\pi }{2}}}{\sqrt{2}}& \frac{e^{-i\frac{\pi }{2}}}{\sqrt{2}}\end{array}\right)=\left(\begin{array}{ll}\frac{1}{4}& \frac{1}{4}\\ \frac{i}{\sqrt{2}}& \frac{-i}{\sqrt{2}}\end{array}\right).$$6.103

6.103

This Jones matrix represents a linear diattenuator oriented at 45° with a diattenuation of 0.778, followed by a 120° retarder oriented at 22.5° and ellipticity = −0.391 ≈ π/8. In the Pauli matrix decomposition, this retarder has a linear retardance of 88° oriented at 22.5° and a circular retardance of 81°. The retarder is then followed by another diattenuator with diattenuation 0.778 oriented at 0°. This particular J provides simple transformation without involving trivial Jones or Mueller matrices.

The tensor product

$$({\mathbf{J}}^{\ast }\otimes \mathbf{J})=\left(\begin{array}{ll}\frac{1}{4}\left(\begin{array}{ll}\frac{1}{4}& \frac{1}{4}\\ \frac{e^{i\frac{\pi }{2}}}{\sqrt{2}}& \frac{e^{-i\frac{\pi }{2}}}{\sqrt{2}}\end{array}\right)& \frac{1}{4}\left(\begin{array}{ll}\frac{1}{4}& \frac{1}{4}\\ \frac{e^{i\frac{\pi }{2}}}{\sqrt{2}}& \frac{e^{-i\frac{\pi }{2}}}{\sqrt{2}}\end{array}\right)\\ \frac{e^{-i\frac{\pi }{2}}}{\sqrt{2}}\left(\begin{array}{ll}\frac{1}{4}& \frac{1}{4}\\ \frac{e^{i\frac{\pi }{2}}}{\sqrt{2}}& \frac{e^{-i\frac{\pi }{2}}}{\sqrt{2}}\end{array}\right)& \frac{e^{i\frac{\pi }{2}}}{\sqrt{2}}\left(\begin{array}{ll}\frac{1}{4}& \frac{1}{4}\\ \frac{e^{i\frac{\pi }{2}}}{\sqrt{2}}& \frac{e^{-i\frac{\pi }{2}}}{\sqrt{2}}\end{array}\right)\end{array}\right),$$6.104

6.104

which contracts to

$$\begin{array}{l}({\mathbf{J}}^{\ast }\otimes \mathbf{J})=\left(\begin{array}{llll}{\left(\frac{1}{4}\right)}^2& {\left(\frac{1}{4}\right)}^2& {\left(\frac{1}{4}\right)}^2& {\left(\frac{1}{4}\right)}^2\\ \frac{e^{i\frac{\pi }{2}}}{4\sqrt{2}}& \frac{e^{-i\frac{\pi }{2}}}{4\sqrt{2}}& \frac{e^{i\frac{\pi }{2}}}{4\sqrt{2}}& \frac{e^{-i\frac{\pi }{2}}}{4\sqrt{2}}\\ \frac{e^{-i\frac{\pi }{2}}}{4\sqrt{2}}& \frac{e^{-i\frac{\pi }{2}}}{4\sqrt{2}}& \frac{e^{i\frac{\pi }{2}}}{4\sqrt{2}}& \frac{e^{i\frac{\pi }{2}}}{4\sqrt{2}}\\ \frac{e^{-i\frac{\pi }{2}}}{\sqrt{2}}\frac{e^{i\frac{\pi }{2}}}{\sqrt{2}}& \frac{e^{-i\frac{\pi }{2}}}{\sqrt{2}}\frac{e^{-i\frac{\pi }{2}}}{\sqrt{2}}& \frac{e^{i\frac{\pi }{2}}}{\sqrt{2}}\frac{e^{i\frac{\pi }{2}}}{\sqrt{2}}& \frac{e^{i\frac{\pi }{2}}}{\sqrt{2}}\frac{e^{-i\frac{\pi }{2}}}{\sqrt{2}}\end{array}\right)\\ \phantom{(J^{\ast }\otimes J)}=\left(\begin{array}{llll}\frac{1}{16}& \frac{1}{16}& \frac{1}{16}& \frac{1}{16}\\ \frac{i}{4\sqrt{2}}& -\frac{i}{4\sqrt{2}}& \frac{i}{4\sqrt{2}}& -\frac{i}{4\sqrt{2}}\\ -\frac{i}{4\sqrt{2}}& -\frac{i}{4\sqrt{2}}& \frac{i}{4\sqrt{2}}& \frac{i}{4\sqrt{2}}\\ \frac{1}{2}& \frac{-1}{2}& \frac{-1}{2}& \frac{1}{2}\end{array}\right).\end{array}$$6.105

6.105

The corresponding Mueller matrix is

$$\mathbf{M}=\mathbf{U}\cdot ({\mathbf{J}}^{\ast }\otimes \mathbf{J})\cdot {\mathbf{U}}^{-1}=\left(\begin{array}{llll}\frac{9}{16}& 0& -\frac{7}{16}& 0\\ -\frac{7}{16}& 0& \frac{9}{16}& 0\\ 0& 0& 0& -\frac{1}{2\sqrt{2}}\\ 0& -\frac{1}{2\sqrt{2}}& 0& 0\end{array}\right).$$6.106

6.106

6.12.2Conversion of Jones Matrices to Mueller Matrices Using Pauli Matrices

An equivalent method to convert Jones matrices to Mueller matrices utilizes dot products with two Pauli matrices to determine each Mueller matrix element, Mi,j,

$$M_{ij}=\frac{1}{2}\text{Tr}({\mathbf{\sigma }}_i\cdot \mathbf{J}*\cdot {\mathbf{\sigma }}_j\cdot {\mathbf{J}}^{\text{T}}),$$6.107

6.107

where Tr is the trace of the matrix and

$$\begin{array}{llll}{\mathbf{\sigma }}_0=\left(\begin{array}{ll}1& 0\\ 0& 1\end{array}\right),& {\mathbf{\sigma }}_1=\left(\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 0& \hfill -1\end{array}\right),& {\mathbf{\sigma }}_2=\left(\begin{array}{ll}0& 1\\ 1& 0\end{array}\right),\text{and}& {\mathbf{\sigma }}_3=\left(\begin{array}{ll}0& -i\\ i& 0\end{array}\right)\end{array}$$6.108

6.108

are the identity matrix and Pauli matrices.

Example 6.12Calculation of a Component in the Jones Matrix

Calculation of the M13 component from the Jones matrix in Equation 6.100.

$$\begin{array}{l}{M}_{13}=\frac{1}{2}\text{Tr}({\sigma }_1\cdot J*\cdot {\sigma }_3\cdot J^{\text{T}})\\ \phantom{M_{13}}=\frac{1}{2}\text{Tr}\left(\left(\begin{array}{ll}1& 0\\ 0& -1\end{array}\right)\cdot \left(\begin{array}{ll}{\rho }_{xx}{e}^{i{\varphi }_{xx}}& {\rho }_{xy}{e}^{i{\varphi }_{xy}}\\ {\rho }_{yx}{e}^{i{\varphi }_{yx}}& {\rho }_{yy}{e}^{i{\varphi }_{yy}}\end{array}\right)\cdot \left(\begin{array}{ll}0& -i\\ i& 0\end{array}\right)\cdot \left(\begin{array}{ll}{\rho }_{xx}{e}^{-i{\varphi }_{xx}}& {\rho }_{yx}{e}^{-i{\varphi }_{yx}}\\ {\rho }_{xy}{e}^{-i{\varphi }_{xy}}& {\rho }_{yy}{e}^{-i{\varphi }_{yy}}\end{array}\right)\right)\\ \phantom{M_{13}}={\rho }_{xx}{\rho }_{xy}\sin ({\varphi }_{xx}-{\varphi }_{xy})+{\rho }_{yx}{\rho }_{yy}\sin ({\varphi }_{yy}-{\varphi }_{yx}),\end{array}$$6.109

6.109

which matches the result from the first method in Equation 6.102.

6.12.3Transforming Mueller Matrices into Jones Matrices

Non-depolarizing Mueller matrices are transformed into the equivalent Jones matrices using the following relations:

$$\mathbf{J}=\left(\begin{array}{ll}{j}_{xx}\hfill & j_{xy}\hfill \\ j_{yx}\hfill & j_{yy}\hfill \end{array}\right)=\left(\begin{array}{l}{\rho }_{xx}e{}^{-i{\varphi }_{xx}}{\rho }_{xy}e{}^{-i{\varphi }_{xy}}\\ {\rho }_{yx}e{}^{-i{\varphi }_{yx}}{\rho }_{yy}e{}^{-i{\varphi }_{yy}}\end{array}\right),$$6.110

6.110

where the amplitudes are

$$\begin{array}{l}{\rho }_{xx}=\sqrt{\frac{M_{00}+M_{01}+M_{10}+M_{11}}{2}},{\rho }_{xy}=\sqrt{\frac{M_{00}-M_{01}+M_{10}-M_{11}}{2}},\\ {\rho }_{yx}=\sqrt{\frac{M_{00}+M_{01}-M_{10}-M_{11}}{2}},{\rho }_{yy}=\sqrt{\frac{M_{00}-M_{01}-M_{10}+M_{11}}{2}},\end{array}$$6.111

6.111

and the relative phases are

$$\{\begin{array}{l}{\varphi }_{xx}-{\varphi }_{xy}={\text{tan}}^{-1}\left(\frac{M_{03}+M_{13}}{M_{02}+M_{12}}\right),\\ {\varphi }_{yx}-{\varphi }_{xx}={\text{tan}}^{-1}\left(\frac{M_{30}+M_{31}}{M_{20}+M_{21}}\right),\\ {\varphi }_{yy}-{\varphi }_{xx}={\text{tan}}^{-1}\left(\frac{M_{32}-M_{23}}{M_{22}+M_{33}}\right).\end{array}$$6.112

6.112

The phase ϕxx is not determined and is the “reference phase” for the other ϕ. Because of the large number of Mueller matrix elements, and the constraints between non-depolarizing elements, these equations are not unique.

A special case occurs when jxx = 0; both the numerator and denominator of the tan−1 are 0 and the phase equations in Equation 6.112 fail. The transformation equations can be recast in closely related forms and use the phase of another Jones matrix element as the “reference phase.”

Example 6.13Cascading Two Linear Diattenuators with a Rotation

Two identical linear diattenuators (partial linear polarizers) have Tmax = 1.0 and Tmin = 0.707. Find the Mueller matrix M(θ) as the transmission axis of the second diattenuator is rotated.

1. What is the maximum and minimum diattenuation as a function of θ?

2. Do any of the combinations form elliptical diattenuators?

3. What is the polarizance as a function of θ?

The mathematical expressions are relatively complex, so it only makes sense to work this problem using a calculator or computer software.

1. By multiplying the two linear diattenuator expressions, it yields the complex expression

   $$\begin{array}{l}M(\theta )=LD(T_{\max },T_{\min },\theta )\cdot LD(T_{\max },T_{\min },0)\hfill \\ \phantom{M(\theta )}=\frac{1}{2}\left(\begin{array}{llll}A& B\cos 2\theta & B\sin 2\theta & 0\\ B\cos 2\theta & A{\cos }^{2}2\theta +C{\sin }^{2}2\theta & A-C\cos 2\theta \sin 2\theta & 0\\ B\sin 2\theta & (A-C)\cos 2\theta \sin 2\theta & C{\cos }^{2}2\theta +A\sin {}^{2}2\theta & 0\\ 0& 0& 0& C\end{array}\right)\frac{1}{2}\left(\begin{array}{llll}A& B& 0& 0\\ B& A& 0& 0\\ 0& 0& C& 0\\ 0& 0& 0& C\end{array}\right)\hfill \\ \phantom{M(\theta )}=\frac{1}{4}\left(\begin{array}{llll}{A}^2+B^2\cos 2\theta & 2AB{\cos }^2\theta & BC\sin 2\theta & 0\\ 2B{\cos }^2\theta (C+(A-C)\cos 2\theta )& B^2\cos 2\theta +A^2{\cos }^{2}2\theta +AC{\sin }^{2}2\theta & \frac{1}{2}(A-2C)\sin 4\theta & 0\\ B(A+(A-C)\cos 2\theta )\sin 2\theta & (B^2+A(A-C)\cos 2\theta )\sin 2\theta & C(C{\cos }^{2}2\theta +A{\sin }^{2}2\theta )& 0\\ 0& 0& 0& C^2\end{array}\right)\hfill \end{array}$$6.113

   6.113

   where

   $$\begin{array}{lll}A=T_{\text{max}}+T_{\text{min}},& B=T_{\text{max}}-T_{\text{min}},& C=2\sqrt{T_{\text{max}}{T}_{\min }}.\end{array}$$6.114

   6.114

   Substituting Tmax = 1.0 and Tmin = 0.707 = $1\text{/}\sqrt{2}$ yields expressions for the 16 elements:

   $$\mathbf{M}=\left(\begin{array}{l}{M}_{00}\\ M_{01}\\ M_{02}\\ M_{03}\\ M_{10}\\ M_{11}\\ M_{12}\\ M_{13}\\ M_{20}\\ M_{21}\\ M_{22}\\ M_{23}\\ M_{30}\\ M_{31}\\ M_{32}\\ M_{33}\end{array}\right)=\frac{1}{8}\left(\begin{array}{l}3+2\sqrt{2}+(3-2\sqrt{2})\text{cos}2\theta \\ 2{\text{cos}}^2\theta \\ -2^{3/4}\left(-2+\sqrt{2}\right)\text{sin}2\theta \\ 0\\ \text{cos}2\theta +{\text{cos}}^{2}2\theta +{22}^{1/4}\left(-1+\sqrt{2}\right){\text{sin}}^{2}2\theta \\ \left(3-2\sqrt{2}\right)\text{cos}2\theta +\left(3+2\sqrt{2}\right){\text{cos}}^{2}2\theta +{22}^{1/4}\left(1+\sqrt{2}\right){\text{sin}}^{2}2\theta \\ -\frac{\left(-2\sqrt{2}+{22}^{3/4}\right)\text{sin}4\theta }{2^{1/4}}\\ 0\\ (1+(1+{22}^{1/4}-{22}^{3/4})\text{cos}2\theta )\text{sin}2\theta \\ -\left(-3+2\sqrt{2}+\left(-3+{22}^{1/4}-2\sqrt{2}+{22}^{3/4}\right)\cos 2\theta \right)\sin 2\theta \\ {22}^{3/4}\left(2^{3/4}{\text{cos}}^{2}2\theta +\left(1+\frac{1}{\sqrt{2}}\right){\text{sin}}^{2}2\theta \right)\\ 0\\ 0\\ 0\\ 0\\ 4\frac{1}{\sqrt{2}}\end{array}\right)$$6.115

   6.115

2. The diattenuation of an individual plate is

   $$D=\frac{1-\frac{1}{\sqrt{2}}}{1+\frac{1}{\sqrt{2}}}=\frac{\sqrt{2}-1}{\sqrt{2}+1}=3-2\sqrt{2}\approx 0.1716,$$6.116

   6.116

   a rather weak partial polarizer. The maximum diattenuation occurs when the transmission axes of the two plates are parallel.

   $$\begin{array}{l}{T}_{\text{max}}=1\times 1=1\text{and}{T}_{\text{min}}=0.7\times 0.7=1\text{/}2,\hfill \\ D_{\text{max}}=(1-1\text{/}2)/(1+1\text{/}2)=1\text{/}3.\hfill \end{array}$$6.117

   6.117

   Likewise, the minimum diattenuation occurs when the transmission axes are perpendicular and the diattenuations cancel,

   $$\begin{array}{l}{T}_{\text{max}}=1\times 0.7=0.7\text{and}{T}_{\text{min}}=0.7\times 1=0.7,\hfill \\ D_{\text{min}}=(0.7-0.7)/(0.7+0.7)=0.\hfill \end{array}$$6.118

   6.118

3. Since M03 = 0, the diattenuation is always linear.

4. The polarizance P(θ) is purely linear since M30 = 0, and is graphed in Figure 6.24.

   $$\begin{array}{l}P(\theta )=\frac{\sqrt{M_{10}^2+M_{20}^2}}{M_{00}}=\frac{\sqrt{{\left(\cos 2\theta +{\cos }^{2}2\theta +{22}^{1/4}\left(-1+\sqrt{2}\right){\sin }^{2}2\theta \right)}^2+{\left(\left(1+\left(1+{22}^{1/4}-{22}^{3/4}\right)\cos 2\theta \right)\sin 2\theta \right)}^2}}{32\left(\frac{1}{4}{\left(1+\frac{1}{\sqrt{2}}\right)}^2+\frac{1}{4}{\left(1-\frac{1}{\sqrt{2}}\right)}^2\cos 2\theta \right)}\\ \phantom{P(\theta )}=\frac{\sqrt{-13+12\sqrt{2}+4\text{cos}2\theta +(17-12\sqrt{2})\text{cos}4\theta }}{8\sqrt{2}\left(\frac{1}{4}{\left(1+\frac{1}{\sqrt{2}}\right)}^2+\left(\frac{1}{4}{\left(1-\frac{1}{\sqrt{2}}\right)}^2\text{cos}2\theta \right)\right)}.\end{array}$$6.119

   6.119

   When unpolarized light is incident, the exiting light is unpolarized for θ = 90° + n 180°.

Figure 6.24Polarizance of the diattenuator combination as the second element is rotated.

6.13Ray Tracing with Mueller Matrices

Mueller matrices are frequently used for ray tracing, particularly incoherent ray tracing. Jones matrices are preferred for ray tracing imaging systems because they contain an absolute phase. Many systems need an incoherent ray trace, such as illumination optics and simulations of scattering systems. In illumination systems, for example, automobile headlamps that are faceted into many lenslets, light rays reach each part of the illuminated surface from many different paths with very different optical path lengths. The interference does not need to be calculated between these different rays, since the quality of a spherical wavefront is not being evaluated. Thus, summing Mueller matrices or Stokes parameters is appropriate. Similarly, in simulating light propagating through aerosols in the atmosphere, animal tissue, turbid media, or scattered light calculations in optical systems, the flux, direction, and polarization state are needed, but the optical path length is not used for determining interference. For these purposes, Mueller matrices are commonly used to describe these interactions.

The s-component leaving one interaction is usually not aligned with the s-component of the next interaction, which often happens with skew rays; Mueller rotation operations (Equation 6.11) need to be applied at each of these interactions. Each ray will generally have a different geometrical transformation, which is explored in Chapter 17, and this will need to be addressed to combine the Mueller matrices from different ray paths, which, for example, combine at a detector pixel.

The Mueller matrices for scattering can also be used in a Mueller matrix ray trace. There are many different models for the Mueller matrices for scattering from various surfaces or volume scattering.22,23 These Mueller matrix calculations will not be described here. A library of Mueller matrices for scattering is available from the National Institute of Standards and Technology.24

6.13.1Mueller Matrices for Refraction

Reflections and refractions at homogenous and isotropic interfaces (e.g., typical glass or metal interfaces) have s- and p-eigenpolarizations. The associated polarization properties are a combination of diattenuation and retardance, whose axes are aligned with the s- and p-planes. Since the light changes direction, a different coordinate system is needed to represent the incident and exiting Stokes parameters in global coordinates.

In what follows, s is aligned with +S1 and p is aligned with −S1. Ts is the s-intensity transmittance and Tp is the p-intensity transmittance. The retardance between the s- and p-states is δ. Ts, Tp, and δ are determined from Fresnel equations (Chapter 8) or from a thin film coating calculation such as Sections 13.2 and 13.3.1.4 The Mueller matrix representing refraction is the product of the diattenuator and retarder Mueller matrices,

$$\begin{array}{l}LDR(D,\delta ,0)=\frac{1}{2}\left(\begin{array}{llll}{T}_s+T_p& T_s-T_p& 0& 0\\ T_s-T_p& T_s+T_p& 0& 0\\ 0& 0& 2\sqrt{T_s{T}_p}\cos \delta & 2\sqrt{T_s{T}_p}\sin \delta \\ 0& 0& -2\sqrt{T_s{T}_p}\sin \delta & 2\sqrt{T_s{T}_p}\cos \delta \end{array}\right)\\ \phantom{LDR(D,\delta ,0)}=\frac{T_s+T_p}{2}\left(\begin{array}{llll}1& D& 0& 0\\ D& 1& 0& 0\\ 0& 0& \sqrt{1-D^2}\cos \delta & \sqrt{1-D^2}\sin \delta \\ 0& 0& -\sqrt{1-D^2}\sin \delta & \sqrt{1-D^2}\cos \delta \end{array}\right),\end{array}$$6.120

6.120

where D is the diattenuation. For transmission at an uncoated interface, the retardance δ is 0. For thin film-coated interfaces, such as anti-reflection coatings or beam splitter coatings, the retardance is non-zero. Refraction Mueller matrices are homogeneous; the eigenpolarizations, the s- and p-polarizations, are orthogonal.

6.13.2Mueller Matrices for Reflection

In most optics notation, including this chapter, a sign change occurs in the coordinate system after reflection to maintain a right-handed coordinate system for changing direction of the propagation vector. Imagine a Stokes polarimeter measuring in transmission. Now, to measure in reflection, the polarimeter rotates around about the vertical y-axis, perpendicular to the z-axis along which the incident light propagates. By moving the polarimeter around the y-axis to measure the beam in reflection, it is seen the 45° component has changed sign. In addition, the helicity (i.e., handedness) of all circular and elliptical states also changes sign upon the reflection, since right circular polarization reflects as left circular polarization, and vice versa.

After reflection, the S2 component of Stokes parameters (linearly polarized light at 45°/135°) and the S3 component (circularly polarized light) change sign. The S2 component changes sign during reflection (diffuse or specular) because the z-component of the light propagation vector (the component parallel to the sample surface normal) changes sign. To maintain a right-handed coordinate system, one of the transverse coordinates must change sign as well. Choosing x, then spatial coordinates (x, y, z) switch to (−x, y, −z) after reflection or backscatter from a sample; z is the direction of propagation before reflection that changes to −z after reflection. The change of coordinates dictates that a beam polarized at an angle of 45° that reflects polarized in the same global plane is described as having a 135° orientation in the coordinates after reflection.

Rs is the s-intensity reflectance, Rp is the p-intensity reflectance, and δ is the retardance as calculated from Fresnel equations or thin film equations. The Mueller matrix for reflection is

$$M_{\text{refl}}(R_s,R_p,\delta )=\frac{1}{2}\left(\begin{array}{llll}{R}_s+R_p& R_s-R_p& 0& 0\\ R_s-R_p& R_s+R_p& 0& 0\\ 0& 0& -2\sqrt{R_s{R}_p}\cos \delta & -2\sqrt{R_s{R}_p}\sin \delta \\ 0& 0& 2\sqrt{R_s{R}_p}\sin \delta & -2\sqrt{R_s{R}_p}\cos \delta \end{array}\right).$$6.121

6.121

With this convention for reflection, the equation for rotating the Mueller matrix, M, of a sample measured by a polarimeter in a reflection configuration about its normal changes to

$$\begin{array}{l}{M}_{\text{refl}}(\theta )=R(\theta )\cdot M_{\text{refl}}\cdot R(\theta )\\ \left(\begin{array}{llll}1& 0& 0& 0\\ 0& \cos \theta & -\sin \theta & 0\\ 0& \sin \theta & \cos \theta & 0\\ 0& 0& 0& 1\end{array}\right)\cdot \left(\begin{array}{llll}{M}_{00}& M_{01}& M_{02}& M_{03}\\ M_{10}& M_{11}& M_{12}& M_{13}\\ M_{20}& M_{21}& M_{22}& M_{23}\\ M_{30}& M_{31}& M_{32}& M_{33}\end{array}\right)\cdot \left(\begin{array}{llll}1& 0& 0& 0\\ 0& \cos \theta & -\sin \theta & 0\\ 0& \sin \theta & \cos \theta & 0\\ 0& 0& 0& 1\end{array}\right),\end{array}$$6.122

6.122

compared to Equation 6.11 for Mueller matrices in transmission. For example, the Mueller matrix of a transmission polarizer with its transmission axis oriented at 20° and the Mueller matrix of a reflection polarizer oriented at 20° (for the incident light) are different since polarized light exits the reflection polarizer oriented at −20° in the reflection coordinates (20° in the incident coordinates). In essence, the reflection polarizer Mueller matrix is analyzing at 20° but polarizing at −20°. For the special cases of linear polarizer matrices oriented at 0° or 90° and linear retarders oriented at 0° or 90°, this transformation results in the same Mueller matrices for transmission and reflection.

The normalized reflection Mueller matrices for weakly polarizing reflecting samples, those with diattenuation, retardance, and depolarization close to zero, are close to the Mueller matrix for an ideal reflector,

$$M_{\text{refl}}=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right).$$6.123

6.123

Mrefl is also the Mueller matrix for reflection or scatter in the absence of polarization effects.

6.14The Origins of the Mueller Matrix

Hans Müller, a Swiss-born professor of physics, developed the Mueller matrix concept in the early 1940s4 and has the Mueller matrix named in his honor. He described it in a classified report,25 later declassified, and in detail in notes for his physics course in 1943.26 His only publication related to the 4 × 4 matrix that bears his name was a short meeting abstract for the Optical Society of America.27 His graduate student Nathan Parke developed the Mueller matrix concept further in his dissertation28 and a related publication.1 R. Clark Jones then compared the Mueller and Jones matrices in one of his series of papers on the Jones calculus29 and made reference to the “recently declassified” report that Müller had authored.

Before all of this work in 1929, Paul Soleillet had developed a set of four linear equations to relate incident and exiting Stokes parameters, equivalent to the Mueller matrix except that a matrix formalism was not employed.30 These linear equations were also placed into a matrix formalism by Francis Perrin in 1942.31

6.15Problem Sets

1. Show that the following Mueller matrices are equal:

   1. LR(δ, 45°) and LR(2π − δ, 135°)

   2. LP(θ) and LP(θ + π)

   3. LD(1, t, π/8) and LD(1, t, 9π/8)

2. 1. Find the Mueller matrices for linear polarizers at 0°, 45°, and 90°: LP(0), LP(π/4), and LP(π/2).

   2. Calculate LP(0)·LP(π/2) and LP(0)·LP(π/4)·LP(π/2).

   3. Find the Mueller matrices for quarter wave linear retarders with fast axes at 0°, 45°, and 90°: LR(π/2, 0), LR(π/2, π/4), and LR(π/2, π/2).

   4. Do LP(0) and LR(π/2, 0) commute, that is, is LP(0)·LR(π/2, 0) = LR(π/2, 0)·LP(0) true?

   5. Do LP(0) and LR(π/2, π/4) commute?

3. Find LR(π/2, 0)·LP(π/4)·LR(π/2, π/2). Show this is a circular polarizer.

4. Show that the Mueller matrix for a quarter wave linear retarder with fast axis orientation θ equals the Mueller matrix for a three-quarter wave linear retarder with fast axis orientation θ ± π/2.

5. Show that the diattenuator Mueller matrix equation (Equation 6.59) reduces to the linear polarizer equation (Equation 6.37) when Tmax = 1 and Tmin = 0.

6. Analyze the properties of the Mueller matrix $M=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& -1\end{array}\right)$.

   1. Identify it as a polarizer or retarder Mueller matrix.

   2. How does the transmitted flux vary with the incident Stokes parameters?

   3. What is the action on circularly polarized light?

   4. Plot the action on linearly polarized light as the axis changes.

   Use the eigenvalues and associated eigenvectors in your description.

7. Show that the elliptically polarized Stokes parameters S(θ, η) = (1, cos 2θ cos η, sin 2θ cos η, sin η) is an eigenpolarization of EP(θ, η).

8. Calculate the Mueller matrix EP(θ, η) for an elliptical polarizer in Equation 6.38 that transmits the Stokes parameters (1,cos 2θ cos η, sin 2θ cos η, sin η).

   1. Find the linear retarder Mueller matrix U that transforms the linearly polarized state (1, cos 2θ, sin 2θ, 0) into the target state (1, cos 2θ cos η, sin 2θ cos η, sin η).

   2. Starting with the linear polarizer Mueller matrix LP(θ), apply a unitary transform using U to transform LP(θ) into EP(θ, η).

   3. Verify the eigenvalues and eigenvectors of EP(θ, η).

9. 1. How are the eigenvalues of an ideal circular retarder Mueller matrix equation (Equation 6.24) related to the retardance δ?

   2. How are the eigenvalues of a linear retarder Mueller matrix related to its retardance δ?

10. Multiply the Mueller matrices for an LR(π/2,0), followed by an LR(π/2, π/4), and finally a CR(π/2). Determine the retardance and the eigenpolarizations.

11. 1. Describe the set of all Mueller matrices that transform arbitrary incident Stokes parameters into unpolarized light. Which elements must be zero? Which elements can be non-zero?

    2. How many degrees of freedom does this set have?

    3. Which of these Mueller matrices completely block certain incident states of polarization?

12. Consider the partially polarized beam with the Stokes parameters S = (1, s1, s2, s3).

    1. Which diattenuator Mueller matrix MD transforms S into unpolarized light?

    2. Explain why this diattenuator can reduce, not increase, the degree of polarization.

    3. What is the relationship between the degree of polarization of S and the diattenuation of MD?

    4. What is the effect of MD on the orthogonal polarization state with the same DoP?

13. What is the periodicity in orientation θ of the equation for the Mueller matrix of a half wave linear retarder in Equation 6.25?

14. Show that the Mueller matrix for a half wave elliptical retarder with retardance components (δH, δ45, δR) in Equation 6.28,

    $$HWR(r_1,r_2,r_3)=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& -1+2r_1^2& 2r_2{r}_1& 2r_1{r}_3\\ 0& 2r_2{r}_1& -1+2r_2^2& 2r_2{r}_3\\ 0& 2r_3{r}_1& 2r_2{r}_3& -1+2r_3^2\end{array}\right),$$

    

    where $\sqrt{r_1^2+r_2^2+r_3^2}=1$, $r_1=\frac{{\delta }_{\text{H}}}{\pi }$, $r_2=\frac{{\delta }_{45}}{\pi }$, and $r_3=\frac{{\delta }_{\text{R}}}{\pi }$, is the same Mueller matrix as for an elliptical retarder with orthogonal retardance components (−δH, −δ45, −δR).

15. Develop a set of equations for transforming Mueller matrices into Jones matrices for the special case where Jones matrix element jxx = 0.

16. Using the equation for the general elliptical retarder in Equation 6.28, show that the following two Mueller matrices are equal, $ER({\delta }_{\text{H}},{\delta }_{45},{\delta }_{\text{R}})=ER\left(\frac{2\pi ({\delta }_{\text{H}},{\delta }_{45},{\delta }_{\text{R}})}{\sqrt{{\delta }_{\text{H}}^2+{\delta }_{45}^2+{\delta }_{\text{R}}^2}}-({\delta }_{\text{H}},{\delta }_{45},{\delta }_{\text{R}})\right)$.

17. Given a polarized beam with major axis orientation θ and latitude on the Poincaré sphere of η, find the orientations for a half wave linear retarder that will convert the state into the orthogonal state.

18. Create an example of a retarder Mueller matrix multiplying a set of Stokes parameters that produces an arc of states propagating through a retarder on the Poincaré sphere following the left-hand rule. With your left thumb aligned along the fast axis of a retarder emerging from the Poincaré sphere, the motion of states around the rotation axis follows the direction of the left-hand fingers.

19. Using the Poincaré sphere, deduce all of the retarders that will convert horizontal linearly polarized light into 45° linearly polarized light. What is the minimum retardance that will perform the transformation? How is the fast axis oriented?

20. A linear retarder with retardance 90° with fast axis oriented at 45° is placed between a horizontal quarter wave retarder and a vertical quarter wave linear retarder. The assembly forms a circular retarder. What is the retardance of the assembly? How do the properties of the assembly vary as the fast axis of the central quarter wave retarder is rotated from 0° to 180°?

21. Show by reasoning with the Poincaré sphere that, when a linear retarder with retardance δ oriented at 45° is placed between a horizontal quarter wave linear retarder and a vertical quarter wave linear retarder, the assembly forms a circular retarder.

    1. What is the retardance of the retarder? Explain the magnitude of the retardance using the Poincaré sphere.

    2. Work the problem with Jones matrices, and show that this sequence performs a unitary transformation on the middle retarder.

22. Consider a linear diattenuator LD(Tmax, Tmin, π/4). What is the condition on the incident and exiting Stokes parameters such that the entering and exiting degrees of polarization are equal? Explain how a non-polarizing element, the diattenuator, can decrease the degree of polarization of certain input states.

23. For a partially linear polarized beam with Stokes parameters, (S0, S1, S2, 0) where $S_0>\sqrt{S_1^2+S_2^2}$, show that the maximum amount of light that can be transmitted through an ideal polarizer is more than the polarized flux $\sqrt{S_1^2+S_2^2}$. How much more flux can be transmitted?

24. Determine the linear diattenuator that converts the state (4, 1, 1, 0) into unpolarized light. Let Tmax = 1 for simplicity. Do diattenuators increase the degree of polarization? Explain why the degree of polarization is reduced here.

25. Derive the Mueller matrix LR(δ,0) for a linear retarder with a retardance δ radians and a fast axis orientation at 0. Set up and solve a set of at least 16 linear equations for the matrix elements Mij. Each set of four equations is of the form

    $$LR(\delta ,0°)\cdot S=S^{\prime }=\left(\begin{array}{llll}{M}_{00}& M_{01}& M_{02}& M_{03}\\ M_{10}& M_{11}& M_{12}& M_{13}\\ M_{20}& M_{21}& M_{22}& M_{23}\\ M_{30}& M_{31}& M_{32}& M_{33}\end{array}\right)\cdot \left(\begin{array}{l}{S}_0\\ S_1\\ S_2\\ S_3\end{array}\right)=\left(\begin{array}{l}{S^{\prime }}_0\\ {S^{\prime }}_1\\ {S^{\prime }}_2\\ {S^{\prime }}_3\end{array}\right),$$

    

    and should relate appropriate incident and exiting Stokes parameters. Jones calculus is needed to generate incident and exiting pairs of states with the needed S2 and S3 components.

26. Calculate and compare the depolarization index and the average degree of polarization for the family of diatteunator Mueller matrices Ms as a function of s, t, and u, where

    $$Ms=\left(\begin{array}{llll}1& s& t& u\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$

    

27. 1. Calculate the depolarization index for the family of Mueller matrices Mα connecting the ideal depolarizer and the identity matrix as a function of the mixing ratio α

       $$\mathbf{M}\mathbf{\alpha }=\alpha I+(1-\alpha )ID=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \alpha & 0& 0\\ 0& 0& \alpha & 0\\ 0& 0& 0& \alpha \end{array}\right).$$

       

    2. Calculate the average degree of polarization for the family of Mueller matrices Mα connecting the ideal depolarizer and the identity matrix as a function of the mixing ratio α.

    3. Calculate the depolarization index for the family of partial depolarizers Mαγ that depolarize linearly polarized light to a DoP of α and circularly polarized light to γ, as a function of α and γ, where

       $$M\mathbf{\alpha }\mathbf{\gamma }=\left(\begin{array}{llll}1& 0& 0& 0\\ 0& \alpha & 0& 0\\ 0& 0& \alpha & 0\\ 0& 0& 0& \gamma \end{array}\right).$$

       

    4. Calculate the average degree of polarization for Mαγ as a function of the mixing ratio α.

    5. For which values of α and γ are the depolarization index and the average degree of polarization equal?

    6. Which is greater, the depolarization index or the average degree of polarization?

    7. In the neighborhood of the ideal depolarizer, which varies linearly and which varies quadratically?

28. Which forms of depolarization are expected due to the following polarization element defects? Add or integrate Mueller matrices to evaluate the following properties:

    1. A crystalline quarter wave linear retarder with a 45° fast axis has a wedge, a thickness variation. For example, add 89° and 91° retarders, or integrate from 89° to 91°.

    2. A polymer linear retarder with a 0° fast axis has a fast axis variation about 0° due to the stretching process.

    3. A sheet polarizer with a 45° transmission axis has transmission axis variation about 45° due to the stretching process.

    4. A sheet polarizer with a 90° transmission axis has diattenuation variations due to thickness variations.

    5. A sheet polarizer with a 45° transmission axis has transmission axis variation about 45° due to the stretching process.

29. A system in production needs a quarter wave linear retarder to convert right circularly polarized light into horizontal linearly polarized light. The final polarization state should lie within a 0.03 radian circle of horizontal linearly polarized light. Use the Poincaré sphere to tolerate the allowable error in retarder magnitude and orientation.

References

1N. G. Parke, Optical algebra, J. Math. Phys. 28.1 (1949): 131–139.

2R. A. Chipman, Mueller matrices, in Handbook of Optics, Vol. I, Chapter 14, ed. M. Bass, McGraw-Hill (2009).

3P. W. Maymon and R. A. Chipman, Linear Polarization Sensitivity Specifications for Spaceborne Instruments, San Diego International Society for Optics and Photonics (1992).

4W. A. Shurcliff, Polarized Light, Production and Use, Cambridge, MA: Harvard University Press (1962).

5C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach, Wiley-Interscience (1998), 228 pp.

6J. J. Gil and E. Bernabeu, Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix, Optik 76.2 (1987): 67–71.

7D. Brewster, Experiments on the depolarisation of light as exhibited by various mineral, animal, and vegetable bodies, with a reference of the phenomena to the general principles of polarization, Philos. Trans. R. Soc. London 105 (1815): 29–53.

8J. Christopher Dainty, ed. Laser Speckle and Related Phenomena, Vol. 9, Springer Science & Business Media (2013).

9R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach, Vol. 10, Springer Science & Business Media (2012).

10J. J. Gil, and E. Bernabeu, A depolarization criterion in Mueller matrices, J. Mod. Opt. 32.3 (1985): 259–261.

11J. J. Gil and E. Bernabeu, Depolarization and polarization indices of an optical system, J. Mod. Opt. 33.2 (1986): 185–189.

12R. A. Chipman, Depolarization index and the average degree of polarization, Appl. Opt. 44.13 (2005): 2490–2495.

13B. DeBoo, J. Sasian, and R. Chipman, Degree of polarization surfaces and maps for analysis of depolarization, Opt. Express, 12(20) (2004): 4941–4958.

14S. R. Cloude, Conditions for the physical realisability of matrix operators in polarimetry, Proc. Soc. Photo - Opt. Instrum. Eng. 1166 (1989): 177–185.

15S. Cloude, Polarisation: Applications in Remote Sensing, Oxford University Press (2010).

16D. Goldstein, Polarized Light, 3rd edition, Section 8.4.2. New York, NY: Marcel Dekker (2011).

17C. R. Givens and A. B. Kostinski, A simple necessary and sufficient condition on physically realizable Mueller matrices, J. Mod. Opt. 40.3 (1993): 471–481.

18H. D. Noble, S. C. McClain, and R. A. Chipman, Mueller matrix roots depolarization parameters, Appl. Opt. 51 (2012): 735–744.

19R. Simon, The connection between Mueller and Jones matrices of polarization optics, Opt. Commun. 42.5 (1982): 293–297.

20K. Kim, L. Mandel et al., Relationship between Jones and Mueller matrices for random media. J. Opt. Soc. Am. A 4 (1987): 433–437.

21D. Goldstein, Polarized Light, 2nd edition New York, NY: Marcel Dekker (2003), 166 pp.

22J. C. Stover, Optical Scattering: Measurement and Analysis, SPIE Press (2012).

23T. Germer, Polarized light diffusely scattered under smooth and rough interfaces, Optical Science and Technology, SPIE’s 48th Annual Meeting, International Society for Optics and Photonics (2003).

24T. Germer (http://www.nist.gov/pml/div685/grp06/scattering_scatmech.cfm, accessed February 29, 2016).

25H. Mueller, Memorandum on the polarization optics of the photoelastic shutter, Report No. 2 of the OSRD project OEMsr-576, Nov. 15, 1943. A declassified report.

26H. Mueller, Informal notes about 1943 on Course 8.26 at Massachusetts Institute of Technology.

27H. Mueller, The foundations of optics, Proceedings of the Winter Meeting of the Optical Society of America, J. Opt. Soc. Am. 38 (1948): 661.

28N. G. Parke III, Matrix Optics, PhD thesis, Department of Physics, Massachusetts Institute of Technology (1948), 181 pp.

29R. Clark Jones, A new calculus for the treatment of optical system, V. A more general formulation, and description of another calculus, J. Opt. Soc. Am 37(2) (1947): 107–110.

30P. Soleillet, Parameters characterising partially polarized light in fluorescence phenomena, Ann. Phys. 12(10) (1929): 23–97.

31F. Perrin, Polarization of light scattered by isotropic opalescent media, J. Chem. Phys. 10.7 (1942): 415–427.

1 Polarcor is a trademark of Corning Inc.

2 The right-handed cross product can be easily verified. By placing your right-hand fingers along +S1 and sweeping them to +S2, your thumb will point toward S3; this test checks that the coordinate system is right handed.

3 In this book for Jones matrices, a positive σ3 is associated with left circular polarization (see Sections 14.6.1 and 14.6.2). In the Stokes parameters, a positive S3 is associated with right circular and elliptical polarization. Thus, a minus sign is needed for conversion between these two components.

4 In some works on Stokes parameters and Mueller matrices, a positive value of the last Stokes parameter S3 indicates left circularly polarized light, not right circularly polarized light, as it does in this book.