Advanced Mueller Ellipsometry Instrumentation and Data Analysis

Abstract

The main object of this chapter is to give an overview the possibilities offered by instruments capable of measuring full Mueller matrices in the field of optical characterization. We have chosen to call these instruments Mueller ellipsometers in order to highlight their close relation with instruments traditionally used in ellipsometry. We want to make clear to the reader the place that Mueller ellipsometry takes with respect to standard ellipsometry by showing the similarities but also the differences among these techniques, both in instrumentation and data treatment. To do so the chapter starts by a review of the optical formalisms used in standard and Mueller ellipsometry respectively. In order to highlight the particularities and the advantages brought by Mueller ellipsometry, a special section is devoted to the algebraic properties of Mueller matrices and to the description of different ways to decompose them. Matrix decompositions are used to unveil the basic polarimetric properties of a the sample when a precise model is not available. Then follows a description of the most common optical configurations used to build standard ellipsometers. Special attention is paid to show what can and what cannot be measured with them. On the basis of this knowledge it is shown the interest of measuring whole Mueller matrices, in particular for samples characterized by complex anisotropy and/or depolarization. Among the numerous optical assemblies able to measure full Mueller matrices, most of them are laboratory prototypes, and only very few have been industrialized so far. Because an extensive and comparative review of all the Mueller ellipsometric instruments developed to date is clearly out of the scope of this chapter, we limit our description to four Mueller ellipsometers, two imaging and two spectroscopic systems that have been developed by us in the past years. The technical description of the Mueller ellipsometers is accompanied by some examples of applications which, without being exhaustive, are representative of the type of analyses performed in ellipsometry, and also illustrate the advantages that can be brought by modern Mueller ellipsometers to optical metrology, materials science and biomedicine.

Appendices

Appendix 1. Mueller Matrices of Some Common Optical Retarders and Diattenuators

In the following we provide specific expressions for Mueller matrices corresponding to different types of general and fundamental optical elements.

Homogeneous Elliptic Diattenuator

The following expression corresponds to the Mueller matrix of a homogeneous elliptic diattenuator oriented with an azimuth angle $\uptheta$ respect to the laboratory axis.

$$\begin{aligned} \mathbf{M}&=\frac{\tau _P }{2}\left( {{\begin{array}{cccc} 1&{C\cos 2\Psi }&{S\cos 2\Psi \cos (\delta )}&{S\cos 2\Psi \sin (\delta )} \\ {C\cos 2\Psi }&{C^{2}+S^{2}\sin 2\Psi }&{CS\cos (\delta )\left( {1-\sin 2\Psi } \right)}&{CS\sin (\delta )\left( {1-\sin 2\Psi } \right)} \\ {S\cos 2\Psi \cos (\delta )}&{CS\cos (\delta )\left( {1-\sin 2\Psi } \right)}&{\left( {S^{2}+C^{2}\sin 2\Psi } \right)\cos ^{2}\left( \delta \right)+\sin 2\Psi \sin ^{2}\left( \delta \right)}&{S^{2}\cos \left( \delta \right)\sin (\delta )\left( {1-\sin 2\Psi } \right)} \\ {S\cos 2\Psi \sin (\delta )}&{CS\sin (\delta )\left( {1-\sin 2\Psi } \right)}&{S^{2}\cos \left( \delta \right)\sin (\delta )\left( {1-\sin 2\Psi } \right)}&{\left( {S^{2}+C^{2}\sin 2\Psi } \right)\sin ^{2}\left( \delta \right)+\sin 2\Psi \cos ^{2}\left( \delta \right)} \\ \end{array} }} \right) \\ C&=\cos \left( {2\theta } \right);\;S=\sin \left( {2\theta } \right) \end{aligned}$$

(A1.1)

In expression () the angles $\Psi$ and $\Delta$ are the ellipsometric angles with its standard meaning, and the angle $\updelta$ represents the ellipticity of the eigenvalues of the elliptic retarder.

The following Table A1.1 provides matrices representing some particular cases of the previous general formula corresponding to ideal circular polarizers, dichroic circular polarizers and an ideal elliptical polarizer elliptical oriented at particular angles respect to the laboratory axis.

Table 2.2 In the table there are summarized the eigenvectors, the Jones matrices, the Mueller matrices and the vector diattenuation for some particular cases of the general expression ()

Table A1.1 provides matrices representing some particular cases of the previous general formula corresponding to ideal linear polarizers and dichroic linear polarizers oriented at particular angles respect to the laboratory axis.

The following expression corresponds to the Mueller matrix of a homogeneous elliptic retarder oriented with an azimuth angle $\uptheta$ respect to the laboratory axis. The retardance provided by the system is $\Delta$. The phase difference between the two linear components needed to build an ellipse is given by the angle $\upvarphi$.

$$\begin{aligned} \mathbf{M}=\frac{\tau _P }{2}\left( {{\begin{array}{cccc} 1&0&0&0 \\ 0&{d^{2}-e^{2}-f^{2}+g^{2}}&{2\left( {de+fg} \right)}&{2\left( {df-eg} \right)} \\ 0&{2\left( {de-fg} \right)}&{-d^{2}+e^{2}-f^{2}+g^{2}}&{2\left( {ef+dg} \right)} \\ 0&{2\left( {df+eg} \right)}&{2\left( {ef-dg} \right)}&{-d^{2}-e^{2}+f^{2}+g^{2}} \\ \end{array} }} \right) \end{aligned}$$

(A1.2)

$$\begin{aligned}&d=\cos \left( {2\theta } \right)\sin \left( {\frac{\Delta }{2}} \right);\;e=\sin \left( {2\theta } \right)\sin \left( {\frac{\Delta }{2}} \right)\cos \left( \varphi \right);\\&f=\sin \left( {2\theta } \right)\sin \left( {\frac{\Delta }{2}} \right)\sin \left( \varphi \right);\;g=\cos \left( {\frac{\Delta }{2}} \right) \end{aligned}$$

Table A1.2 provides Mueller matrices representing some particular cases of the previous general formula corresponding to circular retarders and linear retarders oriented at particular angles respect to the x–y reference coordinate axis.

Appendix 2. Differential Matrices of Fundamental Polarimetric Properties

In this appendix we provide a detailed expression of the differential matrices corresponding to the eight fundamental polarimetric properties, written according to the $4\times 4$ Stokes formalism and the $2\times 2$ Jones formalism. We also provide the way to deduce them from the original Mueller or Stokes matrices.

Table 2.3 List of the eigenvectors, the Jones matrices, the Mueller matrices and the vector diattenuation for some particular cases of the general expression ()

Mueller matrix of an homogeneous elliptic retarder

Table 2.4 This table summarizes the eigenvalues, the Jones matrices, the Mueller matrices and the vector retardances corresponding to some particular cases of the general expression ()

According to [26, 27] the differential matrix m of a given Mueller matrix M is a $4\times 4$ matrix containing simple expressions of the fundamental polarimetric properties: isotropic refraction, $\upvarphi$ isotropic absorption, $\upalpha$, linear birefringence along the coordinate axis x–y, $\upeta$, linear dichroism along the coordinate x–y, $\upbeta$, linear birefringence along the bisectors to the coordinate axis x–y, $\upeta$, linear dichroism along the bisectors to the coordinate axis x–y, $\upgamma$, circular birefringence, $\upmu$, and circular dichroism, $\updelta$. The bisectors to the coordinate axis x–y form a coordinate axes rotated $45^{\circ }$ respect to the x–y. A particular choice of the x–y axis well adapted for ellipsometric measurements in reflection, or in transmission with tilted samples, is the p-s axis defined respect to the plane of incidence. When light propagates along the z direction in an anisotropic medium, which is considered as homogeneous in the x, y directions, the transformation of the Stokes vector at a given position z, S(z) to a Stokes vector at a given position S($\mathrm{z}+\Delta$z) can be described by the Mueller matrix $\mathbf{M}_{\mathbf{z},{{\varvec{\Delta }}}\mathbf{z}}$. The transformation can be written according to the following expression:

$$\begin{aligned} \mathbf{S}(z+\Delta z)=\mathbf{M}_{\mathbf{z},{{\varvec{\Delta }}}\mathbf{z}} \mathbf{S}(z) \end{aligned}$$

(A2.1)

Subtraction of S(z) from both sides of expression () leads to:

$$\begin{aligned} \mathbf{S}(z+\Delta z)-\mathbf{S}(z)=\mathbf{S}(z)\left( {\mathbf{M}_{\mathbf{z},{{\varvec{\Delta }}}\mathbf{z}} -\mathbf{I}} \right) \end{aligned}$$

(A2.2)

where I is the identity matrix. Clearly, if the latter expression is divided by $\Delta$z and then the it is extrapolated to the limiting case of $\Delta \mathrm{z} \rightarrow 0$, then it is possible to obtain the following expression relating the transformation of the Stokes vector:

$$\begin{aligned} \frac{d\mathbf{S}}{dz}=\mathop {\lim }\limits _{\Delta z\rightarrow 0} \frac{\left(\mathbf{M}_{\mathbf{z},{{\varvec{\Delta }}}\mathbf{z}} -\mathbf{I} \right)}{\Delta z}\mathbf{S}(z)=\mathbf{mS}(z) \end{aligned}$$

(A2.3)

The latter equation is the definition of the matrix m in the $4\times 4$ Stokes formalism. The derivation of expression () is valid for either a non-depolarizing or a depolarizing medium. The matrix m is the expression of the effect of the different optical properties of the medium on the Stokes vector when light travels a differential distance $\Delta$z. For this reason the matrix m is called the differential propagation matrix, or simply the differential matrix.

The relation of () with the expression (2.52) given in the text is easily found. The transformation of an initial Stokes vector S(0) of a beam traveling a distance z inside a medium can be written as:

$$\begin{aligned} \mathbf{S}(z)=\mathbf{M}_\mathbf{z} \mathbf{S}(0) \end{aligned}$$

(A2.4)

After differentiating the previous expression respect to z one gets:

$$\begin{aligned} \frac{d\mathbf{S}}{dz}=\frac{d\mathbf{M}_\mathbf{z} }{dz}\mathbf{S}(0) \end{aligned}$$

(A2.5)

Table 2.5 Differential matrices m and n corresponding to the eight fundamental polarimetric properties

According to () the latter expression can be rewritten as:

$$\begin{aligned} \frac{d\mathbf{S}}{dz}=\mathbf{mS}(z)=\mathbf{mM}_\mathbf{z} \mathbf{S}(0)=\frac{d\mathbf{M}_\mathbf{z} }{dz}\mathbf{S}(0) \end{aligned}$$

(A2.6)

Thus giving an analogous definition of the matrix m which is identical to (2.52):

$$\begin{aligned} \mathbf{M}_\mathbf{z}^{-1} \frac{d\mathbf{M}_\mathbf{z} }{dz}=\mathbf{m} \end{aligned}$$

(A2.7)

An analogous procedure may be performed using Jones vectors instead of Stokes vectors leading to the derivation of a $2\times 2$ matrix called n. Obviously the applicability of matrix n is restricted to non-depolarizing media.

As an illustration, of the method to obtain the expression of the differential matrices m and n, let’s consider their detailed derivation for the particular case of a homogeneous medium along the propagation direction, of total thickness z, showing linear birefringence characterized by the parameter $\upeta$. The respective Mueller and Jones matrices associated to this medium are:

$$\begin{aligned} \mathbf{M}=\left( {{\begin{array}{cccc} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&{\cos (\eta z)}&{\sin (\eta z)} \\ 0&0&{-\sin (\eta z)}&{\cos (\eta z)} \\ \end{array} }} \right)\quad \mathrm{{and}}\quad \mathbf{J}=\left( {{\begin{array}{cc} {e^{i\frac{\eta }{2}z}}&0 \\ 0&{e^{-i\frac{\eta }{2}z}} \\ \end{array} }} \right)\quad \mathrm{{where}}\quad \eta =\frac{4\pi {birr}}{\lambda } \end{aligned}$$

(A2.8)

The total retardance created by the medium, commonly expressed by the ellipsometric angle $\Delta$ is given by $\eta$ times the thickness z. The parameter $\upeta$ is the intensive retardance, also called the differential retardance, whereas $\Delta$ is the extensive retardance, which is proportional to the path that light has travelled inside the medium. The differential retardance depends of the wavelength of light, $\lambda$, and the birefringence of the medium, birr.

For a thin section of thickness $\Delta$z of the material, the corresponding Jones and Mueller matrices can be calculated expanding the terms of matrices in () in a Taylor series expansion respect to z and retaining only the first order terms.

$$\begin{aligned} \mathbf{M}_{{\varvec{\Delta }}\mathbf{z}} =\left( {{\begin{array}{cccc} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&{\eta z} \\ 0&0&{-\eta z}&1 \\ \end{array} }} \right)\quad \mathrm{{and}}\quad \mathbf{J}_{{\varvec{\Delta }}\mathbf{z}}=\left( {{\begin{array}{cc} {1+\frac{i\eta z}{2}}&0 \\ 0&{1-\frac{i\eta z}{2}} \\ \end{array} }} \right) \end{aligned}$$

(A2.9)

Then according to () after subtraction of the respective ($2\times 2$) and ($4\times 4$) unit matrices and division by z, one gets:

$$\begin{aligned} \mathbf{m}=\left( {{\begin{array}{cccc} 0&0&0&0 \\ 0&0&0&0 \\ 0&0&0&\eta \\ 0&0&{-\eta }&0 \\ \end{array} }} \right)\quad \mathrm{{and}}\quad \mathbf{n}=\frac{\eta }{2}\left( {{\begin{array}{cc} i&0 \\ 0&{-i} \\ \end{array} }} \right) \end{aligned}$$

(A2.10)

The matrices ($4\times 4$) and ($2\times 2$) m and n matrices grouped in Table A2.2 the following table are the result of applying the same procedure to each one of the eight fundamental properties: