6. Spectroscopic Ellipsometry

Readers who are unfamiliar with the basics of electromagnetic waves and their propagation in different media are suggested to read the following 2 tutorials first:

1. Electromagnetic waves
2. Wave propagation in different media

Spectroscopic ellipsometry is a common technique in material sciences which is used to determine the optical properties of transparent layers and thin metallic layers. The angle- and polarization dependent reflection and/or transmission of a layered structure is recorded and fitted to a theoretical model. In that way the thickness, refractive index (permittivity) and roughness of the investigated layer(s) can be deduced. A schematic overview of reflection based ellipsometry is presented in figure 1. The polarization of the incident wave is modulated between P and S by a rotating polarizer or a photo-elastic modulator. By performing lock-in measurements at the modulation frequency, both the amplitude and phase of the reflected beam are recorded for the two polarization states.

Ellipsometry2
Figure 1: Schematic overview of spectroscopic ellipsometry measurements. A linearly polarized plane wave is converted into an elliptically polarized wave upon reflection from the sample.

At the interfaces between the different layers in the sample, the reflection for P- and S-polarized waves is determined by the Fresnel reflection coefficients,which are in general complex numbers. Therefore, upon reflection at each interface, the amplitude and phase of the reflected wave are different for the two polarization states, resulting in an elliptically polarized wave. This explains the name spectroscopic ellipsometry: the polarization ellipse of the sample is measured for different wavelengths, such that the optical properties of the investigated sample can be quantified.

Measured quantities

The polarization ellipse (figure 2) of an EM wave can be described in different reference frames, depending on the application. For spectroscopic ellipsometry measurements the polarization state is usually decribed in terms of \tan \Psi and \cos \Delta. These numbers represent the amplitude ratio between the reflected P- and S-polarized waves (\tan \Psi) and the phase difference (\Delta) between them (figure 1). The relationship between these two parameters and the sample response is given by the main equation of ellipsometry

(1) \rho = \frac{r_p}{r_s} = \frac{\frac{E_{r,p}}{E_{i,p}}}{\frac{E_{r,s}}{E_{i,s}}} = \tan \Psi e^{i \Delta} = \tan \Psi (\cos \Delta + i \sin \Delta)

The value of \tan \Psi describes the amplitude ratio between P- and S-polarized waves, while their phase difference \Delta determines the polarization state. For positive values of \Delta the polarization vector rotates right-handed while for negative values of \Delta it rotates left-handed as the wave propagates.

delta
Figure 2: The polarization ellipse is confined to the square defined by the electric field magnitude along the X and Y directions, which are equal in this example (\tan \Psi = 1). For changing values of the phase differences \delta_y - \delta_x the wave has a different polarization state. The respective states are: linearly polarized at 45^o, elliptically polarized (right), right circularly polarized, elliptically polarized (right), linearly polarized at -45^o, elliptically polarized (left), left circularly polarized and elliptically polarized (left).

An alternative representation of the ellipsometric parameters can be given in terms of the ellipticity \epsilon = \tan \gamma and the rotation angle \theta, which are not used that often in terms of spectroscopic ellipsometry. These quantities are mainly used for magneto-optic Kerr effect (MOKE) measurements in which the magnetic properties of a sample are investigated by means of polarized light. The tangent of the ratio between the long and short axis of the polarization ellipse gives the ellipticity \epsilon while the orientation of the long axes with respect to the incident wave is given by a rotation angle \theta. In MOKE measurements the anisotropy introduced by the external magnetic field is quantified such that switching of the magnetization upon changes in the direction and amplitude of the external magnetic field can be observed. The two representations can be used interchangeably and the main advantage for (\epsilon,\theta) over (\Psi,\Delta) is that the ellipticity \epsilon describes the polarization state in the most general way, independent of the frame of reference. The relationships between them are given by equations 2-5 and can be deduced directly from the Stokes parameters.

(2) \Psi = \tan^{-1} \sqrt{\frac{1 + \tan^{2} \theta \tan^2 \gamma}{\tan^{2} \theta + \tan^2 \gamma}} for  \Psi \in < 0, \frac{\pi}{2} >

(3) \Delta = 2 \tan^{-1} \frac{\sqrt{\cos^2 2 \gamma \cos^2 2 \theta} - \sin^2 2 \theta \cos^2 2 \gamma}{\sin 2 \gamma} for  \Delta \in < -\pi, \pi >

(4) \theta = \tan^{-1} \frac{\sin 2 \Psi \cos \Delta}{\sqrt{1 - \sin^2 2 \Psi \sin^2 \Delta} - \cos 2 \Psi} for \theta \in < \frac{-\pi}{2}, \frac{\pi}{2} >

(5) \gamma = \frac{1}{2} \tan^{-1} \frac{\sin 2 \Psi \sin \Delta}{\sqrt{1 - \sin^2 2 \Psi \sin^2 \Delta}} for  \gamma \in < \frac{-\pi}{4}, \frac{\pi}{4} >

PolarizationEllipse
Figure 3: The polarization ellipse with the ellipsometric angles \Psi, \gamma and \theta.

Mathematical description of polarized light

Complex optical systems can be described by means of the Jones and Mueller/Stokes matrix formalism, in which each optical component of the system is described by a characteristic matrix. The total transmission through an optical setup is given by the matrix product of all matrices with the polarization vector of the incident wave. The Jones matrix formalism consists of 2 x 2 matrices for optical components and 2-component Jones vectors, while the the Mueller/Stokes formalism consists of 4 x 4 Mueller matrices in conjunction with 4-component Stokes vectors. The Jones matrix formalism allows to describe all the polarization states discussed before, but does not allow to describe unpolarized light. The Mueller/Stokes formalism is more advanced and allows to describe any possible (partial) polarization state. For spectroscopic ellipsometry measurements the Jones formalism is sufficient, as only linear, circular and elliptical polarization states are used. We’ll briefly introduce the Mueller/Stokes formalism as well, as the four Stokes parameters S_0 - S_3 are measured in a spectroscopic ellipsometry measurement.

Jones vectors

All different polarization states of light can be described by their Jones vector. Let’s assume an EM wave which is propagating in the z-direction and which can be written as a superposition of two plane waves oscillating in the x- and y-directions with frequency \omega

(6) {\bf E}(z,t) =  \begin{pmatrix} E_{x,0} e^{i(\omega t - k_z z + \delta_x)} \\ E_{y,0} e^{i(\omega t - k_z z + \delta_y)}\\ \end{pmatrix}  = e^{i(\omega t - k_z z)} \begin{pmatrix} E_{x,0} e^{i \delta_x} \\ E_{y,0} e^{i \delta_y}\\ \end{pmatrix}

Usually the term e^{i(\omega t - {\bf k} z)} is dropped

(7) {\bf E}(z,t) = \begin{pmatrix} E_x \\ E_y \\ \end{pmatrix}

In which E_x and E_y can be written as the product of the amplitude and phase of the electric field along the x- and y-directions

(8) \begin{matrix} E_x = | E_x | e^{i \delta_x}  \\ E_y = | E_y | e^{i \delta_y}  \\ \end{matrix}

The electric field components can be rewritten in function of the phase difference \delta_x - \delta_y

(9) \begin{matrix} E_x = E_{x,0} e^{i (\delta_x - \delta_y)} = | E_x | e^{i (\delta_x - \delta_y)}  \\ E_y = E_{y,0} = | E_y | \\ \end{matrix}

while the total field intensity is given by

(10) I = I_x + I_y = E_{x,0}^2 + E_{y,0}^2 = |E_x|^2 + |E_y|^2 = E_x E_x^* + E_y E_y^*

As we are only interested in the relative phase and amplitude for the different components, the intensity I is usually normalized such that I = 1. Therefore we can write the linear polarization states along the x- and y-axis as

(11) E_{lin,x} = \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix}

(12) E_{lin,y} = \begin{pmatrix} 0 \\ 1 \\ \end{pmatrix}

For any linear polarization state which makes an angle \Psi with the x-axis, the general Jones vector is given by

(13) E_{lin,\Psi} = \begin{pmatrix} \sin \Psi \\ \cos \Psi \\ \end{pmatrix}

For the circular polarization states the phase difference between E_x and E_y is \pi / 2, from which we can write the normalized Jones vectors as

(14) E_{circ,right} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ i \\ \end{pmatrix}

(15) E_{circ,left} = \frac{1}{\sqrt{2}} \begin{pmatrix} i \\ 1 \\ \end{pmatrix}

The most general shape of polarization is the elliptical polarization for which the main axis of the polarization ellipse rotated with an angle \Psi with respect to the x-axis, which can be written as

(16) E_{ellipt,\Psi}  \begin{pmatrix} \sin \Psi e^{i(\Delta)}\\ \cos \Psi \\ \end{pmatrix}

Jones Matrix

Optical components can be described in terms of their Jones matrix, which characterizes their polarization dependent optical response. In this section we give an overview of the different components which are relevant to describe ellipsometry measurements.

Linear polarizers are used in the different types of ellipsometry setups both on the incident side as polarizer (P) and at the detection side as analyzer (A). A linear polarizer with the its transmission axis along the x-direction is described by

(17) P (0^o) = A (0^o) =  \begin{pmatrix} 1 & 0 \\  0 & 0 \\  \end{pmatrix}

For a polarizer under an arbitrary angle \phi with respect to the x-axis, the resulting matrix can be calculated by using the rotation matrix R(\alpha).

(18) R (\phi) = \begin{pmatrix} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \\ \end{pmatrix}

If a complex optical system such as an ellipsometry setup is described, different rotation matrices have to be introduced in order to describe the overall optical response of the setup, as the different components are usually rotated with respect to the optical axes of the previous component.

The response of a sample can be described by the complex reflection and transmission parameters given by the Fresnel equations, which are related to each other by the ellipsometric angles \Psi and \Delta that appear in the sample matrix S

(19)  S = \begin{pmatrix} \sin \Psi e^{i \Delta} & 0 \\ 0 & \cos \Psi \\ \end{pmatrix}

This matrix can be deduced immediately from the definitions of \tan \Psi and \Delta, if we define the x-direction as the P-axis and the y-direction as the S-axis in figure 1:

(20) \tan \Psi = \frac{\sin \Psi}{\cos \Psi} = \frac {|r_p|}{|r_s|}

The phase difference between the P- and S-polarized waves is defined as \Delta = \delta_x - \delta_y = \delta_p - \delta_s as introduced by equation 9. As an example we show the full description of figure 1 in Jones matrices

(21)   \begin{pmatrix}  E_{r,p} \\  E_{r,s} \\ \end{pmatrix}  = \begin{pmatrix}  \sin \Psi e^{i \Delta} & 0 \\  0 & \cos \Psi \\ \end{pmatrix} \begin{pmatrix}  E_{i,p} \\  E_{i,s} \\  \end{pmatrix}  \begin{pmatrix}  \sin \Psi e^{i \Delta} E_{i,p} \\  \cos \Psi E_{i,s} \\ \end{pmatrix}

from which we can see that a linearly polarized wave (at 45^o) is converted into an elliptically polarized wave upon reflection from the sample. To measure the phase difference \Delta in practical applications, we need to apply a modulation of the polarization, in order to perform lock-in measurements at the modulation frequency \omega. Two types of modulation are used most often: a polarizer rotating at a frequency \omega or a photo-elastic modulator oscillating at frequency \omega.

For a rotating linear polarizer, the Jones matrix is expressed as function of the polarizer angle \alpha = \omega t and it is given by the product of R(\alpha) and a linear polarizer matrix (equation 18).

A photo-elastic modulator consists of a fused quartz crystal to which two electrodes are connected. An oscillating electric field is applied to these electrodes at 50kHz, which corresponds to the resonant oscillation frequency of the piezo transducer. The modulation of the crystal introduces stress into the dielectric material which changes the electron density along the stress direction. This effect introduces anisotropy in the crystal, which gives rise to a phase difference between light waves that pass through the crystal with a polarization along or perpendicular to the stress direction. The introduced phase difference is time-dependent and is given by

(22) \delta = F \sin (\omega t)

where \omega = 2 \pi \nu with \nu = 50 kHz. F is the phase amplitude and is proportional to V/ \lambda with V the applied voltage to the crystal and \lambda the wavelength of the incident light. In a spectroscopic ellipsometry measurement \delta is kept constant for different wavelengths by adjusting the applied voltage to the wavelength. The Jones matrix of a PEM is given by

(23)   \begin{pmatrix} 1 & 0 \\ 0 & e^{i \delta} \\ \end{pmatrix}

PEM
Figure 4: (a) Schematic overview of a PEM. (b) Illustration of the polarization states during one modulation cycle with the peak retardation \lambda / 2.

Stokes parameters

Although the Jones formalism provides a very elegant means to describe polarized light, it does not provide the possibility to define partially polarized or unpolarized light. These polarization states can be described by means of Stokes parameters (S_0 - S_3), which make up a 4-component vector. Similar to the Jones matrices defined before, Mueller matrices can be used to describe any optical component and in that way to fully describe an optical setup. For the systems used in this thesis we can describe all properties in terms of Jones matrices, so therefore we only introduce the Stokes parameters here, as they are the parameters that come out of an ellipsometry measurement. If we consider waves with a polarization vector along the x- and y-direction, the corresponding Stokes vectors are given by

(24) S_0 = I_x + I_y

(25) S_1 = I_x - I_y

(26)  S_2 = I_{+45^o} - I_{-45^o}

(27)  S_3 = I_{circ,right} - I_{circ,left}

The S_0 parameter described the total light intensity, while the S_1 - S_3 parameters describe the polarization state in terms of the differences between all possible linear and circular polarization states. By writing the different Stokes parameters in terms of the electric field values, we can relate them to the ellipsometric parameters (\Psi,\Delta) and (\epsilon,\mu).

(28) S_0 = I_x + I_y = E_{x,0}^2 + E_{y,0}^2 = E_x E_x^* + E_y E_y^*

(29) S_1 = I_x - I_y = E_{x,0}^2 - E_{y,0}^2 = E_x E_x^* - E_y E_y^*

For the S_2 parameter we use a rotation of the x- and y-directions over -45^o

(30) \begin{pmatrix} E_{-45^o} \\ E_{+45^o} \\ \end{pmatrix} = \begin{pmatrix} \cos -45^o & \sin -45^o \\ -\sin -45^o & \cos -45^o \\ \end{pmatrix} \begin{pmatrix} E_x \\ E_y \\  \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} E_x - E_y \\ E_x + E_y \\ \end{pmatrix}

From which we can derive S_2

(31)  S_2  = E_{+45^o} E_{+45^o}^* - E_{-45^o} E_{-45^o}^* = \frac{1}{2} [(E_x + E_y)(E_x^* + E_y^*) - (E_x - E_y)(E_x^* - E_y^*)]  = E_x E_y^* + E_x^* E_y

For S_3 we can make a similar derivation as for S_2 where we rewrite E_{circ,left} and E_{circ,right} in function of E_x and E_y

(32)   \begin{pmatrix}  E_{circ,left} \\ E_{circ,right} \\  \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & i \\ 1 & -i \\ \end{pmatrix} \begin{pmatrix} E_x \\ E_y \\ \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} E_x + i E_y \\ E_x - i E_y \\  \end{pmatrix}

From which we obtain S_3

(33)   S_3  = E_{circ,right} E_{circ,right}^* - E_{circ,left} E_{circ,left}^* = \frac{1}{2} [(E_x - i E_y)(E_x^* + i E_y^*) - (E_x + i E_y)(E_x^* - i E_y^*)]  = i ( E_x E_y^* - E_x^* E_y)

Now we can relate the Stokes parameters to the ellipsometric angles (\Psi,\Delta). For S_0 we start from equation 29 and we see from the main equation of ellipsometry (1) and figure 3 that we obtain a normalized vector with magnitude 1.

(34) S_0 = E_{x,0}^2 + E_{y,0}^2 = E_x E_x^* + E_y E_y^* = \sin^2 \Psi + \cos^2 \Psi = 1

Similarly for S_1 we obtain

(35)  S_1 = E_{x,0}^2 - E_{y,0}^2 = E_x E_x^* - E_y E_y^* = \sin^2 \Psi - \cos^2 \Psi = -\cos 2 \Psi

For S_2 we start from equation 31 and we use that (E_x E_y^*)^* = E_x^* E_y such that S_2 can be rewritten to

(36)  S_2 = 2 Re (E_x E_y^*) = 2 Re (E_x^* E_y)

For complex numbers C we can write Re (C) = Re (C^*) and Im (C) = - Im (C^*). From equation 10 we can write E_x^* = E_{x,0} e^{-i(\delta_{x} - \delta_{y})} and E_y = E_{y,0} which yields

(37) S_2 = 2 E_{x,0} E_{y,0} Re (e^{-i(\delta_x - \delta_y)})  = 2 \sin \Psi \cos \Psi \cos (\delta_x - \delta_y) = 2 \sin \Psi \cos \Psi \cos \Delta \\ = \sin 2 \Psi \cos \Delta

Using Im (C) = \frac {C+C^*}{2i} we obtain S_3 from equation 33

(38)  S_3 = - Im(E_x E_y^*) = Im(E_x^* E_y)  = 2 E_{x,0} E_{y,0} Im (e^{-i(\delta_x - \delta_y)})  = 2 \sin \Psi \cos \Psi \sin (\delta_x - \delta_y) = 2 \sin \Psi \cos \Psi \sin \Delta  = \sin 2 \Psi \sin \Delta

Any polarization state can be described by a point on the Poincaré sphere (figure 5), which is a graphical illustration based on the Stokes parameters. The S_0 parameter gives the intensity of the light which corresponds to the radius of the sphere, while S_1 - S_3 are the coordinate axes for the construction of the sphere. The point P on the sphere corresponding to a certain polarization state is constructed by using the Stokes parameters and the corresponding angles of 2 \epsilon and 2 \theta give the polar coordinates of the point P (equation 39). If we consider the Poincaré sphere as a globe then the equator corresponds to linear polarization states and the poles represent circular polarization states. A polarization state in between those corresponds to elliptical polarization and in the northern hemisphere it rotates right while in the southern hemisphere it rotates left.

Poincare
Figure 5: The Poincaré sphere as a representation of various polarization states and the ellipsometric angles (\epsilon,\theta)

(39) S_1 = \cos 2 \epsilon \cos 2 \theta

(40) S_2 = \cos 2 \epsilon \sin 2 \theta

(41) S_3 = \sin 2 \epsilon

Rotating analyzer/polarizer ellipsometry

In a rotating polarizer ellipsometry setup either the polarizer (P) or the analyzer (A) (figure 6) can be rotating in order to modulate the signal and extract the phase difference \Delta between P- and S-polarized waves. Here we consider the case where the analyzer is rotating at an angular frequency \omega such that the rotation angle is given by \phi_A = \omega t.

RotatingAnalyzer
Figure 6: Schematic overview of a rotating analyzer ellipsometry setup.

The transmission through the setup can then be described by Jones matrices

(42) L_{out} = A R(\phi_A) S R(-\phi_P) P L_{in}

In this configuration L_{out} represents the detected signal of the photodetector along the polarization axis (\phi_A) of the analyzer (A) , while L_{in} represents the incident wave with polarization along the polarization axis (\phi_P) of the polarizer (P). This representation is used because the Jones formalism doesn’t allow to describe the unpolarized light source, but the transmitted light after the polarizer (P) contains only linearly polarized light along the polarization axis of P. Similarly, the photodetector will only detect linearly polarized waves with the polarization along the axis of A. With respect to the coordinate system of (E_{i,p},E_{i,s}) we should formally write the Jones matrices R(-\phi_P) P R(\phi_P) and R(-\phi_A) A R(\phi_A) for the polarizer and analyzer respectively. Due to the definitions of L_{out} and L_{in} we can drop the terms R(\phi_P) and R(-\phi_A), which results in the overall matrix formulation for the setup

(43) \begin{pmatrix} E_A \\ 0 \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix} \begin{pmatrix} \cos \phi_A & \sin \phi_A \\ - \sin \phi_A & \cos \phi_A \\ \end{pmatrix} \begin{pmatrix} \sin \Psi e^{i \Delta} & 0 \\ 0 & \cos \Psi \\ \end{pmatrix} \begin{pmatrix} \cos \phi_P & - \sin \phi_P \\ \sin \phi_P & \cos \phi_P \\ \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 0 \\  \end{pmatrix}  \begin{pmatrix} 1 \\ 0 \\  \end{pmatrix}

In typical ellipsometry measurements the polarizer on the incident side is set at 45^o and in that case the matrix expression can be simplified to

(44) \begin{pmatrix} E_A \\ 0 \\ \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix} \begin{pmatrix} \cos \phi_A & \sin \phi_A \\ - \sin \phi_A & \cos \phi_A \\ \end{pmatrix} \begin{pmatrix} \sin \Psi e^{i \Delta} \\ \cos \Psi \\ \end{pmatrix}

From which we obtain the expression for E_A (we drop the constant 1 / \sqrt{2})

(45) E_A = \cos \phi_A \sin \Psi e^{i \Delta} + \sin \phi_A \cos \Psi

The normalized relative light intensity measured by the detector is obtained using equation 10 and dropping the constant factor of 1/2

(46)  I = | E_A |^2  = I_0 ( 1 - \cos 2 \Psi \cos 2 \phi_A + \sin 2 \Psi \cos \Delta \sin 2 \phi_A  = I_0 ( 1 + S_1 \cos 2 \phi_A + S_2 \sin 2 \phi_A)

In which I_0 is the normalized intensity of the incident beam. Moreover, it is important to note that the modulation of the intensity of the transmitted light varies as a function of 2A as a rotation of the analyzer of 180^o yields the same transmitted intensity. By substituting \phi_A = \omega t in the expression above, we obtain the time-dependent transmitted intensity.

(47) I = I_0 (1 + S_1 \cos 2 \omega t + S_2 \sin 2 \omega t)

So far we considered the case where the polarizer angle P was fixed at 45^o, which was the configuration used for all ellipsometry experiments described so far. Depending on the sample structure under investigation, it can be useful to set different polarizer angles to improve the signal-to-noise ratio. The measured signal will also be time dependent and the Fourier components \alpha and \beta can be deduced by plugging in the correct angle for P in equation 43. In their most general form, the normalized Fourier coefficients can be written as

(48) S_1 = \frac{\cos 2 \phi_P - \cos 2 \Psi}{1 - \cos 2 \phi_P \cos 2\Psi} = \frac{\tan^2 \Psi - \tan^2 \phi_P}{\tan^2 \Psi - \tan^2 \phi_P}

(49)  S_2 = \frac{\sin 2 \Psi \cos \Delta \sin 2 \phi_P}{1 - \cos 2 \phi_P \cos 2\Psi} = \frac{2 \tan \Psi \cos \Delta \tan \phi_P}{\tan^2 \Psi - \tan^2 \phi_P}

These normalized Fourier coefficients are measured during an ellipsometry measurement and the ellipsometric angles \Psi and \Delta are calculated from

(50) \tan \Psi = \sqrt{\frac{1+S_1}{1-S_1}}

(51) \cos \Delta = \frac {S_2}{\sqrt{1 - S_1^2}}

A rotating polarizer ellipsometer is mathematically equivalent to the rotating analyzer ellipsometer, and the ellipsometric angles are obtained by replacing \phi_P by \phi_A in the equations above.

Photo-Elastic Modulator (PEM) based ellipsometry

In PEM-based ellipsometry a photo-elastic modulator is used to modulate the polarization state of the probing beam. In the original design, the modulator was placed behind the sample (figure 7), but nowadays the modulator is often placed before the sample for practical reasons.

PEMellipsometer
Schematic overview of a PEM-based ellipsometry setup.

The polarizer (P) sets the polarization (\phi_P = 45^o), after which the beam passes through the PEM (M) which modulates the beam along \phi_M = 0^o. After reflection from the sample (S) the beam passes through the analyzer (A) which defines the polarization state (\phi_A = -45^o) before the beam enters the detector. The entire setup can be described in terms of Jones matrices

(52) L_{out} = A R(\phi_A) S R (-\phi_M) M R(\phi_M) R(-\phi_P) P L_{in} = A R(\phi_A) S R (-\phi_M) M R(\phi_M-\phi_P) P L_{in}

In which we dropped the terms R(-\phi_A) and R(\phi_P) as the only contributing terms will be polarized along the \phi_A and \phi_P polarization directions, and substituted R(\phi_M) R(-\phi_P) by R(\phi_M-\phi_P). We are only interested in the relative intensities, so the terms in 1/\sqrt{2} (for R(\phi_A) and R(\phi_M-\phi_P)) are dropped in the equation of the Jones matrices

(53) \begin{pmatrix} E_A \\ 0 \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix} \begin{pmatrix} 1 & 1 \\ - 1 & 1 \\ \end{pmatrix} \begin{pmatrix} \sin \Psi e^{i \Delta} & 0 \\ 0 & \cos \Psi \\ \end{pmatrix} \begin{pmatrix} 1 & 0 \\  0 & e^{i \delta} \\ \end{pmatrix} \begin{pmatrix} 1 & - 1 \\ 1 & 1 \\  \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix}

The resulting transmission through the setup is given by

(54) E_A = \sin \Psi e^{i \Delta} + \cos \Psi e^{i \delta}

from which we can deduce the intensity

(55)  I = | E_A |^2  = I_0 ( 1 + \sin 2 \Psi (\cos \Delta \cos \delta + \sin \Delta \sin \delta))  = I_0 ( 1 + S_2 \cos \delta + S_3 \sin \delta)

If we now introduce the time dependence \delta = F \sin \omega t we obtain the expressions for \sin \delta and \cos \delta

(56) \begin{matrix} \sin \delta = \sin(F \sin \omega t) = 2 \sum_{m=0}^\infty J_{2m+1} (F) \sin[(2m+1)\omega t]  \cos \delta = \cos(F \sin \omega t) = J_0(F) + 2 \sum_{m=1}^\infty J_{2m} (F) \cos[2m \omega t]  \end{matrix}

In which the terms J_m are Bessel functions with respect to F. In PEM-based ellipsometry we measure at the modulation frequency (50 kHz) and its first harmonic (100 kHz) and drop the higher order terms

(57) \begin{matrix} \sin \delta = 2 J_1 (F) \sin \omega t  \cos \delta = J_0(F) + 2 J_2 (F) \cos 2 \omega t   \end{matrix}

The applied voltage to the PEM is adjusted such that the retardation is fixed at F = 138^o, which sets the values of the Bessel functions to be J_0(F) = 0, 2J_1(F) = 1.04 and 2J_2(F) = 0.86. This simplifies the analysis tremendously and allows to write the measured intensity from lock-in measurements at 50 kHz and 100 kHz as function of the ellipsometric angles \Psi and \Delta

(58)  I(t) = I_0 (1 + \sin 2 \Psi \sin \Delta [2 J_1 (F) \sin \omega t] + \sin 2\Psi \cos \Delta [2 J_2 (F) \cos 2 \omega t])

In that way, we obtain the final expressions for \Psi and \Delta

(59)  \Psi = \frac{1}{2} \sin^{-1} \sqrt{(2 J_1 (F) \sin \omega t)^2 + (2 J_2 (F) \cos 2 \omega t)^2}

(60)  \Delta = \tan^{-1} \frac{2 J_1 (F) \sin \omega t}{2 J_2 (F) \cos 2 \omega t}