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.

*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 and . These numbers represent the amplitude ratio between the reflected P- and S-polarized waves () and the phase difference () between them (figure 1). The relationship between these two parameters and the sample response is given by the main equation of ellipsometry

(1)

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

*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 (). For changing values of the phase differences the wave has a different polarization state. The respective states are: linearly polarized at , elliptically polarized (right), right circularly polarized, elliptically polarized (right), linearly polarized at , 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 and the rotation angle , 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 while the orientation of the long axes with respect to the incident wave is given by a rotation angle . 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 over is that the ellipticity 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) for

(3) for

(4) for

(5) for

*Figure 3: The polarization ellipse with the ellipsometric angles , and .*

# 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 matrices for optical components and 2-component Jones vectors, while the the Mueller/Stokes formalism consists of 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 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

(6)

Usually the term is dropped

(7)

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

(8)

The electric field components can be rewritten in function of the phase difference

(9)

while the total field intensity is given by

(10)

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)

(12)

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

(13)

For the circular polarization states the phase difference between and is , from which we can write the normalized Jones vectors as

(14)

(15)

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

(16)

## 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)

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

(18)

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 and that appear in the sample matrix S

(19)

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

(20)

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

(21)

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

For a rotating linear polarizer, the Jones matrix is expressed as function of the polarizer angle and it is given by the product of 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 , 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)

where with . F is the phase amplitude and is proportional to with V the applied voltage to the crystal and the wavelength of the incident light. In a spectroscopic ellipsometry measurement is kept constant for different wavelengths by adjusting the applied voltage to the wavelength. The Jones matrix of a PEM is given by

(23)

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

## 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 , 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)

(25)

(26)

(27)

The parameter described the total light intensity, while the 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 and .

(28)

(29)

For the parameter we use a rotation of the x- and y-directions over

(30)

From which we can derive

(31)

For we can make a similar derivation as for where we rewrite and in function of and

(32)

From which we obtain

(33)

Now we can relate the Stokes parameters to the ellipsometric angles . For 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)

Similarly for we obtain

(35)

For we start from equation 31 and we use that such that can be rewritten to

(36)

For complex numbers C we can write and . From equation 10 we can write and which yields

(37)

Using we obtain from equation 33

(38)

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 parameter gives the intensity of the light which corresponds to the radius of the sphere, while 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 and 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.

*Figure 5: The Poincaré sphere as a representation of various polarization states and the ellipsometric angles ()*

(39)

(40)

(41)

# 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 between P- and S-polarized waves. Here we consider the case where the analyzer is rotating at an angular frequency such that the rotation angle is given by .

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

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

(42)

In this configuration represents the detected signal of the photodetector along the polarization axis () of the analyzer (A) , while represents the incident wave with polarization along the polarization axis () 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 () we should formally write the Jones matrices and for the polarizer and analyzer respectively. Due to the definitions of and we can drop the terms and , which results in the overall matrix formulation for the setup

(43)

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

(44)

From which we obtain the expression for (we drop the constant )

(45)

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

(46)

In which 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 yields the same transmitted intensity. By substituting in the expression above, we obtain the time-dependent transmitted intensity.

(47)

So far we considered the case where the polarizer angle P was fixed at , 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 and 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)

(49)

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

(50)

(51)

A rotating polarizer ellipsometer is mathematically equivalent to the rotating analyzer ellipsometer, and the ellipsometric angles are obtained by replacing by 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.

*Schematic overview of a PEM-based ellipsometry setup. *

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

(52)

In which we dropped the terms and as the only contributing terms will be polarized along the and polarization directions, and substituted by . We are only interested in the relative intensities, so the terms in (for and ) are dropped in the equation of the Jones matrices

(53)

The resulting transmission through the setup is given by

(54)

from which we can deduce the intensity

(55)

If we now introduce the time dependence we obtain the expressions for and

(56)

In which the terms 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)

The applied voltage to the PEM is adjusted such that the retardation is fixed at , which sets the values of the Bessel functions to be , and . 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 and

(58)

In that way, we obtain the final expressions for and

(59)

(60)