Ellipsometer Working Principle

An ellipsometer measures the change in the polarization state of light after it interacts with a sample. Ellipsometry is an optical technique that can be used to measure the thickness and optical properties of thin films. It can also be used to determine sample properties (including roughness and extinction/absorbance coefficients) at each wavelength of interest.

Changes in the polarization state can be quantified by measuring the amplitude change and the phase shift induced by interaction with a sample. These changes in the polarization state of light can be exploited to study film thickness values that are significantly smaller than the wavelength of the incident light. Ellipsometers can obtain sub-nanometer resolution using visible wavelengths, thus beating the diffraction limit.

What Is Ellipsometry?

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When we refer to the polarization state of light, we are referring to the direction in which the electric field is oscillating. In naturally occurring light (sunlight, etc), this orientation is random - but in polarized light sources, this orientation is controlled.

During reflection from a surface, polarized light can be resolved into two components: one which is parallel (p-polarization) and one which is normal to the plane of incidence (s-polarization). Here, we define the plane of incidence as the plane containing both the beam path and the surface normal.

The relative phases and amplitudes of the p- and s-components determine the polarization state of light:

• If the p- and s-components are synchronized or 180° out of phase, then the light is linearly polarized.
• If the p- and s-components are 90° out of phase and have equal amplitudes, then the light is circularly polarized.
• If the p- and s-components are out of phase by another arbitrary amount, or their amplitudes are completely different, then the light is elliptically polarized. This is the general state of polarization and it is where ellipsometry gets its name. We are measuring changes in the ellipticity of the polarisation state using an ellipsometer.

When a linearly polarized light beam is reflected from a thin film sample, its polarization state will change. This is caused by differences in the reflection coefficients for p and s components at each interface that the light encounters. It is also affected by small phase shifts in the two components as the light traverses the medium comprising the film. These changes in polarization can be characterized by the difference in polarization amplitude (ψ) and phase (Δ) between the s-polarized and p-polarized light. The changes in polarization state, caused by interaction with the film, will change the lights state of elliptical polarisation (or ellipticity).

For light with a given elliptical polarization, reflection from and interaction with the film can cancel out the phase changes in the incident light and result in linearly polarized light upon reflection from the sample. This is the working principle that underpins how nulling ellipsometers work. A linear polariser is used to extinguish the linearly polarised light that is reflected from the sample under these conditions. This is achieved by crossing the polariser axis with the direction of polarisation of the linearly polarised reflected light to create a null signal. If you know of the polarisation state of incident and reflected light, you can find the ellipsometric angles ψ and Δ. A suitable model can then be used to extract the thickness and optical properties of the sample being studied.

Nulling ellipsometry is one type of ellipsometry technique. Another approach uses rotating polarisers (or other rotating optical elements) to directly measure the ellipticity of the light reflected from a sample. Alternatively, active optical elements such as acousto-optic modulators can be used to apply periodic variations to the polarisation state of light. Both of these measurements are dynamic in nature. They are typically faster measurements than those obtained from a nulling instrument but tend to be less accurate.

How Does Ellipsometry Work?

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An ellipsometer begins by directing the light source through a polarizer, creating linearly polarized light. This light is then passed through other optical components to control the incident light’s ellipticity. This creates a beam with a well-defined polarization state. The components needed to control ellipticity depends on many things including the ellipsometer light source and the desired application of the ellipsometer.

This elliptically polarized light then interacts with the sample, usually a thin film or several thin films supported on a substrate. As polarized light interacts with each layer, its polarization state will change.

The reflected light is then passed through a second polarizer (called the analyzer), before this signal is detected with a detector. These measurements can be taken at multiple angles of incidence to enable more accurate measurement of the amplitude difference and phase shifts at each incident wavelength.

Laser Ellipsometer vs. Spectroscopic Ellipsometer

In a single wavelength (or laser) ellipsometer, a monochromatic light source (often a laser) is directed through a polarizer to make linearly polarized light. Ellipsometers also need an additional element to control the ellipticity of the incident light. This is achieved by passing the light through a quarter waveplate (or compensator) before it interacts with the sample.

Spectroscopic ellipsometers use a similar approach except the optics that control the polarisation state need to have a well characterised wavelength response. While many polarizers typically have a broadband response and can be used for many wavelengths, quarter waveplates do not. Quarter waveplates are only optimized for fixed wavelengths. The retardance of these components will be different at different wavelengths. As a result, this wavelength-dependent retardance needs to be characterised.

Some spectroscopic ellipsometers use alternative ways of varying the ellipticity of the incident light. This is where acousto-optic modulators can be useful, or a Fresnel Rhomb can be used as a broadband quarter waveplate.

Ellipsometer Measurement and Analysis

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The main quantity of interest in ellipsometry is something called the complex reflectance ratio, but this is not very meaningful on its own. To extract meaningful information about the samples, an optical model needs to be applied to this data. This allows users to extract physical information such as film thickness and optical parameters. This is both a strength and a weakness of using ellipsometry. When an appropriate model is used, high quality data can be obtained. Conversely, the use of an inappropriate model can result in misleading and often inaccurate values of the thickness and optical coefficients.

A typical ellipsometry study of a thin film sample might look something like this:

• Measure phase shift and amplitude change induced by a thin film at various angles of incidence
• Pick a model that accurately represents your sample and which relates change in polarization states to physical quantities (such as layer thickness or refractive indexes). One such example might be a thin film supported on a semi-infinite substrate.
• Computationally vary the parameters in your model until you get amplitude and phase shift constants which match your measured results.
• If your model is appropriate and the fit is good, you can get very reliable measurements of very thin films

Spectroscopic ellipsometry works best for samples that consist of discrete, well defined layers that are optically homogenous and isotropic. Ideally, there should not be a lot of specular scattering due to roughness or inhomogeneities in the sample as this can affect subsequent measurements. Multiangle ellipsometry can be used to look at particularly inhomogeneous or rough samples.

Example Analysis: Thin film on a Substrate

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A great place to start with examining ellipsometer measurements is with the Fresnel equations. These can be used to relate changes in polarization state to physical quantities, such as film thickness and optical coefficients. As an example, consider a single thin film deposited on a substrate.

The reflection coefficients for p-polarized and s-polarized light reflected at the boundary between materials with reflective indexes n1 and n2 can be represented by r^p and r^s respectively.

Where φ₁ and φ₂ are the angles of reflection and refraction respectively, and n_i is the refractive index of the respective medium.

Similarly, the transmission coefficients for light passing through the boundary between materials with reflective indexes n₁ and n₂ can be represented by t^p and t^s respectively.

Where the angles of incidence and refraction are related by Snell’s law.

The total reflection coefficients, R^p and R^s can be obtained by summing up the contributions from all light rays that exit the sample. Taking the s polarised light as an example the total reflection coefficient is obtained by tracing the path of each ray (as shown in the diagram above) as it reflects and refracts at each boundary and transmits through each layer. This results in an equation for the total reflection coefficient of the form:

Where j is the square root of -1 and δ is a phase factor introduced by transmission through the film.

Where d is the thickness of the film and λ is the wavelength of the light in vacuum. Note that here the superscript “s” has been omitted from all the individual reflection coefficients in the above combined reflection coefficient for the sake of brevity.

The above expression can be simplified to give:

Where the second and successive terms in this expansion represent an infinite geometric series. This can be simplified further to give:

If the substitution r₁₂=-r₂₁ is used along with the law of conservation of energy t₁₂t₂₁=1-r₁₂² then this expression becomes:

Adding the superscript “s” back in we obtain the final expression for the s reflection coefficient from the film/substrate stack:

Repeating a similar process for the p components gives a total reflection coefficient of the form:

The ratio of total p and s reflection coefficients can then be related to ellipsometric angles, ψ and Δ by constructing a value of the complex reflectivity ratio, ρ, based upon the model derived above. This gives the result:

Where tan(ψ) is the amplitude ratio of the reflection coefficients as defined by the equation, and Δ is the difference in phase between the s and p components. Measurements of the polarization amplitude and phase change of the reflected light can be compared to computationally generated values by substituting refractive indexes and film thickness values into the equations given above. These values are varied until good agreement is obtained between the experimental and computationally determined ψ and Δ values is found.

Example Analysis: Multilayer Stacks

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Analysis of the data obtained from multilayers samples requires a different, more complicated approach. Care must be taken to measure and analyse each new layer in the multilayer sample after it has been deposited. This is to determine their thickness and optical properties for accurate modelling.

One exception to this occurs in the case where a known periodic structure exists. If the multilayer consists of repeating layers of the same materials with the same thickness values, measurements are required after the number of layers that form a single period in the structure have been deposited.

When analysing ellipsometry data from a multilayer sample, a relatively simple method, based on Jones matrices, can be used. These matrices are used to track the changes in the state of polarisation of light as it passes through the sample and is reflected or transmitted from/through each of the boundaries within it.

Consider a multilayer stack. Reflection and transmission of light at each of the boundaries in a multilayer containing layers of optically isotropic materials with sharp interfaces can be described by a boundary matrix, B_i,j, of the form

where r_i,i+1 and t_i,i+1 are the Fresnel reflection and transmission coefficients at the interface between layers i and i+1. The above example is for light interacting with the boundary between layers 1 and 2.

Similarly, the phase changes that are introduced by transmission of light through each layer can be represented by transmission matrices according to:

where the phase change in the layer i with thickness d and refractive index, n_i is represented by:

TThe total reflection coefficient for the whole multilayer sample can then be obtained from the matrix elements of M by:

This matrix multiplication approach needs to be applied to both the p and s polarisations and the complex reflectivity ratio constructed

The analysis then proceeds as per a single film on a substrate in that the thickness values and optical constants of the layers can be varied and the calculated values of ψ and Δ obtained above fitted to the experimental values. As ellipsometry only produces two measured quantities (ψ and Δ) at any given wavelength it is only ever possible to extract two pieces of information from the sample (e.g. thickness and refractive index) using a single measurement. This is why each unique layer in the multilayer sample needs to be measured after being deposited and the above model applied.

Benefits and Applications of Ellipsometers

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There are alternative methods for studying surface topography such as atomic force microscopy, or by using a stylus profilometer. However, ellipsometry has some benefits over these other methods including:

Ellipsometers are used in many different industries to characterize thin film coatings including:

• Polymer science
• Semiconductors
• Microelectronics
• Flat panel displays
• Biosensors
• Optical Coatings

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