How Do Ψ and Δ Reveal Thin-Film Changes? An Introduction to Ellipsometry

The reflectance change from a transparent film can be small even when the polarization of the reflected light has changed substantially. Ellipsometry uses that change: Ψ and Δ jointly measure the amplitude ratio and phase difference between p- and s-polarized reflection.

This tutorial builds a first ellipsometry model for an air / SiO₂ / Si structure, then sweeps the SiO₂ thickness from 50 to 150 nm. By the end, you can configure an ellipsometry calculation, read Ψ/Δ spectra, and identify spectral regions that are more sensitive to film thickness.

An ellipsometer measures how oblique reflection changes the polarization state (Source: Guillaume Paumier / Wikimedia Commons · License: CC BY-SA 3.0)

Definition of Ψ and Δ

Incident light can be resolved into p polarization parallel to the plane of incidence and s polarization perpendicular to it. Let $r_p$ and $r_s$ be their complex amplitude reflection coefficients. The ellipsometric ratio is

$$ \rho=\frac{r_p}{r_s}=\tan\Psi,\exp(\mathrm{i}\Delta). $$

Here, $\rho$ is the complex ellipsometric ratio; $r_p$ and $r_s$ are the complex amplitude reflection coefficients for p and s polarization; $\Psi$ describes their amplitude ratio; $\Delta$ describes their phase difference; and $\mathrm{i}$ is the imaginary unit. Ψ and Δ must be used together to describe the change in the polarization ellipse completely.

A typical ellipsometry path: known polarized light reflects from the sample and enters polarization analysis and detection (Source: Buntgarn, Stannered / Wikimedia Commons · License: CC BY-SA 3.0)

Ellipsometry is normally performed at oblique incidence, where the p and s reflection responses differ more strongly. A change in film thickness or refractive index changes the optical phase thickness, shifting or reshaping the Ψ and Δ spectra.

Build the SiO₂ / Si Model

Create an air / SiO₂ / Si structure. Use wavelength-dependent optical constants for a 100 nm SiO₂ film. Treat silicon as the semi-infinite bottom medium rather than adding a millimeter-scale coherent film.

Position	Material	Setting
Top medium	Air	$n=1.00$, $k=0$
Film	SiO₂	100 nm, wavelength-dependent index
Bottom medium	Si	Wavelength-dependent complex index

100 nm SiO₂ / Si ellipsometry model

Configure Oblique-Incidence Ellipsometry

On the Optics page, use 400–800 nm with a 2 nm step and an incidence angle of 70°. Enable Psi and Delta. The ellipsometric parameters are calculated from the ratio of p and s reflection coefficients; pRatio is not an experimental polarizer angle that needs to be swept.

Wavelength, angle, and detector settings for ellipsometry

Read Ψ and Δ for the 100 nm Film

Run the calculation and open Psi first.

Ψ spectrum of 100 nm SiO₂ / Si

Ψ describes the changing ratio of p and s reflection amplitudes. At 550 nm, Ψ is about 0.906 rad (51.90°) for the 100 nm film. The spectral shape combines thin-film interference with the complex refractive index of silicon.

Δ spectrum of 100 nm SiO₂ / Si

Δ is the phase difference between p and s reflection. At 550 nm, it is about 1.653 rad (94.70°). The result page displays phase in a principal range. When a curve crosses the boundary it may jump from a positive to a negative value; this is phase wrapping, not a physical discontinuity in the sample response.

Sweep Thickness and Watch the Spectra Move

On the Sweep page, add SiO2 Film → Thickness from 50 to 150 nm with a 25 nm step.

SiO₂ thickness Sweep

Ψ spectra as SiO₂ thickness changes

As thickness increases from 50 to 150 nm, the Ψ maximum moves progressively toward longer wavelengths. The curves do not simply shift vertically: both peak position and line shape change, so a full spectrum contains more thickness information than one wavelength.

Δ spectra as SiO₂ thickness changes

Δ is also sensitive to thickness, and the location of phase wrapping moves with the film. When comparing or fitting curves, use a consistent phase representation rather than interpreting jumps between $2\pi$-equivalent values as large errors.

At 550 nm, the results are:

SiO₂ thickness	Ψ	Δ
50 nm	25.98°	87.17°
75 nm	35.37°	98.02°
100 nm	51.90°	94.70°
125 nm	83.94°	−42.83°
150 nm	48.22°	−96.36°

The negative Δ values for 125 and 150 nm follow from the selected principal range. Their equivalent unwrapped values are 317.17° and 263.64°.

From a Forward Model to Thickness Measurement

This Sweep establishes the basic ellipsometric distinction: different film thicknesses produce distinguishable Ψ/Δ spectra. In a measurement workflow, experimental curves are compared with a model containing material optical constants, surface layers, and thickness variables, and the joint residual of Ψ and Δ is used to judge the parameters.

Using Ψ alone discards phase information; using Δ alone can leave phase-wrapping ambiguities. Use both curves and prefer spectral regions that are sensitive to the target parameter and supported by reliable material data. This tutorial performs forward sensitivity analysis, not automatic inversion of measured data.

Variation Exercise

Keep SiO₂ at 100 nm and sweep incidence angle from 60° to 75° in 5° steps. Compare Ψ and Δ near 550 nm and decide whether 70° distinguishes 75 nm from 100 nm more clearly than 60°.

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Going further

These tutorials cover the operating steps only. The physics behind them, the full parameter reference for each feature, how to read the results, and the algorithm validation all live on the documentation site:

• Transfer-Matrix Method — the physics and formulas behind these tutorials
• Feature Guide — complete reference for every screen and parameter
• Results — how to read each kind of output
• Open Dreapex TMM — build and simulate in the browser