White light interferometry analysis for measuring thin film thickness down to few nanometers

TL;DR

The paper addresses accurate thickness measurements of non-opaque thin films using white-light interferometry, demonstrated on TTAB foam films. It develops a practical analysis framework that maps spectral interference patterns to film thickness using three dedicated branches: FFT for many fringes, peak-fitting with linear regression and RANSAC for intermediate thickness, and Scheludko renormalization for a few sharp features, with an explicit expression for $h$ in terms of $\Delta$ and $m$. The approach is implemented in the open-source Python library optifik, includes peak-detection, normalization, and time-resolved analysis, and is validated against time-resolved foam-film thinning. The work highlights limitations from spectral range, refractive-index dispersion, and signal quality, while offering a memory-based correction to fill data gaps and extending applicability to other non-opaque thin films, including potential transmission geometries.

Abstract

We present a practical white-light interferometric method, supported by an open-source Python library \textit{optifik} for automated spectrum-to-thickness deduction, enabling foam film measurements down to a few nanometers. We describe three typical spectral scenarii encountered in this method: spectra exhibiting numerous interference fringes, spectra with a moderate number of peaks, and spectra with only a few identifiable features, providing illustrative examples for each case. We also discuss the main limitations of the technique, including spectral range constraints, the necessity of knowing the refractive index, and the influence of spectral resolution and signal quality. Finally, we demonstrate the application of the method in a time-resolved study of a TTAB (tetradecyltrimethylammonium bromide) foam film undergoing elongation and thinning. This method can be adapted to measure any thin non-opaque layer.

Figures (8)

• Figure 1: Variation of $\Delta$ as a function of $\phi$ (blue curve). Intervals shown in orange represent regions where the interference order $p$ lies between a destructive interference order $p_d$ and a constructive interference order $p_c$ that satisfy $p_d < p_c$, and vice versa for the intervals shown in green. Film thickness using Eq. \ref{['eq:h_expression']} with $\lambda = 660$ nm and $n=1.333$ at three consecutive destructive interference orders $p_d \in [0,1,2]$ and three consecutive constructive interference orders $p_c \in [0.5,1.5,2.5]$ is indicated along the top horizontal axis. Periodicity in the phase $\phi$ and symmetry are show with three black vertical lines at $\phi_0$, $2\pi + \phi_0$, and $2\pi - \phi_0$ and the horizontal line $\Delta(\phi_0)$.
• Figure 2: Set-up scheme with two examples of typical spectra with the wavelength $\lambda \in [400, 800]~$nm.
• Figure 3: Smoothed data spectrum in range $[450, 800]~$nm with 18 red dots representing peak detection: ($a$) in the $I^\star(\lambda)$ representation, and ($b$) in the $I^\star(n(\lambda)/\lambda)$ representation. ($c$) Fast Fourier Transform (FFT) on $I^\star (n(\lambda)/\lambda)$. The orange dot represents the dominant spatial frequency $\mathcal{D^\star} = 2h$ with $h$ the film thickness. The inset shows the same FFT spectrum with zero-padding. The green dot represents peak detection with an accuracy improved by a factor 20.
• Figure 4: ($a$) Smoothed data spectrum in the range $[450, 800]$ nm with 7 red dots representing peak detection. ($b$) Least squares linear fit of $n(\lambda)/\lambda$ as a function of $N$ giving the slope $1/4h$. For this case, $h = 1340 \pm 30$ nm.
• Figure 5: ($a$) Smoothed data spectrum in range $[450, 800]$ nm with 11 red dots representing peak detection. ($b$) Least squares linear fit of $n(\lambda)/\lambda$ as a function of $N$ of the subgroup $N \in \{3, \ldots, 10\}$. For this case, $h = 1430 \pm 10$ nm.