Windowing in terahertz time-domain spectroscopy: resolving resonances in thin-film samples

TL;DR

This work addresses the challenge of extracting thin-film complex permittivity from THz-TDS data when measurements occur on thick substrates, where Fabry–Pérot artifacts and finite time windows distort spectra. It formalizes a practical, windowing-based workflow that decides between truncation and apodization, aligns signals, applies time-domain windows, and uses Fourier-domain inversion of a thin-film transfer function to retrieve $\epsilon_\mathrm{film}(\nu)$, demonstrated on PbTe thin films. Key contributions include concrete criteria for truncation versus apodization, demonstration of how window length and shape affect resonance fidelity, and the introduction of simple metrics $D_1$ and $D_2$ to guide window choices. The approach provides a usable experimental workflow that improves spectral fidelity and parameter accuracy for resonant thin-film THz responses, potentially enhancing parameter extraction such as Lorentz oscillator fits and enabling robust comparison across measurements and conditions.

Abstract

Terahertz time-domain spectroscopy (THz-TDS) has become a powerful tool for investigating the optical properties of thin films, offering direct access to the complex permittivity in the terahertz range. However, in transmission-based measurements of thin films on thick substrates, multiple reflections and limited time windows can introduce artifacts that obscure resonant features such as phonon modes. Time-domain windowing remains one of the most widely adopted strategies to mitigate these effects, yet systematic guidelines on its application remain scarce. In this work, we organize a practical routine for extracting the complex permittivity from THz-TDS data, focusing on when and how to apply time-domain windowing. The routine incorporates decision points for truncation versus smooth apodization, and emphasizes tailoring the window length and shape to specific signal conditions. We demonstrate the approach using representative measurements on PbTe thin films, highlighting cases in which truncation suffices, where apodization is essential, and how different window functions and lengths influence the resulting spectra. We also propose simple metrics to assess signal continuity and guide window selection. Although other analysis techniques exist, including parametric spectral estimation, this study focuses on formalizing windowing-based processing into an accessible experimental workflow. Our results show that the choice of window parameters can significantly affect the accuracy of extracted material parameters, particularly for sharply resonant systems. This work provides an accessible framework for improving spectral fidelity in THz-TDS of thin-film samples.

Figures (5)

• Figure 1: Schematic of THz-TDS measurements for extracting the complex permittivity of a thin film. (a) Reference measurement through a bare substrate of thickness $d_\textrm{sub}$, where the incident THz electric field $E_\textrm{i}(t)$ is reshaped into the reference signal $E_\textrm{ref}(t)$ after transmission. (b) Sample measurement through a thin film of thickness $d$ deposited on an identical substrate, yielding the transmitted sample signal $E_\textrm{sam}(t)$. Multiple internal reflections within the substrate and film layers generate Fabry--Pérot echoes in both waveforms, with pronounced oscillations in the sample due to resonant absorption.
• Figure 2: Effect of time-domain truncation on extracted optical properties of a PbTe thin film at 120. (a) Time-domain electric field signals for the reference $E_\textrm{ref}(t)$ (black) and sample $E_\textrm{sam}(t)$ (red). For clarity, all curves were vertically offset and the sample waveform is scaled $\times 10$. Arrows mark truncation points $t^\textrm{refl}$ and main peaks $t^\textrm{max}$. Dashed: original signals. Solid: truncated signals. Inset: final 1ps of truncated waveforms, normalized by their respective peak amplitudes, showing smooth decay. (b) Transmittance spectra calculated with (solid) and without (dashed) time-domain truncation. (c) Real and (d) imaginary parts of the extracted complex permittivity of the film. Truncation suppresses Fabry--Pérot oscillations and improves the clarity of the phonon resonance near 0.75.
• Figure 3: Effect of apodization windowing on extracted optical properties of a PbTe thin film at 10. (a) Sample time-domain waveform $E_\textrm{sam}(t)$ under three processing conditions: untruncated (dashed red), truncated (solid red), and Gaussian-windowed (solid blue). For clarity, all curves are vertically offset. Truncation suppresses internal reflections but introduces a sharp discontinuity. Inset: final 1 of truncated waveform, normalized by its peak amplitude, showing rapid slope change exceeding 10 of peak value. (b) Transmittance spectra calculated from truncated (solid red) and Gaussian-windowed (solid blue) signals. (c) Real and (d) imaginary parts of the extracted complex permittivity of the film. Gaussian windowing suppresses spectral artifacts and yields a more physically realistic resonance near 0.5.
• Figure 4: Effect of Gaussian window length variation on extracted optical properties of a PbTe thin film at 10. (a) Time-domain transmitted electric fields after applying Gaussian windows with standard deviations $\sigma=\qty{1.67}{\ps}$ (green), $\sigma=\qty{3.33}{\ps}$ (yellow), and $\sigma=\qty{6.67}{\ps}$ (blue). The black curve shows the original, untruncated signal ($\sigma=\infty$). Inset: vertically-offset zoomed view near Fabry--Pérot echoes highlighting the critical trade-off between echo suppression and signal preservation. (b) Transmittance spectra derived from the windowed signals. The intermediate $\sigma=\qty{3.33}{\ps}$ case (yellow) balances suppression of reflection-induced spectral modulations with preservation of the phonon absorption feature near 0.5, while shorter windows cause artificial broadening/attenuation (green) and longer windows retain spurious oscillations (blue).
• Figure 5: Effect of window function choice on extracted optical properties of a PbTe thin film at 10. Each curve corresponds to a different window function applied to identical time-domain data, with window lengths individually optimized for echo suppression and spectral fidelity. (a) Real and (b) imaginary parts of the complex permittivity. Gaussian windowing (red) yields sharper, higher-amplitude features; Flattop (gray) and Blackman--Harris (blue) produce broader lineshapes with attenuated peaks; Barthann (green) exhibits intermediate characteristics.