THz-TDS: Fundamentals, Methods & Applications

• THz-TDS is an ultrafast spectroscopic technique that employs femtosecond laser pulses to generate broadband terahertz transients, capturing both amplitude and phase information.
• It utilizes advanced detection schemes such as electro-optic and photoconductive sampling to accurately resolve sub-picosecond temporal field profiles.
• Recent innovations in instrumentation, signal processing, and forward modeling have significantly enhanced bandwidth, SNR, and application-specific imaging capabilities.

Terahertz Time-Domain Spectroscopy (THz-TDS) is a coherent, ultrafast spectroscopic technique that measures the electric field of single-cycle or few-cycle pulses in the terahertz range (0.1–10 THz), providing direct access to both amplitude and phase information. THz-TDS is employed across materials science, condensed matter physics, chemistry, security, and biomedical research for probing low-energy excitations, carrier dynamics, and complex electromagnetic responses in diverse systems. The method has seen rapid advances in instrumentation, modeling, and application domains over the past decade.

1. Fundamental Principles and Detection Schemes

THz-TDS utilizes ultrashort laser pulses (usually femtosecond-duration) to generate broadband THz transients through mechanisms like photoconductive antenna emission, optical rectification in nonlinear crystals, or plasma-based methods (Zhao, 2023). The THz pulse is directed onto the sample, and the transmitted or reflected field is detected by sampling techniques that preserve the electric field’s temporal profile on a sub-picosecond timescale.

Detection Modalities:

• Electro-Optic Sampling (EOS): Based on the Pockels effect, an intense THz field induces birefringence in an EO crystal (e.g., ZnTe or GaP), modulating the polarization of a synchronized probe pulse. The induced phase shift, Δϕ, is given by

$$\Delta \phi = E_{\text{THz}} \cdot d \cdot n_g^3 \cdot r_{41},$$

with $d$ the crystal thickness, $n_g$ the group index, $r_{41}$ the EO coefficient, and $E_{\text{THz}}$ the instantaneous THz field.

• Photoconductive Sampling (PCS): Utilizes a semiconductor PCA gated by an optical probe. The transient photocurrent generated in the antenna gap is proportional to the incident THz field, assuming the carrier lifetime is short relative to the pulse duration (Zhao, 2023).
• Advanced Detection Methods: THz air-breakdown coherent detection (THz-ABCD) enables detection in air via laser-induced plasma, bypassing limitations such as phonon absorption in EO crystals. This promotes bandwidth up to 30 THz without the need for solid detection media (Zhao, 2023).
• Single-Shot and Asynchronous Sampling: Single-shot approaches use chirped probe pulses or reflective echelon optics to encode the entire temporal waveform into a single optical or spatial dimension (II et al., 2016, Zhao, 2023). Asynchronous optical sampling (ASOPS/ECOPS) allows rapid acquisition by employing two lasers with slightly mismatched repetition rates, scanning the delay window electronically (Harris et al., 2022).

2. Data Acquisition, Signal Processing, and Uncertainty Quantification

Accurate THz-TDS measurements require precise synchronization, noise suppression, and careful treatment of systematic effects such as delay line uncertainty and amplitude fluctuations.

• Delay Line Uncertainty: Positional errors in the delay line translate to time-axis noise. The error in the measured electric field $E(t)$ due to a time-shift $\Delta t$ is

$\Delta E(t) = \frac{dE}{dt} \cdot \Delta t.$

This leads to a white noise floor in the frequency domain, independent of signal amplitude—a fundamental limit on achievable SNR (Jahn et al., 2021).

• Noise Reduction and Signal Estimation: Advanced algorithms correct for trace drift, amplitude fluctuations, and time-axis errors using reference-based alignment and frequency-domain shift corrections. For example, corrected traces are computed as

$\tilde{E}_i(f) = e^{j2\pi f \delta_i} E_i(f),$

and optimized for minimal $\ell_2$ distance to a reference (Denakpo et al., 11 Oct 2024). Signal estimation improvements can raise SNR from ~28 dB to 62 dB in typical systems.

• Covariance Matrix Estimation: Reliable uncertainty quantification requires an empirical or regularized estimation of the covariance matrix in the time domain. Fitting models for extracting materials parameters then weight residuals by the inverse square root of the noise covariance, e.g.,

$$C_{\text{model}}\{P\} = \|\ [M_\text{noise}]^{-1/2} \cdot (E_{\text{sample}} - E_{\text{model}}\{P\})\ \|_2,$$

ensuring physical confidence intervals for extracted quantities (Denakpo et al., 11 Oct 2024).

• Windowing and Apodization: In transmission experiments, especially on thin films, time-domain truncation must be tailored to eliminate echoes and avoid artifacts; metrics such as relative amplitude discontinuity and slope at the truncation edge guide the choice between simple truncation and smoother windowing functions (e.g., Gaussian or Blackman-Harris) (Marulanda et al., 1 Jul 2025).

3. Instrumentation Innovations and Application-Specific Platforms

Recent developments have focused on expanding bandwidth, increasing dynamic range, enabling high-speed or high-throughput operation, and optimizing system geometry for specialized applications.

• High-Power and Broadband Systems: Use of organic nonlinear crystals such as MNA in both emission (via optical rectification) and detection (via EOS) has enabled >9 THz bandwidth and >40 dB dynamic range with average THz powers in the mW regime (Mansourzadeh et al., 20 Dec 2024). Repetition rates up to 400 kHz using commercial Yb lasers and tilted-pulse front schemes have produced 24 mW average THz power (Millon et al., 2022).
• Portable and Rapid Imaging Platforms: Heliostat-based beam steering and ECOPS in PHASR scanners enable 2 kHz temporal trace acquisition across extended fields-of-view (up to 40×27 mm²), providing real-time clinical imaging capabilities (Harris et al., 2022).
• Aperture and Geometry Optimization: For quantum materials, the choice of aperture strongly affects low-frequency transmission. The transmitted amplitude through a Gaussian beam and circular aperture is given by

$P(D) = P_0 [1 - e^{-D^2/(2w_0^2)}],$

with $w_0$ the beam waist. Small apertures diminish low-frequency response, distorting phase and amplitude and corrupting parameter extraction (Dias et al., 11 Jul 2025).

• Single-Pixel and Compressed Sensing Imaging: Spatial single-pixel architectures use modulated beams and DMDs to spatially encode images and reconstruct multidimensional (space, time, frequency) information, applying compressive sensing (ℓ₁, e.g., minimize $\|\Psi S(t)\|_1$ s.t. $W(t) = H S(t)$) to reduce acquisition time (Zanotto et al., 2019).

4. Advanced Forward Modeling, Simulation, and Data Interpretation

THz pulse propagation in complex and dispersive media increasingly leverages advanced numerical and machine learning techniques for interpretation and model inversion.

• Physics-Informed Neural Networks (PINNs): Instead of explicit meshing, PINNs embed Maxwell’s equations and Drude-Lorentz or auxiliary material models directly in the loss function:

$$L = L_{\mathrm{PDE}} + L_{\mathrm{IC}} + L_{\mathrm{BC}} + L_{\mathrm{Data}},$$

e.g.,

$$L_{\mathrm{PDE}} = \frac{1}{N_f} \sum_{i} \left(\nabla^2 E(x_i, t_i) - \frac{n^2}{c^2} \frac{\partial^2 E(x_i, t_i)}{\partial t^2} \right)^2.$$

These approaches eliminate meshing constraints, efficiently incorporate broadband dispersion, and can integrate experimental data for hybrid simulation-analysis workflows (Zhu et al., 8 Sep 2025). They show strong agreement with FDTD for both lossless and dispersive cases.

• Constrained Super-Resolution Algorithms: Gas-phase THz-TDS faces fundamental resolution limits due to the time window length. Fitting the time-domain signal to a physically constrained model comprising exponentially damped sinusoids (with line intensities, frequencies, and damping rates as variables) achieves ten-fold improved spectral resolution over the conventional Fourier criterion, enabling detection of ultra-narrow spectral lines (Eliet et al., 2021).

5. Materials Characterization and Specialized Applications

THz-TDS is extensively used for extracting transport and optical parameters in 2D materials, semiconductors, molecular crystals, polymers, and layered heterostructures.

• Optoelectronic Parameters via Model Fitting: For conducting layers (e.g., graphene), the frequency-dependent complex conductivity is extracted and parameters such as DC conductivity, carrier scattering time, carrier density, and mobility are obtained by model inversion (Drude or Drude–Smith). For example, in graphene:

$$\nu_F^* = \nu_F \left(1 + \frac{4\alpha}{1 + (\pi/2)\alpha} \ln \left( \frac{\Lambda}{\sqrt{\pi n}} \right) \right)$$

encapsulates interaction-driven Fermi velocity renormalization (Whelan et al., 2020).

• 2D Materials and Moiré/Twistronic Systems: In bilayer MoS₂, THz-TDS probes temperature-dependent intra-band conductivity and reveals conduction via the Σ_min valley, with analysis via the Drude–Smith model:

$$\sigma(\omega) = \frac{\sigma_0}{1 - i\omega\tau} \left[1 + \frac{c}{1 - i\omega\tau} \right],$$

where $c$ quantifies carrier localization and backscattering (Cheng et al., 2023).

• Defect Metrology in Multilayered Media: Echo labeling algorithms inspired by acoustic room geometry processing use the low-rank properties of Euclidean distance matrices (EDMs) to identify true and virtual image sources, allowing position-resolved defect mapping. Maximum independent set algorithms in graph representations isolate unique scatterers without artifacts (Sevdiren et al., 6 Feb 2025).
• Thin-Film and Sharp-Resonance Systems: Accurate extraction of dielectric function $\varepsilon_{\text{film}}(\nu)$ in thin films requires windowed Fourier transforms and careful assignment of model parameters (resonance frequency, linewidth, oscillator strength), with specific window metrics (discontinuity, slope) guiding data reduction (Marulanda et al., 1 Jul 2025).

6. Emerging Challenges, Systematic Effects, and Future Prospects

• Spectral and Spatial Filtering: Beam truncation by sub-waist apertures or edge effects filters out low-frequency components and introduces phase distortions, which are especially problematic for quantum materials where low-energy physics is central (Dias et al., 11 Jul 2025).
• Standardization and Data Processing: There is an ongoing effort to develop open-source tools, data reduction standards, and collaborative methodologies, especially as ultra-rapid (dual comb, ECOPS) and high-bandwidth systems proliferate (Denakpo et al., 11 Oct 2024).
• Time-Resolved and Nonlinear Phenomena: THz-TDS is advancing into pump–probe and extreme field applications. For example, in time-resolved studies of transient superconductivity, the measured response $\Sigma(\omega;t)$ contains additional oscillatory factors due to the finite observation window, which are particularly pronounced when momentum relaxation time is long (superconducting regimes) (Orenstein et al., 2015).
• Broader Integration: The field is moving toward hybrid platforms that combine THz-TDS with frequency-modulated continuous wave (FMCW) or single-pixel imaging modalities for simultaneous penetration and specificity (Shen et al., 2020, Zanotto et al., 2019).

In conclusion, THz-TDS stands as a highly versatile, information-rich methodology for extracting dynamical, spectral, and structural properties across a wide array of materials and devices. Continuous progress in instrumentation, data analysis, forward modeling, and application-specific system design is expanding both the achievable measurement fidelity and the breadth of accessible physical phenomena.