Terahertz time domain spectroscopy data processing
Analyzing uncertainties to push boundaries
Data and signal processing are fundamental steps in distilling meaningful insights from experimental measurements. In our research, we adopt a Bayesian approach to estimate the underlying information or signal from recorded acquisitions. This processing not only facilitates the extraction of pertinent information regarding the materials and metamaterials studied in time domain spectroscopy but also accounts for the inherent 'noise' inevitably present in experimental data. By effectively addressing this noise, we ensure the accuracy and reliability of our results, ultimately enhancing the depth and validity of our scientific findings.
Time Domain Spectroscopy
TeraHertz Time Domain Spectroscopy (THz-TDS) utilizes an ultrashort pulsed laser to generate short pulses of TeraHertz radiation, as depicted in Figure 1. Our setup employs a photoconductive emission antenna to capture the envelope of the ultrashort laser pulse and convert it into the THz frequency range. A conductive antenna, acting as a receiver, is utilized in the sampling detection scheme. It detects the THz field exclusively when illuminated by the ultrashort laser. The experiment's recorded output comprises the electric field plotted against an equivalent time, determined by the displacement of a mechanical delay line, resulting in what are known as 'time traces.' These time traces can be further analyzed using Fourier transform techniques to extract spectral information and compare it to a reference.
Sources of Noise & Uncertainties
The dynamic range of modern THz-TDS systems has significantly improved compared to early experiments, now exceeding 100 dB. Consequently, the sources of noise and perturbations have evolved and can be found in various components of the system. Adopting a broad definition, noise encompasses any unwanted modification or addition that the signal experiences during capture, storage, transmission, processing, or conversion. It includes noise from the receiver, laser amplitude, and we have identified the most significant source of noise as a slow drift in temporal delay caused by temperature fluctuations in the optical fiber delivering the ultrashort pulse laser to the antennas, as depicted in Figure 2.
This noise not only reduces the signal-to-noise ratio by up to 30dB but also introduces bias to the signal estimator when using a simple averaging method. In a recent publication, we introduced a methodology and accompanying software, Correct@TDS, designed to mitigate several sources of noise in THz TDS setups. This software enables the retrieval of a more accurate signal estimator than the conventional average, along with algorithms capable of estimating noise correlation and precision matrices.
Extracting information by fitting in the time domain
In the process of extracting information through fitting in the time domain, our aim is to derive key parameters that characterize the material or metamaterial under investigation. These parameters could be spectroscopic in nature, such as frequency and the damping of a line representing the motion of charges in the material or photons in the metamaterial. Alternatively, they could be more analytical, such as the concentration of gas or aerosol in atmospheric measurements or solute in solution.
To extract meaningful information from the acquired signal estimator, we fit the data in the time domain. This involves adjusting theoretical models to closely match the experimental data obtained from THz-TDS measurements. By fitting the data in the time domain before applying any further modifications, we can identify any discrepancies between the model and the experimental results. This allows us to easily detect any experimental mistakes or model misrepresentations.
Our fit@tds software streamlines this process, enabling researchers to perform precise fitting of THz-TDS data. In our initial paper, the software enabled fitting in the time domain without additional weighting. In a recent publication, we introduced an updated version of the software that incorporates noise weighting, providing a quantitative comparison between the model and experiment. This feature allows us to quantitatively assess the validity of the model and compare different models using criteria such as the Akaike criteria.
Currently, we are developing a third version of the software, which will incorporate uncertainties into the fit parameters. By accounting for uncertainties, we can derive error bars on the retrieved meaningful magnitudes using the precision matrix provided by correct@tds. This enhancement will further improve the reliability of the extracted information and enhance the robustness of our analyses.
Deeper Understanding, Expanded Boundaries
Innovative data analysis methods not only provide us with insightful and reliable information but also challenge us to view our experiments from new perspectives, opening up new avenues of exploration. One such realization was the discovery that inherent spectral information exists within the time traces, surpassing traditional Fourier Transform limitations.
By implementing a constraint reconstruction algorithm and leveraging the intrinsic periodicity, known as the repetition frequency, of the ultrashort laser, we achieved a remarkable 30-fold super resolution in THz-TDS. This breakthrough allowed us to measure lines as narrow as 30 MHz and resolve doublets with a 400 MHz beating frequency.
We are dedicated to improving analytical techniques and expanding the scope of time domain analysis. We anticipate new discoveries and advancements as we continue our research.