Deterministic Compressed Sensing in Time-Domain Spectroscopy

Since the arrival of short-pulse lasers, time-domain detection has revolutionized the ways spectroscopic measurements in the terahertz (THz) frequency range are performed. As a result, well-established time-domain spectroscopy (TDS) techniques such as THz TDS or time-domain Raman spectroscopy (TDRS) enable to detect infrared-active and Raman-active resonances in materials and rival conventionally employed spectroscopy techniques such as Fourier transform infrared (FT-IR) or Raman spectroscopy.^1–4 The fundamental difference and at the same time fundamental advantage is that the spectroscopic information is achieved by direct sampling of a time-variant optical signal in the near-infrared, from which the spectroscopic information in the THz frequency range is deduced via subsequent Fourier transforms (FT). Thereby complex broadband detectors in the THz frequency range and/or bulky opto-mechanics such as high-resolution monochromators or interferometers can be avoided. A fundamental sketch of the most basic form of the TDS principle is sketched in Figure 1a.

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Figure 1.

(a) Principle of the most basic time-domain spectroscopy (TDS) set-up, and (b) representation of the resonance spectrum of a TDS as a sparse superposition of damped harmonic oscillations.

According to the Whittaker–Nyquist–Shannon (WNS) theorem, assuming equidistant sampling points, the spectral resolution ( d f ) is determined via the length of the sampled time-window T max :

Achieving a set frequency resolution and frequency bandwidth, hence requires a number of N sampling points. ⁵

Here we demonstrate a novel approach based on compressed sensing to determine the THz resonances of materials from a deterministic TDS measurement with a significantly lower number of non-equidistant sampling points M.^6,7 As we show one can define an optimal set of these M sampling points in a deterministic way and with guarantee to achieve a chosen probability of success. The fixed sampling grid of M sampling points can then be utilized in compact chip-integrated photonic delay lines as recently proposed based on, e.g., switchable waveguides. ⁸

Deterministic compressed sensing based time-domain spectroscopy (dCSTDS) exploits the fact that when the unknown signal is a sparse superposition of damped oscillations (see Figure 1b), such a signal is uniquely determined by many fewer samples, in contrast to the requirements of the WNS theorem.^7,9 One of the key results in compressed sensing is that such signals can even be found (recovered) from sub-Nyquist samples by convex algorithms instead of performing an exponentially complex combinatorial search, see, e.g., Xia et al. ¹⁰ for a rigorous theoretical treatment, further review and references. Recovery in the sub-Nyquist regime is possible if the so-called measurement matrix, containing sampled versions of each candidate signal, satisfies certain conditions known in the field as mutual coherence (maximum pairwise correlation of the columns), restricted isometry property, or nullspace properties. Theoretical compressed sensing results in the case of Fourier sparsity have been obtained for random sampling, and, to date, practical demonstrations in the THz regime have likewise been limited to random sampling approaches. ¹¹ The design of deterministic sampling is more related to mutual coherence and has been linked to combinatorial designs such as difference sets. ¹⁰ Here we demonstrate such a deterministic approach based on a concrete difference set. For recovery, we use a version of the square-root LASSO algorithm, which has stable and instance-optimal recovery guarantees even in the case of unknown noise power. ¹² In comparison to Bayesian and recently proposed neural estimators in compressed sensing, the concept of instance optimality, coupled with deterministic sampling patterns, is essential for trustworthy spectrometry implementations.

To demonstrate the applicability of deterministic compressed sensing a recent example for TDRS in the geosciences was chosen. ¹³ Figure 2a shows the TDRS data sampled according to the WNS theorem and according to dCSTDS and Figure 2b shows the corresponding spectra in the frequency domain. As can be seen, an excellent quantitative match can be achieved with dCSTDS with only a quarter of the sampling points. The sampling points are determined by the cyclic difference set (4,4)–Singer set PG(4,4). ¹⁴

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Figure 2.

Benchmarking of time-domain spectroscopy performed with sampling according to WNS and dCSTDS on the example of a TDRS measurement of quartz: (a) time-domain and (b) frequency domain. Note that the N = 330 data points required by the WNS theorem could be reduced to M = 85 data points, The accuracy derived from the reconstructed trace can be quantified to a normalized mean-square error NMSE of 0.004. The following common definition was used NMSE = ∑ i | y i − y ^ i | 2 ∑ i | y i | 2 , see also Sheehy et al. ¹⁵ No further thresholding was applied to the reconstructed spectrum.