Computational electro-optic frequency comb spectroscopy

TL;DR

The paper tackles high-spectral-resolution spectroscopy with a non-interferometric, scan-less approach that leverages a dynamically reconfigurable electro-optic frequency comb to interrogate a sample with a sequence of known spectra. By measuring the integrated power for each probe comb and solving the linear inverse problem $\mathbf{A}\mathbf{x}=\mathbf{b}$—regularized via truncated SVD—the spectral response of the sample is reconstructed without dispersive spectrometers or interferometers. The authors provide theoretical foundations, numerical validations, and experimental demonstrations, including a tunable bandpass and a molecular absorption line of HCN around $1545\,\mathrm{nm}$, achieving acquisition in $\sim10\,\mathrm{ms}$ and RMSEs as low as $0.009$. The work suggests a path toward compact, chip-scale spectrometers that retain the benefits of optical frequency comb sampling for sensing applications like gas detection and hyperspectral imaging.

Abstract

Computational techniques have gained significant traction in photonics, enabling the co-design of hardware and data processing algorithms to drastically simplify optical system architectures and improve their performance. However, their application in optical frequency comb spectroscopy remains considerably underexplored. In this work, we introduce a non-interferometric approach to frequency comb spectroscopy based on dynamically tailored electro-optic modulation. The core of our method is a reconfigurable electro-optic comb generator capable of producing a sequence of known comb spectra to interrogate a spectroscopic sample. Instead of recording spectrally resolved or interferometric data, our system captures a set of integrated optical power measurements--one per probe comb--from which the sample's spectral response is computationally reconstructed by solving an inverse problem. We present the theoretical foundations of this method, assess its limitations, and validate it through numerical simulations. As a proof of concept, we demonstrate the experimental reconstruction of several spectral signatures, including a molecular absorption line at 1545 nm. For these results, we use numerically computed spectra and experimentally measured power values, all acquired within 10 milliseconds. Finally, we discuss potential extensions and improvements of the method, as well as its integration into chip-scale spectroscopic systems.

Figures (9)

• Figure 1: Operational principle. A comb generator, driven by a dynamically reconfigurable signal and fed by a cw laser, generates a set of known comb spectra for interrogating a spectroscopic sample. In each reconfiguration, the power is redistributed among the comb lines but remains constant, as illustrated in the bottom-left inset (const denotes a constant value). In contrast, the power transmitted through (or reflected by) the sample varies sequentially based on the overlap between its spectral response and each generated comb. The sample's response is then reconstructed from the measured power values by solving a linear system $\textbf{Ax}=\textbf{b}$, where $\textbf{A}$ is the spectrum matrix, and $\textbf{x}$ and $\textbf{b}$ represent the spectral response and the measured powers, respectively (see bottom-right inset). In practice, this linear system is ill-posed, and its solution requires formulating and solving an optimization problem.
• Figure 2: Examples of simulated EO-OFCs. (a) Symmetric spectrum about the carrier frequency when the modulated phase is a single-harmonic signal of frequency $f_s$ with a modulation index $\beta_{1}=2\pi$. (b) Asymmetric spectrum when the modulation phase is a signal composed of two harmonics $f_s$, $2f_s$ with modulation indexes $\beta_{1}=2\pi$, $\beta_{2}=\beta_{1}/10$ and a relative phase of $\varphi_{2}=0$. (c) Spectrum generated using the same modulation parameters as in (b) but with a relative phase $\varphi_{2}=\pi$.
• Figure 3: Numerical simulations. (a) Phase modulation to reconstruct a spectral response similar to the reflectivity of a FBG centered at the laser frequency. For the i-th configuration, $\Phi_i(t)$ is a single-tone signal of frequency $f_s$ with a modulation index $\beta_i$. (b) Reconstructed points of the filter response $\textit{x}(\nu)$ (blue squares) from $M=31$ noisy measurements using the method based on the Moore-Penrose matrix. The solid line is the theoretical curve for the FBG reflectivity. (c) Phase modulation employed for the reconstruction of a more complex spectral response, which results from the overlapping of two different Gaussian curves asymmetrically shifted from the laser frequency. For the i-th configuration, $\Phi_i(t)$ is a two-tone signal with modulation indices $\beta_{1i}$ and $\beta_{2i}$ corresponding to the frequencies $f_s$ and $2f_s$, respectively. The angle $\varphi_i$ denotes the relative phase between the two harmonics. (d) Reconstructed points of $\textit{x}(\nu)$ (blue squares) and theoretical curve (solid line). In this case, we consider $41$ measurements and, as before, we employ a reconstruction algorithm based on the pseudoinverse matrix method.
• Figure 4: Sketch of the basic setup for computational EO comb spectroscopy. In the simplest configuration, a phase modulator (PM), fed by a cw laser, is driven by a frequency generator (FG) or an arbitrary waveform generator. The programmed modulation is boosted by an RF amplifier (Amp.). RF filters can be placed at the output of the amplifier to eliminate spurious harmonics (not shown). The sequence of combs generated by the reconfigurable EO modulation interrogates a spectroscopic sample. The light transmitted (or reflected) is captured by a photodetector (PD) and the generated electrical signal is subsequently digitized. To increase the number of comb lines, several PMs can be placed in cascade and the optical signal can be amplified if necessary.
• Figure 5: Experimental results for the transmission $x(\nu)$ of a tunable bandpass filter centered at the laser frequency. (a) Series of measured powers $b_i$ as a function of the modulation index $\beta / \pi$. (b) Reconstructed values of $x(\nu)$ (blue squares) together with a reference curve (red line) obtained by fitting dual-comb measurements (green triangles) with a Gaussian function. The RMSE of the reconstruction is $0.018$.