Multiple-beam Interference Spectroscopy: Instrument Analysis and Spectrum Reconstruction

Abstract

Hyperspectral imaging systems based on multiple-beam interference (MBI), such as Fabry-Perot interferometry, are attracting interest due to their compact design, high throughput, and fine resolution. Unlike dispersive devices, which measure spectra directly, the desired spectra in interferometric systems are reconstructed from measured interferograms. Although the response of MBI devices is modeled by the Airy function, existing reconstruction techniques are often limited to Fourier-transform spectroscopy, which is tailored for two-beam interference (TBI). These methods impose limitations for MBI and are susceptible to non-idealities like irregular sampling and noise, highlighting the need for an in-depth numerical framework. To fill this gap, we propose a rigorous taxonomy of the TBI and MBI instrument description and propose a unified Bayesian formulation which both embeds the description of existing literature works and adds some of the real-world non-idealities of the acquisition process. Under this framework, we provide a comprehensive review of spectroscopy forward and inverse models. In the forward model, we propose a thorough analysis of the discretization of the continuous model and the ill-posedness of the problem. In the inverse model, we extend the range of existing solutions for spectrum reconstruction, framing them as an optimization problem. Specifically, we provide a progressive comparative analysis of reconstruction methods from more specific to more general scenarios, up to employing the proposed Bayesian framework with prior knowledge, such as sparsity constraints. Experiments on simulated and real data demonstrate the framework's flexibility and noise robustness. The code is available at https://github.com/mhmdjouni/inverspyctrometry.

Figures (13)

• Figure 1: The acquisition and inversion pipelines of the reconstruction of spectra from interferograms. It portrays both the role of the instrument for capturing the interferogram, which is modeled by a mathematical transformation of the input, and the necessity of an inversion pipeline for spectral reconstruction.
• Figure 2: Illustration of the operation principle of a Michelson interferometer (left) and a fpi (right).
• Figure 3: Transfer matrices of some (Michelson) and () instruments. Each row in the transfer matrices represents the interferometer response at a given . The spectral support of the instruments is $\Omega \in [1, 2.5] \, \um^{-1}$, falling in the near-infrared and visible domains. The range of is arbitrarily chosen with $L=51$ samples and a step of $\Delta\delta = 0.2 \, \um$, giving a Nyquist wavenumber $\sigma_{\textrm{Nyq}} = \sigma_{\max}$. As such, $\Omega$ covers a ratio $\alpha=0.6$ of the Nyquist range $[0, \sigma_{\textrm{Nyq}}]$. The wavenumbers in are oversampled such that $K > L$ in order to observe the rankness and the sampling limit via the singular values, while those of follow eq. \ref{['eq:sampling_analysis_wavenumbers_mbi']}.
• Figure 4: The change of the condition number of the transfer matrix of a with respect to reflectivity. The range of and wavenumbers is the same as that of \ref{['fig:transmittance_responses']}. The condition number shows a minimum at $\mathcal{R}=0.7$.
• Figure 5: Different reflectivity regimes.