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The next generation of astronomical space-based far-infrared (FIR) missions require ultra-sensitive spectroscopy as a diagnostic tool. These instruments use ultra-sensitive detector technologies to attain unprecedented levels of spectral observing sensitivity. The reception patterns of the individual detectors consist of individually coherent orthogonal field distributions, or equivalently, they are few-mode (5 to 20), to increase the spectral-spatial coupling to the astronomical source. However, the disadvantage of few-mode detectors is an increase in coupling to external (from the sky or warm telescope optics) and internal (from the instrument itself) straylight, which can greatly affect the measurement of the source spectrum. Therefore, understanding the spectral-spatial few-mode behavior of these systems in detail, and developing verification and calibration strategies, are crucial to ensure that the science goals of these future mission are met. Since conventional modelling techniques are less suited to address this problem, we developed a modal framework to model, analyze, and address these issues. In this paper, we use Herschel’s spectral and photometric imaging receiver (SPIRE) as a case study, because its optical design is representative for future FIR missions and illustrative to highlight calibration issues observed in-flight, while including straylight. Our analysis consist out of two part. In the first part, we use our modal framework to simulate the few-mode SPIRE Fourier transform spectrometer (FTS). In the second part, we carry out a end-to-end frequency-dependent partially coherent analysis of Herschel-SPIRE. These simulations offer a qualitative explanation for the few-mode behavior observed in-flight. Furthermore, we use the Herschel-SPIRE case-study to demonstrate how the modelling framework can be used to support the design, verification and calibration of spectrometers for future FIR missions. The modal framework is not only limited to the spectrometers discussed, but it can be used to simulated a wide range of spectrometers, such as low-resolution gratings and high-spectral resolution Fabry-Pérot interferometers.
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