High-resolution spectroscopy can make key science measurements for a variety of astrophysics and planetary targets, including solar system planetary atmospheres, comets, solar wind charge exchange emission, and interstellar and interplanetary medium. With the ability to record adjacent spectral lines simultaneously key isotopic ratios such as D/H, C-12/C-13, O-16/O-18, etc., can be measured precisely. Traditional high spectral resolution spectrometers usually must couple to large optics to compensate for their low throughput, which prohibits achieving compactness, in particular in space and remote field applications. Also, the high cost of construction and maintenance limit their quantity and usage for the long duration temporal measurement of the sources. Spatial heterodyne spectrometers (SHS) are increasingly used in scientific observations and industry. To date, SHS instruments come in two major architectures: Michelson design and cyclical design. Cyclical SHS, also known as reflective SHS, can offer significant advantages over traditional spectrometers in obtaining high-resolution spectra in shorter wavelengths. Although cyclical SHSs have been introduced before, there has been no mathematical or performance characterization of their technique. This paper presents a comprehensive mathematical design and performance expectations of the cyclical tunable SHS technique to enable and expand its usage in a variety of platforms and applications, in the industry and astronomical observations from ground and space telescopes.
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