Toward the Miniaturization of Cross-Dispersed Echelle Spectrographs: Size and Spectral Resolving Power Parametrization of the Cassegrain and Three-fold Configurations

Cross-dispersed echelle spectrographs (CDESs) are a fundamental tool for modern astronomy. The spectral resolution achieved by these instruments is directly related to the size of the echelle grating and the focal length of the collimator. However, when they are used in ground-based observatories, they are typically seeing-limited, meaning that the achievable spectral resolving power is limited by the coherent fraction of the telescope aperture. As a result, achieving high-spectral resolution requires large echelle apertures and long collimator focal lengths, leading to instruments with significant dimensions. These dimensions are often also necessary to ensure the thermal and mechanical stability required by demanding science cases. With the advent of large-aperture telescopes and the unrelenting demand for high resolution, the prevailing trajectory of these instruments leans toward escalating in both size and complexity. However, it is important to explore miniaturization strategies for CDESs, which are particularly valuable, for example, for space-based instruments where size, weight, and power are crucial. In addition, space-based observatories are not affected by atmospheric seeing and can therefore operate in a diffraction-limited regime, allowing the full grating-limited spectral resolution to be exploited. In this work, we demonstrate the working principle of a parametric model that describes two configurations of CDES: the Three-fold (3F) and the Cassegrain (CA) design. The Three-fold CDES is a white-pupil design that is currently used at major observatories, and it is based on a single parabolic mirror for collimation and folding of the optical path. The Cassegrain CDES, a white-pupil design with a novel approach to the collimator, uses an inverted Cassegrain telescope design as a telephoto collimator. This allows for the design to achieve the same collimator focal length as with a single parabolic mirror but in a smaller footprint, paving the way for a possible miniaturization strategy for the 3F design. The presented model evaluates both configurations using two merit figures: the spectral resolving power and the design dimensions without the need to use conventional graphical ray-tracing software. This is used to systematically explore the design space of both configurations. Using Zemax simulations, we validate the model outputs in terms of spectral resolving power and instrument dimensions.