Coupling loss and transmission in a multimode fiber-fed Virtually Imaged Phased Array (VIPA)

The Virtually Imaged Phased Array (VIPA) is a spectral disperser that has seen increasing adoption across various applications, including optical telecommunications, high-resolution spectroscopy, and LiDAR. Although VIPAs are typically fed by single-mode optical fiber in these applications, there is growing interest in using a multimode optical fiber feed to enable higher throughput. However, a multimode fiber feed introduces several challenges, one of which is a fundamental limit on how much light can be coupled into a VIPA. In this work, we derive a closed-form expression for the minimum tilt angle of a multimode fiber-fed VIPA required for no etendue coupling loss, and we show that this angle depends solely on the input fiber's etendue, the focal ratio of the injection cylindrical lens, and the VIPA's intrinsic parameters. Moreover, we derive the 2D and 3D etendue that can be coupled into a VIPA, as well as the VIPA's etendue coupling efficiency and total transmission, given a particular configuration. We validate our mathematical models using ray-tracing simulations in Zemax OpticStudio non-sequential mode. Our results provide practical guidance for the design of high-throughput optical systems based on multimode fiber-fed VIPAs.