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The current state of far-infrared astronomy drives the need to developcompact, sensitive spectrometers for future space and ground-based instruments.Here we present details of the $\rm \mu$-Spec spectrometers currently indevelopment for the far-infrared balloon mission EXCLAIM. The spectrometers aredesigned to cover the $\rm 555 - 714\ \mu$m range with a resolution of $\rm R\=\ \lambda / \Delta\lambda\ =\ 512$ at the $\rm 638\ \mu$m band center. Thespectrometer design incorporates a Rowland grating spectrometer implemented ina parallel plate waveguide on a low-loss single-crystal Si chip, employing Nbmicrostrip planar transmission lines and thin-film Al kinetic inductancedetectors (KIDs). The EXCLAIM $\rm \mu$-Spec design is an advancement upon asuccessful $\rm R = 64\ \mu$-Spec prototype, and can be considered a sub-mmsuperconducting photonic integrated circuit (PIC) that combines spectraldispersion and detection. The design operates in a single $M{=}2$ gratingorder, allowing one spectrometer to cover the full EXCLAIM band withoutrequiring a multi-order focal plane. The EXCLAIM instrument will fly sixspectrometers, which are fabricated on a single 150 mm diameter Si wafer.Fabrication involves a flip-wafer-bonding process with patterning of thesuperconducting layers on both sides of the Si dielectric. The spectrometersare designed to operate at 100 mK, and will include 355 Al KID detectorstargeting a goal of NEP ${\sim}8\times10^{-19}$ $\rm W/\sqrt{Hz}$. We summarizethe design, fabrication, and ongoing development of these $\rm \mu$-Specspectrometers for EXCLAIM.

Developing a New Generation of Integrated Micro-Spec Far Infrared  Spectrometers for the EXperiment for Cryogenic Large-Aperture Intensity  Mapping (EXCLAIM)