  The current state of far-infrared astronomy drives the need to develop
compact, sensitive spectrometers for future space and ground-based instruments.
Here we present details of the $\rm \mu$-Spec spectrometers currently in
development for the far-infrared balloon mission EXCLAIM. The spectrometers are
designed 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. The
spectrometer design incorporates a Rowland grating spectrometer implemented in
a parallel plate waveguide on a low-loss single-crystal Si chip, employing Nb
microstrip planar transmission lines and thin-film Al kinetic inductance
detectors (KIDs). The EXCLAIM $\rm \mu$-Spec design is an advancement upon a
successful $\rm R = 64\ \mu$-Spec prototype, and can be considered a sub-mm
superconducting photonic integrated circuit (PIC) that combines spectral
dispersion and detection. The design operates in a single $M{=}2$ grating
order, allowing one spectrometer to cover the full EXCLAIM band without
requiring a multi-order focal plane. The EXCLAIM instrument will fly six
spectrometers, which are fabricated on a single 150 mm diameter Si wafer.
Fabrication involves a flip-wafer-bonding process with patterning of the
superconducting layers on both sides of the Si dielectric. The spectrometers
are designed to operate at 100 mK, and will include 355 Al KID detectors
targeting a goal of NEP ${\sim}8\times10^{-19}$ $\rm W/\sqrt{Hz}$. We summarize
the design, fabrication, and ongoing development of these $\rm \mu$-Spec
spectrometers for EXCLAIM.
