  The EXperiment for Cryogenic Large-Aperture Intensity Mapping (EXCLAIM) is a
cryogenic balloon-borne instrument that will map carbon monoxide and
singly-ionized carbon emission lines across redshifts from 0 to 3.5, using an
intensity mapping approach. EXCLAIM will broaden our understanding of these
elemental and molecular gases and the role they play in star formation
processes across cosmic time scales. The focal plane of EXCLAIM's cryogenic
telescope features six {\mu}-Spec spectrometers. {\mu}-Spec is a compact,
integrated grating-analog spectrometer, which uses meandered superconducting
niobium microstrip transmission lines on a single-crystal silicon dielectric to
synthesize the grating. It features superconducting aluminum microwave kinetic
inductance detectors (MKIDs), also in a microstrip architecture. The
spectrometers for EXCLAIM couple to the telescope optics via a hybrid planar
antenna coupled to a silicon lenslet. The spectrometers operate from 420 to 540
GHz with a resolving power R={\lambda}/{\Delta}{\lambda}=512 and employ an
array of 355 MKIDs on each spectrometer. The spectrometer design targets a
noise equivalent power (NEP) of 2x10-18W/\sqrt{Hz} (defined at the input to the
main lobe of the spectrometer lenslet beam, within a 9-degree half width),
enabled by the cryogenic telescope environment, the sensitive MKID detectors,
and the low dielectric loss of single-crystal silicon. We report on these
spectrometers under development for EXCLAIM, providing an overview of the
spectrometer and component designs, the spectrometer fabrication process,
fabrication developments since previous prototype demonstrations, and the
current status of their development for the EXCLAIM mission.
