  This work describes the optical design of the EXperiment for Cryogenic
Large-Aperture Intensity Mapping (EXCLAIM). EXCLAIM is a balloon-borne
telescope that will measure integrated line emission from carbon monoxide (CO)
at redshifts z < 1 and ionized carbon ([CII]) at redshifts z = 2.5-3.5 to probe
star formation over cosmic time in cross-correlation with galaxy redshift
surveys. The EXCLAIM instrument will observe at frequencies of 420--540 GHz
using six microfabricated silicon integrated spectrometers with spectral
resolving power R = 512 coupled to kinetic inductance detectors (KIDs). A
completely cryogenic telescope cooled to a temperature below 5 K provides
low-background observations between narrow atmospheric lines in the
stratosphere. Off-axis reflective optics use a $90$-cm primary mirror to
provide 4.2' full-width at half-maximum (FWHM) resolution at the center of the
EXCLAIM band over a field of view of 22.5'. Illumination of the 1.7 K cold stop
combined with blackened baffling at multiple places in the optical system
ensures low (< -40 dB) edge illumination of the primary to minimize spill onto
warmer elements at the top of the dewar.
