  The BINGO telescope was designed to measure the fluctuations of the 21-cm
radiation arising from the hyperfine transition of neutral hydrogen and aims to
measure the Baryon Acoustic Oscillations (BAO) from such fluctuations,
therefore serving as a pathfinder to future deeper intensity mapping surveys.
The requirements for the Phase 1 of the projects consider a large reflector
system (two 40 m-class dishes in a crossed-Dragone configuration), illuminating
a focal plane with 28 horns to measure the sky with two circular polarisations
in a drift scan mode to produce measurements of the radiation in intensity as
well as the circular polarisation. In this paper we present the optical design
for the instrument. We describe the intensity and polarisation properties of
the beams and the optical arrangement of the horns in the focal plane to
produce a homogeneous and well-sampled map after the end of Phase 1. Our
analysis provides an optimal model for the location of the horns in the focal
plane, producing a homogeneous and Nyquist sampled map after the nominal survey
time. We arrive at an optimal configuration for the optical system, including
the focal plane positioning and the beam behavior of the instrument. We present
an estimate of the expected side lobes both for intensity and polarisation, as
well as the effect of band averaging on the final side lobes. The cross
polarisation leakage values for the final configuration allow us to conclude
that the optical arrangement meets the requirements of the project. We conclude
that the chosen optical design meets the requirements for the project in terms
of polarisation purity, area coverage as well as homogeneity of coverage so
that BINGO can perform a successful BAO experiment. We further conclude that
the requirements on the placement and r.m.s. error on the mirrors are also
achievable so that a successful experiment can be conducted.(Abridged)
