  We develop a light-matter interface enabling strong and uniform coupling
between a chain of cold atoms and photons of an optical cavity. This interface
is a fiber Fabry-Perot cavity, doubly resonant for both the wavelength of the
atomic transition and for a geometrically commensurate red-detuned intracavity
trapping lattice. Fulfilling the condition of a strong and uniform atom-photon
coupling requires optimization of the spatial overlap between the two standing
waves in the cavity. In a strong-coupling cavity, where the mode waists and
Rayleigh range are small, we derive the expression of the optimal trapping
wavelength taking into account the Gouy phase. The main parameter controlling
the overlap of the standing waves is the relative phase shift at the reflection
on the cavity mirrors between the two wavelengths, for which we derive the
optimal value. We have built a microcavity optimized according to these
results, employing custom-made mirrors with engineered reflection phase for
both wavelengths. We present a method to measure with high precision the
relative phase shift at reflection, which allows us to determine the spatial
overlap of the two modes in this cavity.
