  The presence of stable, compact circumbinary discs of gas and dust around
post-asymptotic giant branch (post-AGB) binary systems has been well
established. We focus on one such system: IRAS 08544-4431. We present an
interferometric multi-wavelength analysis of the circumstellar environment of
IRAS 08544-4431. The aim is to constrain different contributions to the total
flux in the H, K, L, and N-bands in the radial direction. The data from
VLTI/PIONIER, VLTI/GRAVITY, and VLTI/MATISSE range from the near-infrared,
where the post-AGB star dominates, to the mid-infrared, where the disc
dominates. We fitted two geometric models to the visibility data to reproduce
the circumbinary disc: a ring with a Gaussian width and a flat disc model with
a temperature gradient. The flux contributions from the disc, the primary star
(modelled as a point-source), and an over-resolved component are recovered
along with the radial size of the emission, the temperature of the disc as a
function of radius, and the spectral dependencies of the different components.
The trends of all visibility data were well reproduced with the geometric
models. The near-infrared data were best fitted with a Gaussian ring model
while the mid-infrared data favoured a temperature gradient model. This implies
that a vertical structure is present at the disc inner rim, which we attribute
to a rounded puffed-up inner rim. The N-to-K size ratio is 2.8, referring to a
continuous flat source, analogues to young stellar objects. By combining
optical interferometric instruments operating at different wavelengths we can
resolve the complex structure of circumstellar discs and study the
wavelength-dependent opacity profile. A detailed radial, vertical, and
azimuthal structural analysis awaits a radiative transfer treatment in 3D to
capture all non-radial complexity.
