  Since the advent of experiments with photon Bose-Einstein condensates in
dye-filled microcavities in 2010, many investigations have focused upon the
emerging effective photon-photon interaction. Despite its smallness, it can be
identified to stem from two physically distinct mechanisms. On the one hand, a
Kerr nonlinearity of the dye medium yields a photon-photon contact interaction.
On the other hand, a heating of the dye medium leads to an additional
thermo-optic interaction, which is both delayed and non-local. The latter turns
out to represent the leading contribution to the effective interaction for the
current 2D experiments. Here we analyse theoretically how the effective
photon-photon interaction increases when the system dimension is reduced from
2D to 1D. To this end, we consider an anisotropic harmonic trapping potential
and determine via a variational approach how the properties of the photon
Bose-Einstein condensate in general, and both aforementioned interaction
mechanisms in particular, change with increasing anisotropy. We find that the
thermo-optic interaction strength increases at first linearly with the trap
aspect ratio and lateron saturates at a certain value of the trap aspect ratio.
Furthermore, in the strong 1D limit the roles of both interactions get reversed
as the thermo-optic interaction remains saturated and the contact Kerr
interaction becomes the leading interaction mechanism. Finally, we discuss how
the predicted effects can be measured experimentally.
