  In this work we analyze the physical properties of a sample of 56
spectroscopically selected star-forming (SF) Ly$\alpha$ emitting galaxies at
2.0$\lesssim$z$\lesssim$3.5 using both a spectral energy distribution (SED)
fitting procedure from rest-frame UV to mid-IR and direct 160$\mu$m
observations taken with the Photodetector Array Camera & Spectrometer (PACS)
instrument onboard \emph{Herschel Space Observatory}. We define LAEs as those
Ly$\alpha$ emitting galaxies whose rest-frame Ly$\alpha$ equivalent widths
(Ly$\alpha$ EW$_{rest-frame}$) are above 20\AA, the typical threshold in
narrow-band searches. Ly$\alpha$ emitting galaxies with Ly$\alpha$
EW$_{rest-frame}$ are called non-LAEs. As a result of an individual SED fitting
for each object, we find that the studied sample of LAEs contains galaxies with
ages mostly below 100Myr and a wide variety of dust attenuations, SFRs, and
stellar masses. The heterogeneity in the physical properties is also seen in
the morphology, ranging from bulge-like galaxies to highly clumpy systems. In
this way, we find that LAEs at 2.0$\lesssim$z$\lesssim$3.5 are very diverse,
and do not have a bimodal nature, as suggested in previous works. Furthermore,
the main difference between LAEs and non-LAEs is their dust attenuation,
because LAEs are not as dusty as non-LAEs. On the FIR side, four galaxies of
the sample (two LAEs and two non-LAEs) have PACS-FIR counterparts. Their total
IR luminosity place all of them in the ULIRG regime and are all dusty objects,
with A$_{1200}$$\gtrsim$4mag. This is an indication from direct FIR
measurements that dust and Ly$\alpha$ emission are not mutually exclusive. This
population of red and dusty LAEs is not seen at z$\sim$0.3, suggesting an
evolution with redshift of the IR nature of galaxies selected via their
Ly$\alpha$ emission.
