  We study the second-order Raman process of mono- and few-layer MoTe$_2$, by
combining {\em ab initio} density functional perturbation calculations with
experimental Raman spectroscopy using 532, 633 and 785 nm excitation lasers.
The calculated electronic band structure and the density of states show that
the electron-photon resonance process occurs at the high-symmetry M point in
the Brillouin zone, where a strong optical absorption occurs by a logarithmic
Van-Hove singularity. Double resonance Raman scattering with inter-valley
electron-phonon coupling connects two of the three inequivalent M points in the
Brillouin zone, giving rise to second-order Raman peaks due to the M point
phonons. The predicted frequencies of the second-order Raman peaks agree with
the observed peak positions that cannot be assigned in terms of a first-order
process. Our study attempts to supply a basic understanding of the second-order
Raman process occurring in transition metal di-chalcogenides (TMDs) and may
provide additional information both on the lattice dynamics and optical
processes especially for TMDs with small energy band gaps such as MoTe$_2$ or
at high laser excitation energy.
