  We study the reflection of a Hermite-Gaussian beam at an interface between
two dielectric media. We show that unlike Laguerre-Gaussian beams,
Hermite-Gaussian beams undergo no significant distortion upon reflection. We
report Goos-H\"anchen shift for all the spots of a higher order Hermite
Gaussian beam near the critical angle. The shift is shown to be insignificant
away from the critical angle. The calculations are carried out neglecting the
longitudinal component along the direction of propagation for a spatially
finite, s-polarized, full 3-d vector beam. We briefly discuss the difficulties
associated with the paraxial approximation pertaining to a vector Gaussian
beam.
