  Interaction of axisymmetrically-polarized (radially or
azimuthally-polarized), relativistically intense laser pulses (ALP) with
under-dense plasma is shown experimentally to be different from the interaction
of conventional Gaussian pulses. The difference is clearly observed in distinct
spectra of scattered laser light as well as in appearance of a strong side
emission of second harmonic in the vicinity of focus spot. According 3D
particle-in-cell simulations, this is a result of instability in the
propagation of ALP in under-dense plasma. Laser wakefield acceleration of
electrons by ALP, therefore, is less efficient than that by Gaussian laser
pulses but ALP may be interesting for efficient electron self-injection.
