  This paper presents the method for the first time to generate intense
high-order optical vortices that carry orbital angular momentum in the extreme
ultraviolet region. In three-dimensional particle-in-cell simulation, both the
reflected and transmitted light beams include high-order harmonics of the
Laguerre-Gaussian (LG) mode when a linearly polarized LG laser pulse impinges
on a solid foil. The mode of the generated LG harmonic scales with its order,
in good agreement with our theoretical analysis. The intensity of the generated
high-order vortex harmonics is close to the relativistic region, and the pulse
duration can be in attosecond scale. The obtained intense vortex beam possesses
the combined properties of fine transversal structure due to the high-order
mode and the fine longitudinal structure due to the short wavelength of the
high-order harmonics. Thus, the obtained intense vortex beam may have
extraordinarily promising applications for high-capacity quantum information
and for high-resolution detection in both spatial and temporal scales because
of the addition of a new degree of freedom.
