We report on a gigahertz self-mode-locked high-order Hermite-Gaussian (HG) Nd:GdVO(4) laser. With a pump power of 2.2 W, the average output power for the TEM(0,m) modes from m=9 to m=0 are among 350-780 mW at a repetition rate of 3.5 GHz. The mode-locked pulse width is in the range of 20-25 ps for various HG TEM(0,m) modes. With a simple cylindrical-lens converter, the mode-locked HG beams are converted to generate picosecond optical vortex pulses.