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A new approach to laser particle acceleration in vacuum is proposed that incorporates three ingredients: (i) a radially polarized laser beam; (ii) an aligned axicon focus; and (iii) the approximate matching of particle and laser phase velocities. This combination has several important advantages which are discussed. The critical issue in any vacuum acceleration scheme is the inherent backward slippage of the particle relative to the laser beam phase, causing the particles to experience successive accelerations and decelerations. According to the Lawson–Woodward theorem, which applies for fairly general conditions, the net result is no energy gain. It is shown that this theorem does not apply to the present laser particle acceleration scheme, and this is verified by applying an exact solution for the electric field. Indeed, acceleration gradients approaching 1 GeV/m over an interaction length of a few centimeters are predicted for 1013 W laser peak power at 10.6 μm.