J. Plasma Physics (2007), vol. 73, part 5, pp. 635–647. ľ 2006 Cambridge University Press doi:10.1017/S0022377806006131 Printed in the United Kingdom 635 Hamiltonian formulation of direct laser acceleration in vacuum M. ELOY 1 , A. GUERREIRO 2 , J. T. MENDONC ¸A 3 and R. BINGHAM 4 1 Faculdade de Engenharia da Universidade Cat´ olica Portuguesa, Estrada Oct´ avio Pato, 2635-631 Rio de Mouro, Portugal (meloy@fe.ucp.pt) 2 CLOQ/Faculdade de Ciˆ encias da Universidade do Porto, R. do Campo Alegre, 687, 4169-007 Porto, Portugal 3 GoLP/Centro de F´ ısica de Plasmas, Instituto Superior T´ ecnico, 1049-001 Lisboa, Portugal 4 Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, UK (Received 20 September 2004) Abstract. We present a new formulation for the direct laser acceleration of electrons in vacuum based on the Hamiltonian theory. Two different regimes for the snow- plowed, accelerated electrons are identified and characterized, the first pertaining to high-intensity and the second to low-intensity pulses, both leading to efficient electron acceleration. Particle energy yields are shown to be independent of the exact shape of the laser pulse and energy gains are estimated. 1. Introduction The interaction of a short, intense laser pulse with an electron has become a subject of great interest. Several breakthroughs have been made in producing high energy electron beams up to GeV energies [1, 2, 3, 4] using lasers and up to 85 GeV using electron beams [5]. In these schemes the particles are accelerated by longitudinal relativistic nonlinear electron plasma waves driven by short pulse lasers or electron beams in either gas jets or plasma channels. The most recent paper [5] demonstrates that the acceleration of electrons can be at the high energy frontier [5, 6]. In this article we discuss direct acceleration of the electrons by the laser in vaccum. The possibility of direct acceleration of electrons by an intense (10 19 W cm -2 ) subpicosecond (300 fs) laser pulse in vacuum [7] was recently demonstrated, in which electrons, injected at an angle with respect to the laser pulse, were accelerated up to the MeV range. Direct laser acceleration schemes in a vacuum have been studied extensively, not only theoretically but also experimentally and numerically, and electron energies have been predicted to be raised to the GeV range [8]. In such schemes a net electron energy gain is observed either by including the contribu- tions of the transverse laser vector potential, by considering physical boundaries which limit the region of interaction, or even by considering focussed laser pulses, conditions for which the Lawson–Woodward (LW) theorem [9] does not apply. In fact, the LW theorem is valid only if the laser fields propagate in a vacuum, with no walls or boundaries present, the electron with which the pulse interacts is highly relativistic (v ∼ c) along the acceleration path, no static electric or magnetic fields