Properties of ponderomotive forces, proportional to $\big\langle E^4 \big\rangle$, on a charged particle in an inhomogeneous electromagnetic wave

The characteristic features of the dynamics of particles crossing an inhomogeneous linearly polarised wave are investigated numerically and analytically. The initial particle velocity is perpendicular to the direction of propagation of the wave and its electric field vector. It was established that two ponderomotive forces, proportional to$\big\langle E^4 \big\rangle$, act on a particle crossing the wave. One of the forces, proportional to the initial velocity of the particles, deflects the beam of particles and displaces them in the direction opposed to that of the propagation of the wave. It is found that when the length and width of the inhomogeneous wave are altered, the deflection and the displacement vary periodically. The ponderomotive force is maximal when the time taken by the particle to cross the wave corresponds to an integral number of half-periods of the particle vibrations. The second force has the same direction as the gradient force. Analytic expressions describing the forces acting on particles crossing a weakly inhomogeneous magnetic wave are obtained.