The Pondermotive Force

The Pondermotive Force

This is a good topic if you are thinking about studying laser or plasma physics. The goal is to understand how a laser beam can give a acceleration to a charged particle. That is called the pondermotive force.

We can start by considering a free charge q in the presence of an electromagnetic plane wave. At time t=0 the charge is at position r(0) = 0 with velocity v(0) = 0.

Assuming the charge does not move a large distance, we can take the electric field as constant in space, E(t)=E₀ cos(ωt). If e_k is the unit vector in the direction of propagation k, then E₀ is orthogonal to e_k.

Assuming the charge is moving non-relativistically, v/c << 1, we can at zeroth order ignore the force from the magnetic field part of the wave. Newton's equation of motion, ma=qE, together with the initial conditions at t=0, then give for this zeroth order motion of the charge,

r₀(t) = (q/mω²)E₀[1-cos(ωt)]

Thus the charge oscillates in phase with the same frequency as E. Averaging over one period of oscillation, the charge experiences no net acceleration, <a>=0.

Since v/c<<1, we can now include the efect of the magnetic field of the wave as a small perturbation to the zeroth order motion. The magnetic field is,

B(t)=B₀cos(ωt), with B₀ = e_k×E₀

The force on the charge due to B is, q(v/c)×B. To first order in perturbation, we can replace the velocity v by its zeroth order approximation, dr₀/dt, and solve for the resulting perturbed motion of the charge. You should find that this additional motion is again periodic and gives no net acceleration when averaged over one period of oscillation, <a>=0 [however, if you applied the initial conditions properly, you will see that the charge now has a constant drift velocity in the direction e_k].

Suppose now that the electric field was not constant in planes perpendicular to e_k, as in the above plane wave, but that it varied with position -- this would be the case for a laser beam, where the light is confined within a beam of finite cross-section, and the amplitdue of the E field decays to zero as one moves transverse to the direction of the beam.

Again, assuming the motion of the electron is small, one could approximate this spatial dependence of E by a Taylor expansion, E = E₀ + (r·∇)E, and regard the new term (r·∇)E as a perturbation on the motion r₀(t). If you now compute the motion of the charge under this perturbation, and average over one period of osillation, you will see that this spatial variation in E will exert an average force on (and so impart a net acceleration to) the charge. This is called the pondermotive force, and it can be easily expressed in terms of E(r).