Motion in Oscillating Fields

We have seen that charged particles can be confined by a static magnetic field. A somewhat more surprising fact is that charged particles can also be confined by a rapidly oscillating, inhomogeneous electromagnetic wave-field. In order to demonstrate this, we again employ our averaging technique (Hazeltine and Waelbroeck 2004). To lowest order, a particle executes simple harmonic motion in response to an oscillating wave-field. However, to higher order, any weak inhomogeneity in the field causes the restoring force at one turning point to exceed that at the other. On average, this yields a net force that acts on the center of oscillation of the particle.

Consider a spatially inhomogeneous electromagnetic wave-field oscillating at frequency $\omega$ :

$${\bf E}({\bf r}, t) = {\bf E}_0({\bf r})\,\cos(\omega\, t).$$ (2.121)

The equation of motion of a charged particle placed in this field is written

$$m\,\frac{d{\bf v}}{dt}= e\left[{\bf E}_0({\bf r})\,\cos(\omega\, t) +{\bf v} \times {\bf B}_0({\bf r})\,\sin(\omega\, t)\right],$$ (2.122)

where

$${\bf B}_0 = -\omega^{-1}\,\nabla\times{\bf E}_0,$$ (2.123)

in accordance with Faraday's law.

In order for our averaging technique to be applicable, the electric field ${\bf E}_0$ experienced by the particle must remain approximately constant during an oscillation. Thus,

$$({\bf v}\cdot\nabla)\,{\bf E}_0 \ll \omega\,{\bf E}_0.$$ (2.124)

When this inequality is satisfied, Equation (2.123) implies that the magnetic force experienced by the particle is smaller than the electric force by one order in the expansion parameter.

Let us now apply the averaging technique. We make the substitution $t\rightarrow\tau$ in the oscillatory terms, and seek a change of variables,

$${\bf r} = {\bf R}(t) + \mbox{\boldmath$\xi$} ({\bf R}, {\bf U}, t,\tau),$$ (2.125)

$${\bf v} = {\bf U}(t) + {\bf u}({\bf R}, {\bf U},t,\tau),$$ (2.126)

such that $\xi$ and ${\bf u}$ are periodic functions of $\tau$ with vanishing mean. Averaging $d{\bf r}/dt = {\bf v}$ again yields $d{\bf R}/dt = {\bf U}$ to all orders. To lowest order, the momentum evolution equation, Equation (2.122), reduces to

$$\frac{\partial{\bf u}}{\partial \tau} = \frac{e}{m}\,{\bf E}_0({\bf R})\,\cos(\omega\,\tau).$$ (2.127)

Moreover, the lowest order oscillating component of $d{\bf r}/dt = {\bf v}$ gives

$$\frac{\partial\mbox{\boldmath$\xi$}}{\partial\tau} = {\bf u}.$$ (2.128)

The solutions to the previous two equations, taking into account the constraints $\langle {\bf u}\rangle =\langle$$\xi$ $\rangle = {\bf0}$ , are

$${\bf u} = \frac{e}{m\,\omega}\,{\bf E}_0\,\sin(\omega\,\tau),$$ (2.129)

$$\mbox{\boldmath$\xi$} = -\frac{e}{m\,\omega^2}\,{\bf E}_0\,\cos(\omega\,\tau),$$ (2.130)

respectively. Here, $\langle\cdots\rangle\equiv(2\pi)^{-1}\oint(\cdots)\,d(\omega\,\tau)$ represents an oscillation average.

It follows that, to lowest order, there is no motion of the center of oscillation. To first order, the oscillation average of Equation (2.122) yields

$$\frac{d{\bf U}}{dt} = \frac{e}{m} \left\langle (\mbox{\boldmath$\xi$}\cdot\nabla)\, {\bf E} + {\bf u}\times{\bf B}\right\rangle,$$ (2.131)

which reduces to

$$\frac{d{\bf U}}{dt} = -\frac{e^2}{m^2\,\omega^2} \left[ ({\bf E}_... ... E}_0\times(\nabla\times{\bf E}_0)\,\langle\sin^2(\omega\,\tau)\rangle \right].$$ (2.132)

The oscillation averages of the trigonometric functions are both equal to $1/2$ . Furthermore, we have $\nabla(\vert{\bf E}_0\vert^{\,2}/2)\equiv ({\bf E}_0\cdot\nabla)\,{\bf E}_0 +{\bf E}_0\times(\nabla\times{\bf E}_0)$ . Thus, the equation of motion for the center of oscillation reduces to

$$m\,\frac {d{\bf U}}{dt} = -e\,\nabla{\mit\Phi}_{\rm pond},$$ (2.133)

where

$${\mit\Phi}_{\rm pond} = \frac{1}{4} \frac{e}{m\,\omega^2}\,\vert{\bf E}_0\vert^{\,2}.$$ (2.134)

It follows that the oscillation center experiences a force, known as the ponderomotive force, that is proportional to the gradient in the amplitude of the wave-field. The ponderomotive force is independent of the sign of the charge, so both electrons and ions can be confined in the same potential well.

The total energy of the oscillation center,

$${\cal E}_{\rm osc} = \frac{m}{2}\,U^{\,2} +e\,{\mit\Phi}_{\rm pond},$$ (2.135)

is conserved by its equation of motion, Equation (2.133). However, it follows from Equation (2.129) that the ponderomotive potential energy is equal to the average kinetic energy of the oscillatory motion: that is,

$$e\,{\mit\Phi}_{\rm pond} = \frac{m}{2}\,\langle u^2\rangle.$$ (2.136)

Thus, the force on the center of oscillation originates in a transfer of energy from the oscillatory motion to the average motion.

Most of the important applications of the ponderomotive force occur in laser plasma physics. For instance, a laser beam can propagate in a plasma provided that its frequency exceeds the plasma frequency. If the beam is sufficiently intense then plasma particles are repulsed from the center of the beam by the ponderomotive force. The resulting variation in the plasma density gives rise to a cylindrical well in the index of refraction that acts as a wave-guide for the laser beam (Kruer 2003).