Ponderomotive energy

In strong field laser physics, the term Ponderomotive Energy[1] refers to the cycle averaged quiver energy of a free electron in an E-field.

Equation

The Ponderomotive Energy equation is given by,

$U_p =$$e^2$$E_a^2$$/4m$$\omega_0^2$

Where $e$ is the electron charge, $E_a$ is the linearly polarised electric field amplitude, $\omega_0^2$ is the laser carrier frequency and $m$ is the electron mass.

Description

In terms of the laser intensity $I$, using $I=c\epsilon_0 E_a^2/2$, it reads less simply $U_p=e^2 I/2 c \epsilon_0 m \omega_0^2=2e^2/c \epsilon_0 m \times I/4\omega_0^2$. Now, atomic units provide $e=m=1$, $\epsilon_0=1/4\pi$, $\alpha c=1$ where $\alpha \approx 1/137$. Thus, $2e^2/c \epsilon_0 m=8\pi/137$.

The formula for the ponderomotive energy can be easily derived. A free electron of charge $e$ interacts with an electric field $E \, \exp(-i\omega t)$. The force on the electron is

$F = eE \, \exp(-i\omega t)$.

The acceleration of the electron is

$a_{m} = F/m = (eE/m) \, \exp(-i\omega t)$.

Because the electron executes harmonic motion, the electron's position is

$x = -a /\omega^2 = -\frac{eE}{m\omega^2} \, \exp(-i\omega t) = -\frac{e}{m\omega^2} \sqrt{\frac{2I_0}{c\epsilon_0}} \, \exp(-i\omega t)$.

For a particle experiencing harmonic motion, the time-averaged energy is

$U = \textstyle{\frac{1}{2}}m\omega^2 \langle x^2\rangle = e^2 E^2/ 4 m \omega^2$.

In laser physics, this is called the ponderomotive energy $U_p$.

Atomic units

$U_p =$$\frac{I}{4\omega^2}.$