# 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

Converting between SI units and atomic units is more subtle than the introduction suggests. As presented, the Ponderomotive energy in atomic units appears to have some issues. If one uses the atomic unit of electric field,[2] then the ponderomotive energy is just

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