# Ponderomotive energy

In strong-field laser physics, ponderomotive energy is the cycle-averaged quiver energy of a free electron in an electromagnetic field.[1]

## Equation

The ponderomotive energy is given by

$U_p = {e^2 E_a^2 \over 4m \omega_0^2}$,

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.

In terms of the laser intensity $I$, using $I=c\epsilon_0 E_a^2/2$, it reads less simply:

$U_p={e^2 I \over 2 c \epsilon_0 m \omega_0^2}={2e^2 \over c \epsilon_0 m} \times {I \over 4\omega_0^2}$.

### Atomic units

In atomic units, $e=m=1$, $\epsilon_0=1/4\pi$, $\alpha c=1$ where $\alpha \approx 1/137$. If one uses the atomic unit of electric field,[2] then the ponderomotive energy is just

$U_p = \frac{I}{4\omega_0^2}.$

## Derivation

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

$a_{m} = {F \over m} = {e E \over m} \exp(-i\omega t)$.

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

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

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

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

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