# 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

${\displaystyle U_{p}={e^{2}E_{a}^{2} \over 4m\omega _{0}^{2}}}$,

where ${\displaystyle e}$ is the electron charge, ${\displaystyle E_{a}}$ is the linearly polarised electric field amplitude, ${\displaystyle \omega _{0}}$ is the laser carrier frequency and ${\displaystyle m}$ is the electron mass.

In terms of the laser intensity ${\displaystyle I}$, using ${\displaystyle I=c\epsilon _{0}E_{a}^{2}/2}$, it reads less simply:

${\displaystyle U_{p}={e^{2}I \over 2c\epsilon _{0}m\omega _{0}^{2}}={2e^{2} \over c\epsilon _{0}m}\times {I \over 4\omega _{0}^{2}}}$,

where ${\displaystyle \epsilon _{0}}$ is the vacuum permittivity.

### Atomic units

In atomic units, ${\displaystyle e=m=1}$, ${\displaystyle \epsilon _{0}=1/4\pi }$, ${\displaystyle \alpha c=1}$ where ${\displaystyle \alpha \approx 1/137}$. If one uses the atomic unit of electric field,[2] then the ponderomotive energy is just

${\displaystyle U_{p}={\frac {E_{a}^{2}}{4\omega _{0}^{2}}}.}$

## Derivation

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

${\displaystyle F=qE\,\exp(-i\omega t)}$.

The acceleration of the particle is

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

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

${\displaystyle x={-a \over \omega ^{2}}=-{\frac {qE}{m\omega ^{2}}}\,\exp(-i\omega t)=-{\frac {q}{m\omega ^{2}}}{\sqrt {\frac {2I_{0}}{c\epsilon _{0}}}}\,\exp(-i\omega t)}$.

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

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

In laser physics, this is called the ponderomotive energy ${\displaystyle U_{p}}$.