Ponderomotive energy

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

In terms of the laser intensity $I$, using $I=c\epsilon _{0}E_{a}^{2}/2$, it reads less simply $U_{p}=e^{2}I/2c\epsilon _{0}m\omega _{0}^{2}=2e^{2}/c\epsilon _{0}m\times I/4\omega _{0}^{2}$. 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

 $x=-a/\omega ^{2}=-(eE/m\omega ^{2})\,\exp(-i\omega t)$. 

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

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

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, then the ponderomotive enrgy is just

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

See also

• Ponderomotive force
• Harmonic generation