Ponderomotive energy

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

Equation

The ponderomotive energy is given by

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

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

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

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

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

For typical orders of magnitudes involved in laser physics, this becomes:

$U_{p}(\mathrm {eV} )=9.33\cdot I(10^{14}\ \mathrm {W/cm} ^{2})\cdot \lambda ^{2}(\mathrm {\mu m} ^{2})$,

where the laser wavelength is $\lambda =2\pi c/\omega _{0}$, and $c$ is the speed of light. The units are electronvolts (eV), watts (W), centimeters (cm) and micrometers (μm).

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, then the ponderomotive energy is just

$U_{p}={\frac {E^{2}}{4\omega _{0}^{2}}}.$

Derivation

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

$F=qE\,\cos(\omega t)$.

The acceleration of the particle is

$a_{m}={F \over m}={qE \over m}\cos(\omega t)$.

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

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

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

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

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

See also

• Ponderomotive force
• Electric constant
• Harmonic generation
• List of laser articles