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

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

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}={eE \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 4m\omega ^{2}}

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

References and notes

↑ Highly Excited Atoms. By J. P. Connerade. p. 339
↑ CODATA Value: atomic unit of electric field

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Ponderomotive energy

Equation

Atomic units

Derivation

See also

References and notes