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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^{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 (\mathrm {\mu m} )^{2}

,^[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,^[3] 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 $U_{p}$ .

References and notes

^ Highly Excited Atoms. By J. P. Connerade. p. 339
^ https://www.phys.ksu.edu/personal/cdlin/class/class11a-amo2/atomic_units.pdf ^{[bare URL PDF]}
^ CODATA Value: atomic unit of electric field

[1] Highly Excited Atoms. By J. P. Connerade. p. 339

[2] ttps://www.phys.ksu.edu/personal/cdlin/class/class11a-amo2/atomic_units.pdf ^{[bare URL PDF]}

[3] CODATA Value: atomic unit of electric field

[1]

[2]

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Equation

Atomic units

Derivation

See also

References and notes