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

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

where ${\displaystyle e}$  is the electron charge, ${\displaystyle E}$  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 ${\displaystyle I=c\epsilon _{0}E^{2}/2}$ , it reads less simply:

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

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

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

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

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

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,^[3] then the ponderomotive energy is just

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

Derivation

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

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

The acceleration of the particle is

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

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

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

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

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

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

See also

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

References and notes

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