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 {\displaystyle m} 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:
\( {\displaystyle U_{p}={e^{2}I \over 2c\epsilon _{0}m\omega _{0}^{2}}={2e^{2} \over c\epsilon _{0}m}\cdot {I \over 4\omega _{0}^{2}}}, \)
where \( \epsilon _{0} \) 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
\( {\displaystyle U_{p}={\frac {E_{a}^{2}}{4\omega _{0}^{2}}}.} \)
Derivation
The formula for the ponderomotive energy can be easily derived. A free particle of charge q interacts with an electric field \( {\displaystyle E\,\cos(\omega t)} \). The force on the charged particle is
\( {\displaystyle F=qE\,\cos(\omega t)}. \)
The acceleration of the particle is
a \( {\displaystyle a_{m}={F \over m}={qE \over m}\cos(\omega t)}. \)
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 \( U_{p}. \)
See also
Ponderomotive force
Electric constant
Harmonic generation
List of laser articles
References and notes
Highly Excited Atoms. By J. P. Connerade. p. 339
CODATA Value: atomic unit of electric field
Hellenica World - Scientific Library
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