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 4 m ω 0 2 {\displaystyle U_{p}={e^{2}E^{2} \over 4m\omega _{0}^{2}}} , where e {\displaystyle e} is the electron charge, E {\displaystyle E} is the linearly polarised electric field amplitude, ω 0 {\displaystyle \omega _{0}} is the laser carrier frequency and m {\displaystyle m} is the electron mass. In terms of the laser intensity I {\displaystyle I} , using I = c ϵ 0 E 2 / 2 {\displaystyle I=c\epsilon _{0}E^{2}/2} , it reads less simply:
U p = e 2 I 2 c ϵ 0 m ω 0 2 = 2 e 2 c ϵ 0 m ⋅ I 4 ω 0 2 {\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 ϵ 0 {\displaystyle \epsilon _{0}} is the vacuum permittivity. For typical orders of magnitudes involved in laser physics, this becomes:
U p ( e V ) = 9.33 ⋅ I ( 10 14 W / c m 2 ) ⋅ λ 2 ( μ m 2 ) {\displaystyle 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 λ = 2 π c / ω 0 {\displaystyle \lambda =2\pi c/\omega _{0}} , and c {\displaystyle 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 {\displaystyle e=m=1} , ϵ 0 = 1 / 4 π {\displaystyle \epsilon _{0}=1/4\pi } , α c = 1 {\displaystyle \alpha c=1} where α ≈ 1 / 137 {\displaystyle \alpha \approx 1/137} . If one uses the atomic unit of electric field, then the ponderomotive energy is just
U p = E 2 4 ω 0 2 . {\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
q {\displaystyle q} interacts with an electric field E cos ( ω t ) {\displaystyle E\,\cos(\omega t)} . The force on the charged particle is
F = q E cos ( ω t ) {\displaystyle F=qE\,\cos(\omega t)} . The acceleration of the particle is
a m = F m = q E m cos ( ω t ) {\displaystyle a_{m}={F \over m}={qE \over m}\cos(\omega t)} . Because the electron executes harmonic motion, the particle's position is
x = − a ω 2 = − q E m ω 2 cos ( ω t ) = − q m ω 2 2 I 0 c ϵ 0 cos ( ω t ) {\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 kinetic energy is
U = 1 2 m ω 2 ⟨ x 2 ⟩ = q 2 E 2 4 m ω 2 {\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 {\displaystyle U_{p}} .
See also Ponderomotive force Electric constant Harmonic generation List of laser articles
References and notes