Ponderomotive energy

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

$U_p=\frac{e^2{E}^2}{4m{\omega }_0^2}$,

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 $I=c{ϵ}_0{E}^2/2$, it reads less simply:

$U_p=\frac{e^{2}I}{2c{ϵ}_{0}m{\omega }_0^2}=\frac{2e^2}{c{ϵ}_{0}m}\cdot \frac{I}{4{\omega }_0^2}$,

where ${ϵ}_0$ is the vacuum permittivity.

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

$U_p(\mathrm{e}\mathrm{V})=9.33\cdot I(10^{14}{\mathrm{W}/\mathrm{c}\mathrm{m}}^2)\cdot {\lambda }^2({\mathrm{\mu }\mathrm{m}}^2)$,

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, $e=m=1$, ${ϵ}_0=1/4\pi$, $\alpha c=1$ where $\alpha \approx 1/137$. If one uses the atomic unit of electric field, 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

$a_m=\frac{F}{m}=\frac{qE}{m}\cos (\omega t)$.

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

$x=\frac{-a}{{\omega }^2}=-\frac{qE}{m{\omega }^2}\cos (\omega t)=-\frac{q}{m{\omega }^2}\sqrt{\frac{2I_0}{c{ϵ}_0}}\cos (\omega t)$.

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

$U=\frac{1}{2}m{\omega }^2⟨x^2⟩=\frac{q^2{E}^2}{4m{\omega }^2}$.

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

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