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
 * $$U_p = {e^2 E^2 \over 4m \omega_0^2}$$,

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:
 * $$U_p={e^2 I \over 2 c \epsilon_0 m \omega_0^2}={2e^2 \over c \epsilon_0 m} \cdot {I \over 4\omega_0^2}$$,

where $$\epsilon_0$$ is the vacuum permittivity.

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


 * $$ U_p (\mathrm{eV}) = 9.33 \cdot I(10^{14}\ \mathrm{W/cm}^2) \cdot \lambda^2(\mathrm{\mu m}^2) $$,

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, $$e=m=1$$, $$\epsilon_0=1/4\pi$$, $$\alpha c=1$$ 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
 * $$a_{m} = {F \over m} = {q E \over m} \cos(\omega t)$$.

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

In laser physics, this is called the ponderomotive energy $$U_p$$.