Ponderomotive energy

In strong-field laser physics, ponderomotive energy is the cycle-averaged quiver energy of a free electron in an electromagnetic field.<ref>Highly Excited Atoms. By J. P. Connerade. p. 339</ref>

Equation

The ponderomotive energy is given by

<math>U_p = {e^2 E^2 \over 4m \omega_0^2}</math>,

where <math>e</math> is the electron charge, <math>E</math> is the linearly polarised electric field amplitude, <math>\omega_0</math> is the laser carrier frequency and <math>m</math> is the electron mass.

In terms of the laser intensity <math>I</math>, using <math>I=c\epsilon_0 E^2/2</math>, it reads less simply:

<math>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}</math>,

where <math>\epsilon_0</math> is the vacuum permittivity.

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

<math> U_p (\mathrm{eV}) = 9.33 \cdot I(10^{14} \mathrm{W/cm}^2) \cdot \lambda(\mathrm{\mu m})^2 </math>,<ref>https://www.phys.ksu.edu/personal/cdlin/class/class11a-amo2/atomic_units.pdf</ref>

where the laser wavelength is <math>\lambda= 2\pi c/\omega_0</math>, and <math>c</math> is the speed of light. The units are electronvolts (eV), watts (W), centimeters (cm) and micrometers (μm).

Atomic units

In atomic units, <math>e=m=1</math>, <math>\epsilon_0=1/4\pi</math>, <math>\alpha c=1</math> where <math>\alpha \approx 1/137</math>. If one uses the atomic unit of electric field,<ref>CODATA Value: atomic unit of electric field</ref> then the ponderomotive energy is just

<math>U_p = \frac{E^2}{4\omega_0^2}.</math>

Derivation

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

<math>F = qE \, \cos(\omega t)</math>.

The acceleration of the particle is

<math>a_{m} = {F \over m} = {q E \over m} \cos(\omega t)</math>.

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

<math>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)</math>.

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

<math>U = \textstyle{\frac{1}{2}}m\omega^2 \langle x^2\rangle = {q^2 E^2 \over 4 m \omega^2}</math>.

In laser physics, this is called the ponderomotive energy <math>U_p</math>.

See also

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