Ponderomotive energy explained

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 ^{2} \over 4 m \omega ^{2 }} _{0} ,

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.I , using I = c\epsilon _{0} E ^{2/2} , it reads less simply:U ^{2} _{p = {e} I \over 2 c \epsilon _{0} m \omega ^{2 } = { 2 e} _{0} ^{2} \over c \epsilon _{0} m} ⋅ {I \over 4 \omega ^{2 }} _{0} ,

where

\epsilon _{0} is the vacuum permittivity.

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

U _{p} ( eV ) = 9 . 33 ⋅ I( 10 ^{14} W/cm ^{2)} ⋅ λ ^{2(\mum} ^{2)} ,^[2]

where the laser wavelength is

λ= 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 ≈ 1 / 137 . If one uses the atomic unit of electric field,^[3] then the ponderomotive energy is justU _{p} = (E ^{2})/(4 \omega ^{2} _{0}) .

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}=} - (qE)/(m\omega ^{2}) \cos(\omega t) =- (q)/(\sqrt{ m\omega ^{2}) (2 I _{0})/(c\epsilon _{0}) } \, \cos(\omega t).

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

U = style{ (1)/(2) }m\omega^2 \langle x^2\rangle = .

In laser physics, this is called the ponderomotive energy

U _{p} .

See also

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

Notes and References

1. Highly Excited Atoms. By J. P. Connerade. p. 339
2. https://www.phys.ksu.edu/personal/cdlin/class/class11a-amo2/atomic_units.pdf
3. CODATA Value: atomic unit of electric field