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

Up={e2E2\over4m

2}
\omega
0
,

where

e

is the electron charge,

E

is the linearly polarised electric field amplitude,

\omega0

is the laser carrier frequency and

m

is the electron mass.

I

, using

I=c\epsilon0E2/2

, it reads less simply:
2
U
p={e

I\over2c\epsilon0m

2}={2e
\omega
0

2\overc\epsilon0m}{I\over

2}
4\omega
0
,

where

\epsilon0

is the vacuum permittivity.

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

Up(eV)=9.33I(1014W/cm2)λ2(\mum2)

,[2]

where the laser wavelength is

λ=2\pic/\omega0

, 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

,

\epsilon0=1/4\pi

,

\alphac=1

where

\alpha1/137

. If one uses the atomic unit of electric field,[3] then the ponderomotive energy is just

Up=

E2
2
4\omega
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(\omegat)

. The force on the charged particle is

F=qE\cos(\omegat)

.

The acceleration of the particle is

am={F\overm}={qE\overm}\cos(\omegat)

.

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

x={-a\over\omega2}=-

qE
m\omega2

\cos(\omegat)=-

q\sqrt{
m\omega2
2I0
c\epsilon0
} \, \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

Up

.

See also

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