Ponderomotive energy

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In strong field laser physics, the term Ponderomotive Energy[1] refers to the cycle averaged quiver energy of a free electron in an E-field.


The Ponderomotive Energy equation is given by,

U_p = e^2E_a^2/4m\omega_0^2

Where e is the electron charge, E_a is the linearly polarised electric field amplitude, \omega_0^2 is the laser carrier frequency and m is the electron mass.


In terms of the laser intensity I, using I=c\epsilon_0 E_a^2/2, it reads less simply U_p=e^2 I/2 c \epsilon_0 m \omega_0^2=2e^2/c \epsilon_0 m \times I/4\omega_0^2. Now, atomic units provide e=m=1, \epsilon_0=1/4\pi, \alpha c=1 where \alpha \approx 1/137. Thus, 2e^2/c \epsilon_0 m=8\pi/137.

The formula for the ponderomotive energy can be easily derived. A free electron of charge e interacts with an electric field E \, \exp(-i\omega t). The force on the electron is

F = eE \, \exp(-i\omega t).

The acceleration of the electron is

a_{m} = F/m = (eE/m) \, \exp(-i\omega t).

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

x = -a /\omega^2 = -\frac{eE}{m\omega^2} \, \exp(-i\omega t) = -\frac{e}{m\omega^2} \sqrt{\frac{2I_0}{c\epsilon_0}} \, \exp(-i\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 = e^2 E^2/ 4 m \omega^2 .

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

Atomic units[edit]

See also: Atomic units

Converting between SI units and atomic units is more subtle than the introduction suggests. As presented, the Ponderomotive energy in atomic units appears to have some issues. If one uses the atomic unit of electric field,[2] then the ponderomotive energy is just

U_p = \frac{I}{4\omega^2}.

See also[edit]

References and notes[edit]

  1. ^ Highly Excited Atoms. By J. P. Connerade. p339
  2. ^ CODATA Value: atomic unit of electric field