Physics: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

[math]\displaystyle{ U_p = {e^2 E^2 \over 4m \omega_0^2} }[/math],

where [math]\displaystyle{ e }[/math] is the electron charge, [math]\displaystyle{ E }[/math] is the linearly polarised electric field amplitude, [math]\displaystyle{ \omega_0 }[/math] is the laser carrier frequency and [math]\displaystyle{ m }[/math] is the electron mass.

In terms of the laser intensity [math]\displaystyle{ I }[/math], using [math]\displaystyle{ I=c\epsilon_0 E^2/2 }[/math], it reads less simply:

[math]\displaystyle{ 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]\displaystyle{ \epsilon_0 }[/math] is the vacuum permittivity.

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

[math]\displaystyle{ U_p (\mathrm{eV}) = 9.33 \cdot I(10^{14} \mathrm{W/cm}^2) \cdot \lambda(\mathrm{\mu m})^2 }[/math],

where the laser wavelength is [math]\displaystyle{ \lambda= 2\pi c/\omega_0 }[/math], and [math]\displaystyle{ 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]\displaystyle{ e=m=1 }[/math], [math]\displaystyle{ \epsilon_0=1/4\pi }[/math], [math]\displaystyle{ \alpha c=1 }[/math] where [math]\displaystyle{ \alpha \approx 1/137 }[/math]. If one uses the atomic unit of electric field, then the ponderomotive energy is just

[math]\displaystyle{ 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]\displaystyle{ q }[/math] interacts with an electric field [math]\displaystyle{ E \, \cos(\omega t) }[/math]. The force on the charged particle is

[math]\displaystyle{ F = qE \, \cos(\omega t) }[/math].

The acceleration of the particle is

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ U_p }[/math].

See also

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