Girish Mahajan (Editor)

Ponderomotive energy

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In strong-field laser physics, ponderomotive energy is the cycle-averaged quiver energy of a free electron in an electromagnetic field.

Contents

Equation

The ponderomotive energy is given by

U p = e 2 E a 2 4 m ω 0 2 ,

where e is the electron charge, E a is the linearly polarised electric field amplitude, ω 0 is the laser carrier frequency and m is the electron mass.

In terms of the laser intensity I , using I = c ϵ 0 E a 2 / 2 , it reads less simply:

U p = e 2 I 2 c ϵ 0 m ω 0 2 = 2 e 2 c ϵ 0 m × I 4 ω 0 2 ,

where ϵ 0 is the vacuum permittivity.

Atomic units

In atomic units, e = m = 1 , ϵ 0 = 1 / 4 π , α c = 1 where α ≈ 1 / 137 . If one uses the atomic unit of electric field, then the ponderomotive energy is just

U p = E a 2 4 ω 0 2 .

Derivation

The formula for the ponderomotive energy can be easily derived. A free particle of charge q interacts with an electric field E exp ⁡ ( − i ω t ) . The force on the charged particle is

F = q E exp ⁡ ( − i ω t ) .

The acceleration of the particle is

a m = F m = q E m exp ⁡ ( − i ω t ) .

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

x = − a ω 2 = − q E m ω 2 exp ⁡ ( − i ω t ) = − q m ω 2 2 I 0 c ϵ 0 exp ⁡ ( − i ω t ) .

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

U = 1 2 m ω 2 ⟨ x 2 ⟩ = q 2 E 2 4 m ω 2 .

In laser physics, this is called the ponderomotive energy U p .

References

Ponderomotive energy Wikipedia
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