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

U

p

=

e

2

E

2

4

m

ω

0

2

{\displaystyle U_{p}={e^{2}E^{2} \over 4m\omega _{0}^{2}}}

,

where

e

{\displaystyle e}

is the electron charge,

E

{\displaystyle E}

is the linearly polarised electric field amplitude,

ω

0

{\displaystyle \omega _{0}}

is the laser carrier frequency and

m

{\displaystyle m}

is the electron mass.

In terms of the laser intensity

I

{\displaystyle I}

, using

I

=

c

ϵ

0

E

2

/

2

{\displaystyle I=c\epsilon _{0}E^{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

{\displaystyle U_{p}={e^{2}I \over 2c\epsilon _{0}m\omega _{0}^{2}}={2e^{2} \over c\epsilon _{0}m}\cdot {I \over 4\omega _{0}^{2}}}

,

where

ϵ

0

{\displaystyle \epsilon _{0}}

is the vacuum permittivity.

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

U

p

(

e

V

)

=

9.33

⋅

I

(

10

14

W

/

c

m

2

)

⋅

λ

2

(

μ

m

2

)

{\displaystyle U_{p}(\mathrm {eV} )=9.33\cdot I(10^{14}\ \mathrm {W/cm} ^{2})\cdot \lambda ^{2}(\mathrm {\mu m} ^{2})}

,

where the laser wavelength is

λ

=

2

π

c

/

ω

0

{\displaystyle \lambda =2\pi c/\omega _{0}}

, and

c

{\displaystyle 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

{\displaystyle e=m=1}

,

ϵ

0

=

1

/

4

π

{\displaystyle \epsilon _{0}=1/4\pi }

,

α

c

=

1

{\displaystyle \alpha c=1}

where

α

≈

1

/

137

{\displaystyle \alpha \approx 1/137}

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

U

p

=

E

2

4

ω

0

2

.

{\displaystyle U_{p}={\frac {E^{2}}{4\omega _{0}^{2}}}.}

Derivation

The formula for the ponderomotive energy can be easily derived. A free particle of charge

q

{\displaystyle q}

interacts with an electric field

E

cos

⁡

(

ω

t

)

{\displaystyle E\,\cos(\omega t)}

. The force on the charged particle is

F

=

q

E

cos

⁡

(

ω

t

)

{\displaystyle F=qE\,\cos(\omega t)}

.