Equation
edit
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
(
μ
m
)
2
{\displaystyle U_{p}(\mathrm {eV} )=9.33\cdot I(10^{14}\mathrm {W/cm} ^{2})\cdot \lambda (\mathrm {\mu m} )^{2}}
,[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
edit
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 ,[3] 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
edit
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)}
.
The acceleration of the particle is
a
m
=
F
m
=
q
E
m
cos
(
ω
t
)
{\displaystyle a_{m}={F \over m}={qE \over m}\cos(\omega t)}
.
Because the electron executes harmonic motion, the particle's position is
x
=
−
a
ω
2
=
−
q
E
m
ω
2
cos
(
ω
t
)
=
−
q
m
ω
2
2
I
0
c
ϵ
0
cos
(
ω
t
)
{\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)}
.
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
{\displaystyle U=\textstyle {\frac {1}{2}}m\omega ^{2}\langle x^{2}\rangle ={q^{2}E^{2} \over 4m\omega ^{2}}}
.
In laser physics, this is called the ponderomotive energy
U
p
{\displaystyle U_{p}}
.
See also
edit
References and notes
edit
^ Highly Excited Atoms . By J. P. Connerade. p. 339
^ https://www.phys.ksu.edu/personal/cdlin/class/class11a-amo2/atomic_units.pdf [bare URL PDF ]
^ CODATA Value: atomic unit of electric field