different mass from the mass in the vacuum due to the strong coupling with laser
field and behaves like the electron is dressed. It is called dressed electron in a strong
field. The effective mass of the dressed electron is given in the form, for example, in
[10]:
m à ¼ m 1 þ a
j j
2
1=2
ð5:4:19Þ
where a is the strength parameter of laser field defined in (5.2.46). The effective mass
is obtained with Volkov solution. Volkov solution is the exact solution of Dirac
equation in a plane electromagnetic wave with any amplitude. This solution provides
many QED effects, and it is, for example, used to derive the nonlinear QED cross
section of pair creation via multiphoton absorption in the strong laser and relativistic
electron beam interaction scheme [11].
From the energy conservation relation for n-photon recoil, the minimum energy
of the scattered electron to all angle is calculated for the incident angle of laser θ 0 , for
example from [10], in the form:
E min ¼
1
1 þ 2n b
E 0 b
ω 0 m=m Ã
ð
Þ
2 1 þ cosθ 0
ð
Þ
E 0
ð5:4:20Þ
where E 0 is the initial electron beam energy and
b
E 0 ¼ E 0 =mc
2 , b
ω 0 ¼ ħω 0 =mc
2
It is noted that taking m à ¼ m and n ¼ 1 reproduces the relation for the ordinary,
linear Compton scattering (inverse Compton scattering, ICS).
For the case of a 0 ¼ 0.5, E 0 ¼ 46.6 GeV, and θ 0 ¼ 17
∘ , the scattered electron
energy spectra are plotted theoretically in Fig. 5.16 for the laser with 1.06 μm
wavelength and the number of electron in a bunch of 5 Â 10
9 . It is noted that the
photon energy in the beam frame is 211 keV upshifted from 1 eV and E min is
25.6 GeV for n ¼ 1. In order to calculate the scattering cross section beyond KleinNishina formula, Volkov solution is used to obtain the scattering matrix [11]. These
parameters are taken from the corresponding experiment [10]. In Fig. 5.16, the
dotted line below 25.6 eV is data only including the linear Compton scattering and
MCS effect. The numerical values of NCS signal for n ¼ 1, 2, 3, and 4 in (5.4.20) are
also spotted in Fig. 5.16.
In order to demonstrate NCS, the experiment was done with electron bean in
SLAC [10]. The aim is to identify the electrons below 25.6 GeV to measure the
excess electrons from the dashed line. It is reported that the above NCS theory is
experimentally confirmed at the first time, and up to n ¼ 4 events are experimentally
confirmed as shown details of the experimental data and analysis [10].
The same type of experiment has been done with more intense laser and electron
beam generated by laser wake field acceleration technique [12]. This is also an
200
5 Relativistic Laser-Electron Interactions
field and behaves like the electron is dressed. It is called dressed electron in a strong
field. The effective mass of the dressed electron is given in the form, for example, in
[10]:
m à ¼ m 1 þ a
j j
2
1=2
ð5:4:19Þ
where a is the strength parameter of laser field defined in (5.2.46). The effective mass
is obtained with Volkov solution. Volkov solution is the exact solution of Dirac
equation in a plane electromagnetic wave with any amplitude. This solution provides
many QED effects, and it is, for example, used to derive the nonlinear QED cross
section of pair creation via multiphoton absorption in the strong laser and relativistic
electron beam interaction scheme [11].
From the energy conservation relation for n-photon recoil, the minimum energy
of the scattered electron to all angle is calculated for the incident angle of laser θ 0 , for
example from [10], in the form:
E min ¼
1
1 þ 2n b
E 0 b
ω 0 m=m Ã
ð
Þ
2 1 þ cosθ 0
ð
Þ
E 0
ð5:4:20Þ
where E 0 is the initial electron beam energy and
b
E 0 ¼ E 0 =mc
2 , b
ω 0 ¼ ħω 0 =mc
2
It is noted that taking m à ¼ m and n ¼ 1 reproduces the relation for the ordinary,
linear Compton scattering (inverse Compton scattering, ICS).
For the case of a 0 ¼ 0.5, E 0 ¼ 46.6 GeV, and θ 0 ¼ 17
∘ , the scattered electron
energy spectra are plotted theoretically in Fig. 5.16 for the laser with 1.06 μm
wavelength and the number of electron in a bunch of 5 Â 10
9 . It is noted that the
photon energy in the beam frame is 211 keV upshifted from 1 eV and E min is
25.6 GeV for n ¼ 1. In order to calculate the scattering cross section beyond KleinNishina formula, Volkov solution is used to obtain the scattering matrix [11]. These
parameters are taken from the corresponding experiment [10]. In Fig. 5.16, the
dotted line below 25.6 eV is data only including the linear Compton scattering and
MCS effect. The numerical values of NCS signal for n ¼ 1, 2, 3, and 4 in (5.4.20) are
also spotted in Fig. 5.16.
In order to demonstrate NCS, the experiment was done with electron bean in
SLAC [10]. The aim is to identify the electrons below 25.6 GeV to measure the
excess electrons from the dashed line. It is reported that the above NCS theory is
experimentally confirmed at the first time, and up to n ¼ 4 events are experimentally
confirmed as shown details of the experimental data and analysis [10].
The same type of experiment has been done with more intense laser and electron
beam generated by laser wake field acceleration technique [12]. This is also an
200
5 Relativistic Laser-Electron Interactions