In Fig. 8.18, the electric fields of the laser and plasma waves are shown in (a), and
the spectrum of the plasma waves is shown in (b). In Fig. 8.18a, the normalized
electric field defined in (8.1.10) with laser frequency ω is plotted for t ¼ 2.6 ps with
solid black line and the other three times, while the shorter wavelength and largeamplitude one plotted with gray lines shows the normalized E y [¼a(x,t)] with
reduced value as E x /10; note that its peak is a ¼ 6. It is clearly seen that large
amplitude of about 0.1 ~ 0.15 is induced and the plasma wave is turbulent. Fourier
components of the plasma wave electric field E x is plotted in Fig. 8.18b with broad
spectrum around the peak of FRS matching waves.
8.5.1 Modeling PIC Simulation
The evolution of electron dynamics in the present case can be described by the set of
Eqs. (8.1.7)–(8.1.9). Here, however, we have to model the longitudinal electric field
in the x-direction due to the turbulent plasma waves. In Ref. [8], the multimode
longitudinal waves is modeled as
E x t, x
ð Þ ¼
X N
j¼ÀN
E 0,j θ v g t À x
À
Á
cos ω pe t À k x,j x þ φ 0,j
ð8:5:1Þ
The number of the plasma waves is 2 N + 1, and N ¼ 15 is used, and the
wavenumber k x,j and amplitude E 0,j are roughly fitted to the spectrum in
Fig. 8.18b. In (8.5.1), φ 0,j is random phase and θ is the step function.
In Fig. 8.19, the electron spectrum obtained after integrating (8.1.7)–(8.1.9) is
shown by the dotted line for the case with (8.5.1) and no laser field and by solid line
for both fields included consistently. The number of the test particles is 2.5 Â 10
4 . It
is clear that the hot electrons are dominantly produced via electron plasma wave
interaction. Then, the chaotic acceleration by the nonlinear property of the relativistic laser interaction accelerates the electrons further more. Compared to the
Fig. 8.18 (a) The electric field of laser and plasma waves. (b) Fourier spectra of the electric field Ex
by plasma waves
314
8 Chaos due to Relativistic Effect
the spectrum of the plasma waves is shown in (b). In Fig. 8.18a, the normalized
electric field defined in (8.1.10) with laser frequency ω is plotted for t ¼ 2.6 ps with
solid black line and the other three times, while the shorter wavelength and largeamplitude one plotted with gray lines shows the normalized E y [¼a(x,t)] with
reduced value as E x /10; note that its peak is a ¼ 6. It is clearly seen that large
amplitude of about 0.1 ~ 0.15 is induced and the plasma wave is turbulent. Fourier
components of the plasma wave electric field E x is plotted in Fig. 8.18b with broad
spectrum around the peak of FRS matching waves.
8.5.1 Modeling PIC Simulation
The evolution of electron dynamics in the present case can be described by the set of
Eqs. (8.1.7)–(8.1.9). Here, however, we have to model the longitudinal electric field
in the x-direction due to the turbulent plasma waves. In Ref. [8], the multimode
longitudinal waves is modeled as
E x t, x
ð Þ ¼
X N
j¼ÀN
E 0,j θ v g t À x
À
Á
cos ω pe t À k x,j x þ φ 0,j
ð8:5:1Þ
The number of the plasma waves is 2 N + 1, and N ¼ 15 is used, and the
wavenumber k x,j and amplitude E 0,j are roughly fitted to the spectrum in
Fig. 8.18b. In (8.5.1), φ 0,j is random phase and θ is the step function.
In Fig. 8.19, the electron spectrum obtained after integrating (8.1.7)–(8.1.9) is
shown by the dotted line for the case with (8.5.1) and no laser field and by solid line
for both fields included consistently. The number of the test particles is 2.5 Â 10
4 . It
is clear that the hot electrons are dominantly produced via electron plasma wave
interaction. Then, the chaotic acceleration by the nonlinear property of the relativistic laser interaction accelerates the electrons further more. Compared to the
Fig. 8.18 (a) The electric field of laser and plasma waves. (b) Fourier spectra of the electric field Ex
by plasma waves
314
8 Chaos due to Relativistic Effect