y-direction. When the electron has pulse force due to the Coulomb scattering at the
time with maximum velocity, it jumps to position (2) while preserving the kinetic
energy, meaning the path in the same circle as seen in Fig. 2.17. Then, the pulse force
should satisfy the nonadiabatic condition (Appendix 2). Then, strong force changes
the momentum drastically like the jump to (3). If the jump happens when the electron
has the maximum energy at (2), the average energy of the electron increases.
The laser absorption via Coulomb collision process can be described qualitatively
as follows. Electron motion is the repeat of increase and decrease of kinetic energy
by laser oscillation field. The Coulomb collision due to the fixed ions works as
nonadiabatic force to electrons while keeping the kinetic energy. Change of the
motion direction of electron allows the increase or decrease of energy after the short
time nonadiabatic force. This is nothing without the famous random walk statistical
problem. If we assume that the energy change by each collision is given by the
change of absolute value of the velocity Δv, a kind of diffusion equation is obtained
in velocity space. Then, the random number Δv should satisfy the relation
Àv os Δv v os
where v os is the amplitude of oscillation define din (1.3.10).
2.6.2 Simple Diffusion Model
It is useful to derive the absorption rate based on an intuitive model for collisional
absorption. For simplicity, only the velocity space of electrons is considered. Define
a velocity distribution function of electron in velocity space f(v) then assume that the
velocity distribution change in time because of random chance to obtain an
additional velocity toward forward and backward directions in colliding with the
background ions. At the time step n and the velocity point k, the following finite
difference relation is obtained:
f nþ1,k À f n,k ¼ À f n,k þ
1
2
f n,kþ1 þ f n,kÀ1
À
Á
ð2:6:1Þ
This is the same method to obtain diffusion phenomena in space based on random
walk model. Then, the time step of n and the step of velocity jump k in (2.6.1) are
assumed simply to be
Δv ¼ v os , Δt ¼ ν
À1
ei
ð2:6:2Þ
where v os is the oscillation velocity by laser electric field defined in (2.3.5). It is
reasonable to assume that an electron loses or gain the velocity of oscillation velocity
in colliding with an ion, where it is assumed that the collision event happens in very
2.6 Laser Absorption in Plasma
73
time with maximum velocity, it jumps to position (2) while preserving the kinetic
energy, meaning the path in the same circle as seen in Fig. 2.17. Then, the pulse force
should satisfy the nonadiabatic condition (Appendix 2). Then, strong force changes
the momentum drastically like the jump to (3). If the jump happens when the electron
has the maximum energy at (2), the average energy of the electron increases.
The laser absorption via Coulomb collision process can be described qualitatively
as follows. Electron motion is the repeat of increase and decrease of kinetic energy
by laser oscillation field. The Coulomb collision due to the fixed ions works as
nonadiabatic force to electrons while keeping the kinetic energy. Change of the
motion direction of electron allows the increase or decrease of energy after the short
time nonadiabatic force. This is nothing without the famous random walk statistical
problem. If we assume that the energy change by each collision is given by the
change of absolute value of the velocity Δv, a kind of diffusion equation is obtained
in velocity space. Then, the random number Δv should satisfy the relation
Àv os Δv v os
where v os is the amplitude of oscillation define din (1.3.10).
2.6.2 Simple Diffusion Model
It is useful to derive the absorption rate based on an intuitive model for collisional
absorption. For simplicity, only the velocity space of electrons is considered. Define
a velocity distribution function of electron in velocity space f(v) then assume that the
velocity distribution change in time because of random chance to obtain an
additional velocity toward forward and backward directions in colliding with the
background ions. At the time step n and the velocity point k, the following finite
difference relation is obtained:
f nþ1,k À f n,k ¼ À f n,k þ
1
2
f n,kþ1 þ f n,kÀ1
À
Á
ð2:6:1Þ
This is the same method to obtain diffusion phenomena in space based on random
walk model. Then, the time step of n and the step of velocity jump k in (2.6.1) are
assumed simply to be
Δv ¼ v os , Δt ¼ ν
À1
ei
ð2:6:2Þ
where v os is the oscillation velocity by laser electric field defined in (2.3.5). It is
reasonable to assume that an electron loses or gain the velocity of oscillation velocity
in colliding with an ion, where it is assumed that the collision event happens in very
2.6 Laser Absorption in Plasma
73