30
A. Hu et al.
short laser pulses is absorbed by free electrons due to inverse Bremsstrahlung (Joule
heat). The evolution of the absorbed energy involves thermalization within the electron gas (electron subsystem), energy transfer to the lattice and thermal diffusion in
the lattice. These processes can be expressed as:
C e
∂ T e
∂t
=
∂ Q(z)
∂ Z
− γ (T e − T l ) + S
(1.3.8)
C l
∂ T l
∂t
= γ (T e − T l )
(1.3.9)
Q(Z ) = −k e
∂ T e
∂z
, S = I (t)Aαexp(−α Z )
(1.3.10)
here z is the direction of energy propagation perpendicular to the target surface, Q(z)
is the heat flux, S is the laser source function, I(t) the laser intensity, A = 1 − R is the
surface transmissivity and α is the absorption coefficient. C e and C l are the specific
heat of the electron and lattice subsystems with C e = aT e where a is a constant, γ
is the electron-lattice coupling parameter, and k e is electron thermal conductivity. In
(1.3.8) we should consider three characteristic time scales: τ e , τ i , and τ L . τ e = C e /γ
is the electron cooling time, τ i = C l / γ is the lattice heating time and τ L the laser
pulse width. Following previous studies [9, 115], the laser pulses can be separated
into three kinds of time regimes.
Femtosecond Pulses For a fs pulse, the laser width is much shorter than the electron
cooling time, τ L τ e ~ 1 ps. Then, C e T e / t γ T e , and electron-lattice coupling
can be neglected. If D e τ L < α
−2 , where D e = k e /C e is the electron thermal diffusivity,
the electron heat conduction term can be neglected and (1.3.8) reduces to
C
e
∂ T
2
e
∂t
= 2I α αexp(−αz)
(1.3.11)
and gives
T e (t) =
T
2
0 +
2I α α
C
e
t exp(−αz)
1/2
(1.3.12)
here it is assumed that I(t) = I 0 and I a = AI 0 , while T 0 = T e (0) is the initial
temperature. C
e = C e /T e is a constant when T e remains smaller than the Fermi energy
(in temperature). At the end of the laser pulse the electron temperature is given by
T e (τ L ) ≈
2F α α
C
e
1
2
exp
−
z
δ
(1.3.13)
A. Hu et al.
short laser pulses is absorbed by free electrons due to inverse Bremsstrahlung (Joule
heat). The evolution of the absorbed energy involves thermalization within the electron gas (electron subsystem), energy transfer to the lattice and thermal diffusion in
the lattice. These processes can be expressed as:
C e
∂ T e
∂t
=
∂ Q(z)
∂ Z
− γ (T e − T l ) + S
(1.3.8)
C l
∂ T l
∂t
= γ (T e − T l )
(1.3.9)
Q(Z ) = −k e
∂ T e
∂z
, S = I (t)Aαexp(−α Z )
(1.3.10)
here z is the direction of energy propagation perpendicular to the target surface, Q(z)
is the heat flux, S is the laser source function, I(t) the laser intensity, A = 1 − R is the
surface transmissivity and α is the absorption coefficient. C e and C l are the specific
heat of the electron and lattice subsystems with C e = aT e where a is a constant, γ
is the electron-lattice coupling parameter, and k e is electron thermal conductivity. In
(1.3.8) we should consider three characteristic time scales: τ e , τ i , and τ L . τ e = C e /γ
is the electron cooling time, τ i = C l / γ is the lattice heating time and τ L the laser
pulse width. Following previous studies [9, 115], the laser pulses can be separated
into three kinds of time regimes.
Femtosecond Pulses For a fs pulse, the laser width is much shorter than the electron
cooling time, τ L τ e ~ 1 ps. Then, C e T e / t γ T e , and electron-lattice coupling
can be neglected. If D e τ L < α
−2 , where D e = k e /C e is the electron thermal diffusivity,
the electron heat conduction term can be neglected and (1.3.8) reduces to
C
e
∂ T
2
e
∂t
= 2I α αexp(−αz)
(1.3.11)
and gives
T e (t) =
T
2
0 +
2I α α
C
e
t exp(−αz)
1/2
(1.3.12)
here it is assumed that I(t) = I 0 and I a = AI 0 , while T 0 = T e (0) is the initial
temperature. C
e = C e /T e is a constant when T e remains smaller than the Fermi energy
(in temperature). At the end of the laser pulse the electron temperature is given by
T e (τ L ) ≈
2F α α
C
e
1
2
exp
−
z
δ
(1.3.13)