32
A. Hu et al.
It is obvious that in the ps regime the lattice temperature remains much lower
than the electron temperature. Thus the lattice temperature can be omitted in (1.3.16).
When the condition k e T e α
2
γ T e is fulfilled (1.3.16) and (1.3.11) are very simple.
The electron and lattice temperatures at the end of a ps pulse are given by
T e ≈
I a a
γ
exp(−az)
(1.3.19)
T l ≈
F a a
C l
exp(−az)
(1.3.20)
Note that the obtained lattice temperature is governed by the electron cooling time.
Thus, in fs and ps regimes (1.3.14), (1.3.19) and (1.3.20) give the same expression for
the lattice temperature. This indicates that a logarithmic dependence of the ablation
depth on laser fluence is also found in the ps regime. However, this conclusion is
based on an assumption that the electron heat conduction is negligible. This is a very
crude approximation since the electron heat conduction and the formation of melted
zone must be related in ps ablation.
Nanosecond Pulses Ablation with ns pulses can be modeled with the condition τ i
~ 10 ps τ L . In this case, the electron and lattice temperatures are equal, T e = T l
= T and (1.3.8)–(1.3.10) reduce to
C l ∂ T
∂t
=
∂
∂z
k 0 ∂ T
∂z
+ I a a exp(−az)
(1.3.21)
There are many experimental and theoretical studies on the processes involved in
laser heating and irradiation with long pulses [117]. In this regime the target surface
is first heated to the melting point and then to the vaporization temperature. During
the interaction the dominant energy loss is heat conduction into the solid target.
The heat penetration depth is given by l ~ (Dt)
1/2 , where D = k 0 /C i is the thermal
diffusivity. Note that for a long pulse, D L 1/α
2 . The energy deposited inside the
target per unit mass is given by E m ~ It/ρl. Evaporation occurs when E m ~ L v at t th ,
where L v is the specific heat of evaporation. So, the condition for strong evaporation
becomes, E m > L v (or τ L > t th ) and
I ≥ I th ∼
ρ L v D
1/2
τ 1/2
F > F th ∼ ρ L v D
1/2
τ
1/2
l
(1.3.22)
for the laser intensity and the fluence, respectively. A striking characteristic is that
the threshold laser fluence depends on the square root of the laser pulse width. A
deviation of the damage threshold from the τ
1/2 scaling with short pulses has been
clearly evident by ablation of fused silica by infrared (1053 nm) and visible (526 nm)
laser radiation [118].
A. Hu et al.
It is obvious that in the ps regime the lattice temperature remains much lower
than the electron temperature. Thus the lattice temperature can be omitted in (1.3.16).
When the condition k e T e α
2
γ T e is fulfilled (1.3.16) and (1.3.11) are very simple.
The electron and lattice temperatures at the end of a ps pulse are given by
T e ≈
I a a
γ
exp(−az)
(1.3.19)
T l ≈
F a a
C l
exp(−az)
(1.3.20)
Note that the obtained lattice temperature is governed by the electron cooling time.
Thus, in fs and ps regimes (1.3.14), (1.3.19) and (1.3.20) give the same expression for
the lattice temperature. This indicates that a logarithmic dependence of the ablation
depth on laser fluence is also found in the ps regime. However, this conclusion is
based on an assumption that the electron heat conduction is negligible. This is a very
crude approximation since the electron heat conduction and the formation of melted
zone must be related in ps ablation.
Nanosecond Pulses Ablation with ns pulses can be modeled with the condition τ i
~ 10 ps τ L . In this case, the electron and lattice temperatures are equal, T e = T l
= T and (1.3.8)–(1.3.10) reduce to
C l ∂ T
∂t
=
∂
∂z
k 0 ∂ T
∂z
+ I a a exp(−az)
(1.3.21)
There are many experimental and theoretical studies on the processes involved in
laser heating and irradiation with long pulses [117]. In this regime the target surface
is first heated to the melting point and then to the vaporization temperature. During
the interaction the dominant energy loss is heat conduction into the solid target.
The heat penetration depth is given by l ~ (Dt)
1/2 , where D = k 0 /C i is the thermal
diffusivity. Note that for a long pulse, D L 1/α
2 . The energy deposited inside the
target per unit mass is given by E m ~ It/ρl. Evaporation occurs when E m ~ L v at t th ,
where L v is the specific heat of evaporation. So, the condition for strong evaporation
becomes, E m > L v (or τ L > t th ) and
I ≥ I th ∼
ρ L v D
1/2
τ 1/2
F > F th ∼ ρ L v D
1/2
τ
1/2
l
(1.3.22)
for the laser intensity and the fluence, respectively. A striking characteristic is that
the threshold laser fluence depends on the square root of the laser pulse width. A
deviation of the damage threshold from the τ
1/2 scaling with short pulses has been
clearly evident by ablation of fused silica by infrared (1053 nm) and visible (526 nm)
laser radiation [118].