**3. Laser-material interaction at different laser pulse durations**

Laser ablation of the materials starts with photon absorption, followed by the heating and photoionisation of the subjected area on the target by the laser beam. Subsequently, the ablated materials released from the target surface as solid fragments, vapours, liquid drops or as an expanding plasma plume. The amount of ablated material and phase depends on the absorbed energy by the target material [13]. After laser-material interaction with short pulses and low intensity, due to the inverse Bremsstrahlung, the laser beam energy will be absorbed by free electrons from the material followed by thermalisation within the electrons, and energy transfer to the lattice. Finally, energy will be lost due to electron heat transfer to the target material. The energy transfer from the laser beam to the target material can be described using 1D and 2D diffusion models when this is considered rapid thermalisation in the electron subsystem and if both lattice and electron subsystems are characterised by their temperatures (*Ti* lattice temperature and *Te* electron temperature) [5]:

$$C\_e \frac{\partial T\_e}{\partial t} = -\frac{\partial Q(z)}{\partial z} - \gamma \left(T\_e - T\_i\right) + S \tag{1}$$

$$C\_i \frac{\partial T\_i}{\partial t} = \mathcal{Y} \left(T\_s - T\_i\right) \tag{2}$$

$$\mathcal{Q}(z) = -k\_\circ \left(\frac{\partial T\_\circ}{\partial z}\right) \tag{3}$$

$$S = I\begin{pmatrix} t \end{pmatrix} A\alpha \, e^{-\alpha z} \tag{4}$$

where *Q*(*z*) is the heat flux along the *z*-axis perpendicular to the target material surface, S is the laser heating source, *I*(*t*) is the laser intensity as a function of time, *A* is the surface transmissivity (*A* = 1 − *R*), *R* is the reflectivity of the target material, *α* is the absorption coefficient of the target material, *Ce* is the heat capacity of the electrons per unit volume, *Ci* is the lattice heat capacity per unit volume, *γ* is the parameter characterising the electron-lattice coupling, and *ke* is the thermal conductivity of the electrons. Two non-linear differential Eqs. (1) and (2) are used to model the cooling dynamics for Te and Ti , which account for the electronphonon coupling and thermal conductivity of the sample material [14]. In addition, these equations can be used to model the time evolution of the electron and lattice temperatures, Te and Ti [15]. Eqs. (1)–(4) can be written as:

$$C\_e \frac{\partial T\_e}{\partial t} = k\_e \left(\frac{\partial^2 T\_e}{\partial z^2}\right) - C\_i \frac{\partial T\_i}{\partial t} + I\left(t\right) A a e^{-az} \tag{5}$$

The lattice heat capacity (*Ci* ) is considerably higher than the electronic heat capacity (*Ce*); in this case, the electrons have a very high temperature. When the Fermi energy is higher than the electron energy, the non-equilibrium thermal conductivity and heat capacity of the electron can be written as *ke* = *ko* (*Ti* ).*Te/Ti* and *Ce = C′eTe*, respectively, where *ko(Ti* ) is the conventional equilibrium thermal conductivity of a material and *C′*e is a constant. In Eq. (5), a thermal conductivity in the lattice subsystem (phonon component) is neglected, and it has three characteristic timescales; *τe* is the electron cooling time (*τe* = *Ce/γ*), *τ<sup>i</sup>* is the lattice heating time (*τe* << *τ<sup>i</sup>* ) (*τ<sup>i</sup> = Ci /γ*), and *τL* is the laser pulse duration. In laser-material interaction, these parameters define three different interaction regimes: nanosecond, picosecond and femtosec‐ ond [5].

Zimmer [16] proposed a model for the analytical solution of the laser-induced temperature distribution across internal solid-liquid interfaces (see **Figure 2a**). It was shown that high solid surface temperature can be obtained with short laser pulse durations, sufficient interface absorption and high absorption liquids. As shown in **Figure 2b**, in the case of using a nano‐ second laser (20 ns), the temperature of the liquid environment is quite higher than that of the transparent solid material.

**Figure 2.** Diagram showing laser heating of a solid-liquid interface. *IL* is the laser absorption, *T* is the temperature dis‐ tribution in both materials, and *JI* is the heat conduction across the interface [16].

#### **3.1. Nanosecond laser**

energy by the target material [13]. After laser-material interaction with short pulses and low intensity, due to the inverse Bremsstrahlung, the laser beam energy will be absorbed by free electrons from the material followed by thermalisation within the electrons, and energy transfer to the lattice. Finally, energy will be lost due to electron heat transfer to the target material. The energy transfer from the laser beam to the target material can be described using 1D and 2D diffusion models when this is considered rapid thermalisation in the electron subsystem and if both lattice and electron subsystems are characterised by their temperatures

> ( ) ( ) *<sup>e</sup> e e i T Qz <sup>C</sup> TT S*

g

¶ ¶ (1)

¶ = - ¶ (2)


a a

= -+ ç ÷ ¶¶ ¶ è ø (5)

) is considerably higher than the electronic heat capacity (*Ce*); in

è ø ¶ (3)

, which account for the electron-

) is the conventional

is

=- - - +

( ) *<sup>i</sup> i ei <sup>T</sup> C TT t* g

( ) *<sup>e</sup> e <sup>T</sup> Qz k <sup>z</sup>* æ ö ¶ = - ç ÷

( ) *<sup>z</sup> S ItA e*

a a

where *Q*(*z*) is the heat flux along the *z*-axis perpendicular to the target material surface, S is the laser heating source, *I*(*t*) is the laser intensity as a function of time, *A* is the surface transmissivity (*A* = 1 − *R*), *R* is the reflectivity of the target material, *α* is the absorption coefficient of the target material, *Ce* is the heat capacity of the electrons per unit volume, *Ci*

the lattice heat capacity per unit volume, *γ* is the parameter characterising the electron-lattice coupling, and *ke* is the thermal conductivity of the electrons. Two non-linear differential Eqs.

phonon coupling and thermal conductivity of the sample material [14]. In addition, these equations can be used to model the time evolution of the electron and lattice temperatures,

( ) <sup>2</sup>

this case, the electrons have a very high temperature. When the Fermi energy is higher than the electron energy, the non-equilibrium thermal conductivity and heat capacity of the electron

equilibrium thermal conductivity of a material and *C′*e is a constant. In Eq. (5), a thermal

and *Ce = C′eTe*, respectively, where *ko(Ti*

*e ei z*

2

*T TT C k C ItAe tz t*


*ee i*

*t z*

¶ ¶

lattice temperature and *Te* electron temperature) [5]:

(1) and (2) are used to model the cooling dynamics for Te and Ti

).*Te/Ti*

[15]. Eqs. (1)–(4) can be written as:

(*Ti*

3064 High Energy and Short Pulse Lasers

Te and Ti

The lattice heat capacity (*Ci*

can be written as *ke* = *ko* (*Ti*

It was shown that the laser pulse duration has an effect on both the material ablation thresholds and penetration depths. Long pulse duration or increasing laser pulse duration increases the threshold fluence and decreases the effective energy penetration depth [1]. Low-intensity long laser pulse interaction with a target material firstly heats the surface of the target due to the absorbed energy, which leads to melting and vaporisation. It should be noted that vaporisation of the target requires much more energy than melting. "In case of low laser intensities the created vapour remains transparent for the laser radiation". The electron and lattice (ion) temperatures are equal (*Te* = *Ti* = *T*) [6]. In other words, if the laser pulse duration is long in comparison with the electron-phonon energy-transfer time (*τL* >> *τ<sup>i</sup>* ), the electrons and lattice temperatures will remain at the same thermal equilibrium point (*Te = Ti = T*) [5, 17]; as such, Eq. (5) reduces to the heat equation:

$$C\_i \frac{\partial T}{\partial t} = k\_o \left(\frac{\partial^2 T}{\partial z^2}\right) + I\_a a e^{-az} \tag{6}$$

In nanosecond laser beam interaction with material, the surface of the target material will be heated to melting point and then to vaporisation temperature. During the laser-material interaction, energy will be lost as heat conduction into the target material; the heat penetration depth (*l*) is given by *l* ~ (*Dt* ) 1/2, where *D* is the heat diffusion coefficient and is given by (*D* = *ko/Ci* ). It can be noted that for this regime of lasers, the condition DτLα<sup>2</sup> >> 1 is fulfilled, for example the thermal penetration depth is quite larger than the optical penetration depth [18]. The energy deposited in the target material per unit mass is given by *Em ~ Iαt/pl*; at a specific time (*t = tth*), this energy becomes higher than the specific heat of evaporation Ω, at which point considerable evaporation will occur. When Em ~ Ω, the results are *tth ~ D(*Ωρ*/I)*<sup>2</sup> . Consequently, for strong evaporation conditions, Em > Ω or τL > *fth* can be written for laser intensity as [5, 6]:

$$I > I\_{\rm th} \sim \frac{\rho \Omega D^{1/2}}{\sigma\_L^{1/2}} \tag{7}$$

and for laser fluence as:

$$F > F\_{\mu} \sim \rho \Omega D^{1/2} \times \tau\_{\perp}^{1/2} \tag{8}$$

The threshold laser fluence increases as τ<sup>L</sup> 1/2 . In nanosecond laser ablation regimes, there is enough time for thermal waves to propagate into the target material and to create a relatively large layer of melted material target [5, 6]. Nanosecond laser pulses can ablate the target materials even at low laser intensities in both the vapour and liquid phases, so a recoil pressure that expels the liquid will be created due to the vaporisation process [6]. Evaporation occur‐ rence makes challenge to precise laser processing with nanosecond laser pulses [18].

At long laser pulse duration, interaction with materials usually fulfils the condition *L*th >> α−1, *L*th being the heat-penetration depth which is given by *L*th ≈ (2*Dτ*p) 1/2, where *D = k/ρc, D* is the heat-diffusion coefficient. So, long laser pulse duration creates sufficient time for thermal waves to propagate within the target material, and the absorbed energy will be stored in a layer with a thickness of about *L*th. In this case, the target material needs much more energy to vaporise than to melt; in other words, evaporation will occur, while the energy absorbed per unit volume into the vaporised layer becomes higher than the latent heat of evaporation per unit volume, namely [19].

$$
\Delta \varpi\_{\upsilon} \approx \frac{A \left( F\_L - F\_{\imath h} \right)}{\rho L\_{\upsilon}} \tag{9}
$$

where *F*th is the laser fluence threshold which represents the minimum energy above which appreciable evaporation occurs from liquid metals. This figure is approximately given by the energy required to melt a surface layer of the target material of the order of *L*th [19]:

$$F\_{ih} \approx \frac{\rho c \Delta T\_m L\_{ih}}{A} \tag{10}$$

where Δ*T*m = *T*<sup>m</sup> − *T*o and Tm and To are the melting and initial target temperature, respectively.

#### **3.2. Picosecond laser**

2 2


*T T C k Ie t z*

*i o*

considerable evaporation will occur. When Em ~ Ω, the results are *tth ~ D(*Ωρ*/I)*<sup>2</sup>

*th* ~

*<sup>D</sup> I I* r

1/2 1/2 *FF D* > W´ *th* ~ r

1/2

rence makes challenge to precise laser processing with nanosecond laser pulses [18].

*L*th being the heat-penetration depth which is given by *L*th ≈ (2*Dτ*p)

enough time for thermal waves to propagate into the target material and to create a relatively large layer of melted material target [5, 6]. Nanosecond laser pulses can ablate the target materials even at low laser intensities in both the vapour and liquid phases, so a recoil pressure that expels the liquid will be created due to the vaporisation process [6]. Evaporation occur‐

At long laser pulse duration, interaction with materials usually fulfils the condition *L*th >> α−1,

heat-diffusion coefficient. So, long laser pulse duration creates sufficient time for thermal waves to propagate within the target material, and the absorbed energy will be stored in a layer with a thickness of about *L*th. In this case, the target material needs much more energy to vaporise than to melt; in other words, evaporation will occur, while the energy absorbed per unit volume into the vaporised layer becomes higher than the latent heat of evaporation per

*AF F* ( *L th* ) *<sup>z</sup> <sup>L</sup>*

r u

u

)

depth (*l*) is given by *l* ~ (*Dt*

3086 High Energy and Short Pulse Lasers

and for laser fluence as:

unit volume, namely [19].

The threshold laser fluence increases as τ<sup>L</sup>

*ko/Ci*

*z*

= + ç ÷ ¶ ¶ è ø (6)

1/2, where *D* is the heat diffusion coefficient and is given by (*D* =

> (7)

*<sup>L</sup>* (8)

1/2, where *D = k/ρc, D* is the

. In nanosecond laser ablation regimes, there is


. Consequently,

a

aa

In nanosecond laser beam interaction with material, the surface of the target material will be heated to melting point and then to vaporisation temperature. During the laser-material interaction, energy will be lost as heat conduction into the target material; the heat penetration

). It can be noted that for this regime of lasers, the condition DτLα<sup>2</sup> >> 1 is fulfilled, for example the thermal penetration depth is quite larger than the optical penetration depth [18]. The energy deposited in the target material per unit mass is given by *Em ~ Iαt/pl*; at a specific time (*t = tth*), this energy becomes higher than the specific heat of evaporation Ω, at which point

for strong evaporation conditions, Em > Ω or τL > *fth* can be written for laser intensity as [5, 6]:

*L*

t

1/2 1/2 Ω

> t

At low-intensity, short laser pulse interactions with a target material, due to the inverse Bremsstrahlung most of the laser energy will be absorbed by the free electrons of the target. This result can be described by the difference between the electron and lattice temperatures (*Te > Ti* ) in a transient nano-equilibrium state. In spite of a small energy exchange between the lattice and the electron heat conduction, the electrons are cooled [6]. For picosecond laser ablation, time *t* is much greater than *τ*<sup>e</sup> (*t* >> *τ*e), which is equivalent to Ce*Te/t << γTe*. In addition, when the condition *τ*e << *τ*L << *τ*<sup>i</sup> is fulfilled, Eq. (1) becomes quasi-stationary for the electron temperature [5]. In other words, when the laser pulse duration is shorter than the electronphonon energy-transfer time, then the electron and lattice have different temperatures, meaning that they will be in a non-thermal equilibrium state. In this case, Eq. (5) becomes the following equations [5, 17]:

$$k\_e \left(\frac{\partial^2 T\_e}{\partial z^2}\right) - \gamma \left(T\_e - T\_i\right) + I\_a a e^{(-az)} = 0\tag{11}$$

$$T\_i = T\_o + \frac{1}{\tau\_i} \int\_0^\ell e^{\left(\frac{l-\theta}{\tau\_i}\right)} T\_e(\theta) d\theta \tag{12}$$

This method represents the lattice temperature in integral from. The above equations describe heating of metal targets by the laser pulses when the laser pulse duration *τ*L >> *τ*e. By neglecting *T*o and when *t* << *τ*<sup>i</sup> , because of the quasi-stationary character of the electron temperature, Eq. (12) can be reduced as follows [5, 18]:

$$T\_i = T\_o \left( 1 - e^{\left(-\frac{t}{\tau\_i}\right)} \right) = \left(\frac{t}{\tau\_i}\right) T\_o \tag{13}$$

It can be noted from the last equation that during picosecond laser ablation, the lattice temperature remains notably lower than the electron temperature, and thus, the lattice temperature in Eq. (11) can be neglected.

When *keTe<sup>α</sup>* <sup>2</sup>≪*γTe*, from Eqs. (11) and (13), it can be concluded that the electron cooling is due to an exchange of energy with the lattice of the material target. Finally, both the lattice and electron temperature at the end of the laser pulse can be expressed by the following equation [5]:

$$T\_a = \frac{I\_a a}{\gamma} e^{(-az)} \text{ and } \ T\_l = \frac{F\_a a}{C\_i} e^{(-az)} \tag{14}$$

It can be noted that when the condition τe << τ<sup>L</sup> is fulfilled, at the end of the pulse, both lattice temperature and attainable lattice temperature are approximately equal.

#### **3.3. Femtosecond laser**

For femtosecond lasers, if the laser pulse duration *τL* is assumed to be shorter than the electron cooling time *τe*. (*τ<sup>L</sup> << τe*.) and when *t* << *τe*, it is equivalent to *CeTe* / *t* ≫*γTe*. In this case, the electron-lattice coupling can be neglected and Eq. (1) can be solved easily. When *DeτL* < *α*<sup>2</sup> (where *D*e = *ke/Ce* is the electron thermal diffusivity) is fulfilled, to simplify the solution of the equation, the electron heat conduction term can be neglected. Thus, Eq. (1) can be written as [5]:

$$C\_e \left(\frac{\partial T\_e^2}{\partial t^2}\right) = 2I\_a a e^{(-az)}\tag{15}$$

which gives

$$T\_e\left(t\right) = \left(T\_o^2 + \frac{2I\_\alpha \alpha}{C\_e} t e^{\left(-\alpha z\right)}\right)^{1/2} \tag{16}$$

where *I*(*t*) = *Io* is assumed constant, and *I*α = *Io A*, and *To* = *Te* (0) refer to the initial temperature.

It has been shown that heat conduction of the target material can be neglected at the very short timescales of picosecond and femtosecond laser pulse durations; thus the target temperature at the end of the pulse within the target material can be given by [19].

$$T\left(z, \tau\_{\perp}\right) \approx \left(\frac{aAF\_a}{\rho c}\right)e^{(-az)}\tag{17}$$

where *F<sup>α</sup>* = *IoτL* is the laser pulse fluence, and *τL* is the laser pulse duration.

The evolution of the electron temperature (*Te*) and lattice temperatures (*Ti* ) after the laser pulse is described by Eq. (5) with S = 0. In addition, the electron temperature and lattice temperature initial conditions are given by Eq. (17) and *Ti* = *To*. Due to the energy transfer to the lattice and heat conduction of the bulk material, the electrons are rapidly cooled after the laser pulse. Since the electron cooling time is quite short, then Eq. (2) can be written as *Ti* ≃*Te*(*τ<sup>L</sup>* )*t* / *τ<sup>i</sup>* . It should be noted that the initial lattice temperature is neglected here. On the other hand, the attainable lattice temperature is determined by the average cooling time of the electrons *τ<sup>e</sup> <sup>α</sup>* <sup>=</sup>*Ce* ` *Te*(*τ<sup>L</sup>* ) / 2*γ* and is given by the following equation [5]:

It can be noted from the last equation that during picosecond laser ablation, the lattice temperature remains notably lower than the electron temperature, and thus, the lattice

When *keTe<sup>α</sup>* <sup>2</sup>≪*γTe*, from Eqs. (11) and (13), it can be concluded that the electron cooling is due to an exchange of energy with the lattice of the material target. Finally, both the lattice and electron temperature at the end of the laser pulse can be expressed by the following equation

(- - ) ( ) ; ; *z z*

It can be noted that when the condition τe << τ<sup>L</sup> is fulfilled, at the end of the pulse, both lattice

For femtosecond lasers, if the laser pulse duration *τL* is assumed to be shorter than the electron cooling time *τe*. (*τ<sup>L</sup> << τe*.) and when *t* << *τe*, it is equivalent to *CeTe* / *t* ≫*γTe*. In this case, the electron-lattice coupling can be neglected and Eq. (1) can be solved easily. When *DeτL* < *α*<sup>2</sup> (where *D*e = *ke/Ce* is the electron thermal diffusivity) is fulfilled, to simplify the solution of the equation, the electron heat conduction term can be neglected. Thus, Eq. (1) can be written as [5]:

*I F T e and T e*

*i*

 a

 a

(14)

è ø ¶ (15)

è ø (16)

è ø (17)

*C*

 a

*e i*

temperature and attainable lattice temperature are approximately equal.

2

*<sup>I</sup> T t T te*

*<sup>T</sup> C Ie t*

 ( ) <sup>2</sup> 2 *<sup>e</sup> <sup>z</sup>*

 - æ ö ¶ ç ÷ =

aa

2 ( ) ` 2 *<sup>z</sup>*

 æ ö = + ç ÷

*e*

where *I*(*t*) = *Io* is assumed constant, and *I*α = *Io A*, and *To* = *Te* (0) refer to the initial temperature.

It has been shown that heat conduction of the target material can be neglected at the very short timescales of picosecond and femtosecond laser pulse durations; thus the target temperature

> a

( ) ( ) , *<sup>z</sup>*

r


*C* aa a

a

1/2

'

*e*

( )

*e o*

at the end of the pulse within the target material can be given by [19].

*L AF T z <sup>e</sup> <sup>c</sup>* a a

where *F<sup>α</sup>* = *IoτL* is the laser pulse fluence, and *τL* is the laser pulse duration.

t

a

a

g

a

temperature in Eq. (11) can be neglected.

[5]:

**3.3. Femtosecond laser**

3108 High Energy and Short Pulse Lasers

which gives

$$T\_i = T\_\epsilon^2 \left(\tau\_L\right) \frac{C\_\epsilon}{2C\_i} = \frac{F\_a a}{C\_i} e^{(-az)}\tag{18}$$

Fann et al. [20] and Wang et al. [14] have shown that the time scale for significant energy transfer and fast electron cooling is about 1 ps. In the case of *Ci Ti* >> ρΩ, where ρ is the density and Ω is the specific heat of evaporation per unit mass, considerable evaporation will occur. From Eq. (18), the conditions for strong evaporation can be given in the form:

$$F\_{\alpha} \ge F\_{\alpha} \text{ e}^{(az)} \tag{19}$$

where *Fth* = ρΩ/α is the threshold fluence laser evaporation by femtosecond laser pulses. Then, the ablation depth per laser pulse (or ablation rate) L can be written as [5, 21]:

$$L = \alpha^{-1} \ln(\frac{F\_a}{F\_\hbar}) \tag{20}$$

The logarithmic dependence of the ablation depth on the laser pulse fluence is well known for the laser ablation of organic polymers and metal targets with femtosecond pulse duration [5].

It can be noted that Eqs. (14) and (18) give the same expressions for the lattice temperature in both picosecond and femtosecond laser regimes. Therefore, the condition for strong evapora‐ tion given by (19), the fluence threshold and the ablation depth per pulse given by (20) remain unchanged [5, 18]. Thus, in the picosecond laser range, it is possible that logarithmic depend‐ ence of the ablation depth on the fluence exists. Here, electron heat conduction inside the target material is neglected. In this case, laser ablation is accompanied by the electron heat conduction and production of a melted area in the target material. Even evaporation can be considered as a direct solid-vapour transition process, whereby the existence of a liquid phase in the target material reduces the precision of laser material processing. Femtosecond laser ablation effects a direct solid-vapour transition due to the short timescales in this laser regime; as a result of this, the lattice is heated on a picosecond timescale, leading to the production of vapour and plasma phases followed by a rapid expansion in vacuum. Here, in a first approximation, thermal conduction into the target material can be neglected during all of the aforementioned processes. Due to the advanced properties of picosecond laser ablation, highly precise and pure laser material processing can be achieved, as has been experimentally demonstrated by Chichkov et al. [5].
