**2.1. Introduction**

band-gap materials, such as polymers [1, 2], fused silica [3–6], and silicon [7–10]. However, almost all of these studies employed one-color laser pulses. More recently, coherent waveformsynthesized two-color laser pulses have been successfully used for increasing plasma generation [11], generating high harmonics [12], and producing broadband terahertz radiation [13]. By studying femtosecond laser ablation of polymethylmethacrylate (PMMA), our group demonstrated that the ablated hole areas exhibited clear modulation with a contrast of 22% by varying the relative phase between the *ω* and 2*ω* beams [14]. It was assumed that different peak intensity for the synthesized waveform was responsible for the observed phenomena.

336 Applications of Laser Ablation - Thin Film Deposition, Nanomaterial Synthesis and Surface Modification

In general, ultrafast laser ablation of dielectrics, such as PMMA, has been explained by the photochemical, photothermal, and photophysical models [15]. In the photochemical model, direct bond breaking in PMMA is achieved by exposing it to an ultrashort laser pulse for producing several reaction products, such as CO, CO2, CH4, CH3OH, and HCOOCH3. In the photothermal model, electronic excitation by picosecond laser pulses results in thermal bond breaking, leading to the formation of PMMA monomers. Among these models, the most interesting one is the photophysical one, in which both thermal and nonthermal bond breaking occur simultaneously. In thermal bond breaking, electronic excitation by ultrashort laser pulses results in ultrashort-laser-induced ionization in the picosecond (ps) and fs ranges. The three main processes of photophysical laser-induced breakdown are (i) excitation of conduction band electrons through ionization, (ii) heating of conduction band electrons through irradiation of the dielectric, and (iii) plasma energy transfer to the lattice, which causes bond breaking

The Keldysh formalism, describing electron tunneling through a barrier created by the electric field of a laser, is often employed for modeling laser breakdown of materials by photoionization, including both multiphoton and tunneling cases. The Keldysh parameter can be expressed as the square root of the ratio between the ionization potential and twice the value of the ponderomotive potential of the laser pulse. Alternatively, it can be expressed as the ratio of tunneling frequency to the laser frequency. The tunneling time or the inverse of the tunneling frequency is given by the mean free time of an electron passing through a barrier width, *l*tunneling = *I*p/*eE*(*t, ϕ*), where *I*p is the ionization potential, *e* is the electron charge, and *E*(*t, ϕ*) is the optical

Depending on the laser intensity used for above-threshold ionization [20–22], two regions of photoionization exist: the tunneling ionization region [20, 23, 24] and multiphoton ionization region [25–28]. In tunneling ionization, the electric field is extremely strong. The Coulomb well can be suppressed to cause the bound electron to tunnel through the barrier and be ionized. At lower laser intensities, the electron can absorb several photons simultaneously. The electron makes the transition from the valence band to the conduction band if the total energy of the

The boundary between tunneling ionization and multiphoton ionization is unclear. Schumacher et al. showed that there should be a so-called intermediate region that exhibit both tunneling and multiphoton characteristics. Mazur et al., following the Keldysh formalism, estimated that the intermediate region corresponded to a Keldysh parameter *γ* ≈ 1.5 [29].

absorbed photons is greater than or equal to the band gap of the material.

The physical mechanism was not clear.

[16–19].

field.

In this section, some key concepts of ultrafast laser ablation will be summarized. This includes light-matter interaction mechanisms such as photochemical, photothermal and photophysical. Dielectric breakdown due to ionization by tunneling, multiphoton and avalanche processes are described. Most relevant for this work, the so-called intermediate regime of photoionization, will be formulated by using the Keldysh equation, defining the Keldysh parameter used throughout this chapter.

### **2.2. Photoexcitation processes**

Laser ablation is one of the manifestations of light-matter interactions. As expected, the ablation processes depend on characteristics of the irradiating laser, such as its intensity, wavelength, and polarization. When ultrafast laser are used, ablation mechanism become more complicated. For polymeric materials, not only photoionization but also the direct bond breaking will lead the ablation process. The main mechanisms are photochemical, photothermal, and photophysical. These three effects are located in different regions of laser pulse.

For ultrafast laser ablation of PMMA, there are two dominant mechanisms, i.e., photochemical and photothermal. In photochemical events, absorption of photons by the material being processed lead directly to covalent bond breaking [15]. The polymeric materials, such as PMMA, are generally made of a wide variety of chromophores, which may dissociate into reactive fragments by absorption of energetic UV photons. Absorption of less energetic photons, e.g., those in the visible or near infrared band, can also lead to the above photochemical processes [15]. Photothermal effect is another basic mechanism of laser ablation. Irradiated by ultrashort laser pulses, the irradiated material absorbs photons and transfer energy to electrons such that photoionization of the material can occur. In this case, excited electrons can heat up the lattice and induce bond breaking [15]. Depending on fluence of the irradiating laser, ablation could be originated through either tunneling ionization or above-threshold ionization (ATI). Multiphoton and avalanche ionization are two main mechanisms of ATI.

In a large band gap material, it is difficult to ionize the constituent atoms by absorbing only one photon from commonly available lasers. Theoretically, an atom might absorb two or more photons simultaneously, giving electrons sufficient energy to cross the band gap from the valence band to the conduction band. This is illustrated schematically in **Figure 1**. For multiphoton ionization to occur, the laser intensity needs to be in the range of 1012 –1016 W/cm2 . In contrast, the avalanche ionization mechanism, for which the laser intensity required is in the range of 109 –1012 W/cm2 , depicts the process whereas a small number of initial electrons of the materials are accelerated to a high value of kinetic energy. Afterwards, high-energy electrons will collide with another electron of lower energy, which is shown schematically in **Figure 1**. Afterwards, the two electrons are accelerated by the laser field collide with other electrons in an avalanche-like process, leading to large amount of electrons with high energies to form a plasma.

**Figure 1.** Schematic diagrams illustrating the processes of (a) multiphoton ionization and (b) avalanche ionization of materials.

Finally, in the photophysical mechanism, nonthermal, photochemical, thermal, and photothermal processes all play their respective roles. Two independent mechanisms of bond breaking could be present. Further, bond breaking energies for the ground state and the excited state chromophores are, in general, different. The photophysical mechanism of ablation usually applies for irradiating lasers with short laser pulses, of which the pulse duration is in the ps and fs range.

### **2.3. Basics of femtosecond ablation dynamics of PMMA in the intermediate regime**

In a class paper, Keldysh showed that the total photoionization rate of a material upon irradiation by a laser can be written as [31, 32]:

Laser Ablation of Polymethylmethacrylate (PMMA) by Phase-Controlled Femtosecond Two-Color... http://dx.doi.org/10.5772/65637 339

$$\begin{split} \boldsymbol{\sigma} &= \frac{2\alpha \boldsymbol{\rho}}{9\pi} \bigg( \frac{\sqrt{1+\boldsymbol{\gamma}^{2}}}{\boldsymbol{\gamma}} \frac{m\boldsymbol{\alpha}}{\hbar} \bigg)^{3/2} \mathcal{Q} \left( \boldsymbol{\gamma}, \frac{\tilde{\boldsymbol{\Delta}}}{\hbar \boldsymbol{\alpha}} \right) \exp\left\{ -\boldsymbol{\pi} < \frac{\tilde{\boldsymbol{\Delta}}}{\hbar \boldsymbol{\alpha}} + 1 \right\} \\ &\times \bigg[ K \left( \frac{\boldsymbol{\gamma}}{\sqrt{1+\boldsymbol{\gamma}^{2}}} \right) - E \left( \frac{\boldsymbol{\gamma}}{\sqrt{1+\boldsymbol{\gamma}^{2}}} \right) \bigg] / E \left( \frac{1}{\sqrt{1+\boldsymbol{\gamma}^{2}}} \right) \bigg] \end{split} \tag{1}$$

where *γ* = *ω*(*mI*p)1/2/*eF* is the so-called Keldysh parameter, *ω* is laser frequency, m is the electronic mass, *e* is the electronic charge, ℏ is plank constant, and Δ is the effective ionization potential,

electrons such that photoionization of the material can occur. In this case, excited electrons can heat up the lattice and induce bond breaking [15]. Depending on fluence of the irradiating laser, ablation could be originated through either tunneling ionization or above-threshold ionization (ATI). Multiphoton and avalanche ionization are two main mechanisms of ATI.

338 Applications of Laser Ablation - Thin Film Deposition, Nanomaterial Synthesis and Surface Modification

In a large band gap material, it is difficult to ionize the constituent atoms by absorbing only one photon from commonly available lasers. Theoretically, an atom might absorb two or more photons simultaneously, giving electrons sufficient energy to cross the band gap from the valence band to the conduction band. This is illustrated schematically in **Figure 1**. For multiphoton ionization to occur, the laser intensity needs to be in the range of 1012 –1016 W/cm2

contrast, the avalanche ionization mechanism, for which the laser intensity required is in the

materials are accelerated to a high value of kinetic energy. Afterwards, high-energy electrons will collide with another electron of lower energy, which is shown schematically in **Figure 1**. Afterwards, the two electrons are accelerated by the laser field collide with other electrons in an avalanche-like process, leading to large amount of electrons with high energies to form a

**Figure 1.** Schematic diagrams illustrating the processes of (a) multiphoton ionization and (b) avalanche ionization of

Finally, in the photophysical mechanism, nonthermal, photochemical, thermal, and photothermal processes all play their respective roles. Two independent mechanisms of bond breaking could be present. Further, bond breaking energies for the ground state and the excited state chromophores are, in general, different. The photophysical mechanism of ablation usually applies for irradiating lasers with short laser pulses, of which the pulse duration is in

**2.3. Basics of femtosecond ablation dynamics of PMMA in the intermediate regime**

In a class paper, Keldysh showed that the total photoionization rate of a material upon irradia-

, depicts the process whereas a small number of initial electrons of the

range of 109 –1012 W/cm2

plasma.

materials.

the ps and fs range.

tion by a laser can be written as [31, 32]:

. In

$$
\tilde{\Delta} = \frac{2}{\pi} I\_{\rho} \frac{\sqrt{1 + \chi^2}}{\chi} E\left(\frac{1}{\sqrt{1 + \chi^2}}\right). \tag{2}
$$

The symbol < Δ/ℏ +1> in Eq. (1) is the integer part of the number, Δ/ℏ + 1, while (, Δ/ℏ) is defined by Eq. (3):

$$\begin{split} \mathcal{Q}\left(\boldsymbol{\gamma}, \frac{\tilde{\boldsymbol{\Delta}}}{\hbar \boldsymbol{\alpha}}\right) &= \sqrt{\frac{\pi}{2K\left(1/\sqrt{1+\boldsymbol{\gamma}^{2}}\right)}} \\ \times \sum\_{s=0}^{\infty} \exp\left\{-\pi n \left(K\left(\boldsymbol{\gamma}\,\sqrt{1+\boldsymbol{\gamma}^{2}}\right) - E\left(\boldsymbol{\gamma}\,\sqrt{1+\boldsymbol{\gamma}^{2}}\right)\right) / E\left(1/\sqrt{1+\boldsymbol{\gamma}^{2}}\right)\right\}} \\ \times \Phi\left\{\sqrt{\frac{\pi^{2}\left(2<\frac{\tilde{\boldsymbol{\Delta}}}{\hbar \boldsymbol{\alpha}}+1>-2\frac{\tilde{\boldsymbol{\Delta}}}{\hbar \boldsymbol{\alpha}}+n\right)}{2K\left(1/\sqrt{1+\boldsymbol{\gamma}^{2}}\right)E\left(1/\sqrt{1+\boldsymbol{\gamma}^{2}}\right)}\right\}}, \end{split} \tag{3}$$

where *K* and *E* are first and second kind of the complete elliptic integrals and *Φ* is the Dawson integral.

In the presence of high-intensity or strong electric field of the laser, we are in the region of tunneling ionization or *γ* << 1. The rate of tunneling ionization is given by

$$\begin{split} \boldsymbol{\sigma}\_{\text{unasing}} &= \frac{2}{9} \frac{I\_{\rho}}{\pi^{2}} \frac{I\_{\rho}}{\hbar} \left( \frac{mI\_{\rho}}{\hbar^{2}} \right)^{3/2} \left( \frac{e\hbar F}{m^{1/2}I\_{\rho}^{3/2}} \right)^{3/2} \\ &\times \exp\left\{ -\frac{\pi}{2} \frac{m^{1/2}I\_{\rho}^{3/2}}{e\hbar F} \left( 1 - \frac{1}{8} \frac{m\alpha^{2}I\_{\rho}}{e^{2}F^{2}} \right) \right\}. \end{split} \tag{4}$$

On the other hand, if *γ* >> 1, the ionization is in the regime of multiphoton absorption. The probability of multiphoton absorption is given by

$$\begin{split} \sigma\_{\text{multiphoton}} &= \frac{2}{9\pi} \alpha \left(\frac{m\alpha}{\hbar}\right)^{3/2} \Phi \left[\sqrt{2 < \frac{\tilde{\Lambda}}{\hbar \alpha} + 1 > -\frac{2\tilde{\Lambda}}{\hbar \alpha}}\right] \\ &\times \exp\left\{2 < \frac{\tilde{\Lambda}}{\hbar \alpha} + 1 > \left(1 - \frac{e^2 F^2}{4m\alpha^2 I\_p}\right)\right\} \left(\frac{e^2 F^2}{16m\alpha^2 I\_p}\right)^{\epsilon \tilde{\Lambda}/\hbar \alpha + 1 >} . \end{split} \tag{5}$$

**Figure 2.** The photonionization rate and Keldysh parameter are plotted as a function of laser intensity (*λ* = 800 nm) in PMMA.

**Figure 3.** Photonionization rates are plotted as a function of the Keldysh parameter for NIR (*λ* = 800 nm) ultrafast laser ablation of PMMA.

Below, we have plotted the effective ionization potential Δ =Δ+ 22/4 2 and the Keldysh parameter as a function of laser intensity in PMMA in **Figure 2**. The laser wavelength is assumed to be in the near infrared (*λ* = 800 nm). Besides the total ionization rate, contributions by tunneling and multiphoton ionization, respectively, are also shown. The dependence of the photoionization rate on the Keldysh parameter is also enlightening. This is illustrated in **Figure 3**. Similar curves for ultrafast laser ablation of PMMA using NUV (*λ* = 400 nm) are presented in **Figures 4** and **5**.

On the other hand, if *γ* >> 1, the ionization is in the regime of multiphoton absorption. The

2 2 2 1

æ ö é ù D D <sup>=</sup> F < + >- ê ú ç ÷ è ø ë û

exp 2 1 1 . 4 16 *p p*

**Figure 2.** The photonionization rate and Keldysh parameter are plotted as a function of laser intensity (*λ* = 800 nm) in

**Figure 3.** Photonionization rates are plotted as a function of the Keldysh parameter for NIR (*λ* = 800 nm) ultrafast laser

/ 1 2 2 2 2

% %

w

(5)

<D + >

% h

 w

2 2

ww

w

*eF eF mI mI*

h hh

3/2

ì ü <sup>D</sup> æ öæ ö ï ï ´ < +> - í ý ç ÷ç ÷ ï ï î þ è øè ø

*m*

w

340 Applications of Laser Ablation - Thin Film Deposition, Nanomaterial Synthesis and Surface Modification

probability of multiphoton absorption is given by

multiphoton

v

PMMA.

ablation of PMMA.

9

% h

w

p  w

**Figure 4.** Photonionization rates and Keldysh parameter are plotted as a function of laser intensity for NUV (*λ* = 400 nm) laser ablation of PMMA.

**Figure 5.** Photonionization rates are plotted as a function of the Keldysh parameter for NUV (*λ* =400 nm) ultrafast laser ablation of PMMA.

The solid blue lines in **Figures 2**–**5** correspond to the photoionization rate of PMMA calculated using Eq. (1). The full expression of Keldysh formula (Eq. 1) take into account both tunneling and multiphoton ionization processes. The dashed black line and dotted red line represent the tunneling ionization and multiphoton ionization rates determined from Eqs. (4) and (5), respectively. When the Keldysh parameter, *γ* ≈1.5, tunneling and multiphoton ionization rates overlap each other for PMMA irradiated with either NIR (800 nm) or NUV (400 nm) beams. We defined this overlapping region as an intermediate regime. To predict the ionization rate in the intermediate regime, we assume the process resembles that of tunneling ionization. The ionization rate depends on the instantaneous field amplitude, *F*L(*t*), which is given by [22]:

$$\varpi(t) = 4I\_{\rho}^{5/2} \frac{1}{F\_L(t, \varphi)} \exp\left(-\frac{2}{3} I\_{\rho}^{3/2} \frac{1}{F\_L(t, \varphi)}\right) \tag{6}$$

In order to understand the phase dependence of observed dual-color laser ablation phenomena, we proceed as follows: assume that the irradiating laser consists of beams at two commensurate laser frequencies, i.e., the fundamental (*ω*) laser beam and its second harmonic (2*ω*). The dual-color laser field *F*(*t*) can then be written as

$$F\_L(t, \varphi) = F\_o e^{-2\ln 2^{\frac{2}{\alpha'} / r^2}} \cos \alpha t + F\_{2o} e^{-2\ln 2^{\frac{2}{\alpha'} / r^2}} \cos(2\alpha t + \varphi) \tag{7}$$

where *Fω* and *F*2*<sup>ω</sup>* are the envelope function of the fundamental and second-harmonic laser fields, respectively; *ϕ* is the relative phase of the second-harmonic (2*ω*) beam with respect to that of the fundamental (*ω*) beam.

The Keldysh model above can be used to describe the photoionization phenomenon due to either the multiphoton or tunneling route. Typically, the Keldysh parameter is defined by the square root of the ratio of ionization potential and twice the ponderomotive potential of the laser pulse. Some researchers also define it as the ratio of tunneling frequency to the laser frequency [33]. In the original derivation, the Keldysh parameter was used to describe the phenomenon of an electron tunneling through a barrier created by the optical field. The tunneling time or the inverse of the tunneling frequency is determined by the mean free time of the electron passing through a barrier width *l*.

$$I\_{\text{unassigned}} = I\_p \;/\ eF\_L(t, \varphi) \tag{8}$$

where *Ip* is the ionization potential, *e* is the electron charge and *F*L(*t,ϕ*) is the electric field of the incident laser. The average velocity of an electron can be written as,

$$<\text{v}> = \sqrt{\frac{2I\_{\rho}}{m\_{\text{e}}}},\tag{9}$$

where *me* is the mass of an electron. By combining Eqs. (8) and (9), the tunneling time is given by

$$t\_{\text{unending}}(t,\varphi) = \frac{l}{<\nu>} = \frac{\sqrt{I\_{\rho}m\_{\epsilon}}}{\sqrt{2}eF\_{L}(t,\varphi)} = \frac{1}{\nu\_{\text{unending}}(t,\varphi)}\tag{10}$$

Tunneling can occur if the mean tunneling time, which is given by Eq. (10), is less than half the period of the laser. Taking this into account, we modify the Keldysh parameter,*γ* as appropriate for this study as

$$\gamma = \frac{2t\_{\text{unenging}}}{t\_{\text{laser}}} = \frac{I\_{\text{unnings}}}{I\_{\text{laser}}} = \nu\_{\text{laser}} \frac{\sqrt{2I\_{\rho}m\_{\text{e}}}}{eF\_{L}(t,\rho)}\tag{11}$$

where *t*laser is the period of laser, *l*laser is the mean distance that an electron moves during half of period *t*laser at a mean velocity of <*v*>, and *v*laser is the laser frequency. When the Keldysh parameter *γ* has relative phase dependence at dual-color synthesized waveform condition, the ionization rate can be calculated after we determine the effective frequency of the dual-color laser pulse.

For the purpose of defining the envelope equation for single-cycle pulse of the synthesized waveform, we express the complex electric field as [34]

$$E(t) = \tilde{E}\_a(t)e^{-i\alpha\_0 \gamma\_{r+b\nu}} + c.c. = \tilde{E}(t) + c.c. = \frac{1}{\sqrt{2\pi}} \Big|\_{0}^{a} \tilde{E}(oo)e^{-i\alpha t}do + c.c.,\tag{12}$$

where *ω*0 is the effective carrier frequency denoted as

The solid blue lines in **Figures 2**–**5** correspond to the photoionization rate of PMMA calculated using Eq. (1). The full expression of Keldysh formula (Eq. 1) take into account both tunneling and multiphoton ionization processes. The dashed black line and dotted red line represent the tunneling ionization and multiphoton ionization rates determined from Eqs. (4) and (5), respectively. When the Keldysh parameter, *γ* ≈1.5, tunneling and multiphoton ionization rates overlap each other for PMMA irradiated with either NIR (800 nm) or NUV (400 nm) beams. We defined this overlapping region as an intermediate regime. To predict the ionization rate in the intermediate regime, we assume the process resembles that of tunneling ionization. The ionization rate depends on the instantaneous field amplitude, *F*L(*t*), which is given by [22]:

> 5/2 1 21 3/2 ( ) 4 exp (, ) 3 (, ) *p p L L*

2 2 2 2 2 ln 2 / 2 ln 2 / <sup>2</sup> (, ) cos cos(2 ) *t t F t Fe t F e <sup>L</sup> t*

w

 w

where *Fω* and *F*2*<sup>ω</sup>* are the envelope function of the fundamental and second-harmonic laser fields, respectively; *ϕ* is the relative phase of the second-harmonic (2*ω*) beam with respect to

The Keldysh model above can be used to describe the photoionization phenomenon due to either the multiphoton or tunneling route. Typically, the Keldysh parameter is defined by the square root of the ratio of ionization potential and twice the ponderomotive potential of the laser pulse. Some researchers also define it as the ratio of tunneling frequency to the laser frequency [33]. In the original derivation, the Keldysh parameter was used to describe the phenomenon of an electron tunneling through a barrier created by the optical field. The tunneling time or the inverse of the tunneling frequency is determined by the mean free time

tunneling / (, ) *p L l I eF t* =

incident laser. The average velocity of an electron can be written as,

where *Ip* is the ionization potential, *e* is the electron charge and *F*L(*t,ϕ*) is the electric field of the

<sup>2</sup> , *<sup>p</sup> e*

j

t

*F t F t*

In order to understand the phase dependence of observed dual-color laser ablation phenomena, we proceed as follows: assume that the irradiating laser consists of beams at two commensurate laser frequencies, i.e., the fundamental (*ω*) laser beam and its second harmonic

æ ö <sup>=</sup> ç ÷ -

 j

 t

w j


è ø (6)

(8)

*<sup>I</sup> <sup>v</sup> <sup>m</sup>* < >= (9)

*t I I*

342 Applications of Laser Ablation - Thin Film Deposition, Nanomaterial Synthesis and Surface Modification

j

v

j

of the electron passing through a barrier width *l*.

that of the fundamental (*ω*) beam.

(2*ω*). The dual-color laser field *F*(*t*) can then be written as

w

$$\alpha\_0 = \frac{\int\_0^\alpha \alpha \left| E(\alpha) \right|^2 d\alpha}{\int\_0^\alpha \left| E(\alpha) \right|^2 d\alpha} \tag{13}$$

In Eqs. (12) and (13), *E*(*ω*) is the Fourier transform of *E*(*t*) and *ψ* is the imaginary part of the complex envelope. Substituting the effective carrier frequency into Eq. (1), we can plot the photoionization rate as a function of laser intensity and the relative phase between the fundamental (*ω*) and second-harmonic (2*ω*) beams in our experiment, which is shown in **Figure 6**.

According to **Figure 6**, the ionization rate is predicted to be dependent on the relative phase of the fundamental (800 nm) and second-harmonic (400 nm) beams. Further, the modulation of ionization rate is more pronounced at higher laser intensities.

**Figure 6.** The total ionization rate versus laser intensity and relative phase of fundamental (*ω*) and second-harmonic (2*ω*) laser beams.

Recall that ablation by a laser with low and high intensities would fall into the regimes governed by multiphoton and tunneling ionization mechanisms, respectively. Conventionally, tunneling ionization corresponds to a regime in which the Keldysh parameter *γ* << 1. In this limit, the strength of the field is more than the value necessary to overcome the barrier. For a weaker field such that the Keldysh parameter *γ* >> 1, the main mechanism for ionization is due to the multiphoton ionization. In this regime, the electric filed strength is below the value that required for overcoming the barrier. In order to calculate the Keldysh parameter for the dualcolor case, we need to define the period of the laser *t*laser. It can be easily shown that the period of dual-color synthesized waveform by NIR (800 nm) and NUV (400 nm) beams is essentially that of the period of fundamental (*ω*) beam. Therefore, Eqs. (1) and (11) can be combined to determine the ionization rate as shown in **Figure 6**.

In the intermediate ionization regime, which is defined by *γ* ≈1.5, the ionization rate has a strong phase dependence. Likewise, the electron tunnel time now depends on the phase difference between the two colors. Note that the electric field is actually lower than the value required for electrons to overcome the barrier.

Ablation threshold is an important parameter for laser material processing. It is a function of the laser pulse duration, wavelength, and intensity. According to the simplified Fokker-Planck equation [35]:

$$\frac{\partial n}{\partial t} = \beta(I)n + P(I),\tag{14}$$

where *n* is the free electron number, *β* is the avalanche ionization factor, assuming that the photogenerated electron distribution grows in magnitude without changing its shape. *P*(*I*) is the multiphoton ionization rate. It can be approximated by the tunneling ionization rate by using Keldysh equation.

First of all, we need to calculate the number of free electrons generated by the laser pulse, assumed to be Gaussian in shape, *I*(*t*) = *I*0exp(−4ln2*t*<sup>2</sup> /*τ*<sup>2</sup> ), where *τ* is the pulse duration. By solving the equation above, the free electron number can be obtained as

$$n = n\_0 \exp\left(\int\_0^\alpha \beta dt\right) = n\_0 \exp\left(\alpha \int\_0^\alpha Id dt\right) = n\_0 \exp\left(\alpha I\_0 \int\_0^\alpha e^{-4\ln 2^{\frac{\alpha}{2}}/r^2} dt\right) = n\_0 \exp\left(\frac{\alpha I\_0 \pi}{4} \sqrt{\frac{\pi}{\ln 2}}\right) \tag{15}$$

where *α* is the absorption coefficient of the material and *n*0 is the total number of free electrons which is generated through multiphoton or tunneling ionization mechanism,

$$m\_0 = \bigcap\_{\rightarrow}^{\alpha} P(I)dt \tag{16}$$

The ablation threshold, *F*th, can then be written as

$$F\_{\eta\_1} = \frac{2}{\alpha} \ln \left( \frac{n\_{cr}}{n\_0} \right) \tag{17}$$

where the density of free electrons, *n*cr correspond to the threshold fluence, *F*th. For ablation with ultrafast laser pulses, contribution by the avalanche ionization mechanism is not significant. Dielectric breakdown or ablation is through the processes of photoionization by tunneling and multiphoton ionization mechanisms. When the free electron number increases, the ablation threshold decreases.
