An Experimental Study of Multiphoton Ionization in Fused Silica at IR and Visible Wavelengths

Abstract

We present the results of an experimental study of multiphoton ionization in fused silica, using a linearly polarized femtosecond Satsuma fiber laser with an active medium based on Yb⁺³ ions, at 515 -nm and 1030 -nm wavelengths. The radiation transmission in the fused silica was measured as a function of the laser intensity and wavelength and the data were analyzed using a theoretical model based on the Keldysh theory. We determined the multiphoton absorption cross-sections in the fused silica in the case of four- and eight-photon ionization and analyzed the contribution of avalanche ionization. The obtained results provide insight into the fundamental processes involved in multiphoton ionization and have implications for its applications, such as laser micromachining and material processing.

1. Introduction

The study of the interaction process between high-intensity ultrashort laser pulses and dielectric materials is currently of great interest, both from the standpoint of the fundamental aspects of solid-state physics and for solving a number of applied problems. Knowledge of the material behavior exposed to focused femtosecond laser irradiation is crucial for the understanding and optimization of the mechanisms of the nano- and microscale self-organization of electromagnetic fields, electron-hole plasma, and matter during the laser writing of periodic anisotropic sub wave nanostructures in dielectric volume [1,2,3,4,5]. In particular, the multiphoton ionization process in dielectrics [6,7] plays a key role here, since it is the lever for triggering many other laser–matter interaction processes. However, it should be noted that it is difficult to isolate and unambiguously determine the contribution of individual photoionization processes experimentally, especially considering the contribution of avalanche processes [8,9], which begin to work at high energy inputs. In this paper, a combination of precise experimental knowledge and a theoretical analysis provides valuable information on the multiphoton ionization cross-sections in fused silica in the case of four- and eight-photon ionization, which is necessary for the further construction of a highly predictive theoretical model of absorption and the local energy deposition of femtosecond laser pulses to analyze the thermal, phase, and hydrodynamic effects in its laser nano- and micromachining. The paper also analyzes the conditions under which the contribution of avalanche ionization becomes significant.

2. Keldysh Theory of Photoionization Applied to Dielectric Materials

Experimentally, multiquantum ionization was first observed in [10], where the quantum photoelectric effect on xenon atoms exposed by the ruby laser field was considered. As it is known, the process of the multiquantum photoelectric effect in the field of radiation with frequency ω is characterized by a multiphoton absorption cross-section ${\sigma }^{\left(n\right)}$, defined as [11]:

$$W_i={\sigma }^{\left(n\right)}{F}^n.$$

(1)

Here, $W_i$ is the ionization probability of an atom (molecule) per unit time, $F=I/\hslash \omega$ is the photon flux, $I$ is the intensity of the ionizing radiation of frequency $\omega$, and $n=\left[I_i/\hslash \omega \right]+1$ is the multiphoton order of the process ($I_i$ is the ionization potential of an atom). The multiphoton cross-section ${\sigma }^{\left(n\right)}$depends on the radiation frequency and ionization potential.

The modern theory of multiquantum ionization is based on Keldysh’s work, according to which, the probability of multiphoton ionization per unit time is described by the following expression [12]:

$$W_i~\mathit{exp}\left(-\frac{2\tilde{I}}{\hslash \omega }\Gamma \left(\gamma \right)\right),\Gamma \left(\gamma \right)=arsh \gamma -\gamma \frac{\sqrt{1+{\gamma }^2}}{1+2{\gamma }^2}$$

(2)

where $\tilde{I}=I_i+\frac{e^2{{\rm E}}_0^2}{4m{\omega }^2}=I_i\left(1+\frac{1}{2{\gamma }^2}\right)$ is the effective ionization potential, $\gamma =\frac{\omega \sqrt{2m{I}_i}}{e{E}_0}$ is the Keldysh parameter, and $E_0$ is the amplitude of the wave field strength.

The Keldysh theory distinguishes between two different regimes: $\gamma \ll 1$ and $\gamma \gg 1$. In the first case ($\gamma \ll 1$), the expansion of the function $\Gamma \left(\gamma \right)$ gives $\Gamma \left(\gamma \right)\approx 4{\gamma }^3/3$. As a result, the ionization probability is described by a well-known tunnel formula that does not depend on the radiation frequency [13]:

$$W_i~\mathit{exp}\left(-\frac{I_i}{\hslash \omega }\frac{4}{3}\gamma \right)=\frac{\sqrt{2\pi }\hslash }{2m{a}^2}\sqrt{\frac{E_0}{{{\rm E}}_a}}\mathit{exp}\left(-\frac{4{{\rm E}}_a}{3{{\rm E}}_0}\right),$$

(3)

where ${{\rm E}}_a=\frac{\sqrt{2m{I}_i^3}}{e}$– is the atomic value of the electric field strength and $a=\frac{e_{}^2}{I_i}$ is the characteristic size of an ionized atom. Actually, Keldysh’s work [12] does not have a precise identification of the pre-exponential multiplier in expression (3). Its calculation in the tunnel limit was correctly performed by V. S. Popov with collaborators [14,15]. The use of the pre-exponential factor obtained in these works allows us to rewrite the Keldysh Formula (2) in the form:

$$W_i=\frac{\sqrt{2\pi }\hslash }{2m{a}^2}\sqrt{\frac{E_0}{{{\rm E}}_a}}\mathit{exp}\left(-\frac{2\tilde{I}}{\hslash \omega }\left(arsh\gamma -\gamma \frac{\sqrt{1+{\gamma }^2}}{1+2{\gamma }^2}\right)\right).$$

(4)

On the other hand, the decomposition of the function $\Gamma \left(\gamma \right)$ in the opposite case $\gamma \gg 1$ gives $\Gamma \left(\gamma \right)\approx \ln \left(2\gamma \right)-1/2$, which leads to a power dependence of the photoionization probability on the radiation intensity:

$$W_i=\frac{\sqrt{2\pi }\hslash }{2m{a}^2}\sqrt{\frac{E_0}{{{\rm E}}_a}}\mathit{exp}\left(\tilde{I}/\hslash \omega \right)\times {(4{\gamma }^2)}^{-\tilde{I}/\hslash \omega }~{\left(\frac{e^2{{\rm E}}_0^2}{8m{\omega }^2{I}_i}\right)}^n,$$

(5)

where $n$ characterizes the multiphoton order. In the absence of resonances with intermediate atomic states, Formula (5) allows one to directly determine the multiphoton absorption cross-section (see Expression (1)).

Similar ideas about the processes of tunnel and multiphoton ionization in dielectrics, which govern the transitions of electrons from valence to a conduction band, were also generalized in [12]. In this case, when writing Expressions (2–5), the width of the energy bandgap serves as an analogue of the ionization potential, and instead of electron mass, the effective mass $m^{*}$. is used. Further development of the Keldysh theory for the description of absorption processes in solid dielectrics was conducted in [16,17,18,19,20].

An experimental observation of the radiation absorption in various dielectrics has been carried out in a number of papers, see, for example, [21,22,23,24]. As a rule, these studies were conducted in a regime close to the multiphoton one ($\gamma \ge 1$), which usually corresponds to the intensity range of 1–10 TW/cm². In this regime, absorption in media is mostly characterized by the cross-section of the multiquantum photoelectric effect. In the present work, the cross-sections for the fourth and eighth photon absorptions in fused silica are determined by processing the experimental data on the 515 nm and 1030 nm radiation transmissions in the sample, considering the dependence of the ionization probability according to the Keldysh formula in the multiphoton limit (5). It is important to note that the ionization probability in this case was calculated taking into account the pre-exponential multiplier obtained in [14,15].

3. Experiment

During our experimental studies, the nonlinear transmission $T_{nl}$ of a fused silica sample in the form of a double-sided polished parallel plate with a thickness of 2 mm was evaluated. The diagram of the experimental setup used to measure the transmission of the sample is shown in Figure 1.

A Satsuma fiber laser (Amplitude Systemes, France) with an active medium based on linear polarization and Yb⁺³ ions at 515 nm and 1030 nm was used as a laser radiation source. The pulse duration varied from 0.3 ps to 1.2 ps and was measured using a scanning interferometric autocorrelator AA-20DD (Avesta Project). The energy of the pulses varied in the range from 2 nJ to 460 nJ and the laser repetition rate was 2 Hz. The temporal contrast of the pulses was 10⁷. The sample was mounted on a three-coordinate moving platform (Standa) and aligned perpendicular to the incident laser radiation. The laser radiation was focused by a micro-lens with NA = 0.55 and a focal length of f’ = 5 mm onto the front surface of the sample. This was performed to a depth of 300 μm into the focal spot with a radius at the energy level of 1/e $w_0\approx 0.59$ μm for the 515 nm wavelength and $w_0\approx 1.17$ μm for the 1030 nm wavelength. An Ophir PD10-C energy meter was placed under the bottom surface of the sample to record the radiation passing through the plate. When measuring the transmission, there was a simultaneous shift along the horizontal axis, so that each subsequent laser pulse fell on a fresh section of the sample. For each energy, the values were averaged over 30 laser pulses.

4. Results and Discussion

For the laser pulse parameters under the study, the Keldysh parameter was in the range of $\gamma \ge 1$. Hence, the increase in the electron density $N_e$ can be written in the form:

$$\frac{d{N}_e}{dt}=N_0{\sigma }^{\left(n\right)}{F}^n$$

(6)

Here, $N_0$ is the concentration of the atoms in the medium, $n=4$ for the 515 nm wavelength and $n=8$ for the 1030 nm wavelength, leading to a decrease in the beam intensity while it propagates along the z axis [25]:

$$\frac{dI\left(z,r,t\right)}{dz}=-{\beta }_{n}I{\left(z,r,t\right)}^n,$$

(7)

Here, ${\beta }_n$. is the coefficient of the n-photon absorption in the medium and $I\left(z,r,t\right)$ is the local value of the intensity at length z. The coefficient ${\beta }_n$ is determined via the multiphoton absorption cross-section ${\sigma }^{\left(n\right)}$ introduced above, as follows:

$${\beta }_n=n\times N_0\times \hslash {\omega }^{\left(1-n\right)}{\sigma }^{\left(n\right)},$$

(8)

The integration of (7) with the boundary conditions $I\left(z=0,r,t\right)=I_0\times exp\left(-{\left(t/\tau \right)}^2\right)\times exp\left(-{\left(r/w_0\right)}^2\right)$, where $I_0$ is the intensity at the center of the focused beam and $w_0$ is the focal beam waist radius, gives an expression for the intensity evolution in the fused silica:

$$I\left(z,r,t\right)=\frac{I_{0}exp\left(-{\left(t/\tau \right)}^2\right)exp(-{\left(r/w_0\right)}^2)}{\sqrt[n-1]{1+{\beta }_{n}z\left(n-1\right)\times I_0^{n-1}exp\left(-\left(n-1\right){\left(t/\tau \right)}^2\right)\times exp\left(-\left(n-1\right){\left(r/w_0\right)}^2\right)}},$$

(9)

Figure 2 and Figure 3 show the data on the energy fraction $1-T_{nl}$ absorbed over the length $z_0$, where $T_{nl}=\frac{Q\left(z_0\right)}{Q_0}$ is the measured nonlinear transmission. The corresponding error bars were calculated using the errors of the measured values of $Q\left(z_0\right)$ and $Q_0$. Here, one should note the drastic increase in the absolute errors in the low-energy limit, which is caused by the small values of the transmitted energy against high noises. The experimental curves were superimposed with the simulation results of the absorbed energy, which are based on the numerical integration of Expression (9) over the pulse length and focal spot size for the various pulse durations and wavelengths of 515 (Figure 2) and 1030 nm (Figure 3). The values of ${\beta }_n$ and corresponding values of ${\sigma }^{\left(n\right)}$ (see Expression (8)) for both wavelengths were determined by comparing the ionization probability (5), which characterizes the multiphoton limit of the Keldysh theory, and the ionization probability (1), which explicitly contains the value of the multiphoton cross-section ${\sigma }^{\left(n\right)}$. The aforementioned ionization probability curves, plotted by Formulas (1), (3)–(5) for both considered wavelengths, are depicted in Figure 4. Concretely, to determine the cross-section for the four-photon absorption (in the case of the 515 nm radiation exposure), it is necessary to analyze the data in Figure 2 together with the ionization probability data presented in Figure 4a, while for the eight-photon absorption (the 1030 nm radiation exposure), a joint analysis of Figure 3 and Figure 4b is required.

It can be seen from Figure 2 and Figure 3 that the calculated curves are in good agreement with the experimental data in the region of not very high absorption, especially for the two shortest pulse durations, 0.3 and 0.6 ps. (Here, we do not consider a low-energy limit where there is no absorption of radiation due to the multiphoton process. The partial decrement of the transmitted signal at a level of 1–2% is governed by the reflection at the sample surface and does not allow for an observation of the multiphoton processes in this range.) Indeed, in this region, Expression (9) can be represented in the following form:

$$I\left(z,r,t\right)=I_0\mathit{exp}\left(-{\left(t/\tau \right)}^2\right)\times exp(-{\left(r/w_0\right)}^2)\times (1-{\beta }_{n}z\times I_0^{n-1}exp\left(-\left(n-1\right){\left(t/\tau \right)}^2\right)\times exp\left(-\left(n-1\right){\left(r/w_0\right)}^2\right).$$

(10)

Upon integrating (10) over time and radial field distribution at a length of $z_0$, one can obtain:

$$Q(z_0)=Q_0-{\beta }_n{z}_0\frac{1}{\sqrt{n}}{\left(\frac{1}{\sqrt{\pi }\tau }\right)}^{n-1}{Q}_0^n,$$

(11)

where the initial pulse energy $Q_0$ is related to the peak intensity $I_0$ as:

$$I_0=\frac{Q_0}{\sqrt{\pi }{S}_0\tau }, S_0=\pi w_0^2.$$

(12)

As can be seen from Expression (11), the dependence $ln\left(1-\frac{Q\left(z_0\right)}{Q_0}\right)$ on the logarithm of the initial pulse energy $Q_0$ can be approximated by a linear dependence with a slope factor of $n-1$ and a free term, determining ${\beta }_n$ for a given pulse duration. For longer pulse durations, the agreement between the experimental and theoretical data is seen to be rather poor, especially for the 1030 nm wavelength (see Figure 2c and Figure 3c). From our point of view, this is the result of the presence of an additional mechanism of the radiation absorption that occurs in the case of sufficiently long pulses. Namely, it is absorption due to an avalanche ionization in the volume of the fused silica.

Thus, let us estimate the possible competition between the rates of absorption in the processes of multiphoton and avalanche ionization. Let us introduce the rate of electron transition from the valence band to the conduction one (the rate of the electron impact ionization), according to the equation [26]:

$${\nu }_i=\frac{1}{I_i}\frac{4\pi e^2{I}_0{\nu }_{tr}}{c{m}^{*}\left({\omega }^2+{\nu }_{tr}^2\right)}.$$

(13)

Here, ${\nu }_{tr}$ is the transport collisional frequency for the electrons in the conduction band and $I_i$ stands for the bandgap width. In each multiphoton and electron avalanche ionization process, the absorbed energy from the laser field energy is approximately the same and equal to $I_i$. Hence, to estimate the contribution of the avalanche process to the total absorption, we need to compare the multiphoton and electron impact production rates. Then, the electron dynamics are governed by the equation:

$$\frac{d{N}_e}{dt}=N_0{\sigma }^{\left(n\right)}{F}^n+{\nu }_{tr}{N}_e$$

(14)

A general solution for Equation (14) with a zero initial concentration of electrons in the conduction band reads:

$$N_e\left(t\right)=N_0\frac{{\sigma }^{\left(n\right)}{F}^n}{{\nu }_i}\left(\exp \left({\nu }_{i}t\right)-1\right).$$

(15)

It follows from (15) that impact ionization contributes significantly to the absorption of the radiation for the laser pulse durations.

$$\tau \ge \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\nu }_i$}\right..$$

Assuming that, in fused silica, the bandgap is $I_i\approx 9$ eV and ${\nu }_{tr}\approx 5\times {10}^{13}$ s⁻¹, one obtains ${\nu }_i\approx {10}^{12}$ s⁻¹ from (13). This means that, for pulse durations of ~ 1 ps and more, the electron avalanche will essentially contribute to the radiation absorption, and the multiphoton ionization used in the experiment processing underestimates the absorption in the fused silica. On the other hand, our studies confirm the validity of the applied procedure for pulses of 0.3 and 0.6 ps durations.

Finally, the ${\beta }_n$ values obtained from Figure 2 and Figure 3, taking into account Expression (8), allow us to obtain the multiphoton absorption cross-sections in the cases of four- and eigh-photon ionization:

$$\begin{gathered} {\sigma }^{\left(4\right)}=\frac{{\beta }_4{\left(\hslash \omega \right)}^3}{4N_0}=2.05\times {10}^{-114}{\mathrm{cm}}^8{\mathrm{s}}^3 \\ {\sigma }^{\left(8\right)}=\frac{{\beta }_8{\left(\hslash \omega \right)}^7}{8N_0}=7.6\times {10}^{-240}{\mathrm{cm}}^{16}{\mathrm{s}}^7 \end{gathered}$$

We note that very similar four-photon cross-sections for the fused silica were measured for the 473 nm laser wavelength ${\sigma }^{\left(4\right)}=4.4\times {10}^{-116}$ cm⁸s³ [27]. Although the difference in the cross-sections seems to be large, due to the high nonlinearly of the process, it results in fairly close data for the multiphoton electron production.

5. Conclusions

The cross-sections of the multiquantum photoeffect in fused silica were determined for the cases of four- and eight-photon absorption. It was found that our experimental data are in agreement with the Keldysh theory. Hence, one can use the Keldysh approximation for laser parameters when a transition from the multiphoton limit to the tunnel one occurs. The conditions under which the contribution of avalanche processes becomes essential were revealed. The conducted research showed that the use of the multiquantum effect approximation to determine the rate of electron production in the volume of a solid dielectric in our conditions is acceptable up to peak intensities of $~2÷3\times {10}^{13}$ W/cm². The findings of this study have significant implications for the future modelling of the strong-field electron dynamics in dielectrics and semiconductors, which is of importance for the field of solid-state physics and for solving a number of applied problems related to laser–matter interaction processes, in particular laser nano- and microprocessing.