1. Introduction

Ultrafast laser micromachining ensures ultra-high quality and precision on diverse materials, such as glass, silicon, metals, biological tissue and so on [1,2]. The market of ultrafast laser micromachining is growing rapidly, reaching an annual revenue of about 380 M${\$} $ in 2020, even with the COVID-19 raging around the world [3]. The further expansion of the market requires improvements on the cost-effectiveness of ultrafast lasers. One of such practical considerations is the polarization extinction ratio (PER) of ultrafast lasers. For high power ultrafast fiber lasers and multipass solid state lasers, depolarization always accompanies the photonic crystal gain fibers or main amplifiers [4,5]. Then extra components and efforts are needed to keep the PER high at the cost of laser energies, since it is commonly believed linear polarization laser is more efficient in the ultrafast laser micromachining [6,7]. So it is practically important to understand how the polarization affects the efficiency of ultrafast laser micromachining. In this paper, we focus on the capability of generating free carriers, because the micromachining efficiency is strongly related to the density and distribution of free carriers generation.

During ultrafast laser micromachining, photons are mainly absorbed to generate free carriers via the ionization process, and subsequently energy transfers from carriers to lattice is in tens of picoseconds. The interaction between laser and material is completed before the modifications occur, leaving the material a nonequilibrium state. Then the hydrodynamic motion of the electrons and lattice lead to the modification of materials. Therefore, the ionization processing dominates the micromachining process [8]. On the other hand, measurements of laser pulses including transmission, spectra, etc. will only correspond to the laser pulses propagation and photon-electron interaction process, and not be influenced by the complex breakdown process.

Indeed, various methods have been investigated to improve the ultrafast laser fabrication precision and efficiency by controlling carrier distribution, such as the using of Bessel beams [9], pulse train with picoseconds delay [10] and so on. The optimization of all the ultrafast laser pulses parameters, including methods mentioned above, will result in the modulation of the distribution of free carriers. We propose in this paper that polarization can also be applied to control the distribution of carriers, especially for some circumstances where other methods suffer limitations. For example, in bulk silicon, strong nonlinear and plasma effects in pre-focal region limit the maximum carrier density that can be reached, so it is hard to cause breakdown in such materials. Increasing the pulse energy or decreasing the pulse duration will only lead to a larger distribution area of carriers rather than a higher density of free carriers [11]. So other laser properties should be engineered to overcome this limitation.

Linear and circular polarizations are two extreme conditions, so the practical ultrafast lasers micromachining efficiency and characteristics will behave somewhere between those of the two extreme conditions. Here, we study the density distribution of free carriers generation by the two polarizations of ultrafast laser pulses at different pulse energies and pulse durations, based on the measurements of nonlinear absorption of ultrafast laser pulses in fused silica. By finding the portion of free carriers generated via the multiphoton ionization (MPI) and avalanche ionization (AI) respectively, we analyze the dependence on polarizations for the nonlinear absorption of femtosecond and picosecond laser pulses. In this paper, we limit the laser pulse durations to the "cold avalanche ionization" conditions [12]. Then, we compare the machining profiles of linearly and circularly polarized pulses via simulations and experiments. Finally, we find that the free carriers generated by circularly polarized pulses are more concentrated, compared to that of linearly polarized pulses. Since the ionization processing of ultrafast laser and all kinds of dielectrics are similar [13], our results are also applicable to other materials such as silicon, ceramics and so on, if only the materials are transparent to laser pulses.

2. Experiment setup

In order to examine the impact of polarization on the ionization process, we measure the ultrafast laser pulse transmission. In the experiments, as shown in Fig. 1, the linearly polarized output beam of a femtosecond Ti:sapphire amplifier with a central wavelength of 810 nm is split by a polarizer beam splitter (PBS) cube into two arms, namely, seeding pulses and driving pulses. The maximum pulse energy is about 6 ${\rm \mu} J$. The delay between the seeding and driving pulses can be tuned with a linear translation (Newport ILS150) with the minimum step of about 1 fs. And the zero delay is ensured by the observation of the interference between the two arms of laser. The driving pulse can be stretched from 50 fs to few picoseconds (up-chirped) by various fused silica glasses (SCHOTT, N-BK7). The two pulses are combined by another PBS and then focused 70 $\mu m$ inside a 150 ${\rm \mu} m$ thick fused silica sample glass (Suprasil I, Boston Piezo Optics, Inc.), by a 0.25 NA objective lens. The spot radius of the focused beams is estimated to be 2.5 ${\rm \mu} m$. A quarter-wave plate before the objective is utilized to control the polarizations of the focused beam. The sample glass is moved smoothly during the measuring processing by a three-dimensional translation stage system. The beams are collected by a 0.4 NA objective lens and then incident into the integrating sphere. The transmission data is the average result of 100 measurements for each pulse energy by a boxcar averager (Stanford Research Systems, SR250).

 

Fig. 1. The experiment setup: Ti:S laser is a 810 nm regenerative amplifier Ti:sapphire femtosecond laser operated at 500 Hz; $\lambda /2$ is a half waveplate; $\lambda /4$ is a quarter waveplate; PBS is a polarizer beam splitter; G is a dispersion glass; TS1 is a linear translation stage and TS2 is a three-dimensional translation stage; OB1 and OB2 are objective lenses; S is the fused silica sample under study; IS is a integrating sphere; PD is a photodiode; SP is a spectrometer; BS is a beam-stop.

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3. Models and theory

To model the interaction of ultrafast laser pulses with transparent dielectrics, we take into consideration both the propagation of pulses and the material response. Our model is mainly based on the Forward Maxwell Equation, in which the material response is treated as part of the nonlinear polarization $P^{NL}$. The equation is in the form of the Nonlinear Schrödinger type as

The complex conductivity $\sigma (\omega )$ can be obtained from the Drude model, $\sigma (\omega )=\frac {{\rho }e^2\tau _c}{m_e(1+i\omega \tau _c)}$, where $\tau _c$ is the reciprocal of collision frequency. Here we only need the imaginary part of the conductivity since the real part is equal to the cross section for avalanche absorption. We employ a finite difference scheme, namely the Crank-Nicolson method, to discretize the transverse variables [16] of the three-dimensional simulation. And split-step Fourier algorithm is used to solve Eq. (1). Though the nonlinear Schrödinger equation include some approximations, the difference to the whole nonlinear Maxwell’s equations is acceptable even when the NA of the focused beam is lager than 0.25 [17,18].

In our model, the initial pulse field is calculated from the experiment measured spectrum. The phase of the shortest pulse is derived by the PICASO method [19]. We change the pulse duration by setting the length of dispersion glass in our simulation, for example a 1.6 ps pulse is obtained by setting a SF57 glass to be 90 mm in length.

To test our model, we conduct several experiments. Firstly, we verify the validity of the unidirectional propagation of the pulses by monitoring the reflection spectra of the pulses. We find the reflected spectra are almost the same as the incident pulse spectra while the transmitted spectra are not, which confirms that the reflection is independent with the interaction processing. Secondly, we calibrate the MPI cross section and AI coefficient by fitting the transmission curve, confirming that the laser parameters are consistent with experimental settings. We will discuss the value of MPI cross section and AI coefficient in the next section. Finally, we compare the transmitted spectra (after the fused silica sample) of simulations with the measurements, which agree well with each other, and simulated parameters are consistent with experimental setting too.

In practical applications, when ultrafast laser pulses propagate through materials, the laser energies will be absorbed along the propagation, resulting in a self-limiting effects [20], a phenomenon that energy absorption will clam the laser intensity before the focus, when a high NA microscope objective is employed. So both the laser modification volume and the energy absorption will increase with laser pulse energy [15]. For ultrafast laser micromachining, the extending of modification volumes originate in the MPI, because the initial free carriers are generated via this process [21,22]. Only when the density of free carriers generated by the MPI reaches a certain value, will AI starts absorbing laser energies [23]. So the portions of absorbed laser energies via MPI and AI will change with pulse energies and pulse durations. And from the perspective of time scale, MPI completes within a sub-laser-cycle [24,25], but AI accumulates throughout the laser pulse duration [26]. Thus, the contributions of free carriers generation from these two different processes or mechanism will vary with laser pulse durations.

We simulate the dynamics of contributions of energy absorption by AI and the maximum carrier density as pulse energy increases in Fig. 2. The energy absorption contribution ratio of AI shown in Fig. 2(a) is obtained by dividing the energy absorbed via AI to the total absorbed energy. Clearly, MPI contributes most of the nonlinear absorption at low laser energy fluence, because the density of free carriers is not high enough to trigger AI. However, the portion of energy absorbed via AI grows rapidly with laser pulses energy, especially for longer pulses, when the laser energy approaches the absorption threshold. Once the portion of energy absorbed via AI reaches its maximum, as the laser energy goes far beyond the threshold value, it will decrease gradually, because more laser energy will be absorbed via MPI, since the ionization volume is extended via MPI. So, the maximum portion of AI indicates the saturation of the maximum carrier density with the increasing of laser energies as shown in Fig. 2(b). Namely, the optimized micromachining efficiency is achieved at this pulse energy, without scarifying the spatial quality of ultrafast laser micromachining. Further increasing of laser energy mainly results in the expansion of the laser modification volume.

 

Fig. 2. Energy absorption contribution ratio of AI(a) and maximum of the maximum free carrier density(b) at different pulse durations for single linearly polarized laser pulses. The parameters of the simulation are the same except the durations. The insert figure shows the absorbed energy changing with energy fluence of 50 fs pulses. The dotted line shows the absorption threshold. The dash lines show the energy fluence corresponding to the maximum ratio of AI. The focused beam waist is estimated to be 2.5 ${\rm \mu} m$, and AI coefficient is ${\rm \alpha} =4 J^{-1}cm^2$, the coefficient of MPI is ${\rm \sigma} _6=4\times 10^{-13} (TWcm^{-2})^{-6}cm^{-3}ps^{-1}$.

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Notably, the portion of free carriers generated via AI increases with pulse duration, meaning long (picosecond) pulses could be more efficient in ultrafast laser micromachining. Due to the difference of the contribution between MPI and AI, the micromachining characteristics of femtosecond and picosecond laser pulses are also different, such as the damage threshold [22], the damage area profile [27,28] and also incubation effects [29].

4. Results and discussion

In ultrafast laser micromachining, a significant portion of laser energy should be absorbed to drive chemical reactions and structural changes [20]. It was demonstrated that the nonlinear absorption of femtosecond lasers is significantly higher for a linearly polarized laser pulse than that for a circularly polarized laser pulse [6]. The nonlinear absorption in dielectrics is related to the density distribution of free carriers. And the free carriers come from two different processes for ultrafast laser micromachining, MPI and AI, as explained in Eq. (2). Because MPI is a highly nonlinear process, its ionization rate is determined by the maximum amplitude of the electric field, then a polarization dependent process [30]. On the other hand, AI is mainly determined by the average intensity of the laser pulses, so its ionization rate shows little polarization dependence in the micromachining laser intensity regime [31]. As we show in section 3, MPI dominates during the ionization processing for tens of femtoseconds laser pulses micromachining. So due to the polarization dependency of MPI, energy absorption is stronger for linearly polarized pulses, when femtosecond laser is employed. To verify our assumption, we measure the transmissions of laser pulses in different polarization phases with different durations.

In Fig. 3, we show the ultrafast laser pulse transmission v.s. laser pulse energy for linear polarization (orange solid lines) and circular polarization (blue solid lines) at two pulse durations, 50 fs and 1.6 ps, respectively. Obviously, the nonlinear absorption is higher for linear polarization pulse at both pulse durations. For the short pulses (50 fs), the nonlinear absorption threshold is lower and the nonlinear absorption rate is larger for linear polarization. The nonlinear absorption will saturate at high laser energies because of plasma defocusing. For the long pulses (1.6 ps), the difference of the nonlinear absorption between two polarizations is significantly smaller. When the laser pulse energy is large enough, the nonlinear absorption is almost the same for linear and circular polarization pulses, because the nonlinear absorption is dominated by AI which is polarization independent. This result is consistent with the analysis of contribution of AI shown in Fig. 2, confirming that multiphoton ionization is polarization dependent but avalanche ionization is not.

 

Fig. 3. Transmissions of 50 fs (a) and 1.6 ps pulse (b) varying with energy fluence( or peak intensity). The solid lines show the transmission of single pulse sequence while the doted lines show the pump-probe transmission. The pump pulses have a duration of 50 fs and pulse energy of 65 nJ (0.52 Jcm$^{-2}$) and 70 nJ ($0.56$ Jcm$^{-2}$) for linear and circular polarization respectively. The red dash lines show the fitting of our simulation.

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In Fig. 3, we also show the nonlinear absorption of these laser pulses with presence of free carrier seeding, where pump laser pulses are sent 100 fs ahead of energy deposition laser pulses to avoid self trapped excitons and interference between the two beams. The probe pulses will interact with the free carriers generated by the seeding pulses, and there will only exist AI process for the probe pulses if the energy of the probe pulse is below the nonlinear absorption threshold. There are two advantages of employing a seeding pulse: first it can help to deposit more laser energies into the materials [15]; second it may help to control the modification precision much higher than the diffraction limit of the laser [32]. In order to generate same population of initial carriers, we employed different pump laser energies: 65 nJ (0.52 Jcm$^{-2}$) for linearly polarized pulse and 70 nJ ($0.56$ Jcm$^{-2}$) for circularly polarized pulse, respectively. The maximum of the seeding free carrier density is about ${\rm 5}\times 10^{19} cm^{-3}$.

For the short pulses (50 fs), with the presence of seeding, the nonlinear absorption is the same for both linearly and circularly polarized laser pulses when the laser energy is below the nonlinear absorption threshold. Because at these pulse energies, AI is the only mechanism for free carriers generation. This confirms again that avalanche ionization is polarization independent. When the laser energy is above the nonlinear absorption threshold, more energy can be absorbed for linearly polarized pulse, but the difference is much smaller with the presence of seeding. For the long pulses (1.6 ps), the results are clearer that the nonlinear absorption is almost the same for both polarizations, consistent with that almost all free carriers are generated via AI for picosecond laser micromachining shown in Fig. 2.

We also compare the simulations with experimental results, which are in good agreements, as shown in Fig. 3 with red dash lines. According to the simulations, we obtain the same AI coefficient (${\rm \alpha} =4J^{-1}cm^2$) for both linear and circular polarizations, the same as the value obtained in Ref. [33]. For the cross section of MPI, we obtain different results, which is ${\rm \sigma} _6=4\times 10^{13} (TWcm^{-2})^{-6}cm^{-3}ps^{-1}$ for linear polarization and ${\rm \sigma} _6'= 2\times 10^{13} (TWcm^{-2})^{-6}cm^{-3}ps^{-1}$ for circular polarization, respectively. This ratio of circular- to linear-polarization cross sections ${\sigma _6}'/ \sigma _6 < 1$ is in agreement with the theoretical estimation [30].

In the discussions above, we concentrate on the nonlinear absorption because it is identical to the total free carriers generation (energy conservation). For ultrafast laser micromachining in dielectrics, it is the free carrier density and distribution that determine the machining efficiency [34] and accuracy. However, large energy absorption is always accompanied with large modification volume (low free carrier density), which is not helpful for micromachining efficiency.

With the parameters shown above, the carrier distributions are simulated as shown in Fig. 4(c) and (d). The absorbed energies via AI and their portions to the total absorbed energy for linear and circular polarizations are shown in Fig. 4(a) and (b). The pulse energy used in (c) and (d) for linearly polarized pulses are labeled as $E_2$, $E_3$ for 50 fs, and $E_5$ for 1600 fs. And the pulse energy for circularly polarized pulses is labeled as $E_1$ and $E_4$. $E_1$ is equal to $E_3$, while the absorbed energy is the same for linear polarization with incident pulse energy of $E_2$ and circular polarization with incident pulse energy of $E_1$. In order to have the same amount of energy being absorbed, the laser pulse energy of the circular polarization is larger for 50 fs but almost the same for 1600 fs, consistent with the experiment transmission data.

 

Fig. 4. (a) and (b), Energy absorption contribution ratio of AI of 50 fs and 1.6 ps for linearly polarized and circularly polarized laser pulses, the solid lines are linear polarization and the dash lines are circular polarization. (c) and (d), the difference of the free carrier distribution between circularly and linearly polarized pulses, the pulse energy of each polarization is shown in (a) and (b) as black circles. The insert figure shares the same coordinate with (c).

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As mentioned above, the material modification volumes are determined by MPI. The portion of energy absorbed via MPI at large laser energies is always smaller for circularly polarized pulses than linearly polarized pulses. So the modification area will always be smaller for circularly polarized pulses with the same incident pulse energy as the linearly polarized pulses. The difference of the distributions of the free carrier density at the same absorbed energy for linear and circular polarizations is shown in Fig. 4(c) and (d) for 50 fs and 1600 fs pulses respectively. Since the self-focusing effect is stronger for linear polarization, the maximum carrier density is higher for 50 fs pulses with linear polarization, which is consistent with the results that damage threshold is lower for linear polarization [35].

It may sound counter-intuitive but is true for picosecond pulses: with the same amount of free carriers generation, the density of the free carriers will be higher with circular polarization, meaning a better machining efficiency. Not only the maximum carrier density is larger for circularly polarized pulses, but also the carrier distribution area is more concentrated. So, the employing of circular polarization pulses may help to improve the resolution of micromachining meanwhile increase the processing efficiency.

To verify the simulation results, we compare the width of lines written by circular and linear polarization 1600 fs pulses, shown in Fig. 5. In the experiment, a 0.25 NA objective lens is used to focus the laser pulses 100 ${\rm \mu} m$ inside a 2 mm-thickness fused silica glass. The repetition rate of the laser pulses is set to 1 kHz, and the glass is moved smoothly with a speed of 0.5 ${\rm mm} s^{-1}$. The laser pulse energy and pulse duration of the two polarization are set to be the same, 600 nJ (5.5 ${\rm Jcm}^{-2}$) and 1600 fs respectively, as marked with circle in Fig. 4(b). As shown in Fig. 3(b), the absorbed energy will also be the same for the linearly and circularly polarized pulses. To measure the line widths of the trace, we image them with a 1.2NA microscope objective and then average the widths of each line along the lines as we show in Fig. 5(b). The damage caused by the picosecond laser pulses scatter the illuminating light, so the transmission value represents the degree of laser micromachining efficiency. We did not observe significant difference of the laser micromachining efficiency for the two different polarizations. We attribute this to the incubation effect of the overlapped beams, where multi-shot laser pulses wash out the difference. Further, the line written with linearly polarized pulses are wider than that of circularly polarized pulses, which qualitatively confirms that the modification volume will be larger for linear polarization lasers.

 

Fig. 5. The transmission of the modified area by the circular and linear polarization pulses with duration of 1600 fs and pulses energy of 600 nJ($5.5 {\rm Jcm}^{-2}$). The lines are written 100 $\mu m$ inside fused silica with a 0.25 NA microscope objective lens, and then imaged via a 1.2 NA microscope objective lens. The statistic shown in (b) is the transmission averaged along the lines in (a).

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In our model, we employ a stationary avalanche coefficient which could be viewed as an effective avalanche coefficient. Therefore, we may over-estimate the contribution of the impact ionization and neglect the energy absorbed by those electrons whose energy is not high enough for impact ionization [36]. Because the heating process of free electrons in laser field involves multiple collisions, only when the electrons energy exceeds the critical energy will the impact ionization occur. However, the results obtained here are not significantly affected by this, which can be verified that the curves of the simulations and the experimental measurements agree well. We attribute this to the average effect of the interaction volume in which complex propagation wash out the details of dynamics of the avalanche ionization. In theories, the processes under analysis are isolated in a small region where the laser peak intensity is assumed to be a constant. In our experiments and simulations, it is the integration of the nonlinear absorptions of the whole focal volume that is under study. If a more complex and detailed model is used for the impact ionization process, people may get more accurate parameters and carrier densities. However, it will need more studies on the dynamics of the ionization processes.

5. Conclusion

In conclusion, MPI is strongly polarization dependent, but AI is not. In our experiments, it is confirmed that the coefficient of AI is also independent of the pulse duration (from 50 fs and 1.6 ps). So by analyzing the portion of free carriers generated via MPI or AI, people can find the degrees of polarization dependence of the nonlinear absorption with different pulse durations. In our experiments and simulations, it shows that almost all free carriers are generated via AI for picosecond pulses (at the range of laser micromachining energy fluence). Meaning that in industrial applications, where most employed ultrafast lasers are picosecond lasers, depolarizations of ultrafast laser are not a parameter that will impact the energy deposition efficiency. A positive information for real-life applications, that costs and energies can be saved on the laser front.

We also find the free carrier density is higher for circularly polarized pulse at picosecond pulse duration, showing that depolarization of a linearly polarized laser is not necessary a negative effect for ultrafast laser micromachining. Given the modification volume and micromachining efficiency can be controlled by the polarization, this makes it a degree of freedom for ultrafast laser micromachining. Further, the free carriers generated by circularly polarized pulses are more concentrated, compared to the that of linearly polarized pulses. Given the amplitude of the field will be smaller for the circular polarization, compared with linear polarization at the same laser intensity, circular polarization laser provides a possible method to reduce some problematic nonlinear effects, such as optical Kerr effect [14] and three photon absorption, in micromachining of low energy band and high non-linearity materials, like silicon. Then, circularly polarized ultrafast pulses can help to increase the upper limitation of the free carrier density while keep the micromachining precision.