1. Introduction

Laser materials processing is one of the most important areas of laser physics. It has a big impact on scientific applications and industry [1]. Volumetric modification of materials also inherently belongs to this area, which is used, for example, in the creation of photonic structures [2,3], direct laser writing of waveguides [4–6], the inscription of volume nanogratings [7,8], fabrication of Bragg gratings [9,10] and data storage [11–13]. Fast, easy, and cheap production is of great importance in this field of technology. Research in this area and optimization of processing parameters are still in large demand for advancing existing applications and the development of new ones.

In the volumetric modification process, high precision and reproducibility of the created structures are important. The energy coupling from the laser radiation field into the material must be efficient and localized in a small volume to obtain desirable structures with the required accuracy [1,14]. However, the interaction of intense laser pulses with solids is a very complex phenomenon that involves numerous processes occurring simultaneously and/or successively. Thus, to get control over precise volumetric modification, detailed knowledge of the mechanisms accompanying the interaction is required. Several nonlinear processes occur during the interaction of a powerful laser pulse with dielectrics that includes multiphoton and/or tunneling ionization, electron collisional ionization (avalanche), the Kerr effect, and scattering of laser radiation by generated electron plasma [15–17]. The dynamic development of these processes and the resulting material modification depend on many factors, including material properties, laser wavelength, and pulse duration [18–23].

Recently, it was demonstrated that the spatiotemporal shaping of the laser beam can represent a powerful technique for achieving a desirable level of material modification and its localization toward the nanoscale [24–28]. However, due to the multiparametricity of the material modification phenomenon, more efforts are needed to understand and realize the full potential of laser beam manipulation in material processing [1].

In this experimental and theoretical work, we perform systematic experimental and theoretical studies of the volumetric modification of fused silica with two femtosecond Gaussian laser pulses that couple with the material at different separation times. Some experimental findings on excitation by double time-separated pulses were reported in the past for pulse energies below 400 nJ, showing that a small-energy pre-pulse can considerably influence absorption/transmission of the following pulse [23,29]. However, these works could not recognize the influence of exciton formation on material modification. In our study, we address the modification level at the double-pulse irradiation regimes that, to our knowledge, has not been investigated in the past. Varying the time separation between pulses in the double-pulse experiments allows us to follow the evolution of the excited electron population, which causes shielding of the laser radiation and leads to delocalization of the absorbed energy. From this perspective, such experiments enable us to observe the influence of self-trapped excitons (STEs) in fused silica, having two characteristic decay times of ∼34 ps and ∼338 ps [29], on the volumetric modification of the material. The experimental data are compared with state-of-the-art numerical simulations, which describe the effect of laser energy absorption, its spatial distribution inside the material, and plasma shielding in the focal region upon propagation of laser pulses in a nonlinear media. The results shed new light on the role of laser-generated defect states in the laser inscription of photonic structures in optical glasses.

2. Experimental

Volumetric modification of fused silica samples was performed by using an ultrafast Ti:Sapphire laser Astrella (Coherent), operating at a central wavelength of λ₀ = 800 nm. The pulse duration of τ_FWHM ≈ 40 fs is achieved by controlling the compressor in the laser system. The experimental setup includes an attenuator containing a half-wave plate and a polarizer. The laser beam is divided by a beam-splitter into two parts. One part is guided through a delay line to introduce a time delay between the two pulses. It can also be attenuated independently using a variable neutral density filter, thus controlling the ratio of the energies in the two beams. Both beam arms are combined into collinear beams and focused with an objective (10× magnification, numerical aperture NA = 0.25) approximately 200 µm below the sample surface. A simplified experimental scheme is shown in Fig. 1. In its lower left panel, a grayscale image from an optical microscope is given, from which the degree of material modification was evaluated.

 

Fig. 1. Simplified experimental scheme. BS stands for 50:50 beam splitter, ND filter is a variable neutral density filter. The lower left part shows the real microscopic image in the plane of the modified area in grayscale for different pulse separations and the transmission microscope signal (TMS) acquisition process.

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The studied samples are fused silica plates with a thickness of 1 mm and transverse dimensions of 10 × 10 mm². The volumetric modifications of the material created by the laser were evaluated post-mortem in the focal plane using an optical microscope (Olympus BX43, objective 20×) with white light illumination. By integrating the grayscale in a defined area of the transmitted optical microscope image and extracting the background, we obtain a dimensionless number characterizing the degree of damage called below space-integrated transmission microscope signal (TMS). The area of the evaluation was the same for all regimes and it was larger than the visible modification region in the regime of the most extensive modification.

For each experiment, the accumulation of structural modification (TMS) induced by a sequence of double pulses coupling with the same focal areas was examined (sequences of 1, 2, 3, 5, 10, 20, and 50 double pulses). The laser system allows us to operate in the single-shot regime by controlling the high-voltage on the Pockels cells. Each individual pulse is split into two parts using a beam splitter as described in the caption to Fig. 1. Accumulation of pulses (5, 10, 20, 50) was performed manually by opening the cells for one laser pulse, with a time delay of around 1 s. Although the resulting contrast of the modified area of the sample in the microscope image does not scale linearly with the number of acting pulses, the measured TMSs can be qualitatively compared with the numerical simulation results (see below) as a function of the time delay between pulses. The comparison of the experimental and numerical data enables us to gain insight into the dynamics of electron excitation/recombination processes.

3. Numerical modeling

For the theoretical analysis of the double-pulse action in the regimes of the experiments, we use the model based on nonlinear Maxwell’s equations supplemented with the rate equations for the generation of the conduction band (CB) electrons (often called free-electron plasma), their recombination to the excitonic states, and the STE re-excitation, and with the hydrodynamic-type equations describing the oscillations of CB electrons in the laser field. The details of the model can be found in Supplement 1 and Refs. [28,30–33]. The equations were solved in the axisymmetric frame described by coordinates (z,r), for height and radius, respectively.

A series of simulations were performed for two Gaussian laser pulses with different time delays between them. Within the double pulse, the first pulse starts to propagate to the unperturbed fused silica sample from the boundary of the simulation region (z,r) = (0,r). The second pulse enters the simulation domain collinearly to the first one with a variable delay and propagates through a perturbed area of the sample, which is populated by the CB electrons and the STEs whose density depends on the time elapsed from the first pulse. To describe accurately the tight focusing conditions, the laser beam was considered as focused by a parabolic mirror [28,34,35]. The simulations were performed for laser beam energies and separation times between pulses as in experiments.

For comparison of the simulation results with the experimental data, the spatial distribution of the absorbed laser energy density ${E_{\textrm{ab}}}$ was calculated within the laser-irradiated sample by integration multiphoton absorption and Joule heating over the laser beam propagation through the sample (see Supplement 1). We note that the peak values of the absorbed energy density $E_{\textrm{ab}}^{\textrm{peak}}$ can be considered as a characteristic parameter for evaluation of the threshold and strength of material modification. Indeed, according to the energy balance considerations [28], a local modification of fused silica in the form of compaction (and a corresponding change of the index of refraction) can become observable at single-pulse laser irradiation if ${E_{\textrm{ab}}}$ is larger than ca. 1700 J/cm³. If ${E_{\textrm{ab}}}\; $ exceeds ∼2400 J/cm³, the material melts in such region(s) [28] which can lead to a stronger modification including possible cavitation that depends on $E_{\textrm{ab}}^{\textrm{peak}}$ and associated pressure gradients. Detailed mechanisms of glass matrix rearrangement can be found in the work by Canning et al. [36]. Thus, we accept here the $E_{\textrm{ab}}^{\textrm{peak}}$ value as a parameter that can be correlated with the measured degree of modification (TMS in our notation).

We underline that such a comparison can only serve as a qualitative evaluation. The results of simulations provide information on the absorbed energy distribution by the end of the action of one double laser pulse. In experiments, the modification does not end immediately after the laser beam propagation but can include a chain of processes depending on both the peak absorbed energy and the absorbed energy distribution (temperature and pressure gradients) [24,28,31]. While the thermal conductivity is a slow process and does not play a noticeable role at femto- and picosecond timescales, the generated pressure waves are of the utmost importance for the observation of final modification features as they lead to the redistribution of material within the laser-excited regions.

The situation is even more complicated for multi-pulse irradiation regimes. Each next pulse couples with the material whose properties have been modified by previous laser pulses. The pulse-to-pulse modification can result from the redistributed material density due to laser-generated stress waves [24,28] and the change of the local refractive index. The latter can be a result of both the local density changes and/or accumulated defect states [30,31] that can strongly influence the laser pulse propagation through the already modified zone. However, qualitative tendencies in the laser-induced modification should still be determined by the peak absorbed energy density even if it is somewhat varying from pulse to pulse.

4. Results and discussion

The interaction of an intense ultrashort laser pulse with fused silica is accompanied by the generation of free electrons and holes via multiphoton and/or tunneling ionization [15]. Upon propagation of a focused laser beam in an ionizable material, as soon as the intensity reaches a level sufficient for efficient photoionization, the free-electron plasma is created swiftly by the front part of the beam. The beam energy loss due to photoionization and, additionally, due to bremsstrahlung absorption and light scattering by the conduction electrons, leads to a saturation of the laser intensity upon further beam propagation toward the focus [30,31]. This effect is usually referred to as intensity clamping [37,38] and, according to simulations for Gaussian laser pulses propagating inside fused silica with NA = 0.25, it leads to the maximum intensity level of the order of (2-3) × 10¹³ W/cm² in a wide range of pulse energies [30,31]. However, we use here a more general term ‘shielding’ which is relevant for double or multiple pulses when the previous pulses create a “defect-mediated shield”, screening partially the next pulses from propagation toward the focal zone [30].

For double-pulse experiments with a relatively short separation time between pulses, the situation is even more complicated. The subsequent pulses arriving at the same place of the sample are affected by shielding and delocalization of the absorbed energy [30,39] because of the swiftly changing population of the CB electrons and the formation of the STEs.

The dynamics of the CB electron density excited by single femtosecond Gaussian laser pulses inside silica glass have already been theoretically studied [17,30,31,40–42]. In this work, we are focusing on the shielding effect in fused silica samples during the action of two Gaussian femtosecond pulses with a variable time delay between them in the regimes of volumetric modification.

Figure 2 shows the modeling results on the spatial distribution of the absorbed energy density ${E_{\textrm{ab}}}$ after the propagation of laser light through the focal region for two extreme cases: (a) two separate pulses 2 µJ each with an infinite time delay between them (assuming that the second pulse is not influenced by the excitation created by the first pulse and hence the absorption is the sum for two laser pulses) and (b) single laser pulse with the energy of 4 µJ. NA and pulse duration were the same for the two cases. The shielding (clamping) effect is evidenced in this Figure. The peak absorbed energy density $E_{\textrm{ab}}^{\textrm{peak}}$ is considerably lower for the single pulse with the same energy as the total energy of the two successive 2-µJ pulses. On the contrary, the total absorbed laser energy is only slightly larger for the single 4-µJ pulse (3.45 µJ) than for two 2-µJ pulses (3.25 µJ). Together with the much larger $E_{\textrm{ab}}^{\textrm{peak}}$ value in the case of two separate pulses (Fig. 2), this indicates that using multiple pulses of lower incident energy provides a more efficient laser energy coupling with better localization as compared to more powerful single pulses.

 

Fig. 2. Calculated distributions of the absorbed energy density in fused silica as a result of the action of two laser pulses 2 µJ each with infinite time separation for avoiding the STE-related effects (a) and one laser pulse with the energy 4 µJ (b). All other irradiation parameters are the same. The color scale is the same for (a) and (b).

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Experimentally measured TMS values for accumulated double-pulse sequences (10, 20, and 50) with the energies of E_P = 2 µJ each and the numerical data on the peak absorbed energy density are presented in Fig. 3. For a given number of double pulses, the smallest TMS value in the material was achieved in the regime of a partial temporal overlapping between two successive pulses. With increasing time delay τ_d between the two pulses, the modification gradually increases up to approximately double the value. However, as laser energy coupling at τ_d ≈ 0 ps (±100 fs) is affected by the interference pattern, we do not consider such beam overlaps here. By fitting the obtained TMS values as a function of τ_d with an exponential growth function, we obtain the characteristic decay time of the shielding effect τ_shield ≈ 600 fs caused by the CB electron population. The numerical modeling, which correlates the spatial distribution of the absorbed laser energy with the effect of plasma shielding, describes reasonably well the experimental observation (pink crosses in Fig. 3).

 

Fig. 3. Experimental results on the TMS (space-integrated transmission microscope signal in relative units) in fused silica for the case of irradiation by sequences of double pulses as a function of time separation between them. The data are given for the accumulation of 10, 20, and 50 successive double pulses at a wavelength of ${\lambda _0}$ = 800 nm with the energy in each pulse of 2 µJ. The green dashed line represents the exponential growth fitting function for a series of 50 + 50 pulses that gives the plasma shielding time of ∼600 fs. The fitting parameters are similar for the other series (600–700 fs) with larger standard errors. By pink crosses (a pink dashed line is given to guide the eye), the simulation data on the peak absorbed energy density are presented for one pair of pulses at the energy ${E_\textrm{P}}$ = 2 µJ each.

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It is known that the mean time for the trapping of photoexcited carriers is ∼150 fs [43]. However, a negative phase shift and hence the ${\tau _{\textrm{shield}}}$ value should be variable and depend on the CB electron population excited the laser pulse action as demonstrated in [43]. The authors show that, for 70-fs intensive laser pulses, the phase shift remains negative up to 500 fs after the pulse while, for longer, picosecond timescale, pulses, the plasma defocusing effect is very short. We note that, in the volumetric modification regimes with tightly focused laser beams, the pulse duration effect can be more complicated [30]. It can be expected that, with increasing pulse duration from sub-100-fs pulses to longer ones, the local excitation level around focus may increase [30] and, at even longer pulses towards the ps range, it should inevitably go down as found in [43]. However, this effect calls for further studies. In our case, according to the simulations, the CB electron density reaches its maximum value of 7.72 × 10²⁰cm^-3 at 2-µJ, 40-fs laser pulse, thus providing a strong defocusing/shielding effect.

To better understand the mechanisms that affect the laser energy absorption after the first pulse, we performed another double-pulse experiment with two different energies of pulses in the pulse pair, 0.5 µJ and 5 µJ. The focusing conditions were the same as for Fig. 3. In such a combination of low- and high-energy pulses, the laser energy absorption considerably depends on the pulse sequence. The 3D maps of the spatial distribution of the absorbed energy density ${E_{\textrm{ab}}}$ simulated for the combination of 0.5 µJ and 5 µJ laser pulses for the time delays of ±10 ps are shown in Fig. 4. We present here 3D maps for long time delays (10ps) to show the effect of STEs. Plasma shielding for shorter time delays is highlighted in the previous experiment (see Fig. 3).

 

Fig. 4. Calculated distributions of the absorbed energy density in fused silica as a result of the action of two laser pulses with energies 5 µJ and 0.5 µJ. (a) For -10 ps time delay (first pulse 5 µJ, second pulse 0.5 µJ) and (b) for 10 ps time delay (first pulse 0.5 µJ, second pulse 5 µJ). The color scale is the same for (a) and (b).

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Figure 5 shows the experimental data on the TMS values as a function of the time delay for the case of two pulses of different energies. First of all, a substantial reduction of TMS around the zero-time delay is observed, similar to that reported in Fig. 3. As discussed above, its origin is related to the plasma shielding/defocusing effect and the characteristic time of the effect is also ca. 600 fs. However, a considerable asymmetry in the TMS values is observed depending on which pulse couples with the sample first (Fig. 5). The modification is greater if the sample is first illuminated with a weaker, 0.5-µJ pulse followed by the more energetic one. Based on the simulations and the analysis of the modification spots, we conclude that there are two scenarios affecting the resulting volumetric modification, both related to the formation of STEs.

 

Fig. 5. Experimental data on the TMS (space-integrated transmission microscope signal in relative units) for the case of irradiation by double pulses as a function of time separation between two pulses (accumulation of 5, 10, 20, and 50 and double pulses) at ${\lambda _0}$ = 800 nm. The total energy in the double pulse is 5.5 µJ which is divided into two pulses unequally, 5 µJ in one pulse and 0.5 µJ in another one. Negative separation time corresponds to the case when the more energetic pulse couples with the sample first. A slight offset of the time zero may be due to an experimental error. By pink crosses (a dashed line is given to guide the eye), the calculated maxima of the absorbed energy density are presented for one double pulse. The self-trapped exciton decay time of 34 ps [29] was used in simulations.

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When the low-energy pulse comes first, its energy is absorbed in a localized region around the geometrical focus. As soon as CB electrons are trapped in the excitonic states and the plasma shielding effect does not affect anymore the laser beam propagation, the localized excitonic population then serves as a precursor for free electron generation by the front part of the following pulse associated with the localized energy absorption. We remind that the electrons are easier excited from the STE states by laser light than the valence electrons due to the lower STE binding energy (5.2 vs. 9 eV respectively for fused silica) [29,44]. Another important point is that the STE decay in fused silica is described by a double exponent with characteristic times of ∼34 ps and ∼338 ps [29] (in the simulations, we applied the shortest decay time). Thus, when first-pulse-generated excitons are formed mostly in a localized focal region, this region represents a reservoir for a localized absorption of the laser light from the second pulse, provided that the latter couples with the STE-seeded region within the STE lifetime.

The situation overturns when a more energetic pulse comes first (negative delays). In such a case, the laser intensity of the pulse reaches the level, which is sufficient for efficient ionization, well before the focal region. This leads to a considerable delocalization of the absorbed laser energy inside the bulk of a transparent material with the generated CB electrons spread over an extended pre-focal region [30,33]. Consequently, the STE population created by CB electron trapping is also delocalized. In such a situation, re-excitation of the electrons from the STE states by the front part of the second pulse leads to the creation of an “umbrella-like” shield that can partially screen the focal region from the remaining part of the beam [30], thus reducing the maximum of the absorbed energy density and, hence, the space-integrated transmission microscope signal. We admit that long-living defects (color centers and non-bridged oxygen hole centers) were shown to be responsible for the asymmetric writing of waveguide structures in fused silica [33] although the defect density is much smaller as compared to that of STEs formed due to trapping of laser-generated CB electron (only approximately 0.1% of the STEs are converted to permanent defects [45]). Overall, the control over the generation of the STEs and other defects in fused or crystalline silica as well as in other optical materials can represent a valuable tool for the optimization of direct laser writing of 3D structures in the material bulk [30,46,47] that calls for further studies.

5. Conclusion

We have investigated the dynamics of free electron generation and the associated shielding effect under the conditions of ultrashort-pulse laser modification of fused silica. It has been shown that using double pulses with a time separation of up to hundreds of femtoseconds provides a significantly smaller volumetric modification as compared to pulses coupling with the material with a larger time delay. The origin is related to the shielding of the focal zone by laser-generated free-electron plasma. Based on the experimentally measured dependence of the volumetric modification on the time delay between two successive pulses, the duration of the shielding effect was estimated as τ_shield ≈ 600 fs. Furthermore, using sequences of two pulses of different energies, it has been shown that absorption of laser light depends essentially on which pulse couples with the sample first, of lower or higher energy. This difference is explained by two scenarios, both related to the formation of self-trapped excitons. Depending on the energy of the first laser pulse in the pulse pair, the generated STE population can serve for enhancing or reducing the modification level (measured as TMS) and its localization after the action of the second pulse. The experimental results were supported by numerical simulations based on Maxwell's equations that helped to understand the dynamics of the conduction band electrons and STEs. Thus, achieving control over the generation of various short- and long-living defect states in optical materials can represent a valuable tool for the optimization of direct laser writing of 3D structures in the material bulk.