1. Introduction

Recent advances in ultrashort laser technology pave the way to new non-invasive laser material processing techniques in many industrial fields including aircraft, medical devices, luxury and watch industries, as well as optics manufacturing, electronics or smart electronics. This laser technology also enables one to process transparent dielectric materials with an outstanding quality and precision. Several applications have been reported in the literature as for instance refractive index change [1], 3D data storage [2–5], wave guide and diffraction gratings generation [6–9], phase mask manufacturing [10,11], bottom-up glass drilling [12] and ablation-free glass cutting [13]. However, processing time is often too long to meet industrial requirements. In order to overcome this limitation, beam shaping has been proposed as one of the possible strategies to improve the laser energy deposition and the overall process throughput [14].

The nature of material modifications results from the laser energy absorption which is driven by the laser-induced electron dynamics. At the early stages of the laser-matter interaction, the latter includes the following main processes: (i) electrons excitation by non-linear absorption [15] and impact ionization [16], (ii) lattice heating by electron-phonon coupling, and (iii) thermal diffusion and structural changes [17,18]. These processes depend on the laser pulse energy, duration, wavelength, polarization, etc. Therefore, the laser energy absorption, linked to the laser ablation efficiency, can be improved by tuning laser parameters, i.e. by carrying out either spatial or temporal beam shaping.

Both temporal and spatial pulse shaping approaches have been widely reported in the scientific literature. For instance, the use of a Bessel beam enables an elongated and narrow energy deposition and ablation [19–23], which has shown to be favorable for laser cutting processes. Temporal pulse shaping allows one to control the electron dynamics [24–26] involved in the laser modification, and therefore to tailor the material structural change. Spatial and temporal pulse shaping can also be employed simultaneously, as for instance by using a Bessel beam with several sub-pulses [27,28]. Double-pulse ultrafast laser irradiation of dielectric materials has also been investigated. Such experiments have been reported in fused silica [24–29], sapphire [24,29] and borosilicate glass [28]. Most papers mention a slightly positive effect on removal rate [24–28] and optical breakdown threshold [28] related to the second pulse for short pulse-to-pulse delay (from 0 to 600 fs). According to Hernandez-Ruela et al. [25] and Gedvilas et al. [30], the energy distribution between the two pulses has a significant influence on the ablated volume. Reyne et al. have shown that the effect of double-pulse laser irradiation on a birefringent crystal is polarization and wavelength-dependent [31]. The influence of polarization on the modification threshold of dielectric materials has been also reported in case of single-pulse laser irradiation at wavelength of 800 nm and pulse duration shorter than 100 fs [32–35]. Nevertheless, to our knowledge, the role of the polarization state in the optimization of the ablation efficiency in case of double-pulse irradiation of dielectrics materials has not yet been investigated.

In the present work, results on double femtosecond pulses irradiation at wavelength of 1030 nm of fused silica are reported in both sub-ablation and ablation regimes with two polarization configurations: (i) an orthogonally-crossed linear polarization (CLP) where the two pulses have linear polarization orthogonal to each other, and, (ii) a counter-rotating circular polarization (CRCP) where the two pulses have circular polarization rotating in opposite direction to each other. The purpose of this work is to get a better understanding of the laser-matter interaction on one hand, and to pave the way for improving the ablation rate, on the other hand. Results are studied with respect to the variation of the laser fluence, polarization configuration and pulse-to-pulse delay, and are discussed in terms of optical transmission, modification threshold and material removal rate. Results demonstrate that the pulse-to-pulse delay has a major influence on ablated volume, and modification and ablation threshold. Polarization state has also, to a lesser extent, a significant influence on ablation efficiency. To support the interpretation of experimental results, a model describing the laser electron dynamics and the energy absorption has been developed. The model parameters have been set according to the results of single-pulse study: the ablated volume has been compared with the theoretical absorbed energy density required to heat the target material up to the glass transition temperature of fused silica [36]. These parameters have then been kept for the double-pulse laser irradiation case. The theoretical curves are compared with experimental results allowing one a better understanding of the underlying physical processes.

The paper is organized as follows. The experimental setup as well as the analyses protocol are described in Sec. 2. The numerical model is described in Sec. 3. The transmission measurement and modification threshold analyses are described in Sec. 4.1. The ablated volume measurements performed in CRCP and CLP configurations are presented in Sec. 4.2. Finally, Sec. 5.1, Sec. 5.2 and Sec. 5.3 are dedicated to the discussion on the variation of the cumulative double-pulse intensity and resulting polarization configuration with the pulse-to-pulse delay, and their cooperative effect on the final ablation efficiency, respectively.

2. Experimental setup and protocol

The experimental setup employed for all tests includes an ultrafast laser, a delay line, a beam expander, a 3-axis translation stage, a 20x focusing objective (NA=0.42) and a power-meter. A short description of the setup is provided below; more detailed explanations can be found in [37].

The ultrafast laser is an amplified Yb-doped hybrid femtosecond laser (Yuja from Amplitude Systemes). The operating wavelength is 1030 nm. The main laser parameters are summarized in Table 1. The spectral width is 3 nm. The coherence time is about 1.2 ps. The maximum average power is 10 W. The repetition rate ranges from 100 kHz up to 2 MHz. Nevertheless, an internal pulse picker enables to generate lower repetition rates down to 500 Hz. The laser pulse exhibits a Gaussian shape with a Full Width at Half Maximum (FWHM) of 480 fs. The focused spot size, measured by a WinCam-D beam analyzer, is 7.5 ± 0.3 µm at 1/e². The effective numerical aperture, calculated from the measured focused spot size, is about 0.17. Energy measurement is performed after the sample owing to a photodiode. Uncertainty on the power measurement is about 5%. For all experiments, an energy ratio between the two sub-pulses of 50:50 is chosen in order to be in the farthest away condition from the single-pulse irradiation case (that is an energy ratio of 0:100) and not to favor the role of one pulse with respect to the other. The total fluence, defined as the sum of each single pulse fluence, is varied from 0.5 to 22 J/cm² with an uncertainty which considers both spot size and power uncertainties. All experiments were performed on 1 mm-thick uncoated fused silica. The beam is focused on the sample surface.

Table 1. Main parameters of the laser system.

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Figure 1 shows the scheme of the experimental setup employed for all tests. A delay line is used to split the initial pulse from the laser output into two sub-pulses with a controlled time delay. A half-wave plate and a polarizer cube are used to tune the polarization in order to split the beam and to set the energy ratio between the two sub-pulses. A micrometric translation stage is introduced on one arm for the fine tuning of the optical path difference between the two sub-pulses. Finally, the two beams are recombined using a polarizer cube, after which the two sub-pulses are polarized linearly and orthogonally to each other. For experiments carried in CRCP configuration, a quarter-wave phase plate is additionally introduced to obtain a circular polarized wave. Thus, an either clockwise or counterclockwise circular polarization is obtained. The accuracy on the zero-delay position is about ±130 fs. The pulse-to-pulse delay was varied from 0 to 5 ps; six values were considered in the study: 0, 0.133, 0.267, 0.5, 1.0 and 5.0 ps. For each value of delay, the total fluence was varied from 0.5 to 22 J/cm². Each experiment was repeated three times for each set of parameters in order to reduce the uncertainty of the results. For all experiments, a repetition rate of 500 Hz and a scanning speed of 10 mm/s were used in order to maintain a distance between successive double-pulses of 20 µm. Doing so, the spot size being 7.5 µm, there was no spatial overlap between successive double-pulse, i.e each crater was produced by one double-pulse irradiation only. Transmission and removal rate were measured for each set of parameters. Topography measurements were carried out using a Leica DCM 3D confocal microscope based on LED technology with a 50x/0.90-objective. The volume of an ablation crater was then calculated from its 3D profile by integrating the volume below the mean surface level. For each crater, four to six measurements were considered for the ablated volume calculation in order to reduce uncertainties. Removal rate uncertainty takes into account both fluence and volume uncertainties (about 10%).

 

Fig. 1. Scheme of the laser workstation with the general setup (left) and the detail of the components of the delay line (right).

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3. Modeling of the electron dynamics for energy deposition calculation

This section is devoted to the modeling of absorbed laser energy to support the interpretations of experimental observations. Hereafter, we interpret the evolution of the ablated volume in terms of absorbed energy density. Indeed, a sufficient, absorbed energy density (that can be associated to a material temperature) can lead to material modifications and ultimately to material ablation. Four processes are considered to describe the electron dynamics in dielectric materials. First, the valence electrons are promoted into the conduction band through (i) photoionization, i.e. multiphoton absorption or tunneling depending on the laser parameters. These produced conduction electrons further absorb photons from the incoming laser pulse; this phenomenon is called (ii) electron heating. When the electron kinetic energy is of the order of the energy bandgap, they may collide with valence electrons leading to the so-called (iii) impact ionization. At the same time, free electrons (iv) recombine, a process by which electrons return to their initial state, or be turned into defects as self-trapped excitons [38]. First, the evolution of the electron populations in the various allowed conduction states is calculated from the Rethfeld model [16]:

 

Fig. 2. The blue line represents the theorical absorbed energy density as a function of fluence for a single-laser pulse laser irradiation obtained with the modelling parameters provided in Table 2. The horizontal red dashed line shows the absorbed energy density required to reach the glass transition temperature.

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Table 2. Simulation parameters.

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4. Results

4.1 Optical transmission and modification threshold fluence

This section is devoted to the transmission measurements in both CLP and CRCP configurations. Figures 3(a)–3(b) shows the optical transmission as a function of total fluence for different pulse-to-pulse delays ranging from 0 to 5 ps in CLP and CRCP polarization states, respectively. At fluences below 0.9 J/cm², there is no surface modification and the optical transmission is about 94%, which corresponds to the experimental value of the optical transmission without material modification after reflection on both front and rear side of the sample. This value is different from the theoretical one of 93.4% obtained with Fresnel equations due to experimental uncertainties. For higher fluences, the optical transmission drops due to defects or ablation crater appearance. The surface-modification threshold fluence is here defined as the fluence value at which the transmission becomes lower than 94%. Figures 3(c)–3(d) depicts the evolution of the modification threshold fluence as a function of pulse-to-pulse delay.

 

Fig. 3. Optical transmission as a function of the total fluence in (a) CLP and (b) CRCP configuration. Modification threshold fluences, corresponding to fluence values where the transmission becomes lower than 94%, are extracted from (a) and (b) and are plotted in (c) and (d) as a function of the temporal delay for CLP and CRCP configurations, respectively. The full dots correspond to measured data. The dashed lines are guides to the eyes.

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The modification threshold fluence increases with the pulse-to-pulse delay with an evolution that depends on the polarization configuration. Indeed, the drop in optical transmission occurs after 133 fs in CLP configuration and after 500 fs in CRCP configuration. Noticeably, in CRCP configuration double pulse laser irradiation (pulse-to-pulse delay = 0 ps) leads to a lower optical transmission compared to single pulse one. A more detailed explanation of these observations is given in the discussion Sec. 5.2.

4.2 Ablated volume

Figure 4 presents the evolution of the ablated volume as a function of the fluence for pulse-to-pulse delays ranging from 0 to 5 ps, in both CLP and CRCP configurations. Single-pulse laser irradiation result is also shown as a reference. The ablated volume increases linearly with the fluence up to 20 J/cm² for both single and double pulses laser irradiation [Figs. 4(a)–4(b)]. The linear behavior has been verified by a fitting procedure: the correlation coefficient R² is about 0.96 in CRCP configuration and 0.98 in CLP configuration. As shown in Figs. 4(c)–4(d), the ablated volume exhibits a drastic drop with increasing pulse-to-pulse delay regardless the fluence, in line with the behavior of both modification and ablation threshold fluences observed in Figs. 3(c)–3(d). Although both polarization configurations present similar ablated volume values for short delays, the drop in ablated volume occurs for delays longer than 133 fs in CLP configuration and long than 500 fs in CRCP configuration. These specific delay values are defined hereafter as the transition delay for each polarization configuration. It is also noticeable that for delays ranging from 133 fs to 1 ps, ablated volume values related to CLP configuration present a minimum value as opposite to the trend of ablated volume in CRCP configuration [see Fig. 4(c)]. Finally, the ablated volume remains almost constant for delays longer than 1 ps, when the two sub-pulses do not longer overlap.

 

Fig. 4. Ablated volume as a function of the fluence for pulse-to-pulse delay ranging from 0 to 5 ps for a fused silica target in (a) CLP and (b) CRCP configuration. In (c) and (d) the ablated volume extracted from (a) and (b) is shown as a function of pulse-to-pulse delay for fluences ranging from 4.7 to 22.2 J/cm² in CLP and CRCP configuration, respectively.

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5. Discussion

5.1 Influence of pulse-to-pulse delay

In the early stage of the laser-matter interaction a strong increase of free electron density takes place due to the nonlinear dependence of photoionization on laser intensity. Once the critical electron density (∼10²¹ cm⁻³) is reached, the matter exhibits a metallic behavior and the free electron density no longer significantly increases. In this condition, both free electron absorption and free electron reflection become significant, and both absorption and ablated volume increase linearly with respect to an increase of fluence. Note that the higher the laser fluence, the earlier the critical density is reached.

When the incoming pulse is split into two independent sub-pulses, the loss of intensity translates into a detrimental influence in terms of ablation efficiency: the photoionization rate is decreased, and thus free electrons production and energy deposition are smaller. This detrimental effect is mainly connected to the pulse-to-pulse delay which determines the cumulative intensity of the two sub-pulses. As the pulse-to-pulse delay increases from zero, where the cumulative intensity is at its maximum, up to 1 ps, the detrimental influence becomes more and more important and therefore, both optical transmission and ablated volume, which results from electronic excitation, decrease as observed in Fig. 3 and Fig. 4. For pulse-to-pulse delay longer than 1 ps, when the two sub-pulses do not longer overlap, the optical transmission and ablated volume do not longer evolve.

The decrease in cumulative intensity of the two sub-pulses with the increase in pulse-to-pulse delay also explains the increment of both modification and ablation threshold fluences observed in Figs. 3(c)–3(d). Consequently, the shorter the pulse-to-pulse delay, the higher the ablated volume.

The saturation phenomenon observed above 20 J/cm² in Figs. 4(a)–4(b) is most probably due to shielding effect due to air breakdown. At peak intensity of about 4×10¹³ W/cm² [45], which corresponds to a fluence of about 20 J/cm² for a single 500 fs-pulse, air breakdown may take place, leading to a significant decrease in the incoming laser pulse energy on the target. This energy loss thus does not contribute to material ablation.

The previous explanations are supported by modeling of energy deposition. Calculations have been performed with the model described in Sec. 3 in double-pulse configuration, with an energy distribution of 50:50 between the two sub-pulses and with a pulse-to-pulse delay ranging from 0 to 5 ps as in experiments. The modeling parameters are the same as the ones determined in single pulse configuration (see Table 2). Figure 5 shows the absorbed energy density [Fig. 5(a)] and the electron density normalized with respect to the critical electron density [Fig. 5(b)] evolutions as a function of total fluence and delay between the two sub-pulses. In Fig. 5(a) the absorbed energy density decreases with increasing the delay up to a value between 1 ps and 2 ps, depending on the fluence, and remains constant for longer delays. In Fig. 5(b) the electron density decreases rapidly with increasing pulse-to-pulse delay up to 2 ps and then no longer evolves. The critical electron density is reached at 6.5 J/cm² regardless the delay.

 

Fig. 5. Theoretical calculation of (a) the absorbed energy density and (b) the electron density normalized to the critical electron density as a function of the total fluence of the two sub-pulses and the pulse-to-pulse delay. The wavelength is 1030 nm, the pulse duration at FHWM is 480 fs and the energy ratio between the two sub-pulses is set to 50:50 as in experiments.

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In order to compare the theoretical predictions and the experimental results presented in Figs. 4(c)–4(d), extractions of the absorbed energy density from Fig. 5(a) as function of pulse-to-pulse delay for various fluences are shown in Fig. 6. Three peculiar behaviors can be identified: (i) an optimum of absorbed energy density for zero delay, also observed in Figs. 4(c)–4(d), (ii) a significant drop of absorbed energy density for pulse-to-pulse delay of the order of 1-1.5 ps. This behavior is due to the decrease in the maximum cumulative intensity with increasing delay, and (iii) a stabilization of the absorbed energy density to a constant value after 1-1.5 ps delay. These theoretical calculations confirm the drop of the ablation rate with respect to the increase of the pulse-to-pulse delay above 1 ps, when the two sub-pulses no longer overlap.

 

Fig. 6. Theoretical calculation of absorbed energy density as a function of pulse-to-pulse delay for various total fluence values. The wavelength is 1030 nm, the pulse duration at FHWM is 480 fs and the energy ratio between the two pulses is set to 50:50.

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5.2 Influence of polarization state

As shown in Figs. 4(a)–4(b), the effect of the polarization configuration on the ablated volume is significant for delay longer than the transition delay (defined as the delay after which a drop in ablated volume is observed for each polarization configuration, see Sec. 4.2. This effect is observed for both single and double pulse laser irradiation. The influence of polarization on ablated volume most probably lies in the difference in absorption cross section between linear and circular polarization: the ratio of 6-photons absorption cross sections for linear and circular polarization is about ${\; \sigma }_6^{\textrm{lin}}/{\sigma }_6^{\textrm{circu}} \approx 3.7{\; }$ (see Table 3). Thus, linear polarization leads to a higher photoionization rate, higher free electron density, and higher energy deposition compared to circular polarization. This trends is also supported by Reiss et al. [46] which predicts larger cross sections for linear polarization than circular polarization for large multiphoton order. Ye et al. [33] have as well obtained a stronger bulk modification with a linear polarization compared to circular polarization.

Table 3. Absorption cross section values of 6-photon absorption in fused silica [35].

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The polarization configuration has also an effect on the morphology of the ablation craters. This effect is barely visible for short delays and becomes quite significant for long delays. Indeed, for 1 ps delay, the variation between CLP and CRCP configurations in the measurements of the diameter and depth is less than 10% and about 40%, respectively. Figures 7(a) and 7(b) presents representative 3D profiles of two craters obtained at pulse-to-pulse delay of 1 ps, a total fluence of 9.8 J/cm² in CLP and CRCP configuration, respectively. The significant effect of pulse-to-pulse delay, that is of the polarization state (see Sec. 5.3), on the crater depth, and to a lesser extent on the crater diameter, could be attributed to the resulting cumulative intensity of the two sub-pulses which is maximum at the center of the beam and lower at its periphery. Moreover, an increase of fluence from 2 to 22 J/cm² corresponds to an increase of crater depth by a factor of nine whereas the crater diameter increases by a factor of two only.

 

Fig. 7. 3D profiles carried out using a Leica DCM 3D confocal microscope using a 50x/0.90-objective with the CLP configuration in (a) and CRCP configuration in (b) with a pulse-to-pulse delay of 1ps and a total fluence of 9.8 J/cm².

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5.3 Cooperating effect between pulse-to-pulse delay and polarization state

In the following paragraph, the cooperating effect between pulse-to-pulse delay and polarization state is discussed. The influence of the pulse-to-pulse delay on the cumulative intensity and therefore on the energy deposition was presented in Sec. 5.1. Appendix A presents the calculation of the resulting polarization state as a function of pulse-to-pulse delay between the two sub-pulses by considering the interferences between both electric fields.

For both polarization configurations, as soon as the two sub-pulses start temporally overlapping, the resulting polarization state is, as a first approximation, elliptical, until the overlap is completed. As shown in Fig. 4, the ablated volume is similar for short pulse-to-pulse delays for both polarization configurations and exhibits from a drastic drop with increasing pulse-to-pulse delay. This drop occurs after the transition delay value, that is after 133 fs in CLP configuration and 500 fs in CRCP configuration. Why is this transition delay shorter in CLP configuration compared to CRCP configuration?

In case of CLP configuration, the polarization state calculation shows that the ellipticity ratio between the two sub-pulses depends on their phase difference whereas the polarization orientation depends on their amplitude ratio. However, the phase difference cannot be determined due to the uncertainty on the definition of the zero-delay position which is about ±133 fs. The latter value corresponds to 40 optical cycles, and therefore it may take any value between 0 and 2π. Thus, in case of CLP configuration the ellipticity ratio cannot be accurately calculated.

On the other hand, in case of CRCP configuration the polarization state calculation shows that the ellipticity ratio depends on the amplitude ratio between the two sub-pulses while the polarization orientation depends on their phase difference. In this case, the resulting polarization state for a given pulse-to-pulse delay is mainly elliptical except in a temporal interval between the two sub-pulses, where it tends to be linear. This temporal interval becomes larger as the pulse-to-pulse delay decreases, that is the higher the temporal overlap is between the two sub-pulses, the more linear the resulting polarization state. The existence of a time interval where the resulting polarization state is mainly linear has a major contribution on the non-linear ablation behavior because, as the pulse-to-pulse delay decreases, the cumulative intensity also increases. These two effects act together to enhance the ablation at short pulse-to-pulse delays as shown in Fig. 4(d). Contrary to CLP configuration, in CRCP configuration this cooperation phenomenon is significant enough to counter-balance the loss in intensity due to the pulse splitting and postpone the transition delay to 500 fs. This effect also explains the higher ablated volume and the lower modification threshold observed for zero-delay double-pulse laser irradiation in CRCP configuration compared to the case of single pulse irradiation, as shown in Fig. 4(b) and Fig. 3(b), respectively. Consequently, the effect of polarization state on ablated volume is delay-dependent.

This effect is found also for CLP configuration and allows explaining the opposite behavior observed in the ablated volume in this configuration where a minimum can be observed for delays ranging from 133 fs to 1 ps. Figure 8 presents the evolution of percentage of linear polarisation and maximum normalised cumulative intensity with the pulse-to-pulse delay for both polarisation configurations. A more detailed explanation of the calculations is given in Appendix A. As previously mentioned, in CRCP configuration it is possible to precisely define a resulting polarization state for all pulse-to-pulse delay values which, together with the cumulative intensity evolution, brings to a favourable cooperative effect on the ablated volume [Fig. 8(a)]. In CLP configuration, due to the uncertainty on the phase difference between the two sub-pulses, it is not possible to obtain a single theoretical curve representing the precise resulting polarization states for each pulse-to-pulse delay as for CRCP configuration: all possible curves lie in the blue region highlighted in Fig. 8(b). Nevertheless, an experimental curve exhibiting a minimum for 267 fs-delay can be plotted from the experimental data presented in Fig. 4(c). By comparing the results on ablated volumes in the two polarization configurations for the same maximum normalised cumulative intensity, it is possible to retrieve the resulting polarisation state for CLP configuration: the lowest ablated volume is obtained at the lowest cumulative intensity when the resulting polarisation state is consequently mostly circular.

 

Fig. 8. Evolution of percentage of linear polarization (blue dotted line) and maximum normalized cumulative intensity (red dotted line) with the pulse-to-pulse delay for (a) CRCP and (b) CLP configuration. Calculations are performed by implementing Eq. (4). presented in Appendix A.

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It is interesting to notice that delay-dependent effect of polarization state on ablated volume is consistent also for long pulse-to-pulse delays where the two sub-pulses no longer overlap. Figure 9 presents the evolution of the ablated volume with the fluence for long pulse-to-pulse delays (1 and 5 ps) in both polarization configurations (CLP and CRCP) and for single-pulse irradiation in linear and circular polarization. With respect to the cases in CRCP configuration and single-pulse irradiation in circular polarization, the increase of the ablated volume is about 1.2, 2 and 3 times higher in linear configuration, in CLP configuration for a 1 ps-delay, and in CLP configuration for a 5 ps-delay, respectively.

 

Fig. 9. Ablated volume as a function of the fluence for 1 and 5 ps pulse-to-pulse delays in both polarization configurations (CLP, full dots and CRCP, crosses) and for single-pulse irradiation in linear (full dots, blue) and circular polarization (crosses, blue).

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6. Conclusion

A single-wavelength double femtosecond pulses laser irradiation of fused silica in both sub-ablation and ablation regimes is reported in view of an optimization of the ablation process. Two different polarization configurations are compared: orthogonally-crossed linear polarization (CLP) and counter-rotating circular polarization (CRCP). The influence of total fluence and polarization state are investigated for pulse-to-pulse delays in the range 0-5 ps. The results are discussed in terms of optical transmission, modification threshold fluence and ablated volume.

It is demonstrated that pulse-to-pulse delay has a major influence on ablated volume, modification and ablation threshold. Indeed, when the incoming pulse is split into two sub-pulses, a decrease in maximum intensity, photoionization rate and energy deposition is observed. Thus, the overall trend is a drastic drop of ablated volume with increasing pulse-to-pulse delay. Polarization state has also, to a lesser extent, a significant influence on ablated volume: linear polarization leads to a higher absorption cross section compared to circular polarization and therefore to a more efficient ablation. The elliptical polarization results in an intermediate behavior: for short pulse-to-pulse delays, when the two sub-pulses overlap, the resulting polarization is mainly elliptical leading to similar ablated volume for both polarization configurations. However, in this range of pulse-to-pulse delays, it is demonstrated that in CRCP configuration, specific conditions exist for which the cooperative effect between the resulting linear polarization state and the increasing cumulative intensity leads to an enhancement of ablated volume. Therefore, the effect of polarization on ablated volume is delay-dependent.

Overall, these experimental investigations have shown that the ablation efficiency results from the combination of two parameters, the polarization state and the cumulative intensity which depends on, the pulse-to-pulse delay. Depending on conditions, these two parameters could be either antagonist or cooperating leading to a decrease or increase of the ablated volume, respectively. Experiment perform with a ratio energy of 70:30 were also performed. The experimental results are similar but the effects on ablated volume are less pronounced and close to single pulse laser irradiation experiment. Although no significant improvement of the ablation rate was observed upon double-pulse irradiation with respect to the single pulse case; The presented results allow one an improved understanding of the laser-matter interaction leading to dielectric material ablation and of the influence of key parameters, such as the polarization state and the pulse-to-pulse delay, on the ablation behavior.

Previous interpretations of experimental results have been supported by a numerical model which considers the laser induced electron dynamics and energy deposition. The model parameters have been set according to the results of single-pulse study: the ablated volume has been compared with the theoretical absorbed energy density required to heat the target material up to the glass transition temperature for fused silica. These parameters have been kept for the double-pulse laser irradiation case. The theoretical predictions are in a good agreement with experimental data. Calculations of the resulting polarization of the two overlapping sub-pulses have been also performed to understand the influence of pulse-to-pulse delay and initial polarization configuration on the ablated volume.

As a perspective, to further improve the ablation efficiency the results presented in this work will be completed by further experiments, by using a shorter wavelength for the first pulse (515 nm) on one hand, and using a longer wavelength for the second pulse (1030 nm) on the other hand. Contrary to the present single-color results where each sub-pulse has the same rates for both ionization and free electron heating, we expect that using a two-color configuration will render it possible to manage efficiently both processes and thus improve the energy deposition and ablation efficiency. Indeed, the short wavelength sub-pulse will produce a large free electron density which will be efficiently heated by the long-wavelength sub-pulse.

Appendix A: Calculations of resulting polarization in CLP and CRCP configurations.

A1 Counter-rotating circular polarization (CRCP) configuration

For zero pulse-to-pulse delay, the resulting polarization is always linear for any value of ${\Delta }\varphi \in [{0,{\; }2{\pi }} ]$. The direction of the linear polarization depends of the phase $\varphi $ value of each sub-pulse. Nevertheless, since the value of the phase does not affect the state of polarization, the ellipticity degree can be defined as the ratio between the intensity ${I_1}(t )$ and ${I_2}(t )\; $of the two sub-pulses. The resulting polarization state ${P_{\textrm{state}}}(t )\; $at any time can be calculated with the equations presented in Table 4 and the corresponding boundary conditions.

Table 4. Calculation of resulting polarization state in CRCP configuration. With ${P_{\textrm{state}}}(t )= 0$, the polarization is circular and with ${P_{\textrm{state}}}(t )= 1$ it is linear.

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It follows that the resulting polarization is linear when ${I_1}(t )= {I_2}(t )$, whereas it is circular when either ${I_1}(t )< < {I_2}(t )$ or ${I_2}(t )< < {I_1}(t )$ and elliptical otherwise. Figure 10 presents the evolution of the normalized intensity of each sub-pulse, the cumulative intensity and the resulting polarization state as a function of time for given pulse-to-pulse delays: 0, 267, 500 and 1000 fs. Hence, by introducing a delay between the two sub-pulses, the resulting polarization (i) tends towards a linear polarization when the amplitudes of the two sub-pulses tends to the same value, (ii) becomes mainly elliptical as the pulse-to-pulse delay becomes more important and finally (iii) tends to a circular polarization when the two sub-pulses do not longer overlap.

 

Fig. 10. Normalised intensity and resulting polarization state as a function of time for pulse-to-pulse delays of 0, 267, 500, 1000 fs.

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The resulting polarization state ${P_{state}}(t )\; $defined in Eq. (4) allows one calculating the average ratio of ellipticity for a given pulse-to-pulse delay as follow:

(4)$$\frac{{\sum ({{P_{state}}(t )\times {I_{tot}}(t )} )}}{{\sum {I_{tot}}(t )}}$$

With ${I_{tot}}(t )$ the total intensity from the two sub-pulses. Any resulting polarization state with an ellipticity degree higher than 70% is considered as a linear polarization state. The results of this calculation lead to the values of percentage of linear polarization presented in Fig. 8(a).

A2 Orthogonally-crossed linear polarization (CLP) configuration

For zero pulse-to-pulse delay in this case, for $\varDelta \varphi = 0{\; }[{2{\pi }} ],\varDelta \varphi = {\pi \; }[{2{\pi }} ]{\; }or{\; }\varDelta \varphi = 2{\pi \; }[{2{\pi }} ]$, the resulting polarization is linear. For $\varDelta \varphi = \frac{{\pi }}{2}{\; }[{2{\pi }} ]{\; }$ the resulting polarization is counterclockwise circular. For $\varDelta \varphi ={-} \frac{{\pi }}{2}{\; }[{2{\pi }} ]$, the resulting polarization is clockwise circular. Apart from these four particular cases, with $\varDelta \varphi \in {\; }\left] {0,\frac{{\pi }}{2}} \right[{\; } \cup {\; }\left] {\frac{{\pi }}{2},{\pi }} \right[{\cup} {\; }\left] {\frac{{\pi }}{2},\frac{{3{\pi }}}{2}} \right[{\cup} {\; }\left] {\frac{{3{\pi }}}{2},2{\pi }} \right[$ the resulting polarization is elliptical and the ellipticity degree depends on the intensity ratio of the two sub-pulses. Thus, for a zero pulse-to-pulse delay, from a statistical point of view the resulting polarization is mainly elliptical. When a delay is introduced between the two sub-pulses, the resulting polarization depends on the phase difference between the two sub-pulses as long as there is an overlap. For $\varDelta \varphi = 0{\; }[{2{\pi }} ]{\; },$ $\varDelta \varphi = {\pi \; }[{2{\pi }} ]{\; }or{\; }\varDelta \varphi = 2{\pi \; }[{2{\pi }} ]$ the resulting polarization is linear, whereas for $\varDelta \varphi \in {\; }\left] {0,{\pi }} \right[{\; } \cup \left] {{\pi },2{\pi }} \right[$ the polarization is elliptical. Thus, the resulting polarization is mainly elliptical when there is an overlap regardless the pulse-to-pulse delay. The ellipticity degree depends on the ratio between the intensities of the two sub-pulses and their phase difference $\varphi $. The polarization state ${P_{\textrm{state}}}(t )\; $can be calculated at any time with the equations and the conditions presented in Table 5.

Table 5. Calculation of resulting polarization state in CLP configuration. A(t) is given by Eq. (5).

View Table | View all tables in this article

The ellipticity coefficient $A(t )\; $is calculated as follow:

(5)$$A(t )= \; {E_1}{(t )^2} - 2{E_1}(t ){E_2}(t )+ \; {E_2}{(t )^2} = \sin {({{\Delta }\varphi (t )} )^2}$$

Where ${E_1}(t )\; and\; {E_2}(t )$ are the electric fields of the first and second sub-pulse, respectively.

As for CRCP configuration, the resulting polarization state ${P_{state}}(t )\; $defined in Table 5 allows calculating the average ratio of ellipticity for a given pulse-to-pulse delay with Eq. (4). Any resulting polarization state with an ellipticity degree higher than 70% is considered as a linear polarization state. The results of this calculation lead to the values of percentage of linear polarization presented in Fig. 8(b). Since the zero-delay uncertainty is about 133fs, which corresponds to 40 optical cycles, it is not possible to determine the phase difference value accurately. Thus, the phase difference could have any value between 0 and 2${\pi }$, corresponding to the blue area in Fig. 8(b).

Funding

French Government Defence (172906052, 172906053); French National Association of Research and Technology (2017/0506).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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