Ultrashort-pulse laser surface and bulk nano- and micromachining of dielectrics have multiple promising applications in micro-optics, microfluidics, and memory storage. The fundamental principles relate intrinsic inter-band multi-photon (MPA) and laser-induced intra-band free-carrier absorption (FCA) to particular ablation mechanisms and features. These principles are yet to be quantified into a complete set of basic experimental laser-matter interaction parameters, describing photoexcitation, relaxation, and final ablation. In this study, we considered the characteristic double-crater structure of single-shot ablation spots on dielectric surfaces and single-shot transmission spectra to extract crucial information about the underlying basic processes of ultrafast photoexcitation and laser energy deposition. Specifically, energy-dependent crater profiles and accompanying prompt self-phase modulation (SPM) spectral broadening were studied in single-shot surface ablation experiments on fluorite (CaF2) surface photo-excited by tightly focused 515- or 1030-nm, 300-fs laser pulses. Crater size dependence demonstrated two slopes, scaling proportionally to the squared focal 1/e-radius at higher energies (intensities) for larger ablated spots, and a much smaller squared 1/e-radius at lower energies (intensities) for (sub) micron-wide ablated spots, indicating a transition from 1D to 3D-ablation. As a result, these slopes were related to lower-intensity wavelength-dependent multi-photon inter-band transitions and wavelength-independent higher-intensity linear absorption in the emerging near-critical electron-hole plasma (EHP), respectively. Crater depth dependences on the local laser intensity fitted in the corresponding ranges by multi- and one-photon absorption provided the corresponding absorption coefficients. Spectral broadening measurements indicated even values for the red and blue shoulders of the laser pulse spectrum, representing the SPM effect in the weakly excited fluorite at the leading pulse front and providing the corresponding Kerr coefficient. In the second regime, the blue-shoulder broadening value saturated, indicating the appearance of near-critical plasma screening at the trailing pulse front, which is consistent with our calculations. These complementary experiments and related analysis provided an important set of key basic parameters, characterizing not only surface ablation, but also propagation of high-intensity ultrashort laser pulses in bulk fluorite, and enabling precise forecasting of optimal energy deposition for high-efficiency ultrashort-laser micro-structuring of this dielectric material.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Ultrashort-pulse (mostly, femtosecond/fs) laser ablative nano- and micromachining on dielectric surfaces and in volume materials is employed both in traditional applications, such as waveguide writing , fabrication of birefringent elements , 3D optical memory based on nano-void arrays , as well as innovative emerging fields such as optical vortex generation , holographic recording , and 5D optical memory storage . These applications differ in the volume density of the deposited laser energy, enabling structural rearrangement , periodic nanograting fabrication , hollow nanovoid formation , or microchannel drilling , with varying amount of transfer/removal of the ablative material. Fundamental femtosecond-laser energy deposition mechanisms are well-known [11–19] and are related to multi-photon (MPI) or tunnel inter-band intrinsic photoionization (TI), or avalanche ionization (AI) via inverse bremsstrahlung heating of photo-generated free carriers. However, unambiguous or straightforward quantification of their key parameters, which change both in spatial and temporal domains, directly from experimental optical transmission [20–21], reflection , interferometric , or holographic [24,25] measurements is yet to be achieved. This hinders predictive linking between the input laser, focusing, material parameters, and the resulting volume energy densities and microstructures.
During femtosecond-laser ablation on dielectric surface crater depth [26–28], ablated volumes per pulse [29,30] and crater diameters [31–33] appear to be the most crucial machining parameters, which affect the efficiency, as well as spatial, longitudinal, and transversal (lateral) resolutions of ablation. Usually, these parameters are analyzed separately, without a comprehensive analysis of crater profiles that involve the diameter of the external ablation rim, neighboring above-threshold shallow crater, and the central deep depressions . Specifically, numerous double-crater structures of single-shot ablated spots have been observed [35–39], with their external shallow, even flat (“gentle” ablation ) and central ultra-deep (“strong” ablation ) craters). Meanwhile, direct relationships between such important ablation features and specific regimes of photoexcitation/electron dynamics are yet to be established.
Recently, a two-step analysis procedure was proposed [40,41] for single-shot crater profiles on dielectric surfaces. Both radii and depths are simultaneously considered, covering the entire range of surface ablation modifications, from the characteristic nanometer-deep craters to submicron or even multi-micron depths. First, characteristic ranges of non-linear energy deposition were revealed through the Liu analysis . For local ablation related to the energy deposition spot above the ablation threshold, the Gaussian energy distribution over the focal spot with a focal 1/e-radius Wfoc results in energy deposition, which is proportional to the Nth power of laser intensity, IN, providing the squared characteristic ablation radius Rabl2 = Wfoc2/N [28,40,41]. This procedure enables the determination of multi(N)-photon absorption processes in qualitative agreement with the conventional femtosecond-laser ionization dynamics [11–23,43–46]. Moreover, at higher femtosecond-laser intensities, linear free-carrier (near-critical electron-ion plasma) absorption could be identified according to Rabl2 ≈ Wfoc2 [28,40,41]. Second, the identified intensity ranges for different absorption processes were used for non-linear fitting of the corresponding crater depths across their radial profiles as a function of radial laser intensity [28,40,41]. This is a well-defined manner of deriving absorption coefficients [28,40,41] compared to multi-photon absorption (MPA) approximations [26,27].
In this study, we investigated in detail a two-crater structure on fluorite surfaces CaF2(111) photo-excited by tightly focused 515- or 1030-nm, 300-fs laser pulses. Comparing to multi-component silica glasses, this “clean” material was chosen for its crystalline character with the well-defined optical, electronic, lattice, and transport parameters, with its electronic band and absorption spectra  presented in Fig. 1. In this material, lack of any considerable density of defect or impurity states in the bandgap ensures no their effect on not only near-IR, visible, or UV photoexcitation, but also carrier trapping, recombination, and transport dynamics. Determining the two-slope dependence of crater size and depth on pulse energy (intensity) enabled the identification of key laser energy deposition processes in the material in the context of 3D-1D ablation transition, while inhomogeneous spectral broadening indicated the plasma dynamics during the pump pulse. Based on these results, we explain the different effective squared beam radii as fluence calibration slopes in their relationship to actual energy deposition mechanisms, and derived their numerical laser absorption and transport parameters, which are crucial for sub-wavelength ultrashort-laser nanomachining.
2. Experiment details
Laser ablation and spectral transmittance studies were performed to characterize the high-intensity non-linear interactions between focused femtosecond laser pulses, having a plane phase wave-front and well-defined waist peak intensity, on the CaF2 surface. The schematic of the experimental setup is shown in Fig. 2(a).
In these experiments, we used 300-fs full-width at half maximum (FWHM), quasi-monochromatic 515-nm (FWHM Δλ ≈ 1.7 nm) or 1030-nm (FWHM Δλ ≈ 7 nm) laser pulses (TEM00-mode, M2 ≤ 1.07, 1/e-radii σ(1030 nm) ≈ 10 mm, and σ(515 nm) ≈ 8 mm). The pulses were tightly focused onto a 2-mm-thick optically-polished UV-grade CaF2 slab by a microscope objective with numerical apertures (NA) of 0.25 (effective focal distance f0.25 = 16 mm) into the wavelength-dependent focal spots with the 1/e-radii wfoc(515 nm) ≈ 2 μm and wfoc(1030 nm) ≈ 4 μm  (the calculated 1/e-radii w0(515 nm) = λf0.25/(2πσ*) ≈ 1.7 μm and w0(1030 nm) = λf0.25/(2πσ*) ≈ 3.5 μm) owing to the limited objective aperture. The corresponding calculated wavelength-dependent Rayleigh lengths in fluorite, which are relevant for our SPM spectral broadening studies as the corresponding effective interaction lengths, were zCaF2(515 nm) = λf0.252/(4πnCaF2σ2) ≈ 12 μm and zCaF2(1030 nm) = λf0.252/(4πnCaF2σ*2) ≈ 24 μm, where nCaF2(500 nm) = 1.44 and nCaF2(1.0 μm) = 1.43 are the corresponding refractive indices . For 0.65-NA focusing at the 1030-nm wavelength (f0.65 = 4 mm), the employed focusing parameters were wfoc(1030 nm) ≈ 0.9 μm (the calculated 1/e-radii w0(1030 nm) = λf0.65/(2πσ*) ≈ 0.84 μm) due to the limited objective aperture in air and zCaF2(1030 nm) = λf0.652/(4πnCaF2σ*2) ≈ 1.5 μm in fluorite. All these focal parameters are summarized in Table 1.
Single-shot laser exposition occurred at the variable incident pulse energies E = 0.6-3.6 μJ (1030 nm) and 0.1-1.3 μJ (515 nm), providing the NA-dependent peak fluence F0 = E/(πwfoc2) ≈ 1.2-110 J/cm2 (1030 nm) and 0.8-11 J/cm2 (515 nm), peak intensity I0 = F0/τ≈ 4-370 TW/cm2 (1030 nm) and 3-37 TW/cm2 (515 nm).
The CaF2 slab was arranged on a computer-driven motorized 3D-positioning platform. It was scanned from shot to shot to produce a linear pattern of ten 120-μm spaced single-shot micro-craters at each pulse energy, which were flat and shallow (≈25-nm deep) near the crater edge and deep in the center (Figs. 2(b) and 2(c)). The crater profiles were characterized by atomic force microscopy (AFM). AFM topography scans were performed over 10 μm × 10 μm CaF2 surface spots at 0.04-μm steps through a microscope Solver Pro P6 (NT-MDT) in a semi-contact mode with a 10-μm long high-resolution silicon probe HA_NC (tip curvature – 10 nm, tip angle – 30°) mounted at the angle of 20° in the scan head “Smena”.
Similarly, spectral acquisition of the pump-pulse transmission was performed on fresh CaF2 surface spots in a scanning mode over 50 laser pulses at each pulse energy (Fig. 2(d)). A collecting silica-glass optical fiber was arranged just behind the rear sample surface without any additional collecting optics and coupled to a PC-controlled, single-grating spectrometer ASP-150SF.
3. Results and discussion
3.1 Crater profiles and their interpretation
The profiles of the two-crater single-shot ablated spots provide two important characteristics of laser-matter interactions, namely energy-dependent radii of the external shallow crater and the central deep crater. Further, we acquired radial depth variations across the focal spots, which can be converted into local intensity- or fluence dependence of depth. Because the rather narrow and deep central craters could be perturbed during material removal by non-local effects such as lateral melt displacements and redeposition, the control curves of the maximal crater depth in the central peak intensity points were used for comparison.
3.1.1 a) Analysis of crater diameters
We used the squared radii Rabl2 of the external shallow and central deep craters versus the natural logarithm of incident pulse energy, lnE, to characterize the focusing and energy deposition conditions [28,40,41]. According to the Liu analysis , the surface modification features appearing above the threshold IM for the Gaussian beam with energy E and 1/e-radius wfoc,
This expression can be employed for the linear absorption/energy deposition on the surface, directly replicating the focal energy distribution, resulting in49] and, later, till the picosecond- or multi-picosecond-scale ablation onset (depending on femtosecond-laser ablation mechanisms ).
In dielectrics, where multi(N)-photon absorption of optical photons is the key initial process of energy deposition at low and moderate intensities of ultrashort laser pulses, the effective focal spot size ∼ wfoc/√N is determined by the main energy deposition region
However, for threshold-like modification processes such as ablation, the squared modification radius again reads similarly to Eq. (3).
Our quantitative examination of the experimental data for both the 1030-nm and 515-nm wavelengths indicates that Rabl2-lnE curves in Fig. 3 exhibit two characteristic ranges: range #1 for small and range #2 for large pulse energies, coinciding with small and large ablated spots, respectively. At smaller pulse energies, the derived slopes w12(1030 nm) ≈ 0.40 μm2 at NA = 0.65 and w12(515 nm) ≈ 0.87 μm2 at NA = 0.25 are unexpectedly almost two-fold and five-fold lower than their corresponding focal 1/e-radii -radius wfoc(1030 nm)2 ≈ 0.92 μm2 and wfoc(515 nm)2 ≈ 22 μm2 (Fig. 3), respectively. In contrast, at higher pulse energies for large ablated spots, the curve slopes w22 (1030 nm) ≈ 0.95 μm2 at NA = 0.65 and w22(515 nm) ≈ 4.3 μm2 at NA = 0.25 are in good agreement with their corresponding regular squared values wfoc2.
The observed integer reduction of the curve slopes in the low-energy (low-intensity) range could indicate multi-photon — 2, 3, or 5-photon — absorption in the material at these particular wavelengths, while the linear free-carrier absorption at high energies (intensities) is in agreement with previous interferometric observations . However, in Eqs. (4), and (5) we have revealed that threshold-like multi-photon processes do not change the curve slopes. Hence, the corresponding low-energy crater sizes appear to be enormously large ablation features.
A possible explanation is as follows. For very small ablation spots (in both Figs. 3(a) and 3(b) - Rabl2 <1 μm2), a transition is expected versus the decreasing pulse energy E from 1D in-depth ambipolar diffusion of EHP and the following thermal diffusion over the energy deposition distance LD, to their 3D hemispherical diffusion within the ultra-narrow, (sub) micron-wide focal and ablation spots. The transition occurs during the characteristic ablation onset time , which is a characteristic of broad focal and ablation spots. In a phase explosion regime, implying a hydrodynamic expulsion of supercritical fluid, the characteristic ablation onset time is determined by Z/Cl ∼10-100 ps, where Z is the thickness of the material in the supercritical state (volume energy density ε ≥ εabl) and Cl is the longitudinal sonic velocity. Then, for large focal spots wfoc » LD with the effective nonlinear energy deposition/ablation scale LD « Rabl « wfoc in the axial coordinates,
As a result, the low-energy values w1(1030 nm) ≈ 0.63 μm at NA = 0.65 and w1(515 nm) ≈ 0.94 μm at NA = 0.25 can be related to the energy deposition distance LD,S along the surface during the characteristic ablation onset time, which could be represented as follows :
At 1030-nm wavelength and NA = 0.65, for N = 9 (Fig. 1) the focal contribution to LD,S ≈ w1≈ 0.63 μm (LD,S2 = 0.40 ± 0.04 μm2) in the MPA assumption tends to wfoc/√9 ≈ 0.3 μm, where the rest value = 0.54 ± 0.02 μm is the transport length σS (1030 nm). At 515-nm wavelength for N = 5 (Fig. 1), the focal contribution to LD,S ≈ w1≈ 0.93 μm (LD,S2 = 0.87 ± 0.28 μm2) in the MPA assumption tends to wfoc/√5 ≈ 0.9 μm, where the rest value as the transport length σS(515 nm) = 0.3 ± 0.3 μm is of the same order as at the 1030-nm laser wavelength. Typically, these lengths could be different at different focusing conditions realizing different EHP and temperature gradients. These transport lengths are expected to determine also the in-depth laser energy deposition through the EHP and thermal diffusion processes, and the final ablation depth, as considered next in crater depth analysis for both small and large craters in fluorite.
3.1.2 b) Analysis of crater depths
Purely “optical” analysis of the crater depth dependence on laser intensity in the ranges Iabl,1 ≤ I0 ≤ Iabl,2 and I0 ≥ Iabl,2 could be performed in approximations of pure MPA and free-carrier absorption (FCA) regimes, enabling quantification of their corresponding coefficients. This was done via analysis of the acquired crater profiles, assuming that, across the ablated spot, the ablation threshold is achieved not only on the surface, but also at a certain depth Z via the nonlinear transmission of the incident intensity I [28,40,41]. The crater depth dependence on local intensity Z(I) is demonstrated in Fig. 4 for both these ranges, where the depth increases slowly in the MPA regime for lower I and much faster in the FCA regime for higher I. Following previous studies [28,40,41], in the first range, the crater depth variation versus I was fitted considering the N-photon absorption as follows:
In the high-intensity range (I0 ≥ Iabl,2) the logarithmic approximation of the linear absorption regime,4(a)) and (1.8 ± 0.2) × 104 cm-1 at 515 nm (Fig. 4(b)), potentially indicating the strong FCA process in the near-critical EHP, which is more pronounced for the near-IR laser radiation.
Our analysis according to Eq. (10), including the energy transport depths σB(1030 nm) = 0.54 ± 0.02 μm and σB(515 nm) = 0.3 ± 0.3 μm, indicates their magnitudes to be comparable to the abovementioned values 1/α, including the transport length σB. Moreover, since the maximal (plateau) crater depth at higher intensities approaches rather similar wavelength-independent magnitudes ≈ 350-400 nm in the high-intensity range (Fig. 4), this may indicate the predominant contribution of the EHP/thermal transport in the energy deposition. This implies that the optical absorption depth δabs « 1/α ∼ 0.1–1 μm-1 could be realized as a strong FCA process in near-critical EHP. Despite the much stronger FCA for 1030-nm femtosecond laser pulses, the much higher threshold intensities in Fig. 4 demonstrate the much lower efficiency of overall photoionization for the longer-wavelength photons, which is in good agreement with previous studies [11,12].
Finally, in the low-intensity range, the shallow craters exhibited nearly constant depths Z ≤ 50 nm within the experimental error bars, representing a slightly violent ablation character (see (Figs. 2(b), 2(d), 2(e)). Such a double-crater structure with the external shallow and central ultra-deep crater is rather universal and is well known for many dielectric materials [11,35].
3.2 SPM spectral broadening
Kerr self-focusing coefficients n2 and cumulative EHP appearance at the trailing pump-pulse part were revealed during the 1030-nm and 515-nm femtosecond-laser pump pulses in our spectral broadening studies (see Figs. 2(f) and 2(g)). The broadening magnitude in the spectral inflection points follows the time-derivative of the pulse envelope, according to the following equation for the spectral component spectral component at a wavelength λ0 over the propagation length L :
In the range #1 in Fig. 5, representing lower femtosecond-laser intensities the “red” and “blue” broadening is uniform; however, at higher intensities in the range #2, the “blue” component saturates first and even decreases, while the “red” component persists at much higher intensities. Since the “blue” shoulder broadens at the trailing pulse front, its saturation and drop could indicate the screening effect of the accumulated opaque near-critical EHP, emerging in CaF2 at the trailing edge of the transmitted femtosecond-laser pulses and decreasing their laser intensity. The same saturation and drop for the “red” shoulder exhibits dense EHP formation already at the leading 1030-nm pulse front, screening the four-wave interactions, related to the Kerr effect, with much stronger screening for the “blue” shoulder (Fig. 5(a)).
3.3 Plasma parameters
Numerical analysis was performed to relate the measured effective linear FCA coefficient α ≈ (1.8 ± 0.4) × 104 cm-1 to the peak EHP density during the pulse. Prompt optical constants of the photo-excited fluorite versus EHP density ρ were described using the common expression for the dielectric function with the inter-band and intra-band (Drude) contributions (the first and second terms, respectively) 
The electronic high-frequency dielectric constant εHF ≈ εIB, and e–h pair relaxation time in the Fermi-liquid approximation adapted from  is taken in the form
The electronic bandgap renormalization (shrinkage, ω* ≥ ω)  and band filling effect (ρ ≤ ρbf)  were neglected. The calculated magnitudes of ε* were converted into plasma density-dependent prompt true absorption and reflection coefficients, α0 and R, respectively, assuming Fresnel reflection from a homogeneous photo-excited surface layer of fluorite to yield the corresponding effective absorption coefficients α(ρ) = (1 − R(ρ))α0(ρ).
The calculated effective absorption coefficients α(ρ) rapidly increased with respect to ρ (Fig. 6) for near-critical EHP densities (according to Eq. (14), ρcrit(515 nm) ≈ 8 × 1021 cm-3 in CaF2), saturating for ρ > ρcrit (515 nm) at the level ≈ 1.2 × 104 cm-1, which is reasonably close to the experimental value α (515 nm) ≈ (1.8 ± 0.4) × 104 cm-1. Such self-consistent saturated effective absorption coefficient of the near- and supercritical EHP results from the strongly increasing true absorption coefficient α0 and reflection coefficient R (Fig. 6).
3.4 Characteristic electron dynamics and related absorption mechanisms
Our experimental determination of narrow-crater size dependence on incident pulse energy enabled the evaluation of the optical nonlinearity, describing the femtosecond-laser energy deposition rate η in fluorite to facilitate ablation. This is generally not possible for broad, multi-micron craters. Below, we will discuss a few main regimes of femtosecond-laser generation EHP evolution and energy deposition dynamics, considering the standard rate equation  to provide useful insights into these dynamics.
Here, the first term describes the saturable N-photon absorption/ionization (MPA(I), saturation threshold ρ0); the second term represents the avalanche ionization (AI) transforming into free-carrier absorption (FCA) at sub/near-critical EHP densities ρ, and the third term indicates the reverse Auger recombination (AR) rate with its coefficient γ. Since the quantitative values of the material parameters ρ0 and γ are not known, as for many other dielectrics, we will next perform a qualitative analysis. For the given dielectric bandgap, it is based on incident femtosecond-laser intensity magnitudes and characteristic ranges of EHP density, which related to the critical density ρcrit at the given laser wavelength and the characteristic Auger-recombination density ρauger; this makes the Auger-relaxation time ∼1/(γρauger2) comparable to the laser pulse width.
Specifically, we can find several characteristic cases.
- ii) Medium NIR-laser intensities ∼ 10 TW/cm2, ρ ≤ ρcrit < ρauger (unlimited MPI-seeded FCA):
- iv) Low visible/UV laser intensities « 10 TW/cm2, ρauger < ρ < ρcrit (Auger recombination-limited MPI):
In this study, particularly for the narrow craters in Fig. 3, we observed the slope ≈2 at 1030-nm pumping and ≈5 at 515-nm pumping, respectively, describing the femtosecond-laser energy deposition relationships with intensity during the ablation. At 515-nm pumping, the 5-photon dependence represents simply either case i) (ρ < ρauger, ρcrit), or case iv) (ρauger < ρ < ρcrit), describing the sub-critical EHP generation regime via the multi-photon absorption. In contrast, at 1030-nm pumping, the square dependence represents the interplay between the low-intensity MPI-dominated regimes i) and iv) and high-intensity FCA-dominated regime v) (ρauger < ρ < ρcrit), which are temporally convoluted over the laser pulse.
In conclusion, single-shot femtosecond-laser surface ablation studies at both 1030- and 515-nm wavelengths revealed a double-crater structure, where the external shallow crater demonstrates its characteristic radius, considerably smaller than the focal radius, while the central deep crater follows the focal radius in its extension versus increasing femtosecond-laser pulse energy. The ablation depth varies differently in the corresponding intensity ranges, indicating potentially not only different ablation mechanisms – low-intensity spallation and high-intensity phase explosion, but also the corresponding multi-photon and free-carrier absorption mechanisms. For smaller intensities, the different size dependence for the shallow craters was related to their 3D energy deposition (absorption + transport) character, as compared to the broader high-intensity craters with their strong FCA absorption, resulting in 1D energy deposition. These finding are qualitatively justified by our spectral broadening studies for the 1030-nm and 515-nm pump pulses, enabling also to evaluate their wavelength-dependent Kerr coefficients. Our analysis of ultrafast laser energy deposition and electron dynamics in fluorite photo-excited by tightly focused near-IR and visible-range femtosecond laser pulses in the framework of the general rate equation supports the observed trends in terms of incident intensity, laser wavelength, and characteristic plasma density ranges.
Ministry of Science and Higher Education of the Russian Federation (0705-2020-0041).
The authors declare no conflicts of interest.
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