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Abstract
We explore, both by numerical simulations and experimentally, the flexibility in controlling Bessel beam parameters by re-imaging it into transparent material with a demagnifying collimator for the formation of high-aspect ratio nanochannels. Analysis of nanochannels produced by in-house precision-made axicon with 275 fs pulses in sapphire reveals the intensity threshold of ∼7.2 × 1013 W/cm2 required to create the cylindrical microexplosion. We estimate that the maximum applied pressure during the process was 1.5 TPa and that the resulting density of compressed sapphire in the nanochannel’s shells are ∼1.19 ± 0.02 times higher than the pristine crystal, and higher than what was achieved before in spherical microexplosion with Gaussian pulses.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
During the last few decades, it was demonstrated that tightly focused ultrashort laser pulses deep inside a transparent material generate energy densities beyond the MJ/cm3 level [1–9]. Such extreme energy concentration offers unique opportunities. The extremely high heating and quenching rates lead to unusual solid-plasma-solid transformation paths overcoming kinetic barriers to nucleation and allowed the phase transition to previously unknown high-pressure material phases [10–13]. All transformed material remains confined inside the bulk of the undamaged crystal for further studies of their unusual structures and properties, and new physical phenomena have been uncovered. Few examples are: the spatial separation of ions with different mass and formation of molecular oxygen inside the voids in oxides [10,14,15]; switching the valence in Fe-atoms in olivine [16]; the bright photoluminescence from voids formed in SiO2 [3]; and the formation of N-vacancies in c-BN crystals [17]. Moreover, recent theoretical investigations, using numerical and modified DFT-studies, searched for the possible paths of material transformations under high pressure from the initially chaotic state [12]. These studies uncovered many physically allowed paths for formation of multiple novel phases from the initially chaotic state, including incommensurable phases.
There is a major limitation using tightly focused Gaussian beam in confined micro-explosion on the amount of laser-affected material generated. Imposed by the laser pulse diffraction, the radius of the focal spot is thus limited [5]. The diffraction-limited spot size and the very short absorption depth of the laser energy in the plasma formed, which constitutes 30-60 nm in solid density plasma, affect only a sub-micron size volume of the material. This modified volume represents less than a nanogram of the restructured phase and only around 1010–1012 atoms per single laser shot. While the absorbed energy density is in the order of 107 J/cm3, culminating in ∼10 TPa achievable pressure, the very small volume of the new atomic arrangements makes structural studies extremely complicated.
It was demonstrated recently that Bessel-shaped pulses transform a much larger amount of material in a single pulse [8,18]. Due to the cylindrical geometry of the shock wave expansion, it allegedly creates higher energy density than previously achieved [10–15]. The goal is to achieve the longest possible length of the beam with the available pulse energy. Control and prediction of the ultrashort laser pulse parameters and their Bessel-shaped geometries for material modification and channel formation are key challenges to open new technological opportunities. It requires the precise knowledge of the intensity distribution along the beam inside the bulk of a material. To achieve the required length, the correct intensity distribution and the precise control of the depth where the beam should be formed, the re-imaging optical system must be properly constructed. There is a number of factors to be taken into account: the influence of imperfections of the axicon’s surface on the resulted Bessel beam shape [19–21]; the propagation of optical waves through the material with particular refractive index; the potential influence of limiting factors – the aperture of the re-imaging optics; the level of laser fluence at the sample surface; and beam instabilities induced by Kerr-related or ionisation nonlinearity of the material [22,23].
In this paper, we explore by numerical simulation and experimentally, the advantage of the Bessel beam for the single-pulse laser energy deposition deep inside the bulk of transparent media. The zero-order Bessel-Gaussian beam is generated using an in-house made axicon with precise conical shape and reimaged by a 4F-collimator in the bulk of a sapphire sample. This combination provides a high degree of flexibility in constructing the Bessel pulses and can achieve the desirable beam geometry with sufficient intensity inside the material. The conical-shaped refractive axicon is the central piece due to its ability to withstand high power of ultrashort laser pulses. It is well established that imperfections of conical shape, namely, the lens-like round tip at the apex and the curved shape of the conical surface, distort the Bessel pulses [19–22]. The influence of the axicon’s geometry imperfections is modelled and compared to the experimentally generated Bessel-Gaussian beam using an in-house precision-made axicon and a commercially available axicon. The difference in their performance is demonstrated.
After characterising the Bessel beam intensity distribution inside the sapphire sample at low intensity, we generated high-aspect ratio nanochannels using powerful single ultrashort laser pulses. The nanochannels were opened by focused ion beam milling and analysed with scanning electron microscopy. By comparing the nanochannel length with the measured intensity distribution we evaluated the intensity threshold required for the nanochannel formation. The presented analysis forms the base for explicit Bessel pulse design via combination of axicon angle and demagnifying collimator to achieve the threshold pulse intensity and to meet the required beam configuration. The analysis of the Bessel pulse geometry in materials with various refractive indices further expands their application. These practical considerations for the laser pulse energy, intensity distribution, energy concentration, and the beam configuration allow to produce nanochannels of a desired size and at a desired depth under the surface with ultrashort Bessel pulses in any transparent material with known optical properties. Moreover, most of the previous works on using Bessel beams formed by ultrashort laser pulses concentrated on glass processing and modification techniques such as cutting, drilling and formation of channels through the transparent sample and formation of 3D photonic structures [1,3,9,19,24–27]. Precise control and knowledge of the Bessel beams will be very attractive for laser applications in 3D-micromachinning and intra-volume modifications.
2. Modelling the Bessel-Gaussian beam
2.1 Gaussian beam propagation and Bessel-Gaussian beam formation
An axicon, a refractive element characterised by a conical shape, introduces a phase shift which transforms an incoming Gaussian beam into a zero-order Bessel-Gaussian beam [23]. Applying the Bessel-shaped needle-like beam concept to powerful ultrashort laser pulses for the modification of transparent materials requires careful consideration of the beam’s geometry in the bulk of the media and its intensity distribution to predict the length of the beam as well as the laser pulse power to reach the threshold intensity for the nanochannel formation. To accomplish this task, we used a two-fold approach to simulate the beam and predict the optimal parameters for experiments. First, scalar diffraction theory was used to simulate the wave propagation from the exit of the laser output until the interaction region inside the sapphire. This allowed inclusion of imperfections of the axicon. Second, we used analytical expressions from Jarutis et al. [28] for the estimation of Bessel beam intensity and important beam parameters, such as lateral diameter and axial length. These analytical expressions do not allow calculation of beam parameters in the case of imperfect axicons. Noteworthy is that the results from both approaches are very similar in the precision-made axicon case. Moreover, the numerical implementation of the analytical expression is faster compared to the scalar diffraction theory, as it is not needed to simulate the entire propagation train. Hereby, the analytical expressions are used to confirm and extend the results given by the scalar diffraction model.
The numerical simulations of intensity distribution are based on a free-space propagation method using Fourier Optics with phase shifts caused by diverse optical elements. The numerical analysis before the conical lens starts with the description of the incoming Gaussian beam at the entrance of the axicon. The complex amplitude ${U_g}$ of the beam at the laser output cross-section is given, in cylindrical coordinates, by:
Fig. 1. Formation of a Bessel beam from a Gaussian beam with waist w0 by an axicon with a wedge angle $\alpha_{\it 0}$ and re-imaging it into transparent media using a two-lens de-magnifying collimator. The plane wave front is refracted by the axicon, transmitting into a new beam, whose plane waves cover a surface of a cone with an angle β. The characteristic of the imperfect axicon’s lens-like oblate tip a (inset) is included to describe the intensity oscillations on its axial propagation (see Section 3.1 below). The beam is further re-imaged by a two-lens collimator into a bulk of sapphire crystal.
When the axicon is evenly illuminated, it refracts the incoming plane waves into wave that cover a conical surface with an angle $\beta $ relative to the optical axis. After passing the axicon the interfering wavefronts create an intensity profile described by the initial beam parameters as well as the axicon geometry and its refractive index.
The angle between the optical axis and the normal of the refracted wave front, $\beta $, is given by the axicon parameters for large axicon’s apex angle [20]:
2.2. Bessel beam propagation
To give a fully quantitative description of the Bessel beam axial propagation, it is required to provide expressions for the beam length $\Delta z$ at FWHM intensity level and the distance of maximum intensity of the beam from the axicon ${z_f}$, often termed as the focus of the beam (Fig. 1). Following the expressions which can be found in the work of Jarutis et al. [28], for the zero-order Bessel beam:
For the intensity characterisation of the beam, it is convenient to use the diameter of the Bessel beam at the FWHM, dFWHM, of its intensity distribution: where the numerical coefficients are from the characteristics of the first kind zero order Bessel function.An analytical expression for the zero-order Bessel like beam intensity can be obtained from the theoretical treatment from the work of Jarutis et al. [28] as follows:
2.3. Phase modulation produced by imperfect axicon
In this section we describe, following Refs. [19,20], the formation of the quasi-Bessel beam with imperfections of the axicon’s round-tip. In addition, we introduce the deviation of its conical surface away from the perfect cone by replacing it, for the sake of simplicity, with the radius of curvature of the front surface (Fig. 2). To derive an expression for the transmitted complex amplitude we treat the axicon as a thin optical element. Then its phase modulation can be described by finding its thickness function [30]. To do this, we split the axicon geometry in three sections, a curved surface with thickness function ${\Delta _1}$, a region with constant thickness ${\Delta _2}$ and conical surface with thickness ${\Delta _3}$, where the thickness on the optical axis will be ${\Delta _{convex}} = {\Delta _1} + {\Delta _2} + {\Delta _3}$ for a convex curved surface and ${\Delta _{concave}} ={-} {\Delta _1} + {\Delta _2} + {\Delta _3}$ for a concave surface, as shown in Fig. 2. Regardless the difference on the sum of the thickness, both cases have the same thickness function, given the convention of signs for the radius of curvature R. Now, considering $|R |\gg {\Delta _0}$, rays passing through the centre of the axicon do not observe a thickness difference between convex and concave surface, which is also what happens when the curvature is added in the conical surface.
Fig. 2. Schematics representation of the thickness function calculation for imperfect axicons with curved entrance surface. A radius of curvature $|R |\gg {w_0}$ of the first surface of the axicon is included in order to analyse the effect of the curved surface on the intensity distribution in the re-imaged Bessel beam.
The thickness function for an imperfect axicon is based on a modification of the expression derived by Brzobohaty et al. [20], assuming its conical surface as a hyperboloid of revolution of two sheets:
2.4. Re-imaging Bessel-Gaussian beam
The constructed Bessel-Gaussian beam is further reimaged using a 4F-collimator comprised by two lenses ${L_1}$ and ${L_2}$, with focal lengths ${f_1}$ and ${f_2}$, respectively, and separated by a distance ${d_2} = {f_1} + {f_2}$. Figure 3 shows the collimator setup, where an object plane at a distance ${d_1}$ from ${L_1}$ will be reimaged at an image plane located at a distance of ${d_3}$ from the second lens and demagnified by $M = {f_2}/{f_1}$, since ${f_1} > {f_2}$. This demagnification will have direct effect on the Bessel beam parameters, where the core radius of the reimaged beam will be $d{^{\prime}_{FWHM}} = M{d_{FWHM}}$, the length will be $\mathrm{\Delta }z^{\prime} = {M^2}\mathrm{\Delta }z$ and the intensity will be $I{^{\prime}_{ax}} = {I_{ax}}/{M^2}$. The purpose of demagnification is to reduce the central core volume, increase the intensity in the central core, and to increase the deposited energy density to the level sufficient for the optical breakdown and creating the micro-explosion conditions for formation of cylindrical cavity. The new angle $\beta ^{\prime}$ for the reimaged Bessel-Gaussian beam inside the material could be found from the relation $sin\beta ^{\prime} = ({sin\beta } )/M$.
Fig. 3. Imaging Bessel-Gaussian beam by a Keplerian telescope system comprised of two lenses ${L_1}$ and ${L_2}$ with focal lengths ${f_1}$ and ${f_2}$ respectively and separated by distance ${d_2} = {f_1} + {f_2}$.
The analytical considerations, described above, provide a simple method to characterize the re-imaged Bessel-Gaussian beam parameters. However, when we need a comparison with real axicons, distortion of the wave field due to morphology imperfections needs to be accounted. To do that, the propagation of the beam through the collimator can be modelled taking in account the phase modulation induced by each of the lenses and its free-space propagation (See Section 2.1) using scalar diffraction theory. Each of the two lenses in the imaging setup with focal length ${f_i}$ re-shapes the incoming beam adding a phase equal to $- ik{r^2}/2{f_i}$. The resulting complex amplitude just after the lens is obtained from:
where distances ${z_{Lj}}$ denotes the position of the j-th lens (j = 1, 2) relative to the previous optical element. Thus, the new lateral intensity distribution at a distance ${d_3}$ from ${L_2}$, reimaged from the object plane of a distance ${d_1}$ from lens ${L_1}$, can be numerically simulated taking in account the phase modulation of each lens (Eq. (13)) and a set of Fourier transforms to simulate the free-space propagation between optical elements in the same way as presented in Section 2.1. Finally, the numerically simulated 3-D micro-Bessel beam is obtained by stacking several cross-section distributions along a range of axial positions.Modelling of the beam re-imaged into media with refractive index ${n_m} > {n_0}\; ({n_0}$ = 1 in air) considers, according to Snell’s law, the changes of the wavelength and the angle between the refracted wavevector and the axis of propagation. The micro-Bessel beam formed in the media with refractive index ${n_m}$ will have a wavelength of ${\lambda _m} = {\lambda _0}/{n_m}$. This can be calculated using the wavevector Snell’s law $\vec{k}_m^x = \vec{k}_0^x$, where x denotes the projection of $\vec{k}$ on the interface boundary, from which the angle of propagation in the media, ${\beta _m}$ with refractive index ${n_m}$ is:
where $\beta ^{\prime}$ is the angle between the wavevector and the optical axis of the de-magnified beam in air after the second lens. Eq.(6) shows that the Bessel-Gaussian beam diameter $d_{FWHM}^m\; $ does not depend on the refractive index of the media:A good reference for the estimation of the peak intensity of the beam inside the sapphire sample can be taken from previous studies. Given that the increase of peak intensity in the central core $I_0^{BB}$, formed after the axicon, relative to the initial peak intensity of the Gaussian beam $I_0^{GB}$ can be evaluated as [28]:
3. Influence of axicon imperfections and focusing media refractive index
3.1 Effects of axicon geometrical imperfections on beam intensity
The following parameters are used to demonstrate the beam propagation through the optical system, and to compare the simulated results with the measured beam parameters formed in experimental setup presented in Fig. 1: λ = 1029 nm, w0 = 1.925 mm, the axicon of 25 mm in diameter and thickness Δ0 = 5.2 mm made of fused silica (nax = 1.4585 at 1029 nm) [31], and with the base angle α0 = 1.0°. The Bessel beam is reimaged with a de-magnifying collimator with positive lenses f1 = 200 mm and f2 = 3.0 mm, located at 203.0 mm apart.
The resulted intensity distribution in the re-imaged zero-order micro-Bessel beam is depicted below in Fig. 4 for the round-tip imperfection a varying from 0 µm (sharp round-tip) to 20 µm. It can be seen that with the increase of a, the intensity oscillations of the central core are growing, the length of the beam shrinks, and the whole beam is shifted away along the z-axis from the initial position. The reason for the modulation of intensity in z-direction is due to the different length of the k-vectors, the one created by the conical surface and another one formed by the round-tip acting as a lens [20].
Fig. 4. Calculated zero-order micro-Bessel beam propagation in air for different round-tip imperfection parameter a of the axicon. (a) – a = 0 (perfect axicon); (b) – a = 3 µm; (c) – a = 5 µm; (d) – a = 10 µm; and (e) – a = 20 µm. A plot with the central core axial intensity distribution for all cases is shown in (f) for comparison.
To simulate influence of the axicon’s imperfection due to bending of conical surface away from its perfect shape, the field curvature is included to calculate the propagation of the beam (Fig. 2). The results for the concave curvature ($|R |\gg {w_0}$) added to the plane surface of the axicon are shown for the ideal axicon (a = 0) in Fig. 5(a), and for an axicon with a = 10 µm – in Fig. 5(b). As can be seen in Fig. 5, both imperfections lead to shrinking the Bessel beam along the z-axis. This leads to an axially shrinking of the length of the beam, which can be clearly seen in both plots.
Fig. 5. (a). Central core axial intensity distribution for different concave curvatures from R = ∞ (perfect conical surface) to R = 0.5 m for an axicon with a sharp tip (a = 0 µm); and (b) – same as in (a) and including the rounded tip of $a = 10\; \mu m$.
3.2. Beam length focused into media with different refractive indices
Following Eqs. (4,14,16), the expected micro-Bessel beam axial lengths in the media with different refractive indices ${n_m}$ and re-imaged with the collimator in our experimental conditions are presented in Fig. 6, for axicon’s wedge angles of $0.3^\circ ,\,0.5^\circ $ and $1.0^\circ $. The micro-beam axial length increases linearly with ${n_m}$, and the length change is faster for smaller angle ${\alpha _0}$ of the axicon forming the beam.
Fig. 6. Example of the calculated dependence of length $\Delta {z_m}$ of the micro-Bessel beam formed by axicons with α0 = $0.5^\circ $ (red triangles), $1.0^\circ $ (green dots), and $1.5$° (blue crosses) and re-imaged by the de-magnifying telescope with M = 1:67 into media with various refractive indices ${n_m} \ge {n_{air}}$. The refractive indices at 1.029 µm are: ${n_{air}} = 1$, water nm = 1.32, lithium fluoride nm =1.39, fused silica nm = 1.45, sapphire nm = 1.755, diamond nm = 2.39 and chalcogenide glass As2S3, nm = 2.48 [32].
With these parameters in hand, we are now able to estimate the maximum intensity in the Bessel beam by following Eqs. (7–9) and taking the size of the beam at FWHM level of the 2.5 µJ, 275 fs laser pulse. The maximum intensity at the focal point of the central core of the Bessel beam in air is ∼9.13 × 1013 W/cm2. From the geometrical law of energy conservation and the Snell’s law, the power of laser pulse spreads out in sapphire in accordance with the Fresnel equations. Following Eq. (17) the reflectivity of sapphire at the angle of incidence of R = 0.076, thus the intensity in the focus of the central core of the Bessel beam inside the sapphire sample is 8.57 × 1013 W/cm2. Such level of intensity is sufficient to create nanochannels inside the sapphire crystal [7,8,18,19,25]. The changes in material parameters under the intense laser pulse due to optical breakdown, transition to plasma state and formation of nanochannels by the resulting shock wave, are considered in Section 4 below.
4. Experimental: Bessel beam formation and re-imaging
4.1. Fabrication and characterisation of a precision-made axicon
The refractive axicon of a conical shape is the central piece in the realisation of the Bessel beam for high-intensity microexplosion experiments in transparent media with high-power laser pulses. For our studies we fabricated a positive axicon using a diamond turning with ultra-precision lathe (Moore Nanotech UPL250). The workpiece, a 5-mm thick 1-inch diameter calcium fluoride (CaF2) from Thorlabs with a refractive index of 1.4280 at 1029 nm [31,32], was held on the spindle by a vacuum chuck. The cutting process consists of a series of roughing and fine finishing steps. The cut depth was set to 20 µm, with feed rate of 7.5 µm per revolution by using a 0.5 mm radius diamond cutter. For the finishing steps, the cut depth was reduced to 2 µm, with feed rate of 2.5 µm per revolution using a 0.6 mm radius special edged cutter. After each pass, the surface was monitored using an on-machine white-light interferometer until the form accuracy ∼0.1 µm and surface quality requirements λ/10 were obtained. The axicon was then analysed with a surface profiler Wyko from Veeco to characterise its geometry.
Figure 7 presents the 3D contours and the measured profile over the tip of the in-house made positive axicon and a commercially available (surface flatness λ/2 at 632 nm) positive axicon for comparison. The commercially available AR-coated positive axicon was made of UV fused silica with ${\alpha _0}$ = 1.0°, and its transverse profile is represented in blue in Fig. 7(c). The in-house made axicon was designed to have a positive apex angle of 178° with an optical quality of λ/10 for the 1029 nm ultrashort pulsed laser experiments. The high contour accuracy and the sharp tip with radius of curvature ∼0.6 µm demonstrate the high-precision of the fabrication process. The measured profiles were compared to an ideal cone with a 178° apex angle and the variation from the ideal tip position is defined as the parameter a. The distance from the actual profile to the cone surface was evaluated to obtain the shape error dependence on the transverse distance. The a-parameter of the in-house made axicon (red curve in Fig. 7(c)) was less than 1 µm (∼0.8 µm). Our in-house manufactured axicon is compared with a commercially available AR-coated positive axicon made of UV fused silica with ${\alpha _0}$ = 1.0°. Its transverse profile is represented in blue in Fig. 7(c). It is noticed a clear difference in the sharpness of the tips, the commercially available axicon presenting a more rounded tip with a larger a-parameter. Moreover, it was noticed that the slope was not perfectly straight, and bumps were observed along its profile, as it can be observed in the right section of its profile. The geometry of the axicon and particularly its tip are critical parameters of the axicon for constructing the micro-Bessel beam [20]. The experimental intensity axial profiles resulted from these axicons, and after passing through the demagnification collimator, were compared with the calculated from our numerical model. Figure 7(d) shows the comparison between the intensity on the axis formed by the commercial axicon (blue dotted line) and the simulated one for an axicon with R = -1 m and a = 10 µm (blue solid line) and the comparison between the intensity on the axis formed by the in-house made axicon (red dotted line) and the simulated one for an axicon with a = 10 µm, without curve surface (red solid line). In addition, it is shown the axial propagation for both case in the inset for the commercial axicon (blue contour line) and for the in-house made axicon (red contour line). From the comparison, we related the commercial axicon imperfection on the round tip and its conical surface to the intensity oscillation on its propagation. Moreover, for the in-house made axicon we reached a good agreement between the calculated and the measures profiles which will be discuss in detail in Section 4.2.
Fig. 7. Optical profilometer test results of axicon tips and intensity along the z-axis in the Bessel beams formed by the axicons and re-imaged by the telescope in air. (a) – 3D plot of the in-house made axicon; and (b) –for a commercially available axicon. (c) – related profiles in x-cross-section (solid lines) and y-cross-section (dashed lines) from the profiles shown in (a) and (b). (d) – calculated (solid lines) and measured (dotted lines) intensity profiles obtained with 2.5 µJ pulses for the in-house made (red curve) and the commercial (blue curve) axicons. Numerical modelling done for an axicon with a = 0 µm and without curved surface and with R = -1 m and a = 10 µm, for the in-house made and the commercial, respectively. The insets show the related 2D experimental profile of the beams formed by the axicons for comparison, the red-framed – by the in-house made and the blue-framed – by the commercial one.
4.2. Demagnified Bessel beam intensity in air
Bessel beam analysis is performed using a 40 W Carbide laser (CB3-40W, Light Conversion) with a pulse duration 275 fs. A Gaussian beam of $\lambda $ = 1029 nm with diameter 2w0 = 3.85 mm at the entrance of the in-house made axicon described earlier. The generated Bessel beam is then de-magnified using a telescopic setup consisting of a plano-convex lens with a focal length f1 = 200 mm and an infinity corrected microscope objective (Olympus UPLFLN 60x, focal length f2 = 3.0 mm) with numerical aperture NA= 0.9 and working distance WD = 0.2 mm. To characterize the Bessel beam, an imaging setup was inserted after the beam with an identical microscope objective as the one in the collimator setup and mounted on a motorized translation stage. The focal plane of this objective was observed using a plano-convex lens with a focal length of 150 mm and imaged on a CCD camera (Allied Vision Marlin). The images of the beam in transverse cross-sections were taken at a 1-µm steps and reconstructed using an in-house designed software based on LabView programming. It was noted that a deviation of the beam away from the center of the tip, thus away from the ideal conical surface, for just a few microns introduced significant changes in the intensity profile and the length of beam.
Figure 8 presents the intensity distribution in the micro-Bessel beam along the z-axis in x-z and y-z planes and the intensity cross-sections at selected z-positions. The beam images and its characteristics are presented in Table 1. The results demonstrate very close correlations between the simulated (see Section 3.1 and 4.2) and experimental data, justifying the near-perfect conical shape of the in-house made axicon for the formation of Bessel beam with a predictable length and smooth intensity distribution.
Fig. 8. Intensity distribution in micro-Bessel beam formed by the in-house made axicon and re-imaged by a collimator in air. (a) – intensity profiles in x-z and y-z planes; (b) – measured central core axial intensity profile (red dotted line) compared with the simulated intensity profile (solid line); (c) – central cross sections of the formed Bessel beam taken for different z positions. The intensity colours of the images are in arbitrary unit in each image. The related intensity, normalised for the highest value along the beam propagation, is represented in each image (in white) with the corresponding axis on the left.
Table 1. Computational and Experimental Results on Formation of the Gaussian-Bessel Beam with In-House Made Axicon and Re-Imaging in Air and in Sapphire; Beam Diameter at FWHM Intensity Level ${{\boldsymbol d}_{{\boldsymbol FWHM}}}$, Length at FWHM and 10% ($\Delta {{\boldsymbol z}_{10{\boldsymbol \%}}}$) Levels, and Magnification in Lateral Mx,y and Axial Mz Directions.
4.3. Intensity of the Bessel beam in sapphire
The above considerations allow us to calculate the beam parameters we expect to form in our experiments and compare those with the experimental results.
The experiments have been conducted with a Gaussian beam waist of w0 = 1.925 mm entering the in-house made axicon manufactured out of CaF2 with refractive index 1.458 at λ0 = 1.029 µm. The beam is further re-imaged with the two-lens collimator with magnification M = 67:1, described above. The angle of convergence of the rays after the collimator, which is the angle of incidence in sapphire, is $\beta ^{\prime}$ = 25.8°, and the image of the Bessel beam is $\mathrm{\Delta }z{^{\prime}_{FWHM}} \cong $ 57.3 µm long while the beam diameter $d{^{\prime}_{FWHM}}$ = 0.82 µm. Reimaging the beam into the sapphire sample changes the angle of the transmitted rays propagating inside the sample to ${\beta _m}$ = 14.83°, with the beam length increasing to $\Delta z_{FWHM}^m$ = 100.6 µm. The central core diameter stays unchanged inside the sample.
Intensity distribution is acquired inside the sapphire crystal using the 275 fs multiple pulses at low energy level, well below the threshold of any material modifications induced by laser. The sample is monocrystalline sapphire, c-cut, density of dsapph = 3.98 g/cm3 [31], double side polished with a thickness of 200 µm, and its position is controlled by motorised stages with sub-micron precision. Figure 9 shows the results of the low-energy measurements and computations for Bessel beam propagation inside the sapphire, and the beam lengths at FWHM and 10% of maximum intensity are presented in Table 1.
Fig. 9. Bessel beam re-imaged inside the sapphire sample with low-intensity beam and normalised to the simulated intensity produced with 2.5 µJ laser pulse. (a) – measured intensity of the beam in y-z plane; (b) – numerical simulations in y-z plane, (c) – measured (red dots) and simulated (blue solid line) axial intensity distribution in the central core of the Bessel beam. The peak intensity is 8.57 × 1013 W/cm2; the dashed lines indicate intensity level at 10%, at FWHM and at the channel formation threshold of ∼7.2 × 1013 W/cm2 deduced from the nanochannel length of ∼55 µm via focused ion beam milling (see next Section).
The direct measurement of intensity inside the sapphire sample at the pulse energy above the optical breakdown and plasma formation is a formidable task. However, close correlation between the measured and computed intensities at low pulse energy level, the level below any material modifications, gives us an opportunity to evaluate the intensity of the beam at the pulse energy above the optical breakdown. To evaluate the intensity distribution along the axis of the Bessel beam at high pulse energy we applied the intensity measurements at low energy and normalised the intensity distribution to the maximum computed intensity point at 2.5 µJ pulse energy used in experiments for void formation. The resulted intensity profile presented in Fig. 9(a) and Fig. 9(c), it accounts for the reflectance of light from the sapphire surface of ∼8% at the angle of incidence $\beta ^{\prime}$ = 25.8°. We note here that the pulse energy was measured just before the sapphire sample. The measured axial intensity distribution in the central core along the z-axis generated by the in-house made axicon coincide well with the numerical simulations, both in air (Fig. 7(c)) and in sapphire (Fig. 9(c)). We should note here that the full length of the Bessel beam depends on two independent parameters, whichever is larger. One is the distance determined by the pulse duration tp: ${l_{BB}} = ({{t_p} \times c/n} )\cos {\beta _m}$, where $c/n$ is the speed of laser pulse propagation in sapphire. The other is determined by the width of the conically shaped converging ring of laser energy, formed by the re-imaging collimator, which depends on the aperture of the beam and the reimaging optics.
The close correlation between the computations and the experimentally measured intensity distribution presented in Table 1 and Fig. 9(c), the corresponding peak intensity of the Bessel beam inside sapphire can be defined as ∼8.57 × 1013 W/cm2. We note that the observed deviation between the calculated and measured intensity in the falling slope of the intensity distribution, but that does not influence the maximum intensity at the peak of the distribution.
5. Nanochannels produced with ultrashort laser pulses in confined microexplosion
5.1. Experimental results
Nanochannels were produced inside the sapphire sample using single 275 fs, Bessel-shaped laser pulses with 2.5 µJ energy at 1,029 nm. The arrays of nanochannels were positioned 2-µm apart in normal to surface direction. After the laser processing, the structures were opened up by focused ion beam (FIB) milling (FEI Helios 600i) and observed by scanning electron microscopy (SEM), the results are presented in Figs. 10,11. All channels are formed inside the sapphire to guarantee the conservation of mass in these experiments.
Fig. 10. Scanning electron microscopy images of nanochannels in sapphire generated by single Bessel pulses of 2.5 µJ at 275 fs and opened with the focused ion beam. (a) – the array of nanochannel positioned 2 µm apart; (b) – a magnified image of a ∼33 µm long part of a 52 µm long channel demonstrating high-aspect ratio continues void produced with the near-perfectly shaped in-house made axicon; (c) – upper part of the nanochannel, the arrows indicate the diameter of the channel and the shock-wave affected area. The scale bars are 10 µm in (a) and 1 µm in (c).
Fig. 11. Nanochannel sizes opened with focused ion beam milling: (a) – top (second row) and side-view (first row) of arrays of the nanochannels imaged at the 52° tilted sample; (b) – magnified nanochannel cross-section with indication of size for the nanovoid and shock-wave affected areas; (c) – side-view of the nanochannel cross-section with indication of nanovoid and shock-wave affected areas. The scale bars are 1 µm in (a), 250 nm in (b) and 500 nm in (c).
Analysis of the SEM images show that the length nanochannel voids is in the range 52.0-56.5 µm. Each of the cylindrical void is surrounded by a shock-wave compressed area of ∼550 nm – 650 nm in diameter. It is noteworthy that the image is taken at 52° tilt of the sample, so that vertical scale is (sin52°)-1 = 1.27 times shorter than the horizontal one.
For better visualisation and measurements of the channel parameter and the surrounding compressed area we milled across the array of channels parallel to the channel axis – see Fig. 10. Magnifications of one of the channels are also presented (Figs. 10(a),10(c)). Figure 11 presents two rows of nanochannels focused on different position inside the crystal. The first row was opened by a parallel FIB milling to observe the channel over their length. The second row, on the top of the Figure, was laser focused on a higher level and the sample was cut open perpendicularly to the nanochannels, just above the beginning of the first row, to display the nanochannels transversally. The magnified channel cross-sections allowed us to measure accurately the channel diameter along with the diameter of the shock wave compressed area. The measurements show 200 µm – 250 µm diameter of the channels close to the top part and 250 µm – 270 µm at the middle of the channels where the laser intensity reaches maximum value.
5.2. Analysis of nanochannel formation
The process of ultrashort pulse induced microexplosion proceeds in several consecutive steps [4–6,13]. It starts with propagation of ultrashort laser pulses through the transparent media towards the beam axis and reaching high intensity at the axis where the Bessel-Gaussian beam formed. If intensity at the axis is above the ionisation threshold when sufficient number of valence electrons absorb multiple photons, the electrons are transferred from the valence to the conduction band, and intense incident laser light starts absorbing due to the inverse bremsstrahlung process. The ionization breakdown occurs at the early stage of the pulse duration and at the axis where the intensity is the highest. The solid matter at the high-intensity core of the Bessel-Gaussian beam transforms to a solid-density plasma with electron temperature reaching several tens of eV [4–6].
After the absorbed energy equilibrates between the electrons and ions, which typically takes a few picoseconds after the pulse, a strong shock wave emerges and starts propagating away from the axis. The shock wave compresses the material against the surrounding cold crystal and the following behind rarefaction wave forms a void [4–6,10,11]. While expanding, the shock wave loses energy, decelerates and finally stops and converts into a sound wave.
There are several important deductions that could be made from the observations of FIB-opened nanochannels in sapphire and from the comparisons between their measured parameters and the results of numerical simulations. First, the measured channel length of 52 µm – 56.5 µm set against the Bessel pulse intensity distribution in sapphire presented in Fig. 9(c) provides information on the intensity threshold of (7.2 ± 0.05) × 1013 W/cm2 for channel formation in cylindrical microexplosion with 1029 nm ultrashort laser pulse. When compared with the threshold of nanovoid formation of ∼1.1 × 1014 W/cm2 produced with the spherical microexplosion using Gaussian pulses [3], this threshold is ∼1.5 times lower. The intensity threshold of 7.2 × 1013 W/cm2 with 275 fs pulse duration results in the laser fluence of ∼20 J/cm2.
Furthermore, formation of a long nanovoid lv ≅ 55 µm with 2rv = 250 nm in diameter indicates that a volume of sapphire of ${V_v} = {l_v}r_v^2$ = 2.7 × 10−12 cm3 (∼1.1 × 10−11 g) was moved away from the axis and compressed into a circumjacent cylindrical shell against the solid crystal around the axis. The resulting void is about two orders of magnitude larger than voids generated by tightly focused Gaussian beams [4–6,13]. By analogy with the spherically induced microexplosion in sapphire [5], the condition of conservation of mass in confined microexplosion allows one to estimate the density of compressed sapphire. The diameter of a cylinder surrounding the channel of ∼650 nm is an indication of the distance of expansion of the shock wave away from the axis, where the expanding shock wave pressure is reduced to the Young’s modulus of sapphire, Ysapph ≅ 4 × 105 J/cm3 = 400 GPa, and the shock wave converts into an acoustic wave. Taking the volume of a shock-wave affected cylinder Vsw with 2rsw = (600–650) nm in diameter, the corresponding volume of compressed sapphire is ${V_{sw}} - {V_v} = {l_v}\pi ({r_{sw}^2 - r_v^2} )\; = {l_v}\pi ({r_{sw}^2 - r_v^2} )\; $ = (12.85–15.55) µm3. Following these geometry reasonings and applying the conservation of mass, the density of compressed sapphire in cylindrical microexplosion is ${V_{sw}}/({{V_{sw}} - {V_v}} )$ = 1.17–1.21 times higher than the crystalline sapphire density of 3.98 g/cm3, it is in the range of (4.66-4.82) g/cm3. This value is higher than ×1.14 density increase deduced from the spherical shock wave expansion in confined microexplosion experiments with Gaussian pulses [5].
Let us estimate the absorbed energy density and the maximum shock wave pressure to form nanochannels in our experiments by following the same logic as presented for spherical microexplosion [4–6]. To form a void in solids, the material from the void should be expelled by the energy absorbed in the void volume. The required amount of work is Ysapph×Vv, as the Young’s bulk modulus, the ratio of stress to a corresponding strain, is a measure of material stiffness or rigidity, the strength the material. Taking Ysapph = 4×105 J/cm3 and Vv = 2.7 × 10−12 cm3 this energy comprises Eabs = 1.08 μJ, which is the evidence of strong, about 45%, absorption of the 2.5 µJ total pulse energy concentrated in the central spike on the axis. By taking the average length of the channels as 55 µm and the absorption depth (skin layer) in sapphire plasma as ls = 65 nm [5], the volume where the pulse energy is absorbed is ${V_{abs}} = {l_{v\; }}\pi l_s^2 = \; $0.73 × 10−12 cm3, and the absorbed energy density is ${E_{abs}}/{V_{abs}}$ ≅ 1.5 MJ/cm3. This deposited laser energy is several times higher than the Young modulus of sapphire of 0.4 MJ/cm3, indicating that the maximum pressure as high as 1.5 TPa can be achieved in the presented experiments on cylindrical microexplosion in confined conditions.
5.3 Reimaging Bessel-Gaussian beam with ‘zoom’ collimator
Re-imaging of the Bessel-Gaussian beam with variable demagnification provides high degree of flexibility in the beam parameters, in addition to variation on axicon geometry, laser pulse energy and the Gaussian beam waist at the axicon entrance. Demagnifying collimator offers the way of optimisation of the beam geometry and intensity on the beam axis inside the transparent sample for most effective channel formation with desirable length using available pulse power. Below we demonstrate the results of calculations of Bessel-Gaussian beam formation with the fixed waist w0 = 1.925 mm of the Gaussian beam at the axicon entrance, λ = 1029 nm, and the axicon with the base angle α0 = 1.0°; all are the same parameters as in Section 3.1 earlier in the paper. The derived intensity threshold for the formation of nanochannels inside sapphire provides a valuable information of the required collimation and pulse energy necessary to produce nano-and micro-voids with desirable length and size.
Figure 12 shows several examples of the influence of demagnification on intensity and the Bessel beam geometry inside sapphire. Figure 12(a) shows peak intensity of the beam depending on the level of demagnification for various pulse energies. The dashed line denotes the intensity threshold to produce nanochannels in the sapphire sample at the intensity level of 7.2 × 1013 W/cm2. From the curve intersection with the threshold level, the required pulse energy at different demagnification can be determined to reach the nanochannel formation threshold inside sapphire.
Fig. 12. (a) - Effect of the collimator demagnification on the re-imaged beam peak intensity inside sapphire for different pulse energies. The dashed line denotes the intensity threshold of 7.2´1013 W/cm2 to produce nanochannels inside the sapphire sample. (b) – Collimator’s demagnification influence on the length and diameter of the central core of the re-imaged Bessel-Gaussian beam.
The geometry of the beam, i.e. the beam length and its diameter at different demagnification of the re-imaging collimator, are presented in Fig. 12(b). A word of caution is required here about the presented length of the Bessel beam shown here. Theoretical studies of the intense Bessel-Gaussian beam in transparent media at high intensity indicate that the originally stable diffraction-free beam may become unstable due to the development of strong ionization nonlinearity, Kerr instability or non-linear four-wave mixing [18,22,23,33,34]. These possible effects in sapphire, have not been studied to our knowledge, and have to be accounted for long, more than tens of mm beam length and at high beam intensities.
6. Conclusions
We presented here the numerical simulations and experimental results of Bessel-Gauss beams formed by an axicon and reimaged inside a sapphire sample with a 4F-optical collimator. we demonstrated, both with numerical modelling and experimentally, that the axicon’s shape is of crucial importance for the formation of cylindrical voids inside the material.
Thoroughly analysed beam characteristics and close correlation between numerical predictions and experimentally constructed beam using an in-house axicon with precise conical shape allowed us, for the first time in our knowledge, to determine the nanochannel formation intensity threshold of 7.2 × 1013 W/cm2 in sapphire. The analysis of the nanovoids diameter and their length, together with the radius of the shock-wave affected material indicates that the pressure of the compressed sapphire is within the range 1.17–1.21 times the pristine sapphire crystal. Applying the energy and mass conservation laws allowed us to evaluate the energy concentration up to 1.5×106 J/cm3. A pressure as high as 1.5 TPa, which is much higher than the strength of any material, was generated by a single Bessel-shaped ultrashort laser pulse with 2.5 µJ energy. The volume of the produced nanochannels in sapphire is almost two orders of magnitude higher than that made by tightly-focused Gaussian pulses [4–6,10,11,13].
The numerical simulations and experimental observation give compelling evidence that cylindrical microexplosion in confined geometry using Bessel-Gaussian beams is the way to achieve higher energy concentration and significantly larger amount of high-pressure material phase than that produced with Gaussian ultrashort laser pulses [10–16].
The numerical simulations developed in this study allow to gain knowledge of the parameters of the Bessel beam and the formation of channels through transparent materials, giving precision and better control for ultrashort laser micromachining applications, which requires smooth and homogeneous intensity distribution along the Bessel beam propagation and control over the length of the channels, such as cutting, drilling, texturizing or intra-volume modifications.
Funding
Australian Research Council (DP170100131); Agencia Nacional de Investigación y Desarrollo (72170255).
Acknowledgements
The authors acknowledge many fruitful discussions with E. G. Gamaly. This research was supported by the Australian Government through the Australian Research Discovery Project funding scheme (Project DP170100131). This work was performed in part at the ANFF ACT node of the Australian National Fabrication Facility, a company established under the National Collaborative Research Infrastructure Strategy to provide nano- and micro-fabrication facilities for Australia’s researchers. S.L-V. acknowledges support by Chilean Government trough Becas Chile ANID 72170255 PhD fellowship.
Disclosures
The Authors declare no conflicts of interests.
L.R. and A.V.R initiated the project and planned the experiments. L.R. designed and constructed the experimental setup and performed the microexplosion experiments. K.Y. and S.M. designed and produced home-made axicons. S.M. developed the software for beam images acquisition. L.R. performed the ion beam milling and SEM measurements. S.L-V. and M.P. developed the numerical simulations. L.R., A.V.R. and S.L-V. analysed the data and wrote the manuscript, with input from all authors. All authors discussed the results and contributed to the manuscript preparation.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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