1. Introduction

When ultrashort laser pulses with enough peak power propagate in a nonlinear medium, the interplay between self-focusing, natural diffraction and plasma divergence will give place to a long self-guided structure called filament [1] that spans for distances longer than the Rayleigh length [2]. Its potential applications include the creation of channels for guided transmission of microwave radiation in the atmosphere [3–5], the control of lightning in storm clouds [6, 7] or increasing the speed of techniques available for the fabrication of two- and three-dimensional photonic devices inside a glass [8–11].

Track writing in transparent media is realized by multiphoton absorption that seeds avalanche ionization, eventually resulting in permanent modification of the index of refraction. The characteristics of the structure produced can be manipulated by selecting the appropriate laser and focusing parameters [12]. A popular method of track writing consists of tightly focusing femtosecond pulses through the side of a sample mounted on a 3D translation stage that moves at a controlled speed. Alternatively, the fabrication of good quality waveguides and couplers can be also achieved by a technique called parallel writing [13], which consists of irradiating for a longer period of time very intense loosely focused pulses that form filaments in glass and produce circular micrometer sized regions of altered index of refraction [11,14].

Control of the multiple filamentation (MF), consequently, allows for a considerable speed up of the above mentioned fabrication process. However, it is widely known that by increasing the laser power the beam eventually undergoes a break-up into multiple narrow filaments that are spatially distributed according to random noise in the beam profile. The success of parallel writing by using MF depends critically on the effective control of the position, the number and the length of the filaments during the total exposition time, which cannot be realized automatically by only increasing laser power. Deterministic MF has been realized through modification of the spatial profile into an elliptical beam [15, 16], adjustment of the numerical aperture [17], interference between two replicas of the laser beam [18] introduction of a phase mask [19, 20] and amplitude modulation with a periodic mesh [11, 21]. In particular, periodic elements are well known to introduce a diffraction pattern that modulates the MF distribution and varies along the propagation axis [11, 21–23]. Several studies [24–26] have chosen an array of microlenses to generate MF in solids and liquids. This is an attractive optical element that both provides a high-efficiency collection of light [27] and generates diffraction patterns typical of periodic components, which enable the formation of many filaments per lenslet. Liu et al. [23] observed that an amplitude mesh is preferable over a microlens array because of the strong divergence ultimately introduced by the lenslets to the beam. In our experiment parallel MF is realized outside the focal region by controlling the transverse energy modulation induced by a microlens array. We overcome the disadvantage pointed out by [23] by means of a loosely focusing geometry and obtain MF patterns of comparative length and filaments number. We also demonstrate a strong dependence of the MF distribution on the incident wavelength and sample-to-lens distance as well as a direct effect of pulse energy and polarization on the number of filaments.

2. Experimental setup

A schematic of our setup is sketched in Fig. 1 . The laser source is a commercial system (Libra, Coherent Inc.), consisting of the pulses from a Ti: Sapphire oscillator, amplified through a CPA system that provides linearly polarized 50-fs pulses centered on 800 nm, at a 1 kHz repetition rate The laser power instability from shot to shot is less than 0.3%. By using a beam splitter and optical density filters, we can control the input at fundamental frequency to a maximum of 480 µJ per pulse. An iris opened 11 mm is placed before the focusing element. We have adjusted a quarter-wave plate for nearly circular polarization and used a switching mount to easily change the incident polarization. A BBO crystal and a color filter can be added to generate the second harmonic and filter out the fundamental light. A quarter-wave plate for 400 nm light was also used in order to change the polarization of the second harmonic into circular.

 

Fig. 1 Sketch of the experimental setup. F, filter; WP, quarter-wave plate; PM, power meter; MLA, multilens array; FSB, fused silica block; L, imaging lens; S, screen. The inset Fig. is a representation of the commercial microlens array utilized.

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The focusing element is a commercial 10 × 10 mm array of fused silica parabolic plano-convex microlenses of focal 218.3 mm and pitch 1.015 mm arranged in a square geometry (Edmund Optics, part number #64-487). The radius of curvature of the single lenslets that form the element is 100 mm as provided by manufacturer, and the thickness of the element is 1.20 mm. Controlled MF is achieved inside a 20-mm-long fused silica sample mounted in a translation stage that enables us to control its distance from the lens array. The operating laser energy and focusing geometry were chosen in order to avoid permanent modifications in the sample. Behind the silica, by using an f = 50.8 mm lens (L) we image the MF pattern on a white screen located at a distance of 5.02 m from the microlens array, and take pictures of it using a digital camera. Such lens L can be moved in the direction of the z-axis to select the filamentation pattern in different transversal planes along the filaments propagation inside silica.

3. Results and discussion

When a microlens array is used to modulate the MF distribution in a solid sample placed in the focal region, a single filament per lenslet is formed [25]. In Fig. 2 we show two typical transverse profiles of the fundamental and second harmonic imaged on the screen by lens L1 respectively. Observe that although the array contains a total of 100 lenslets, due to the Gaussian profile of the beam only the central ones carry enough power to generate a filament.

 

Fig. 2 Typical MF patterns formed by (a) 800-nm and (b) 400-nm laser pulses when the sample is placed in the focal region.

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However, this is not always the case, and many filaments per lenslet can be generated if the sample is moved away from the focal region. Figure 3 represents four typical cross-sectional pictures of the MF distributions at different propagation distances inside the fused silica, when the sample was placed before the focus, at a fixed distance of z = 144 mm from the lens array. The pulse energy was set to 315 μJ for the IR pulse and 80 μJ for the second harmonic, which correspond to about the same multiple of the critical power for self-focusing in fused silica. Figures 3(a)–3(d) correspond to incident fundamental frequency and Figs. 3(e)–3(h) to incidence of the second harmonic. In both cases each of the four patterns imaged on the screen were obtained by carefully moving the imaging lens L along the z-axis to select the desired object plane inside the 20-mm fused silica block. For a better intelligibility, only the 9 central filament regions were selected and the distance between their centers is equal to the pitch of the lens array, which is about 1 mm.

 

Fig. 3 Filament patterns at different laser propagation distances inside the fused silica block formed from incidence of fundamental (a-e) and second harmonic laser (e-h). The images are taken at (a, e) 10 mm, (b, f) 12 mm, (c, g) 15 mm and (d, h) 18 mm from the front surface of the sample respectively.

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For both wavelengths we observe the formation of several filaments per lenslet in different distributions. The filaments propagate for a long distance inside the silica, preserving their pattern until the end of the sample. The nonlinearity of the medium is the dominating mechanism once the beam enters the sample, allowing the MF pattern to be conserved, although the effect of linear diffraction continues to affect the beam along its subsequent propagation. The images corresponding to fundamental laser [Figs. 3(a)–3(d)] depict formation of three filaments per lenslet, where the central one carries most of the energy. However, at the same sample position, the filament distribution formed by the second harmonic is totally different [Figs. 3(e)–3(h)], manifesting four filaments per lenslet. The reason for such dissimilarity between the two MF distributions is the wavelength-dependence of the diffraction pattern induced by the microlens array on the entrance face of the sample. Such diffraction is what determines in which positions intense spots will favor nonlinear buildup and therefore filament formation.

It is known that the energy distribution of the incident laser pulse on the entrance face of the sample determines the positions where filaments appear [11, 17–23]. By adjusting the distance of the sample from the focusing element, the MF can be modified into completely different distributions. We have imaged the MF patterns at three different lens-to-sample distances as shown in Fig. 4 . The pulse energy of 480 μJ and 110 μJ for fundamental and second harmonic are used respectively. Figures 4 (a)–4(b) shows the patterns corresponding to the fundamental laser case while Figs. 4(c)–4(e) corresponds to the second harmonic case. Notice how changing the position of the sample from z = 144 mm [Fig. 4(a)] to z = 180 mm [Fig. 4(b)] leads to a drastic transformation of the distribution and number of filaments obtained from fundamental laser light. Filament patterns from incidence of second harmonic also suffer a radical modification when the sample is moved from 144 mm [Fig. 4(c)] to 180 mm [Fig. 4(e)]. However, it can be seen that in the case of second harmonic the filaments formed are more stable and better organized than those appearing from diffraction of the fundamental laser frequency. In addition, as seen from the Fig. 4, the filaments obtained from incidence of second harmonic appear smaller (diameter in the range of 45 – 50 μm) than those formed by the fundamental laser (diameter in the range of 50 – 90 μm).

 

Fig. 4 MF dependence on the relative position of the sample. Filament patterns from the fundamental laser case are depicted on (a-b) while patterns from the second harmonic case in (c-e). Images (a) and (c) correspond to sample position z = 144 mm, image (d) to z = 164 mm and images (b) and (e) to z = 180 mm.

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In order to provide an explanation to this effect, we have performed simulations of the diffraction-pattern’s evolution along the z-axis for three different wavelengths, which allow us to predict the diffraction that determines the MF distributions and contrast them to the experimental results. The simulation was carried out using the ray tracing software Zemax, considering a 10-mm Gaussian beam that passes through a lens array which has the same configuration with that in our experiments, and propagates through the same distance used in our experiments without including nonlinear effects. It was performed for 800-nm, 400-nm and 248-nm light at 5 different sample positions. The results are shown in Fig. 5 , in which each image represents a 1 × 1mm diffraction pattern at the entrance face of the sample. The first row of images in Fig. 5 represent the evolution of the pattern that corresponds to incidence of fundamental light when the sample is moved from position z = 144 mm [Fig. 5(a)] to position z = 180 mm [Fig. 5(e)]. Figures 5(f)–5(j) and 5(k)–5(o) represent the evolution over the same region for the wavelengths 400 nm and 248 nm respectively.

 

Fig. 5 Zemax simulations of the diffraction pattern at the entrance face of the sample for three different wavelengths at five different sample positions. Images (a-e) represent the central spot for 800-nm, (f-j) for 400 nm and (k-o) for 248 nm. Columns from left to right represent sample position at respectively z = 144 mm, 155 mm, 164 mm, 172 mm and the rightmost at 180 mm.

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The simulations depicted in Fig. 5 confirm that the experimental MF distributions shown in Figs. 3 and 4 are indeed determined by the diffraction patterns on the entrance face of the sample. We observe that diffraction from second harmonic [Figs. 5(f) –5(j)] provides intensity patterns that benefit the appearance of more filaments in better organized structures compared to diffraction from fundamental light [Figs. 5(a)–5(e)], in agreement with the experimental results shown in Fig. 4. Also, for all wavelengths the diffraction at distances closer to the lens array (not shown) forms patterns where the energy is distributed over a higher number of intense spots within a bigger area, preventing the diffraction pattern from generating filaments when the sample is too far from the focus. Several studies [11, 21] have attributed the characteristic patterns induced by meshes, gratings and arrays to the Talbot effect [28], which states that such distributions are interference patterns appearing from overlapping copies of the periodic element. Talbot theory predicts modifications and periodic revivals of the resulting pattern as the sample is moved to different fractions of a distance Z_T [28]

Another important feature of the pattern’s evolution is that for all wavelengths the spot size tends to be smaller as the sample approaches the geometric focus of the microlenses. Interestingly, this fact causes that the patterns in Figs. 5(l), 5(n) and 5(o) that correspond to diffraction of 248 nm light are miniaturized versions of those in Figs. 5 (f), 5(h) and 5(i) that correspond to diffraction of light at 400 nm. Such reproduction of the same structure in a smaller scale suggests that by properly tuning the input wavelength and sample position, the distance between individual filaments formed by each lenslet can be regulated. The condensation of the pattern when the sample approaches the focus can lead to filament interaction that originates chaotic distributions like the one appearing in Fig. 4(b) and reduce the filament length.

Besides, some other variables that enable control of the filament number in the MF pattern are the pulse energy and polarization. In Fig. 4(c), where the energy of the second harmonic pulses is higher than that in Figs. 3(e)–3(h), we observe more filaments, although both cases correspond to the same wavelength at the same sample position. However, observe that the fundamental structure of the pattern was not notably affected by the energy. A very clear picture of this effect is shown in Fig. 6 where we irradiate with four different pulse energies fundamental 800-nm pulses on the sample fixed at z = 144 mm. Notice that more filaments corresponding to the position of the higher order diffraction of laser beam appear when the incident energy is increased.

 

Fig. 6 Effect of input energy on the number of filaments formed per lenslet. Pictures (a–d) represent respectively energies of 250 µJ, 300 µJ, 370 µJ and 480 µJ.

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In the experiment we rotated laser polarization from linear polarization (LP) to circular polarization (CP). The filament pattern is as shown in Fig. 7 , which is obtained for 800-nm input pulses of 481 µJ at z = 144 mm. It is found that a MF pattern with a higher number of filaments is obtained under the incidence of LP pulses than that of CP pulses. This is mainly because the critical power for circular polarization (CP) is greater than that for linear polarization (LP) [29]. Figures 6 and 7 confirm the fundamental structure of the pattern is independent from laser energy and polarization, whose effect is only a stronger or weaker completion of the diffraction pattern per lenslet that induces the filaments.

 

Fig. 7 Effect of laser polarization in the number of filaments formed per lenslet. Picture (a) represents LP and (b) CP.

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4. Conclusions

We have shown deterministic control of multifilamentation induced by the diffraction pattern from an array of microlenses by changing the incident wavelength and the sample’s position with respect to the focusing element. By simply moving the sample along the z-axis, different diffraction patterns at its entrance face modulate the MF distribution, which is preserved along its nonlinear propagation. Energy and polarization have shown to influence the total number of filaments but not the fundamental structure of the pattern. By choosing a phase mask and a loose focusing geometry (f-number = 215 for each lenslet) we achieve both high efficiency and long filaments in controllable distributions. Diffraction of second harmonic light proved to enable in general more organized filament patterns per lenslet than those from incidence of fundamental light. There is a growing interest in studying higher photon energy sources for applications of filamentation in glass [30, 31], so the superiority of blue and UV filaments proved in experiments and suggested by simulations is opportune in terms of their characteristics in the processes of index modification and track writing in transparent media.