Linear diffraction was used to modulate intensity distribution across the femtosecond laser beam to create quasi regular arrays of filaments in fused silica. A fringe type of filament distributions (filament-fringe) were formed that could be controlled and observed over a distance of several millimeters. The difference of supercontinuum (SC) emission between individual filaments was also observed.
©2011 Optical Society of America
Refractive index modification in transparent dielectric materials by femtosecond laser pulses has been a topic of extensive research over the past years (see, e.g., [1–7]). Such laser induced modification has been used for fabrication of miniature Bragg gratings , waveguides [2,9,10] or even arrays of waveguides , beam splitters  and Photonic Crystal Fibers (PCFs) , however, manufacturing of these is rather labor-intensive and typically requires shifting of the laser’s spot position inside the substrate in two mutually perpendicular x-, y- directions and its successive translation along the z-direction .
Alternatively, under comparatively loose geometrical focusing condition femtosecond laser pulses can form electron plasma strings that extend far beyond the focal spot, have constant diameters over many Rayleigh lengths of the laser and their intensity is to a large extent not affected by fluctuations of the laser pulse energy. That is well-known as femtosecond filamentation. Compared to the case of tight focusing, the filamentation-based inscription of photonic devices is more precise and requires less mechanical manipulation. Furthermore, compared to the cases of longer pulses, filamentary structures in transparent solids of the femtosecond laser pulses have much less collateral damage to the substrate which mainly results from heating by picosecond or longer laser beam leading to a melt of some volume of the material, and a better controllability by, for example, controlling the initial pulse chirp, beam profiles, focusing conditions.
However, the filamentation is initiated by modulational instability of the beam and in most of the cases leads to random distribution of plasma channels across the beam profile. In order to meet requirements of applications, different methods have been proposed to organize the filamentation. For example, O. G. Kosareva et al. created an array of filaments in fused silica by using a periodic mesh . Input beam ellipticity has also been proved to be able to induce periodic filament arrays [15,16]. Our previous work has shown that the linear diffraction can be used to seed modulation instability across a supercritical laser beam and force its multifilamentary break-up in air  and J. P. Berube et al. demonstrated this in fused silica by using a transmission phase mask . By using a variable circular diaphragm around the plasma zone to control the filaments background  or in the laser beam initially to block a wide outer part of the beam , stronger and longer filaments were obtained in air. In Refs [20–22], the diffraction effect was used to induce regular filaments distribution and supercontnuum generation. Here, we go further to use a similar approach to create regular and controllable arrangements of filaments inside a fused silica sample. We obtained filament-fringes with different shapes, by simply adjusting the size of diaphragms.
2. Experimental setup
The experiments were performed in a fused silica block by using a femtosecond chirped pulse amplification (CPA) laser system to generate filaments. The laser was operating at 800 nm, a repetition rate of 1 kHz and had a pulse duration of 120 fs. The schematic layout of the setup is shown in Fig. 1 . After transmitted through a controllable diaphragm (a circular hole or a slit), the laser beam was focused by a lens with a focal length f = 200 mm onto the entrance face of the block. The distances between the diaphragm and the lens, the lens and the entrance surface of the block were 180 mm and 170 mm respectively. The block has a length of 25 mm and was situated in such a way that the geometrical focus was behind the output surface of it. In our experiments, 190 µJ per pulse before the diaphragm was used. Note that this laser power of 1.58 GW is much lower than the critical power for self-focusing in air which is more than 5 GW for 800 nm laser pulses , however it is high enough for self-focusing in fused silica. Take the energy 15 µJ after a 2 mm circular hole as an example. The laser power is 125 MW which is 70 times larger than the critical power for self-focusing in fused silica (~1.82 MW), not to mention that in bight fringes induced by the diffraction the peak power is even much higher. Filamentation in the block was imaged onto a white screen located at 7.85 m away from the block by means of an imaging lens (f = 75 mm) with a 103 × magnification. A digital camera and a CCD were used respectively to register the image of the beam pattern on the white screen, and monitor the filamentation in the bulk of fused silica, as indicated in the figure. A green filter (BG39) was used before the imaging lens, and a 0° incidence 800 nm high reflecting (HR) dielectric mirror was placed before the CCD to reject the fundamental laser light.
3. Experimental results and discussion
The digital camera image shown in Fig. 2 shows typical filament distribution in transverse plane, when no diaphragm was used. Around 30 filaments were formed at this position and, as clearly seen on the image, they were distributed randomly. However, when a diaphragm was introduced into the beam the distribution changed immediately. As shown in Fig. 3 and Fig. 6(c) , the filaments became to be organized into very regular patterns when different kinds of diaphragms were used. For a circular hole as the diaphragm, the number and the rings of filaments in the block can be controlled by adjusting its diameter. When the diameter was relative small, there was only single filament with multiple refocusing produced in the bulk of fused silica as shown in Fig. 5(a) . The x-coordinate of the figure is just a distance indicator to the filamentation and does not mean the real distance. The single filament propagated up to 6 mm of distance and the refocusing length (peak to peak) was about from 0.6 mm up to 2 mm, which is similar to the observation in transparent condensed media, for example BaF2 crystal and ZK7 glass [24,25], and one order of magnitude shorter than that in liquid . The differences in liquid should be due to different laser parameters, experimental setup and nonlinear optical medium used. This method to generate single filament will be a very convenient way to study some effects, for example spectral broadening versus filament length and intensity, or conical emission interference from one filament . As the hole was increased from 3 mm to 7 mm, more and more interference fringes and a larger number of filaments occurred as concentring rings within the beam profile. Finally, when the hole diameter was increased to 9 mm, comparable to the size of our laser beam, which has a 1/e2 diameter of 6 mm, the hole had no observable influence on the filamentation. As shown in the Fig. 2 and the most right image of the Fig. 3(a), the two filament patterns are quite similar.
We observed the similar effect of filamentation organization when the circular hole was replaced by a slit (Fig. 3(b)). In this case the filaments were assigned in line, and the exact number of lines (one up to six) could be controlled by changing the slit width. Furthermore, when a square aperture was introduced, a matrix-type filaments distribution was achieved (Fig. 6(c)). Besides, the size of diaphragm determines the amount of transmitted laser energy into the fused silica sample, leading to a difference of number of generated filaments. As shown in Fig. 4 , the filaments number changed as increasing the diameter of the circular hole. The error of the number is from the uncertainty in the filaments counting. It should be noted that because the filaments number was counted only from a transverse slice of laser beam, it does not mean the total number of filaments generated in the fused silica sample (Fig. 5(b)). It is well known that the number of filaments should approximate linearly increase as the increase of input laser power assuming that the power is well above critical . However, the filaments became reduced on this plane when the diaphragm was relative big (Fig. 4). It indicates that the diaphragm had influences on the filamentation not only transversely but also longitudinally, which is qualitatively in good agreement with filamentation organization in air by using the similar diaphragm [18,19].
In geometrical optics, the light linearly diffracted by a circular hole, slit or other apertures, as we know, will form a series of bright and dark fringes. This rule applies as we arrange the laser filaments in our fused silica sample into rings, lines and other shapes as well. As soon as there are bright fringes in the laser beam that concentrate more laser energy the nonlinear Kerr response of the material is becoming stronger and consequently the filamentary breakup of the beam will be likely to occur exactly at this position. As a result, the bright laser fringes induced linearly by apertures in air are transformed into a nonlinear type of fringes in fused silica which are constituted by filaments, which we call filament-fringes. As increasing or decreasing the diameter of the circular hole, we observed that in the center of the laser beam filaments appeared and then gone alternately, and simultaneously the surrounding filament-fringes enlarged or reduced their diameters continuously (see supplementary multimedia files: Media 1, for the dynamic changes. In the Media 1, the diameter of the circular hole was changed continuously from 1.5 mm to 9 mm.). It is qualitatively coincident with radially symmetric Fresnel patterns caused by circular apertures in geometrical optics. The slit also induced similar phenomena in our experiments with what the linear light diffraction by a slit have (see supplementary multimedia files: Media 2, in which the width of the slit was changed continuously from 1.5 mm to 4.5 mm.).
However, when the aperture diameter was relative large and comparable to the laser beam diameter, filaments in the center part of the beam became disordered and no more filament-fringes were formed (see the most right two images of Fig. 3(a)). In this case, the diffraction-induced fringes in the beam center were not strong enough in contrast to the initial Gaussian-like power distribution of laser beam. As a result, the diffraction has so weak influence on the self-focusing and thus filamentation that the filament-fringe has no chance to occur in the center part. In our experiments, 4 of maximum regular and distinct filament-fringes were formed by using a 6 mm circular hole and 6 filament-fringes by a 4.5 mm slit. Besides, the spacing of neighboring filaments was typically several tens of micrometers, but in each fringe filaments were not uniformly distributed resulting from the initial energy distribution of our laser beam and nonlinear filaments competitive interactions in the fused silica block.
We further investigated the filaments evolution in the bulk of fused silica along the laser propagation direction by adjusting the imaging lens. Figure 6 show the evolutions when a 5 mm circularly, a 3 mm slit and a 3 × 3 mm square shaped apertures were introduced respectively into the beam. Note that the influence of the refraction occurred through the interface of the block and air on the object distance of the imaging has already considered in the calculation of the object position. It is shown that in the multiple-filament regime (Fig. 5(b) and Fig. 6), most of the filaments could not propagate longer than 1 mm. However, in all three cases in the Fig. 6 the shapes of filament-fringes did not collapse but survived several millimeters, although some filaments terminated and some new ones formed inside the fringes, which is one of advantages of the filament-fringe for applying to fabrication of micro-optical devices. In the case of the circular aperture, three concentric filament-fringes formed and propagated up to 2 mm. However, the diameter of the fringes was getting slightly smaller and smaller along the propagation direction, which is probably harmful to many waveguides fabrication. The reason is that in the experiment a focused laser beam was used into the fused silica sample. One can expect that by using this technique a collimated laser beam will generate parallel propagating filament-fringes. On the other hand, as shown in Fig. 6(a) the inner filament-fringes came into being earlier and correspondingly they faded away also earlier than the outer fringes, which attributes to the initial Gaussian power distribution of the laser beam which has a higher energy or power in the inner part of the beam than that in the outer part. One can use some laser phase modulator or adaptive optics to control the filamentation and thus the starting position of filament-fringes. In the case of the slit aperture, three distinct lines of filaments propagated up to 3 mm. The number of filament-fringes did not change during the propagation. The fringes were parallel and their spacing was keeping at 90-100 µm. By using a 3 × 3 mm square aperture, a matrix doted by filaments was formed and propagated more than 2 mm, thus forming a kind of volume grating .
Note that in our experiment the filaments had their own color. We measured the spectra of some filaments by using a fiber spectrometer (BLUE-Wave, VIS-25, StellarNet Inc.). We replaced the biconvex imaging lens by an achromatic lens in the spectra measurements to reduce possible optical aberrations, although we did not observed obvious differences of filaments images by using the two kinds of lenses. Fiber input was placed directly into the core of the imaged filament at the position of the white screen which was used to take images in above experiments. Two KG3 bandpass filters and neutral density filters were used before the achromatic lens, resulting in a relative high transmission bandwidth (higher than 10% of transmission) of 360-760 nm. Figure 7 show two typical spectra from a greeny and a reddish filaments respectively. We call them green filament and red filament for convenience. In the visible range, the spectrum of the green filament had a big peak between 480 and 580 nm and centered at approximately 530 nm, whereas the spectrum of the red filament had a peak which covered from 540 to 640 nm centered at 590 nm, so that the green filament spectrum went 60 nm deeper into UV than the red one’s spectrum. Filaments have different degrees of spectrum broadening, which probably attributes to that they experienced different distances in the block bearing different levels of laser intensity. As Fig. 5(b) indicates, some filaments were relative longer and some were brighter, which indicates that they had different degrees of nonlinearities including self-focusing, multiphoton ionization, defocusing, self- and cross-phase modulations etc. Therefore, it can be expected that the spectra broadening from different filaments will be much different. Further experimental and theoretical work is needed to quantitatively analysis the effects. Finally, every filament contributed to the output of laser emission and as a result, a colorful and bright laser beam emitted from the block with a very broad spectrum which is known as SC emission [25,30,31].
We experimentally demonstrated that the filament-fringes can be formed in fused silica by using a simple circularly-, slit-, or square- shaped diaphragms and can be precisely controlled by continuously adjusting the size of the diaphragms. The filament-fringes can propagate several millimeters of distance. These 3-dimensional filaments structures will particularly benefit to the fabrication of laser waveguides for example, where non-plane and possibly long structures are desired. Moreover, in order to have more uniform filament-fringes, the initial parameters of laser could be improved, by using some techniques, piShaper  of a laser beam shaping device for example, to have a flat-top beam profile. Besides, we observed that different filaments had individual spectra with different amounts of spectral broadening, so that visually they showed different colors.
Z. Q. Hao acknowledges the support by a Research Fellowship from Alexander von Humboldt Foundation. W. M. Nakaema acknowledges the support by DAAD.
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