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The formation of permanent periodic structural changes in fused silica induced by the multifilamentation process was investigated. A cylindrical lens was used to focus 800 nm 50 fs pulses with 0.5 - 3mJ energy down to a line, resulting in a quasi-periodic linear self-arrangement of multiple filaments (MF). The quasi-period of multiple filaments is shown to be uniquely defined by the critical power of the material and the peak intensity on the sample entrance surface. A novel technique to control this spatial self-arrangement of MF is demonstrated based on the use of a binary phase mask. This technique allowed us to decrease the relative variation of spacing between the adjacent tracks of refractive index modifications by a factor of 4 as compared with the case without the phase mask. 3D + time numerical simulations qualitatively reproduce the main features of multiple filament formation obtained in the experiment.
©2010 Optical Society of America
1. Introduction
For some years now, light filamentation produced by high peak power laser sources has been drawing considerable interest. The phenomenon has given rise to several applications, most notably in atmospheric remote sensing [1,2], white light generation [3], direct writing of waveguides in bulk glass [4–7] and generation of optical pulses down to few cycle regimes [8,9]. Now, a better understanding of the so-called multi-filamentation (MF) process which occurs when the input peak power exceeds by orders of magnitude the medium critical power for self-focusing Pcr, is of utmost importance for further development of the applications mentioned above. The spatial organization of MF builds up on noise in the initial beam or instabilities introduced by the medium, including atmospheric turbulence [10,11]. Further research of MF formation in turbulence has demonstrated different scenarios of the longitudinal wandering of filaments where, depending on the pulse peak power, either modulational instability or self-focusing of the whole beam constitutes the prevailing effect in forming a filament [12]. However, it is possible to control random MF formation by means of an elliptical input beam [13–15]. Another reported way of controlling multiple filamentation in 2-D is through the use of a periodic mesh [16] or strong light field gradients [17]. Interestingly, controlled filamentation of high peak power laser beams has led recently to the formation of permanent structural changes in bulk transparent medium [18–20].
In this paper, we present a new approach to controlling the formation of MF in bulk condensed material. By using a cylindrical lens and a diffractive element (a phase mask) we have induced periodic arrays composed of multiple tracks of refractive index changes. We first show experimentally and justify numerically the formation of quasiperiodic tracks by using a cylindrical lens only. Thus, the self-organization of tracks is demonstrated for initial pulse energy in the range 0.5 – 3.75 mJ and various geometrical focusing conditions. The transverse position, spacing and induced refractive index modifications are compared in the case of the presence and the absence of the phase mask. The forced periodic arrangement of tracks induced by the phase mask is shown to be more advantageous in terms of the more regular spacing in the direction perpendicular to the pulse propagation, better transverse shape of the refractive index modification profile and the larger total amount of induced refractive index change.
2. Experimental set-up and procedure
A Ti-sapphire regenerative amplifier system (Coherent Legend-HE) that produces 35 fs laser pulses of 4.0mJ per pulse at 1kHz repetition rate and a central wavelength of λ = 806nm was used. The laser chain was thus producing pulses with peak power of the order of 200 GW that is several orders of magnitude larger than the critical power for self-focusing in fused silica (~2MW). Such pulses were able to initiate the multi-filamentation process over large areas, resulting in extensive arrays of refractive index tracks.
Our experimental procedures were divided into two parts. In the first, the free multi-filamentation process was initiated along the focal line produced by a cylindrical lens. In the second part, a diffractive element (a phase mask) was introduced between the lens and the sample so as to regularize the spacing of the filaments along this line. The two cases were subsequently analyzed and compared to each other on the basis of the resulting permanent refractive index changes induced in the glass. The set-up presented in Fig. 1 was used in both parts except for the insertion of the phase mask in the second part.
Fig. 1 Schematic representation of the set-up used for the investigation of the formation of multiple filaments in bulk glass. The inset on the right side of the figure shows an actual image of the interference zone measured using a CCD camera 3.5mm in front of the focal plane along the z axis.
The input beam of 8.5mm in diameter (measured at e−2 in intensity) was focused inside a 7mm thick fused silica sample using a 5mm thick cylindrical lens with a focal length of 112mm. The resulting beam waist is 13μm along the x axis. Pulses were stretched to 50fs after propagation through the steering optical components, including the cylindrical lens. Pulse energy was varied between 0.5 and 3.75mJ during the experiment. In the second part of our experiment, a one dimensional binary phase grating with a period of 47.4μm was inserted 8mm in front of the sample with its lines oriented along the x axis. The phase mask is inscribed on a 3mm thick fused silica substrate. The grating period being relatively long with respect to the wavelength, the diffraction angle of the orders −1 and + 1 was ~1° and those two orders were carrying 80% of the incident energy whereas the order 0 was carrying only ~1%. The resulting interference zone (i.e. overlapping between the order −1 and + 1) was therefore much longer (i.e. ~25 cm) than the block thickness (7 mm).
An Olympus IX71 phase contrast microscope was used to visualize the modified zone inscribed inside the bulk sample. The system actually allows us to observe small variations of the refractive index in the sample. Positioning accuracy of the viewing system is ± 0.5μm in x, y and z. Figure 2 depicts a typical schematic of the observed modification zone induced by the propagation of the ultrashort pulses in the sample and defines the coordinate system that will be used for the remainder of the paper.
The origin of the coordinate system is defined as the position of the center of the modified zone for the x and y transverse directions and as the front surface of the sample for the z axis. In the y-z plane, the modified zone is taking the shape of a crescent curving towards the input beam direction z. The use of high pulse energy allows us to modify a large volume of the sample. Typically the modified zone is about 100 x 8000 x 1500 μm along the x, y, and z axis respectively. That volume is composed of a large number of parallel tracks spaced by a distance of a few microns in the transverse (x,y) plane (Fig. 2).
3. Experimental results
3.1 Qualitative description of multi-filamentation process
3.1.1 Direct inscription without binary phase mask
The multi-filamentation process was first studied without the diffractive element. The samples were exposed to 60,000 femtosecond pulses (60s @ 1kHz) with energy ranging from 0.5mJ to 3.75mJ. (Note that the energy threshold at which the tracks were becoming visible at the microscope was of the order of 0.5mJ). Figure 3 shows images in the x-y plane of the glass modifications obtained for two positions of the geometrical focus zgf: on the sample surface zgf = 0 and inside the sample zgf = 3.8 mm.
Fig. 3 Images (in the x-y plane) of the modified zone inscribed for two different values of the geometrical focus of the Ti:Sapphire femtosecond pulse zgf = 0 mm (a-c) and 3.8 mm (d-f) and different values of the pulse energy ranging from 1mJ to 3.75 mJ. The microscope image (observation plane) was set at the z position indicated in the upper right corner.
The depth z of the image plane inside the sample corresponds to the position, where the largest number of refractive index change tracks is visible under the microscope. The characteristic view of material modification changes essentially with the variation of both geometrical focus position and the initial pulse energy. If the cylindrical lens focuses the radiation on the front surface, we can observe the refractive index modulation in the form of a series of dots along the y-direction at lower energy of 1 mJ (Fig. 3a). This modulation disappears becoming a continuous line at higher energy per pulse, i.e. 1.5 mJ, (Fig. 3b) leading to an ablation crater at 3.75 mJ (Fig. 3c).
A similar refractive index change evolution was observed for a fixed pulse energy larger than 2mJ while moving along the focal line (i.e. along y) from the beam periphery towards the beam center. The ablation crater is observed at y = 0 (i.e. in the vicinity of the fluence distribution maximum and in the beam center). The crater extends for a few mm until it is followed by an un-modulated focal line, which is itself replaced by a modulated one until it disappears. Such a behavior points to a local fluence dependence of the phenomenon. Indeed, the degree of material modification or damage is associated with the total amount of free electrons generated due to both multiphoton and avalanche ionization at a given position in the sample [21]. This total number of electrons depends on the peak intensity and duration of a femosecond pulse, yielding fluence as the measure for the induced modification. The quantitative study of fused silica material modifications induced by either multiphoton or avalanche ionization has been performed in [22] under tight focusing conditions.
We also positioned the geometrical focus in the middle of the sample (zgf = 3.8mm, Fig. 3 d-e) and near the back surface (zgf = 6.9mm) (not shown due to the similarity of the results). No surface ablation was observed for those zgf positions. With increasing pulse energy the tracks formed are more regularly organized, especially along the line focus in the y direction. In this direction quasiperiodicity is observed in a way that the period between the tracks increases from 5 to 20 μm. The period increase is associated with either decrease in the initial pulse energy, while remaining at a fixed position on the y-axis, or moving from the beam center to the beam periphery along the y-axis at a fixed pulse energy. Note, that for this particular experiment not any regularizing device (including a binary phase mask) has been used. Thus, the quasiperiodicity was self-induced. It was best pronounced for the case when the geometrical focus was positioned in the middle of the sample at zgf = 3.8 mm. Therefore, the rest of the paper will be devoted to the results in this particular geometry of the experiment. The physical origin and quantitative analysis of the quasiperiodicity is presented later in the numerical simulation section.
Figure 4a shows a series of successive refractive index modification images in the x-y plane centered at the (x,y) = (0,0) in the form of (Media 1). The sequence starts at 100 μm after the entrance surface inside the sample and finishes at z = 1610μm with the frame spacing 6.5μm. As we advance deeper into the sample the line focus experiences modulation followed by disintegration into multiple filaments with quasi equidistant spacing. The presence of the two parallel lines of foci instead of one is due to the super-Gaussian rather than Gaussian beam distribution on the focusing lens, as reported by Grow et al. [23]. Figure 4b (Media 2) shows a lateral scan along the y axis of the y-z plane view. The scan starts at the center (x,y = 0,0) and moves towards the edge of the exposed zone (y = 4000μm). We had to move progressively the field of view along z (by about 500μm) in order to follow the crescent shape of the modified zone (cf. Figure 2).
Fig. 4 Movies of track arrays inscribed with a pulse energy of 2.5mJ and with the geometrical focus positioned in the middle of the bulk sample (z = 3.8mm) a) In the x-y plane; b) In the z-y plane. In (a) the image plane is moved along z (i.e. along the tracks) (Media 1). In (b) the view is moved along the y-axis (Media 2).
For a fixed energy of 3mJ we exposed the bulk sample to several different numbers of pulses. We noted that 2,000 pulses are needed to induce an observable refractive index change in the bulk sample. More exposure time lead to a build-up in the refractive index changes. Indeed, for a greater number of pulses, more tracks are formed and those are longer in length and appear more defined on the microscope images. In both x and y transverse directions, the dimensions of the whole modified zone remain unchanged independently of the exposure time. This points to the fact that the modified zone dimensions depend mainly on focusing conditions and pulse energy. Also, the spacing between adjacent tracks is not affected by the number of pulses. For more than 30,000 pulses, the writing process saturates and no further change is observed in the array of tracks.
3.1.2 Inscription with the binary phase mask
In the second part of our work, we took advantage of the interference pattern produced by a phase mask to force a periodic arrangement of the MF in bulk glass. The grating period is 47.4μm. We are using the set-up depicted in Fig. 1, with the phase mask inserted 8mm in front of the fused silica samples. Again, the sample was exposed to 60 000 femtosecond pulses (60s at 1kHz) for a few different pulse energies ranging from 500μJ to 3.75mJ. The geometric focus was positioned at zgf = 3.8mm for all inscriptions. Results are shown in Fig. 5 .
Fig. 5 Sample image of movies of the evolution of the refractive index changes tracks along the z direction. The tracks are induced under the following conditions: (Media 3) (a) Pulse energy of 1mJ. (Media 4) (b) Pulses energy of 1.5mJ. (Media 5) (c) Pulse energy of 2mJ.
The three movies of Fig. 5 (Meda 3, Media 4 and Media 5) are composed of series of pictures of the x-y plane, centered at the (x,y) = (0,0) position, taken for several z positions in the sample. The first sequence (Media 3) starts 305μm below the front surface and finishes at z = 1530μm. The second movie (Media 4) starts at z = 130μm and finishes at z = 1000μm. Finally, the last one (Media 5) starts at z = 90μm and finishes at z = 875μm. The z spacing between frames is of 6.5μm or less. Those sequences were chosen because they are representative of the results that we have obtained. The difference between the multiple filamentation development with and without the phase mask resides mainly in the fact that with the mask the well pronounced adjacent tracks are separated by distances predefined by the mask. These distances are either 47.4 or half of this value (23.7μm) (see the details in Fig. 11 ). A closer look at the image sequence of Fig. 5c allows the observation of additional tracks, the spacing between which is of the order of 3 μm. Examples of those closely spaced tracks are highlighted in Fig. 6 . These additional tracks are due to multiple filaments formed within the mask-predefined period mainly due to the linear partitioning effect [24]. In terms of the energy balance, the formation of around 5-8 additional filaments per phase mask period is possible, since the peak power flowing through the square with the side 47.4 μm is around 20Pcr taking the peak intensity on the sample surface equal to ~3x1012 W/cm2 (see the Numerical Simulation section for the detailed estimate). The tracks corresponding to these additional filaments are less pronounced since the filaments compete [25] and may merge into a single one.
Fig. 11 The relative variation (Eq. (2) of the track spacing along the y-axis after the inscription made with the initial pulse energy of 3mJ a) without the phase mask and b) with the binary phase mask.
Fig. 6 Close-up images of closely spaced tracks induced by pulse energy of over 2mJ and inscribed with the phase mask.
3.2 Quantitative description of the multi-filamentation process
3.2.1 Refractive index change
We investigated the refractive index change induced by the laser pulses in the fused silica sample by using quantitative optical phase microscopy (QPM) [26]. The technique involves taking several bright field images taken at different depth in the sample and separated by 1μm. We used the Iatia Ltd. qpm software to yield high resolution digital phase images of the tracks. The refractive index change can then be estimated by using the following
Where Δφ is the phase delay induced by a track section of thickness d, Δn is the refractive index difference between the track region and the surrounding unmodified fused silica and λ is the central wavelength transmitted by a neutral filter (600nm ± 20 nm) inserted in the Olympus IX71 microscope. For the sake of comparison we chose to measure only the refractive index of tracks induced at beam maximum (i.e. near y = 0).
First, we measured and averaged the refractive index of 15 adjacent tracks along the y direction starting from the center (y = 0) of the modified zone inscribed with and without the phase mask for different initial pulse energy as shown in Fig. 7 .
Fig. 7 Refractive index change of tracks inscribed with and without the phase mask as a function of the pulse energy. Error bars actually correspond to standard deviation of Δn computed for 15 tracks.
Without a phase mask the light-induced change Δn to the refractive index marked by squares in Fig. 7 is relatively constant and varies between 0.0008 and 0.0020 for low pulse energies up to 2mJ. For pulse energy more than 2mJ, the refractive index amplitude begins to increase and reaches a maximum of 0.003. With a phase mask the refractive index change increases rather linearly from around 0.001 to 0.005 for pulse energies increasing from 0.5 to 3mJ. For a given pulse energy, the induced refractive index change appears to be larger when using a phase mask than for a direct inscription. This is because the filaments originating from the mask-predefined perturbations have less shot-to-shot deviation of their transverse position as compared with the filaments born without the mask [16,27]. This is associated with the irregularities in the beam spatial profile leading to local fluence fluctuations of the beam impinging on the silica block, observed especially for high energy pulses. Fluctuations of the order of 10-15% were actually observed in the beam spatial profile for 3mJ. This explains the increased variance exhibited by refractive index measurement at high pulse energy. To illustrate this, Fig. 8 shows the measured Δn corresponding to 26 consecutive tracks.
Fig. 8 Relative refractive index amplitude of adjacent tracks inscribed with a binary phase mask and a fixed pulse energy of 3mJ. The spacing between tracks is 24μm.
The refractive index changes of adjacent tracks are shown to vary between 0.003 and 0.008 on a spatial scale of ~100-200μm which turned out to be correlated with the beam spatial profile fluctuations.
3.2.2 Tracks periodicity
The self-organization of multiple filaments can be studied through the light-induced permanent refractive index changes in the sample. To quantify the phenomenon, we measured the positions of several adjacent tracks along the y-axis. Figure 9 shows measurements taken for a few different pulse energies with the transverse position accuracy equal to 0.5μm.
Fig. 9 (a)Position of adjacent tracks along the y-axis of a given line of modification inscribed without the phase mask for different pulse energies. (b) Analysis of the measured quasiperiodicity of tracks shown in panel (a). Squares – the average and the standard deviation of a quasiperiod d for the given initial pulse energy. Solid line – the fit to the quasi period d according to the Eq. (9). Circles show the ratio Ppeak/Pcr and its standard deviation per a cell with a side equal to the corresponding quasiperiod d.
For all initial pulse energies studied the track position changes quasi – linearly along the y-axis. With increasing energy the average value of the quasiperiod decreases and reaches 10.8μm (@3mJ) with a corresponding standard deviation of 2.0 μm (Fig. 9b) (see the detailed analysis in the Numerical simulation section). At the smaller energies of 0.5 and 1.5 mJ the quasi-linear dependence of the track spacing on the track number becomes piece-wise linear with the ‘jumps’ between the tracks 10 and 11; then 13 and 14 for 0.5 mJ and between the tracks 12 and 13 for 1.5 mJ. This long spacing between adjacent tracks is due to the instability in the initial beam intensity profile leading to ‘missing’ tracks along the y-line measured.
The quasi – linear dependence of the track positions in a wide energy range clearly proves the periodicity in the formation of multiple filaments in glass induced by the propagation of a highly elliptical fs-laser beam. A periodic organization of MF in water has already been reported [28] and more recently, propagation of a highly elliptic beam through fused silica has yielded similar results [29].
The periodicity of the tracks is better pronounced closer to the sample front surface. Indeed, with increasing propagation distance up to 40% of the initial pulse energy can be absorbed in the laser produced plasma [21]. The initially created filaments can merge or die out due to the peak power decrease in naturally created quasi-periodic cells. Therefore, for the fixed initial pulse energy the quasi-period increases with the propagation distance z along the sample.
We now turn to the inscription made with the binary phase mask, which forces multiple filaments to be born at the predefined positions [16,19]. In order to quantify the periodicity, we measured the positions of several adjacent tracks along the y-axis similarly to the case without the mask and plotted the track spacing value for the given initial pulse energy as shown in Fig. 10 .
Fig. 10 (a)Position of adjacent tracks along the y-axis of a given line of modification inscribed with the phase mask for different pulse energies. (b) The average and the standard deviation of a track period d for the given initial pulse energy. Solid line corresponds to the full and a half of the phase mask period 47.4 μm.
Comparing Figs. 9(a) and 10(a) we see that the track periodicity is better pronounced if the refractive index inscription is made with the mask. Indeed, the average track period takes only two mask-predefined values, the full and one half of the mask period (23.7 μm). The latter takes place for higher initial pulse energy range 2-3 mJ or closer to the beam center if the energy decreases to 1.5 mJ. With decreasing energy or stepping further from the beam center, the period doubles to the initial mask period of 47.4μm. Since the cell with the size of the mask period (47.4μm x 47.4μm or 23.7μm x 23.7μm) contains more than 6 critical powers for self-focusing, further disintegration into multiple filaments is inevitable as discussed earlier and shown in Fig. 6. These additional multiple light filaments are more randomly spaced in the transverse (x,y) plane and therefore do not leave well-pronounced material modifications along the z-axis.
To compare multiple filament regularization with and without the phase mask, let us introduce the relative variation as
where Δyi is the spacing between the adjacent tracks and is the calculated mean spacing. The relative variation is plotted in Fig. 11 for two inscriptions made with and without the phase mask for the initial pulse energy of 3mJ and for the tracks located in a high intensity zone of the beam. The largest relative variation of track spacing for the inscription without the mask reaches 39%, while for the inscription made with the phase mask the relative variation is kept below 10%. This demonstrates a precise control over the spatial formation of multiple filamentation in bulk glass using a diffractive element. By a careful tailoring of the experimental conditions, it is possible to induce a periodic array of refractive index tracks with a specific spacing.4. Numerical simulations
Our experimental analysis is based on the observation of the refractive index change in the fused silica sample for the initial pulse peak power thousand time larger than the critical power for self-focusing in the material. The refractive index changes and track formation are formed mainly due to the laser-produced plasma relaxation in the bulk of our sample. Therefore, our propagation model should be formulated in the full (x,y,z,t) form and include both the time-dependent plasma formation and radial symmetry breaking due to the multiple filament formation in the high peak power conditions [19,21]. Previous simulations of a highly elliptic beam propagation in fused silica by Majus et al [29] concentrated on the nucleation and spatial positions of filaments along the line focus in fused silica. The effect of the plasma and time – domain transformation was not considered.
The system of equations for the light field complex amplitude is given in the slowly varying envelope approximation:
where is the transverse Laplacian, k = n02π/λ = n0ω/c is a wavenumber corresponding to the fundamental laser wavelength λ = 800 nm, n0 = 1.45 is the refractive index and vg is the group velocity in fused silica at the wavelength λ. The terms on the right-hand side of the Eq. (3) describe diffraction, group velocity dispersion with the coefficient k”ω = 390 fs2/cm [30], nonlinear refraction due to the Kerr nonlinearity and the plasma, optical-field-induced ionization energy loss: , where = 6 is the number of photons necessary for the electron transfer to the conduction band with the band-gap energy Wg = 9 eV. The frequency is the plasma frequency. The electron density N e is calculated according to the kinetic equation:where R(|E|2) is the optical-field-induced ionization rate [31]. This model gives reasonable agreement between the experimental and simulation results on the size of the plasma zone derived from the material modification in [22]. The avalanche ionization frequency νi is given by and the elastic collision frequency νc scales as a square root of the local intensity and has the value of at the intensity of about 1013 W/cm2. As the characteristic recombination constant τr = 170 fs we chose the electron plasma lifetime measured after the propagation of a femtosecond pulse in fused silica [32].As an initial condition on the sample entrance surface we take a Gaussian pulse with Gaussian transverse intensity distribution focused by a cylindrical lens:
where τ = t-z/vg is the time in the co-moving coordinate system, vg is the group velocity of the pulse, a 0 = 47 μm, τ0 = 30 fs (corresponding to 50 fs FWHM as used in the experiment) and geometrical focusing distance Rf = 3.8 mm corresponding to the distance between the sample entrance surface and geometrical focusing distance in the experiment. The peak intensity I0 ≈4.5x1012 W/cm2 on the sample surface in the experiment is close to I0 ≈3x1012 W/cm2 in the simulations for the case of 1.5 mJ and 10 μJ energy, respectively. The transverse diameter of the simulated beam (94 μm at e−1 intensity level) is of the same order as the experimental one along the smaller axis of the ellipse (140 μm at e−1 intensity level). The critical power for self-focusing in fused silica is Pcr = 2.7 MW, as used in Ref [19]. Thus, the pulse peak power Ppeak exceeds the critical power by Ppeak/Pcr ≈10000 and Ppeak/Pcr ≈60 in the experiment and the simulations, respectively. There were two sets of the simulations: with and without the phase mask φ(y) “inserted” at the entrance to the sample. The periodic phase φ(y) is given bywhere s = 6 is the degree of the super-Gaussian function, is the total number of phase mask units, dmask_sim = 12.5 μm is the mask period in the simulations, h = 0.1dmask_sim is the transition region between the mask units. The phase mask period dmask_sim was chosen so that the ratio of Ppeak/Pcr flowing through the square with the side dmask_sim at the entrance peak intensity of I0 ≈3 1012 W/cm2 is Ppeak/Pcr ≈1.7. This ratio makes it possible to form at least one filament per a cell with the side equal to the mask period.Thus, the simulation parameters are well adjusted for qualitative comparison with the experiment since the key parameters as the peak intensity and the beam size on the sample surface, the ratio Ppeak >> Pcr and the pulse duration of 50 fs FWHM are in agreement with each other. To reproduce nonstationary multiple filamentation with high quality, the simulations were performed on 460 × 460 spatial grid with nonequidistant steps in the (x,y) transverse plane. In time domain (10τ0) there were 1024 steps and in the propagation direction z the grid steps were varied adaptively with increasing peak intensity, the minimum step equal to 1 micron.
We start the discussion of the simulation results from the distribution of electron density along the sample at the end of the pulse Ne(y,z,τ = τ end) and its cross sections in the (x,y) plane as shown in Fig. 12 .
Fig. 12 (b)The simulated plasma distribution along the fused silica sample in (y,z) plane and (a) in the successive transverse sections shown by the vertical dashed lines in panel (b). The sample entrance surface corresponds to z = 0.
There was no phase mask in this case. The longitudinal and transverse plasma zones reproduce all the major features of the refractive index change evolution observed in the experiment. Due to focusing with the elliptical lens, a plasma “line” appears, which is clearly pronounced by a distance of z = 1500 μm. This plasma line can inscribe the line-shaped refractive index change in the bulk of the sample similar to the one shown at the beginning of the movie in Fig. 4a. As we move deeper into the sample to z = 1750 μm, the line breaks up into separate filaments, which then form the longitudinal tracks of refractive index changes (Fig. 12b). The quasi-periodicity of plasma channels in the y-axis direction becomes obvious starting from z ≈2000 μm. Later on, by z ≈2325 μm, additional rows of filaments appear in the direction of x-axis (perpendicular to the elliptic focus line). In order to study the origin of this quasi-periodicity, let us surround each plasma hot spot as shown in Fig. 13(a) with a square cell with the side d ≈10μm and integrate the energy flowing through this square cell
Fig. 13 (a,b)The distribution of the simulated plasma hot spots in the (x,y) plane at the distance z = 2325 μm without (a) and with (b) a phase mask. White squares show the natural quasiperiod (a) or phase mask period (b). The energy flowing through several particular rectangular reservoirs surrounding the filaments is shown near the corresponding rectangles in microjoules. (c) Energy and amount of critical powers in the rectangular reservoirs, which surround the hot spots indicated by black numbers in panels (a) and (b).
The time interval is τend - τstart = 10τ0, that means the whole grid extension in the temporal domain. The results of the integration without the mask are shown in Fig. 13(a) by the amount of microjoules near several selected plasma hot spots and by the curve marked by squares in Fig. 13(c).
The range of the energy change in the cells with the side of the quasiperiod is 0.14 – 0.30 μJ, so that the ratio Ppeak_cell/Pcr ≈2 - 4 as shown on the right vertical axis of Fig. 13(c). In the estimation of the peak power Ppeak we took into account at least twice decrease of the pulse duration due to the pulse self – compression in the filament [33].
Natural subdivision of the beam into the quasiperiodic cells containing a filament per a cell is associated with the background energy reservoir [34] needed to support the existence of an extended in the longitudinal direction but transversely localized plasma region. The peak power flowing through a square cell with the side equal to a quasi-period is given by:
where I s is the radiation intensity on the sample entrance surface. The intensity I s changes with the initial pulse energy according to the expressionwhere are the half-axes of the ellipse on the sample entrance measured at e−1 intensity level in the experiment. The area does not depend on the initial pulse energy. The average and standard deviation of the quasi-period of the tracks obtained experimentally without a phase mask is shown in Fig. 9b. Averaging was performed over interspacing of 50 tracks shown for five different pulse energies W in Fig. 9a. For each average quasiperiod value d the corresponding peak power Ppeak_cell is calculated according to the Eq. (7) and the ratio Ppeak_cell/Pcr is shown by circles in Fig. 9b together with its average value Ppeak_cell/Pcr ≈3.18 (the horizontal solid line). Finally, the black solid curve in Fig. 9b given by the equationyields a good fit to the dependence d(W) obtained from the experimental data with Ppeak_cell = 3.18Pcr . This value is in agreement with the best critical power per a filament in the case of multiple filament regularization obtained in [27]. Besides, the experimentally obtained range of peak powers Ppeak_cell/Pcr ≈2 - 4 is in agreement with the similar range obtained in the nonstationary (3D + time) simulations as shown in Fig. 13.Thus, we have shown the physical origin of the natural quasiperiodicity arising in both the experiment and the simulations. The quasi-period size is uniquely defined by the critical power of the material and the peak intensity on the sample entrance surface. The ratio Ppeak_cell/Pcr originates from the optimum power for multiple filament regularization.
Comparison of the plasma hot spots formed with and without the phase mask is shown in Fig. 13a,b. Note, that the hot spots formed with the mask, on average, require less energy per a filament than the ones formed without the mask (compare the curves marked by circles and squares in Fig. 13c). Similar to the experiment, the spacing between the hot spots is defined by the phase mask period dmask_sim = 12.5 μm. Further analysis of the relative position of adjacent tracks in the simulations is shown in Fig. 14 .
Fig. 14 (a,b) Positions of all the simulated local plasma maxima in the (y,z) plane of the sample without (a) and with (b) the phase mask. Vertical solid lines in panels (a),(b) indicate the propagation positions z = 2063 and 1887 μm, at which the plasma maximum transverse position is plotted as the function of the hot spot number in panel (c).
In both panels with (Fig. 14b) and without (Fig. 14a) the phase mask we show all the plasma maxima located along the transverse y-direction for each position z from 1500 to 2400 μm. In qualitative agreement with the experiment, in the region of higher fluence the spacing between the tracks is smaller, than in the region of the decreasing fluence. For example, the spacing of adjacent tracks without a phase mask is analyzed at z = 2060 μm (vertical solid line in Fig. 14a) and plotted as a function of the hot spot number in Fig. 14c (squares). The hot spot number increases as we go further from the beam axis. The spacing between the hot spots is around 3 μm in the near-axis region and increases till 10 μm in the off-axis region. With a phase mask two distances z = 1887 μm and z = 2060 μm are analyzed as shown by the vertical solid lines in Fig. 14b as well as on Fig. 14c by closed and open circles, respectively. Near the beginning of the plasma channel the fluence is high and the spacing between the adjacent tracks is of the order of d = 3.5 μm. From the simulations we know that the maximum intensity reached in the positions of plasma hot spots is 3x1013 W/cm2. Therefore, the power flowing through this cell is ~1.4 Pcr, meaning that a single filament can indeed be created and supported for some propagation distance. This is in agreement with the minimum cell size found in the experiment in the presence of the phase mask (of the order of 2.5 μm in Fig. 6).
Further analysis of Fig. 14b shows that by the propagation distance z = 2060 μm the distance between the adjacent tracks increases towards the phase mask period dmask_sim = 12.5 μm. The period increase in the simulations is in agreement with the increase obtained in the experiment: in Fig. 10 the track spacing increases from 23.7 to 47.4 μm as the initial pulse energy decreases from 3 to 1.5 mJ. Note that at each particular propagation position z the number of the plasma hot spots extended in the transverse y direction of the beam is larger in the case of the phase mask than without a phase mask (e.g. compare the curves marked by closed circles and rectangles in Fig. 14c at z = 1887 μm). This demonstrates more efficient energy redistribution in the case of the phase mask. Breakup and fusion of filaments [35] is limited as compared with the case without the mask. As the result, the larger number of tracks with the predefined transverse position persists along the propagation direction.
The simulations reproduce the main features of multiple filament formation obtained in the experiment. The beam focused with a cylindrical lens breaks up into quasiperiodically spaced multiple filaments without any regularizing devices. The size of the natural quasi-period obtained in the simulations is in agreement with the quasi-period size obtained in the experiment (~10μm). Insertion of the phase mask redistributes the initial pulse energy in such a way that a single filament tends to be formed per a phase mask period. Since both in the experiment and simulations the phase mask period contains several critical powers for self-focusing in the material, further disintegration was obtained with a typical spacing between the filaments (refractive index tracks) of the order of 3 μm corresponding to ~1 critical power for self-focusing flowing through the cell. However, with the propagation distance in the simulations (or energy decrease in the experiment) the spacing between filaments is restored to the phase mask period.
5. Conclusions
We have shown self-arrangement of multiple filaments in fused silica induced by geometrical focusing of a 0.5 – 3 mJ, 50 fs pulse with a cylindrical lens. In the experiment the transverse and longitudinal arrangement of filaments has been studied by means of exploring the permanent refractive index changes remaining after multi-shot illumination of the sample. In the simulations multiple filament organization was studied based on the 3D + time model of the pulse propagation, which allows us to follow the nonstationary plasma production responsible for the refractive index changes.
The quasi-period of multiple filaments formed without a diffractive element (a phase mask) is uniquely defined by the critical power of the material and the peak intensity on the sample entrance surface. If the geometry of the experiment is preserved, the quasi-period is scaled as an inverse square root of the initial pulse energy.
Insertion of the phase mask into the beam path forces the filaments to be spaced with the period equal to the phase mask period. The relative variation of spacing between the adjacent tracks of refractive index modifications decreases by a factor of 4 as compared with the case without the phase mask. In the high fluence regions further disintegration of the beam into multiple filaments is possible, so that the peak power within the area surrounding a single filament decreases down to one critical power for self-focusing in the material. A robust array of multiple filaments formed in the presence of the phase mask induces larger amount of permanent refractive index modification as compared with the case without a phase mask.
Finally, the refractive index change measurements reported in this section is suggesting the possibility of using the tracks inscribed as waveguides. Careful tailoring of experimental condition could leads to the large scale direct inscription of waveguides arrays.
Acknowledgements
O. G. Kosareva, N. A. Panov and V. P. Kandidov thank the support of the Russian Foundation for Basic Research through grants No. 09-02-01200-а, 08-02-00517-а, 09-02-01522-а.
J-P. Bérubé, M. Bernier, R. Vallée and S. L. Chin wish to thank M. David Hélie for his precious help making the movies. The experimental part of the paper was supported by funding from the Canadian Institute for Photonic Innovation (CIPI) and the Conseil de Recherches en Sciences Naturelles et en Génie du Canada (CRSNG).
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