A systematic study was performed, both experimentally and theoretically, to investigate the structural periodicity of nanogratings inscribed by focused femtosecond pulses on the surface of dielectrics. The results surprisingly show that although nanogratings generally appear periodic they are in fact intrinsically aperiodic. In the perpendicular writing scheme, the groove spacing gradually decreases from the middle part towards both sides. In the parallel writing scheme, the groove spacing varies quasiperiodically and the variation differs with respect to pulse-to-pulse spacing. Constant groove spacing was obtained only for a particular pulse-to-pulse spacing. These Gaussian-apodized and quasiperiodic variations are found to be intrinsic. They arise from the fact that the grooves are created, depending on the writing scheme, either by a series of local-lobes with different amplitudes and material feedbacks or by a repeated generated leading side-lobe with similar but not identical amplitudes and material feedbacks. The production of each single grooves result from a nonlinear and localized process from which arises its aperiodic nature. All these intrinsically Gaussian-apodized and quasiperiodic variations can be well interpreted based on the incubation-based nanoplasmonic model.
© 2017 Optical Society of America
Femtosecond laser direct writing of nanogratings on the surface and in the volume of various materials has resulted in number of applications in photonics , biosensing , optical data storage . Yet the understanding of their mechanism of formation is still a matter of debate. Nanograting structural features were investigated as a function of various writing parameters such as laser energy/fluence, pulse duration, repetition rate, polarization, and scan speed or pulse overlap [4–7]. Two-color double-pulse experiments for investigation of dynamics of nanograting formation on the surface of various materials  or in the bulk of silicate glass  were also carried out as well as nanograting formation in a confined environment . In particular, the effect of using different objective lenses  and the effect of pre-distributed laser-induced defect on nanograting formation  were also systematically studied. A few models such as self-organization , interference [14–16] and nanoplasmonics [17, 18] have been proposed based on empirical evidences and/or theoretical simulations. However, to the best of our knowledge, no experimental study has ever been performed for investigating the uniformity of groove spacing. So far, nanograting is generally assumed to be periodic as described by the self-organization model , the interference model [14–16] and the nanoplasmonic model based on memory mechanism and mode selection . Its period is simply obtained by measuring an average value over a certain number of grooves or by a FFT analysis on a SEM image, and in most cases, the period is obtained from nanogratings written in the parallel writing scheme. To better understand the nanograting periodicity, separated groove spacing should be measured and analyzed. It has been found that the groove spacing varies from 100 nm to 720 nm in part of a femtosecond laser irradiation spot on ZnO due to spatial intensity variation . In this paper, we investigate the nanograting periodicity by analyzing the separated groove spacing of nanograting formed with various parameters. We demonstrate that nanograting actually consists of an intrinsically aperiodic structure and the groove spacing varies either Gaussian-apodizedly or quasiperiodically depending on the writing scheme. These variations are imprinted in the nanogratings, which can be revealed directly by measuring the individual groove spacing. Among all the proposed models, the incubation-based nanoplasmonic model  can satisfactorily interpret such aperiodic nature of the nanograting formation.
2. Experiments and discussion
The experiment was carried out with a femtosecond Ti:Sapphire laser. The laser wavelength is 800 nm and the pulse duration is 80 fs. The pulse energy was controlled by a combination of a half waveplate and a polarizer. Two writing schemes were adopted. One is parallel writing in which laser polarization is parallel to scan direction, the other is perpendicular writing in which the laser polarization is perpendicular to the scan direction. In the parallel writing scheme, the laser beam was focused by a 25 X Objective lens (Melles Griot, N.A. = 0.5); while in the other, a telescope consisting of a pair of cylindrical lenses was added into the laser path, then the shaped beam was focused by another objective lens (Mitutoyo 20X, N.A. = 0.4) with focus major axis perpendicular to the scan direction. This beam shaping allows much more grooves to form resulting in more separate groove spacings that can be measured and thus better accuracy for the periodicity analysis. Following laser irradiation, the samples were analyzed with a scanned electron microscope (SEM, FEI Quanta 3D FEG). In some cases, the sample was cleaned in an ultrasonic bath with 1% HF acid for 2 minutes to remove the laser-ablated debris before applying focused ion beam etching.
As shown in Fig. 1(a) the nanograting was first inscribed on the surface of a PYREX glass. The laser was running at 10 kHz and the pulse energy and scan speed were set to 230 nJ and 400 µm/s, respectively. A cross-section profile (based on picture grayscale variation) was extracted along the yellow line as plotted in Fig. 1(b). Two-types of grooves appear on this cross-section. Side-grooves are numbered from 1 to 9 while the so-called inner-grooves are labeled from i1 to i4. One notes that the side-grooves are produced at the peaks of the side-lobes which are successively generated in pairs on both sides of the laser peak (peak of the Gaussian intensity profile) as the number of overlapped pulses increases, while the inner-grooves are produced by the inner-lobes which are generated in between the side-lobes except for the first pair which is generated in between the laser peak and the first pair of side-lobes. Details about how the side-lobes and inner-lobes are evolved as a function of the number of overlapped pulses can be found in the incubation-based nanoplasmonic model  and the work on shot-to-shot groove evolution . The spacings between the adjacent side-grooves (1 to 9) were measured and plotted in Fig. 1(c) (the x-axis tick value 1 corresponds to the spacing between the grooves 1 and 2, and so on). It is obvious that the measured groove spacings are not identical and they gradually decrease from the middle towards both sides generally following a Gaussian profile. The red curve is a Gaussian fitting.
With the increase of pulse energy to 260 nJ, the number of grooves increases significantly as shown in Fig. 2(a). Accordingly, the spacings between adjacent side-grooves were measured and plotted in Fig. 2(b). Again, the groove spacing varies more or less along a Gaussian profile although the variation is less obvious in this case (as compared to the near-threshold case shown in Fig. 1(c)). But the gradual decrease of the groove spacing on both sides of the distribution is evident. Nevertheless, a false impression that the nanograting is periodic comes into being when taking into account those inner-grooves. Accordingly, the groove spacing is reduced to 110 nm (i.e. by a factor of 2) and the curvature becomes undetectable as shown in the Fig. 2(c). The coexistence of two periods in the nanostructure created by femtosecond laser irradiation in glass was also demonstrated by its Bragg reflection . That is somehow explaining why this slight aperiodicity is often not detected based on a casual observation.
Similar results were obtained on the surface of a fused silica substrate. After irradiation, the sample was treated in ultrasonic bath with 1% HF acid for only 2 minutes before SEM analysis. A in-depth cross-section view of a nanograting was shown in Fig. 3(a). It is obvious that the grating structure is interleaved with two sets of grooves (i.e. side-grooves and inner-grooves). The groove spacing and groove depth of the side-grooves were plotted in Fig. 3(b) and 3(c), respectively. Both of them clearly appear to vary according to a Gaussian function.
For the sake of clarity, a summary of the incubation-based nanoplasmonic model  is present in the following. When a tightly-focused femtosecond laser pulse is interacting with a stationary pristine sample near the ablation threshold, a nanoplama zone is created at the peak of the pulse. The field/intensity distribution of the pulse is then modified due to the presence of the laser-induce plasma (i.e., a pair of side-lobe is generated in the direction parallel to the laser polarization). This modification is then imprinted into the material through laser-induced defects which are generated in accordance with the modified distribution. This in turn causes new nanoplasma zones to develop symmetrically when interacting with successive pulses. During shot-to-shot interaction, the laser-induced defects are accumulated which gradually lowers the ablation threshold (i.e., incubation effect [21, 22]). This process leads to the evolution of intensity distribution which ultimately produces pairs of side-grooves and inner-grooves.
A numerical simulation based on the nanoplasmonic model was performed to account for the preceding results. The plasma density was assumed to vary only as a function of lobe intensity without considering the influence from material removal, and the groove width was an experimental average value due to the fact that the reduced ablation thresholds for each individual lobe are not well known. The grooves were produced pair by pair at the peaks of the side-lobes and inner-lobes except for the central groove which was first created at the laser peak, exactly the same as the stationary case. The simulated local intensity distribution is shown in Fig. 4(a). Those highrise rectangles are the enhanced peak intensities due to local field enhancement, and the dips are the consequence of local field suppression. The groove spacings between the adjacent side-lobes were extracted from the simulated structure and are shown in Fig. 4(b). Accordingly, groove spacing appears to vary as in the experiment according to a Gaussian fit. The small deviation from the Gaussian fitting is due to the fact that the incubation effect was not fully taken into account in the modeling because of our limited knowledge about the precise defect density induced by each individual lobe. However, the basic mechanism leading to the decrease of the groove spacing from the peak towards the edges can be simply understood, as illustrated in Fig. 4(c) where two local intensity distributions which were simulated independently for a single groove created at the peak and on the side of the same Gaussian profile are shown side by side. The groove width was assumed to be the same and the plasma density was slightly higher for the peak. For direct comparison, the central intensity distribution (appearing in blue) was replicated and shifted to overlap with the side one, and the green dashed lines mark the positions of the peaks of the respective side-lobes. Obviously, the side-lobe associated to the central distribution (i.e. at the laser peak) is farther apart from the groove. The main reason is not because of the difference of plasma density but the Gaussian envelope. The field suppression has to extend farther in order to reach a higher amplitude when the side-lobe is closer to the laser peak. Therefore, the groove spacing gradually decreases from the laser peak towards the edges.
When it comes to parallel writing, the production of grooves relies on a different mechanism than what has been described for the perpendicular case. The grooves, instead of being created by a series of paired side-lobes and inner-lobes, are now induced by a repetitively generated leading side-lobe and extended perpendicularly to the scan direction. The grating structure seems to be periodic at first glance, and the average period increases with the pulse-to-pulse spacing (defined as ratio of the scan speed to the laser repetition rate), and decreases with the increase of pulse fluence . As shown in Fig. 5, three nanogratings were inscribed with the same pulse energy of 110 nJ and pulse duration of 45 fs but with three different pulse-to-pulse spacings set to 40, 50 and 60 nm, respectively. The corresponding measured groove spacing for each nanograting was plotted in the right column. It is obvious that the average groove spacing or nanograting period increases with the pulse-to-pulse spacing. But surprisingly, the groove spacing seems to vary somewhat periodically with a fluctuation pattern depending on the pulse-to-pulse spacing. Simulations based on the incubation-based nanoplasmonic model were performed and results were also shown in the same plots. For each nanograting, both plasma density and incubation effect were assumed to be identical for all the grooves (i.e. all the grooves were created at the same reduced ablation threshold). This is a fairly reasonable assumption providing sufficiently quantitative agreement with the experimental measurements. Based on this approximation, the groove spacing varies periodically and the variation pattern is obviously dependent on the pulse-to-pulse spacing, which suggests that the nanograting is intrinsically aperiodic. A special case that the simulated nanograting for a particular pulse-to-pulse spacing is perfectly periodic does exist as shown in the right column of Fig. 5(b). One might argue that this general groove spacing variation could come from experimental imperfection such as laser instability, surface defects, and/or mechanical vibration. However, this kind of experimental imperfection by no means can account for the pulse-to-pulse spacing dependent periodically varying patterns as shown by both simulation and experimental results. Therefore, experimental imperfection cannot be responsible for the aperiodic or quasiperiodic grating structure.
Such intrinsic aperiodicity in the parallel writing case can also be understood based on the nanoplasmonic model. To that purpose, an illustrative drawing on how a new groove is created is shown in Fig. 6(a). A schematic based on simulation is also shown on the right. The vertical black line represents a groove just created. As the laser moves forward (to the right in the drawing), a new leading side-lobe is generated and its amplitude increases as the laser peak moves closer to it. The spacing between the just created groove and the laser peak (short red line) is defined as x, and the spacing between the laser peak and the leading side-lobe is defined as y which is not only a function of x but also a function of the local plasma density and incubation effect. After a few pulses (e.g. 2), the laser peak arrives at a position (long red line) where its corresponding leading side-lobe (the second blue line on the right or the purple one in the plot) reaches the reduced ablation threshold resulting in the formation of a new groove (green line). At this moment, x reaches its maximum value, xg, while y is at its minimum value, yg, and the sum of them is the groove spacing. As shown in Fig. 6(b) the simulated results were plotted as functions of the laser peak position. The initial peak position was set to 0 and a portion of the evolution process was extracted only after the laser matter interaction reaches the steady state (i.e. the number of overlapped pulses per unit length becomes constant). The red bars mark the laser peak positions which are separated by the pulse-to-pulse spacing. Those with a higher amplitude are the peak positions at which x reaches its maximum, y reaches its minimum and new grooves are created at the leading side-lobes and marked by the green bars. The spacings between each two successive grooves are calculated and shown in the plot. The special case that the nanograting is perfectly periodic is also presented in Fig. 6(c). By comparison of the general case with the special case side by side, one can immediately tell the difference which is the number of overlapped pulses required for producing a new groove. In the general case, this number varies by one more or one less for the successive grooves suggesting that the xg represented by an open red circle in the plot also varies. This leads to the variation of groove spacing. However, in the special case, this number remains the same for all grooves, which means that the xg is a constant and thus the groove spacing does not vary. Nevertheless, this constant groove spacing must be equal to nd (xg + yg = nd) where n is an integer number and d is the pulse-to-pulse spacing as indicated in Fig. 6(a).
The physics behind the nanograting formation process relies on the dynamics of both local intensity distribution and incubation effect. As the laser peak moves away from the just created groove, both amplitude and position of the leading side-lobes change (see the plot in Fig. 6(a)). The combination of these changes gives rise to a particular incubation process and thus create a new groove at a very specific distance from the previous one. However, the distances from the successive laser peaks to the just created groove are most likely not identical in comparison with the previous due to the fact that the laser pulses always arrive in a discrete manner separated by the pulse-to-pulse spacing. Consequently, the leading side-lobes are different in terms of amplitude and position. This results in a different incubation process and thus the number of overlapped pulses required for producing a groove varies by one more or one less. The new groove is thus created at anther distance resulting in a different groove spacing. Therefore, the grating structure is aperiodic. However, it is also very straightforward that theoretically a particular pulse-to-pulse spacing does exist for the whole process to be identical. This means that the leading side-lobes can always be repeatedly distributed with the same amplitudes and positions for all the grooves. The incubation process is thus unique and the number of required overlapped pulses is a constant. Therefore, a perfectly periodic grating structure is obtainable from a theoretical point of view. For a specific pulse fluence, there is always and only a particular pulse-to-pulse spacing that can lead to a perfectly periodic grating structure whose period must be a multiple of that particular pulse-to-pulse spacing.
We have demonstrated experimentally and theoretically that nanograting is an intrinsically aperiodic structure. Depending on the writing scheme, the groove spacing can vary Gaussian-apodizedly or quasi-periodically. Perfectly periodic grating structure can also be obtained for a particular pulse-to-pulse spacing. Theoretical simulations based on the incubation-based nanoplasmonic model which is able to reveal the groove formation process on a shot-to-shot basis show that the Gaussian intensity profile is the main cause of the Gaussian-apodized variation, and the inconsistent spacing between the laser peak and the just created groove is responsible for the quasi-periodic structure. Both are in good agreement with the experimental results. All this satisfactorily and undoubtedly shows that the nanograting is intrinsically aperiodic.
Fonds de recherche Québec - Nature et technologies (FRQ-NT); Natural Sciences and Engineering Research Council of Canada; Canada Foundation for Innovation; and Canadian Institute for Photonic Innovations.
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