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Abstract
Subwavelength-scale surface structures have many important engineering and nanotechnology applications, e.g., superhydrophobicity and light-trapping. However, an effective and economic nanofabrication solution for general engineering materials, e.g., metals or silicon, is still not available to date. In this paper, we present an experimental and theoretical study of the nanostructure formation mechanism based on double time-delayed femtosecond laser beams and the coupled mode theory (CMT), demonstrating the use of an optical analogue of massless Dirac particles for high-throughput nanofabrication for the first time. In the experiments, a variety of complex periodic structures, including hexagonally arranged nanoholes, nano-square array, and periodic ripples, have been fabricated. The formation mechanisms of these nanostructures are explained by the CMT, where a transient plasmonic waveguide array (TPWA) is formed by the interference between the preceding laser and the induced surface plasmon polaritons (SPPs). The SPPs induced by the subsequent laser propagates through the TPWA, resulting in conical diffraction. This result shows the first practical application of the massless Dirac dynamics in nanofabrication.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Parallization of ultrafast lasers has been a challenging and important way to realize economic nanofabrication [1,2]. Laser induced periodic surface structures (LIPSS) have attracted much attention in recent years as an effective high-throughput laser nanopatterning method for creating nanoscale structured and functional surfaces, e.g., superhydrophobic surface [3], antireflective surface [4], structural colored surface [5], antibacterial surface [6] etc. LIPSS also have been demonstrated to produce dielectric optical metasurface elements with a variety of functions [7]. In recent years, although improvements have been made in the structure quality [8] and minimal feature size [9], LIPSS still face the challenges of limited patterns variation. Besides 1D periodic ripples with subwavelength or deep subwavelength periods [8,9], at present only a few 2D nanostructures, e.g., curved ripples [10], rectangle array [11], triangle array [12], circular dot array [13], spike array [14] etc., have been fabricated via LIPSS by manipulating the incident laser induced surface plasmon polaritons (SPPs) [10–12] or by employing different ambient media [13,14].
A well-established physical model suggests that low spatial frequency LIPSS (LSFL) [15] is generated due to the interference between the incident laser and SPPs [16]. Therefore, the control of the generation, propagation, and diffraction of the SPPs at the metal-dielectric interface can not only improve the pattern diversity of LIPSS, but also provide insight about the SPPs’ propagation characteristics on the excited metal-dielectric interface. One convenient way to manipulate SPPs is to apply double time-delayed femtosecond lasers with different polarizations. Some 2D nanopatterns [12,13,17,18] and the control of ripples’ orientation [19] have been demonstrated by double femtosecond lasers, but the related formation mechanisms are not well understood, which hinders the further development of complex LIPSS patterns. Although the nonlinear convection flow may interpret the formation mechanisms of LIPSS generated by double femtosecond lasers, the origin of the convection flow remains unclear [17]. A more recent study on the formation mechanism of 2D nanopatterns also provides a phenomenological explanation by combining the SPPs and the self-organization process [18].
In this paper, we employed double collinear femtosecond laser beams with orthogonal linear polarizations and different time delays on silicon targets. Using double laser beams with different parameters, nanostructrues formed by 1-3 set(s) of different oriented parallel etching lines are fabricated on the silicon surface, which respectively are periodic ripples, square array, and hexagonally arranged nanohole array. The formation mechanisms of all these nanostructures are quantitatively explained by analytical models derived based on the coupled mode theory (CMT) [20]. In the model, the nanostructures are generated by the interaction between the SPPs induced by the second laser and the transient plasmonic waveguide array (TPWA), formed due to the interference between the first laser and its corresponding SPPs. For the first time, we experimentally demonstrated conical diffraction of the SPPs induced by the TPWA, i.e., an optical analogue for massless Dirac dynamics in waveguide systems [20–22]. This fundamental understanding may help improve the pattern diversity and range of engineering applications of LIPSS.
The silicon nanohole array fabricated in this paper, for example, can reduce reflectance by 50% over the 400 nm – 900 nm wavelength range and may improve the efficiency of photovoltaic cells [23,24]. Silicon surface with ordered nanohole array can also enhance the hydrogen generation efficiency in the water splitting reaction [25]. Furthermore, silicon nanohole array has been used as silicon-based mid-infrared optical metasurface elements [26]. Presently, such nanostructures are fabricated by a combination of deep ultraviolet lithography (UVL) and metal-assisted silicon etching [23]; polystyrene nanosphere lithography and reactive-ion etching [24] or metal-assisted electroless etching [25] etc. These methods are limited by the complex procedures and high cost. Our new LIPSS process based on the conical diffraction of SPPs may be a viable solution to address the issue with one simple step to fabricate ordered silicon nanohole array and many other complex 2D patterns.
2. Experimental setup
Figure 1 presents the experimental setup. In the experiments, the light source is a Ti: sapphire femtosecond laser amplifier system (Spitfire Pro, Spectra-Physics Inc.) that generates 100 fs, 1 kHz, 800 nm laser pulses. The experimental setup is shown in Fig. 1. The half-wave plate (HWP) and polarization beam splitter (PBS) are used to continuously adjust the laser power. The femtosecond laser beam reflected by PBS has a vertical linear polarization. Next, the laser beam enters a Michelson interferometer. A quarter-wave plate (QWP) is inserted in one arm to control the polarization. The reflective mirror (M3) is mounted on a linear stage to adjust the optical path difference between the two arms. Accordingly, the interferometer outputs double collinear femtosecond laser beams with orthogonal polarizations and adjustable time delay. The first and second laser beams are vertically (E1) and horizontally polarized (E2), respectively. A plano-convex lens (L1, f = 300 mm) focuses the double beams for nanofabrication. The target is a monocrystalline silicon (100) wafer (thickness = 650 μm), mounted on a precision XYZ stage (PLS-85, Physik Instrumente Inc.). The beam spots on the silicon surface are imaged by a CCD camera (BFS-U3-13Y3C-C, FLIR System Inc.) via a plano-convex lens (L2, f = 50 mm) to ensure the spatial overlapping of the double laser beams and to monitor the fabrication process in situ.
Fig. 1. Experimental setup of double collinear femtosecond laser beams for fabricating nanostructures on silicon targets. HWP: half-wave plate, PBS: polarization beam splitter, M1-M4: reflective mirror, NPBS: 50/50 non-polarization beam splitter, QWP: quarter-wave plate, L1 and L2: plano-convex lens.
The fabricated nanostructures are characterized by a scanning electron microscope (SEM, JSM-7800F, JOEL Inc.) and an atomic force microscope (AFM, Dimension Icon, Bruker Inc.). The geometrical properties of the nanostructures are obtained using ImageJ [27]. The angle measurement error is ±3° based on the statistics of 30 repeated measurements.
3. Experimental results
To identify the optimal conditions for generating uniform nanostructures, we experimentally explored the parametric space when the power ratio (1:1) for the dual beams is kept constant. During experiments, the time delay (Δt) between double laser beams varies from 0.5 ps to 16 ps; for each Δt, the target scans at speeds (v) from 0.2 mm s−1 to 10 mm s−1 along the vertical direction, as indicated by the dashed arrow in Fig. 2(b); and the energy fluence (F) of a single laser beam varies from 0.07 J cm−2 to 0.31 J cm−2. SEM measurements are performed and representative nanostructures are shown in Figs. 2(a)–2(e). In Figs. 2(a)–2(c), the nanoholes appear in the intersecting points of three sets of parallel lines, labeled by ① - ③. We define line set ① as the lines with the minimal angle ($\boldsymbol{\theta}$) relative to the horizontal direction, i.e., y axis in Fig. 2(d). The periods of all three sets of parallel lines are 760 nm. Therefore, the geometry of nanohole array can be characterized by $\boldsymbol{\theta}$ and the angles α, β, and γ between any two of the three parallel line sets, as shown in Figs. 2(a)–2(c). The nanoholes, which have an average diameter of 380 nm, are arrayed hexagonally as indicated by the circles in Fig. 2(a); the side length of the hexagon is 880 nm. We demonstrate the fabrication of this nanohole array along a 30 µm-wide stripe with good uniformity in Fig. 3. The depth profile of the nanohole array is characterized by an AFM and shown in Fig. 2(f). The nanohole depth is not less than 300 nm. By increasing v and decreasing Δt, the nano-square array can be fabricated, as shown in Fig. 2(d). The square array is formed by two sets of orthogonal parallel lines, labeled by ① and ②, and perpendicular to the laser beams’ polarizations E1 and E2, respectively. Figure 2(e) presents the fabrication result of 1D periodic nanostructures. Accordingly, one may conclude that double femtosecond laser beams with orthogonal polarizations can produce nanostructures composed of one, two or three sets of periodic etching lines.
Fig. 2. Nanostructures on silicon fabricated by double orthogonally polarized femtosecond laser beams, where the power ratio between the first and second beams is 1:1 in all experiments. (a) nanohole array: Δt = 6 ps; v = 1.1 mm s−1; F = 0.10 J cm−2; (b) nanohole array: Δt = 6 ps; v = 1.2 mm s−1; F = 0.11 J cm−2; (c) nanohole array: Δt = 4 ps; v = 1.8 mm s−1; F = 0.11 J cm−2; (d) nano-sqaure array: Δt = 2 ps; v = 2.8 mm s−1; F = 0.12 J cm−2; (e) 1-D periodic structure: Δt = 4 ps; v = 2.2 mm s−1; F = 0.12 J cm−2; and (f) color-coded depth profile of the nanohole array in (a). The dashed arrow in (b) indicates the target scanning direction for all the patterns in (a) – (f).
Fig. 3. Nanohole array on silicon fabricated by double femtosecond laser beams with orthogonal polarizations: (a) SEM image of the nanohole array; and (b) zoom-in view of (a). During fabrication, the two laser beams have identical laser fluences on the target surface (∼0.11 J cm−2) and a time interval of 1 ps. The target scanning speed is 1.5 mm s−1, i.e., the fabrication rate is approximately 1 cm2h−1.
Figure 4 presents the dependence of the nanostructure morphology on the fabrication parameters, i.e., Δt, v, and F. Data points of different colors represent different morphologies formed on the silicon wafer. Blue and green points respectively indicate square array (Fig. 2(d)) and 1D periodic ripples (Fig. 2(e)). Gray points indicate no obvious surface modifications. Black points indicate over ablation with no LIPSS formed. From Fig. 4, it is observed that nanohole arrays of identical morphologies can be formed in two different parametric regions represented by the red and orange points respectively. The nanohole array can only be produced in a small parametric space with F ∼ 0.1 J cm−2 and v ≤ 2 mm s−1. As Δt gradually increases from 0.5 ps to 8 ps, the parametric region that can form nanohole array also extends; however, when Δt further increases to 16 ps, the parametric region for producing nanoholes shrinks slightly compared with that of 8 ps. Besides, it is also found that the square array always appears at large v and near the onset threshold of LIPSS. When Δt < 2 ps or Δt = 16 ps, the square array cannot be formed. The relationship between the orientations of nanohole arrays ($\boldsymbol{\theta}$) and the fabrication parameters, i.e., Δt, F, and v, are investigated. From the results it is found that when different fabrication parameters are used, $\boldsymbol{\theta}$ fluctuates in the range of ±25°. We also examine the relationship between the intersection angles (α, β, γ) and the fabrication parameters, and find that α, β, and γ all have an angle close to 60° with a fluctuation of ±5°. Considering the measurement error (measurements are performed via ImageJ [27]; the angle measurement error is ±3°.), we conclude that the nanohole arrays are well arranged hexagonally regardless of the fabrication parameters.
Fig. 4. Effects of laser fluence and scanning speed on silicon morphology generated by double femtosecond laser beams with time delays of 0.5 ps - 16 ps: (a) Δt = 0.5 ps, 1 ps; (b) Δt = 2 ps, 4 ps; (c) Δt = 6 ps, 8 ps; and (d) Δt = 16 ps. BT: below modification threshold; PR: periodic ripples; SA: square array; NA: nanohole array; and EA: excessive ablation without LIPSS.
We performed Raman spectroscopy measurements (RM 1000B, Renishaw Inc.) from 200 cm−1 to 1635 cm−1 on the fabricated nanohole array, square array, and 1D periodic ripples as well as the unprocessed silicon target as a reference. The results are presented in Fig. 5(a). Comparing with the unprocessed silicon target, the Raman spectral curves of all the nanostructured silicon have shoulders in the range of 400 cm−1 - 500 cm−1, indicating the presence of the amorphous silicon after the irradiation of femtosecond laser beams [28]. By peak-fitting and integrating the peak-fitted curves, the amorphous ratio is estimated to be 15% for silicon surface covered by the nanohole arrays. (Methods for peak-fitting and amorphous ratio calculation are described in Ref. [29]) The reflectance spectra of the nanostructures in Fig. 2(a) are measured via an optical microscope (BX51, Olympus Inc.) and spectrometer (S2000, Ocean Optics Inc.). The results are presented in Fig. 5(b), which show a ∼50% decrease in reflectance from 400 nm to 900 nm. This shows the nanohole array can be used to enhance surface absorption. Further reduction in reflectance can be achieved by optimizing the size and aspect ratio of the nano-structures via fine-tuning the processing parameters, e.g., laser fluence, wavelength, scanning speed etc.
Fig. 5. (a): Raman spectra of unprocessed (primitive) silicon and different nanostructures shown in Fig. 2. The wavelength of the excitation laser is 514.6 nm. (b): Reflectance of primitive silicon and silicon with patterned nanohole arrays shown in Fig. 2(a).
4. Formation mechanisms of nanostructures
In the experiments, we fabricated different nanostructures (which are formed by 1 to 3 sets of parallel lines) on silicon by double time-delayed femtosecond laser beams. The parallel lines all have a period of ∼760 nm which is identical to those formed by a single laser beam as shown in the insets of Fig. 6(a). This result agrees with previous studies that LSFL on silicon produced by single 800 nm femtosecond laser have periods in the range of 560 nm – 770 nm [15]. It is generally believed that LSFL are generated by the interference between the incident laser and SPPs [16], where phase matching between the wave vectors of the incident laser and SPPs is achieved via scattering from the surface roughness or parametric scattering etc. [30] Our calculation shows that the lifetime of SPPs [31] with an 800 nm femtosecond laser at the air-excited silicon interface is not longer than 0.63 ps even for the extreme case that the valance band electrons become totally excited [32] Hence, it infers that the nanopatterns presented in Fig. 2 are not genereated by the interaction between the second laser beam (E2) and the SPPs excited by the first laser beam (E1). On the other hand, the sub-surface SPPs in the plasma-substrate model [33] only accounts for the generation of high spatial frequency LIPSS (HSFL). Therefore, we hypothesize that the line sets ① - ③ in Fig. 2 are respectively formed by the interferences between the incident laser and SPPs with different wave vectors at the air-excited silicon interface.
Fig. 6. (a) Relationship between the ripple orientation and the fabrication parameters (F and v) for a single laser E1 to irradiate the silicon target. Orientation angles of 60° and -60° represent the cases for excess ablation and no ablation respectively. (b) Dependence of the diffraction angles π/2-α (triangles) and −(π/2-γ) (squares) on the incident angle $\boldsymbol{\theta}$ of SPPs2, measured from the nanohole array SEM images. The blue solid and dashed lines are iso-frequency curves of the TPWA for η = 1 (η is defined in Eq. (7)); and the dotted-dash lines are the asymptotic lines. The black solid and dashed lines are iso-frequency curves of the TPWA for η = -1. The two dotted lines respectively denote φ = ±32.5°.
Next, we employ the CMT to explain the generation mechanisms of SPPs with different wave vectors, which induce the generation of etching line sets with different orientations. When double femtosecond laser beams (E1 and E2) fabricate nanopatterns on silicon, the target surface is first irradiated by the preceding beam (E1), which excites the silicon surface to a metallic state enabling the air-silicon interface to support SPPs; the interference between E1 and E1 induced SPPs (SPPs1) generates a transient periodic surface permittivity variation which can be considered as a 2D transient plasmonic waveguide array (TPWA).When the second beam (E2), which has a polarization perpendicular to E1, irradiates silicon, another SPPs (SPPs2) is excited and propagates through the TPWA. As SPPs2 diffract due to the TPWA, the diffraction orders may induce new SPPs with different wave vectors, thereby forming nanostructures in Fig. 2.
To mathematically describe the propagation of SPPs in a waveguide array, the coupled mode equation that considers the coupling between adjacent waveguides is employed, which is presented in Eq. (1) [20,34]:
Using Eq. (3) the propagation directions of the diffraction orders of SPPs2 can be predicted.
In experiments, when the laser E2 is blocked and the target is scanned along the x axis, the periodic ripples fabricated by the laser E1 are not often perpendicular to E1; the angle between the ripples and y axis fluctuates over a range of ±25° when different F and v are used, as shown in Fig. 6(a). Two representative images of the periodic ripples produced by the laser E1 are shown in Fig. 6(a). This is expected as previous studies [35] also suggest that the orientation of the periodic ripples on silicon is not only determined by the laser polarization, but also the scanning speed, which may be attributed to the seeding effect of the initial structures. From the results, it is observed that the angle $\boldsymbol{\theta}$ between line set ① and the y axis also distributes in a similar range. Meanwhile, in Fig. 4(c), when Δt = 8 ps and F = 0.14 J cm−2, the nanohole array is formed at v = 1.2 mm s−1 or 1.3 mm s−1; and the square array is formed at v = 3.0 - 3.6 mm s−1. As it is commonly believed E1 can induce horizontal ripples and forms line set ① in the square array at v ≥ 3.0 mm s−1, E1 must be able to generate ripples at a slower scanning speed when the nanohole array is formed. Therefore, we may deduce line set ① in Fig. 2(a) is generated by E1. For nanohole arrays, the angle $\boldsymbol{\theta}$ between line set ① and the y axis is just the incident angle of SPPs2 on the TPWA; and the angles π/2-α and –(π/2-γ) (see Figs. 2(a)–2(c)) are thus the diffraction angles of SPPs2 through the TPWA.
By measuring the orienting angle $\boldsymbol{\theta}$ and the intersecting angles α and γ of the nanohole arrays fabricated via different parameters, the relation between the incident angle $\boldsymbol{\theta}$ and diffraction angle φ of SPPs2 is plotted in triangles and squares in Fig. 6(b). Note that the error bars are not shown in the horizontal axis for figure clarity; the measurement error of $\boldsymbol{\theta}$ is on the same scale as φ. The blue solid and dashed lines in Fig. 6(b) represent the iso-frequency curves of the TPWA, calculated via Eq. (2), where d = 760 nm, k0 = 2π/λSPPs2 ≈ 2π/d, and C = 4.2×105 m−1. The dotted-dash lines are asymptotic lines of the iso-frequency curves which indicate the slopes of the curves for $\boldsymbol{\theta}$ ∼ ±15°, i.e., φ is weakly dependent on $\boldsymbol{\theta}$ around this point. The slopes of the asymptotic lines are ±0.637, meaning φ = ±32.5°. The two dotted horizontal lines in Fig. 6(b) respectively indicate φ = ±32.5°. Therefore, Fig. 6(b) shows that Eq. (3) can precisely predict the diffraction angle for 6° < $\boldsymbol{\theta}$ < 24° or -24° < $\boldsymbol{\theta}$ < -6°.
Fig. 7. Schematic of the formation process of the nanohole array by the conical diffraction of SPPs2 propagating through the TPWA: (a) SPPs1 appears due to the laser beam E1; (b) SPPs1 interferes with E1 to form TPWA, i.e. the line set ① in the final state; it should be noted that based on the experimental results in Ref. [36], the highly-excited region in each period of TPWA may consist of two closely adjacent sub-regions at the specific time interval, forming a binary waveguide array (see Fig. 7(e)); (c) SPPs2 appears due to E2 and diffracts into SPPs2-1 and SPPs2-2; (d) the two diffraction orders interfere with E2 to form line sets ② and ③; accordingly the nanohole array is formed on the target surface; and (e) schematic of the TPWA configuration that achieves the alternating positive and negative coupling coefficients and conical diffraction.
When $\boldsymbol{\theta}$ = 0°, the diffraction angle φ = 0° according to Eq. (3). Therefore, the square array is fabricated. As discussed previously, Eq. (1) describes a waveguide array whose coupling coefficients between any adjacent waveguides are equal. However, in experiments, the TPWA induced by E1 may not always meet this condition. The transient reflectance variation due to the interference between the incident femtosecond laser and SPPs on silicon has been measured using the pump-probe technique [36], showing a binary periodic reflectance distribution, which may make the coupling coefficient between adjacent waveguides vary alternately [20]. Besides, E2 may interfere with SPPs2 locally, generating a periodic intensity distribution, inducing a periodically varying permittivity and propagation constants along the waveguide. This may also induce alternating coupling coefficients between adjacent waveguides [37]. To describe these more general cases, the coupled mode equations with alternating coupling coefficients need to be used [22]:
As is shown in Fig. 2(c), line set ① produced by E1 is nearly perpendicular to E1. The SPPs2 induced by E2 is normally incident upon the TPWA. As such, two diffraction orders (SPPs2-1 and SPPs2-2) are generated due to the propagation of SPPs2 through the TPWA, which forms line sets ② and ③. This phenomenon is the conical diffraction at normal incidence which is analogous to the ones described in waveguide arrays with alternating negative and positive coupling coefficients [20–22]. Figure 7 illustrates the conical diffraction of SPPs2 and the formation of nanohole array.
The conical diffraction at normal incidence is an optical analogue of massless Dirac dynamics, which emulates the quantum dynamic effect via a controllable system. Numerous studies have been performed to design proper structures to generate the non-diffracting beam splitting phenomenon [20–22,38]. However, to date only one experimental demonstration was achieved in the waveguide system [21], where the conical diffraction cannot be directly observed due to the sinusoidally curved geometry of the waveguide. Importantly, the nanostructures in Fig. 2(c) may be a 2D optical analogue of massless Dirac dynamics, where the SPPs2 induced by E2 experiences the conical diffraction due to the TPWA formed by E1. Theoretical analyses show that waveguide arrays with alternating distances between successive waveguides (i.e., binary waveguide array) [20,21] with alternating metallic and dielectric layers [22,38] or with periodically varying permittivity along the waveguide [37] can generate alternating positive and negative coupling coefficients. As a large parametric space is explored to fabricate honeycomb nanhole arrays in the experiments, all aforementioned cases may occur in our system. Experimental and numerical evidence [6,36] shows that the binary waveguide array is the most probable reason.The results in Ref. [36] (See Fig. 3 in [36]) suggest that when the silicon surface is irradiated by a single femtosecond laser beam to form periodic ripples, the transient metallic region, i.e. the region with high reflectance, exhibits binary charateristics. Therefore, we hypothesize that the TPWA induced by the femtosecond laser E1 is a binary waveguide array, as illustrated in Fig. 7(e). The SPPs2 induced by the subsequent laser E2 will propagate in the interface between the metallic region of the TPWA and the air. Based on the numrical analysis in Ref. [20], the binary plasmonic waveguide array can induce conical diffraction due to the alternate seperations between metallic regions in the waveguide array. In Fig. 4, it is seen that with a moderate time delay, e.g., 8 ps, the parametric regions forming the nanohole array is larger than that with a smaller or larger time delay. This is becuase as time elapses, the TPWA gradually varnishes, which makes large time delay suboptimal for forming the nanohole array; likewise, as a short time period is needed for the TPWA to develop to a state to diffract SPPs2, a small time delay is also not optimal for forming the nanohole array.
From the results, we may conclude that a small laser dose (determined by F and v) can weaken the SPPs2 itself, leading to weaker SPPs2’s diffraction orders and no clear etching lines (formed by the SPPs2). In this case, only the etching lines generated by the laser E1 are produced, as shown in Fig. 2(e). This agrees well with the experimental results in Fig. 4, where the periodic ripples are always generated at fast scanning speeds. In addition, weak coupling can result in the weak SPPs2 diffractions, leading to a relatively intense zero order of SPPs2, which may be the possible cause that square arrays are produced. As can be seen in Fig. 4, the square array is very sensitive to the fabrication parameters and the parametric region forming the square array locates inside the one of the periodic ripples; as such, a small or large time delay, which hinders weak coupling states, will not form square arrays. The reasoning proposed here uses a universal model to explain a wide variety of nanostructures fabricated by double femtosecond laser beams, including not only the nanostructures shown in Fig. 2, but also those in the literature, such as the triangle array [12], rhombic array [18], dot array [39] etc. By controlling the incident angle of SPPs on the TPWA, the diffraction orders’ propagation directions and the orientations of the etching lines can be adjusted, which expands the variety of LIPSS-enabled nanopatterns.
Lastly, it is worth to note that although the Sipe’s model well explains various LIPSS patterns [40], it is based on the calculation of diffractional intensity of LIPSS. As the Sipe’s model does not consider surface-scattered waves during the formation of LIPSS, it cannot be used to analyze the formation dynamics of the fabrication results in Fig. 2, which are attributed to the complex dynamics and interactions of the surface-scattered waves.
5. Conclusions
In conclusion, we have applied the double time-delayed collinear femtosecond laser beams to create 2D periodic nanostructures on silicon, including periodic ripples, square array, and nanohole array that are respectively formed by one, two and three sets of parallel lines. We also demonstrate the fabrication of large-area hexagonal nanohole array on silicon, which greatly reduces the fabrication cost and improves the throughput of various functional nanostructures [23–26]. Based on the CMT, a universal explanation about the formation mechanism of the different periodic nanostructures induced by double femtosecond laser beams is proposed and experimentally verified. The 2D CMT well describes and predicts the different nanostructures in our experiments. It is believed that the understanding of the nanostructures’ formation mechanisms can benefit the improvement of the LIPSS pattern’s diversity. Notably, the nanohole array presents the first intuitive experimental demonstration of the optical analogue of the massless Dirac dynamics in waveguide systems – this is also the first optical application of the massless Dirac dynamics.
Funding
National Key Research and Development Program of China (2018YFB0504400); Innovation and Technology Commission, Innovation Technology Fund (ITS/196/19FP); Research Grants Council, General Research Fund (14203020, 14209018).
Acknowledgments
The authors would like to thank Dr. Wei Zhang for his help in AFM measurements, Tingyuan Wang for his help in reflectance spectra measurements, and Yuelin Wang for his help in SEM measurements.
Disclosures
The authors declare no conflicts of interest.
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