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Abstract
We demonstrate a prominent spatial frequency shift (SFS) for the femtosecond laser-induced periodic structures by only changing the polarization ellipticity of the working laser. The nanostructures are fabricated on the surfaces of silicon (Si) and zinc selenide (ZnSe) using elliptically polarized femtosecond laser pulses, with the pulse duration of 35 fs, the central wavelength of 800nm, and the repetition rate of 1kHz. The experimental results show that the red- and blue-shift trends of the SFS are individually represented on silicon and zinc selenide with the increased polarization ellipticity, where low- and high-spatial-frequency nano-ripples are fabricated, respectively. These unique phenomena are explained by using the laser-surface plasmon polariton interference mechanism and the effective medium theory. The proposed nanostructures with regulatable period are further used for creating nano-gratings on silicon which perform chirped characteristics.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
For decades, one of the booming topics in laser precision microfabrication field is the formation principle and application of laser-induced periodic surface structures (LIPSS), also termed as “ripples” [1–4]. These nanostructures have been induced in many materials under the irradiation of linearly polarized laser pulses, of which the orientation depends on the laser polarization direction and the spatial period is less than the laser wavelength [5,6]. Owing to the convenience in processing large-area periodic structures, LIPSS have been used to create functional components [7], such as polarizing optical elements [8,9], superhydrophobic surfaces [10], and structural coloring surfaces [11,12], etc.
The formation mechanism of LIPSS has been intensively studied since it was first reported in 1965 [13], and several theoretical models have been proposed. The early theoretical models developed in 1980s suggested that the formation of LIPSS was attributed to the interference between incident laser and surface-scattered light [14,15]. This model deduced that the period of surface structures on metals and semiconductors or dielectrics equals to laser wavelength λ or λ/n for normal incident lasers, respectively, here n is the refractive index of dielectrics, which is inconsistent with the experiment results. Afterwards the model was modified with the assumption that the LIPSS is generated by the interference between incident laser and surface plasmon polariton (SPP) [16–18]. On this basis, the period of surface structures equals to the SPP wavelength, which is smaller than that of the input laser. Since then, the laser-SPP interference model is commonly used, and some ameliorations were proposed to conform specific experiment results, such as multi-pulse feedback mechanism [19], surface plasma parametric decay process [20], and effective medium approximation [21], etc.
The spatial period and ripple orientation are the most concerned features of LIPSS for laser fabrication, because they are related to the specific function of processed products, such as the diffraction wavelength and polarization response [7]. Usually, the period Λ depends on the utilized laser wavelength λ and refractive index n of the target material. For strong absorbing materials, Λ≈ λ, named low-spatial frequency LIPSS (LSFL), including two types: LSFL-I and LSFL-II with ripple orientation perpendicular or parallel to the polarization direction, respectively. While for large bandgap materials, Λ≈ λ/(2n), named high-spatial frequency LIPSS (HSFL), including two types: HSFL-I and HSFL-II with ripple orientation perpendicular or parallel to the polarization direction, respectively [22]. There are a few ways to control the LIPSS period on a specific material. For semiconductors and metals, the incident laser pulse energy can adjust the spatial frequency (∼1/ Λ) of LIPSS [16,23], by changing the material refractive index through the absorption of laser energy [16]. Increasing the incident laser pulse number can bring a spatial frequency-shift (SFS) to LIPSS [19,24], due to the multi-pulse feedback mechanism [19]. The interference angle between incident light and SPPs can also changes the SFS [25]. Comparing with the spatial frequency, the influencing factors of the orientation of LIPSS are simpler. A number of works have shown that the orientation of LIPSS is decided by the polarization state of incident laser, i.e., it is perpendicular or parallel to the polarization direction [5]. In our recent work, we have demonstrated that elliptical polarization can also induce the LIPSS on silicon (Si) surface [26], with the orientation perpendicular to the long axis of polarization ellipse. However, there has been less previous evidence for the relation between spatial frequency and laser polarization state.
In this work, we experimentally and theoretically demonstrate the SFS of LIPSS with polarization ellipticity of incident laser beam. Under the irradiation of elliptically polarized femtosecond laser pulses, two types of ripples, LSFL-I and HSFL-I, are fabricated on the surface of Si and zinc selenide (ZnSe), respectively. With the increase of polarization ellipticity, LSFL-I performs a red-shift trend of spatial frequency, while HSFL-I performs a blue-shift trend. These phenomena are explained by using the modified laser-SPP interference model, considering the two-photon absorption effect, the dispersion relations of SPP, and the effective medium theory. The theoretical calculation results are conformed to the experimental data. Based on the SFS phenomenon, nano-grating components are fabricated on Si by regulating spatial period with polarization ellipticity.
2. Experimental details
2.1 Experimental setup
To reveal the relationship between the spatial frequency of LIPSS and the polarization ellipticity, we fabricated a set of structures on the surface of a crystalline Si (refractive index nSi=3.669 at wavelength of 800 nm) and a ZnSe (refractive index nZnSe=2.5115 at wavelength of 800 nm) by using elliptically polarized femtosecond laser beams. The experimental setup schematic is shown in Fig. 1. A Ti:sapphire regenerative amplifier system (Spectra Physics, Spitfire ACE-35F) is used as the laser source to generate linearly polarized femtosecond laser pulses, with the pulse duration of 35 fs, the central wavelength of 800 nm, and the repetition rate of 1 kHz. The irradiated laser pulse number is controlled by an electro-mechanic shutter (Thorlabs, SH05). The laser pulse energy is controlled by the combination of a half-wave plate HWP1 and a Glan-Taylor prism GTP. Another half-wave plate HWP2 is used together with a quarter-wave plate QWP to modulate the polarization ellipticity of the femtosecond laser pulse irradiated on the target, which is defined as the ratio of short axis to long axis of polarization ellipse [27]. The elliptically polarized laser beam is focused upon the Si/ ZnSe surface via a ten-/ twenty-fold microscope objective (Daheng Optics, GCO-2102, NA=0.25 and GCO-2107, NA=0.4), which show a constant transmission factor with different polarization ellipticities. The laser power (P) is measured before the microscope objective with a power meter (Thorlabs, PM100D), and the laser pulse energy (E) irradiated on the sample is calculated with the transmittance (T) of objective and laser repetition rate (R) as E = PT/R. The average pulse energy density (F0) is calculated with focus area (S) as F0=E/S. The sample is mounted on a three-axis translation stage (Thorlabs, MAX302), which can be moved underneath the focused laser beam. The fabricated samples are examined by a scanning electron microscope (SEM, FEI, Verios G4) to show the detailed structure morphologies.
Fig. 1. Femtosecond laser micro machining setup. HWP1 and HWP2, zero-order half-wave plates; QWP, zero-order quarter-wave plate; GTP, Glan-Tylor prism; MO, ten-/twenty- fold microscope objective of NA=0.25.
2.2 Red-shift of LSFL-I on Si
In experiments, normally incident laser pulses with different polarization ellipticity were used to irradiate the Si surface in ambient environment, and the derived periodic structures are shown in Figs. 2(a1), 2(b1), and 2(c1), of which the corresponding polarization ellipticities were tanχ=0, 0.18, and 0.36, respectively. The single pulse energy was 140nJ, which corresponds to an energy density of 0.27 J/cm2 (focus diameter was 8.1μm, measured at 1/e of maximum intensity). The laser pulse number was optimized as 100 per site. As displayed in Figs. 2(a1)-(c1), ripples are patterned on the Si surface, with their orientation orthogonal to the long axes of polarization ellipses, and the spatial period is in the range of 620 nm<Λ<690 nm, which can be considered as LSFL-I (Λ≈ λ). In order to measure the spatial period accurately, we introduce the autocorrelation method to further analyze the morphology.
Fig. 2. (a1)- (c1) SEM images and (a2)- (c2) the corresponding autocorrelation images of LIPSS on Si. The solid arrows in the top right-hand corner represent the polarization ellipses of incident beams, and the dotted lines mark the perpendicular direction of ripple orientation. (d) Autocorrelation curves along the corresponding dotted lines in (a2)- (c2). A, B and C are the positions of corresponding first peaks, which indicate the calculated spatial frequencies of the LIPSS.
Autocorrelation is one of the typical tools for period inspection and signal extraction in digital signal processing. The autocorrelation function can be written as [28]
Figures 2(a2)-(c2) display the autocorrelation responses of the morphology images corresponding to Figs. 2(a1)-(c1), which are calculated with the discretization of Eq. (1) by considering the SEM images as the original functions. As shown in Figs. 2(a2)-(c2), the grating-like images clearly visualize the periodic characteristics of fabricated structures and the distance between two peaks represents the average period of corresponding ripple. For comparison, Fig. 2(d) depicts the autocorrelation curves along the perpendicular directions (doted lines) of the ripple orientation. It can be seen that the spatial period of LIPSS increases with the polarization ellipticity. We name this increase of ripple period (decrease of spatial frequency) with polarization ellipticity as “red-shift”. Points A, B, and C in Fig. 2(d) mark the positions of the first peaks in autocorrelation curves, which indicate the spatial periods of corresponding nanostructures. In these three cases, the measured spatial periods are Λ=624 nm, 649 nm, and 689 nm, respectively.
To ensure the feasibility and repeatability of the experimental data, we take 21 samples of the polarization ellipticities between 0 and 0.36, with each sample repeated 8 times under the same condition. The statistical results are shown in Fig. 3. The average period of LIPSS increased from 641 nm to 668 nm with the increase of polarization ellipticity of incident laser. Namely, the results reveal red-shift of spatial frequency of LSFL-I.
Fig. 3. Spatial period of LIPSS on Si versus polarization ellipticity of incident laser. The laser pulse energy is 140 nJ (the energy density is 0.27 J/cm2), and pulse number is 100.
2.3 Blue-shift of HSFL-I on ZnSe
With the same experimental configuration, the periodic structures with higher spatial frequency were fabricated on ZnSe, as displayed in Figs. 4(a1)-(c1), of which the polarization ellipticities were tanχ=0, 0.10, and 0.19, respectively. The laser pulse number kept 100, and the single pulse energy was 65 nJ, the energy density thus was 0.33J/cm2 (focus diameter was 5.0 μm, measured at 1/e of maximum intensity). The corresponding autocorrelation results are shown in Figs. 4(a2)-(c2), and the corresponding autocorrelation curves along the perpendicular directions are shown in Fig. 4(d). As shown, the spatial periods of LIPSS are Λ=192 nm, 176 nm, and 163 nm in the cases of tanχ=0, 0.10 and 0.19, respectively. Consequently, this kind of ripple is the HSFL-I type [Λ≈λ/(2n)].
Fig. 4. (a1)- (c1) SEM images and (a2)- (c2) the corresponding autocorrelation images of LIPSS on ZnSe. The solid arrows in the top right-hand corner represent the polarization ellipses of incident lasers. (d) Autocorrelation curves along the corresponding dotted lines in (a2)- (c2). A, B and C are the positions of first peaks. (e) Spatial period of LIPSS on ZnSe versus polarization ellipticity of incident laser.
Figure 4(e) displays the trend of spatial period of LIPSS on ZnSe with the increase of polarization ellipticity. Considering that the laser-induced structures exhibit inferior periodicity at higher ellipticity values, which would introduce extra errors, the polarization ellipticity is sampled in the range [0, 0.19]. Similarly, the experiments were repeated 8 times under the same condition. In contrast to the results on Si, the spatial period of LIPSS on ZnSe decreases from 185 nm to 168 nm with the increase of polarization ellipticity. Namely, the results reveal blue-shift of spatial frequency.
3. Results and discussions
3.1 Theoretical analysis of SFS on Si
The above experiment results indicate that polarization ellipticity significantly affects the spatial period of LIPSS. To reveal the SFS on Si, we developed a theoretical view based on the laser-SPP interference model, by combining two-photon absorption effect and dispersion relations of SPP. For the sake of brevity, we list the physical quantities and their values to be used in theoretical analysis in Table 1.
Table 1. Values of physical quantities used in the theoretical analysis.
The origin of the LIPSS is commonly considered as the interference of the incident laser and SPP. SPPs can only propagate along the interface between two media that the dielectric functions ε1 and ε2 satisfy the conditions: ε1'+ε2'<0 and ε1'ε2'<0 (ε1’ and ε2’ are the real parts of ε1 and ε2, respectively, the imaginary parts are concerned with the decay of SPP along the surface) [30]. Therefore, for semiconductors (ε'>0), the first step to generate LIPSS is the plasma (ε'<0) excitation at the sample surface by the incident femtosecond laser pulses. In the cases of Si (the bandgap is Eg=1.11 eV), the valence band electrons can jump into conduction band by the two-photon absorption (TPA) effect [31], forming an electron-hole plasma layer on sample surface. TPA pertains to the third-order nonlinear effect, which is influenced by the polarization ellipticity of pump laser. The coefficient of TPA can be expressed as [32]
The blue curve in Fig. 5 displays the variation of TPA coefficient with the polarization ellipticity for Si, calculated with Eq. (3). Obviously, the absorption coefficient decreases with the increase of polarization ellipticity, which is considered as the origin of SFS of LIPSS in our theoretical framework. The TPA coefficient βel is affected by the polarization ellipticity of incident laser due to the third-order nonlinear effect, as expressed in Eq. (2), and the laser-induced electron plasma density is affected by βel duet to nonlinear absorption. Owing to the variant TPA coefficients, the excited plasmas in sample surfaces exhibit different dielectric properties under the irradiation of laser beams with different polarization ellipticity. In semiconductors, the carrier density in laser excited plasmas is dependent on the absorption of laser energy. The carrier density Nc in laser excited electron-hole plasmas can be expressed as [33]
Fig. 5. TPA coefficient and the real part of dielectric function of laser excited Si versus polarization ellipticity.
The dielectric function εp of laser excited plasma can be described by Drude model as follows [34],
Under the irradiation of femtosecond laser, a plasma layer is excited on the sample surface, forming a sandwich structure composed of air-plasma-Si [35], as sketched in Fig. 6(a). Based on the calculation results of dielectric functions in Fig. 5(b), we can identify which interface support the propagation of SPPs. According to the criterion ε1'+ε2'<0 and ε1'ε2'<0, SPPs can only propagate along the air-plasma interface for Si, as illustrated in Fig. 6(a).
Fig. 6. (a) Sketches of laser excited SPP on Si. (b) Normalized SPP wavevector versus relative frequency. The curve is individually divided into three segments by the relative frequencies Ω1=0.212 and Ω2=0.310. Point A represents the experimental condition under the irradiation of linearly polarized laser. The arrow points the drift direction of SPP wavevector with the increase of polarization ellipticity.
The wavevector of SPP propagates along an interface is given by [34],
According to the experimental condition, we seek out the corresponding theoretical values of SPP wavevectors from Fig. 6(b). Point A corresponds to the experimental condition on Si under the irradiation of linearly polarized laser pulses, where the relative frequency is ω/ωp=0.227 and SPP wavevector is kSPP=1.026k0. With the increase of polarization ellipticity, the plasma frequency ωp decreases but the relative frequency ω/ωp increases, leading to the decrease of the wavevector on Si, as depicted by the red arrow in Fig. 6(b).
The laser-SPP interference model has indicated that the spatial period of LIPSS is equal to the SPP wavelength for the normal incidence of laser beams, i.e., Λ=λSPP=2π/kSPP. Therefore, the SFS of LIPSS can be inferred from the drift of SPP wavevectors, as indicated by the arrow in Fig. 6(b). The spatial frequency of LIPSS on Si exhibits a red-shift with the increase of polarization ellipticity.
In above discussions, the SFS trend of LSFL-I on Si is consistent with experiment results. However, there is a discrepancy between the calculated LIPSS period and the experimental one. Combination with Eqs. (3)–(6), the LIPSS period (Λ=λSPP=2π/kSPP) on Si is ΛSi=780 nm under the irradiation of linearly polarized laser beams. Yet for the experiment results, the average periods are 641 nm and 185 nm, respectively. This discrepancy might be caused by the laser-induced nano-defects, so we introduce the effective medium theory (EMT) [37] to modify the theoretical model.
The origin of laser excited SPPs is considered as the scattering of incident light by laser-induced nano-defects (particles or cavities) [38]. The nano-defects should not be ignored for the propagation of surface electromagnetic waves, and have an influence on the SPP wavelength. Considering the ensemble of laser-induced nano-defects, EMT describes the mesoscopic and macroscopic electromagnetic properties of the composite. As an approximation, the nano-defects are assumed to be spheres with radius a. We denote the characteristic distance between the defects as b, and assume that a<<b<<λ, where λ is the laser wavelength. Then the effective dielectric function εeff of the medium-defects composite can be quantitatively evaluated with the Maxwell-Garnett formula of effective medium [37],
Figure 7 illustrates that the approximation of laser excited surface systems on Si, SPPs exist on the plasma-air interface, and the laser-induced defects is considered as nano-particles inlaid on the plasma surface, as displayed in Fig. 7(a). Consequently, the effective medium is composed of air doped with plasma particles (εh=ε0 and εi=εp), as shown in Fig. 7(b) [39].
Utilizing the EMT approximation, the spatial period of LIPSS on Si can be calculated as ΛSi=λSPP=λ[(εp+εeff)/εpεeff]1/2. The experimental data and the corresponding theoretically calculated results for different polarization ellipticities are displayed in Fig. 8. For Si, the volume fraction values are set to 0.105, 0.115 and 0.125. As shown in the curve, the experimental periods of LIPSS are mainly located within the calculated curves that 0.105<f<0.125, so the average curve that f=0.115 is selected as the optimum theoretical results, which is consistent with experimental results.
Fig. 8. Theoretical calculation and experimental measurement results on Si with the increase of polarization ellipticity.
3.2 Fabrication of nano-gratings
The ellipticity-dependent SFS of LIPSS could be used to regulate the spatial period of the fabricated structures. Following this method, we processed nano-gratings with different periods by adjusting the polarization ellipticity of incident laser beams. Using the experimental setup in Fig. 1, nano-gratings are produced on Si by laser scanning, the laser pulse energy density is 0.28 J/cm2, and the scanning speed is optimized as 18 μm/s. The produced components are shown in Figs. 9(a1)- (a3), of which the utilized polarization ellipticities were tanχ=0, 0.18, and 0.36, respectively. Figures 9(b1)- (b3) display the SEM images of corresponding nano-gratings, the measured periods are Λ=645 nm, 659 nm, and 672 nm, respectively. With the increase of polarization ellipticity, the period of nano-gratings increases, which make the components exhibit different colors under white light.
Fig. 9. (a1)- (a3) Nano-gratings processed by laser scanning and (b1)- (b3) the corresponding SEM images. (c) Diffraction spectrum of nano-gratings in (a). A, B and C are the central wavelength of the spectrum. (d) Experimental setup for measuring diffraction spectrum. (e1)- (e3) Nano-gratings which perform chirped characteristics. The blue curves on the right of picture represent the laser polarization ellipticities used for fabricating nano-gratings, of which the horizontal axis represents polarization ellipticity and the vertical axis represents position. The value range in horizontal axis is [0, 0.36].
The performed colors of the samples can be clearly identified in the diffraction spectrum as in Fig. 9(c), which are measured using the setup shown in Fig. 9(d). The fabricated nano-gratings were illuminated by a supercontinuum laser (YSL Photonics, SC-PRO), and the spectrum was measured at a diffraction angle of 80° using a spectrometer (Ocean Optics, USB4000). As shown in Fig. 9(c), the diffraction spectra are staggered and the central wavelength increases from 629 nm to 667 nm. By adjusting the polarization ellipticity of fabrication laser, we can spatially regulate the period of LIPSSs to fabricate nano-gratings which perform chirped characteristics, as shown in Figs. 9(e1)- (e3). The utilized laser energy density and scanning speed are 0.28 J/cm2 and 18 μm/s, respectively. The blue curves in the right of picture represent the laser polarization ellipticity used in fabricating nano-gratings. As shown in the figures, the polarization ellipticity is set to linearly (e1), abruptly (e2), and sinusoidally (e3) changed in [0, 0.36], the processed nano-gratings with the period vary between 640 nm and 690 nm. The fabricated nano-gratings could find its potential application as a chirped grating coupler for efficiently coupling light into silicon waveguide [40].
4. Conclusion
The period of LIPSS have been demonstrated to be polarization dependent. The spatial frequency of the fabricated structures on Si and ZnSe exhibit red- and blue-shift with the increase of laser polarization ellipticity, respectively. Based on the laser-SPP interference model, combining the two-photon absorption effect and the effective medium approximation, a theoretical model is proposed to explain the experimental phenomena on Si. The ellipticity-dependent property of LIPSS is further used to regulate the spatial period of the fabricated structures to create chirped nano-gratings on Si. The variation of LIPSS period is mainly related with the changes in laser pulse energy, pulse number or incident angle in previous studies. Our results confirm that the polarization ellipticity is also applicable to shift the spatial frequency of LIPSSs firstly, as far as we know. Our results would extend the scope of laser polarization states to the domain of laser-matter interaction, and predict fabrication of several potential optoelectronic devices with spectral response.
Funding
Natural Science Basic Research Program of Shaanxi Province (2020JM-104); National Natural Science Foundation of China (11634010, 11774289, 12074312, 91850118); National Key Research and Development Program of China (2017YFA0303800); NSAF Joint Fund (U1630125).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper can be obtained from the authors upon reasonable request.
References
1. H. M. van Driel, J. E. Sipe, and J. F. Young, “Laser-Induced Periodic Surface-Structure on Solids - a Universal Phenomenon,” Phys. Rev. Lett. 49(26), 1955–1958 (1982). [CrossRef]
2. S. Höhm, A. Rosenfeld, J. Krüger, and J. Bonse, “Femtosecond laser-induced periodic surface structures on silica,” J. Appl. Phys. 112(1), 014901 (2012). [CrossRef]
3. J. Bonse, S. Hohm, S. V. Kirner, A. Rosenfeld, and J. Kruger, “Laser-Induced Periodic Surface Structures— A Scientific Evergreen,” IEEE J. Sel. Top. Quantum Electron. 23(3), 9000615 (2017). [CrossRef]
4. J. Bonse, “Quo Vadis LIPSS? -Recent and Future Trends on Laser-Induced Periodic Surface Structures,” Nanomaterials 10(10), 1950 (2020). [CrossRef]
5. J. Bonse, J. Krüger, S. Höhm, and A. Rosenfeld, “Femtosecond laser-induced periodic surface structures,” J. Laser Appl. 24(4), 042006 (2012). [CrossRef]
6. A. Borowiec and H. K. Haugen, “Subwavelength ripple formation on the surfaces of compound semiconductors irradiated with femtosecond laser pulses,” Appl. Phys. Lett. 82(25), 4462–4464 (2003). [CrossRef]
7. J. Bonse, S. V. Kirner, S. Höhm, N. Epperlein, D. Spaltmann, A. Rosenfeld, and J. Krüger, “Applications of laser-induced periodic surface structures (LIPSS),” Proc. SPIE 10092, 100920N (2017). [CrossRef]
8. A. Cerkauskaite, R. Drevinskas, A. Solodar, I. Abdulhalim, and P. G. Kazansky, “Form-Birefringence in ITO Thin Films Engineered by Ultrafast Laser Nanostructuring,” ACS Photonics 4(11), 2944–2951 (2017). [CrossRef]
9. E. Skoulas, A. C. Tasolamprou, G. Kenanakis, and E. Stratakis, “Laser induced periodic surface structures as polarizing optical elements,” Appl. Surf. Sci. 541, 148470 (2021). [CrossRef]
10. B. Wu, M. Zhou, J. Li, X. Ye, G. Li, and L. Cai, “Superhydrophobic surfaces fabricated by microstructuring of stainless steel using a femtosecond laser,” Appl. Surf. Sci. 256(1), 61–66 (2009). [CrossRef]
11. G. Li, J. Li, Y. Hu, C. Zhang, X. Li, J. Chu, and W. Huang, “Femtosecond laser color marking stainless steel surface with different wavelengths,” Appl. Phys. A 118(4), 1189–1196 (2015). [CrossRef]
12. S. Maragkaki, C. A. Skaradzinski, R. Nett, and E. L. Gurevich, “Influence of defects on structural colours generated by laser-induced ripples,” Sci. Rep. 10(1), 53 (2020). [CrossRef]
13. M. Birnbaum, “Semiconductor Surface Damage Produced by Ruby Lasers,” J. Appl. Phys. 36(11), 3688–3689 (1965). [CrossRef]
14. J. F. Young, J. E. Sipe, J. S. Preston, and H. M. van Driel, “Laser-induced periodic surface damage and radiation remnants,” Appl. Phys. Lett. 41(3), 261–264 (1982). [CrossRef]
15. J. E. Sipe, J. F. Young, J. S. Preston, and H. M. van Driel, “Laser-induced periodic surface structure. I. Theory,” Phys. Rev. B 27(2), 1141–1154 (1983). [CrossRef]
16. J. Bonse, A. Rosenfeld, and J. Krüger, “On the role of surface plasmon polaritons in the formation of laser-induced periodic surface structures upon irradiation of silicon by femtosecond-laser pulses,” J. Appl. Phys. 106(10), 104910 (2009). [CrossRef]
17. G. Miyaji, M. Hagiya, and K. Miyazaki, “Excitation of surface plasmon polaritons on silicon with an intense femtosecond laser pulse,” Phys. Rev. B 96(4), 045122 (2017). [CrossRef]
18. Y. Fuentes-Edfuf, J. A. Sánchez-Gil, C. Florian, V. Giannini, J. Solis, and J. Siegel, “Surface Plasmon Polaritons on Rough Metal Surfaces: Role in the Formation of Laser-Induced Periodic Surface Structures,” ACS Omega 4(4), 6939–6946 (2019). [CrossRef]
19. M. Huang, F. L. Zhao, Y. Cheng, N. S. Xu, and Z. Z. Xu, “Origin of Laser-Induced Near-Subwavelength Ripples: Interference between Surface Plasmons and Incident Laser,” ACS Nano 3(12), 4062–4070 (2009). [CrossRef]
20. S. Sakabe, M. Hashida, S. Tokita, S. Namba, and K. Okamuro, “Mechanism for self-formation of periodic grating structures on a metal surface by a femtosecond laser pulse,” Phys. Rev. B 79(3), 033409 (2009). [CrossRef]
21. H. U. Lim, J. Kang, C. Guo, and T. Y. Hwang, “Manipulation of multiple periodic surface structures on metals induced by femtosecond lasers,” Appl. Surf. Sci. 454, 327–333 (2018). [CrossRef]
22. J. Bonse and S. Gräf, “Maxwell Meets Marangoni—A Review of Theories on Laser-Induced Periodic Surface Structures,” Laser Photonics Rev. 14(10), 2000215 (2020). [CrossRef]
23. K. Okamuro, M. Hashida, Y. Miyasaka, Y. Ikuta, S. Tokita, and S. Sakabe, “Laser fluence dependence of periodic grating structures formed on metal surfaces under femtosecond laser pulse irradiation,” Phys. Rev. B 82(16), 165417 (2010). [CrossRef]
24. J. Bonse and J. Krüger, “Pulse number dependence of laser-induced periodic surface structures for femtosecond laser irradiation of silicon,” J. Appl. Phys. 108(3), 034903 (2010). [CrossRef]
25. T. Y. Hwang and C. Guo, “Angular effects of nanostructure-covered femtosecond laser induced periodic surface structures on metals,” J. Appl. Phys. 108(7), 073523 (2010). [CrossRef]
26. H. C. Cheng, P. Li, S. Liu, H. Lu, L. Han, and J. L. Zhao, “Polarization-switchable nanoripples fabricated on a silicon surface by femtosecond-laser-assisted nanopatterning,” Appl. Opt. 59(24), 7211–7216 (2020). [CrossRef]
27. D. H. Goldstein, Polarized Light, third ed. (CRC, 2017). [CrossRef]
28. J. W. Goodman, Introduction to Fourier optics, third ed. (Roberts & Company, 2004).
29. C. Schinke, P. C. Peest, J. Schmidt, R. Brendel, K. Bothe, M. R. Vogt, I. Kroger, S. Winter, A. Schirmacher, S. Lim, H. T. Nguyen, and D. MacDonald, “Uncertainty analysis for the coefficient of band-to-band absorption of crystalline silicon,” AIP Adv. 5(6), 067168 (2015). [CrossRef]
30. T. J. Y. Derrien, J. Kruger, and J. Bonse, “Properties of surface plasmon polaritons on lossy materials: lifetimes, periods and excitation conditions,” J. Opt. 18(11), 115007 (2016). [CrossRef]
31. R. W. Boyd, Nonlinear optics, third ed. (Academic, 2008).
32. Z. B. Liu, X. Q. Yan, W. Y. Zhou, and J. G. Tian, “Evolutions of polarization and nonlinearities in an isotropic nonlinear medium,” Opt. Express 16(11), 8144–8149 (2008). [CrossRef]
33. K. Sokolowski-Tinten and D. von der Linde, “Generation of dense electron-hole plasmas in silicon,” Phys. Rev. B 61(4), 2643–2650 (2000). [CrossRef]
34. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer Science and Business Media, 2007). [CrossRef]
35. L. Wang, Q. D. Chen, X. W. Cao, R. Buividas, X. Wang, S. Juodkazis, and H. B. Sun, “Plasmonic nano-printing: large-area nanoscale energy deposition for efficient surface texturing,” Light: Sci. Appl. 6(12), e17112 (2017). [CrossRef]
36. E. T. Arakawa, M. W. Williams, R. N. Hamm, and R. H. Ritchie, “Effect of Damping on Surface Plasmon Dispersion,” Phys. Rev. Lett. 31(18), 1127–1129 (1973). [CrossRef]
37. T. C. Choy, Effective Medium Theory: Principles and Applications, second ed. (Oxford University, 2016).
38. A. Abou Saleh, A. Rudenko, L. Douillard, F. Pigeon, F. Garrelie, and J.-P. Colombier, “Nanoscale Imaging of Ultrafast Light Coupling to Self-Organized Nanostructures,” ACS Photonics 6(9), 2287–2294 (2019). [CrossRef]
39. A. Rudenko, C. Mauclair, F. Garrelie, R. Stoian, and J.-P. Colombier, “Amplification and regulation of periodic nanostructures in multipulse ultrashort laser-induced surface evolution by electromagnetic-hydrodynamic simulations,” Phys. Rev. B 99(23), 235412 (2019). [CrossRef]
40. C. Xia, L. Chao, and T. Hon Ki, “Fabrication-Tolerant Waveguide Chirped Grating Coupler for Coupling to a Perfectly Vertical Optical Fiber,” IEEE Photon. Technol. Lett. 20(23), 1914–1916 (2008). [CrossRef]
