domain: spi.gproc spi phys

text:
Maksim Sergeev

Yaroslava Andreeva
email: andreeva.ym@gmail.com

Vladislav Koval

Maxim Sergeev

Vadim P Veiko

Nathalie Destouches

Francis Vocanson

Hongfeng Ma

Anton Loshachenko

Tatiana E Itina
email: tatiana.itina@univ-st-etienne.fr

Maxim Sergeev

Picosecond laser writing of -2 nanocomposite nanogratings for optical filtering

Keywords: Laser surface structuring, laser fabrication, nanoparticles

nanocomposites, silica glass

Introduction

Various surface patterns as well as micro-and nanostructures are traditionally of great practical interest in optics and photonics, where they are used for light manipulation , encoding and decoding optical information, for optical switching, interferometry, SERS, sensors, etc. [START_REF] Hülsenberg | Microstructuring of glasses[END_REF][START_REF] Masuda | 3-d microstructuring inside photosensitive glass by femtosecond laser excitation[END_REF][START_REF] Otanicar | Filtering light with nanoparticles: a review of optically selective particles and applications[END_REF][START_REF] Gu | Optical storage arrays: a perspective for future big data storage[END_REF]. Additionally, the unique optical properties of the periodic surface structures, such as gratings and metasurfaces, further extend the range of their applications to security, chemistry, biology, and medicine [START_REF] Holloway | An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials[END_REF].

Numerous studies focused on a surface nanostructures design and considerable progress was achieved [START_REF] Marqués | Metamaterials with negative parameters: theory, design, and microwave applications[END_REF][START_REF] Bilotti | Equivalent-circuit models for the design of metamaterials based on artificial magnetic inclusions[END_REF]. However, a well-controlled fabrication of such structures still represents a great challenge. For this, many methods were proposed ranging from bottom-up, such as colloidal and cluster chemistry [START_REF] Gazoni | Designed nanoparticle-mesoporous multilayer nanocomposites as tunable plasmonic-photonic architectures for electromagnetic field enhancement[END_REF], to top-down, including nanoparticle-assisted photo-lithography and multi-photon polymerization. However, nanoparticle manipulation at nanoscale and fabricating well-organized or periodic nanoparticle-containing structures are still challenging. In fact, many bottom-up procedures are particularly hard to control because of self-assembly or auto-organization that is often involved.

To solve this issue, a well-optimized laser treatment is particularly promising, namely because of the control possibilities that lasers can provide. In fact, lasers are known to be a powerful surface structuring tool [START_REF] Fukushima | Microfabrication of gold dots in sio2/tio2 glass films by two-photon absorption[END_REF] that can also be used for growing and tailoring the properties of nano-objects [START_REF] Xu | High efficiency integration of three-dimensional functional microdevices inside a microfluidic chip by using femtosecond laser multifoci parallel microfabrication[END_REF][START_REF] Jonušauskas | Plasmon assisted 3d microstructuring of gold nanoparticle-doped polymers[END_REF], and, particularly, for nanoparticle formation in solid matrices [START_REF] Baraldi | Polarization-driven self-organization of silver nanoparticles in 1d and 2d subwavelength gratings for plasmonic photocatalysis[END_REF].

Laser processing of transparent materials by beam interference further extended the possibilities of both surface and volume nanostructuring. In this case, additional possibilities related with nanoparticle presence [START_REF] Natoli | Localized pulsed laser interaction with sub-micronic gold particles embedded in silica: a method for investigating laser damage initiation[END_REF][START_REF] Podlipensky | Femtosecond laser assisted production of dichroitic 3d structures in composite glass containing ag nanoparticles[END_REF] and with laser beam focusing in the in bulk were explored [START_REF] Juodkazis | Three-dimensional microfabrication of materials by femtosecond lasers for photonics applications[END_REF]. The photocontrolled organization of metallic NPs in T iO 2 and SiO 2 films was also demonstrated [START_REF] Baraldi | Polarization-driven self-organization of silver nanoparticles in 1d and 2d subwavelength gratings for plasmonic photocatalysis[END_REF]. Additionally, self-organized nanostructures were shown to be formed by scanning CW laser irradiation [START_REF] Liu | Understanding the growth mechanisms of ag nanoparticles controlled by plasmon-induced charge transfers in ag-tio2 films[END_REF] and even femtosecond laser pulses [START_REF] Liu | Three-dimensional self-organization in nanocomposite layered systems by ultrafast laser pulses[END_REF].

Laser beam interference had already been shown to be promising for a controlled synthesis of silver nanoparticles (Ag NPs) in mesoporous T iO 2 films [START_REF] Destouches | Photodirected organization of silver nanoparticles in mesostructured silica and titania films[END_REF]. As a result, periodic structures consisting of highly ordered arrays of NPs were formed with period, comparable to laser wavelength. The advantage of such 2D ordered nanoparticle arrays is in their sensitivity to the incident light, which depends on both the angle of incidence and the polarization direction [START_REF] Destouches | Self-organized growth of metallic nanoparticles in a thin film under homogeneous and continuouswave light excitation[END_REF][START_REF] Loeschner | Self-organized, gratinglike nanostructures in polymer films with embedded metal nanoparticles induced by femtosecond laser irradiation[END_REF].

However, well-controlled synthesis of metallic NPs in nanostructures as well as laser modifications SiO 2 surface relief remain unstudied. In fact, the concentration of free electrons is very small in mesoporous SiO 2 matrices. These electrons are essential for the efficient ion reduction that allows nanocluster formation. As electron donors, polyatomic organic compounds or metallic salts can be used. For instance, a solution of silver nitrate in ethylene glycol was dissolved in such matrices [START_REF] Luo | The role of poly (ethylene glycol) in the formation of silver nanoparticles[END_REF][START_REF] Sudeep | Photosensitized growth of silver nanoparticles under visible light irradiation: a mechanistic investigation[END_REF]. Upon laser irradiation, ethylene glycol is decomposed, so that free electrons are released and can be involved in the reduction of silver ions. It should be emphasized that the synthesis of Ag NPs in SiO 2 films becomes extremely difficult in the absence of such an electron donor.

This work aims to demonstrate the possibilities of a controlled synthesis of periodic structures in porous SiO 2 films prepared by the sol-gel method and impregnated with silver nitrate. For this, we use a laser beam interference set-up with two picosecond lasers operating in a scanning regime [START_REF] Burrow | Multi-beam interference advances and applications: nano-electronics, photonic crystals, metamaterials, subwavelength structures, optical trapping, and biomedical structures[END_REF]. In the performed experiments, laser irradiation not only leads to silver nanoparticle formation and manipulation, but also to the periodic surface structuring of the nanocoposite films. The resulted periodic surface structures have several potential applications, namely as sensors, polarization splitters, couplers, etc. Additionally, such structures can be useful for molecular detection, photocatalysis [START_REF] Baraldi | Polarization-driven self-organization of silver nanoparticles in 1d and 2d subwavelength gratings for plasmonic photocatalysis[END_REF][START_REF] Crespo-Monteiro | Microstructuring of mesoporous titania films loaded with silver salts to enhance the photocatalytic degradation of methyl blue under visible light[END_REF] as well as for diffracting deflector used in solar cells [START_REF] Gâté | Dynamic interferometry lithography on a tio2 photoresist sol-gel for diffracting deflector module[END_REF]. The revealed structure formation mechanisms are examined and explained. Finally, interesting optical properties of the obtained structures are discussed.

Materials and methods

Samples preparation

Mesoporous silica films were prepared following an evaporation induced self assembly (EISA) route as described in [START_REF] Bakhti | Switchable silver nanostructures controlled with an atomic force microscope[END_REF][START_REF] Destouches | Photodirected organization of silver nanoparticles in mesostructured silica and titania films[END_REF]. Briefly, the sol-gel formulation was prepared by mixing tetraetoxysilane (TEOS) as silica precursor, with hydrochloric acid solution as catalyst, with tribloc copolymer F127 as structuring agent dissolved in ethanol. The films were deposited by dipcoating them on cleaned glass substrates before calcination at 673.15 K to remove the copolymer templates and form interconnected mesopores. Next, the films were soaked in an ionic silver nitrate solution 0.5 M (water/EtOH 1:1) for 1h before rinsing and drying at room temperature. The resulting film thickness is around 180 nm. The average pore size in the films is ∼ 7 ± 3 nm; the pore volume is ∼10%.

Laser treatment

An experimental setup used for laser micro processing by a two-beam interference field is shown in figure 1. The processing schematics consists of a laser source, a block of mirrors, a phase diffraction grating with a period p=30 µm, a lens with a focal length f 1 , a spatial filter, an aspherical lens with a focal length f 2 , and a threecoordinate positioning system. Regarding the laser source, we used the third harmonic of picosecond Nd:YAG laser λ=355 nm at the repetition rate of f =10 Hz and a pulse duration equal to τ =30 ps. The average energy was varied from 3 to 60 µJ, and the number of pulses for one laser spot N was taken in the range from 1 to 1000. Optical transmission of the optical scheme was about 17%

To determine the modification threshold, laser processing was conducted in a stationary mode (without movements of the sample along X and Y axis). Taking into account the pulse repetition rate ν = 10 Hz and the spot diameter d=98 µm, the scanning speed corresponding to this processing mode was determined as follows: V =dν/N .

To create a stable, double-beam interference pattern in the focal plane, a laser beam was split by a phase grating, and two identical beams of the first interference order were subtracted by a spatial filter. Next, the beams were focused by the second lens creating an interference pattern at the intersection [START_REF] Veiko | Picosecond laser registration of interference pattern by oxidation of thin cr films[END_REF][START_REF] Gedvilas | Thermo-chemical microstructuring of thin metal films using multi-beam interference by short (nano-& picosecond) laser pulses[END_REF]. In the case of a confocal scheme, the angle between two beams is changed to manage the period of the pattern, which is determined as [START_REF] Srikanth | Measurement of period of interference patterns with submicron period[END_REF]:

Λ = λ/2sinθ ( 1
)
where Λ is the period of the interference pattern, λ is the wavelength, θ is the angle between two beams. The angle θ is expressed through the distance between the diffraction maxima of ± 1 order and focal length f 2 as follows:

θ = arctg(L ±1 /2f 2 ) (2)
where L ± 1 is the distance between the diffraction maximums of ± 1 order and f 2 is a focal length of the second lens. Hence, the easiest way to change the angle between the interfering beams is to manipulate it with focal distance of the lenses. In our setup, the first lens was changed to obtain different periods of the interference field. To create interference patterns with the period of about 400 and 500 nm, we used lenses with focal distances of 8 and 9,7 mm, respectively. In both cases, the first micro lens with 300 mm focal distance was used. The expected (calculated) periods of nanostructures were 415 and 516 nm, but due to non-ideal optical scheme and optical losses on the components the actual nanopatterns with the periods of 410 and 540 nm occurred.

Characterization

To investigate the resulting structures on the nanoscale, a CrossBeam workstation Zeiss AURIGA as a scanning electron microscope (SEM) was used. The microscope has relatively high special resolution (images were carried out using SE2 detector at acceleration voltage 10-15 kV, beam current 600 pA) and good image contrast. Optical properties were characterized by using a microscope-spectrophotometer MSFU LOMO. The transmittance and reflectance spectra in visible wave range as well as diffusion reflectance spectra were measured in circular areas of about 50 µm in diameter. Carl Zeiss Axio Imager A1M optical microscope with the magnification up to 160× was used for visual analysis of the obtained structures in both reflected and transmitted light modes. The microscope is equipped with additional λ/4 plate allowing observations in the polarization contrast mode.

Results and discussion

Nanograting fabrication

The mesoporous SiO 2 thin films are activated by the presence of small Ag clusters and subjected to the scanning laser interference field. The direction of the interference pattern is parallel to the scanning direction. We use two identical laser beams and the absorption is enhanced by the formation and growth of silver nanoparticles (NPs). Consequently, different nanogratings were recorded at a constant scanning speed of V sc =1.6 µm/s and various laser fluences. Figure 2 demonstrates SEM microimages of the nanogratings. The obtained nanostructures turned out to be oriented differently as a function of laser fluence. For lower laser fluences, structures are aligned along the scanning direction, as expected for the interference pattern (Fig. 2b). With the increase in laser fluence, however, instead of the interference pattern based structure, another one appears and is oriented along laser polarization (Fig. 2c). Further increase in laser fluence above a certain threshold obviously leads to film damage (Fig. 2d).

An interference pattern is formed at smaller laser energy. With the increase in energy, the periodic structure appears to be parallel to laser polarization, contrary to our laser interference pattern. For instance, this effect can be observed in the central part of the laser track if peak laser fluence is too large. Such periodic self-organization is often observed in laser processing of various surfaces, including metals, semiconductors, dielectrics, ceramics, polymers, and nano-composites [START_REF] Huang | Origin of laser-induced near-subwavelength ripples: interference between surface plasmons and incident laser[END_REF][START_REF] Tsibidis | Convection roll-driven generation of supra-wavelength periodic surface structures on dielectrics upon irradiation with femtosecond pulsed lasers[END_REF][START_REF] Gräf | Formation and properties of laser-induced periodic surface structures on different glasses[END_REF][START_REF] Golosov | Femtosecond laser writing of subwave one-dimensional quasiperiodic nanostructures on a titanium surface[END_REF][START_REF] Höhm | Laser-induced periodic surface structures on titanium upon single-and two-color femtosecond double-pulse irradiation[END_REF][START_REF] Bonse | On the role of surface plasmon polaritons in the formation of laser-induced periodic surface structures upon irradiation of silicon by femtosecond-laser pulses[END_REF][START_REF] Bonse | Femtosecond laser-induced periodic surface structures[END_REF][START_REF] Rudenko | Spontaneous periodic ordering on the surface and in the bulk of dielectrics irradiated by ultrafast laser: a shared electromagnetic origin[END_REF]. Possible formation scenarios range from wave interference [START_REF] Huang | Origin of laser-induced near-subwavelength ripples: interference between surface plasmons and incident laser[END_REF] to various instabilities and thermo-capillary waves [START_REF] Ursu | On the role of the periodical structures induced by powerful laser irradiation of metallic surfaces in the energy coupling process[END_REF]. The prevailing formation mechanisms depend on the laser-irradiated material, as well as on the fabricated structure orientation and periodicity. On one hand, periodic sub-wavelength structures, such as ripples, are typically formed at lower laser fluence and mostly oriented perpendicular to laser polarization. These small structures are attributed to the plasmonic effects [START_REF] Bonse | On the role of surface plasmon polaritons in the formation of laser-induced periodic surface structures upon irradiation of silicon by femtosecond-laser pulses[END_REF][START_REF] Derrien | Formation of femtosecond laser induced surface structures on silicon: Insights from numerical modeling and single pulse experiments[END_REF]. On the other hand, larger "grooves" are formed at higher fluences and are typically oriented parallel to laser polarization. These grooves are formed due to thermo-capillary waves, instabilities, or other thermodynamic phenomena [START_REF] Tsibidis | Convection roll-driven generation of supra-wavelength periodic surface structures on dielectrics upon irradiation with femtosecond pulsed lasers[END_REF].

To optimize the proposed laser processing technology, experiments were carried out revealing a strong dependency of the periodic structure orientation on laser fluence. In the experiments, we varied both the average laser energy E 0 and the number of laser pulses N from 1 to 1000. For structures with Λ=540 nm, E 0 was in the range from 3 to 60 µJ, and N varied from 1 to 1000.

To explain the obtained thresholds, formation mechanisms of the observed structures should be better understood. Figure 3 indicates the role of the applied pulses per laser spot. Even if laser repetition rate, laser spot diameter and nanoparticle concentration are small, several accumulation affects remain possible. Thus, nanoparticles are known to help enhancing electromagnetic field, promoting multi-photon ionization (MPI) and refractive index changes [START_REF] Rudenko | From random inhomogeneities to periodic nanostructures induced in bulk silica by ultrashort laser[END_REF][START_REF] Duchateau | Strong nonlinear electron multiplication without impact ionization in dielectric nanoparticles embedded in optical materials[END_REF]. More importantly, NPs serve as localized sources of electrons, ions and heat.

Nanoparticle heating process strats with electron energy absorption and electron-phonon relaxation in NP that takes up to several picoseconds, depending on nanoparticle material and size. The glass matrix is heated by the energy transfer from nanoparticles via convection and conduction, which typically takes up to several nanoseconds. Contrary to the heating, material cooling takes a longer, up to microseconds [START_REF] Veiko | Effective time of thermal effect of ultrashort laser pulses on dielectrics[END_REF]. These processes are accompanied by a set of phase transitions, such as softening, densification, Figure 3: Laser processing regimes: zone I corresponds to the desirable interference pattern with Λ 1 =540 nm recorded at F =0.15 J/cm 2 (Fig. 2b); zone II corresponds to the selforganized grooves (Fig. 2c); in the point 3, film destruction is observed (Fig. 2d).

. crystallization, etc. [START_REF] Papazoglou | Physical mechanisms of fused silica restructuring and densification after femtosecond laser excitation[END_REF]. These modifications also take a shorter time than the inter-pulse time in the present study (0.1 s).

In the case of fused silica, however, electrons can be also "stored" in selftrapped excitons [START_REF] Ashkenasi | Surface damage threshold and structuring of dielectrics using femtosecond laser pulses: the role of incubation[END_REF][START_REF] Itoh | Materials modification by electronic excitation[END_REF], considerably decreasing the material modification threshold in ultra-short multi-pulse irradiation regimes. In addition, surface relief and reflectivity/absorptivity changes are also accumulated [START_REF] Bonse | Pulse number dependence of laser-induced periodic surface structures for femtosecond laser irradiation of silicon[END_REF][START_REF] Miyamoto | Characteristics of laser absorption and welding in foturan glass by ultrashort laser pulses[END_REF]. As a result, both structures' formation threshold's decay with N . Commonly, incubation and accumulation effects lead to the following pulse number dependency of the threshold fluence: F = F 1 N S-1 , where N is the number of laser pulses, F 1 is the single pulse threshold, and S is the slope. The results shown in Figure 3, line I, give F 1 = 0.4 J/cm 2 , and S is approximately 4e -5 .

The periodic structures observed in the central part of the laser track (Fig. 2) better correspond to the grooves. For their formation, the laser-irradiated material should turn to a liquid state and its viscosity should typically reduce to allow convective liquid motion. Thus, three observed thresholds (Figure 3) can be explained as follows:

(i) The formation threshold of the interference pattern should be connected to silica glass softening point at around T 1 =1986 K.

(ii) For the grooves formation, glass should be heated up to the liquid state and its viscosity, η(T ), should drop down, so the condition P e >1 is satisfied, where P e is Peclet number. This is known to take place at around T 2 =3000 K [START_REF] Rudenko | Nanopore-mediated ultrashort laser-induced formation and erasure of volume nanogratings in glass[END_REF].

(iii) The upper laser fluence limit is defined by the nanocomposite destruction that can be associated with boiling point, around T 3 =3220 K.

Here we analyze the connections of laser processing parameters and optical properties via geometry of nanocomposite nanogratings. Furthermore, nanocomposite optical properties, are commonly calculated either based on Mie theory accounting for nanoparticle sizes and concentrations (σ i • n i ) [START_REF] Mie | Beiträge zur optik trüber medien, speziell kolloidaler metallösungen[END_REF][START_REF] Banerjee | Optical absorption of composites of nanocrystalline silver prepared by electrodeposition[END_REF] or on a suitable effective medium approximation [START_REF] Golovan | Optical properties of porous-system-based nanocomposites[END_REF][START_REF] Srivastava | Optical properties of polymer nanocomposites[END_REF], as will be shown below (Section 3.2).

Optical properties

The optical reflection R(λ) and transmission T(λ) of a periodic nanocomposite structure can be calculated as follows [START_REF] Smith | The optical properties of metals[END_REF]:

R(λ) = Re f ull (λ) -1 √ 2 1 cos 2 φ 2 + Im f ull (λ) 2 Re ef f (λ) -1 √ 2 1 cos 2 φ 2 + Im f ull (λ) 2 (3)
T (λ) = 1 -R ef f (λ) 2 exp -4π λ h cos φ Im f ull (λ) 1 -R ef f (λ) 2 exp -8π λ h cos φ Im f ull (λ) , (4)
where f ull is the dielectric function of the effective medium, and φ is the angle between nanograting orientation and the direction of the incident light direction. The optical density of such materials is given by: OD(λ) = ln 1/T (λ) .

(

) 5
More details about the calculation procedure can be found in the Appendix A.

The OD defines the total losses accounting for absorption, reflection, and scattering. Figure 4 shows the experimental and the calculated spectra of optical density. For both nanograting periods, a maximum near 430 nm is clearly observed. This maximum can be attributed to the surface plasmon resonance (SPR) of silver nanoparticles in the nanograting. The presence of the resonant NPs is also confirmed by SEM analysis (Fig. 5 b,c). The optical properties of the fabricated nanostructures are then examined. For this, 1 × 1 mm regions were recorded without overlapping along the X axis. Scanning along Y-axis was performed for the revealed optimum laser irradiation conditions, that is at N =1000 and for F =0.21 and 0.33 J/cm 2 for Λ 1 =410 nm and Λ 2 =540 nm respectively.

The measured reflection spectra of the modified regions are presented in Figure 5. In the initial/non treated film, the reflection maximum is observed at 560 nm, which is typical for silver nanoparticle containing porous glass [START_REF] Boni | Femtosecond third-order nonlinear spectra of leadgermanium oxide glasses containing silver nanoparticles[END_REF]. Upon the formation of the periodic structures on the surface, the peak is shifted. Interestingly, the shift strongly depends on the period of the formed structures: the maximum is observed at 480 nm for structures with periods Λ 1 = 410 nm, while for the ones with Λ 2 = 540 nm it is located at 530 nm. These shifts can be attributed to both laser-induced nanoparticles modifications and to the formation of the surface relief, as discussed below. and two different incident plane orientations: parallel and perpendicular to the grating direction. We emphasize that in both cases the created surface structures exhibit considerably different characteristics than the initial surface.

The diffuse scattering spectra reveal two distinctive peaks. The maximum positions depend on the period of nanograting and on the orientation of the incidence plane with respect to the periodic structures. For the direction parallel to the periodic surface structures, a significant increase in scattering is observed. The amount of the scattered light is much lower if the light incident plane is in the perpendicular direction.

The peaks observed near 402 and 498±1 nm for parallel direction correspond fairly well to the period of the nanograting Λ 1 = 410nm and Λ 2 = 540 nm. The scattering maximums observed for the orthogonal polarization direction are much less pronounced and exhibit slight red shift with the increase of period. This peak can be explained by the Relay-Wood effect. The intensity and the position of this peak are related to the projection of the incident wave on the nanograting. Thus, the reflection of the resulting nanostructure is mostly in the range above 400 nm, and it can be controlled by rotating of the laser-induced surface structures.

Additionally, one can see a minimum around 350-400 nm corresponding to the plasmonic effect on the silver NPs in the dielectric matrix. The peak position does not depend on the illumination angle and slightly changes with a period of nanograting due to the differences in the NPs size and distribution. These results are confirmed by the simulation of the reflection presented in figure 6(b).

The described grating properties lead to an interesting visual effect, which can be observed in the crossed-polarized mode by using optical microscope (Figure B.7 in the Appendix B). The image acquires greater contrast while observed at angles of 0 • and 90 • , while at 45 • the structure almost disappears.

Conclusion

To summarize, we have considered laser-based fabrication of a periodic gratin-like relief on the surface of a mesoporous silica film activated by the presence of silver nanoparticles using dual beam interference.

The obtained results demonstrate that the formation of the periodic structure strongly depends on laser fluence: in the case of lower laser fluence, the grating formation correlates perfectly well with the beam interference pattern, whereas at larger laser fluence; periodic grooves are formed in the direction parallel to laser polarization. This second structure is attributed to thermo-hydrodynamic effects that erase the interference pattern in our experiments. At higher laser fluences, film damage is observed.

As a result of the performed experiments, a novel laser-based procedure of the controlled grating fabrication is proposed. Particularly, the fluence thresholds corresponding to both regimes have been shown to decay with the number of applied laser pulses indicating incubation and accumulation effects. This effect is attributed to electrons injected from nanoparticles as well as created by multi-photon ionization around nanoparticles, and stored in self-trapped excitons promoting structural changes and phase transitions. Surface reflectivity also changes in multi-pulse regimes.

Additionally, interesting optical properties of the interference-induced surface structures were revealed. This structure reflects mostly in the wavelength range above 400 nm, but differently when it is rotated parallel or perpendicularly to a light incident plane. Thus, it can be used for instance, as an optical filter and can be useful for second harmonic generation.

Acknowledgments

This research was performed in the frame of the PHC Kolmogorov France-Russia collaborative project Formalas operated by Campus France (38091TJ The obtained nanostructures were investigated both by SEM and optical spectrophotometry. The obtained experimental data was used in the modelling. Before and after laser irradiation, the average diameter, D av , the relative concentration of NPs in composite C rN P , and their dispersion σ N P 2 were determined by SEM. Additionally, period Λ g , depth h g , and grating's coefficient L g were also measured. The surface relief was then considered in the modeling of the grating's optical properties.

Firstly, the volume fraction ν N P of all NPs and their concentrations were calculated. All the mentioned parameters are presented in Table A Then, the effective dielectric function ε ef f of the nanocomposite Ag -SiO 2 film was determined. This function included the dielectric properties of metallic phase in the form of Ag NPs, and the matrix medium, namely the silica sol-gel porous film. The dielectric function of metallic phase, which depends on a wavelength of incident light, was described as the equation [START_REF] Sancho-Parramon | Surface plasmon resonance broadening of metallic particles in the quasi-static approximation: a numerical study of size confinement and interparticle interaction effects[END_REF][START_REF] Cavaliere | Exploring the optical and morphological properties of ag and ag/tio2 nanocomposites grown by supersonic cluster beam deposition[END_REF] N

P (λ) = Bulk (λ) + s λ 2 k 2 τ + ikλ -z λ 2 1 + iaλ + λ 2 1 + i(a + b)λ , (A.1)
where ε Bulk -is the dielectric permeability of bulk material of NPs, τ =9.2 fs -time of electron relaxation.

The optical constants of silver were taken from [START_REF] Zhang | Plasmonic photocatalysis[END_REF]: ε Bulk , n = 5.86 • 10 28 is the free electron concentration, m = 4.56 • 10 -31 kg is the effective mass of electron [START_REF] Zhang | Plasmonic photocatalysis[END_REF] with charge e = 1.6 • 10 19 Kl, υ F = 1.4 • 10 6 m/s is the Fermi velocity, ε 0 = 8.85 • 10 -12 F/m is the dielectric constant, c = 3 • 10 8 m/s is the light velocity in vacuum, r N P = 0.5D av -radius of spherical particles, ρ = 10500 kg/m 3 is the silver density, N A = 6.02•10 23 1/Mol is the Avogadro constant, m a =0.108 kg/Mol is the atomic mass of silver. The coefficients s, k, a and b were described by following equations:

s = 4 3 n 2 e r N P mν F , k = 2, z = 1 12 N A 0 m a m e πc 2 , a = 1 2π 2 cτ , b = 0.75 ν F 2πcr N P , (A.
2) where ε Bulk is the dielectric permeability of bulk material of NPs, and τ =9.2 fs is electron relaxation time. The influence of transparent matrix on the optical properties of the metallic NPs was accounted for by using Bruggeman approximation with Bergman equation ε ef f = f (ε N P , ε M , ν N P , σ N P ) [START_REF] Sturm | Effective dielectric functions of alkali halide composites and their spectral representation[END_REF]. The dielectric permeability ε M of sol-gel matrix SiO 2 was determined from optical constants of fused silica [START_REF] Smith | The optical properties of metals[END_REF].

The surface relief in the shape of grating changed the optical properties of the composite film. To account for this, effective dielectric permeability should be corrected for composite layer with the thickness equal to h g . New effective medium of this layer was determined as an average dielectric function between the composite and the environment by using the following shape function. The dielectric permeability of effective medium in the shape of the grating depended on grating orientation and on the incident light direction by using the the angel ψ between them. Finally, the dielectric permeability of grating in the air was determined as follows:

ε g = [1 + (ε ef f -1) ν gF ] 2 cos 2 ψ + ε ef f ε ef f -(ε ef f -1) ν gF 2 sin 2 ψ 1/2 , (A.3)
where ν gF -is the volume fraction of grating as effective composite layer. The total effective dielectric medium consists of two layers: the first of them with h g thickness is the grating, the second layer with h f ilm -h g thickness is the composite. The material of top layer was characterized by the dielectric permeability ε g , and the next layer under it by ε ef f . The effective dielectric permeability of the two-layer material was determined as:

ε f ull = h g h f ilm ε g + 1 - h g h f ilm ε ef f . (A.4)
Therefore, the optical reflection R(λ), transmission T(λ) and optical density OD(λ) of two-layers composite structure can be calculated using the equations 34.

After interaction with the grating, the reflected and transmitted lights were exposed to diffraction also known as Rayleigh-Wood effect [START_REF] Pommet | Limits of scalar diffraction theory for diffractive phase elements[END_REF]. As a result, the initiated light was split ed into the main maximum of 0 order and ±1, ±2. . . ±m orders. The diffraction efficiency of such grating for reflected and transmitted light with amplitude A F = R T for m diffracted order was determined as

η (m) = 4A 2 F L 2 g • Sinc 2 (L g m) cos 2 2 π λ h g 2 √ ε f ull cos ϕ t -cos ϕ + π 2 m ,
(A.5) where ϕ t -is the angel of refracted light. At the registration of reflected or transmitted light after it diffraction should be considered the sum intensity of all orders.

Based on the experimental data for the optical density of Ag-SiO 2 film and of the gratings (Λ 1 =410 and Λ 2 = 540 nm), the optical spectra were simulated (fig. 5).

Based on the comparison between the simulated optical density with the experimental spectra, the average diameter D av , concentration C N P and volume fraction ν N P were then corrected (Tab. A.1). The angel ϕ and its component ϕ r were equal to 0. At this angel, the main maximum of diffracted light was propagated and the all orders m was equal 0.

After the correction of size and concentration of silver NPs, the spectra of diffused reflection at the angel of incident light ϕ = 45 • were calculated by using equation A.5 (fig. 6). The differences between the experimental and simulated spectra is caused by the imperfect grating shape. Thus, small variations of depth h g and grating coefficient L g caused fluctuations of the grating's optical properties.

Figure 1 :

 1 Figure 1: Experimental setup used for controlled laser writing of the two-beam interference patterns.

Figure 2 :

 2 Figure 2: Nanograting formation. SEM image of the boundary zone and a schematics of radial distribution of laser energy (a). SEM images of the nanogratings with Λ 1 =540 nm obtained for different laser fluences: interference pattern recorded at F = 0.15 J/cm 2 (b); urface grooves obtained at F = 0.35 J/cm 2 (c); destruction of the film occurred at F = 0.68 J/cm 2 (d). White double arrow corresponds to the polarization direction, the green arrow shows the scanning direction. Here, nanostructure depth varies with laser fluence, but does not exceed initial film depth.

Figure 4 :

 4 Figure 4: Measured and calculated optical density. Here, the incident light is normal to gratings surface. The inset and table show parameters used in the simulations.

Figure 5 :

 5 Figure 5: Reflectance spectra of the nanograings with different periods (a). SEM images of the nanogratings: structure with the period of Λ 1 = 410 nm obtained by laser scanning with the processing parameters N =1000, F =0.21 J/cm 2 (b), structure with the period of Λ 2 = 540 nm obtained by N =1000, F = 0.33 J/cm 2 (c). White double arrow shows the laser polarization direction, green arrow is a scanning direction

Figure 6 :

 6 Figure 6: (a) Normalized experimental spectra of diffuse scattering of the gratings with Λ 1 = 410 nm and Λ 2 = 540 nm. (b) Simulated reflection spectra of the nanogratings.

  ). ITMO Professorship and Fellowship program is gratefully acknowledged by TEI. H.M is grateful to The French Ministry of Science and Education for his PhD Scholarship. SEM measurements were done on the base of Interdisciplinary Resource Centre for Nanotechnology, Research Park, St. Petersburg State University.

	Appendix A. Calculation of the optical properties of diffraction
	gratings with Ag NPs

  .1.

	Properties			Ag-SiO 2	Grating Λ 1	Grating Λ 2
				Film
	average	diameter	25	19.3	8/ 30
	D av , nm
	relative concentra-	0.019	0.074	0.066/0.019
	tion C rN P
	concentration C N P ,	0.218	0.028	0.036
	Mol/m 3
	dispersion σ N P	2	0.7	0.5	0.6
	volume fraction ν N P 1.43	1.463	0.283/ 2.295
	period Λ g , nm		-	410	540
	depth h g , nm		180	105	160
	grating's coefficient	-	0.46	0.46
	L g

Table A.1: Properties of composite film before and after laser writing of nanogratings

Appendix B. Optical microimages of the obtained nanogratings