Probing nonperturbative third and fifth harmonic generation on silicon without and with thermal oxide layer

We examine Si with and without additional SiO2 thin film coating as a candidate for producing powerful 3rd and 5th harmonics of Ti:sapphire laser pulses for future spectroscopic application. Polarization rotation experiments have been performed at different incident angles to determine the origin of the generated harmonics and a strong polarization-dependency of the harmonic signals was observed. A simplified tensor formalism is introduced to reproduce the measurements with high accuracy. Comparing the measurements with the Oh symmetry of the bulk crystal, the C2v structural symmetry for the uncoated Si sample and a C4v symmetry for the SiO2 coated sample, we conclude that the polarization anisotropies are determined by the surface/interface symmetries.


Introduction
Converting the wavelength of the ultrashort pulses of laser frequency combs to the ultraviolet (UV) and vacuum ultraviolet (VUV) wavelengths makes them potential sources for several new applications. It can be used for high-resolution electronic transition spectroscopy, to measure new high energy atomic transitions with high precision and to develop next-generation atomic clocks. The ultrashort pulses in the UV and VUV * Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. wavelengths regime would be useful for tracking fast chemical and biological processes. High harmonic (HH) generation has been proven to be a suitable technique for the conversion to shorter wavelengths, reaching wavelength even below 100 nm. In recent years, different solid crystalline materials demonstrated the potential to replace noble gases as nonlinear medium for frequency conversion, avoiding the technical difficulties accompanied with high gas loads and differential pumping of vacuum chambers.
Among the practical solid media, silicon (Si) was tested successfully in recent years. Silicon is a key material in the semiconductor industry and widely used in modern information technology. It is available in very high quality and has high damage threshold [1], which is an important factor for HH generation because good conversion efficiency requires high laser intensities in the GW cm −2 or even in the TW cm −2 .
Obviously, generation of harmonics in Si even at high orders is in the interest of several research groups recently and it is studied experimentally and theoretically [2][3][4][5]. Experiments typically used laser pulses in the infrared, at wavelength of 1.3 µm-3.8 µm, below the bandgap in which Si was transparent and the duration of the used laser pulses was in the range of 25 fs up to 400 fs [6][7][8]. Harmonics were observed in transmission (forward geometry) and also appeared in reflection from the samples (backward geometry). The harmonic spectrum extending from the visible even into the extreme UV spectral range was successfully generated in silicon crystals. As a promising direction, forming nanostructures, nano-antennas on the surface of Si, plasmon-assisted harmonics were produced to enhance the conversion efficiency or to lower the required laser intensities [9][10][11].
Silicon does not only represent a suitable medium for the generation of optical harmonics thus constituting a prime material to investigate, but it can also be used to further understand the process of harmonic generation. Generation of harmonics can be described similarly to metal surfaces [12] or based on the band structure [3,4,6]. The harmonics are expected to be generated in a thin surface/interface layer containing contribution from the surface/interface [13][14][15][16][17] and from the bulk crystal [8,18,19]. Studying 2nd harmonic (H2) generation in Si [20,21], dangling bonds and surface Si-Si bonds were proposed as the sources of the surface contribution.
In this study, 3rd and 5th harmonics are generated using Si samples and the ultrashort pulses of a Ti:sapphire frequency comb. The laser operates at 800 nm, where Si is not transparent, therefore the experiments are performed in reflection geometry. We developed a deep ultraviolet and VUV frequency comb, that can be used as a source for high precision measurements in these hardly reachable wavelengths. Furthermore, harmonic generation on a thin film of SiO 2 grown onto the Si substrate is investigated. Experiments based on rotating the laser polarization have been performed, which yields information about the symmetry of the harmonic source by comparing the measurement with the symmetry of the bulk crystal and the symmetry of the surface/interface.

Experimental procedure
To generate harmonics on the Si sample, the surface was illuminated by a linearly polarized laser beam at different oblique angles and the direction of the polarization was rotated using a half-wave-plate. The plane of the incidence was the xz plane defined in figures 1(a) and (b) and the experimental setup is presented in figure 1(c). Figure 1(a) shows the silicon crystal surface (001) from above and figure 1(b) shows it from the side visualized by the software of [22]. Atoms in figure 1(a) represented in dark blue color are on the surface and others in lighter blue are below the surface. Taking all the atoms, the silicon crystal has a tetragonal fourfold symmetry. The setup is designed by considering the observation that the harmonic beams co-propagate with the reflected laser beam. The laser source was a Ti:sapphire frequency comb (FC8004, Menlo Systems). It delivered laser pulses at repetition rate of 108 MHz and with pulse duration of 28 fs at the central wavelength of 800 nm. The carrier offset frequency and the repetition rate can be stabilized and the later scanned when performing high precision spectroscopic measurements. The typical pulse energy was 8 nJ, which was focused onto the surface of the sample using a lens with focal length of 10 mm. It produced a focus with diameter (full witdth at half maximum, FWHM) of 6.0 ± 0.5 µm, and an on-axis peak intensity of 0.7 ± 0.2 TW cm −2 . In some measurements, a lens with focal length of 15 mm was used resulting in a 9.0 ± 0.5 µm focus (FWHM) and an on-axis peak intensity of 0.3 ± 0.1 TW cm −2 .
The reflected laser beam together with the generated beams of harmonics were directed to the entrance of the VUV spectrometer (McPherson 234/302) using a VUV mirror. Both the sample and the mirror were placed onto rotation stages to hold them parallel while measurements were performed at different angles of incidence (θ). The distance (≈60 mm) of the VUVgrade MgF 2 lens from the sample was set to focus the generated 5th harmonic (H5) beam to the input slit of a VUV spectrometer, which was located at about 500 mm from the lens. Under this condition, the 3rd harmonic (H3) and especially the fundamental laser beam is not focused to the input slit, giving a better contrast for the measurements by decreasing the scattered background in the spectrometer. Furthermore, it was possible to insert a VUV bandpass filter (160-BB-1D, Pelham Research Optical L.L.C.) to further suppress the fundamental and the H3 beams. The spectrally resolved beam was detected with a VUV photomultiplier (Hamamatsu R6836), sensitive in the 115-320 nm spectral range. The setup of the source and the VUV spectrometer were in vacuum with a background pressure of 10 −3 mbar to avoid the reabsorption of the H5 by air. Both the VUV mirror and the lens were placed onto translation stages to ensure the focusing of the H5 beam into the input slit of the spectrometer at different angle of incidences.

Sample description
Using the experimental setup presented in figure 1(c), harmonics were generated in two sample configurations. In the first one, a Si substrate was used (n Si = 3.694 @ 800 nm) [23]. The Si (100) substrate characteristics were P-type (boron), 10-20 Ωcm, 500 µm-thick, 2 sides-polished. In the second case, a SiO 2 film was grown on the Si substrate (n SiO2 = 1.461 @ 800 nm) [24]. The SiO 2 layer was thermally grown at 1100 • C (experimental thickness was 389 ± 1 nm, measured by optical reflectometry (Nanospec 6100)). This thermally grown SiO 2 layer is typically an amorphous layer (more specifically considered as a vitreous non-crystalline solid) with a stoichiometric relation between Si and O, except in the first few nanometers (less than 10 nm) of interface with the underlying silicon substrate. Prior to SiO 2 thermal growth, both Silicon substrates underwent a chemical cleaning procedure (HF plus piranha solution) to remove the native silicon oxide. Since no further chemical etching was performed preceding optical measurements (some days after), differing from the 400 nm thermally grown stoichiometric SiO 2 , a very thin amorphous silicon oxide is expected onto the Si (100) substrate. Previous studies have shown that silicon is one of few materials whose native oxide will self-limit its growth at a thickness of about 2 nm in a matter of hours [25]. Furthermore, this native amorphous oxide onto Si (100) typically exhibits a varying composition with depth, gradually changing from a highly oxidized, near stoichiometric state at the surface to a silicon rich phase near the interface. High depth resolution medium energy ion scattering spectrometry analysis has shown an outer oxide layer of about 0.7 nm of stoichiometric SiO 2 , and an underlying sub-oxide SiO x one (x < 2) of approximately 0.6 nm [26].

Generated spectra
The spectra of generated harmonics have been recorded at different angles of incidence (θ) and are presented in figure 2.
The possible values of the θ were limited: using a lens with f = 10 mm, it was not possible to set angles below 40 • because the reflected beams hit the lens. The upper value was limited to 60 • because the distance between the sample and the VUV lens should have been about 60 mm to image the sample onto the input slit of the VUV spectrometer. Therefore, for the measurement at 20 • , a lens with f = 15 mm was used. For all spectral measurements, the polarization direction of the linearly polarized laser beam was set (ε = 0) to be parallel to the plane of incidence, when the intensity of the generated harmonics was maximal. From the Si sample, one can observe the generation of strong H3 (267 nm, 4.6 eV) and the weaker H5 (160 nm, 7.75 eV). The intensity of both decreases with the increase of θ, consistent with other observations [7,12,17]. A third peak between the harmonics was also observed, sometimes only at shorter wavelength than H3 and sometimes at both sides of H3. The origin of these peaks can be fluorescence from the sample. Their positions change in an interval of about 1.1 eV, which is close to the bandgap of the Si. Thus, the lower state can even be a dangling bond state, which is situated in the bandgap and whose energy is direction dependent [27]. In a material with inversion symmetry, such as silicon, only odd harmonics can occur. It is consistent with our observation, because even harmonics as H4 and H6, which would be in the observable spectral range of the VUV spectrometer, have not been generated or are below our detection limit.
Further, we examined the effect of a thin SiO 2 film grown onto the Si substrate on the generation of harmonics. Crystalline SiO 2 has high band gap of 8.1 eV [28], so it would be fully transparent even at the wavelength of H5 (7.75 eV). In our case, the SiO 2 film was amorphous as mentioned above. Amorphous SiO 2 has a smaller bandgap [28] of 7.5 eV meaning an absorption [24] α = 1.4 µm −1 , and consequently the film absorbs about half of the H5 signal.
From the presence of the film, one can expect contradictory behaviors. On one hand, the actual angle of incidence at the Si/SiO 2 boundary (θ i ) is smaller, 32 • and 36 • in the case of θ = 50 • and 60 • , which predicts larger harmonic signal according to the previous measurements [7,12,17]. On the other hand, earlier measurements [13,16] obtained smaller harmonic signal from the interface of two materials compared to material-vacuum surfaces. To address this question, we measured spectra for a Si substrate containing about 400 nm-thick SiO 2 film, shown in figure 2. The obtained spectra look somewhat different from the pure Si case. The H3s are similar; however, the H5s are weaker, and at θ = 60 • , it is even below the detection threshold. It was not possible to perform a measurement at θ = 40 • because the sample was overheated and damaged within the time interval of the spectral measurement even when using a lens with a longer f = 15 mm focal length.

Intensity of the harmonic lines
The non-perturbative character of the harmonic generation process was checked by directly measuring the dependence of the harmonic intensities on the laser intensity. The measurements were performed by moving the sample out of the focus by means of moving the lens away from the sample. The measurement results can be seen in figure 3 using θ = 40 • for Si and θ = 60 • (θ i = 36 • ) for Si/SiO 2 , respectively. For the Si sample, I (3ω) ∝ a · I 4 (ω) + b · I 1.95 (ω) and I (5ω) ∝ I 5 (ω), furthermore for the Si/SiO 2 sample I (3ω) ∝ a · I 3 (ω) + b · I 1.9 (ω) has been found. Because of the too small H5 signal on the Si/SiO 2 sample, it was not possible to perform a reliable measurement there. From the measurements, it is possible to estimate the power of the corresponding generated harmonics. Considering the detector gain and spectral efficiency, the reflectivity of the spectrometer grating and the transmission of the used spectral filter, we obtain an about 0.4 nW H5 power and an about 8 µW H3 power (θ = 40 • ) from the Si sample. The spectral widths (FWHM) of harmonics H3 and H5 are about 9 nm and 7.3 nm, meaning that their relative spectral widths (∆λ/λ) are approximately preserving that of the driving laser with width of 32 nm.

Polarization dependence of the harmonic signal
To gain information about the origin of the harmonics of the Si and Si/SiO 2 samples, the polarization direction (ε) of the linearly polarized incident laser beam was rotated using a half-wave plate ( figure 1(c)), and at few θs, the intensity of the generated H3 and H5 was measured, selected by the used monochromator. The measurement results (continuous yellow lines) compared to different calculations (dashed/dotted lines) are presented in figure 4.
Considering that Silicon is a crystalline material with cubic O h symmetry (point group m3m), the third-order susceptibility tensor has 21 nonzero elements [29]. To simplify the calculation of the polarization dependence of the H3 signal, we introduced a 2-dimensional susceptibility tensor (3 × 3) instead of the original 4-dimensional one, a simplification which is possible for this crystal symmetry (and also for several others like isotropic, cubic, orthorhombic, few hexagonal and few tetragonal crystals): The linearly polarized electric field with unit vector ⃗ e, polarization angle ε and angle of propagation θ can be written in the general form: The practical choice of the coordinate system is different for the case inside the bulk crystal or for the case of surface. Figures 1(a) and (b) show both cases. The difference can be handled by choosing azimuthal angle of incidence φ differently. In the case of the surface φ = 0 • and equation (2) can be directly applied. Inside Si crystal, the coordinate system must be oriented parallel to the crystal axis: φ = 45 • , furthermore, the refracted beam must be considered. It can be done by replacing the electric field by its transform ⃗ e (ω) ←φt⃗ e (ω), wherê φ is the rotation tensor with φ andt calculates the transmitted beam though the surface using the Fresnel formulas. If a SiO 2 layer is also involved, transmission of the incident beam through the silica surfacet 10 and the silica-silicon interfacê t 21 : ⃗ e (ω) ←φt 21t10 ⃗ e (ω) and the transmission loss of the generated harmonics through the silica surface⃗ e (qω) ←t 01 ⃗ e (qω) must be considered.
Inside the crystal, the generated H3 field than can be expressed in a simple form for m3m point group: Here, the first column vector on the right side means element-wise multiplication. According to [30] for silicon a = 3χ In the case of assuming surface generation, equation (3) needs to be modified. One must consider that the laser beam is a transversal wave, and the generated harmonic beam should be also transversal. Taking the example of the silicon surface, the harmonic beam leaves the surface at the same angle as the incidence angle and for the beams of the laser and of the backward harmonics: |e z /e x |= tan(θ). It means that the first and the last rows of the tensor in equation (3) should be the same (or multiplied by −1). In isotropic media (such as gases) a = 1, and H3 and other odd harmonics can be routinely generated. It can be observed that the source of the harmonics has a different symmetry compared to the bulk Si and the surface might essentially contributes to the generation. The surface has an orthorhombic C 2v (mm2) symmetry according to the reported observations by atomic-force microscopy [31][32][33] and the same type of formalism as equation (3) can be applied but with 5 independent tensor elements a 1 . . . a 5 . A similar formalism is also suitable to calculate H5. In general, and considering that the first and third rows should be equal, for harmonic q = 3 and q = 5:  Here, χ np is the qth order nonperturbative susceptibility, which might differ in its unit from the perturbative one, however as tensor further represents the crystal symmetry. Furthermore, 'r' (which is applied as elementwise power) is used as the rank of the contribution of the nonlinear refractive index, n = n lin + n nl I r , which also defines the rank of the nonperturbative process as 2r + 1. For H3 and H5 generation, r = 1 and r = 2 can be expected, however, as it has been concluded in several publications [7-9, 11, 16, 18, 34-37], '2r + 1' should not be necessarily equal to the harmonic order 'q' and it should not be integer eighter in the strong field regime.
Equation (4) was successfully used to describe the polarization dependence of the harmonics and the calculated curves (black dashed) in figure 4. The calculated curves were obtained by using the next tensors by fitting the normalized calculated intensity I (ε, q) = E 2 x (qω) + E 2 y (qω) + E 2 z (qω) to the measured normalized intensity: Silicon-Vacuum (SV), figure 4(a), for H3 r = 1.5 and for H5 r = 2: Silicon-Silica (SS), figure 4(b), for H3 r = 1 and for H5 r = 2: To fit the calculated curves in figure 4, the 5 independent tensor elements of equation (4) and 'r' were chosen. Considering fourfold symmetry of the Si as can be seen in figure 1(a), a 4 = 1 should be chosen meaning only four independent elements, however, in the case of the Si, it was not possible to obtain a correct fit and a 4 became a free parameter. Tensor elements a 2 and a 4 mainly determine the rate of the signal along the x and y axis. But a 1 , a 3 and a 5 cross-polarization elements mainly affect the shape of the calculated curves. The values of the obtained tensor element and 'r' are unique and well determined within the accuracy of about ±0.1. The only exception is a 4 which required an accuracy of about ±0.01.
Using the values of equations (5) and (6), good agreement between the calculated and measured curved in figure 4 has been found.
While the '0' and '1' tensor elements in equations (5) and (6) cannot be considered as independent, the polarization dependence of H3 and H5 for the Si sample can be described by four independent elements, which further decreases to two independent elements for the Si/SiO 2 sample.

Optical etching the surface of silicon
In the experiments, the surface of the silicon crystal is illuminated from x-direction under certain θs (figures 1(a) and (b)). The intensity of the laser is high enough to remove the native, few atoms thick oxidized layer and possibly absorbed gases and contaminations from the surface easily. The surface is heated up by the laser and the generated harmonics themselves or multiphoton absorption can also break the most common bonds [38] like Si-O bonds (5-6 photons); Si-C, O-H bonds (3 photons) and the Si-H bonds can even be broken by 2 laser photons [20,21,39]. As we mentioned earlier, even much thicker SiO 2 film could have been damaged at the used high laser intensity. Beyond removal of thin native oxidized layer, further degradation of the surface is not observed even after illumination for long time intervals.

Decreased harmonic signal from the oxidized sample
For measuring the intensity dependence of the harmonic signal in figure 3, the polarization direction of the laser beam was chosen to be in the incident plane (ε = 0) and the incident angle θ = 60 • for the Si/SiO 2 sample was near the Brewster angles for both the laser and harmonic beams (laser: 55 • ; H3: 56 • ; H5: 59 • ). Thus, the harmonics generated from the Si should be near independent of the presence of the SiO 2 film. Furthermore, because the laser beam arrived at θ i = 36 • onto the Si/SiO 2 interface, the θ = 60 • spectrum in figure 2(b) should be directly comparable to the spectra of 40 • or 60 • of Si in figure 2(a) depending on whether the incident or the refracted beam is considered. However, as it can be observed in figure 3, in the range of high intensities, in the non-perturbative regime, H3 is stronger and H5 is much stronger when the Si surface does not contain the SiO 2 layer. This contradiction suggests that the Si surface or the Si/SiO 2 interface itself (not the Si bulk material or the film) plays an essential role in the generation process. It is consistent with others observation and suggestion [20,21], namely that dangling bonds and surface Si-Si bonds can be the sources of the harmonics. Because the density of the dangling bonds is lower in the Si/SiO 2 interface compared to the Si surface [39], it can explain the weaker harmonics from the SiO 2 containing sample.

Symmetry difference between Si and Si/SiO 2
Measuring and calculating the polarization-dependent harmonic signals for H3 and H5 provides the opportunity to determine the symmetries of the source of the harmonics. Si crystal is a cubic crystal with O h (m3m) symmetry. Therefore, one can expect χ zz using the notation introduced in equation (1) or 1 = a 4 = a 2 using the notation from equation (4). Contrary that expectation, the measurements exhibit strong polarization anisotropy. For Si, χ zz is observed, representing a twofold C 2v (mm2) symmetry, which is the symmetry of the Si surface. For Si/SiO 2 , the measurements show a fourfold C 4v (4mm) symmetry with χ zz , which can be the symmetry of the Si/SiO 2 interface or the surface of the SiO 2 film. A reference measurement from bulk SiO 2 sample produced much weaker H3 and no H5, therefore, Si/SiO 2 interface remains as the reason of the observed symmetry.
Regarding to the literature, on a cleaned silicon surface, Si dimers can form c (4 × 2) and p (2 × 2) local structures accompanied by dangling bonds [31,[39][40][41]. The photon energy of the laser (1.54 eV) is large enough to transfer these local structures to each other or even enough to break the surface bond of the Si-Si dimers. The c (4 × 2) local structure would produce a hexagonal arrangement [31]. The measured tetragonal/orthorhombic symmetry in figure 4, therefore suggests that the laser illuminated surface of Si contain mainly single Si atoms and/or dimers in p (2 × 2) local structures, while single atoms represent fourfold symmetry and p (2 × 2) local structure represents twofold symmetry, which can explain the observed twofold C 2v symmetry. As mentioned earlier, the density of the Si dimers and dangling bonds is lower in the Si/SiO 2 interface compared to the Si surface, which can explain the fourfold C 4v symmetry of the Si/SiO 2 sample.

Large polarizability perpendicular to surface
The difference between χ  (5) and (6). The χ (q) zz tensor element (a 2 ) represents the nonlinearity and polarizability in the perpendicular direction to the surface caused by the E z component of the laser electric field. Such large tensor element should not be observed inside the bulk Si crystal; it can be the effect of the surface. It indicates that the surface takes a large contribution to the harmonic generation process. This observation can be derived by expecting the present of the out-of-surface electron orbitals [42], which are oriented perpendicular to the silicon surface and consequently can be easily polarized in z-direction. This might explain the large polarizability and large tensor values in that direction. Additionally, dangling bounds can also contribute.
Another important observation is that the χ (q) yz element (a 5 ) of the nonlinear susceptibility tensors of the Si/SiO 2 sample vanished in contrast to Si, where it is large. This yz crosspolarization element is the one, which can be attributed to dangling bonds, because of their oblique angles. The vanishing of these tensor components in the case of the silicon-silica interface can be an indication of the essential reduction of the dangling bonds by passivation with the oxygen atoms [43].

Summary
We examined, in backward (reflection) geometry, the generation of the 3rd and 5th harmonics on the surface of silicon and on the interface between silicon and silica when a thin silica film was grown on a silicon substrate. In both cases, a strong anisotropy of the harmonic signals on the polarization direction of the driving laser beam was observed. Furthermore, a strong dependence of the shape of the polarization curves on the angle of incidence of the laser beam was found. The differences, observed in the harmonic signals, can be qualitatively explained by the formation of the out-of-surface electron orbitals and the activation of dangling bonds on the surface of the silicon. Furthermore, a simplified tensor formalism to describe the polarization anisotropy was introduced, which determined the C 2v symmetry of the Si surface and the C 4v symmetry of the Si/SiO 2 interface and described the polarization dependence with high accuracy. It also reveals that, in the nonperturbative regime, the source of the harmonics was the surface and interface instead of bulk of the Si crystal with its O h symmetry.
In the same geometry, stronger H3 and weaker H5 were generated on the silicon-silica interface than on the silicon surface. The weaker observed H5 might be partly a consequence of the reabsorption of the signal in the silica film, but the main reason should be the passivation and reduced number of the dangling bonds on the interface. To explore this phenomenon, further studies are necessary with films of different thickness, material, and crystallization quality.
Silicon has been found to serve as a suitable material to convert the short pulses of a Ti:sapphire oscillator to its 3rd and 5th harmonics with an efficient manner. Because of the about one order of magnitude higher damage threshold [1] of Si, comparing to the used intensity of the recent experiments, further increase of the generated harmonic power appears feasible.
Our experimental results compared to calculations suggest that the source of the generating harmonics H3 and H5 is the surface of the silicon sample for Si substrate, or the source is the Si/SiO 2 interface, when a silica film was grown on the silicon substrate. In the high-field regime and assuming nonperturbative harmonic generation, dangling bonds and out-ofsurface electron orbitals seems to be the actual (microscopic) sources of the harmonics. These orbitals, especially the outof-surface orbitals are located dominantly on the surface.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.

Funding
This work is part of the thorium nuclear clock project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 856415). Work has been performed within the project 20FUN01 TSCAC, which has received funding from the EMPIR programme co-financed by the participating States and from the European Union's Horizon 2020 research and innovation programme. We acknowledge support from the Österreichische Nationalstiftung für Forschung, Technologie und Entwicklung (AQUnet Project). This research has used the Spanish ICTS Network 'MICRONANOFABS', partially funded by FEDER funds through 'MINATEC-PLUS-2' project FICTS2019-02-40.