Structure and Nonlinear Optical Properties of TeO2–WO3–PbO Thin Film Glasses

Structure and nonlinear optical properties of TeO 2 –WO 3 –PbO thin film glasses G. Gorni a,1 , D. Munoz-Martin a,2 , A. Ruiz de la Cruz a,3 , V. Martin-Diaconescu b , L. Simonelli b , J. Garcia-Lopez c , J.M. Fernandez Navarro a , J. Solis a , J. Gonzalo a,* a Laser Processing Group, Instituto de Optica (IO-CSIC), Serrano 121, 28006, Madrid, Spain b ALBA Synchrotron Light Facility, Carrer de la Llum 2-26, 08290, Cerdanyola del Vall` es, Spain c Centro Nacional de Aceleradores, (CNA, US-Junta de Andalucía-CSIC), 41092, Sevilla, Spain ARTICLE INFO Keywords: Tellurite glasses Tungsten oxide Thin films Third order optical susceptibility Glass structure Pulsed laser deposition ABSTRACT Lead tungsten tellurite thin film glasses have been produced by pulsed laser deposition in a broad TeO 2 compositional range (50–85 mol%). Films present high transparency with an optical band gap E g =3.3–3.4 eV, large refractive index (n >2) and reduced absorption (k <10 −4 ) in the visible and near infrared ranges, whereas their nonlinear optical response (∣ χ (3) ∣) is found to be ≥10 −12 esu at 1.3 μ m. The dependence of n, E g and ∣ χ (3) ∣ with film composition is analyzed and compared with the values determined for the parent bulk glasses. ∣ χ (3) ∣ increases as the TeO 2 content in the film glasses decreases, while it remains constant for bulk glasses. This behaviour is analyzed in the frame of Line’s model and correlated to the structural differences between bulk and film glasses. In the case of bulk glasses, X-ray absorption spectroscopy results at the W L 3 and L 1 -edges clearly indicate the presence of W 6+ ions in a distorted octahedral coordination, with the coordination environment and W–O bond distance remaining unchanged regardless of the WO 3 content. Raman analysis suggests that films have a structure close to that of bulk glasses, with a moderate WO 3 and PbO enrichment, and a relative increase of the non-bridging oxygen fraction that are proposed to be responsible for the observed increase of ∣ χ (3) ∣ in the films. 1. Introduction Tellurite glasses are of great interest for the development of integrated photonic applications, such as ultrafast optical switching [1–6] or Raman gain applications [7,8], due to their excellent nonlinear optical properties [9–11] that are related to different factors, such as the empty 5d orbitals of Te 4+ ions, its non-bonding electron lone pair 5s 2 or the presence of Te–O–Te bridges in (TeO) 2 chain like structures [12–15]. In particular, tellurite glasses have a broad range of transparency from the visible up to mid IR (from 400 nm to 6 μ m), high refractive index, low phonon energy and finally, they show a nonlinear optical response which is orders of magnitude larger than that of conventional silicate or borate glasses. Unfortunately, the synthesis of pure TeO 2 glasses presents limitations due to their tendency to devitrification, which is overcome by adding other oxides to stabilize the glass structure and to improve their optical properties [1,7,16]. In particular, the addition of heavy metal oxides having lone s 2 electron pairs (PbO, Bi 2 O 3 ) or transition metal oxides with empty d orbitals, such as TiO 2 , WO 3 or Nb 2 O 5 , preserves the framework-type structure of tellurite glasses, while it enhances their nonlinear optical response [1,7,16–18]. However, their use for the development of integrated applications has been hindered by the difficulty to produce high optical quality thin film glasses. To this purpose different thin film preparation methods have been attempted, such as sol-gel, sputtering, electron beam deposition or pulsed laser deposition (PLD) [19–28]. In particular, PLD has proven to be an excellent technique to produce complex oxide films [29, 30], which has been successfully applied to the elaboration of multicomponent heavy metal oxide film glasses with compositions outside the bulk glass formation region [31], as well as to the production of good quality Er-doped tellurite thin film glasses with propagation losses as This article is part of a special issue entitled: 80th birthday of Giancarlo C. Righini published in Optical Materials. * Corresponding author. E-mail address: [email protected] (J. Gonzalo). 1 Present Address: Instituto de Cer´ amica y Vidrio (ICV, CSIC), Kelsen 5, 28049 Madrid, Spain. 2 Present address: Centro L´ aser, Universidad Polit´ ecnica de Madrid, Alan Turing 1, 28038 Madrid, Spain. 3 Present address: Tenaris-TAMSA, Tubos de Acero de M´ exico S.A., Carr. M´ exico – Veracruz, Km 433.7, Vía Xalapa, 91697 Veracruz, Mexico. Contents lists available at ScienceDirect Optical Materials journal homepage: www.elsevier.com/locate/optmat https://doi.org/10.1016/j.optmat.2025.117206 Received 19 February 2025; Received in revised form 7 May 2025; Accepted 24 May 2025 Optical Materials 167 (2025) 117206 Available online 10 June 2025 0925-3467/© 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/bync-nd/4.0/ ). low as 0.1 dB/cm [32]. However, the experimental deposition parameters must be carefully selected in order to obtain good quality film glasses as they determine the structure and composition of the deposited films, and thus their optical properties [27,28,33–36]. In the present work we exploit the characteristics of PLD to produce TeO 2 –WO 3 –PbO thin film glasses in a broad compositional range. This system is suitable for the development of nonlinear optical integrated devices since the three oxides can act as glass network formers, whereas their high polarisability could contribute to large high refractive index and non-linear optical properties [1–3,18,37]. We compare the structure and the optical response of the deposited films with that of parent bulk glasses. Their nonlinear optical response is evaluated in the frame of Line’s semiempirical model [12,18], and compared with the experimental results obtained by Degenerate Forward Mixing (DFWM). The observed results are discussed in terms of the structural differences observed between film glasses deposited by PLD and the corresponding parent bulk glasses. X-ray absorption spectroscopy of the bulk glasses evidences the presence of W 6+ ions in a distorted octahedral coordination, with the coordination environment and W–O bond distance remaining unchanged regardless of the WO 3 content, whereas thin film glasses deposited from these bulk glasses by PLD in a gas background present a structure close to that of bulk glasses; but with a reduced density, a moderate WO 3 and PbO enrichment and a much larger concentration of non-bridging oxygen bonds (NBO). These factors lead to a reduced linear refractive index and to an enhancement of the nonlinear optical response in the case of film glasses. 2. Experimental TeO 2 –WO 3 –PbO thin film glasses were produced by PLD using an ArF excimer laser (λ =193 nm, τ =20 ns FWHM, rep. rate =20 Hz). The laser beam was focused on the surface of bulk TeO 2 ⋅ WO 3 ⋅ PbO (TWP) glass targets to lead to an energy density of 1.7 J cm −2 . Glass targets with decreasing TeO 2 contents were prepared using standard melting methods as described elsewhere [38]. Table 1 summarizes their composition and density. Film deposition process took place in a vacuum chamber first evacuated to a residual pressure of 5 ×10 −4 Pa and then filled with oxygen up to a dynamic pressure of PO2 =4–5 Pa. Films 3–4 μ m thick were deposited on silicon, fused silica and borosilicate glass substrates held at room temperature and placed in front of the glass target at a distance, d TS =4 cm from its surface. 2.1. Compositional and structural characterization The atomic composition of the films was measured by ion beam techniques. Nuclear reaction analysis (NRA) was used to determine the oxygen content of the films through the nuclear reaction 16 O(d, p) 17 O at 0.878 MeV. The absolute oxygen content was determined within 10 % using a Ta 2 O 5 /Ta reference. Cation contents were measured by Rutherford backscattering spectrometry (RBS) using a 4 He 2+ beam at 1.98 MeV. The composition and thickness of the films were determined by simulation of the RBS spectra using the SIMNRA code [39]. X-ray absorption spectroscopy (XAS) measurements were performed at the CLAESS beamline [40] of the ALBA synchrotron source using a Si (311) double crystal monochromator and a spot size of ~200 μ m ×200 μ m. Analysis of X-ray absorption near edge structure (XANES) and extended X-ray absorption fine structure (EXAFS) were carried out to study the oxidation state and the local environment of W ions for representative bulk glass samples. Specific amounts of glass powders were mixed with cellulose and pressed into 9 mm pellets. Measurements of the W L 3 -edge (10,207 eV) and W L 1 -edge (12,100 eV) were carried out in transmission mode and the incident and transmitted intensities were measured with two ionization chambers (IC) filled with a N 2 /Kr mixture at 1 bar, providing ~10 % and ~80 % absorption in the first and second IC, respectively. A W foil was used for energy calibration and all measurements were performed at RT. The monochromator used for XAS has an energy resolution of 0.8-1 eV in the 10–12 keV range. The pre-edge, XANES and EXAFS regions were scanned with 1, 0.3 and 1 eV per point, respectively. Data analysis was carried out with ATHENA and ARTEMIS software of the DEMETER package [41]. The glass network structure of bulk and thin film glasses was analyzed by Raman spectroscopy. Raman spectra were recorded with a confocal Raman microscope equipped with an Ar + laser emitting at 514.5 nm. The output laser power was 150 mW and the beam was focused on the sample using a 50×objective. The elastic scattering was eliminated using a notch filter and the signal was detected through an electrically refrigerated CCD camera. In the case of film glasses the contribution of the substrate was carefully subtracted. 2.2. Linear optical characterization The linear refractive index has been determined from spectroscopic ellipsometry [42] using a Woollam VASE ellipsometer (Woollam Co. Inc.) set in photometry mode. A single monochromator was used to spectrally filter the white light generated by a Xe lamp, then collimated and directed through a polarizer to generate either sor p-linear polarized light. The ellipsometric parameters tanΨ and cosΔ of bulk and film glasses have been measured at three different angles of incidence (60◦, 65◦and 70◦) in the wavelength range from 300 to 1700 nm. Both the real (n) and imaginary (k) parts of the refractive index have been obtained from fitting the experimental ellipsometric data using Cauchy-type dispersion for the real part of the linear refractive index (n) and an exponential decay for the imaginary part (k): n(λ) = An+Bn λ2+Cn λ4(1) k(λ) = Akexp[Bk(1 λ−1 γ)] (2) where A n , B n , C n , A k , B k are adjustable parameters and γ is the optical gap expressed in nanometers. The experimental data were fitted to the model described using the Marquardt-Levenberg iterative algorithm [43] to minimize the mean square error. The experimental error for n is 0.01, while is 1 % in the case of k for values above our experimental resolution limit (10 −4 ). Transmittance spectra of bulk glass samples and films deposited on fused silica substrates were measured at Room Temperature using a Varian Cary 5000 spectrophotometer within the wavelength range of 250–2500 nm, employing a step size of 1 nm. The optical energy gap can be determined from the expression [1,2,44]: ( α ℏ ω )1 r=A(ℏ ω −Eg)(3) where α is the absorption coefficient, A is a constant related to the steepness of the transition edge, and r has different values depending upon the type of transition between the valence and the conduction bands [44,45]. In the case of tellurite glasses it has been shown that r =2 [1,46], which indicates that electronic transitions are indirectly allowed, and expression (3) becomes the well-known Tauc formula. Then, the optical energy gap (E g ) of tellurite thin film glasses can be determined by extrapolating ( α ħ ω ) 1/2 to 0 [1,2,44,45]. However, the use Table 1 Glass composition (mol%) and room temperature density of TeO 2 ⋅WO 3 ⋅PbO bulk glasses. Density data are taken form Ref. [38]. Glass TeO 2 WO 3 PbO ρ (g cm −3 ) TWP1 85 15 0 5.890 TWP3 80 15 5 6.056 TWP6 70 20 10 6.313 TWP8 60 30 10 6.519 TWP9 60 20 20 6.663 TWP10 50 30 20 6.831 G. Gorni et al. Optical Materials 167 (2025) 117206 2 of this relation in the case of mm-thick bulk glasses leads to an underestimation of E g [12,46,47]. Thus, in the case of bulk glasses we have estimated E g using a simple alloying model [48]: EgB =x Eg(TeO2) + y Eg(WO3) + z Eg(PbO)(4) where E gB is the optical energy gap of a bulk glass of composition: x TeO 2 - y WO 3 - z PbO mol%. E g (PbO)=2.8 eV [37] is the optical bandgap of PbO, whereas E g (TeO 2 )=3.5 [48], and E g (WO 3 )=2.8 eV [49] are the average optical bandgaps of amorphous TeO 2 and WO 3 , respectively. 2.3. Nonlinear optical characterization The modulus of the diagonal component of the third order optical susceptibility tensor (∣ χ (3) xxxx ∣) was measured by Degenerate Forward Mixing (DFWM) in the forward folded box configuration [50]. As thoroughly discussed in Ref. [51], the nonlinear refractive index and absorption coefficient are particularly difficult to determine in thin films, where the contribution of the thick substrate can dominate the material response. This is particularly relevant for the case of Z-scan measurements using short pulsed lasers at high repetition rates. In these cases, in addition to the presence of the strong substrate contribution to the non-linear response, “thermal lensing” artefacts can strongly affect the measured values even when a tiny amount of linear or non-linear absorption in the film and substrate ensemble gets involved. In this respect, DFWM measurements are much more reliable and versatile and can be considered free of spurious thermal contributions, although the true non-linear contribution of the substrate to the conjugated signal must be carefully analyzed as described in Appendix A. The excitation source for the DFWM measurements was an optical parametric amplifier operating at 1300 nm pumped with a 1 kHz repetition rate femtosecond Ti:Sapphire regenerative amplifier operating at λ =800 nm. The pulse compressor of the amplifier was adjusted to precompensate the dispersion caused by the optical elements located in the beam path in order to produce the shortest possible pulse (≈100 fs) at the sample. The excitation laser beam was split in three parallel arms with equal power (I p1 =I p2 =I pr =I 0 /3), allowing separate control of the beam polarization and the relative delay between pulses. In the case of bulk glasses a few mm thick, the beams were overlapped at the sample by means of a 75 mm focal length lens leading to a beam waist of ≈40 μ m. The intensity of the conjugated beam, I C , follows then the characteristic cubic dependence as a function of the pump beam intensity, whereas the use of parallel polarizations for all the beams allows accessing ∣ χ (3) xxxx ∣ that for simplicity we will refer to from now as ∣ χ (3) ∣ [50]: IC≅(3 π L ε 0n2 0cλ) χ (3)Ip1Ip2Ipr (5) where L is the overlapping length of the beams in the focus of the lens and n0 is the real part of the linear refractive index of the sample. The absolute value of ∣ χ (3) ∣ was finally evaluated by using a fused silica plate as reference material (∣ χ (3) ∣ SiO2 =1.5 ±0.5 ×10 −14 esu) [50]. In the case of film glasses, their reduced thickness (typically ~ μ m) makes extremely complex the accurate determination of their χ (3) due to the not negligible χ (3) of the substrate and the large thickness difference between sample and reference. Thus, we have developed the procedure described in Appendix A to increase the film-substrate signal ratio. 3. Results & discussion 3.1. Composition and structure of film glasses An O 2 pressure close to 8 Pa led to films with good optical properties and free of elemental Te 0 in the case of TeO 2 –TiO 2 –Nb 2 O 5 thin film glasses deposited by PLD [28]. Since the target-substrate distance (d TS = 4 cm) used here is longer than in the previous work (d TS =3 cm), the optimum O 2 pressure (P O2 ≈5 Pa) was found to be lower in this work than the value determined in Ref. [28] (see Fig. S1 of the Supplementary Material). The molar composition of the film glasses determined from RBS and NRA analysis is shown in Fig. 1, where the composition of the parent bulk glasses has been included for comparison. A decrease of TeO 2 and an increase of WO 3 and PbO contents with respect to bulk glasses are observed in all cases. The relative WO 3 enrichment in the film glasses increases with the WO 3 molar content up to a value of ≈16 % for the film having the lowest TeO 2 content, while Pb enrichment is similar in all cases ≈4–6 %. This behavior is related to the different angular distribution for species of different masses present in the laser generated plasma, which is characteristic of PLD of multicomponent oxides in a background O 2 atmosphere of a few Pa [29,52]. In this pressure range, the ablated species present in the plasma are scattered by the background O 2 gas molecules. This preferentially broadens the angular distribution of light atoms or molecules and leads to an enrichment on the heaviest elements along the axis of the plasma expansion [29], which corresponds to the area where the substrate was placed. Thus, a decrease of the light (Te; m Te =127.6 amu) to heavy (W, m W =183.8 amu; Pb, m Pb =207.2 amu) relative cation concentration is expected as it is shown in Fig. 1. In addition to these compositional changes, PLD is known to lead to multicomponent oxide thin film glasses having structural differences with respect to the parent bulk glasses [28,31,36]. Thus, we have analyzed in the first place the structure of TWP bulk glasses using XAS, to compare later on their structure with that of film glasses by means of Raman analysis. Fig. 2a shows the XANES spectra at the W L 3 -edge of representative bulk glass samples and the WO 3 reference. The edge position, determined as the maximum of the first derivative spectrum, is around 10,208 eV (see Fig. S2a of the Supplementary Material) and is consistent across all the glasses and the WO 3 reference. This indicates the presence of W 6+ ions, in agreement with X-ray photoelectron spectroscopy (XPS) results (see Section S1 of the Supplementary Material), but with bulk sensitivity, characteristic of XAS. The white-line transition of the WO 3 reference appears at 10,212 eV and is associated with the excitation of 2p 3/2 electrons to empty 5d levels. Its shape reveals two distinct contributions: A and B (as visualized also in Fig. S2b of the Supplementary Material), corresponding to transitions to the t 2g (A) and e g (B) levels of W 6+ in a distorted octahedral structure [53]. For the glasses, the overall shape of the white line is similar to the WO 3 reference, however the higher intensity of the t 2g transition suggests a different p-d mixing for the W 6+ ions in glasses. Moreover, a more asymmetrical peak shape suggests a higher variation in the local structure with slight deviations in bond metrics and centrosymmetry from the WO 3 reference. Despite this difference, the overall spectral envelope and features of W 6+ in the glasses closely resemble those of the WO 3 reference. To further investigate the site symmetry of W 6+ ions in these glasses, spectra were also measured at the W L 1 -edge (see Fig. 2b). Here, the absorption process involves the excitation of electrons from the 2s orbital to empty 6p energy levels. A prominent Fig. 1. Molar composition in mol % of ( ) bulk and ( ) film TWP glasses determined from RBS and NRA analysis. G. Gorni et al. Optical Materials 167 (2025) 117206 3 shoulder is observed around 12,106 eV in all the glasses, with a very similar feature present in the WO 3 reference. In contrast, a comparison with a BaWO 4 reference from the Materials Data Repository operated by the National Institute for Materials Science (NIMS), Japan, reveals a sharp, distinct peak. These transitions, referred to as pre-edge peaks, arise from the excitation of 2s electrons into the p component of mixed p-d orbitals and are highly sensitive to the local site symmetry of the ions. Their intensity is influenced by both the oxidation state and the geometry of the absorbing ion, with much higher intensity observed in low-symmetry sites, such as tetrahedral coordination [54], as exemplified by the BaWO 4 reference. Furthermore, the white-line transition in the glasses comprises two resonances around 12,120 eV and 12,140 eV, with a shape nearly identical to that of WO 3 . Considering the known structure of WO 3 , which consists of a distorted octahedron with four oxygen atoms at 1.80 Å and two oxygen atoms at around 2.11 Å [55], the coordination of W 6+ in these glasses is inferred to be a distorted octahedral geometry, rather than a regular octahedral or tetrahedral configuration [49]. The XANES spectra at both L 3 and L 1 -edge reveal that the local structure around W 6+ in these glasses is not affected by the initial WO 3 content and in all cases, we can confirm the presence of distorted octahedral units. The analysis of the EXAFS spectra at the W L 3 -edge was performed to confirm this result and to obtain quantitative information on the bond distance and the coordination of W 6+ ions in these glasses. The EXAFS spectra were analyzed after spectra normalization and background subtraction. A Fourier Transform (FT) was then applied to the k 2 - weighted EXAFS oscillations in the 3-10 Å −1 range, using a Hanning window. The fit was performed in the R-range 1–2.2 Å (in the phase uncorrected scale), see Fig. S3 of the Supplementary Material. The result of the fit shows that all the signals cannot be reproduced with W in tetrahedral coordination but with W in a distorted octahedron with 4 O at 1.81 Å and 2 O at 2.11 Å, see Table SII of the Supplementary Material. These results are similar to those reported by other authors for W-containing glasses [56] and different W compounds [57,58], thus excluding again the formation of tetrahedral units in these glasses. The second shell contributions, accounting for W–O–W and W–O–Te linkages, have not be fitted but are visible on Fig. S3a in the 2.3–3.7 Å range. In this range, TPW1 and TPW10 present a similar FT intensity distribution, which is slightly different from that of TPW9 and TPW3, which are similar to each other. Having this information in mind, we have analyzed the structure of the TWP bulk glasses through Raman analysis. Fig. 3 shows the normalized reduced Raman spectra [59] in the region from 200 to 1100 cm −1 for representative glasses with decreasing TeO 2 molar contents. The spectrum of the binary TeO 2 -WO 3 (TWP1) glass shows three main bands at 300–550 cm −1 , 550–725 cm −1 , 725–890 cm −1 and a strong peak centered around 925 cm −1 . The network of binary TeO 2 -WO 3 glasses is mainly built by chains of [TeO 4 ] trigonal bipyramids (tbp) coordination polyhedra, similar to that of the crystalline structure of α -TeO 2 , some of which are transformed in [TeO 3+δ ] polyhedra and [TeO 3 ] trigonal pyramids (tp) in the case of glasses [1,7,59,60]. Upon addition of WO 3 , which acts as conditional network former, XAS analysis has shown that WO 3 forms distorted [WO 6 ] octahedral units, with W 6+ cations in a sixfold coordination, with four W–O bonds at 1.80 Å and 2 longer W–O bonds at 2.11 Å. These units break the Te–O–Te linkages to form new W–O–W and W–O–Te linkages that contribute to the glass network [38,59,60]. Within this picture, we have assigned the bands observed in Fig. 3 to vibration modes of these units according to previous reports on the structure of TeO 2 -WO 3 glasses and to XAS results described before [59–62]. We have performed a peak fitting of Raman spectra and, as an example, the deconvolution of Raman spectrum of TWP1 with 8 Gaussian bands is shown in Fig. 3. The Fig. 2. XANES spectra at the (a) W L 3 -and (b) W L 1 -edge of bulk glasses and reference compounds. Fig. 3. Raman spectra of representative TWP (black line) bulk glasses. Spectra have been normalized by the intensity of the band peaking at ≈925 cm −1 and shifted vertically to ease comparison. They are ordered according to their TeO 2 molar content from TWP1 (85 %) to TWP10 (50 %). The deconvolution in Gaussian peaks is shown for TWP1 bulk glass: Green and blue lines correspond, respectively, to the different Te–O and W–O vibration modes. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) G. Gorni et al. Optical Materials 167 (2025) 117206 4 band at 550–725 cm −1 corresponds to the continuous network of [TeO 4 ] tbp. In particular, the peak at 605 cm −1 is assigned to the antisymmetric stretching of the continuous network of [TeO 4 ] tbp, while the peak at 660 cm −1 relates to the antisymmetric vibration of unequivalent Te–O–Te linkages. The band at 725–890 cm −1 is mainly related to the vibration of non-bridging oxygens (NBO) associated to [TeO 3+δ ] and [TeO 3 ] coordination polyhedra, depending if the interaction between NBO and adjacent atoms is strong (peak at 730 cm −1 ) or weak (peak at 805 cm −1 ). The remaining peaks are assigned to vibrations of [WO 6 ] distorted octahedra, consistently to what was observed in the XAS analysis. The intense peak at 925 cm −1 has been associated with different vibrations, such as W – – O bonds in O – – WO 5 units or W–O - bonds with unshared electron pairs [60]. In our case, considering the similarity of the XAS spectra of the glasses to that of anhydrous WO 3 , we associate this band with W–O - vibrations, as no W – – O double bonds are observed in anhydrous WO 3 . Moreover, the presence of W – – O is more common in WO 4 tetrahedral units than in WO 6 octahedral units. On the other side, stretching vibrations of W–O single bonds lead to a much weak peak at 860 cm −1 that appears as a shoulder of the intense band at 725-890 cm −1 in the TWP1 spectrum. Finally, we assign the peak centered at 350 cm −1 to deformation vibrations of [WO 6 ] units. Furthermore, there should be a non-negligible contribution of the bending vibration of [TeO 3 ] tps and the vibration of Te–O–W linkages to the Raman spectra around 350 cm −1 and 730 cm −1 , respectively [59,60] although we have not considered additional Gaussian peaks for simplicity. Table 2 summarizes the band assignment made for TWP bulk glasses. As it is shown in Fig. 3, the addition of PbO and the reduction of the TeO 2 have a clear effect on the glass structure. The positions of the bands related to the vibration of W–O - bonds with unshared electron pairs (925 cm −1 ) and [TeO 4 ] tbp (605 cm −1 ) shift when we decrease (increase) the TeO 2 (PbO) molar content (TWP9), which is due to the lower field intensity of Pb 2+ when compared to Te 4+ and W 6+ [1,38]. Simultaneously, the intensity of the peaks at 490 cm −1 and 660 cm −1 decreases, whereas the intensity of peaks related to NBO increases. In particular, the peak at 730 cm −1 , becomes the most intense peak related to Te–O vibration modes. This trend suggests that the increase of the PbO molar content induces a cleavage of Te–O–Te linkages, decreases the amount of [TeO 4 ] tbp and therefore increases the NBO fraction. These effects are more pronounced for TWP10, with only a 50 mol % of TeO 2 , and for which the intensity of the bands related to the continuous Te-based glass network decreases even further. Raman spectra obtained for films produced from targets with the largest (TWP1) and lowest (TWP10) TeO 2 contents are included in Fig. 4. They are close to their parent bulk glasses spectra, which suggests that films mostly preserve the glass structure, although two main differences are clearly observed. First, peaks in the 700-800 cm −1 range, assigned to vibrations of NBO, are more intense and second, the position of the peak related to the vibration modes of [WO 6 ] distorted octahedra (925 cm −1 ), slightly shifts towards higher wavenumbers in the case of films, which is related to the relative WO 3 enrichment of the films with respect to the parent bulk glasses, in a similar way to what it occurs in the case of TWP bulk glasses as we increase the WO 3 content. Since the NBO concentration influences the glass optical properties, we have qualitatively estimated the fraction of NBO in the glasses from the ratio of the intensities of the deconvoluted bands related to NBO (730 and 805 cm −1 ) to the peak assigned to the [TeO 4 ] tbp continuous network (660 cm −1 ) [7,36,63]. This ratio IR=I[NBO]/I[TeO4]= [I730 + I805]/I660) is shown in Fig. 5 for film and bulk glasses. I R increases for decreasing TeO 2 contents due to the increase of conditional glass former (WO 3 ) and modifier (PbO) fractions, which break the glass network. In addition, I R is always larger for film glasses when compared to the parent bulk glasses. This difference increases as TeO 2 molar content decreases to reach a maximum value I R(film) ≈1.5 I R(bulk) for TWP10. This increase can only be partially related to W and Pb enrichment in the films with respect to bulk targets and it is most likely related to the intrinsic characteristics of PLD, such as a very high instantaneous deposition rate in a very short time interval (≤1 μ s), a high atom mobility at the substrate and very fast cooling times [29]. Thus, film glasses are synthesized in experimental conditions far from that of traditional melting synthesis in which melting takes place under thermodynamic equilibrium Table 2 Assignment of vibrational bands observed in the Raman spectra of bulk TeO 2 ⋅WO 3 ⋅PbO glasses. Bands that appear as shoulders of intense bands are indicated by (s). Position (cm −1 ) Assignment 350 Deformation vibrations of [WO 6 ] (s) 485 Symmetrical stretching and bending vibrations of Te–O–Te linkages 605 Antisymmetrical vibration of a continuous [TeO 4 ] network (s) 660 Antisymmetrical vibration of Te eq -O ax -Te linkages 730 Stretching modes of NBO with strong interaction with adjacent Te atoms 805 Stretching modes of NBO with weak interaction with adjacent Te atoms 860 Stretching of W–O bonds in [WO 6 ] (s) 925 Stretching vibration of W–O - bonds in [WO 6 ] Fig. 4. Raman spectra corresponding to ( ) the films with the largest (TWP1) and lowest (TWP10) TeO 2 contents. Raman spectra measured for the parent bulk glasses ( ) have been included in the figure for comparison. Spectra have been normalized by the intensity of the band peaking at ≈925 cm −1 . Spectra corresponding to TWP1 film and bulk glasses have been shifted vertically to ease comparison. Fig. 5. Intensity ratio (I NBO /I [TeO4] ) of the deconvoluted Gaussian peaks related to NBO (730 and 805 cm −1 ) to the 660 cm −1 peak assigned to the [TeO 4 ] tbp continuous network for ( ) bulk and ( ) film TWP glasses. Dashed lines are linear fits of the experimental data. G. Gorni et al. Optical Materials 167 (2025) 117206 5 conditions with, typically, slow cooling rates and, as a consequence, the vitreous network of film glasses deposited by PLD is more disordered with a larger NBO concentration [28,36], similar to what is observed in the case of liquid TeO 2 [64]. 3.2. Linear optical properties Deposited film glasses have excellent optical properties. As it is shown in Fig. S5 of the supplementary material for a representative film glass, they have a large refractive index (n >2 in the studied spectral range that it is close to 2.1 for λ ≥800 nm, while k <10 −4 for λ>450 nm). In addition, they present a sharp absorption edge and a high transmission at wavelengths larger than that of the UV absorption edge. Yet, the structural peculiarities observed in the film glasses have an impact on their optical response when compared to that of parent bulk glasses. Fig. 6a shows the evolution of n with composition at λ =1.3 μ m both for film (n f ) and bulk (n B ) glasses. The value of n slightly increases in both cases as the TeO 2 molar content decreases, n f being a ≈3 % lower than n B in all cases. The increase of n as we increase PbO and WO 3 contents is expected since their oxide polarizabilities are larger than that of TeO 2 [37] and the fraction of highly polarisable NBO increases as the TeO 2 molar content decreases, as it is shown in Fig. 5 [1–3]. However, for a similar glass composition n B >n f despite the lower NBO fraction in bulk glasses. This behavior has been observed previously in sputtered amorphous TeO 2 films [21] and TeO 2 –TiO 2 –Nb 2 O 5 film glasses produced by PLD [28], and it is related to the effect of the experimental growth conditions on the films structure. In particular, PLD of complex oxides requires a O 2 gas environment, P O2 ≈5 Pa in this work, to avoid oxygen deficiency that leads to the presence of reduced elements such as Te 0 [27,28]. This modifies the film growth kinetic and leads to less dense film glasses [28], which decreases n f . The results presented in Fig. 6a suggest that this decrease of film density cannot be compensated by the increase of NBO fraction, neither by the enrichment of film glasses in PbO and WO 3 . The presence of O 2 -rich defects in the structure of the film glasses [35], the opening of the glass network to accommodate the excess of NBO [24] for films deposited in a O 2 environment, or the formation of complex (TeO 2 ) polymerized structures [15] could be responsible for this behavior. The optical energy gap for film glasses (E gf ) calculated from Eq. (3) is presented in Fig. 6b along with the values for bulk glasses (E gB ) calculated from the simple alloying model [48] using Eq. (4) and those of TWP1, TWP6 and TWP8 bulk glasses previously published in literature [46,47]. E gf is in the 3.3–3.4 eV range, whereas E gB seems to decrease from ≈3.4 to ≈3.2 eV when TeO 2 molar content decreases from 85 to 50 mol%. However, it is difficult to establish a general trend on the evolution of E g with TeO 2 molar content, since we have considered glasses in which we have modified simultaneously the molar content of the three oxides, and while the addition of WO 3 tends to decrease E g , that of PbO has the opposite effect [38]. Nonetheless, the evolution of E gB with TeO 2 content in the case of bulk glasses is in good agreement with the behavior of n B shown in Fig. 6a: As TeO 2 molar content decreases and PbO and WO 3 molar contents increase, n B increases, which is associated to the larger oxide polarizabilities of PbO and WO 3 and the increase of the fraction of highly polarisable NBO (Fig. 5), while E gB decreases (Fig. 6b) [37]. However, the evolution of E gf has a different behavior. E gf shows a slight increase in the considered TeO 2 molar content range, and it is larger than E gB values for TeO 2 molar contents lower than 70 %. Yet, if we consider the cases in which we have a equimolar substitution of TeO 2 by PbO (continuous black lines in Fig. 6b) and TeO 2 by WO 3 (dashed black lines in Fig. 6b) we observe, respectively, an increase and a decrease of E gf as it occurs in the case of bulk glasses [38]. The fact E gf ≥E gB may be related to the reduced density of film glasses, which translates in smaller refractive indices than bulk glasses (Fig. 6a) and thus, it could compensate the effect of larger concentration of NBO in the films on E gf. . These results reflects the complex dependence of E g and n with cations and oxygen polarisabilities in the case of glasses combining oxides having lone s 2 electron pairs (TeO 2 , PbO) and empty d orbitals (WO 3 ). Indeed, these results suggest that the film glass structure has a relevant role on the optical properties of films. 3.3. Non-linear optical response The non-linear optical susceptibility determined from Eq. (5) for bulk (∣ χ (3) B ∣) and film (∣ χ (3) f ∣) glasses at λ =1300 nm is shown in Fig. 7. ∣ χ (3) B ∣ is close to 10 −12 esu for all the compositional range and no significant variations are observed when adding WO 3 or PbO to the glass composition. The measured∣ χ (3) B ∣ values are comparable to that of bulk TeO 2 glass and TeO 2 based oxide glasses, either modelled [12,15] or measured using different techniques, such as DFWM, Z-Scan or Third Harmonic Generation, [1,12–14,16,65]. This suggests similar mechanisms responsible for the nonlinear optical response, such as the Fig. 6. (a) Evolution of n at 1.3 μ m with glass composition for ( ) bulk and ( ) film TWP glasses. (b) ( ) Optical energy gap (E g ) determined from Tauc plots for TWP film glasses ( ); E g values taken from refs. 46, 47 for TWP1, TWP6 and TWP8 bulk glasses and ( ) E g values calculated using the simple alloying model for TWP bulk glasses. Error bars for experimentally determined E gf have been included in Fig. 6b. Lines are guides for the eyes indicating the evolution of () bulk and ( ) film E g , and the evolution of E gf in the case of equimolar substitution of (——) TeO 2 by PbO and (– - -) TeO 2 by WO 3 . Fig. 7. Dependence of ∣ χ (3) ∣ with TeO 2 molar content for ( ) film and ( ) bulk TWP glasses. Error bars have been included in the figure. The shaded area correspond to the ∣ χ (3) ∣ values, including errors, calculated for bulk and film glasses using the Lines’ bond-orbital theory. G. Gorni et al. Optical Materials 167 (2025) 117206 6 hyperpolarizability of oxide bonds or the contribution of NBOs [15,16, 36]. As stated in the introduction, the addition of PbO or WO 3 to tellurite glasses leads to an enhancement of their nonlinear optical response. Yet, in the case of ternary glasses, there is not a straightforward dependence with the composition and while the increase of the [WO 3 ]/[TeO 2 ] ratio may lead to an increase of ∣ χ (3) B ∣ [66,67], its dependence with the [WO 3 ]/[TeO 2 +PbO] ratio is not so clear. However, a detailed study of the structural and compositional factors contributing to the nonlinear response of TeO 2 –WO 3 –PbO glasses would require dedicated experiments involving the synthesis of a much larger number of compositions as in Ref. [38], which is out of the scope of the present work. Measured ∣ χ (3) f ∣ values are similar than ∣ χ (3) B ∣ for large TeO 2 molar fractions (>60 mol%). However, ∣ χ (3) f ∣ increases, and ∣ χ (3) f ∣ >∣ χ (3) B ∣ as we further decrease the TeO 2 content, to reach a value of ∣ χ (3) f ∣ ≈4 ×10 −12 esu for the film having the smallest TeO 2 molar content (50 mol%). According to empirical models,∣ χ (3) ∣ should increase with n [2,3] and thus, ∣ χ (3) B ∣ should be larger than ∣ χ (3) f ∣ for all the compositional range, which is not the case. In order to investigate this issue in detail we have evaluated the nonlinear response using Lines’ bond-orbital theory that models the optical response in terms of the perturbations induced by the applied electric fields on the local bonding orbitals [2,12,18]. In the case of single oxides this model allows considering the contribution of the empty d orbitals to n 2 according to [2,12,16,18]: n2(esu) = 25l2 B(n2−1)f3 LE2 S n(E2 S−E2)4×10−13 (6) where f L =(n 2 +2)/3 is Lorentz local-field correction factor, E is the photon energy, l B (in Å) is the bond length, and E S (in eV) is the Sellmeier gap determined from the expression for the dispersion of n obtained using the single oscillator Wemple’s model [16,68]: 1 (n2−1)=E0 Ed −E2 E0Ed (7) where E is the photon energy, E d is the dispersion energy that is a measure of the electronic oscillator strength of interband optical transitions and finally, (E 0 ≈E S ) is the average oscillator energy, which is related to the average excitation energy for electronic transitions between the valence and conduction bands. The values of E 0 and E d can be obtained from the linear plot of 1/(n 2 -1) vs. E 2 (see section S4 of the Supplementary Material) [1,12,16,18,21,22,65,66,68,69]. E S values obtained for TWP bulk and film glasses are included in Table 3. Since at λ =1300 nm we are well below the two photon absorption threshold (i.e E g >2E) we can assume a pure refractive nonlinearity, therefore ∣ χ (3) ∣ values correspond to Re χ (3) [70] and then Eq. (6) becomes [18,50]: χ (3) Lines(esu) = 25l2 B(n2−1)f3 LE2 S 3 π (E2 S−E2)4×10−13 (8) Even if Lines’ model has been extended to multicomponent glasses [71], yet it is difficult to evaluate χ (3) in these cases due to the complexity in the assignation of a single characteristic l B for a multicomponent glass [65,72]. We have estimated l B using the approximation [12,73]: lB=(∑xil3 Bi)1/3, where x i is the molar fraction of component i (with Σ x i =1) and l Bi is the corresponding bond length (2.24 Å (Pb–O), 1.91 Å (Te–O)) [12,65], while for W–O bond length we have considered 1.91 Å that is the average value calculated in this work. The calculated l B , f L (@ 1300 nm) values are also included in Table 3. The values of χ (3) at 1300 nm determined from Eq. (8) using the data included in Table 3 for bulk and film glasses have been included in Fig. 7 as a shaded area. They do not show any appreciable variation in the studied compositional range, and they are in all cases in the range ≈0.9 ±0.2 ×10 −12 esu. Thus, Lines’ model predicts well the experimental values measured for bulk glasses and film glasses having TeO 2 molar contents above 60 %; but it predicts χ (3) values smaller than those measured experimentally in the case of film glasses with lower TeO 2 contents. As mentioned in the introduction, the addition of WO 3 and PbO increases the nonlinear optical response of binary tellurite glasses due to their large hyperpolarizability [16]. However, we do not observe a significant variation of ∣ χ (3) B ∣ even in the case of bulk glasses with large content of WO 3 and PbO (TWP10 having 30 mol% of WO 3 and 20 mol% of PbO). Thus, as it happens in the case of the linear optical properties, this result suggests the existence of additional factors contributing to the nonlinear response of the film glasses. Ab-initio calculations [15] have suggested that the intense nonlinear optical response of tellurite glasses is related to the high hyperpolarizability of (TeO 2 ) n chainlike structures characterized by the presence of Te–O–Te bridges that in our case are evidenced by the Raman band at 485 cm −1 (Table 2 and Fig. 3). As the concentration of WO 3 and PbO increases, the intensity of the 485 cm −1 band decreases due to the formation of Te–O–W linkages and the breaking of the network by the introduction of lead. Both factors decrease the length of the TeO 2 chainlike structures, which should lead to a decrease of χ (3) . However, the incorporation of W 6+ and Pb 2+ to the glass and the associated increase of very polarisable NBOs (Fig. 5) may have the opposite effect as they increase χ (3) . In the case of bulk glasses, these contributions might compensate each other to lead to an approximately constant ∣ χ (3) B ∣ value in the considered compositional range considered. However, this is not the case in the case of film glasses as ∣ χ (3) f ∣ increases for TeO 2 molar contents below 60 %. The observed enrichment in WO 3 and PbO cannot be responsible for such an increase, as TWP10 bulk glass has higher WO 3 and PbO molar contents than TWP9 film glass but its∣ χ (3) B ∣ does not show any increase. Alternatively, in a previous work we evidenced the importance that an excess of NBO could have in the nonlinear optical response of TeO 2 –TiO 2 –Nb 2 O 5 thin film glasses [36]. In the present case we observe a similar behavior for film glasses with reduced (<60 %) TeO 2 molar content (Fig. 5): The increase of the IR= I[NBO]/I[TeO4]ratio, related to the concentration of NBO, is much larger than that of the corresponding parent bulk glasses. Thus, this excess of NBO could not only compensate for the aforementioned decrease of χ (3) , which would be related to the shortening of the TeO 2 chains [15] in the films with the lowest TeO 2 contents, but also induce the increase of ∣ χ (3) f ∣ in these films as it is shown in Fig. 7. 4. Conclusions TeO 2 –WO 3 –PbO ternary tellurite thin film glasses have been produced by PLD at room temperature in a moderate (5 Pa) O 2 pressure. The structural analysis of bulk glasses shows the presence of W 6+ ions in a distorted octahedral coordination. Films show a moderate (small) enrichment of WO 3 (PbO) when compared to the parent bulk glasses, which is related to the characteristics of the laser generated plasma expansion. In addition, films deposited from parent bulk glasses with reduced TeO 2 molar content (<60 %) present a large concentration of NBO. Both film and bulk glasses present excellent optical properties. Table 3 Refractive index, n, at 1300 nm, and calculated l B , f L (@ 1300 nm), and E S values for bulk and film glasses. Sample l B (Å) f L n@1300 E S (eV) TWP1 Bulk 1.91 10.15 2.12 6.52 TWP3 Bulk 1.93 10.15 2.12 6.97 TWP6 Bulk 1.95 10.33 2.13 6.81 TWP8 Bulk 1.95 10.51 2.14 6.44 TWP9 Bulk 1.98 10.74 2.15 6.57 TWP10 Bulk 1.98 10.74 2.15 6.32 TWP1 Film 1.91 9.38 2.08 6.33 TWP3 Film 1.93 9.38 2.08 5.45 TWP6 Film 1.95 9.56 2.09 6.70 TWP8 Film 1.95 9.38 2.08 6.40 TWP9 Film 1.99 9.56 2.09 6.42 TWP10 Film 1.99 9.76 2.10 6.48 G. Gorni et al. Optical Materials 167 (2025) 117206 7 Bulk glasses present almost constant values of E g ≈3.3-3.2, n ≈ 2.12–2.15 and a ∣ χ (3) B ∣≈0.8–1.1 ±0.2 ×10 −12 in all the considered compositional range. Instead, film glasses show slightly higher (lower) E g ≈3.3–3.4 (n ≈2.05–2.10) and a nonlinear third order optical susceptibility that increases from ∣ χ (3) f ∣≈1.2 ±0.6 ×10 −12 up to 3.9 ±1,9 ×10 −12 esu when decreasing the TeO 2 molar content. These differences between bulk and film glasses reveal that their optical constants are not determined by the cations only, but that structural factors related to the deposition of the films by PLD have a strong influence on them. In particular, the lower refractive index of the films is most likely related to the expected reduced density of the films associated to their growth in a gas background, while the much larger concentration of NBO in the case of films with reduced TeO 2 (<60 mol%) with respect to the parent bulk glasses would contribute to the observed enhancement of ∣ χ (3) f ∣ for these film glasses. CRediT authorship contribution statement G. Gorni: Writing – review & editing, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation. D. Munoz-Martin: Writing – review & editing, Investigation. A. Ruiz de la Cruza: Software, Methodology, Investigation, Formal analysis. V. Martin-Diaconescu: Validation, Software, Investigation. L. Simonelli: Validation, Software, Investigation. J. Garcia-Lopez: Software, Investigation, Formal analysis. J.M. Fernandez Navarro: Supervision, Methodology, Investigation, Conceptualization. J. Solis: Writing – review & editing, Writing – original draft, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. J. Gonzalo: Writing – review & editing, Writing – original draft, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was funded by the Spanish Research Agency (AEI, Ministry of Research and Innovation) and the European Regional Development Fund (ERDF) under grants PID2020-112770RB-C21 and PID2021123190OB-I00, and by the CSIC (PIE-202050E195). G. Gorni acknowledges Grant FJC2020-044866-I funded by MCIN/AEI/10.13039/ 501100011033, by Plan de Recuperaci´ on, Transformaci´ on y Resiliencia, and by the “European Union NextGeneration EU/PRTR”. A. Jha (Institute for Materials Research, Univ. of Leeds) is thanked for helpful discussions. Appendix B. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.optmat.2025.117206. APPENDIX A Two major problems must be solved to achieve reliable χ (3) measurements in the case of film glasses a few μ m thick [51]. First, thin film glasses are deposited on substrates 10 2 –10 3 μ m thick with small but not negligible χ (3) . Thus, the nonlinear optical response of the films, ∣ χ (3) f ∣, can be hidden by that of the substrate, ∣ χ (3) s ∣. Second, the measurement of ∣ χ (3) f ∣ is based on the comparison between the conjugated signals of sample and reference. Thus, calculated ∣ χ (3) f ∣ values may yield wrong results if we do not consider the large thickness difference between them, and the overlapping of the beams at different depths of them. According to Eq. (5) the ratio between the conjugated signal intensity originated in the film glass (ICf) and that of the substrate (ICs) will be given by: Icf Ics ≅A χ (3) f  χ (3) s (A1) where A=n2 SLf n2 fLS (A2) and n f , n s and L f , L s are the refractive indexes and overlapping lengths of the beams for film (f) and substrate (s), respectively. The typical values of these parameters for the glasses and substrates considered in the present work are n f ≈2, n s ≈1.5, L f ≈2–3 μ m, and L s ≈100 μ m which leads to A≈60–90. In the case of heavy metal oxide and tellurite glasses,∣ χ (3) ∣ is in the range ~10 −13 -10 −11 esu, which is several orders of magnitude larger than that of fused silica or corning glass substrates (~10 −14 esu) [50]. Thus, the deconvolution of the film glass signal from the conjugated signal generated by the film-substrate ensemble (sample) should be in principle possible. However, a decrease of the substrate contribution to the total conjugated signal is desirable to improve the precision of the measurement. This is achieved through the use of optics with large numerical apertures, such as microscope objectives; yet, their small entrance pupil makes the alignment of the experimental system extremely complicated. We have used in this work an alternative approach that is based in locating the sample in a position with respect to the focusing lens that helps to increase the film-substrate signal ratio. G. Gorni et al. Optical Materials 167 (2025) 117206 8 Fig. A1. (a) ( ) Intensity of the conjugated signal of a 100 μ m thick bare substrate measured at different distances from the lens focus. The position Δx =0 mm corresponds to the maximum intensity detected. The dashed line corresponds to fitting of the experimental data with a Gaussian function. (b) Scheme of the relative positions of the film (green line) - substrate (pale blue rectangle) ensemble with respect to the conjugated signal for (1) maximum contribution of the substrate to the measured ∣ χ (3) ∣ and (2) the actual position where DFWM measurements were made. (c) Comparison of the SiO 2 reference thickness (dark blue rectangle) with respect to the spatial intensity distribution of the conjugated signal. Fig. A1a shows the intensity of the conjugated signal of a bare substrate as a function of its position along the beam propagation axis in the vicinity of the focus of the lens used to overlap the beams in the DFWM experiment. The confocal parameter of the lens is much larger than the substrate thickness and therefore, the intensity of the conjugated signal mirrors the expected Gaussian distribution of intensities associated to the focal volume of a Gaussian beam. Fig. A1a clearly shows that the maximum contribution of the substrate to the conjugated signal occurs when the substrate is placed right at the focus of the lens (Δx =0 mm) and it decreases as we move away from it. We can take advantage of this fact to improve the film-substrate signal ratio. In the position 1 indicated in Figure A1b, both film and substrate contribute with the same weight to the DFWM signal. However, if we move the sample several hundreds of microns away from the focus (position 2), the relative contribution of the film will increase with respect to that of the substrate due to the decrease of the overlapping of the beams as we move away from the focus. We can then estimate an effective overlapping length by assuming that the total conjugated signal of the substrate is the sum of the conjugated signals generated by differential thicknesses of the substrate weighted by the overlapping distribution Gaussian function, f(x): Ic=∫Δx0+Ls Δx0 I(x)d(x) = I0∫Δx0+Ls Δx0 f(x)d(x)(A3) where L s is the substrate thickness, Δx 0 the distance from the maximum overlapping to the surface of the substrate, and I 0 the intensity of the conjugated signal at Δx 0 . We can then define an effective substrate thickness, Leff s, as the thickness of an equivalent substrate which generates the same total conjugated signal IC if the overlapping distribution was constant and equal to f(Δx0): Ic=I0∫Δx0+Ls Δx0 f(x)d(x) = I0∫Leff s 0 f(Δx0)d(x)(A4) Leff s=1 f(Δx0)∫Δx0+Ls Δx0 f(x)d(x)(A5) Leff s can be considered as the overlapping length inside the substrate, while in the case of the film, we assume that f(x) is constant due to its small thickness, and therefore, the effective film thickness is similar to the real thickness. To illustrate the effect on the film-substrate signal ratio, we have calculated that by moving the sample away from the focus 300 μ m, as it is shown in position 2 of Fig. A1b, Leff s≈76 μ m which leads to a ~25 % increase of the film-substrate signal ratio. The difference between the total thickness and the effective thickness is much larger in the case of the reference, since it is 1 mm thick while the pumping beams are only effectively overlapped over a few hundreds of microns inside the sample (Fig. A1c). For example, in the same conditions than those described above with the reference moved away 300 μ m from the focus of the lens, the effective reference thickness, Leff r, is ≈150 μ m. In this case, the use of the total reference thickness instead of the effective one for the calculation of ∣ χ (3) f ∣ can lead to serious mistakes, due to the large difference between the two values. We have estimated that our approach allows the determination of ∣ χ (3) f ∣ with an accuracy close to 50 % by moving away the sample from the center of the overlapping region and subtracting the contribution of the substrate to the conjugated signal. G. Gorni et al. Optical Materials 167 (2025) 117206 9