^{1}, F. Di Fonzo

^{2}, S. Pietralunga

^{3}, C. Ubaldi

^{3}and C. E. Bottani

^{1}

### Abstract

The interest in the measurement of the elastic properties of thin films is witnessed by a number of new techniques being proposed. However, the precision of results is seldom assessed in detail. Brillouin spectroscopy (BS) is an established optical, contactless, non-destructive technique, which provides a full elastic characterization of bulk materials and thin films. In the present work, the whole process of measurement of the elastic moduli by BS is critically analyzed: experimental setup, data recording, calibration, and calculation of the elastic moduli. It is shown that combining BS with ellipsometry a fully optical characterization can be obtained. The key factors affecting uncertainty of the results are identified and discussed. A procedure is proposed to discriminate factors affecting the precision from those affecting the accuracy. By the characterization of a model transparent material, silica in bulk and film form, it is demonstrated that both precision and accuracy of the elastic modulimeasured by BS can reach 1% range, qualifying BS as a reference technique.

I. INTRODUCTION

II. INELASTIC LIGHT SCATTERING BY BULK AND SURFACE ELASTIC WAVES

III. EXPERIMENTS

IV. ASSESSMENT OF THE UNCERTAINTIES

V. RESULTS AND DISCUSSION

VI. CONCLUSIONS

### Key Topics

- Silica
- 21.0
- Velocity measurement
- 21.0
- Refractive index
- 16.0
- Backscattering
- 15.0
- Elastic moduli
- 14.0

## Figures

The main Brillouin scattering geometries (thin arrows: optical wavevectors; thick arrows: acoustic wavevectors). (a) backscattering, at incidence angle , direct backscattering by a surface acoustic wave of wavevector **k** _{s} or a bulk acoustic wave of wavevector **k** _{b} ^{(1)}; for a transparent film on a reflecting substrate, indirect backscattering by a bulk acoustic wave of wavevector **k** _{b} ^{(2)}. (b) Forward scattering, with incidence angle _{ i } and light collection at angle _{ s } and (c) transmission scattering, with incidence angle _{ i } and light collection at angle _{ s }.

The main Brillouin scattering geometries (thin arrows: optical wavevectors; thick arrows: acoustic wavevectors). (a) backscattering, at incidence angle , direct backscattering by a surface acoustic wave of wavevector **k** _{s} or a bulk acoustic wave of wavevector **k** _{b} ^{(1)}; for a transparent film on a reflecting substrate, indirect backscattering by a bulk acoustic wave of wavevector **k** _{b} ^{(2)}. (b) Forward scattering, with incidence angle _{ i } and light collection at angle _{ s } and (c) transmission scattering, with incidence angle _{ i } and light collection at angle _{ s }.

Experimental setup for Brillouin spectroscopy. BeS: beam splitter; M1, M2: mirrors; Sp: sample; L1–L5: lenses; P: entrance pinhole of the spectrometer; FP: Fabry-Perot interferometer; D: light detector. With BeS in place the backscattering and the 90º scattering geometries are superposed; 90º scattering alone is obtained removing BeS, backscattering alone is obtained substituting BeS by M1. Additional details, such as the mirrors which steer the scattered beam, are not shown.

Experimental setup for Brillouin spectroscopy. BeS: beam splitter; M1, M2: mirrors; Sp: sample; L1–L5: lenses; P: entrance pinhole of the spectrometer; FP: Fabry-Perot interferometer; D: light detector. With BeS in place the backscattering and the 90º scattering geometries are superposed; 90º scattering alone is obtained removing BeS, backscattering alone is obtained substituting BeS by M1. Additional details, such as the mirrors which steer the scattered beam, are not shown.

Brillouin spectra from the bulk silica sample. Bottom to top (see also Fig. 2): backscattering alone; 90º forward scattering (Fig. 1(b)) together with backscattering; 90º transmission scattering (Fig. 1(c)) together with backscattering. Incident light polarized normal to the incidence plane. For clarity, the second and third spectra are shifted by two and four decades, and the strong peak at null frequency shift, due to elastically scattered light, is removed. Peak labels: Gh: instrumental “ghosts” due to elastically scattered light; R: Rayleigh wave of probed wavevector **k** _{s}; BL and BT: bulk longitudinal and bulk transversal waves, of probed wavevectors **k** _{b} ^{(1} ^{)}, **k** _{b} ^{(3} ^{)}, and **k** _{b} ^{(4} ^{)} (see Fig. 1). Lines with square marker connect some Stokes/anti-Stokes doublets; lines with a circle marker connect the peaks due to the BL and BT waves probed by the same wavevector.

Brillouin spectra from the bulk silica sample. Bottom to top (see also Fig. 2): backscattering alone; 90º forward scattering (Fig. 1(b)) together with backscattering; 90º transmission scattering (Fig. 1(c)) together with backscattering. Incident light polarized normal to the incidence plane. For clarity, the second and third spectra are shifted by two and four decades, and the strong peak at null frequency shift, due to elastically scattered light, is removed. Peak labels: Gh: instrumental “ghosts” due to elastically scattered light; R: Rayleigh wave of probed wavevector **k** _{s}; BL and BT: bulk longitudinal and bulk transversal waves, of probed wavevectors **k** _{b} ^{(1} ^{)}, **k** _{b} ^{(3} ^{)}, and **k** _{b} ^{(4} ^{)} (see Fig. 1). Lines with square marker connect some Stokes/anti-Stokes doublets; lines with a circle marker connect the peaks due to the BL and BT waves probed by the same wavevector.

Brillouin spectra measured in backscattering from the film silica sample. Bottom to top: 45° and 60° incidence angles, with incident light polarized normal to the incidence plane; 60° incidence angle, with incident light polarized in the incidence plane. Scattered light collected without polarization analysis. For clarity, the second and third spectra are shifted, and the peak due to elastically scattered light is removed. Peak labels and markers: same of Fig. 3.

Brillouin spectra measured in backscattering from the film silica sample. Bottom to top: 45° and 60° incidence angles, with incident light polarized normal to the incidence plane; 60° incidence angle, with incident light polarized in the incidence plane. Scattered light collected without polarization analysis. For clarity, the second and third spectra are shifted, and the peak due to elastically scattered light is removed. Peak labels and markers: same of Fig. 3.

The two analysis procedures for one of the subsets of 15 doublets from the bulk sample. For the measured value , the central curve corresponds to the value , while couples of curves delimit three nested bands, which correspond to the three terms of Eq. (D.6). The inner band is determined by the precision of , the next band includes the inaccuracy due to *n* _{ b }, the outer band also includes the inaccuracy due to ρ_{ b }. Analogous curves are drawn for (for which the third band coincides with the second) and (for which the two outer bands coincide with the inner one). The approximately elliptical curves are the isolevel curves of the normalized estimator *S’*, at the 68%, 90%, 95%, 99%, and 99.9% confidence level. The thicker curve is the 95% confidence region and the two translated replicas are the 95% confidence regions obtained by modified values of *n* _{ b } and ρ_{ b }. The nominal values *E* = 72 GPa, *ν* = 0.17, supplied with the specimen to calibrate the indenters, are also shown.

The two analysis procedures for one of the subsets of 15 doublets from the bulk sample. For the measured value , the central curve corresponds to the value , while couples of curves delimit three nested bands, which correspond to the three terms of Eq. (D.6). The inner band is determined by the precision of , the next band includes the inaccuracy due to *n* _{ b }, the outer band also includes the inaccuracy due to ρ_{ b }. Analogous curves are drawn for (for which the third band coincides with the second) and (for which the two outer bands coincide with the inner one). The approximately elliptical curves are the isolevel curves of the normalized estimator *S’*, at the 68%, 90%, 95%, 99%, and 99.9% confidence level. The thicker curve is the 95% confidence region and the two translated replicas are the 95% confidence regions obtained by modified values of *n* _{ b } and ρ_{ b }. The nominal values *E* = 72 GPa, *ν* = 0.17, supplied with the specimen to calibrate the indenters, are also shown.

The two analysis procedures for one of the subsets of 15 doublets from the film sample. For the measured value the central curve corresponds to the value , then couples of curves delimit three nested bands, which correspond to the three terms of Eq. (D.8). The inner band is determined by the precision of , the next band includes the inaccuracy due to *n* _{ f } (also including the indirect effect by the mass density, see Part D of the Supplementary material in the supplementary material), the outer band also includes the inaccuracy due to ρ_{ b }. Analogous curves are drawn for , , and . For , the two outer bands coincide with the inner one. The approximately elliptical curves are the isolevel curves of the normalized estimator *S* ^{ ′ }, at the 68%, 90%, 95%, 99%, and 99.9% confidence level. The thicker curve is the 95% confidence region and the two translated and partially overlapped replicas are the 95% confidence regions obtained by modified values of *n* _{ f } and ρ_{ b }.

The two analysis procedures for one of the subsets of 15 doublets from the film sample. For the measured value the central curve corresponds to the value , then couples of curves delimit three nested bands, which correspond to the three terms of Eq. (D.8). The inner band is determined by the precision of , the next band includes the inaccuracy due to *n* _{ f } (also including the indirect effect by the mass density, see Part D of the Supplementary material in the supplementary material), the outer band also includes the inaccuracy due to ρ_{ b }. Analogous curves are drawn for , , and . For , the two outer bands coincide with the inner one. The approximately elliptical curves are the isolevel curves of the normalized estimator *S* ^{ ′ }, at the 68%, 90%, 95%, 99%, and 99.9% confidence level. The thicker curve is the 95% confidence region and the two translated and partially overlapped replicas are the 95% confidence regions obtained by modified values of *n* _{ f } and ρ_{ b }.

## Tables

For the velocities *v* _{ l }, *v* _{ t }, and *v* _{ R } measured in the various geometries, contributions to the relative uncertainties *σ* _{ v }/*v* due to the various causes discussed in the text. Null contributions are indicated by zero when they are null only in the specified conditions, and by a dash when they are intrinsically null. The values of *σ* _{ p }/*n* _{ p } report the ranges of values found for the various cases. For *σ* _{ g }/*n* _{ g } the range of observed value is ≈(1 ÷ 8) × 10^{−4}, for all the cases. The reported typical value results from the combination (Eq. (7)) with *σ* _{ FSR }/*FSR* ≈ 2 × 10^{−4}, which also is same for all the cases.

For the velocities *v* _{ l }, *v* _{ t }, and *v* _{ R } measured in the various geometries, contributions to the relative uncertainties *σ* _{ v }/*v* due to the various causes discussed in the text. Null contributions are indicated by zero when they are null only in the specified conditions, and by a dash when they are intrinsically null. The values of *σ* _{ p }/*n* _{ p } report the ranges of values found for the various cases. For *σ* _{ g }/*n* _{ g } the range of observed value is ≈(1 ÷ 8) × 10^{−4}, for all the cases. The reported typical value results from the combination (Eq. (7)) with *σ* _{ FSR }/*FSR* ≈ 2 × 10^{−4}, which also is same for all the cases.

For the bulk and the film samples, for the moduli *E*, *B*, *G*, and *C* _{11} and for Poisson's ratio *ν*, for the values obtained from the 12 subsets of 15 doublets each: the values associated to the narrowest and the widest uncertainty intervals; the uncertainties due to finite precision alone (these are the same values of Table F.I), and those which also include the possible inaccuracies due to *n* and *ρ*; the relative difference between the highest and the lowest estimates from the 12 subsets, the best estimate, given by the weighted average of the values given by the 12 subsets.

For the bulk and the film samples, for the moduli *E*, *B*, *G*, and *C* _{11} and for Poisson's ratio *ν*, for the values obtained from the 12 subsets of 15 doublets each: the values associated to the narrowest and the widest uncertainty intervals; the uncertainties due to finite precision alone (these are the same values of Table F.I), and those which also include the possible inaccuracies due to *n* and *ρ*; the relative difference between the highest and the lowest estimates from the 12 subsets, the best estimate, given by the weighted average of the values given by the 12 subsets.

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