RESEARCH ARTICLE | APRIL 25 2023 Contactless characterization of the elastic properties of glass microspheres Jeremie Maire ; Tomasz Necio ; Emigdio Chávez-Ángel ; Martín F. Colombano; Juliana Jaramillo-Fernández ; Clivia M. Sotomayor-Torres ; Nestor E. Capuj; Daniel Navarro-Urrios  APL Mater. 11, 041128 (2023) https://doi.org/10.1063/5.0146969 CrossMark   View Export Online Citation Articles You May Be Interested In Electrical contactless microfluidic flow quantification Appl. Phys. Lett. (January 2022) Novel contactless mode of electroreflectance Appl. Phys. Lett. (October 1991) New contactless measurement of photoeffects in semiconductors 26 September 2023 14:03:19 Journal of Applied Physics (October 1989) APL Materials ARTICLE scitation.org/journal/apm Contactless characterization of the elastic properties of glass microspheres Cite as: APL Mater. 11, 041128 (2023); doi: 10.1063/5.0146969 Submitted: 17 February 2023 • Accepted: 7 April 2023 • Published Online: 25 April 2023 Jeremie Maire,1,a) Tomasz Necio,1 Emigdio Chávez-Ángel,1 Martín F. Colombano,2 3 Juliana Jaramillo-Fernández, Clivia M. Sotomayor-Torres,1,4 Nestor E. Capuj,5,6 and Daniel Navarro-Urrios3,b) AFFILIATIONS 1 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra 08193, Barcelona, Spain 2 Université Paris Cité, CNRS, Laboratoire Matériaux et Phénomènes Quantiques, 75013 Paris, France 3 MIND-IN2UB, Departament d’Electrònica, Facultat de Física, Universitat de Barcelona, Martí i Franquès 1 08028, Barcelona, Spain 4 ICREA—Institució Catalana de Recerca i Estudis Avançats, Barcelona 08010, Spain 5 Depto. Física, Universidad de La Laguna, 38200 San Cristóbal de La Laguna, Spain 6 Instituto Universitario de Materiales y Nanotecnología, Universidad de La Laguna, 38071 Santa Cruz de Tenerife, Spain 26 September 2023 14:03:19 a) Current address: Univ. Bordeaux, CNRS, Bordeaux INP, I2M, UMR 5295, F-33400 Talence, France and YaaaArts et Metiers Institute of Technology, CNRS, Bordeaux INP, Hesam Universite, I2M, UMR 5295, F-33400 Talence, France. b) Author to whom correspondence should be addressed: dnavarro@ub.edu ABSTRACT Glass microspheres are of great interest for numerous industrial, biomedical, or standalone applications, but it remains challenging to evaluate their elastic and optical properties in a non-destructive way. In this work, we address this issue by using two complementary contactless tech- niques to obtain elastic and optical constants of glass microspheres with diameters ranging from 10 to 60 μm. The first technique we employ is Brillouin Light Scattering, which yields scattering with longitudinal acoustic phonons, the frequency of which is found to be 5% lower than that measured in the bulk material. The second technique involves exciting the optical whispering gallery modes of the microspheres, which allows us to transduce some of their vibrational modes. The combined data allow for extracting the refractive index and the elastic constants of the material. Our findings indicate that the values of those properties are reduced with respect to their bulk material counterpart due to an effective decrease of the density, resulting from the fabrication process. We propose the use of this combined method to extract elastic and optical parameters of glass materials in microsphere geometries and compare them with the values of the pristine material from which they are formed. © 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0146969 INTRODUCTION intermolecular binding detection limits for biosensing applications,1 ultrahigh optical quality factors,9 optical lasing at low pump power High quality spherical microspheres are one of the earliest thresholds,10 and even mechanical lasing.6 Microspheres can also forms of optical microcavities, and, as such, they have been widely be exploited in the form of colloidal crystals with applications in studied and utilized for a variety of applications, such as sensing,1–3 the fields of organic electronics, photonics, phononics,11,12 lithog- magnetometry,4 or optomechanical oscillators.5,6 In this context, raphy,13 and thermal management.14–16 They can be produced in a light is confined inside the microcavity via total internal reflec- variety of materials, such as polymers, crystalline and glassy com- tion, therefore, leading to whispering gallery modes resonances.7 pounds, and sizes ranging from a few hundred nanometers to These resonances can lead, for instance, to a strong enhancement millimeters. They can be easily fabricated, either individually or of the spontaneous emission rate in light emitting media.8 The in large quantities by a variety of methods, which include polish- microsphere shape itself has also been used to demonstrate low ing, chemical etching, and rapid quenching of liquid droplets. Their APL Mater. 11, 041128 (2023); doi: 10.1063/5.0146969 11, 041128-1 © Author(s) 2023 APL Materials ARTICLE scitation.org/journal/apm elastic properties are of crucial importance for most applications. RESULTS AND DISCUSSION For example, in phononic colloidal crystals, the phonon dispersion BLS and speed of sound relation is directly dependent on the elastic properties of the spheres as well as on their size.11,17 In composite materials, spheres can be The sound velocity and the phonon frequency of the micro- added to reinforce mechanically a matrix,18 requiring precise char- spheres were determined from Brillouin light scattering (BLS) mea- acterization of their elastic properties. However, it is well known surements. BLS is a widely used technique for measuring sound that in many cases, the fabrication method can significantly alter velocities and acoustic phonon frequencies in solids, liquids, and the optical and elastic properties of the material in the microsphere gases.22–25 It probes the light frequency shift, resulting from the shape with respect to the bulk values. Extracting these properties inelastic scattering of the light induced by thermally populated below the millimeter scale without damaging the spheres remains a acoustic waves in the medium. During this inelastic scattering pro- challenge. cess, the incident photon either loose or gain energy so that a phonon Here, we combine two optical non-destructive techniques, is emitted or absorbed. This results in the energy of inelastically namely, inelastic Brillouin light scattering (BLS) and optomechan- backscattered photons shifting to lower (Stokes) or higher (Anti- ical coupling (OM), to extract the elastic and optical constants Stokes) values, respectively. The difference between incident and of Barium–Titanium-Silicate glass material in a microsphere with backscattered photon energy corresponds to the phonon energy. To diameters ranging from 10 to 60 μm. We have observed that the detect this shift of a few GHz, much smaller than that of Raman longitudinal phonon frequencies detected in the BLS experiment scattering, a six-pass tandem Fabry–Perot interferometer is widely decrease in the microsphere when compared to those measured used.26 in the pristine bulk material by as much as 5%; information that More recently, BLS has been used to measure the disper- is considered alone can be attributed either to a reduction of the sion relation in phononic crystals27 and mechanical properties in material refractive index, to a relaxation of the elastic properties, or microsphere-based colloidal crystals.28 To that effect, the system was a combination of both. The analysis of the frequency of the sphere mounted with a microscope objective, enabling precise selection of vibrational modes detected with the OM technique allows extracting the area and depth of material to be characterized, which is also ade- the elastic constants of the material in the microsphere in a reliable quate for measuring a single microsphere with diameters larger than way, leading to longitudinal and transversal sound velocities that are a few micrometers. Our implementation comprises a 532 nm laser 4957 m/s (2% less than the value extracted from the BLS measure- focused on a microsphere by means of a ×50 microscope objective 26 September 2023 14:03:19 ment of the bulk material) and 2655 m/s. The material refractive (NA = 0.45). The position of the laser beam on the microsphere is index at 532 nm in the microsphere as extracted from the BLS adjusted by a two-axis translation stage. After being backscattered, experiment is thus 1.73, which is 3% lower than that of the bulk. This the light is collected and directed toward a Tandem Fabry–Perot set of values also allows extracting an effective mass density value for Interferometer.29 This device consists of two Fabry–Perot cavities the glass BTS material in the microsphere form of 4126 kg/m3 , which placed in series, each one of slightly different lengths, which allows is about 8% lower than the bulk value. Thus, the experimental results for a very narrow bandwidth and a wide free spectral range. The indicate that the fabrication process induces an effective decrease in mirrors on one side of each cavity oscillate with an amplitude of density, which explains the lowering in the refractive index and the 490 nm around the position in resonance with the elastic peak, elastic constants of the material. which, combined with the total length of the tandem cavity, deter- mines the range of frequency shifts of the inelastically scattered light that can be detected. Each mirror’s position scanned during SAMPLE FABRICATION the sweeps corresponds to a single frequency shift that is detected The Nd-doped Barium–Titanium-Silicate (BTS) microspheres by a photodiode. The back-and-forth movement of the mirrors is with a composition of BaO(40%)–TiO2 (20%)–SiO2 (40%) and then repeated thousands of times, and the photons counted at each 1.5%mol Nd2 O3 were fabricated by quenching liquid droplets. We frequency are summed to form the spectra. To minimize contam- first prepare a precursor by mixing and melting commercial pow- ination of the signal by the substrate, the spheres were deposited ders of ACS reagent grade (purity ≥99.9%) BaCO3 , TiO2 , SiO2 , on a flat surface and illuminated by the excitation laser from the and Nd2 O3 in a crucible at 1500 ○ C for one hour. The precur- side. A schematic representation of the configuration is shown in sor is then crushed using a pestle and mortar to obtain par- Fig. 1(a). An optical microscope image of a sphere is shown in the ticles close in size to the desired microspheres. These particles inset of Fig. 1(a). The measurements are performed at ambient tem- are then dropped through a vertical furnace and melt during the perature in the air. The extremely strong signal at the origin of the fall. The surface tension pulls them into spheres, which quench spectrum [leftmost part of Fig. 1(b)] corresponds to elastic light scat- into an amorphous state when reaching the cooler region out- tering. Both Stokes (negative frequency shift) and Anti-Stokes peaks side the furnace. With this method, microspheres with diame- (positive frequency shift) are fitted with a Lorentzian function. An ters D ranging from 10 to 100 μm are obtained. This method average of their peak position is used to compensate for the slight has been used before to fabricate microspheres, possessing opti- asymmetry of the spectrum with respect to the laser wavelength, cal modes with high quality factors6,8 up to 108 .4 Its principle is which is due to the non-perfect tuning of the Fabry–Perot inter- explained in more detail elsewhere.19 The density and refractive ferometer. The experimental error in determining the frequencies, index of the bulk material are ρ = 4465 kg/m3 20 and η = 1.78,21 including the Lorentz peak fitting, remains below 0.6%. Figure 1(b) respectively. shows representative Brillouin spectra fitted with the Lorentzian. In APL Mater. 11, 041128 (2023); doi: 10.1063/5.0146969 11, 041128-2 © Author(s) 2023 APL Materials ARTICLE scitation.org/journal/apm 26 September 2023 14:03:19 i FIG. 1. BLS of microspheres. (a) Light with energy Ephoton is focused on the microsphere and excites acoustic waves in the microsphere of energy E phonon . Scattered light s of energy Ephoton is then collected by the same microscope objective and sent to be detected (see main text). Brillouin frequency shift doublet is defined as f = ±Ephonon / h = (Ephoton i − Ephoton s )/h, where h is the Planck constant. The acoustic wave vector q is selected geometrically and has a magnitude of ±q = k s − k i . Symbols k i , k s , and q denote the incident light, scattered light, and phonon wave vectors, respectively. (Inset) Optical microscope image of the microsphere obtained with a ×50 objective. The green light corresponds to the laser used for the experiment. (b) Typical BLS spectra for bulk BTS (blue line) and a microsphere (black line). For comparison, the signal for bulk is scaled down by a factor of ×20. The red curve corresponds to a Lorentzian fit to the peak associated with the microsphere. Only the anti-stokes part of the spectra is shown for clarity. (Inset) Close-up view of the region with the Brillouin peaks. (c) Brillouin shift as a function of the diameter. the backscattering geometry, the selection rules favor the longitu- as well as the error during the measurement of the spectrum. If we dinal mode, and unless significant light depolarization occurs, no consider the refractive index of the bulk material given above, then coupling with the acoustic shear modes takes place.30 we can calculate the speed of longitudinal sound waves for the bulk In the backscattering configuration, the Brillouin shift is related material, which is, therefore, v long,BTS = 5036 m/s. to the frequency of the scattering acoustic wave through the The decrease in phonon frequency as measured by BLS may following expression: result from a reduction in elastic constants or the refractive index of the material at 532 nm or both. However, from this set of data 2 ⋅ η ⋅ vlong alone, it is not possible to isolate each contribution. Therefore, a fBLS = , (1) λ complementary technique based on the OM coupling of optical and vibrational modes of the microspheres has been used to determine where vlong is the speed of longitudinal sound waves in the material, the physical origin of the phonon frequency shift observed in BLS. λ is the photon wavelength (532 nm), and η is the refractive index of the material at that wavelength. Figure 1(c) shows the Brillouin shift as a function of sphere diameter. We note a clear difference between MECHANICAL MODE bulk BTS and microspheres, with a smaller frequency detected in the microspheres independently of their diameter, which corresponds to The determination of the elastic properties of the material is a 5% difference. The error bars consider the error in the Lorentz fit, achieved through the analysis of the vibrational eigenmodes of the APL Mater. 11, 041128 (2023); doi: 10.1063/5.0146969 11, 041128-3 © Author(s) 2023 APL Materials ARTICLE scitation.org/journal/apm microspheres, commonly referred to as the Lamb modes. The Lamb signal is then detected by a 12 GHz bandwidth InGaAs detector con- modes were measured using the experimental setup sketched in nected to a signal analyzer with a bandwidth of 13.5 GHz. Similarly Fig. 2(a), which is based on the coupling of evanescent light from to the BLS experiments, OM measurements are performed at room a tapered fiber to the whispering gallery modes of the spheres.6 temperature and atmospheric pressure. An infrared driving laser with tunable wavelength (1440–1640 nm) With the evanescent fiber coupling system, we first acquire an and power up to 20 mW passes through a polarization controller optical transmission spectrum [see Fig. 2(b)] in which each sharp before being coupled to a tapered microlooped fiber. The micro- dip corresponds to an optical resonance. Details about optical reso- loop is then positioned in contact with the microsphere as shown nances in these microspheres are given elsewhere.6 The mechanical in the microscope image in Fig. 2(a). When the laser wavelength spectrum is obtained by processing the transmitted optical signal matches an optical resonance of the microsphere, the mechanical with a spectrum analyzer. In that spectrum, the thermal motion modes activated by the thermal Langevin force modulate the trans- of vibrational modes with a large enough degree of OM coupling mission of the tapered fiber around its static value. The transmitted appears as Lorentzian peaks with quality factors limited to around 26 September 2023 14:03:19 FIG. 2. Optomechanical study of microspheres. (a) Schematic of the evanescent fiber coupling measurement system. The picture is a microscope image of a microsphere with the tapered fiber aligned. (b) Optical spectrum of a microsphere as measured by the IR detector. (c) Simulated optical whispering gallery mode at 1530 nm for a microsphere of diameter 36.6 μm. The white arrows indicate the direction of the electric field. (d) Radio frequency (RF) spectra of a microsphere of diameter D = 36.6 μm (red curve) and simulated optomechanical coupling rate with vibrational modes of the sphere. (e) RF spectra of microspheres of different diameters. The vertical offset represents the increase in diameter. The dashed gray lines are guides for the eyes for the two first modes. (f) Frequencies of the two first modes of each sphere as a function of the inverse of the diameter, where the lines are linear fits to the data. The color schemes represent the electric field in panel c and the deformation in panels d and f. APL Mater. 11, 041128 (2023); doi: 10.1063/5.0146969 11, 041128-4 © Author(s) 2023 APL Materials ARTICLE scitation.org/journal/apm 100. The red curve of Fig. 2(d) shows a representative example of In the case of l = 0, the correct equation to use is35 the registered mechanical spectra that corresponds to a sphere of diameter D = 36.6 μm. Several peaks are visible, which correspond 2 −1 to several vibrational modes of the spheres. βl=0 β 1 β tan ( ) − l=0 (1 − ( l=0 ) ) = 0. (3) Finite element methods simulations computed with COMSOL 2vlong 2vlong 4 2vtrans Multiphysics are used to identify which of the vibrational modes supported by the microsphere can be detected in the experiment, i.e., When the experimental values of βl=2 and βl=0 are plugged into can display enough degree of OM coupling with the optical whisper- Eqs. (2) and (3), respectively, the resulting solutions are the (vtrans , ing gallery modes to overcome the noise level. Figure 2(c) illustrates vlong ) curves plotted in Fig. 3. The intersection of the two curves the optical mode of a sphere that has been used to perform the sim- allows extracting the pair of values that are fitting both vibrational ulations, which is a fundamental WGM with azimuthal polarization modes. In our case, we obtain vlong = 4957 m/s and vtrans = 2655 m/s. emphasized by the white arrows. It should be noted that comparable An equivalent analysis including the {l = 4, n = 1} and {l = 6, results can be achieved using other optical modes. We then com- n = 1} vibrational modes is provided in the supplementary material puted all the vibrational modes of the microsphere in the frequency S2, where we verify the model since the crossing point of the range of the experiment and finally calculated the optomechani- extracted {vlong , vtrans } curves is almost the same. The value of vlong cal coupling considering only the moving boundary contribution.31 is 2% lower than the value extracted from the BLS measurement The results are shown as black dots in Fig. 2(d), allowing us to of the bulk material. It is also worth noting that FEM simulations conclude that the observed vibration modes are essential of the sec- agree well with the OM experiment by using these pair of acous- ond class (spheroidal) type,32 which are univocally described by the tic velocities as input material parameters [see Fig. 2(d)]. Since the angular momentum index (l) and the overtone number (n).33 In the acoustic velocities of the material within the spheres are accurately spheroidal type of modes, the sphere expands and contracts along its determined, it is then possible to extract the refractive index at radial axis, with the lower frequency mode having l = 2 sometimes λ = 532 nm from Eq. (1), which gives a value of ηsph = 1.733, i.e., called “American football” mode and the second mode fulfilling 3% lower than that of the bulk material. Finally, we can use the well- l = 0, which corresponds with the “breathing” mode of the sphere. known Newton–Drude formula36,37 to extract the density ρsph of the It is worth noting that the fundamental mode with l = 4 appears very material of the sphere, close to the breathing mode. 26 September 2023 14:03:19 Figure 2(e) shows the mechanical spectra of spheres with increasing diameter, from bottom to top, respectively. We observe (η2 − 1) α′ = , (4) that the frequency of each mechanical mode decreases with the ρsph Wεo increasing diameter of the sphere. In Fig. 2(f), we plot the value of the mechanical frequency as a function of 1/D for the l = 2 and l = 0 vibrational modes. The apparent fitting using a straight line where the right-hand side is a constant that can be determined from provides a slope of βl=2 = 2233 m/s and βl=0 = 4210 m/s, respec- the bulk values (α′ is the mean molar polarizability, εo the dielectric tively, which can be understood as a normalized eigenfrequency. permittivity, and W is the molecular weight). By substituting ηsph in The analytical calculation of the vibrational eigenmodes of elastic Eq. (4), we get that the density is ρsph = 4126 kg/m3 , which is about isotopic spheres indeed shows a dependence of mode frequencies 8% lower than the bulk value. proportional to 1/D in the absence of geometrical stress effects in the microspheres, so we can conclude that the elastic properties of the material composing the microspheres do not depend on their diameter. The frequency of the vibrational modes of the spheroidal type is calculated by finding the roots of the following equation valid for l > 0:32,34 2 2 1 β 1 β β β − ( l ) (2l2 − l − 1 − ( l ) ) jl ( l ) jl ( l ) 2 2vtrans 2 2vtrans 2vlong 2vtrans 2 βl β β β + (l3 + 2l2 − l − 2 −( ) )( l ) jl+1 ( l ) jl ( l ) 2vtrans 2vlong 2vlong 2vtrans 2 1 β β β β + (l3 + l2 − 2l − ( l ) )( l ) jl ( l ) jl+1 ( l ) 2 2vtrans 2vtrans 2vlong 2vtrans β2l β β + (2 − l2 − l)( ) jl+1 ( l ) jl+1 ( l ) = 0, (2) 4vtrans vlong 2vlong 2vtrans FIG. 3. Extraction of the elastic parameters. Transversal and longitudinal sound velocity curves that satisfy Eqs. (2) (orange curve) and (3) (blue curve) when including the experimental values of the vibrational frequencies. The blue curve where jl are the spherical Bessel functions of the first kind and vtrans corresponds to the {l = 0, n = 1} mode, and the orange curve corresponds to the {l = 2, n = 1} mode. is the transversal sound velocity. APL Mater. 11, 041128 (2023); doi: 10.1063/5.0146969 11, 041128-5 © Author(s) 2023 APL Materials ARTICLE scitation.org/journal/apm DISCUSSION ON THE ELASTIC AND OPTICAL that are also visible in some of the RF spectra. This analysis verifies PROPERTIES OF THE MICROSPHERES the accuracy of the method. Section S3 studies the linewidth of the Brillouin peaks as a function of the sphere diameter. Section S4 is The fabrication process used for the microspheres involves a dedicated to the analysis of the sphericity of the microscope using thermal cycle that can directly affect the density of the resulting glass, an image analysis technique. which has been shown to directly depend on the quenching tem- perature and quenching rate, i.e., the cooling speed.38 Due to the characteristics of the solidification process of the microsphere, we indeed expect the sphere to have a lower density, which we can esti- ACKNOWLEDGMENTS mate simply. When heated up and melted, the volume occupied by This work was supported by the MICINN projects ALLE- the material increases by 3 × α × (Tmelt − Tinit ) ≈ 3%, where we have GRO (Grant No. PID2021-124618NB-C22) and MOCCASIN-2D used the thermal expansion coefficient of the solid glass phase α, (Grant No. TED2021-132040B-C21). TN acknowledges the support which is considered to be constant in the range of temperature used of the European Commission through the Erasmus + program. J.M., for the thermal treatment. T melt is the temperature used to melt the E.C.A., M.F.C., and C.M.S.T. acknowledge the support of the Span- chip, and T init is the initial temperature. When cooling, the sphere ish projects S. Ochoa 2017-0706, MINECO Project No. FIS2015- will contract until the outermost layer reaches the glass transition 70862-P, and the SGR Project No. SGR1238. J.M. and C.M.S.T. temperature, at which temperature that layer will solidify. It is then acknowledge support by the EC Future Emerging Technologies possible to calculate an upper limit of the average density of the project Grant Agreement No. 753450 (PHENOMEN). sphere material, which is smaller than that of the bulk material by at least 1.9%. It is important to note that the estimate provided is a conservative one, due to the existence of a temperature increase AUTHOR DECLARATIONS toward the center of the sphere and the fact that the expansion coef- ficient of melted glass can be significantly higher than that of its solid Conflict of Interest counterpart.39 The authors have no conflicts to disclose. CONCLUSIONS Author Contributions 26 September 2023 14:03:19 In conclusion, we have used two non-destructive contactless Jeremie Maire: Data curation (equal); Formal analysis (equal); optical techniques, namely, Brillouin Light Scattering and optome- Investigation (equal); Visualization (equal); Writing – original draft chanical evanescent coupling, to determine the elastic and optical (equal); Writing – review & editing (equal). Tomasz Necio: Data properties of glass microspheres with diameters ranging from 10 to curation (equal); Formal analysis (equal); Investigation (equal); 60 μm. Our observations indicate that the Brillouin shift displayed Methodology (lead); Software (equal). Emigdio Chávez-Ángel: by the microspheres is independent of sphere diameter and lower Investigation (supporting); Writing – review & editing (equal). than that observed in the bulk material. This decrease has been Martín F. Colombano: Investigation (supporting); Software (equal). quantitatively attributed to a reduction in the refractive index and Juliana Jaramillo-Fernández: Investigation (equal); Writing – the elastic constants of the material in the microsphere. All these review & editing (equal). Clivia M. Sotomayor-Torres: Funding material parameters have been univocally determined thanks to the acquisition (equal); Project administration (equal). Nestor E. Capuj: independent extraction of the elastic constants of the material using Conceptualization (equal); Investigation (equal); Resources (lead). the optomechanical coupling technique and fitting the experimental Daniel Navarro-Urrios: Conceptualization (equal); Data cura- results to an analytical model of the vibrations of an elastic sphere. tion (equal); Formal analysis (equal); Funding acquisition (equal); We attribute the observed changes with respect to the bulk material Investigation (equal); Methodology (equal); Project administration to a lower density in the spheres originating from the thermal treat- (equal); Software (equal); Supervision (lead); Validation (equal); ment during the fabrication process, which is widely employed in the Visualization (equal); Writing – review & editing (equal). field. The advantage of our characterization method lies in its con- tactless nature and the wide range of spheres and materials that can DATA AVAILABILITY be characterized. It is particularly useful for high Q-factor optical The data that support the findings of this study are available cavities without damaging them and can be adapted to other simple from the corresponding author upon reasonable request. geometries. REFERENCES SUPPLEMENTARY MATERIAL 1 F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: Label-free detection down to single molecules,” Nat. Methods 5(7), 591–596 (2008). The supplementary material file contains four sections and six 2 N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, figures: Section S1 contains a double check of the analytical model “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. using finite-element-method simulations of the vibrational modes 87(20), 201107 (2005). of the spheres. Section S2 repeats the analysis performed in the main 3 C.-H. Dong, L. He, Y.-F. Xiao, V. R. Gaddam, S. K. Ozdemir, Z.-F. 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