# Cascade Brillouin Lasing in a Tellurite-Glass Microsphere Resonator with Whispering Gallery Modes

^{1}

^{2}

^{*}

## Abstract

**:**

^{7}), a high Brillouin gain coefficient (compared to standard materials, e.g., silica glasses), and a Brillouin frequency shift of 9 ± 0.5 GHz. The high density of excited resonance modes and high loaded Q-factors allowed us to achieve experimentally cascade Stokes-Brillouin lasing up to the 4th order inclusive. The experimental results are supported by the results of the theoretical analysis. We also theoretically obtained the dependences of the output Brillouin powers on the pump power and found the pump-power thresholds for the first five Brillouin orders at different values of pump frequency detuning and Q-factors, and showed a significant effect of these parameters on the processes under consideration.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Fabrication and Characterization of Tellurite Glass

_{2}–21.5WO

_{3}–10La

_{2}O

_{3}–4Bi

_{2}O

_{3}(TWLB) composition. Tungsten-tellurite glasses modified by lanthanum oxide and bismuth oxide demonstrate excellent optical, physical and chemical properties which make them suitable for fiber-based optical devices [40,41]. The choice of additives to a binary tungsten-tellurite glass was made for the following reasons: lanthanum oxide increases the glass transition temperature and the crystallization resistance of tellurite glasses, and bismuth oxide increases the values of linear (n) and nonlinear refractive indices. The TWLB glass was prepared by melting a mixture of initial oxides TeO

_{2}, WO

_{3}, La

_{2}O

_{3}, and Bi

_{2}O

_{3}in a platinum crucible inside a sealed quartz-glass reactor. The glass synthesis was carried out in an atmosphere of purified oxygen. The low concentration of metal impurities was attained by using high-purity initial oxides and selecting synthesis conditions that minimized contamination of the glass-forming melt with reactor materials [42]. Samples of glasses were made from super-high-purity tellurium oxide obtained by vacuum distillation according to the original method using the high-purity, commercially available reagents WO

_{3}, La

_{2}O

_{3}, and Bi

_{2}O

_{3}. The use of high-purity oxides allowed us to obtain glasses with a total content of 3d transition metals of the order of 1–2 ppm wt. To reduce the content of hydroxyl groups in the glass, the samples were synthesized and molded in a sealed quartz-glass reactor equipped with manipulators for mixing the melt and molding samples (for details please see [40,41,42]). A systematic approach to preventing the penetration of impurities guarantees a high purity of synthesized samples [43]. After cooling and annealing at the glass transition temperature, the casting was cut, and the segments were ground and polished for research. The prepared samples were visually optically homogeneous and free from defects (Figure 1a, inset).

_{0}/I) = β + αL. We obtained α < 0.001 cm

^{−1}, which corresponds to the extremely low OH concentration of about 5 × 10

^{15}cm

^{−3}[41]. We drew a single-index fiber with a diameter of 100 µm from the synthesized TWLB glass without any coating. The low content of impurities and OH groups is very important for the fabrication of microresonators with high Q-factors.

#### 2.2. Fabrication of Tellurite Microsphere Resonators

_{2}laser (Coherent Diamond C-40A), which allowed us to improve the quality of the samples and achieve record loaded Q-factors for tellurite microspheres (≥2.5 × 10

^{7}). As far as we know, the highest previously measured value of the loaded Q-factor in a tellurite microsphere was 1.07 × 10

^{7}and corresponded to an intrinsic recalculated Q-factor of 1.3 × 10

^{7}[39].

_{2}laser radiation with adjustable pulse duration (0.5–1 s) and instantaneous power of 2 W was forwarded to the fiber, as a result of which a taper was formed (II) and then broke (III). After that, the CO

_{2}laser pulsed with a 100 ms duration and 50–100 mJ energy were focused on the formed fiber end (IV), which was heated and melted with the formation of a sphere under the action of the surface-tension force (V). The final size of the microsphere at the end of the fiber stem was controlled by changing the laser parameters and could range from ~30 µm to a few hundred µm. In this study, we used a microsphere with a diameter of 75 μm.

#### 2.3. Theoretical Description of Microsphere Characteristics

_{0}= 2πf/c, R is the microsphere radius, n is the glass refractive index, f is the frequency, c is the vacuum speed of light, l is the polar mode index, J

_{ν}is the Bessel function of order ν, H

_{ν}

^{(1)}is the Hankel function of the 1st order ν, and the prime denotes the derivative with respect to the argument in the parentheses. Each equation has multiple roots f

_{q}, and when sorted in ascending order, q ≥ 1 corresponds to the radial mode index. The characteristic equations were solved numerically using a home-made computer python code; the glass dispersion n(f) was taken into account. The roots were iteratively localized using high-order approximations [47]. For the fundamental WGM (q = 1, l = m), the free spectral range (FSR) was FSR = f

_{l}

_{+1}− f

_{l}≈ 600 GHz near λ = 1.55 μm.

_{eff}(below they will be used for estimating intracavity Brillouin gain coefficients):

_{l,m}is the split eigenfrequency, f

_{l}

^{(0)}is the unperturbed eigenfrequency from Equation (1), $\eta =\left({R}_{z}-{R}_{x}\right)/R$ is the shape-deformation parameter, R

_{z}and R

_{x}are the spheroid semiaxes (z is the symmetry axis).

#### 2.4. Theoretical Model of Cascade Brillouin Lasing

^{th}), dependence of output powers on pump power, and the number of Brillouin cascades, we performed the theoretical analysis. The schematic diagram of the considered cascade Brillouin laser is shown in Figure 4, where the pump laser initiates the generation of a backward Stokes–Brillouin wave that acts as a pump for the Stokes–Brillouin wave of the second cascade. The 2nd-order Brillouin wave propagates in the opposite direction with respect to the 1st-order wave. Further, the 2nd-order Brillouin wave pumps the 3rd-order Brillouin wave, and so on. In this case, even-order Brillouin waves propagate in the forward direction, while odd-order Brillouin waves propagate in the backward direction relative to the pump.

_{0}is the intracavity field amplitude at the pump frequency; A

_{j}is the intracavity field amplitude of the generated Brillouin wave of the jth order (j = 1,…N); N is the maximum order of the generated Brillouin cascade; t is time; Δω

_{0}is the detuning of the pump frequency from the exact resonance; S is the amplitude of the incident pump wave, P

_{p}= |S|

^{2}is the pump power; τ

_{j}is the effective photon lifetime (related to the loaded Q-factor by Q

_{j}= ω⸱τ

_{j}) with allowance for the intrinsic lifetime τ

_{j}

^{0}and coupling lifetime τ

_{j}

^{c}(1/τ

_{j}= 1/τ

_{j}

^{0}+ 1/τ

_{j}

^{c}) for the WGM in which the jth-order Brillouin lasing arises (hereinafter the subscript j = 0 corresponds to the pumped WGM); κ

_{j}= 1/τ

_{j}

^{c}is the coupling coefficient; ${g}_{\mathrm{j}}={\Gamma}_{\mathrm{j}}{g}_{Te}{c}^{2}/\left(2{n}^{2}{V}_{\mathrm{j}}\right)$ is the intracavity Brillouin gain coefficient; ${g}_{Te}$ is the Brillouin gain for bulk tellurite glass (${g}_{Te}$ ≈ 1.7⸱10

^{−10}m/W [33]); n is the refractive index; Vj is the effective mode volume of the WGM in which the jth-order Brillouin lasing arises; and Γ

_{j}is the overlap integral between the mode fields corresponding the jth and (j − 1)th Brillouin cascade. The output power is P

_{j}= κ|A

_{j}|

^{2}.

_{j}

^{c}= 4τ

_{j}based on our experimental estimates. The effective mode volumes were set V

_{j}= 10

^{4}µm

^{3}(based on the results presented in Figure 3b) and Γ

_{j}= 0.05, therefore, ${g}_{\mathrm{j}}$ ≈ 9⸱10

^{18}1/(W⸱s

^{2}).

_{j}/dt = 0 in Equations (4)–(6) and the system of equations describing cascade Brillouin lasing is algebraic. For the successively analyzed N = 1, …, N = 5, the system of Equations (4)–(6) was easily solved analytically and expressions for |A

_{j}|

^{2}and threshold pump powers P

_{j}

^{th}were found.

## 3. Results

#### 3.1. Experimental Results

^{7}, and 20% of the resonances had a loaded Q-factor of more than 2.5 × 10

^{7}. The resonance curves were symmetrical and well approximated by the Lorentz function. An example of a resonance with a Q-factor of 2.5 × 10

^{7}is shown in Figure 6c. The accuracy of Q-factor measurements was estimated to be 5% for most resonances, which was limited by resonance-curve fitting uncertainties that originated from the noise and slight deviations of measured shapes from the perfect Lorentz curves.

_{B}. For example, at a pump wavelength of 1529.8 nm, we observed 2nd-order Stokes–Brillouin wave generation shifted by 18 ± 2 GHz relative to the pump frequency (Figure 7a,b). The best experimental result was attained at a pump wavelength of 1564.9 nm. For this case, we recorded the spectrum of generated Stokes–Brillouin waves of the 2nd and 4th orders to be shifted, respectively, by 18 ± 2 GHz and 36 ± 2 GHz relative to the pump (Figure 7c,d). Thus, the value of the Brillouin shift corresponded to Δ

_{B}= 9 ± 0.5 GHz. In [39], the Brillouin shift was Δ

_{B}= 8.2 GHz for zinc–tellurite glass. In our case, we used glass of the tungsten-tellurite system with slightly different physical properties, which explains the difference in the Δ

_{B}values. Note that the achievement of cascade Brillouin generation of the 4th order is possible here due to the high WGM density and the high quality of the fabricated microsphere, which provides high values of the loaded Q-factors for most of the resonant modes. The high WGM density makes it possible to select a few modes with a spectral distance of Δ

_{B}required for cascade Brillouin lasing.

#### 3.2. Theoretical Results

_{2}for different modes. For the steady-state Brillouin lasing investigated here, the dispersion itself is not important. However, the dispersion can strongly influence other nonlinear processes occurring in the microsphere, for example, due to the Kerr nonlinearity. Therefore, the calculation of the dispersion is important for the complete characterization of the microsphere properties. We found that the dispersion was normal for all considered modes (Figure 9a–d).

_{0}= 0) and loaded Q-factor Q = 2.5 × 10

^{7}. When N is even, the powers of odd Brillouin orders are constant and do not depend on the pump power for P

_{N−1}

^{st}< P

_{P}< P

_{N}

^{th}, while the powers of even Brillouin orders increase linearly with increasing pump power. When N is odd, the powers of even Brillouin orders are constant and do not depend on the pump power for P

_{N−1}

^{st}< P

_{P}< P

_{N}

^{th}, while the powers of odd Brillouin orders increase according to the law $~\sqrt{{P}_{p}}-const$. Analytical dependences presented in Figure 10b,d also demonstrate that the power in the 2nd Brillouin cascade is twice as high as in the 4th cascade. This result agrees with the experimental data presented in Figure 7c,d.

_{0}= 0).

_{2}

^{nd}− P

_{1}

^{st}= const and P

_{4}

^{th}− P

_{3}

^{rd}= const for any detuning. For each Q-factor, the larger the detuning, the higher the threshold for a certain cascade, which agrees with the theoretical results on Brillouin lasing obtained for ring resonators [50]. The Q-factor and detuning significantly affect the number of generated cascades. Indeed, to generate high-order cascade Brillouin waves, a large Q-factor is required. So, the calculations confirmed that the experimentally attained 4th order Brillouin lasing (Figure 7c,d) was possible due to the record high Q-factors of the produced microsphere (for microresonators made of tellurite glasses).

## 4. Discussion

^{7}and 60% of resonances had loaded Q-factors Q ≥ 1 × 10

^{7}. The use of a specially synthesized high-quality ultra-dry glass (64.5TeO

_{2}–21.5WO

_{3}–10La

_{2}O

_{3}–4Bi

_{2}O

_{3}) and the fabrication of microspheres employing a CO

_{2}laser instead of a microheater allowed us to increase the Q-factors by an order of magnitude compared to our earlier results [45]. In the produced microresonator, cascade Stokes–Brillouin generation up to the 4th order inclusive was attained. To the best of our knowledge, stimulated Brillouin scattering of the 1st order in a tellurite microsphere was previously reported only in one article [39], where a sample with a loaded Q-factor of 1.07 × 10

^{7}was made. Our experimental result, confirmed by the theoretical analysis, was achieved due to the high Q-factors. Cascade Brillouin lasing was observed in different families of WGMs, and a large number of modes with the required spectral interval of 9 GHz corresponding to the Brillouin frequency shift Δ

_{B}was observed experimentally and can be mainly explained by the splitting of degenerate frequencies under a small deformation of the microsphere. As a result of the theoretical analysis, we found the pump-power thresholds for the first five Brillouin orders at different values of detuning Δω

_{0}and Q-factors and showed a significant influence of these parameters on the processes under consideration. We obtained the dependences of the output Brillouin powers on the pump power. The theoretical analysis of steady-state generation was carried out within the framework of the equations for the mean fields and the theory of coupled modes [50,51]. Note that similar equations are also used to describe laser generation and Raman generation in microresonators and provide a good agreement with the corresponding experimental results [14,44,45,54]. So, the results of our work demonstrate the prospects of using tellurite-glass microspheres for cascade Brillouin lasing, which can expand the scope of microresonator sensors. The obtained results on generating Brillouin cascades can be useful for rotation sensors requiring counter-propagating waves.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Reynolds, T.; Riesen, N.; Meldrum, A.; Fan, X.; Hall, J.M.M.; Monro, T.M.; François, A. Fluorescent and lasing whispering gallery mode microresonators for sensing applications. Laser Photonics Rev.
**2017**, 11, 1600265. [Google Scholar] [CrossRef] - Yu, D.; Humar, M.; Meserve, K.; Bailey, R.C.; Nic Chormaic, S.; Vollmer, F. Whispering-gallery-mode sensors for biological and physical sensing. Nat. Rev. Methods Primers
**2021**, 1, 83. [Google Scholar] [CrossRef] - Jiang, X.; Qavi, A.J.; Huang, S.H.; Yang, L. Whispering-gallery sensors. Matter
**2020**, 3, 371–392. [Google Scholar] [CrossRef] [PubMed] - Tan, T.; Yuan, Z.; Zhang, H.; Yan, G.; Zhou, S.; An, N.; Peng, B.; Soavi, G.; Rao, Y.; Yao, B. Multispecies and individual gas molecule detection using stokes solitons in a graphene over-modal microresonator. Nat. Commun.
**2021**, 12, 6716. [Google Scholar] [CrossRef] [PubMed] - Liu, W.; Chen, Y.-L.; Tang, S.-J.; Vollmer, F.; Xiao, Y.-F. Nonlinear sensing with whispering-gallery mode microcavities: From label-free detection to spectral fingerprinting. Nano Lett.
**2020**, 21, 1566–1575. [Google Scholar] [CrossRef] - Brice, I.; Grundsteins, K.; Draguns, K.; Atvars, A.; Alnis, J. Whispering gallery mode resonator temperature compensation and refractive index sensing in glucose droplets. Sensors
**2021**, 21, 7184. [Google Scholar] [CrossRef] - Zhivotkov, D.; Ristić, D.; Romanova, E.; Ivanda, M. Refractometric gas sensing using a whispering gallery mode microresonator coated with a supra-micron sol-gel layer. Opt. Mater.
**2021**, 118, 111286. [Google Scholar] [CrossRef] - Lin, G.; Song, Q. Kerr frequency comb interaction with Raman, Brillouin, and second order nonlinear effects. Laser Photonics Rev.
**2021**, 16, 2100184. [Google Scholar] [CrossRef] - Shen, X.; Beltran, R.C.; Diep, V.M.; Soltani, S.; Armani, A.M. Low-threshold parametric oscillation in organically modified microcavities. Sci. Adv.
**2018**, 4, eaao450. [Google Scholar] [CrossRef] [Green Version] - Frigenti, G.; Cavigli, L.; Ratto, F.; Centi, S.; Murzina, T.V.; Farnesi, D.; Pelli, S.; Soria, S.; Nunzi Conti, G. Microbubble resonators for scattering-free absorption spectroscopy of nanoparticles. Opt. Express
**2021**, 29, 31130–31136. [Google Scholar] [CrossRef] - Bochek, D.; Toropov, N.; Vatnik, I.; Churkin, D.; Sumetsky, M. SNAP Microresonators introduced by strong bending of optical fibers. Opt. Lett.
**2019**, 44, 3218–3221. [Google Scholar] [CrossRef] [PubMed] - Shitikov, A.E.; Bilenko, I.A.; Kondratiev, N.M.; Lobanov, V.E.; Markosyan, A.; Gorodetsky, M.L. Billion Q-factor in silicon WGM resonators. Optica
**2018**, 5, 1525–1528. [Google Scholar] [CrossRef] - Smirnov, S.; Andryushkov, V.; Podivilov, E.; Sturman, B.; Breunig, I. Soliton based χ(2) combs in high-Q optical microresonators. Opt. Express
**2021**, 29, 27434–27449. [Google Scholar] [CrossRef] [PubMed] - Min, B.; Kippenberg, T.J.; Yang, L.; Vahala, K.J.; Kalkman, J.; Polman, A. Erbium-implanted high-qsilica toroidal microcavity laser on a silicon chip. Phys. Rev. A
**2004**, 70, 033803. [Google Scholar] [CrossRef] [Green Version] - Spillane, S.M.; Kippenberg, T.J.; Vahala, K.J. Ultralow-threshold raman laser using a spherical dielectric microcavity. Nature
**2002**, 415, 621–623. [Google Scholar] [CrossRef] [PubMed] - Zhu, S.; Xiao, B.; Jiang, B.; Shi, L.; Zhang, X. Tunable brillouin and raman microlasers using hybrid microbottle resonators. Nanophotonics
**2019**, 8, 931–940. [Google Scholar] [CrossRef] - Yao, B.; Yu, C.; Wu, Y.; Huang, S.-W.; Wu, H.; Gong, Y.; Chen, Y.; Li, Y.; Wong, C.W.; Fan, X.; et al. Graphene-enhanced brillouin optomechanical microresonator for ultrasensitive gas detection. Nano Lett.
**2017**, 17, 4996–5002. [Google Scholar] [CrossRef] - Yang, J.; Qin, T.; Zhang, F.; Chen, X.; Jiang, X.; Wan, W. Multiphysical sensing of light, sound and microwave in a microcavity Brillouin laser. Nanophotonics
**2020**, 9, 2915–2925. [Google Scholar] [CrossRef] - Qin, G.-Q.; Wang, M.; Wen, J.-W.; Ruan, D.; Long, G.-L. Brillouin cavity optomechanics sensing with enhanced dynamical Backaction. Photon. Res.
**2019**, 7, 1440–1446. [Google Scholar] [CrossRef] - Eggleton, B.J.; Poulton, C.G.; Rakich, P.T.; Steel, M.J.; Bahl, G. Brillouin integrated photonics. Nat. Photonics
**2019**, 13, 664–677. [Google Scholar] [CrossRef] - Li, J.; Suh, M.-G.; Vahala, K. Microresonator brillouin gyroscope. Optica
**2017**, 4, 346–348. [Google Scholar] [CrossRef] [Green Version] - Wang, H.; Lai, Y.-H.; Yuan, Z.; Suh, M.-G.; Vahala, K. Petermann-factor sensitivity limit near an exceptional point in a Brillouin ring laser gyroscope. Nat. Commun.
**2020**, 11, 1610. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Enzian, G.; Szczykulska, M.; Silver, J.; Del Bino, L.; Zhang, S.; Walmsley, I.A.; Del’Haye, P.; Vanner, M.R. Observation of Brillouin optomechanical strong coupling with an 11 GHz mechanical Mode. Optica
**2018**, 6, 7–14. [Google Scholar] [CrossRef] - Feng, C.; Schneider, T. Benefits of spectral property engineering in distributed Brillouin fiber sensing. Sensors
**2021**, 21, 1881. [Google Scholar] [CrossRef] [PubMed] - Hu, D.J.J.; Humbert, G.; Dong, H.; Zhang, H.; Hao, J.; Sun, Q. Review of specialty fiber based Brillouin optical time domain analysis technology. Photonics
**2021**, 8, 421. [Google Scholar] [CrossRef] - Gorshkov, B.G.; Yüksel, K.; Fotiadi, A.A.; Wuilpart, M.; Korobko, D.A.; Zhirnov, A.A.; Stepanov, K.V.; Turov, A.T.; Konstantinov, Y.A.; Lobach, I.A. Scientific applications of distributed acoustic sensing: State-of-the-art review and perspective. Sensors
**2022**, 22, 1033. [Google Scholar] [CrossRef] - Tkachenko, A.Y.; Lobach, I.A.; Kablukov, S.I. All-fiber Brillouin optical spectrum analyzer based on self-sweeping fiber laser. Opt. Express
**2017**, 25, 17600–17605. [Google Scholar] [CrossRef] - Lopez-Mercado, C.A.; Korobko, D.A.; Zolotovskii, I.O.; Fotiadi, A.A. Application of dual-frequency self-injection locked DFB laser for Brillouin optical time domain analysis. Sensors
**2021**, 21, 6859. [Google Scholar] [CrossRef] - Kovach, A.; Chen, D.; He, J.; Choi, H.; Dogan, A.H.; Ghasemkhani, M.; Taheri, H.; Armani, A.M. Emerging material systems for integrated optical kerr frequency combs. Adv. Opt. Photonics
**2020**, 12, 135–222. [Google Scholar] [CrossRef] [Green Version] - Tao, G.; Ebendorff-Heidepriem, H.; Stolyarov, A.M.; Danto, S.; Badding, J.V.; Fink, Y.; Ballato, J.; Abouraddy, A.F. Infrared fibers. Adv. Opt. Photonics
**2015**, 7, 379–458. [Google Scholar] [CrossRef] - Deroh, M.; Beugnot, J.-C.; Hammani, K.; Finot, C.; Fatome, J.; Smektala, F.; Maillotte, H.; Sylvestre, T.; Kibler, B. Comparative analysis of stimulated Brillouin scattering at 2 µm in various infrared glass-based optical fibers. J. Opt. Soc. Am. B
**2020**, 37, 3792–3800. [Google Scholar] [CrossRef] - Wolff, C.; Smith, M.J.A.; Stiller, B.; Poulton, C.G. Brillouin scattering—Theory and experiment: Tutorial. J. Opt. Soc. Am. B
**2021**, 38, 1243–1269. [Google Scholar] [CrossRef] - Qin, G.; Sotobayashi, H.; Tsuchiya, M.; Mori, A.; Suzuki, T.; Ohishi, Y. Stimulated Brillouin scattering in a single-mode tellurite fiber for amplification, lasing, and slow light generation. J. Lightwave Technol.
**2008**, 26, 492–498. [Google Scholar] [CrossRef] - Domachuk, P.; Wolchover, N.A.; Cronin-Golomb, M.; Wang, A.; George, A.K.; Cordeiro, C.M.B.; Knight, J.C.; Omenetto, F.G. Over 4000 nm bandwidth of mid-IR supercontinuum generation in sub-centimeter segments of highly nonlinear tellurite PCFs. Opt. Express
**2008**, 16, 7161–7168. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kibler, B.; Lemière, A.; Gomes, J.-T.; Gaponov, D.; Lavoute, L.; Désévédavy, F.; Smektala, F. Octave-spanning coherent supercontinuum generation in a step-index tellurite fiber and towards few-cycle pulse compression at 2 µm. Opt. Commun.
**2021**, 488, 126853. [Google Scholar] [CrossRef] - Kedenburg, S.; Steinle, T.; Mörz, F.; Steinmann, A.; Nguyen, D.; Rhonehouse, D.; Zong, J.; Chavez-Pirson, A.; Giessen, H. Solitonic supercontinuum of femtosecond mid-IR pulses in W-type index tellurite fibers with two zero dispersion wavelengths. APL Photonics
**2016**, 1, 086101. [Google Scholar] [CrossRef] [Green Version] - Okhrimchuk, A.G.; Pryamikov, A.D.; Gladyshev, A.V.; Alagashev, G.K.; Smayev, M.P.; Likhov, V.V.; Dorofeev, V.V.; Motorin, S.E.; Yatsenko, Y.P. Direct laser written waveguide in tellurite glass for supercontinuum generation in 2 µm spectral range. J. Lightwave Technol.
**2020**, 38, 1492–1500. [Google Scholar] [CrossRef] - Anashkina, E.A. Laser sources based on rare-earth ion doped tellurite glass fibers and microspheres. Fibers
**2020**, 8, 30. [Google Scholar] [CrossRef] - Guo, C.; Che, K.; Zhang, P.; Wu, J.; Huang, Y.; Xu, H.; Cai, Z. Low-threshold stimulated brillouin scattering in high-Q whispering gallery mode tellurite microspheres. Opt. Express
**2015**, 23, 32261–32266. [Google Scholar] [CrossRef] - Dorofeev, V.V.; Koltashev, V.V.; Motorin, S.E.; Plekhovich, A.D.; Kim, A.V. Thermal, optical, and IR-emission properties of extremely low Hydroxyl TeO
_{2}-WO_{3}-Bi_{2}O_{3}-La_{2}O_{3}-xEr_{2}O_{3}glasses for mid-infrared photonics. Photonics**2021**, 8, 320. [Google Scholar] [CrossRef] - Anashkina, E.A.; Dorofeev, V.V.; Koltashev, V.V.; Kim, A.V. Development of Er
^{3+}-doped high-purity tellurite glass fibers for gain-switched laser operation at 2.7 µm. Opt. Mater. Express**2017**, 7, 4337–4351. [Google Scholar] [CrossRef] - Moiseev, A.N.; Dorofeev, V.V.; Chilyasov, A.V.; Pimenov, V.G.; Kotereva, T.V.; Kraev, I.A.; Ketkova, L.A.; Kosolapov, A.F.; Plotnichenko, V.G.; Koltashev, V.V. Low-loss, high-purity (TeO
_{2})0.75(WO_{3})0.25 glass. Inorg. Mater.**2011**, 47, 665–669. [Google Scholar] [CrossRef] - Dorofeev, V.V.; Moiseev, A.N.; Churbanov, M.F.; Snopatin, G.E.; Chilyasov, A.V.; Kraev, I.A.; Lobanov, A.S.; Kotereva, T.V.; Ketkova, L.A.; Pushkin, A.A.; et al. High-purity TeO
_{2}–WO_{3}–(La_{2}O_{3},Bi_{2}O_{3}) glasses for fiber-optics. Opt. Mater.**2011**, 33, 1911–1915. [Google Scholar] [CrossRef] - Anashkina, E.A.; Andrianov, A. Erbium-doped tellurite glass microlaser in C-band and L-band. J. Lightwave Technol.
**2021**, 39, 3568–3574. [Google Scholar] [CrossRef] - Anashkina, E.A.; Dorofeev, V.V.; Andrianov, A.V. In-band pumped thulium-doped tellurite glass microsphere laser. Appl. Sci.
**2021**, 11, 5440. [Google Scholar] [CrossRef] - Oraevsky, A.N. Whispering-gallery waves. Quantum Electron.
**2002**, 32, 377–400. [Google Scholar] [CrossRef] - Schiller, S. Asymptotic expansion of morphological resonance frequencies in mie scattering. Appl. Opt.
**1993**, 32, 2181–2185. [Google Scholar] [CrossRef] - Teraoka, I.; Arnold, S. Whispering-gallery modes in a microsphere coated with a high-refractive index layer: Polarization-dependent sensitivity enhancement of the resonance-shift sensor and TE-TM resonance matching. J. Opt. Soc. Am. B
**2007**, 24, 653–659. [Google Scholar] [CrossRef] [Green Version] - Kher-Alden, J.; Maayani, S.; Martin, L.L.; Douvidzon, M.; Deych, L.; Carmon, T. Microspheres with atomic-scale tolerances generate hyperdegeneracy. Phys. Rev. X
**2020**, 10, 031049. [Google Scholar] [CrossRef] - Korobko, D.A.; Zolotovskii, I.O.; Svetukhin, V.V.; Zhukov, A.V.; Fomin, A.N.; Borisova, C.V.; Fotiadi, A.A. Detuning effects in Brillouin ring microresonator laser. Opt. Express
**2020**, 28, 4962–4972. [Google Scholar] [CrossRef] - Che, K.; Tang, D.; Ren, C.; Xu, H.; Chen, L.; Jin, C.; Cai, Z. Thermal characteristics of Brillouin microsphere lasers. IEEE J. Quantum Electron.
**2018**, 54, 1000108. [Google Scholar] [CrossRef] - Anashkina, E.A.; Marisova, M.P.; Salgals, T.; Alnis, J.; Lyashuk, I.; Leuchs, G.; Spolitis, S.; Bobrovs, V.; Andrianov, A.V. Optical frequency combs generated in silica microspheres in the telecommunication C-, U-, and E-bands. Photonics
**2021**, 8, 345. [Google Scholar] [CrossRef] - Spolitis, S.; Murnieks, R.; Skladova, L.; Salgals, T.; Andrianov, A.V.; Marisova, M.P.; Leuchs, G.; Anashkina, E.A.; Bobrovs, V. IM/DD WDM-PON communication system based on optical frequency comb generated in silica whispering gallery mode resonator. IEEE Access
**2021**, 9, 66335–66345. [Google Scholar] [CrossRef] - Suzuki, R.; Kubota, A.; Hori, A.; Fujii, S.; Tanabe, T. Broadband gain induced raman comb formation in a silica microresonator. J. Opt. Soc. Am. B
**2018**, 35, 933–938. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(

**a**) Transmittance spectra of produced tellurite-glass samples with photo of 0.23 cm sample in the inset. (

**b**) Absorption spectrum within OH band of 6 cm sample.

**Figure 2.**Schematic representation of the successive stages of fabrication of microresonators from single-index tellurite fiber.

**Figure 3.**(

**a**) Examples of calculated electric fields of eigenmodes with different indices for 75 µm tellurite microsphere resonator. (

**b**) Calculated effective mode volume for eigenmodes with different indices for this resonator.

**Figure 4.**Schematic diagram of cascade Brillouin lasing in microsphere. Even-order Brillouin waves propagate in forward direction and odd-order Brillouin waves propagate in backward direction relative to pump wave. Aj is intracavity field amplitude, Pj is output power, subscript j indicates the jth order of the Brillouin cascade, j = 0 corresponds to pump wave. Only Brillouin orders considered in this study are indicated.

**Figure 5.**Simplified schematic diagram of the experimental setup. Inset: microphoto of the used tellurite microsphere and silica fiber taper.

**Figure 6.**(

**a**) Experimental resonance dips of eigenmodes of the produced 75 µm tellurite microsphere resonator recorded for the output pump power of 0.3 µW with the oscilloscope at the pump-laser-sweeping rate of 10 GHz/s. (

**b**) Statistics of Q-factors for these resonances. (

**c**) Resonance dip on magnified scale and its Lorentz approximation demonstrating loaded Q-factor.

**Figure 7.**Experimental spectra measured for waves propagating co-directionally with pump. Spectra demonstrating cascade Stokes–Brillouin lasing: (

**a**,

**b**) of the 2nd order; (

**c**,

**d**) up to the 4th order.

**Figure 8.**Eigenfrequencies of ideal 75 µm tellurite microsphere near λ = 1.55 μm for TE (

**a**) and TM (

**b**) modes with different radial indices q; vertical lines show resonance positions. Resulting splitting of the fundamental TE mode for microresonator with the shape-deformation parameter η defined based on Equation (3); η = 0.001 (

**c**), η = 0.005 (

**d**).

**Figure 9.**(

**a**,

**b**) 2nd-order dispersion of TE (

**a**) and TM modes (

**b**) of ideal microsphere as a function of frequency; only modes with odd q are shown. (

**c**,

**d**) 2nd-order dispersion of TE modes with one radial variation for microresonator with η = 0.001 ((

**c**), every 20th mode is shown), η = 0.005 ((

**d**), every 10th mode is shown); red line marks the dispersion of the corresponding modes of ideal microsphere.

**Figure 10.**Theoretically calculated output powers of generated Stokes–Brillouin waves of the 1st order (

**a**), 2nd order (

**b**); 3rd order (

**c**); 4th order (

**d**); and 5th order (

**e**) as functions of pump power for zero detuning (Δω

_{0}= 0) and loaded Q-factors Q = 2.5 × 10

^{7}.

**Figure 11.**Theoretically calculated threshold pump powers as functions of Q-factors for cascade Brillouin waves of the 1st–5th orders for zero detuning (Δω

_{0}= 0). Thresholds for waves of the 6th and higher orders are not shown.

**Figure 12.**Theoretically calculated diagram demonstrating the number of Brillouin laser cascades for different pump powers and detuning for loaded Q-factors: Q = 2.5 × 10

^{7}(

**a**) and Q = 1 × 10

^{7}(

**b**).

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## Share and Cite

**MDPI and ACS Style**

Anashkina, E.A.; Marisova, M.P.; Dorofeev, V.V.; Andrianov, A.V.
Cascade Brillouin Lasing in a Tellurite-Glass Microsphere Resonator with Whispering Gallery Modes. *Sensors* **2022**, *22*, 2866.
https://doi.org/10.3390/s22082866

**AMA Style**

Anashkina EA, Marisova MP, Dorofeev VV, Andrianov AV.
Cascade Brillouin Lasing in a Tellurite-Glass Microsphere Resonator with Whispering Gallery Modes. *Sensors*. 2022; 22(8):2866.
https://doi.org/10.3390/s22082866

**Chicago/Turabian Style**

Anashkina, Elena A., Maria P. Marisova, Vitaly V. Dorofeev, and Alexey V. Andrianov.
2022. "Cascade Brillouin Lasing in a Tellurite-Glass Microsphere Resonator with Whispering Gallery Modes" *Sensors* 22, no. 8: 2866.
https://doi.org/10.3390/s22082866