The whispering gallery mode （WGM）resonator can confine the light in a cavity by total internal reflections around its inner surface，which can enhance the light power intensity in a very small mode volume and store the energy for a long time，in another word it means high Q factor or ultra-high Q factor can be provided^［1-4］. Such resonators have attracted great attention for decades，due to it provides an ideal platform to investigate linear applications or nonlinear effects between light and matter，such as resonator sensing applications，cavity optomechanical research，light storage，and cavity quantum dot investigation^［5-8］.

Crystal WGM resonators have been demonstrated as an ideal platform for nonlinear effects research^［9］. It can be fabricated easily with the chemical mechanical polishing method，which has been investigated by Maleki and ultra-high Q factor can exceed in the order of 10^9［10］. This ultra-high Q ensures a long photo lifetime and incredibly enhance the nonlinear effects. This is the reason why over the past two decades，applications into the mid-infrared and ultraviolet frequency ranges based on crystal WGM resonators have attracted so many optical researchers^［11-12］.

Stimulated Brillouin scattering is the process that a pump photon interacts with lattice oscillation and is scattered into a Stokes photo and an acoustical photo^［13］. During the SBS process，the pump wave is scattered into a Brillouin wave by the acoustic wave，and the effect of the acoustic wave is reinforced by the beating between the two optical waves through the electrostriction effect. Thus，SBS requires strict phase-matching conditions，considering both photo and phonon modes^［14］. In the SBS process，it can generate two types of SBS，namely backward and forward scattering，due to the acoustic wave travel through different directions （Clockwise and Count-Clockwise）. The schematic description of SBS process in WGM resonator is shown in Fig. 1. Stokes light can be amplified when it resonates with a resonance frequency in WGM resonators since the SBS is a coherent process. Due to the SBS threshold power is proportional to the mode volume and inversely proportional to the square of the Q factor，it can be easily generated at a low input power in WGM resonators which have high or ultra-high Q factor and a small mode volume.

SBS process can act as an ideal platform to investigate ultra-narrow linewidth lasers and ultra-sensitive gyroscopes since it has the advantage to combine the high signal level and low noise^［15-17］. However，there is a limitation to achieve SBS in a single WGM resonator that the frequency of two optical modes must separate exactly to match the Brillouin frequency shift. Up to date，there are usually two methods to satisfy this condition. One is to precisely control the FSR in a large scale resonator，which is usually not an easy task for fabrication procedure. The other is to use different transverse modes in a resonator for the Stokes and the pump，which is hard to choose different modes when using prism or inclination fiber coupling methods^［18-19］.

In this article，we present the observation results of the cascaded SBS in a CaF₂ WGM resonator with an mm-sized 12.6 mm diameter and ultra-high Q factor 1.16×108. To lower the resonator fabrication difficulties to precisely control the FSR，we choose to employ different transverse modes to match the Brillouin frequency shift by using different positions of the optical tapered fiber to select different modes when coupling light into the resonator. During the experimental procedure，we found that the SBS stable state affected seriously by the fluctuation of the environment. To solve this problem，we devised a stable packaged platform for packaging the CaF₂ WGM resonator and the tapered fiber. After packaging，the unstable problem can be perfectly solved and the packaged device can be used out of the laboratory. The result of the cascaded SBS and our packaged platform can be used in further investigation of SBS based resonator gyroscopes and SBS based lasers，which can significantly enhance the stability and sensitivity benefited from Brillouin nonlinear effect.

The CaF₂ crystal WGM resonator used in this experiment was fabricated through the chemical mechanical polishing method instead of the wet etching method mentioned in the literature，which ensures the resonator acquiring ultra-high Q factor and large scale^［20-21］. The CaF₂ resonator with an initial 15 mm diameter and 0.5 mm thickness was firstly mounted on an air rotation spindle，then the rough polishing procedure was used to get the initial circle shape，at last，the fine grinding procedure was adopted to acquire the ultra-smooth surface. During the procedure，the diamond abrasive paste and diamond water-based suspension was used as the polishing material. The ultra-high Q factor in a WGM resonator usually requires the root mean square （RMS）of the optically smooth surface to the sub-nanometer level，which can be measured by the white light interferometer.

The experimental setup about the generation of SBS is depicted in Fig. 2. It contains a tunable laser in a 1550 nm band，a function generator，a polarization controller，a three-dimensional coupling platform，a photodetector （PD），a spectrometer，and high-resolution digital oscilloscope. The tunable laser （New focus TLB 6728）was derived by a function generator. The laser beam polarization was adjusted by a polarization controller，then through into the tapered fiber and coupling into the micro-disk resonator. The transmission light was separated into two light beams，and one sent to an Optical spectrum analyzer （Yokogawa AQ6370D），the other sent to a photodetector where the transmission light transferred to an electrical signal. Finally，the electrical signal was recorded by a high-resolution digital oscilloscope （Keysight Infiniium MSOS604A）.

To solve the unstable problem，we designed a packaging platform which can package the CaF₂ resonator and the tapered fiber in an N₂ atmosphere. The packaging platform not only can protect the resonator and waveguide disturbed by the environmental fluctuation，but also ensure the coupling state adjustment. This is important for the further integrated SBS laser investigation. Figure 3 is the actual package platform with the resonator and a tapered fiber.

The WGM resonator has an attractive property that is their ability to confine the light into a small volume for a long time. This means the light phono has enough time to interact with the cavity matter. Usually，one method to calculate the Q factor of the WGM resonator is measuring the linewidth of the half maximum of the cavity mode at the condition of the weakly damped oscillator.，the other is the ring-down spectroscopy. The former method is based on the formation Q≈v/△v，which consists of the linewidth of resonance △v and the resonance frequency v. The Q factor measured using the former method is an approximate result which is usually half of the actual value. The later method to measure the Q factor is through measuring the photo lifetime τ of resonance through the cavity ring-down since Q=2πvτ. It is more accurate than the former method，but it should at the critical coupling condition which requires the more accurately adjust moving station and measure equipment. Take the deep consideration based on our measurement platform，we employ the former method to acquire the Q factor of our fabricated micro-disk CaF₂ resonator.

Figure 4（a）is the actual CaF₂ micro-disk resonator fabricated through our laboratory designed fabricating equipment^［22］. The resonator has a 12.6 mm diameter after the polishing procedures and with a 0.1 mm rim thickness under the approximately ‘V’ shape condition. Figure 4（b）is the simulation result of the intrinsic mode in COMSOL software，which shows the electric field component distribution. Figure 4（c）is the sweeping power transmission spectrum between 1 550 nm to 1 550.2 nm. Figure 4（d）is the zoom in on a selected resonance mode from which the measured ultra-high Q factor is 1.16×108，which is based on the linewidth method. The calculation method based on the following equation，

where，λ0 is the central wavelength of the resonance，△λ is the wavelength width of full width at half height. These two parameters can be calculated from the Lorentz fitting result of the resonance which is indicated by the red line in Fig. 4（d）.

The Brillouin frequency shift is determined by the following equation^［19］，

where the vB is the Brillouin frequency shift，n is the refractive index，va is the speed of the acoustic wave and the λ is the wavelength of the light wave. Through the equation （2），we get the Brillouin frequency shift ΩB=12.1GHz in our CaF₂ resonator at 1 550 nm wavelength，which matches with literature^［21］.

The SBS can appear as long as the pump power over the Brillouin threshold at the phase-matching condition. The threshold power is given by^［23］，

where the n=1.33 is the refractive index of CaF₂ at 1.55 µm wavelength，Vm=12.6×10-6cm3 is the effective mode volume. B is the mode overlap which is close to 1 when the FSR is close to the Brillouin frequency shift.gB=2.8×10-9cm/Wis the Brillouin gain of CaF₂. QP=QB=1.16×108，λP=λB=1.55μm are the Q factor and the wavelength of the pump and Brillouin lights，respectively. Based on these parameters，we obtained the Pth=0.243mW theoretical for our CaF₂ resonator.

WGM resonators with ultra-high Q factors are suitable for lower the SBS threshold power. But the SBS is usually realized in the fiber ring resonators，as they can easily have an FSR which is smaller than the SBS gain bandwidth to satisfy the double resonance condition，which means the pump and the scatted Stokes light frequency perfectly match the resonator resonance frequency in one FSR. The SBS typically has a bandwidth of tens of MHz and a frequency shift of the order of a few GHz. The calcium fluoride which we used for our resonator has an SBS bandwidth of 12 MHz. It is difficult to realize the Brillouin laser based on a single-mode ring resonator. This problem can be solved by the multimode structure of CaF₂ WGM resonator，which can facilitate the SBS between the different WGM family.

To investigate the SBS in our CaF₂ resonator and choose different WGM families，we imply the adiabatically tapered fiber with an approximate 1 μm diameter twist as the waveguide，which was fabricated through the oxyhydrogen flame heating method. During the tapered fiber fabrication procedure，the original 125 μm diameter commercial optical fiber （Coning SFM-28）was gradually shrinking to 1 μm twist，thus the tapered fiber has different propagation constant at different coupling positions，which ensure exciting the different mode families in our CaF₂ resonator. Figure 5 shows the result of stimulated Brillouin lights in the forward direction which matches the Brillouin frequency shift ΩB=12.1GHz or multi ΩB. The S1 and S3 represent the first order and third order Stokes light respectively，and the AS2 represents the second order anti-Stokes light.

During the procedure of increasing input power and detuning，the cascaded SBS occurred which up to the 5th Stokes in the forward direction. The corresponding result is depicted in Fig. 6. The position of each order Stokes light is indicated in the enlarged picture. However，it should be noted that the cascaded Brillouin lights have lower power. We believe that this is due to other nonlinear effects strongly happening around first and second Brillouin lights，which disperse the cascaded Brillouin light power. The nonlinear effects including stimulated Raman scattering and four-wave-mixing，which occur easily in WGM resonator as long as the pump power exceeds the threshold.

In conclusion，we have reported the stimulated Brillouin scattering and the cascaded SBS in a millimeter size CaF₂ WGM micro-disk resonator with ultra-high Q factor up to 1.16×108. Due to the large scale of the resonator，we observe multimode resonances between the resonator and the tapered fiber，which can perfectly match the SBS frequency shift. This eliminates the exact requirement of the diameter of a WGM resonator to generate stimulated Brillouin lights. To solve the problem of environment fluctuation，we designed a coupling platform for coupling light from tapered fiber to CaF₂ WGM resonator and a packaged box that can seal the coupling platform in N₂ atmosphere. Based on this packaged platform，we realized the cascaded Brillouin scattering with up to 5th Stokes （60.5 GHz frequency shift）. This work and packaged resonator can further be used to investigate optical applications in infrared fields，such as packaged multi-wavelength Brillouin narrow linewidth laser and the packaged Brillouin gyroscope.