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Figure 11. Cavity ring down profile of a MBR resonance.
Figure
10. Sketch of MBR fabrication technique:( a) a capillary with a sealed end,( b) microbubble formation by heating,( c) picture of the arc discharge unit in our labs,( d) picture of an MBR after the arch discharge and inflation process.
are the ones connected with the quality of the fabrication process and take into account the material absorption, the scattering produced by the roughness of the MBR surface and the surface contamination by impurities such as dust. Coupling losses( j 2), instead, quantify the energy exchange between the coupler and the resonator. These quantities can be retrieved by performing a fast wavelength scan of the WGM resonance and analysing the resulting cavity ring down( CRD) profile [ 50 ]. Figure 11 shows a CRD profile, whose analysis leads to g 2 = 2.8 10 �5 and j 2 = 6.4 10 �6, proving the overall small values. In addition, the intrinsic losses g 2 can be used to compute an intrinsic quality factor Q 0 = 3.3 10 8, proving the high quality of the the arc discharge fabrication process.
2.2.4 Microtoroids
Ultrahigh-Q toroidal microcavities were fabricated first by Kerry Vahala’ s group [ 51 ]. The fabrication technique is based on several steps. First, the toroid is fabricated in an oxide-coated Si wafer using photolithography, then wet and dry etching, and finally laser reflowing. The laser reflowing is used to remove lithographic flaws and create a very smooth surface, and also to fabricate an optical isolated toroidal structure as the disk shrinks. Microtoroids are hybrid structures, in the sense that they can be fabricated in planar arrays, but the coupling system is still external which reduces the multiplexing capabilities.
3 Modeling of microbubble WGMR
For modeling the microbubble WGMs, we used a formal approach, using analytical expressions as much as possible and resorting to numerical methods only in a few instances. This makes the theory versatile and enables finalization that can be different from other models. This formal derivation follows the argument of Balac and Féron [ 52 ], where the WGMs of a microsphere are deduced, and constitutes an original contribution, since no similar dissertation about the MBR system was found in literature [ 53 ]. Guigot et al. [ 54 ] have published a numerical study of a microsphere with a cladding layer in order to classify the different family modes( core, cladding and composed modes) and their potential use in sensing. For more detailed numerical solutions such as global solution search procedures based on particle swarm optimization approach, we refer to the results published by [ 55, 56 ].
We will present here a summary of the different modeling published in literature as a help for better understanding the different sensing mechanisms. Teraoka et al. [ 57 ] published a first-order perturbation theory similar to the one widely used in quantum mechanics is developed for transverse-electric and transverse-magnetic photonic resonance modes in a dielectric microsphere. The same group published a general formula for the shifts in TE and TM polarization by adsorption of another dielectric medium, using a vector wave equation [ 58 ]. Yariv and Yeh [ 59 ] defined the common strategy to solve the different field components. In our modeling, the microbubble system can be divided into three dielectric sectors, each representing a different part of the microbubble and each having spherical symmetry. The first sector represents the material hosted inside the MBR and it is a core sphere of refractive index n c and radius R c. The second sector represents the MBR walls and it is a spherical shell of refractive index n w with internal radius R c and thickness W, or equivalently with external radius R e = R c + W. Finally, the third sector represents the medium surrounding the MBR and it is a spherical shell of refractive index n e, internal radius R e and infinite thickness. The refractive indexes n c, n w, and n w are assumed to be uniform within each sector and therefore the index changes abruptly at the sectors boundaries.
Figures 12a and 12b show this modeling through a sagittal cut of the microbubble, using a 3D and 2D prospective, respectively. In these panels the dielectric sectors are depicted with different colors, the geometrical parameters