Fig. 1. Structural characterization of a thin-walled microbubble by FIB milling and SEM imaging. (a) The microbubble’s support stem on the left side was initially removed through FIB milling, while half of the support stem on the right side was cut, creating a gap. (b) A third FIB milling was conducted from the center of the left side towards the center of the gap on the right side, resulting in the removal of half of the microbubble. (c) After rotating the microbubble’s left half, the wall structure of the microbubble is clearly visible under SEM imaging. (d) Due to the high SEM imaging resolution, the wall thickness variation along the cavity axis can be determined with accuracy down to the nanometer scale (upper panel). To describe such a wall structure, Gaussian profiles were used to fit the outer and inner boundaries of the microbubble (lower panel). (e) The dependence of the wall thickness t on the outer diameter d of the thin-walled microbubble is shown. A reciprocal linear relation is satisfied near the center of the microbubble at larger outer diameters.

Fig. 2. Theoretical model of a thin-walled microbubble. (a) Reconstructed 3D geometry of the microbubble from the SEM images shown in Fig. 1. Due to the geometrical symmetries, the electric field distribution of WGMs in the microbubble can be calculated by solving the three coupled differential equations, Eqs. (3)–(5). Three mode indices, i.e., n, m, and l, are resolved in the end to identify the WGMs, in addition to their polarization state (either TM or TE). (b) Radial field distribution. Owing to the wavelength-scale wall thickness, only the fundamental radial mode with n=1 is considered. The inset illustrates the physical meaning of the effective permittivity εeff. (c) Axial field distribution. The quasipotential of the microbubble forms a quantum well—due mainly to diameter variation—that confines the axial motion such that different axial modes emerge. (d) Azimuthal field distribution. Only a small portion of the azimuthal field is shown for better visibility. (e) Dependence of εeff on wall thickness. (f) Illusory example showing a quantum-barrier-like quasipotential (lower panel) formed solely by wall thickness variation (upper panel) where a constant outer diameter was used. (g) Dependence of kcirc on the outer diameter.

Fig. 3. Verification of the theoretical model of the thin-walled microbubble by finite element method simulations. (a) Resonant wavelengths for different azimuthal modes. Simulated field distribution in the cross section is shown in the inset. (b) Resonant wavelengths for different axial modes. The inset shows the axial field distribution. n=1 for all cases. Note that the proposed theoretical model can also be used for higher-order radial modes with n>1.