Fig. 5: Structural property of the fabricated curved surface on GaAs substrate and measurement.

a top view of the SEM image, in which the green dashed lines label the position of the sections in c–e, in which y = 5.1 μm corresponds to the diameter of the surface. b The AFM image shown in 3D form. c Fit of the surface of the section at y = 5.1 μm from the AFM image in b, with a circle graduate changing its radius with the height axis Z. The fit line follows the equation \(\left( {x - x_c} \right)^2 + \left( {z - z_c} \right)^2 = R^2\left( {1 + a \cdot \sin \theta + {{{\mathrm{b}}}} \cdot \sin ^2\theta } \right)^2,\tan \theta = (z - z_c)/(x - x_c)\), where \({{{\mathrm{x}}}}_{{{\mathrm{c}}}} = 6.35\,{{{\mathrm{\mu m}}}},z_c = 0.56507\,{{{\mathrm{\mu m}}}},a = - 0.62026,b = 0.39503,R = 3.70538\,{{{\mathrm{\mu m}}}}\). d Circle fit of the same data in c, but with the equation \(\left( {x - x_c} \right)^2\, +\, \left( {z - z_c} \right)^2 = R^2\), where \(x_c = 6.37678\,{{{\mathrm{\mu m}}}},z_c = - 1.06306\,{{{\mathrm{\mu m}}}},R = 3.96216\,{{{\mathrm{\mu m}}}}\). The fitting works well for the region of z = 1~2 μm, in which the WGM and periodic modes actually forms. e Circle fit of the section at y = 2.6 μm of the AFM image in b. The fit line follow the equation \(\left( {x - x_c} \right)^2 \,+\, \left( {z - z_c} \right)^2 = R^2\), where \(x_c = 6.32893\,{{{\mathrm{\mu m}}}},z_c = - 1.01123\,{{{\mathrm{\mu m}}}},R = 2.97804\,{{{\mathrm{\mu m}}}}\); f. The radii of the fit circles, obtained similarly as in d and e, as a function of the cross-section point on the Y axis