leading to δ$_{1}$ = 13 nm. In Fig. 3(a) we show an AFM scan of the corrugations imprinted on the sphere used in the first set of our measurements. The amplitude of these corrugations over a 30 × 30 µ m 2 area on the sphere was measured with an AFM to be A$_{2}$ = 13 . 7 ± 0 . 4 nm (i.e., an increase by a factor of 1.7 as compared with Refs. [37, 38]). Figure 3(b) is a typical section of Fig. 3(a) along a y = const plane. It illustrates a symmetry between the top and bottom of the imprinted corrugations. The variance of the stochastic roughness of the imprinted grooves δ$_{2}$ was found using the same procedure as for a grating with the result δ$_{2}$ = 2 . 7 nm. 

In Fig. 4 the AFM scan of a 5 × 5 µ m 2 area of imprinted corrugations on the sphere used in the first set of measurements is demonstrated. Here, the lighter tone shows higher points, as indicated in the additional scale. As is seen in Fig. 4, the imprinting procedure did not destroy the sphericity of the imprinted surface. An AFM scan of the sphere surface with imprinted corrugations used in the second set of our measurements was also performed. The amplitude of these corrugations was measured to be ˜ A$_{2}$ = 25 . 5 ± 0 . 6 nm (increase by a factor of 3.2 as compared with Refs. [37, 38]). The respective variance of the stochastic roughness in this case was ˜ δ$_{2}$ = 8 . 8 nm. Note that the relatively large values of the variance of the stochastic roughness presented above (in comparison with the case of flat surfaces [43]) are explained by the fact that the r.m.s. deviation is calculated along the z -axis, i.e., not perpendicular to the corrugated surface at most of the intermediate points. In Fig. 5 we show a 5 × 5 µ m 2 area of the corrugations imprinted on the sphere used in the second set of our measurements. As can be seen in this figure, the grooves are deeper than for the sphere used in the first set of measurements, and the sphericity of the surface is preserved. The diameters of the spheres with the imprinted corrugations used in the first and second set of measurements were measured to be 2 R = 194 . 0 ± 0 . 3 µ m and 2 ˜ R = 194 . 8 ± 0 . 3 µ m, respectively, using a scanning electron microscope (SEM). The SEM was calibrated with a NIST-traceable calibration grating which was independently cross calibrated with our interferometrically calibrated AFM. 

The distinctive feature of our setup is that the lateral Casimir force acting along the x -axis tangential to the corrugated sphere and a grating leads to the vertical bending of the cantilever. This bending is measured using bicell photodiodes A and B in Fig. 1. Whereas a force acting normal to the test bodies (the normal Casimir force) leads to a torsional deflection of the cantilever. The torsional spring constant of the used cantilever K$_{tor}$ was 

