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It is important that corrugated surfaces move parallel to each other as the phase is changed with the x -piezo. The grating template was mounted as parallel as possible to the x -axis on the AFM piezo. However, experimentally there is always a finite nonzero angle between the grating template and the x -axis. When the phase change is introduced by moving the x -piezo, this tilt will lead to changes in the separation distance between the corrugated surfaces, which in turn would result in a systematic error of decreasing or increasing force with phase. This nonparallelity should be corrected before start of the experiments using the lateral electrostatic force. A voltage is applied to the plate and the lateral electrostatic force is monitored as a function of the phase over many periods of the grating. A correcting voltage adapted from that applied to the x -piezo using an adjustable voltage divider was synchronously applied to the z -piezo. The applied voltage to the z -piezo was changed till the amplitude of the lateral electrostatic force become independent of the phase. As an example, in Fig. 6 we show the deflection signal due to the lateral electrostatic force versus the phase difference x$_{0}$ between the corrugated surfaces (a) before and (b) after the tilt correction was introduced. This was done approximately 25 nm away from contact with a 0.2 V applied voltage. It can be seen that the amplitudes of each peak are more identical in Fig. 6(b). This procedure experimentally verifies that the separation distance between the corrugations remains fixed as the phase was changed with the x -piezo. 

The cantilever deflection signal S tot lat corresponding to the total force acting at the sep- aration a = z$_{0}$ + 3 . 96 nm (sum of the lateral electrostatic and lateral Casimir forces) was recorded at 8192 evenly spaced data points. The sphere was moved further away from the grating by 5.40 nm to a separation 9.36 nm from z$_{0}$ , and the measurement was repeated. The cantilever deflection signal S tot lat was measured at many other a at the same voltage and also for a second voltage of 101.202 mV applied to the grating. To obtain the deflection signal due to the lateral electrostatic force alone, it is necessary to subtract from the obtained signal S tot lat the deflection signal S C lat due to the lateral Casimir force (the signal S C lat was measured before S tot lat and the respective measurement data are reported in Sec. IV). To perform this subtraction, S C lat was fitted to the following sum of harmonics: 

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which takes proper account of the fact that the lateral Casimir force is asymmetric (see Sec. IV). As the sphere-grating separations in the measurements of the lateral Casimir force 

