content_image/356352.jpg
values of the separation on contact and of the lateral force calibration constant 

<img src='content_image/356352.jpg'>

Typically in our second set of measurements with the larger amplitude corrugations the calibration errors are a bit larger than in the first, where smaller amplitude corrugations were used. 

## IV. MEASUREMENT DATA FOR THE LATERAL CASIMIR FORCE AND ER- ROR ANALYSIS 

Now we describe how the measurements of the lateral Casimir force were performed. In this case the residual voltage V$_{0}$ , as determined in the electrostatic calibration using the normal electric force, was applied to the grating in order to make the electric force equal to zero. The x -piezo was used to move the grating along the x -axis and thus change ϕ . The z -piezo, which was independently controlled by an external voltage source, was used to change a . The piezo extensions with applied voltage in both directions were calibrated using optical interferometry [51]. 

Initially the corrugated sphere was positioned 3.79 nm from the separation on contact between the two surfaces z$_{0}$ determined by the corrugations and the highest roughness peaks. The thermomechanical drift of the separation distance was measured to be 0.14 nm/min from the difference in the z -piezo voltage to bring about contact of the two corrugated surfaces after a time interval around 30 minutes. A phase shift was introduced by moving the x -piezo continuously for a total distance of 3 . 3 µ m at 0.103 Hz. The photodiode signal corresponding to the cantilever deflection was filtered with a low-pass filter with a 30 ms time constant and recorded at each of the 8192 points corresponding to x -changes of 0.4 nm. The effect of the scattered laser light which would lead to a linear modification of the signal with phase was found to be negligible in this experiment. 

Then the separation from z$_{0}$ was increased by 3.6 nm from 3.79 nm to 7.39 nm and the deflection signal S C lat was similarly measured as a function of ϕ and recorded. After this, the separation from z$_{0}$ was increased by 3.96 nm from 7.39 nm to 11.35 nm and the measurements repeated. Next S C lat due to the lateral Casimir force as a function of ϕ was measured at separations of 20.05, 32.48, 45.30, 58.01 and 70.86 nm from z$_{0}$ . 

