Experimental Investigation of the Casimir Force beyond the Proximity-Force Approximation

The analysis of all Casimir force experiments using a sphere-plate geometry requires the use of the proximity-force approximation (PFA) to relate the Casimir force between a sphere and a flat plate to the Casimir energy between two parallel plates. Because it has been difficult to assess the PFA’s range of applicability theoretically, we have conducted an experimental search for corrections to the PFA by measuring the Casimir force and force gradient between a gold-coated plate and five gold-coated spheres with different radii using a microelectromechanical torsion oscillator. For separations $z<300\mathrm{nm}$, we find that the magnitude of the fractional deviation from the PFA in the force gradient measurement is, at the 95% confidence level, less than $0.4z/R$, where $R$ is the radius of the sphere.

Figure 1

Measured values of the force (a) and effective pressure (b), as defined by Eq. (3), obtained using the $148.2\mu \mathrm{m}$ sphere. They have been normalized to the force and pressure of idealized bodies as described in the text. The deviations from unity result from finite conductivity and roughness corrections. The inset in (a) exhibits the plot of the normalized force as a function of $1/R$ for $z=170\mathrm{nm}$. The value of $\beta$ is obtained from the best fit line as described in the text. The inset in (b) is a similar plot of the normalized pressure for the measurements obtained at $z=170\mathrm{nm}$ from which ${\beta }^{\prime }$ is obtained.

Figure 2

(a) $\beta$ vs $z$ determined from the force measurements. (b) ${\beta }^{\prime }$ values obtained from the force gradient measurements for the entire range of separations. (c) Same as (b), but for $164\mathrm{nm}<z<400\mathrm{nm}$. In all cases the error bars denote statistical 95% confidence level intervals.