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A$_{2}$ = 0, and from Eq. (8) it follows that b = A$_{1}$ . As a result 

The respective Yukawa-type force is given by 

The constraints on α, λ can be found from the inequality 

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Here, the confidence interval [ − Ξ$_{F}$ ( a ) , Ξ$_{F}$ ( a )] for the difference between theoretical and mean experimental Casimir forces, F $^{theor}$( a ) − ¯ F $^{expt}$( a ), was found at different separations in Ref. [37] at a 95% confidence level. As an example, at separations a = 100, 110 and 120 nm, the half-width of the confidence interval is equal to 9.17, 8.35, and 7.96 pN, respectively. 

The computations of the constraints were performed by the substitution of Eq. (13) into Eq. (14) for the realistic values of the parameters given below. The corrugated plate (grating) was precisely the same as described in Sec. II basing on the experimental configuration of Ref. [38]. The smooth polystyrene sphere of radius R = 100 µ m was assumed to be coated with only one layer of Au (as in the experiment of Ref. [28]) of thickness Δ$_{Au}$$_{,s}$ = 100 nm. This means that in the expression (5) for the function Ψ( λ ) one must put Δ$_{Cr}$ = 0. The obtained constraints are shown by the grey line in Fig. 2 (as in Fig. 1, the region of α, λ below the line is allowed and above the line is prohibited). For comparison purposes, we reproduce in Fig. 2 the solid and the short-dashed lines of Fig. 1, which show the constraints obtained by us in Sec. II from the measurement of the lateral Casimir force and in Ref. [36] from the experiment using a micromachined oscillator, respectively. As is seen in Fig. 2, the largest strengthening of already obtained constraints given by the solid line up to a factor of 4 . 5 × 10 4 occurs at λ = 1 . 6 nm. It is obtained from the measurement data at the shortest separation a = 100 nm. 

It is instructive also to compare the prospective constraints given by the grey line in Fig. 2 with those obtained from the experiment [28] in Ref. [36] using the configuration of a smooth sphere above a plane plate (the long-dashed line in Fig. 1). Here, at λ = 1 . 6 nm the largest strengthening of constraints by a factor of 1 . 1 × 10 12 is possible. So strong improvement of 

