The integral in Eq. (9) can be calculated using the formula 2.5.40(3) in Ref. [44], with the result where I$_{n}$ ( z ) is the Bessel function of imaginary argument. The lateral Yukawa force between corrugated surfaces is obtained from Eq. (10) by taking the negative differentiation with respect to the phase shift: Now we are in a position to determine the constraints on the parameters of the Yukawa- type interaction following from the measurement data of experiment [38]. For this purpose we substitute Eq. (11) into Eq. (2) and find the allowed values of α and λ . The computational results are shown in Fig. 1, where the allowed values of α, λ lie below the solid line and the prohibited values of α, λ lie above the solid line. These constraints were obtained in the interaction region ranging from λ = 1 . 6 nm to λ = 35 . 5 nm using the measurement data of Ref. [38] at different separations a$_{i}$ . Thus, at λ ≤ 4 . 5 nm the strongest constraints follow from Eq. (2) with a$_{1}$ = 121 . 1 nm and Δ$_{1}$ = 11 . 1 pN. For λ in the range from 4.5 to 22.4 nm, the separation a$_{2}$ = 124 . 7 nm has been used with Δ$_{2}$ = 4 . 7 pN. Finally, for λ from 22.4 to 35.5 nm the strongest constraints were obtained for a$_{3}$ = 137 . 3 nm and Δ$_{3}$ = 2 . 5 pN [38]. Note that our constraints indicated by the solid line are determined with the same 95% confidence level as the absolute errors Δ$_{i}$ on the right-hand side of Eq. (2). For com- parison purposes in the same figure we have plotted the previously known strongest con- straints on α, λ determined with 95% confidence level. The long-dashed line follows from the measurement data of the experiment [28] reanalyzed using rigorous statistical methods in Refs. [36, 37]. The short-dashed line represents constraints obtained from the experiment [36]. As can be seen in Fig. 1, our constraints indicated by the solid line are the strongest over the interaction range from 1.6 to 14 nm. The largest strengthening of previously known constraints shown by the long-dashed line by a factor of 2 . 4 × 10 7 is achieved at λ = 1 . 6 nm. (Note that the confidence level of slightly stronger constraints obtained in Ref. [33] from the measurement of the Casimir force between two crossed cylinders [32] cannot be determined because of several uncertainties inherent to this experiment [21, 22].) The physical reason 8 F Yu , lat ps, cor ( a, ϕ ) = − 2 π Λ ∂E Yu ps, $_{cor}$( a, ϕ ) ∂ϕ = 8 π $^{3}$Gαλ$^{3}$Ψ( λ ) e − $_{a/λ}$A$_{1}$ A$_{2}$ b Λ I$_{1}$ ( b/λ ) sin ϕ. (11) E Yu ps, $_{cor}$( a, ϕ ) = − 4 π $^{2}$Gαλ$^{4}$Ψ( λ ) e − $^{a/λ}$I$_{0}$ ( b/λ ) , (10)