where Φ( x, λ ) ≡ x − λ + ( x + λ ) e − 2 $^{x/λ}$. (4) Applying the first equality in (3) to the layer structure of the experimental test bodies, as described above, one arrives at E Yu $_{ps,l}$( a ) = − 4 π $^{2}$Gαλ$^{4}$e − $^{a/λ}$Ψ( λ ) , (5) Ψ( λ ) ≡ [ ρ$_{Au}$ − ( ρ$_{Au}$ − ρ$_{g}$ ) e − Δ Au$_{,g}$ $^{/λ}$] × [ ρ$_{Au}$ Φ( R, λ ) − ( ρ$_{Au}$ − ρ$_{Cr}$ )Φ( R − Δ$_{Au}$$_{,s}$ , λ ) e − Δ Au$_{,s}$ /λ − ( ρ$_{Cr}$ − ρ$_{s}$ )Φ( R − Δ$_{Au}$$_{,s}$ − Δ$_{Cr}$ , λ ) e − (Δ Au$_{,s}$ +Δ Cr ) $^{/λ}$] . As shown below, significant strengthening of constraints on α from the data of this exper- iment holds at λ below 10 nm. Because of this, keeping in mind that Δ$_{Au}$$_{,g}$ = 300 nm, not only the thickness of Pyrex substrate but the Au layer on the grating as well can be considered as infinitely thick. Now we need to calculate the effect of aligned corrugations on the Yukawa-type energy (5). This can be done by using the method of geometrical averaging [21, 22], i.e., by replacing the closest separation a between the smooth surfaces in Eq. (5) with respective separation between corrugated surfaces and averaging over the period of corrugations. In order for such an approximate method works properly, in addition to the conditions indicated above, one more condition, namely, Λ ≪ R , should be valid. It is satisfied with large supply for the listed above experimental parameters of Ref. [38]. The separation distance between the closest points of corrugated surfaces is given by It can be identically represented as where the following notations are introduced: z$_{2}$ − z$_{1}$ = a + A$_{2}$ sin(2 πx/ Λ + ϕ ) − A$_{1}$ sin(2 πx/ Λ) . (6) z$_{2}$ − z$_{1}$ = a + b cos(2 πx/λ − ˜ ϕ ) , (7) b ≡ b ( ϕ ) = ( A 2 1 + A 2 − 2 A$_{1}$A$_{2}$ cos ϕ ) 1 / $^{2}$, (8) tan ˜ ϕ = ( A$_{2}$ cos ϕ − A$_{1}$ ) / ( A$_{1}$ sin ϕ ) . E Yu ps, $_{cor}$( a, ϕ ) = − 4 π $^{2}$Gαλ$^{4}$Ψ( λ ) e − a/λ Λ ∫ Λ 0 dx e − b cos(2 πx/ Λ − ˜ ϕ ) $^{/λ}$. (9) Substituting Eq. (7) into Eq. (5) in place of a and performing the geometrical averaging, we arrive at 7