for so strong strengthening of the constraints obtained from the experiment with corrugated surfaces is that at a separation, for instance, a = 121 . 4 nm between the mean levels of cor- rugations, the distance between two closest points of the surfaces can be as small as only 22 nm. 

The strongest constraints on the Yukawa-type interaction shown in Fig. 1 place limits on the parameters of gauge barions and strange moduli [7, 45]. The existence of such particles, which are predicted in many extra-dimensional models, would result in a Yukawa- type interaction with a very large | α | and an interaction range λ from 10 − $^{8}$m to 3 × 10 − $^{6}$m. The obtained results can be also used to constrain the predictions of chameleon theories which introduce scalar fields with masses depending on the local background matter density [46, 47]. 

## III. YUKAWA FORCE BETWEEN A SMOOTH SPHERE AND A CORRUGATED PLATE 

The constraints on α, λ shown by the long-dashed line in Fig. 1 are obtained from the measurement of the normal Casimir force (i.e., directed perpendicular to the surface) acting between a smooth sphere and a plane plate [28]. The question arises whether the use of a corrugated plate (a grating) could lead to stronger constraints. The normal Casimir forces acting between a smooth sphere and a corrugated plate with sinusoidal [48] and rectangular [49] corrugations have been measured. However, due to the absence of sufficiently exact theory of the Casimir force applicable to corrugated surfaces at that time, it was not possible to investigate the strength of constraints which might be imposed using the results of these measurements. Presently the exact theory of the Casimir force applicable to corrugated surfaces of test bodies made of real materials, at the laboratory temperature, is available [38]. Because of this, it is pertinent to verify the potentialities of different configurations with corrugated surfaces for the strengthening of constraints on the Yukawa-type interaction. In this section we consider the configuration of a smooth sphere above a sinusoidally corrugated plate (grating). 

The energy of the Yukawa-type interaction for the configuration of our interest is obtain- able from Eq. (10), where the amplitude of corrugations on a sphere is put equal to zero, 

