have helped to strengthen the previously known constraints on the Yukawa interaction in the submicrometer range up to ten thousand times. 

In this paper we obtain stronger constraints on the hypothetical interaction of Yukawa- type from recent measurement of the lateral Casimir force between sinusoidally corrugated surfaces of a sphere and a plate [38]. As compared with the previously known constraints in the interaction range below 14 nm obtained at the same high confidence (a 95% confidence level), a strengthening up to a factor of 2 . 4 × 10 7 is achieved. We also discuss respective limits on the parameters of hypothetical light elementary particles. The use of corrugated surfaces opens new prospective opportunities for constraining the Yukawa-type hypothetical interactions from the Casimir effect. In view of this fact in the present paper we propose several experiments of this type and calculate the strength of constraints that can be ob- tained in future in different interaction ranges. Specifically we consider the measurement of the normal Casimir force acting between a smooth sphere and a sinusoidally corrugated plate. We also explore the potentials of dynamic experiments where the separation distance between the test bodies is varied harmonically and the measured quantities are the gradients of the Casimir force [36] (Casimir-Polder force [39]), or of the Casimir pressure [40]. 

The structure of the paper is as follows. In Sec. II we calculate the lateral Yukawa force acting between sinusoidally corrugated surfaces of a sphere and a plate and obtain constraints on its parameters from the measurement results of Ref. [38]. Section III is devoted to the calculation of the normal Yukawa force in the configuration of a smooth sphere above a sinusoidally corrugated plate. Prospective constraints on the parameters of this force obtainable in such a configuration are presented. In Sec. IV the gradient of the Yukawa force acting between a smooth sphere and a corrugated plate in the dynamic regime is calculated and used to estimate prospective constraints. The possibility of using the gradient of the Yukawa pressure between two parallel plates in the dynamic regime for obtaining stronger constraints is considered in Sec. V. In Sec. VI the gradient of the Yukawa force in the configuration of an atom oscillating near a substrate is calculated and applied for constraining hypothetical interactions from the measurement data for the Casimir-Polder force. In Sec. VII the reader will find our conclusions and discussions. 

