1 arXiv:1002.2141v1 [hep-th] 10 Feb 2010 We report stronger constraints on the parameters of Yukawa-type corrections to Newtonian gravity from measurements of the lateral Casimir force between sinusoidally corrugated surfaces of a sphere and a plate. In the interaction range from 1.6 to 14 nm the strengthening of previously known high confidence constraints up to a factor of 2 . 4 × 10 7 is achieved using these measurements. It is shown that the replacement of a plane plate with a corrugated one in the measurements of the normal Casimir force by means of an atomic force microscope would result in the strengthening of respective high confidence constraints on the Yukawa-type interaction by a factor of 1 . 1 × 10 $^{12}$. The use of a corrugated plate instead of a plane plate in the experiment by means of a micromachined oscillator also leads to strengthening of the obtained constraints. We further obtain constraints on the parameters of Yukawa-type interaction from the data of experiments measuring the gradient of the Casimir pressure between two parallel plates and the gradient of the Casimir-Polder force between an atom and a plate. The obtained results are compared with the previously known constraints. The possibilities of how to further strengthen the constraints on non-Newtonian gravity are discussed. PACS numbers: 14.80.-j, 04.50.-h, 04.80.Cc, 12.20.Fv Advance and prospects in constraining the Yukawa-type corrections to Newtonian gravity from the Casimir effect $^{1}$Department of Physics, Federal University of Para´ıba, C.P.5008, CEP 58059–970, Jo˜ ao Pessoa, Pb-Brazil $^{2}$North-West Technical University, Millionnaya St. 5, St.Petersburg, 191065, Russia $^{3}$Noncommercial Partnership “Scientific Instruments”, Tverskaya St. 11, Moscow, 103905, Russia V. B. Bezerra, 1 G. L. Klimchitskaya, 2 V. M. Mostepanenko, 3 and C. Romero 1 Abstract