Fig. 1: The Casimir force between perfectly rectangular gratings.

a Schematic of a part of the perfectly rectangular silicon grating. Initially, the displacement d of the movable grating (blue) along the y-direction is zero. The inset shows the top-view schematic. h represents the length of a grating finger. The lateral separation between adjacent grating fingers is \(g = p/2 - w\sim 92\,{\mathrm{nm}}\) and the initial separation in y is \(s\sim 430\,{\mathrm{nm}}\). b Top-view schematic for the interpenetration of the two gratings. I–IV panels depict the four stages of the interpenetration. The black dotted line denotes the initial location of the bottom edge of the blue movable grating. d is defined as the displacement of the movable grating from this initial position. The dashed frame encloses a unit cell. The bars measure 1 μm. c Calculated Casimir force per unit cell in the y-direction as a function of displacement d. The black line is the force calculated using the PFA. The red circles and purple squares are the Casimir force calculated by SCUFF-EM and the scattering theory respectively. Inset: The ratio ρ of the Casimir force to the force obtained by the PFA. The black dashed line marks where interpenetration occurs. d The gradient of the Casimir force.