Corrugated Surfaces

Normal force

A common method for dealing with non-flat geometries is the  Proximity Force Approximation, which in effect assumes an effective Casimir interactions between locally parallel pairs on points on the two surfaces, and sums over all such pairs.

For a sinusoidally corrugated plate:

"Demonstration of the Nontrivial Boundary Dependence of the Casimir Force,"

A. Roy and U. Mohideen, Phys. Rev. Lett. 82, 4380 (1999)

Path-Integral formulation

Integrate over all configurations of the field in the space between deformed plates (or other boundaries)

Thermal fluctuations: Scalar field with Dirichlet boundary conditions

Quantum fluctuations: EM field is equivalent to Dirichlet + Neumann in certain geometries,

TM modes (Dirichlet)  +  TE modes (Neumann)

"Probing the Strong Boundary Shape Dependence of the Casimir Force,"

T. Emig, A. Hanke, R. Golestanian, and M. Kardar, Phys. Rev. Lett. 87, 260402 (2001)

  Rauno Büscher and Thorsten Emig, ("reduced distance" limit)

"Nonperturbative approach to Casimir interactions in periodic geometries,"  

Phys. Rev. A 69, 062101 (2004)

Lateral force

A sideways force to "align" the two plates at  b=λ/2 :

Comparison with the result of pair-wise summation:

"Demonstration of the Lateral Casimir Force,"

F. Chen, U. Mohideen, et. al , Phys. Rev. Lett. 88, 101801 (2002)