**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,"

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)

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)