Analytic theory of curvature effects for wave problems with general boundary conditions

A formalism based on a combination of differential geometry and perturbation theory is used to obtain analytic expressions for confined eigenmode changes due to general curvature effects. In cases of circular-shaped and helix-shaped structures, where alternative analytic solutions can be found, the perturbative solution is shown to yield the same result. The present technique allows the generalization of earlier results to arbitrary boundary conditions. The power of the method is illustrated using examples based on Maxwell’s and Schrödinger’s equations for applications in photonics and nanoelectronics.

Figure 1

Squared curvature for a Möbius-shaped nanowire structure with parameters $c_1=3$ nm, $c_2=2$ nm, and $c_3=0.5$ nm.

Figure 2

Ground-state (top) and first-excited (bottom) states of a Möbius-shaped nanowire with parameters $c_1=3$ nm, $c_2=2$ nm, and $c_3=0.5$ nm. Solid and dashed lines indicate the unperturbed and perturbed eigenstates, respectively.