### Topological effects

The study of electromagnetic wave propagation in helical structures began in earnest in the 1940s, with the invention of the travelling-wave tube amplifier. In this device a microwave signal is guided along a helical wire that spirals around an axially propagating electron beam **[Kompfner (1947)]**. Since the physical distance over which the spiralling microwave signal travels is longer than the distance directly along the axis, its group and phase velocities are both effectively reduced. By appropriate design the velocity difference between the two waves can be adjusted, permitting the microwave signal to be amplified with power from the electron beam. In a similar manner, the geometrical stretching of the cladding structure in a t-PCF causes the effective optical path-length along the axis, and thus the effective refractive index, to increase topologically with radius ρ following the relationship *n*_{eff}(ρ) = *n*_{0}√(1 + α^{2}ρ^{2}) where *n*_{0} is the index in the untwisted case and α the twist rate in rad/m.