Topology and curvature effects in the photonics of ring – split ring – cuboid transitions

Topological transitions in various materials are actively being studied, including topological quantum phase transitions, going beyond the Landau theory and the concept of the order parameter. Here we propose the concept of a transition between two structures with different topology using the example of the transition between a flat dielectric ring and a split ring and its further unbending into a rectangular Fabry-Pérot resonator. Experimentally and theoretically, we discovered the lifting of the degeneracy of the CW-CCW photonic modes of the ring and the formation of two families: topological, which acquire an additional phase $\pi$, equal to the Berry phase in a thin Möbius strip, and ordinary ones, which do not acquire an additional phase. Topological modes arise due to the gradual “cutting” of one antinode of the field by a gap into two antinodes as the angular size of the gap increases from zero to one degree. Thus, using a topological Fabry-Pérot resonator with variable curvature and fixed length, resonant modes with an arbitrary non-integer number of waves are realized and a new generation of resonators is created with the prospect of unique classical and quantum applications.