Gouy and spatial-curvature-induced phase shifts of light in two-dimensional curved space

Gouy phase is the axial phase anomaly of converging light waves discovered over one century ago, and is so far widely studied in various systems. In this work, we have theoretically calculated Gouy phase of light beams in both paraxial and nonparaxial regime on two-dimensional curved surface by generalizing angular spectrum method. We find that curvature of surface will also introduce an extra phase shift, which is named as spatial curvature-induced (SCI) phase. The behaviors of both phase shifts are illustrated on two typical surfaces of revolution, circular truncated cone and spherical surface. Gouy phase evolves slower on surface with greater spatial curvature on circular truncated cone, which is however opposite on spherical surface, while SCI phase evolves faster with curvature on both surfaces. On circular truncated cone, both phase shifts approach to a limit value along propagation, which does not happen on spherical surface due to the existence of singularity on the pole. An interpretation is presented to explain this peculiar phenomenon. Finally we also provide the analytical expression of paraxial Gaussian beam on general SORs. By comparing the result with the exact method we find the analytical expression is valid under the approximation that beam waist and scale of surface are beyond order of wavelength. We expect this work will enhance the comprehension about the behavior of electromagnetic wave in curved space, and further contribute to the study of general relativity phenomena in laboratory.