Generalization of Wolf effect of light on arbitrary two-dimensional surface of revolution

Abstract

Investigation of physics on two-dimensional curved surface has significant meaning in study of general relativity, inasmuch as its realizability in experimental analogy and verification of faint gravitational effects in laboratory. Several phenomena about dynamics of particles and electromagnetic waves have been explored on curved surfaces. Here we consider Wolf effect, a phenomenon of spectral shift due to the fluctuating nature of light fields, on an arbitrary surface of revolution (SOR).The general expression of the propagation of partially coherent beams propagating on arbitrary SOR is derived and the corresponding evolution of light spectrum is also obtained. We investigate the extra influence of surface topology on spectral shift by defining two quantities, effective propagation distance and effective transverse distance, and compare them with longitudinal and transverse proper lengths. Spectral shift is accelerated when the defined effective quantities are greater than real proper lengths, and vice versa. We also employ some typical SORs, cylindrical surfaces, conical surfaces, SORs generated by power function and periodic peanut-shell shapes, as examples to provide concrete analyses. This work generalizes the research of Wolf effect to arbitrary SORs, and provides a universal method for analyzing properties of propagation compared with that in flat space for any SOR whose topology is known.

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

“…, , 0. is the effective propagation distance in curved space whose physical meaning is concretely demonstrated in our previous work [35] is the Gouy phase under paraxial approximation, which is rather similar to that in flat space, except for the propagation distance being replaced by the effective propagation distance, and the existence of factor 1/2 which is however because of one rather than two dimensions transversely. Meanwhile, SCI phase is known as .…”

“…Experimentally it can be realized by covering a thin layer of waveguide on surface [27]. In the latest decade various concepts have been reconsidered and reported, such as solitons [28], evolution of speckle pattern [29], spatially accelerating wave packets following nongeodesic trajectories [30,31], topological phases in curved space photonic lattices [32], phase and group velocity of wave packets [33], Wolf effect of light spectrum [34,35], etc. Specially, in a pioneering work [26], Schrödinger equation for linear propagation on curved space with constant Gaussian curvature is derived, which is in essence the wave equation under paraxial approximation, and one of whose solutions is the fundamental notion in optics, Gaussian beam.…”

“…When transverse distribution of beam is Gaussian, á ñ x k 2 is supposed to be inverse square of beam radius w(z) [2]. It is interesting to note that the term x ( ) r k r z 0 can be regarded as a proper transverse wave number, since the real or proper length at different position z is indeed stretched or shrunk by an expansion coefficient r(z)/r 0 , owing to geometry of surface [35]. In equation (22), compared with flat space, the additional part of wave number will certainly give rise to a phase shift…”

Theory of light propagation in arbitrary two-dimensional curved space

“…In this manner, the remaining 2D spatial metric tensor can be depicted as a 2D curved surface embedded in 3D space, and the interplay between EM waves and spatial curvature can be revealed by investigating light propagation on such appropriately fabricated surfaces. Ever since this notion was put forward by Batz and Peschel [10] in 2008, various optical phenomena have been reported both theoretically [10][11][12][13][14][15][16][17][18] and experimentally [19][20][21][22][23]. Besides optics and photonics, similar studies on curved surfaces have also been extended to surface plasmon polaritons [24], acoustic topological insulators [25] and quantum particles [26].…”

“…Intriguing questions naturally arise, for instance, which pathway would the light beam take and how would the curvature of surface affect its divergence? Besides, it has also been revealed in prior studies [10,15,17,20] that the existed method for calculating light fields on closed SORs collapsed at artificial singularities (such as both the north and south poles on spherical surfaces), leading to an artificial "infinite intensity" thereat.…”