In this paper, ABCD matrix is introduced to study the paraxial transmission of light on a constant gaussian curvature surface (CGCS), which is the first time to our knowledge. It is also proved that the curved surface can be used as the implementation of fractional Fourier transform, we further generalize that we can obtain the transfer matrix of an arbitrary surface with gently varying curvature by matrix optics. As a beam propagation example, based on the Collins integral, an analytical propagation formula for the hollow Gaussian beams (HGBs) on the CGCS is derived. The propagation characteristics of HGBs on one CGCS are illustrated graphically in detail, including the change of dark spots size and splitting rays. Besides its propagation periodicity and diffraction properties, a criterion for convergence and divergence of the spot size is proposed. The area of the dark region of the HGBs can easily be controlled by proper choice of the beam parameters and the shape of CGCS. In addition, we also study the special propagation properties of the hollow beam with a fractional order. Compared with HGBs in flat space, these novel characteristics of HGBs propagation on curved surface may further expand the application range of hollow beam.

Analogous coherent states are deduced from classical optical fields on curved surface in this paper. The Gaussian laser beam, as a fundamental mode, cannot be adequately simulated by coherent states due to their inherent diffraction in flat space. But things will be different when it propagates on a surface with uniform curvature called the constant Gaussian curvature surface (CGCS). By generalizing the method of Feynman path integral, an equivalent coherent states solution is demonstrated here to describe the beam propagation. The temporal evolution of the Schrodinger equation is analogously translated into a spatial transmission in this derivation, we obtain the expression of quantized momentum transmitted on curved surface, which is proportional to the square root of the Gaussian curvature $K$. In addition, we build a beam propagation picture identical to the squeezed state. We hope this research can give a new view on the quantum field in curved space.

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In this paper, an analytical expression of Airy beams near a black hole is derived by general relativity concepts. This paper demonstrates the self‐acceleration and the self‐healing properties of Airy beams near a black hole with different Schwarzschild radii. It shows, during transmission, that the equivalent acceleration near a black hole decreases to a minimum negative value, then increases and eventually approaches zero. After propagating a certain distance, the trajectories of Airy beams approaching a black hole may no longer travel along parabolas, but rather almost straight lines due to the existence of the strong gravitational field. The shapes of the wave structure of Airy beams remain unvarying during the transmission, which indicates that the nondiffraction characteristic is still present. Moreover, the self‐healing property of Airy beams near a black hole gradually disappears with the increase of the strength of the gravitational field, because the energy flow to the major lobe is prevented by the gravitational field of the black hole. These intriguing features may open new prospects in the fields of nanophotonic optics, relativistic effects, transformation optics, and so on.

Electrodynamics on curved surfaces, as a developed theory, has analogously become a new experimental verification of light transmission in general relativity. The thermal effect of an optical field on a specific 2D surface with constant Gaussian curvature is described in this paper. By considering the analogy between Schrodinger equation and Helmholtz equation under the paraxial approximation, the “quantized” momentum field is generated from the light transmitting on a curved surface by using the effective potential approach, and when decreasing the number of photons until n=0$n = 0$ is thought about, a temperature of Hawking‐like radiation is obtained. The creation process of radiation is also investigated, which is the scattering of light as it travels from a surface of positive curvature to a surface of negative curvature. The derived temperature of radiation field is also equivalent to the event horizon scattering explanation of Hawking radiation. The research may provide new perspectives for Hawking radiation and thermal lens.