The hollow Gaussian beam propagation on curved surface based on matrix optics method

Abstract

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.

‘Classical’ coherent state generated by curved surface

“…where ρ = ρ b − ρ a . It is worth noting that this analytic solution is consistent with that obtained by the optical matrix and Collins formula in reference [26]. Equation ( 2) is a linear differential equation whose solutions satisfy the linear superposition property, thus by assuming eigen-solutions u n (h, ρ), a special set of solutions, we can separated it into the following form:…”

“…where η σ p h = u σ e −ip h h dh is the Fourier transform of u σ . According to equation (12), it is found that the transmission of the light beam along axis can be regarded as the rotation in the position-momentum (h − p h ) space, which coincides with our previous discussion that CGCS can be regarded as a fractional Fourier transform system [26,27]. Furthermore, the coordinate uncertainty (spot size) and momentum uncertainty can be obtained by the second moment:…”

“…This paper principally illustrates the light paraxial transmission on a CGCS and the line element of the light on CGCS can be expressed as follows [26]:…”

The Optical Perspective of Hawking‐Like Radiation on a Curved Surface

“…The 2D surface light transmission simulation of the general relativistic effect is largely based on the related theories of the 3+1 membrane paradigm [9] and embedding diagrams. In recent years, great progress has been made in theory [10][11][12][13][14][15] and…”

“…[8] Unlike other fields that require relativistic laboratory tests of strong gravitational fields, optical studies can be carried out under a 2D analog.The 2D surface light transmission simulation of the general relativistic effect is largely based on the related theories of the 3+1 membrane paradigm [9] and embedding diagrams. In recent years, great progress has been made in theory [10][11][12][13][14][15] and…”