Light rays in Fourier domain

Abstract

Establishing a universal model to characterize the relationship between light rays and optical waves is of great significance in optics. The ray model provides us with an intuitive way to study the propagation of beams as well as their interaction between objects. Traditional ray model is based on the normal of a beam wave front. The normal vector is defined as the direction of ray. However, it fails to describe the relationship between light ray and optical wave in the neighborhood of focus or caustic lines/surface since light ray in those regions are no longer perpendicular to the wavefront. In this work, the ray model of a light beam is built according to its Fourier angular spectrum, where the positions of rays can be determined by the gradient of the phase of the Fourier angular spectrum. On the other hand, the Fourier angular spectrum of a light beam can be reconstructed through the ray model. Using Fourier angular spectra, we construct the ray model of two typical beams including the Airy beam and the Cusp beam. It is hard to construct ray model directly from the optical field of these beams. In this ray model, the information about ray including direction and position involves the propagation properties of light beams such as self-accelerating. In addition, we demonstrate that the optical field of the focused plane wave can be reconstructed by the ray model in Fourier regime, and the optical field in spatial domain can be obtained by inverse Fourier transform. Simulation results are consistent with the results from Debye’s method. Finally, the high-dimensional ray model of light beams is elaborated in both spatial and spectral regime. Combined with focused plane wave, Airy beam and rays in quadratic gradient-index waveguide, our results show that the ray model actually carries the information about optical field in both spatial and Fourier domain. Actually, the traditional ray model is just a spatial projection of the high-dimensional ray model. Hence, when traditional ray model fails at the focus or caustic lines/surface, it is able to obtain the spectrum of the corresponding optical field from the Fourier domain, and then obtain the field distribution in spatial domain by inverse Fourier transform.