Broadband Vortex Waves Generation by All-Dielectric Vogel Metasurface

Optical vortices are electromagnetic waves with a helical wavefront and associated orbital angular momentum. They have diverse applications in optics, such as ultrafast singular optics, quantum optics, microscopy, as well as optical trapping [1]. These applications require compact, easily integrated and high-performance devices, so the development of highly efficient broadband single-layer structures for vortex generation and control is an extremely relevant research topic. In this report we demonstrate direct generation of high-quality vortex waves with azimuthal order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$l=-2$</tex> by a transmissive aperiodic metasurface based on a Vogel spiral array, which in the future can be easily integrated into fiber or laser facets. The Vogel metasurface composed of an aperiodic array of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> subwavelength TiO<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> nanopillars which are placed on the isotropic substrate (air-like material). All nanopillars forming the metasurface are arranged in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x-y$</tex> plane and have the length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$a$</tex>, width <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$b$</tex>, and height <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$h$</tex> as shown in Fig. 1(a). Coordinates <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(x_{k},y_{k})$</tex> of the center of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> -th <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(k=1,2,\ \ldots,N)$</tex> nanopillar in the Vogel structure can be obtained as follow <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x_k=a_0 \sqrt{ k \cos (k \alpha)}, y_k=a_0 \sqrt{ k \sin (k \alpha)}$</tex> [2]. Here ao is a constant scaling parameter, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex> is an irrational number known as the divergence angle, which determines the constant aperture between adjacent nanopillars of the spiral array. In this paper, we focus on the optical properties of an aperiodic metasurfaces based on GA Vogel spiral array. The phase distribution <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\varphi(x_{k},y_{k})$</tex> of the transmissive metasurface for generating focused OAM mode with the topological charge <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$l=-2$</tex> is obtained in the standard way [3]. This phase profile is implemented in the metasurface design by rotation of each nanopillars at a given coordinate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(x_{k},y_{k})$</tex> by an angle <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\theta\left(x_k, y_k\right)=-\varphi\left(x_k, y_k\right) / 2$</tex>, that correspond to the left-handed circularly polarized incident wave.