International Journal of Optics

Volume 2017, Article ID 6852019, 10 pages

https://doi.org/10.1155/2017/6852019

## Generation of Perfect Optical Vortices by Using a Transmission Liquid Crystal Spatial Light Modulator

GOTS, Grupo de Óptica y Tratamiento de Señales, Escuela de Física, Universidad Industrial de Santander, Carrera 27 Calle 9, A.A. 678, 680002 Bucaramanga, Colombia

Correspondence should be addressed to Cristian H. Acevedo; moc.oohay@dvr_naitsirc

Received 21 November 2016; Revised 22 January 2017; Accepted 20 February 2017; Published 30 March 2017

Academic Editor: Stefan Wabnitz

Copyright © 2017 Nelson Anaya Carvajal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We have experimentally created perfect optical vortices by the Fourier transformation of holographic masks with combination of axicons and spiral functions, which are displayed on a transmission liquid crystal spatial light modulator. We showed theoretically that the size of the annular vortex in the Fourier plane is independent of the spiral phase topological charge but it is dependent on the axicon. We also studied numerically and experimentally the free space diffraction of a perfect optical vortex after the Fourier back plane and we found that the size of the intensity pattern of a perfect optical vortex depends on the topological charge and the propagation distance.

#### 1. Introduction

As is well known, an optical vortex beam is an electromagnetic wave with a helical wavefront due to phase singularities [1]. These phase singularities are threads of darkness embedded within light fields in their spatial distribution, points for 2D, and lines for 3D [2, 3]. Allen and his collaborators proved that the complex amplitude of an optical vortex possessing an azimuthal phase factor exp () carries an orbital angular momentum of m, where is the topological charge and is the azimuthal angle [4]. The unique optical properties of the optical vortices have been widely used in applications such as optical tweezers [5–8], image processing [9–11], communication systems in free space [12–14], and optical fibers [15, 16]. Motivated by these applications several methods for generating the optical vortex beam have been proposed [17–27]; however the diameter of these optical vortices is related to their topological charges. This property causes difficulties to achieve a high spatial accuracy and high orbital angular momentum coupling optical vortices into a fiber.

To solve these requirements, Ostrovsk et al. have introduced the perfect optical vortex (POV) concept [28]. The perfect optical vortices are electromagnetic waves whose ring-width size and average ring-diameter (the arithmetic average of the inner and outer ring-diameters) are both independent of the topological charge. To experimentally generate the POVs or POV array a Gaussian beam (or a wave plane) is directed toward a special phase mask [28, 29], an axicon [30], or a phase mask by combining an axicon and a spiral phase function [31, 32]. The modulation of these phase masks is programmed onto a reflection liquid crystal spatial light modulator (RLC-SLM) working in phase only mode. The masks are created with the discretization of the phase into levels depending on the characteristics of the RLC-SLM. The most of RLC-SLMs operate with an 8-bit dynamic range or 256 phase levels.

In this work, we present an experimental approach for generating perfect optical vortices by using of phase masks of three levels with shape of axicon and spiral phase functions, which are displayed in a transmission liquid crystal spatial light modulator. The intensity and vorticity of the POVs are measured with a CMOS camera and the far-field diffraction pattern through an equilateral triangular slit, respectively. We showed numerically and experimentally that the free space propagation diffraction patterns of the POVs are dependent on the distance and they are variant to ring-width size and to average ring-diameter.

#### 2. Theory

As mentioned earlier, perfect optical vortices are beams whose diameter and width-ring are independent of its topological charge. POVs can be approximately generated by means of the Fourier transformation from Bessel-Gauss (BG) beams [31]. The complex field amplitude of a BG beam with amplitude unit can be described in cylindrical coordinates , as [33]where is the waist width of the Gaussian beam, is the radial wave vector, and is an th order Bessel function of first kind.

The field of a POV is obtained in the back focal plane of a converging lens by substituting (1) in the Fourier transform diffraction spectrum yielding the resultwhere is the back focal length of the lens. By analytic solving of the integral in (2), we can obtain the Fourier diffraction spectrum for BG beams with different topological charges at the back focal plane of lens, which reads with and , the radius and half width-ring of the perfect optical vortices, respectively. is an th order modified Bessel function of first kind and it can be written as [34]The radius in (3) can be expressed approximately aswhere we had considered the experimental fact that (axicon period). It can see from (5) that the average radius of a POV is independent of the topological charge of the spiral phase and it basically governed Fourier transform of the axicon [35, 36]. The columns three and four of Figure 1 provide simulations of the theoretical results for the intensity distribution of POVs with topological charges , , , , and , obtained by Fourier transform diffraction of BG beams with axicon period [mm] and [mm], respectively. From Figure 1, we can observe that the diameters of BG beams increase with the topological charge (first column) however; the diameters of POVs apparently do not change (third and fourth column). Also in Figure 1, one can see that the radii of the POVs of the fourth column are bigger than the radii of the POVs of the third column. Last result can be easily explained by decrease of the axicon period in (5) because is inverse to The phase profiles of the BG beams on first column of Figure 1 are shown in second column of the same figure, while for the POVs on third and fourth column in Figure 1, the phase structures are equal and they are shown on fifth column of Figure 1. It is clear from each phase profile that the number of central dislocations is equal to the value of for each used BG beam and each generated POV.