DOE-generated laser beams with given orbital angular moment: application for micromanipulation

TLDR: In this paper, a countable set of linearly independent solutions of the paraxial wave (Schroedinger-type) equation is derived and given the name hyper-geometric modes.

Abstract: A countable set of linearly independent solutions of the paraxial wave (Schroedinger-type) equation is derived and given the name hyper-geometric modes. These solutions describe pure optical vortices that can be generated when a spiral phase plate is illuminated with a plane wave. The distinction between these modes and the familiar paraxial modes is that in propagation the radius of the former increases as a square root of distance and the phase velocity is the same for all modes. In the present work experimental results on trapping and rotation of 5-10 micron-sized biological objects (yeast cells) and polysty rene beads of diameter 5 P m using various laser beams are discussed. Keywords: diffractive optical elements, pure op tical vortices, orbital angular moment, hyper-geometric modes, optical microparticle manipulation 1. INTRODUCTION The higher-order Bessel and Laguerre-Gaussian (LG) modes contain optical vortices providing screw character and presence of orbital angular moment. A microparticle, trapped in such a beam, receives a rotary movement. The new types of laser beams having orbital angular moment - optical vortices "imbedded" in a plane or a Gaussian beam, are considered. After passing some distance, such fields get rather stable configuration, reminding of LG modes, and are distributed under the similar law. In optics, the Hermite-Gauss (HG) and LG modes, which are partial solutions of the paraxial wave equation (PWE) or Schroedinger equation in the Cartesian or cylindrical coordinates, have long been in wide use [1]. They represent the transverse modes of stable laser resonators. Such modes preserve their structure (cross-section intensity distribution), changing only the scale along the propagation axis. Because these modes form an orthogonal basis it is possible to use their linear combinations for constr ucting other solutions of the PWE. In the cylindrical coordinates, the PWE has other modal solutions that, similar to the HG and LG modes, preserve their structure, changing only in scale. These are referred to as paraxial diffracted Bessel modes [2] and should be distinguished from the paraxial diffraction-free Bessel beams [3 ], which will be reffered to as the Durnin-Bessel modes, to distinguish them from the diffracting Bessel modes. As di stinct from the Gaussian mode s, both Bessel modes possess the infinite energy (their intensity being finite at every space point). The effective diameter of the diffracted Bessel beam increases linearly along the optical axis with increa sing distance from the initial plane. The Durnin-Bessel (DB) beam have a constant diameter. Recently introduced [4-8] new modal solutions of the PWE have been studied theoretically [4-7] and experimentally [8]. These are the Ince-Gaussian modes derived as a solution of the PWE in the elliptic coordinates. In these coordinates, the PWE is solved via separation of variab les, with the solution found as a product of the Gaussian function by the Ince polynomials. Note that the Ince pol ynomials are properly a solution of the Whitteker-Hill equation [2]. The Ince-Gaussian (IG) modes represent an orthogonal basis that generalizes the HG and LG modes. When the elliptic coordinates change to cylindrical (the ellipses change to the circumferences) the IG modes change to the LG modes. With the ellipse eccentric ity tending to infinity (the ellipse changing to a line segment), the IG modes change to the HG modes.