Abstract. The Gaussian beam diffraction by a thin amplitude grating with a dislocation shifted relative to the beam axis is considered. The érst-order diffracted beam with an off-axis optical vortex is represented as a superposition of Laguerre ë Gauss modes. The optical scheme permitting the spatial sepa- ration of modes with even and odd mode indices is proposed and realised experimentally. The method of separation is based on the difference in Gouy phase shifts for focused beams. The possibility of separating photons with zero and nonzero orbi- tal angular momentum is discussed. Keywords: optical vortices, Laguerre ë Gauss modes, Gouy phase shift, photon orbital angular momentum Physical optics has been recently enriched by the concept of optical vortex (OV) [1] used for describing the structure of a wave éeld with phase defects (wave-front dislocations) [2]. The interest in OVs is due to their peculiar properties as well as possible applications in problems of manipulation with microparticles [3]. A distinguishing feature of an OV is that the éeld amplitude vanishes at its axis, while the phase becomes indeénite, or singular due to a jump by p or mp in the case of an m-fold vortex (a review of the OV properties is given in [4, 5]). If the OV axis coincides with the beam axis (as, for example, for Laguerre ë Gauss modes with a non- zero azimuthal index), the integer value of m is called the topological charge. The structure of beams with OVs can be determined from the expression for the Laguerre ë Gauss modes LG l  p (in the cylindrical coordinates r, j, z): ELG l p  E LG w 0 w   2 p r w  j l j exp   r 2 w 2  L j l j p  2r 2 w 2   exp  i  kz  kr 2 2Rz  lj  Q arctan  z z R  , (1) where E LG is the amplitude parameter; w 0 is transverse size of the beam waist; w  w 0 (1  z 2 =z 2 R ) 1=2 is the transverse size of the beam at a distance z from the waist; R(z)  z(1  z 2 R =z 2 ) is the radius of curvature of the wave front at the beam axis; z R  kw 2 0 =2 is the Rayleigh length; k is the wave number; L j l j p is the associated Laguerre polynomial; l is the azimuthal mode index; p is the radial mode index [6]. For beams with a nonzero index l, the amplitude vanishes at the beam axis, while the phase is an exponential function of azimuth, exp (ilj), which corresponds to an OV with the topological charge m  l. The number Q  2p j l j 1 is known as the mode index determining the aféliation to a family of modes with the same Q. The mode index deter- mines the small correction to the phase velocity of the mode, associated with the Gouy phase shift arctan (z=z R ). For a Gaussian beam (p; l  0) Q  1 and the additional phase incursion (relative to a plane wave) over the distance between the waist and the far-éeld zone due to the Gouy phase shift is  p=2. The `doughnut' mode LG 1 0 ( p  0, l  1) has the index Q  2, and the corresponding addi- tional phase incursion is equal to p. The expression for the 3D equiphase surface (wave front) of the LG l  0 mode at the waist can be obtained from Eqn (1) putting R(z) !1 and z 5 z R : kz  lj  const. (2) The equation for the wave front (2) describes a helicoidal surface with a step equal to ll (l is the wavelength). As the wave front leaves the waist, its shape remains helicoidal with a singularity (phase discontinuity of lp) at the axis. Such a wave-front structure leads to the interference fringe splitting (emergence of l new fringes) during the interference of the beam containing OVs with a plane wave, which is the main indication of the presence of OVs in interference detection [7]. Another important consequence is the presence of the orbital angular momentum of the beam [8] due to the circulation of the optical êux around the OV axis. The orbital angular momentum L z for an axisymmetric OV beam with energy W, frequency o, and topological charge m is deéned as [8] L z  mW o , (3) which gives the value m h when recalculated per photon. The quantised value of the orbital angular momentum of a photon either can be random or may reêect the physical reality of the existence of the corresponding orbital quantum number. At the present time, there are arguments in favour of the quantum origin of the orbital angular momentum as well as of a purely classical description [9]. For instance, the coaxial interference of Laguerre ë Gauss modes leads to a M V Vasnetsov, V V Slyusar, M S Soskin Institute of Physics, National Academy of Science of Ukraine, prosp. Nauki 46, 03039 Kyiv, Ukraine; e-mail: mvas@iop.kiev.ua Received 30 August 2000; revision received 31 January 2001 Kvantovaya Elektronika 31 (5) 464 ë 466 (2001) Translated by Ram Wadhwa PACS numbers: 42.25.Fx DOI:10.1070/QE2001v031n05ABEH001980 Mode separator for a beam with an off-axis optical vortex M V Vasnetsov, V V Slyusar, M S Soskin 612/952 ë MB ë 27/vii-01 ë SVERKA ë 3 ÒÑÎÑÔ ÍÑÏÒ. å 3 Quantum Electronics 31 (5) 464 ë 466 (2001) ß2001 Kvantovaya Elektronika and Turpion Ltd