0021-3640/05/8111- $26.00 © 2005 Pleiades Publishing, Inc. 0567 JETP Letters, Vol. 81, No. 11, 2005, pp. 567–570. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 81, No. 11, 2005, pp. 699–702. Original Russian Text Copyright © 2005 by Vasnetsov, Pas’ko, Soskin. The vibrational frequency in a monochromatic opti- cal beam, e.g., the lowest transverse mode of laser radi- ation (Gaussian beam), is independent of the spatial coordinate in the beam cross section. However, if the beam is set in rotation by means of a deflecting element (in this case, the beam moves in space over a conical or cylindrical surface), the optical frequency is split into a symmetric spectrum due to the rotational Doppler effect (RDE) [1]. The splitting between the neighboring spectral components is equal to the rotational frequency Ω of the deflecting element; in other words, compo- nents with frequencies ϖ ± Ω , ±2Ω , ±3Ω , … appear in the spectrum around the optical frequency ϖ . The RDE is associated with the existence of the orbital angular momentum (OAM) [2] for beams with a helicoidal wavefront, for which the phase is expressed in terms of the azimuth angle ϕ as mϕ, where m is an integer (orbital number) [3]. The OAM per photon has a quan- tized value m [4]. In the optical range, the RDE was detected when a Gaussian beam was transformed into a beam with an analogous phase dependence on a rotat- ing spiral zone plate [5]. Since the beam deflected from the rotational axis can be represented in the form of a superposition of axial azimuthal harmonics, each of these harmonics acquires the corresponding frequency shift as a result of beam rotation. Azimuthal harmonics are solutions of the scalar wave equation in the paraxial approximation (e.g., in the form of the Laguerre–Gauss (LG) modes). A pecu- liar feature of the RDE spectrum is that its shape is determined by the radial coordinate measured from the rotational axis, because each harmonic has its own radial amplitude distribution, and its contribution to the spectrum being measured is determined by the radial position of the measurement point [6]. In this work, we consider the azimuthal dependence of the RDE spectrum of a rotating displaced beam. The beam under study is taken in the form of a superposi- tion of two LG modes with the initially nonzero OAM of one of the components. It should be noted that the RDE spectrum of such a beam (but without displace- ment from the rotational axis) was measured experi- mentally in [6]. Beam rotation is shown schematically in Fig. 1. The beam is displaced parallel to itself when it passes through an inclined transparent plane-parallel plate. The rotation of the plate about the axis of the incident beam leads to the motion of the transmitted beam over a cylindrical surface. An analogous scheme is shown for a version with beam reflection in an optical element that displaces and rotates the beam. The parallel displacement of the beam (LG mode) in its constriction can be written in the form of the trans- formation (1) where E LG is the amplitude parameter of the mode, l is the azimuthal mode index, w 0 is the beam dimension in the waist, and x 0 and y 0 are the coordinates of the beam axis displacement. Expanding the exponential term in polar coordinates (ρ, ϕ), we obtain (2) Exy , ( 29 E LG x x 0 – ( 29 iy y 0 – ( 29 + w 0 -------------------------------------------- l = × x x 0 – ( 29 2 y y 0 – ( 29 2 + w 0 2 ----------------------------------------------- – , exp E ρϕ , ( 29 E LG w 0 l -------- ρ e i ϕ δ e i θ – ( 29 l ρ 2 δ 2 + w 0 2 ---------------- –       exp = × I m 2 ρδ w 0 2 ---------     im ϕ θ – ( 29 [ ] , exp m ∞ – = ∞ ∑ Spatial Dependence of the Frequency Spectrum of a Rotating Optical Beam M. V. Vasnetsov*, V. A. Pas’ko, and M. S. Soskin Institute of Physics, National Academy of Sciences of Ukraine, Kiev, 03028 Ukraine * e-mail: mvas@iop.kiev.ua Received April 25, 2005 A rotating optical beam displaced relative to the rotational axis becomes polychromatic due to the rotational Doppler effect. The case where the initial beam has the form of a superposition of two Laguerre–Gauss modes and carries an elementary image in the form of an asymmetric intensity distribution is considered. The spatial distribution of the monochromatic components in the beam cross section is determined. © 2005 Pleiades Publishing, Inc. PACS numbers: 42.15.Dp, 42.25.Fx