Simferopol State University. Submitted December 16, 1996. Pis'ma Zh. Tekh. Fiz. 23, 76â81 August 26, 1997. A theoretical and experimental analysis is made of ...
Vortex optical Magnus effect in multimode fibers V. V. Butkovskaya, A. V. Volyar, and T. A. Fadeeva Simferopol State University
~Submitted December 16, 1996! Pis’ma Zh. Tekh. Fiz. 23, 76–81 ~August 26, 1997! A theoretical and experimental analysis is made of the optical Magnus effect in multimode optical fibers excited by a laser beam whose wavefront has a pure screw dislocation and carries the topological charge 6l, where l is the azimuthal quantum number. It is found that the angular rotation of the plane of propagation of a local wave depends on the magnitude and sign of the topological charge and changes qualitatively when the circulation of the polarization is reversed. The phase mechanism is attributed to spin-orbit interaction in the photon ensemble. It is demonstrated experimentally that the optical Magnus effect in a few-mode fiber for the CP11 mode at the beat length is observed as a rotation of the axis of the pure edge dislocation field through an angle proportional to the beat length. © 1997 American Institute of Physics. @S1063-7850~97!03108-X#
The propagation of a polarized meridional local plane wave through a multimode fiber is associated with rotation of the plane of propagation of the wave about the axis of symmetry of the fiber. In optical fibers with a parabolic refractive index profile, which conserve the polarization state of the wave, the plane of propagation of the wave undergoes a circular rotation, which has been described as the optical Magnus effect.1 Optical fibers with a stepped refractive index profile do not conserve the polarization state of the wave and thus the plane of propagation of a local wave oscillates about some arbitrary equilibrium state.2,3 It is known that the optical vortices of an electromagnetic wave carry an angular momentum additional to the wave spin.4 It is expected that the specific rotation of the plane of polarization of the wave carrying the topological charge will respond to the magnitude and sign of the topological charge l of the vortex of the exciting beam. Here we report an experimental investigation of the rotation of the wave caustic in a multimode optical fiber as a function of the magnitude and sign of the topological charge l and we also study the characteristic features of the optical Magnus effect in few-mode graded-index fibers. 1. An experimental investigation was made of the specific rotation ¸ of the wave caustic formed at the exit end of a multimode fiber as a function of the magnitude and sign of the topological charge l of a pure screw dislocation of the exciting-beam wavefront. The experimental apparatus used in Ref. 2 was used as the basis for this study. Linearly polarized laser radiation was passed through a optical polarization modulator, after which the polarization state varied from linearly polarized to right circularly polarized at frequencies between 0.1 and 10 Hz. The modulated laser beam was incident on a phase transparency with a computer-generated hologram of a screw wavefront dislocation, having the topological charge l. The computer-generated holograms were created by the technique described in Ref. 5. After the hologram, the light was focused onto the entry end of the fiber by a 203 microscope objective such that predominantly meridional rays propagated in the fiber. The sample used was a straight multimode fiber with a stepped refractive index pro649
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file, having diameter D50.5 mm, length l57 cm, and numerical aperture A50.65. The near-field pattern of the exit end of the fiber was projected into the plane of the screen by an f 53 cm short-focus lens. The meridional excitation of the fiber was tuned using the pattern of the wave caustics. The angular displacement ¸ of an arbitrary wave caustic was measured for two successive polarization states of the light after the modulator. For given l the angular displacement ¸ was then averaged for different meridional caustics. The results of the measurements are plotted in Fig. 1 as a family of curves giving the specific angular displacement ¸ as a function of the magnitude and sign of the topological charge l of an optical vortex of the exciting laser beam. For the same directions of s and l, the angular displacement ¸ increases with increasing modulus u l u of the topological charge, whereas for opposite directions of s and l, the angular displacement ¸ decreases monotonically with increasing u l u . A simultaneous reversal of the signs of s and l does not alter the nature of the curve ¸ (l) and merely corresponds to an angular rotation of ¸ (l) about the origin. The angular displacement ¸ for a Gaussian beam carrying no optical vortex (l50) is typically less than that for the topological charge u l u 51. The behavior of the angular displacement ¸ of the caustics in a multimode fiber as a function of the topological charge of the exciting optical field is treated physically in terms of the vector nature of the spin s and orbital angular momentum l of the optical vortex. The field propagating in the fiber carries the z component of the total angular momentum, which is defined4 per photon as L z 5(l s )/ v ~here s 561 is the spin number!. As the topological charge increases, the intrinsic angular momentum of the photon obviously increases. This angular momentum is added to the spin angular momentum with allowance for the sign of the circulation of the electric vector and causes additional twisting of the caustics. This explains the different branches of the curves in Fig. 1 for the different signs of l and s . We did not obtain a linear relation between the angular rotations ¸ and the azimuthal number l. This is evidently because of the
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© 1997 American Institute of Physics
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FIG. 2. Photographs of the radiation field of the circularly polarized CP11 mode at half the beat length: a — excited by right circularly polarized light, b — excited by left circularly polarized light. FIG. 1. Angular displacement ¸ versus topological charge l of vortex ( s is the spin number and l is the azimuthal quantum number of the vortex!.
nonuniform distribution of the angular momentum over the fiber cross section,4 since the field in the fiber cannot be represented as a paraxial beam and a cross term appears in the description of the angular momentum of the field. 2. An experimental investigation was made of the optical Magnus effect in a few-mode fiber excited by a smooth Gaussian beam. We selected a stepped-index fiber having a core diameter D53.5 m m, which supported the HE11 , HE21 , TE01 and TM01 modes. The maximum measured beat length was 3.8 m. The fiber was excited by radiation which had been passed through a phase mask with the profile of the LP11 mode to suppress the excitation of the HE11 mode as far as possible. The fiber was broken off 2 cm at a time until the pattern of the radiation field reproduced the field of the LP11 mode turned through an angle of 38° ~Fig. 2a!. Then the polarization state was changed to left circularly polarized by means of an electrooptic modulator. As a result, the radiation field pattern of the mode was rotated through 242° ~Fig. 2b!. In a few-mode fiber excited by a smooth wave, pure edge and pure screw disclinations of the vector field are superposed to form optical vortices. It can be shown that the field of a pure edge dislocation of the circularly polarized CP11 mode at the beat length D b z 0 5m p ~where D b 52 d b 1 /2, d b 1 is the polarization correction to the HE11 mode! has the form: 650
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S D
1 E⇒cos~ f 2 s z d b 21z 0 ! s i . z Here the ‘‘1’’ and ‘‘-’’ signs correspond to excitation of the fiber by right and left circularly polarized light, respectively. Reversal of the direction of circulation of the circular polarization rotates the axis of the pure edge dislocation through the angle D f 52 d b z 0 . These results show that the optical Magnus effect in a fiber is determined by the polarization correction to the HE21 mode. This work was partially supported by the International Soros Program for the Support of Education in the Exact Sciences ~ISSEP!, Grant N PSU062108.
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A. V. Dugin, B. Ya. Zel’dovich, N. D. Kundikova, and V. S. Liberman, Zh. E´ksp. Teor. Fiz. 100, 1474 ~1991! @Sov. Phys. JETP 73, 816 ~1991!#. 2 A. V. Volyar and S. N. Lapaeva, Pis’ma Zh. Tekh. Fiz. 18~8!, 53 ~1992! @Sov. Tech. Phys. Lett. 18, 261 ~1992!#. 3 A. V. Volyar, S. N. Lapaeva, and Yu. N. Mitsa, Pis’ma Zh. Tekh. Fiz. 20~5!, 32 ~1994! @Tech. Phys. Lett. 20, 190 ~1994!#. 4 B. Zeldovich, Phys. Rev. A 5, 7980 ~1991! @sic#. 5 V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, J. Mod. Opt. 39, 985 ~1992!. Translated by R. M. Durham Butkovskaya et al.
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