Abstract

We have theoretically studied long-period spun l-helicoidal fibers and their ability to generate singular beams from regular ones. On the basis of perturbation theory in the presence of degeneracy, applied to the scalar waveguide equation, we obtained the structure of coupled modes of such fibers and their spectra. It is shown that the coupled modes consist of the fields, which taken separately bear topological charges that differ by l units. We have numerically studied the process of the passage of a Gaussian beam through such a fiber and demonstrated that long-period l-helicoidal fibers have the ability to change—in a certain wavelength range—the topological charge of the incoming Gaussian beam by l units, generating in this way charge-l optical vortex.

© 2012 Optical Society of America

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    [CrossRef]
  3. N. O. Eckerskorn, N. Zeng, V. G. Shvedov, W. Krolikowski, and A. V. Rode, “Effect of polarization on transport of particles in air by optical vortex beam,” J. Opt. 14, 055302 (2012).
    [CrossRef]
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    [CrossRef]
  5. M. Vasnetsov and K. Staliunas, Optical Vortices, Vol. 228 of Horizons of World Physics (Nova Science, 1999).
  6. T. Su, R. P. Scott, S. S. Djordjevic, N. K. Fontaine, D. J. Geisler, X. Cai, and S. J. B. Yoo, “Demonstration of free space coherent optical communication using integrated silicon photonic orbital angular momentum devices,” Opt. Express 20, 9396–9402 (2012).
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  7. F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
    [CrossRef]
  8. M. W. Beijersbergen, L. Allen, H.E.L.O. van der Ween, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. C. D. Poole, C. D. Townsend, and K. T. Nelson, “Helical-grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9, 598–604 (1991).
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  17. C. N. Alexeyev, B. P. Lapin, and M. A. Yavorsky, “The effect of spin–orbit coupling on the structure of the stopband in helical-core optical fibres,” J. Opt. A 10, 085006 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2012 (7)

2011 (2)

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation of optical vortices in layered helical waveguides,” Phys. Rev. A 83, 063820 (2011).
[CrossRef]

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13, 064001 (2011).
[CrossRef]

2008 (3)

C. N. Alexeyev, B. P. Lapin, and M. A. Yavorsky, “Helical core optical fibers maintaining propagation of a solitary optical vortex,” Phys. Rev. A 78, 013813 (2008).
[CrossRef]

C. N. Alexeyev, B. P. Lapin, and M. A. Yavorsky, “The effect of spin–orbit coupling on the structure of the stopband in helical-core optical fibres,” J. Opt. A 10, 085006 (2008).
[CrossRef]

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Narrowband reflective mode converter on the basis of a twisted elliptical core fibre,” J. Opt. A 10, 015301 (2008).
[CrossRef]

2007 (1)

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Multihelix chiral fibre filters of higher-order optical vortices,” J. Opt. A 9, 537–542 (2007).
[CrossRef]

2006 (1)

K. J. Webb, and M.-C. Yang, “Generation and control of optical vortices using left-handed materials,” Phys. Rev. E 74, 016601 (2006).
[CrossRef]

2001 (2)

1994 (1)

M. W. Bejersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

1993 (1)

M. W. Beijersbergen, L. Allen, H.E.L.O. van der Ween, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

1992 (1)

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

1991 (1)

C. D. Poole, C. D. Townsend, and K. T. Nelson, “Helical-grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9, 598–604 (1991).
[CrossRef]

Alexeyev, C. N.

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period twisted elliptical fibers,” Appl. Opt. 51, C193–C197 (2012).
[CrossRef]

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation of optical vortices in layered helical waveguides,” Phys. Rev. A 83, 063820 (2011).
[CrossRef]

C. N. Alexeyev, B. P. Lapin, and M. A. Yavorsky, “Helical core optical fibers maintaining propagation of a solitary optical vortex,” Phys. Rev. A 78, 013813 (2008).
[CrossRef]

C. N. Alexeyev, B. P. Lapin, and M. A. Yavorsky, “The effect of spin–orbit coupling on the structure of the stopband in helical-core optical fibres,” J. Opt. A 10, 085006 (2008).
[CrossRef]

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Narrowband reflective mode converter on the basis of a twisted elliptical core fibre,” J. Opt. A 10, 015301 (2008).
[CrossRef]

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Multihelix chiral fibre filters of higher-order optical vortices,” J. Opt. A 9, 537–542 (2007).
[CrossRef]

Allen, L.

M. W. Beijersbergen, L. Allen, H.E.L.O. van der Ween, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Augustyniak, I.

Barbieri, C.

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H.E.L.O. van der Ween, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Bejersbergen, M. W.

M. W. Bejersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Bianchini, A.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

E. Mari, F. Tamburini, G. A. Swartzlander, A. Bianchini, C. Barbieri, F. Romanato, and B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20, 2445–2451 (2012).
[CrossRef]

Cai, X.

Coerwinkel, R. P. C.

M. W. Bejersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Davydov, A. S.

A. S. Davydov, Quantum Mechanics (Pergamon, 1976).

Djordjevic, S. S.

Drobczynski, S.

Eckerskorn, N. O.

N. O. Eckerskorn, N. Zeng, V. G. Shvedov, W. Krolikowski, and A. V. Rode, “Effect of polarization on transport of particles in air by optical vortex beam,” J. Opt. 14, 055302 (2012).
[CrossRef]

Erdogan, T.

Fadeyeva, T. A.

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period twisted elliptical fibers,” Appl. Opt. 51, C193–C197 (2012).
[CrossRef]

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation of optical vortices in layered helical waveguides,” Phys. Rev. A 83, 063820 (2011).
[CrossRef]

Fontaine, N. K.

Geisler, D. J.

Karimi, E.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13, 064001 (2011).
[CrossRef]

Kristensen, M.

M. W. Bejersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Krolikowski, W.

N. O. Eckerskorn, N. Zeng, V. G. Shvedov, W. Krolikowski, and A. V. Rode, “Effect of polarization on transport of particles in air by optical vortex beam,” J. Opt. 14, 055302 (2012).
[CrossRef]

Krupych, O.

Lapin, B. P.

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period twisted elliptical fibers,” Appl. Opt. 51, C193–C197 (2012).
[CrossRef]

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation of optical vortices in layered helical waveguides,” Phys. Rev. A 83, 063820 (2011).
[CrossRef]

C. N. Alexeyev, B. P. Lapin, and M. A. Yavorsky, “Helical core optical fibers maintaining propagation of a solitary optical vortex,” Phys. Rev. A 78, 013813 (2008).
[CrossRef]

C. N. Alexeyev, B. P. Lapin, and M. A. Yavorsky, “The effect of spin–orbit coupling on the structure of the stopband in helical-core optical fibres,” J. Opt. A 10, 085006 (2008).
[CrossRef]

Lee, K. S.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1985).

Mari, E.

E. Mari, F. Tamburini, G. A. Swartzlander, A. Bianchini, C. Barbieri, F. Romanato, and B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20, 2445–2451 (2012).
[CrossRef]

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

Marrucci, L.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13, 064001 (2011).
[CrossRef]

Masajada, J.

Nagali, E.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13, 064001 (2011).
[CrossRef]

Nelson, K. T.

C. D. Poole, C. D. Townsend, and K. T. Nelson, “Helical-grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9, 598–604 (1991).
[CrossRef]

Piccirillo, B.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13, 064001 (2011).
[CrossRef]

Poole, C. D.

C. D. Poole, C. D. Townsend, and K. T. Nelson, “Helical-grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9, 598–604 (1991).
[CrossRef]

Popiolek-Masajada, A.

Rode, A. V.

N. O. Eckerskorn, N. Zeng, V. G. Shvedov, W. Krolikowski, and A. V. Rode, “Effect of polarization on transport of particles in air by optical vortex beam,” J. Opt. 14, 055302 (2012).
[CrossRef]

Romanato, F.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

E. Mari, F. Tamburini, G. A. Swartzlander, A. Bianchini, C. Barbieri, F. Romanato, and B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20, 2445–2451 (2012).
[CrossRef]

Santamato, E.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13, 064001 (2011).
[CrossRef]

Savaryn, V.

Sciarrino, F.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13, 064001 (2011).
[CrossRef]

Scott, R. P.

Shvedov, V. G.

N. O. Eckerskorn, N. Zeng, V. G. Shvedov, W. Krolikowski, and A. V. Rode, “Effect of polarization on transport of particles in air by optical vortex beam,” J. Opt. 14, 055302 (2012).
[CrossRef]

Skab, I.

Slussarenko, S.

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13, 064001 (2011).
[CrossRef]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1985).

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Sponselli, A.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

Staliunas, K.

M. Vasnetsov and K. Staliunas, Optical Vortices, Vol. 228 of Horizons of World Physics (Nova Science, 1999).

Su, T.

Swartzlander, G. A.

Tamburini, F.

E. Mari, F. Tamburini, G. A. Swartzlander, A. Bianchini, C. Barbieri, F. Romanato, and B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20, 2445–2451 (2012).
[CrossRef]

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

Thidé, B.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

E. Mari, F. Tamburini, G. A. Swartzlander, A. Bianchini, C. Barbieri, F. Romanato, and B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20, 2445–2451 (2012).
[CrossRef]

Townsend, C. D.

C. D. Poole, C. D. Townsend, and K. T. Nelson, “Helical-grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9, 598–604 (1991).
[CrossRef]

van der Ween, H.E.L.O.

M. W. Beijersbergen, L. Allen, H.E.L.O. van der Ween, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Vasnetsov, M.

M. Vasnetsov and K. Staliunas, Optical Vortices, Vol. 228 of Horizons of World Physics (Nova Science, 1999).

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Vasylkiv, Y.

Vlokh, R.

Volyar, A. V.

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Narrowband reflective mode converter on the basis of a twisted elliptical core fibre,” J. Opt. A 10, 015301 (2008).
[CrossRef]

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Multihelix chiral fibre filters of higher-order optical vortices,” J. Opt. A 9, 537–542 (2007).
[CrossRef]

Webb, K. J.

K. J. Webb, and M.-C. Yang, “Generation and control of optical vortices using left-handed materials,” Phys. Rev. E 74, 016601 (2006).
[CrossRef]

Woerdman, J. P.

M. W. Bejersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H.E.L.O. van der Ween, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Yang, M.-C.

K. J. Webb, and M.-C. Yang, “Generation and control of optical vortices using left-handed materials,” Phys. Rev. E 74, 016601 (2006).
[CrossRef]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984).

Yavorsky, M. A.

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period twisted elliptical fibers,” Appl. Opt. 51, C193–C197 (2012).
[CrossRef]

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation of optical vortices in layered helical waveguides,” Phys. Rev. A 83, 063820 (2011).
[CrossRef]

C. N. Alexeyev, B. P. Lapin, and M. A. Yavorsky, “Helical core optical fibers maintaining propagation of a solitary optical vortex,” Phys. Rev. A 78, 013813 (2008).
[CrossRef]

C. N. Alexeyev, B. P. Lapin, and M. A. Yavorsky, “The effect of spin–orbit coupling on the structure of the stopband in helical-core optical fibres,” J. Opt. A 10, 085006 (2008).
[CrossRef]

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Narrowband reflective mode converter on the basis of a twisted elliptical core fibre,” J. Opt. A 10, 015301 (2008).
[CrossRef]

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Multihelix chiral fibre filters of higher-order optical vortices,” J. Opt. A 9, 537–542 (2007).
[CrossRef]

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984).

Yoo, S. J. B.

Zeng, N.

N. O. Eckerskorn, N. Zeng, V. G. Shvedov, W. Krolikowski, and A. V. Rode, “Effect of polarization on transport of particles in air by optical vortex beam,” J. Opt. 14, 055302 (2012).
[CrossRef]

Appl. Opt. (3)

J. Lightwave Technol. (1)

C. D. Poole, C. D. Townsend, and K. T. Nelson, “Helical-grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9, 598–604 (1991).
[CrossRef]

J. Mod. Opt. (1)

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

J. Opt. (2)

N. O. Eckerskorn, N. Zeng, V. G. Shvedov, W. Krolikowski, and A. V. Rode, “Effect of polarization on transport of particles in air by optical vortex beam,” J. Opt. 14, 055302 (2012).
[CrossRef]

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13, 064001 (2011).
[CrossRef]

J. Opt. A (3)

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Multihelix chiral fibre filters of higher-order optical vortices,” J. Opt. A 9, 537–542 (2007).
[CrossRef]

C. N. Alexeyev, B. P. Lapin, and M. A. Yavorsky, “The effect of spin–orbit coupling on the structure of the stopband in helical-core optical fibres,” J. Opt. A 10, 085006 (2008).
[CrossRef]

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Narrowband reflective mode converter on the basis of a twisted elliptical core fibre,” J. Opt. A 10, 015301 (2008).
[CrossRef]

J. Opt. Soc. Am. A (1)

New J. Phys. (1)

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[CrossRef]

Opt. Commun. (2)

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Opt. Express (2)

Phys. Rev. A (2)

C. N. Alexeyev, T. A. Fadeyeva, B. P. Lapin, and M. A. Yavorsky, “Generation of optical vortices in layered helical waveguides,” Phys. Rev. A 83, 063820 (2011).
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Phys. Rev. E (1)

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Figures (4)

Fig. 1.
Fig. 1.

Scheme of generation of the optical vortex (OV) from the incident Gaussian beam (GB) by a multihelicoidal (l=4) optical fiber.

Fig. 2.
Fig. 2.

Zero-approximation spectra of multihelicoidal (l=3) fiber modes vs lattice vector q. The type of the mode is indicated at the corresponding curve. Insets show repulsion of spectral branches due to the effect of mode coupling.

Fig. 3.
Fig. 3.

Transmission coefficients |Pi|2 for the outcoming modes (a) |1,0 and (b) |1,3 vs wavelength of the incoming field |1,0; d=d0=0.003m, Δ=103, δ=5·102, r0=10λ0, λ0=6.328·107m, H=7.4·104m. Insets show superfine structure due to interference effects.

Fig. 4.
Fig. 4.

Transmission coefficients for the outcoming mode |1,3 vs wavelength of the incoming field |1,0; the fiber’s length dd40=0.24673m. Other parameters are the same as in Fig. 3.

Equations (25)

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n2(r,φ,z)=nco2(12Δf(r))2nco2Δδrfrcos(l(φqz)),
2Et+k2n2Et=0,
{2r˜2+1r˜r˜+1r˜22φ˜2+(iβqφ˜)2+k2n˜22k2nco2r˜Δδfr˜cos(lφ˜)}et(r˜,φ˜)=0.
(H^0+V^)|e=β2|e,
H^0=(2r˜2+1r˜r˜+1r˜22φ˜2+k2n˜22iβqφ˜+q22φ˜2),
V^=2k2nco2r˜Δδfr˜cos(lφ˜).
|σ,m=(1iσ)exp(imφ˜)Fm(r),
(2r2+1rr+k2n˜2m2r2β˜m2)Fm(r)=0.
βm(1,2)=±β˜m+mq.
|mm|=l,σ=σ.
V12=V21=σ,0|V^|σ,l=k2nco2ΔδN0NlA,
(β˜02β2AAβ˜l2(β±lq)2)xa=0,
β1,2=±β˜0,β3,4=±β˜l+lq,β5,6=±β˜llq.
a(q0,β˜0),b(q0,β˜0),
β=β˜0+δa,q=q0+ε,β=β˜0+δb,
(2β˜0δaAA2β˜l(δa+lε))xa=0,
(2β˜0δbAA2β˜l(δb+lε))xb=0.
δ1,2a=0.5(lε±l2ε2+Q2),δ1,2b=0.5(lε±l2ε2+Q2),
|ψ1a={c1|1,0ei(β˜0+0.5lε)z+c2|1,lei(β˜l0.5lε)z}exp(iz2l2ε2+Q2),|ψ2a={c2|1,0ei(β˜0+0.5lε)z+c1|1,lei(β˜l0.5lε)z}exp(iz2l2ε2+Q2),
|ψ1b={c2|1,0ei(β˜0+0.5lε)z+c1|1,lei(β˜l0.5lε)z}exp(iz2l2ε2+Q2),|ψ2b={c1|1,0ei(β˜0+0.5lε)z+c2|1,lei(β˜l0.5lε)z}exp(iz2l2ε2+Q2).
|Φ1(z0)=|1,0eikz+(R1|1,0+R2|1,l+R3|1,l)eikz.
|Φ2(0<z<d)=T1|ψ1a+T2|ψ2a+T3|ψ1b+T4|ψ2b+T5|1,leiβ˜lz+T6|1,leiβ˜lz,
|Φ3(zd)=(P1|1,0+P2|1,l+P3|1,l)eik(zd).
dm=π(2m+1)Q,
Δn22nco2Δδf˜rl=1(alcoslφ+blsinlφ),

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