Abstract

We considered the propagation of Bessel beams through the biaxially induced birefringent crystal implemented by the mechanical torsion of the uniaxial crystal around its optical axis. Analyzing the solutions to the wave equation in the form of eigenmodes, we found that the system enables us to convert the beams with a uniform distribution of the linear polarization at the beam cross section into radially, azimuthally, and spirally polarized beams. Moreover, we revealed that the above system permits us to convert the beams with the space-variant linear polarization in accordance with the rule ss+1, where s is the topological index of the centered polarization singularity.

© 2012 Optical Society of America

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  1. I. Skab, Y. Vasylkiv, V. Savaryn, and R. Vlokh, “Optical anisotropy induced by torsion stresses in LiNbO3 crystals: appearing of optical vortex,” J. Opt. Soc. Am. A 28, 633–640 (2011).
    [CrossRef]
  2. I. Skab, Y. Vasylkiv, V. Savaryn, and R. Vlokh, “Relations for optical indicatrix parameters in the condition of crystal torsion,” Ukr. J. Phys. Opt. 11, 193–240 (2010).
    [CrossRef]
  3. I. Skab, Y. Vasylkiv, B. Zapeka, V. Savaryn, and R. Vlokh, “Appearance of singularities of optical fields under torsion of crystals containing threefold symmetry axes,” J. Opt. Soc. Am. A 28, 1331–1340 (2011).
    [CrossRef]
  4. Y. Vasylkiv, I. Skab, and R. Vlokh, “Measurements of piezooptic coefficients π14 and π25 in Pb5Ge3O11 crystals using torsion induced optical vortex,” Ukr. J. Phys. Opt. 12, 101–108 (2011).
    [CrossRef]
  5. Y. Vasylkiv, V. Savaryn, I. Smaga, I. Skab, and R. Vlokh, “On determination of sign of piezo-optic coefficients using torsion method,” Appl. Opt. 50, 2512–2518 (2011).
    [CrossRef]
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    [CrossRef]
  8. C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express 17, 12891–12899 (2009).
    [CrossRef]
  9. R. Vlokh, A. Volyar, O. Mys, and O. Krupych, “Appearance of optical vortex at conical refraction. Examples of NANO2 and YFeO3 crystals,” Ukr. J. Phys. Opt. 4, 90–93 (2003).
    [CrossRef]
  10. N. S. Kazak, N. A. Khio, and A. A. Ryzhevich, “Generation of Bessel light beams under the condition of internal conical refraction,” Quantum Electron. 29, 1020–1024 (1999).
    [CrossRef]
  11. V. Belyi, T. King, N. Kazak, N. Khio, E. Katranji, and A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE 4403, 229–240(2001).
    [CrossRef]
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  13. M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” Prog. Opt. 50, 13–50 (2007).
    [CrossRef]
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  15. M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt. 42, 219–276 (2001).
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  19. T. Fadeyeva and A. Volyar, “Nondiffracting vortex-beams in a birefringent chiral crystal,” J. Opt. Soc. Am. A 27, 13–20 (2010).
    [CrossRef]
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    [CrossRef]
  21. T. Fadeyeva, V. Shvedov, N. Shostka, C. Alexeyev, and A. Volyar, “Natural shaping of the cylindrically polarized beams,” Opt. Lett. 35, 3787–3789 (2010).
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    [CrossRef]
  28. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
  29. M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A 7, 685–690 (2005).
    [CrossRef]
  30. A. V. Volyar, V. Z. Zhilaitis, and V. G. Shvedov, “Optical eddies in small-mode fibers: II. The spin-orbit interaction,” Opt. Spectrosc. 86, 593–598 (1999).
    [CrossRef]
  31. C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A 6, S162–S165 (2004).
    [CrossRef]
  32. A. Yariv and P. Yuh, Optical Waves in Crystals (Wiley-Interscience, 1984).
  33. A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman & Hall, 1983).
  34. L. Marrucci, E. Karinni, S. Slussarenko, B. Piecirillo, 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]
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    [CrossRef]

2011 (6)

2010 (4)

2009 (2)

T. Fadeyeva, A. Rubass, and A. Volyar, “Transverse shift of high-order paraxial vortex-beam induced by a homogeneous anisotropic medium,” Phys. Rev. A 79, 05381(2009).
[CrossRef]

C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express 17, 12891–12899 (2009).
[CrossRef]

2007 (1)

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” Prog. Opt. 50, 13–50 (2007).
[CrossRef]

2006 (2)

2005 (1)

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A 7, 685–690 (2005).
[CrossRef]

2004 (4)

2003 (2)

R. Vlokh, A. Volyar, O. Mys, and O. Krupych, “Appearance of optical vortex at conical refraction. Examples of NANO2 and YFeO3 crystals,” Ukr. J. Phys. Opt. 4, 90–93 (2003).
[CrossRef]

A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Formation of linearly polarized light with axial symmetry by use of space-variant subwavelength gratings,” Opt. Lett. 28, 510–512 (2003).
[CrossRef]

2002 (1)

K. Volke-Sepulveda, V. Garres-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
[CrossRef]

2001 (2)

V. Belyi, T. King, N. Kazak, N. Khio, E. Katranji, and A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE 4403, 229–240(2001).
[CrossRef]

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

1999 (3)

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun. 167, 1–5 (1999).
[CrossRef]

N. S. Kazak, N. A. Khio, and A. A. Ryzhevich, “Generation of Bessel light beams under the condition of internal conical refraction,” Quantum Electron. 29, 1020–1024 (1999).
[CrossRef]

A. V. Volyar, V. Z. Zhilaitis, and V. G. Shvedov, “Optical eddies in small-mode fibers: II. The spin-orbit interaction,” Opt. Spectrosc. 86, 593–598 (1999).
[CrossRef]

Alexeyev, C.

T. Fadeyeva, V. Shvedov, N. Shostka, C. Alexeyev, and A. Volyar, “Natural shaping of the cylindrically polarized beams,” Opt. Lett. 35, 3787–3789 (2010).
[CrossRef]

C. Alexeyev, A. Volyar, and M. Yavorsky, “Fiber optical vortices,” in Lasers, Optics and Electro-Optics Research Trends, L. I. Chen, ed. (Nova Science, 2007) 131–223.

Alexeyev, C. N.

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A 6, S162–S165 (2004).
[CrossRef]

Allen, L.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).

Angelsky, O. V.

Arlt, J.

K. Volke-Sepulveda, V. Garres-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
[CrossRef]

Barnett, S. M.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).

Bekshaev, A. Ya.

Belsky, A. M.

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun. 167, 1–5 (1999).
[CrossRef]

Belyi, V.

V. Belyi, T. King, N. Kazak, N. Khio, E. Katranji, and A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE 4403, 229–240(2001).
[CrossRef]

Berry, M. V.

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” Prog. Opt. 50, 13–50 (2007).
[CrossRef]

M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observation and theory,” Proc. R. Soc. A 462, 1629–1642 (2006).
[CrossRef]

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A 7, 685–690 (2005).
[CrossRef]

M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A 6, 289–300 (2004).
[CrossRef]

Biener, G.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Chavez-Cerda, S.

K. Volke-Sepulveda, V. Garres-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
[CrossRef]

Desyatnikov, A.

Dholakia, K.

K. Volke-Sepulveda, V. Garres-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
[CrossRef]

Dichburn, R. W.

R. W. Dichburn, Light.3rd ed. (Academic, 1976).

Donegan, J.

Donegan, J. F.

Fadeyeva, T.

Garres-Chavez, V.

K. Volke-Sepulveda, V. Garres-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
[CrossRef]

Hanson, S. G.

Hasman, E.

Jeffrey, M. R.

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” Prog. Opt. 50, 13–50 (2007).
[CrossRef]

M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observation and theory,” Proc. R. Soc. A 462, 1629–1642 (2006).
[CrossRef]

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A 7, 685–690 (2005).
[CrossRef]

Karinni, E.

L. Marrucci, E. Karinni, S. Slussarenko, B. Piecirillo, 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]

Katranji, E.

V. Belyi, T. King, N. Kazak, N. Khio, E. Katranji, and A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE 4403, 229–240(2001).
[CrossRef]

Kazak, N.

V. Belyi, T. King, N. Kazak, N. Khio, E. Katranji, and A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE 4403, 229–240(2001).
[CrossRef]

Kazak, N. S.

N. S. Kazak, N. A. Khio, and A. A. Ryzhevich, “Generation of Bessel light beams under the condition of internal conical refraction,” Quantum Electron. 29, 1020–1024 (1999).
[CrossRef]

Khio, N.

V. Belyi, T. King, N. Kazak, N. Khio, E. Katranji, and A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE 4403, 229–240(2001).
[CrossRef]

Khio, N. A.

N. S. Kazak, N. A. Khio, and A. A. Ryzhevich, “Generation of Bessel light beams under the condition of internal conical refraction,” Quantum Electron. 29, 1020–1024 (1999).
[CrossRef]

King, T.

V. Belyi, T. King, N. Kazak, N. Khio, E. Katranji, and A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE 4403, 229–240(2001).
[CrossRef]

Kivshar, Y.

Kleiner, V.

Krolikowski, W.

Krupych, O.

R. Vlokh, A. Volyar, O. Mys, and O. Krupych, “Appearance of optical vortex at conical refraction. Examples of NANO2 and YFeO3 crystals,” Ukr. J. Phys. Opt. 4, 90–93 (2003).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman & Hall, 1983).

Lunney, J.

Lunney, J. G.

Maksimyak, A. P.

Maksimyak, P. P.

Mansuripur, M.

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A 7, 685–690 (2005).
[CrossRef]

Marrucci, L.

L. Marrucci, E. Karinni, S. Slussarenko, B. Piecirillo, 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]

Mys, O.

R. Vlokh, A. Volyar, O. Mys, and O. Krupych, “Appearance of optical vortex at conical refraction. Examples of NANO2 and YFeO3 crystals,” Ukr. J. Phys. Opt. 4, 90–93 (2003).
[CrossRef]

Nagali, E.

L. Marrucci, E. Karinni, S. Slussarenko, B. Piecirillo, 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]

Neshev, D.

Niv, A.

Nye, J. F.

J. F. Nye, Natural Focusing and Fine Structure of Light(Institute of Physics, 1999).

O’Dwyer, D. P.

Padgett, M. J.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).

Phelan, C. F.

Phelan, G.

Piecirillo, B.

L. Marrucci, E. Karinni, S. Slussarenko, B. Piecirillo, 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]

Rakovich, Y. P.

Rubass, A.

T. Fadeyeva, A. Rubass, and A. Volyar, “Transverse shift of high-order paraxial vortex-beam induced by a homogeneous anisotropic medium,” Phys. Rev. A 79, 05381(2009).
[CrossRef]

Ryzhevich, A.

V. Belyi, T. King, N. Kazak, N. Khio, E. Katranji, and A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE 4403, 229–240(2001).
[CrossRef]

Ryzhevich, A. A.

N. S. Kazak, N. A. Khio, and A. A. Ryzhevich, “Generation of Bessel light beams under the condition of internal conical refraction,” Quantum Electron. 29, 1020–1024 (1999).
[CrossRef]

Savaryn, V.

Sciarrino, F.

L. Marrucci, E. Karinni, S. Slussarenko, B. Piecirillo, 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]

Shostka, N.

Shvedov, V.

Shvedov, V. G.

A. V. Volyar, V. Z. Zhilaitis, and V. G. Shvedov, “Optical eddies in small-mode fibers: II. The spin-orbit interaction,” Opt. Spectrosc. 86, 593–598 (1999).
[CrossRef]

Skab, I.

Slussarenko, S.

L. Marrucci, E. Karinni, S. Slussarenko, B. Piecirillo, 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]

Smaga, I.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman & Hall, 1983).

Soskin, M. S.

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

Stepanov, M. A.

A. M. Belsky and M. A. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun. 167, 1–5 (1999).
[CrossRef]

Vasnetsov, M. V.

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

Vasylkiv, Y.

Vlokh, R.

Y. Vasylkiv, I. Skab, and R. Vlokh, “Measurements of piezooptic coefficients π14 and π25 in Pb5Ge3O11 crystals using torsion induced optical vortex,” Ukr. J. Phys. Opt. 12, 101–108 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, B. Zapeka, V. Savaryn, and R. Vlokh, “Appearance of singularities of optical fields under torsion of crystals containing threefold symmetry axes,” J. Opt. Soc. Am. A 28, 1331–1340 (2011).
[CrossRef]

Y. Vasylkiv, V. Savaryn, I. Smaga, I. Skab, and R. Vlokh, “On determination of sign of piezo-optic coefficients using torsion method,” Appl. Opt. 50, 2512–2518 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, V. Savaryn, and R. Vlokh, “Optical anisotropy induced by torsion stresses in LiNbO3 crystals: appearing of optical vortex,” J. Opt. Soc. Am. A 28, 633–640 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, V. Savaryn, and R. Vlokh, “Relations for optical indicatrix parameters in the condition of crystal torsion,” Ukr. J. Phys. Opt. 11, 193–240 (2010).
[CrossRef]

R. Vlokh, A. Volyar, O. Mys, and O. Krupych, “Appearance of optical vortex at conical refraction. Examples of NANO2 and YFeO3 crystals,” Ukr. J. Phys. Opt. 4, 90–93 (2003).
[CrossRef]

Volke-Sepulveda, K.

K. Volke-Sepulveda, V. Garres-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
[CrossRef]

Volyar, A.

T. Fadeyeva, V. Shvedov, N. Shostka, C. Alexeyev, and A. Volyar, “Natural shaping of the cylindrically polarized beams,” Opt. Lett. 35, 3787–3789 (2010).
[CrossRef]

T. Fadeyeva and A. Volyar, “Extreme spin-orbit coupling in crystal-travelling paraxial beams,” J. Opt. Soc. Am. A 27, 381–389 (2010).
[CrossRef]

T. Fadeyeva and A. Volyar, “Nondiffracting vortex-beams in a birefringent chiral crystal,” J. Opt. Soc. Am. A 27, 13–20 (2010).
[CrossRef]

T. Fadeyeva, A. Rubass, and A. Volyar, “Transverse shift of high-order paraxial vortex-beam induced by a homogeneous anisotropic medium,” Phys. Rev. A 79, 05381(2009).
[CrossRef]

A. Volyar, V. Shvedov, T. Fadeyeva, A. Desyatnikov, D. Neshev, W. Krolikowski, and Y. Kivshar, “Generation of single-charge optical vortices with an uniaxial crystal,” Opt. Express 14, 3724–3729 (2006).
[CrossRef]

R. Vlokh, A. Volyar, O. Mys, and O. Krupych, “Appearance of optical vortex at conical refraction. Examples of NANO2 and YFeO3 crystals,” Ukr. J. Phys. Opt. 4, 90–93 (2003).
[CrossRef]

C. Alexeyev, A. Volyar, and M. Yavorsky, “Fiber optical vortices,” in Lasers, Optics and Electro-Optics Research Trends, L. I. Chen, ed. (Nova Science, 2007) 131–223.

Volyar, A. V.

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A 6, S162–S165 (2004).
[CrossRef]

A. V. Volyar, V. Z. Zhilaitis, and V. G. Shvedov, “Optical eddies in small-mode fibers: II. The spin-orbit interaction,” Opt. Spectrosc. 86, 593–598 (1999).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Yariv, A.

A. Yariv and P. Yuh, Optical Waves in Crystals (Wiley-Interscience, 1984).

Yavorsky, M.

C. Alexeyev, A. Volyar, and M. Yavorsky, “Fiber optical vortices,” in Lasers, Optics and Electro-Optics Research Trends, L. I. Chen, ed. (Nova Science, 2007) 131–223.

Yavorsky, M. A.

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A 6, S162–S165 (2004).
[CrossRef]

Yuh, P.

A. Yariv and P. Yuh, Optical Waves in Crystals (Wiley-Interscience, 1984).

Zapeka, B.

Zenkova, C. Yu.

Zhilaitis, V. Z.

A. V. Volyar, V. Z. Zhilaitis, and V. G. Shvedov, “Optical eddies in small-mode fibers: II. The spin-orbit interaction,” Opt. Spectrosc. 86, 593–598 (1999).
[CrossRef]

Appl. Opt. (1)

J. Opt. (1)

L. Marrucci, E. Karinni, S. Slussarenko, B. Piecirillo, 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, “Vortex-preserving weakly guiding anisotropic twisted fibres,” J. Opt. A 6, S162–S165 (2004).
[CrossRef]

M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A 6, 289–300 (2004).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Sketches of the mechanical torsion of the (a) uniaxial crystal and (b) the distribution of the polarization states in the output field Eout.

Fig. 2.
Fig. 2.

Sketch of the optical axes’ positions in the twisting uniaxial crystal.

Fig. 3.
Fig. 3.

Polarization distributions in the eigenmodes with m=0 on the background of the intensity distribution. The envelopes of the directions of the polarization axes are represented by the solid lines.

Fig. 4.
Fig. 4.

Dispersive curves Δβ(U):Mz=0.162N×m, no=ε=2.3, ρ=0.5cm|π14|=8.87·1013m2/N.

Fig. 5.
Fig. 5.

Evolution of the intensity and phase distribution in the E+ component of the Bessel beam with the initial linear x-polarization: U=3·103m1, m=6.

Fig. 6.
Fig. 6.

Shaping of the (TE) azimuthally, (SP) spirally, and (TM) radially mode beams, U=3·103m1.

Fig. 7.
Fig. 7.

Polarization distributions against the background of the intensity distributions in the mode beams (26) with m=1, m=2, m=3; U=3·103m1.

Equations (26)

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ε̑=(ε11π14ε112σ32π14ε112σ31π44εε33σ31π44ε112σ31ε11+π14ε112σ32π44εε33σ32π14εε33σ31π44εε33σ32ε33),
2E+k2D=(E),
2Ex+k2(εγx)Exk2γyEy=0,
2Ey+k2(ε+γx)Eyk2γyEx=0,
2E++k2εE+k2γreiφE=0,
2E+k2εEk2γreiφE+=0,
E+=F+(r)exp{imφ}eiβz,E=F(r)exp{i(m+1)φ}eiβzm=0,±1,±2
{d2dr2+1rddrm2r2+k2ε}F+(r)+k2γrF=β2F+,
{d2dr2+1rddr(m+1)2r2+k2ε}F(r)+k2γrF+=β2F.
H^0(F˜+F˜)=β˜2(F˜+F˜),
|F=C1|1+C2|2.
C=k2γSrJm(Ur)Jm+1(Ur)dSSJm2(Ur)dSSJm+12(Ur)dS,
Cσ^xx=Δβ2x.
Δβ1,22=±C
(F+F)1=(Jm(Ur)Jm+1(Ur))forΔβ12and(F+F)2=(Jm(Ur)Jm+1(Ur))forΔβ22.
(E+E)1=(Jm(Ur)Jm+1(Ur)exp(iφ))exp{imφ}exp{i(β˜+Δβ)z}
(E+E)2=(Jm(Ur)Jm+1(Ur)exp(iφ))exp{imφ}exp{i(β˜Δβ)z}.
E(in)(r,φ,z=0)=e+Jm(Ur)exp(imφ).
E=2eiβ˜z{(Jm(Ur)0)eimφcos(Δβz)i(0Jm+1(Ur))ei(m+1)φsin(Δβz)}.
E(in)(r,φ,z=0)=(eiψ1)eimφJm(Ur),
E+=2{Jm(Ur)ei(mφ+ψ)cos(Δβz)iJm1(Ur)ei(m1)φsin(Δβz)}eiβ˜z,E=2{Jm(Ur)eimφcos(Δβz)iJm+1(Ur)ei[(m+1)φ+ψ]sin(Δβz)}eiβ˜z.
ETM(r,φ,z=π/Δβ)=i(eiφeiφ)J1(Ur)eiπβ˜/Δβ.
ETE(r,φ,z=π/Δβ)=i(eiφeiφ)J1(Ur)eiπβ˜/Δβ.
ESP(r,φ,z=π/Δβ)=i(eiφei(φ+ψ))J1(Ur)eiπβ˜/Δβ.
Ein(r,φ,z=0)=(eimφei(mφ+α))Jm(Ur)
E+=2{Jm(Ur)eimφcos(Δβz)iJm+1(Ur)ei((m+1)φα)sin(Δβz)}eiβ˜z,E=2{Jm(Ur)ei(mφ+α)cos(Δβz)+iJm+1(Ur)ei(m+1)φsin(Δβz)}eiβ˜z.

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