Abstract

We show that a system of glass disks compressed along their diameters enables one to induce a doubly charged vortex beam in the emergent light when the incident light is circularly polarized. Using such a disk system, one can control the efficiency of conversion of the spin angular momentum to the orbital angular momentum by a loading force. The consideration presented here can be extended for the case of crystalline materials with high optical damage thresholds in order to induce high-power vortex beams.

© 2012 Optical Society of America

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  1. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
    [CrossRef]
  2. D. P. DiVincenzo, “Quantum computation,” Science 270, 255–261 (1995).
    [CrossRef]
  3. S. Y. Kilin, “Quantum information,” Sov. Phys. Usp. 42, 435–452 (1999).
    [CrossRef]
  4. D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
    [CrossRef]
  5. O. V. Angelsky, A. P. Maksimyak, P. P. Maksimyak, and S. G. Hanson, “Biaxial crystal-based optical tweezers,” Ukr. J. Phys. Opt. 11, 99–106 (2010).
    [CrossRef]
  6. 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]
  7. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef]
  8. V. G. Shvedov, “Nonparaxial singular beams inside the focal region of a high numerical-aperture lens,” Ukr. J. Phys. Opt. 12, 109–116 (2011).
    [CrossRef]
  9. K. Watanabe, N. Horiguchi, and H. Kano, “Optimized measurement probe of the localized surface plasmon microscope by using radially polarized illumination,” Appl. Opt. 46, 4985–4990 (2007).
    [CrossRef]
  10. N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6341 (2004).
    [CrossRef]
  11. D. Biss and T. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28, 923–925 (2003).
    [CrossRef]
  12. E. Yew and C. Sheppard, “Second harmonic generation polarization microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. 275, 453–457 (2007).
    [CrossRef]
  13. B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, “Photon spin-to-orbital angular momentum conversation via an electrically tunable q-plate,” Appl. Phys. Lett. 97, 241104 (2010).
    [CrossRef]
  14. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
    [CrossRef]
  15. L. Marrucci, “Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals,” Mol. Cryst. Liq. Cryst. 488, 148–162(2008).
    [CrossRef]
  16. A. Volyar, V. Shvedov, T. Fadeyeva, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Generation of single-charge optical vortices with an uniaxial crystal,” Opt. Express 14, 3724–3729 (2006).
    [CrossRef]
  17. M. Beresna, M. Gecevičius, and P. G. Kazansky, “Polarization sensitive elements fabricated by femtosecond laser nanostructuring of glass,” Opt. Mater. Express 1, 783–795 (2011).
    [CrossRef]
  18. M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
    [CrossRef]
  19. I. Skab, Y. Vasylkiv, V. Savaryn, and R. Vlokh, “Optical anisotropy induced by torsion stresses in LiNbO3 crystals: appearance of an optical vortex,” J. Opt. Soc. Am. A. 28, 633–640 (2011).
    [CrossRef]
  20. Y. Vasylkiv, I. Skab, and R. Vlokh, “Measurements of piezooptic coefficients π14 and π52 in Pb5Ge3O11 crystals using torsion induced optical vortex,” Ukr. J. Phys. Opt. 12, 101–108 (2011).
    [CrossRef]
  21. I. Skab, Y. Vasylkiv, B. Zapeka, V. Savaryn, and R. Vlokh, “On the appearance of singularities of optical field under torsion of crystals containing three-fold symmetry axes,” J. Opt. Soc. Am. A 28, 1331–1340 (2011).
  22. I. P. Skab, Y. Vasylkiv, and R. O. Vlokh, “On the possibility of electrooptic operation by orbital angular momentum of light beams via Pockels effect in crystals,” Ukr. J. Phys. Opt. 12, 127–136 (2011).
    [CrossRef]
  23. Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “On the spin-to-orbit momentum conversion operated by electric field in optically active Bi12GeO20 crystals,” Ukr. J. Phys. Opt. 12, 171–179 (2011).
    [CrossRef]
  24. I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electrooptic Pockels effect in crystals,” Phys. Rev. A 84, 043815 (2011).
    [CrossRef]
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  27. M. R. Dennis, “Local phase structure of wave dislocation lines: twist and twirl,” J. Opt. A: Pure Appl. Opt. 6, S202–S208 (2004).
    [CrossRef]
  28. I. Skab, Y. Vasylkiv, I. Smaga, V. Savaryn, and R. Vlokh, “On the method for measuring piezooptic coefficients π25 and π14 in the crystals belonging to point symmetry groups 3 and 3¯,” Ukr. J. Phys. Opt. 12, 28–35 (2011).
    [CrossRef]
  29. Y. Vasylkiv, V. Savaryn, I. Smaga, O. Krupych, I. Skab, and R. Vlokh, “Studies of piezooptic coefficients in LiNbO3 crystals using a crystalline disk compressed along its diameter,” Ukr. J. Phys. Opt. 12, 180–190 (2011).
    [CrossRef]
  30. I. Martynyuk-Lototska, O. Mys, O. Krupych, V. Adamiv, Ya. Burak, R. Vlokh, and W. Schranz, “Elastic, piezooptic and acoustooptic properties of borate crystals (BaB2O4, Li2B4O7 and CsLiB6O10),” Integr. Ferroelectr. 63, 99–103 (2004).
    [CrossRef]
  31. C. T. Chen, B. C. Wu, A. D. Jiang, and G. M. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235–243 (1985).
  32. R. Vlokh, Y. Pyatak, and I. Skab, “Elasto-optic effect in LiNbO3 under the crystal bending,” Ferroelectrics 126, 239–242 (1992).
    [CrossRef]

2011 (12)

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]

V. G. Shvedov, “Nonparaxial singular beams inside the focal region of a high numerical-aperture lens,” Ukr. J. Phys. Opt. 12, 109–116 (2011).
[CrossRef]

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[CrossRef]

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

Y. Vasylkiv, I. Skab, and R. Vlokh, “Measurements of piezooptic coefficients π14 and π52 in Pb5Ge3O11 crystals using torsion induced optical vortex,” Ukr. J. Phys. Opt. 12, 101–108 (2011).
[CrossRef]

I. P. Skab, Y. Vasylkiv, and R. O. Vlokh, “On the possibility of electrooptic operation by orbital angular momentum of light beams via Pockels effect in crystals,” Ukr. J. Phys. Opt. 12, 127–136 (2011).
[CrossRef]

Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “On the spin-to-orbit momentum conversion operated by electric field in optically active Bi12GeO20 crystals,” Ukr. J. Phys. Opt. 12, 171–179 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electrooptic Pockels effect in crystals,” Phys. Rev. A 84, 043815 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, I. Smaga, V. Savaryn, and R. Vlokh, “On the method for measuring piezooptic coefficients π25 and π14 in the crystals belonging to point symmetry groups 3 and 3¯,” Ukr. J. Phys. Opt. 12, 28–35 (2011).
[CrossRef]

Y. Vasylkiv, V. Savaryn, I. Smaga, O. Krupych, I. Skab, and R. Vlokh, “Studies of piezooptic coefficients in LiNbO3 crystals using a crystalline disk compressed along its diameter,” Ukr. J. Phys. Opt. 12, 180–190 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, B. Zapeka, V. Savaryn, and R. Vlokh, “On the appearance of singularities of optical field under torsion of crystals containing three-fold symmetry axes,” J. Opt. Soc. Am. A 28, 1331–1340 (2011).

M. Beresna, M. Gecevičius, and P. G. Kazansky, “Polarization sensitive elements fabricated by femtosecond laser nanostructuring of glass,” Opt. Mater. Express 1, 783–795 (2011).
[CrossRef]

2010 (2)

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, “Photon spin-to-orbital angular momentum conversation via an electrically tunable q-plate,” Appl. Phys. Lett. 97, 241104 (2010).
[CrossRef]

O. V. Angelsky, A. P. Maksimyak, P. P. Maksimyak, and S. G. Hanson, “Biaxial crystal-based optical tweezers,” Ukr. J. Phys. Opt. 11, 99–106 (2010).
[CrossRef]

2008 (1)

L. Marrucci, “Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals,” Mol. Cryst. Liq. Cryst. 488, 148–162(2008).
[CrossRef]

2007 (2)

E. Yew and C. Sheppard, “Second harmonic generation polarization microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. 275, 453–457 (2007).
[CrossRef]

K. Watanabe, N. Horiguchi, and H. Kano, “Optimized measurement probe of the localized surface plasmon microscope by using radially polarized illumination,” Appl. Opt. 46, 4985–4990 (2007).
[CrossRef]

2006 (2)

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

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef]

2004 (3)

M. R. Dennis, “Local phase structure of wave dislocation lines: twist and twirl,” J. Opt. A: Pure Appl. Opt. 6, S202–S208 (2004).
[CrossRef]

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6341 (2004).
[CrossRef]

I. Martynyuk-Lototska, O. Mys, O. Krupych, V. Adamiv, Ya. Burak, R. Vlokh, and W. Schranz, “Elastic, piezooptic and acoustooptic properties of borate crystals (BaB2O4, Li2B4O7 and CsLiB6O10),” Integr. Ferroelectr. 63, 99–103 (2004).
[CrossRef]

2003 (2)

D. Biss and T. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28, 923–925 (2003).
[CrossRef]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

2001 (1)

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

1999 (1)

S. Y. Kilin, “Quantum information,” Sov. Phys. Usp. 42, 435–452 (1999).
[CrossRef]

1998 (1)

D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
[CrossRef]

1995 (1)

D. P. DiVincenzo, “Quantum computation,” Science 270, 255–261 (1995).
[CrossRef]

1992 (1)

R. Vlokh, Y. Pyatak, and I. Skab, “Elasto-optic effect in LiNbO3 under the crystal bending,” Ferroelectrics 126, 239–242 (1992).
[CrossRef]

1985 (1)

C. T. Chen, B. C. Wu, A. D. Jiang, and G. M. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235–243 (1985).

Adamiv, V.

I. Martynyuk-Lototska, O. Mys, O. Krupych, V. Adamiv, Ya. Burak, R. Vlokh, and W. Schranz, “Elastic, piezooptic and acoustooptic properties of borate crystals (BaB2O4, Li2B4O7 and CsLiB6O10),” Integr. Ferroelectr. 63, 99–103 (2004).
[CrossRef]

Angelsky, O. V.

O. V. Angelsky, A. P. Maksimyak, P. P. Maksimyak, and S. G. Hanson, “Biaxial crystal-based optical tweezers,” Ukr. J. Phys. Opt. 11, 99–106 (2010).
[CrossRef]

Beresna, M.

M. Beresna, M. Gecevičius, and P. G. Kazansky, “Polarization sensitive elements fabricated by femtosecond laser nanostructuring of glass,” Opt. Mater. Express 1, 783–795 (2011).
[CrossRef]

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[CrossRef]

Biss, D.

Boschi, D.

D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
[CrossRef]

Branca, S.

D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
[CrossRef]

Brown, T.

Burak, Ya.

I. Martynyuk-Lototska, O. Mys, O. Krupych, V. Adamiv, Ya. Burak, R. Vlokh, and W. Schranz, “Elastic, piezooptic and acoustooptic properties of borate crystals (BaB2O4, Li2B4O7 and CsLiB6O10),” Integr. Ferroelectr. 63, 99–103 (2004).
[CrossRef]

Chen, C. T.

C. T. Chen, B. C. Wu, A. D. Jiang, and G. M. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235–243 (1985).

D’Ambrosio, V.

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, “Photon spin-to-orbital angular momentum conversation via an electrically tunable q-plate,” Appl. Phys. Lett. 97, 241104 (2010).
[CrossRef]

Dennis, M. R.

M. R. Dennis, “Local phase structure of wave dislocation lines: twist and twirl,” J. Opt. A: Pure Appl. Opt. 6, S202–S208 (2004).
[CrossRef]

Desyatnikov, A. S.

DiVincenzo, D. P.

D. P. DiVincenzo, “Quantum computation,” Science 270, 255–261 (1995).
[CrossRef]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Fadeyeva, T.

Frocht, M. M.

M. M. Frocht, Photoelasticity (Wiley, 1965).

Gecevicius, M.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[CrossRef]

M. Beresna, M. Gecevičius, and P. G. Kazansky, “Polarization sensitive elements fabricated by femtosecond laser nanostructuring of glass,” Opt. Mater. Express 1, 783–795 (2011).
[CrossRef]

Gertus, T.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[CrossRef]

Hanson, S. G.

O. V. Angelsky, A. P. Maksimyak, P. P. Maksimyak, and S. G. Hanson, “Biaxial crystal-based optical tweezers,” Ukr. J. Phys. Opt. 11, 99–106 (2010).
[CrossRef]

Hardy, L.

D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
[CrossRef]

Hayazawa, N.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6341 (2004).
[CrossRef]

Horiguchi, N.

Jiang, A. D.

C. T. Chen, B. C. Wu, A. D. Jiang, and G. M. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235–243 (1985).

Kano, H.

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]

Kawata, S.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6341 (2004).
[CrossRef]

Kazansky, P. G.

M. Beresna, M. Gecevičius, and P. G. Kazansky, “Polarization sensitive elements fabricated by femtosecond laser nanostructuring of glass,” Opt. Mater. Express 1, 783–795 (2011).
[CrossRef]

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[CrossRef]

Kilin, S. Y.

S. Y. Kilin, “Quantum information,” Sov. Phys. Usp. 42, 435–452 (1999).
[CrossRef]

Kivshar, Y. S.

Krolikowski, W.

Krupych, O.

Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “On the spin-to-orbit momentum conversion operated by electric field in optically active Bi12GeO20 crystals,” Ukr. J. Phys. Opt. 12, 171–179 (2011).
[CrossRef]

Y. Vasylkiv, V. Savaryn, I. Smaga, O. Krupych, I. Skab, and R. Vlokh, “Studies of piezooptic coefficients in LiNbO3 crystals using a crystalline disk compressed along its diameter,” Ukr. J. Phys. Opt. 12, 180–190 (2011).
[CrossRef]

I. Martynyuk-Lototska, O. Mys, O. Krupych, V. Adamiv, Ya. Burak, R. Vlokh, and W. Schranz, “Elastic, piezooptic and acoustooptic properties of borate crystals (BaB2O4, Li2B4O7 and CsLiB6O10),” Integr. Ferroelectr. 63, 99–103 (2004).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Maksimyak, A. P.

O. V. Angelsky, A. P. Maksimyak, P. P. Maksimyak, and S. G. Hanson, “Biaxial crystal-based optical tweezers,” Ukr. J. Phys. Opt. 11, 99–106 (2010).
[CrossRef]

Maksimyak, P. P.

O. V. Angelsky, A. P. Maksimyak, P. P. Maksimyak, and S. G. Hanson, “Biaxial crystal-based optical tweezers,” Ukr. J. Phys. Opt. 11, 99–106 (2010).
[CrossRef]

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[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]

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, “Photon spin-to-orbital angular momentum conversation via an electrically tunable q-plate,” Appl. Phys. Lett. 97, 241104 (2010).
[CrossRef]

L. Marrucci, “Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals,” Mol. Cryst. Liq. Cryst. 488, 148–162(2008).
[CrossRef]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef]

Martini, F. De

D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
[CrossRef]

Martynyuk-Lototska, I.

I. Martynyuk-Lototska, O. Mys, O. Krupych, V. Adamiv, Ya. Burak, R. Vlokh, and W. Schranz, “Elastic, piezooptic and acoustooptic properties of borate crystals (BaB2O4, Li2B4O7 and CsLiB6O10),” Integr. Ferroelectr. 63, 99–103 (2004).
[CrossRef]

Mys, O.

I. Martynyuk-Lototska, O. Mys, O. Krupych, V. Adamiv, Ya. Burak, R. Vlokh, and W. Schranz, “Elastic, piezooptic and acoustooptic properties of borate crystals (BaB2O4, Li2B4O7 and CsLiB6O10),” Integr. Ferroelectr. 63, 99–103 (2004).
[CrossRef]

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]

Neshev, D. N.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[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]

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, “Photon spin-to-orbital angular momentum conversation via an electrically tunable q-plate,” Appl. Phys. Lett. 97, 241104 (2010).
[CrossRef]

Popescu, S.

D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
[CrossRef]

Pyatak, Y.

R. Vlokh, Y. Pyatak, and I. Skab, “Elasto-optic effect in LiNbO3 under the crystal bending,” Ferroelectrics 126, 239–242 (1992).
[CrossRef]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Saito, Y.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6341 (2004).
[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]

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, “Photon spin-to-orbital angular momentum conversation via an electrically tunable q-plate,” Appl. Phys. Lett. 97, 241104 (2010).
[CrossRef]

Savaryn, V.

I. Skab, Y. Vasylkiv, I. Smaga, V. Savaryn, and R. Vlokh, “On the method for measuring piezooptic coefficients π25 and π14 in the crystals belonging to point symmetry groups 3 and 3¯,” Ukr. J. Phys. Opt. 12, 28–35 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, B. Zapeka, V. Savaryn, and R. Vlokh, “On the appearance of singularities of optical field under torsion of crystals containing three-fold symmetry axes,” J. Opt. Soc. Am. A 28, 1331–1340 (2011).

Y. Vasylkiv, V. Savaryn, I. Smaga, O. Krupych, I. Skab, and R. Vlokh, “Studies of piezooptic coefficients in LiNbO3 crystals using a crystalline disk compressed along its diameter,” Ukr. J. Phys. Opt. 12, 180–190 (2011).
[CrossRef]

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

Schranz, W.

I. Martynyuk-Lototska, O. Mys, O. Krupych, V. Adamiv, Ya. Burak, R. Vlokh, and W. Schranz, “Elastic, piezooptic and acoustooptic properties of borate crystals (BaB2O4, Li2B4O7 and CsLiB6O10),” Integr. Ferroelectr. 63, 99–103 (2004).
[CrossRef]

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]

Sheppard, C.

E. Yew and C. Sheppard, “Second harmonic generation polarization microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. 275, 453–457 (2007).
[CrossRef]

Shvedov, V.

Shvedov, V. G.

V. G. Shvedov, “Nonparaxial singular beams inside the focal region of a high numerical-aperture lens,” Ukr. J. Phys. Opt. 12, 109–116 (2011).
[CrossRef]

Skab, I.

Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “On the spin-to-orbit momentum conversion operated by electric field in optically active Bi12GeO20 crystals,” Ukr. J. Phys. Opt. 12, 171–179 (2011).
[CrossRef]

Y. Vasylkiv, V. Savaryn, I. Smaga, O. Krupych, I. Skab, and R. Vlokh, “Studies of piezooptic coefficients in LiNbO3 crystals using a crystalline disk compressed along its diameter,” Ukr. J. Phys. Opt. 12, 180–190 (2011).
[CrossRef]

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

I. Skab, Y. Vasylkiv, B. Zapeka, V. Savaryn, and R. Vlokh, “On the appearance of singularities of optical field under torsion of crystals containing three-fold symmetry axes,” J. Opt. Soc. Am. A 28, 1331–1340 (2011).

I. Skab, Y. Vasylkiv, I. Smaga, V. Savaryn, and R. Vlokh, “On the method for measuring piezooptic coefficients π25 and π14 in the crystals belonging to point symmetry groups 3 and 3¯,” Ukr. J. Phys. Opt. 12, 28–35 (2011).
[CrossRef]

Y. Vasylkiv, I. Skab, and R. Vlokh, “Measurements of piezooptic coefficients π14 and π52 in Pb5Ge3O11 crystals using torsion induced optical vortex,” Ukr. J. Phys. Opt. 12, 101–108 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electrooptic Pockels effect in crystals,” Phys. Rev. A 84, 043815 (2011).
[CrossRef]

R. Vlokh, Y. Pyatak, and I. Skab, “Elasto-optic effect in LiNbO3 under the crystal bending,” Ferroelectrics 126, 239–242 (1992).
[CrossRef]

Skab, I. P.

I. P. Skab, Y. Vasylkiv, and R. O. Vlokh, “On the possibility of electrooptic operation by orbital angular momentum of light beams via Pockels effect in crystals,” Ukr. J. Phys. Opt. 12, 127–136 (2011).
[CrossRef]

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]

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, “Photon spin-to-orbital angular momentum conversation via an electrically tunable q-plate,” Appl. Phys. Lett. 97, 241104 (2010).
[CrossRef]

Smaga, I.

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electrooptic Pockels effect in crystals,” Phys. Rev. A 84, 043815 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, I. Smaga, V. Savaryn, and R. Vlokh, “On the method for measuring piezooptic coefficients π25 and π14 in the crystals belonging to point symmetry groups 3 and 3¯,” Ukr. J. Phys. Opt. 12, 28–35 (2011).
[CrossRef]

Y. Vasylkiv, V. Savaryn, I. Smaga, O. Krupych, I. Skab, and R. Vlokh, “Studies of piezooptic coefficients in LiNbO3 crystals using a crystalline disk compressed along its diameter,” Ukr. J. Phys. Opt. 12, 180–190 (2011).
[CrossRef]

Soskin, M. S.

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

Vasnetsov, M. V.

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

Vasylkiv, Y.

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

Y. Vasylkiv, V. Savaryn, I. Smaga, O. Krupych, I. Skab, and R. Vlokh, “Studies of piezooptic coefficients in LiNbO3 crystals using a crystalline disk compressed along its diameter,” Ukr. J. Phys. Opt. 12, 180–190 (2011).
[CrossRef]

Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “On the spin-to-orbit momentum conversion operated by electric field in optically active Bi12GeO20 crystals,” Ukr. J. Phys. Opt. 12, 171–179 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, B. Zapeka, V. Savaryn, and R. Vlokh, “On the appearance of singularities of optical field under torsion of crystals containing three-fold symmetry axes,” J. Opt. Soc. Am. A 28, 1331–1340 (2011).

I. Skab, Y. Vasylkiv, I. Smaga, V. Savaryn, and R. Vlokh, “On the method for measuring piezooptic coefficients π25 and π14 in the crystals belonging to point symmetry groups 3 and 3¯,” Ukr. J. Phys. Opt. 12, 28–35 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electrooptic Pockels effect in crystals,” Phys. Rev. A 84, 043815 (2011).
[CrossRef]

Y. Vasylkiv, I. Skab, and R. Vlokh, “Measurements of piezooptic coefficients π14 and π52 in Pb5Ge3O11 crystals using torsion induced optical vortex,” Ukr. J. Phys. Opt. 12, 101–108 (2011).
[CrossRef]

I. P. Skab, Y. Vasylkiv, and R. O. Vlokh, “On the possibility of electrooptic operation by orbital angular momentum of light beams via Pockels effect in crystals,” Ukr. J. Phys. Opt. 12, 127–136 (2011).
[CrossRef]

Vlokh, R.

Y. Vasylkiv, I. Skab, and R. Vlokh, “Measurements of piezooptic coefficients π14 and π52 in Pb5Ge3O11 crystals using torsion induced optical vortex,” Ukr. J. Phys. Opt. 12, 101–108 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electrooptic Pockels effect in crystals,” Phys. Rev. A 84, 043815 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, I. Smaga, V. Savaryn, and R. Vlokh, “On the method for measuring piezooptic coefficients π25 and π14 in the crystals belonging to point symmetry groups 3 and 3¯,” Ukr. J. Phys. Opt. 12, 28–35 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, B. Zapeka, V. Savaryn, and R. Vlokh, “On the appearance of singularities of optical field under torsion of crystals containing three-fold symmetry axes,” J. Opt. Soc. Am. A 28, 1331–1340 (2011).

Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “On the spin-to-orbit momentum conversion operated by electric field in optically active Bi12GeO20 crystals,” Ukr. J. Phys. Opt. 12, 171–179 (2011).
[CrossRef]

Y. Vasylkiv, V. Savaryn, I. Smaga, O. Krupych, I. Skab, and R. Vlokh, “Studies of piezooptic coefficients in LiNbO3 crystals using a crystalline disk compressed along its diameter,” Ukr. J. Phys. Opt. 12, 180–190 (2011).
[CrossRef]

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

I. Martynyuk-Lototska, O. Mys, O. Krupych, V. Adamiv, Ya. Burak, R. Vlokh, and W. Schranz, “Elastic, piezooptic and acoustooptic properties of borate crystals (BaB2O4, Li2B4O7 and CsLiB6O10),” Integr. Ferroelectr. 63, 99–103 (2004).
[CrossRef]

R. Vlokh, Y. Pyatak, and I. Skab, “Elasto-optic effect in LiNbO3 under the crystal bending,” Ferroelectrics 126, 239–242 (1992).
[CrossRef]

Vlokh, R. O.

I. P. Skab, Y. Vasylkiv, and R. O. Vlokh, “On the possibility of electrooptic operation by orbital angular momentum of light beams via Pockels effect in crystals,” Ukr. J. Phys. Opt. 12, 127–136 (2011).
[CrossRef]

Volyar, A.

Watanabe, K.

Wu, B. C.

C. T. Chen, B. C. Wu, A. D. Jiang, and G. M. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235–243 (1985).

Yew, E.

E. Yew and C. Sheppard, “Second harmonic generation polarization microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. 275, 453–457 (2007).
[CrossRef]

You, G. M.

C. T. Chen, B. C. Wu, A. D. Jiang, and G. M. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235–243 (1985).

Zapeka, B.

Appl. Opt. (1)

Appl. Phys. Lett. (3)

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239–6341 (2004).
[CrossRef]

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, “Photon spin-to-orbital angular momentum conversation via an electrically tunable q-plate,” Appl. Phys. Lett. 97, 241104 (2010).
[CrossRef]

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[CrossRef]

Ferroelectrics (1)

R. Vlokh, Y. Pyatak, and I. Skab, “Elasto-optic effect in LiNbO3 under the crystal bending,” Ferroelectrics 126, 239–242 (1992).
[CrossRef]

Integr. Ferroelectr. (1)

I. Martynyuk-Lototska, O. Mys, O. Krupych, V. Adamiv, Ya. Burak, R. Vlokh, and W. Schranz, “Elastic, piezooptic and acoustooptic properties of borate crystals (BaB2O4, Li2B4O7 and CsLiB6O10),” Integr. Ferroelectr. 63, 99–103 (2004).
[CrossRef]

J. Opt. (1)

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: Pure Appl. Opt. (1)

M. R. Dennis, “Local phase structure of wave dislocation lines: twist and twirl,” J. Opt. A: Pure Appl. Opt. 6, S202–S208 (2004).
[CrossRef]

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

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

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

Mol. Cryst. Liq. Cryst. (1)

L. Marrucci, “Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals,” Mol. Cryst. Liq. Cryst. 488, 148–162(2008).
[CrossRef]

Opt. Commun. (1)

E. Yew and C. Sheppard, “Second harmonic generation polarization microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. 275, 453–457 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Mater. Express (1)

Phys. Rev. A (1)

I. Skab, Y. Vasylkiv, I. Smaga, and R. Vlokh, “Spin-to-orbital momentum conversion via electrooptic Pockels effect in crystals,” Phys. Rev. A 84, 043815 (2011).
[CrossRef]

Phys. Rev. Lett. (3)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef]

D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 80, 1121–1125 (1998).
[CrossRef]

Prog. Opt. (1)

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

Sci. Sin. Ser. B (1)

C. T. Chen, B. C. Wu, A. D. Jiang, and G. M. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235–243 (1985).

Science (1)

D. P. DiVincenzo, “Quantum computation,” Science 270, 255–261 (1995).
[CrossRef]

Sov. Phys. Usp. (1)

S. Y. Kilin, “Quantum information,” Sov. Phys. Usp. 42, 435–452 (1999).
[CrossRef]

Ukr. J. Phys. Opt. (7)

O. V. Angelsky, A. P. Maksimyak, P. P. Maksimyak, and S. G. Hanson, “Biaxial crystal-based optical tweezers,” Ukr. J. Phys. Opt. 11, 99–106 (2010).
[CrossRef]

V. G. Shvedov, “Nonparaxial singular beams inside the focal region of a high numerical-aperture lens,” Ukr. J. Phys. Opt. 12, 109–116 (2011).
[CrossRef]

Y. Vasylkiv, I. Skab, and R. Vlokh, “Measurements of piezooptic coefficients π14 and π52 in Pb5Ge3O11 crystals using torsion induced optical vortex,” Ukr. J. Phys. Opt. 12, 101–108 (2011).
[CrossRef]

I. P. Skab, Y. Vasylkiv, and R. O. Vlokh, “On the possibility of electrooptic operation by orbital angular momentum of light beams via Pockels effect in crystals,” Ukr. J. Phys. Opt. 12, 127–136 (2011).
[CrossRef]

Y. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “On the spin-to-orbit momentum conversion operated by electric field in optically active Bi12GeO20 crystals,” Ukr. J. Phys. Opt. 12, 171–179 (2011).
[CrossRef]

I. Skab, Y. Vasylkiv, I. Smaga, V. Savaryn, and R. Vlokh, “On the method for measuring piezooptic coefficients π25 and π14 in the crystals belonging to point symmetry groups 3 and 3¯,” Ukr. J. Phys. Opt. 12, 28–35 (2011).
[CrossRef]

Y. Vasylkiv, V. Savaryn, I. Smaga, O. Krupych, I. Skab, and R. Vlokh, “Studies of piezooptic coefficients in LiNbO3 crystals using a crystalline disk compressed along its diameter,” Ukr. J. Phys. Opt. 12, 180–190 (2011).
[CrossRef]

Other (2)

M. M. Frocht, Photoelasticity (Wiley, 1965).

GOST 51130-86, Silica Optical Glass. General Specification (Izd. Standartov, 1999).

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Figures (7)

Fig. 1.
Fig. 1.

Maps of (a) optical birefringence and (b) optical indicatrix rotation angle for the glass disk compressed along its diameter.

Fig. 2.
Fig. 2.

XY maps of (a) optical birefringence and (b) optical indicatrix rotation angle induced in the glass disk by the loading force 20 N (λ=632.8nm).

Fig. 3.
Fig. 3.

Experimental dependences of optical indicatrix rotation angle on the angle φ for ρ=R/5 (circles), ρ=2R/5 (triangles), ρ=3R/5 (crosses), and ρ=4R/5 (diamonds).

Fig. 4.
Fig. 4.

Simulated XY maps of (a) optical indicatrix rotation angle and (b) optical birefringence induced in the two glass disks loaded by the force 20 N along their common diameter (λ=632.8nm), and (c) coordinate dependence of the birefringence under the same conditions.

Fig. 5.
Fig. 5.

Two pairs of glass disks successively placed in the optical path.

Fig. 6.
Fig. 6.

Spatial distribution of phase difference (a) caused by a system of four glass disks loaded along their diameters by the force 100 N, (b) corresponding map of optical indicatrix rotation angle, and (c) dependence of this angle on the angle φ for ρ=R/5 (circles), ρ=2R/5 (triangles), ρ=3R/5 (crosses), and ρ=4R/5 (diamonds).

Fig. 7.
Fig. 7.

Appearance of a doughnut mode for different loading forces: (a) 20, (b) 50, and (c) 100 N.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

σ1=2Pπd[(RY)X2(X2+(RY)2)2+(R+Y)X2(X2+(R+Y)2)212R],
σ2=2Pπd[(RY)3(X2+(RY)2)2+(R+Y)3(X2+(R+Y)2)212R],
σ6=2Pπd[(RY)2X(X2+(RY)2)2(R+Y)2X(X2+(R+Y)2)2].
(B1+π11σ1+π12σ2)X2+(B1+π12σ1+π11σ2)Y2+2(π11π12)σ6XY=1,
Δn12=12no3(π11π12)(σ1σ2)2+4σ62,
tan2ζ3=2σ6(σ1σ2).
σ1=2Pπd[YX2(X2+Y2)2+(2R+Y)X2(X2+(2R+Y)2)212R],
σ2=2Pπd[Y3(X2+Y2)2+(2R+Y)3(X2+(2R+Y)2)212R],
σ6=2Pπd[Y2X(X2+Y2)2(2R+Y)2X(X2+(2R+Y)2)2].
ΔΓ=2πdΔn12/λ=4no3P(π11π12)λ(2R2Y+3RY2+Y3X2RYX2)(X2+4R2+4RY+Y2)(X2+Y2).
ΔΓ=4no3P(π11π12)λ(2R2sinφ+3Rρ4Rρcos2φ+ρ2sinφ2ρ2cos2φsinφ)ρ(4R2+4Rρsinφ+ρ2).
tan2ζ3=4XR(X2+Y2+2RY)4R2Y24X2R2+4RY3+4X2RY+X4+2X2Y2+Y4,
tan2ζ3=4Rcosφ(2Rsinφ+ρ)8R2cos2φ4R24Rρsinφρ2.
I=I02{1+sinΔΓsin[2(αζ3)]}=C1+C2sin[2(αC3)].
C1=I02,C2=I02sinΔΓ,C3=ζ3.
sinΔΓ=C2/C1,

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