Abstract

We describe a method for generation of optical vortices that relies on bending of transparent parallelepiped-shaped samples fabricated from either glass or crystalline solid materials. It is shown that the induced singularity of optical indicatrix rotation leads in general to appearance of a mixed screw-edge dislocation of the phase front of outgoing optical beam. At the same time, some specified geometrical parameters of the sample can ensure generation of a purely screw dislocation of the phase front and, as a result, a singly charged canonical optical vortex.

© 2012 Optical Society of America

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References

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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]
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    [CrossRef]
  9. S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express 19, 4085–4090 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. 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]
  15. 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]
  16. 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]
  17. I. Skab, Y. Vasylkiv, O. Krupych, V. Savaryn, and R. Vlokh, “Generation of doubly charged vortex beam by concentrated loading of glass disks along their diameter,” Appl. Opt. 51, 1631–1637 (2012).
    [CrossRef]
  18. B. G. Mytsyk, A. S. Andrushchak, N. M. Demyanyshyn, Y. P. Kost, A. V. Kityk, P. Mandracci, and W. Schranz, “Piezo-optic coefficients of MgO-doped LiNbO3 crystals,” Appl. Opt. 48, 1904–1911 (2009).
    [CrossRef]
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  20. S. P. Timoshenko, Strength of Materials. Part II: Advanced Theory and Problems (Nauka, 1965).
  21. R. Vlokh, Y. Pyatak, and I. Skab, “Elasto-optic effect in LiNbO3 under the crystal bending,” Ferroelectrics 126, 239–242 (1992).
    [CrossRef]
  22. N. D. Mermin, “Topological theory of defects,” Rev. Mod. Phys. 51, 591–648 (1979).
    [CrossRef]

2012

2011

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]

S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express 19, 4085–4090 (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).
[CrossRef]

2010

K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18, 4518–4525 (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]

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]

2009

2008

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]

2001

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

1999

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

1998

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

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

1994

1992

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

1979

N. D. Mermin, “Topological theory of defects,” Rev. Mod. Phys. 51, 591–648 (1979).
[CrossRef]

Andrushchak, A. S.

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]

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]

Chigrinov, V.

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]

De Martini, F.

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]

Demyanyshyn, N. M.

Desyatnikov, A.

DiVincenzo, D. P.

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

Du, T.

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]

Houghton, J. K.

Jacobs, S. D.

Kilin, S. Y.

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

Kitamura, K.

Kityk, A. V.

Kivshar, Y.

Korenic, E. M.

Kost, Y. P.

Kreuzer, F.

Krolikowski, W.

Krupych, O.

I. Skab, Y. Vasylkiv, O. Krupych, V. Savaryn, and R. Vlokh, “Generation of doubly charged vortex beam by concentrated loading of glass disks along their diameter,” Appl. Opt. 51, 1631–1637 (2012).
[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]

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]

Mandracci, P.

Marrucci, L.

S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express 19, 4085–4090 (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]

Mermin, N. D.

N. D. Mermin, “Topological theory of defects,” Rev. Mod. Phys. 51, 591–648 (1979).
[CrossRef]

Murauski, A.

Mytsyk, B. G.

Noda, S.

Piccirillo, B.

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]

Rode, A.

Sakai, K.

Santamato, E.

S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express 19, 4085–4090 (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.

Schmid, A.

Schranz, W.

Shvedov, V.

Skab, I.

I. Skab, Y. Vasylkiv, O. Krupych, V. Savaryn, and R. Vlokh, “Generation of doubly charged vortex beam by concentrated loading of glass disks along their diameter,” Appl. Opt. 51, 1631–1637 (2012).
[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).
[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, 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]

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.

S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express 19, 4085–4090 (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]

Soskin, M. S.

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

Timoshenko, S. P.

S. P. Timoshenko, Strength of Materials. Part I: Elementary Theory and Problems (Nauka, 1965).

S. P. Timoshenko, Strength of Materials. Part II: Advanced Theory and Problems (Nauka, 1965).

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, O. Krupych, V. Savaryn, and R. Vlokh, “Generation of doubly charged vortex beam by concentrated loading of glass disks along their diameter,” Appl. Opt. 51, 1631–1637 (2012).
[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. 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]

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, 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).
[CrossRef]

Vlokh, R.

I. Skab, Y. Vasylkiv, O. Krupych, V. Savaryn, and R. Vlokh, “Generation of doubly charged vortex beam by concentrated loading of glass disks along their diameter,” Appl. Opt. 51, 1631–1637 (2012).
[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).
[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, 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]

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]

Zapeka, B.

Appl. Opt.

Appl. Phys. Lett.

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]

Ferroelectrics

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

J. Opt. Soc. Am. A

Mol. Cryst. Liq. Cryst.

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. Express

Phys. Rev. A

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.

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.

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

Rev. Mod. Phys.

N. D. Mermin, “Topological theory of defects,” Rev. Mod. Phys. 51, 591–648 (1979).
[CrossRef]

Science

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

Sov. Phys. Usp.

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

Ukr. J. Phys. Opt.

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]

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]

Other

S. P. Timoshenko, Strength of Materials. Part I: Elementary Theory and Problems (Nauka, 1965).

S. P. Timoshenko, Strength of Materials. Part II: Advanced Theory and Problems (Nauka, 1965).

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Figures (9)

Fig. 1.
Fig. 1.

Schematic illustrating the concentrated loading of the crystalline sample.

Fig. 2.
Fig. 2.

(a) Simulated dependences of optical indicatrix rotation angle on the tracing angle φ for different dimensionless coordinate ρ˜ (ρ˜=0.5, balls; 0.25, full circles; 0.1, semiopen circles; 0.05, open circle; 0.01, crosses); (b) theoretical XY maps of optical indicatrix rotation simulated for the case of LiNbO3 crystals subjected to concentrated loading.

Fig. 3.
Fig. 3.

Schematic representing the application of the load distributed over the entire upper-sample surface.

Fig. 4.
Fig. 4.

Spatial distributions of (a) birefringence and (b) optical indicatrix rotation angle simulated for the LiNbO3 crystal bent by a load distributed over the entire upper-sample surface (P1=58.86N; hereafter, the light wavelength is equal to λ=632.8nm).

Fig. 5.
Fig. 5.

Schematic illustrating the application of the load distributed over the distance d on the upper-sample surface.

Fig. 6.
Fig. 6.

Spatial distributions of (a) birefringence and (b) optical indicatrix rotation angle simulated for the LiNbO3 crystal bent by a load distributed over the distance d=0.66mm on the upper-sample surface.

Fig. 7.
Fig. 7.

Comparison of (a) and (b) experimental and (c) and (d) theoretical spatial distributions of (a) and (c) birefringence and (b) and (d) optical indicatrix rotation angle for the LiNbO3 crystals bent by a concentrated load P1=14.72N.

Fig. 8.
Fig. 8.

Dependences of optical indicatrix rotation angle on the tracing angle φ for the (a) first (lower) singularity and (b) second (upper) singularity calculated for different dimensionless coordinate ρ˜ (ρ˜=0.01, crosses; 0.025, triangles; 0.05, open circles; 0.075, stars; 0.1, semiopen circles; 0.25, full circles; 0.281, balls).

Fig. 9.
Fig. 9.

Experimental distributions of (a) birefringence and (b) optical indicatrix rotation angle for the LiNbO3 crystals bent by a load distributed over the entire upper-sample surface (q=3234N/m). (c) View of elliptical optical vortex observed behind the system of crossed circular polarizers and a bent sample in between.

Equations (44)

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

σ2=6P1bh3X(l2+Y),
σ6=3P1bh3(h24X2),
σ2=MX/JZ=6P1bh3X(l2Y),
σ6=3P1bh3(h24X2),
Δn12=no32π666P1bh3X2(l2+Y)2+(h24X2)2,
tan2ζZ=h2/4X2X(l/2+Y),
Δn12=no32π666P1bh3X2(l2Y)2+(h24X2)2,
tan2ζZ=h2/4X2X(l/2Y),
tan2ζZ=h2/4ρ2cos2φρcosφ(l/2ρsinφ)for0<φ<π,
tan2ζZ=h2/4ρ2cos2φρcosφ(l/2+ρsinφ)forπ<φ<2π.
tan2ζZ=hsinφ(lhsinφ)tanφ.
Q=qY,
M=q2(l24Y2),
σ2=MJZX=6qbh3(h24Y2)X,
σ6=Q2JZ(h24X2)=6qbh3(h24X2)Y.
tan2ζZ=2σ6σ2=2(h24X2)Y(l24Y2)X=2(h24ρ2cos2φ)(l24ρ2sin2φ)tanφ,
Δn12=3no3qπ66bh3(l24Y2)2X2+4(h24X2)2Y2.
tan2ζZ=2h2l2tanφ,
Δn12=3no3P1π6622bh3X2+Y2,
tan2ζZ=tanφ(orζZ=φ/2).
Q=P12,M=P12(l2+Y).
σ2=MJZX=6P1bh3(l2+Y)X,
σ6=Q2JZ(h24X)=3P1bh3(h22+X2).
Δn12=3no3P1bh3π66(l2+Y)2X2+(h24+X2)2,
tan2ζZ=(h2/4X2)(l/2+Y)X.
Q=P12,M=P12(l2Y),
σ2=MJZX=6P1bh3(l2+Y)X,
σ6=Q2JZ(h24X)=3P1bh3(h22+X2),
Δn12=3no3P1bh3π66(l2Y)2X2+(h24X2)2,
tan2ζZ=(h2/4X2)(l/2Y)X.
Q=P1dY,M=P12(l2+Y14d(2Y+d)2).
σ2=MJZX=6P1bh3d(2ld)4Y24dX,
σ6=Q2JZ(h24X)=6P1Ybh3d(h24X2),
tan2ζZ=2(h2/4X2)Y(d(2ld)/4Y2)X=2(h2/4ρ2cos2φ)(d(2ld)/4ρ2sin2φ)tanφ,
Δn12=3no3P1bh3dπ66(d(2ld)4Y2)2X2+4(h24X2)2Y2.
tan2ζZ=2h2d(2ld)tanφ,
tan2ζZ=tanφ(orζZ=φ/2);
Δn12=3no3P1l2bh3π66X2+Y2.
σ1=2P1cos4(arctanYh/2+X)πb(h/2+X).
Δn12=no32π66(σ1σ2)2+4σ62,
tan2ζZ=2σ6σ1σ2.
Δn12=no3π66P1b(cos4Θπ(h/2+X)+3h3(l/2±Y)X)2+9h6(h2/4+X2)2,
tan2ζZ=3(h2/4X2)h3cos4Θπ(h/2+X)+3(l/2±Y)X,
Δn12=no32π66P1b(2π(h/2+X)+3lh3X),tan2ζZ=0.

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