Abstract

We have revealed topological defects (TDs) of the angle of optical indicatrix (OI) orientation in a glass Li2B4O7 sample, which originate from the specific spatial distribution of optical birefringence caused by residual mechanical stresses. It has been found that the strength of TDs of the OI rotation angle is equal to ±1/2. Following from the experimental results, we have shown that the regions around the TDs of OI orientation are those of a 3D stressed state. We have formulated criteria for determining whether 2D or 3D distributions of the optical anisotropy parameters appear, based on the TDs of OI orientation. It has been shown that, in some particular cases, the regions with the TDs can testify the availability of a 2D stressed state. Besides, we have demonstrated experimentally that the TDs can appear for more than one projection in the case of 3D distributions of the OI parameters, which appear under bending of glass plates by a distributed load.

© 2014 Optical Society of America

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References

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  1. C. A. Sciammarella and F. M. Sciammarella, Experimental Mechanics of Solids (Wiley, 2012).
  2. H. Aben, Integrated Photoelasticity (McGraw-Hill, 1979).
  3. I. Romanishin, “Tomography of stress tensor field by acoustic elasticity,” Nondestr. Test. Eval. 15, 361–371 (1998).
    [Crossref]
  4. Y.-C. Hung, J. A. Bennett, F. A. Garcia-Pastor, M. Di Michiel, J.-Y. Buffiere, T. J. A. Doel, P. Bowen, and P. J. Withers, “Fatigue crack growth and load redistribution in Ti/SiC composites observed in-situ,” Acta Mater. 57, 590–599 (2009).
    [Crossref]
  5. F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten order ungleichförmig erwärmten unkrystellinischen Körpern,” Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin II, 1–254 (1841).
  6. E. G. Coker and L. N. G. Filon, A Treatise on Photo-Elasticity (Cambridge University, 1931).
  7. V. A. Sharafutdinov, Integral Geometry of Tensor Fields (VSP, 1994).
  8. M. Defrise and G. T. Gullberg, “3D reconstruction of tensors and vectors,” 02-17-2005, Lawrence Berkeley National Laboratory. http://escholarship.org/uc/item/8df222vs .
  9. H. Aben, A. Errapart, L. Ainola, and J. Anton, “Photoelastic tomography for residual stress measurement in glass,” Opt. Eng. 44, 093601 (2005).
    [Crossref]
  10. P. M. Sutton, “Stress measurement in circular cylinders,” J. Am. Ceram. Soc. 41, 103–109 (1958).
    [Crossref]
  11. H. K. Aben, “Magnetophotoelasticity—photoelasticity in a magnetic field,” Exp. Mech. 10, 97–105 (1970).
    [Crossref]
  12. L. Ainola and H. Aben, “Alternative equations of magnetophotoelasticity and approximate solution of the inverse problem,” J. Opt. Soc. Am. A 19, 1886–1893 (2002).
    [Crossref]
  13. I. Skab, Yu. Vasylkiv, and R. Vlokh, “Induction of optical vortex in the crystals subjected to bending stresses,” Appl. Opt. 51, 5797–5805 (2012).
    [Crossref]
  14. I. Skab, Yu. 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]
  15. I. Skab, Yu. 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]
  16. I. Skab, Yu. 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]
  17. V. Savaryn, Yu. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “Polarization singularities of optical fields caused by structural dislocations in crystals,” J. Opt. 15, 044023 (2013).
    [Crossref]
  18. O. Krupych, Yu. Vasylkiv, O. Kvasnyuk, I. Skab, and R. Vlokh, “Appearance of optical singularities at the light propagation through glasses with residual stresses,” Ukr. J. Phys. Opt. 13, 170–176 (2012).
    [Crossref]
  19. H. Aben and J. Josepson, “Strange interference blots in the interferometry of inhomogeneous birefringent object,” Appl. Opt. 36, 7172–7179 (1997).
    [Crossref]
  20. V. T. Adamiv, Ya. V. Burak, R. V. Gamernyk, M. M. Romanyuk, and I. M. Teslyuk, “Optical properties of alkali and alkaline earth tetraborate glasses prepared in the alumina crucible,” Funct. Mater. 18, 298–303 (2011).
  21. V. Adamiv, I. Teslyuk, Ya. Dyachok, G. Romanyuk, O. Krupych, O. Mys, I. Martynyuk-Lototska, Ya. Burak, and R. Vlokh, “Synthesis and optical characterization of LiKB4O7, Li2B6O10, and LiCsB6O10 glasses,” Appl. Opt. 49, 5360–5365 (2010).
    [Crossref]
  22. R. Vlokh, O. Krupych, M. Kostyrko, V. Netolya, and I. Trach, “Gradient thermooptical effect in LiNbO3 crystals,” Ukr. J. Phys. Opt. 2, 154–158 (2001).
    [Crossref]
  23. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw-Hill, 1959).

2013 (1)

V. Savaryn, Yu. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “Polarization singularities of optical fields caused by structural dislocations in crystals,” J. Opt. 15, 044023 (2013).
[Crossref]

2012 (3)

2011 (3)

2010 (1)

2009 (1)

Y.-C. Hung, J. A. Bennett, F. A. Garcia-Pastor, M. Di Michiel, J.-Y. Buffiere, T. J. A. Doel, P. Bowen, and P. J. Withers, “Fatigue crack growth and load redistribution in Ti/SiC composites observed in-situ,” Acta Mater. 57, 590–599 (2009).
[Crossref]

2005 (1)

H. Aben, A. Errapart, L. Ainola, and J. Anton, “Photoelastic tomography for residual stress measurement in glass,” Opt. Eng. 44, 093601 (2005).
[Crossref]

2002 (1)

2001 (1)

R. Vlokh, O. Krupych, M. Kostyrko, V. Netolya, and I. Trach, “Gradient thermooptical effect in LiNbO3 crystals,” Ukr. J. Phys. Opt. 2, 154–158 (2001).
[Crossref]

1998 (1)

I. Romanishin, “Tomography of stress tensor field by acoustic elasticity,” Nondestr. Test. Eval. 15, 361–371 (1998).
[Crossref]

1997 (1)

1970 (1)

H. K. Aben, “Magnetophotoelasticity—photoelasticity in a magnetic field,” Exp. Mech. 10, 97–105 (1970).
[Crossref]

1958 (1)

P. M. Sutton, “Stress measurement in circular cylinders,” J. Am. Ceram. Soc. 41, 103–109 (1958).
[Crossref]

1841 (1)

F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten order ungleichförmig erwärmten unkrystellinischen Körpern,” Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin II, 1–254 (1841).

Aben, H.

Aben, H. K.

H. K. Aben, “Magnetophotoelasticity—photoelasticity in a magnetic field,” Exp. Mech. 10, 97–105 (1970).
[Crossref]

Adamiv, V.

Adamiv, V. T.

V. T. Adamiv, Ya. V. Burak, R. V. Gamernyk, M. M. Romanyuk, and I. M. Teslyuk, “Optical properties of alkali and alkaline earth tetraborate glasses prepared in the alumina crucible,” Funct. Mater. 18, 298–303 (2011).

Ainola, L.

H. Aben, A. Errapart, L. Ainola, and J. Anton, “Photoelastic tomography for residual stress measurement in glass,” Opt. Eng. 44, 093601 (2005).
[Crossref]

L. Ainola and H. Aben, “Alternative equations of magnetophotoelasticity and approximate solution of the inverse problem,” J. Opt. Soc. Am. A 19, 1886–1893 (2002).
[Crossref]

Anton, J.

H. Aben, A. Errapart, L. Ainola, and J. Anton, “Photoelastic tomography for residual stress measurement in glass,” Opt. Eng. 44, 093601 (2005).
[Crossref]

Bennett, J. A.

Y.-C. Hung, J. A. Bennett, F. A. Garcia-Pastor, M. Di Michiel, J.-Y. Buffiere, T. J. A. Doel, P. Bowen, and P. J. Withers, “Fatigue crack growth and load redistribution in Ti/SiC composites observed in-situ,” Acta Mater. 57, 590–599 (2009).
[Crossref]

Bowen, P.

Y.-C. Hung, J. A. Bennett, F. A. Garcia-Pastor, M. Di Michiel, J.-Y. Buffiere, T. J. A. Doel, P. Bowen, and P. J. Withers, “Fatigue crack growth and load redistribution in Ti/SiC composites observed in-situ,” Acta Mater. 57, 590–599 (2009).
[Crossref]

Buffiere, J.-Y.

Y.-C. Hung, J. A. Bennett, F. A. Garcia-Pastor, M. Di Michiel, J.-Y. Buffiere, T. J. A. Doel, P. Bowen, and P. J. Withers, “Fatigue crack growth and load redistribution in Ti/SiC composites observed in-situ,” Acta Mater. 57, 590–599 (2009).
[Crossref]

Burak, Ya.

Burak, Ya. V.

V. T. Adamiv, Ya. V. Burak, R. V. Gamernyk, M. M. Romanyuk, and I. M. Teslyuk, “Optical properties of alkali and alkaline earth tetraborate glasses prepared in the alumina crucible,” Funct. Mater. 18, 298–303 (2011).

Coker, E. G.

E. G. Coker and L. N. G. Filon, A Treatise on Photo-Elasticity (Cambridge University, 1931).

Di Michiel, M.

Y.-C. Hung, J. A. Bennett, F. A. Garcia-Pastor, M. Di Michiel, J.-Y. Buffiere, T. J. A. Doel, P. Bowen, and P. J. Withers, “Fatigue crack growth and load redistribution in Ti/SiC composites observed in-situ,” Acta Mater. 57, 590–599 (2009).
[Crossref]

Doel, T. J. A.

Y.-C. Hung, J. A. Bennett, F. A. Garcia-Pastor, M. Di Michiel, J.-Y. Buffiere, T. J. A. Doel, P. Bowen, and P. J. Withers, “Fatigue crack growth and load redistribution in Ti/SiC composites observed in-situ,” Acta Mater. 57, 590–599 (2009).
[Crossref]

Dyachok, Ya.

Errapart, A.

H. Aben, A. Errapart, L. Ainola, and J. Anton, “Photoelastic tomography for residual stress measurement in glass,” Opt. Eng. 44, 093601 (2005).
[Crossref]

Filon, L. N. G.

E. G. Coker and L. N. G. Filon, A Treatise on Photo-Elasticity (Cambridge University, 1931).

Gamernyk, R. V.

V. T. Adamiv, Ya. V. Burak, R. V. Gamernyk, M. M. Romanyuk, and I. M. Teslyuk, “Optical properties of alkali and alkaline earth tetraborate glasses prepared in the alumina crucible,” Funct. Mater. 18, 298–303 (2011).

Garcia-Pastor, F. A.

Y.-C. Hung, J. A. Bennett, F. A. Garcia-Pastor, M. Di Michiel, J.-Y. Buffiere, T. J. A. Doel, P. Bowen, and P. J. Withers, “Fatigue crack growth and load redistribution in Ti/SiC composites observed in-situ,” Acta Mater. 57, 590–599 (2009).
[Crossref]

Hung, Y.-C.

Y.-C. Hung, J. A. Bennett, F. A. Garcia-Pastor, M. Di Michiel, J.-Y. Buffiere, T. J. A. Doel, P. Bowen, and P. J. Withers, “Fatigue crack growth and load redistribution in Ti/SiC composites observed in-situ,” Acta Mater. 57, 590–599 (2009).
[Crossref]

Josepson, J.

Kostyrko, M.

R. Vlokh, O. Krupych, M. Kostyrko, V. Netolya, and I. Trach, “Gradient thermooptical effect in LiNbO3 crystals,” Ukr. J. Phys. Opt. 2, 154–158 (2001).
[Crossref]

Krupych, O.

V. Savaryn, Yu. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “Polarization singularities of optical fields caused by structural dislocations in crystals,” J. Opt. 15, 044023 (2013).
[Crossref]

O. Krupych, Yu. Vasylkiv, O. Kvasnyuk, I. Skab, and R. Vlokh, “Appearance of optical singularities at the light propagation through glasses with residual stresses,” Ukr. J. Phys. Opt. 13, 170–176 (2012).
[Crossref]

I. Skab, Yu. 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]

V. Adamiv, I. Teslyuk, Ya. Dyachok, G. Romanyuk, O. Krupych, O. Mys, I. Martynyuk-Lototska, Ya. Burak, and R. Vlokh, “Synthesis and optical characterization of LiKB4O7, Li2B6O10, and LiCsB6O10 glasses,” Appl. Opt. 49, 5360–5365 (2010).
[Crossref]

R. Vlokh, O. Krupych, M. Kostyrko, V. Netolya, and I. Trach, “Gradient thermooptical effect in LiNbO3 crystals,” Ukr. J. Phys. Opt. 2, 154–158 (2001).
[Crossref]

Kvasnyuk, O.

O. Krupych, Yu. Vasylkiv, O. Kvasnyuk, I. Skab, and R. Vlokh, “Appearance of optical singularities at the light propagation through glasses with residual stresses,” Ukr. J. Phys. Opt. 13, 170–176 (2012).
[Crossref]

Martynyuk-Lototska, I.

Mys, O.

Netolya, V.

R. Vlokh, O. Krupych, M. Kostyrko, V. Netolya, and I. Trach, “Gradient thermooptical effect in LiNbO3 crystals,” Ukr. J. Phys. Opt. 2, 154–158 (2001).
[Crossref]

Neumann, F. E.

F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten order ungleichförmig erwärmten unkrystellinischen Körpern,” Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin II, 1–254 (1841).

Romanishin, I.

I. Romanishin, “Tomography of stress tensor field by acoustic elasticity,” Nondestr. Test. Eval. 15, 361–371 (1998).
[Crossref]

Romanyuk, G.

Romanyuk, M. M.

V. T. Adamiv, Ya. V. Burak, R. V. Gamernyk, M. M. Romanyuk, and I. M. Teslyuk, “Optical properties of alkali and alkaline earth tetraborate glasses prepared in the alumina crucible,” Funct. Mater. 18, 298–303 (2011).

Savaryn, V.

Sciammarella, C. A.

C. A. Sciammarella and F. M. Sciammarella, Experimental Mechanics of Solids (Wiley, 2012).

Sciammarella, F. M.

C. A. Sciammarella and F. M. Sciammarella, Experimental Mechanics of Solids (Wiley, 2012).

Sharafutdinov, V. A.

V. A. Sharafutdinov, Integral Geometry of Tensor Fields (VSP, 1994).

Skab, I.

Sutton, P. M.

P. M. Sutton, “Stress measurement in circular cylinders,” J. Am. Ceram. Soc. 41, 103–109 (1958).
[Crossref]

Teslyuk, I.

Teslyuk, I. M.

V. T. Adamiv, Ya. V. Burak, R. V. Gamernyk, M. M. Romanyuk, and I. M. Teslyuk, “Optical properties of alkali and alkaline earth tetraborate glasses prepared in the alumina crucible,” Funct. Mater. 18, 298–303 (2011).

Timoshenko, S.

S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw-Hill, 1959).

Trach, I.

R. Vlokh, O. Krupych, M. Kostyrko, V. Netolya, and I. Trach, “Gradient thermooptical effect in LiNbO3 crystals,” Ukr. J. Phys. Opt. 2, 154–158 (2001).
[Crossref]

Vasylkiv, Yu.

Vlokh, R.

V. Savaryn, Yu. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “Polarization singularities of optical fields caused by structural dislocations in crystals,” J. Opt. 15, 044023 (2013).
[Crossref]

O. Krupych, Yu. Vasylkiv, O. Kvasnyuk, I. Skab, and R. Vlokh, “Appearance of optical singularities at the light propagation through glasses with residual stresses,” Ukr. J. Phys. Opt. 13, 170–176 (2012).
[Crossref]

I. Skab, Yu. 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, Yu. Vasylkiv, and R. Vlokh, “Induction of optical vortex in the crystals subjected to bending stresses,” Appl. Opt. 51, 5797–5805 (2012).
[Crossref]

I. Skab, Yu. 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, Yu. 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]

V. Adamiv, I. Teslyuk, Ya. Dyachok, G. Romanyuk, O. Krupych, O. Mys, I. Martynyuk-Lototska, Ya. Burak, and R. Vlokh, “Synthesis and optical characterization of LiKB4O7, Li2B6O10, and LiCsB6O10 glasses,” Appl. Opt. 49, 5360–5365 (2010).
[Crossref]

R. Vlokh, O. Krupych, M. Kostyrko, V. Netolya, and I. Trach, “Gradient thermooptical effect in LiNbO3 crystals,” Ukr. J. Phys. Opt. 2, 154–158 (2001).
[Crossref]

Withers, P. J.

Y.-C. Hung, J. A. Bennett, F. A. Garcia-Pastor, M. Di Michiel, J.-Y. Buffiere, T. J. A. Doel, P. Bowen, and P. J. Withers, “Fatigue crack growth and load redistribution in Ti/SiC composites observed in-situ,” Acta Mater. 57, 590–599 (2009).
[Crossref]

Woinowsky-Krieger, S.

S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw-Hill, 1959).

Zapeka, B.

Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin (1)

F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichts in komprimierten order ungleichförmig erwärmten unkrystellinischen Körpern,” Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin II, 1–254 (1841).

Acta Mater. (1)

Y.-C. Hung, J. A. Bennett, F. A. Garcia-Pastor, M. Di Michiel, J.-Y. Buffiere, T. J. A. Doel, P. Bowen, and P. J. Withers, “Fatigue crack growth and load redistribution in Ti/SiC composites observed in-situ,” Acta Mater. 57, 590–599 (2009).
[Crossref]

Appl. Opt. (4)

Exp. Mech. (1)

H. K. Aben, “Magnetophotoelasticity—photoelasticity in a magnetic field,” Exp. Mech. 10, 97–105 (1970).
[Crossref]

Funct. Mater. (1)

V. T. Adamiv, Ya. V. Burak, R. V. Gamernyk, M. M. Romanyuk, and I. M. Teslyuk, “Optical properties of alkali and alkaline earth tetraborate glasses prepared in the alumina crucible,” Funct. Mater. 18, 298–303 (2011).

J. Am. Ceram. Soc. (1)

P. M. Sutton, “Stress measurement in circular cylinders,” J. Am. Ceram. Soc. 41, 103–109 (1958).
[Crossref]

J. Opt. (1)

V. Savaryn, Yu. Vasylkiv, O. Krupych, I. Skab, and R. Vlokh, “Polarization singularities of optical fields caused by structural dislocations in crystals,” J. Opt. 15, 044023 (2013).
[Crossref]

J. Opt. Soc. Am. A (3)

Nondestr. Test. Eval. (1)

I. Romanishin, “Tomography of stress tensor field by acoustic elasticity,” Nondestr. Test. Eval. 15, 361–371 (1998).
[Crossref]

Opt. Eng. (1)

H. Aben, A. Errapart, L. Ainola, and J. Anton, “Photoelastic tomography for residual stress measurement in glass,” Opt. Eng. 44, 093601 (2005).
[Crossref]

Ukr. J. Phys. Opt. (2)

O. Krupych, Yu. Vasylkiv, O. Kvasnyuk, I. Skab, and R. Vlokh, “Appearance of optical singularities at the light propagation through glasses with residual stresses,” Ukr. J. Phys. Opt. 13, 170–176 (2012).
[Crossref]

R. Vlokh, O. Krupych, M. Kostyrko, V. Netolya, and I. Trach, “Gradient thermooptical effect in LiNbO3 crystals,” Ukr. J. Phys. Opt. 2, 154–158 (2001).
[Crossref]

Other (6)

S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw-Hill, 1959).

C. A. Sciammarella and F. M. Sciammarella, Experimental Mechanics of Solids (Wiley, 2012).

H. Aben, Integrated Photoelasticity (McGraw-Hill, 1979).

E. G. Coker and L. N. G. Filon, A Treatise on Photo-Elasticity (Cambridge University, 1931).

V. A. Sharafutdinov, Integral Geometry of Tensor Fields (VSP, 1994).

M. Defrise and G. T. Gullberg, “3D reconstruction of tensors and vectors,” 02-17-2005, Lawrence Berkeley National Laboratory. http://escholarship.org/uc/item/8df222vs .

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Figures (12)

Fig. 1.
Fig. 1.

(a) Schematic orientation of a chain of TDs of the OI orientation inside a sample in the case of 2D spatial distribution of the OI parameters (dashed line; double-side arrows correspond to the long axis of a cross-section ellipse of OI). (b) Schematic representation of a glass sample mounted onto rotation stage (1, sample; 2, sample mounting; and 3, rotation stage with a step motor).

Fig. 2.
Fig. 2.

Schematic representation of our imaging polarimeter (I, light source section; II, polarization generator; III, sample section; IV, polarization analyzer; and V, controlling unit): 1, He–Ne laser; 2, ray shutter; 3, 8, polarizers; 4, 9, quarter-wave plates; 5, coherence scrambler; 6, beam expander; 7, spatial filter; 10, analyzer; 11, objective lens; 12, CCD camera; 13, TV monitor; 14, frame grabber; 15, PC; 16, shutter’s controller; 17, step motors’ controllers; 18, step motors; 19, reference position controller.

Fig. 3.
Fig. 3.

Schematic presentation of 3D bending of a silicate glass plate (1), using a distributed load (2), and ball supports (3) placed in the corners of the plate.

Fig. 4.
Fig. 4.

XY distribution of the OI rotation angle measured for the case when light propagates along the Z axis (locations of TDs of OI orientation are indicated by arrows).

Fig. 5.
Fig. 5.

Angular dependences of phase difference measured under rotation of sample around the X (open circles) and Y (full circles) axes.

Fig. 6.
Fig. 6.

XY coordinates of TDs at the front and back sample faces, and orientation of chains of the defects inside our glass sample derived in the assumption of a 2D distribution of optical anisotropy parameters. The defect chains are depicted by dashed lines.

Fig. 7.
Fig. 7.

XY maps of light intensity in the case of transmission minimum for our glass sample placed between crossed polarizers, as observed in the vicinity of the (a) “bottom left” and (b) “central top” defects.

Fig. 8.
Fig. 8.

Dependences of phase difference on the X [panels (a), (c)] and Y [panels (b), (d)] coordinates for the (a), (b) “central top” and (c), (d) “bottom left” defects: circles correspond to the experimental data and lines to the results of linear fitting.

Fig. 9.
Fig. 9.

Map of integrated phase difference measured when the light propagates along the Z axis: locations of zeros of the phase difference that correspond to the TDs of OI orientation are indicated by arrows.

Fig. 10.
Fig. 10.

XZ map of (a) integrated phase difference and (b) OI rotation for the case when light propagates along the Y axis: locations of zeros of the phase difference corresponding to the defects are indicated by arrows.

Fig. 11.
Fig. 11.

(a) Intersection of the lines of chains of the defects inside a sample, and (b) schematic spatial distributions of OI parameters for the XY and XZ projections around the region where the lines of chains of the TDs are intersected.

Fig. 12.
Fig. 12.

(a), (b) YZ and (c), (d) XZ maps of OI orientation (a), (c) and integrated phase difference (b), (d), which correspond to bending of a silicate-glass plate by the distributed loading force P=19.6N (λ=632.8nm): locations of the TDs of OI orientations are indicated by arrows.

Equations (5)

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

δ=πn3λ(π11π12)0l(σ1σ2)dl,
I=I02{1+sinΔΓsin[2(αζ)]}=C1+C2sin[2(αC3)],
C1=I02,C2=I02sinΔΓ,C3=ζ.
sinΔΓ=C2/C1,
{Δn=12n3(π11π12)(σ1σ2),tan2ζ=σ6σ1σ2.

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