Singularities of interference of three waves with different polarization states

Piotr Kurzynowski; Władysław A. Woźniak; Marzena Zdunek; Monika Borwińska

doi:10.1364/OE.20.026755

Abstract

We presented the interference setup which can produce interesting two-dimensional patterns in polarization state of the resulting light wave emerging from the setup. The main element of our setup is the Wollaston prism which gives two plane, linearly polarized waves (eigenwaves of both Wollaston’s wedges) with linearly changed phase difference between them (along the x-axis). The third wave coming from the second arm of proposed polarization interferometer is linearly or circularly polarized with linearly changed phase difference along the y-axis. The interference of three plane waves with different polarization states (LLL – linear-linear-linear or LLC – linear-linear-circular) and variable change difference produce two-dimensional light polarization and phase distributions with some characteristic points and lines which can be claimed to constitute singularities of different types. The aim of this article is to find all kind of these phase and polarization singularities as well as their classification. We postulated in our theoretical simulations and verified in our experiments different kinds of polarization singularities, depending on which polarization parameter was considered (the azimuth and ellipticity angles or the diagonal and phase angles). We also observed the phase singularities as well as the isolated zero intensity points which resulted from the polarization singularities when the proper analyzer was used at the end of the setup. The classification of all these singularities as well as their relationships were analyzed and described.

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References

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J. F. Nye and M. V. Berry, “Dislocation in Wave Trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]
M. Soskin and M. V. Vasnetov, “Singular Optics,” in Progress in Optics, (Elsevier, 2001), Vol. 42, Chap.4.
T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115(3-4), 339–346 (1995).
[Crossref]
J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[Crossref]
P. Senthilkumaran, “Interferometric array illuminator with analysis of nonobservable fringes,” Appl. Opt. 38(8), 1311–1316 (1999).
[Crossref] [PubMed]
F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95(25), 253901 (2005).
[Crossref]
P. Kurzynowski, W. A. Woźniak, and E. Frą Czek, “Optical vortices generation using the Wollaston prism,” Appl. Opt. 45(30), 7898–7903 (2006).
[Crossref] [PubMed]
P. Kurzynowski and M. Borwińska, “Generation of vortex-type markers in a one-wave setup,” Appl. Opt. 46(5), 676–679 (2007).
[Crossref] [PubMed]
I. Freund, “Polarization singularities in optical lattices,” Opt. Lett. 29(8), 875–877 (2004).
[Crossref] [PubMed]
M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002).
[Crossref]
I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
[Crossref] [PubMed]
O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207(1-6), 57–65 (2002).
[Crossref]
K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express 16(2), 695–709 (2008).
[Crossref] [PubMed]
P. Kurzynowski, W. A. Woźniak, and M. Borwińska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12(3), 035406 (2010).
[Crossref]
R. Azzam and N. Bashara, Ellipsometry and Polarized Light, (North-Holland Publishing Company, 1977).
W. A. Woźniak, P. Kurzynowski, and S. Drobczyński, “Adjustment method of an imaging Stokes polarimeter based on liquid crystal variable retarders,” Appl. Opt. 50(2), 203–212 (2011).
[Crossref] [PubMed]

2011 (1)

W. A. Woźniak, P. Kurzynowski, and S. Drobczyński, “Adjustment method of an imaging Stokes polarimeter based on liquid crystal variable retarders,” Appl. Opt. 50(2), 203–212 (2011).
[Crossref] [PubMed]

2010 (1)

P. Kurzynowski, W. A. Woźniak, and M. Borwińska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12(3), 035406 (2010).
[Crossref]

2005 (1)

F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95(25), 253901 (2005).
[Crossref]

2004 (1)

I. Freund, “Polarization singularities in optical lattices,” Opt. Lett. 29(8), 875–877 (2004).
[Crossref] [PubMed]

2002 (3)

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002).
[Crossref]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
[Crossref] [PubMed]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207(1-6), 57–65 (2002).
[Crossref]

2001 (1)

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[Crossref]

1999 (1)

P. Senthilkumaran, “Interferometric array illuminator with analysis of nonobservable fringes,” Appl. Opt. 38(8), 1311–1316 (1999).
[Crossref] [PubMed]

1995 (1)

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115(3-4), 339–346 (1995).
[Crossref]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocation in Wave Trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Ackemann, T.

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115(3-4), 339–346 (1995).
[Crossref]

Angelsky, O. V.

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207(1-6), 57–65 (2002).
[Crossref]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
[Crossref] [PubMed]

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocation in Wave Trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express 16(2), 695–709 (2008).
[Crossref] [PubMed]

Borwinska, M.

P. Kurzynowski, W. A. Woźniak, and M. Borwińska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12(3), 035406 (2010).
[Crossref]

P. Kurzynowski and M. Borwińska, “Generation of vortex-type markers in a one-wave setup,” Appl. Opt. 46(5), 676–679 (2007).
[Crossref] [PubMed]

Dennis, M.

F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95(25), 253901 (2005).
[Crossref]

Dennis, M. R.

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002).
[Crossref]

Drobczynski, S.

W. A. Woźniak, P. Kurzynowski, and S. Drobczyński, “Adjustment method of an imaging Stokes polarimeter based on liquid crystal variable retarders,” Appl. Opt. 50(2), 203–212 (2011).
[Crossref] [PubMed]

Dubik, B.

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[Crossref]

Flossmann, F.

F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95(25), 253901 (2005).
[Crossref]

Fra Czek, E.

P. Kurzynowski, W. A. Woźniak, and E. Frą Czek, “Optical vortices generation using the Wollaston prism,” Appl. Opt. 45(30), 7898–7903 (2006).
[Crossref] [PubMed]

Freund, I.

I. Freund, “Polarization singularities in optical lattices,” Opt. Lett. 29(8), 875–877 (2004).
[Crossref] [PubMed]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
[Crossref] [PubMed]

Hasman, E.

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express 16(2), 695–709 (2008).
[Crossref] [PubMed]

Kleiner, V.

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express 16(2), 695–709 (2008).
[Crossref] [PubMed]

Kriege, E.

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115(3-4), 339–346 (1995).
[Crossref]

Kurzynowski, P.

W. A. Woźniak, P. Kurzynowski, and S. Drobczyński, “Adjustment method of an imaging Stokes polarimeter based on liquid crystal variable retarders,” Appl. Opt. 50(2), 203–212 (2011).
[Crossref] [PubMed]

P. Kurzynowski, W. A. Woźniak, and M. Borwińska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12(3), 035406 (2010).
[Crossref]

P. Kurzynowski and M. Borwińska, “Generation of vortex-type markers in a one-wave setup,” Appl. Opt. 46(5), 676–679 (2007).
[Crossref] [PubMed]

P. Kurzynowski, W. A. Woźniak, and E. Frą Czek, “Optical vortices generation using the Wollaston prism,” Appl. Opt. 45(30), 7898–7903 (2006).
[Crossref] [PubMed]

Lange, W.

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115(3-4), 339–346 (1995).
[Crossref]

Maier, M.

F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95(25), 253901 (2005).
[Crossref]

Masajada, J.

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[Crossref]

Mokhun, A. I.

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
[Crossref] [PubMed]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207(1-6), 57–65 (2002).
[Crossref]

Mokhun, I. I.

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207(1-6), 57–65 (2002).
[Crossref]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
[Crossref] [PubMed]

Niv, A.

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express 16(2), 695–709 (2008).
[Crossref] [PubMed]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocation in Wave Trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Schwarz, U.

F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95(25), 253901 (2005).
[Crossref]

Senthilkumaran, P.

P. Senthilkumaran, “Interferometric array illuminator with analysis of nonobservable fringes,” Appl. Opt. 38(8), 1311–1316 (1999).
[Crossref] [PubMed]

Soskin, M. S.

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207(1-6), 57–65 (2002).
[Crossref]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
[Crossref] [PubMed]

Wozniak, W. A.

W. A. Woźniak, P. Kurzynowski, and S. Drobczyński, “Adjustment method of an imaging Stokes polarimeter based on liquid crystal variable retarders,” Appl. Opt. 50(2), 203–212 (2011).
[Crossref] [PubMed]

P. Kurzynowski, W. A. Woźniak, and M. Borwińska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12(3), 035406 (2010).
[Crossref]

P. Kurzynowski, W. A. Woźniak, and E. Frą Czek, “Optical vortices generation using the Wollaston prism,” Appl. Opt. 45(30), 7898–7903 (2006).
[Crossref] [PubMed]

Appl. Opt. (4)

P. Senthilkumaran, “Interferometric array illuminator with analysis of nonobservable fringes,” Appl. Opt. 38(8), 1311–1316 (1999).
[Crossref] [PubMed]

P. Kurzynowski, W. A. Woźniak, and E. Frą Czek, “Optical vortices generation using the Wollaston prism,” Appl. Opt. 45(30), 7898–7903 (2006).
[Crossref] [PubMed]

P. Kurzynowski and M. Borwińska, “Generation of vortex-type markers in a one-wave setup,” Appl. Opt. 46(5), 676–679 (2007).
[Crossref] [PubMed]

W. A. Woźniak, P. Kurzynowski, and S. Drobczyński, “Adjustment method of an imaging Stokes polarimeter based on liquid crystal variable retarders,” Appl. Opt. 50(2), 203–212 (2011).
[Crossref] [PubMed]

J. Opt. (1)

P. Kurzynowski, W. A. Woźniak, and M. Borwińska, “Regular lattices of polarization singularities: their generation and properties,” J. Opt. 12(3), 035406 (2010).
[Crossref]

Opt. Commun. (4)

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002).
[Crossref]

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. S. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207(1-6), 57–65 (2002).
[Crossref]

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115(3-4), 339–346 (1995).
[Crossref]

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[Crossref]

Opt. Express (1)

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express 16(2), 695–709 (2008).
[Crossref] [PubMed]

Opt. Lett. (2)

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
[Crossref] [PubMed]

I. Freund, “Polarization singularities in optical lattices,” Opt. Lett. 29(8), 875–877 (2004).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95(25), 253901 (2005).
[Crossref]

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

J. F. Nye and M. V. Berry, “Dislocation in Wave Trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Other (2)

M. Soskin and M. V. Vasnetov, “Singular Optics,” in Progress in Optics, (Elsevier, 2001), Vol. 42, Chap.4.

R. Azzam and N. Bashara, Ellipsometry and Polarized Light, (North-Holland Publishing Company, 1977).

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Fig. 1

Scheme of the setup realizing the interference of three plane waves with different polarization states and different phase distributions.

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Fig. 2

a) Intensity $I$ and b) $V_{1}$ , c) $V_{2}$ , d) $V_{3}$ Stokes vectors parameters distributions for the LLL interference – theoretical simulations.

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Fig. 3

a) $α, ϑ$ and b) $δ, β$ distributions for the LLL interference – theoretical simulations. Different colors denote the $α$ or $δ$ distributions while black lines denotes the $ϑ$ or $β$ distributions, respectively. $P_{0}$ – the points in which the zero intensity of the light occur. $P_{R}$ and $P_{L}$ – the points representing right- and left-handed circular polarization states of the light.

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Fig. 4

a) Intensity $I$ and b) $V_{1}$ , c) $V_{2}$ , d) $V_{3}$ Stokes vectors parameters distributions for the LLC interference – theoretical simulations.

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Fig. 5

a) $α, ϑ$ and b) $δ, β$ distributions for the “LLC interference” – theoretical simulations. Different colors denote the $α$ or $δ$ distributions while black lines – $ϑ$ or $β$ distributions, respectively. $P_{0}$ – the points in which the zero intensity of the light occur. $P_{p}$ and $P_{m}$ – the points representing linear polarization states of the light with the azimuth angles equal to −45°, and + 45°, respectively.

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Fig. 6

Properties of the light field in LLC interference with the linear analyzer at the end of the setup: the intensity distributions $I$ for the analyzer’s azimuth angle equal to a) −45° and b) + 45°; c) the phase distribution for the azimuth angle of the analyzer equal to −45°; d) the scheme of the sublattices shift caused by the analyzer’s rotation.

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Fig. 7

a) Intensity $I$ and b) $V_{1}$ , c) $V_{2}$ , d) $V_{3}$ Stokes vectors parameters distributions for the LLL interference – experimental results.

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Fig. 8

a) Intensity $I$ and b) $V_{1}$ , c) $V_{2}$ , d) $V_{3}$ Stokes vectors parameters distributions for the LLC interference – experimental results.

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Fig. 9

Intensity $I$ distributions for the LLC interference with the linear analyzer at the end of the setup – experimental results for two different analyzer’s azimuthal orientation: a) the azimuth angle of the analyzer equal to −45° and b) equal to + 45°.

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Equations (8)

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

E_{A} = \exp (+ i K x) [\begin{matrix} 1 \\ 0 \end{matrix}]

E_{B} = \exp (- i K x) [\begin{matrix} 0 \\ 1 \end{matrix}]

E_{C} = \exp (+ i K y) [\begin{matrix} 1 \\ \exp (i γ) \end{matrix}],

E \equiv [\begin{matrix} E_{x} \\ E_{y} \end{matrix}] = E_{A} + E_{B} + E_{C} = [\begin{matrix} \exp (+ i K x) + \exp (+ i K y) \\ \exp (- i K x) + \exp (i γ) \cdot \exp (+ i K y) \end{matrix}]

I = {| E_{x} |}^{2} + {| E_{y} |}^{2}

\tan β = \frac{| E_{y} |}{| E_{x} |}

δ = \arg (E_{x} \cdot E_{y}^{*})

V = [\begin{matrix} V_{1} \\ V_{2} \\ V_{3} \end{matrix}] = [\begin{matrix} \cos 2 α \cdot \cos 2 ϑ \\ \sin 2 α \cdot \cos 2 ϑ \\ \sin 2 ϑ \end{matrix}] = [\begin{matrix} \cos 2 β \\ \sin 2 β \cdot \cos δ \\ \sin 2 β \cdot \sin δ \end{matrix}]

Abstract

References

2011 (1)

2010 (1)

2008 (1)

2007 (1)

2006 (1)

2005 (1)

2004 (1)

2002 (3)

2001 (1)

1999 (1)

1995 (1)

1974 (1)

Ackemann, T.

Angelsky, O. V.

Berry, M. V.

Bliokh, K. Y.

Borwinska, M.

Dennis, M.

Dennis, M. R.

Drobczynski, S.

Dubik, B.

Flossmann, F.

Fra Czek, E.

Freund, I.

Hasman, E.

Kleiner, V.

Kriege, E.

Kurzynowski, P.

Lange, W.

Maier, M.

Masajada, J.

Mokhun, A. I.

Mokhun, I. I.

Niv, A.

Nye, J. F.

Schwarz, U.

Senthilkumaran, P.

Soskin, M. S.

Wozniak, W. A.

Appl. Opt. (4)

J. Opt. (1)

Opt. Commun. (4)

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

Other (2)

Cited By

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Equations (8)

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