Abstract

A new application of the optical vortex birefringence compensator has been presented. Two parameters of elliptically birefringent medium, an ellipticity angle and an optical path difference, can be measured simultaneously in the setup based on the C polarization type singularities generated using two Wollaston compensators. The theory and numerical investigations of the proposed method have been presented as well as some experiments verifying our theoretical predictions.

© 2008 Optical Society of America

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References

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  1. M. Soskin and M. V. Vasnetov, “Singular optics,” in Progress in Optics (Elsevier, 2001), Vol. 42, Chap. 4.
  2. J. Masajada, A. Popiołek-Masajada, and D. Wieliczka, “The interferometric system using optical vortices as a phase markers,” Opt. Commun. 207, 85-93 (2002).
    [CrossRef]
  3. P. Kurzynowski, W. A. Woźniak, and E. Frączek, “Optical vortices generation using the Wollaston prism,” Appl. Opt. 45, 7898-7903 (2006).
    [CrossRef] [PubMed]
  4. M. Borwińska, A. Popiołek-Masajada, and B. Dubik, “Reconstruction of the plane wave tilt and its orientation using optical vortex interferometer,” Opt. Eng. 46, 073604 (2007).
    [CrossRef]
  5. A. Popiołek-Masajada, P. Kurzynowski, W. A. Woźniak, and M. Borwińska, “Measurements of the small wave tilt using the optical vortex interferometer with the Wollaston compensator,” Appl. Opt. 46, 8039-8044 (2007).
    [CrossRef] [PubMed]
  6. W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. Hanson, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120-127 (2006).
    [CrossRef] [PubMed]
  7. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201-221 (2002).
    [CrossRef]
  8. F. Flossman, U. T. Schwarz, and M. Maier, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
    [CrossRef]
  9. M. Borwińska, A. Popiołek-Masajada, and P. Kurzynowski, “Measurements of birefringent media properties using optical birefringence compensator,” Appl. Opt. 46, 6419-6426 (2007).
    [CrossRef] [PubMed]
  10. J. Kobayashi and Y. Uesu, “A new optical method and apparatus 'HAUP' for measuring simultaneously optical activity and birefringence of crystals. I. Principles and construction,” J. Appl. Crystallogr. 16, 204-211 (1983).
    [CrossRef]
  11. C. C. Montarou and T. K. Gaylord, “Two-wave-plate compensator for single-point retardation measurement,” Appl. Opt. 43, 6580-6595 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
  15. I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164-5172(1994).
  16. P. Gomez and C. Hernandez, “High-accuracy universal polarimeters measurement of optical activity and birefringence of α-quartz in the presence of multiple reflections,” J. Opt. Soc. Am. B 15, 1147-1154 (1998).
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    [CrossRef]
  18. J. Masajada, A. Popiołek-Masajada, E. Frączek, and W. Frączek, “Vortex point localization problem in optical vortices Interferometry,” Opt. Commun. 234, 23-28 (2004).
    [CrossRef]

2007 (4)

2006 (3)

2005 (1)

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

2004 (2)

C. C. Montarou and T. K. Gaylord, “Two-wave-plate compensator for single-point retardation measurement,” Appl. Opt. 43, 6580-6595 (2004).
[CrossRef]

J. Masajada, A. Popiołek-Masajada, E. Frączek, and W. Frączek, “Vortex point localization problem in optical vortices Interferometry,” Opt. Commun. 234, 23-28 (2004).
[CrossRef]

2002 (2)

J. Masajada, A. Popiołek-Masajada, and D. Wieliczka, “The interferometric system using optical vortices as a phase markers,” Opt. Commun. 207, 85-93 (2002).
[CrossRef]

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

2001 (2)

1998 (1)

1994 (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164-5172(1994).

1983 (1)

J. Kobayashi and Y. Uesu, “A new optical method and apparatus 'HAUP' for measuring simultaneously optical activity and birefringence of crystals. I. Principles and construction,” J. Appl. Crystallogr. 16, 204-211 (1983).
[CrossRef]

1982 (1)

Berezhna, S. Y.

Berezhnyy, I. V.

Borwinska, M.

Dennis, M. R.

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

Drobczynski, S.

P. Kurzynowski, W. A. Woźniak, S. Drobczyński, “A new phase difference compensation method for elliptically birefringent media,” Opt. Commun. 267, 44-49 (2006).
[CrossRef]

Dubik, B.

M. Borwińska, A. Popiołek-Masajada, and B. Dubik, “Reconstruction of the plane wave tilt and its orientation using optical vortex interferometer,” Opt. Eng. 46, 073604 (2007).
[CrossRef]

Flossman, F.

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

Fraczek, E.

P. Kurzynowski, W. A. Woźniak, and E. Frączek, “Optical vortices generation using the Wollaston prism,” Appl. Opt. 45, 7898-7903 (2006).
[CrossRef] [PubMed]

J. Masajada, A. Popiołek-Masajada, E. Frączek, and W. Frączek, “Vortex point localization problem in optical vortices Interferometry,” Opt. Commun. 234, 23-28 (2004).
[CrossRef]

Fraczek, W.

J. Masajada, A. Popiołek-Masajada, E. Frączek, and W. Frączek, “Vortex point localization problem in optical vortices Interferometry,” Opt. Commun. 234, 23-28 (2004).
[CrossRef]

Freund, I.

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164-5172(1994).

Gaylord, T. K.

Gomez, P.

Hanson, S.

Hernandez, C.

Ina, H.

Ishijima, R.

Kobayashi, H.

Kobayashi, J.

J. Kobayashi and Y. Uesu, “A new optical method and apparatus 'HAUP' for measuring simultaneously optical activity and birefringence of crystals. I. Principles and construction,” J. Appl. Crystallogr. 16, 204-211 (1983).
[CrossRef]

Kurzynowski, P.

Maier, M.

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

Masajada, J.

J. Masajada, A. Popiołek-Masajada, E. Frączek, and W. Frączek, “Vortex point localization problem in optical vortices Interferometry,” Opt. Commun. 234, 23-28 (2004).
[CrossRef]

J. Masajada, A. Popiołek-Masajada, and D. Wieliczka, “The interferometric system using optical vortices as a phase markers,” Opt. Commun. 207, 85-93 (2002).
[CrossRef]

Miyamoto, Y.

Montarou, C. C.

Popiolek-Masajada, A.

A. Popiołek-Masajada, P. Kurzynowski, W. A. Woźniak, and M. Borwińska, “Measurements of the small wave tilt using the optical vortex interferometer with the Wollaston compensator,” Appl. Opt. 46, 8039-8044 (2007).
[CrossRef] [PubMed]

M. Borwińska, A. Popiołek-Masajada, and P. Kurzynowski, “Measurements of birefringent media properties using optical birefringence compensator,” Appl. Opt. 46, 6419-6426 (2007).
[CrossRef] [PubMed]

M. Borwińska, A. Popiołek-Masajada, and B. Dubik, “Reconstruction of the plane wave tilt and its orientation using optical vortex interferometer,” Opt. Eng. 46, 073604 (2007).
[CrossRef]

J. Masajada, A. Popiołek-Masajada, E. Frączek, and W. Frączek, “Vortex point localization problem in optical vortices Interferometry,” Opt. Commun. 234, 23-28 (2004).
[CrossRef]

J. Masajada, A. Popiołek-Masajada, and D. Wieliczka, “The interferometric system using optical vortices as a phase markers,” Opt. Commun. 207, 85-93 (2002).
[CrossRef]

Schwarz, U. T.

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

Shvartsman, N.

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164-5172(1994).

Soskin, M.

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

Takashi, M.

Takeda, M.

Uesu, Y.

J. Kobayashi and Y. Uesu, “A new optical method and apparatus 'HAUP' for measuring simultaneously optical activity and birefringence of crystals. I. Principles and construction,” J. Appl. Crystallogr. 16, 204-211 (1983).
[CrossRef]

Vasnetov, M. V.

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

Wada, A.

Wang, W.

Wieliczka, D.

J. Masajada, A. Popiołek-Masajada, and D. Wieliczka, “The interferometric system using optical vortices as a phase markers,” Opt. Commun. 207, 85-93 (2002).
[CrossRef]

Wozniak, W. A.

Yokozeki, T.

Appl. Opt. (5)

J. Appl. Crystallogr. (1)

J. Kobayashi and Y. Uesu, “A new optical method and apparatus 'HAUP' for measuring simultaneously optical activity and birefringence of crystals. I. Principles and construction,” J. Appl. Crystallogr. 16, 204-211 (1983).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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

Opt. Commun. (4)

P. Kurzynowski, W. A. Woźniak, S. Drobczyński, “A new phase difference compensation method for elliptically birefringent media,” Opt. Commun. 267, 44-49 (2006).
[CrossRef]

J. Masajada, A. Popiołek-Masajada, E. Frączek, and W. Frączek, “Vortex point localization problem in optical vortices Interferometry,” Opt. Commun. 234, 23-28 (2004).
[CrossRef]

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

J. Masajada, A. Popiołek-Masajada, and D. Wieliczka, “The interferometric system using optical vortices as a phase markers,” Opt. Commun. 207, 85-93 (2002).
[CrossRef]

Opt. Eng. (1)

M. Borwińska, A. Popiołek-Masajada, and B. Dubik, “Reconstruction of the plane wave tilt and its orientation using optical vortex interferometer,” Opt. Eng. 46, 073604 (2007).
[CrossRef]

Opt. Express (1)

Phys. Rev. (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164-5172(1994).

Phys. Rev. Lett. (1)

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

Other (1)

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

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Figures (8)

Fig. 1
Fig. 1

Scheme of the interferometer setup: L, laser; C, collimator; P, polarizer ( α P = 0 ); W1, first Wollaston compensator ( α W 1 = 45 ); M, measured medium ( α = 45 ); W2, second Wollaston compensator ( α W 2 = 0 ); A, analyzer ( α A = 45 ); and CCD, camera.

Fig. 2
Fig. 2

Numerically generated intensity distribution for the measurement setup: dashed lines, elementary lattice cells; solid lines, the directions of constant phase lines given by each Wollaston compensator separately; Λ 1 and Λ 2 , fringe distances for both Wollaston compensators; circles denote the positions of OVs.

Fig. 3
Fig. 3

Measurement method of the quantities ψ + and ψ : Λ + and Λ , fringe distances for the equivalent Wollaston compensators; Φ + and Φ , measured OV’s displacements. The positions of the OVs in elementary cell without and with birefringent medium in the setup are marked by black and white circles, respectively.

Fig. 4
Fig. 4

Behavior of the single optical vortex for variable phase difference γ for some constant ellipticity angles ϑ (from 45 to 45 ).

Fig. 5
Fig. 5

Behavior of the single optical vortex for variable ellipticity angle ϑ for some constant phase differences γ.

Fig. 6
Fig. 6

Scheme of the experimental setup: L, laser; C, collimator; P, polarizer ( α P = 0 ); W1, first Wollaston compensator ( α W 1 = 45 ); LCM1, first liquid crystal phase modulator ( α LCM 1 = 0 ); E, Ehringhaus compensator ( α E = 45 ); LCM2, second liquid crystal phase modulator ( α LCM 2 = 90 ); W2, second Wollaston compensator ( α W 2 = 0 ); A, analyzer ( α A = 45 ); and CCD, camera (here, rotated by angle α CCD = 22.5 ).

Fig. 7
Fig. 7

Behavior of the vortex lattice for some particular ellipticity angles ϑ of the medium (simulated by some particular phase shifts γ 0 = γ 90 introduced by liquid crystal modulators): (a)  ϑ = 0 , (b)  ϑ = 10 , (c)  ϑ = 20 , (d)  ϑ = 30 , (e)  ϑ = 40 , and (f)  ϑ = 20 ; the points correspond to the sequence of the vortices positions for increasing the phase difference γ.

Fig. 8
Fig. 8

Changes of vortices positions for three different ellipticity angles ϑ ( 0 , 2 0 , and 45 ° ) and phase difference γ changed from 0 to 36 0 with a constant step equal to 2 0 .

Tables (1)

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Table 1 Results of Errors Estimations

Equations (16)

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[ M ] = [ 1 0 0 0 0 * * * 0 m 32 * m 34 0 m 42 * m 44 ] ,
m 32 = sin ( γ ) · sin ( 2 ϑ ) , m 34 = sin ( 2 ϑ ) · cos ( 2 ϑ ) · ( 1 cos ( γ ) ) , m 42 = sin ( γ ) · cos ( 2 ϑ ) , m 44 = sin 2 ( 2 ϑ ) · ( 1 cos ( γ ) ) + cos ( γ ) ,
I out ( x , y ) = 1 2 ( 1 + m 32 · cos δ 1 · cos δ 2 + m 34 · sin δ 1 · cos δ 2 + m 42 · cos δ 1 · sin δ 2 + m 44 · sin δ 1 · sin δ 2 ) .
δ ± = δ 1 ± δ 2 .
2 a · cos ψ + m 44 m 32 , 2 a · sin ψ + m 42 + m 34 ,
2 b · cos ψ m 44 + m 32 , 2 b · sin ψ m 42 m 34 ,
I out ( x , y ) = a · sin 2 ( δ + + ψ + 2 ) + b · cos 2 ( δ + ψ 2 ) .
ψ ± = 2 π Φ ± Λ ± .
tan ( 2 ϑ ) = sin ( ψ + ψ 2 ) 2 · sin ( ψ + 2 ) · sin ( ψ 2 ) ,
sin ( γ ) = sin ( ψ + ψ 2 ) · sin ( ψ + 2 ) · sin ( ψ 2 ) cos ( ψ + + ψ 2 ) · cos ( ψ + ψ 2 ) 1 .
ψ + ψ = 0 ,
F = sin ( ψ + 2 ) · sin ( ψ 2 )
tan ψ + = m 42 + m 34 m 44 m 32 , tan ψ = m 42 m 34 m 44 + m 32 .
I out ( x , y ) = 0 { δ + + ψ + = M · π δ + ψ = N · π ,
[ M ] = [ 1 0 0 0 0 X Y S Y C 0 Y S C 2 + S 2 X C S ( 1 X ) 0 Y C C S ( 1 X ) S 2 + C 2 X ] ,
[ M ] = [ T 90 ] · [ L ] · [ T 0 ] [ 1 0 0 0 0 1 0 0 0 0 C S 0 0 S C ] · [ 1 0 0 0 0 X 0 Y 0 0 1 0 0 Y 0 X ] · [ 1 0 0 0 0 1 0 0 0 0 C S 0 0 S C ] ,

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