Abstract

We propose a novel birefringent interferometer setup for the study of unfolding points, and obtain for the first time to our knowledge the spatial polarization structure very near the unfolding point of a uniformly polarized optical vortex beam propagating in a birefringent crystal. The unfolding point is reconstructed by folding back the two separated eigen-beams at the output of the birefringent crystal into a single beam using another identical birefringent crystal, resulting in a birefringent interferometer of Mach-Zehnder type. We also demonstrate that the separation near the unfolding point can be varied by a small rotation of the second crystal.

© 2012 OSA

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  1. M. S. Soskin and M. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
  2. J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (IoP Publishing, 1999).
  3. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002).
  4. A. Nesci, R. Dändliker, M. Salt, and H. P. Herzig, “Measuring amplitude and phase distribution of fields generated by gratings with sub-wavelength resolution,” Opt. Commun. 205(4-6), 229–238 (2002).
  5. M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).
  6. A. Desyatnikov, T. A. Fadeyeva, V. G. Shvedov, Y. V. Izdebskaya, A. V. Volyar, E. Brasselet, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Spatially engineered polarization states and optical vortices in uniaxial crystals,” Opt. Express 18(10), 10848–10863 (2010).
  7. F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95(25), 253901 (2005).
  8. F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14(23), 11402–11411 (2006).
  9. X. D. Xu, P. K. Kuo, S. Y. Zhang, X. J. Shui, and Z. N. Zhang, “Application of an optical birefringence interferometer to photothermal detection,” Microw. Opt. Technol. Lett. 35(2), 140–143 (2002).
  10. C. Cheng, “The signal processing approach for the birefringent material based Mach-Zehnder interferometer design,” Proc. of IEEE, 48th Midwest Symposium on Circuits and Systems (Covington, Kentucky, 2005), 211–214, 10.1109/MWSCAS.2005.1594076.
    [CrossRef]
  11. J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426(6964), 264–267 (2003).
  12. A. Yariv and P. Yeh, Optical Waves in Crystals (John Wiley & Sons, 1984).
  13. E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1993).
  14. Y. Miyamoto, M. Masuda, A. Wada, and M. Takeda, “Electron-beam lithography fabrication of phase holograms to generate Laguerre-Gaussian beams,” Proc. SPIE 3740, 232–235 (1999).
  15. http://www.castech.com/products_detail/&productId=61213567-1e08-41a0-9ac2-0d61b8c01db1.html.
  16. T. Fadeyeva, Y. Egorov, A. Rubass, G. A. Swartzlander, Jr., and A. Volyar, “Indistinguishability limit for off-axis vortex beams in uniaxial crystals,” Opt. Lett. 32(21), 3116–3118 (2007).
  17. T. Kihara, “Measurement method of Stokes parameters using a quarter-wave plate with phase difference errors,” Appl. Opt. 50(17), 2582–2587 (2011).
  18. U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica 2, 006 (2009),
    [CrossRef]

2011 (1)

2010 (1)

2009 (1)

U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica 2, 006 (2009),
[CrossRef]

2007 (1)

2006 (1)

2005 (1)

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

2004 (1)

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).

2003 (1)

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426(6964), 264–267 (2003).

2002 (3)

X. D. Xu, P. K. Kuo, S. Y. Zhang, X. J. Shui, and Z. N. Zhang, “Application of an optical birefringence interferometer to photothermal detection,” Microw. Opt. Technol. Lett. 35(2), 140–143 (2002).

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

A. Nesci, R. Dändliker, M. Salt, and H. P. Herzig, “Measuring amplitude and phase distribution of fields generated by gratings with sub-wavelength resolution,” Opt. Commun. 205(4-6), 229–238 (2002).

2001 (1)

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

1999 (1)

Y. Miyamoto, M. Masuda, A. Wada, and M. Takeda, “Electron-beam lithography fabrication of phase holograms to generate Laguerre-Gaussian beams,” Proc. SPIE 3740, 232–235 (1999).

Berry, M. V.

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).

Branning, D.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426(6964), 264–267 (2003).

Brasselet, E.

Dändliker, R.

A. Nesci, R. Dändliker, M. Salt, and H. P. Herzig, “Measuring amplitude and phase distribution of fields generated by gratings with sub-wavelength resolution,” Opt. Commun. 205(4-6), 229–238 (2002).

Dennis, M. R.

U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica 2, 006 (2009),
[CrossRef]

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14(23), 11402–11411 (2006).

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

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).

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

Desyatnikov, A.

Egorov, Y.

Fadeyeva, T.

Fadeyeva, T. A.

Flossmann, F.

U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica 2, 006 (2009),
[CrossRef]

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14(23), 11402–11411 (2006).

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

Herzig, H. P.

A. Nesci, R. Dändliker, M. Salt, and H. P. Herzig, “Measuring amplitude and phase distribution of fields generated by gratings with sub-wavelength resolution,” Opt. Commun. 205(4-6), 229–238 (2002).

Izdebskaya, Y. V.

Kihara, T.

Kivshar, Y. S.

Krolikowski, W.

Kuo, P. K.

X. D. Xu, P. K. Kuo, S. Y. Zhang, X. J. Shui, and Z. N. Zhang, “Application of an optical birefringence interferometer to photothermal detection,” Microw. Opt. Technol. Lett. 35(2), 140–143 (2002).

Lee, Jr., R. L.

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).

Maier, M.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14(23), 11402–11411 (2006).

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

Masuda, M.

Y. Miyamoto, M. Masuda, A. Wada, and M. Takeda, “Electron-beam lithography fabrication of phase holograms to generate Laguerre-Gaussian beams,” Proc. SPIE 3740, 232–235 (1999).

Miyamoto, Y.

Y. Miyamoto, M. Masuda, A. Wada, and M. Takeda, “Electron-beam lithography fabrication of phase holograms to generate Laguerre-Gaussian beams,” Proc. SPIE 3740, 232–235 (1999).

Nesci, A.

A. Nesci, R. Dändliker, M. Salt, and H. P. Herzig, “Measuring amplitude and phase distribution of fields generated by gratings with sub-wavelength resolution,” Opt. Commun. 205(4-6), 229–238 (2002).

Neshev, D. N.

O’Brien, J. L.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426(6964), 264–267 (2003).

Pryde, G. J.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426(6964), 264–267 (2003).

Ralph, T. C.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426(6964), 264–267 (2003).

Rubass, A.

Salt, M.

A. Nesci, R. Dändliker, M. Salt, and H. P. Herzig, “Measuring amplitude and phase distribution of fields generated by gratings with sub-wavelength resolution,” Opt. Commun. 205(4-6), 229–238 (2002).

Schwarz, U. T.

U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica 2, 006 (2009),
[CrossRef]

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14(23), 11402–11411 (2006).

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

Shui, X. J.

X. D. Xu, P. K. Kuo, S. Y. Zhang, X. J. Shui, and Z. N. Zhang, “Application of an optical birefringence interferometer to photothermal detection,” Microw. Opt. Technol. Lett. 35(2), 140–143 (2002).

Shvedov, V. G.

Soskin, M. S.

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

Swartzlander, Jr., G. A.

Takeda, M.

Y. Miyamoto, M. Masuda, A. Wada, and M. Takeda, “Electron-beam lithography fabrication of phase holograms to generate Laguerre-Gaussian beams,” Proc. SPIE 3740, 232–235 (1999).

Vasnetsov, M.

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

Volyar, A.

Volyar, A. V.

Wada, A.

Y. Miyamoto, M. Masuda, A. Wada, and M. Takeda, “Electron-beam lithography fabrication of phase holograms to generate Laguerre-Gaussian beams,” Proc. SPIE 3740, 232–235 (1999).

White, A. G.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426(6964), 264–267 (2003).

Xu, X. D.

X. D. Xu, P. K. Kuo, S. Y. Zhang, X. J. Shui, and Z. N. Zhang, “Application of an optical birefringence interferometer to photothermal detection,” Microw. Opt. Technol. Lett. 35(2), 140–143 (2002).

Zhang, S. Y.

X. D. Xu, P. K. Kuo, S. Y. Zhang, X. J. Shui, and Z. N. Zhang, “Application of an optical birefringence interferometer to photothermal detection,” Microw. Opt. Technol. Lett. 35(2), 140–143 (2002).

Zhang, Z. N.

X. D. Xu, P. K. Kuo, S. Y. Zhang, X. J. Shui, and Z. N. Zhang, “Application of an optical birefringence interferometer to photothermal detection,” Microw. Opt. Technol. Lett. 35(2), 140–143 (2002).

Appl. Opt. (1)

Microw. Opt. Technol. Lett. (1)

X. D. Xu, P. K. Kuo, S. Y. Zhang, X. J. Shui, and Z. N. Zhang, “Application of an optical birefringence interferometer to photothermal detection,” Microw. Opt. Technol. Lett. 35(2), 140–143 (2002).

Nature (1)

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426(6964), 264–267 (2003).

New J. Phys. (1)

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., “Polarization singularities in the clear sky,” New J. Phys. 6, 162 (2004).

Opt. Commun. (2)

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

A. Nesci, R. Dändliker, M. Salt, and H. P. Herzig, “Measuring amplitude and phase distribution of fields generated by gratings with sub-wavelength resolution,” Opt. Commun. 205(4-6), 229–238 (2002).

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

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

Proc. SPIE (1)

Y. Miyamoto, M. Masuda, A. Wada, and M. Takeda, “Electron-beam lithography fabrication of phase holograms to generate Laguerre-Gaussian beams,” Proc. SPIE 3740, 232–235 (1999).

Prog. Opt. (1)

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

Topologica (1)

U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica 2, 006 (2009),
[CrossRef]

Other (5)

J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (IoP Publishing, 1999).

http://www.castech.com/products_detail/&productId=61213567-1e08-41a0-9ac2-0d61b8c01db1.html.

A. Yariv and P. Yeh, Optical Waves in Crystals (John Wiley & Sons, 1984).

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1993).

C. Cheng, “The signal processing approach for the birefringent material based Mach-Zehnder interferometer design,” Proc. of IEEE, 48th Midwest Symposium on Circuits and Systems (Covington, Kentucky, 2005), 211–214, 10.1109/MWSCAS.2005.1594076.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Experimental setup. P1,2: Polarizers, HG: Hologram, A: Aperture, Cr1,2: YVO4 crystals, HWP: Half Wave Plate, QWP: Quarter Wave Plate, R: Rotational stage, T: Translation stage, CCD: Charge coupled device connected to a personal computer.

Fig. 2
Fig. 2

Experimental results; (a) Normalized total intensity (S0); (b), (c) and (d) represents s1, s2 and s3 ; (e) and (f) represent 2α and δ respectively of the input beam.

Fig. 3
Fig. 3

(a) Position of O-ray and E-ray on CCD (b) Separation d between the two eigen-beams as a function of rotation angle of Cr2.

Fig. 4
Fig. 4

Experimental results; (a) Normalized total intensity (S0); (b), (c) and (d) represents s1, s2 and s3; (e) and (f) represent 2α and δ respectively of the beam at the output of the interferometer for the relative separation of 1.7% in y-direction

Fig. 5
Fig. 5

Simulation results using Eq. (3) and Eq. (6); (a) Normalized total intensity (S0); (b), (c) and (d) represents s1, s2 and s3; (e) and (f) represent 2α and δ respectively of the beam at the output of the interferometer for the relative separation of 1.7% in y-direction

Fig. 6
Fig. 6

Experimental results; (a) Normalized total intensity (S0); (b), (c) and (d) represents s1, s2 and s3; (e) and (f) represent 2α and δ respectively of the beam at the output of the interferometer for the −42% separation in y-direction

Fig. 7
Fig. 7

Computer simulations; (a) Normalized total intensity (S0); (b), (c) and (d) represents s1, s2 and s3; (e) and (f) represent 2α and δ respectively of the beam at the output of the interferometer for the −42% separation in y-direction

Equations (6)

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E out = E e e ^ H + E o e ^ V ,
E e (x,y)=[x+i(yd)]exp{[ x 2 + (yd) 2 ]/ w 0 2 } E o (x,y)=(x +iy)exp[-(x 2 +y 2 )/w 0 2 ] ,
S 0 = | E e (x,y) | 2 + | E o (x,y) | 2 S 2 = E e (x,y) E o * (x,y)+ E o (x,y) E e * (x,y) S 1 = | E e (x,y) | 2 - | E o (x,y) | 2 S 3 =-i[ E e (x,y) E o * (x,y)- E o (x,y) E e * (x,y)].
S 0 =I( 0 0 , 0 0 )+I( 90 0 , 90 0 ) S 1 =I( 0 0 , 0 0 )-I( 90 0 , 90 0 ) S 2 =I( 45 0 , 45 0 )-I( 135 0 , 135 0 ) S 3 =I( 0 0 , 45 0 )-I( 0 0 , 135 0 ),
d r = d (2 w 0 ) .
E e =[(x d 1 )+i(y d 2 )]exp{[ (x d 1 ) 2 + (y d 2 ) 2 ]/ w 0 2 } ×exp{i[ 2π λ (xsinθcosϕ+ysinθsinϕ)+Δ]} E o =(x+iy)exp[( x 2 + y 2 )/ w 0 2 ]

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