Abstract

The interest on the conical refraction (CR) phenomenon in biaxial crystals has revived in the last years due to its prospective for generating structured polarized light beams, i.e. vector beams. While the intensity and the polarization structure of the CR beams are well known, an accurate experimental study of their phase structure has not been yet carried out. We investigate the phase structure of the CR rings by means of a Mach-Zehnder interferometer while applying the phase-shifting interferometric technique to measure the phase at the focal plane. In general the two beams interfering correspond to different states of polarization (SOP) which locally vary. To distinguish if there is an additional phase added to the geometrical one we have derived the appropriate theoretical expressions using the Jones matrix formalism. We demonstrate that the phase of the CR rings is equivalent to that one introduced by an azimuthally segmented polarizer with CR-like polarization distribution. Additionally, we obtain direct evidence that the Poggendorff dark ring is an annular singularity, with a π phase change between the inner and outer bright rings.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]

2015 (2)

2014 (4)

2013 (4)

2012 (3)

2011 (2)

2010 (2)

2009 (3)

2008 (2)

2007 (1)

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamiltons diabolical point at the heart of crystal optics,” Prog. Opt. 50, 13–50 (2007).

2006 (1)

2004 (4)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

2002 (2)

A. M. Belskii and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun. 204(1-6), 1–6 (2002).
[Crossref]

Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[Crossref] [PubMed]

2000 (1)

1978 (2)

A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 436–439 (1978).

A. M. Belskii and A. P. Khapalyuk, “Propagation of confined light beams along the beam axes (axes of single ray velocity) of biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).

Ahufinger, V.

Albero, J.

J. L. Martínez-Fuentes, J. Albero, and I. Moreno, “Analysis of optical polarization modulation systems through the Pancharatnam connection,” Opt. Commun. 285(4), 393–401 (2012).
[Crossref]

Alfano, R. R.

Ballantine, K. E.

Barnett, S.

Belskii, A. M.

A. M. Belskii and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun. 204(1-6), 1–6 (2002).
[Crossref]

A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 436–439 (1978).

A. M. Belskii and A. P. Khapalyuk, “Propagation of confined light beams along the beam axes (axes of single ray velocity) of biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).

Berry, M. V.

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamiltons diabolical point at the heart of crystal optics,” Prog. Opt. 50, 13–50 (2007).

M. V. Berry, “Conical diffraction asymptotics: Fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. 6(4), 289–300 (2004).
[Crossref]

Birkl, G.

Brown, T.

Brown, T. G.

Bursukova, M.

T. Kalkandjiev and M. Bursukova, “Conical refraction: an experimental introduction,” Proc. SPIE 6994, 69940B (2008).

Campos, J.

Cardano, F.

Choudhury, A.

Courtial, J.

de Lisio, C.

De Zela, F.

J. C. Loredo, O. Ortíz, R. Weingärtner, and F. De Zela, “Measurement of Pancharatnam’s phase by robust interferometric and polarimetric methods,” Phys. Rev. 80(1), 012113 (2009).
[Crossref]

Denisenko, V. G.

Desyatnikov, A. S.

Donegan, J. F.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Dudley, A.

Fernández, E.

Forbes, A.

Franke-Arnold, S.

Gibson, G.

Grier, D.

Hnatovsky, C.

Huguenin, J. A. O.

Jeffrey, M. R.

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamiltons diabolical point at the heart of crystal optics,” Prog. Opt. 50, 13–50 (2007).

Kalkandjiev, T.

T. Kalkandjiev and M. Bursukova, “Conical refraction: an experimental introduction,” Proc. SPIE 6994, 69940B (2008).

Kalkandjiev, T. K.

Kalkandkiev, T. K.

Karimi, E.

Khapalyuk, A. P.

A. M. Belskii and A. P. Khapalyuk, “Propagation of confined light beams along the beam axes (axes of single ray velocity) of biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).

A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 436–439 (1978).

Khoury, A. Z.

Kivshar, Y. S.

Kristensen, P.

Krolikowski, W.

Küber, J.

Kurzynowski, P.

Ladavac, K.

Leger, J.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Lizana, A.

Loiko, Y.

Loiko, Y. V.

Loredo, J. C.

J. C. Loredo, O. Ortíz, R. Weingärtner, and F. De Zela, “Measurement of Pancharatnam’s phase by robust interferometric and polarimetric methods,” Phys. Rev. 80(1), 012113 (2009).
[Crossref]

Lunney, J. G.

Márquez, A.

X. Zheng, A. Lizana, A. Peinado, C. Ramírez, J. L. Martínez, A. Márquez, I. Moreno, and J. Campos, “Compact LCOS-SLM based polarization pattern beam generator,” J. Lightwave Technol.in press.

Marrucci, L.

Martínez, J. L.

X. Zheng, A. Lizana, A. Peinado, C. Ramírez, J. L. Martínez, A. Márquez, I. Moreno, and J. Campos, “Compact LCOS-SLM based polarization pattern beam generator,” J. Lightwave Technol.in press.

Martínez-Fuentes, J. L.

J. L. Martínez-Fuentes, J. Albero, and I. Moreno, “Analysis of optical polarization modulation systems through the Pancharatnam connection,” Opt. Commun. 285(4), 393–401 (2012).
[Crossref]

Milione, G.

Minovich, A.

Mompart, J.

A. Turpin, J. Polo, Y. V. Loiko, J. Küber, F. Schmaltz, T. K. Kalkandjiev, V. Ahufinger, G. Birkl, and J. Mompart, “Blue-detuned optical ring trap for Bose-Einstein condensates based on conical refraction,” Opt. Express 23(2), 1638–1650 (2015).
[Crossref] [PubMed]

A. Turpin, Y. V. Loiko, A. Peinado, A. Lizana, T. K. Kalkandjiev, J. Campos, and J. Mompart, “Polarization tailored novel vector beams based on conical refraction,” Opt. Express 23(5), 5704–5715 (2015).
[Crossref] [PubMed]

A. Turpin, Y. V. Loiko, T. K. Kalkandkiev, H. Tomizawa, and J. Mompart, “Super-Gaussian conical refraction beam,” Opt. Lett. 39(15), 4349–4352 (2014).
[PubMed]

Y. V. Loiko, A. Turpin, T. K. Kalkandjiev, E. U. Rafailov, and J. Mompart, “Generating a three-dimensional dark focus from a single conically refracted light beam,” Opt. Lett. 38(22), 4648–4651 (2013).
[Crossref] [PubMed]

A. Turpin, V. Shvedov, C. Hnatovsky, Y. V. Loiko, J. Mompart, and W. Krolikowski, “Optical vault: a reconfigurable bottle beam based on conical refraction of light,” Opt. Express 21(22), 26335–26340 (2013).
[Crossref] [PubMed]

A. Peinado, A. Turpin, A. Lizana, E. Fernández, J. Mompart, and J. Campos, “Conical refraction as a tool for polarization metrology,” Opt. Lett. 38(20), 4100–4103 (2013).
[Crossref] [PubMed]

A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, H. Tomizawa, and J. Mompart, “Wave-vector and polarization dependence of conical refraction,” Opt. Express 21(4), 4503–4511 (2013).
[Crossref] [PubMed]

A. Turpin, Y. Loiko, T. K. Kalkandjiev, and J. Mompart, “Free-space optical polarization demultiplexing and multiplexing by means of conical refraction,” Opt. Lett. 37(20), 4197–4199 (2012).
[Crossref] [PubMed]

Moreno, I.

J. L. Martínez-Fuentes, J. Albero, and I. Moreno, “Analysis of optical polarization modulation systems through the Pancharatnam connection,” Opt. Commun. 285(4), 393–401 (2012).
[Crossref]

X. Zheng, A. Lizana, A. Peinado, C. Ramírez, J. L. Martínez, A. Márquez, I. Moreno, and J. Campos, “Compact LCOS-SLM based polarization pattern beam generator,” J. Lightwave Technol.in press.

O’Dwyer, D. P.

Ortíz, O.

J. C. Loredo, O. Ortíz, R. Weingärtner, and F. De Zela, “Measurement of Pancharatnam’s phase by robust interferometric and polarimetric methods,” Phys. Rev. 80(1), 012113 (2009).
[Crossref]

Otani, Y.

Padgett, M.

Pas’ko, V.

Peinado, A.

Phelan, C. F.

Polo, J.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Rafailov, E. U.

Rakovich, Y. P.

Ramachandran, S.

Ramírez, C.

X. Zheng, A. Lizana, A. Peinado, C. Ramírez, J. L. Martínez, A. Márquez, I. Moreno, and J. Campos, “Compact LCOS-SLM based polarization pattern beam generator,” J. Lightwave Technol.in press.

Rodríguez-Herrera, O. G.

Santamato, E.

Schmaltz, F.

Sheppard, C. J. R.

Shvedov, V.

Slussarenko, S.

Soskin, M. S.

Souza, C. E. R.

Stepanov, M. A.

A. M. Belskii and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun. 204(1-6), 1–6 (2002).
[Crossref]

Szarycz, M.

Sztul, H. I.

Tomizawa, H.

Torres, J. P.

J. P. Torres, “Quantum engineering of light,” Opt. Pura Apl. 44, 309–314 (2011).

Turpin, A.

A. Turpin, Y. V. Loiko, A. Peinado, A. Lizana, T. K. Kalkandjiev, J. Campos, and J. Mompart, “Polarization tailored novel vector beams based on conical refraction,” Opt. Express 23(5), 5704–5715 (2015).
[Crossref] [PubMed]

A. Turpin, J. Polo, Y. V. Loiko, J. Küber, F. Schmaltz, T. K. Kalkandjiev, V. Ahufinger, G. Birkl, and J. Mompart, “Blue-detuned optical ring trap for Bose-Einstein condensates based on conical refraction,” Opt. Express 23(2), 1638–1650 (2015).
[Crossref] [PubMed]

A. Turpin, Y. V. Loiko, T. K. Kalkandkiev, H. Tomizawa, and J. Mompart, “Super-Gaussian conical refraction beam,” Opt. Lett. 39(15), 4349–4352 (2014).
[PubMed]

Y. V. Loiko, A. Turpin, T. K. Kalkandjiev, E. U. Rafailov, and J. Mompart, “Generating a three-dimensional dark focus from a single conically refracted light beam,” Opt. Lett. 38(22), 4648–4651 (2013).
[Crossref] [PubMed]

A. Turpin, V. Shvedov, C. Hnatovsky, Y. V. Loiko, J. Mompart, and W. Krolikowski, “Optical vault: a reconfigurable bottle beam based on conical refraction of light,” Opt. Express 21(22), 26335–26340 (2013).
[Crossref] [PubMed]

A. Peinado, A. Turpin, A. Lizana, E. Fernández, J. Mompart, and J. Campos, “Conical refraction as a tool for polarization metrology,” Opt. Lett. 38(20), 4100–4103 (2013).
[Crossref] [PubMed]

A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, H. Tomizawa, and J. Mompart, “Wave-vector and polarization dependence of conical refraction,” Opt. Express 21(4), 4503–4511 (2013).
[Crossref] [PubMed]

A. Turpin, Y. Loiko, T. K. Kalkandjiev, and J. Mompart, “Free-space optical polarization demultiplexing and multiplexing by means of conical refraction,” Opt. Lett. 37(20), 4197–4199 (2012).
[Crossref] [PubMed]

Tyo, J. S.

Vasnetsov, M.

Wakayama, T.

Weingärtner, R.

J. C. Loredo, O. Ortíz, R. Weingärtner, and F. De Zela, “Measurement of Pancharatnam’s phase by robust interferometric and polarimetric methods,” Phys. Rev. 80(1), 012113 (2009).
[Crossref]

Wozniak, W. A.

Yan, M. F.

Yonemura, M.

Yoshizawa, T.

Youngworth, K.

Zhan, Q.

Zheng, X.

X. Zheng, A. Lizana, A. Peinado, C. Ramírez, J. L. Martínez, A. Márquez, I. Moreno, and J. Campos, “Compact LCOS-SLM based polarization pattern beam generator,” J. Lightwave Technol.in press.

Adv. Opt. Photon. (1)

Appl. Opt. (2)

J. Opt. A, Pure Appl. Opt. (1)

M. V. Berry, “Conical diffraction asymptotics: Fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. 6(4), 289–300 (2004).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

J. L. Martínez-Fuentes, J. Albero, and I. Moreno, “Analysis of optical polarization modulation systems through the Pancharatnam connection,” Opt. Commun. 285(4), 393–401 (2012).
[Crossref]

A. M. Belskii and M. A. Stepanov, “Internal conical refraction of light beams in biaxial gyrotropic crystals,” Opt. Commun. 204(1-6), 1–6 (2002).
[Crossref]

Opt. Express (12)

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[Crossref] [PubMed]

Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[Crossref] [PubMed]

K. Ladavac and D. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12(6), 1144–1149 (2004).
[Crossref] [PubMed]

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[Crossref] [PubMed]

T. G. Brown and Q. Zhan, “Introduction: unconventional polarization states of light,” Opt. Express 18, 10775–10776 (2010).
[Crossref] [PubMed]

D. P. O’Dwyer, C. F. Phelan, K. E. Ballantine, Y. P. Rakovich, J. G. Lunney, and J. F. Donegan, “Conical diffraction of linearly polarised light controls the angular position of a microscopic object,” Opt. Express 18(26), 27319–27326 (2010).
[PubMed]

A. Dudley, G. Milione, R. R. Alfano, and A. Forbes, “All-digital wavefront sensing for structured light beams,” Opt. Express 22(11), 14031–14040 (2014).
[PubMed]

A. Turpin, V. Shvedov, C. Hnatovsky, Y. V. Loiko, J. Mompart, and W. Krolikowski, “Optical vault: a reconfigurable bottle beam based on conical refraction of light,” Opt. Express 21(22), 26335–26340 (2013).
[Crossref] [PubMed]

A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, H. Tomizawa, and J. Mompart, “Wave-vector and polarization dependence of conical refraction,” Opt. Express 21(4), 4503–4511 (2013).
[Crossref] [PubMed]

A. Turpin, J. Polo, Y. V. Loiko, J. Küber, F. Schmaltz, T. K. Kalkandjiev, V. Ahufinger, G. Birkl, and J. Mompart, “Blue-detuned optical ring trap for Bose-Einstein condensates based on conical refraction,” Opt. Express 23(2), 1638–1650 (2015).
[Crossref] [PubMed]

A. Turpin, Y. V. Loiko, A. Peinado, A. Lizana, T. K. Kalkandjiev, J. Campos, and J. Mompart, “Polarization tailored novel vector beams based on conical refraction,” Opt. Express 23(5), 5704–5715 (2015).
[Crossref] [PubMed]

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Figures (9)

Fig. 1
Fig. 1 Diagrams for the first (a) and second (b) experiment in the Mach-Zehnder interferometer analyzed in the text. Thin arrows (in red) represent the polarization state. BSi: Non polarizing beam splitters. LPi: Linear polarizers oriented at βi with respect to the vertical.
Fig. 2
Fig. 2 “Conical refraction polarizer” of eight sectors applied in the sample beam of the interferometric setup. Green arrows indicate the orientation of the linear polarizer in each sector.
Fig. 3
Fig. 3 Diagrams for the first (a) and second (b) experiment in the Mach-Zehnder setup, where the conical polarizer is indicated by the circle divided in eight sectors. The red arrows represent the Polarization state. The green arrows represent the transmission axis orientation of the conical polarizer.
Fig. 4
Fig. 4 Results for the experiment displayed in Fig. 3(a). (a) Sample interferogram. (b) Processed phase map pattern, where a constant phase-difference is obtained across the aperture.
Fig. 5
Fig. 5 Results for the experiment displayed in Fig. 3(b). (a) Sample interferogram where discontinuity in the fringes between the two bottom sectors is clearly visible. (b) CRP sector structure where the thick purple arrow indicates the two sectors at the bottom, whose transmission axis are almost orthogonal to the SOP in the reference beam, vertically polarized.
Fig. 6
Fig. 6 Experimental setup for the Mach-Zehnder used to measure the phase distribution in the focal plane. An analogous setup is used for the CRP study. The CRP is illuminated with a collimated beam, and by means of lens L2 its image is formed at the CCD sensor plane.
Fig. 7
Fig. 7 Intensity distribution at the focal plane when a biaxial crystal is illuminated with linearly polarized light oriented (a) along the vertical, and (b) at −45°. In (c) we indicate the orientation of the linearly polarized eigenpolarizations along the azimuth of the ring.
Fig. 8
Fig. 8 Results at the focal plane for the experiment analogous to the one in Fig. 3(a). (a) Sample interferogram. (b) Experimentally processed and (c) numerically calculated phase map, where a constant phase-difference is obtained in the azimuthal direction, but a phase jump along the radial direction between the inner and outer rings is clearly visible. (d) Instantaneous transverse intensity and polarization distribution of the CR rings showing a change in the sense of the polarization between the inner and outer rings.
Fig. 9
Fig. 9 Results at the focal plane for the experiments corresponding to the arrangement sketched in Fig. 3(b). (a) Experimentally processed and (b) numerically calculated phase map, where a discontinuity in the azimuthal direction near the top is visible. A second discontinuity, this one in the radial direction, is located in the null intensity ring between the inner and outer bright rings. (c) Orientation of the linearly polarized eigenpolarizations along the azimuth of the ring, where a thick purple arrow indicates where, approximately, the orientation is almost orthogonal to the SOP of the reference beam, horizontally polarized.

Equations (16)

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I(ρ,Z)=| B 0 | 2 +| B 1 | 2 ,
B 0 (ρ,Z)= 1 2π 0 ηa(η) e i Z η 2 2n J 0 (ηρ)cos(η ρ 0 )dη ,
B 1 (ρ,Z)= 1 2π 0 ηa(η) e i Z η 2 2n J 1 (ηρ)sin(η ρ 0 )dη ,
a(η)= 0 ρE( ρ ) J 0 (ηρ)dρ ,
δ=arg( J 1 * J 2 ),
J out =R( β ) P X R( β ) J in ,
R( β )=( cosβ sinβ sinβ cosβ ),
P X =( 1 0 0 0 ).
J out =exp( iβ )( cosβ sinβ )= J LP ( β ),
J RHC * J LP ( β 2 )=1,
δ=arg( J RHC * J LP ( β 2 ) )=0.
J LP * ( β 1 ) J LP ( β 2 )= e i( β 2 β 1 ) cos( β 2 β 1 ),
δ={ β 2 β 1 ;cos( β 2 β 1 )0 β 2 β 1 +π;cos( β 2 β)<0 } .
J * LP ( β 1 =0 ) J LP ( β 2,± )=exp( iπi( π 2 +α ) )sinα,
δ ± =π( π 2 +α ).
Δδ= δ δ + =π+2α,

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