Abstract

The polarization evolution of vector beams (VBs) generated by q-plates is investigated theoretically and experimentally. An analytical model is developed for the VB created by a general quarter-wave q-plate based on vector diffraction theory. It is found that the polarization distribution of VBs varies with position and the value q. In particular, for the incidence of circular polarization, the exit vector vortex beam has polarization states that cover the whole surface of the Poincaré sphere, thereby constituting a full Poincaré beam. For the incidence of linear polarization, the VB is not cylindrical but specularly symmetric, and exhibits an azimuthal spin splitting. These results are in sharp contrast with those derived by the commonly used model, i.e., regarding the incident light as a plane wave. By implementing q-plates with dielectric metasurfaces, further experiments validate the theoretical results.

© 2017 Chinese Laser Press

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2016 (5)

2015 (5)

2014 (3)

I. Moreno, J. A. Davis, D. M. Cottrell, and R. Donoso, “Encoding high-order cylindrically polarized light beams,” Appl. Opt. 53, 5493–5501 (2014).
[Crossref]

X. Ling, X. Zhou, W. Shu, H. Luo, and S. Wen, “Realization of tunable photonic spin Hall effect by tailoring the Pancharatnam-Berry phase,” Sci. Rep. 4, 5557 (2014).

Y. Liu, X. Ling, X. Yi, X. Zhou, H. Luo, and S. Wen, “Realization of polarization evolution on higher-order Poincaré sphere with metasurface,” Appl. Phys. Lett. 104, 191110 (2014).
[Crossref]

2013 (3)

G. Li, M. Kang, S. Chen, S. Zhang, E. Y. B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13, 4148–4151 (2013).
[Crossref]

D. Maluenda, I. Juvells, R. Martínez-Herrero, and A. Carnicer, “Reconfigurable beams with arbitrary polarization and shape distributions at a given plane,” Opt. Express 21, 5424–5431 (2013).
[Crossref]

V. D’Ambrosio, N. Spagnolo, L. D. Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

2012 (7)

X. Li, T. H. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).
[Crossref]

V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961 (2012).
[Crossref]

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref]

M. Kang, J. Chen, B. Gu, Y. Li, L. T. Vuong, and H. T. Wang, “Spatial splitting of spin states in subwavelength metallic microstructures via partial conversion of spin-to-orbital angular momentum,” Phys. Rev. A 85, 035801 (2012).
[Crossref]

X. Ling, X. Zhou, H. Luo, and S. Wen, “Steering far-field spin-dependent splitting of light by inhomogeneous anisotropic media,” Phys. Rev. A 86, 053824 (2012).
[Crossref]

F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. 51, C1–C6 (2012).
[Crossref]

E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, “Poincaré-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light,” Appl. Opt. 51, 2925–2934 (2012).
[Crossref]

2011 (4)

M. Beresna, M. Gecevičius, and P. G. Kazansky, “Polarization sensitive elements fabricated by femtosecond laser nanostructuring of glass,” Opt. Mater. Express 1, 783–795 (2011).
[Crossref]

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[Crossref]

X. Li, Y. Cao, and M. Gu, “Superresolution-focal-volume induced 3.0 Tbytes/disk capacity by focusing a radially polarized beam,” Opt. Lett. 36, 2510–2512 (2011).
[Crossref]

C. Hnatovsky, V. G. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[Crossref]

2010 (3)

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. 105, 030407 (2010).
[Crossref]

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010).
[Crossref]

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

2009 (3)

2007 (3)

2006 (3)

2005 (2)

Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30, 3063–3065 (2005).
[Crossref]

E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005).

2004 (2)

M. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57(5), 35–40 (2004).
[Crossref]

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
[Crossref]

2003 (2)

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

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref]

2002 (2)

T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203, 1–5 (2002).
[Crossref]

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002).
[Crossref]

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

2000 (3)

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal and radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[Crossref]

1997 (1)

1996 (1)

1990 (1)

Alfano, R. R.

G. Milione, T. A. Nguyen, J. Leach, D. A. Nolan, and R. R. Alfano, “Using the nonseparability of vector beams to encode information for optical communication,” Opt. Lett. 40, 4887–4890 (2015).
[Crossref]

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[Crossref]

Allen, L.

M. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57(5), 35–40 (2004).
[Crossref]

Alonso, M. A.

Aolita, L.

V. D’Ambrosio, N. Spagnolo, L. D. Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961 (2012).
[Crossref]

Baccari, F.

V. D’Ambrosio, F. Baccari, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Arbitrary, direct and deterministic manipulation of vector beams via electrically-tuned q-plates,” Sci. Rep. 5, 7840 (2015).
[Crossref]

Badham, K.

Barreiro, J. T.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. 105, 030407 (2010).
[Crossref]

Beckley, A. M.

Beresna, M.

M. Beresna, M. Gecevičius, and P. G. Kazansky, “Polarization sensitive elements fabricated by femtosecond laser nanostructuring of glass,” Opt. Mater. Express 1, 783–795 (2011).
[Crossref]

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[Crossref]

Bernet, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[Crossref]

Beversluis, M.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref]

Biener, G.

Biss, D. P.

Bliokh, K. Y.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

Blit, S.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal and radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[Crossref]

Bomzon, Z.

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002).
[Crossref]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal and radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[Crossref]

Bouhelier, A.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref]

Brown, T. G.

Cao, Y.

Cardano, F.

Carnicer, A.

Cheah, K. W.

G. Li, M. Kang, S. Chen, S. Zhang, E. Y. B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13, 4148–4151 (2013).
[Crossref]

Chen, J.

M. Kang, J. Chen, B. Gu, Y. Li, L. T. Vuong, and H. T. Wang, “Spatial splitting of spin states in subwavelength metallic microstructures via partial conversion of spin-to-orbital angular momentum,” Phys. Rev. A 85, 035801 (2012).
[Crossref]

X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Chen, S.

G. Li, M. Kang, S. Chen, S. Zhang, E. Y. B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13, 4148–4151 (2013).
[Crossref]

Chen, Z.

Cheng, H.

Cottrell, D. M.

Courjon, D.

T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203, 1–5 (2002).
[Crossref]

Courtial, J.

M. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57(5), 35–40 (2004).
[Crossref]

Cui, Y.

D’Ambrosio, V.

V. D’Ambrosio, F. Baccari, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Arbitrary, direct and deterministic manipulation of vector beams via electrically-tuned q-plates,” Sci. Rep. 5, 7840 (2015).
[Crossref]

V. D’Ambrosio, N. Spagnolo, L. D. Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[Crossref]

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Figures (11)

Fig. 1.
Fig. 1. Polarization distributions of vector fields generated by q -plates for different incident waves. The first column shows the theoretical results for an incident plane wave, while the others show an incident Gaussian beam. The four rows correspond to incidences of right, left, and right circular and linear polarizations ( σ = + 1 , 1 , + 1 , 0 ) , respectively. The q -plates used have q = 1 and α 0 = π / 4 in the upper two rows and q = 1 and α 0 = 0 in the other rows. The dimension for all images is w 0 × w 0 , where w 0 = 1.75    mm .
Fig. 2.
Fig. 2. Polarization evolution on the Poincaré sphere. (a) For the vector vortex beam in Fig. 1(j), the polarization states along a radial direction (from r = 0 to for a given θ ) evolve from the north pole to the south pole. For the vector field in Fig. 1(i), the states in a radial direction correspond to a single point that does not change with propagation. (b) For the VB in Fig. 1(n), the polarization states on circles with different radii around the beam center evolve along different 8-shaped curves. Here, r 0 = 0.3    mm is the radius of the first circle in Fig. 1. For the vector field in Fig. 1(m), the states on all circles evolve along the same 8-shaped curve.
Fig. 3.
Fig. 3. Schematic of experimental setup to generate VBs. The inset is a schematic drawing of the q -plate with q = 1 and α 0 = π / 4 . White short lines denote the orientations of local optical axes.
Fig. 4.
Fig. 4. Intensities and polarizations measured on the transverse plane z = 50    cm . The top and bottom rows result from the incidence of right and left circularly polarized Gaussian beams, respectively. The left and right columns are the theoretical and experimental results, respectively. Here the q -plate with q = 1 and α 0 = π / 4 is used. The size for all images is 2.4    mm × 2.3    mm .
Fig. 5.
Fig. 5. For the radially polarized beam in Fig. 4(a), (a) is the intensity through a linear polarizer with the transmission axis indicated by the arrow; (b) and (c) are the intensities of the left and right circularly polarized components, respectively. The bottom row shows the experimental results corresponding to the top one.
Fig. 6.
Fig. 6. For the azimuthally polarized beam Fig. 4(c), (a) is the intensity through a linear polarizer with the transmission axis indicated by the arrow; (b) and (c) are the intensities of the left and right circularly polarized components, respectively. In the bottom row are the experimental results corresponding to the top one.
Fig. 7.
Fig. 7. Transverse intensities and polarizations for a spirally polarized VB at different propagation distances. The top, middle, and bottom rows correspond to z = 50 , 100, and 150 cm, respectively. The left and right columns are the theoretical and experimental results, respectively. Here the incident Gaussian beam is right circularly polarized and the q -plate with q = 1 and α 0 = 0 is used. The size for all images is 2.4    mm × 2.3    mm .
Fig. 8.
Fig. 8. (a) Theoretical and (b) experimental results of the radial intensity distributions for the spirally polarized beam in Fig. 7. The intensities are normalized by the center intensity of the incident Gaussian beam. (c) Theoretical and (d) experimental results of the radial intensity distributions for the spirally polarized beam and its two circular polarization components measured at z = 50    cm . On top are the local polarization states. (e) is the evolution of polarization states along a radial line ( 0 r 2.5    mm and θ = π / 4 ).
Fig. 9.
Fig. 9. Transverse intensity and polarization distribution for the VB generated by a linearly polarized Gaussian beam passing through a q -plate. (a) Theoretical results calculated by Eq. (16) and (b) experimental results measured at z = 50    cm . Here, the q -plate with q = 1 and α 0 = 0 is used. The size for all images is 2.4    mm × 2.3    mm .
Fig. 10.
Fig. 10. (a) and (b) are the transverse intensities for two circularly polarized components, respectively, and (c) the Stokes parameter S 3 , corresponding to the VB in Fig. 9. The second rows are correspondent experimental results.
Fig. 11.
Fig. 11. Polarization evolution on the Poincaré sphere. Shown are the theoretical and experimental results for the third string of polarization states (with a radius r = 3 r 0 ) on the beam cross section in Fig. 9.

Equations (17)

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E i ( x , y , 0 ) = E 0 ( x , y ) ( cos ϑ e i ϕ | + + sin ϑ e i ϕ | ) ,
T = [ cos δ 2 i sin δ 2 cos 2 α i sin δ 2 sin 2 α i sin δ 2 sin 2 α cos δ 2 + i sin δ 2 cos 2 α ] .
T 1 / 4 = 1 2 [ 1 i cos 2 α i sin 2 α i sin 2 α 1 + cos 2 α ] = 1 2 ( I + T 1 / 2 ) ,
T 1 / 2 = i [ cos 2 α sin 2 α sin 2 α cos 2 α ] .
α = q θ + α 0 ,
E ( x , y , 0 ) = E 0 2 [ cos ϑ e i ϕ | + + sin ϑ e i ϕ | ] i E 0 2 [ sin ϑ e i ( 2 α ϕ ) | + + cos ϑ e i ( 2 α ϕ ) | ] .
E ( r , z ) = e i k z i λ z 0 0 2 π ρ d ρ d φ E ( ρ , 0 ) × exp [ i k ρ 2 2 z i k r ρ z cos ( φ θ ) ] ,
E ( r , z ) = 1 2 [ E i ( r , z ) i E 1 / 2 ( r , z ) ] .
E i ( r , z ) = A 1 [ cos ϑ e i ϕ | + + sin ϑ e i ϕ | ]
E 1 / 2 ( r , z ) = A 2 [ sin ϑ e i ( 2 α ϕ ) | + + cos ϑ e i ( 2 α ϕ ) | ] .
A 1 = e i k z k r 2 k w 0 2 + i 2 z k w 0 2 k w 0 2 + i 2 z ,
A 2 = e i k z + i k r 2 2 z k w 0 2 k w 0 2 + i 2 z ( i k 2 r 2 w 0 2 2 k w 0 2 z + i 4 z 2 ) | q | × 2 2 | q | π Γ ( 1 / 2 + | q | ) F 1 1 ( 1 + | q | , 1 + 2 | q | , i k 2 r 2 w 0 2 2 k w 0 2 z + i 4 z 2 ) .
E ± ( x , y , z ) = 1 2 [ A 1 | ± i A 2 e ± i 2 α | ]
= A e ± i β [ cos χ e i β | + + sin χ e ± i β | ] ,
E p ± ( x , y , 0 ) = E 0 2 [ | ± i e ± i 2 α | ] = E 0 e ± i ( α π / 4 ) [ cos ( α π / 4 ) sin ( α π / 4 ) ] .
E ( r , z ) = 1 2 [ A 1 cos ϕ i A 2 cos ( 2 α ϕ ) A 1 sin ϕ i A 2 sin ( 2 α ϕ ) ] .
E p ( r , z ) = E 0 2 [ cos ϕ i cos ( 2 α ϕ ) sin ϕ i sin ( 2 α ϕ ) ] .

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