Abstract

We show theoretically that by tightly focusing a full Poincaré beam one can create states of light whose average spin angular momentum (SAM) is purely transverse. Moreover, the transverse SAM is significantly increased if a narrow annular aperture is inserted into the Poincaré beam. In this case the state with transverse SAM can exist over an extremely long distance along the optical axis. To avoid diffraction on the aperture and increase the light conversion efficiency we employ the phenomenon of conical diffraction in biaxial crystals to convert a Gaussian beam into a ring-shaped beam, with a high ratio of the ring radius to its width.

© 2015 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  34. C. Junge, D. O’Shea, J. Volz, and A. Rauschenbeutel, “Strong coupling between single atoms and nontransversal photons,” Phys. Rev. Lett. 110(21), 213604 (2013).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  37. 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] [PubMed]
  38. A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 436–439 (1978).
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    [Crossref]
  40. M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. R. Soc. A 462(2070), 1629–1642 (2006).
    [Crossref]
  41. V. Peet, “Biaxial crystal as a versatile mode converter,” J. Opt. 12(9), 095706 (2010).
    [Crossref]
  42. V. Peet, “Conical refraction and formation of multiring focal image with Laguerre-Gauss light beams,” Opt. Lett. 36(15), 2913–2915 (2011).
    [Crossref] [PubMed]
  43. C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express 17(15), 12891–12899 (2009).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  45. A. Turpin, Y. Loiko, T. K. Kalkandjiev, H. Tomizawa, and J. Mompart, “On the dual-cone nature of the conical refraction phenomenon,” Opt. Lett. 40(8), 1639–1642 (2015).
    [Crossref] [PubMed]
  46. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems: structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
    [Crossref]
  47. M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Santos, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68(2), 187–197 (1999).
    [Crossref]

2015 (9)

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
[Crossref]

A. Bekshaev, K. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5(1), 011039 (2015).
[Crossref]

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref] [PubMed]

V. Shvedov, P. Karpinski, Y. Sheng, X. Chen, W. Zhu, W. Krolikowski, and C. Hnatovsky, “Visualizing polarization singularities in Bessel-Poincaré beams,” Opt. Express 23(9), 12444–12453 (2015).
[Crossref] [PubMed]

D. Colas, L. Dominici, S. Donati, A. A. Pervishko, T. C. H. Liew, I. A. Shelykh, D. Ballarini, M. de Giorgi, A. Bramati, G. Gigli, E. del Valle, F. P. Laussy, A. V. Kavokin, and D. Sanvitto, “Polarization shaping of Poincaré beams by polariton oscillations,” Light Sci. Appl. 4(11), e350 (2015).
[Crossref]

Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23(14), 17701–17710 (2015).
[Crossref] [PubMed]

C. Wei, D. Wu, C. Liang, F. Wang, and Y. Cai, “Experimental verification of significant reduction of turbulence-induced scintillation in a full Poincaré beam,” Opt. Express 23(19), 24331–24341 (2015).
[Crossref] [PubMed]

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] [PubMed]

A. Turpin, Y. Loiko, T. K. Kalkandjiev, H. Tomizawa, and J. Mompart, “On the dual-cone nature of the conical refraction phenomenon,” Opt. Lett. 40(8), 1639–1642 (2015).
[Crossref] [PubMed]

2014 (4)

A. Canaguier-Durand and C. Genet, “Transverse spinning of a sphere in a plasmonic field,” Phys. Rev. A 89(3), 033841 (2014).
[Crossref]

T. Ehmke, T. H. Nitzsche, A. Knebl, and A. Heisterkamp, “Molecular orientation sensitive second harmonic microscopy by radially and azimuthally polarized light,” Biomed. Opt. Express 5(7), 2231–2246 (2014).
[Crossref] [PubMed]

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photonics 8(11), 846–850 (2014).
[Crossref]

W. Zhu and W. She, “Transverse angular momentum and transverse barycenter shift of a focused light field due to nonuniform input angular momentum,” Opt. Lett. 39(6), 1337–1340 (2014).
[Crossref] [PubMed]

2013 (8)

W. Zhu and W. She, “Generation of tunable three-dimensional polarization in 4Pi focusing system,” Opt. Express 21(14), 17265–17274 (2013).
[Crossref] [PubMed]

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonics wheel- demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. 8, 13032 (2013).
[Crossref]

C. Hnatovsky, V. G. Shvedov, and W. Krolikowski, “The role of light-induced nanostructures in femtosecond laser micromachining with vector and scalar pulses,” Opt. Express 21(10), 12651–12656 (2013).
[Crossref] [PubMed]

K. Lou, S.-X. Qian, Z.-C. Ren, C. Tu, Y. Li, and H.-T. Wang, “Femtosecond laser processing by using patterned vector optical fields,” Sci. Rep. 3, 2281 (2013).
[Crossref] [PubMed]

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, “Direct electron acceleration with radially polarized laser beams,” Appl. Sci. 3(1), 70–93 (2013).
[Crossref]

F. Cardano, E. Karimi, L. Marrucci, C. de Lisio, and E. Santamato, “Generation and dynamics of optical beams with polarization singularities,” Opt. Express 21(7), 8815–8820 (2013).
[Crossref] [PubMed]

C. Junge, D. O’Shea, J. Volz, and A. Rauschenbeutel, “Strong coupling between single atoms and nontransversal photons,” Phys. Rev. Lett. 110(21), 213604 (2013).
[Crossref] [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]

2012 (2)

2011 (3)

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[Crossref] [PubMed]

V. Peet, “Conical refraction and formation of multiring focal image with Laguerre-Gauss light beams,” Opt. Lett. 36(15), 2913–2915 (2011).
[Crossref] [PubMed]

2010 (3)

V. Peet, “Biaxial crystal as a versatile mode converter,” J. Opt. 12(9), 095706 (2010).
[Crossref]

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18(10), 10777–10785 (2010).
[Crossref] [PubMed]

B. K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010).
[Crossref]

2009 (4)

C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80(6), 063814 (2009).
[Crossref]

M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. 11(9), 094001 (2009).
[Crossref]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express 17(15), 12891–12899 (2009).
[Crossref] [PubMed]

2006 (1)

M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. R. Soc. A 462(2070), 1629–1642 (2006).
[Crossref]

2004 (1)

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

2003 (1)

V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91(9), 093602 (2003).
[Crossref] [PubMed]

2000 (1)

1999 (1)

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Santos, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68(2), 187–197 (1999).
[Crossref]

1998 (1)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[Crossref]

1997 (1)

1978 (1)

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

1975 (1)

M. Lax, W. H. Louisell, and B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems: structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

1940 (2)

F. Belinfante, “On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields,” Physica 7(5), 449–474 (1940).
[Crossref]

F. J. Belinfante, “On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields,” Physica 7(5), 449–474 (1940).
[Crossref]

Aguiló, M.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Santos, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68(2), 187–197 (1999).
[Crossref]

Aiello, A.

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref] [PubMed]

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonics wheel- demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. 8, 13032 (2013).
[Crossref]

B. K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010).
[Crossref]

Allen, L.

Alonso, M. A.

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18(10), 10777–10785 (2010).
[Crossref] [PubMed]

B. K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010).
[Crossref]

April, A.

C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, “Direct electron acceleration with radially polarized laser beams,” Appl. Sci. 3(1), 70–93 (2013).
[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] [PubMed]

Ballarini, D.

D. Colas, L. Dominici, S. Donati, A. A. Pervishko, T. C. H. Liew, I. A. Shelykh, D. Ballarini, M. de Giorgi, A. Bramati, G. Gigli, E. del Valle, F. P. Laussy, A. V. Kavokin, and D. Sanvitto, “Polarization shaping of Poincaré beams by polariton oscillations,” Light Sci. Appl. 4(11), e350 (2015).
[Crossref]

Banzer, P.

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref] [PubMed]

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonics wheel- demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. 8, 13032 (2013).
[Crossref]

Bauer, T.

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref] [PubMed]

P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonics wheel- demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. 8, 13032 (2013).
[Crossref]

Beckley, A. M.

Bekshaev, A.

A. Bekshaev, K. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5(1), 011039 (2015).
[Crossref]

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

Belinfante, F.

F. Belinfante, “On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields,” Physica 7(5), 449–474 (1940).
[Crossref]

Belinfante, F. J.

F. J. Belinfante, “On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields,” Physica 7(5), 449–474 (1940).
[Crossref]

Belskii, A. M.

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

Berry, M. V.

M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. 11(9), 094001 (2009).
[Crossref]

M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. R. Soc. A 462(2070), 1629–1642 (2006).
[Crossref]

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

Bliokh, B. K. Y.

B. K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010).
[Crossref]

Bliokh, K.

A. Bekshaev, K. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5(1), 011039 (2015).
[Crossref]

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

Bliokh, K. Y.

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Krolikowski, W.

V. Shvedov, P. Karpinski, Y. Sheng, X. Chen, W. Zhu, W. Krolikowski, and C. Hnatovsky, “Visualizing polarization singularities in Bessel-Poincaré beams,” Opt. Express 23(9), 12444–12453 (2015).
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P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, “The photonics wheel- demonstration of a state of light with purely transverse angular momentum,” J. Eur. Opt. Soc. 8, 13032 (2013).
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C. Varin, S. Payeur, V. Marceau, S. Fourmaux, A. April, B. Schmidt, P.-L. Fortin, N. Thiré, T. Brabec, F. Légaré, J.-C. Kieffer, and M. Piché, “Direct electron acceleration with radially polarized laser beams,” Appl. Sci. 3(1), 70–93 (2013).
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Qian, S.-X.

K. Lou, S.-X. Qian, Z.-C. Ren, C. Tu, Y. Li, and H.-T. Wang, “Femtosecond laser processing by using patterned vector optical fields,” Sci. Rep. 3, 2281 (2013).
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K. Lou, S.-X. Qian, Z.-C. Ren, C. Tu, Y. Li, and H.-T. Wang, “Femtosecond laser processing by using patterned vector optical fields,” Sci. Rep. 3, 2281 (2013).
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V. Shvedov, C. Hnatovsky, N. Eckerskorn, A. Rode, and W. Krolikowski, “Polarization-sensitive photophoresis,” Appl. Phys. Lett. 101(5), 051106 (2012).
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Santos, X.

M. C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Santos, M. Aguiló, and F. Díaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68(2), 187–197 (1999).
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V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91(9), 093602 (2003).
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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).
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Shelykh, I. A.

D. Colas, L. Dominici, S. Donati, A. A. Pervishko, T. C. H. Liew, I. A. Shelykh, D. Ballarini, M. de Giorgi, A. Bramati, G. Gigli, E. del Valle, F. P. Laussy, A. V. Kavokin, and D. Sanvitto, “Polarization shaping of Poincaré beams by polariton oscillations,” Light Sci. Appl. 4(11), e350 (2015).
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Shvedov, V.

V. Shvedov, P. Karpinski, Y. Sheng, X. Chen, W. Zhu, W. Krolikowski, and C. Hnatovsky, “Visualizing polarization singularities in Bessel-Poincaré beams,” Opt. Express 23(9), 12444–12453 (2015).
[Crossref] [PubMed]

V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photonics 8(11), 846–850 (2014).
[Crossref]

V. Shvedov, C. Hnatovsky, N. Eckerskorn, A. Rode, and W. Krolikowski, “Polarization-sensitive photophoresis,” Appl. Phys. Lett. 101(5), 051106 (2012).
[Crossref]

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106(12), 123901 (2011).
[Crossref] [PubMed]

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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] [PubMed]

K. Lou, S.-X. Qian, Z.-C. Ren, C. Tu, Y. Li, and H.-T. Wang, “Femtosecond laser processing by using patterned vector optical fields,” Sci. Rep. 3, 2281 (2013).
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Figures (6)

Fig. 1
Fig. 1 (a) The polarization and intensity distribution of the full Poincaré beam given by Eq. (7) for wf = 1.30. Right-handed circular, linear and left-handed circular polarization states are shown in red, green, and black, respectively. The blue circle represents polarization states along the meridian of the Poincaré sphere. (b) Schematic of tight focusing with a high NA lens.
Fig. 2
Fig. 2 Average transverse SAM [Sy]on the minimum focal angle θl and the normalized beam waist wf. The black line indicates the maximum value of SAM.
Fig. 3
Fig. 3 The distributions of electric EE*(first and third rows) and magnetic HH* (second and forth rows) parts of field energy (a,e) and the SAM densities [sx (b,f), sy (c,g), and sz (d,h)] of focal field in the focal plane for θl = 0° (a-d) and θl = 64° (e-h), when wf = 1.30. All quantities are normalized to the maximum electric field energy.
Fig. 4
Fig. 4 The distributions of electric EE* (left column) and magnetic HH* (right column) parts of field energy (a,e) and the SAM densities [sx (b,f), sy (c,g), and sz (d,h)] of focal field in xz plane for θl = 0° (a-d) and θl = 64° (e-h),, when wf = 1.30. All quantities are normalized to the maximum electric field energy. The size of the figures are 5μm × 2μm
Fig. 5
Fig. 5 The schematic for the creation of narrow ring-shaped intensity pattern with anti-symmetrically distributed Stokes parameter. A circularly polarized Gaussian beam is focused by the focal lens (FL) with NA = 0.3 into BC. After collimated by a collimating lens (CL) (NA = 0.65), the conically diffracted beam is filtered by a pin hole. The ring-shaped intensity pattern is obtained in the focal plane of lens 1 (L1). The lens 2 (L2) and CCD forms an image system. The QWP convert the polarization into the desired one. (b) The intensity distribution of total intensity near the image focal plane, where red line and two white lines indicate the focal and front and back spots planes, respectively. Two white vertical lines indicate positions of two Raman spots. (c, d) The experimentally measured intensity distributions in front of pin hole (c) and in the CCD plane (d).
Fig. 6
Fig. 6 (a) The polarization states of conically diffracted fields after QWP with φ0 = π/2. The red, green, and pink lines represent s3>0, s3<0, and s3 = 0, respectively. (b~c) The experimentally recorded intensity distributions of the CR ring structure after a polarizer with diagonal (b), horizontal (c), and vertical (d) directions.

Equations (18)

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sIm[( E * ×E)+( H * ×H)].
s E Im[( E y * E z ) e x +( E z * E x ) e y +( E x * E y ) e z ], s H Im[( H y * H z ) e x +( H z * H x ) e y +( H x * H y ) e z ],
U= U x (x,y,z) e x + U y (x,y,z) e y
E=[ U+i k 1 ( U) e z ] e ikz , H=[ ( e z ×U)+i k 1 ( ( e z ×U) e z ] e ikz .
U x (ρ,z=0) u x (ρ)=4 W/( w 0 2 c) exp( ρ 2 / w 0 2 ), U y (ρ,φ,z=0) u y (ρ) e iφ =4 2W/( w 0 2 c) (ρ/ w 0 )exp( ρ 2 / w 0 2 ) e iφ ,
u x (θ)=4 W/c 1 fNA w f exp{ sin 2 θ/ (NA w f ) 2 }, u y (θ)=4 2W/c sinθ fN A 2 w f 2 exp[ sin 2 θ/ (NA w f ) 2 ] e iφ .,
E in (θ,φ)=[ u x (θ) e x + u y (θ) e iφ e y ] e ikz , H in (θ,φ)=[ u y (θ) e iφ e x + u x (θ) e y ] e ikz .
E x =[ I x 0 +cos2ϕ K x 2 (i e iϕ K y 1 i e i3ϕ K y 3 )/2], E y =[sin2ϕ K x 2 + e iϕ I y 1 ( e iϕ K x 1 + e i3ϕ K y 3 )/2], E z =[2cosϕ L x 1 i L y 0 +i e i2ϕ L y 2 ],
H x =[sin2ϕ K x 2 e iϕ I y 1 ( e iϕ K x 1 + e i3ϕ K y 3 )/2], H y =[ I x 0 cos2ϕ K x 2 +(i e iϕ K y 1 i e i3ϕ K y 3 )/2], H z =[2sinϕ L x 1 L y 0 e i2ϕ L y 2 ],
I x,y n = i n C θ 1 θ 2 cosθ sinθ J n ( k 0 rsinθ) e i k 0 zcosθ (cosθ+1) u x,y (θ)dθ,
K x,y n = i n C θ 1 θ 2 cosθ sinθ J n ( k 0 rsinθ) e i k 0 zcosθ (cosθ1) u x,y (θ)dθ,
L x,y n = i n C θ 1 θ 2 cosθ sin 2 θ J n ( k 0 rsinθ) e i k 0 zcosθ u x,y (θ)dθ,
S= θ 1 θ 2 u x u y sin 2 θdθ θ 1 θ 2 [ u x 2 + u y 2 ]sinθdθ e ^ y .
U in =4 W/c 1 w 0 ξ(z) exp( ρ 2 w 0 2 ξ(z) ) σ ± ,
U=( B 0 σ ± + B 1 e ±iφ σ ),
B 0 = 1 2π 0 ηa(η) e iz η 2 /4 z 0 cos(η ρ 0 ) J 0 (η ρ 1 )dη, B 1 = 1 2π 0 ηa(η) e iz η 2 /4 z 0 sin(η ρ 0 ) J 1 (η ρ 1 )dη,
T= 2 2 [ 1icos2 φ 0 isin2 φ 0 isin2 φ 0 1+icos2 φ 0 ],
U= 1/2 {[ B 0 i B 1 e ±i(φ2 φ 0 ) ] σ ± +[i B 0 e ±i2 φ 0 + B 1 e ±iφ ] σ }.

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