Abstract

Nonlinear propagation of focused axisymmetrically-polarized ultrashort optical pulses along the optic axis in a uniaxial crystal is investigated experimentally and theoretically. The energy transfer between an azimuthally-polarized pulse and a radially-polarized pulse is observed. To analyze the nonlinear propagation, a general paraxial equation with a third-order nonlinearity for axisymmetrically-polarized pulses in a uniaxial crystal is derived and the extended Stokes parameters (ESPs) based on cylindrical coordinates are newly-introduced. The simulation results by using this equation, providing the calculated ESPs, well explain our experimental observations: 1) the energy transfer is attributed to the four-wave-mixing effect, reflecting the overlapping between the axisymmetrically polarized modes, 2) the variations of the polarization defined from the ESPs are clarified to be affected by the self- and the cross-phase modulations, which make the effective propagation length long or short.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. Hnatovsky, V Shvedov, W Krolikowski, and A Rode, “Revealing local field structure of focused ultrashort pulses,” Phys. Rev. Lett. 106, 123901 (2011).
    [CrossRef] [PubMed]
  2. J. Hamazaki, R. Morita, K. Chujo, Y. Kobayashi, S. Tanda, and T. Omatsu, “Optical-vortex laser ablation,” Opt. Express 18, 2144–2151 (2010).
    [CrossRef] [PubMed]
  3. T. Omatsu, K. Chujo, K. Miyamoto, M. Okida, K. Nakamura, N. Aoki, and R. Morita, “Metal microneedle fabrication using twisted light with spin,” Opt. Express 18, 17967–17973 (2010).
    [CrossRef] [PubMed]
  4. K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012).
    [CrossRef] [PubMed]
  5. N. M. Litchinitser, “Structured light meets structured matter,” Science 337, 1054–1055 (2012).
    [CrossRef] [PubMed]
  6. K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity in nanostructures,” Phys. Rev. Lett. 110, 143603 (2013).
    [CrossRef]
  7. Y. Tokizane, K. Shimatake, Y. Toda, K. Oka, M. Tsubota, S. Tanda, and R. Morita, “Global evaluation of closed-loop electron dynamics in quasi-one-dimensional conductors using polarization vortices,” Opt. Express 17, 24198–24207 (2009).
    [CrossRef]
  8. 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] [PubMed]
  9. Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
    [CrossRef] [PubMed]
  10. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
    [CrossRef] [PubMed]
  11. Y. S. Rumala, G. Milione, T. A. Nguyen, S. Pratavieira, Z. Hossain, D. Nolan, S. Slussarenko, E. Karimi, L. Marrucci, and R. R. Alfano, “Tunable supercontinum light vector vortex beam generator using a q-plate,” Opt. Lett. 38, 5083–5086 (2013).
    [CrossRef] [PubMed]
  12. S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34, 2525–2527 (2009).
    [CrossRef] [PubMed]
  13. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
    [CrossRef] [PubMed]
  14. 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]
  15. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef] [PubMed]
  16. Y. Tokizane, K. Oka, and R. Morita, “Supercontinuum optical vortex pulse generation without spatial or topological-charge dispersion,” Opt. Express 17, 14517–14525 (2009).
    [CrossRef] [PubMed]
  17. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948–1950 (1996).
    [CrossRef] [PubMed]
  18. V. G. Shvedov, C. Hnatovsky, W. Krolikowski, and A. V. Rode, “Efficient beam converter for the generation of high-power femtosecond vortices,” Opt. Lett. 35, 2660–2662 (2010).
    [CrossRef] [PubMed]
  19. Y. Izdebskaya, E. Brasselet, V. Shvedov, A. Desyatnikov, W. Krolikowski, and Y. Kivshar, “Dynamics of linear polarization conversion in uniaxial crystals,” Opt. Express 17, 18196–18208 (2009).
    [CrossRef] [PubMed]
  20. A. Volyar, V. Shvedov, T. Fadeyeva, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Generation of single-charge optical vortices with an uniaxial crystal,” Opt. Express 14, 3724–3729 (2006).
    [CrossRef]
  21. T. Fadeyeva, A. Rubass, Y. Egorov, A. Volyar, and G. Swartzlander, “Quadrefringence of optical vortices in a uniaxial crystal,” J. Opt. Soc. Am. A 25, 1634–1641 (2008).
    [CrossRef]
  22. T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-traveling paraxial beams,” J. Opt. Soc. Am. A 27, 381–389 (2010).
    [CrossRef]
  23. V. Shvedov, W. Krolikowski, A. Volyar, D. N. Neshev, A. S. Desyatnikov, and Y. S. Kivshar, “Focusing and correlation properties of white-light optical vortices,” Opt. Express 13, 7393–7398 (2005).
    [CrossRef] [PubMed]
  24. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).
  25. C. Shang and H. Hsu, “The spatial symmetric forms of third-order nonlinear susceptibility,” IEEE J. Quantum Electron. 23, 177–179 (1987).
    [CrossRef]
  26. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
    [CrossRef]
  27. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
    [CrossRef]
  28. E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. 34, 1021–1023 (2009).
    [CrossRef] [PubMed]
  29. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
    [CrossRef] [PubMed]
  30. K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19, 26132–26149 (2011).
    [CrossRef]
  31. G. Zhan, K. Oka, T. Ishigaki, and N. Baba, “Birefringent imaging spectrometer,” Appl. Opt. 41, 734–738 (2002).
    [CrossRef] [PubMed]
  32. W. J. Tropf and M. E. Thomas, “Properties of Crystals and Glasses,” in Optical Properties of Materials, Nonlinear Optics, Quantum Optics, Vol. IV of Handbook of Optics, M. Bass, G. Li, and E. V. Stryland, eds. (McGraw-Hill, 2010), Chapter 2.
  33. J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express 14, 8382–8392 (2006).
    [CrossRef] [PubMed]
  34. A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010).
    [CrossRef] [PubMed]
  35. E. J. Weniger, “On the analyticity of Laguerre series,” J. Phys. A: Math. Theor. 41, 425207 (2008).
    [CrossRef]

2013 (2)

2012 (2)

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012).
[CrossRef] [PubMed]

N. M. Litchinitser, “Structured light meets structured matter,” Science 337, 1054–1055 (2012).
[CrossRef] [PubMed]

2011 (3)

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

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19, 26132–26149 (2011).
[CrossRef]

2010 (5)

2009 (5)

2008 (2)

2007 (1)

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]

2006 (3)

2005 (1)

2004 (1)

2002 (3)

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

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

G. Zhan, K. Oka, T. Ishigaki, and N. Baba, “Birefringent imaging spectrometer,” Appl. Opt. 41, 734–738 (2002).
[CrossRef] [PubMed]

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] [PubMed]

2000 (1)

1996 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

1987 (1)

C. Shang and H. Hsu, “The spatial symmetric forms of third-order nonlinear susceptibility,” IEEE J. Quantum Electron. 23, 177–179 (1987).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

Alfano, R. R.

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Alonso, M. A.

Aoki, N.

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012).
[CrossRef] [PubMed]

T. Omatsu, K. Chujo, K. Miyamoto, M. Okida, K. Nakamura, N. Aoki, and R. Morita, “Metal microneedle fabrication using twisted light with spin,” Opt. Express 18, 17967–17973 (2010).
[CrossRef] [PubMed]

Baba, N.

Beckley, A. M.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

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. 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] [PubMed]

Bliokh, K. Y.

Brasselet, E.

Brown, T. G.

Chujo, K.

Dainty, C.

Dennis, M. R.

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

Desyatnikov, A.

Desyatnikov, A. S.

Egorov, Y.

Fadeyeva, T.

Fadeyeva, T. A.

Fürhapter, 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]

Hamazaki, J.

Hnatovsky, C.

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

V. G. Shvedov, C. Hnatovsky, W. Krolikowski, and A. V. Rode, “Efficient beam converter for the generation of high-power femtosecond vortices,” Opt. Lett. 35, 2660–2662 (2010).
[CrossRef] [PubMed]

Hossain, Z.

Hsu, H.

C. Shang and H. Hsu, “The spatial symmetric forms of third-order nonlinear susceptibility,” IEEE J. Quantum Electron. 23, 177–179 (1987).
[CrossRef]

Ishigaki, T.

Izdebskaya, Y.

Jesacher, A.

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]

Karimi, E.

Keitel, C. H.

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

Kivshar, Y.

Kivshar, Y. S.

Kobayashi, Y.

Kristensen, P.

Krolikowski, W

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

Krolikowski, W.

Lara, D.

Litchinitser, N. M.

N. M. Litchinitser, “Structured light meets structured matter,” Science 337, 1054–1055 (2012).
[CrossRef] [PubMed]

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Marrucci, L.

Maurer, C.

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]

Milione, G.

Mineta, Y.

Miyamoto, K.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity in nanostructures,” Phys. Rev. Lett. 110, 143603 (2013).
[CrossRef]

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012).
[CrossRef] [PubMed]

T. Omatsu, K. Chujo, K. Miyamoto, M. Okida, K. Nakamura, N. Aoki, and R. Morita, “Metal microneedle fabrication using twisted light with spin,” Opt. Express 18, 17967–17973 (2010).
[CrossRef] [PubMed]

Morita, R.

Nakamura, K.

Neshev, D. N.

Nguyen, T. A.

Nolan, D.

Nolan, D. A.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

Novotny, L.

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] [PubMed]

Oka, K.

Okida, M.

Omatsu, T.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity in nanostructures,” Phys. Rev. Lett. 110, 143603 (2013).
[CrossRef]

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012).
[CrossRef] [PubMed]

J. Hamazaki, R. Morita, K. Chujo, Y. Kobayashi, S. Tanda, and T. Omatsu, “Optical-vortex laser ablation,” Opt. Express 18, 2144–2151 (2010).
[CrossRef] [PubMed]

T. Omatsu, K. Chujo, K. Miyamoto, M. Okida, K. Nakamura, N. Aoki, and R. Morita, “Metal microneedle fabrication using twisted light with spin,” Opt. Express 18, 17967–17973 (2010).
[CrossRef] [PubMed]

Ostrovskaya, E. A.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Pratavieira, S.

Ramachandran, S.

Ritsch-Marte, M.

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]

Rode, A

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

Rode, A. V.

Rodríguez-Herrera, O. G.

Rubass, A.

Rumala, Y. S.

Salamin, Y. I.

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

Schadt, M.

Shang, C.

C. Shang and H. Hsu, “The spatial symmetric forms of third-order nonlinear susceptibility,” IEEE J. Quantum Electron. 23, 177–179 (1987).
[CrossRef]

Shimatake, K.

Shvedov, V

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

Shvedov, V.

Shvedov, V. G.

Slussarenko, S.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Stalder, M.

Swartzlander, G.

Sztul, H. I.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

Takahashi, F.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity in nanostructures,” Phys. Rev. Lett. 110, 143603 (2013).
[CrossRef]

Takizawa, S.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity in nanostructures,” Phys. Rev. Lett. 110, 143603 (2013).
[CrossRef]

Tanda, S.

Thomas, M. E.

W. J. Tropf and M. E. Thomas, “Properties of Crystals and Glasses,” in Optical Properties of Materials, Nonlinear Optics, Quantum Optics, Vol. IV of Handbook of Optics, M. Bass, G. Li, and E. V. Stryland, eds. (McGraw-Hill, 2010), Chapter 2.

Toda, Y.

Tokizane, Y.

Toyoda, K.

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity in nanostructures,” Phys. Rev. Lett. 110, 143603 (2013).
[CrossRef]

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012).
[CrossRef] [PubMed]

Tropf, W. J.

W. J. Tropf and M. E. Thomas, “Properties of Crystals and Glasses,” in Optical Properties of Materials, Nonlinear Optics, Quantum Optics, Vol. IV of Handbook of Optics, M. Bass, G. Li, and E. V. Stryland, eds. (McGraw-Hill, 2010), Chapter 2.

Tsubota, M.

Volyar, A.

Volyar, A. V.

Weniger, E. J.

E. J. Weniger, “On the analyticity of Laguerre series,” J. Phys. A: Math. Theor. 41, 425207 (2008).
[CrossRef]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Yan, M. F.

Youngworth, K. S.

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] [PubMed]

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

Zhan, G.

Zhan, Q.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

C. Shang and H. Hsu, “The spatial symmetric forms of third-order nonlinear susceptibility,” IEEE J. Quantum Electron. 23, 177–179 (1987).
[CrossRef]

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

J. Phys. A: Math. Theor. (1)

E. J. Weniger, “On the analyticity of Laguerre series,” J. Phys. A: Math. Theor. 41, 425207 (2008).
[CrossRef]

Nano Lett. (1)

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012).
[CrossRef] [PubMed]

New J. Phys. (1)

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]

Opt. Commun. (1)

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

Opt. Express (12)

V. Shvedov, W. Krolikowski, A. Volyar, D. N. Neshev, A. S. Desyatnikov, and Y. S. Kivshar, “Focusing and correlation properties of white-light optical vortices,” Opt. Express 13, 7393–7398 (2005).
[CrossRef] [PubMed]

J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express 14, 8382–8392 (2006).
[CrossRef] [PubMed]

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

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

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19, 26132–26149 (2011).
[CrossRef]

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

Y. Tokizane, K. Oka, and R. Morita, “Supercontinuum optical vortex pulse generation without spatial or topological-charge dispersion,” Opt. Express 17, 14517–14525 (2009).
[CrossRef] [PubMed]

Y. Izdebskaya, E. Brasselet, V. Shvedov, A. Desyatnikov, W. Krolikowski, and Y. Kivshar, “Dynamics of linear polarization conversion in uniaxial crystals,” Opt. Express 17, 18196–18208 (2009).
[CrossRef] [PubMed]

A. Volyar, V. Shvedov, T. Fadeyeva, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Generation of single-charge optical vortices with an uniaxial crystal,” Opt. Express 14, 3724–3729 (2006).
[CrossRef]

J. Hamazaki, R. Morita, K. Chujo, Y. Kobayashi, S. Tanda, and T. Omatsu, “Optical-vortex laser ablation,” Opt. Express 18, 2144–2151 (2010).
[CrossRef] [PubMed]

T. Omatsu, K. Chujo, K. Miyamoto, M. Okida, K. Nakamura, N. Aoki, and R. Morita, “Metal microneedle fabrication using twisted light with spin,” Opt. Express 18, 17967–17973 (2010).
[CrossRef] [PubMed]

Y. Tokizane, K. Shimatake, Y. Toda, K. Oka, M. Tsubota, S. Tanda, and R. Morita, “Global evaluation of closed-loop electron dynamics in quasi-one-dimensional conductors using polarization vortices,” Opt. Express 17, 24198–24207 (2009).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (6)

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] [PubMed]

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

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

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity in nanostructures,” Phys. Rev. Lett. 110, 143603 (2013).
[CrossRef]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

Science (1)

N. M. Litchinitser, “Structured light meets structured matter,” Science 337, 1054–1055 (2012).
[CrossRef] [PubMed]

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

W. J. Tropf and M. E. Thomas, “Properties of Crystals and Glasses,” in Optical Properties of Materials, Nonlinear Optics, Quantum Optics, Vol. IV of Handbook of Optics, M. Bass, G. Li, and E. V. Stryland, eds. (McGraw-Hill, 2010), Chapter 2.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Poincaré sphere corresponding to the ESPs based on cylindrical coordinates and typical axisymmetrically polarized states (blue points).

Fig. 2
Fig. 2

(a) Trajectories of linear propagations in positive (α > 0) uniaxial crystals on a plane of the second and third normalized ESPs. Blue points indicate initial states. (b) DOP V space E as a function of the propagation length = |α|z/(z0no) (α = 0.237; calcite at 800 nm) in the linear propagation case [Eq. (22)].

Fig. 3
Fig. 3

(a) Setup for nonlinear propagation of axisymmetrically polarized pulses. BP, a bandpass filter; QWP1,2, quarter-wave plates; HWP, a half-wave plate; SPP, a spiral phase plate (l = −1); L1,2, convex lenses; CR, a nonlinear crystal (a 2 mm- or 5 mm-thick c-cut calcite crystal); PBS, a polarizing beam splitter; ND, a neutral density filter. zF is the relative position between the input surface of CR and the focal point of the input beam (without CR). (b) The flow chart to obtain the ESPs. x, y, +45° and −45° represent intensity distributions of linearly x-, y-, +45°-, −45°-polarized components, respectively. LCP and RCP represent intensity distributions of LCP and RCP components, respectively. Spatial-dependent, conventional Stokes parameters (S1(x, y), S2(x, y), S3(x, y)) obtained from these six intensity profiles provide ESPs. (c) Schematic definitions of zF, z F , out A and z F , out R. The red, green and orange lines stand for the beam paths of |s = +1〉|l = −1〉 OV, the RP and AP modes, respectively.

Fig. 4
Fig. 4

Experimental results. Δ S ˜ 1 E after the nonlinear propagation of (a) 2mm(≃ 4z0) and (b) 5mm(≃ 10z0) c-cut calcite. Δ V space E after the nonlinear propagation of (c) 2mm and (d) 5mm c-cut calcite. Each graph has results for two pulse energy: 0.39 μJ and 0.81 μJ. The cyan bars represent the focus position where the focus of the input beam is at the input facet (zF = 0). The orange and green bars correspond to the focus positions zF where the foci of the AP and the RP modes are at the output facet, respectively.

Fig. 5
Fig. 5

Trajectories for |s = +1〉|l = −1〉 OV (red line), RP mode (green line) and AP mode (orange line) inputs. (a) The focus of input beam corresponds to the input facet of the crystals (zF = 0mm). The focus of the AP mode corresponds to the output facet of (b) 2 mm-thick CR (zF ∼ 1.2mm) and (c) that of 5 mm-thick CR (zF ∼ 3.0mm).

Fig. 6
Fig. 6

Simulation results for focus position dependence of Δ S ˜ 1 E and Δ V space E at two standardized crystal lengths. (a) and (c) z = 4z0; (b) and (d) z = 10z0 (γ = 2/3). The cyan bars represent the focus position where the input beam is at the input facet (zF = 0). The orange and green bars correspond to the focus positions zF where the foci of the AP and the RP modes are at the output facet, respectively.

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

D ( r , t ) = ε E ( r , t ) + P NL ( r , t ) .
P ˜ x ( r , t ) = 2 ε o n o n 2 E [ ( | E ˜ x | 2 + γ | E ˜ y | 2 ) E ˜ x + ( 1 γ ) E ˜ x * E ˜ y 2 ] ,
P ˜ y ( r , t ) = 2 ε o n o n 2 E [ ( γ | E ˜ x | 2 + | E ˜ y | 2 ) E ˜ y + ( 1 γ ) E ˜ y * E ˜ x 2 ] ,
n o , e = ( ε o , e / ε 0 ) 1 / 2 ,
n 2 E n o 8 3 = χ x x x x ( 3 ) ,
γ n 2 E n o 8 3 = χ x x y y ( 3 ) + χ xyxy ( 3 ) = 2 χ x x y y ( 3 ) .
2 ( E ˜ e i k z ) + [ ( E ˜ e i k z ) ] + k 2 ε E ˜ e i k z = k 2 ε 0 P ˜ NL e i k z .
( E ˜ e i k z ) = e i k z ( 1 ε o ε e ) E ˜ e i k z 1 ε e P ˜ NL ,
2 i k 0 n o z E ˜ + 2 E ˜ ( 1 ε o ε e ) ( E ˜ ) + k 2 ε 0 P ˜ NL = 0 .
E ˜ = ( E ˜ x E ˜ y ) = E ˜ r e r + E ˜ ϕ e ϕ .
e r = ( cos ϕ sin ϕ ) , e ϕ = ( sin ϕ cos ϕ ) ,
2 i k n o z E ˜ r = n o 2 n e 2 ( r 2 + 1 r r 1 r 2 ) E ˜ r 2 k 2 n 0 n 2 E [ ( | E ˜ r | 2 + γ | E ˜ ϕ | 2 ) E ˜ r + ( 1 γ ) E ˜ ϕ 2 E ˜ r * ] ,
2 i k n o z E ˜ ϕ = ( r 2 + 1 r r 1 r 2 ) E ˜ ϕ 2 k 2 n 0 n 2 E [ ( γ | E ˜ r | 2 + | E ˜ ϕ | 2 ) E ˜ ϕ + ( 1 γ ) E ˜ r 2 E ˜ ϕ * ] .
S 0 E ( z ) = | E r | 2 + | E ϕ | 2 d x d y = S 0 d x d y , S 1 E ( z ) = | E r | 2 | E ϕ | 2 d x d y = ( S 1 cos ( 2 ϕ ) + S 2 sin ( 2 ϕ ) ) d x d y , S 2 E ( z ) = E r * E ϕ + E r E ϕ * d x d y = ( S 1 sin ( 2 ϕ ) + S 2 cos ( 2 ϕ ) ) d x d y , S 3 E ( z ) = i E r * E ϕ E r E ϕ * d x d y = S 3 d x d y
V E ( z ) = [ ( S 1 E ( z ) ) 2 + ( S 2 E ( z ) ) 2 + ( S 3 E ( z ) ) 2 ] S 0 E ( z ) 1 / 2 .
V space E ( z ) = [ ( S 1 E ( z ) ) 2 + ( S 2 E ( z ) ) 2 + ( S 3 E ( z ) ) 2 ] 1 / 2 S 0 ( r ) V ( r ) d x d y ,
E ˜ ( r , ϕ , z 0 ) = ( A r e r + A ϕ e ϕ ) r w 0 σ 2 ( z ) exp ( r 2 w 0 2 σ ( z ) ) ,
E ˜ ( r , ϕ , z 0 ) = A r r w 0 σ r 2 ( z ) exp ( r 2 w 0 2 σ r ( z ) ) e r + A ϕ r w 0 σ ϕ 2 ( z ) exp ( r 2 w 0 2 σ ϕ ( z ) ) e ϕ ,
S ˜ 1 E ( z ˜ ) = | A r | 2 | A ϕ | 2 | A r | 2 + | A ϕ | 2 = const . ,
S ˜ 1 E ( z ˜ ) = 4 [ 1 ( S ˜ 1 E ) 2 ] 1 / 2 ( 4 z ˜ 2 ) cos δ r ϕ + 4 sgn ( α ) z ˜ sin δ r ϕ ( 4 + z ˜ 2 ) 2 ,
S ˜ 3 E ( z ˜ ) = 4 [ 1 ( S ˜ 1 E ) 2 ] 1 / 2 ( 4 z ˜ 2 ) sin δ r ϕ + 4 sgn ( α ) z ˜ cos δ r ϕ ( 4 + z ˜ 2 ) 2 ,
V space E ( z ˜ ) = { [ 1 ( S ˜ 1 E ) 2 ] ( 4 4 + z ˜ 2 ) 2 + ( S ˜ 1 E ) 2 } 1 / 2 ,
Δ S ˜ 1 E = S ˜ 1 E S ˜ 1 E , Linear ,
Δ V space E = V space E V space E , Linear .
n o z F , out A = 1.65 z F , out A = L , n e 2 n o z F , out R = 1.33 z F , out R = L ,
z ˜ E ˜ r = i { n o 4 n e 2 ( r ˜ 2 + 1 r ˜ r ˜ 1 r ˜ 2 ) E ˜ r + k z 0 n 2 E [ ( | E ˜ r | 2 + γ | E ˜ ϕ | 2 ) E ˜ r + ( 1 γ ) E ˜ ϕ 2 E ˜ r * ] } ,
z ˜ E ˜ ϕ = i { 1 4 n o ( r ˜ 2 + 1 r ˜ r ˜ 1 r ˜ 2 ) E ˜ ϕ + k z 0 n 2 E [ ( γ | E ˜ r | 2 + | E ˜ ϕ | 2 ) E ˜ ϕ + ( 1 γ ) E ˜ r 2 E ˜ ϕ * ] } .
E ˜ ( gen ) = A ( gen ) p = 0 π 1 / 2 ( 2 p 1 ) ! ! 2 ( p + 1 ) ( 2 p ) ! ! u 1 p ,
u m p = ( 2 r w 0 | σ ( z ) | ) | m | L p | m | ( 2 r 2 w 0 2 | σ ( z ) | 2 ) 1 | σ ( z ) | exp ( r 2 w 0 2 σ ( z ) + i m ϕ i Ψ G ( z ) ) ,
Ψ G ( z ) = ( 2 p + | m | + 1 ) arctan ( z / z 0 ) .

Metrics