Abstract

We investigate the far-field vectorial self-diffraction behavior of a cylindrical vector field passing though an optically thin Kerr medium. Theoretically, we obtain the analytical expression of the focal field of the cylindrical vector field with arbitrary integer topological charge based on the Fourier transform under the weak-focusing condition. Considering the additional nonlinear phase shift photoinduced by a self-focusing medium, we simulate the far-field vectorial self-diffraction patterns of the cylindrical vector field using the Huygens-Fresnel diffraction integral method. Experimentally, we observe the vectorial self-diffraction rings of the femtosecond-pulsed radially polarized field and high-order cylindrical vector field in carbon disulfide, which is in good agreement with the theoretical simulations. Our results benefit the understanding of the related spatial self-phase modulation effects of the vector light fields, such as spatial solitons, self-trapping, and self-guided propagation.

© 2011 OSA

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  1. W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–105 (1967).
    [CrossRef]
  2. S. D. Durbin, S. M. Arakelian, and Y. R. Shen, “Laser induced diffraction rings from a nematic liquid crystal film,” Opt. Lett. 6, 411–413 (1981).
    [PubMed]
  3. P. F. Wu, B. Zou, X. Wu, J. Xu, X. Gong, G. Zhang, G. Tang, and W. Chen, “Biphotonic self-diffraction in azo-doped polymer film,” Appl. Phys. Lett. 70, 1224–1226 (1997).
    [CrossRef]
  4. R. G. Harrison, L. Dambly, D. J. Yu, and W. P. Lu, “A new self-diffraction pattern formation in defocusing liquid media,” Opt. Commun. 139, 69–72 (1997).
    [CrossRef]
  5. W. Ji, W. Z. Chen, S. H. Lim, J. Y. Lin, and Z. X. Guo, “Gravitation-dependent, thermally-induced self-diffraction in carbon nanotube solutions,” Opt. Express 14, 8958–8966 (2006).
    [CrossRef] [PubMed]
  6. M. Trejo-Durán, J. A. Andrade-Lucio, A. Martinez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
    [CrossRef]
  7. A. B. Villafranca and K. Saravanamuttu, “Spontaneous and sequential transitions of a Gaussian beam into diffraction rings, single ring and circular array of filaments in a photopolymer,” Opt. Express 19, 15560–15573 (2011).
    [CrossRef] [PubMed]
  8. E. Santamato and Y. R. Shen, “Field-curvature effect on the diffraction ring pattern of a laser beam dressed by spatial self-phase modulation in a nematic film,” Opt. Lett. 9, 564–566 (1984).
    [CrossRef] [PubMed]
  9. D. J. Yu, W. P. Lu, and R. G. Harrison, “Analysis of dark spot formation in absorbing liquid media,” J. Mod. Opt. 45, 2597–2606 (1998).
    [CrossRef]
  10. L. G. Deng, K. N. He, T. Z. Zhou, and C. D. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A: Pure Appl. Opt. 7, 409–415 (2005).
    [CrossRef]
  11. C. M. Nascimento, M. A. R. C. Alencar, S. Chávez-Cerda, M. G. A. da Silva, M. R. Meneghetti, and J. M. Hickmann, “Experimental demonstration of novel effects on the far-field diffraction patterns of a Gaussian beam in a Kerr medium,” J. Opt. A: Pure Appl. Opt. 8, 947–951 (2006).
    [CrossRef]
  12. E. V. G. Ramirez, M. L. A. Carrasco, M. M. M. Otero, S. C. Cerda, and M. D. I. Castillo, “Far field intensity distributions due to spatial self phase modulation of a Gaussian beam by a thin nonlocal nonlinear media,” Opt. Express 18, 22067–22079 (2010).
    [CrossRef] [PubMed]
  13. Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
    [CrossRef]
  14. A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98, 851–855 (2010).
    [CrossRef]
  15. S. Y. Yang and Q. W. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A: Pure Appl. Opt. 10, 152103(2008).
    [CrossRef]
  16. A. A. Ishaaya, L. T. Vuong, T. D. Grow, and A. L. Gaeta, “Self-focusing dynamics of polarization vortices in Kerr media,” Opt. Lett. 33, 13–15 (2008).
    [CrossRef]
  17. X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
    [CrossRef] [PubMed]
  18. W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613–1615 (1993).
    [CrossRef]
  19. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18, 10828–10833 (2010).
    [CrossRef] [PubMed]
  20. T. Züchner, A. V. Failla, and A. J. Meixner, “Light microscopy with doughnut modes: a concept to detect, characterize, and manipulate individual nanoobjects,” Angew. Chem. Int. Ed. 50, 5274–5293 (2011).
    [CrossRef]
  21. X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
    [CrossRef] [PubMed]
  22. X. Q. Yan, X. L. Zhang, S. Shi, Z. B. Liu, and J. G. Tian, “Third-order nonlinear susceptibility tensor elements of CS2 at femtosecond time scale,” Opt. Express 19, 5559–5564 (2011).
    [CrossRef] [PubMed]
  23. B. Gu, Y. Wang, J. Wang, and W. Ji, “Femtosecond third-order optical nonlinearity of polycrystalline BiFeO3,” Opt. Express 17, 10970–10975 (2009).
    [CrossRef] [PubMed]

2011 (3)

2010 (4)

2009 (2)

2008 (2)

S. Y. Yang and Q. W. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A: Pure Appl. Opt. 10, 152103(2008).
[CrossRef]

A. A. Ishaaya, L. T. Vuong, T. D. Grow, and A. L. Gaeta, “Self-focusing dynamics of polarization vortices in Kerr media,” Opt. Lett. 33, 13–15 (2008).
[CrossRef]

2007 (2)

X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
[CrossRef] [PubMed]

M. Trejo-Durán, J. A. Andrade-Lucio, A. Martinez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
[CrossRef]

2006 (2)

W. Ji, W. Z. Chen, S. H. Lim, J. Y. Lin, and Z. X. Guo, “Gravitation-dependent, thermally-induced self-diffraction in carbon nanotube solutions,” Opt. Express 14, 8958–8966 (2006).
[CrossRef] [PubMed]

C. M. Nascimento, M. A. R. C. Alencar, S. Chávez-Cerda, M. G. A. da Silva, M. R. Meneghetti, and J. M. Hickmann, “Experimental demonstration of novel effects on the far-field diffraction patterns of a Gaussian beam in a Kerr medium,” J. Opt. A: Pure Appl. Opt. 8, 947–951 (2006).
[CrossRef]

2005 (1)

L. G. Deng, K. N. He, T. Z. Zhou, and C. D. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A: Pure Appl. Opt. 7, 409–415 (2005).
[CrossRef]

1998 (1)

D. J. Yu, W. P. Lu, and R. G. Harrison, “Analysis of dark spot formation in absorbing liquid media,” J. Mod. Opt. 45, 2597–2606 (1998).
[CrossRef]

1997 (2)

P. F. Wu, B. Zou, X. Wu, J. Xu, X. Gong, G. Zhang, G. Tang, and W. Chen, “Biphotonic self-diffraction in azo-doped polymer film,” Appl. Phys. Lett. 70, 1224–1226 (1997).
[CrossRef]

R. G. Harrison, L. Dambly, D. J. Yu, and W. P. Lu, “A new self-diffraction pattern formation in defocusing liquid media,” Opt. Commun. 139, 69–72 (1997).
[CrossRef]

1993 (1)

W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613–1615 (1993).
[CrossRef]

1984 (1)

1981 (1)

1967 (1)

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–105 (1967).
[CrossRef]

Alencar, M. A. R. C.

C. M. Nascimento, M. A. R. C. Alencar, S. Chávez-Cerda, M. G. A. da Silva, M. R. Meneghetti, and J. M. Hickmann, “Experimental demonstration of novel effects on the far-field diffraction patterns of a Gaussian beam in a Kerr medium,” J. Opt. A: Pure Appl. Opt. 8, 947–951 (2006).
[CrossRef]

Andrade-Lucio, J. A.

M. Trejo-Durán, J. A. Andrade-Lucio, A. Martinez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
[CrossRef]

Arakelian, S. M.

Callen, W. R.

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–105 (1967).
[CrossRef]

Carrasco, M. L. A.

Castaño, V. M.

M. Trejo-Durán, J. A. Andrade-Lucio, A. Martinez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
[CrossRef]

Castillo, M. D. I.

Cerda, S. C.

Chávez-Cerda, S.

C. M. Nascimento, M. A. R. C. Alencar, S. Chávez-Cerda, M. G. A. da Silva, M. R. Meneghetti, and J. M. Hickmann, “Experimental demonstration of novel effects on the far-field diffraction patterns of a Gaussian beam in a Kerr medium,” J. Opt. A: Pure Appl. Opt. 8, 947–951 (2006).
[CrossRef]

Chen, J.

Chen, W.

P. F. Wu, B. Zou, X. Wu, J. Xu, X. Gong, G. Zhang, G. Tang, and W. Chen, “Biphotonic self-diffraction in azo-doped polymer film,” Appl. Phys. Lett. 70, 1224–1226 (1997).
[CrossRef]

Chen, W. Z.

da Silva, M. G. A.

C. M. Nascimento, M. A. R. C. Alencar, S. Chávez-Cerda, M. G. A. da Silva, M. R. Meneghetti, and J. M. Hickmann, “Experimental demonstration of novel effects on the far-field diffraction patterns of a Gaussian beam in a Kerr medium,” J. Opt. A: Pure Appl. Opt. 8, 947–951 (2006).
[CrossRef]

Dambly, L.

R. G. Harrison, L. Dambly, D. J. Yu, and W. P. Lu, “A new self-diffraction pattern formation in defocusing liquid media,” Opt. Commun. 139, 69–72 (1997).
[CrossRef]

Deng, L. G.

L. G. Deng, K. N. He, T. Z. Zhou, and C. D. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A: Pure Appl. Opt. 7, 409–415 (2005).
[CrossRef]

Ding, J.

Ding, J. P.

Durbin, S. D.

Failla, A. V.

T. Züchner, A. V. Failla, and A. J. Meixner, “Light microscopy with doughnut modes: a concept to detect, characterize, and manipulate individual nanoobjects,” Angew. Chem. Int. Ed. 50, 5274–5293 (2011).
[CrossRef]

Gaeta, A. L.

Gong, X.

P. F. Wu, B. Zou, X. Wu, J. Xu, X. Gong, G. Zhang, G. Tang, and W. Chen, “Biphotonic self-diffraction in azo-doped polymer film,” Appl. Phys. Lett. 70, 1224–1226 (1997).
[CrossRef]

Grow, T. D.

Gu, B.

Guo, C. S.

Guo, Z. X.

Harrison, R. G.

D. J. Yu, W. P. Lu, and R. G. Harrison, “Analysis of dark spot formation in absorbing liquid media,” J. Mod. Opt. 45, 2597–2606 (1998).
[CrossRef]

R. G. Harrison, L. Dambly, D. J. Yu, and W. P. Lu, “A new self-diffraction pattern formation in defocusing liquid media,” Opt. Commun. 139, 69–72 (1997).
[CrossRef]

He, K. N.

L. G. Deng, K. N. He, T. Z. Zhou, and C. D. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A: Pure Appl. Opt. 7, 409–415 (2005).
[CrossRef]

Hickmann, J. M.

C. M. Nascimento, M. A. R. C. Alencar, S. Chávez-Cerda, M. G. A. da Silva, M. R. Meneghetti, and J. M. Hickmann, “Experimental demonstration of novel effects on the far-field diffraction patterns of a Gaussian beam in a Kerr medium,” J. Opt. A: Pure Appl. Opt. 8, 947–951 (2006).
[CrossRef]

Huth, B. G.

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–105 (1967).
[CrossRef]

Ishaaya, A. A.

Ji, W.

Kozawa, Y.

A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98, 851–855 (2010).
[CrossRef]

Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18, 10828–10833 (2010).
[CrossRef] [PubMed]

Li, C. D.

L. G. Deng, K. N. He, T. Z. Zhou, and C. D. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A: Pure Appl. Opt. 7, 409–415 (2005).
[CrossRef]

Li, Y. N.

Lim, S. H.

Lin, J. Y.

Liu, Z. B.

Lu, W. P.

D. J. Yu, W. P. Lu, and R. G. Harrison, “Analysis of dark spot formation in absorbing liquid media,” J. Mod. Opt. 45, 2597–2606 (1998).
[CrossRef]

R. G. Harrison, L. Dambly, D. J. Yu, and W. P. Lu, “A new self-diffraction pattern formation in defocusing liquid media,” Opt. Commun. 139, 69–72 (1997).
[CrossRef]

Martinez-Richa, A.

M. Trejo-Durán, J. A. Andrade-Lucio, A. Martinez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
[CrossRef]

Meixner, A. J.

T. Züchner, A. V. Failla, and A. J. Meixner, “Light microscopy with doughnut modes: a concept to detect, characterize, and manipulate individual nanoobjects,” Angew. Chem. Int. Ed. 50, 5274–5293 (2011).
[CrossRef]

Meneghetti, M. R.

C. M. Nascimento, M. A. R. C. Alencar, S. Chávez-Cerda, M. G. A. da Silva, M. R. Meneghetti, and J. M. Hickmann, “Experimental demonstration of novel effects on the far-field diffraction patterns of a Gaussian beam in a Kerr medium,” J. Opt. A: Pure Appl. Opt. 8, 947–951 (2006).
[CrossRef]

Nascimento, C. M.

C. M. Nascimento, M. A. R. C. Alencar, S. Chávez-Cerda, M. G. A. da Silva, M. R. Meneghetti, and J. M. Hickmann, “Experimental demonstration of novel effects on the far-field diffraction patterns of a Gaussian beam in a Kerr medium,” J. Opt. A: Pure Appl. Opt. 8, 947–951 (2006).
[CrossRef]

Ni, W. J.

Ohtsu, A.

A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98, 851–855 (2010).
[CrossRef]

Otero, M. M. M.

Palffy-Muhoray, P.

W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613–1615 (1993).
[CrossRef]

Pantell, R. H.

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–105 (1967).
[CrossRef]

Ramirez, E. V. G.

Santamato, E.

Saravanamuttu, K.

Sato, S.

A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98, 851–855 (2010).
[CrossRef]

Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18, 10828–10833 (2010).
[CrossRef] [PubMed]

Shen, Y. R.

Shi, S.

Tang, G.

P. F. Wu, B. Zou, X. Wu, J. Xu, X. Gong, G. Zhang, G. Tang, and W. Chen, “Biphotonic self-diffraction in azo-doped polymer film,” Appl. Phys. Lett. 70, 1224–1226 (1997).
[CrossRef]

Tian, J. G.

Trejo-Durán, M.

M. Trejo-Durán, J. A. Andrade-Lucio, A. Martinez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
[CrossRef]

Vera-Graziano, R.

M. Trejo-Durán, J. A. Andrade-Lucio, A. Martinez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
[CrossRef]

Villafranca, A. B.

Vuong, L. T.

Wang, H. T.

Wang, J.

Wang, X. L.

Wang, Y.

Wu, P. F.

P. F. Wu, B. Zou, X. Wu, J. Xu, X. Gong, G. Zhang, G. Tang, and W. Chen, “Biphotonic self-diffraction in azo-doped polymer film,” Appl. Phys. Lett. 70, 1224–1226 (1997).
[CrossRef]

Wu, X.

P. F. Wu, B. Zou, X. Wu, J. Xu, X. Gong, G. Zhang, G. Tang, and W. Chen, “Biphotonic self-diffraction in azo-doped polymer film,” Appl. Phys. Lett. 70, 1224–1226 (1997).
[CrossRef]

Xu, J.

P. F. Wu, B. Zou, X. Wu, J. Xu, X. Gong, G. Zhang, G. Tang, and W. Chen, “Biphotonic self-diffraction in azo-doped polymer film,” Appl. Phys. Lett. 70, 1224–1226 (1997).
[CrossRef]

Yan, X. Q.

Yang, S. Y.

S. Y. Yang and Q. W. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A: Pure Appl. Opt. 10, 152103(2008).
[CrossRef]

Yu, D. J.

D. J. Yu, W. P. Lu, and R. G. Harrison, “Analysis of dark spot formation in absorbing liquid media,” J. Mod. Opt. 45, 2597–2606 (1998).
[CrossRef]

R. G. Harrison, L. Dambly, D. J. Yu, and W. P. Lu, “A new self-diffraction pattern formation in defocusing liquid media,” Opt. Commun. 139, 69–72 (1997).
[CrossRef]

Zhan, Q. W.

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

S. Y. Yang and Q. W. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A: Pure Appl. Opt. 10, 152103(2008).
[CrossRef]

Zhang, G.

P. F. Wu, B. Zou, X. Wu, J. Xu, X. Gong, G. Zhang, G. Tang, and W. Chen, “Biphotonic self-diffraction in azo-doped polymer film,” Appl. Phys. Lett. 70, 1224–1226 (1997).
[CrossRef]

Zhang, X. L.

Zhao, W.

W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613–1615 (1993).
[CrossRef]

Zhou, T. Z.

L. G. Deng, K. N. He, T. Z. Zhou, and C. D. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A: Pure Appl. Opt. 7, 409–415 (2005).
[CrossRef]

Zou, B.

P. F. Wu, B. Zou, X. Wu, J. Xu, X. Gong, G. Zhang, G. Tang, and W. Chen, “Biphotonic self-diffraction in azo-doped polymer film,” Appl. Phys. Lett. 70, 1224–1226 (1997).
[CrossRef]

Züchner, T.

T. Züchner, A. V. Failla, and A. J. Meixner, “Light microscopy with doughnut modes: a concept to detect, characterize, and manipulate individual nanoobjects,” Angew. Chem. Int. Ed. 50, 5274–5293 (2011).
[CrossRef]

Adv. Opt. Photon. (1)

Angew. Chem. Int. Ed. (1)

T. Züchner, A. V. Failla, and A. J. Meixner, “Light microscopy with doughnut modes: a concept to detect, characterize, and manipulate individual nanoobjects,” Angew. Chem. Int. Ed. 50, 5274–5293 (2011).
[CrossRef]

Appl. Phys. B (1)

A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B 98, 851–855 (2010).
[CrossRef]

Appl. Phys. Lett. (4)

M. Trejo-Durán, J. A. Andrade-Lucio, A. Martinez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
[CrossRef]

W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. 63, 1613–1615 (1993).
[CrossRef]

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–105 (1967).
[CrossRef]

P. F. Wu, B. Zou, X. Wu, J. Xu, X. Gong, G. Zhang, G. Tang, and W. Chen, “Biphotonic self-diffraction in azo-doped polymer film,” Appl. Phys. Lett. 70, 1224–1226 (1997).
[CrossRef]

J. Mod. Opt. (1)

D. J. Yu, W. P. Lu, and R. G. Harrison, “Analysis of dark spot formation in absorbing liquid media,” J. Mod. Opt. 45, 2597–2606 (1998).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (3)

L. G. Deng, K. N. He, T. Z. Zhou, and C. D. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A: Pure Appl. Opt. 7, 409–415 (2005).
[CrossRef]

C. M. Nascimento, M. A. R. C. Alencar, S. Chávez-Cerda, M. G. A. da Silva, M. R. Meneghetti, and J. M. Hickmann, “Experimental demonstration of novel effects on the far-field diffraction patterns of a Gaussian beam in a Kerr medium,” J. Opt. A: Pure Appl. Opt. 8, 947–951 (2006).
[CrossRef]

S. Y. Yang and Q. W. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A: Pure Appl. Opt. 10, 152103(2008).
[CrossRef]

Opt. Commun. (1)

R. G. Harrison, L. Dambly, D. J. Yu, and W. P. Lu, “A new self-diffraction pattern formation in defocusing liquid media,” Opt. Commun. 139, 69–72 (1997).
[CrossRef]

Opt. Express (7)

W. Ji, W. Z. Chen, S. H. Lim, J. Y. Lin, and Z. X. Guo, “Gravitation-dependent, thermally-induced self-diffraction in carbon nanotube solutions,” Opt. Express 14, 8958–8966 (2006).
[CrossRef] [PubMed]

B. Gu, Y. Wang, J. Wang, and W. Ji, “Femtosecond third-order optical nonlinearity of polycrystalline BiFeO3,” Opt. Express 17, 10970–10975 (2009).
[CrossRef] [PubMed]

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[CrossRef] [PubMed]

Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18, 10828–10833 (2010).
[CrossRef] [PubMed]

E. V. G. Ramirez, M. L. A. Carrasco, M. M. M. Otero, S. C. Cerda, and M. D. I. Castillo, “Far field intensity distributions due to spatial self phase modulation of a Gaussian beam by a thin nonlocal nonlinear media,” Opt. Express 18, 22067–22079 (2010).
[CrossRef] [PubMed]

X. Q. Yan, X. L. Zhang, S. Shi, Z. B. Liu, and J. G. Tian, “Third-order nonlinear susceptibility tensor elements of CS2 at femtosecond time scale,” Opt. Express 19, 5559–5564 (2011).
[CrossRef] [PubMed]

A. B. Villafranca and K. Saravanamuttu, “Spontaneous and sequential transitions of a Gaussian beam into diffraction rings, single ring and circular array of filaments in a photopolymer,” Opt. Express 19, 15560–15573 (2011).
[CrossRef] [PubMed]

Opt. Lett. (4)

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Figures (6)

Fig. 1
Fig. 1

Intensity patterns (top row) and the intensity profiles along the diameter (lower row) of the focused vector fields with different topological charges m.

Fig. 2
Fig. 2

Far-field intensity patterns without (top row) and with (middle row) a vertical polarizer (lower row) of the vector fields with different topological charges m by taking ΔΦm = π, φ0 = 0, λ = 804 nm, ω0 = 65 μm, d = 180 mm, and ΔΦm = π. The bottom row gives the intensity profiles along the diameter of the far-field intensity patterns shown in the top row.

Fig. 3
Fig. 3

Theoretically simulated far-field intensity patterns without (top row) and with (middle row) a vertical polarizer of the vector fields with m = 2 at different nonlinear phase shifts ΔΦ2, by taking φ0 = 0, λ = 804 nm, ω0 = 65 μm, and d = 180 mm. The bottom row is the intensity profiles along the diameter of the far-field intensity patterns shown in the top row.

Fig. 4
Fig. 4

Experimental scheme for investigating the self-diffraction behaviors of the femtosecond vector fields.

Fig. 5
Fig. 5

Experimentally observed far-field intensity patterns (top row) and theoretically simulated far-field intensity patterns (middle row) of the femtosecond vector field with m = 1 and φ0 = 0. The solid (dotted) lines in the bottom row give the corresponding intensity profiles alone the horizontal center line of the intensity patterns shown in the top (middle) row. The former two columns and the latter two columns are the cases of without and with the nonlinear sample at the focal plane, respectively.

Fig. 6
Fig. 6

Experimentally observed far-field intensity patterns (top row) and theoretically simulated far-field intensity patterns (middle row) of the femtosecond vector field with m = 2 and φ0 = 0. The solid (dotted) lines in the bottom row give the corresponding intensity profiles alone the horizontal center line of the intensity patterns shown in the top (middle) row. The former two columns and the latter two columns are the cases of without and with the nonlinear sample at the focal plane, respectively.

Tables (1)

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Table 1 Coefficients Am for different topological charges m.

Equations (6)

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E ( ρ , φ ) = A ( ρ ) P ( φ ) = A ( ρ ) [ cos ( m φ + φ 0 ) e ^ x + sin ( m φ + φ 0 ) e ^ y ] ,
E f ( r , ψ ) = A 0 0 a ρ d ρ 0 2 π P ( φ ) exp [ j k ρ r f cos ( φ ψ ) ] d φ ,
E f ( r , ψ ) = B m ( r ) [ cos ( m ψ + φ 0 ) e ^ x + sin ( m ψ + φ 0 ) e ^ y ] ,
B m ( r ) = E 0 j 3 m A m ( m + 2 ) m ! ( π r 2 ω 0 ) m F 1 2 [ m 2 + 1 , ( m 2 + 2 , m + 1 ) ; ( π r 2 ω 0 ) 2 ] .
E e ( r , ψ ) = E f ( r , ψ ) exp [ j k n 2 | E f ( r , ψ ) | 2 L ] ,
E a ( r a , ϕ ) = k j 3 m 1 exp ( j k d ) d exp ( j k r a 2 2 d ) [ cos ( m ϕ + φ 0 ) e ^ x + sin ( m ϕ + φ 0 ) e ^ y ] × 0 B m ( r ) exp [ j k n 2 | B m ( r ) | 2 L ] exp ( j k r 2 2 d ) J m ( k r r a d ) r d r .

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