## Vectorial self-diffraction effect in optically Kerr medium |

Optics Express, Vol. 20, Issue 1, pp. 149-157 (2012)

http://dx.doi.org/10.1364/OE.20.000149

Acrobat PDF (1336 KB)

### 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

## 1. Introduction

*et al*. [1

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]

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]

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]

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]

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]

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]

## 2. Theory

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

*ρ*and

*φ*are the polar radius and azimuthal angle in the polar coordinate system, respectively. Here

**P**(

*φ*) is the unit vector describing the distribution of the states of polarization of the vector field,

*m*is the topological charge, and

*φ*

_{0}is the initial phase [17

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]

**ê**

_{x}and

**ê**

_{y}are the unit vectors in the Cartesian coordinate system, respectively. Interestingly, two extreme cases of vector fields describing by Eq. (1) are the radially and azimuthally polarized vector fields when

*m*= 1 with

*φ*

_{0}= 0 and

*π*/2, respectively. In the case of

*m*= 0, Eq. (1) describes the horizontal and vertical linearly-polarized fields, for

*φ*

_{0}= 0 and

*φ*

_{0}=

*π*/2, respectively.

*A*(

*ρ*) stands for the amplitude distribution in the cross-section of the cylindrical vector field. Under the uniform-intensity illumination, we have

*A*(

*ρ*) =

*A*

_{0}within the region of 0 ≤

*ρ*≤

*a*, where

*a*is the radius of the cylindrical vector field.

*f*under the weak-focusing condition can be written, according to the Fourier transform, as follows where

*k*= 2

*π*/

*λ*is the wave vector and

*λ*is the wavelength of the used laser in free space. A Cartesian system (x′,y′) and a corresponding polar coordinate system (

*r,*

*ψ*) are attached in the rear focal plane of the lens. After integrating Eq. (2) over

*φ*for an integer

*m*≥ 0, we yield the focused field distribution where Here

*ω*

_{0}=

*λf*/2

*a*is the beam waist,

*E*

_{0}denotes the electric field amplitude at the focus,

_{1}

*F*

_{2}[·] is the generalized hypergeometric function, and

*A*is a normalized constant obtained by the condition of (|

_{m}*E*|

_{f}^{2}/|

*E*

_{0}|

^{2})

_{max}= 1. The first five coefficients of

*A*are listed in Table 1. Setting

_{m}*m*= 0 in Eq. (3), the field distribution has the well-known Airy spot profile that describes the focused field of the top-hat beam [18

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]

*m*of the cylindrical vector field increases.

*ɛ*is the incident energy and

*τ*is the full width at half maximum of the pulse duration.

_{F}**E**

*(*

_{e}*r,*

*ψ*) at the exit plane of the sample can be given as where

*n*

_{2}is the third-order nonlinear refraction index, and

*L*is the sample thickness. We now define the peak nonlinear refractive phase shift at the focus, as ΔΦ

*=*

_{m}*kn*

_{2}

*I*

_{0m}

*L*.

*r*), which has a distance

_{a},ϕ*d*from the exit plane of the sample

*J*(·) is the Bessel function of the first kind of

_{m}*m*th order. In the above theoretical analysis, we only consider the optical field under the steady-state condition. The temporal profile of the laser pluses have been omitted. In fact, we can easily extend the steady-state results to transient effects induced by a pulse train by using the time-averaging approximation. For the cylindrical vector field with a temporally Gaussian pulses, we yield the average nonlinear refractive phase shift at the focus as

*φ*

_{0}= 0,

*λ*= 804 nm,

*ω*

_{0}= 65

*μ*m, and

*d*= 180 mm. The top row of Fig. 2 presents the numerical simulations of the far-field patterns of the vector fields for different topological charges

*m*at a fixed value ΔΦ

*=*

_{m}*π*. For the sake of comparison, the far-field pattern of the scalar linearly-polarized top-hat beam (i.e., the case of

*m*= 0) is also shown in the first column of Fig. 2. Compared with the single ring structures in the far-field patterns of the vector fields in the absence of the nonlinearity, as shown in Fig. 1, the far-field patterns induced the nonlinearity exhibit the multiple concentric ring structures, originating from the refractive-index changes self-induced by the nonlinearity. To identify the polarization features of the far-field multiple ring patterns induced by the nonlinearity, a vertical polarizer is used, and the corresponding patterns are displayed in the middle row of Fig. 2. To more clearly show the the properties of the far-field multiple ring patterns, the bottom row also shows the corresponding intensity profiles along the horizontal center lines of the intensity patterns in the top (or middle) row. For the case of the scalar top-hat beam, as shown in the first column, we find the complete extinction. For the case of vector fields, the far-field intensity patterns induced by the nonlinearity behind the vertical polarizer appear the radial extinction and exhibit the the radially-modulated fan-shaped structures. Moreover, the number of the extinction directions is the same as the topological charge

*m*. Evidently, such a extinction nature is the same input vector fields as reported in Ref. 21

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]

*n*(

*r*) =

*γ*

*I*(

*r*). As a result, the different locations of the field cross-section in the radial direction experience the different nonlinear phase shifts of ΔΦ(

*r*) =

*kn*

_{2}

*I*(

*r*)

*L*, resulting in the self-phase modulation. If the phase difference between the two locations in the far-field plane is ΔΦ(

*r*) =

*h*

*π*,

*h*being an even or odd integer, the constructive or destructive interference takes place, respectively, giving rise to the appearance of self-diffraction patterns with the multiple concentric ring structures.

*m*= 2 and

*φ*

_{0}= 0 at different values of ΔΦ

_{2}, under the conditions of

*λ*= 804 nm,

*ω*

_{0}= 65

*μ*m, and

*d*= 180 mm. In principle, the change of the nonlinear phase shift can originate from the different nonlinearity (i.e. different nonlinear medium) for a given laser intensity or the different laser intensity for a given nonlinear medium and both. One can see that there occurs always a dark spot in the central zones of the far-field patterns and its size is almost independent of the nonlinear phase shift ΔΦ

_{2}. As ΔΦ

_{2}increases, the number of bright diffraction rings around the dark spot increases and the more light energy is diffracted into the outer rings.

## 3. Experiment

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]

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]

*m*with the fixed pulse energy of

*ɛ*= 1.3

*μ*J, the pulse duration of

*τ*≃ 70 fs, and the repetition rate of 1 kHz at the central wavelength of

_{F}*λ*= 804 nm. In addition, it should be emphasized that the generated femtosecond-pulsed cylindrical vector field has a top-hat spatial distribution with a diameter of 2

*a*= 3.7 mm and a near-Gaussian temporal profile. The cylindrical vector field was focused by a lens with a focal length of

*f*= 300 mm, producing the beam waist of

*ω*

_{0}= 65

*μ*m at the focus (the Rayleigh length

*z*

_{0}= 16.5 mm). As the nonlinear medium, the carbon disulfide was contained in 5 mm thick quartz cell at room temperature and standard atmosphere. The wall thickness of the quartz cell was 1 mm. The sample was located at the focal plane. A detector (Beamview, Coherent Inc.) was placed at the observation plane with a distance of

*d*= 180 mm from the sample to probe the transmitted intensity distribution. Considering the interface losses of the light energy, we determine the optical intensities for vector field with

*m*= 1 and 2 within the solution as

*I*

_{01}= 74 and

*I*

_{02}= 42 GW/cm

^{2}, respectively.

*m*= 1 and

*φ*

_{0}= 0) without and with the nonlinear sample at the focal plane are shown in the middle row of Fig. 5. To identify the vectorial properties of the far-field field, a horizontal polarizer was used. Correspondingly, the results are also displayed in Fig. 5. Most importantly, as shown in Fig. 5, we experimentally detected the vectorial self-diffraction ring, for the first time to our knowledge. For a high-order cylindrical vector field with

*m*= 2 and

*φ*

_{0}= 0, the corresponding experimental results are displayed in the middle row of Fig. 6. It should be pointed out that the saturated signals exist in the observed experimental results due to the energy saturation of the detector.

*n*

_{2}= 3×10

^{−6}cm

^{2}/GW and

*n*= 3.3×10

_{q}^{−7}cm

^{2}/GW at around 800 nm under the femtosecond pulse excitation [22

22. X. Q. Yan, X. L. Zhang, S. Shi, Z. B. Liu, and J. G. Tian, “Third-order nonlinear susceptibility tensor elements of CS_{2} 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 BiFeO_{3},” Opt. Express **17**, 10970–10975 (2009). [CrossRef] [PubMed]

*m*= 1 and 2 arising from the carbon disulfide were estimated to be 〈ΔΦ〉 = 1.93

*π*and 1.10

*π*, respectively. The phase shifts originating from the quartz walls were so small, ∼ 4%, that it should not induce a significant change of the far-field intensity. Hence, the contribution of the quartz walls was not taken into consideration in the analysis. Using Eq. (6) and the known experimental parameters, we simulate the far-field self-diffraction patterns without and with the nonlinear sample at the focal plane, as shown in the lower rows of Figs. 5 and 6. Clearly, the theoretical simulations are consistent with the experimental observations, implying that our theoretical analysis is reasonable and could gain an insight on the underlying mechanisms for the observed vectorial self-diffraction effect. It should be noted that the discrepancy between the theory and experiment in the periphery of the self-diffraction patterns is apparent. This difference is anticipated for the following reason. A monochromatic field is considered for a single defined wavelength and a unique peak intensity in the theoretical simulations; whereas a femtosecond pulse train at a central wavelength with a spectral bandwidth of tens-of-nanometers is used in the measurements. Furthermore, the peak intensity considerably changes over the pulse temporal envelope.

## 4. Conclusion

## Acknowledgments

## References and links

1. | W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. |

2. | S. D. Durbin, S. M. Arakelian, and Y. R. Shen, “Laser induced diffraction rings from a nematic liquid crystal film,” Opt. Lett. |

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. |

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. |

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 |

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. |

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 |

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. | D. J. Yu, W. P. Lu, and R. G. Harrison, “Analysis of dark spot formation in absorbing liquid media,” J. Mod. Opt. |

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. |

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. |

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 |

13. | Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. |

14. | A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B |

15. | S. Y. Yang and Q. W. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A: Pure Appl. Opt. |

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. |

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. | W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. |

19. | Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express |

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. |

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. |

22. | X. Q. Yan, X. L. Zhang, S. Shi, Z. B. Liu, and J. G. Tian, “Third-order nonlinear susceptibility tensor elements of CS |

23. | B. Gu, Y. Wang, J. Wang, and W. Ji, “Femtosecond third-order optical nonlinearity of polycrystalline BiFeO |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(190.3270) Nonlinear optics : Kerr effect

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(260.7120) Physical optics : Ultrafast phenomena

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: October 6, 2011

Revised Manuscript: November 17, 2011

Manuscript Accepted: December 3, 2011

Published: December 19, 2011

**Citation**

Bing Gu, Fan Ye, Kai Lou, Yongnan Li, Jing Chen, and Hui-Tian Wang, "Vectorial self-diffraction effect in optically Kerr medium," Opt. Express **20**, 149-157 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-1-149

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### References

- 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]
- 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]
- 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]
- 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. Express14, 8958–8966 (2006). [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]
- 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. Express19, 15560–15573 (2011). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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. Express18, 22067–22079 (2010). [CrossRef] [PubMed]
- Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1, 1–57 (2009). [CrossRef]
- A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B98, 851–855 (2010). [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]
- 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]
- 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. Express18, 10786–10795 (2010). [CrossRef] [PubMed]
- W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett.63, 1613–1615 (1993). [CrossRef]
- Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express18, 10828–10833 (2010). [CrossRef] [PubMed]
- 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]
- 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]
- 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. Express19, 5559–5564 (2011). [CrossRef] [PubMed]
- B. Gu, Y. Wang, J. Wang, and W. Ji, “Femtosecond third-order optical nonlinearity of polycrystalline BiFeO3,” Opt. Express17, 10970–10975 (2009). [CrossRef] [PubMed]

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