## Elliptic-symmetry vector optical fields |

Optics Express, Vol. 22, Issue 16, pp. 19302-19313 (2014)

http://dx.doi.org/10.1364/OE.22.019302

Acrobat PDF (4427 KB)

### Abstract

We present in principle and demonstrate experimentally a new kind of vector fields: elliptic-symmetry vector optical fields. This is a significant development in vector fields, as this breaks the cylindrical symmetry and enriches the family of vector fields. Due to the presence of an additional degrees of freedom, which is the interval between the foci in the elliptic coordinate system, the elliptic-symmetry vector fields are more flexible than the cylindrical vector fields for controlling the spatial structure of polarization and for engineering the focusing fields. The elliptic-symmetry vector fields can find many specific applications from optical trapping to optical machining and so on.

© 2014 Optical Society of America

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

2. G. M. Lerman, Y. Lilach, and U. Levy, “Demonstration of spatially inhomogeneous vector beams with elliptical symmetry,” Opt. Lett. **34**, 1669–1671 (2009). [CrossRef] [PubMed]

3. X. L. Wang, J. P. 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]

4. 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]

3. X. L. Wang, J. P. 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]

4. 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]

*δ*(

*x*,

*y*); (iii) the two parts must pass through different optical paths (not only is the separation in space), making the two parts have orthogonal states of polarization by using a pair of wave plates (such as 1/2 and 1/4 wave plates) behind the spatial filter (SF) placed in the Fourier plane of the 4f system; (iv) the two orthogonally polarized parts are combined by the Ronchi phase grating (G) placed in the output plane of the 4f system.

3. X. L. Wang, J. P. 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]

**32**, 3549–3551 (2007). [CrossRef] [PubMed]

4. 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]

*x*and

*y*components are always in phase. Arbitrary vector fields can be generated by changing the additional phase

*δ*, because

*δ*is allowed to possess arbitrary spatial distribution which can be flexibly realized by SLM.

5. J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, “Mathieu functions, a visual approach,” Am. J. Phys. **71**, 233–242 (2003). [CrossRef]

*ε*,

*η*), which has the following relation with a Cartesian coordinate system (

*x*,

*y*)

*ε*,

*η*) has two foci,

*F*

_{1}and

*F*

_{2}, which are located at (−

*f*, 0) and (

*f*, 0) in the system (

*x*,

*y*), respectively. The curves of constant

*ε*are a series of confocal ellipses (solid lines), while the curves of constant

*η*are a series of confocal hyperbolas (dotted lines). These two kinds of curves can be written as

*δ*in the transmission function of the holographic grating displayed on SLM in Eq. (1) has the following form in the polar coordinate system [6

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

*φ*and

*r*are the azimuthal and radial coordinates in the polar coordinate system,

*r*

_{0}is the radius of the circular vector field, and

*δ*

_{0}is the initial phase. Therefore, if the additional phase

*δ*has the following form as the vector field described by Eq. (2) should be the elliptic-symmetry vector field.

*ε*and

*η*in terms of

*x*and

*y*as

*ε*is limited within a range of [0, +∞) and

*η*has a range of [0, 2

*π*), respectively.

*m*≠ 0 and

*n*≡ 0, implying that

*δ*in Eq. (6) is a function of

*η*independent of

*ε*in the elliptic coordinate system (

*ε*,

*η*), the polarization distributions of the elliptic-symmetry vector fields are schematically shown in Fig. 3(a) for

*m*= 0.5 and

*δ*

_{0}= 0, in Fig. 3(b) for

*m*= 1 and

*δ*

_{0}= 0, and in Fig. 3(c) for

*m*= 1 and

*δ*

_{0}=

*π*/2, respectively. For the case of (

*m*,

*n*,

*δ*

_{0}) = (0.5, 0, 0) in Fig. 3(a), its polarization map has a singular ray along the +

*x*direction, where the polarization has the uncertainty, therefore this kind of vector field will occur a zero-intensity dark ray in the +

*x*direction. This is similar to the case when

*m*= 0.5 reported in [3

**32**, 3549–3551 (2007). [CrossRef] [PubMed]

*m*= 1 and

*n*= 0, the two cases of

*δ*

_{0}= 0 and

*δ*

_{0}=

*π*/2 in Eq. (6) are similar to the radially polarized (RP) and azimuthally polarized (AP) fields [3

**32**, 3549–3551 (2007). [CrossRef] [PubMed]

*m*> 1, the polarization map occurs more spots of singularity. For the case when (

*m*,

*n*,

*δ*

_{0}) = (3, 0, 0), as shown in Fig. 3(d), there occur three points of singularity. For the above four cases, the polarization distributions are a function of

*η*independent of

*ε*. When

*m*= 0,

*n*= 1 and

*δ*

_{0}= 0, as shown in Fig. 3(e), the polarization map has no polarization singularity, and the polarization distribution is a function of

*ε*independent of

*η*. When

*m*=

*n*= 1 and

*δ*

_{0}= 0, as shown in Fig. 3(f), the polarization distribution depends on both

*ε*and

*η*.

*m*when

*n*≡ 0 and

*δ*

_{0}= 0. We can see from the first row that the total intensity pattern has some singular spots for the integer

*m*, while that has also a singular ray besides some singular spots for the non-integer

*m*, due to the polarization uncertainty. The simulated intensities of the

*x*(

*y*) components in the second (fourth) row are in good agreement with the measured those in the third (fifth) row. Indeed, the polarization distribution depends on

*η*independent of

*ε*, implying that all positions in the curves of constant

*η*(hyperbolic curves) have the same polarization. In particular, we find from the second and third rows that the zero-intensity singularities for the non-integer

*m*originate from the contribution of the

*x*component only, while from the fourth and fifth rows that the zero-intensity singularities for the integer

*m*are from the contribution of the

*y*component only.

*n*when

*m*≡ 0 and

*δ*

_{0}= 0. We can see from the first row that any total intensity pattern exhibits a homogeneous distribution, while has no singularity for the integer

*n*. In fact, the elliptic-symmetry vector field has always no singularity for any

*n*(is not limited to the integer

*n*) provided that

*m*≡ 0 (i.e.

*δ*is independent of

*η*). The simulated intensities of the

*x*(

*y*) components in the second (fourth) row are in good agreement with the measured ones in the third (fifth) row. Indeed, the polarization distribution depends on

*ε*independent of

*η*, implying that positions with the same polarization form the elliptic curves of constant

*ε*. We can also see from the first and second rows that there is a bright line formed by the

*x*component between two foci. For the

*x*(or

*y*) component, the number of the bright (or extinction) ellipses is equal to 2

*n*.

*δ*is a function of both

*ε*and

*η*, i.e.

*m*and

*n*must be nonzero simultaneously. Figure 6 shows the elliptic-symmetry vector fields for different

*n*= (0.5, 1, 1.5, 2, 2.5, 3) when

*m*≡ 1 and

*δ*

_{0}= 0. We can see from the first row that any total intensity pattern exists a zero-intensity singular segment between the two foci. The intensity patterns of the

*x*component exhibit a non-common-origin dual-spiral structure, and the origins of the two spiral curves are located at the two foci. In contrast, the intensity patterns of the

*y*component has also a dual-spiral structure, and the origins of the two spiral curves are connected by a bright segment between the two foci, which is different from the

*x*component. In particular, the zero-intensity singular segment in any total intensity pattern is from the contribution of the

*y*component only. Figure 7 shows the elliptic-symmetry vector fields for different

*m*= (0.5, 1, 1.5, 2, 2.5, 3) when

*n*≡ 2 and

*δ*

_{0}= 0. Figure 8 shows the elliptic-symmetry vector fields for different combinations of

*m*and

*n*with

*m*=

*n*when

*δ*

_{0}= 0. As shown in the first rows in Figs. 7 and 8, in the total intensity patterns there have the zero-intensity singular spots due to the polarization uncertainty caused by the nonzero

*m*in the cases of the integer

*m*. However, in the total intensity patterns there have also the zero-intensity singular ray besides the zero-intensity singular spots in the cases of the half-integer

*m*. The intensity pattern of the

*x*or

*y*component exhibits a single- or multi-spiral structure depending on

*m*. The singular ray originates from the contributions of both the

*x*and

*y*components for the half-integer

*m*. From Figs. 6 and 8, it is clear that the arms of the spiral curves in the intensity pattern of the

*x*or

*y*component is equal to 2|

*m*| independent of

*n*, while the separation between turns of the spiral curves decreases as |

*n*| increases.

*m*and

*n*, the interval 2

*f*between the two foci as an important degree of freedom can also be used to manipulate the elliptic-symmetry vector fields. Figure 9 shows the elliptic-symmetry vector fields for five different

*f*= (0.1, 0.2, 0.3, 0.4, 0.5) in the two cases of (

*m*,

*n*,

*δ*

_{0}) = (2, 0, 0) and (

*m*,

*n*,

*δ*

_{0}) = (0, 1.5, 0). The total intensity patterns have two zero-intensity singular spots located at two foci for the case of (

*m*,

*n*,

*δ*

_{0}) = (2, 0, 0). The intensity patterns of the

*x*and

*y*components exhibit the confocal hyperbolic shapes. As

*f*decreases to zero, the elliptic coordinate system degenerates into the traditional polar coordinate system (two foci coincide to an origin). Correspondingly, the hyperbolic curves describing the trajectory with the same polarization become to the radial rays in the polar coordinate system.

*m*,

*n*,

*δ*

_{0}) = (0, 1.5, 0). The intensity patterns of the

*x*and

*y*components exhibit the confocal elliptic shapes because the trajectory drawn by the positions with the same polarization. As

*f*decreases to zero, the ellipticity of the elliptically-shaped intensity patterns of the

*x*and

*y*components becomes small until to zero, implying that the confocal ellipses degenerate into the concentric circles. In fact, when

*f*= 0, all the cases in the elliptic coordinate system degenerate into those in the polar coordinate system reported in [3

**32**, 3549–3551 (2007). [CrossRef] [PubMed]

2. G. M. Lerman, Y. Lilach, and U. Levy, “Demonstration of spatially inhomogeneous vector beams with elliptical symmetry,” Opt. Lett. **34**, 1669–1671 (2009). [CrossRef] [PubMed]

2. G. M. Lerman, Y. Lilach, and U. Levy, “Demonstration of spatially inhomogeneous vector beams with elliptical symmetry,” Opt. Lett. **34**, 1669–1671 (2009). [CrossRef] [PubMed]

7. M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian beams,” Opt. Lett. **29**, 144–146 (2004). [CrossRef] [PubMed]

8. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. **25**, 1493–1495 (2000). [CrossRef]

9. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A **22**, 289–298 (2005). [CrossRef]

*f*of the elliptic coordinate system). Here we focus on their tight focusing property by an objective with high numerical-aperture (NA). The tightly focused vector fields have been already studied carefully [10

10. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A **253**, 358–379 (1959). [CrossRef]

13. M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A **11**, 1–7 (2009). [CrossRef]

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

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

*ρ*and

*ϕ*are the radial and azimuthal coordinates, and

*E*and

_{ρ}*E*are the radial and azimuthal components of the incident vector fields in the polar coordinate system (

_{ϕ}*ρ*,

*ϕ*) attached on the input plane, respectively.

*r*,

*φ*and

*z*are the radial, azimuthal and longitudinal coordinates in the cylindric coordinate system (

*r*,

*φ*,

*z*) attached on the focal plane.

*k*= 2

*π/λ*is the wavenumber and

*λ*is the wavelength in free space, and

*F*is the focal length of the objective.

*P*(

*θ*) is the pupil plane apodization function, which can be chosen to be

*θ*=

*ρ/F*.

*θ*is the maximum ray angle passing through the objective, defined as sin

_{m}*θ*=

_{m}*NA*. The incident field is a round field has a radius of

*ρ*=

_{m}*F*sin

*θ*=

_{m}*F*·

*NA*. Applying Eq. (8) to the elliptic-symmetry vector fields we presented above, we can yield the tight focusing fields.

*f*= 0, the elliptic coordinate system becomes the traditional polar coordinate system, correspondingly, the elliptic-symmetry vector fields degenerates into the cylindric vector fields. We consider only the case of

*n*= 0 here. As an example, we give a comparison between the cylindric- and elliptic-symmetry vector fields in two cases

*δ*

_{0}= 0 and

*δ*

_{0}=

*π*/2 when

*m*= 3. The first column in Fig. 10 shows the intensity distributions of the tight focusing fields for the cylindric vector fields, which exhibit clearly the fourfold rotation symmetry. As

*f*increases, the cylindric symmetry is broken. This results show the redistribution of the intensity of the tight focusing field in the focal plane. The fourfold rotation symmetry of the tight focusing field intensity pattern becomes to the mirror symmetry (twofold rotation symmetry), as shown from the first column to the sixth column. For instance, for the total intensity in the case of

*δ*

_{0}= 0, the zero intensity at the centre of the pattern shown in the first column and in the top row becomes the nonzero intensity at the centre shown in the sixth column and in the top row. For the total intensity in the case of

*δ*

_{0}=

*π*/2, as

*f*increases, the tight focusing field is evolved to a pattern composed of a pair of “ears” when

*f*= 0.9

*F*. The transverse and longitudinal components of the tight focusing fields are also shown in Fig. 10. For the transverse components, four strong spots become to two strong spots as

*f*increases for both cases of

*δ*

_{0}= 0 and

*δ*

_{0}=

*π*/2. We can also find that as

*f*increases, for the longitudinal components, four relatively strong spots become to one strong spot when

*δ*

_{0}= 0, while only the intensity weakens when

*δ*

_{0}=

*π*/2 (the intensity pattern is changed slightly). Clearly, due to the flexibly engineerable focal field, the elliptic-symmetry vector fields can be useful in many areas such as optical trapping, optical tweezers, laser machining and so on [1

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

12. G. M. Lerman and U. Levy, “Tight focusing of spatially variant vector optical fields with elliptical symmetry of linear polarization,” Opt. Lett. **32**, 2194–2196 (2007). [CrossRef] [PubMed]

14. X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. **282**, 3421–3425 (2009). [CrossRef]

19. C. Hnatovsky, V. G. Shvedov, N. Shostka, A. V. Rode, and W. Krolikowski, “Polarization-dependent ablation of silicon using tightly focused femtosecond laser vortex pulses,” Opt. Lett. **37**, 226–228 (2012). [CrossRef] [PubMed]

*f*is introduced. As the researches of cylindrical vector fields have been very mature, it is necessary to expand the family of vector fields. We present the tight focusing of the elliptic-symmetry vector fields in order to illustrate that the applications of cylindrical vector fields can be expanded to the elliptic-symmetry vector fields, and the new vector fields can also have new properties and be more flexible than the cylindrical vector fields. We hope more conclusions can be gotten by using elliptic-symmetry vector fields.

## Acknowledgments

## References and links

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

2. | G. M. Lerman, Y. Lilach, and U. Levy, “Demonstration of spatially inhomogeneous vector beams with elliptical symmetry,” Opt. Lett. |

3. | X. L. Wang, J. P. 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. |

4. | 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 |

5. | J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, “Mathieu functions, a visual approach,” Am. J. Phys. |

6. | X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. |

7. | M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian beams,” Opt. Lett. |

8. | J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. |

9. | J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A |

10. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A |

11. | K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express |

12. | G. M. Lerman and U. Levy, “Tight focusing of spatially variant vector optical fields with elliptical symmetry of linear polarization,” Opt. Lett. |

13. | M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A |

14. | X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. |

15. | Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express |

16. | W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. |

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

18. | M. G. Donato, S. Vasi, P. H. Jones, R. Sayed, F. Bonaccorso, A. C. Ferrari, P. G. Gucciardi, and O. M. Maragò, “Optical trapping of nanotubes with cylindrical vector beams,” Opt. Lett. |

19. | C. Hnatovsky, V. G. Shvedov, N. Shostka, A. V. Rode, and W. Krolikowski, “Polarization-dependent ablation of silicon using tightly focused femtosecond laser vortex pulses,” Opt. Lett. |

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(260.5430) Physical optics : Polarization

(260.6042) Physical optics : Singular optics

(070.6120) Fourier optics and signal processing : Spatial light modulators

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 13, 2014

Revised Manuscript: July 16, 2014

Manuscript Accepted: July 16, 2014

Published: August 4, 2014

**Virtual Issues**

Vol. 9, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Yue Pan, Yongnan Li, Si-Min Li, Zhi-Cheng Ren, Ling-Jun Kong, Chenghou Tu, and Hui-Tian Wang, "Elliptic-symmetry vector optical fields," Opt. Express **22**, 19302-19313 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-16-19302

Sort: Year | Journal | Reset

### References

- Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1, 1–57 (2009). [CrossRef]
- G. M. Lerman, Y. Lilach, and U. Levy, “Demonstration of spatially inhomogeneous vector beams with elliptical symmetry,” Opt. Lett.34, 1669–1671 (2009). [CrossRef] [PubMed]
- X. L. Wang, J. P. 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. 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]
- J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, “Mathieu functions, a visual approach,” Am. J. Phys.71, 233–242 (2003). [CrossRef]
- X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105, 253602 (2010). [CrossRef]
- M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian beams,” Opt. Lett.29, 144–146 (2004). [CrossRef] [PubMed]
- J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett.25, 1493–1495 (2000). [CrossRef]
- J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A22, 289–298 (2005). [CrossRef]
- B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A253, 358–379 (1959). [CrossRef]
- K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7, 77–87 (2000). [CrossRef] [PubMed]
- G. M. Lerman and U. Levy, “Tight focusing of spatially variant vector optical fields with elliptical symmetry of linear polarization,” Opt. Lett.32, 2194–2196 (2007). [CrossRef] [PubMed]
- M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A11, 1–7 (2009). [CrossRef]
- X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun.282, 3421–3425 (2009). [CrossRef]
- Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express12, 3377–3382 (2004). [CrossRef] [PubMed]
- W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun.265, 411–417 (2006). [CrossRef]
- Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express18, 10828–10833 (2010). [CrossRef] [PubMed]
- M. G. Donato, S. Vasi, P. H. Jones, R. Sayed, F. Bonaccorso, A. C. Ferrari, P. G. Gucciardi, and O. M. Maragò, “Optical trapping of nanotubes with cylindrical vector beams,” Opt. Lett.37, 3381–3383 (2012). [CrossRef]
- C. Hnatovsky, V. G. Shvedov, N. Shostka, A. V. Rode, and W. Krolikowski, “Polarization-dependent ablation of silicon using tightly focused femtosecond laser vortex pulses,” Opt. Lett.37, 226–228 (2012). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.