## Vector optical fields with polarization distributions similar to electric and magnetic field lines |

Optics Express, Vol. 21, Issue 13, pp. 16200-16209 (2013)

http://dx.doi.org/10.1364/OE.21.016200

Acrobat PDF (4002 KB)

### Abstract

We present, design and generate a new kind of vector optical fields with linear polarization distributions modeling to electric and magnetic field lines. The geometric configurations of “electric charges” and “magnetic charges” can engineer the spatial structure and symmetry of polarizations of vector optical field, providing additional degrees of freedom assisting in controlling the field symmetry at the focus and allowing engineering of the field distribution at the focus to the specific applications.

© 2013 OSA

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

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

7. S. Liu, P. Li, T. Peng, and J. L. Zhao, “Generation of arbitrary spatially variant polarization beams with a trapezoid Sagnac interferometer,” Opt. Express **20**, 21715–21721 (2012) [CrossRef] [PubMed] .

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

18. L. X. Yang, X. S. Xie, S. C. Wang, and J. Y. Zhou, “Minimized spot of annular radially polarized focusing beam,” Opt. Lett. **38**, 1331–1333 (2013) [CrossRef] [PubMed] .

10. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. **91**, 233901 (2003) [CrossRef] [PubMed] .

15. H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photon. **2**, 501–505 (2008) [CrossRef] .

18. L. X. Yang, X. S. Xie, S. C. Wang, and J. Y. Zhou, “Minimized spot of annular radially polarized focusing beam,” Opt. Lett. **38**, 1331–1333 (2013) [CrossRef] [PubMed] .

13. W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. **265**, 411–417 (2006) [CrossRef] .

14. N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. **279**, 229–234 (2007) [CrossRef] .

16. 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. 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] .

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

*λ*× 2

*λ*[19

19. 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] .

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

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

6. 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] .

16. 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] .

*N*charges in the two-dimensional Cartesian coordinate system (

*x,y*). The induced electric field at a point

*P*(

*x,y*) can be written as

*p*is the amount of

_{j}*j*th charge located at (

*x*,

_{j}*y*).

_{j}**ê**

*and*

_{x}**ê**

*are the unit vectors along the*

_{y}*x*and

*y*directions.

*x,y*) is taken as the SoP distribution of L-LP-VF is the same as the electric field lines described by Eq. (1).

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

6. 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] .

16. 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] .

*λ*/4 wave plates placed in the Fourier plane of the 4f system should be used. In principle, the created L-LP-VF could be written as where

*δ*(

*x,y*) is the additional phase distribution of the transmission function

*t*(

*x,y*) = {1 +

*γ*cos[2

*πf*

_{0}

*x*+

*δ*(

*x,y*)]}/2 of the holographic grating displayed in a spatial light modulator [5

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

6. 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] .

**282**, 3421–3425 (2009) [CrossRef] .

*δ*(

*x,y*) is set as

*δ*(

*x,y*) = Δ(

*x,y*), the spatial SoP distribution of the created L-LP-VF is indeed the same as the structure of electric field lines described by Eq. (1).

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

**18**, 10786–10795 (2010) [CrossRef] [PubMed] .

**282**, 3421–3425 (2009) [CrossRef] .

*t*(

*x,y*) = {1 +

*γ*cos[2

*πf*

_{0}

*x*+

*δ*(

*x,y*)]}/2, where

*δ*(

*x,y*) = Δ(

*x,y*) is described by Eq. (2b). The input field is diffracted into the

*±*1th orders carrying the respective wavefronts of exp[±

*j*Δ(

*x,y*)], by the computer-controlled holographic grating displayed at SLM. In the Fourier plane of the 4f system, the focused±1th orders by L1 are spatially filtered by SF and then converted into the right- and left-handed circularly polarized light by two

*λ*/4 waveplates, respectively. The demanded L-LP-VFs are generated in the output plane of the 4f system, by combining the ±1th orders by L2 and G.

*p*

_{1}= +1) and unit negative charge (

*p*

_{2}= −1), locate at (

*x*

_{1},

*y*

_{1}) = (−

*d*, 0) and (

*x*

_{2},

*y*

_{2}) = (

*d*, 0), respectively; (ii) two unit positive charges of (

*p*

_{1}= +1) and (

*p*

_{2}= +1), locate also at (

*x*

_{1},

*y*

_{1}) = (−

*d*, 0) and (

*x*

_{2},

*y*

_{2}) = (

*d*, 0), respectively. The directions of electric field lines of both cases are shown in the first column of Fig. 2. The simulated SoP distributions of the L-LP-VFs corresponding to the electric field lines of the electric dipole (

*p*

_{1},

*p*

_{2}) = (+1, −1) and the dual unit positive charges (

*p*

_{1},

*p*

_{2}) = (+1, +1) are indeed in complete agreement with the field lines in the first column of Fig. 2. Here we only show the directions of SoP distributions and the fields lines by the arrows.

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

18. L. X. Yang, X. S. Xie, S. C. Wang, and J. Y. Zhou, “Minimized spot of annular radially polarized focusing beam,” Opt. Lett. **38**, 1331–1333 (2013) [CrossRef] [PubMed] .

19. 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] .

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

*p*

_{2}≡ −1 and different

*p*

_{1}= +2, +3, +4 and +5, in which

*p*

_{1}and

*p*

_{2}are still located at (

*x*

_{1},

*y*

_{1}) = (−

*d*, 0) and (

*x*

_{2},

*y*

_{2}) = (

*d*, 0). All the intensity exhibit the uniform distribution excluding some dark spots which correspond to the polarization singularities. The electric field lines and the SoP distributions have two “intrinsic singularities” located at (−

*d*, 0) and (

*d*, 0) for all the four situations. In addition, there is also a “derivative singularity” located at (

*d*, 0) for

*p*

_{1}= +2, (3.73

*d*, 0) for

*p*

_{1}= +3, (3

*d*, 0) for

*p*

_{1}= +4 and (2.62

*d*, 0) for

*p*

_{1}= +5. Since any picture has a dimension of 6

*d*× 6

*d*, the “derivative singularity” in the first row disappear for both

*p*

_{1}= +2 and

*p*

_{1}= +3. The intensity patterns behind the horizonal and vertical polarizers exhibit the peculiar patterns with a mirror symmetry about

*y*= 0. Clearly, the simulated results in the second and fourth rows are in well agreement with the experimental ones in the third and fifth rows. The dark spots in the intensity patterns correspond to the polarization singularities.

*p*

_{1},

*p*

_{2}and

*p*

_{3}, located at (−

*d*, 0), (

*d*, 0), (0,

*d*), are shown in the first and second columns of Fig. 4. The two situations of four “charges”

*p*

_{1},

*p*

_{2},

*p*

_{3}and

*p*

_{4}, located at (−

*d*, 0), (

*d*, 0), (0,

*d*) and (0, −

*d*), are shown in the third and fourth columns of Fig. 4. As shown in the first row, besides three (four) “intrinsic singularities” for the three (four) “charges”, the electric field lines or the corresponding SoP distributions have also two “derivative singularities” located at near (−

*d*/2,

*d*/2) between

*p*

_{1}and

*p*

_{3}and near (

*d*/2,

*d*/2) between

*p*

_{2}and

*p*

_{3}for (

*p*

_{1},

*p*

_{2},

*p*

_{3}) = (+1, +1, +1) in the first column, respectively. There is only one “derivative singularity” located at near (0, −0.33

*d*) for (

*p*

_{1},

*p*

_{2},

*p*

_{3}) = (+1, +1, −1) in the second column. There are five “derivative singularities” distributed in four corners and center of a square with a side length of ∼0.77

*d*and its center located at (0, 0), for (

*p*

_{1},

*p*

_{2},

*p*

_{3},

*p*

_{4}) = (+1, +1, +1, +1) in the third column. In contrast, there is only one “derivative singularity” located at (0, 0), for (

*p*

_{1},

*p*

_{2},

*p*

_{3},

*p*

_{4}) = (+1, +1, −1, −1) in the fourth column. Behind the horizonal and vertical polarizers, the extinction patterns exhibit a mirror symmetry about

*x*= 0 only for the three “charges” in the first and second columns, while have a mirror symmetry about

*x*= 0 and

*y*= 0 simultaneously for the four “charges” in the third and fourth columns, respectively. The results reveals that the SoP distributions of the generated L-LP-VFs are in agreement with the theoretical designs.

*d*is the radius of the current loop, which is in the

*xz*plane and centered on the origin. Referencing Eq. (2b) as the electric field lines, if only we set the additional phase as

*x*

_{1},

*y*

_{1}) = (−

*d*, 0) and (

*x*

_{2},

*y*

_{2}) = (

*d*, 0). These are another kind of representative L-LP-VFs with dual polarization singularities, which are different from the cylindrical- and elliptical-symmetric L-LP-VFs with a single polarization singularity and the above L-LP-VFs with multiple polarization singularities. It should be pointed out that the L-LP-VFs with the SoP distributions imitating to the magnetic field lines have the two “intrinsic singularities”, whereas have no “derivative singularities”. To enrich this kind of L-LP-VFs, we introduce two new parameters

*m*

_{1}and

*m*

_{2}into Eq. (4), thus Eq. (4) can be rewritten as follows

*m*

_{1},

*m*

_{2}) = (1, 1), (

*m*

_{1},

*m*

_{2}) = (1, 3), (

*m*

_{1},

*m*

_{2}) = (1, 5), (

*m*

_{1},

*m*

_{2}) = (3, 1), (

*m*

_{1},

*m*

_{2}) = (5, 1), and (

*m*

_{1},

*m*

_{2}) = (5, 5). As shown in the first row of Fig. 5, the magnetic field lines or the corresponding SoP distributions have always two singularities located at (

*x*= 0 and a twofold rotation inversion symmetry. As shown in the second to fifth rows, the extinction patterns behind horizontal and vertical polarizers have a mirror symmetry about both

*x*= 0 and

*y*= 0, which are higher than the symmetry of the SoP distributions. As

*m*

_{1}and/or

*m*

_{2}increase, the

*x*- and

*y*-component fractions of the L-LP-VFs decrease and increase, respectively. In particular, the experimental results in the third and fifth rows are in well agreement with the simulations in the second and fourth row.

21. 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] .

*NA*= 0.9, a radius

*R*

_{0}, and a focal length

*f*=

*R*

_{0}/

*NA*. All the length units are normalized by the light wavelength

*λ*. As shown in Fig. 6, we first simulate the tight focusing fields of the two typical L-LP-VFs with their SoP distributions similar to the field lines of the electric dipole and the dual-positive charges shown in Fig. 2. One can see that the focal fields exhibit no cylindrical and elliptical symmetry, whereas the mirror symmetry and central inversion symmetry caused by the symmetry of the SoP distribution. The focal fields can be engineered not only by the “charges” but also by the interval between them. For the case of the “electric dipole” in the top row, when

*d*= 0.1

*f*the focal field has two strong spots separated by a sharp dark line. When

*d*is increased to

*d*= 0.3

*f*, the two strong spots are close to each other. When

*d*= 0.5

*f*, the sharp dark line is completely cut to become two dark spots, while the two strong spots are connected. When

*d*= 0.7

*f*, the focal field exhibits an “H” shape, in which its strong area is a round rectangle. When

*d*= 0.9

*f*, the focal field becomes an elliptical strong area inside the background of a weak “H”. For the case of the dual positive charges in the bottom row, the focal field exhibits a nearly circular strong spot with a full width at half maximum (FWHM) of 0.74

*λ*× 0.82

*λ*when

*d*= 0.1

*f*, a date-pit-like strong spot with a size of 0.58

*λ*× 1.22

*λ*(FWHM) when

*d*= 0.3

*f*, a sharp line with a size of 0.50

*λ*× 1.60

*λ*(FWHM) when

*d*= 0.5

*f*, and two separated strong spots when

*d*= 0.7

*f*. When

*d*= 0.9

*f*, there occur a pair of nearly circular strong spots in the vertical direction and a pair of nearly circular secondary strong spots in the horizontal direction, respectively.

*m*

_{1},

*m*

_{2}) = (1, 1), for different values of

*d*. Two dark spots are surrounded by a strong racetrack-like shape when

*d*= 0.1

*f*. The two dark spots have been connected each other when

*d*= 0.2

*f*. The focal fields exhibit a pattern that a dark area is sandwiched between a pair of “lune” strong areas when

*d*= 0.3

*f*and

*d*= 0.4

*f*. A relatively dark elliptical area is surrounded by an ellipse-like strong ring when

*d*= 0.5

*f*.

*m*

_{1}when

*d*= 0.3

*f*and

*m*

_{2}≡ 1. As

*m*

_{1}increases from

*m*

_{2}= 1 to

*m*

_{1}= 5, the pattern of the focal field experiences an evolution as follows: when

*m*

_{1}is changed from

*m*

_{1}= 1 to

*m*

_{1}= 3, the central dark line is cut in the horizontal direction to become two dark spots and a pair of “ear” strong areas. When

*m*

_{1}= 4 and

*m*

_{1}= 5, the focal field patterns are composed of a relatively strong elliptical spot at the center and a pair of strong “ears”, just the central elliptical spot is stronger when

*m*

_{1}= 5.

*λ*/4 waveplates in the Fourier plane of the 4f system should probably be replaced by two

*λ*/2 waveplates and the additional phase distribution

*δ*should also be redesigned. The peculiar patterns of the focal fields can be engineered by designing the SoP distributions.

## Acknowledgments

## References and links

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

2. | J. Arlt and M.J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. |

3. | Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. |

4. | C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. |

5. | X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector fields with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. |

6. | 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 |

7. | S. Liu, P. Li, T. Peng, and J. L. Zhao, “Generation of arbitrary spatially variant polarization beams with a trapezoid Sagnac interferometer,” Opt. Express |

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

9. | Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express |

10. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. |

11. | N. Bokor and N. Davidson, “Generation of a hollow dark spherical spot by 4 |

12. | Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. |

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

14. | N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. |

15. | H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photon. |

16. | 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. |

17. | W. Zhang, S. Liu, P. Li, X. Y. Jiao, and J. L. Zhao, “Controlling the polarization singularities of the focused azimuthally polarized beams,” Opt. Express |

18. | L. X. Yang, X. S. Xie, S. C. Wang, and J. Y. Zhou, “Minimized spot of annular radially polarized focusing beam,” Opt. Lett. |

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

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

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

**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: May 10, 2013

Revised Manuscript: June 16, 2013

Manuscript Accepted: June 17, 2013

Published: June 28, 2013

**Citation**

Yue Pan, Si-Min Li, Lei Mao, Ling-Jun Kong, Yongnan Li, Chenghou Tu, Pei Wang, and Hui-Tian Wang, "Vector optical fields with polarization distributions similar to electric and magnetic field lines," Opt. Express **21**, 16200-16209 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-16200

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### References

- Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1, 1–57 (2009). [CrossRef]
- J. Arlt and M.J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett.25, 191–193 (2000). [CrossRef]
- Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett.27, 285–287 (2002). [CrossRef]
- 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]
- X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector fields 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]
- S. Liu, P. Li, T. Peng, and J. L. Zhao, “Generation of arbitrary spatially variant polarization beams with a trapezoid Sagnac interferometer,” Opt. Express20, 21715–21721 (2012). [CrossRef] [PubMed]
- K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express7, 77–87 (2000). [CrossRef] [PubMed]
- Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express10, 324–331 (2002). [CrossRef] [PubMed]
- R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003). [CrossRef] [PubMed]
- N. Bokor and N. Davidson, “Generation of a hollow dark spherical spot by 4π focusing of a radially polarized Laguerre-Gaussian beam,” Opt. Lett.31, 149–151 (2006). [CrossRef] [PubMed]
- Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett.31, 820–822 (2006). [CrossRef] [PubMed]
- W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun.265, 411–417 (2006). [CrossRef]
- N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun.279, 229–234 (2007). [CrossRef]
- H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photon.2, 501–505 (2008). [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]
- W. Zhang, S. Liu, P. Li, X. Y. Jiao, and J. L. Zhao, “Controlling the polarization singularities of the focused azimuthally polarized beams,” Opt. Express21, 974–983 (2013). [CrossRef] [PubMed]
- L. X. Yang, X. S. Xie, S. C. Wang, and J. Y. Zhou, “Minimized spot of annular radially polarized focusing beam,” Opt. Lett.38, 1331–1333 (2013). [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]
- G. Lerman, Y. Lilach, and U. Levy, “Demonstration of spatially inhomogeneous vector beams with elliptical symmetry,” Opt. Lett.34, 1669–1671 (2009). [CrossRef] [PubMed]
- 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]

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