## Controlling the polarization singularities of the focused azimuthally polarized beams |

Optics Express, Vol. 21, Issue 1, pp. 974-983 (2013)

http://dx.doi.org/10.1364/OE.21.000974

Acrobat PDF (8204 KB)

### Abstract

We mainly investigate the polarization singularities of the focused azimuthally polarized (AP) beams modulated by spiral phase and sector obstacles. The results reveal that either the spiral phase or sector obstacle can convert the central *V*-point to *C*-points, *C*-point dipoles, or even double *V*-points under certain conditions. The conversion can be selectively controlled by appropriately setting the topological charge of the spiral phase and the sector angle of the obstacle. These results may have implications for the researches on polarization, focal field manipulation, or even angular momentum of the focused cylindrically polarized beams.

© 2013 OSA

## 1. Introduction

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

8. X. Jiao, S. Liu, Q. Wang, X. Gan, P. Li, and J. Zhao, “Redistributing energy flow and polarization of a focused azimuthally polarized beam with rotationally symmetric sector-shaped obstacles,” Opt. Lett. **37**(6), 1041–1043 (2012). [CrossRef] [PubMed]

2. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express **7**(2), 77–87 (2000). [CrossRef] [PubMed]

3. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics **2**(8), 501–505 (2008). [CrossRef]

4. W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commum. **265**(2), 411–417 (2006). [CrossRef]

5. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. **31**(6), 820–822 (2006). [CrossRef] [PubMed]

6. Y. Zhao, Q. Zhan, Y. Zhang, and Y.-P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. **30**(8), 848–850 (2005). [CrossRef] [PubMed]

7. X.-L. Wang, K. Lou, J. Chen, B. Gu, Y. Li, and H.-T. Wang, “Unveiling locally linearly polarized vector fields with broken axial symmetry,” Phys. Rev. A **83**(6), 063813 (2011). [CrossRef]

8. X. Jiao, S. Liu, Q. Wang, X. Gan, P. Li, and J. Zhao, “Redistributing energy flow and polarization of a focused azimuthally polarized beam with rotationally symmetric sector-shaped obstacles,” Opt. Lett. **37**(6), 1041–1043 (2012). [CrossRef] [PubMed]

9. I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. **27**(7), 545–547 (2002). [CrossRef] [PubMed]

11. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. **201**(4-6), 251–270 (2002). [CrossRef]

11. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. **201**(4-6), 251–270 (2002). [CrossRef]

12. K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express **16**(2), 695–709 (2008). [CrossRef] [PubMed]

13. O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **65**(33 Pt 2B), 036602 (2002). [CrossRef] [PubMed]

17. M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of Polarization Singularities at the Nanoscale,” Phys. Rev. Lett. **102**(3), 033902 (2009). [CrossRef] [PubMed]

18. R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express **14**(12), 5733–5745 (2006). [CrossRef] [PubMed]

*V*-points) emerge [8

8. X. Jiao, S. Liu, Q. Wang, X. Gan, P. Li, and J. Zhao, “Redistributing energy flow and polarization of a focused azimuthally polarized beam with rotationally symmetric sector-shaped obstacles,” Opt. Lett. **37**(6), 1041–1043 (2012). [CrossRef] [PubMed]

## 2. Polarizations and singularities

18. R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express **14**(12), 5733–5745 (2006). [CrossRef] [PubMed]

*E*and

_{x}*E*are the two polarization components along

_{y}*x*- and

*y*-axes, respectively, and the symbol “*” denotes the complex conjugated value. The normalized Stokes parameters are

*s*

_{1}

*= S*

_{1}/

*S*

_{0},

*s*

_{2}

*= S*

_{2}/

*S*

_{0}and

*s*

_{3}

*= S*

_{3}/

*S*

_{0}, which meet

*s*

_{1}

^{2}+

*s*

_{2}

^{2}+

*s*

_{3}

^{2}= 1.

*Φ*and tan

*χ*denote the orientation of the major axis of the ellipse and the ellipticity, and can be expressed with Stokes parameters as: tan2

*Φ*=

*s*

_{2}/

*s*

_{1}, and sin2

*χ*=

*s*

_{3}/

*s*

_{0}, respectively. The ellipticity tan

*χ =*0, + 1 and −1 mean linear polarization, right- and left-hand circular polarizations, respectively.

9. I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. **27**(7), 545–547 (2002). [CrossRef] [PubMed]

18. R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express **14**(12), 5733–5745 (2006). [CrossRef] [PubMed]

*C*-point (point of circular polarization), where the orientation of the major (or minor) axis of the polarization ellipse is undefined and the Stokes parameters satisfy the condition of circular polarization:

*s*

_{1}=

*s*

_{2}= 0, and

*s*

_{3}= ± 1, as depicted by the two points of the north and south poles, respectively; (2)

*L*-line (line of linear polarization), where the handedness of the polarization ellipse is undefined and the Stokes parameters satisfy the condition of linear polarization:

*s*

_{3}= 0, as depicted by the equatorial line; (3)

*V*-point (vector point singularities), which is an isolated and stationary point with the orientation of the linear polarization undefined and the Stokes parameters satisfy

*s*

_{1}=

*s*

_{2}=

*s*

_{3}= 0. The

*V*-point dose not locate on the Poincare sphere, and the full Stokes parameter

*S*

_{0}= 0. To manifest the polarization singularities of a light field, the crossing curves of the zero contours of Stokes parameters can be employed [9

9. I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. **27**(7), 545–547 (2002). [CrossRef] [PubMed]

10. I. Freund, “Ordinary polarization singularities in three-dimensional optical fields,” Opt. Lett. **37**(12), 2223–2225 (2012). [CrossRef] [PubMed]

14. M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett. **28**(16), 1475–1477 (2003). [CrossRef] [PubMed]

## 3. Singularity of the focused azimuthally polarized (AP) beams

2. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express **7**(2), 77–87 (2000). [CrossRef] [PubMed]

20. B. Richards and E. Wolf, “Electomagnetic Diffraction in Optical Systems: Structure of the Image Field in an Aplanatic System,” Proc. Roy. Soc. London Series A **253**(1274), 358–379 (1959). [CrossRef]

*x*,

*y*,

*z*) denote the Cartesian coordinates in the focal plane,

*k =*2π/

*λ*is the wave number,

*λ*is the wavelength in vacuum,

*θ*is the polar angle,

*φ*is the azimuthal angle,

*θ*

_{max}= arcsin(NA), and

*A*describes the complex amplitude of the incident beam. In consideration of the modulation on the AP beam with a sector obstacle, the incident beam is given aswhere Δ(

*φ*) is the transmission function of the obstacle.

*ϑ*(

*φ*,

*θ*) denotes the additional phase shift, and

*J*

_{1}() denotes the first order of the Bessel function of the first kind. In this paper, we set the parameters as following: NA = 0.95,

*λ*= 632.8nm, and Δ(

*φ*) = 1 and

*ϑ*(

*φ*,

*θ*) = 0 correspond to the AP beam in the absence of obstacle and additional phase shift, respectively. The incident beam, in fact, is a type of Bessel-Gaussian beam [21

21. K. Huang, P. Shi, G. W. Cao, K. Li, X. B. Zhang, and Y. P. Li, “Vector-vortex Bessel-Gauss beams and their tightly focusing properties,” Opt. Lett. **36**(6), 888–890 (2011). [CrossRef] [PubMed]

*s*

_{1}and

*s*

_{2}, respectively. Figure 2(b) shows the ellipticity (tan

*χ*) and the orientation (

*Φ*) of the major axis (marked by short lines) of polarization ellipse. In Fig. 2(a), there is a crossing point (point A) of the zero contours of

*s*

_{1}and

*s*

_{2}, where the orientation of the major axis of polarization ellipse is undefined. It is easily seen from Fig. 2(b) that

*s*

_{3}≡0, i.e. the focal field of the AP beam is linearly polarized. As a result, point A denotes the

*V*-point, which represents the PS with undefined direction of linear polarization. It is the sole

*V*-point in the focal field of the AP beam. Actually, the focal field of the AP beam still keeps azimuthal polarization [see Fig. 2(b)]. Furthermore,

*S*

_{0}at the

*V*-point is zero, namely, the intensity at point A is zero, as also shown as the hollow ring distribution of the focal field in Fig. 2(a).

*ϑ*(

*φ*,

*θ*) =

*lφ*, the polarization state in the focal plane changes dramatically. Figure 3 represents the calculated results of the intensity (top) and polarization (bottom) distributions of the focused AP beams carrying spiral phases with different topological charges. Where, Figs. 3(a)-3(c) correspond to

*l*= + 1, + 2, and + 3, respectively; the dotted, dashed and solid lines in the top row denote the zero contours of

*s*

_{1},

*s*

_{2}, and

*s*

_{3}, respectively; the background and short lines in the bottom row denote the ellipticity (tan

*χ*) and the orientation (

*Φ*) of polarization ellipse, respectively.

*χ =*± 1) or elliptically (1>|tan

*χ*|>0) polarized field, similar to the case of AP beam obstructed by odd-fold symmetric obstacles [8

**37**(6), 1041–1043 (2012). [CrossRef] [PubMed]

*l =*+ 1, as shown in Fig. 3(a), the AP beam is focused into a solid light spot with a smaller size (about 0.5

*λ*) [22

22. X. Hao, C. Kuang, T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. **35**(23), 3928–3930 (2010). [CrossRef] [PubMed]

*s*

_{1}and

*s*

_{2}in the center of the spot (where

*s*

_{3}

*=*+ 1≠0), presenting as a

*C*-point with right-hand circular polarization. Namely, the spiral phase exp(i

*φ*) converts the central

*V*-point of the focal field of the AP beam to a

*C*-point. And due to this conversion, the ring-like focal field of the AP beam is shrunk to a solid spot. In this case, the orbital angular momentum (OAM) carried by the spiral phase is partially converted into the spin angular momentum (SAM). Moreover, around point A, the solid spot presents right-hand elliptical polarization (tan

*χ*>0) with radial-ward major axis, resembling the RP beam.

*l*≠ + 1, the focal field of AP beam keeps ring-like profile, although elliptical polarizations are introduced into the center region. From the crossed Stokes contours [see the top of Figs. 3(b) and 3(c)] and ellipticity distributions of polarization [see the bottom of Fig. 3(b) and 3(c)], it seems that a

*C*-point is also formed at the crossing of zero contours of

*s*

_{1}and

*s*

_{2}[see points B and C in Figs. 3(b) and 3(c), respectively]. However, at points B and C,

*s*

_{3}is calculated as zero, i.e. points B and C are both

*V*-points. Actually, it can be seen that the ellipticity distributions are discontinuous at the center points, as shown in the insets (the zoom views of the center regions) of the bottom in Figs. 3(b) and 3(c). The values of these two discontinuous points have no physical meaning because the ellipticity of the polarization ellipse of

*V*-point is undefined. In a word, the spiral phase exp(i

*lφ*) never changes the central

*V*-point in the focal field of the AP beam when

*l*≠ + 1. Therefore, the type of the central singularity can be selectively converted to another one by merely changing the topological charge

*l*of the spiral phase.

*l*of the spiral phase is negative, the intensity and polarization of the focal field of AP beam are also calculated. We find that the sign of

*l*does not influence the intensity of the focal field at all, but changes the handedness of the polarization, as shown in Fig. 4 . Figures 4(a)-4(c) correspond to

*l*= −1, −2, and −3, and the background and short lines denote the distribution of tan

*χ*and orientation of

*Φ*, respectively. Comparing with the bottom of Fig. 3, it reveals that the ellipticity of the polarization ellipse corresponding to

*l*= −1 (−2, −3) is just opposite in sign to that of

*l*= + 1 ( + 2, + 3). The center points (A, B and C) in Figs. 4(a)-4(c) are

*C*-point (left-hand circularly polarized),

*V*-point, and

*V*-point, respectively, same as that in Fig. 3.

*A*=

*l*

_{0}(

*θ*)exp(

*ilφ*) [where

*l*

_{0}(

*θ*) denotes the amplitude], the focal field can be expressed aswhere (

*ρ*,

*ϕ*) denote the cylindrical coordinates in the focal plane,

**e**and

_{L}**e**denote the unit vectors of left- and right-hand circular polarizations, respectively. From Eq. (4), the focal field can be considered as the composing of the two circularly polarized components. To make the central intensity (

_{R}*ρ*= 0) of the focal plane nonzero, at least one of the Bessel functions in Eq. (4) needs to be 0th order, i.e.,

*l*= ± 1. When

*l*= 1, for example, the central intensity of the left-hand polarized component turns to zero, and the right-hand circular polarization manifests itself at the central point. When

*l*≠ ± 1, the central point with the zero-intensity presents as a

*V*-point.

*nφ*)

**e**, cos(

_{x}*nφ*)

**e**, 0

_{y}**e**]

_{z}^{T}. The nonzero integer

*n*denotes the topological charge of the polarization. The influence of the spiral phase on the focused high-order AP beam is similar to that on the single-charge one (

*n*= 1). We find that the focal field can be also decomposed into

**e**and

_{L}**e**components, containing four Bessel function terms

_{R}*J*

_{l}_{±}

*, and*

_{n}*J*

_{l}_{± (}

_{n}_{-2)}in the integral formula. That is to say, when

*l*= ±

*n*or ± (

*n*-2), the central intensity is nonzero. Thus, the central

*V*-point would be converted into a

*C*-point, and the ring-like focal field transforms to a solid spot. By performing a series of calculations, we prove that this conclusion works except in the case of

*l*= 0. This is because that the incident beam with

*l*= 0 carry no angular momentum, and the circular or elliptical polarization would not arise due to the conservation of angular momentum. Above all, when

*l*= ±

*n*or ± (

*n*-2), and

*l*≠0, the

*V*-point is converted into

*C*-point; when

*l*≠ ±

*n*and

*l*≠ ± (

*n*-2), the

*V*-point remains unchanged. This conclusion can expressed as:

## 4. Singularity transition associated with sector amplitude obstacle

*Ψ*. For an amplitude obstacle, the transmission function of the obstacle can be expressed as

*Ψ =*45°, 90°, 135°, 180°, 225°, and 270°, respectively. The results reveal that by introducing the sector obstacle, the cylindrical symmetry of the focal field of the AP beam is broken, and elliptical and circular polarizations are introduced to the focal field. As the sector angle increasing, the focal field splits vertically into two spots firstly [

*Ψ*<180°, as shown in Figs. 6(a)-6(c)], and then combines gradually to an elliptical spot [

*Ψ*≥180°, as shown in Figs. 6(d)-6(f)]. From the crossed zero contours of

*s*

_{1},

*s*

_{2}, and

*s*

_{3}, it can be seen that the

*V*-point in Fig. 2 is divided into two

*C*-points (see the two white circle points in the top of Fig. 6), which are separated by a horizontal

*L*-line (

*s*

_{3}= 0, see the solid lines in Fig. 6). It is specially noticed that the

*L*-line depicts the boundary of elliptical polarizations with opposite handedness. Thus, the two separated

*C*-points have opposite handedness, i.e. the upper and lower points are right- (tan

*χ*>0) and left-hand (tan

*χ*<0) circularly polarized, respectively. The two

*C*-points can be considered as a dipole, around which the orientations of the major axis of the polarization ellipse are much similar to the electric field lines of the electric dipole, as shown in the bottom of Fig. 6. Moreover, with increasing the sector angle, the two C-points depart away from each other gradually, and finally locate beyond the focal spot [see Fig. 6(f)].

*C*-points emerge in the focal field, just as the crossing points of zero contours of

*s*

_{1}and

*s*

_{2}in Fig. 6. However, these points locate away from the focal field, where the intensity is too weak to be taken into consideration. Here, we only consider these two crucial

*C*-points.

*C*-point dipole is also investigated, as shown in Fig. 7 , where the sector angle

*Ψ =*180°, and the topological charges in Figs. 7(a)-7(d) are

*l =*−1, + 1, + 2, and + 3, respectively. It is clearly seen that due to the OAM carried by spiral phase, the focal spot of the half-blocked AP beam is dramatically moved away from the focus (see the top of Fig. 7). The sign and value of the OAM can be characterized separately by that of the topological charge. Thus the moving direction and displacement of the focal spot are determined by

*l*: when

*l*<0 (>0), focal spot moves upward (downward) and the displacement increases with |

*l*|. Meanwhile, the spiral phase affects the

*C*-point dipole little. Especially with the variation of

*l*, the

*C*-point dipole just moves with the focal spot, but the dipole spacing is almost unchanged.

## 5. Singularity transition associated with sector phase obstacle

*Ψ =*30°, 60°, 83.5°, 90°, 110° and 180°, respectively. It reveals that with the increasing of the sector region of phase modulation, the intensity profile of the focal field gradually changes to petal-like distribution, with a pair of dark cores horizontally locating on the both side of the focus [see the top of Fig. 8(f)]. From the bottom of Fig. 8, it can be seen that the sector phase obstacle also introduces elliptical and circular polarizations into the focal field. However, when the phase is modulated on half plane (

*Ψ =*180°), the focal field maintains locally linear polarizations, resembling the case with even-fold symmetric amplitude obstacles [8

**37**(6), 1041–1043 (2012). [CrossRef] [PubMed]

*s*

_{1},

*s*

_{2}, and

*s*

_{3}in Fig. 8, it reveals that the polarization singularities are dramatically changed. With the modulation of the sector phase obstacle, the central

*V*-point in Fig. 2 is converted into a

*C*-point dipole, composed by two

*C*-points with opposite handedness [see Fig. 8(a)]. With increasing the sector angle

*Ψ*, the two

*C*-points depart away from each other gradually [see Figs. 8(b) and 8(c)], along the directions marked by the arrowheads in Fig. 8(a). The angle

*Ψ =*83.5° is a critical angle for the singularity transition, at which the zero contour of

*s*

_{1}just intersects the two self-crossing points of the zero contour of

*s*

_{2}, as shown in Fig. 8(c). When

*Ψ*>83.5°, each point of the

*C*-point dipole splits into three

*C*-points with unchanged handedness, as shown in Fig. 8(d). With the further increase of

*Ψ*, the six

*C*-points move along the arrowheads in Fig. 8(d). Especially for the

*C*-points on the sides, they can be considered two

*C*-point dipoles, and gradually join together on each side as

*Ψ*increasing. Interestingly, the six

*C*-points are merged and converted into four

*V*-points when

*Ψ*increases to 180°.

*Ψ =*180°, due to the modulation of the phase obstacle, the ring-like focal field of the AP beam splits into a double-dark-core distribution with the maximum intensity residing the central point; meanwhile, the central

*V*-point of the focused AP beam splits into two

*V*-points, each of which occupies a dark core of the focal field.

*Ψ =*180°), and the corresponding results are depicted in Fig. 9 . In Figs. 9(a)-9(d), the topological charges of the attached spiral phases are

*l =*−1, + 1, + 2, and + 3, respectively. Comparing with Fig. 8(f), it can be seen that when |

*l*|

*=*1, the central dark core and

*V*-point of the focal field are recalled by the spiral phase. Instead, the two dark cores and the

*V*-points in Fig. 8(f) are vanished. In addition, two left-hand (right-hand)

*C*-points emerge when

*l =*−1 ( + 1), as shown in Figs. 9(a) and 9(b). While for |

*l*|≠1, the focal field becomes confusing, and more

*C*-points arise. Furthermore, when

*Ψ*≠180°, the focal field of the AP beam modulated with phase obstacle and spiral phase becomes also complicated as Figs. 9(c) and 9(d).

## 6. Conclusions

*V*-point to a

*C*-point, just if the topological charges of the spiral phase and azimuthal polarization are equal in absolute value, i.e. |

*l*| =

*n*. The sector obstacles, including the amplitude and phase ones, can also convert the

*V*-point to one or more

*C*-point dipoles, with the position of the singularities changed with the sector angle

*Ψ*. In the case of the amplitude obstacle, the spiral phase moves the

*C*-point dipole totally. While in the case of the phase obstacle, especially when the phase of the AP beam is modulated on half plane (

*Ψ =*180°), the central

*V*-point of the focal field splits into double

*V*-points, and then can be recalled by the single charged spiral phase (

*l =*± 1). Based on these conclusions, the PSs of the focused AP beam can be selectively converted and controlled. We hope these results may have some useful implications for the researches on polarization, focal field manipulation, or even angular momentum of the focused CV beams.

## Acknowledgments

## References and links

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

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

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

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

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

6. | Y. Zhao, Q. Zhan, Y. Zhang, and Y.-P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. |

7. | X.-L. Wang, K. Lou, J. Chen, B. Gu, Y. Li, and H.-T. Wang, “Unveiling locally linearly polarized vector fields with broken axial symmetry,” Phys. Rev. A |

8. | X. Jiao, S. Liu, Q. Wang, X. Gan, P. Li, and J. Zhao, “Redistributing energy flow and polarization of a focused azimuthally polarized beam with rotationally symmetric sector-shaped obstacles,” Opt. Lett. |

9. | I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. |

10. | I. Freund, “Ordinary polarization singularities in three-dimensional optical fields,” Opt. Lett. |

11. | I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. |

12. | K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express |

13. | O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

14. | M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett. |

15. | F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. |

16. | F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization singularities in 2D and 3D speckle fields,” Phys. Rev. Lett. |

17. | M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of Polarization Singularities at the Nanoscale,” Phys. Rev. Lett. |

18. | R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express |

19. | M. Born and E. Wolf, |

20. | B. Richards and E. Wolf, “Electomagnetic Diffraction in Optical Systems: Structure of the Image Field in an Aplanatic System,” Proc. Roy. Soc. London Series A |

21. | K. Huang, P. Shi, G. W. Cao, K. Li, X. B. Zhang, and Y. P. Li, “Vector-vortex Bessel-Gauss beams and their tightly focusing properties,” Opt. Lett. |

22. | X. Hao, C. Kuang, T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(260.5430) Physical optics : Polarization

(260.6042) Physical optics : Singular optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: November 8, 2012

Revised Manuscript: December 9, 2012

Manuscript Accepted: December 12, 2012

Published: January 9, 2013

**Citation**

Wei Zhang, Sheng Liu, Peng Li, Xiangyang Jiao, and Jianlin Zhao, "Controlling the polarization singularities of the focused azimuthally polarized beams," Opt. Express **21**, 974-983 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-1-974

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### References

- Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1(1), 1–57 (2009). [CrossRef]
- K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000). [CrossRef] [PubMed]
- H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008). [CrossRef]
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