## Generation of tunable three-dimensional polarization in 4Pi focusing system |

Optics Express, Vol. 21, Issue 14, pp. 17265-17274 (2013)

http://dx.doi.org/10.1364/OE.21.017265

Acrobat PDF (2257 KB)

### Abstract

We show that, by uniformly modulating the amplitude or polarization of one half of the input beam, a tunable three-dimensional (3D) polarization field near the focus of a 4Pi focusing system can be generated. If the input field is radially polarized and modulated by an amplitude-phase modulator, the longitudinal component of the focused field will partially convert to the transversal one according to the modulation factor and a 3D linear polarization state is created. If the input field is circularly polarized in one half and elliptically polarized in another half, the focal field will have elliptical polarization with the normal to the polarization ellipse being 3D controllable, corresponding to a 3D controllable spin angular momentum.

© 2013 OSA

## 1. Introduction

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

3. S. Hell and E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A **9**(12), 2159–2166 (1992). [CrossRef]

4. Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. **37**(8), 1286–1288 (2012). [CrossRef] [PubMed]

6. J. Wang, W. Chen, and Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt. **14**(5), 055004 (2012). [CrossRef]

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

7. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. **85**(21), 4482–4485 (2000). [CrossRef] [PubMed]

8. X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun **3**, 998 (2012). [CrossRef] [PubMed]

7. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. **85**(21), 4482–4485 (2000). [CrossRef] [PubMed]

8. X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun **3**, 998 (2012). [CrossRef] [PubMed]

9. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser trapped microscopic particles,” Nature **394**(6691), 348–350 (1998). [CrossRef]

10. A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. **12**(8), 083053 (2010). [CrossRef]

12. M. Mansuripur, “Spin and orbital angular momenta of electromagnetic waves in free space,” Phys. Rev. A **84**(3), 033838 (2011). [CrossRef]

10. A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. **12**(8), 083053 (2010). [CrossRef]

12. M. Mansuripur, “Spin and orbital angular momenta of electromagnetic waves in free space,” Phys. Rev. A **84**(3), 033838 (2011). [CrossRef]

13. A. F. Abouraddy and K. C. Toussaint Jr., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. **96**(15), 153901 (2006). [CrossRef] [PubMed]

15. W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. **284**(1), 52–56 (2011). [CrossRef]

13. A. F. Abouraddy and K. C. Toussaint Jr., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. **96**(15), 153901 (2006). [CrossRef] [PubMed]

14. W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. **12**(4), 045707 (2010). [CrossRef]

15. W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. **284**(1), 52–56 (2011). [CrossRef]

2. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. **29**(17), 1968–1970 (2004). [CrossRef] [PubMed]

4. Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. **37**(8), 1286–1288 (2012). [CrossRef] [PubMed]

16. C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A **80**(6), 063814 (2009). [CrossRef]

## 2. The generation of 3D linear polarization

17. 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]

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

*C*is the normalized factor,

*α*= arcsin(NA/

*n*) with

*n*= 1 in the free space,

*X*(

*θ*) = (cos

*θ*)

^{1/2}is pupil apodization function for an aplanatic lens. The function

*l*(

_{0}*θ*) describes the amplitude distribution of the Bessel-Gaussian beam, which has the formwith

*β*= 1.5 in our configuration.

*T*(

*ϕ*) is the modulation function, with

*T*(

*ϕ*) =

*T*

_{L}_{/}

*(*

_{R}*ϕ*) for the modulators at the left and right sides, respectively.

*T*(

_{R}*ϕ*) =

*T*(

_{L}*ϕ +*π), andwhere

*a*is the modulator factor ranging from −1 to 1,

*ϕ*

_{0}is the angle between the amplitude jump line and

*x*-axis,. The electric field near the focus is the superposition of the fields from the left and right objectives. That is [4

4. Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. **37**(8), 1286–1288 (2012). [CrossRef] [PubMed]

*χ*) to describe the orientation of 3D linear polarization, where

_{l},ψ_{l}*χ*ranges from 0 to

_{l}*π*/2, and

*ψ*takes value between 0 and 2

_{l}*π*. At the focus, the polarization orientation can be deduced by superposing the fields of plane waves in the angular spectrum from diffractions at two opposing objectives [18

18. S. N. Khonina and I. Golub, “Optimization of focusing of linearly polarized light,” Opt. Lett. **36**(3), 352–354 (2011). [CrossRef] [PubMed]

*a*= 1, the APMs will not affect the input radially polarized beams, which result in the focal field being longitudinally polarized [4

**37**(8), 1286–1288 (2012). [CrossRef] [PubMed]

*a*decreases from 1 to −1, the polarization vector of the focal field will vary gradually form longitudinal direction to the transverse one perpendicular to the amplitude jump line. That means the polar angle of 3D linear polarization

*χ*depends on

_{l}*a*. The numerical result is shown in Fig. 2(g), from which one sees that

*χ*and

_{l}*a*almost have a linear relation. The intensity and polarization distributions in the focal plane are plotted in Figs. 2(a)-2(d) for the case of

*ϕ*

_{0}=

*π*/2,

*a*= 1, 0.23, −0.3 and −1, which correspond to the polar angles of 0,

*π*/6,

*π*/3, and

*π*/2, respectively. Due to the rotational symmetry of the input radial polarization, rotating APMs in both sides will cause a rotation of the focal field around

*z*axis. After simple analysis, one can easily get that the azimuthal angle of 3D linear polarization

*ψ*varies with the rotation angle

_{l}*ϕ*

_{0}according to the law

*ψ*=

_{l}*ϕ*

_{0}+

*π*/2. In Figs. 2(d)-2(f),

*a*is fixed to be −1 and

*ϕ*

_{0}=

*π*/2,

*π*, and 5

*π*/4, which give

*ψ*= –

_{l}*π*, –

*π*/2, and –

*π*/4. Therefore, by controlling the configure factors

*a*and

*ϕ*

_{0}, arbitrary 3D linear polarization can be created.

*x*,

*y*, and

*z*axes rely on the modulator factor

*a*, and are equal respectively to 0.56λ and 0.56λ, and 0.3λ at

*a*= 1. When

*a*decreases, The FWHM along

*x*axis will slightly increases, while the FWHMs along

*y*and

*z*axes almost keep the same. It should be noted that the spot size can be significantly reduced by adding an annular aperture [4

**37**(8), 1286–1288 (2012). [CrossRef] [PubMed]

*a*= −1 and

*ϕ*

_{0}= 5

*π*/4, where the polarization vectors are nearly uniformly directed at an angle of −45° to

*x*axis and 90° to

*z*axis. Within the FWHM region, the average intensity ratio between the field component along the dominating polarization direction and the total light field is 95.24%. The ratio is always more than 91% for an arbitrary

*a*. It is also found that the peak intensity of the focal field always occurs at geometrical focus. As demonstrated in Fig. 3, the intensity at geometrical focus only depends on the modulator factor

*a*, and takes its maximal value at

*a*= 1. When

*a*decreases, the intensity decreases gradually; and reaches its minimum at

*a*= −0.07 instead of

*a*= 0 because of the vectorial interference between the modulated and unmodulated parts of the input light field. For the same reason, the intensity at geometrical focus for a positive modulator factor |

*a*| is always larger than that for a negative modulator factor -|

*a*|. This kind of tunable 3D polarization may find important applications in optical imaging [19

19. E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. **82**(20), 4014–4017 (1999). [CrossRef]

9. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser trapped microscopic particles,” Nature **394**(6691), 348–350 (1998). [CrossRef]

## 3. Generation of 3D elliptical polarization

20. J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A, Pure Appl. Opt. **2**(3), 216–222 (2000). [CrossRef]

*x*axis; the LCVR with its fast axis being 45° to

*x*axis leads to a phase retarder

*δ*. The field from the LCVR is cos(

*δ*/2)

**e**

*+*

_{x}*i*sin((

*δ*/2)

**e**

*[21]. When*

_{y}*δ*= ±

*π*/2, it is right (left) handed circular polarizations

**e**

_{±}= 2

^{-1/2}(

**e**

*±*

_{x}*i*

**e**

*). Now, we consider a LCVR, which is divided into two regions (0<*

_{y}*ϕ*<

*π*and

*π*<

*ϕ*< 2

*π*) having different phase retarders

*δ*

_{1}and

*δ*

_{2}, where

*δ*

_{1}= ±

*π*/2 and -

*π*/2≤

*δ*

_{2}≥

*π*/2, respectively. If the polarizer and LCVR have been rotated around

*z*axis by an angle of

*ϕ*

_{0}, the final field becomes

*l*

_{0}(

*θ*)[

*A*(

_{L}*ϕ*)

**e**+

_{+}*B*

_{L}_{(}

*ϕ*)

**e**] and

_{+}*l*

_{0}(

*θ*)[

*A*(

_{R}*ϕ*)

**e**+

_{+}*B*

_{R}_{(}

*ϕ*)

**e**], respectively, where

_{+}*A*(

_{L}*ϕ*) =

*A*(

_{R}*ϕ*+

*π*) =

*A*(

*ϕ*),

*B*(

_{L}*ϕ*) =

*B*(

_{R}*ϕ*+

*π*) =

*B*(

*ϕ*). Following [22

22. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. **99**(7), 073901 (2007). [CrossRef] [PubMed]

*z*-axis. In contrary, the normal vector can take any 3D direction for a nonparaxial field with 3D elliptical polarization. The 3D orientation of elliptical polarization can be characterized by the local SAM vector, which is in the form ofThe local SAM vector is normal to the plane of polarization ellipse; and its length, after normalized to the local intensity, is equal to the “third” Stokes parameter, which characterizes the degree of circular polarization [21]. The average spin part of angular momentum per photon is

*ћ*

**withwhere the integral region Ω is the whole central spot. Equation (8) is equal to the angular spectrum form given in [16**

*σ*16. C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A **80**(6), 063814 (2009). [CrossRef]

**| = 1 associates with circular polarization, while |**

*σ***| = 0 means linear polarization. As we will show below, the normalized local SAM vector near the geometrical focus and the average SAM vector of the central spot almost have same orientation but different length.**

*σ**δ*

_{1},

*δ*

_{2}, and

*ϕ*

_{0}. For example, a transverse SAM vector is created at focus with

*δ*

_{1}= -

*δ*

_{2}=

*π*/2 and

*ϕ*

_{0}= -

*π*/2, as shown in Fig. 4. A light field carrying 3D SAM has 3D elliptical polarization. Figures 5(a)-5(c) are the intensity and polarizationdistributions within the central spot, which are projected on three orthogonal planes. When the configure factors (

*δ*

_{1},

*δ*

_{2},

*ϕ*

_{0}) are equal to (

*π*/2,

*π*/2,-

*π*/2), (

*π*/2,-

*π*/5,0), and (

*π*/2,-

*π*/2,

*π*/4), the fields at geometrical focus are circularly polarized with normalized local SAM vector (

*χ*,

_{s}*ψ*) = (0,0), elliptically polarized with (

_{s}*χ*,

_{s}*ψ*) = (π/4,-π/2), and elliptically polarized with (

_{s}*χ*,

_{s}*ψ*) = (

_{s}*π*/2,-

*π*/4), respectively. One sees from Figs. 5(a)-5(c) that, the focal fields within the central spot have very similar intensity distributions. The FWHMs of the central spot are 0.64λ and 0.3λ along transverse and longitudinal directions, respectively, although with different configure factors. That means the spot size is always as small as that generated by a circular polarized input beam. In the focal plane, the light field is almost uniformly polarized. However, the polarization will deviate slowly from that at the geometrical focus when the observed point moves away from the focal plane. But the average SAM vectors within the whole central spot almost keep the same direction as that of local one at focus. The difference between polar angle of the average SAM vector in central spot and that of local SAM vector at focus (

*χ*and

_{σ}*χ*) is small, which is evident in Fig. 5(d), where the dependences of

_{s}*χ*and

_{σ}*χ*on

_{s}*δ*

_{2}are plotted for

*δ*

_{1}= ±

*π*/2. The azimuthal angles of the average and local SAM vectors (

*ψ*and

_{σ}*ψ*) are also similar, and

_{s}*δ*

_{1}= ±

*π*/2. The dependences of |

**s**| and |

**| on**

*σ**δ*

_{2}for

*δ*

_{1}= ±

*π*/2 are shown in Fig. 5(e), from which one sees that, |

**s**(

*δ*

_{1},

*δ*

_{2})| = |

**s**(-

*δ*

_{1},-

*δ*

_{2})|, |

**(**

*σ**δ*

_{1},

*δ*

_{2})| = |

**(-**

*σ**δ*

_{1},-

*δ*

_{2})|, and |

**| is always smaller than |**

*σ***s**| because of the polarization deviation. The average SAM |

**| will be much closer to |**

*σ***s**| within FWHM region, where the degree of polarization deviation is much smaller. It is well known that, under tightly focusing condition, the longitude component of angular momentum conserves, and SAM will convert to orbital angular momentum [22

22. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. **99**(7), 073901 (2007). [CrossRef] [PubMed]

*δ*

_{1}= -

*δ*

_{2}= ±

*π*/2, an average zero angular momentum of input field can produce a nonzero transverse SAM in the focal field.

23. A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. **103**(10), 100401 (2009). [CrossRef] [PubMed]

24. H. Shpaisman, D. B. Ruffner, and D. G. Grier, “Light-driven three-dimensional rotational motion of dandelion-shaped microparticles,” Appl. Phys. Lett. **102**(7), 071103 (2013). [CrossRef]

25. P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. **78**(2), 249–251 (2001). [CrossRef]

## 4. Conclusion

## Appendix: the local SAM vector

26. A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. **13**(5), 053001 (2011). [CrossRef]

27. R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt. **52**, 1045–1052 (2005). [CrossRef]

*δ*= 0, Δ

_{x}*δ*

_{1}=

*δ*-

_{z}*δ*, Δ

_{y}*δ*

_{2}=

*δ*-

_{z}*δ*, and Δ

_{x}*δ*

_{3}=

*δ*-

_{y}*δ*, and rewrite Eq. (10) aswhere

_{x}**e**

_{1}and

**e**

_{2}are two vectors. A new unit vector

**e**

_{3}orthogonal to

**e**

_{2}is found to bewhich locates at the plane determined by

**e**

_{1}and

**e**

_{2}. Now, the electric field can be written in the basis of

**e**

_{2}and

**e**

_{3}:According to [21], the “third” Stokes parameter is

**| is equal to**

*s**s*

_{3}. And

**,**

*s***e**

_{2}, and

**e**

_{3}form a right-handed system.

## Acknowledgments

## References and links

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

2. | N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. |

3. | S. Hell and E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A |

4. | Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. |

5. | 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 |

6. | J. Wang, W. Chen, and Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt. |

7. | B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. |

8. | X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun |

9. | M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser trapped microscopic particles,” Nature |

10. | A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. |

11. | W. Zhu and W. She, “Electrically controlling spin and orbital angular momentum of a focused light beam in a uniaxial crystal,” Opt. Express |

12. | M. Mansuripur, “Spin and orbital angular momenta of electromagnetic waves in free space,” Phys. Rev. A |

13. | A. F. Abouraddy and K. C. Toussaint Jr., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. |

14. | W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. |

15. | W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. |

16. | C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A |

17. | 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 |

18. | S. N. Khonina and I. Golub, “Optimization of focusing of linearly polarized light,” Opt. Lett. |

19. | E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. |

20. | J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A, Pure Appl. Opt. |

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

22. | Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. |

23. | A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. |

24. | H. Shpaisman, D. B. Ruffner, and D. G. Grier, “Light-driven three-dimensional rotational motion of dandelion-shaped microparticles,” Appl. Phys. Lett. |

25. | P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. |

26. | A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. |

27. | R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(180.6900) Microscopy : Three-dimensional microscopy

(260.5430) Physical optics : Polarization

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 3, 2013

Revised Manuscript: July 1, 2013

Manuscript Accepted: July 2, 2013

Published: July 11, 2013

**Virtual Issues**

Vol. 8, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Wenguo Zhu and Weilong She, "Generation of tunable three-dimensional polarization in 4Pi focusing system," Opt. Express **21**, 17265-17274 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-17265

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### References

- K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000). [CrossRef] [PubMed]
- N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett.29(17), 1968–1970 (2004). [CrossRef] [PubMed]
- S. Hell and E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A9(12), 2159–2166 (1992). [CrossRef]
- Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett.37(8), 1286–1288 (2012). [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]
- J. Wang, W. Chen, and Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt.14(5), 055004 (2012). [CrossRef]
- B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett.85(21), 4482–4485 (2000). [CrossRef] [PubMed]
- X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun3, 998 (2012). [CrossRef] [PubMed]
- M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser trapped microscopic particles,” Nature394(6691), 348–350 (1998). [CrossRef]
- A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys.12(8), 083053 (2010). [CrossRef]
- W. Zhu and W. She, “Electrically controlling spin and orbital angular momentum of a focused light beam in a uniaxial crystal,” Opt. Express20(23), 25876–25883 (2012). [CrossRef] [PubMed]
- M. Mansuripur, “Spin and orbital angular momenta of electromagnetic waves in free space,” Phys. Rev. A84(3), 033838 (2011). [CrossRef]
- A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett.96(15), 153901 (2006). [CrossRef] [PubMed]
- W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt.12(4), 045707 (2010). [CrossRef]
- W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun.284(1), 52–56 (2011). [CrossRef]
- C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A80(6), 063814 (2009). [CrossRef]
- B. Richards and E. Wolf, “Electomagnetic diffraction in optical systems: Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London Series A253(1274), 358–379 (1959). [CrossRef]
- S. N. Khonina and I. Golub, “Optimization of focusing of linearly polarized light,” Opt. Lett.36(3), 352–354 (2011). [CrossRef] [PubMed]
- E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett.82(20), 4014–4017 (1999). [CrossRef]
- J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A, Pure Appl. Opt.2(3), 216–222 (2000). [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Pergamon Press, 6th edition, 1983).
- Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett.99(7), 073901 (2007). [CrossRef] [PubMed]
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