## Nonparaxial and paraxial focusing of azimuthal-variant vector beams |

Optics Express, Vol. 20, Issue 16, pp. 17684-17694 (2012)

http://dx.doi.org/10.1364/OE.20.017684

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### Abstract

Based on the vectorial Rayleigh-Sommerfeld formulas under the weak nonparaxial approximation, we investigate the propagation behavior of a lowest-order Laguerre-Gaussian beam with azimuthal-variant states of polarization. We present the analytical expressions for the radial, azimuthal, and longitudinal components of the electric field with an arbitrary integer topological charge *m* focused by a nonaperturing thin lens. We illustrate the three-dimensional optical intensities, energy flux distributions, beam waists, and focal shifts of the focused azimuthal-variant vector beams under the nonparaxial and paraxial approximations.

© 2012 OSA

## 1. Introduction

1. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A **15**, 2705–2711 (1998). [CrossRef]

2. J. S. Ahn, H. W. Kihm, J. E. Kihm, D. S. Kim, and K. G. Lee, “3-dimensional local field polarization vector mapping of a focused radially polarized beam using gold nanoparticle functionalized tips,” Opt. Express **17**, 2280–2286 (2009). [CrossRef] [PubMed]

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

4. 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]

5. V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A **27**, 372–380 (2010). [CrossRef]

6. X. Jia and Y. Wang “Vectorial structure of far field of cylindrically polarized beams diffracted at a circular aperture,” Opt. Lett. **36**, 295–297 (2011). [CrossRef] [PubMed]

7. D. M. Deng, Q. Guo, S. Lan, and X. B. Yang, “Application of the multiscale singular perturbation method to nonparaxial beam propagation in free space,” J. Opt. Soc. Am. A **24**, 3317–3325 (2007). [CrossRef]

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

1. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A **15**, 2705–2711 (1998). [CrossRef]

9. R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A **21**, 2029–2037 (2004). [CrossRef]

10. Z. Mei and D. Zhao, “Nonparaxial propagation of controllable dark-hollow beams,” J. Opt. Soc. Am. A **25**, 537–542 (2008). [CrossRef]

11. D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. B **26**, 2044–2049 (2009). [CrossRef]

12. Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express **15**, 11942–11951 (2007). [CrossRef] [PubMed]

13. G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B **26**, 141–147 (2009). [CrossRef]

5. V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A **27**, 372–380 (2010). [CrossRef]

14. X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B **102**, 205–213 (2011). [CrossRef]

15. B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett. **28**, 2440–2442 (2003). [CrossRef] [PubMed]

17. X. Jia, Y. Wang, and B. Li, “Nonparaxial analyses of radially polarized beams diffracted at a circular aperture,” Opt. Express **18**, 7064–7075 (2010). [CrossRef] [PubMed]

18. K. Duan and B. Lü, “Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture,” J. Opt. Soc. Am. A **21**, 1613–1620 (2004). [CrossRef]

19. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. **86**, 5251–5254 (2001). [CrossRef] [PubMed]

20. C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. **106**, 123901 (2011). [CrossRef] [PubMed]

*et al*. [21

21. P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express **4**, 411–419 (1999). [CrossRef] [PubMed]

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

*et al*. [23

23. D. M. Deng, Q. Guo, L. J. Wu, and X. B. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B **24**, 636–643 (2007). [CrossRef]

*et al*. [24

24. M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A: Pure Appl. Opt. **11**, 065204 (2009). [CrossRef]

*et al*. [25

25. D. M. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B **23**, 1228–1234 (2006). [CrossRef]

*et al*. [26

26. P. P. Banerjee, G. Cook, and D. R. Evans, “A *q*-parameter approach to analysis of propagation, focusing, and waveguiding of radially polarized Gaussian beams,” J. Opt. Soc. Am. A **26**, 1366–1374 (2009). [CrossRef]

*et al*. [27

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

*m*. The vectorial Rayleigh-Sommerfeld formulas under the weak nonparaxial approximation developed by Kotlyar

*et al*. [5

5. V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A **27**, 372–380 (2010). [CrossRef]

28. V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt. **13**, 075703 (2011). [CrossRef]

*m*focused by a nonaperturing thin lens. By numerical illustration, we investigate the three-dimensional optical intensities, energy flux distributions, beam waists, and focal shifts of the focused azimuthal-variant vector beams under the nonparaxial and paraxial approximations. For the special case of

*m*= 1, the results are in agreement with the ones reported previously [5

**27**, 372–380 (2010). [CrossRef]

26. P. P. Banerjee, G. Cook, and D. R. Evans, “A *q*-parameter approach to analysis of propagation, focusing, and waveguiding of radially polarized Gaussian beams,” J. Opt. Soc. Am. A **26**, 1366–1374 (2009). [CrossRef]

## 2. Theory

*z*= 0 can be expressed by [1

1. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A **15**, 2705–2711 (1998). [CrossRef]

29. B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express **20**, 149–157 (2012). [CrossRef] [PubMed]

**ê**

*and*

_{r}**ê**

*are the unit vectors in the polar coordinate system (*

_{ϕ}*r*,

*ϕ*),

*A*(

*r*) represents the radial-dependent amplitude,

*m*is the azimuthal topological charge, and

*φ*

_{0}is the initial phase of the vector beam. Two extreme cases of vector beams are the radially and azimuthally polarized vector beams for

*m*= 1 with

*φ*

_{0}= 0 and

*π*/2, respectively. Note that the azimuthal-variation vector beam belongs to a kind of local linearly polarized vector field. And that the spatial distribution of states of polarization is dependent on the azimuthal angle

*ϕ*only.

*z*direction can be given by [5

**27**, 372–380 (2010). [CrossRef]

*γ*=

*kρ*/

*ξ*,

*k*= 2

*π*/

*λ*, and

*λ*is the wavelength. Note that the weak nonparaxial approximation takes the form

**27**, 372–380 (2010). [CrossRef]

28. V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt. **13**, 075703 (2011). [CrossRef]

*ω*<

*λ*and

*z*≥

*ω*

^{2}/(2

*λ*) [30

30. A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. **202**, 17–20 (2002). [CrossRef]

*ω*is the waist width. This approximation to describe the nonparaxial propagation of light beams is well known [12

12. Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express **15**, 11942–11951 (2007). [CrossRef] [PubMed]

17. X. Jia, Y. Wang, and B. Li, “Nonparaxial analyses of radially polarized beams diffracted at a circular aperture,” Opt. Express **18**, 7064–7075 (2010). [CrossRef] [PubMed]

23. D. M. Deng, Q. Guo, L. J. Wu, and X. B. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B **24**, 636–643 (2007). [CrossRef]

*ϕ*for an integer

*m*≥ 0 can be accomplished using the identities where

*J*(·) is the Bessel function of

_{m}*m*th-order.

23. D. M. Deng, Q. Guo, L. J. Wu, and X. B. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B **24**, 636–643 (2007). [CrossRef]

25. D. M. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B **23**, 1228–1234 (2006). [CrossRef]

*ω*

_{0}is the waist radius of the Gaussian beam, and

*E*

_{0}is an amplitude constant. One takes

*f*, we have

**27**, 372–380 (2010). [CrossRef]

*m*] > −1, Re[

*β*] > 0,

*γ*> 0,

*t*=

*γ*

^{2}/(8

*β*), and

*I*(·) is the modified Bessel function of

_{m}*m*th-order, we yield the analytical results

*β*=

*α*−

*ik*/(2

*ξ*) and

*t*=

*γ*

^{2}/(8

*β*). Equation (8), which is the basic result of the present work, gives a general three-dimensional electric field of the nonparaxial propagation of azimuthal-variant vector beams.

*m*= 1 and

*φ*

_{0}= 0, we deduce the nonparaxial propagation of a radially polarized vector beam from Eq. (8) as follows where

*q*=

*ξ*−

*ik*/(2

*α*). The obtained result is coincident with the one reported previously [5

**27**, 372–380 (2010). [CrossRef]

*m*= 1 and

*φ*

_{0}=

*π*/2), we yield

*z*

^{2}+

*ρ*

^{2})

^{1/2}≈

*z*+

*ρ*

^{2}/(2

*z*) ≈

*z*. Accordingly, we obtain the propagation expressions for the azimuthal-variant vector beam under the paraxial approximation as

*γ*′ =

*kρ*/

*z*,

*β*′ =

*α*−

*ik*/(2

*z*), and

*t*′ =

*γ*′

^{2}/(8

*β*′). In particular, for a radially polarized vector beam (

*m*= 1 and

*φ*

_{0}= 0), one gets [5

**27**, 372–380 (2010). [CrossRef]

26. P. P. Banerjee, G. Cook, and D. R. Evans, “A *q*-parameter approach to analysis of propagation, focusing, and waveguiding of radially polarized Gaussian beams,” J. Opt. Soc. Am. A **26**, 1366–1374 (2009). [CrossRef]

*q*≃

*z*−

*ik*/(2

*α*). For the case of

*m*= 1 and

*φ*

_{0}=

*π*/2, we obtain the paraxial propagation of an azimuthally polarized vector beam as

24. M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A: Pure Appl. Opt. **11**, 065204 (2009). [CrossRef]

29. B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express **20**, 149–157 (2012). [CrossRef] [PubMed]

*ρ*

_{0}of a vector beam as that radius within 80% of the beam’s power is enclosed. Accordingly, the width

*ρ*

_{0}of a focused vector beam at the propagation distance

*z*satisfies [21

21. P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express **4**, 411–419 (1999). [CrossRef] [PubMed]

*I*= |

_{G}*E*|

_{r}^{2}+ |

*E*|

_{ϕ}^{2}+ |

*E*|

_{z}^{2}is the nonparaxial intensity of the vector beam. For the case of the paraxial propagation, one should replace

*I*by

_{G}*I*= |

_{GP}*E*|

_{rp}^{2}+ |

*E*|

_{ϕp}^{2}+ |

*E*|

_{zp}^{2}in Eq. (14).

*z*plane can be given by the time-average of the

*z*component of the Poynting vector, Here Re[·] is the real part and the asterisk denotes the complex conjugate. The magnetic fields

**H**(

*ρ*,

*θ*,

*z*) under the nonparaxial and paraxial approximations are easily obtained by taking the curl of Eqs. (8) and (11), respectively.

## 3. Numerical results and discussions

*λ*= 633 nm,

*E*

_{0}= 1 (a.u.),

*ω*

_{0}= 1

*μ*m, and

*f*= 4

*μ*m [5

**27**, 372–380 (2010). [CrossRef]

*m*= 1 for

*φ*

_{0}=

*π*/4 at the lens’ geometrical focus (

*x*−

*y*plane). The intensity patterns are normalized by the maximum of the global intensity

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

*m*= 1 and

*φ*

_{0}=

*π*/4 are quite different from those of radial polarized vector beam (

*m*= 1 and

*φ*

_{0}= 0) exhibiting the radial and longitudinal components only or of azimuthal polarized vector beam (

*m*= 1 and

*φ*

_{0}=

*π*/2) just having the azimuthal component [5

**27**, 372–380 (2010). [CrossRef]

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

*φ*

_{0}. Consequently, the

*φ*

_{0}-dependent energy flux distribution is predictable. To more clearly show the nonparaxial intensity patterns, the middle row in Fig. 1 displays the corresponding intensity profiles alone the

*x*-axis for

*y*= 0. For the sake of comparison, the paraxial intensity profiles obtained by Eq. (11) are also illustrated by circles in Figs. 1(e)–1(h). It should be noted that there is a little difference between the nonparaxial and paraxial optical intensity distributions. As Jia

*et al*. [17

17. X. Jia, Y. Wang, and B. Li, “Nonparaxial analyses of radially polarized beams diffracted at a circular aperture,” Opt. Express **18**, 7064–7075 (2010). [CrossRef] [PubMed]

*ω*

_{0}/

*λ*decreasing. Figures 1(i)–1(

*l*) display the nonparaxial intensity patterns through the focus (

*x*−

*z*plane and

*y*= 0), which are normalized by the maximum of the global intensity

*l*), the optical power is somewhat more concentrated before

*z*=

*f*, and therefore the beam is likely to be narrower there. Apparently, the true focus occurs not at the lens’ geometric focus but rather closer to the lens. This is a well-known focal shift that has been discussed previously [21

21. P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express **4**, 411–419 (1999). [CrossRef] [PubMed]

*ρ*

_{0}of nonparaxial focused vector beam with

*m*= 1 and

*φ*

_{0}=

*π*/4 against the propagation distance

*z*, with the same parameters as used in Fig. 1. As shown in Fig. 2, the focused beam converges gradually, subsequently reaches the true focus with a minimum beam waist at

*z*≃ 0.55

*f*, and then diverges strongly. The asymmetry in the beam waist on either side of the true focus is obvious, completely different from that of the paraxial result (circles in Fig. 2). This difference between the nonparaxial and paraxial results is anticipated because the paraxial approximation is inapplicable when the beam waist is comparable to the wavelength. In the case of

*λ*= 633 nm,

*ω*

_{0}= 1

*μ*m, and

*f*= 4

*μ*m, we estimate the focal shift of the vector beam with

*m*= 1 for

*φ*

_{0}=

*π*/4 to be |(

*z*−

*f*)/

*f*| ≃ 0.45, which is very close to that of a radially polarized beam [5

**27**, 372–380 (2010). [CrossRef]

*φ*

_{0}.

*m*= 1 and

*φ*

_{0}=

*π*/4) at the planes of the lens’ geometrical focus and the true focus, respectively, with the same parameters as used in Fig. 1. Solid lines (circles) in Figs. 3(c) and 3(d) corresponding to Figs. 3(a) and 3(b) are the nonparaxial (paraxial) cross-section energy flux profiles at

*y*= 0 for

*z*=

*f*and

*z*= 0.55

*f*, respectively. As shown in Fig. 3, the energy flux distributions of the vector beam have the so-called doughnut patterns with the on-axis energy null and annular energy distribution. Besides, the energy flux is most concentrated just at the true focus than any other place. The difference between the nonparaxial and paraxial energy flux is observable, as shown in Figs. 3(c) and 3(d). This is because that the paraxial propagation approximately describes the beam propagation in our case of

*ω*

_{0}∼ 2

*λ*. Nevertheless, the results obtained by the nonparaxial and paraxial theories are identical under the conditions of

*ω*

_{0}≫

*λ*and

*z*≫

*λ*.

*λ*= 532 nm,

*E*

_{0}= 1 (a.u.),

*ω*

_{0}= 2.5 mm,

*f*= 8 mm [4

4. 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*

_{0}= 0 for

*m*= 1, 3, and 5 at focus (top row) and though focus when

*y*= 0 (lower row). All intensity patterns are normalized by the maximum of

*I*(

_{G}*x*, 0,

*f*). Interestingly, the focused field of the vector beam has a doughnut-shaped intensity profile with the characteristic of axially symmetric profile, as shown in Fig. 4. Furthermore, the radius of the doughnut field increases with increasing the topological charge

*m*of the vector beam. These results are comparable with the well-known results of a high order cylindrical vector beams [21

**4**, 411–419 (1999). [CrossRef] [PubMed]

24. M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A: Pure Appl. Opt. **11**, 065204 (2009). [CrossRef]

29. B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express **20**, 149–157 (2012). [CrossRef] [PubMed]

*ρ*

_{0}of paraxial focused vector beam with different values of

*m*through the lens’ geometrical focus, with the same parameters as used in Fig. 4. The distinct symmetry in the beam waist of the beam with different

*m*on either side of the geometric focus can be seen, indicating that the vector beam reaches a minimum waist just at the lens’ geometric focus. As displayed in Fig. 5, the change of the beam waist near focus weakens gradually as the value of

*m*increases. When the beam goes far away from focus, the beam waists of vector beams with different topological charges are nearly identical. It is noteworthy that the focal shift is negligible because the Fresnel number of our system is very large. Actually, the focal shift would be significant for the system with a narrow beam, long wavelength, and long lens focal length [21

**4**, 411–419 (1999). [CrossRef] [PubMed]

*x*–

*y*plane) by taking

*λ*= 532 nm,

*E*

_{0}= 1 (a.u.),

*ω*

_{0}= 2.5 mm,

*f*= 8 mm, and

*z*=

*f*. It is found that the paraxial energy flux distributions have the cylindrical symmetry and dark center, which are similar to the intensity distributions (see Fig. 4). Figure 6(a) shows the normalized cross-section energy flux profiles of the vector beams with

*φ*

_{0}= 0 for

*m*= 1 (solid line), 3 (dashed line), and 5 (dotted line). As illustrated in Fig. 6(a), the ring of the energy flux distributions at the focal plane becomes larger and thicker when the topological charge of the vector beam increases. Moreover, the size of dark center without energy flux is larger with increasing the topological charge. The effect of the parameter

*φ*

_{0}on the paraxial energy flux distribution is analyzed, as illustrated in Fig. 6(b). From Fig. 6(b), one finds that the thickness of the energy flux ring becomes narrower when the initial phase of the vector beam changes from

*φ*

_{0}= 0 to

*π*/2, indicating that the vector beam with the azimuthal polarization concentrates much more energy flux than that of the radial polarization.

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A |

2. | J. S. Ahn, H. W. Kihm, J. E. Kihm, D. S. Kim, and K. G. Lee, “3-dimensional local field polarization vector mapping of a focused radially polarized beam using gold nanoparticle functionalized tips,” Opt. Express |

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

4. | 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. |

5. | V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A |

6. | X. Jia and Y. Wang “Vectorial structure of far field of cylindrically polarized beams diffracted at a circular aperture,” Opt. Lett. |

7. | D. M. Deng, Q. Guo, S. Lan, and X. B. Yang, “Application of the multiscale singular perturbation method to nonparaxial beam propagation in free space,” J. Opt. Soc. Am. A |

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

9. | R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A |

10. | Z. Mei and D. Zhao, “Nonparaxial propagation of controllable dark-hollow beams,” J. Opt. Soc. Am. A |

11. | D. Deng and Q. Guo, “Exact nonparaxial propagation of a hollow Gaussian beam,” J. Opt. Soc. Am. B |

12. | Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express |

13. | G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B |

14. | X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B |

15. | B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett. |

16. | S. Guha and G. D. Gillen, “Description of light propagation through a circular aperture using nonparaxial vector diffraction theory,” Opt. Express |

17. | X. Jia, Y. Wang, and B. Li, “Nonparaxial analyses of radially polarized beams diffracted at a circular aperture,” Opt. Express |

18. | K. Duan and B. Lü, “Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture,” J. Opt. Soc. Am. A |

19. | L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. |

20. | C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. |

21. | P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express |

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

23. | D. M. Deng, Q. Guo, L. J. Wu, and X. B. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B |

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

25. | D. M. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B |

26. | P. P. Banerjee, G. Cook, and D. R. Evans, “A |

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

28. | V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Lensless focusing of hypergeometric laser beams,” J. Opt. |

29. | B. Gu, F. Ye, K. Lou, Y. Li, J. Chen, and H. T. Wang, “Vectorial self-diffraction effect in optically Kerr medium,” Opt. Express |

30. | A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. |

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(260.5430) Physical optics : Polarization

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: May 9, 2012

Revised Manuscript: July 1, 2012

Manuscript Accepted: July 1, 2012

Published: July 19, 2012

**Citation**

Bing Gu and Yiping Cui, "Nonparaxial and paraxial focusing of azimuthal-variant vector beams," Opt. Express **20**, 17684-17694 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-17684

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### References

- A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A15, 2705–2711 (1998). [CrossRef]
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