Abstract

In this paper, we introduce a new kind of partially coherent vector beam with special correlation function and vortex phase named radially polarized Laguerre-Gaussian-correlated Schell-model (LGCSM) vortex beam as a natural extension of scalar LGCSM vortex beam. The realizability conditions for such beam are derived. The tight focusing properties of a radially polarized LGCSM vortex beam passing through a high numerical aperture (NA) objective lens are investigated numerically based on the vectorial diffraction theory. We find that not only the transverse component but also the longitudinal component of the focal field distributions can be shaped by regulating the structures of the correlation functions, which is quite different from that of the conventional radially polarized partially coherent beam. Moreover, a series of wildly used focal field with novel structure, e.g., focal spot, flat-topped or doughnut beam profiles, needle-like focal field and controllable three-dimensional (3D) optical cage, were obtained. These results indicate that the focus shaping can be achieved by combining the regulation of the structures of the correlation functions with the regulation of beam parameters effectively. Our results may be useful for potential applications in optical trapping, optical high-resolution microscopy and optical data storage.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam

Yahong Chen, Fei Wang, Chengliang Zhao, and Yangjian Cai
Opt. Express 22(5) 5826-5838 (2014)

Focus shaping of partially coherent radially polarized vortex beam with tunable topological charge

Hua-Feng Xu, Rui Zhang, Zong-Qiang Sheng, and Jun Qu
Opt. Express 27(17) 23959-23969 (2019)

Elliptical Laguerre-Gaussian correlated Schell-model beam

Yahong Chen, Lin Liu, Fei Wang, Chengliang Zhao, and Yangjian Cai
Opt. Express 22(11) 13975-13987 (2014)

References

  • View by:
  • |
  • |
  • |

  1. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  2. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
    [Crossref]
  3. L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
    [Crossref]
  4. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
    [Crossref] [PubMed]
  5. R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112(2), 247–259 (2013).
    [Crossref]
  6. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
    [Crossref] [PubMed]
  7. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010).
    [Crossref] [PubMed]
  8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
    [Crossref] [PubMed]
  9. Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
    [Crossref]
  10. L. Huang, H. Guo, J. Li, L. Ling, B. Feng, and Z. Y. Li, “Optical trapping of gold nanoparticles by cylindrical vector beam,” Opt. Lett. 37(10), 1694–1696 (2012).
    [Crossref] [PubMed]
  11. Y. Zhang and J. Bai, “Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams,” Opt. Express 17(5), 3698–3706 (2009).
    [Crossref] [PubMed]
  12. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
    [Crossref]
  13. K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
    [Crossref] [PubMed]
  14. F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
    [Crossref] [PubMed]
  15. Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
    [Crossref]
  16. L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
    [Crossref]
  17. Y. X. Ren, M. Li, K. Huang, J. G. Wu, H. F. Gao, Z. Q. Wang, and Y. M. Li, “Experimental generation of Laguerre-Gaussian beam using digital micromirror device,” Appl. Opt. 49(10), 1838–1844 (2010).
    [Crossref] [PubMed]
  18. Y. Ren, R. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527(7-8), 447–470 (2015).
    [Crossref]
  19. Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
    [Crossref]
  20. Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33(1), 49–51 (2008).
    [Crossref] [PubMed]
  21. Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
    [Crossref]
  22. L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
    [Crossref]
  23. L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
    [Crossref]
  24. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [Crossref] [PubMed]
  25. F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11(8), 085706 (2009).
  26. Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
    [Crossref] [PubMed]
  27. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [Crossref] [PubMed]
  28. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
    [Crossref] [PubMed]
  29. Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
    [Crossref]
  30. Z. Mei, “Two types of sinc Schell-model beams and their propagation characteristics,” Opt. Lett. 39(14), 4188–4191 (2014).
    [Crossref] [PubMed]
  31. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
    [Crossref] [PubMed]
  32. X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
    [Crossref]
  33. Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
    [Crossref] [PubMed]
  34. C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
    [Crossref]
  35. H. Xu, Z. Zhang, J. Qu, and W. Huang, “The tight focusing properties of Laguerre-Gaussian-correlated Schell-model beams,” J. Mod. Opt. 63(15), 1429–1437 (2016).
    [Crossref]
  36. Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
    [Crossref] [PubMed]
  37. G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
    [Crossref] [PubMed]
  38. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
    [Crossref]

2017 (2)

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

2016 (1)

H. Xu, Z. Zhang, J. Qu, and W. Huang, “The tight focusing properties of Laguerre-Gaussian-correlated Schell-model beams,” J. Mod. Opt. 63(15), 1429–1437 (2016).
[Crossref]

2015 (4)

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref] [PubMed]

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Y. Ren, R. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527(7-8), 447–470 (2015).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

2014 (6)

2013 (2)

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112(2), 247–259 (2013).
[Crossref]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

2012 (4)

2011 (3)

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[Crossref]

2010 (4)

2009 (4)

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

Y. Zhang and J. Bai, “Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams,” Opt. Express 17(5), 3698–3706 (2009).
[Crossref] [PubMed]

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11(8), 085706 (2009).

2008 (3)

2007 (1)

2005 (1)

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

2004 (1)

2002 (1)

Bai, J.

Baykal, Y.

Cai, Y.

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

Y. Chen and Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[Crossref] [PubMed]

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112(2), 247–259 (2013).
[Crossref]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
[Crossref] [PubMed]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[Crossref]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16(11), 7665–7673 (2008).
[Crossref] [PubMed]

Chen, B.

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

Chen, R.

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112(2), 247–259 (2013).
[Crossref]

Chen, Y.

Chen, Z.

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Chong, C.

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

Ding, B.

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Dong, Y.

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112(2), 247–259 (2013).
[Crossref]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[Crossref]

Eyyuboglu, H. T.

Fan, H.

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Feng, B.

Gao, H. F.

Gong, L.

Y. Ren, R. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527(7-8), 447–470 (2015).
[Crossref]

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

Gori, F.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11(8), 085706 (2009).

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

Gu, J.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Guo, H.

Guo, L.

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

Hong, M.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref] [PubMed]

Hua, L.

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

Huang, K.

Huang, L.

Huang, W.

H. Xu, Z. Zhang, J. Qu, and W. Huang, “The tight focusing properties of Laguerre-Gaussian-correlated Schell-model beams,” J. Mod. Opt. 63(15), 1429–1437 (2016).
[Crossref]

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Jiao, J.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref] [PubMed]

Kang, X. L.

Korotkova, O.

Kozawa, Y.

Leger, J.

Li, J.

Li, M.

Li, Y.

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

Li, Y. M.

Li, Y. P.

Li, Z. Y.

Liang, C.

Lin, Q.

Ling, L.

Liu, L.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

Liu, W.

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

Liu, X.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

Lu, R.

Y. Ren, R. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527(7-8), 447–470 (2015).
[Crossref]

Lukyanchuk, B.

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

Luo, X.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref] [PubMed]

Mei, Z.

Peng, X.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

Ping, C.

Pu, J.

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33(1), 49–51 (2008).
[Crossref] [PubMed]

Qin, F.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref] [PubMed]

Qiu, C.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref] [PubMed]

Qu, J.

H. Xu, Z. Zhang, J. Qu, and W. Huang, “The tight focusing properties of Laguerre-Gaussian-correlated Schell-model beams,” J. Mod. Opt. 63(15), 1429–1437 (2016).
[Crossref]

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Rao, L.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Ren, Y.

Y. Ren, R. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527(7-8), 447–470 (2015).
[Crossref]

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

Ren, Y. X.

Sahin, S.

Sanchez, V. R.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11(8), 085706 (2009).

Santarsiero, M.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11(8), 085706 (2009).

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

Sato, S.

Shchepakina, E.

Sheppard, C.

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

Shi, L.

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

Shi, P.

Shirai, T.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11(8), 085706 (2009).

Suyama, T.

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Tan, Z.

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

Tang, Z.

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

Wang, F.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited],” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[Crossref] [PubMed]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112(2), 247–259 (2013).
[Crossref]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
[Crossref] [PubMed]

Wang, H.

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

Wang, M.

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

Wang, X.

Wang, Z.

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

Wang, Z. Q.

Wolf, E.

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

Wu, G.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

G. Wu, F. Wang, and Y. Cai, “Coherence and polarization properties of a radially polarized beam with variable spatial coherence,” Opt. Express 20(27), 28301–28318 (2012).
[Crossref] [PubMed]

Wu, J.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref] [PubMed]

Wu, J. G.

Xu, H.

H. Xu, Z. Zhang, J. Qu, and W. Huang, “The tight focusing properties of Laguerre-Gaussian-correlated Schell-model beams,” J. Mod. Opt. 63(15), 1429–1437 (2016).
[Crossref]

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Yei, P.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Zhan, Q.

Zhang, X.

Zhang, Y.

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Y. Zhang and J. Bai, “Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams,” Opt. Express 17(5), 3698–3706 (2009).
[Crossref] [PubMed]

Zhang, Z.

H. Xu, Z. Zhang, J. Qu, and W. Huang, “The tight focusing properties of Laguerre-Gaussian-correlated Schell-model beams,” J. Mod. Opt. 63(15), 1429–1437 (2016).
[Crossref]

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33(1), 49–51 (2008).
[Crossref] [PubMed]

Zhao, C.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[Crossref]

Zhong, M.

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

Adv. Opt. Photonics (1)

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

Ann. Phys. (1)

Y. Ren, R. Lu, and L. Gong, “Tailoring light with a digital micromirror device,” Ann. Phys. 527(7-8), 447–470 (2015).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (2)

Y. Dong, Y. Cai, and C. Zhao, “Degree of polarization of a tightly focused partially coherent dark hollow beam,” Appl. Phys. B 105(2), 405–414 (2011).
[Crossref]

R. Chen, Y. Dong, F. Wang, and Y. Cai, “Statistical properties of a cylindrical vector partially coherent beam in turbulent atmosphere,” Appl. Phys. B 112(2), 247–259 (2013).
[Crossref]

Appl. Phys. Lett. (1)

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

J. Appl. Phys. (1)

L. Gong, Y. Ren, W. Liu, M. Wang, M. Zhong, Z. Wang, and Y. Li, “Generation of cylindrically polarized vector vortex beams with digital micromirror device,” J. Appl. Phys. 116(18), 183105 (2014).
[Crossref]

J. Mod. Opt. (1)

H. Xu, Z. Zhang, J. Qu, and W. Huang, “The tight focusing properties of Laguerre-Gaussian-correlated Schell-model beams,” J. Mod. Opt. 63(15), 1429–1437 (2016).
[Crossref]

J. Opt. (3)

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. 11(8), 085706 (2009).

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Photonics (1)

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

Opt. Commun. (1)

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005).
[Crossref]

Opt. Express (8)

Opt. Laser Technol. (2)

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Opt. Lett. (8)

Phys. Rev. A (3)

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010).
[Crossref]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

Sci. Rep. (1)

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref] [PubMed]

Other (1)

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1 Scheme of tight focusing of a light beam focused by a high NA objective lens.
Fig. 2
Fig. 2 Intensity distributions of the total intensity I total , transverse intensity I tra , and longitudinal intensity I z of a tightly focused radially polarized LGCSM vortex beam with coherence width δ 0 =1mmand topological charge l =1 in the focal plane for different values of mode order n. The green solid line denotes the corresponding cross line of the intensity distribution at ρ x ( ρ y )=0.
Fig. 3
Fig. 3 Intensity distributions of the total intensity I total , transverse intensity I tra , and longitudinal intensity I z of a tightly focused radially polarized LGCSM vortex beam with mode order n =3 and topological charge l =1 in the focal plane for different values of coherence width δ 0 . The green solid line denotes the corresponding cross line of the intensity distribution at ρ x ( ρ y )=0.
Fig. 4
Fig. 4 Intensity distributions of the total intensity I total , transverse intensity I tra , and longitudinal intensity I z of a tightly focused radially polarized LGCSM vortex beam with coherence width δ 0 =1mmand mode order n =3 in the focal plane for different values of topological charge l. The green solid line denotes the corresponding cross line of the intensity distribution at ρ x ( ρ y )=0.
Fig. 5
Fig. 5 Intensity distributions of the total intensity I total of a tightly focused radially polarized LGCSM vortex beam with coherence width δ 0 =1mm and topological charge l =1 in the ρ-z plane for different values of mode order n.
Fig. 6
Fig. 6 Intensity distributions of the total intensity I total of a tightly focused radially polarized LGCSM vortex beam with mode order n =3 and topological charge l =3 in the ρ-z plane for different values of coherence width δ 0 .
Fig. 7
Fig. 7 Intensity distributions of the total intensity I total of a tightly focused radially polarized LGCSM vortex beam with mode order n =3 and coherence width δ 0 =1mm in the ρ-z plane for different values of topological charge l.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

W αβ ( r 1 , r 2 ,ω,0 )= E α * ( r 1 ,ω,0 ) E β ( r 2 ,ω,0 ) , ( α,β=x,y )
W αβ ( 0 ) ( r 1 , r 2 )= p αβ ( v ) H α * ( r 1 ,v ) H β ( r 2 ,v ) d 2 v, ( α,β=x,y )
p ( v )=( p xx ( v ) p xy ( v ) p yx ( v ) p yy ( v ) )
p xx ( v )0, p yy ( v )0, p xx ( v ) p yy ( v ) | p xy ( v ) | 2 0.
H α ( r,v )= α w 0 exp( r 2 4 w 0 2 )exp( i rv ), ( α=x,y ),
p αβ ( v )= B αβ ( π 2n+1 δ 0αβ 2n+2 2 n+1 /n! ) v 2n exp( 2 π 2 δ 0αβ 2 v 2 ).
W αβ ( r 1 , r 2 )= α 1 β 2 w 0 2 exp[ r 1 2 + r 2 2 4 w 0 2 ] μ αβ ( r 1 r 2 ), ( α,β=x,y )
μ αβ ( r 1 r 2 )= B αβ exp[ ( r 1 r 2 ) 2 2 δ 0αβ 2 ] L n 0 [ ( r 1 r 2 ) 2 2 δ 0αβ 2 ]
B xx = B yy =1, B xy = B yx * , δ 0xy = δ 0yx
δ 0xx 2n+2 δ 0yy 2n+2 exp[ 2 π 2 ( δ 0xx 2 + δ 0yy 2 ) v 2 ] | B xy | 2 ( δ 0xy 2n+2 ) 2 exp( 4 π 2 δ 0xy 2 v 2 )
( δ 0xx 2 + δ 0yy 2 )/2 δ 0xy δ 0xy δ 0xy / | B xy | 1/( n+1 )
θ( r,0 )= 1 2 arctan[ 2Re[ W xy ( r,r,0 ) ] W xx ( r,r,0 ) W yy ( r,r,0 ) ],
ε( r,0 )= A ( r,0 )/ A + ( r,0 ), ( 0ε1 ).
A ± ( r,0 )= 1 2 { [ W xx ( r,r,0 ) W yy ( r,r,0 ) ] 2 +4 | W xy ( r,r,0 ) | 2 ± [ W xx ( r,r,0 ) W yy ( r,r,0 ) ] 2 +4 ( Re[ W xy ( r,r,0 ) ] ) 2 } .
Re[ B xy ]=1, | B xy |=1
B xx = B yy = B xy = B yx =1
δ 0xx = δ 0yy = δ 0xy = δ 0yx = δ 0
W αβ ( r 1 , r 2 )= α 1 β 2 w 0 2 exp[ r 1 2 + r 2 2 4 w 0 2 ( r 1 r 2 ) 2 2 δ 0 2 ] L n 0 [ ( r 1 r 2 ) 2 2 δ 0 2 ] ×exp[ il( φ 1 φ 2 ) ], ( α,β=x,y )
E f ( r,φ,z )=[ E fx E fy E fz ]= i k 1 f 2π 0 θ max 0 2π [ l x ( θ,ϕ )[ cosθ+ sin 2 ϕ( 1cosθ ) ]+ l y ( θ,ϕ )cosϕsinϕ( cosθ1 ) l x ( θ,ϕ )cosϕsinϕ( cosθ1 )+ l y ( θ,ϕ )[ cosθ+ sin 2 ϕ( 1cosθ ) ] l x ( θ,ϕ )cosϕsinθ l y ( θ,ϕ )sinϕsinθ ] × cosθ sinθexp[ i k 1 ( zcosθ+rsinθcos( ϕφ ) ) ]dϕdθ
W fαβ ( r 1 , φ 1 , r 2 , φ 2 ,z )= E fα * ( r 1 , φ 1 ,z ) E fβ ( r 2 , φ 2 ,z ) , ( α,β=x,y,z )
W fxx ( r 1 , φ 1 , r 2 , φ 2 ,z )= f 2 n 1 λ 2 0 θ max 0 θ max 0 2π 0 2π W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) ×exp[ i k 1 ( ζ 2 ζ 1 ) ] × ( cos θ 1 cos θ 2 ) 3/2 ( sin θ 1 sin θ 2 ) 2 cos ϕ 1 cos ϕ 2 d θ 1 d θ 2 d ϕ 1 d ϕ 2 ,
W fyy ( r 1 , φ 1 , r 2 , φ 2 ,z )= f 2 n 1 λ 2 0 θ max 0 θ max 0 2π 0 2π W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) ×exp[ i k 1 ( ζ 2 ζ 1 ) ] × ( cos θ 1 cos θ 2 ) 3/2 ( sin θ 1 sin θ 2 ) 2 sin ϕ 1 sin ϕ 2 d θ 1 d θ 2 d ϕ 1 d ϕ 2 ,
W fzz ( r 1 , φ 1 , r 2 , φ 2 ,z )= f 2 n 1 λ 2 0 θ max 0 θ max 0 2π 0 2π W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) ×exp[ i k 1 ( ζ 2 ζ 1 ) ] × ( cos θ 1 cos θ 2 ) 1/2 ( sin θ 1 sin θ 2 ) 3 d θ 1 d θ 2 d ϕ 1 d ϕ 2
ζ i =z cos θ i + r i sin θ i cos( ϕ i φ i ), ( i=1,2 )
W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 )= f 2 w 0 2 exp[ ( f 2 4 w 0 2 + f 2 2 δ 0 2 )( sin 2 θ 1 + sin 2 θ 2 )il( ϕ 1 ϕ 2 ) ] × L n 0 [ f 2 2 δ 0 2 ( sin 2 θ 1 + sin 2 θ 2 2sin θ 1 sin θ 2 cos( ϕ 1 ϕ 2 ) ) ]
I tra ( r,φ,z )= W fxx ( r,φ,r,φ,z )+ W fyy ( r,φ,r,φ,z ),
I z ( r,φ,z )= W fzz ( r,φ,r,φ,z ),
I total ( r,φ,z )= I tra ( r,φ,z )+ I z ( r,φ,z ),

Metrics