Abstract

The analytical vectorial structure of HGB is investigated in the far field based on the vector plane wave spectrum and the method of stationary phase. The energy flux distributions of HGB in the far-field, which is composed of TE term and TM term, are demonstrated. The physics pictures of HGB is illustrated from the vectorial structure, which is important to understand the theoretical aspects of both scalar and vector HGB propagation.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Jianping Yin, Wenbao Wang, and Yifu Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed., North-Holland, 44, 119–204, (2003).
  2. Zhengling Wang, Meng Dai, and Jianping Yin, “Atomic (or molecular) guiding using a blue-detuned doughnut mode in a hollow metallic waveguide,” Opt. Express 13, 8406–8423, (2005).
    [Crossref] [PubMed]
  3. V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [JETP Lett. 52, 429–431 (1990).
  4. M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610, (1999).
    [Crossref]
  5. Min Yan, Jianping Yin, and Yifu Zhu, “Dark-hollow-beam guiding and splitting of a low-velocity atomic beam,” J. Opt. Soc. Am B. 17, 1817 (2000).
    [Crossref]
  6. Xiao Wang and Michael G. Littman, “Laser cavity for generation of variable-radius rings of light,” Opt. Lett. 18, 767–768, (1993).
    [Crossref] [PubMed]
  7. I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70, (1998).
    [Crossref]
  8. Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr. “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am B 15, 2226–2234, (1998).
    [Crossref]
  9. Zhengjun Liu, Haifa Zhao, Jianlong Liu, Jie Lin, Muhammad Ashfaq Ahmad, and Shutian Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. 32, 2076–2078, (2007).
    [Crossref] [PubMed]
  10. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasa, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.,  78, 4713–4716, (1997).
    [Crossref]
  11. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
    [Crossref]
  12. Yangjian Cai, Xuanhui Lu, and Qiang Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084–1086, (2003).
    [Crossref] [PubMed]
  13. Yangjian Cai and Sailing He, “Propagation of hollow Gaussian beams through apertured paraxial optical systems,” J. Opt. Soc. Am. A 23, 1410–1418, (2006).
    [Crossref]
  14. Yangjian Cai and Sailing He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353–1367, (2006).
    [Crossref] [PubMed]
  15. Degang Deng, Hua Yu, Shiqing Xu, Guanglei Tian, and Zhengxiu Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am B. 25, 83–87, (2008).
    [Crossref]
  16. Chengliang Zhao, Ligang Wang, and Xuanhui Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A. 363, 502–506, (2007).
    [Crossref]
  17. Rosario Martínez-Herrero, Pedro M. Mejías, Salvador Bosch, and Arturo Carnicer, “Vectorical structure of nonparaxial electromagenetic beams,” J. Opt. Soc. Am A. 18, 1678–1680, (2001).
    [Crossref]
  18. Hanming Guo, Jiabi Chen, and Songlin Zhuang, “Vector plane wave spectrum of an arbitrary polarized electromagnetic wave,” Opt. Express 14, 2095–2100, (2006).
    [Crossref] [PubMed]
  19. Guoquan Zhou, “Analytical vectorial structure of Laguerre-Gaussian beam in the far field,” Opt. Lett. 31, 2616–2618, (2006).
    [Crossref] [PubMed]
  20. Guoquan Zhou, Yongzhou Ni, and Zhongwei Zhang, “Analytical vectorial structure of non-paraxial nonsymmetrical vector Gaussian beam in the far field,” Opt. Commun. 272, 32–39, (2007).
    [Crossref]
  21. Dongmei Deng and Qi Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. 32, 2711–2713, (2007).
    [Crossref] [PubMed]
  22. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  23. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201, (1972).
    [Crossref]
  24. J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol and Boston, 1986).

2008 (1)

Degang Deng, Hua Yu, Shiqing Xu, Guanglei Tian, and Zhengxiu Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am B. 25, 83–87, (2008).
[Crossref]

2007 (4)

Chengliang Zhao, Ligang Wang, and Xuanhui Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A. 363, 502–506, (2007).
[Crossref]

Guoquan Zhou, Yongzhou Ni, and Zhongwei Zhang, “Analytical vectorial structure of non-paraxial nonsymmetrical vector Gaussian beam in the far field,” Opt. Commun. 272, 32–39, (2007).
[Crossref]

Zhengjun Liu, Haifa Zhao, Jianlong Liu, Jie Lin, Muhammad Ashfaq Ahmad, and Shutian Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. 32, 2076–2078, (2007).
[Crossref] [PubMed]

Dongmei Deng and Qi Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. 32, 2711–2713, (2007).
[Crossref] [PubMed]

2006 (4)

2005 (1)

2003 (1)

2001 (1)

Rosario Martínez-Herrero, Pedro M. Mejías, Salvador Bosch, and Arturo Carnicer, “Vectorical structure of nonparaxial electromagenetic beams,” J. Opt. Soc. Am A. 18, 1678–1680, (2001).
[Crossref]

2000 (2)

Min Yan, Jianping Yin, and Yifu Zhu, “Dark-hollow-beam guiding and splitting of a low-velocity atomic beam,” J. Opt. Soc. Am B. 17, 1817 (2000).
[Crossref]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[Crossref]

1999 (1)

1998 (2)

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70, (1998).
[Crossref]

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr. “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am B 15, 2226–2234, (1998).
[Crossref]

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasa, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.,  78, 4713–4716, (1997).
[Crossref]

1993 (1)

1990 (1)

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [JETP Lett. 52, 429–431 (1990).

1972 (1)

Ahmad, Muhammad Ashfaq

Arlt, J.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[Crossref]

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [JETP Lett. 52, 429–431 (1990).

Bosch, Salvador

Rosario Martínez-Herrero, Pedro M. Mejías, Salvador Bosch, and Arturo Carnicer, “Vectorical structure of nonparaxial electromagenetic beams,” J. Opt. Soc. Am A. 18, 1678–1680, (2001).
[Crossref]

Cai, Yangjian

Carnicer, Arturo

Rosario Martínez-Herrero, Pedro M. Mejías, Salvador Bosch, and Arturo Carnicer, “Vectorical structure of nonparaxial electromagenetic beams,” J. Opt. Soc. Am A. 18, 1678–1680, (2001).
[Crossref]

Carter, W. H.

Chen, Jiabi

Dai, Meng

Deng, Degang

Degang Deng, Hua Yu, Shiqing Xu, Guanglei Tian, and Zhengxiu Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am B. 25, 83–87, (2008).
[Crossref]

Deng, Dongmei

Dholakia, K.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[Crossref]

Fan, Zhengxiu

Degang Deng, Hua Yu, Shiqing Xu, Guanglei Tian, and Zhengxiu Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am B. 25, 83–87, (2008).
[Crossref]

Grimm, R.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70, (1998).
[Crossref]

Guo, Hanming

Guo, Qi

Haist, T.

He, Sailing

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasa, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.,  78, 4713–4716, (1997).
[Crossref]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasa, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.,  78, 4713–4716, (1997).
[Crossref]

Lin, Jie

Lin, Qiang

Littman, Michael G.

Liu, Jianlong

Liu, Shutian

Liu, Zhengjun

Lu, Xuanhui

Chengliang Zhao, Ligang Wang, and Xuanhui Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A. 363, 502–506, (2007).
[Crossref]

Yangjian Cai, Xuanhui Lu, and Qiang Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084–1086, (2003).
[Crossref] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Manek, I.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70, (1998).
[Crossref]

Martínez-Herrero, Rosario

Rosario Martínez-Herrero, Pedro M. Mejías, Salvador Bosch, and Arturo Carnicer, “Vectorical structure of nonparaxial electromagenetic beams,” J. Opt. Soc. Am A. 18, 1678–1680, (2001).
[Crossref]

Mejías, Pedro M.

Rosario Martínez-Herrero, Pedro M. Mejías, Salvador Bosch, and Arturo Carnicer, “Vectorical structure of nonparaxial electromagenetic beams,” J. Opt. Soc. Am A. 18, 1678–1680, (2001).
[Crossref]

Ni, Yongzhou

Guoquan Zhou, Yongzhou Ni, and Zhongwei Zhang, “Analytical vectorial structure of non-paraxial nonsymmetrical vector Gaussian beam in the far field,” Opt. Commun. 272, 32–39, (2007).
[Crossref]

Ovchinnikov, Y. B.

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70, (1998).
[Crossref]

Reicherter, M.

Rozas, D.

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr. “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am B 15, 2226–2234, (1998).
[Crossref]

Sacks, Z. S.

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr. “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am B 15, 2226–2234, (1998).
[Crossref]

Sasa, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasa, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.,  78, 4713–4716, (1997).
[Crossref]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasa, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.,  78, 4713–4716, (1997).
[Crossref]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasa, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.,  78, 4713–4716, (1997).
[Crossref]

Soskin, M. S.

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [JETP Lett. 52, 429–431 (1990).

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol and Boston, 1986).

Swartzlander, Jr., G. A.

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr. “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am B 15, 2226–2234, (1998).
[Crossref]

Tian, Guanglei

Degang Deng, Hua Yu, Shiqing Xu, Guanglei Tian, and Zhengxiu Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am B. 25, 83–87, (2008).
[Crossref]

Tiziani, H. J.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasa, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.,  78, 4713–4716, (1997).
[Crossref]

Vasnetsov, M. V.

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [JETP Lett. 52, 429–431 (1990).

Wagemann, E. U.

Wang, Ligang

Chengliang Zhao, Ligang Wang, and Xuanhui Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A. 363, 502–506, (2007).
[Crossref]

Wang, Wenbao

Jianping Yin, Wenbao Wang, and Yifu Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed., North-Holland, 44, 119–204, (2003).

Wang, Xiao

Wang, Zhengling

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Xu, Shiqing

Degang Deng, Hua Yu, Shiqing Xu, Guanglei Tian, and Zhengxiu Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am B. 25, 83–87, (2008).
[Crossref]

Yan, Min

Min Yan, Jianping Yin, and Yifu Zhu, “Dark-hollow-beam guiding and splitting of a low-velocity atomic beam,” J. Opt. Soc. Am B. 17, 1817 (2000).
[Crossref]

Yin, Jianping

Zhengling Wang, Meng Dai, and Jianping Yin, “Atomic (or molecular) guiding using a blue-detuned doughnut mode in a hollow metallic waveguide,” Opt. Express 13, 8406–8423, (2005).
[Crossref] [PubMed]

Min Yan, Jianping Yin, and Yifu Zhu, “Dark-hollow-beam guiding and splitting of a low-velocity atomic beam,” J. Opt. Soc. Am B. 17, 1817 (2000).
[Crossref]

Jianping Yin, Wenbao Wang, and Yifu Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed., North-Holland, 44, 119–204, (2003).

Yu, Hua

Degang Deng, Hua Yu, Shiqing Xu, Guanglei Tian, and Zhengxiu Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am B. 25, 83–87, (2008).
[Crossref]

Zhang, Zhongwei

Guoquan Zhou, Yongzhou Ni, and Zhongwei Zhang, “Analytical vectorial structure of non-paraxial nonsymmetrical vector Gaussian beam in the far field,” Opt. Commun. 272, 32–39, (2007).
[Crossref]

Zhao, Chengliang

Chengliang Zhao, Ligang Wang, and Xuanhui Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A. 363, 502–506, (2007).
[Crossref]

Zhao, Haifa

Zhou, Guoquan

Guoquan Zhou, Yongzhou Ni, and Zhongwei Zhang, “Analytical vectorial structure of non-paraxial nonsymmetrical vector Gaussian beam in the far field,” Opt. Commun. 272, 32–39, (2007).
[Crossref]

Guoquan Zhou, “Analytical vectorial structure of Laguerre-Gaussian beam in the far field,” Opt. Lett. 31, 2616–2618, (2006).
[Crossref] [PubMed]

Zhu, Yifu

Min Yan, Jianping Yin, and Yifu Zhu, “Dark-hollow-beam guiding and splitting of a low-velocity atomic beam,” J. Opt. Soc. Am B. 17, 1817 (2000).
[Crossref]

Jianping Yin, Wenbao Wang, and Yifu Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed., North-Holland, 44, 119–204, (2003).

Zhuang, Songlin

J. Opt. Soc. Am A. (1)

Rosario Martínez-Herrero, Pedro M. Mejías, Salvador Bosch, and Arturo Carnicer, “Vectorical structure of nonparaxial electromagenetic beams,” J. Opt. Soc. Am A. 18, 1678–1680, (2001).
[Crossref]

J. Opt. Soc. Am B (1)

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr. “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am B 15, 2226–2234, (1998).
[Crossref]

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

Min Yan, Jianping Yin, and Yifu Zhu, “Dark-hollow-beam guiding and splitting of a low-velocity atomic beam,” J. Opt. Soc. Am B. 17, 1817 (2000).
[Crossref]

Degang Deng, Hua Yu, Shiqing Xu, Guanglei Tian, and Zhengxiu Fan, “Nonparaxial propagation of vectorial hollow Gaussian beams,” J. Opt. Soc. Am B. 25, 83–87, (2008).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (3)

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[Crossref]

Guoquan Zhou, Yongzhou Ni, and Zhongwei Zhang, “Analytical vectorial structure of non-paraxial nonsymmetrical vector Gaussian beam in the far field,” Opt. Commun. 272, 32–39, (2007).
[Crossref]

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70, (1998).
[Crossref]

Opt. Express (3)

Opt. Lett. (6)

Phys. Lett. A. (1)

Chengliang Zhao, Ligang Wang, and Xuanhui Lu, “Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams,” Phys. Lett. A. 363, 502–506, (2007).
[Crossref]

Phys. Rev. Lett. (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasa, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett.,  78, 4713–4716, (1997).
[Crossref]

Pis’ma Zh. Eksp. Teor. Fiz. (1)

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [JETP Lett. 52, 429–431 (1990).

Other (3)

Jianping Yin, Wenbao Wang, and Yifu Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed., North-Holland, 44, 119–204, (2003).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol and Boston, 1986).

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 (3)

Fig. 1.
Fig. 1.

Normalized intensity distribution of hollow Gaussian beams as a function of ρ/w 0 with different beam order n at z=0 plane based on Eq. (7).

Fig. 2.
Fig. 2.

Normalized energy flux distribution of HGB at the plane z=600λ for beam order n=4. (a) TE term, (b) TM term, (c) whole beam.

Fig. 3.
Fig. 3.

Normalized energy flux distribution of HGB at the plane z=600λ for beam order n=10. (a) TE term, (b) TM term, (c) whole beam.

Equations (39)

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

E x ( x , y , z ) = + A x ( p , q ) exp [ i k ( px + qy + γ z ) ] d p d q ,
E y ( x , y , z ) = + A y ( p , q ) exp [ i k ( px + qy + γ z ) ] d p d q ,
E z ( x , y , z ) = + [ p γ A x ( p , q ) + q γ A y ( p , q ) ] exp [ i k ( px + qy + mz ) ] d p d q ,
γ = { ( 1 p 2 q 2 ) 1 2 , if p 2 + q 2 1 i ( p 2 + q 2 1 ) 1 2 , if p 2 + q 2 > 1 .
A x ( p , q ) = 1 λ 2 + E x ( x , y , 0 ) exp [ i k ( p x + q y ) ] d x d y ,
A y ( p , q ) = 1 λ 2 + E y ( x , y , 0 ) exp [ i k ( p x + q y ) ] d x d y .
E x ( x , y , 0 ) = G 0 [ ρ 2 w 0 2 ] n exp [ ρ 2 w 0 2 ] ,
E y ( x , y , 0 ) = 0 ,
A x ( p , q ) = G 0 n ! π f 2 2 n + 2 m = 0 n ( n m ) L m [ p 2 + q 2 2 f 2 ] exp [ p 2 + q 2 4 f 2 ] ,
A y ( p , q ) = 0 ,
E ( r ) = E TE ( r ) + E TM ( r ) ,
E TE ( r ) = + 1 p 2 + q 2 [ q A x ( p , q ) p A y ( p , q ) ] ( q e ̂ x p e ̂ y ) exp [ i k u ] d p d q ,
E TM ( r ) = + 1 p 2 + q 2 [ q A x ( p , q ) + p A y ( p , q ) ] ( q e ̂ x + p e ̂ y b 2 γ e ̂ z ) exp [ i k u ] d p d q ,
H ( r ) = H TE ( r ) + H TM ( r ) ,
H TE ( r ) = ε μ + 1 p 2 + q 2 [ q A x ( p , q ) p A y ( p , q ) ]
× [ p γ e ̂ x + q γ e ̂ y ( p 2 + q 2 ) e ̂ z ] exp ( i k u ) d p d q ,
H T M ( r ) = ε μ + [ p A x ( p , q ) + q A y ( p , q ) ] 1 b 2 γ ( q e ̂ x p e ̂ y ) exp ( i k u ) d p d q .
E T E ( r ) = G 0 n ! 2 n m = 0 n ( n m ) i Z R y z r 2 ρ 2 L m [ ρ 2 2 f 2 r 2 ] exp [ ρ 2 4 f 2 r 2 + i k r ] ( y e ̂ x x e ̂ y ) ,
H T E ( r ) = G 0 ε μ n ! 2 n m = 0 n ( n m ) i Z R y z r 3 ρ 2 L m [ ρ 2 2 f 2 r 2 ]
× exp [ ρ 2 4 f 2 r 2 + i k r ] ( x z e ̂ x + y z e ̂ y ρ 2 e ̂ z ) ,
E TM ( r ) = G 0 n ! 2 n m = 0 n ( n m ) i Z R x r 2 ρ 2 L m [ ρ 2 2 f 2 r 2 ]
× exp [ ρ 2 4 f 2 r 2 + i k r ] ( x z e ̂ x + y z e ̂ y ρ 2 e ̂ z ) ,
H TM ( r ) = G 0 ε μ n ! 2 n m = 0 n ( n m ) i Z R x r ρ 2 L m [ ρ 2 2 f 2 r 2 ] exp [ ρ 2 4 f 2 r 2 + i k r ] ( y e ̂ x x e ̂ y ) .
S z = 1 2 Re [ E * ( r ) × H ( r ) ] z
= S z TE + S z TM ,
S z TE = G 0 2 ε μ ( n ! ) 2 2 2 n + 1 Z R 2 y 2 z 3 r 5 ρ 2
× m = 0 n b = 0 n ( n m ) ( n b ) L m [ ρ 2 2 f 2 r 2 ] L b [ ρ 2 2 f 2 r 2 ] exp [ ρ 2 2 f 2 r 2 ] .
S z TM = G 0 2 ε μ ( n ! ) 2 2 2 n + 1 Z R 2 x 2 z r 3 ρ 2
× m = 0 n b = 0 n ( n m ) ( n b ) L m [ ρ 2 2 f 2 r 2 ] L b [ ρ 2 2 f 2 r 2 ] exp [ ρ 2 2 f 2 r 2 ] .
S z = G 0 2 ε μ ( n ! ) 2 2 2 n + 1 Z R 2 z r 3 ρ 2 ( y 2 z 2 r 2 + x 2 )
× m = 0 n b = 0 n ( n m ) ( n b ) L m [ ρ 2 2 f 2 r 2 ] L b [ ρ 2 2 f 2 r 2 ] exp [ ρ 2 2 f 2 r 2 ] .
I ( k ) = S f ( x , y ) e ikg ( x , y ) dxdy ,
g ( x , y ) x x 1 , y 1 = g ( x , y ) y x 1 , y 1 = 0 ,
g l = 0 .
I = 2 π i ε k α 1 β 1 γ 1 2 f ( x 1 , y 1 ) e ikg ( x 1 , y 1 ) as k ,
α 1 = 2 g ( x , y ) x 2 x 1 , y 1 , β 1 = 2 g ( x , y ) y 2 x 1 , y 1 , γ 1 = 2 g ( x , y ) x y x 1 , y 1 .
ε = { 1 , if α 1 β 1 > γ 1 2 and α 1 > 0 , 1 , if α 1 β 1 > γ 1 2 and α 1 < 0 , i , if α 1 β 1 < γ 1 2 .
E ( x , y , z ) = + A ( p , q ) e ik ( px + qy + mz ) dpdq
= 2 π i z k r 2 A ( x r , y r ) e ikr as kr .

Metrics