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

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  1. J. Yin, W. Wang, and Y. Zhu,"Generation of dark hollow beams and their applications," in Progress in Optics, E. Wolf, ed., (North-Holland, 2003) 44. 119-204
  2. Z. Wang, M. Dai, and J. 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, M. S. Soskin, "Laser beams with screw dislocations in their wavefronts, " Pis,maZh. 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. M. Yan, J. Yin, and Y. Zhu, "Dark-hollow-beam guiding and splitting of a low-velocity atomic beam," J. Opt. Soc. Am B. 17, 1817 (2000).
    [CrossRef]
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    [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. Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. 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. Y. Cai, X. Lu, and Q. Lin, "Hollow Gaussian beams and their propagation properties," Opt. Lett. 28, 1084-1086 (2003).
    [CrossRef] [PubMed]
  13. Y. Cai and S. He, "Propagation of hollow Gaussian beams through apertured paraxial optical systems," J. Opt. Soc. Am. A 23, 1410-1418 (2006).
    [CrossRef]
  14. Y. Cai and S. He, "Propagation of various dark hollow beams in a turbulent atmosphere," Opt. Express 14, 1353-1367 (2006).
    [CrossRef] [PubMed]
  15. D. Deng, H. Yu, S. Xu, G. Tian, Z. Fan, "Nonparaxial propagation of vectorial hollow Gaussian beams," J. Opt. Soc. Am B. 25, 83-87 (2008).
    [CrossRef]
  16. C. Zhao, L. Wang, X. Lu, "Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams," Phys. Lett. A. 363, 502-506 (2007).
    [CrossRef]
  17. R. Martınez-Herrero, P. M. Mejıas, S. Bosch and A. Carnicer, "Vectorical structure of nonparaxial electromagenetic beams," J. Opt. Soc. Am A. 18, 1678-1680 (2001).
    [CrossRef]
  18. H. Guo, J. Chen, and S. Zhuang, "Vector plane wave spectrum of an arbitrary polarized electromagnetic wave," Opt. Express 14, 2095-2100 (2006).
    [CrossRef] [PubMed]
  19. G. Zhou, "Analytical vectorial structure of Laguerre-Gaussian beam in the far field," Opt. Lett. 31, 2616- 2618 (2006).
    [CrossRef] [PubMed]
  20. G. Zhou, Y. Ni, and Z. Zhang, "Analytical vectorial structure of non-paraxial nonsymmetrical vector Gaussian beam in the far field," Opt. Commun. 272, 32-39 (2007).
    [CrossRef]
  21. D. Deng and Q. 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).
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    [CrossRef]
  24. J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol and Boston, 1986).

2008

D. Deng, H. Yu, S. Xu, G. Tian, Z. Fan, "Nonparaxial propagation of vectorial hollow Gaussian beams," J. Opt. Soc. Am B. 25, 83-87 (2008).
[CrossRef]

2007

C. Zhao, L. Wang, X. Lu, "Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams," Phys. Lett. A. 363, 502-506 (2007).
[CrossRef]

G. Zhou, Y. Ni, and Z. Zhang, "Analytical vectorial structure of non-paraxial nonsymmetrical vector Gaussian beam in the far field," Opt. Commun. 272, 32-39 (2007).
[CrossRef]

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, "Generation of hollow Gaussian beams by spatial filtering," Opt. Lett. 32, 2076-2078 (2007).
[CrossRef] [PubMed]

D. Deng and Q. Guo, "Analytical vectorial structure of radially polarized light beams," Opt. Lett. 32, 2711-2713 (2007).
[CrossRef] [PubMed]

2006

2005

2003

2001

R. Martınez-Herrero, P. M. Mejıas, S. Bosch and A. Carnicer, "Vectorical structure of nonparaxial electromagenetic beams," J. Opt. Soc. Am A. 18, 1678-1680 (2001).
[CrossRef]

2000

M. Yan, J. Yin, and Y. 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

1998

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

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

1990

V. Yu. Bazhenov, M. V. Vasnetsov, M. S. Soskin, "Laser beams with screw dislocations in their wavefronts, " Pis,maZh. Eksp. Teor. Fiz. 52, 1037-1039 (1990) [JETP Lett. 52, 429-431 (1990)].

1972

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, M. S. Soskin, "Laser beams with screw dislocations in their wavefronts, " Pis,maZh. Eksp. Teor. Fiz. 52, 1037-1039 (1990) [JETP Lett. 52, 429-431 (1990)].

Bosch, S.

R. Martınez-Herrero, P. M. Mejıas, S. Bosch and A. Carnicer, "Vectorical structure of nonparaxial electromagenetic beams," J. Opt. Soc. Am A. 18, 1678-1680 (2001).
[CrossRef]

Cai, Y.

Carnicer, A.

R. Martınez-Herrero, P. M. Mejıas, S. Bosch and A. Carnicer, "Vectorical structure of nonparaxial electromagenetic beams," J. Opt. Soc. Am A. 18, 1678-1680 (2001).
[CrossRef]

Carter, W. H.

Chen, J.

Dai, M.

Deng, D.

D. Deng, H. Yu, S. Xu, G. Tian, Z. Fan, "Nonparaxial propagation of vectorial hollow Gaussian beams," J. Opt. Soc. Am B. 25, 83-87 (2008).
[CrossRef]

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, Z.

D. Deng, H. Yu, S. Xu, G. Tian, Z. 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, H.

Haist, T.

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, Q.

Liu, Z.

Lu, X.

C. Zhao, L. Wang, X. Lu, "Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams," Phys. Lett. A. 363, 502-506 (2007).
[CrossRef]

Y. Cai, X. Lu, and Q. Lin, "Hollow Gaussian beams and their propagation properties," Opt. Lett. 28, 1084-1086 (2003).
[CrossRef] [PubMed]

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]

Mart??inez-Herrero, R.

R. Martınez-Herrero, P. M. Mejıas, S. Bosch and A. Carnicer, "Vectorical structure of nonparaxial electromagenetic beams," J. Opt. Soc. Am A. 18, 1678-1680 (2001).
[CrossRef]

Mej??ias, P. M.

R. Martınez-Herrero, P. M. Mejıas, S. Bosch and A. Carnicer, "Vectorical structure of nonparaxial electromagenetic beams," J. Opt. Soc. Am A. 18, 1678-1680 (2001).
[CrossRef]

Ni, Y.

G. Zhou, Y. Ni, and Z. 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, M. S. Soskin, "Laser beams with screw dislocations in their wavefronts, " Pis,maZh. Eksp. Teor. Fiz. 52, 1037-1039 (1990) [JETP Lett. 52, 429-431 (1990)].

Swartzlander, 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, G.

D. Deng, H. Yu, S. Xu, G. Tian, Z. 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, M. S. Soskin, "Laser beams with screw dislocations in their wavefronts, " Pis,maZh. Eksp. Teor. Fiz. 52, 1037-1039 (1990) [JETP Lett. 52, 429-431 (1990)].

Wagemann, E. U.

Wang, L..

C. Zhao, L. Wang, X. Lu, "Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams," Phys. Lett. A. 363, 502-506 (2007).
[CrossRef]

Wang, Z.

Xu, S.

D. Deng, H. Yu, S. Xu, G. Tian, Z. Fan, "Nonparaxial propagation of vectorial hollow Gaussian beams," J. Opt. Soc. Am B. 25, 83-87 (2008).
[CrossRef]

Yan, M.

M. Yan, J. Yin, and Y. Zhu, "Dark-hollow-beam guiding and splitting of a low-velocity atomic beam," J. Opt. Soc. Am B. 17, 1817 (2000).
[CrossRef]

Yin, J.

Z. Wang, M. Dai, and J. Yin, "Atomic (or molecular) guiding using a blue-detuned doughnut mode in a hollow metallic waveguide," Opt. Express 13, 8406-8423 (2005).
[CrossRef] [PubMed]

M. Yan, J. Yin, and Y. Zhu, "Dark-hollow-beam guiding and splitting of a low-velocity atomic beam," J. Opt. Soc. Am B. 17, 1817 (2000).
[CrossRef]

Yu, H.

D. Deng, H. Yu, S. Xu, G. Tian, Z. Fan, "Nonparaxial propagation of vectorial hollow Gaussian beams," J. Opt. Soc. Am B. 25, 83-87 (2008).
[CrossRef]

Zhang, Z.

G. Zhou, Y. Ni, and Z. Zhang, "Analytical vectorial structure of non-paraxial nonsymmetrical vector Gaussian beam in the far field," Opt. Commun. 272, 32-39 (2007).
[CrossRef]

Zhao, C.

C. Zhao, L. Wang, X. Lu, "Radiation forces on a dielectric sphere produced by highly focused hollow Gaussian beams," Phys. Lett. A. 363, 502-506 (2007).
[CrossRef]

Zhou, G.

G. Zhou, Y. Ni, and Z. Zhang, "Analytical vectorial structure of non-paraxial nonsymmetrical vector Gaussian beam in the far field," Opt. Commun. 272, 32-39 (2007).
[CrossRef]

G. Zhou, "Analytical vectorial structure of Laguerre-Gaussian beam in the far field," Opt. Lett. 31, 2616- 2618 (2006).
[CrossRef] [PubMed]

Zhu, Y.

M. Yan, J. Yin, and Y. Zhu, "Dark-hollow-beam guiding and splitting of a low-velocity atomic beam," J. Opt. Soc. Am B. 17, 1817 (2000).
[CrossRef]

Zhuang, S.

J. Opt. Soc. Am A.

R. Martınez-Herrero, P. M. Mejıas, S. Bosch and A. Carnicer, "Vectorical structure of nonparaxial electromagenetic beams," J. Opt. Soc. Am A. 18, 1678-1680 (2001).
[CrossRef]

J. Opt. Soc. Am B

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.

D. Deng, H. Yu, S. Xu, G. Tian, Z. Fan, "Nonparaxial propagation of vectorial hollow Gaussian beams," J. Opt. Soc. Am B. 25, 83-87 (2008).
[CrossRef]

M. Yan, J. Yin, and Y. Zhu, "Dark-hollow-beam guiding and splitting of a low-velocity atomic beam," J. Opt. Soc. Am B. 17, 1817 (2000).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

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

G. Zhou, Y. Ni, and Z. 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

Opt. Lett.

Phys. Lett. A.

C. Zhao, L. Wang, X. 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.

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]

Zh. Eksp. Teor. Fiz.

V. Yu. Bazhenov, M. V. Vasnetsov, M. S. Soskin, "Laser beams with screw dislocations in their wavefronts, " Pis,maZh. Eksp. Teor. Fiz. 52, 1037-1039 (1990) [JETP Lett. 52, 429-431 (1990)].

Other

J. Yin, W. Wang, and Y. Zhu,"Generation of dark hollow beams and their applications," in Progress in Optics, E. Wolf, ed., (North-Holland, 2003) 44. 119-204

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).

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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 .

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