Abstract

The analytical far-field expressions for the TE and TM terms and energy flux distributions in two off-axis superimposed nonparaxial Laguerre–Gaussian beams with azimuthal and radial indices l1=l2=+1, p1=p2=0 are derived and used to study the far-field properties including phase singularities and energy flux distributions of the resulting beam, where our main attention focuses on the dependence of phase singularities on the controlling parameters such as the off-axis distance, relative phase, amplitude ratio, and waist widths of superimposed beams, and the symmetry property of edge dislocations and energy flux distributions. The results are interpreted and compared with previous work.

© 2010 Optical Society of America

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  1. E. F. Yelden, H. J. J. Seguin, C. E. Capjack, and S. K. Nikumb, “Multichannel slab discharge for CO2 laser excitation,” Appl. Phys. Lett. 58, 693-695 (1991).
    [CrossRef]
  2. H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400-407 (1996).
    [CrossRef]
  3. K. M. Abramski, A. D. Colley, H. J. Baker, and D. R. Hall, “High-power two-dimensional waveguide CO2 laser arrays,” IEEE J. Quantum Electron. 32, 340-349 (1996).
    [CrossRef]
  4. J. D. Strohschein, H. J. J. Seguin, and C. E. Capjack, “Beam propagation contrasts for a radial laser array,” Appl. Opt. 37, 1045-1048 (1998).
    [CrossRef]
  5. G. Hergenhan, B. Lücke, and U. Branch, “Coherent coupling of vertical-cavity surface-emitting laser arrays and efficient beam combining by diffractive optical elements: concept and experimental verification,” Appl. Opt. 42, 1667-1680 (2003).
    [CrossRef] [PubMed]
  6. R. Xiao, J. Zhou, M. Liu, and Z. F. Jiang, “Coherent combining technology of master oscillator power amplifier fiber arrays,” Opt. Express 16, 2015-2022 (2008).
    [CrossRef] [PubMed]
  7. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73-87 (1993).
    [CrossRef]
  8. V. Pyragaite and A. Stabinis, “Free-space propagation of overlapping light vortex beams,” Opt. Commun. 213, 187-191 (2002).
    [CrossRef]
  9. G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
    [CrossRef]
  10. I. D. Maleev and G. A. Swartzlander, Jr., “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169-1176 (2003).
    [CrossRef]
  11. K. Cheng and B. Lü, “Composite coherent vortices in coherent and incoherent superpositions of two off-axis partially coherent vortex beams,” J. Mod. Opt. 55, 2751-2764 (2008).
    [CrossRef]
  12. K. Cheng, H. Yan, and B. Lü, “Composite coherence vortices in the superimposed field of partially coherent vortex beams and their propagation dynamics,” Acta Phys. Sin. 57, 4911-4920 (2008).
  13. J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11, 045710-1-9 (2009).
    [CrossRef]
  14. R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A 18, 1678-1680 (2001).
    [CrossRef]
  15. 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]
  16. G. Zhou, “Analytical vectorial structure of Laguerre-Gaussian beam in the far field,” Opt. Lett. 31, 2616-2618 (2006).
    [CrossRef] [PubMed]
  17. 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]
  18. G. Zhou, L. Chen, and Y. Ni, “Investigation in the far field characteristics of TM polarized Gaussian beam from the vectorial structure,” Opt. Laser Technol. 39, 1473-1477 (2007).
    [CrossRef]
  19. G. Wu, Q. Lou, and J. Zhou, “Analytical vectorial structure of hollow Gaussian beams in the far field,” Opt. Express 16, 6417-6424 (2008).
    [CrossRef] [PubMed]
  20. G. Zhou, X. Chu, and J. Zheng, “Investigation in hollow Gaussian beam from vectorial structure,” Opt. Commun. 281, 5653-5658 (2008).
    [CrossRef]
  21. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge, 1995).
  22. X. Zeng, C. Liang, Y. An, “Far-field radiation of planar Gaussian sources and comparison with solutions based on the parabolic approximation,” Appl. Opt. 36, 2042-2047 (1997).
    [CrossRef] [PubMed]

2009 (1)

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11, 045710-1-9 (2009).
[CrossRef]

2008 (5)

K. Cheng and B. Lü, “Composite coherent vortices in coherent and incoherent superpositions of two off-axis partially coherent vortex beams,” J. Mod. Opt. 55, 2751-2764 (2008).
[CrossRef]

K. Cheng, H. Yan, and B. Lü, “Composite coherence vortices in the superimposed field of partially coherent vortex beams and their propagation dynamics,” Acta Phys. Sin. 57, 4911-4920 (2008).

G. Zhou, X. Chu, and J. Zheng, “Investigation in hollow Gaussian beam from vectorial structure,” Opt. Commun. 281, 5653-5658 (2008).
[CrossRef]

R. Xiao, J. Zhou, M. Liu, and Z. F. Jiang, “Coherent combining technology of master oscillator power amplifier fiber arrays,” Opt. Express 16, 2015-2022 (2008).
[CrossRef] [PubMed]

G. Wu, Q. Lou, and J. Zhou, “Analytical vectorial structure of hollow Gaussian beams in the far field,” Opt. Express 16, 6417-6424 (2008).
[CrossRef] [PubMed]

2007 (2)

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, L. Chen, and Y. Ni, “Investigation in the far field characteristics of TM polarized Gaussian beam from the vectorial structure,” Opt. Laser Technol. 39, 1473-1477 (2007).
[CrossRef]

2006 (2)

2003 (2)

2002 (1)

V. Pyragaite and A. Stabinis, “Free-space propagation of overlapping light vortex beams,” Opt. Commun. 213, 187-191 (2002).
[CrossRef]

2001 (2)

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A 18, 1678-1680 (2001).
[CrossRef]

1998 (1)

1997 (1)

1996 (2)

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400-407 (1996).
[CrossRef]

K. M. Abramski, A. D. Colley, H. J. Baker, and D. R. Hall, “High-power two-dimensional waveguide CO2 laser arrays,” IEEE J. Quantum Electron. 32, 340-349 (1996).
[CrossRef]

1993 (1)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73-87 (1993).
[CrossRef]

1991 (1)

E. F. Yelden, H. J. J. Seguin, C. E. Capjack, and S. K. Nikumb, “Multichannel slab discharge for CO2 laser excitation,” Appl. Phys. Lett. 58, 693-695 (1991).
[CrossRef]

Abramski, K. M.

K. M. Abramski, A. D. Colley, H. J. Baker, and D. R. Hall, “High-power two-dimensional waveguide CO2 laser arrays,” IEEE J. Quantum Electron. 32, 340-349 (1996).
[CrossRef]

An, Y.

Baker, H. J.

K. M. Abramski, A. D. Colley, H. J. Baker, and D. R. Hall, “High-power two-dimensional waveguide CO2 laser arrays,” IEEE J. Quantum Electron. 32, 340-349 (1996).
[CrossRef]

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400-407 (1996).
[CrossRef]

Bosch, S.

Branch, U.

Capjack, C. E.

J. D. Strohschein, H. J. J. Seguin, and C. E. Capjack, “Beam propagation contrasts for a radial laser array,” Appl. Opt. 37, 1045-1048 (1998).
[CrossRef]

E. F. Yelden, H. J. J. Seguin, C. E. Capjack, and S. K. Nikumb, “Multichannel slab discharge for CO2 laser excitation,” Appl. Phys. Lett. 58, 693-695 (1991).
[CrossRef]

Carnicer, A.

Chen, J.

Chen, L.

G. Zhou, L. Chen, and Y. Ni, “Investigation in the far field characteristics of TM polarized Gaussian beam from the vectorial structure,” Opt. Laser Technol. 39, 1473-1477 (2007).
[CrossRef]

Cheng, K.

K. Cheng and B. Lü, “Composite coherent vortices in coherent and incoherent superpositions of two off-axis partially coherent vortex beams,” J. Mod. Opt. 55, 2751-2764 (2008).
[CrossRef]

K. Cheng, H. Yan, and B. Lü, “Composite coherence vortices in the superimposed field of partially coherent vortex beams and their propagation dynamics,” Acta Phys. Sin. 57, 4911-4920 (2008).

Chu, X.

G. Zhou, X. Chu, and J. Zheng, “Investigation in hollow Gaussian beam from vectorial structure,” Opt. Commun. 281, 5653-5658 (2008).
[CrossRef]

Colley, A. D.

K. M. Abramski, A. D. Colley, H. J. Baker, and D. R. Hall, “High-power two-dimensional waveguide CO2 laser arrays,” IEEE J. Quantum Electron. 32, 340-349 (1996).
[CrossRef]

Guo, H.

Hall, D. R.

K. M. Abramski, A. D. Colley, H. J. Baker, and D. R. Hall, “High-power two-dimensional waveguide CO2 laser arrays,” IEEE J. Quantum Electron. 32, 340-349 (1996).
[CrossRef]

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400-407 (1996).
[CrossRef]

Hergenhan, G.

Hornby, A. M.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400-407 (1996).
[CrossRef]

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73-87 (1993).
[CrossRef]

Jiang, Z. F.

Li, J.

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11, 045710-1-9 (2009).
[CrossRef]

Liang, C.

Liu, M.

Lou, Q.

Lü, B.

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11, 045710-1-9 (2009).
[CrossRef]

K. Cheng, H. Yan, and B. Lü, “Composite coherence vortices in the superimposed field of partially coherent vortex beams and their propagation dynamics,” Acta Phys. Sin. 57, 4911-4920 (2008).

K. Cheng and B. Lü, “Composite coherent vortices in coherent and incoherent superpositions of two off-axis partially coherent vortex beams,” J. Mod. Opt. 55, 2751-2764 (2008).
[CrossRef]

Lücke, B.

Maleev, I. D.

Mandel, L.

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

Martínez-Herrero, R.

Mejías, P. M.

Molina-Terriza, G.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Morley, R. J.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400-407 (1996).
[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]

G. Zhou, L. Chen, and Y. Ni, “Investigation in the far field characteristics of TM polarized Gaussian beam from the vectorial structure,” Opt. Laser Technol. 39, 1473-1477 (2007).
[CrossRef]

Nikumb, S. K.

E. F. Yelden, H. J. J. Seguin, C. E. Capjack, and S. K. Nikumb, “Multichannel slab discharge for CO2 laser excitation,” Appl. Phys. Lett. 58, 693-695 (1991).
[CrossRef]

Pyragaite, V.

V. Pyragaite and A. Stabinis, “Free-space propagation of overlapping light vortex beams,” Opt. Commun. 213, 187-191 (2002).
[CrossRef]

Recolons, J.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Seguin, H. J. J.

J. D. Strohschein, H. J. J. Seguin, and C. E. Capjack, “Beam propagation contrasts for a radial laser array,” Appl. Opt. 37, 1045-1048 (1998).
[CrossRef]

E. F. Yelden, H. J. J. Seguin, C. E. Capjack, and S. K. Nikumb, “Multichannel slab discharge for CO2 laser excitation,” Appl. Phys. Lett. 58, 693-695 (1991).
[CrossRef]

Stabinis, A.

V. Pyragaite and A. Stabinis, “Free-space propagation of overlapping light vortex beams,” Opt. Commun. 213, 187-191 (2002).
[CrossRef]

Strohschein, J. D.

Swartzlander, G. A.

Taghizadeh, M. R.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400-407 (1996).
[CrossRef]

Torner, L.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Wolf, E.

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

Wright, E. M.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Wu, G.

Xiao, R.

Yan, H.

K. Cheng, H. Yan, and B. Lü, “Composite coherence vortices in the superimposed field of partially coherent vortex beams and their propagation dynamics,” Acta Phys. Sin. 57, 4911-4920 (2008).

Yelden, E. F.

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400-407 (1996).
[CrossRef]

E. F. Yelden, H. J. J. Seguin, C. E. Capjack, and S. K. Nikumb, “Multichannel slab discharge for CO2 laser excitation,” Appl. Phys. Lett. 58, 693-695 (1991).
[CrossRef]

Zeng, X.

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]

Zheng, J.

G. Zhou, X. Chu, and J. Zheng, “Investigation in hollow Gaussian beam from vectorial structure,” Opt. Commun. 281, 5653-5658 (2008).
[CrossRef]

Zhou, G.

G. Zhou, X. Chu, and J. Zheng, “Investigation in hollow Gaussian beam from vectorial structure,” Opt. Commun. 281, 5653-5658 (2008).
[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]

G. Zhou, L. Chen, and Y. Ni, “Investigation in the far field characteristics of TM polarized Gaussian beam from the vectorial structure,” Opt. Laser Technol. 39, 1473-1477 (2007).
[CrossRef]

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

Zhou, J.

Zhuang, S.

Acta Phys. Sin. (1)

K. Cheng, H. Yan, and B. Lü, “Composite coherence vortices in the superimposed field of partially coherent vortex beams and their propagation dynamics,” Acta Phys. Sin. 57, 4911-4920 (2008).

Appl. Opt. (3)

Appl. Phys. Lett. (1)

E. F. Yelden, H. J. J. Seguin, C. E. Capjack, and S. K. Nikumb, “Multichannel slab discharge for CO2 laser excitation,” Appl. Phys. Lett. 58, 693-695 (1991).
[CrossRef]

IEEE J. Quantum Electron. (2)

H. J. Baker, D. R. Hall, A. M. Hornby, R. J. Morley, M. R. Taghizadeh, and E. F. Yelden, “Propagation characteristics of coherent array beams from carbon dioxide waveguide lasers,” IEEE J. Quantum Electron. 32, 400-407 (1996).
[CrossRef]

K. M. Abramski, A. D. Colley, H. J. Baker, and D. R. Hall, “High-power two-dimensional waveguide CO2 laser arrays,” IEEE J. Quantum Electron. 32, 340-349 (1996).
[CrossRef]

J. Mod. Opt. (2)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73-87 (1993).
[CrossRef]

K. Cheng and B. Lü, “Composite coherent vortices in coherent and incoherent superpositions of two off-axis partially coherent vortex beams,” J. Mod. Opt. 55, 2751-2764 (2008).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A, Pure Appl. Opt. 11, 045710-1-9 (2009).
[CrossRef]

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

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

Opt. Commun. (3)

G. Zhou, X. Chu, and J. Zheng, “Investigation in hollow Gaussian beam from vectorial structure,” Opt. Commun. 281, 5653-5658 (2008).
[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]

V. Pyragaite and A. Stabinis, “Free-space propagation of overlapping light vortex beams,” Opt. Commun. 213, 187-191 (2002).
[CrossRef]

Opt. Express (3)

Opt. Laser Technol. (1)

G. Zhou, L. Chen, and Y. Ni, “Investigation in the far field characteristics of TM polarized Gaussian beam from the vectorial structure,” Opt. Laser Technol. 39, 1473-1477 (2007).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Contour lines of phase of the x component E TE x for different values of the off-axis parameter a: (a) a = 40 λ , (b) a = 50 λ , (c) a = 70 λ . The calculation parameters may be found in the text.

Fig. 2
Fig. 2

Contour lines of phase of the x component E TE x for different values of the relative phase β: (a) β = π 3 , (b) β = π .

Fig. 3
Fig. 3

Contour lines of phase of the x component E TE x for different values of the amplitude ratio η: (a) η = 0.5 , (b) η = 2 . Contour lines of the phase of the x component E TE x for different values of the waist widths w 01 , w 02 : (c) w 01 = 0.5 λ , w 02 = 0.4 λ , and (d) w 01 = 0.5 λ , w 02 = 0.6 λ .

Fig. 4
Fig. 4

Energy flux distributions: (a) S z TE , β = 0 ; (b) S z TM , β = 0 ; (c) S z whole , β = 0 ; (d) S z TE , β = π 2 ; (e) S z TM , β = π 2 ; (f) S z whole , β = π 2 .

Equations (33)

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E x ( ρ , θ , z ) = E 0 w 0 w ( z ) ( 2 ρ w ( z ) ) | l | exp ( ρ 2 w ( z ) 2 ) L p | l | ( 2 ρ 2 w ( z ) 2 ) exp { i [ k z + k ρ 2 2 R ( z ) + l θ ( 2 p + | l | + 1 ) arctan ( z z R ) ] } ,
E y ( ρ , θ , z ) = 0 ,
E x ( ρ , θ , 0 ) = E x 1 ( ρ 1 , θ 1 , 0 ) exp ( i β ) + E x 2 ( ρ 2 , θ 2 , 0 ) ,
E y ( ρ , θ , 0 ) = 0 ,
E x ( x 0 , y 0 , 0 ) = E 01 2 w 01 ( x 0 a + i y 0 ) exp [ ( x 0 a ) 2 + y 0 2 w 01 2 + i β ] + E 02 2 w 02 ( x 0 + a i y 0 ) exp [ ( x 0 + a ) 2 + y 0 2 w 02 2 ] ,
E y ( x 0 , y 0 , 0 ) = 0 .
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 .
A x ( p , q ) = 2 π 2 λ 3 { E 01 w 01 3 ( q i p ) exp [ i β i a k p 1 4 k 2 w 01 2 ( p 2 + q 2 ) ] E 02 w 02 3 ( q + i p ) exp [ i a k p 1 4 k 2 w 02 2 ( p 2 + q 2 ) ] } ,
A y ( p , q ) = 0.
E ( r ) = E TE ( r ) + E TM ( r ) ,
H ( r ) = H TE ( r ) + H TM ( r ) ,
E TE ( r ) = q p 2 + q 2 A x ( p , q ) ( q i p j ) exp [ i k ( p x + q y + γ z ) ] d p d q ,
H TE ( r ) = ε μ q p 2 + q 2 A x ( p , q ) [ p γ i + q γ j ( p 2 + q 2 ) k ] exp [ i k ( p x + q y + γ z ) ] d p d q ,
E TM ( r ) = p γ ( p 2 + q 2 ) A x ( p , q ) [ p γ i + q γ j ( p 2 + q 2 ) k ] exp [ i k ( p x + q y + γ z ) ] d p d q ,
H TM ( r ) = ε μ p γ ( p 2 + q 2 ) A x ( p , q ) [ q i p j ] exp [ i k ( p x + q y + γ z ) ] d p d q ,
E TE ( r ) = 2 i π 2 y z λ 2 r 3 ( x 2 + y 2 ) exp ( i k r ) { E 01 w 01 3 ( y i x ) exp [ i β i k a x r k 2 w 01 2 ( x 2 + y 2 ) 4 r 2 ] E 02 w 02 3 ( y + i x ) exp [ i k a x r k 2 w 02 2 ( x 2 + y 2 ) 4 r 2 ] } ( y i x j ) ,
H TE ( r ) = 2 ε i π 2 y z λ 2 r 4 μ ( x 2 + y 2 ) exp ( i k r ) { E 01 w 01 3 ( y i x ) exp [ i β i k a x r k 2 w 01 2 ( x 2 + y 2 ) 4 r 2 ] E 02 w 02 3 ( y + i x ) exp [ i k a x r k 2 w 02 2 ( x 2 + y 2 ) 4 r 2 ] } [ x z i + y z j ( x 2 + y 2 ) k ] ,
E TM ( r ) = 2 π 2 x λ 2 r 3 ( x 2 + y 2 ) exp ( i k r ) { E 01 w 01 3 ( x + i y ) exp [ i β i k a x r k 2 w 01 2 ( x 2 + y 2 ) 4 r 2 ] + E 02 w 02 3 ( x i y ) exp [ i k a x r k 2 w 02 2 ( x 2 + y 2 ) 4 r 2 ] } [ x z i + y z j ( x 2 + y 2 ) k ] ,
H TM ( r ) = 2 ε π 2 x λ 2 r 2 μ ( x 2 + y 2 ) exp ( i k r ) { E 01 w 01 3 ( x + i y ) exp [ i β i k a x r k 2 w 01 2 ( x 2 + y 2 ) 4 r 2 ] + E 02 w 02 3 ( x i y ) exp [ i k a x r k 2 w 02 2 ( x 2 + y 2 ) 4 r 2 ] } [ y i x j ] .
φ E TE x = arctan { Im [ E TE x ( x , y , z ) ] Re [ E TE x ( x , y , z ) ] } = const . ,
Re [ E TE x ( x , y , z ) ] = 0 ,
Im [ E TE x ( x , y , z ) ] = 0 .
η w 02 3 x + w 01 3 exp [ π 2 ( x 2 + y 2 ) ( w 02 2 w 01 2 ) λ 2 r 2 ] [ x cos ( β 4 π x a λ r ) y sin ( β 4 π x a λ r ) ] = 0 ,
η w 02 3 y + w 01 3 exp [ π 2 ( x 2 + y 2 ) ( w 02 2 w 01 2 ) λ 2 r 2 ] [ y cos ( β 4 π x a λ r ) + x sin ( β 4 π x a λ r ) ] = 0 .
x + x cos ( 4 π x a λ r ) + y sin ( 4 π x a λ r ) = 0 ,
y + y cos ( 4 π x a λ r ) x sin ( 4 π x a λ r ) = 0 .
S z TE = 1 2 Re [ E TE ( r ) × H TE * ( r ) ] z ,
S z TM = 1 2 Re [ E TM ( r ) × H TM * ( r ) ] z ,
S z whole = S z TE + S z TM .
S z TE = k 2 π 2 y 2 z 3 4 λ 2 r 7 ( x 2 + y 2 ) exp [ 3 k 2 ( x 2 + y 2 ) ( w 01 2 + w 02 2 ) 4 r 2 ] { ( x 2 + y 2 ) [ E 01 2 w 01 6     exp ( k 2 ( x 2 + y 2 ) ( w 01 2 + 3 w 02 2 ) 4 r 2 ) + E 02 2 w 02 6 exp ( k 2 ( x 2 + y 2 ) ( 3 w 01 2 + w 02 2 ) 4 r 2 ) ] + 2 E 01 E 02 w 01 3 w 02 3     exp [ k 2 ( x 2 + y 2 ) ( w 01 2 + w 02 2 ) 2 r 2 ] [ ( x 2 y 2 ) cos ( β + 2 a k x r ) + 2 x y sin ( β + 2 a k x r ) ] } ,
S z TM = x 2 r 2 y 2 z 2 S z TE ,
S z whole = ( 1 + x 2 r 2 y 2 z 2 ) S z TE .

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