Abstract

The property of self-healing at the focal plane for both scalar and vector Bessel–Gauss (BG) beams is investigated in the tight focusing condition. For the BG beam, which is partially obstructed at the pupil plane, the spatial intensity distribution at the focal plane is well recovered. Furthermore, recovery of not only intensity but also polarization distribution is observed for an obstructed vector BG beam. This self-healing effect for both the intensity and polarization components is recognized even when the half of the beam is obstructed by a semicircular obstacle. The effect of the size of the obstacle on recovery of polarization and intensity distribution is studied. The role of the beam size at the pupil plane is also discussed.

© 2011 Optical Society of America

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2010 (5)

F. O. Farrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785 (2010).
[Crossref]

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, 965–967 (2010).
[Crossref] [PubMed]

J. Li, Y. Fang, S. Zhou, and Y. Ye, “Focusing of concentric piecewise vector Bessel-Gauss beam,” Opt. Laser Eng. 48, 1247–1251 (2010).
[Crossref]

J. Chen and Y. Yu, “The focusing property of vector Bessel-Gauss beams by a high-numerical aperture objective,” Opt. Commun. 283, 1655–1660 (2010).
[Crossref]

A. Ito, Y. Kozawa, and S. Sato, “Generation of hollow scalar and vector beams using a spot-defect mirror,” J. Opt. Soc. Am. A 27, 2072–2077 (2010).
[Crossref]

2009 (1)

2008 (2)

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008).
[Crossref] [PubMed]

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

2007 (3)

2004 (2)

S. H. Tao and X. Yuan, “Self-reconstruction property of fractional Bessel beams,” J. Opt. Soc. Am. A 21, 1192–1197 (2004).
[Crossref]

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Splading, G. Cristobal, and K. Dholakia, “The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization,” J. Opt. A: Pure Appl. Opt. 6, S235–S238 (2004).
[Crossref]

2003 (1)

J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. 50, 1573–1580 (2003).
[Crossref]

2002 (2)

Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
[Crossref]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147(2002).
[Crossref] [PubMed]

2001 (1)

J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant fields by polarization-grating axicons,” Opt. Commun. 192, 13–18 (2001).
[Crossref]

2000 (2)

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” JETP Lett. 71, 130–133(2000).
[Crossref]

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

1999 (1)

1998 (1)

1996 (4)

D. G. Hall, “Vector-beam solutions of Mazwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
[Crossref] [PubMed]

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. 122, 169–177 (1996).
[Crossref]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
[Crossref]

C. Palma, G. Cincotti, G. Guattari, and M. Santarsiero, “Imaging of generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[Crossref]

1995 (2)

B. Lu, W. Huang, and B. Zhang, “Fraunhofer diffraction of a Bessel bam focused by an aperture lens,” Opt. Commun. 119, 6–12 (1995).
[Crossref]

Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[Crossref]

1990 (1)

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435 (1990).
[Crossref]

1987 (2)

1959 (1)

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

Anguiano-Morales, M.

Arlt, J.

J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. 50, 1573–1580 (2003).
[Crossref]

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
[Crossref]

Boothroyd, S. A.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. 122, 169–177 (1996).
[Crossref]

Bouchal, Z.

Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
[Crossref]

Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[Crossref]

Broky, J.

Brown, T. G.

Chavez-Cerda, S.

Chen, J.

J. Chen and Y. Yu, “The focusing property of vector Bessel-Gauss beams by a high-numerical aperture objective,” Opt. Commun. 283, 1655–1660 (2010).
[Crossref]

Chong, C. T.

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

Christodoulides, D. N.

Chrostowski, J.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. 122, 169–177 (1996).
[Crossref]

Cincotti, G.

C. Palma, G. Cincotti, G. Guattari, and M. Santarsiero, “Imaging of generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[Crossref]

Cristobal, G.

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Splading, G. Cristobal, and K. Dholakia, “The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization,” J. Opt. A: Pure Appl. Opt. 6, S235–S238 (2004).
[Crossref]

Dholakia, K.

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Splading, G. Cristobal, and K. Dholakia, “The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization,” J. Opt. A: Pure Appl. Opt. 6, S235–S238 (2004).
[Crossref]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147(2002).
[Crossref] [PubMed]

Dogariu, A.

Durnin, J.

Erdogan, T.

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435 (1990).
[Crossref]

Fang, Y.

J. Li, Y. Fang, S. Zhou, and Y. Ye, “Focusing of concentric piecewise vector Bessel-Gauss beam,” Opt. Laser Eng. 48, 1247–1251 (2010).
[Crossref]

Farrbach, F. O.

F. O. Farrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785 (2010).
[Crossref]

Fernandez-Nieves, A.

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Splading, G. Cristobal, and K. Dholakia, “The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization,” J. Opt. A: Pure Appl. Opt. 6, S235–S238 (2004).
[Crossref]

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
[Crossref]

Garces-Chavez, V.

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Splading, G. Cristobal, and K. Dholakia, “The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization,” J. Opt. A: Pure Appl. Opt. 6, S235–S238 (2004).
[Crossref]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147(2002).
[Crossref] [PubMed]

Gori, F.

F. Gori and G. Guattari, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Greene, P.

Greene, P. L.

Guattari, G.

C. Palma, G. Cincotti, G. Guattari, and M. Santarsiero, “Imaging of generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[Crossref]

F. Gori and G. Guattari, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Hall, D. G.

Huang, K.

Huang, W.

B. Lu, W. Huang, and B. Zhang, “Fraunhofer diffraction of a Bessel bam focused by an aperture lens,” Opt. Commun. 119, 6–12 (1995).
[Crossref]

Ito, A.

Iturbe-Castillo, M. D.

Kang, X.-L.

Kozawa, Y.

Li, J.

J. Li, Y. Fang, S. Zhou, and Y. Ye, “Focusing of concentric piecewise vector Bessel-Gauss beam,” Opt. Laser Eng. 48, 1247–1251 (2010).
[Crossref]

Li, Y. P.

Lu, B.

B. Lu, W. Huang, and B. Zhang, “Fraunhofer diffraction of a Bessel bam focused by an aperture lens,” Opt. Commun. 119, 6–12 (1995).
[Crossref]

Lukyanchuk, B.

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

MacDonald, R. P.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. 122, 169–177 (1996).
[Crossref]

Marienko, I. G.

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” JETP Lett. 71, 130–133(2000).
[Crossref]

Martinez, A.

McGloin, D.

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Splading, G. Cristobal, and K. Dholakia, “The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization,” J. Opt. A: Pure Appl. Opt. 6, S235–S238 (2004).
[Crossref]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147(2002).
[Crossref] [PubMed]

Melville, H.

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147(2002).
[Crossref] [PubMed]

Okamoto, T.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. 122, 169–177 (1996).
[Crossref]

Olivik, M.

Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[Crossref]

Palma, C.

C. Palma, G. Cincotti, G. Guattari, and M. Santarsiero, “Imaging of generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[Crossref]

Richards, B.

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

Rohrbach, A.

F. O. Farrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785 (2010).
[Crossref]

Santarsiero, M.

C. Palma, G. Cincotti, G. Guattari, and M. Santarsiero, “Imaging of generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[Crossref]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
[Crossref]

Sato, S.

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
[Crossref]

Sheppard, C.

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

Sheppard, C. J. R.

Shi, L.

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

Shi, P.

Sibbett, W.

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147(2002).
[Crossref] [PubMed]

Simon, P.

F. O. Farrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785 (2010).
[Crossref]

Siviloglou, G. A.

Soskin, M. S.

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” JETP Lett. 71, 130–133(2000).
[Crossref]

Spagnolo, G. S.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
[Crossref]

Splading, G. C.

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Splading, G. Cristobal, and K. Dholakia, “The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization,” J. Opt. A: Pure Appl. Opt. 6, S235–S238 (2004).
[Crossref]

Summers, M. D.

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Splading, G. Cristobal, and K. Dholakia, “The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization,” J. Opt. A: Pure Appl. Opt. 6, S235–S238 (2004).
[Crossref]

Syrett, B. A.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. 122, 169–177 (1996).
[Crossref]

Tao, S. H.

Tervo, J.

J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant fields by polarization-grating axicons,” Opt. Commun. 192, 13–18 (2001).
[Crossref]

Turunen, J.

J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant fields by polarization-grating axicons,” Opt. Commun. 192, 13–18 (2001).
[Crossref]

Vasnetsov, M. V.

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” JETP Lett. 71, 130–133(2000).
[Crossref]

Wang, H.

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

Wolf, E.

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

Ye, Y.

J. Li, Y. Fang, S. Zhou, and Y. Ye, “Focusing of concentric piecewise vector Bessel-Gauss beam,” Opt. Laser Eng. 48, 1247–1251 (2010).
[Crossref]

Yew, E. Y. S.

Youngworth, K. S.

Yu, Y.

J. Chen and Y. Yu, “The focusing property of vector Bessel-Gauss beams by a high-numerical aperture objective,” Opt. Commun. 283, 1655–1660 (2010).
[Crossref]

Yuan, X.

Zhan, Q.

Zhang, B.

B. Lu, W. Huang, and B. Zhang, “Fraunhofer diffraction of a Bessel bam focused by an aperture lens,” Opt. Commun. 119, 6–12 (1995).
[Crossref]

Zhang, X.

Zhou, S.

J. Li, Y. Fang, S. Zhou, and Y. Ye, “Focusing of concentric piecewise vector Bessel-Gauss beam,” Opt. Laser Eng. 48, 1247–1251 (2010).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Opt. (1)

J. Appl. Phys. (1)

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435 (1990).
[Crossref]

J. Mod. Opt. (4)

Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[Crossref]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
[Crossref]

C. Palma, G. Cincotti, G. Guattari, and M. Santarsiero, “Imaging of generalized Bessel-Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[Crossref]

J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. 50, 1573–1580 (2003).
[Crossref]

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

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Splading, G. Cristobal, and K. Dholakia, “The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization,” J. Opt. A: Pure Appl. Opt. 6, S235–S238 (2004).
[Crossref]

J. Opt. Soc. Am. A (5)

JETP Lett. (1)

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” JETP Lett. 71, 130–133(2000).
[Crossref]

Nat. Photon. (2)

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Figures (6)

Fig. 1
Fig. 1

Calculated intensity distribution and polarization components for a radially polarized BG beam for three values of β: 0.00005 k , 0.000275 k , and 0.0005 k . Field distribution at the pupil plane: (a) total intensity, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane: (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component.

Fig. 2
Fig. 2

Calculated intensity distribution and polarization components for azimuthally polarized BG beam for three values of β: 0.00005 k , 0.000275 k , and 0.0005 k . Field distribution at the pupil plane: (a) total intensity, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane: (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component.

Fig. 3
Fig. 3

Calculated intensity distribution and polarization components for radially and azimuthally polarized BG beams with β = 0.0005 k and m = 0 . Top and bottom rows correspond to radially and azimuthally polarized beams, respectively. Field distribution at the pupil plane: (a) total intensity, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane: (d) total intensity distribution, (e) radial component, (f) azimuthal component, and (g) longitudinal component.

Fig. 4
Fig. 4

Calculated intensity distribution and polarization components for a radially polarized BG beam for m = 0 and for three different values of β. Field distribution at the pupil plane: (a) total intensity distribution, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane: (d) total intensity, (e) radial component, (f) azimuthal component, and (g) longitudinal component.

Fig. 5
Fig. 5

Calculated intensity distribution and polarization components for a vector BG beam with β = 0.0005 k and m = 3 for different-sized obstacles. Field distribution at the pupil plane: (a) total intensity distribution, (b) radial component, and (c) azimuthal component. Field distribution at the focal plane: (d) total intensity, (e) radial component, (f) azimuthal component, and (g) longitudinal component.

Fig. 6
Fig. 6

Calculated intensity distribution and polarization components for a linearly polarized (x-polarized) BG beam with β = 0.0005 k and n = 1 for different-sized obstacles. Field distribution at the pupil plane: (a) total intensity. Field distribution at the focal plane: (b) total intensity, (c) x-polarized component, (d) y-polarized component, and (e) longitudinal component.

Equations (13)

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E ( s ) = U ( r , ϕ , z ) exp ( i ω t ) i ^ x ,
U ( r , ϕ , z ) = E 0 ω 0 ω ( z ) i n exp [ i k z i ψ ( z ) ] × exp { r 2 [ 1 ω 2 ( z ) i k 2 R ( z ) ] } Q ( z ) × J n ( u ) exp ( i n ϕ ) ,
E ( v ) = U ( r , ϕ , z ) exp ( i ω t ) ,
U ( r , ϕ , z ) = u radial i ^ r + u azimutal i ^ ϕ ,
U ( r , ϕ , z ) = E 0 ω 0 ω ( z ) exp [ i k z i ψ ( z ) ] × exp { r 2 [ 1 ω 2 ( z ) i k 2 R ( z ) ] } Q ( z ) T ( r , ϕ , z ) ,
T e ( r , ϕ , z ) = [ J m 1 ( u ) J m + 1 ( u ) ] [ sin ( m ϕ ) cos ( m ϕ ) ] i ^ ϕ + [ J m 1 ( u ) + J m + 1 ( u ) ] [ cos ( m ϕ ) sin ( m ϕ ) ] i ^ r ,
T m ( r , ϕ , z ) = [ J m 1 ( u ) + J m + 1 ( u ) ] [ cos ( m ϕ ) sin ( m ϕ ) ] i ^ ϕ + [ J m 1 ( u ) J m + 1 ( u ) ] [ sin ( m ϕ ) cos ( m ϕ ) ] i ^ r ,
U r , radial ( r 0 , ϕ 0 , z 0 ) = i A π 0 α 0 2 π cos ( θ ) sin ( θ ) cos ( θ ) cos ( ϕ ϕ 0 ) u radial exp { i k [ z 0 cos ( θ ) + r 0 sin ( θ ) cos ( ϕ ϕ 0 ) ] } d ϕ d θ ,
U ϕ , radial ( r 0 , ϕ 0 , z 0 ) = i A π 0 α 0 2 π cos ( θ ) sin ( θ ) cos ( θ ) sin ( ϕ ϕ 0 ) u radial exp { i k [ z 0 cos ( θ ) + r 0 sin ( θ ) cos ( ϕ ϕ 0 ) ] } d ϕ d θ ,
U z , radial ( r 0 , ϕ 0 , z 0 ) = i A π 0 α 0 2 π cos ( θ ) sin 2 ( θ ) × u radial exp { i k [ z 0 cos ( θ ) + r 0 sin ( θ ) cos ( ϕ ϕ 0 ) ] } d ϕ d θ ,
U r , azimutal ( r 0 , ϕ 0 , z 0 ) = i A π 0 α 0 2 π cos ( θ ) sin ( θ ) sin ( ϕ ϕ 0 ) × u azimutal exp { i k [ z 0 cos ( θ ) + r 0 sin ( θ ) cos ( ϕ ϕ 0 ) ] } d ϕ d θ ,
U ϕ , azimutal ( r 0 , ϕ 0 , z 0 ) = i A π 0 α 0 2 π cos ( θ ) sin ( θ ) cos ( ϕ ϕ 0 ) u azimutal exp { i k [ z 0 cos ( θ ) + r 0 sin ( θ ) cos ( ϕ ϕ 0 ) ] } d ϕ d θ ,
U z , azimutal ( r 0 , ϕ 0 , z 0 ) = 0 ,

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