Abstract

Starting from the vectorial Rayleigh–Sommerfeld formulas, the nonparaxial propagation of radially polarized light beams in free space is investigated analytically and numerically. The paraxial case can be treated as a special case of the general result. The exact expression of the radially polarized light beams, which is valid for the fields to be predicted for an arbitrary transverse beam size, has been derived in closed-form terms for any on-axis point. The validity of the analytical results is confirmed by the numerical results.

© 2006 Optical Society of America

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2005 (3)

2004 (1)

2003 (1)

2002 (4)

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, "Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings," Opt. Lett. 27, 285-287 (2002).
[CrossRef]

F. Treussart, R. Allaume, V. Le Floch, L. T. Xiao, J. M. Courty, and J.-F. Roch, "Direct measurement of the photon statistics of a triggered single photon source," Phys. Rev. Lett. 89, 093601 (2002).
[CrossRef] [PubMed]

C. Varin and M. Piché, "Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams," Appl. Phys. B 74, S83-S88 (2002).
[CrossRef]

A. Ciattoni, B. Crosignami, and P. D. Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17-20 (2002).
[CrossRef]

2001 (3)

Z. Bomzon, V. Kleiner, and E. Hasman, "Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating," Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

J. Azoulay, A. Débarre, R. Jaffiol, and P. Tchénio, "Original tools for single-molecule spectroscopy," Single Mol. 2, 241-249 (2001).
[CrossRef]

2000 (3)

B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

A. V. Nesterov and V. G. Niziev, "Laser beams with axially symmetric polarization," J. Phys. D 33, 1817-1822 (2000).
[CrossRef]

R. Oron, S. Blit, N. Davidson, and A. A. Friesem, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

1999 (3)

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (1999).
[CrossRef]

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

K. T. Gahagan and G. A. Swartzlander Jr., "Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap," J. Opt. Soc. Am. B 16, 533-537 (1999).
[CrossRef]

1998 (1)

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

1996 (1)

1990 (1)

1974 (1)

G. N. Vinokurov, A. A. Mak, and V. M. Mitkin, "Generating azimuthal and radial modes in optical resonators," Kvantovaya Elektron. (Kiev) 8, 1890-1 (1974) (in Russian).

1972 (1)

D. Pohl, "Operation of a ruby laser in the purely transverse electric mode," Appl. Phys. Lett. 20, 266-7 (1972).
[CrossRef]

Aït-Ameur, K.

Allaume, R.

F. Treussart, R. Allaume, V. Le Floch, L. T. Xiao, J. M. Courty, and J.-F. Roch, "Direct measurement of the photon statistics of a triggered single photon source," Phys. Rev. Lett. 89, 093601 (2002).
[CrossRef] [PubMed]

Armstrong, D. J.

Azoulay, J.

J. Azoulay, A. Débarre, R. Jaffiol, and P. Tchénio, "Original tools for single-molecule spectroscopy," Single Mol. 2, 241-249 (2001).
[CrossRef]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Biener, G.

Blit, S.

R. Oron, S. Blit, N. Davidson, and A. A. Friesem, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Bomzon, Z.

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, "Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings," Opt. Lett. 27, 285-287 (2002).
[CrossRef]

Z. Bomzon, V. Kleiner, and E. Hasman, "Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating," Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

Borghi, R.

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown, "Inhomogeneous polarization in scanning optical microscopy," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing VII, J.Conchello, C.J.Cogswell, A.G.Tescher, and T.Wilson, eds., Proc. SPIE 3919, 75-85 (2000).

Ciattoni, A.

A. Ciattoni, B. Crosignami, and P. D. Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17-20 (2002).
[CrossRef]

Courty, J. M.

F. Treussart, R. Allaume, V. Le Floch, L. T. Xiao, J. M. Courty, and J.-F. Roch, "Direct measurement of the photon statistics of a triggered single photon source," Phys. Rev. Lett. 89, 093601 (2002).
[CrossRef] [PubMed]

Crosignami, B.

A. Ciattoni, B. Crosignami, and P. D. Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17-20 (2002).
[CrossRef]

Davidson, N.

R. Oron, S. Blit, N. Davidson, and A. A. Friesem, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Débarre, A.

J. Azoulay, A. Débarre, R. Jaffiol, and P. Tchénio, "Original tools for single-molecule spectroscopy," Single Mol. 2, 241-249 (2001).
[CrossRef]

Denis, R. de S.

Dong, B.-Z.

C.-H. Niu, B.-Y. Gu, B.-Z. Dong, and Y. Zhang, "A new method for generating axially symmetric and radially polarized beams," J. Phys. D 38, 827-832 (2005).
[CrossRef]

Ford, D. H.

Friesem, A. A.

R. Oron, S. Blit, N. Davidson, and A. A. Friesem, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Gahagan, K. T.

Glur, H.

Gu, B.-Y.

C.-H. Niu, B.-Y. Gu, B.-Z. Dong, and Y. Zhang, "A new method for generating axially symmetric and radially polarized beams," J. Phys. D 38, 827-832 (2005).
[CrossRef]

Hasman, E.

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, "Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings," Opt. Lett. 27, 285-287 (2002).
[CrossRef]

Z. Bomzon, V. Kleiner, and E. Hasman, "Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating," Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

Hecht, B.

B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Hierle, R.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Jaffiol, R.

J. Azoulay, A. Débarre, R. Jaffiol, and P. Tchénio, "Original tools for single-molecule spectroscopy," Single Mol. 2, 241-249 (2001).
[CrossRef]

Joannopoulis, J. D.

J. D. Joannopoulis, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U., 1995).

Kimura, W. D.

Kleiner, V.

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, "Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings," Opt. Lett. 27, 285-287 (2002).
[CrossRef]

Z. Bomzon, V. Kleiner, and E. Hasman, "Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating," Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Le Floch, V.

F. Treussart, R. Allaume, V. Le Floch, L. T. Xiao, J. M. Courty, and J.-F. Roch, "Direct measurement of the photon statistics of a triggered single photon source," Phys. Rev. Lett. 89, 093601 (2002).
[CrossRef] [PubMed]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1966).

Mak, A. A.

G. N. Vinokurov, A. A. Mak, and V. M. Mitkin, "Generating azimuthal and radial modes in optical resonators," Kvantovaya Elektron. (Kiev) 8, 1890-1 (1974) (in Russian).

Meade, R. D.

J. D. Joannopoulis, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U., 1995).

Mitkin, V. M.

G. N. Vinokurov, A. A. Mak, and V. M. Mitkin, "Generating azimuthal and radial modes in optical resonators," Kvantovaya Elektron. (Kiev) 8, 1890-1 (1974) (in Russian).

Nesterov, A. V.

A. V. Nesterov and V. G. Niziev, "Laser beams with axially symmetric polarization," J. Phys. D 33, 1817-1822 (2000).
[CrossRef]

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (1999).
[CrossRef]

Niu, C.-H.

C.-H. Niu, B.-Y. Gu, B.-Z. Dong, and Y. Zhang, "A new method for generating axially symmetric and radially polarized beams," J. Phys. D 38, 827-832 (2005).
[CrossRef]

Niziev, V. G.

A. V. Nesterov and V. G. Niziev, "Laser beams with axially symmetric polarization," J. Phys. D 33, 1817-1822 (2000).
[CrossRef]

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (1999).
[CrossRef]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Oron, R.

R. Oron, S. Blit, N. Davidson, and A. A. Friesem, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

Passilly, N.

Philips, M. C.

Piché, M.

C. Varin and M. Piché, "Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams," Appl. Phys. B 74, S83-S88 (2002).
[CrossRef]

Pohl, D.

D. Pohl, "Operation of a ruby laser in the purely transverse electric mode," Appl. Phys. Lett. 20, 266-7 (1972).
[CrossRef]

Porto, P. D.

A. Ciattoni, B. Crosignami, and P. D. Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17-20 (2002).
[CrossRef]

Roch, J.-F.

N. Passilly, R. de S. Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J.-F. Roch, "Simple interferometric technique for generation of a radially polarized light beam," J. Opt. Soc. Am. A 22, 984-991 (2005).
[CrossRef]

F. Treussart, R. Allaume, V. Le Floch, L. T. Xiao, J. M. Courty, and J.-F. Roch, "Direct measurement of the photon statistics of a triggered single photon source," Phys. Rev. Lett. 89, 093601 (2002).
[CrossRef] [PubMed]

Roth, M. S.

Santarsiero, M.

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Schadt, M.

Shih, C.-C.

C.-C. Shih and P. V. Estates, CA, "Radial polarization laser resonator," U. S. patent5,359,622 (1994).

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "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. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Sick, B.

B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000).
[CrossRef] [PubMed]

Smith, A. V.

Stalder, M.

Swartzlander, G. A.

Tchénio, P.

J. Azoulay, A. Débarre, R. Jaffiol, and P. Tchénio, "Original tools for single-molecule spectroscopy," Single Mol. 2, 241-249 (2001).
[CrossRef]

Tidwell, S. C.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Tovar, A. A.

Treussart, F.

N. Passilly, R. de S. Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J.-F. Roch, "Simple interferometric technique for generation of a radially polarized light beam," J. Opt. Soc. Am. A 22, 984-991 (2005).
[CrossRef]

F. Treussart, R. Allaume, V. Le Floch, L. T. Xiao, J. M. Courty, and J.-F. Roch, "Direct measurement of the photon statistics of a triggered single photon source," Phys. Rev. Lett. 89, 093601 (2002).
[CrossRef] [PubMed]

V. Estates, P.

C.-C. Shih and P. V. Estates, CA, "Radial polarization laser resonator," U. S. patent5,359,622 (1994).

Varin, C.

C. Varin and M. Piché, "Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams," Appl. Phys. B 74, S83-S88 (2002).
[CrossRef]

Vinokurov, G. N.

G. N. Vinokurov, A. A. Mak, and V. M. Mitkin, "Generating azimuthal and radial modes in optical resonators," Kvantovaya Elektron. (Kiev) 8, 1890-1 (1974) (in Russian).

Weber, H. P.

Winn, J. N.

J. D. Joannopoulis, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U., 1995).

Wyss, E. W.

Xiao, L. T.

F. Treussart, R. Allaume, V. Le Floch, L. T. Xiao, J. M. Courty, and J.-F. Roch, "Direct measurement of the photon statistics of a triggered single photon source," Phys. Rev. Lett. 89, 093601 (2002).
[CrossRef] [PubMed]

Yakunin, V. P.

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (1999).
[CrossRef]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown, "Inhomogeneous polarization in scanning optical microscopy," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing VII, J.Conchello, C.J.Cogswell, A.G.Tescher, and T.Wilson, eds., Proc. SPIE 3919, 75-85 (2000).

Zhang, Y.

C.-H. Niu, B.-Y. Gu, B.-Z. Dong, and Y. Zhang, "A new method for generating axially symmetric and radially polarized beams," J. Phys. D 38, 827-832 (2005).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

C. Varin and M. Piché, "Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams," Appl. Phys. B 74, S83-S88 (2002).
[CrossRef]

Appl. Phys. Lett. (3)

R. Oron, S. Blit, N. Davidson, and A. A. Friesem, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000).
[CrossRef]

D. Pohl, "Operation of a ruby laser in the purely transverse electric mode," Appl. Phys. Lett. 20, 266-7 (1972).
[CrossRef]

Z. Bomzon, V. Kleiner, and E. Hasman, "Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating," Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

J. Opt. Soc. Am. A (3)

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

J. Phys. D (4)

C.-H. Niu, B.-Y. Gu, B.-Z. Dong, and Y. Zhang, "A new method for generating axially symmetric and radially polarized beams," J. Phys. D 38, 827-832 (2005).
[CrossRef]

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D 32, 2871-2875 (1999).
[CrossRef]

A. V. Nesterov and V. G. Niziev, "Laser beams with axially symmetric polarization," J. Phys. D 33, 1817-1822 (2000).
[CrossRef]

V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

Kvantovaya Elektron. (Kiev) (1)

G. N. Vinokurov, A. A. Mak, and V. M. Mitkin, "Generating azimuthal and radial modes in optical resonators," Kvantovaya Elektron. (Kiev) 8, 1890-1 (1974) (in Russian).

Opt. Commun. (1)

A. Ciattoni, B. Crosignami, and P. D. Porto, "Vectorial analytical description of propagation of a highly nonparaxial beam," Opt. Commun. 202, 17-20 (2002).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. Lett. (4)

F. Treussart, R. Allaume, V. Le Floch, L. T. Xiao, J. M. Courty, and J.-F. Roch, "Direct measurement of the photon statistics of a triggered single photon source," Phys. Rev. Lett. 89, 093601 (2002).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Irradiance distribution of the radially polarized light beam at the plane z = z R for (a) nonparaxial propagation and (b) paraxial propagation versus x λ and y λ . Solid and dashed curves in (c) are the cross-section irradiance distributions at y = 0 corresponding to Figs. 1a, 1b, respectively. The parameters are chosen such that p = 0 , f = 0.1 .

Fig. 2
Fig. 2

Irradiance distribution of the radially polarized light beam at the plane z = z R for (a) nonparaxial propagation and (b) paraxial propagation versus x λ and y λ . Solid and dashed curves in (c) are the cross-section irradiance distributions at y = 0 corresponding to Figs. 2a, 2b, respectively. The parameters are chosen such that p = 0 , f = 0.5 .

Fig. 3
Fig. 3

Irradiance distribution of the radially polarized light beam at the plane z = z R for (a) nonparaxial propagation and (b) paraxial propagation versus x λ and y λ . Solid and dashed curves in (c) are the cross-section irradiance distributions at y = 0 corresponding to Figs. 3a, 3b, respectively. The parameters are chosen such that p = 3 , f = 0.1 .

Fig. 4
Fig. 4

Irradiance distribution of the radially polarized light beam at the plane z = z R for (a) nonparaxial propagation and (b) paraxial propagation versus x λ and y λ . Solid and dashed curves in (c) are the cross-section irradiance distributions at y = 0 corresponding to Figs. 4a, 4b, respectively. The parameters are chosen such that p = 3 , f = 0.5 .

Fig. 5
Fig. 5

Modulus of the relative error as a function of z z R for a radially polarized light beam having (a) f = 0.1 and for different values of the order p and (b) the beam order p = 2 and for different values of f.

Fig. 6
Fig. 6

Exact normalized on-axis field amplitude as a function of z z R for a radially polarized light beam having (a) f = 0.1 and for different values of the orders p = 0 (solid curve) and p = 3 (dotted curve), together with the paraxial prediction p = 0 (dashed curve) and p = 3 (dash-dotted curve), and (b) f = 0.2 and for different values of the orders p = 0 (solid curve) and p = 3 (dotted curve), together with the paraxial prediction p = 0 (dashed curve) and p = 3 (dash-dotted curve).

Equations (27)

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E p 1 ( r , 0 ) = E 0 2 r w 0 L p 1 ( 2 r 2 w 0 2 ) exp ( r 2 w 0 2 ) e ̂ r ,
E p 1 ( x , y , 0 ) = E 0 2 w 0 L p 1 [ 2 ( x 2 + y 2 ) w 0 2 ] exp [ ( x 2 + y 2 w 0 2 ) ] [ x e ̂ x + y e ̂ y ] = E p 1 x ( x , y , 0 ) e ̂ x + E p 1 y ( x , y , 0 ) e ̂ y ,
E p 1 x ( x , y , 0 ) = E 0 2 w 0 L p 1 [ 2 ( x 2 + y 2 ) w 0 2 ] exp [ ( x 2 + y 2 w 0 2 ) ] x ,
E p 1 y ( x , y , 0 ) = E 0 2 w 0 L p 1 [ 2 ( x 2 + y 2 ) w 0 2 ] exp [ ( x 2 + y 2 w 0 2 ) ] y .
E p 1 x ( r ) = 1 2 π + E p 1 x ( x 0 , y 0 , 0 ) G ( r , r 0 ) z d x 0 d y 0 ,
E p 1 y ( r ) = 1 2 π + E p 1 y ( x 0 , y 0 , 0 ) G ( r , r 0 ) z d x 0 d y 0 ,
E p 1 z ( r ) = 1 2 π + [ E p 1 x ( x 0 , y 0 , 0 ) G ( r , r 0 ) x + E p 1 y ( x 0 , y 0 , 0 ) G ( r , r 0 ) y ] d x 0 d y 0 ,
G ( r , r 0 ) = exp ( i k r r 0 ) r r 0 ,
r r 0 r + x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r .
E p 1 x ( r ) = ( 1 ) p + 1 E 0 2 k 2 z x 4 w 0 r 3 [ 1 w 0 2 i k ( 2 r ) ] 2 [ 1 w 0 2 + i k ( 2 r ) 1 w 0 2 i k ( 2 r ) ] p × exp [ i k r k 2 ( x 2 + y 2 ) ( 4 r 2 ) 1 w 0 2 i k ( 2 r ) ] L p 1 [ k 2 ( x 2 + y 2 ) ( 2 r 2 ) 1 w 0 2 + k 2 w 0 2 ( 4 r 2 ) ] ,
E p 1 y ( r ) = ( 1 ) p + 1 E 0 2 k 2 z y 4 w 0 r 3 [ 1 w 0 2 i k ( 2 r ) ] 2 [ 1 w 0 2 + i k ( 2 r ) 1 w 0 2 i k ( 2 r ) ] p × exp [ i k r k 2 ( x 2 + y 2 ) ( 4 r 2 ) 1 w 0 2 i k ( 2 r ) ] L p 1 [ k 2 ( x 2 + y 2 ) ( 2 r 2 ) 1 w 0 2 + k 2 w 0 2 ( 4 r 2 ) ] ,
E p 1 z ( r ) = i 2 E 0 k ( 2 w 0 r 2 ) [ 1 w 0 2 i k ( 2 r ) ] 2 exp [ i k r k 2 ( x 2 + y 2 ) ( 4 r 2 ) 1 w 0 2 i k ( 2 r ) ] × { ( 1 ) p + 1 i k ( x 2 + y 2 ) 2 r [ 1 w 0 2 + i k ( 2 r ) 1 w 0 2 i k ( 2 r ) ] p × L p 1 [ k 2 ( x 2 + y 2 ) ( 2 r 2 ) 1 w 0 2 + k 2 w 0 2 ( 4 r 2 ) ] + m = 0 p ( 1 ) m + 1 ( p + 1 ) ! ( p m ) ! m ! × [ 1 w 0 2 1 w 0 2 i k ( 2 r ) ] m L m + 1 [ k 2 ( x 2 + y 2 ) ( 4 r 2 ) 1 w 0 2 i k ( 2 r ) ] } ,
E 01 x ( r ) = E 0 2 k 2 z x 4 w 0 r 3 [ 1 w 0 2 i k ( 2 r ) ] 2 exp [ i k r k 2 ( x 2 + y 2 ) ( 4 r 2 ) 1 w 0 2 i k ( 2 r ) ] ,
E 01 y ( r ) = E 0 2 k 2 z y 4 w 0 r 3 [ 1 w 0 2 i k ( 2 r ) ] 2 exp [ i k r k 2 ( x 2 + y 2 ) ( 4 r 2 ) 1 w 0 2 i k ( 2 r ) ] ,
E 01 z ( r ) = i 2 E 0 k ( 2 w 0 r 2 ) [ 1 w 0 2 i k ( 2 r ) ] 2 exp [ i k r k 2 ( x 2 + y 2 ) ( 4 r 2 ) 1 w 0 2 i k ( 2 r ) ] × { i k ( x 2 + y 2 ) 2 r + L 1 [ k 2 ( x 2 + y 2 ) ( 4 r 2 ) 1 w 0 2 i k ( 2 r ) ] } .
E p 1 x p ( r ) = E 0 2 x w 0 ( 1 + i z z R ) 2 ( 1 i z z R 1 + i z z R ) p exp [ i k z ( x 2 + y 2 ) w 0 2 1 + i z z R ] L p 1 [ 2 ( x 2 + y 2 ) w 0 2 1 + z 2 z R 2 ] ,
E p 1 y p ( r ) = E 0 2 y w 0 ( 1 + i z z R ) 2 ( 1 i z z R 1 + i z z R ) p exp [ i k z ( x 2 + y 2 ) w 0 2 1 + i z z R ] L p 1 [ 2 ( x 2 + y 2 ) w 0 2 1 + z 2 z R 2 ] ,
E p 1 z p ( r ) = i 2 2 E 0 ( k w 0 ) ( 1 + i z z R ) 2 exp [ i k z ( x 2 + y 2 ) w 0 2 1 + i z z R ] × { i k ( x 2 + y 2 ) 2 z ( 1 i z z R 1 + i z z R ) p L p 1 [ 2 ( x 2 + y 2 ) w 0 2 1 + z 2 z R 2 ] + m = 0 p ( 1 ) m ( p + 1 ) ! ( p m ) ! m ! × ( 2 1 i z R z ) m L m + 1 [ ( x 2 + y 2 ) w 0 2 z 2 z R 2 ( 1 i z R z ) ] } ,
E 01 x p ( r ) = E 0 2 x w 0 ( 1 + i z z R ) 2 exp [ i k z ( x 2 + y 2 ) w 0 2 1 + i z z R ] ,
E 01 y p ( r ) = E 0 2 y w 0 ( 1 + i z z R ) 2 exp [ i k z ( x 2 + y 2 ) w 0 2 1 + i z z R ] ,
E 01 z p ( r ) = E 0 2 2 i k w 0 ( 1 + i z z R ) 2 exp [ i k z ( x 2 + y 2 ) w 0 2 1 + i z z R ] L 1 [ ( x 2 + y 2 ) w 0 2 1 + i z z R ] .
E p 1 x ( 0 , 0 , z ) = E 0 2 2 π w 0 0 2 π cos θ 0 d θ 0 0 + ρ 0 2 L p 1 ( 2 ρ 0 2 w 0 2 ) exp ( ρ 0 2 w 0 2 ) z [ exp [ i k ( ρ 0 2 + z 2 ) 1 2 ] ( ρ 0 2 + z 2 ) 1 2 ] d ρ 0 ,
E p 1 y ( 0 , 0 , z ) = E 0 2 2 π w 0 0 2 π sin θ 0 d θ 0 0 + ρ 0 2 L p 1 ( 2 ρ 0 2 w 0 2 ) exp ( ρ 0 2 w 0 2 ) z [ exp ( i k ρ 0 2 + z 2 ) ρ 0 2 + z 2 ] d ρ 0 ,
E p 1 z ( 0 , 0 , z ) = E 0 2 w 0 z 0 + ρ 0 3 L p 1 ( 2 ρ 0 2 w 0 2 ) exp ( ρ 0 2 w 0 2 ) z [ exp ( i k ρ 0 2 + z 2 ) ρ 0 2 + z 2 ] d ρ 0 ,
E p 1 z ( 0 , 0 , z ) = E 0 2 w 0 m = 0 p ( p + 1 ) ! ( m + 1 ) ! ( p m ) ! m ! ( 2 w 0 2 ) m × m + 1 t m + 1 z 2 0 + exp ( t ρ 0 2 ) [ exp ( i k ρ 0 2 + z 2 ) ρ 0 2 + z 2 ] d ρ 0 2 ,
E p 1 z ( 0 , 0 , z ) = E 0 2 w 0 m = 0 p ( p + 1 ) ! ( m + 1 ) ! ( p m ) ! m ! ( 2 w 0 2 ) m × m + 1 t m + 1 { π t exp ( k 2 4 t + z 2 t ) [ 1 e r f ( z t i k 2 t ) ] } ,
E 01 z ( 0 , 0 , z ) = E 0 2 exp ( i k z ) { π ( z 2 + w 0 2 2 + z R 2 ) w 0 2 exp [ ( z i z R ) 2 w 0 2 ] [ 1 Erf ( z i z R w 0 ) ] z + i z R w 0 } .

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