Abstract

A radially polarized elegant beam (RPEB) in which the argument of the Laguerre part is complex is introduced and studied. The nonparaxial and paraxial propagation properties of the RPEB are demonstrated analytically and numerically in free space in terms of the vectorial Rayleigh–Sommerfeld formulas. The electric field distribution of the RPEB preserves the radial polarization at any propagation position. For the lowest-order RPEB and the lowest-order radially polarized beam cases, the general expressions for the nonparaxial propagation field are identical in free space. The exact on-axis expression of the RPEB has been provided in closed-form terms for an arbitrary transverse beam size.

© 2007 Optical Society of America

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2006

2005

2004

2003

2002

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]

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]

2001

Z. Bomzon, V. Kleiner, and E. Hasman, "Formation of radially and azimuthally polarized light using space-variant 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]

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, "Characterising elegant and standard Hermite-Gaussian beam modes," Opt. Commun. 191, 173-179 (2001).
[CrossRef]

2000

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]

A. Ciattoni, P. D. Porto, B. Crosignani, and A. Yariv, "Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation," J. Opt. Soc. Am. B 17, 809-819 (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

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]

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]

1998

1997

1996

1990

1988

1986

1985

1977

1973

A. E. Siegman, "Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions," J. Opt. Soc. Am. A 63, 1093-1094 (1973).
[CrossRef]

Aït-Ameur, K.

Armstrong, D. J.

Arscott, F. M.

F. M. Arscott, Periodic Differential Equations (Pergamon, 1964).

Bandres, M. A.

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 space-variant 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).

Casperson, L. W.

Chaumet, P. C.

Chen, X. Z.

D. M. Deng, H. Guo, X. Z. Chen, and H. J. Kong, "Characteristics of coherent and incoherent off-axis elegant Hermite-Gaussian beam combinations," J. Opt. A Pure Appl. Opt. 5, 489-494 (2003).
[CrossRef]

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]

A. Ciattoni, P. D. Porto, B. Crosignani, and A. Yariv, "Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation," J. Opt. Soc. Am. B 17, 809-819 (2000).
[CrossRef]

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]

Crosignani, B.

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]

Deng, D. M.

D. M. Deng, "Propagation of elegant Hermite cosine Gaussian laser beams," Opt. Commun. 259, 409-414 (2006).
[CrossRef]

D. M. Deng, "Nonparaxial propagation of radially polarized light beams," J. Opt. Soc. Am. B 23, 1228-1234 (2006).
[CrossRef]

D. M. Deng, H. Guo, X. Z. Chen, and H. J. Kong, "Characteristics of coherent and incoherent off-axis elegant Hermite-Gaussian beam combinations," J. Opt. A Pure Appl. Opt. 5, 489-494 (2003).
[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]

Duan, K. L.

Fainman, Y.

Feit, M. D.

Felsen, L. B.

Fleck, J. A.

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]

Fukumitsu, O.

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]

Guo, H.

D. M. Deng, H. Guo, X. Z. Chen, and H. J. Kong, "Characteristics of coherent and incoherent off-axis elegant Hermite-Gaussian beam combinations," J. Opt. A Pure Appl. Opt. 5, 489-494 (2003).
[CrossRef]

Gutierrez-Vega, J. C.

Hall, D. G.

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 space-variant 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]

Khonina, S. N.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, "Generalized Hermite beams in free space," Optik (Stuttgart) 108, 20-26 (1998).

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 space-variant subwavelength metal strip grating," Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

Kong, H. J.

D. M. Deng, H. Guo, X. Z. Chen, and H. J. Kong, "Characteristics of coherent and incoherent off-axis elegant Hermite-Gaussian beam combinations," J. Opt. A Pure Appl. Opt. 5, 489-494 (2003).
[CrossRef]

Kostenbauder, A.

Kotlyar, V. V.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, "Generalized Hermite beams in free space," Optik (Stuttgart) 108, 20-26 (1998).

Kozawa, Y. C.

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]

Levy, U.

Lü, B. D.

Luneburg, R. K.

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

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]

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]

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]

Okamotoa, M.

M. Okamotoa and H. Sadada, "Generation of optical vortices by converting elegant Hermite Gaussian beams," Jpn. J. Appl. Phys. Part 1 44, 1743-1747 (2005).
[CrossRef]

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]

Pang, L.

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]

Piper, J. A.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, "Characterising elegant and standard Hermite-Gaussian beam modes," Opt. Commun. 191, 173-179 (2001).
[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]

A. Ciattoni, P. D. Porto, B. Crosignani, and A. Yariv, "Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation," J. Opt. Soc. Am. B 17, 809-819 (2000).
[CrossRef]

Roch, J.-F.

Roth, M. S.

Sadada, H.

M. Okamotoa and H. Sadada, "Generation of optical vortices by converting elegant Hermite Gaussian beams," Jpn. J. Appl. Phys. Part 1 44, 1743-1747 (2005).
[CrossRef]

Saghafi, S.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, "Characterising elegant and standard Hermite-Gaussian beam modes," Opt. Commun. 191, 173-179 (2001).
[CrossRef]

Salamin, Y. I.

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]

Sato, S. C.

Savchenko, A. Yu.

Seshadri, S. R.

Sheppard, C. J. R.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, "Characterising elegant and standard Hermite-Gaussian beam modes," Opt. Commun. 191, 173-179 (2001).
[CrossRef]

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]

Shin, S. Y.

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]

Siegman, A. E.

A. Kostenbauder, Y. Sun, and A. E. Siegman, "Eigenmode expansions using biorthogonal functions: complex-valued Hermite-Gaussians," J. Opt. Soc. Am. A 14, 1780-1790 (1997).
[CrossRef]

A. E. Siegman, "Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions," J. Opt. Soc. Am. A 63, 1093-1094 (1973).
[CrossRef]

A. E. Siegman, Lasers (University Science, 1986).

Smith, A. V.

Soifer, V. A.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, "Generalized Hermite beams in free space," Optik (Stuttgart) 108, 20-26 (1998).

Sun, Y.

Swartzlander, G. A.

Takenaka, T.

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.

Tover, A. A.

Treussart, F.

Tsai, C.-H.

Varin, C.

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[CrossRef]

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[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Relative irradiance distribution of the RPEB versus x λ and y λ in the z = 20 z R plane for (a) nonparaxial and (b) paraxial propagation. Solid and dashed curves in (c) are the cross-section irradiance distributions at y = 0 corresponding to (a) and (b), respectively. The parameters are chosen as n = 2 and f = 0.1 .

Fig. 2
Fig. 2

Relative irradiance distribution of the RPEB versus x λ and y λ in the plane z = 20 z R for (a) nonparaxial and (b) paraxial propagation. Solid and dashed curves in (c) are the cross-section irradiance distributions at y = 0 corresponding to (a) and (b). The parameters are chosen such that n = 3 , f = 0.1 .

Fig. 3
Fig. 3

Energy flux distribution of the RPEB versus x λ and y λ for (a) nonparaxial and (b) paraxial propagation in the z = 20 z R plane. Solid and dashed curves in (c) are the cross-section energy flux distributions at y = 0 corresponding to (a) and (b). The parameters are chosen such that n = 2 , f = 0.1 .

Fig. 4
Fig. 4

Energy flux distribution of the RPEB versus x λ and y λ for (a) nonparaxial and (b) paraxial propagation in the z = 20 z R plane. Solid and dashed curves in (c) are the cross-section energy flux distributions at y = 0 in (a) and (b). The parameters are chosen to be n = 3 , f = 0.1 .

Fig. 5
Fig. 5

Behaviors of the modulus of the relative error as a function of z z R for the RPEB with (a) f = 0.1 , the beam order n = 0 (solid curve), n = 1 (dashed curve), n = 2 (dotted curve) and n = 3 (dashed–dotted curve); (b) the beam order n = 2 , f = 0.1 (solid curve), f = 1 ( 2 π ) (dashed curve), f = 0.2 (dotted curve), and f = 2 ( 3 π ) (dashed–dotted curve).

Fig. 6
Fig. 6

Behaviors of the exact normalized on-axis field amplitudes as a function of z z R for the RPEB, for different values of the orders: n = 2 (solid curves) and n = 3 (dotted curves), together with the paraxial prediction: n = 2 (dashed curves) and n = 3 (dashed–dotted curves) with (a) f = 0.1 and (b) f = 0.2 .

Equations (20)

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E n 1 ( r , 0 ) = E 0 r w 0 L n 1 ( r 2 w 0 2 ) exp ( r 2 w 0 2 ) e ̂ r ,
E n 1 ( x , y , 0 ) = E 0 w 0 L n 1 [ x 2 + y 2 w 0 2 ] exp [ x 2 + y 2 w 0 2 ] [ x e ̂ x + y e ̂ y ] = E n 1 x ( x , y , 0 ) e ̂ x + E n 1 y ( x , y , 0 ) e ̂ y ,
E n 1 x ( x , y , 0 ) = E 0 w 0 L n 1 [ x 2 + y 2 w 0 2 ] exp [ x 2 + y 2 w 0 2 ] x ,
E n 1 y ( x , y , 0 ) = E 0 w 0 L n 1 [ x 2 + y 2 w 0 2 ] exp [ x 2 + y 2 w 0 2 ] y .
E n 1 x ( r ) = 1 2 π + E n 1 x ( x 0 , y 0 , 0 ) z [ exp ( i k R ) R ] d x 0 d y 0 ,
E n 1 y ( r ) = 1 2 π + E n 1 y ( x 0 , y 0 , 0 ) z [ exp ( i k R ) R ] d x 0 d y 0 ,
E n 1 z ( r ) = 1 2 π + { E n 1 x ( x 0 , y 0 , 0 ) x [ exp ( i k R ) R ] + E n 1 y ( x 0 , y 0 , 0 ) y [ exp ( i k R ) R ] } + d x 0 d y 0 ,
R r + x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r .
E n 1 ( r ) = E 0 ( i z R r 1 i z R r ) 2 exp [ i k r ρ 2 ( 4 r 2 f 2 ) 1 i z R r ] × { z w 0 r ( i z R r 1 i z R r ) n L n 1 [ i k ρ 2 ( 2 r ) 1 i z R r ] ( x e ̂ x + y e ̂ y ρ 2 z e ̂ z ) + 2 i f e ̂ z m = 0 n ( 1 ) m ( n + 1 ) ! ( n m ) ! m ! × ( 1 1 i z R r ) m L m + 1 [ ρ 2 ( 4 f 2 r 2 ) 1 i z R r ] } ,
H n 1 ( r ) = ϵ μ i E 0 z R w 0 8 r ( i z R r ) 4 exp [ i k r ρ 2 ( 4 r 2 f 2 ) 1 i z R r ] × { ( i z R r 1 i z R r ) n [ 2 k ρ 2 ( 2 z R + i r ) L n 1 2 [ i k ρ 2 ( 2 r ) 1 i z R r ] + ( i k ( 2 ( n 2 ) w 0 2 r + 4 r 3 2 z R 2 ( ρ 2 + 2 z 2 ) r ) 4 n r 2 + ( 2 w 0 2 + 4 r 2 ) f 2 ) L n 1 [ i k ρ 2 ( 2 r ) 1 i z R r ] ] i k m = 0 n ( 1 ) m r 2 × ( n + 1 ) ! ( n m ) ! m ! ( 1 1 i z R r ) m [ [ k ( 4 z R 2 ρ 2 4 ( 2 + m ) w 0 2 r 2 + 8 r 4 ) + 2 i r ( 8 r 2 + ( m w 0 2 4 ρ 2 2 z 2 ) f 2 ) ] × L m + 1 [ ρ 2 ( 4 f 2 r 2 ) 1 i z R r ] + ( 4 i r z 2 + 2 z R ( ρ 2 + 2 z 2 ) ) f 2 × L m 1 [ ρ 2 ( 4 f 2 r 2 ) 1 i z R r ] ] } ( y e ̂ x x e ̂ y ) .
E 01 ( r ) = E 0 ( i z R r 1 i z R r ) 2 exp [ i k r ρ 2 ( 4 r 2 f 2 ) 1 i z R r ] × { z w 0 r ( x e ̂ x + y e ̂ y ρ 2 z e ̂ z ) + 2 i f e ̂ z × L 1 [ ρ 2 ( 4 f 2 r 2 ) 1 i z R r ] } ,
H 01 ( r ) = ϵ μ i E 0 w 0 3 64 r 4 ( i z R + r ) 5 exp [ i k r ρ 2 ( 4 r 2 f 2 ) 1 i z R r ] × { 2 i r [ 32 r 4 + ( 2 w 0 2 ρ 2 ρ 4 9 ρ 2 z 2 6 z 4 ) f 4 + 8 k 2 r 2 ( w 0 4 + r 4 w 0 2 ( 5 ρ 2 + 2 z 2 ) ) ] + k [ w 0 2 z 2 ( 3 ρ 2 2 z 2 ) f 4 + 32 r 6 32 z R 2 r 2 ( 5 ρ 2 + z 2 ) + 8 w 0 2 r 4 ( 8 + k 2 ( 2 ρ 2 + 3 z 2 ) ) ] } ( y e ̂ x x e ̂ y ) .
E n 1 p ( r ) = E 0 w 0 ( 1 + i z z R ) 2 exp [ i k z ρ 2 w 0 2 1 + i z z R ] × { 1 ( 1 + i z z R ) n L n 1 [ ρ 2 w 0 2 1 + i z z R ] ( x e ̂ x + y e ̂ y ρ 2 z e ̂ z ) + 2 i f e ̂ z m = 0 n ( 1 ) m ( n + 1 ) ! ( n m ) ! m ! × 1 ( 1 i z R z ) m L m + 1 [ i ρ 2 w 0 2 z z R ( 1 + i z z R ) ] } ,
H n 1 p ( r ) = ϵ μ i E 0 z R w 0 8 z ( i z R z ) 4 exp [ i k z ρ 2 w 0 2 1 + i z z R ] × { 1 ( 1 + i z z R ) n [ 2 k ρ 2 ( 2 z R + i z ) L n 1 2 [ ρ 2 w 0 2 1 + i z z R ] + ( i k ( 2 ( n 2 ) w 0 2 z + 4 z 3 2 z R 2 ( ρ 2 + 2 z 2 ) z ) 4 n z 2 + ( 2 w 0 2 + 4 z 2 ) f 2 ) L n 1 [ ρ 2 w 0 2 1 + i z z R ] ] i k m = 0 n ( 1 ) m z 2 × ( n + 1 ) ! ( n m ) ! m ! 1 ( 1 i z R z ) m [ [ k ( 4 z R 2 ρ 2 4 ( 2 + m ) w 0 2 z 2 + 8 z 4 ) + 2 i z ( 8 z 2 + ( m w 0 2 4 ρ 2 2 z 2 ) f 2 ) ] × L m + 1 [ i ρ 2 w 0 2 z z R ( 1 + i z z R ) ] + ( 4 i z 3 + 2 z R ( ρ 2 + 2 z 2 ) ) f 2 × L m 1 [ i ρ 2 w 0 2 z z R ( 1 + i z z R ) ] ] } ( y e ̂ x x e ̂ y ) .
E n 1 x ( 0 , 0 , z ) = E 0 2 π w 0 0 2 π cos θ 0 d θ 0 0 + ρ 0 2 L n 1 ( 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 n 1 y ( 0 , 0 , z ) = E 0 2 π w 0 0 2 π sin θ 0 d θ 0 0 + ρ 0 2 L n 1 ( 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 n 1 z ( 0 , 0 , z ) = E 0 w 0 z 0 + ρ 0 3 L n 1 ( 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 n 1 z ( 0 , 0 , z ) = E 0 w 0 m = 0 n ( n + 1 ) ! ( m + 1 ) ! ( n m ) ! m ! ( 1 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 n 1 z ( 0 , 0 , z ) = E 0 w 0 m = 0 n ( n + 1 ) ! ( m + 1 ) ! ( n m ) ! m ! ( 1 w 0 2 ) m × m + 1 t m + 1 { π t exp ( k 2 4 t + z 2 t ) [ 1 Erf ( z t i k 2 t ) ] } ,
E 01 z ( 0 , 0 , z ) = E 0 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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