Abstract

We derive a full field solution for Laguerre–Gaussian beams consistent with the Helmholtz equation using the angular spectrum method. Field components are presented as an order expansion in the ratio of the wave length to the beam waist, f=λ/(2πw0), which is typically small. The result is then generalized to a beam of arbitrary polarization. This result is then used to reproduce the signature angular momentum properties of Laguerre–Gaussian beams in the paraxial limit. The subsequent higher-order term is similarly obtained, which does not display a clear separation of orbital and spin angular momentum components.

© 2011 Optical Society of America

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  1. L. Allen, M. W. Beijersbergen, R. J.C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef] [PubMed]
  2. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
    [CrossRef]
  3. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601–053604 (2002).
    [CrossRef] [PubMed]
  4. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B: Quantum Semiclass. Opt. 4, S7–S16 (2002).
    [CrossRef]
  5. M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009).
    [CrossRef]
  6. K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momentum and spin–orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
    [CrossRef]
  7. C. F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80, 063814 (2009).
    [CrossRef]
  8. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 32–100 (1950).
  9. P. C. Clemmow, The Plane Wave Spectrum Representations of Electromagnetic Fields (Pergamon, 1966).
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  11. G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
    [CrossRef]
  12. L. Cicchitelli, H. Hora, and R. Postle, “Longitudinal field components for laser beams in vacuum,” Phys. Rev. A 41, 3727–3732(1990).
    [CrossRef] [PubMed]
  13. B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E 58, 3719–3732 (1998).
    [CrossRef]
  14. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A 19, 404–412 (2002).
    [CrossRef]
  15. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  16. S. Yan and B. Yao, “Description of a radially polarized Laguerre–Gaussian beam beyond the paraxial approximation,” Opt. Lett. 32, 3367–3369 (2007).
    [CrossRef] [PubMed]
  17. G. Zhou, “Analytically vectorial structure of an apertured Laguerre–Gaussian beam in the far-field,” Opt. Lett. 31, 2616–2618 (2006).
    [CrossRef] [PubMed]
  18. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007).
  19. F.W. J.Olver, D.W.Lozier, R.F.Boisvert, and C.W.Clark (eds.), in NIST Handbook of Mathematical Functions(Cambridge University, 2010), p. 260.
  20. A. Cerjan and C. Cerjan, “Analytic solution of flat-top Gaussian and Laguerre–Gaussian laser field components,” Opt. Lett. 35, 3465–3467 (2010).
    [CrossRef] [PubMed]
  21. S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
    [CrossRef]
  22. C. F. Li, T. T. Wang, and S. Y. Yang, “Comment on ‘Orbital angular momentum and nonparaxial light beams’,” Opt. Commun. 283, 2787–2788 (2010).
    [CrossRef]
  23. E. D. Rainville, “Laguerre polynomials,” in Special Functions (Chelsea, 1960), p. 209.
  24. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre–Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
    [CrossRef]

2010 (4)

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momentum and spin–orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[CrossRef]

F.W. J.Olver, D.W.Lozier, R.F.Boisvert, and C.W.Clark (eds.), in NIST Handbook of Mathematical Functions(Cambridge University, 2010), p. 260.

A. Cerjan and C. Cerjan, “Analytic solution of flat-top Gaussian and Laguerre–Gaussian laser field components,” Opt. Lett. 35, 3465–3467 (2010).
[CrossRef] [PubMed]

C. F. Li, T. T. Wang, and S. Y. Yang, “Comment on ‘Orbital angular momentum and nonparaxial light beams’,” Opt. Commun. 283, 2787–2788 (2010).
[CrossRef]

2009 (2)

C. F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80, 063814 (2009).
[CrossRef]

M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009).
[CrossRef]

2007 (2)

2006 (1)

2002 (3)

C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A 19, 404–412 (2002).
[CrossRef]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601–053604 (2002).
[CrossRef] [PubMed]

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B: Quantum Semiclass. Opt. 4, S7–S16 (2002).
[CrossRef]

2000 (1)

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre–Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

1998 (1)

B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E 58, 3719–3732 (1998).
[CrossRef]

1994 (1)

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J.C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

1990 (1)

L. Cicchitelli, H. Hora, and R. Postle, “Longitudinal field components for laser beams in vacuum,” Phys. Rev. A 41, 3727–3732(1990).
[CrossRef] [PubMed]

1979 (1)

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1968 (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

1966 (1)

P. C. Clemmow, The Plane Wave Spectrum Representations of Electromagnetic Fields (Pergamon, 1966).

1960 (1)

E. D. Rainville, “Laguerre polynomials,” in Special Functions (Chelsea, 1960), p. 209.

1950 (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 32–100 (1950).

1936 (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[CrossRef]

Agrawal, G. P.

Aiello, A.

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momentum and spin–orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[CrossRef]

Allen, L.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601–053604 (2002).
[CrossRef] [PubMed]

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre–Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J.C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Alonso, M. A.

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momentum and spin–orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[CrossRef]

Barnett, S. M.

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B: Quantum Semiclass. Opt. 4, S7–S16 (2002).
[CrossRef]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[CrossRef]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J.C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009).
[CrossRef]

Beth, R. A.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[CrossRef]

Bliokh, K. Y.

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momentum and spin–orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[CrossRef]

Bouwkamp, C. J.

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 32–100 (1950).

Cerjan, A.

Cerjan, C.

Chen, C. G.

Cicchitelli, L.

L. Cicchitelli, H. Hora, and R. Postle, “Longitudinal field components for laser beams in vacuum,” Phys. Rev. A 41, 3727–3732(1990).
[CrossRef] [PubMed]

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representations of Electromagnetic Fields (Pergamon, 1966).

Ferrera, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007).

Heilmann, R. K.

Hora, H.

L. Cicchitelli, H. Hora, and R. Postle, “Longitudinal field components for laser beams in vacuum,” Phys. Rev. A 41, 3727–3732(1990).
[CrossRef] [PubMed]

Konkola, P. T.

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Li, C. F.

C. F. Li, T. T. Wang, and S. Y. Yang, “Comment on ‘Orbital angular momentum and nonparaxial light beams’,” Opt. Commun. 283, 2787–2788 (2010).
[CrossRef]

C. F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80, 063814 (2009).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

MacVicar, I.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601–053604 (2002).
[CrossRef] [PubMed]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mora, P.

B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E 58, 3719–3732 (1998).
[CrossRef]

O’Neil, A. T.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601–053604 (2002).
[CrossRef] [PubMed]

Ostrovskaya, E. A.

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momentum and spin–orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[CrossRef]

Padgett, M. J.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601–053604 (2002).
[CrossRef] [PubMed]

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre–Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

Pattanayak, D. N.

Postle, R.

L. Cicchitelli, H. Hora, and R. Postle, “Longitudinal field components for laser beams in vacuum,” Phys. Rev. A 41, 3727–3732(1990).
[CrossRef] [PubMed]

Quesnel, B.

B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E 58, 3719–3732 (1998).
[CrossRef]

Rainville, E. D.

E. D. Rainville, “Laguerre polynomials,” in Special Functions (Chelsea, 1960), p. 209.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007).

Schattenburg, M. L.

Spreeuw, R. J.C.

L. Allen, M. W. Beijersbergen, R. J.C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Wang, T. T.

C. F. Li, T. T. Wang, and S. Y. Yang, “Comment on ‘Orbital angular momentum and nonparaxial light beams’,” Opt. Commun. 283, 2787–2788 (2010).
[CrossRef]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J.C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Yan, S.

Yang, S. Y.

C. F. Li, T. T. Wang, and S. Y. Yang, “Comment on ‘Orbital angular momentum and nonparaxial light beams’,” Opt. Commun. 283, 2787–2788 (2010).
[CrossRef]

Yao, B.

Zhou, G.

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

M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009).
[CrossRef]

J. Opt. B: Quantum Semiclass. Opt. (1)

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B: Quantum Semiclass. Opt. 4, S7–S16 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (3)

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[CrossRef]

C. F. Li, T. T. Wang, and S. Y. Yang, “Comment on ‘Orbital angular momentum and nonparaxial light beams’,” Opt. Commun. 283, 2787–2788 (2010).
[CrossRef]

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre–Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[CrossRef]

Phys. Rev. A (5)

L. Allen, M. W. Beijersbergen, R. J.C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momentum and spin–orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[CrossRef]

C. F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80, 063814 (2009).
[CrossRef]

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

L. Cicchitelli, H. Hora, and R. Postle, “Longitudinal field components for laser beams in vacuum,” Phys. Rev. A 41, 3727–3732(1990).
[CrossRef] [PubMed]

Phys. Rev. E (1)

B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E 58, 3719–3732 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601–053604 (2002).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 32–100 (1950).

Other (5)

P. C. Clemmow, The Plane Wave Spectrum Representations of Electromagnetic Fields (Pergamon, 1966).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007).

F.W. J.Olver, D.W.Lozier, R.F.Boisvert, and C.W.Clark (eds.), in NIST Handbook of Mathematical Functions(Cambridge University, 2010), p. 260.

E. D. Rainville, “Laguerre polynomials,” in Special Functions (Chelsea, 1960), p. 209.

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Equations (58)

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E x ( x , y , 0 ) = E 0 ( 2 ρ w 0 ) l L p ( l ) ( 2 ρ 2 w 0 2 ) exp [ ρ 2 w 0 2 ] exp [ i l ϕ ]
A ( b , θ ) = E 0 ( k 2 π ) 2 0 0 2 π E x ( x , y , 0 ) exp [ i k b · ρ ] ρ d ρ d ϕ ,
A ( b , θ ) = E 0 ( k 2 π ) 2 0 0 2 π ( 2 ρ w 0 ) l L p ( l ) ( 2 ρ 2 w 0 2 ) exp [ ρ 2 w 0 2 ] exp [ i l ϕ ] × exp [ i k b ρ cos ( ϕ θ ) ] ρ d ρ d ϕ ,
0 2 π exp [ i l ϕ + i k b ρ cos ( ϕ θ + π ) ] d ϕ = 2 π exp [ i π 2 l ] J l ( k b ρ ) exp [ i l θ ] ,
L p ( l ) ( 2 ρ 2 w 0 2 ) = α = 0 p ( 1 ) α ( p + l p α ) 1 α ! ( 2 ρ w 0 ) 2 α ,
A ( b , θ ) = E 0 k 2 2 π exp [ i l θ ] exp [ i l π 2 ] α = 0 p ( 1 ) α ( p + l p α ) 1 α ! 0 ( 2 ρ w 0 ) 2 α + l exp [ ρ 2 w 0 2 ] J l ( k b ρ ) ρ d ρ .
A ( b , θ ) = E 0 2 π f 2 exp [ i l θ ] exp [ i l π 2 ] α = 0 p ( 1 ) α ( p + l p α ) ( 2 ) 2 α + l 2 × ( b 2 f ) l exp [ b 2 4 f 2 ] L α ( l ) ( b 2 4 f 2 ) ,
E x ( x , y , z ) = + + A ( b , θ ) exp [ i k ( b · ρ + m z ) ] d 2 b ,
E x ( x , y , z ) = E 0 2 π f 2 exp [ i l π 2 ] α = 0 p ( 1 ) α ( p + l p α ) ( 2 ) 2 α + l 2 0 1 0 2 π ( b 2 f ) l exp [ b 2 4 f 2 ] L α ( l ) ( b 2 4 f 2 ) exp [ i l θ ] exp [ i k b · ρ ] exp [ i k m z ] b d b d θ .
L α ( l ) ( b 2 4 f 2 ) = β = 0 α ( 1 ) β ( α + l α β ) 1 β ! ( b 2 f ) 2 β ,
exp [ i k z 1 b 2 ] = n = 0 1 n ! ( b 2 2 ) n ( k z ) n + 1 h n 1 ( 1 ) ( k z ) ,
E x ( x , y , z ) = E 0 exp [ i l ϕ ] n = 0 ( k z ) n + 1 h n 1 ( 1 ) ( k z ) α = 0 p β = 0 α ( 1 ) α + β ( p + l p α ) ( α + l α β ) × 1 n ! β ! I n , l , α , β ( ρ ) ,
I n , l , α , β ( ρ ) = ( 2 ) 2 α + 2 n + l 2 1 f 2 β + l + 2 0 1 ( b 2 ) l + 2 n + 2 β exp [ b 2 4 f 2 ] J l ( k b ρ ) b d b .
I n , l , α , β ( ρ ) = ( 2 ) 2 α + 2 n + l 2 1 f 2 β + l + 2 ( 0 ( b 2 ) l + 2 n + 2 β exp [ b 2 4 f 2 ] J l ( k b ρ ) b d b 1 ( b 2 ) l + 2 n + 2 β exp [ b 2 4 f 2 ] J l ( k b ρ ) b d b ) ,
I n , l , α , β ( ρ ) = f 2 n 2 n + α ( n + β ) ! ( 2 ρ w 0 ) l L n + β ( l ) ( ρ 2 w 0 2 ) exp [ ρ 2 w 0 2 ] .
E x ( x , y , z ) = E 0 n = 0 f 2 n ( k z ) n + 1 h n 1 ( 1 ) ( k z ) α = 0 p β = 0 α ( 1 ) α + β ( p + l p α ) ( α + l α β ) ( n + β n ) × 2 n + α ( 2 ρ w 0 ) l L n + β ( l ) ( ρ 2 w 0 2 ) exp [ ρ 2 w 0 2 ] exp [ i l ϕ ] .
I error ( r ) = ( 2 ) 2 α + 2 n + l 2 1 f 2 β + l + 2 1 ( b 2 ) l + 2 n + 2 β exp [ b 2 4 f 2 ] J l ( k b r ) b d b ( 2 ) 2 p + 2 n + l 2 1 f 2 p + l + 2 1 ( b 2 ) l + 2 n + 2 p exp [ b 2 4 f 2 ] b d b ,
I error = 1 2 p + n + l / 2 f p + l / 2 + 1 E n p l / 2 ( 1 4 f 2 ) ,
exp [ i k z 1 b 2 ] exp [ i k z i k z b 2 2 ] ,
0 1 ( b 2 ) l + 2 β exp [ b 2 ( 1 4 f 2 + i k z 2 ) ] J l ( k ρ b ) b d b
( 1 2 i k z ) ( 1 2 ) l + 2 β exp [ i k z ] exp [ ( 1 4 f 2 + i k z 2 ) ] J l ( k ρ ) exp [ i k z 2 ] J l ( k ρ ) 2 i k z
E z ( x , y , z ) = i k x ( I m ) ,
B ( x , y , z ) = 1 ω k [ 2 x y ( I m ) x ^ + ( k 2 m I 2 x 2 ( I m ) ) y ^ + i k y ( I ) z ^ ] ,
I = + + A ( b , θ ) exp [ i k ( b · ρ + m z ) ] d 2 b ,
I m = + + A ( b , θ ) exp [ i k ( b · ρ + m z ) ] m d 2 b ,
m I = + + A ( b , θ ) m exp [ i k ( b · ρ + m z ) ] d 2 b .
exp [ i k z 1 b 2 ] 1 b 2 = i n = 0 1 n ! ( b 2 2 ) n ( k z ) n + 1 h n ( 1 ) ( k z ) ,
1 b 2 exp [ i k z 1 b 2 ] = i n = 0 1 n ! ( b 2 2 ) n ( k z ) n [ k z h n ( 1 ) ( k z ) 2 n h n 1 ( 1 ) ( k z ) ] ,
E ( x , y , z ) = ( α z x ^ + β z y ^ ) E ( x , y , z ) + E z ( x , y , z ) z ^ ,
E z ( x , y , z ) = α z i k x ( I m ) + β z i k y ( I m ) ,
B ( x , y , z ) = α z 1 ω k [ 2 x y ( I m ) x ^ + ( k 2 m I 2 x 2 ( I m ) ) y ^ + i k y ( I ) z ^ ] + β z 1 ω k [ ( 2 y 2 ( I m ) k 2 m I ) x ^ 2 x y ( I m ) y ^ i k x ( I ) z ^ ] .
v 0 , n ( ρ ) = E 0 ( 2 ρ w 0 ) l exp [ ρ 2 w 0 2 ] α = 0 p β = 0 α ( 1 ) α + β ( p + l p α ) ( α + l α β ) ( n + β n ) × 2 n + α L n + β ( l ) ( ρ 2 w 0 2 ) ,
v 1 , n ( ρ ) = E 0 ( 2 ρ w 0 ) l exp [ ρ 2 w 0 2 ] α = 0 p β = 0 α ( 1 ) α + β ( p + l p α ) ( α + l α β ) ( n + β n ) × 2 n + α l + 1 n + l + β + 1 L n + β ( l + 1 ) ( ρ 2 w 0 2 ) ,
v 2 , n ( r ) = E 0 ( 2 ρ w 0 ) l exp [ ρ 2 w 0 2 ] α = 0 p β = 0 α ( 1 ) α + β ( p + l p α ) ( α + l α β ) ( n + β n ) × 2 n + α ( l + 1 ) ( l + 2 ) ( n + l + β + 1 ) ( n + l + β + 2 ) L n + β ( l + 2 ) ( ρ 2 w 0 2 ) ,
E x ( x , y , z ) = α z exp [ i l ϕ ] n = 0 f 2 n ( k z ) n + 1 h n 1 ( 1 ) ( k z ) v 0 , n ( ρ ) ,
E y ( x , y , z ) = β z exp [ i l ϕ ] n = 0 f 2 n ( k z ) n + 1 h n 1 ( 1 ) ( k z ) v 0 , n ( ρ ) ,
E z ( x , y , z ) = exp [ i l ϕ ] n = 0 f 2 n + 1 ( k z ) n + 1 h n ( 1 ) ( k z ) [ ( α z + i β z ) l w 0 ( i y x ρ 2 ) v 0 , n ( ρ ) + ( α z 2 x w 0 + β z 2 y w 0 ) v 1 , n ( ρ ) ] ,
B x ( x , y , z ) = i k ω β z exp [ i l ϕ ] n = 0 f 2 n ( k z ) n [ 2 n h n 1 ( 1 ) ( k z ) k z h n ( 1 ) ( k z ) ] v 0 , n ( ρ ) + i k ω exp [ i l ϕ ] n = 0 f 2 n + 2 ( k z ) n + 1 h n ( 1 ) ( k z ) [ ( α z 4 x y w 0 2 + β z 4 y 2 w 0 2 ) v 2 , n ( ρ ) + ( α z + i β z ) l ( l 1 ) w 0 2 ρ 4 ( i x 2 + 2 x y i y 2 ) v 0 , n ( ρ ) α z 2 l ρ 2 ( i x 2 + 2 x y i y 2 ) v 1 , n ( ρ ) β z ( 4 l ρ 2 ( y 2 + i x y ) + 2 ) v 1 , n ( ρ ) ] ,
B y ( x , y , z ) = i k ω α z exp [ i l ϕ ] n = 0 f 2 n ( k z ) n [ k z h n ( 1 ) ( k z ) 2 n h n 1 ( 1 ) ( k z ) ] v 0 , n ( ρ ) i k ω exp [ i l ϕ ] n = 0 f 2 n + 2 ( k z ) n + 1 h n ( 1 ) ( k z ) [ ( α z 4 x 2 w 0 2 + β z 4 x y w 0 2 ) v 2 , n ( ρ ) + ( β z i α z ) l ( l 1 ) w 0 2 ρ 4 ( i x 2 + 2 x y i y 2 ) v 0 , n ( ρ ) β z 2 l ρ 2 ( i x 2 + 2 x y i y 2 ) v 1 , n ( ρ ) + α z ( 4 l ρ 2 ( i x y x 2 ) 2 ) v 1 , n ( ρ ) ] ,
B z ( x , y , z ) = i k ω exp [ i l ϕ ] n = 0 f 2 n + 1 ( k z ) n + 1 h n 1 ( 1 ) ( k z ) [ ( i α z β z ) l w 0 ( x i y ρ 2 ) v 0 , n ( ρ ) + ( β z 2 x w 0 α z 2 y w 0 ) v 1 , n ( ρ ) ] .
L p ( l ) ( c r 2 w 0 2 ) = α = 0 p β = 0 α ( 1 ) α + β ( p + l p α ) ( α + l α β ) c α L β ( l ) ( r 2 w 0 2 ) ,
u j ( r ) = E 0 exp [ i k z ] exp [ i l ϕ ] ( 2 ρ w 0 ) l exp [ ρ 2 w 0 2 ] L p j ( l + j ) ( 2 ρ 2 w 0 2 ) ,
E x ( x , y , z ) = α z u 0 ( r ) ,
E y ( x , y , z ) = β z u 0 ( r ) ,
E z ( x , y , z ) = i f [ ( α z + i β z ) l w 0 ( x i y ) ρ 2 u 0 ( r ) 2 w 0 ( α z x + β z y ) u 0 ( r ) 4 w 0 ( α z x + β z y ) u 1 ( r ) ] ,
B x ( x , y , z ) = β z k ω u 0 ( r ) ,
B y ( x , y , z ) = α z k ω u 0 ( r ) ,
B z ( x , y , z ) = i f k ω [ ( α z + i β z ) l w 0 ( y + i x ) ρ 2 u 0 ( r ) 2 w 0 ( α z y β z x ) u 0 ( r ) 4 w 0 ( α z y β z x ) u 1 ( r ) ] ,
S x = σ z f k μ ω [ ( l w 0 y ρ 2 2 y w 0 ) | u 0 | 2 4 y w 0 u 0 * u 1 ] l y μ ω ρ 2 | u 0 | 2 ,
S y = σ z f k μ ω [ ( l w 0 x ρ 2 2 x w 0 ) | u 0 | 2 4 x w 0 u 0 * u 1 ] + l x μ ω ρ 2 | u 0 | 2 ,
S z = k μ ω | u 0 | 2 .
j z = ε l ω | u 0 | 2 ε σ z ω [ ( l 2 ρ 2 w 0 2 ) | u 0 | 2 4 ρ 2 w 0 2 u 0 * u 1 ] .
j z = ε [ l ω σ z ρ 2 ω ρ ] | u 0 | 2 ,
j z d ρ S z d ρ = ε μ ( l + σ z ω ) ,
j z S z = j z , 0 + j z , 2 S z , 0 + S z , 2 .
j z S z = j z , 0 S z , 0 + j z , 2 S z , 0 j z , 0 S z , 0 S z , 2 S z , 0 ,
j z , 2 d ρ S z , 0 d ρ = ε μ f 2 ω [ ( 1 σ z ) l 2 ( l 1 ) w 0 2 0 1 ρ v 00 2 d ρ 2 ( 1 + σ z ) l ( l 1 ) 0 v 00 v 10 ρ d ρ + 2 ( 1 + σ z ) l w 0 2 0 v 10 2 ρ 3 d ρ + σ z w 0 2 0 [ v 11 v 00 v 10 v 01 ] ρ 3 d ρ ] ,
S z , 2 d ρ S z , 0 d ρ = f 2 [ ( 1 + 2 l ) 0 v 00 v 10 ρ d ρ 0 v 00 v 01 ρ d ρ 2 w 0 2 0 v 00 v 20 ρ 3 d ρ ] .

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