Abstract

In this paper we develop a rigorous formulation of Gauss-Laguerre beams in terms of Mie scattering coefficients which permits us to quasi-analytically treat the interaction of a spherical particle located in the focal region of a possibly high numerical aperture lens illuminated by a Gauss-Laguerre beam. This formalism is used to study the scattered field as a function of the radius of a spherical scatterer, as well as the translation of a spherical scatterer through the Gauss-Laguerre illumination in the focal plane. Knowledge of the Mie coefficients provides a deeper insight to understanding the scattering process and explaining the oscillatory behaviour of the scattered intensity distribution.

© 2007 Optical Society of America

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  1. M.E.J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N.R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles", Phys. Rev. A 54, 1593-1596 (1996).
    [CrossRef] [PubMed]
  2. J. Tempere, J.T. Devreese and E.R.I. Abraham, "Vortices in Bose-Einstein condensates confined in a multiply connected Laguerre-Gaussian optical trap", Phys. Rev. A 64, 023603 (2001)
    [CrossRef]
  3. A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of orbital angular momentum states of photons", Nature (London),  412, 3123-3316 (2001).
    [CrossRef]
  4. K. O’Holleran, M.R. Dennis and M.J. Padgett, "Illustrations of optical vortices in three dimensions", J. Europ. Opt. Soc. Rap. Public. 1, 06008 (2006).Q1
    [CrossRef]
  5. A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 1998).
  6. P. Monk, Finite Element Methods for Maxwell’s equations (Oxford University Press, Oxford, 2003).
    [CrossRef]
  7. G. Gouesbet, B. Maheu and G. Gr’ehan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation", J. Opt. Soc. Am. A 5, 1427-1443 (1988).
    [CrossRef]
  8. C.F. Bohren and D.R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 1983).
  9. J.E. Molloy and M.J. Padgett, "Light, action: optical tweezers", Contemp. Phys. 43, 241-258 (2002).
    [CrossRef]
  10. S.M. Barnett and L. Allen, "Orbital angular momentum and non-paraxial light beams", Opt. Commun. 110, 670-678 (1994).
    [CrossRef]
  11. P. Török and P.R.T. Munro, "The use of Gauss-Laguerre vector beams in STED microscopy", Opt. Express 12, 3605-3617 (2004).Q2
    [CrossRef] [PubMed]
  12. A.S. van de Nes, S.F. Pereira and J.J.M. Braat, "On the conservation of fundamental optical quantities in nonparaxial imaging systems", J. Mod. Opt. 53, 677-687 (2006).
    [CrossRef]
  13. A.E. Siegman, Lasers (University Science Books, Sausalito, CA, 1986).
  14. G.N. Watson, A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1966).
  15. S.M. Barnett, "Optical angular-momentum flux", J. Opt. B: Quantum and Semiclass. Opt. 4, S7-S16 (2002).Q3
    [CrossRef]
  16. L. Allen, S.M. Barnett and M.J. Padgett, Optical Angular Momentum (Institute of Physics Publishing, Bristol, 2003).
    [CrossRef]
  17. A. Stratton, Electromagnetic Theory (McGraw-Hill book company, Inc., New York, 1941).
  18. P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill book company, Inc., New York, 1953).
  19. H.C. van de Hulst, Light scattering by small particles (Dover publications, New York, 1981).
  20. G. Mie, "Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen", Ann. Phys. 330, 377-445 (1908).
    [CrossRef]
  21. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970).
  22. P. Török, P.D. Higdon, R. Ju¡skaitis and T.Wilson, "Optimising the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers", Opt. Commun. 155, 335-341 (1998).Q4
    [CrossRef]
  23. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).

2006 (2)

K. O’Holleran, M.R. Dennis and M.J. Padgett, "Illustrations of optical vortices in three dimensions", J. Europ. Opt. Soc. Rap. Public. 1, 06008 (2006).Q1
[CrossRef]

A.S. van de Nes, S.F. Pereira and J.J.M. Braat, "On the conservation of fundamental optical quantities in nonparaxial imaging systems", J. Mod. Opt. 53, 677-687 (2006).
[CrossRef]

2004 (1)

2002 (2)

J.E. Molloy and M.J. Padgett, "Light, action: optical tweezers", Contemp. Phys. 43, 241-258 (2002).
[CrossRef]

S.M. Barnett, "Optical angular-momentum flux", J. Opt. B: Quantum and Semiclass. Opt. 4, S7-S16 (2002).Q3
[CrossRef]

2001 (2)

J. Tempere, J.T. Devreese and E.R.I. Abraham, "Vortices in Bose-Einstein condensates confined in a multiply connected Laguerre-Gaussian optical trap", Phys. Rev. A 64, 023603 (2001)
[CrossRef]

A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of orbital angular momentum states of photons", Nature (London),  412, 3123-3316 (2001).
[CrossRef]

1998 (1)

P. Török, P.D. Higdon, R. Ju¡skaitis and T.Wilson, "Optimising the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers", Opt. Commun. 155, 335-341 (1998).Q4
[CrossRef]

1996 (1)

M.E.J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N.R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles", Phys. Rev. A 54, 1593-1596 (1996).
[CrossRef] [PubMed]

1994 (1)

S.M. Barnett and L. Allen, "Orbital angular momentum and non-paraxial light beams", Opt. Commun. 110, 670-678 (1994).
[CrossRef]

1988 (1)

1908 (1)

G. Mie, "Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen", Ann. Phys. 330, 377-445 (1908).
[CrossRef]

Abraham, E.R.I.

J. Tempere, J.T. Devreese and E.R.I. Abraham, "Vortices in Bose-Einstein condensates confined in a multiply connected Laguerre-Gaussian optical trap", Phys. Rev. A 64, 023603 (2001)
[CrossRef]

Allen, L.

S.M. Barnett and L. Allen, "Orbital angular momentum and non-paraxial light beams", Opt. Commun. 110, 670-678 (1994).
[CrossRef]

Barnett, S.M.

S.M. Barnett, "Optical angular-momentum flux", J. Opt. B: Quantum and Semiclass. Opt. 4, S7-S16 (2002).Q3
[CrossRef]

S.M. Barnett and L. Allen, "Orbital angular momentum and non-paraxial light beams", Opt. Commun. 110, 670-678 (1994).
[CrossRef]

Braat, J.J.M.

A.S. van de Nes, S.F. Pereira and J.J.M. Braat, "On the conservation of fundamental optical quantities in nonparaxial imaging systems", J. Mod. Opt. 53, 677-687 (2006).
[CrossRef]

Dennis, M.R.

K. O’Holleran, M.R. Dennis and M.J. Padgett, "Illustrations of optical vortices in three dimensions", J. Europ. Opt. Soc. Rap. Public. 1, 06008 (2006).Q1
[CrossRef]

Devreese, J.T.

J. Tempere, J.T. Devreese and E.R.I. Abraham, "Vortices in Bose-Einstein condensates confined in a multiply connected Laguerre-Gaussian optical trap", Phys. Rev. A 64, 023603 (2001)
[CrossRef]

Enger, J.

M.E.J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N.R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles", Phys. Rev. A 54, 1593-1596 (1996).
[CrossRef] [PubMed]

Friese, M.E.J.

M.E.J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N.R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles", Phys. Rev. A 54, 1593-1596 (1996).
[CrossRef] [PubMed]

Gouesbet, G.

Gr’ehan, G.

Heckenberg, N.R.

M.E.J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N.R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles", Phys. Rev. A 54, 1593-1596 (1996).
[CrossRef] [PubMed]

Maheu, B.

Mair, A.

A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of orbital angular momentum states of photons", Nature (London),  412, 3123-3316 (2001).
[CrossRef]

Mie, G.

G. Mie, "Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen", Ann. Phys. 330, 377-445 (1908).
[CrossRef]

Molloy, J.E.

J.E. Molloy and M.J. Padgett, "Light, action: optical tweezers", Contemp. Phys. 43, 241-258 (2002).
[CrossRef]

O’Holleran, K.

K. O’Holleran, M.R. Dennis and M.J. Padgett, "Illustrations of optical vortices in three dimensions", J. Europ. Opt. Soc. Rap. Public. 1, 06008 (2006).Q1
[CrossRef]

Padgett, M.J.

K. O’Holleran, M.R. Dennis and M.J. Padgett, "Illustrations of optical vortices in three dimensions", J. Europ. Opt. Soc. Rap. Public. 1, 06008 (2006).Q1
[CrossRef]

J.E. Molloy and M.J. Padgett, "Light, action: optical tweezers", Contemp. Phys. 43, 241-258 (2002).
[CrossRef]

Pereira, S.F.

A.S. van de Nes, S.F. Pereira and J.J.M. Braat, "On the conservation of fundamental optical quantities in nonparaxial imaging systems", J. Mod. Opt. 53, 677-687 (2006).
[CrossRef]

Rubinsztein-Dunlop, H.

M.E.J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N.R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles", Phys. Rev. A 54, 1593-1596 (1996).
[CrossRef] [PubMed]

Tempere, J.

J. Tempere, J.T. Devreese and E.R.I. Abraham, "Vortices in Bose-Einstein condensates confined in a multiply connected Laguerre-Gaussian optical trap", Phys. Rev. A 64, 023603 (2001)
[CrossRef]

van de Nes, A.S.

A.S. van de Nes, S.F. Pereira and J.J.M. Braat, "On the conservation of fundamental optical quantities in nonparaxial imaging systems", J. Mod. Opt. 53, 677-687 (2006).
[CrossRef]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of orbital angular momentum states of photons", Nature (London),  412, 3123-3316 (2001).
[CrossRef]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of orbital angular momentum states of photons", Nature (London),  412, 3123-3316 (2001).
[CrossRef]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of orbital angular momentum states of photons", Nature (London),  412, 3123-3316 (2001).
[CrossRef]

Ann. Phys. (1)

G. Mie, "Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen", Ann. Phys. 330, 377-445 (1908).
[CrossRef]

Contemp. Phys. (1)

J.E. Molloy and M.J. Padgett, "Light, action: optical tweezers", Contemp. Phys. 43, 241-258 (2002).
[CrossRef]

J. Europ. Opt. Soc. Rap. Public. (1)

K. O’Holleran, M.R. Dennis and M.J. Padgett, "Illustrations of optical vortices in three dimensions", J. Europ. Opt. Soc. Rap. Public. 1, 06008 (2006).Q1
[CrossRef]

J. Mod. Opt. (1)

A.S. van de Nes, S.F. Pereira and J.J.M. Braat, "On the conservation of fundamental optical quantities in nonparaxial imaging systems", J. Mod. Opt. 53, 677-687 (2006).
[CrossRef]

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

S.M. Barnett, "Optical angular-momentum flux", J. Opt. B: Quantum and Semiclass. Opt. 4, S7-S16 (2002).Q3
[CrossRef]

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

Nature (London) (1)

A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of orbital angular momentum states of photons", Nature (London),  412, 3123-3316 (2001).
[CrossRef]

Opt. Commun. (2)

P. Török, P.D. Higdon, R. Ju¡skaitis and T.Wilson, "Optimising the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers", Opt. Commun. 155, 335-341 (1998).Q4
[CrossRef]

S.M. Barnett and L. Allen, "Orbital angular momentum and non-paraxial light beams", Opt. Commun. 110, 670-678 (1994).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (2)

M.E.J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N.R. Heckenberg, "Optical angular-momentum transfer to trapped absorbing particles", Phys. Rev. A 54, 1593-1596 (1996).
[CrossRef] [PubMed]

J. Tempere, J.T. Devreese and E.R.I. Abraham, "Vortices in Bose-Einstein condensates confined in a multiply connected Laguerre-Gaussian optical trap", Phys. Rev. A 64, 023603 (2001)
[CrossRef]

Other (11)

A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 1998).

P. Monk, Finite Element Methods for Maxwell’s equations (Oxford University Press, Oxford, 2003).
[CrossRef]

C.F. Bohren and D.R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 1983).

L. Allen, S.M. Barnett and M.J. Padgett, Optical Angular Momentum (Institute of Physics Publishing, Bristol, 2003).
[CrossRef]

A. Stratton, Electromagnetic Theory (McGraw-Hill book company, Inc., New York, 1941).

P.M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill book company, Inc., New York, 1953).

H.C. van de Hulst, Light scattering by small particles (Dover publications, New York, 1981).

A.E. Siegman, Lasers (University Science Books, Sausalito, CA, 1986).

G.N. Watson, A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1966).

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970).

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).

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Figures (11)

Fig. 1.
Fig. 1.

(a) Schematic of the optical system using a Gauss-Laguerre beam for illumination. The sphere is initially placed in focus but is later allowed to be translated along the x-axis. Four detectors are located around the sphere: detector Da measures the transmitted light, detector Db the reflected light, detector Dc the light reflected along the x-direction and detector Dd the light reflected along the y-direction. (b) Each detector consists of four segments with axes x 1 = x and x 2 = y for Da and Db , x 1 = z and x 2 = y for Dc , and x 1 = z and x 2 = x for Dd .

Fig. 2.
Fig. 2.

The x-, y- and z-component of the electric field distribution in the focal plane for a Gauss-Laguerre beam with l = 0,1 and 2 are depicted in the first, second and last row, respectively. The phase is shown as an inset in each figure, where the colour scale changes linearly from blue to red corresponding to [-π, π⟩.

Fig. 3.
Fig. 3.

Logarithmic intensity distribution due to a spherical particle of 1.82 μm radius, illuminated by a focused p = 0, l = 1 Gauss-Laguerre beam. Indicated by black circles are the detector for transmission Da , and the transversal detectors Dc and Dd corresponding to TM and TE, respectively.

Fig. 4.
Fig. 4.

(1st col.) The total intensity as a function of sphere radius for the detector (Da ) placed in transmission, (2nd col.) placed in reflection (Db ), and (3rd col.) placed in the transversal plane with the blue line for TM (Dc ) and red for TE (Dd ). Each row corresponds with illumination of increasing order, GL00, GL01 and GL02, respectively.

Fig. 5.
Fig. 5.

(1st col.) Difference intensity as a function of sphere radius for the detector (Da ) placed in transmission, (2nd col.) placed in reflection (Db ), and (3rd col.) placed in the transversal plane with the blue line for TM (Dc ) and red for TE (Dd ). Each row corresponds with illumination of increasing order, GL00, GL01 and GL02, respectively.

Fig. 6.
Fig. 6.

Split-z (left) and split-x/y (right) intensity as a function of sphere radius for detector placed in the transversal plane with the blue line for TM (Dc ) and red for TE (Dd ). Each row corresponds to illumination with increasing order, GL00, GL01 and GL02, respectively.

Fig. 7.
Fig. 7.

Absolute value of the bh nm coefficients for m = l -1 (left) and m = l + 1 (right) as a function of the sphere radius rs and the mode number n, obtained by illumination with the three Gauss-Laguerre beams GL00, GL01 and GL02.

Fig. 8.
Fig. 8.

(a) Absolute value of the bh nm coefficients for m = ±1 of the plane wave illumination as a function of the sphere radius rs and the mode number n. (b) Absolute value of the bnm coefficient with n = 15 and m = 0 for GL01 illumination as a function of sphere radius, with blue corresponding to the TM coefficient bh and red to the TE coefficient be .

Fig. 9.
Fig. 9.

Transmitted integrated intensity for three different Gauss-Laguerre modes, GL00 blue, GL01 red and GL02 green. (a) The sum signal (b) the difference signal, (c) the split-x configuration and (d) the split-y configuration.

Fig. 10.
Fig. 10.

Reflected integrated intensity for three different Gauss-Laguerre modes, GL00 blue, GL01 red and GL02 green. (a) The sum signal (b) the difference signal, (c) the split-x configuration and (d) the split-y configuration.

Fig. 11.
Fig. 11.

Integrated intensity scattered to the transversal plane for three different Gauss-Laguerre modes, GL00 blue, GL01 red and GL02 green. The top row corresponds to TM and the bottom to TE. (a) The sum signal (b) the difference signal, (c) the split-z configuration and (d) the split-y configuration for TM and split-x configuration for TE.

Equations (26)

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E ( r ) = 1 2 E pl , 0 ( r ; α , β ) + 1 4 E pl , 2 ( r ; α + , β ) + 1 4 E pl , 2 ( r ; α β ) ,
H ( r ) = 1 2 H pl , 0 ( r ; α , β ) + 1 4 H pl , 2 ( r ; α + , β ) + 1 4 H pl , 2 ( r ; α β ) ,
E pl , j ( r ; α , β ) = 0 k NA E pl , j ( k ρ ) e i ( l + j ) ϕ + i k z z ( ( α x ̂ + β y ̂ ) J l + j ( k ρ ρ ) + k ρ 2 k z z ̂ × [ ( + β ) e J l + j 1 ( k ρ ρ ) ( + β ) e J l + j + 1 ( k ρ ρ ) ] ) d k ρ ,
H pl , j ( r ; α , β ) = ε μ 0 k NA E pl , j ( k ρ ) 2 k k z e i ( l + j ) ϕ + i k z z ( ( β x ̂ + α y ̂ ) [ 2 k 2 k ρ 2 ] J l + j ( k ρ ρ ) + k ρ 2 2 [ ( x ̂ + i y ̂ ) ( β ) J l + j 2 ( k ρ ρ ) e 2 ( x ̂ i y ̂ ) ( + β ) J l + j + 2 ( k ρ ρ ) e 2 ] k ρ k z z ̂ [ ( α + ) J l + j 1 ( k ρ ρ ) e + ( α ) J l + j + 1 ( k ρ ρ ) e ] ) d k ρ ,
E pl , j ( k ρ ) = ( 1 ) p i l 1 u pl k ρ R f k z z r ( 1 + ( 1 j ) k z k ) ( k ρ 2 R f 2 k z r ) l 2 L p l ( k ρ 2 R f 2 k z r ) exp [ k ρ 2 R f 2 2 k z r ] ,
r Π e , h ( r ) = n = 0 m = n n a nm e , h r j n ( kr ) P n m ( cos θ ) e imϕ ,
E r ( r ) = n ( n + 1 ) r 2 r Π h = E x ( r ) cos ϕ sin θ + E y ( r ) sin ϕ sin θ + E z ( r ) cos θ ,
H r ( r ) = n ( n + 1 ) r 2 r Π e = H x ( r ) cos ϕ sin θ + H y ( r ) sin ϕ sin θ + H z ( r ) cos θ ,
c nm e = i μ 2 ( k 1 r s ) μ 1 h n ( k 1 r s ) [ k 2 r s j n 1 ( k 2 r s ) n j n ( k 2 r s ) ] μ 2 j n ( k 2 r s ) [ k 1 r s h n 1 ( k 1 r s ) n h n ( k 1 r s ) ] a nm e ,
b nm e = μ 2 j n ( k 2 r s ) [ k 1 r s j n 1 ( k 1 r s ) n j n ( k 1 r s ) ] μ 1 j n ( k 1 r s ) [ k 2 r s j n 1 ( k 2 r s ) n j n ( k 2 r s ) ] μ 1 h n ( k 1 r s ) [ k 2 r s j n 1 ( k 2 r s ) n j n ( k 2 r s ) ] μ 2 j n ( k 2 r s ) [ k 1 r s h n 1 ( k 1 r s ) n h n ( k 1 r s ) ] a nm e ,
c nm h = i μ 2 ε 2 ( μ 1 k 2 r s ) ε 1 h n ( k 1 r s ) [ k 2 r s j n 1 ( k 2 r s ) n j n ( k 2 r s ) ] ε 2 j n ( k 2 r s ) [ k 1 r s h n 1 ( k 1 r s ) n h n ( k 1 r s ) ] a nm h ,
b nm h = ε 2 j n ( k 2 r s ) [ k 1 r s j n 1 ( k 1 r s ) n j n ( k 1 r s ) ] ε 1 j n ( k 1 r s ) [ k 2 r s j n 1 ( k 2 r s ) n j n ( k 2 r s ) ] ε 1 h n ( k 1 r s ) [ k 2 r s j n 1 ( k 2 r s ) n j n ( k 2 r s ) ] ε 2 j n ( k 2 r s ) [ k 1 r s h n 1 ( k 1 r s ) n h n ( k 1 r s ) ] a nm h ,
a nm h = 1 S r S θ S ϕ S a E inc , r r j n ( kr ) P n m ( cos θ ) e imϕ ,
a nm e = 1 S r S θ S ϕ S a H inc , r r j n ( kr ) P n m ( cos θ ) e imϕ ,
S r = n ( n + 1 ) r a j n ( k r a ) , S θ = 2 ( n + m ) ! ( 2 n + 1 ) ( n m ) ! , S ϕ = 2 π .
a nm h = { i n + 1 ( 2 n + 1 ) 2 kn ( n + 1 ) m = ± 1 0 m ± 1 , a nm e = { i ( m 1 ) ε μ i n + 1 ( 2 n + 1 ) 2 ikn ( n + 1 ) m = ± 1 0 m ± 1 .
E r = 0 k E pl ( k ρ ) 2 e il ϕ + i k z z ( [ ( α + ) e + ( α ) e ] J l ( k ρ ρ ) sin θ + k ρ k z [ ( β ) e J l 1 ( k ρ ρ ) ( + β ) e J l + 1 ( k ρ ρ ) ] cos θ ) d k ρ ,
H r = ε μ 0 k E pl ( k ρ ) 2 k k z e il ϕ + i k z z [ 1 2 ( ( 2 k 2 k ρ 2 ) [ ( β ) e ( + β ) e ] J l ( k ρ ρ ) + k ρ 2 [ ( + β ) J l 2 ( k ρ ρ ) e ( + β ) J l + 2 ( k ρ ρ ) e ] ) sin θ k ρ k z [ ( α ) e J l 1 ( k ρ ρ ) + ( α ) e J l + 1 ( k ρ ρ ) ] cos θ ] d k ρ ,
E pl ( k ρ ) = ( 1 ) p ( k z r 2 ) ( p + l + 1 ) 2 k k z ( k ρ 2 k 2 k ρ 2 ) ( 2 p + l + 1 ) 2 exp [ k z r k ρ 2 2 ( k 2 k ρ 2 ) ] .
E r = 0 k NA E pl ( k ρ ) e ilϕ + i k z z ( 1 2 [ ( 1 + k z k ) [ ( α + ) e + ( α + ) e ] J l ( k ρ ρ ) + ( 1 + k z k ) [ ( α + ) e J l 2 ( k ρ ρ ) + ( α + ) e J l + 2 ( k ρ ρ ) ] ] sin θ + k ρ k [ ( + β ) e J l 2 ( k ρ ρ ) ( + ) e J l + 1 ( k ρ ρ ) ] cos θ ) d k ρ ,
H r = ε μ 0 k NA E pl ( k ρ ) k k z e ilϕ + i k z z [ 1 2 ( ( k 2 k ρ 2 + kk z ) [ ( α + ) e ( + β ) e ] J l ( k ρ ρ ) ( k 2 k ρ 2 + k k z ) [ ( α β ) J l 2 ( k ρ ρ ) e ( + ) J l + 2 ( k ρ ρ ) e ] ) sin θ k ρ k z [ ( α + ) e J l 2 ( k ρ ρ ) + ( α + ) e J l + 1 ( k ρ ρ ) ] cos θ ) d k ρ ,
E pl ( k ρ ) = ( 1 ) p i l 1 u pl k ρ R f k z z r ( k ρ 2 R f 2 k z r ) l 2 L p l ( k ρ 2 R f 2 k z r ) exp [ k ρ 2 R f 2 2 k z r ] .
E r = 0 k NA E pl ( k ρ ) e ilϕ + i k z z ( 1 2 [ ( 1 + k z k ) [ ( α + ) e + ( α + ) e ] J l ( k ρ ρ ) + ( 1 + k z k ) [ ( α + ) e J l 2 ( k ρ ρ ) + ( α + ) e i ( 2 ϕ ϕ ) J l + 2 ( k ρ ρ ) ] ] sin θ + k ρ k [ ( + β ) e J l 1 ( k ρ ρ ) ( + β ) e J l + 1 ( k ρ ρ ) ] cos θ ) d k ρ ,
H r = ε μ 0 k NA E pl ( k ρ ) k k z e ilϕ + i k z z [ 1 2 ( ( k 2 k ρ 2 + kk z ) [ ( + β ) e ( + β ) e ] J l ( k ρ ρ ) ( k 2 k ρ 2 + k k z ) [ ( β ) J l 2 ( k ρ ρ ) e i ( 2 ϕ ϕ ) ( + β ) J l + 2 ( k ρ ρ ) e i ( 2 ϕ ϕ ) ] ) sin θ k ρ k z [ ( α + ) e J l 2 ( k ρ ρ ) + ( α + ) e J l + 1 ( k ρ ρ ) ] cos θ ) d k ρ ,
ρ ' = [ ( r a sin θ cos ϕ x off ) 2 + ( r a sin θ sin ϕ y off ) 2 ] 1 2 ,
tan ϕ = r a sin θ sin ϕ y off r a sin θ cos ϕ x off .

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