Abstract

The vector Gaussian beam with high-order corrections is used to describe accurately the laser beam up to numerical aperture NA=1.20 in the optical tweezers for trapping nanoparticles. The beam is then expanded in the T-matrix method into the vector spherical wave function (VSWF) series using the point matching method with a new selection of the matching points. The errors in the beam description and in the VSWF expansion are much lower than those that occur in the paraxial Gaussian beam model.

© 2012 Optical Society of America

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References

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  1. D. Ganic, X. Gan, and M. Gu, “Exact radiation trapping force calculation based on vectorial diffraction theory,” Opt. Express 12, 2670–2675 (2004).
    [CrossRef]
  2. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
    [CrossRef]
  3. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 65, 2900–2906 (1988).
    [CrossRef]
  4. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  5. P. B. Bareil and Y. Sheng, “Angular and position stability of a nanorod trapped in an optical tweezers,” Opt. Express 18, 26388–26398 (2010).
    [CrossRef]
  6. T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017(2003).
    [CrossRef]
  7. N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, “Accurate description of Gaussian laser beams and electron dynamics,” Opt. Commun. 204, 7–15(2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. H. Luo and S. Liu, “Comment on calculation of electromagnetic field components for a fundamental Gaussian beam,” Phys. Rev. E 75, 3–4 (2007).
  12. G. Wang and J. Webb, “Calculation of electromagnetic field components for a fundamental Gaussian beam,” Phys. Rev. E 72, 046501 (2005).
  13. S. Y. Zhang, “Accurate correction field of circularly polarized laser and its acceleration effect,” J. At. Mol. Sci. 1, 308–317 (2010).
    [CrossRef]
  14. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9S196–S203 (2007).
    [CrossRef]
  15. B. C. Brock, “Using vector spherical harmonics to compute antenna mutual impedance from measured or computed fields,” SANDIA Report, SAND2000-2217, Sandia National Laboratories (2000).
  16. N. A. Gumerov and R. Duraiswami, “Recursions for the computation of multipole translation and rotation coefficients for the 3-D Helmholtz equation,” SIAM J. Sci. Comput. 25, 1344–1381 (2003).
    [CrossRef]
  17. L. Boyde, K. J. Chalut, and J. Guck, “Exact analytical expansion of an off-axis Gaussian laser beam usinolumeg the translation theorems for the vector spherical harmonics,” Appl. Opt. 50, 1023–1033 (2011).
    [CrossRef]

2011 (1)

2010 (2)

P. B. Bareil and Y. Sheng, “Angular and position stability of a nanorod trapped in an optical tweezers,” Opt. Express 18, 26388–26398 (2010).
[CrossRef]

S. Y. Zhang, “Accurate correction field of circularly polarized laser and its acceleration effect,” J. At. Mol. Sci. 1, 308–317 (2010).
[CrossRef]

2007 (2)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9S196–S203 (2007).
[CrossRef]

H. Luo and S. Liu, “Comment on calculation of electromagnetic field components for a fundamental Gaussian beam,” Phys. Rev. E 75, 3–4 (2007).

2005 (1)

G. Wang and J. Webb, “Calculation of electromagnetic field components for a fundamental Gaussian beam,” Phys. Rev. E 72, 046501 (2005).

2004 (1)

2003 (2)

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017(2003).
[CrossRef]

N. A. Gumerov and R. Duraiswami, “Recursions for the computation of multipole translation and rotation coefficients for the 3-D Helmholtz equation,” SIAM J. Sci. Comput. 25, 1344–1381 (2003).
[CrossRef]

2002 (1)

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, “Accurate description of Gaussian laser beams and electron dynamics,” Opt. Commun. 204, 7–15(2002).
[CrossRef]

1999 (1)

S. P. Smith, S. R. Bhalotra, A. L. Brody, B. L. Brown, E. K. Boyda, and M. Prentiss, “Inexpensive optical tweezers for undergraduate laboratories,” Am. J. Phys. 67, 26–35 (1999).
[CrossRef]

1989 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

1988 (2)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 65, 2900–2906 (1988).
[CrossRef]

G. Gouesbet, G. Grehan, and B. Maheu, “Computations of the gn coefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef]

1979 (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

1971 (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 65, 2900–2906 (1988).
[CrossRef]

Bareil, P. B.

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 65, 2900–2906 (1988).
[CrossRef]

Bhalotra, S. R.

S. P. Smith, S. R. Bhalotra, A. L. Brody, B. L. Brown, E. K. Boyda, and M. Prentiss, “Inexpensive optical tweezers for undergraduate laboratories,” Am. J. Phys. 67, 26–35 (1999).
[CrossRef]

Boyda, E. K.

S. P. Smith, S. R. Bhalotra, A. L. Brody, B. L. Brown, E. K. Boyda, and M. Prentiss, “Inexpensive optical tweezers for undergraduate laboratories,” Am. J. Phys. 67, 26–35 (1999).
[CrossRef]

Boyde, L.

Brock, B. C.

B. C. Brock, “Using vector spherical harmonics to compute antenna mutual impedance from measured or computed fields,” SANDIA Report, SAND2000-2217, Sandia National Laboratories (2000).

Brody, A. L.

S. P. Smith, S. R. Bhalotra, A. L. Brody, B. L. Brown, E. K. Boyda, and M. Prentiss, “Inexpensive optical tweezers for undergraduate laboratories,” Am. J. Phys. 67, 26–35 (1999).
[CrossRef]

Brown, B. L.

S. P. Smith, S. R. Bhalotra, A. L. Brody, B. L. Brown, E. K. Boyda, and M. Prentiss, “Inexpensive optical tweezers for undergraduate laboratories,” Am. J. Phys. 67, 26–35 (1999).
[CrossRef]

Cao, N.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, “Accurate description of Gaussian laser beams and electron dynamics,” Opt. Commun. 204, 7–15(2002).
[CrossRef]

Chalut, K. J.

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Duraiswami, R.

N. A. Gumerov and R. Duraiswami, “Recursions for the computation of multipole translation and rotation coefficients for the 3-D Helmholtz equation,” SIAM J. Sci. Comput. 25, 1344–1381 (2003).
[CrossRef]

Gan, X.

Ganic, D.

Gouesbet, G.

Grehan, G.

Gu, M.

Guck, J.

Gumerov, N. A.

N. A. Gumerov and R. Duraiswami, “Recursions for the computation of multipole translation and rotation coefficients for the 3-D Helmholtz equation,” SIAM J. Sci. Comput. 25, 1344–1381 (2003).
[CrossRef]

Heckenberg, N. R.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9S196–S203 (2007).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017(2003).
[CrossRef]

Ho, Y. K.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, “Accurate description of Gaussian laser beams and electron dynamics,” Opt. Commun. 204, 7–15(2002).
[CrossRef]

Ito, H.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, “Accurate description of Gaussian laser beams and electron dynamics,” Opt. Commun. 204, 7–15(2002).
[CrossRef]

Kong, Q.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, “Accurate description of Gaussian laser beams and electron dynamics,” Opt. Commun. 204, 7–15(2002).
[CrossRef]

Liu, S.

H. Luo and S. Liu, “Comment on calculation of electromagnetic field components for a fundamental Gaussian beam,” Phys. Rev. E 75, 3–4 (2007).

Loke, V. L. Y.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9S196–S203 (2007).
[CrossRef]

Luo, H.

H. Luo and S. Liu, “Comment on calculation of electromagnetic field components for a fundamental Gaussian beam,” Phys. Rev. E 75, 3–4 (2007).

Maheu, B.

Nieminen, T. A.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9S196–S203 (2007).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017(2003).
[CrossRef]

Nishida, Y.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, “Accurate description of Gaussian laser beams and electron dynamics,” Opt. Commun. 204, 7–15(2002).
[CrossRef]

Prentiss, M.

S. P. Smith, S. R. Bhalotra, A. L. Brody, B. L. Brown, E. K. Boyda, and M. Prentiss, “Inexpensive optical tweezers for undergraduate laboratories,” Am. J. Phys. 67, 26–35 (1999).
[CrossRef]

Rubinsztein-dunlop, H.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9S196–S203 (2007).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017(2003).
[CrossRef]

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 65, 2900–2906 (1988).
[CrossRef]

Sheng, Y.

Smith, S. P.

S. P. Smith, S. R. Bhalotra, A. L. Brody, B. L. Brown, E. K. Boyda, and M. Prentiss, “Inexpensive optical tweezers for undergraduate laboratories,” Am. J. Phys. 67, 26–35 (1999).
[CrossRef]

Stilgoe, A. B.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9S196–S203 (2007).
[CrossRef]

Wang, G.

G. Wang and J. Webb, “Calculation of electromagnetic field components for a fundamental Gaussian beam,” Phys. Rev. E 72, 046501 (2005).

Wang, P. X.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, “Accurate description of Gaussian laser beams and electron dynamics,” Opt. Commun. 204, 7–15(2002).
[CrossRef]

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Webb, J.

G. Wang and J. Webb, “Calculation of electromagnetic field components for a fundamental Gaussian beam,” Phys. Rev. E 72, 046501 (2005).

Yuan, X. Q.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, “Accurate description of Gaussian laser beams and electron dynamics,” Opt. Commun. 204, 7–15(2002).
[CrossRef]

Yugami, N.

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, “Accurate description of Gaussian laser beams and electron dynamics,” Opt. Commun. 204, 7–15(2002).
[CrossRef]

Zhang, S. Y.

S. Y. Zhang, “Accurate correction field of circularly polarized laser and its acceleration effect,” J. At. Mol. Sci. 1, 308–317 (2010).
[CrossRef]

Am. J. Phys. (1)

S. P. Smith, S. R. Bhalotra, A. L. Brody, B. L. Brown, E. K. Boyda, and M. Prentiss, “Inexpensive optical tweezers for undergraduate laboratories,” Am. J. Phys. 67, 26–35 (1999).
[CrossRef]

Appl. Opt. (2)

J. Appl. Phys. (2)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 65, 2900–2906 (1988).
[CrossRef]

J. At. Mol. Sci. (1)

S. Y. Zhang, “Accurate correction field of circularly polarized laser and its acceleration effect,” J. At. Mol. Sci. 1, 308–317 (2010).
[CrossRef]

J. Opt. A (1)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9S196–S203 (2007).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017(2003).
[CrossRef]

Opt. Commun. (1)

N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, “Accurate description of Gaussian laser beams and electron dynamics,” Opt. Commun. 204, 7–15(2002).
[CrossRef]

Opt. Express (2)

Phys. Rev. A (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Phys. Rev. D (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Phys. Rev. E (2)

H. Luo and S. Liu, “Comment on calculation of electromagnetic field components for a fundamental Gaussian beam,” Phys. Rev. E 75, 3–4 (2007).

G. Wang and J. Webb, “Calculation of electromagnetic field components for a fundamental Gaussian beam,” Phys. Rev. E 72, 046501 (2005).

SIAM J. Sci. Comput. (1)

N. A. Gumerov and R. Duraiswami, “Recursions for the computation of multipole translation and rotation coefficients for the 3-D Helmholtz equation,” SIAM J. Sci. Comput. 25, 1344–1381 (2003).
[CrossRef]

Other (1)

B. C. Brock, “Using vector spherical harmonics to compute antenna mutual impedance from measured or computed fields,” SANDIA Report, SAND2000-2217, Sandia National Laboratories (2000).

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Figures (9)

Fig. 1.
Fig. 1.

Transversal distribution of the electric field E2 at the focal plane. Black curve, paraxial Gaussian beam; red curve, fifth-order corrections; green curve, seventh-order corrections; blue curve, ninth-order corrections; NA= (a) 1.25, (b) 1.15, and (c) 1. Right circular polarization.

Fig. 2.
Fig. 2.

Transversal profile of E2 at the focal plane for the vector Gaussian beam with seventh-order correction NA=1.25 (blue curve), 1.20 (green curve), and 1.15 (red curve), right circular polarization.

Fig. 3.
Fig. 3.

Transversal distribution of E2 in the focal plane for NA=1.25. (a) Circular polarization, (b) linear x polarization, and (c) linear x polarization with NA=0.9.

Fig. 4.
Fig. 4.

Distributions of the components of E⃗ in the focal plane with NA=0.90. (a) Ex, (b) Ey, and (c) Ez. Linear polarization in x.

Fig. 5.
Fig. 5.

Distributions of the components of E⃗ in the focal plane with NA=1.20. (a) Ex, (b) Ey, and (c) Ez. Linear polarization in x.

Fig. 6.
Fig. 6.

Transversal profile of the electric field E2. Circular polarization. NA=1.15 (a) z=0nm, and (b) z=500nm. Blue curves, reconstructed from VSWF expansion; green curves, vector Gaussian beam with seventh-order correction.

Fig. 7.
Fig. 7.

Transversal profile of E2. Turquoise curves, vector Gaussian beam with seventh-order correction; blue curves, point matching at far field; red curves, point matching at focal plane. Right circular polarization, NA=1.15. (a) z=0nm, and (b) z=500nm.

Fig. 8.
Fig. 8.

Transversal distribution of the electric field E2. Circular polarization. NA=1.20; (a) z=0nm, and (b) z=500nm. Blue curves, recovered from the VSWF expansion; green curves, vector Gaussian beam with seventh-order correction.

Fig. 9.
Fig. 9.

Distribution of E2 in the focal plane. Beam is linearly x polarized at NA=1.25. (a) Blue curve, reconstructed from VSWF expansion; green curve, vector Gaussian beam with seventh-order correction shifted to y0=250nm. (b) Reconstructed field from its VSWF expansion for a beam shifted to x0=y0=250nm.

Tables (1)

Tables Icon

Table 1. FWHM of a Right Circularly Polarized Vector Gaussian Beam as a Function of NA, λ=1.064nm

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

2A⃗+k2A⃗=0,
A⃗=Ax^=ψ(x,y,z)eikzx^,
w0=1.22λ0n(nNA)21,
(2ξ2+2η2+s22ς2)ψ2iψς=0,
ψ=ψ0+s2ψ2+s4ψ4+s6ψ6+,
(2ξ2+2η22iς)ψ0=0,
(2ξ2+2η22iς)ψ2n+2=2ψ2nς2,
Ex=E0ψ0eiζ/s2{1+s2(ρ2Q2+iρ4Q32Q2ξ2)+s4[2ρ4Q43iρ6Q50.5ρ8Q6+(8ρ2Q42iρ4Q5)ξ2]+s6[5ρ6Q6+9iρ8Q7+2.5iρ10Q8i/6ρ12Q9ξ2(30ρ4Q612iρ6Q7ρ8Q8)+s8[(112Q8ρ656iQ9ρ88Q10ρ10+13iQ11ρ12)ξ2+14Q8ρ828iQ9ρ1010Q10ρ12+76iQ11ρ14+124Q12ρ16]},Ey=E0ψ0eiζ/s2{2s2Q2ξη+s4[(8ρ2Q42iρ4Q5)ξη]+s6[ξη(30ρ4Q612iρ6Q7ρ8Q8)+s8(112Q8ρ656iQ9ρ88Q10ρ10+iQ11ρ12/3)]},Ez=E0ψ0eiζ/s2{2sQξ+s3[(6ρ2Q32iρ4Q4)ξ]+s5[(20ρ4Q5+10iρ6Q6+ρ8Q7)ξ]+s7(70ρ6Q742iρ8Q87ρ10Q9+iρ12Q10/3)ξ+s9(252Q9ρ8+168iQ10ρ10+36Q11ρ123iQ12ρ14+112Q13ρ16)ξ}.
Ecx=Ex±iEy,Ecy=Ey±iEx.
E0=(4Pcnϵnπw02(1+s2+1.5s4+3s6+7.5s8)),
Ex=E0[1+s2O(ρ4)+s4O(ρ8)+s6O(ρ12)+s8O(ρ16)+],
E⃗inc(r⃗)=n=1Nmaxm=nnanmRgM⃗nm(kr⃗)+bnmRgN⃗nm(kr⃗),
RgM⃗nm(k)=Nnjn(kr)C⃗nm(θ,φ),RgN⃗nm(1,2)(kr⃗)=jn(1,2)(kr)krNnP⃗nm(θ,ϕ)+Nn(jn1(1,2)(kr)njn(1,2)(kr)kr⃗)B⃗nm(θ,ϕ),
[E⃗(r,θ,ϕ)]=[[RgM⃗nm(r,θ,ϕ)][RgN⃗nm(r,θ,ϕ)]][anmbnm].
Rx,θ=[cosθ0sinθ010sinθ0cosθ],Ry,γ=[1000cosγsinγ0sinγcosγ],Rz,ϕ=[cosϕsinϕ0sinϕcosϕ0001],

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