Abstract

The vertical radiation force on an absorbing micrometer-sized dielectric sphere situated in an evanescent field is calculated by use of electromagnetic wave theory. The present study is a continuation of an earlier paper [J. Opt. Soc. Am. B 12, 2429 (1995)] in which both the horizontal and the vertical radiation forces were calculated with the constraint that the sphere was nonabsorbing. Whereas the horizontal force can be well accounted for within this constraint, there is no possibility of describing the repulsiveness of the vertical force that was so distinctly demonstrated in the Kawata–Sugiura experiment [Opt. Lett. 17, 772 (1992)] unless a departure from the theory of pure nondispersive dielectrics is made in some way. Introduction of absorption, i.e., of a complex refractive index, is one natural way to generalize the previous theory. We work out general expressions for the vertical force for this case and illustrate the calculations by numerical computations. It turns out that, when it is applied to the Kawata–Sugiura case, the repulsive radiation force caused by absorption is not strong enough to account for the actual lifting of the polystyrene latex or glass spheres. The physical reason for the experimental outcome is, in this case, most probably the presence of surfactants, which make the surfaces of the spheres partially conducting.

© 2003 Optical Society of America

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References

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  1. S. Kawata and T. Sugiura, “Movement of micrometer-sized particles in the evanescent field of a laser beam,” Opt. Lett. 17, 772–774 (1992).
    [CrossRef] [PubMed]
  2. E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12, 2429–2438 (1995).
    [CrossRef]
  3. M. Vilfan, I. Muševič, and M. Čopič, “AFM observation of force on a dielectric sphere in the evanescent field of totally reflected light,” Europhys. Lett. 43, 41–46 (1998).
    [CrossRef]
  4. M. Lester and M. Nieto-Vesperinas, “Optical forces on microparticles in an evanescent laser field,” Opt. Lett. 24, 936–938 (1999).
    [CrossRef]
  5. V. G. Levich, Physicochemical Hydrodynamics (Prentice-Hall, Englewood Cliffs, N.J., 1962).
  6. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Pergamon, Oxford, 1987), Sec. 63.
  7. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).
  8. 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. 64, 1632–1639 (1988).
    [CrossRef]
  9. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  10. H. R. Philipp, in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), p. 749.
  11. 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]
  12. Ø. Farsund and B. U. Felderhof, “Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnetic radiation field,” Physica A 227, 108–130 (1996).
    [CrossRef]
  13. M. Kerker, The Scattering of Light (Academic, New York, 1969).
  14. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1991).
  15. I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52, 133–201 (1979).
    [CrossRef]
  16. P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
    [CrossRef]
  17. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [CrossRef] [PubMed]
  18. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984), Sec. 93.
  19. I. Brevik and V. N. Marachevsky, “Casimir surface force on a dilute dielectric ball,” Phys. Rev. D 60, 085006 (1999).
    [CrossRef]
  20. P. C. Chaumet and M. Nieto-Vesperinas, “Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,” Phys. Rev. B 61, 14119–14127 (2000).
    [CrossRef]
  21. P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62, 11185–11191 (2000).
    [CrossRef]

2000 (3)

P. C. Chaumet and M. Nieto-Vesperinas, “Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,” Phys. Rev. B 61, 14119–14127 (2000).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62, 11185–11191 (2000).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

1999 (2)

M. Lester and M. Nieto-Vesperinas, “Optical forces on microparticles in an evanescent laser field,” Opt. Lett. 24, 936–938 (1999).
[CrossRef]

I. Brevik and V. N. Marachevsky, “Casimir surface force on a dilute dielectric ball,” Phys. Rev. D 60, 085006 (1999).
[CrossRef]

1998 (1)

M. Vilfan, I. Muševič, and M. Čopič, “AFM observation of force on a dielectric sphere in the evanescent field of totally reflected light,” Europhys. Lett. 43, 41–46 (1998).
[CrossRef]

1996 (1)

Ø. Farsund and B. U. Felderhof, “Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnetic radiation field,” Physica A 227, 108–130 (1996).
[CrossRef]

1995 (1)

1992 (1)

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 (1)

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. 64, 1632–1639 (1988).
[CrossRef]

1986 (1)

1979 (1)

I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52, 133–201 (1979).
[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. 64, 1632–1639 (1988).
[CrossRef]

Almaas, E.

Ashkin, A.

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. 64, 1632–1639 (1988).
[CrossRef]

Bjorkholm, J. E.

Brevik, I.

I. Brevik and V. N. Marachevsky, “Casimir surface force on a dilute dielectric ball,” Phys. Rev. D 60, 085006 (1999).
[CrossRef]

E. Almaas and I. Brevik, “Radiation forces on a micrometer-sized sphere in an evanescent field,” J. Opt. Soc. Am. B 12, 2429–2438 (1995).
[CrossRef]

I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52, 133–201 (1979).
[CrossRef]

Chaumet, P. C.

P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, “Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,” Phys. Rev. B 61, 14119–14127 (2000).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62, 11185–11191 (2000).
[CrossRef]

Chu, S.

Copic, M.

M. Vilfan, I. Muševič, and M. Čopič, “AFM observation of force on a dielectric sphere in the evanescent field of totally reflected light,” Europhys. Lett. 43, 41–46 (1998).
[CrossRef]

Dziedzic, J. M.

Farsund, Ø.

Ø. Farsund and B. U. Felderhof, “Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnetic radiation field,” Physica A 227, 108–130 (1996).
[CrossRef]

Felderhof, B. U.

Ø. Farsund and B. U. Felderhof, “Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnetic radiation field,” Physica A 227, 108–130 (1996).
[CrossRef]

Kawata, S.

Lester, M.

Marachevsky, V. N.

I. Brevik and V. N. Marachevsky, “Casimir surface force on a dilute dielectric ball,” Phys. Rev. D 60, 085006 (1999).
[CrossRef]

Muševic, I.

M. Vilfan, I. Muševič, and M. Čopič, “AFM observation of force on a dielectric sphere in the evanescent field of totally reflected light,” Europhys. Lett. 43, 41–46 (1998).
[CrossRef]

Nieto-Vesperinas, M.

P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, “Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,” Phys. Rev. B 61, 14119–14127 (2000).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62, 11185–11191 (2000).
[CrossRef]

M. Lester and M. Nieto-Vesperinas, “Optical forces on microparticles in an evanescent laser field,” Opt. Lett. 24, 936–938 (1999).
[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. 64, 1632–1639 (1988).
[CrossRef]

Sugiura, T.

Vilfan, M.

M. Vilfan, I. Muševič, and M. Čopič, “AFM observation of force on a dielectric sphere in the evanescent field of totally reflected light,” Europhys. Lett. 43, 41–46 (1998).
[CrossRef]

Europhys. Lett. (1)

M. Vilfan, I. Muševič, and M. Čopič, “AFM observation of force on a dielectric sphere in the evanescent field of totally reflected light,” Europhys. Lett. 43, 41–46 (1998).
[CrossRef]

J. Appl. Phys. (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. 64, 1632–1639 (1988).
[CrossRef]

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. Opt. Soc. Am. B (1)

Opt. Lett. (4)

Phys. Rep. (1)

I. Brevik, “Experiments in phenomenological electrodynamics and the electromagnetic energy-momentum tensor,” Phys. Rep. 52, 133–201 (1979).
[CrossRef]

Phys. Rev. B (2)

P. C. Chaumet and M. Nieto-Vesperinas, “Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,” Phys. Rev. B 61, 14119–14127 (2000).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62, 11185–11191 (2000).
[CrossRef]

Phys. Rev. D (1)

I. Brevik and V. N. Marachevsky, “Casimir surface force on a dilute dielectric ball,” Phys. Rev. D 60, 085006 (1999).
[CrossRef]

Physica A (1)

Ø. Farsund and B. U. Felderhof, “Force, torque, and absorbed energy for a body of arbitrary shape and constitution in an electromagnetic radiation field,” Physica A 227, 108–130 (1996).
[CrossRef]

Other (8)

M. Kerker, The Scattering of Light (Academic, New York, 1969).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1991).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

H. R. Philipp, in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), p. 749.

V. G. Levich, Physicochemical Hydrodynamics (Prentice-Hall, Englewood Cliffs, N.J., 1962).

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Pergamon, Oxford, 1987), Sec. 63.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, Oxford, 1984), Sec. 93.

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Figures (5)

Fig. 1
Fig. 1

Spherical particle of radius a with complex refractive index n¯3 situated in an evanescent field with its center at a height h above the plane substrate. A laser beam is incident from below at angle of incidence θ1>θcrit. Refractive indices in transparent media 1 and 2 are, respectively, n1 and n2.

Fig. 2
Fig. 2

Total absorptive force versus increasing cutoff in Eq. (39) for four values of α: 2.5, 5.0, 7.5, and 10.0, with n1=1.75, n2=1.33, and n3=1.50. Inset, absolute value of the difference in the total force with cutoffs l and (l-1), respectively.

Fig. 3
Fig. 3

Nondimensional vertical absorptive force Qxabs=Fxabs/(0E02a2) versus particle size parameter α=k2a for the set of refractive indices shown. The two states of polarization for the incident beam are distinguished. Here, as well as in the subsequent figures, θ1=51°. Also, all figures refer to a sphere resting upon a plate, i.e., to h=a. The figure is scaled against conductivity σ of the sphere.

Fig. 4
Fig. 4

Same as for Fig. 3 but with a higher (dominant real part of) refractive index (n3=1.60) in the sphere.

Fig. 5
Fig. 5

Same as for Fig. 3 but with vacuum (n2=1) surrounding the sphere. 

Tables (1)

Tables Icon

Table 1 Numerical Values of Real and Imaginary Parts of Alm and Blm

Equations (92)

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f=-½E2,
¯3=3+iσ/ω;
¯(ω)=3+i 0ωp2ω(γ-iω);
σ(ω)=0ωp2γ-iω.
σ=0.21S/m.
σ=550S/m.
α=k2a=n2ωa/c,r˜=r/a,
ψl(x)=xjl(x),ξl(1)(x)=xhl(1)(x).
Alm=(b/a)2E0l(l+1)ψl(k2b)ΩEr(i)(b, θ, φ)Ylm*(θ, φ)dΩ,
Blm=(b/a)2H0l(l+1)ψl(k2b)ΩHr(i)(b, θ, φ)Ylm*(θ, φ)dΩ,
β=n1ωc (sin2 θ1-n212)1/2,γ=n1ωcsin θ1,
n21=n2n1
T=2n21cos θ1n212cos θ1+i(sin2 θ1-n212)1/2,
T=2 cos θ1cos θ1+i(sin2 θ1-n212)1/2.
Er(i)=1n21 TE(1)[sin θ1sin θ cos φ-i(sin2 θ1-n212)1/2cos θ]+TE(1)sin θ sin φ×exp[-β(x+h)+iγz],
Hr(i)={TH(1)[-sin θ1sin θ cos φ+i(sin2 θ1-n212)1/2cos θ]+n21TH(1)sin θ sin φ}×exp[-β(x+h)+iγz],
H(2)/H(1)=n21T,H(2)/H(1)=n21T
Alm=Tn2T Blm,
Blm=-n2TT Alm;
Alm=α1(l, m)n21 Texp(-βh)[sin θ1Q1(l, m)-i(sin2 θ1-n212)1/2Q2(l, m)],
Blm=n2α1(l, m)Texp(-βh)Q3(l, m),
α1(l, m)=2l+14π(l-m)!(l+m)!1/2(b/a)2l(l+1)ψl(k2b),
Q1(l, m)=2π(-1)m-1×0π/2sin2 θcosi sin(γb cos θ)Plm(cos θ)×[I|m-1|(βb sin θ)+I|m+1|(βb sin θ)]dθ,
Q2(l, m)=4π(-1)m×0π/2sin θ cos θi sincos(γb cos θ)Plm(cos θ)×I|m|(βb sin θ)dθ,
Q3(l, m)=4πi(-1)mmβb×0π/2sin θcosi sin(γb cos θ)Plm(cos θ)×I|m|(βb sin θ)dθ,
F=Fsurf+Fabs,
Fisurf=-SiknkdS,
F=6πa3(+20)2 0k|E(i)|2;
Fxsurf+iFysurf0E02a2
=iα24l=1m=-ll(l+m+2)(l+m+1)(2l+1)(2l+3)1/2×l(l+2)(2n22almal+1,m+1*+n22almAl+1,m+1*+n22Almal+1,m+1*+2blmbl+1,m+1*+blmBl+1,m+1*+Blmbl+1,m+1*)+(l-m+1)(l-m+2)(2l+1)(2l+3)1/2×l(l+2)(2n22al+1,m-1alm*+n22al+1,m-1Alm*+n22Al+1,m-1alm*+2bl+1,m-1blm*+bl+1,m-1Blm*+Bl+1,m-1blm*)-[(l+m+1)(l-m)]1/2×n2(-2almbl,m+1*+2blmal,m+1*-almBl,m+1*+blmAl,m+1*+Blmal,m+1*-Almbl,m+1*),
Fzsurf0E02a2
=-α22l=1m=-ll(l-m+1)(l+m+1)(2l+1)(2l+3)1/2×l(l+2)I[2n22al+1,malm*+n22al+1,mAlm*+n22Al+1,malm*+2bl+1,mblm*+bl+1,mBlm*+Bl+1,mblm*+n2m(2almblm*+almBlm*+Almblm*)].
f=ρE+J×B-½E2,
Fabs=12RJ×B*dV=σμ02RE×H*dV,
Fxabs=σμ02 a3R02πdφ0πsin θdθ×01r˜2dr˜[(E×H*)rsin θ cos φ+(E×H*)θcos θ cos φ-(E×H*)φsin φ],=σμ02 a3Ri=16Ii,
I1=02πcos φdφ0πsin2 θdθ01r˜2dr˜EθHφ*,
I2=-02πcos φdφ0πsin2 θdθ01r˜2dr˜EφHθ*,
I3=02πcos φdφ0πcos θ sin θdθ01r˜2dr˜EφHr*,
I4=-02πcos φdφ0πcos θ sin θdθ01r˜2dr˜ErHφ*,
I5=-02πsin φdφ0πsin θdθ01r˜2dr˜ErHθ*,
I6=02πsin φdφ0πsin θdθ01r˜2dr˜EθHr*,
L=l=1j=10n32αdu,
Mlj(±)=m=-llk=-jj(δm,k-1±δm,k+1).
Ii=iπαn1c0E02k=14Iik,i=1, 2,
I11=-n32L{ψl(u)ψj(u)Mlj(+)×[kclmdjk*ClmCjkR1(l, m, j, k)]},
I12=-n322n2L{ψl(u)ψj(u)Mlj(+)×[clmcjk*ClmCjkR2(l, m, j, k)]},
I13=1n2L{ψl(u)ψj(u)Mlj(+)×[mkdlmdjk*ClmCjkR3(l, m, j, k)]},
I14=n32L{ψl(u)ψj(u)Mlj(+)×[mdlmcjk*ClmCjkR1(j, k, l, m)]},
I21=-n32L{ψl(u)ψj(u)Mlj(+)×[mclmdjk*ClmCjkR1(j, k, l, m)]},
I22=-n322L{ψl(u)ψj(u)Mlj(+)×[mkclmcjk*ClmCjkR3(l, m, j, k)]},
I23=1n2L{ψl(u)ψj(u)Mlj(+)×[dlmdjk*ClmCjkR2(l, m, j, k)]},
I24=n32L{ψl(u)ψj(u)Mlj(+)×[kdlmcjk*ClmCjkR1(l, m, j, k)]},
Clm=(2l+1)(l-m)!4π(l+m)!1/2
R1(l, m, j, k)=0πsin θ dPlm(cos θ)dθ Pjk(cos θ)dθ,
R2(l, m, j, k)=0πsin2 θ dPlm(cos θ)dθdPjk(cos θ)dθdθ,
R3(l, m, j, k)=0πPlm(cos θ)Pjk(cos θ)dθ,
R4(l, m, j, k)=0πcos θPlm(cos θ)Pjk(cos θ)dθ,
R5(l, m, j, k)=0πcos θ sin θ dPlm(cos θ)dθ Pjk(cos θ)dθ.
Ii=iπαn1c0E02k=12Iik,i=3, 4, 5, 6.
I31=n32Lψl(u)u ψj(u)j(j+1)Mlj(+)×[mclmdjk*ClmCjkR4(l, m, j, k)],
I32=-1n2Lψl(u)u ψj(u)j(j+1)Mlj(+)×[dlmdjk*ClmCjkR5(l, m, j, k)],
I41=n32Lψl(u)u ψj(u)l(l+1)Mlj(+)×[kclmdjk*ClmCjkR4(l, m, j, k)],
I42=n322n2Lψl(u)u ψj(u)l(l+1)Mlj(+)×[clmcjk*ClmCjkR5(j, k, l, m)],
I51=-n32Lψl(u)u ψj(u)l(l+1)Mlj(-)×[clmdjk*ClmCjkR1(j, k, l, m)],
I52=-n322n2Lψl(u)u ψj(u)l(l+1)Mlj(-)×[kclmcjk*ClmCjkR3(l, m, j, k)],
I61=n32Lψl(u)u ψj(u)j(j+1)Mlj(-)×[clmdjk*ClmCjkR1(l, m, j, k)],
I62=-1n2Lψl(u)u ψj(u)j(j+1)Mlj(-)×[mdlmdjk*ClmCjkR3(l, m, j, k)].
Qxabs=Fxabs0E02a2,
Qxabs=-σμ0cλ n12α24Ii=12k=14Iik+i=36k=12Iik.
Qxabs=mg0E02a2+|Qxsurf|.
Er(i)=E0r˜2l=1m=-lll(l+1)Almψl(αr˜)Ylm,
Eθ(i)=αE0r˜l=1m=-llAlmψl(αr˜) Ylmθ-mn2 Blmψl(αr˜) Ylmsin θ,
Eφ(i)=αE0r˜l=1m=-llimAlmψl(αr˜) Ylmsin θ-in2 Blmψl(αr˜) Ylmθ,
Hr(i)=H0r˜2l=1m=-lll(l+1)Blmψl(αr˜)Ylm,
Hθ(i)=αH0r˜l=1m=-llBlmψl(αr˜) Ylmθ+mn2Almψl(αr˜) Ylmsin θ,
Hφ(i)=αH0r˜l=1m=-llimBlmψl(αr˜) Ylmsin θ+in2Almψl(αr˜) Ylmθ.
Er(s)=E0r˜2l=1m=-lll(l+1)almξl(1)(αr˜)Ylm,
Eθ(s)=αE0r˜l=1m=-llalmξl(1)(αr˜) Ylmθ-mn2 blmξl(1)(αr˜) Ylmsin θ,
Eφ(s)=αE0r˜l=1m=-llimalmξl(1)(αr˜) Ylmsin θ-in2 blmξl(1)(αr˜) Ylmθ,
Hr(s)=H0r˜2l=1m=-lll(l+1)blmξl(1)(αr˜)Ylm,
Hθ(s)=αH0r˜l=1m=-llblmξl(1)(αr˜) Ylmθ+mn2almξl(1)(αr˜) Ylmsin θ,
Hφ(s)=αH0r˜l=1m=-llimblmξl(1)(αr˜) Ylmsin θ+in2almξl(1)(αr˜) Ylmθ.
Er(w)=E0r˜2l=1m=-lll(l+1)clmψl(n¯32αr˜)Ylm,
Eθ(w)=αE0r˜l=1m=-lln¯32clmψl(n¯32αr˜) Ylmθ-mn2 dlmψl(n¯32αr˜) Ylmsin θ,
Eφ(w)=αE0r˜l=1m=-llimn¯32clmψl(n¯32αr˜) Ylmsin θ-in2 dlmψl(n¯32αr˜) Ylmθ,
Hr(w)=H0r˜2l=1m=-lll(l+1)dlmψl(n¯32αr˜)Ylm,
Hθ(w)=αH0r˜l=1m=-lln¯32dlmψl(n¯32αr˜) Ylmθ+mn2n¯322clmψl(n¯32αr˜) Ylmsin θ,
Hφ(w)=αH0r˜l=1m=-llimn¯32dlmψl(n¯32αr˜) Ylmsin θ+in2n¯322clmψl(n¯32αr˜) Ylmθ.
alm=ψl(n¯32α)ψl(α)-n¯32ψl(n¯32α)ψl(α)n¯32ψl(n¯32α)ξl(1)(α)-ψl(n¯32α)ξl(1)(α) Alm,
blm=n¯32ψl(n¯32α)ψl(α)-ψl(n¯32α)ψl(α)ψl(n¯32α)ξl(1)(α)-n¯32ψl(n¯32α)ξl(1)(α) Blm,
clm=in¯322ψl(n¯32α)ξl(1)(α)-n¯32ψl(n¯32α)ξl(1)(α) Alm,
dlm=iψl(n¯32α)ξl(1)(α)-n¯32ψl(n¯32α)ξl(1)(α) Blm.

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