Abstract

The theory of Aden and Kerker is used to compute the resonant response of large layered spheres to an incident linearly polarized plane wave. Cases considered include hollow spheres and those with transparent and absorbing cores whose diameters range from zero up to that of the particle. Both core and layer resonances are studied. The results are presented as partial wave amplitudes, energy densities, and scattered light intensities for a typical resonant mode, TE39.

© 1988 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. M. Kerker, The Scattering of Light (Academic, New York, 1969).
  3. C. E. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).
  4. A. Ashkin, J. M. Dziedzic, “Observations of Resonances in the Radiation Pressure on Dielectric Spheres,” Phys. Rev. Lett. 38, 1351 (1977).
    [CrossRef]
  5. R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of Structure Resonances in the Fluorescence from Microspheres,” Phys. Rev. Lett. 44, 475 (1980).
    [CrossRef]
  6. A. Ashkin, J. M. Dziedzic, “Observation of Optical Resonances of Dielectric Spheres by Light Scattering,” Appl. Opt. 20, 1803 (1981).
    [CrossRef] [PubMed]
  7. R. Thurn, W. Kiefer, “Structural Resonances Observed in the Raman Spectra of Optically Levitated Liquid Droplets,” Appl. Opt. 24, 1515 (1985).
    [CrossRef] [PubMed]
  8. P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical Levitation and Partial-Wave Resonances,” Phys. Rev. A 18, 2229 (1978).
    [CrossRef]
  9. A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
    [CrossRef]
  10. J. R. Wait, “Electromagnetic Scattering from a Radially Inhomogeneous Sphere,” Appl. Sci. Res. Sect. B 10, 441 (1967).
    [CrossRef]
  11. R. Bhandari, “Scattering Coefficients for a Multilayered Sphere: Analytic Expressions and Algorithms,” Appl. Opt. 24, 1960 (1985).
    [CrossRef] [PubMed]
  12. R. W. Fenn, H. Oser, “Scattering Properties of Concentric Soot–Water Spheres for Visible and Infrared Light,” Appl. Opt. 4, 1504 (1965).
    [CrossRef]
  13. G. W. Kattawar, D. A. Hood, “Electromagnetic Scattering from a Spherical Polydispersion of Coated Spheres,” Appl. Opt. 15, 1996 (1976).
    [CrossRef] [PubMed]
  14. O. B. Toon, T. P. Ackerman, “Algorithms for the Calculation of Scattering by Stratified Spheres,” Appl. Opt. 20, 3657 (1981).
    [CrossRef] [PubMed]
  15. S. R. Aragon, M. Elwenspoeke, “Mie Scattering from Thin Spherical Bubbles,” J. Chem. Phys. 77, 3406 (1982).
    [CrossRef]
  16. R. Bhandari, “Tiny Core or Thin Layer as a Perturbation in Scattering by a Single-Layered Sphere,” J. Opt. Soc. Am. A 3, 319 (1986).
    [CrossRef]
  17. T. W. Chen, “Scattering of Light by a Stratified Sphere in High Energy Approximation,” Appl. Opt. 26, 4155 (1987).
    [CrossRef] [PubMed]
  18. A. B. Pulchino, “Surface Waves and the Radiative Properties of Micron-Sized Particles,” Appl. Opt. 20, 2986 (1981).
    [CrossRef]
  19. R. L. Hightower, C. B. Richardson, H.-B. Lin, J. D. Eversole, A. J. Campillo, “Measurements of Scattering of Light from Layered Microspheres,” Opt. Lett.13, in press (1988).
    [CrossRef] [PubMed]
  20. Two spellings of Riccati are in the literature; J. F. Riccati was an Italian mathematician (1676–1754).
  21. W. J. Wiscombe, Mie Scattering Calculations: Advances in Technique and Fast Vector-Speed Codes, Document PB301388 (NTIS, Springfield, VA 22176, 1979).
  22. C. B. Richardson, R. L. Hightower, A. L. Pigg, “Optical Measurement of the Evaporation of Sulfuric Acid Droplets,” Appl. Opt. 25, 1226 (1986); C. B. Richardson, R. L. Hightower, “Evaporation of Ammonium Nitrate Particles,” Atmos. Environ. 21, 971 (1987).
    [CrossRef] [PubMed]

1987

1986

1985

1982

S. R. Aragon, M. Elwenspoeke, “Mie Scattering from Thin Spherical Bubbles,” J. Chem. Phys. 77, 3406 (1982).
[CrossRef]

1981

1980

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of Structure Resonances in the Fluorescence from Microspheres,” Phys. Rev. Lett. 44, 475 (1980).
[CrossRef]

1978

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical Levitation and Partial-Wave Resonances,” Phys. Rev. A 18, 2229 (1978).
[CrossRef]

1977

A. Ashkin, J. M. Dziedzic, “Observations of Resonances in the Radiation Pressure on Dielectric Spheres,” Phys. Rev. Lett. 38, 1351 (1977).
[CrossRef]

1976

1967

J. R. Wait, “Electromagnetic Scattering from a Radially Inhomogeneous Sphere,” Appl. Sci. Res. Sect. B 10, 441 (1967).
[CrossRef]

1965

1951

A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

Ackerman, T. P.

Aden, A. L.

A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

Aragon, S. R.

S. R. Aragon, M. Elwenspoeke, “Mie Scattering from Thin Spherical Bubbles,” J. Chem. Phys. 77, 3406 (1982).
[CrossRef]

Ashkin, A.

A. Ashkin, J. M. Dziedzic, “Observation of Optical Resonances of Dielectric Spheres by Light Scattering,” Appl. Opt. 20, 1803 (1981).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Observations of Resonances in the Radiation Pressure on Dielectric Spheres,” Phys. Rev. Lett. 38, 1351 (1977).
[CrossRef]

Barber, P. W.

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of Structure Resonances in the Fluorescence from Microspheres,” Phys. Rev. Lett. 44, 475 (1980).
[CrossRef]

Benner, R. E.

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of Structure Resonances in the Fluorescence from Microspheres,” Phys. Rev. Lett. 44, 475 (1980).
[CrossRef]

Bhandari, R.

Bohren, C. E.

C. E. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Campillo, A. J.

R. L. Hightower, C. B. Richardson, H.-B. Lin, J. D. Eversole, A. J. Campillo, “Measurements of Scattering of Light from Layered Microspheres,” Opt. Lett.13, in press (1988).
[CrossRef] [PubMed]

Chang, R. K.

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of Structure Resonances in the Fluorescence from Microspheres,” Phys. Rev. Lett. 44, 475 (1980).
[CrossRef]

Chen, T. W.

Chylek, P.

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical Levitation and Partial-Wave Resonances,” Phys. Rev. A 18, 2229 (1978).
[CrossRef]

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, “Observation of Optical Resonances of Dielectric Spheres by Light Scattering,” Appl. Opt. 20, 1803 (1981).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Observations of Resonances in the Radiation Pressure on Dielectric Spheres,” Phys. Rev. Lett. 38, 1351 (1977).
[CrossRef]

Elwenspoeke, M.

S. R. Aragon, M. Elwenspoeke, “Mie Scattering from Thin Spherical Bubbles,” J. Chem. Phys. 77, 3406 (1982).
[CrossRef]

Eversole, J. D.

R. L. Hightower, C. B. Richardson, H.-B. Lin, J. D. Eversole, A. J. Campillo, “Measurements of Scattering of Light from Layered Microspheres,” Opt. Lett.13, in press (1988).
[CrossRef] [PubMed]

Fenn, R. W.

Hightower, R. L.

Hood, D. A.

Huffman, D. R.

C. E. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Kattawar, G. W.

Kerker, M.

A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

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

Kiefer, W.

Kiehl, J. T.

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical Levitation and Partial-Wave Resonances,” Phys. Rev. A 18, 2229 (1978).
[CrossRef]

Ko, M. K. W.

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical Levitation and Partial-Wave Resonances,” Phys. Rev. A 18, 2229 (1978).
[CrossRef]

Lin, H.-B.

R. L. Hightower, C. B. Richardson, H.-B. Lin, J. D. Eversole, A. J. Campillo, “Measurements of Scattering of Light from Layered Microspheres,” Opt. Lett.13, in press (1988).
[CrossRef] [PubMed]

Oser, H.

Owen, J. F.

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of Structure Resonances in the Fluorescence from Microspheres,” Phys. Rev. Lett. 44, 475 (1980).
[CrossRef]

Pigg, A. L.

Pulchino, A. B.

Richardson, C. B.

Thurn, R.

Toon, O. B.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Wait, J. R.

J. R. Wait, “Electromagnetic Scattering from a Radially Inhomogeneous Sphere,” Appl. Sci. Res. Sect. B 10, 441 (1967).
[CrossRef]

Wiscombe, W. J.

W. J. Wiscombe, Mie Scattering Calculations: Advances in Technique and Fast Vector-Speed Codes, Document PB301388 (NTIS, Springfield, VA 22176, 1979).

Appl. Opt.

Appl. Sci. Res. Sect. B

J. R. Wait, “Electromagnetic Scattering from a Radially Inhomogeneous Sphere,” Appl. Sci. Res. Sect. B 10, 441 (1967).
[CrossRef]

J. Appl. Phys.

A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

J. Chem. Phys.

S. R. Aragon, M. Elwenspoeke, “Mie Scattering from Thin Spherical Bubbles,” J. Chem. Phys. 77, 3406 (1982).
[CrossRef]

J. Opt. Soc. Am. A

Phys. Rev. A

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical Levitation and Partial-Wave Resonances,” Phys. Rev. A 18, 2229 (1978).
[CrossRef]

Phys. Rev. Lett.

A. Ashkin, J. M. Dziedzic, “Observations of Resonances in the Radiation Pressure on Dielectric Spheres,” Phys. Rev. Lett. 38, 1351 (1977).
[CrossRef]

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of Structure Resonances in the Fluorescence from Microspheres,” Phys. Rev. Lett. 44, 475 (1980).
[CrossRef]

Other

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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

C. E. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

R. L. Hightower, C. B. Richardson, H.-B. Lin, J. D. Eversole, A. J. Campillo, “Measurements of Scattering of Light from Layered Microspheres,” Opt. Lett.13, in press (1988).
[CrossRef] [PubMed]

Two spellings of Riccati are in the literature; J. F. Riccati was an Italian mathematician (1676–1754).

W. J. Wiscombe, Mie Scattering Calculations: Advances in Technique and Fast Vector-Speed Codes, Document PB301388 (NTIS, Springfield, VA 22176, 1979).

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Figures (9)

Fig. 1
Fig. 1

Partial wave coefficient Re{b39} vs core radius r1 and wavenumber k. Particle radius is r2 = 3.1010 μm and layer index is m2 = 1.437. Left branch: m1 = 1.438; center: m1 = 1.437; right: m1 = 1.436.

Fig. 2
Fig. 2

Partial wave coefficient Re{b39} for the particles of Fig. 1 when the core is slightly absorbing, with index m1i × 10−5.

Fig. 3
Fig. 3

Loci of TE39 resonance wavenumbers vs core radius. Particle radius is r2 = 3.1010 μm and layer index is m2 = 1.437. Curve a, m1 = 1.5500; curve b, m1 = 1.4370; curve c, m1 = 1.3000; curve d, m1 = 1.0000.

Fig. 4
Fig. 4

Riccati-Bessel function expressions B39(m1,m2,kr1) (dashed) and β39(m2,kr2) (solid) vs x = kr for m1 = 1.300 and m2 = 1.437. At resonance the two expressions are equal.

Fig. 5
Fig. 5

Partial wave coefficient Re{b39} vs core radius r1 and wavenumber k for the hollow sphere. Particle radius is r2 = 3.1010 μm and layer index is m2 = 1.437. Four cycles of Eq. (6b) are plotted with the resonant first cycle in the foreground.

Fig. 6
Fig. 6

Angle-averaged radial distribution of energy in the particle at resonance. For each, r1 = 2.819 μm, r2 = 3.101 μm, and m2 = 1.437. The maximum value (arbitrary units) and the resonant wavenumber are shown.

Fig. 7
Fig. 7

Intensity of light scattered 83° from a hollow sphere of inner radius 2.8187 μm, outer radius 3.1010 μm, and index 1.437. The polarization is perpendicular to the scattering plane.

Fig. 8
Fig. 8

Amplitude of the TE39 partial wave |c39| when the core is at resonance. Here r1 = 3.101 μm, m1 = 1.437, and m2 = 1.100.

Fig. 9
Fig. 9

Intensity of light scattered 83° from the particle of Fig. 8 when the layer has radius r2 = 5.000 μm.

Equations (38)

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E θ = ( i / k r ) E 0 exp i ( k r - ω t ) cos ϕ S 2 ( θ ) ,
E ϕ = ( i / k r ) E 0 exp i ( k r - ω t ) sin ϕ S 1 ( θ ) .
S 1 ( θ ) = n = 1 ( 2 n + 1 ) / n ( n + 1 ) [ a n π n ( θ ) + b n τ n ( θ ) ] ,
S 2 ( θ ) = n = 1 ( 2 n + 1 ) / n ( n + 1 ) [ a n τ n ( θ ) + b n π n ( θ ) ] ,
a n = ψ n ( x 2 ) [ ψ n ( m 2 x 2 ) - A n χ n ( m 2 x 2 ) ] - m 2 ψ n ( x 2 ) [ ψ n ( m 2 x 2 ) - A n χ n ( m 2 x 2 ) ] ζ n ( x 2 ) [ ψ n ( m 2 x 2 ) - A n χ n ( m 2 x 2 ) ] - m 2 ζ n ( x 2 ) [ ψ n ( m 2 x 2 ) - A n χ n ( m 2 x 2 ) ] ,
b n = m 2 ψ n ( x 2 ) [ ψ n ( m 2 x 2 ) - B n χ n ( m 2 x 2 ) ] - ψ n ( x 2 ) [ ψ n ( m 2 x 2 ) - B n χ n ( m 2 x 2 ) ] m 2 ζ n ( x 2 ) [ ψ n ( m 2 x 2 ) - B n χ n ( m 2 x 2 ) ] - ζ n ( x 2 ) [ ψ n ( m 2 x 2 ) - B n χ n ( m 2 x 2 ) ] .
ψ n ( s ) = s j n ( s ) ,
χ n ( s ) = - s n n ( s ) ,
ζ n ( s ) = s h n ( 1 ) ( s ) ,
A n = m 2 ψ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) - m 1 ψ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) m 2 χ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) - m 1 χ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) ,
B n = m 2 ψ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) - m 1 ψ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) m 2 χ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) - m 1 χ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) .
2 a n = 1 - exp - i 2 Ω a , n ,
2 b n = 1 - exp - i 2 Ω b , n ,
- tan Ω a , n = ψ n ( x 2 ) [ ψ n ( m 2 x 2 ) - A n χ n ( m 2 x 2 ) ] - m 2 ψ n ( x 2 ) [ ψ n ( m 2 x 2 ) - A n χ n ( m 2 x 2 ) ] χ n ( x 2 ) [ ψ n ( m 2 x 2 ) - A n χ n ( m 2 x 2 ) ] - m 2 χ n ( x 2 ) [ ψ n ( m 2 x 2 ) - A n χ n ( m 2 x 2 ) ] ,
- tan Ω b , n = m 2 ψ n ( x 2 ) [ ψ n ( m 2 x 2 ) - B n χ n ( m 2 x 2 ) ] - ψ n ( x 2 ) [ ψ n ( m 2 x 2 ) - B n χ n ( m 2 x 2 ) ] m 2 χ n ( x 2 ) [ ψ n ( m 2 x 2 ) - B n χ n ( m 2 x 2 ) ] - χ n ( x 2 ) [ ψ n ( m 2 x 2 ) - B n χ n ( m 2 x 2 ) ] .
α n = χ n ( x 2 ) ψ n ( m 2 x 2 ) - m 2 χ n ( x 2 ) ψ n ( m 2 x 2 ) χ n ( x 2 ) χ n ( m 2 x 2 ) - m 2 χ n ( x 2 ) χ n ( m 2 x 2 ) ,
β n = χ n ( x 2 ) ψ n ( m 2 x 2 ) - m 2 χ n ( x 2 ) ψ n ( m 2 x 2 ) χ n ( x 2 ) χ n ( m 2 x 2 ) - m 2 χ n ( x 2 ) χ n ( m 2 x 2 ) .
P = E 0 2 ω S 1 ( θ ) 2 / μ 0 c k 2 .
ζ n ( s ) = ψ n ( s ) + i χ n ( s ) ,
1 = ψ n ( s ) χ n ( s ) - χ n ( s ) ψ n ( s ) ,
χ n ( s ) = ( n + 1 ) s - 1 χ n + 1 ( s ) - χ n + 2 ( s ) .
ψ n - 2 ( s ) = ( 2 n - 1 ) s - 1 ψ n - 1 ( s ) - ψ n ( s ) ,
χ n + 2 ( s ) = ( 2 n + 3 ) s - 1 χ n + 1 ( s ) - χ n ( s ) .
ψ n ( s ) = D n ( s ) ψ n ( s ) .
D n ( s ) = n s - 1 + [ n s - 1 - D n - 1 ( s ) ] - 1 .
D 1 ( s ) = - s - 1 + [ tan ( s ) / s - 1 tan ( s ) - 1 ] .
E r = - i sin θ cos ϕ k 1 2 r 2 n = 1 n ( n + 1 ) E n d n π n ( θ ) ψ n ( k 1 r ) , E θ = cos θ k 1 r n = 1 E n [ c n π n ( θ ) ψ n ( k 1 r ) - i d n τ n ( θ ) ψ n ( k 1 r ) ] , E ϕ = - sin ϕ k 1 r E n [ c n τ n ( θ ) ψ n ( k 1 r ) - i d n π n ( θ ) ψ n ( k 1 r ) ] .
U = m 1 ɛ 2 0 E · E * .
U ¯ = ( ɛ 0 E 0 2 / 2 k 2 r 2 ) n = 1 ( 2 n + 1 ) { c n 2 ψ n 2 ( k 1 r ) + d n 2 [ n ( n + 1 ) ( k 1 r ) - 2 ψ n 2 ( k 1 r ) + ψ n 2 ( k 1 r ) ] } .
c n = [ m 1 ψ n ( k 2 r 1 ) - B n m 1 χ n ( k 2 r 1 ) m 2 ψ n ( k 1 r 1 ) ] [ m 2 ψ n ( k r 2 ) - b n m 2 ζ n ( k r 2 ) ψ n ( k 2 r 2 ) - B n χ n ( k 2 r 2 ) ] ,
d n = [ ψ n ( k 2 r 1 ) - A n χ n ( k 2 r 1 ) ψ n ( k 1 r 1 ) ] [ ψ n ( k r 2 ) - a n ζ n ( k r 2 ) ψ n ( k 2 r 2 ) - A n χ n ( k 2 r 2 ) ] .
E r = i sin θ cos ϕ k 2 2 r 2 n = 1 E n n ( n + 1 ) π n ( θ ) g n ψ n ( k 2 r ) - w n χ n ( k 2 r ) , E θ = cos ϕ k 2 r n = 1 E n { π n ( θ ) [ f n ψ n ( k 2 r ) - v n χ n ( k 2 r ) ] - i τ n ( θ ) [ g n ψ n ( k 2 r ) - w n χ n ( k 2 r ) ] } , E ϕ = - sin ϕ k 2 r n = 1 E n { τ n ( θ ) [ f n ψ n ( k 2 r ) - v n χ n ( k 2 r ) ] - i π n ( θ ) [ g n ψ n ( k 2 r ) - w n χ n ( k 2 r ) ] } .
U ¯ = ( ɛ 0 E 0 2 / 2 k 2 r 2 ) n = 1 ( 2 n + 1 ) [ f n 2 ψ n 2 ( k 2 r ) + v n 2 χ n 2 ( k 2 r ) - 2 Re { f n v n * } ψ n ( k 2 r ) χ n ( k 2 r ) + g n 2 [ n ( n + 1 ) ( k 2 r ) - 2 ψ n 2 ( k 2 r ) + ψ n 2 ( k 2 r ) ] + w n 2 [ n ( n + 1 ) ( k 2 r ) - 2 χ n 2 ( k 2 r ) + χ n 2 ( k 2 r ) ] - 2 Re { g n w n * } [ n ( n + 1 ) ( k 2 r ) - 2 ψ n ( k 2 r ) χ n ( k 2 r ) + ψ n ( k 2 r ) χ n ( k 2 r ) ] .
f n = m 2 [ ψ n ( k r 2 ) - b n ζ n ( k r 2 ) ] ψ n ( k r 2 ) - B n χ n ( k 2 r 2 ) ,
g n = ψ n ( k r 2 ) - a n ζ n ( k r 2 ) ψ n ( k 2 r 2 ) - A n χ n ( k 2 r 2 ) ,
v n = B n f n ,
w n = A n g n .
U ¯ = ( ɛ 0 E 0 2 / 2 k 2 r 2 ) n = 1 ( 2 n + 1 ) { b n 2 ζ n ( k r ) 2 + a n 2 [ n ( n + 1 ) ( k r ) - 2 ζ n ( k r ) 2 + ζ n ( k r ) 2 ] } .

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