Abstract

Morphology-dependent resonances in a coated sphere are investigated by computation of the volume-averaged source function obtained from Lorenz–Mie theory. Analytic expressions for the source function in absorbing and nonabsorbing spheres are given in a suitable form for computations. An advantage of the investigation of the source function is that core and shell contributions can be computed and examined independently. Furthermore the influences of the refractive index and the thickness of the outer layer on resonance positions and height are studied in detail. These influences are presented for the a 104 partial-wave example.

© 1994 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
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    [CrossRef]
  12. S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), Chap. 1, pp. 3–61.
  13. P. Chýlek, “Partial-wave resonances and the ripple structure in the Mie normalized extinction cross section,” J. Opt. Soc. Am. 66, 285–287 (1976).
    [CrossRef]
  14. P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
    [CrossRef]
  15. R. E. Benner, P. W. Barber, J. F. Owne, R. K. Chang, “Observation of structure resonances in the fluoresence from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
    [CrossRef]
  16. R. Thurn, W. Kiefer, “Structural resonances observed in the Raman spectra of optically levitated liquid droplets,” Appl. Opt. 24, 1515–1519 (1985).
    [CrossRef] [PubMed]
  17. G. Schweiger, “Observation of input and output structural resonances in the Raman spectrum of a single spheroidal dielectric microparticle,” Opt. Lett. 15, 156–158 (1990).
    [CrossRef] [PubMed]
  18. G. Schweiger, “Observation of morphology-dependent resonances caused by the input field in the Raman spectrum of microdroplets,” J. Raman Spectrosc. 21, 165–168 (1990).
    [CrossRef]
  19. M. Essien, R. L. Armstrong, J. B. Gillespie, “Lasing emission from an evaporating layered microdroplet,” Opt. Lett. 18, 762–764 (1993).
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  22. A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–605 (1951).
    [CrossRef]
  23. G. Schweiger, “Raman scattering on microparticles: size dependence,” J. Opt. Soc. Am. B 8, 1770–1778 (1991).
    [CrossRef]
  24. P. Chýlek, J. D. Pendleton, R. G. Pinnick, “Internal and near-surface scattered field of a spherical particle at resonant conditions,” Appl. Opt. 24, 3940–3942 (1985).
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  25. D. Q. Chowdhury, D. H. Leach, R. K. Chang, “Effect of the Goos–Hänchen shift on the geometrical-optics model for spherical-cavity mode spacing,” J. Opt. Soc. Am. A 11, 1110–1116. (1994)
    [CrossRef]
  26. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966), p. 134, equation 7.

1994 (1)

1993 (2)

T. Kaiser, G. Schweiger, “Stable algorithm for the computation of Mie coefficients for scattered and transmitted fields of a coated sphere,” Comput. Phys. 7, 682–686 (1993).
[CrossRef]

M. Essien, R. L. Armstrong, J. B. Gillespie, “Lasing emission from an evaporating layered microdroplet,” Opt. Lett. 18, 762–764 (1993).
[CrossRef] [PubMed]

1991 (1)

1990 (3)

1988 (2)

1986 (1)

1985 (3)

1981 (2)

1980 (1)

R. E. Benner, P. W. Barber, J. F. Owne, R. K. Chang, “Observation of structure resonances in the fluoresence from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[CrossRef]

1978 (1)

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

1976 (1)

1965 (1)

1951 (2)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–605 (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, 1242–1246 (1951).
[CrossRef]

A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–605 (1951).
[CrossRef]

Armstrong, R. L.

Barber, P. W.

R. E. Benner, P. W. Barber, J. F. Owne, R. K. Chang, “Observation of structure resonances in the fluoresence from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[CrossRef]

Benner, R. E.

R. E. Benner, P. W. Barber, J. F. Owne, R. K. Chang, “Observation of structure resonances in the fluoresence from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[CrossRef]

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), Chap. 1, pp. 3–61.

Bhandari, R.

Bohren, C. F.

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

Campillo, A. J.

Chang, R. K.

D. Q. Chowdhury, D. H. Leach, R. K. Chang, “Effect of the Goos–Hänchen shift on the geometrical-optics model for spherical-cavity mode spacing,” J. Opt. Soc. Am. A 11, 1110–1116. (1994)
[CrossRef]

R. E. Benner, P. W. Barber, J. F. Owne, R. K. Chang, “Observation of structure resonances in the fluoresence from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[CrossRef]

Chowdhury, D. Q.

Chýlek, P.

Essien, M.

Eversole, J. D.

Fenn, R. W.

Gillespie, J. B.

Hightower, R. L.

Hill, S. C.

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), Chap. 1, pp. 3–61.

Huffman, D. R.

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

Kaiser, T.

T. Kaiser, G. Schweiger, “Stable algorithm for the computation of Mie coefficients for scattered and transmitted fields of a coated sphere,” Comput. Phys. 7, 682–686 (1993).
[CrossRef]

Kerker, M.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Kiefer, W.

Kiehl, J. T.

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

Ko, M. K. W.

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

Leach, D. H.

Lin, H.-B.

Lock, J. A.

Oser, H.

Owne, J. F.

R. E. Benner, P. W. Barber, J. F. Owne, R. K. Chang, “Observation of structure resonances in the fluoresence from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[CrossRef]

Pendleton, J. D.

Pinnick, R. G.

Pluchino, A. B.

Richardson, C. B.

Schweiger, G.

T. Kaiser, G. Schweiger, “Stable algorithm for the computation of Mie coefficients for scattered and transmitted fields of a coated sphere,” Comput. Phys. 7, 682–686 (1993).
[CrossRef]

G. Schweiger, “Raman scattering on microparticles: size dependence,” J. Opt. Soc. Am. B 8, 1770–1778 (1991).
[CrossRef]

G. Schweiger, “Observation of input and output structural resonances in the Raman spectrum of a single spheroidal dielectric microparticle,” Opt. Lett. 15, 156–158 (1990).
[CrossRef] [PubMed]

G. Schweiger, “Observation of morphology-dependent resonances caused by the input field in the Raman spectrum of microdroplets,” J. Raman Spectrosc. 21, 165–168 (1990).
[CrossRef]

Thurn, R.

Toon, O. B.

van de Hulst, H. C.

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

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966), p. 134, equation 7.

Appl. Opt. (8)

Comput. Phys. (1)

T. Kaiser, G. Schweiger, “Stable algorithm for the computation of Mie coefficients for scattered and transmitted fields of a coated sphere,” Comput. Phys. 7, 682–686 (1993).
[CrossRef]

J. Appl. Phys. (2)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–605 (1951).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

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

J. Raman Spectrosc. (1)

G. Schweiger, “Observation of morphology-dependent resonances caused by the input field in the Raman spectrum of microdroplets,” J. Raman Spectrosc. 21, 165–168 (1990).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

Phys. Rev. Lett. (1)

R. E. Benner, P. W. Barber, J. F. Owne, R. K. Chang, “Observation of structure resonances in the fluoresence from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[CrossRef]

Other (5)

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

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

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

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), Chap. 1, pp. 3–61.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966), p. 134, equation 7.

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Figures (6)

Fig. 1
Fig. 1

Logarithm of the volume-averaged source function in the shell and in the core for a constant size parameter ratio of x 2/x 1 = 1.3. The inserts show the radial dependence of the angle-averaged field distribution inside the sphere in arbitrary units for a typical resonance in the shell and in the core. The dashed lines represent the boundary between the core and the shell.

Fig. 2
Fig. 2

Ratio of the volume-averaged source function in the core and in the shell for different refractive indices in the layer. The curves are shifted upward by adding a constant factor of 8. The dashed lines represent the baselines for the different traces.

Fig. 3
Fig. 3

Ratio of the source functions in the core and in the shell versus the refractive index in the layer at the resonance mode a 104. The variation of the inner size parameter x 1 is plotted in the upper trace, and x 2 is kept constant. Curves (a), (b), (c), (d) at the top show the radial dependence of the angle-averaged field distributions for the x 1, x 2, m 2 combinations (a), (b), (c), (d) at the bottom, respectively.

Fig. 4
Fig. 4

Combination of the core and the shell size parameters (a–e) for the excitation of various resonance modes of the a 104 partial wave. The dotted curves indicate the ratios of x 2/x 1 for which the resonance condition is fulfilled. The dashed lines are lines with a constant size parameter ratio x 2/x 4.

Fig. 5
Fig. 5

Angle-averaged electric field computed for the core–shell size parameter combinations (a)–(e) in Fig. 4.

Fig. 6
Fig. 6

Radial dependence of the angle-averaged source function computed at five different size parameter combinations [(a)–(e)] with the a 104 partial wave in resonance as indicated in Fig. 4. The dashed lines represent the boundary between the core and the shell.

Equations (44)

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c ^ n = c n ψ n ( m 2 x 1 ) ,
d ^ n = d n ξ n ( m 2 x 1 ) ,
g ^ n = g n ψ n ( m 1 x 1 ) ,
e ^ n = e n ψ n ( m 2 x 1 ) ,
f ^ n = f n ξ n ( m 2 x 1 ) ,
h ^ n = h n ψ n ( m 1 x 1 ) ,
a n = ( D ˜ n m 2 + n x 2 ) ψ n ( x 2 ) - ψ n - 1 ( x 2 ) ( D ˜ n m 2 + n x 2 ) ξ n ( x 2 ) - ξ n - 1 ( x 2 ) ,
b n = ( m 2 G ˜ n + n x 2 ) ψ n ( x 2 ) - ψ n - 1 ( x 2 ) ( m 2 G ˜ n + n x 2 ) ξ n ( x 2 ) - ξ n - 1 ( x 2 ) ,
D ˜ n = D n ( m 2 x 2 ) - ξ n ( m 2 x 2 ) ξ n ( m 2 x 1 ) ψ n ( m 2 x 1 ) ψ n ( m 2 x 2 ) F n ( m 2 x 2 ) A ^ n 1 - ξ n ( m 2 x 2 ) ξ n ( m 2 x 1 ) ψ n ( m 2 x 1 ) ψ n ( m 2 x 2 ) A ^ n ,
G ˜ n = D n ( m 2 x 2 ) - ξ n ( m 2 x 2 ) ξ n ( m 2 x 1 ) ψ n ( m 2 x 1 ) ψ n ( m 2 x 2 ) F n ( m 2 x 2 ) B ^ n 1 - ξ n ( m 2 x 2 ) ξ n ( m 2 x 1 ) ψ n ( m 2 x 1 ) ψ n ( m 2 x 2 ) B ^ n ,
A ^ n = m 2 m 1 D n ( m 1 x 1 ) - D n ( m 2 x 1 ) m 2 m 1 D n ( m 1 x 1 ) - F n ( m 2 x 1 ) ,
B ^ n = m 2 m 1 D n ( m 2 x 1 ) - D n ( m 1 x 1 ) m 2 m 1 F n ( m 2 x 1 ) - D n ( m 1 x 1 ) ,
c ^ n = m 2 i ψ n ( m 2 x 1 ) ψ n ( m 2 x 2 ) ξ n ( x 2 ) [ D n ( m 2 x 2 ) - A ^ n F n ( m 2 x 2 ) ψ n ( m 2 x 1 ) ξ n ( m 2 x 2 ) ψ n ( m 2 x 2 ) ξ n ( m 2 x 1 ) ] - m 2 ξ n ( x 2 ) [ 1 - A ^ n ξ n ( m 2 x 2 ) ψ n ( m 2 x 1 ) ξ n ( m 2 x 1 ) ψ n ( m 2 x 2 ) ] ,
d ^ n = - A ^ n c ^ n ,
g ^ n = G ^ n c ^ n ,
e ^ n = m 2 i ψ n ( m 2 x 1 ) ψ n ( m 2 x 2 ) m 2 ξ n ( x 2 ) [ D n ( m 2 x 2 ) - B ^ n F n ( m 2 x 2 ) ψ n ( m 2 x 1 ) ξ n ( m 2 x 2 ) ψ n ( m 2 x 2 ) ξ n ( m 2 x 1 ) ] - ξ n ( x 2 ) [ 1 - B ^ n ξ n ( m 2 x 2 ) ψ n ( m 2 x 1 ) ξ n ( m 2 x 1 ) ψ n ( m 2 x 2 ) ] ,
f ^ n = - B ^ n e ^ n ,
h ^ n = H ^ n e ^ n ,
G ^ n = D n ( m 2 x 1 ) - F n ( m 2 x 1 ) m 2 m 1 D n ( m 1 x 1 ) - F n ( m 2 x 1 ) ,
H ^ n = F n ( m 2 x 1 ) - D n ( m 2 x 1 ) m 2 m 1 F n ( m 2 x 1 ) - D n ( m 1 x 1 ) .
E r = cos ϕ m 1 2 x 2 n = 1 i n - 1 ( 2 n + 1 ) ψ n ( m 1 x ) ψ n ( m 1 x 1 ) g ^ n π n sin ( θ ) ,
E θ = cos ϕ m 1 x n = 1 i n - 1 2 n + 1 n ( n + 1 ) × [ D n ( m 1 x ) ψ n ( m 1 x ) ψ n ( m 1 x 1 ) g ^ n τ n + i ψ n ( m 1 x ) ψ n ( m 1 x 1 ) h ^ n π n ] ,
E ϕ = sin ϕ m 1 x n = 1 i n - 1 2 n + 1 n ( n + 1 ) × [ D n ( m 1 x ) ψ n ( m 1 x ) ψ n ( m 1 x 1 ) g ^ n π n + i ψ n ( m 1 x ) ψ n ( m 1 x 1 ) h ^ n τ n ] ,
E r = cos ϕ m 2 2 x 2 n = 1 i n - 1 ( 2 n + 1 ) × [ ψ n ( m 2 x ) ψ n ( m 2 x 1 ) c ^ n + ξ n ( m 2 x ) ξ n ( m 2 x 1 ) d ^ n ] π n sin ( θ ) ,
E θ = cos ϕ m 2 x n = 1 i n - 1 2 n + 1 n ( n + 1 ) { [ D n ( m 2 x ) ψ n ( m 2 x ) ψ n ( m 2 x 1 ) c ^ n + F n ( m 2 x ) ξ n ( m 2 x ) ξ n ( m 2 x 1 ) d ^ n ] τ n + i [ ψ n ( m 2 x ) ψ n ( m 2 x 1 ) e ^ n + ξ n ( m 2 x ) ξ n m 2 x 1 ) f ^ n ] π n } ,
E ϕ = - sin ϕ m 2 x n = 1 i n - 1 2 n + 1 n ( n + 1 ) { [ D n ( m 2 x ) ψ n ( m 2 x ) ψ n ( m 2 x 1 ) c ^ n + F n ( m 2 x ) ξ n ( m 2 x ) ξ n ( m 2 x 1 ) d ^ n ] τ n + i [ ψ n ( m 2 x ) ψ n ( m 2 x 1 ) e ^ n + ξ n ( m 2 x ) ξ n m 2 x 1 ) f ^ n ] π n } ,
E 2 d Ω = 2 π m 1 x 2 n = 1 ( 2 n + 1 ) { [ n ( n + 1 ) m 1 x 2 | ψ n ( m 1 x ) ψ n ( m 1 x 1 ) | 2 + D n ( m 1 x ) 2 | ψ n ( m 1 x ) ψ n ( m 1 x 1 ) | 2 ] g ^ n 2 + | ψ n ( m 1 x ) ψ n ( m 1 x 1 ) | 2 h ^ n 2 } ,
E 2 d Ω = 2 π m 2 x 2 n = 1 ( 2 n + 1 ) [ n ( n + 1 ) m 2 x 2 | ψ n ( m 2 x ) ψ n ( m 2 x 1 ) c ^ n + ξ n ( m 2 x ) ξ n ( m 2 x 1 ) d ^ n | 2 + | D n ( m 2 x ) ψ n ( m 2 x ) ψ n ( m 2 x 1 ) c ^ n + F n ( m 2 x ) ξ n ( m 2 x ) ξ n ( m 2 x 1 ) d ^ n | 2 + | ψ n ( m 2 x ) ψ n ( m 2 x 1 ) e ^ n + ξ n ( m 2 x ) ξ n ( m 2 x 1 ) f ^ n | 2 ]
1 V c E 2 d V = ¾ ( m 1 x 1 ) 2 n = 1 ( 2 n + 1 ) { [ 1 - n ( n + 1 ) ( m 1 x 1 ) 2 + D n 2 ( m 1 x 1 ) + 1 m 1 x 1 D n ( m 1 x 1 ) ] g ^ n 2 + [ 1 - n ( n + 1 ) ( m 1 x 1 ) 2 + D n 2 ( m 1 x 1 ) - 1 m 1 x 1 D n ( m 1 x 1 ) ] h ^ n 2 } ,
1 V s E 2 d V = x 3 x 2 3 - x 1 3 ¾ m 2 x 2 n = 1 ( 2 n + 1 ) ( [ 1 - n ( n + 1 ) ( m 2 x ) 2 ] × [ | ψ n ( m 2 x ) ψ n ( m 2 x 1 ) c ^ n + ξ n ( m 2 x ) ξ n ( m 2 x 1 ) d ^ n | 2 + | ψ n ( m 2 x ) ψ n ( m 2 x 1 ) e ^ n + ξ n ( m 2 x ) ξ n ( m 2 x 1 ) f ^ n | 2 ] + | D n ( m 2 x ) ψ n ( m 2 x ) ψ n ( m 2 x 1 ) c ^ n + F n ( m 2 x ) ξ n ( m 2 x ) ξ n ( m 2 x 1 ) d ^ n | 2 + | D n ( m 2 x ) ψ n ( m 2 x ) ψ n ( m 2 x 1 ) e ^ n + F n ( m 2 x ) ξ n ( m 2 x ) ξ n ( m 2 x 1 ) f ^ n | 2 + 1 m 2 x Re { [ ψ n ( m 2 x ) ψ n ( m 2 x 1 ) c ^ n + ξ n ( m 2 x ) ξ n ( m 2 x 1 ) d ^ n ] * × [ D n ( m 2 x ) ψ n ( m 2 x ) ψ n ( m 2 x 1 ) c ^ n + F n ( m 2 x ) ξ n ( m 2 x ) ξ n ( m 2 x 1 ) d ^ n ] - [ ψ n ( m 2 x ) ψ n ( m 2 x 1 ) e ^ n + ξ n ( m 2 x ) ξ n ( m 2 x 1 ) f ^ n ] * [ D n ( m 2 x ) × ψ n ( m 2 x ) ψ n ( m 2 x 1 ) e ^ n + F n ( m 2 x ) ξ n ( m 2 x ) ξ n ( m 2 x 1 ) f ^ n ] } ) | x = x 1 x = x 2 ,
1 V c E 2 d V = ¾ m 1 x 1 2 Re ( m 1 x 1 ) Im ( m 1 x 1 ) n = 1 ( 2 n + 1 ) × { Im [ m 1 x 1 D n * ( m 1 x 1 ) ] g ^ n 2 - Im [ m 1 x 1 D n ( m 1 x 1 ) ] h ^ n 2 } ,
1 V s E 2 d V = x 3 x 2 3 - x 1 3 ¾ Re ( m 2 x ) Im ( m 2 x ) m 2 x 2 n = 1 ( 2 n + 1 ) × Im { m 2 x [ ψ n ( m 2 x ) ψ n ( m 2 x 1 ) c ^ n + ξ n ( m 2 x ) ξ n ( m 2 x 1 ) d ^ n ] × [ D n ( m 2 x ) ψ n ( m 2 x ) ψ n ( m 2 x 1 ) c ^ n + F n ( m 2 x ) ξ n ( m 2 x ) ξ n ( m 2 x 1 ) d ^ n ] * - m 2 x [ ψ n ( m 2 x ) ψ n ( m 2 x 1 ) e ^ n + ξ n ( m 2 x ) ξ n ( m 2 x 1 ) f ^ n ] * × [ D n ( m 2 x ) ψ n ( m 2 x ) ψ n ( m 2 x 1 ) e ^ n + F n ( m 2 x ) ξ n ( m 2 x ) ξ n ( m 2 x 1 ) f ^ n ] } x = x 1 x = x 2 .
p ν , n ( ρ ) = ψ ν ( ρ ) ψ n ( m 2 x 1 ) c ^ n + ξ ν ( ρ ) ξ n ( m 2 x 1 ) d ^ n ,
q ν , n ( ρ ) = ψ ν ( ρ ) ψ n ( m x x 1 ) e ^ n + ξ ν ( ρ ) ξ n ( m 2 x 1 ) f ^ n .
p ν , n ( ρ ) = ρ 2 ν + 1 [ p ν - 1 , n ( ρ ) + p ν + 1 , n ( ρ ) ] ,
p v , n ( ρ ) = p v - 1 , n ( ρ ) - ν ρ p ν , n ( ρ ) ,
p ν , n ( ρ ) = ν + 1 ρ p ν , n ( ρ ) - p ν + 1 , n ( ρ ) ,
p ν , n ( ρ ) = 1 2 ν + 1 [ ( ν + 1 ) p ν - 1 , n ( ρ ) - ν p ν + 1 , n ( ρ ) ] .
E 2 d Ω = 2 π m 2 x 2 n = 1 ( 2 n + 1 ) [ n ( n + 1 ) m 2 x 2 p n , n ( m 2 x ) 2 + p n , n ( m 2 x ) 2 + q n , n ( m 2 x ) 2 ] .
E d Ω = 2 π m 2 x 2 n = 1 ( n + 1 ) p n - 1 , n ( m 2 x ) 2 + n p n + 1 , n ( m 2 x ) 2 + ( 2 n + 1 ) q n , n ( m 2 x ) 2 .
P ν , n ( m x ) = p ν , n ( m x ) 2 d x = { - x 2 Im [ m x p ν , n * ( m x ) p ν , n ( m x ) ] Re ( m x ) Im ( m x )             for Im ( m ) 0 x 2 { [ 1 - ν ( ν + 1 ) ( m x ) 2 ] p ν , n ( m x ) 2 + p ν , n ( m x ) 2 - 1 m x Re [ p ν , n * ( m x ) p ν , n ( m x ) ] }             for Im ( m ) = 0
Q ν , n ( m x ) = q ν , n ( m x ) 2 d x
1 V s E 2 d V = / 2 3 m 2 2 ( x 2 3 - x 1 3 ) n = 1 ( n + 1 ) P n - 1 , n ( m 2 x ) + n P n + 1 , n ( m 2 x ) + ( 2 n + 1 ) Q n , n ( m 2 x ) x = x 1 x = x 2
( n + 1 ) P n - 1 , n ( m x ) + n P n + 1 , n ( m x ) = { x 2 2 n + 1 Re ( m x ) Im ( m x ) Im [ mxp n , n * ( m x ) p n , n ( m x ) ]             for Im ( m ) 0 x 2 ( 2 n + 1 ) { [ 1 - n ( n + 1 ) ( m x ) 2 ] × p n , n ( m x ) 2 + p n , n ( m x ) 2 + 1 m x Re [ p n , n * ( m x ) p n , n ( m x ) ] }             for Im ( m ) = 0

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