Abstract

The properties of morphology-dependent resonances observed in the scattering of electromagnetic waves from dielectric spheres have recently been investigated intensively, and a second-order perturbative expansion for these resonances has also been derived. Nevertheless, it is still desirable to obtain higher-order corrections to their eigenfrequencies, which will become important for strong enough perturbations. Conventional explicit expressions for higher-order corrections inevitably involve multiple sums over intermediate states, which are computationally cumbersome. In this analysis an efficient iterative scheme is developed to evaluate the higher-order perturbation results. This scheme, together with the optimal truncation rule and the Padé resummation, yields accurate numerical results for eigenfrequencies of morphology-dependent resonances even if the dielectric sphere in consideration deviates strongly from a uniform one. It is also interesting to find that a spatial discontinuity in the refractive index, say, at the edge of the dielectric sphere, is crucial to the validity of the perturbative expansion.

© 1998 Optical Society of America

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References

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  1. See, e.g., 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), p. 3.
  2. G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metaalosungen,” Ann. Phys. (Leipzig) 25, 377–442 (1908).
    [CrossRef]
  3. R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
    [CrossRef]
  4. J. B. Snow, S.-X. Qian, R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985).
    [CrossRef] [PubMed]
  5. H. M. Tzeng, K. F. Wall, M. B. Long, R. K. Chang, “Laser emission from individual droplets at wavelengths corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984).
    [CrossRef] [PubMed]
  6. J. D. Eversole, H. B. Lin, A. L. Huston, A. J. Campillo, P. T. Leung, S. Y. Liu, K. Young, “High-precision identification of morphology-dependent resonances in optical processes in microdroplets,” J. Opt. Soc. Am. B 10, 1955–1968 (1993).
    [CrossRef]
  7. M. L. Goldberger, K. M. Watson, Collision Theory (Wiley, New York, 1964).
  8. P. T. Leung, K. M. Pang, “Completeness and time-independent perturbation of morphology-dependent resonances in dielectric spheres,” J. Opt. Soc. Am. B 13, 805–817 (1996).
    [CrossRef]
  9. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  10. A. L. Fetter, J. D. Walecka, Quantum Theory of Many Body Systems (McGraw-Hill, New York, 1971).
  11. D. Q. Chowdhury, S. C. Hill, P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702–1705 (1991).
    [CrossRef]
  12. M. M. Mazumder, S. C. Hill, P. W. Barber, “Morphology-dependent resonances in inhomogeneous spheres: comparison of layered T-matrix and time-independent perturbation method,” J. Opt. Soc. Am. A 9, 1844–1853 (1992).
    [CrossRef]
  13. A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242 (1951).
    [CrossRef]
  14. O. B. Toon, T. P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. 20, 3657–3660 (1981).
    [CrossRef] [PubMed]
  15. R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960–1967 (1985).
    [CrossRef] [PubMed]
  16. A similar calculation has been carried out previously. See P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984).
    [CrossRef]
  17. See, e.g., C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), p. 122.
  18. See, e.g., G. A. Baker, P. Graves-Morris, Padé Approximants, in Vol. 59 of Encyclopedia of Mathematics and Its Applications, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1996).

1996 (1)

1993 (1)

1992 (1)

1991 (1)

1985 (2)

1984 (2)

1981 (1)

1980 (1)

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

1951 (1)

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

1908 (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metaalosungen,” Ann. Phys. (Leipzig) 25, 377–442 (1908).
[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]

Baker, G. A.

See, e.g., G. A. Baker, P. Graves-Morris, Padé Approximants, in Vol. 59 of Encyclopedia of Mathematics and Its Applications, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1996).

Barber, P. W.

Bender, C. M.

See, e.g., C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), p. 122.

Benner, R. E.

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

See, e.g., 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), p. 3.

Bhandari, R.

Campillo, A. J.

Chang, R. K.

Chowdhury, D. Q.

Conwell, P. R.

Eversole, J. D.

Fetter, A. L.

A. L. Fetter, J. D. Walecka, Quantum Theory of Many Body Systems (McGraw-Hill, New York, 1971).

Goldberger, M. L.

M. L. Goldberger, K. M. Watson, Collision Theory (Wiley, New York, 1964).

Graves-Morris, P.

See, e.g., G. A. Baker, P. Graves-Morris, Padé Approximants, in Vol. 59 of Encyclopedia of Mathematics and Its Applications, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1996).

Hill, S. C.

Huston, A. L.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Kerker, M.

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

Leung, P. T.

Lin, H. B.

Liu, S. Y.

Long, M. B.

Mazumder, M. M.

Mie, G.

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metaalosungen,” Ann. Phys. (Leipzig) 25, 377–442 (1908).
[CrossRef]

Orszag, S. A.

See, e.g., C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), p. 122.

Owen, J. F.

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

Pang, K. M.

Qian, S.-X.

Rushforth, C. K.

Snow, J. B.

Toon, O. B.

Tzeng, H. M.

Walecka, J. D.

A. L. Fetter, J. D. Walecka, Quantum Theory of Many Body Systems (McGraw-Hill, New York, 1971).

Wall, K. F.

Watson, K. M.

M. L. Goldberger, K. M. Watson, Collision Theory (Wiley, New York, 1964).

Young, K.

Ann. Phys. (Leipzig) (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metaalosungen,” Ann. Phys. (Leipzig) 25, 377–442 (1908).
[CrossRef]

Appl. Opt. (2)

J. Appl. Phys. (1)

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

J. Opt. Soc. Am. A (3)

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

Opt. Lett. (2)

Phys. Rev. Lett. (1)

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

Other (6)

M. L. Goldberger, K. M. Watson, Collision Theory (Wiley, New York, 1964).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

A. L. Fetter, J. D. Walecka, Quantum Theory of Many Body Systems (McGraw-Hill, New York, 1971).

See, e.g., C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), p. 122.

See, e.g., G. A. Baker, P. Graves-Morris, Padé Approximants, in Vol. 59 of Encyclopedia of Mathematics and Its Applications, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1996).

See, e.g., 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), p. 3.

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Figures (8)

Fig. 1
Fig. 1

Plot of the eigenfrequencies of a uniform dielectric sphere (the unperturbed system) and a layered sphere (n1=1.1, b=0.75a). MDR’s with l=10 are considered. MDR’s of the unperturbed system are denoted by filled circles. The exact positions of MDR’s of the layered sphere are shown by the open circles, and the perturbative results are shown by the pluses. Note that the perturbation does not work very well for the higher-frequency modes.

Fig. 2
Fig. 2

Error in the eigenfrequencies of a layered sphere plotted against the mode order. The parameters are the same as in Fig. 1. The circles represent the differences between the unperturbed frequencies and the exact answers. Squares, diamonds, triangles, and pluses are, respectively, the errors resulting from the second-, fourth-, sixth-, and eighth-order perturbations.

Fig. 3
Fig. 3

Relative error in the eigenfrequencies of a layered sphere (with the parameters n0=1.33 and b=0.75a) plotted against the refractive index n1 of the coating. The fifth TE mode is considered. The results are chosen according to the optimal truncation rule. The solid curve is the result obtained from direct summation of the perturbation series, whereas the dashed curve is for the Padé resummed series.

Fig. 4
Fig. 4

Dielectric constant distribution of the second system that we consider. α=0 corresponds to the unperturbed system; for α=1, the dielectric constant is smooth everywhere.

Fig. 5
Fig. 5

Plot of the eigenfrequencies of the system with a small step in the refractive index. b=0.75a, and α=0.95. The exact answers are given by the open circles, and results of the optimally truncated perturbation series are given by the pluses. The MDR’s of the unperturbed system, denoted by filled circles, are also shown for purposes of comparison.

Fig. 6
Fig. 6

Plot of the eigenfrequencies of the system with a small step in the refractive index. b=0.75a, and α=0.95. The exact answers are shown by the circles, and the results obtained from the Padé resummation of the perturbation series are given by the pluses. The MDR’s of the unperturbed system, denoted by filled circles, are also shown for purposes of comparison. The Padé resummation is accurate, and the imaginary part of the eigenfrequencies asymptotically approaches a constant in the high-frequency limit.

Fig. 7
Fig. 7

Similar to Fig. 6, with one crucial difference: The refractive index is smooth in this case. b=0.75a, and α=1.0. Note that the Padé resummation is not accurate for Re ωa>20.

Fig. 8
Fig. 8

Similar to Fig. 7, only now b=0.5a. The Padé resummation is not accurate for Re ωa>12.

Equations (57)

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×(×E)+(r) 2Et2=0.
×(×e)-(r)ω2e=0.
e=lmϕlm(r)Xlm+1(r)×[ψlm(r)Xlm],
d2Φlmdr2+(r)ω2-l(l+1)r2Φlm=0
ddr1(r)Ψlmdr+ω2-l(l+1)(r)r2Ψlm=0
-ddrρ(r) ddr+ρ(r) l(l+1)r2-ρ(r)(r)ω2φ=0,
limr[φ(r)exp(-inωr)]=const.,
fj|fklimX0Xdrρ(r)(r)fj(r)fk(r)+iωj+ωkρ(X)(X)1/2fj(X)fk(X).
-ddrρ(r) ddr+ρ(r) l(l+1)r2-ρ(r)(r)ω2D˜(r, r; ω)
=δ(r-r).
j ρ(r)(r)fj(r)fj(r)2fj|fj=δ(r-r),
D˜(r, r; ω)=-12j fj(r)fj(r)ωj(ω-ωj)
-ddrρ(0) ddr+ρ(0) l(l+1)r2-ρ(0)(0)ω2D˜(r, r; ω)
=δ(r-r).
-ρ(0)(0)ω2-ddrρ(0) ddr+ρ(0) l(l+1)r2G˜(r, r; ω)
=δ(r-r)+ΔG˜(r, r; ω),
Δ[(0)δρ+ρ(0)δ+δρδ]ω2+ddrδρ ddr-δρ l(l+1)r2.
G˜=D˜+D˜ΔD˜+D˜ΔD˜ΔD˜+.
[D˜ΔD˜](r, r; ω)=0adr1D˜(r, r1; ω)ΔD˜(r1, r; ω).
Djk=-12δjkωj(0)[ω-ωj(0)],
Δjk=0adrfj(0)(r)Δfk(0)(r),
G˜(r, r; ω)=jkfj(0)(r)Gjkfk(0)(r),
G=D+DΔD+DΔDΔD+,
G˜(r, r; ω)=-12m fm(r)fm(r)ωm(ω-ωm),
Wjk{m}Δjk+lmΔjlDllΔlk+lmpmΔjlDllΔlpDppΔpk+.
Gjk=Djk+DjjWjk{m}Dkk+DjjWjm{m}Dmm×n=0[Wmm{m}Dmm]nWmk{m}Dkk.
G˜(r, r; ω)=jkfj(0)(r)fk(0)(r)Djk+DjjWjk{m}Dkk-DjjWjm{m}Wmk{m}Dkk2ωm(0)[ω-ωm(0)]+Wmm{m}.
Fm(ω)2ωm(0)[ω-ωm(0)]+Wmm{m}(ω).
Fkm(ω)l{2ωk(0)[ω-ωk(0)]δkl+Δkl}alm(ω),
Wjk{m}=Δjk+lmΔjlDllWlk{m}.
Fkm(ω)=l{2ωk(0)[ω-ωk(0)]δkl+Δkl}DllWlm{m}=2ωk(0)[ω-ωk(0)]DkkWkm{m}+ΔkmDmmWmm{m}+lmΔklDllWlm{m}=-Wkm{m}+ΔkmDmmWmm{m}+Wkm{m}-Δkm=ΔkmDmmFm(ω),
l{2ωk(0)[ω-ωk(0)]δkl+Δkl}alm(ω)=0,
ωm=ωm(0)-12ωm(0)lΔmlalm
akm=12ωk(0)[ωk(0)-ωm]Δkm+lmΔklalm,
ωm=ωm(0)+s=1μsωm(s),
akm=1,k=ms=1μsakm(s),km .
μΔjk=ωm20adrfj(0)(r)δfk(0)(r).
Δjk=ωm2V1jk.
ωm(s)=-12ωm(0)t=0s-1ωm(t)ωm(s-t-1)V1mm+u=0tωm(u)ωm(t-u)kmV1mkakm(s-t-1),
akm(s)=-1[ωm(0)-ωk(0)]t=1s-1ωm(s-t)akm(t)-12ωk(0)[ωm(0)-ωk(0)]t=0s-1ωm(t)ωm(s-t-1)V1km-12ωk(0)[ωm(0)-ωk(0)]t=0s-1u=0tωm(u)ωm(t-u)×imaim(s-t-1)V1ki.
ωm(1)=-ωm(0)2V1mm,
akm(1)=-12[ωm(0)]2ωk(0)[ωm(0)-ωk(0)]V1km.
ωm(2)=ωm(0)2V1mm2+ωm(0)4kmV1mk [ωm(0)]2ωk(0)[ωm(0)-ωk(0)]V1km,
akm(2)=12V1mm [ωm(0)]2ωk(0)[ωm(0)-ωk(0)]V1km-14ωm(0)ωm(0)-ωk(0)V1mm [ωm(0)]2ωk(0)[ωm(0)-ωk(0)]V1km+14im [ωm(0)]2ωk(0)[ωm(0)-ωk(0)]V1ki×[ωm(0)]2ωi(0)[ωm(0)-ωi(0)]V1im.
μΔjk=-0adrδρddrfj(0)(r)ddrfk(0)(r)+l(l+1)r2fj(0)(r)fk(0)(r).
V2jk=1ωj(0)ωk(0)0adr[ρ(0)(r)]2V2(r)ddrfj(0)(r)×ddrfk(0)(r)+l(l+1)r2fj(0)(r)fk(0)(r).
ωm(s)=-12ωm(0)V2mmδs,1-12kmωk(0)V2mkakm(s-1),
akm(s)=-1[ωm(0)-ωk(0)]t=1s-1ωm(s-t)akm(t)-ωm(0)2[ωm(0)-ωk(0)]V2kmδs,1-12[ωm(0)-ωk(0)]imωi(0)aim(s-1)V1ki.
ωm(1)=-ωm(0)2V2mm,
akm(1)=-12ωm(0)[ωm(0)-ωk(0)]V2km,
ωm(2)=ωm(0)4kmV2mk ωk(0)[ωm(0)-ωk(0)]V2km,
akm(2)=-14ωm(0)[ωm(0)-ωk(0)]V2mm ωm(0)[ωm(0)-ωk(0)]V2km+14im ωi(0)ωm(0)ωm(0)[ωm(0)-ωk(0)]V2ki×ωm(0)[ωm(0)-ωi(0)]V2im.
ρ(a-)rjl(n0ωr)d[rjl(n0ωr)]drr=a=1rhl(1)(ωr)d[rhl(1)(ωr)]drr=a.
(r)=011ifr<bifbr<aotherwise.
ωmi=0sμiωm(i).
PMN(μ)=i=0Nbiμi1+i=1Mciμi,
(r)=0ifr<b0-α(0-1)sin2π2r-ba-bifbr<a1otherwise,

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