Abstract

The magnetic (or electric) fields of morphology-dependent resonances of a dielectric sphere are shown to form an orthogonal complete set for expanding divergence-free vectorial functions inside the dielectric sphere, provided that there is a spatial discontinuity in its refractive index, say, at the edge of the sphere. A transverse projection dyad that picks up the divergence-free part (or its generalization) of a vector is defined and shown to be expandable in terms of the magnetic (or electric) fields of these morphology-dependent resonances. Moreover, the transverse dyadic Green’s function in these dielectric spheres is in turn expressed as a sum of tensor products of relevant morphology-dependent resonance fields. Each term in the sum manifests itself as a resonant response to external perturbations. Thus the morphology-dependent resonance expansion provides a powerful tool to analyze various optical phenomena in dielectric spheres.

© 1999 Optical Society of America

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References

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  1. See, e.g., M. Kerker, ed., Selected Papers on Light Scattering, Proc. SPIE 951 (1988), and references therein.
  2. G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metaalosungen,” Ann. Phys. (Leipzig) 25, 377–442 (1908).
    [CrossRef]
  3. P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).
  4. R. K. Chang and A. J. Campillo, eds., Optical Processes in Microcavities (World Scientific, Singapore, 1996).
  5. R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
    [CrossRef]
  6. J. B. Snow, S.-X. Qian, and R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985).
    [CrossRef] [PubMed]
  7. J.-Z. Zhang and R. K. Chang, “Generation and suppression of stimulated Brillouin scattering in single liquid droplets,” J. Opt. Soc. Am. B 6, 151–153 (1989).
    [CrossRef]
  8. H. M. Tzeng, K. F. Wall, M. B. Long, and R. K. Chang, “Laser emission from individual droplets at wavelengths corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984).
    [CrossRef] [PubMed]
  9. L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
    [CrossRef]
  10. See, e.g., P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).
  11. M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964).
  12. E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
    [CrossRef]
  13. P. T. Leung and 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]
  14. K. M. Lee, P. T. Leung, and K. M. Pang, “Iterative perturbation scheme for morphology-dependent resonances in dielectric spheres,” J. Opt. Soc. Am. A 15, 1383–1393 (1998).
    [CrossRef]
  15. C.-T. Tai, Dyadic Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, New York, 1993).
  16. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, New York, 1995).
  17. A. Q. Howard, “On the longitudinal component of the Green’s functions in bounded media,” Proc. IEEE 62, 1704–1705 (1974); W. A. Johnson, A. Q. Howard, and D. G. Dudley, “On the irrotational component of the electric Green’s dyadic,” Radio Sci. 14, 961–967 (1979).
    [CrossRef]
  18. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  19. O. D. Kellogg, Foundations of Potential Theory (Dover, New York, 1953).
  20. In the present paper we consider the divergence-free term ∇×(4πj/ε) as the source of the magnetic field, whereas in the conventional treatment of the dyadic Green’s function of Maxwell equations15 the current j itself is used instead.
  21. K. M. Pang, “Completeness and perturbation of morphology-dependent resonances in dielectric spheres,” Ph.D. dissertation (Chinese University of Hong Kong, Hong Kong, 1999).
  22. K. C. Ho, “A study of field fluctuation in open optical cavities,” M. Phil. thesis (Chinese University of Hong Kong, Hong Kong, 1997).
  23. K. M. Lee, P. T. Leung, and K. M. Pang, “Dyadic formulation of morphology-dependent resonances. II. Perturbation theory,” J. Opt. Soc. Am. B 16, 1418–1430 (1999).
    [CrossRef]

1999 (1)

1998 (2)

K. M. Lee, P. T. Leung, and K. M. Pang, “Iterative perturbation scheme for morphology-dependent resonances in dielectric spheres,” J. Opt. Soc. Am. A 15, 1383–1393 (1998).
[CrossRef]

E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

1996 (1)

1989 (1)

1985 (2)

1984 (1)

1980 (1)

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

1908 (1)

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

Arnold, S.

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

Barber, P. W.

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

Benner, R. E.

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

Chang, R. K.

Druger, D.

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

Emily, E. S. C.

E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Folan, L. M.

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

Lee, K. M.

Leung, P. T.

Long, M. B.

Maassen van den Brink, A.

E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Mie, G.

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

Owen, J. F.

R. E. Benner, P. W. Barber, J. F. Owen, and 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.

Snow, J. B.

Suen, W. M.

E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Tong, S. S.

E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Tzeng, H. M.

Wall, K. F.

Young, K.

E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Zhang, J.-Z.

Ann. Phys. (Leipzig) (1)

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

Chem. Phys. Lett. (1)

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

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

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

Opt. Lett. (2)

Phys. Rev. Lett. (1)

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

Rev. Mod. Phys. (1)

E. S. C. Emily, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Other (13)

C.-T. Tai, Dyadic Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, New York, 1993).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, New York, 1995).

A. Q. Howard, “On the longitudinal component of the Green’s functions in bounded media,” Proc. IEEE 62, 1704–1705 (1974); W. A. Johnson, A. Q. Howard, and D. G. Dudley, “On the irrotational component of the electric Green’s dyadic,” Radio Sci. 14, 961–967 (1979).
[CrossRef]

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

O. D. Kellogg, Foundations of Potential Theory (Dover, New York, 1953).

In the present paper we consider the divergence-free term ∇×(4πj/ε) as the source of the magnetic field, whereas in the conventional treatment of the dyadic Green’s function of Maxwell equations15 the current j itself is used instead.

K. M. Pang, “Completeness and perturbation of morphology-dependent resonances in dielectric spheres,” Ph.D. dissertation (Chinese University of Hong Kong, Hong Kong, 1999).

K. C. Ho, “A study of field fluctuation in open optical cavities,” M. Phil. thesis (Chinese University of Hong Kong, Hong Kong, 1997).

See, e.g., M. Kerker, ed., Selected Papers on Light Scattering, Proc. SPIE 951 (1988), and references therein.

P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).

R. K. Chang and A. J. Campillo, eds., Optical Processes in Microcavities (World Scientific, Singapore, 1996).

See, e.g., P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

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

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Figures (1)

Fig. 1
Fig. 1

Locations of the leading TE and TM MDR’s for l=10 of a uniform dielectric sphere with n=1.33 in the complex ωa plane.

Equations (91)

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·B=0,
·D=4πρ,
×E+1cBt=0,
×H=4πcJ+1cDt,
ρt+·J=0.
×1(r)×b-ω2b=0.
b=lm×[ϕlm(r)Xlm]+ψlm(r)Xlm,
d2Φlmdr2+(r)ω2-l(l+1)r2Φlm=0
ddr1(r)dΨlmdr+ω2+l(l+1)(r)r2Ψlm=0
-ddrρ(r)ddr+ρ(r)l(l+1)r2-ρ(r)(r)ω2φ=0,
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).
jρ(r)(r)fj(r)fj(r)2=δ(r-r),
D˜(r, r; ω)=-12jfj(r)fj(r)ωj(ω-ωj)
jfj(r)fj(r)ωj=0,
b1jlm=1iω1jl×f1jl(r)rXlm
b2jlm=f2jl(r)rXlm
b1jlmiω1jl×f1jl(r)rXlm*,
b2jlmf2jl(r)rXlm*,
bνjlm|bνjlm
limXr<Xd3rbνjlm·bνjlm+iωνjl+ωνjl(X)-1/2×d3rδ(r-X)bνjlm·bνjlm.
Vt=lm×v1lm(r)rXlm+v2lm(r)rXlm.
Vt(r)=νjlmaνjlmbνjlm(r)
V(r)=Vt(r)+Vl(r),
I¯t(r, r)·V(r)d3r=Vt(r).
P¯t(r, r)νjlmbνjlm(r)bνjlm(r)2.
V(r)=lm×v1lm(r)Xlm(Ω)r+v2lm(r)Xlm(Ω)r+v3lm(r)Ylm(Ω)rlm×[g1lm(r)Xlm(Ω)]+g2lm(r)Xlm(Ω)+[g3lm(r)Ylm(Ω)],
×[gXlm]=dgdr+grrˆ×Xlm+ir[l(l+1)]1/2gYlmrˆ,
[gYlm]=dgdrYlmrˆ-i[l(l+1)]1/2rgrˆ×Xlm,
rlv1lm(r)=-i(1+1/l)1/2rlv3lm(r)=clm,
r<aP¯t(r, r)·{r×[g1lm(r)Xlm(Ω)]
+r[g3lm(r)Ylm(Ω)]}d3r
=×[g1lm(r)Xlm(Ω)],
r<aP¯t(r, r)·g2lm(r)Xlm(Ω)d3r
=g2lm(r)Xlm(Ω).
fXlm*·gXlmdΩ
=fgδllδmm,
fXlm*·×(gXlm)dΩ
=0,
×(fXlm*)·×(gXlm)dΩ
=(r)ω2fg+1r2r[rgr(rf)]δllδmm,
r<abνjlm·r×(g1lmXlm)d3r
=δν1δllδm,m0ar2ω1jl×ω1jl2f1jlrg1lm+1r2r(rg1lmrf1jl)dr
=δν1δllδm,m0aω1jlf1jlv1lmdr
+v1lm(a)rf1jl(r)|r=aω1jl.
r<abνjlm·r(g3lmYlm)d3r
=δν1δllδm,m-i[l(l+1)1/2f1jl(a)v3lm(a)ω1jla=δν1δllδm,m-f1jl(a)rv1lm(r)|r=aω1jl,
r<aP¯t(r, r)·{r×[g1lm(r)Xlm(Ω)]
+r[g3lm(r)Ylm(Ω)]}d3r
=νjlmbνjlm(r)2δν1δllδm,mω1jl×0a(r)f1jl(r)v1lm(r)dr=r×0ajf1jl(r)2r(r)f1jl(r)v1lm(r)drXlm(Ω)=r×0aδ(r-r)rv1lm(r)drXlm(Ω)=×[g1lm(r)Xlm(Ω)].
νjlmbνjlm(r)2δν1δllδm,mω1jlv1lm(a)rf1jl(r)-f1jl(a)rv1lm(r)|r=aω1jl2
=r×jf1jl(r)2rv1lm(a)rf1jl(r)-f1jl(a)rv1lm(r)|r=aω1jlXlm(Ω)=0.
×1μ×E+c22Et2=-4πc2Jt,
×1μ×1D+1c22Dt2=-4πc2Jt,
×1×H+μc22Ht2=4πc×J,
×1×1μB+1c22Bt2=4πc×J.
×1×b(r)-ω2b(r)=×4πj.
×1×G¯tb(r, r; ω)-ω2G¯tb(r, r; ω)=I¯t(r, r).
b(r)=G¯tb(r, r; ω)·r×4πj(r)(r)d3r.
G¯tb(r, r; ω)=νjlm-bνjlm(r)bνjlm(r)2ωνjlm(ω-ωνjlm),r, r<a.
r×1r×-ω2νjlm-bνjlm(r)bνjlm(r)2ωνjlm(ω-ωνjlm)
=νjlm-(ωνjlm2-ω2)2ωνjlm(ω-ωνjlm)bνjlm(r)bνjlm(r)=νjlmωωνjlm+1bνjlm(r)bνjlm(r)2=νjlmbνjlm(r)bνjlm(r)2.
νjlmbνjlm(r)bνjlm(r)ωνjlm=0,r, r<a.
e1jlm=f1jl(r)rXlm,
e2jlm=i(r)ω2jl×f2jl(r)rXlm.
eνjlm|eνjlm
limXr<Xd3r(r)eνjlm·eνjlm+iωνjl+ωνjl(X)1/2×d3rδ(r-X)eνjlm·eνjlm,
e1jlm=f1jl(r)rXlm*,
e2jlm=-i(r)ω2jl×f2jl(r)rXlm*.
eνjlm=i(r)ωνjl×bνjlm,
eνjlm=-i(r)ωνjl×bνjlm,
bνjlm=1iωνjl×eνjlm,
bνjlm=-1iωνjl×eνjlm.
×V(r)=½νjlmbνjlm(r)bνjlm(r)·r×V(r)d3r
×V(r)=½νjlm×eνjlm(r)eνjlm(r)·[(r)V(r)]d3r.
V(r)=Kt(r, r)·[(r)V(r)]d3r+ψ(r),
Kt(r, r)½νjlmeνjlm(r)eνjlm(r).
××ET(r)+(r)2ET(r)t2=-4πJ˜Tt,
r×r×G¯te(r, r; ω)-(r)ω2G¯te(r, r; ω)
=(r)Kt(r, r).
eT(r)=G¯te(r, r; ω)·4πiωj(r)d3r.
G¯te(r, r; ω)=νjlm-eνjlm(r)eνjlm(r)2ωνjlm(ω-ωνjlm),
νjlmeνjlm(r)eνjlm(r)ωνjlm=0,r, r<a.
ρrjl(nωr)d[rjl(nωr)]drr=a=1rhl(1)(ωr)d[rhl(1)(ωr)]drr=a,
b1jlm=1N1jlω1j×[jl(nω1jlr)Xlm],
b2jlm=1N2jljl(nω2jlr)Xlm,
N1jl2=(n2-1)(a3/2)jl2(nω1jla),
N2jl2=1-1n2jl(nω2jla)jl(nω2jla)+1nω2jla2+l2+l(ω2jla)2×a32jl2(nω2jla).
I¯t(r, r)=jlm12Nljl2ω1j2×[jl(nω1jlr)Xlm(Ω)]×[jl(nω1jlr)Xlm*(Ω)]+jlm12N2jl2jl(nω2jlr)Xlm(Ω)jl(nω2jlr)×Xlm*(Ω).
G¯tb(r, r; ω)=jlm-12N1jl2ωlj3(ω-ω1j)×[jl(nω1jlr)Xlm(Ω)]×[jl(nω1jlr)Xlm*(Ω)]+jlm-12N2jl2ω2j(ω-ω2j)jl(nω2jlr)×Xlm(Ω)jl(nω2jlr)Xlm(Ω).

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