Abstract

The geometrical optics model of Mie resonances is presented. The ray path geometry is given and the resonance condition is discussed with special emphasis on the phase shift that the rays undergo at the surface of the dielectric sphere. On the basis of this model, approximate expressions for the positions of first-order resonances are given. Formulas for the cavity mode spacing are rederived in a simple manner. It is shown that the resonance linewidth can be calculated regarding the cavity losses. Formulas for the mode density of Mie resonances are given that account for the different width of resonances and thus may be adapted to specific experimental situations.

© 2000 Optical Society of America

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  1. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  2. S. C. Hill, R. E. Benner, “Morphology-dependant resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988).
  3. R. K. Chang, A. J. Campillo, eds., Optical Processes in Microcavities (World Scientific, Singapore, 1996).
  4. S. C. Hill, R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Quantum Nonlinear Optics, O. Keller, ed. (Nova Science, New York, 1995).
  5. H. B. Lin, A. J. Campillo, “Radial profiling of microdroplets using cavity-enhanced Raman spectroscopy,” Opt. Lett. 20, 1589–1591 (1995).
    [CrossRef] [PubMed]
  6. T. Kaiser, G. Roll, G. Schweiger, “Investigation of coated droplets in an optical trap: Raman scattering, elastic light scattering and evaporation characteristics,” Appl. Opt. 35, 5918–5924 (1996).
    [CrossRef] [PubMed]
  7. J. L. Huckaby, A. K. Ray, B. Das, “Determination of size, refractive index, and dispersion of single droplets from wavelength-dependent scattering spectra,” Appl. Opt. 33, 7112–7125 (1994).
    [CrossRef] [PubMed]
  8. G. Chen, M. M. Mazumder, R. K. Chang, J. C. Swindal, W. P. Acker, “Laser diagnostics for droplet characterization: application of morphology dependent resonances,” Prog. Energy Combust. Sci. 22, 163–188 (1996).
    [CrossRef]
  9. J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (New York) 9, 24–75 (1960).
    [CrossRef]
  10. S. J. Maurer, L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
    [CrossRef]
  11. G. Roll, T. Kaiser, S. Lange, G. Schweiger, “Ray interpretation of multipole fields in spherical dielectric cavities,” J. Opt. Soc. Am. A 15, 2879–2891 (1998).
    [CrossRef]
  12. G. Roll, T. Kaiser, G. Schweiger, “Eigenmodes of spherical dielectric cavities: coupling of internal and external rays,” J. Opt. Soc. Am. A 16, 882–895 (1999).
    [CrossRef]
  13. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  14. This statement holds only for light with the frequency of the illuminating plane wave; secondary fields, e.g., excited Raman or fluorescence fields, are not restricted to modes with azimuthal mode number m=1.
  15. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993).
  16. Throughout this paper we mean the trigonometric expansions of Bessel functions [compare expressions (18) and (19)] when we use the term Debye expansion. This must not be confused with the Debye series expansion of the scattering coefficients of Mie theory. Both the Debye expansion of Bessel functions and the Debye series expansion of the Mie coefficients play an important role in the ray description of the interaction of light with dielectric spheres. The former expansions are approximations to the exact functions, which break down in regions where the argument approaches the order. These expansions are found when the field within an illuminated sphere is investigated by means of geometrical optics.11 The Debye series expansion, on the other hand, denotes a mathematically exact technique to formulate the scattering process in terms of surface interactions of multipole waves. The Debye series expansion is equivalent to Mie theory but shows many similarities to the multiple-interference description of geometrical optics.17
  17. J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
    [CrossRef]
  18. N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, London, 1972), p. 126.
  19. P. Chýlek, “Resonance structure of Mie scattering: distance between resonances,” J. Opt. Soc. Am. A 7, 1609–1613 (1990).
    [CrossRef]
  20. J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A 1, 822–830 (1984).
    [CrossRef]
  21. P. M. Aker, P. A. Moortgat, J. X. Zhang, “Morphology-dependent stimulated Raman scattering imaging. I. Theoretical aspects,” J. Chem. Phys. 105, 7268–7275 (1996).
    [CrossRef]
  22. C. C. Lam, P. T. Leung, K. Young, “Explicit asymptotic formulas for the positions, widths, and strength of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585–1592 (1992).
    [CrossRef]
  23. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343–352 (1993).
    [CrossRef]
  24. At first glance this may be surprising, since we always assumed the rays to be confined by total internal reflection, which should mean T=0. However, owing to the curved surface, the energy confinement is not perfect, but there is a so-called evanescent leakage. Consequently the energy loss—although small—is not zero.
  25. G. Roll, T. Kaiser, G. Schweiger, “Controlled modification of the expansion order as a tool in Mie computations,” Appl. Opt. 37, 2483–2492 (1998).
    [CrossRef]
  26. The fact that the transmission coefficient t is complex in the case of total reflection indicates the fact that the evanescent field is phase shifted with respect to the incident wave. The phase shift is half as large as that of the reflected wave.
  27. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

1999 (1)

1998 (2)

1996 (3)

T. Kaiser, G. Roll, G. Schweiger, “Investigation of coated droplets in an optical trap: Raman scattering, elastic light scattering and evaporation characteristics,” Appl. Opt. 35, 5918–5924 (1996).
[CrossRef] [PubMed]

G. Chen, M. M. Mazumder, R. K. Chang, J. C. Swindal, W. P. Acker, “Laser diagnostics for droplet characterization: application of morphology dependent resonances,” Prog. Energy Combust. Sci. 22, 163–188 (1996).
[CrossRef]

P. M. Aker, P. A. Moortgat, J. X. Zhang, “Morphology-dependent stimulated Raman scattering imaging. I. Theoretical aspects,” J. Chem. Phys. 105, 7268–7275 (1996).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

1992 (1)

1990 (1)

1988 (1)

1984 (1)

1967 (1)

S. J. Maurer, L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[CrossRef]

1960 (1)

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (New York) 9, 24–75 (1960).
[CrossRef]

Acker, W. P.

G. Chen, M. M. Mazumder, R. K. Chang, J. C. Swindal, W. P. Acker, “Laser diagnostics for droplet characterization: application of morphology dependent resonances,” Prog. Energy Combust. Sci. 22, 163–188 (1996).
[CrossRef]

Aker, P. M.

P. M. Aker, P. A. Moortgat, J. X. Zhang, “Morphology-dependent stimulated Raman scattering imaging. I. Theoretical aspects,” J. Chem. Phys. 105, 7268–7275 (1996).
[CrossRef]

Benner, R. E.

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

Bohren, C. F.

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993).

Burke, J. J.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, London, 1972), p. 126.

Campillo, A. J.

Chang, R. K.

G. Chen, M. M. Mazumder, R. K. Chang, J. C. Swindal, W. P. Acker, “Laser diagnostics for droplet characterization: application of morphology dependent resonances,” Prog. Energy Combust. Sci. 22, 163–188 (1996).
[CrossRef]

S. C. Hill, R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Quantum Nonlinear Optics, O. Keller, ed. (Nova Science, New York, 1995).

Chen, G.

G. Chen, M. M. Mazumder, R. K. Chang, J. C. Swindal, W. P. Acker, “Laser diagnostics for droplet characterization: application of morphology dependent resonances,” Prog. Energy Combust. Sci. 22, 163–188 (1996).
[CrossRef]

Chýlek, P.

Das, B.

Felsen, L. B.

S. J. Maurer, L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[CrossRef]

Hill, S. C.

S. C. Hill, R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Quantum Nonlinear Optics, O. Keller, ed. (Nova Science, New York, 1995).

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

Huckaby, J. L.

Huffman, D. R.

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

Jackson, J. D.

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

Johnson, B. R.

Kaiser, T.

Kapany, N. S.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, London, 1972), p. 126.

Keller, J. B.

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (New York) 9, 24–75 (1960).
[CrossRef]

Lam, C. C.

Lange, S.

Leung, P. T.

Lin, H. B.

Lock, J. A.

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Maurer, S. J.

S. J. Maurer, L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[CrossRef]

Mazumder, M. M.

G. Chen, M. M. Mazumder, R. K. Chang, J. C. Swindal, W. P. Acker, “Laser diagnostics for droplet characterization: application of morphology dependent resonances,” Prog. Energy Combust. Sci. 22, 163–188 (1996).
[CrossRef]

Moortgat, P. A.

P. M. Aker, P. A. Moortgat, J. X. Zhang, “Morphology-dependent stimulated Raman scattering imaging. I. Theoretical aspects,” J. Chem. Phys. 105, 7268–7275 (1996).
[CrossRef]

Probert-Jones, J. R.

Ray, A. K.

Roll, G.

Rubinow, S. I.

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (New York) 9, 24–75 (1960).
[CrossRef]

Schweiger, G.

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Swindal, J. C.

G. Chen, M. M. Mazumder, R. K. Chang, J. C. Swindal, W. P. Acker, “Laser diagnostics for droplet characterization: application of morphology dependent resonances,” Prog. Energy Combust. Sci. 22, 163–188 (1996).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993).

Young, K.

Zhang, J. X.

P. M. Aker, P. A. Moortgat, J. X. Zhang, “Morphology-dependent stimulated Raman scattering imaging. I. Theoretical aspects,” J. Chem. Phys. 105, 7268–7275 (1996).
[CrossRef]

Ann. Phys. (New York) (1)

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (New York) 9, 24–75 (1960).
[CrossRef]

Appl. Opt. (3)

J. Chem. Phys. (1)

P. M. Aker, P. A. Moortgat, J. X. Zhang, “Morphology-dependent stimulated Raman scattering imaging. I. Theoretical aspects,” J. Chem. Phys. 105, 7268–7275 (1996).
[CrossRef]

J. Opt. Soc. Am. A (6)

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

Opt. Lett. (1)

Proc. IEEE (1)

S. J. Maurer, L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[CrossRef]

Prog. Energy Combust. Sci. (1)

G. Chen, M. M. Mazumder, R. K. Chang, J. C. Swindal, W. P. Acker, “Laser diagnostics for droplet characterization: application of morphology dependent resonances,” Prog. Energy Combust. Sci. 22, 163–188 (1996).
[CrossRef]

Other (12)

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

This statement holds only for light with the frequency of the illuminating plane wave; secondary fields, e.g., excited Raman or fluorescence fields, are not restricted to modes with azimuthal mode number m=1.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993).

Throughout this paper we mean the trigonometric expansions of Bessel functions [compare expressions (18) and (19)] when we use the term Debye expansion. This must not be confused with the Debye series expansion of the scattering coefficients of Mie theory. Both the Debye expansion of Bessel functions and the Debye series expansion of the Mie coefficients play an important role in the ray description of the interaction of light with dielectric spheres. The former expansions are approximations to the exact functions, which break down in regions where the argument approaches the order. These expansions are found when the field within an illuminated sphere is investigated by means of geometrical optics.11 The Debye series expansion, on the other hand, denotes a mathematically exact technique to formulate the scattering process in terms of surface interactions of multipole waves. The Debye series expansion is equivalent to Mie theory but shows many similarities to the multiple-interference description of geometrical optics.17

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, London, 1972), p. 126.

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-dependant resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988).

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

S. C. Hill, R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Quantum Nonlinear Optics, O. Keller, ed. (Nova Science, New York, 1995).

At first glance this may be surprising, since we always assumed the rays to be confined by total internal reflection, which should mean T=0. However, owing to the curved surface, the energy confinement is not perfect, but there is a so-called evanescent leakage. Consequently the energy loss—although small—is not zero.

The fact that the transmission coefficient t is complex in the case of total reflection indicates the fact that the evanescent field is phase shifted with respect to the incident wave. The phase shift is half as large as that of the reflected wave.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

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Figures (7)

Fig. 1
Fig. 1

Ray propagating within a sphere in the case of an eigenmode with mode number l=50 and azimuthal mode number m=24. The orbit of the ray is a plane annular region. The radius of the inner edge (a circular ray caustic) of this orbit is connected to the mode number by nk0rci=l+1/2. The orbit plane is inclined and makes an angle of π/2+ϑc=π/2+arcsin[m/(l+1/2)], where m is the azimuthal mode number, with the z axis. The incident plane wave is assumed to propagate in the z direction.

Fig. 2
Fig. 2

Path of a ray within its orbit. In the upper part the size of the cavity is such that consecutive congruences of the same kind are not in phase; this represents a nonresonant case. In the lower part the cavity size is such that consecutive congruences are in phase; this state represents a resonance.

Fig. 3
Fig. 3

Boundary phase shift δB,eff as a function of the angle of incidence for different mode numbers l (solid curves) for the TE case and n=1.5. The dashed curve shows the plane-wave–plane-boundary phase shift δB. The deviations between these curves can be understood if what are considered to propagate within the sphere are not rays but beams with a finite spectral width depending on l. The inset shows δB for the TE and TM cases for n=1.5 and the full angular range.

Fig. 4
Fig. 4

Comparison of the resonance positions predicted by geometrical optics with exact positions for 770 TE resonances with 75l150 and x<Λ. Resonances with identical mode order line up along smooth curves. Two sets were computed, with different expressions used for the boundary phase shift.

Fig. 5
Fig. 5

Pictorial representation of the considered situation to derive an expression for the resonance width by means of cavity-loss arguments. A beam is (almost) totally reflected at the inner cavity surface. As a result of the evanescent leakage (tunneling, indicated by wavy arrows) a small amount of energy is coupled to the exterior and radiates away. This leakage ultimately limits the resonator quality factor.

Fig. 6
Fig. 6

Positions of 131 TE resonances in x, l space indicated by diamonds. The vertical lines enclose an interval of width Δxv. In total a number of vtrans (=10) sharp (x<Λ) resonances can be found in this interval. There are vtrans-vΔx(=10-4=6) resonances in this interval that are broader than 10-10; the line of constant resonance widths is indicated by the dashed line. This diagram serves to make visible the strategy for deriving Eq. (53).

Fig. 7
Fig. 7

Number of resonances per size parameter interval that are broader than . Comparison of our predictions (solid curves) with data from the literature (dashed curves) for different values of . The graph with =0 presents the total mode density; i.e., it takes all resonances into account.

Equations (57)

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jl(nx)jl(nx)-α2 yl(x)yl(x)=0,
kir=±(n2k02r2-Λ2)1/2r.
rci=Λnk0.
Λ=k0n(rci)rci,
et=dr(s)ds=1ds drrdη=1nk0r (n2k02r2-Γ2)1/2Λ=kink0,
η(r)=η(rci)±rcircir ncin dρρ[ρ2-rci2(nci/n)2]1/2,
η(r)=η(rci)±arccosrcir,
nk02(a2-rci2)1/2-rci2 arccosrcia+δB-π2
=(ν-1)2π,
2nk0(a2-rci2)1/2=2ABkdr=2r(A)=rcir(B)=a kir(r)dr+2η(A)η(B) kηrdη.
nk0rci=kηr=Λ,
η(A)η(B) kηrdη=Λ arccosrcia,
2rcia kir(r)dr+δB-π2=(ν-1)2π.
ϕr(nk0r)=rcir kir(ρ)dρ=(n2k02r2-Λ2)1/2Λ arccosΛnk0r;
δB=-2 arctanα2 |ken|kin=-2 arctanα2 (sin2 β-sin2 βcrit)1/2cos β,
ker(r)=±i (Λ2-k02r2)1/2r.
sin β=rcia=Λnx
δB=-2 arctanα2Λ2-x2n2x2-Λ21/2.
jl(nk0r)cos[ϕr(nk0r)-π/4][nk0r(n2k02r2-Λ2)1/2]1/2,
yl(k0r)-exp[-ψr(k0r)][k0r(Λ2-k02r2)1/2]1/2,
ψr(k0r)=rcer |ker(ρ)|dρ=(Λ2-k02r)1/2-Λ Λk0r,
δB,eff=-2 arctan-α2 yl(x)yl(x) x(n2x2-Λ2)1/2.
cos γ=Λnx,
Λ(tan γ-γ)=3 π4.
x˜l,1=Λn [1+1.84Λ-2/3]
δ˜B=-2 arctanα2 (n2-1)1/2n Λ1/31.92
xl,1=Λn Λ+1.84Λ-2/31-23π (π+δ˜B)
Δϕr(nx)=ϕr(nx)x Δxν+ϕr(nx)l Δl=0,
ϕr(nx)x=(n2x2-Λ2)1/2x,
ϕr(nx)l=-arccosΛnx.
Δxν=xΛ arctan[(nx/Λ)2-1]1/2[(nx/Λ)2-1]1/2
Δϕr(nx)=ϕr(nx)x Δxl=π,
Δxl=xΛ π[(nx/Λ)2-1]1/2
ϕr(nk0rmax)=π/4.
rmaxaΛnx [1+0.885Λ-2/3],
Δx=2(n2-1)px2Yl2(x),
p=1forTEmodes(Λ/x)2+(Λ/nx)2-1forTMmodes,
ΔxΔt=x/ω=a/c;
Δx=xT2(n2x2-Λ2)1/2.
T=4(n2x2-Λ2)1/2(Λ2-x2)1/2(n2-1)x2p exp[2ψr(x)],
Δx=2(Λ2-x2)1/2(n2-1)xp exp[2ψr(x)],
Δx=Δxtrans exp[2ψr(x)],
Δxtrans=2.12α2(n2-1)Λ1/3=2.12α2(n2-1)x1/3,
tf=2αkinkin+α2ken=2α(n2x2-Λ2)1/2(n2x2-Λ2)1/2+α2(Λ2-x2)1/2i,
Tf=tftf* kenkin=4(n2x2-Λ2)1/2(Λ2-x2)1/2(n2x2-Λ2)/α2+(Λ2-x2)α2.
ker(r)=(k02r2-Λ2)1/2r,
Ee(r)=tf exp-|arker(ρ)|dρ.
t=tf exp[ψr(x)],
νtrans=Λπ (n2-1)1/2-arccos1n+34.
N2 xπ [(n2-1)1/2-arctan(n2-1)1/2],
dNtotdx=x 0.416(n2-1)1/2(n-1),
l(nx)-(nx)1/3 12 3ν-14π2/3,
lx+x1/3 12 32 lnΔxtransΔx2/3.
νΔx=xπ 89 (n-1)3n1/21-12(n-1)×3[ln(Δxtrans)-ln(Δx)]2x2/33/2+14.
νtrans=xπ 89 (n-1)3n1/2+14,
d Nd x=d Ntotd x F
F=1-1-12(n-1) 3[ln(Δxtrans)-ln()]2x2/33/2.

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