Abstract

We found that the resonance positions for a weakly absorbing sphere are much different from those of a nonabsorbing sphere. The shifts of resonance positions for Qext, Qsca, and Qabs from their original resonance positions for a real refractive index are different, and the formulas to describe these different kinds of shift are given. Calculations of 12 resonance cases show that our results are quite close to those given by exact Lorenz–Mie theory. We also derive the line shapes of the different resonances and calculate values for the resonance maxima and half-width as a function of absorption.

© 1995 Optical Society of America

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References

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  1. P. Chýlek, “Partial wave resonances and the ripple structure of the Mie normalized extinction cross section,” J. Opt. Soc. Am. 66, 285–287 (1976).
    [CrossRef]
  2. A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
    [CrossRef]
  3. A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–1814 (1981).
    [CrossRef] [PubMed]
  4. P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
    [CrossRef]
  5. P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978).
    [CrossRef] [PubMed]
  6. H. S. Bennett, G. J. Rosasco, “Resonances in the efficiency factor for absorption: Mie scattering theory,” Appl. Opt. 17, 491–493 (1978).
    [CrossRef] [PubMed]
  7. P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984).
    [CrossRef]
  8. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  9. B. A. Hunter, M. A. Box, B. Maier, “Resonance structure in weakly absorbing spheres,” J. Opt. Soc. Am. A 5, 1281–1286 (1988).
    [CrossRef]
  10. P. Chýlek, “Large-sphere limits of the Mie scattering functions,” J. Opt. Soc. Am. 63, 699–716 (1973).
    [CrossRef]
  11. J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A 1, 822–829 (1984).
    [CrossRef]
  12. H. S. Bennett, G. J. Rosasco, “Internal field resonance structure: implications for optical absorption and scattering by microscopic particles,” J. Opt. Soc. Am. A 1, 62–67 (1984).
  13. G. Videen, W. S. Bickel, “Light-scattering resonances in small spheres,” Phys. Rev. A 45, 6008–6012 (1992).
    [CrossRef] [PubMed]

1992 (1)

G. Videen, W. S. Bickel, “Light-scattering resonances in small spheres,” Phys. Rev. A 45, 6008–6012 (1992).
[CrossRef] [PubMed]

1988 (1)

1984 (3)

1981 (1)

1978 (3)

1977 (1)

A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[CrossRef]

1976 (1)

1973 (1)

Ashkin, A.

A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–1814 (1981).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[CrossRef]

Barber, P. W.

Bennett, H. S.

Bickel, W. S.

G. Videen, W. S. Bickel, “Light-scattering resonances in small spheres,” Phys. Rev. A 45, 6008–6012 (1992).
[CrossRef] [PubMed]

Box, M. A.

Chýlek, P.

Conwell, P. R.

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–1814 (1981).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[CrossRef]

Hunter, B. A.

Kiehl, J. T.

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978).
[CrossRef] [PubMed]

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]

P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978).
[CrossRef] [PubMed]

Maier, B.

Probert-Jones, J. R.

Rosasco, G. J.

Rushforth, C. K.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Videen, G.

G. Videen, W. S. Bickel, “Light-scattering resonances in small spheres,” Phys. Rev. A 45, 6008–6012 (1992).
[CrossRef] [PubMed]

Appl. Opt. (3)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Phys. Rev. A (2)

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

G. Videen, W. S. Bickel, “Light-scattering resonances in small spheres,” Phys. Rev. A 45, 6008–6012 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

A. Ashkin, J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354 (1977).
[CrossRef]

Other (1)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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Figures (4)

Fig. 1
Fig. 1

For a real refractive index the maxima of Re ( b 40 2 ) and | b 40 2 | 2 and the zero value of Im ( b 40 2 ) are in the same position.

Fig. 2
Fig. 2

For a weakly absorbing sphere the maxima of Re ( b 40 2 ) and | b 40 2 | 2 and the zero value of Im ( b 40 2 ) shift different distances.

Fig. 3
Fig. 3

Lorenz–Mie calculations and Taylor-series calculations of the shift of d 38 1 the resonances as a function of absorption. The real part of the pole position does not correspond to any of the values and, for very small mi (mi < 3 × 10−4), is even of different sign.

Fig. 4
Fig. 4

Lorenz–Mie calculations and Taylor-series calculations of (a) the half-widths and (b) the heights of the a 38 1 resonances as a function of absorption.

Tables (2)

Tables Icon

Table 1 Terms in Series Expansion

Tables Icon

Table 2 Values of Resonance Height Decay and Initial Half-Width

Equations (67)

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a n = A n A n + i C n ,
b n = B n B n + i D n .
Re ( a n ) = A n 2 A n 2 + C n 2 ,
Im ( a n ) = A n C n A n 2 + C n 2 ,
Re ( b n ) = B n 2 B n 2 + D n 2 ,
Im ( b n ) = B n D n B n 2 + D n 2 ,
d d x Re ( a n ) = 0 ,
A n = 0 ,
C n = 0 ,
A n C n A n C n = 0 ,
D n = 0.
Re ( a n ) = | a n | 2 = 1 ,
Im ( a n ) = 0
Re ( b n ) = | b n | 2 = 1 ,
Im ( b n ) = 0
a n = A n r + i A n i ( A n r + C n r ) + i ( A n i + C n i ) ,
b n = B n r + i B n i ( B n r + D n r ) + i ( B n i + D n i ) ,
ψ n ( m x ) = ψ n [ ( m r + i m i ) ( x 0 + δ ) ] ψ n ( m r x 0 ) + ( i m i x 0 + m r δ ) ψ n ( m r x 0 ) + ½ ( i m i x 0 + m r δ ) 2 ψ n ( m r x 0 ) ,
η n ( m x ) = η n [ ( m r + i m i ) ( x 0 + δ ) ] η n ( m r x 0 ) + ( i m i x 0 + m r δ ) η n ( m r x 0 ) + ½ ( i m i x 0 + m r δ ) 2 η n ( m r x 0 ) ,
A n r = A n r o + δ A n r δ + δ 2 A n r s + m i 2 A n r m ,
A n i = m i A n i m + δ m i A n i δ ,
C n i = C n i o + δ C n i δ + δ 2 C n i s + m i 2 C n i m ,
C n r = m i C n r m + δ m i C n r δ .
Re ( a n ) = A n r ( A n r + C n r ) + A n i ( A n i + C n i ) ( A n r + C n r ) 2 + ( A n i + C n i ) 2 .
δ = Δ 1 = p 1 m i + q 1 m i 2 ,
p 1 = A n r δ C n r m A n r o C n r δ A n i m C n i δ 2 ( C n i δ ) 2 ,
q 1 = A n r δ C n r m 2 A n r o C n r δ C n r m + A n i m C n i δ C n r m 2 A n r o C n i δ C n i m 2 A n r o ( C n i δ ) 2 .
| a n | 2 = ( A n r ) 2 + ( A n i ) 2 ( A n r + C n r ) 2 + ( A n i + C n i ) 2 .
δ = Δ 2 = p 2 m i + q 2 m i 2 ,
p 2 = A n r δ C n r m A n r o C n r δ A n i m C n i δ C n i δ 2 ,
q 2 = A n r δ ( C n r m ) 2 A n r o C n r δ C n r m 2 A n r o C n i δ C n i m A n r o ( C n i δ ) 2 .
Δ 3 = A n i m C n r m A n r o C n i m A n r o C n i δ m i 2 .
Re ( a n ) = ( A n r o ) 2 + m i A n r o C n r m ( A n r o ) 2 + 2 m i A n r o C n r m + ( C n i δ ) 2 ( δ Δ 1 ) 2 .
H Re ( a n ) = A n r o + m i C n r m A n r o + 2 m i C n r m
HWHM Re ( a n ) = ( A n r o + 2 m i C n r m C n i δ 2 ) 1 / 2 .
| a n | 2 = ( A n r o ) 2 ( A n r o ) 2 + 2 m i A n r o C n r m + ( C n i δ ) 2 ( δ Δ 2 ) 2 .
H | a n | 2 = A n r o A n r o + 2 m i C n r m .
Q abs n = 2 ( 2 n + 1 ) x 2 m i A n r o C n r m A n r o 2 + 2 m i A n r o C n r m + C n i δ 2 ( δ Δ 1 ) 2 .
Area Re ( a n ) = π A n r o C n i δ ,
Area | a n | 2 = π A n r o C n i δ ( 1 m i C n r m A n r o ) ,
Area abs = m i π C n r m C n i δ .
a 20 1
a 20 2
a 38 1
a 38 1
b 40 1
b 40 2
a 60 1
a 60 2
a 60 3
b 60 1
b 60 2
b 60 3
C n r m / A n r o
A n r o / C n i δ
A n r o = m r ψ n ( x 0 ) ψ n ( m r x 0 ) ψ n ( x 0 ) ψ n ( m r x 0 ) ,
A n r δ = ( m r 2 1 ) ψ n ( x 0 ) ψ n ( m r x 0 ) m r ψ n ( x 0 ) ψ n ( m r x 0 ) + m r ψ n ( x 0 ) ψ n ( m r x 0 ) ,
A n r s = ( m r 2 1 2 ) ψ n ( x 0 ) ψ n ( m r x 0 ) + m r ( m r 2 2 1 ) ψ n ( x 0 ) ψ n ( m r x 0 ) + m r 2 ψ n ( x 0 ) ψ n ( m r x 0 ) m r 2 2 ψ n ( x 0 ) ψ n ( m r x 0 ) ,
A n r m = x 0 ψ n ( x 0 ) ψ n ( m r x 0 ) + x 0 2 2 ψ n ( x 0 ) ψ n ( m r x 0 ) m r x 0 2 2 ψ n ( x 0 ) ψ n ( m r x 0 ) ,
A n i m = ψ n ( x 0 ) ψ n ( m r x 0 ) + m r x 0 ψ n ( x 0 ) ψ n ( m r x 0 ) x 0 ψ n ( x 0 ) ψ n ( m r x 0 ) ,
A n i δ = m r ψ n ( x 0 ) ψ n ( m r x 0 ) + ψ n ( x 0 ) ψ n ( m r x 0 ) + m r x 0 ψ n ( x 0 ) ψ n ( m r x 0 ) m r x 0 ψ n ( x 0 ) ψ n ( m r x 0 ) + x 0 ( m r 2 1 ) ψ n ( x 0 ) ψ n ( m r x 0 ) .
B n r o = ψ n ( x 0 ) ψ n ( m r x 0 ) m r ψ n ( x 0 ) ψ n ( m r x 0 ) ,
B n r δ = ψ n ( x 0 ) ψ n ( m r x 0 ) m r 2 ψ n ( x 0 ) ψ n ( m r x 0 ) ,
B n r s = 1 2 ψ n ( x 0 ) ψ n ( m r x 0 ) + m r 2 ψ n ( x 0 ) ψ n ( m r x 0 ) m r 2 2 ψ n ( x 0 ) ψ n ( m r x 0 ) m r 3 2 ψ n ( x 0 ) ψ n ( m r x 0 ) ,
B n r m = x 0 ψ n ( x 0 ) ψ n ( m r x 0 ) + m r x 0 2 2 ψ n ( x 0 ) ψ n ( m r x 0 ) x 0 2 2 ψ n ( x 0 ) ψ n ( m r x 0 ) ,
B n i m = ψ n ( x 0 ) ψ n ( m r x 0 ) m r x 0 ψ n ( x 0 ) ψ n ( m r x 0 ) + x 0 ψ n ( x 0 ) ψ n ( m r x 0 ) ,
B n i δ = m r ψ n ( x 0 ) ψ n ( m r x 0 ) ψ n ( x 0 ) ψ n ( m r x 0 ) m r 2 x 0 ψ n ( x 0 ) ψ n ( m r x 0 ) + x 0 ψ n ( x 0 ) ψ n ( m r x 0 ) .

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