Abstract

In the framework of Mie theory the involved electromagnetic fields are expanded in an infinite series of multipoles. In numerical computations the summation has to be terminated after a finite number of terms (the expansion order N), which unavoidably produces errors. On the other hand, it is known that the contributions of terms of order l with x < l < N, where x is the dimensionless size parameter, are highly localized, i.e., these contributions appear as sharp peaks in resonance spectra. We show that it is possible to specify the expansion order in a controlled manner to extract certain features from Mie spectra. This controlled modification of the expansion order can be used as a high-pass, low-pass or bandpass filter. Formulas that serve as linewidth (frequency) and resonance-order filters are given, and their usage is demonstrated.

© 1998 Optical Society of America

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References

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  1. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  2. V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1976).
  3. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A.
  4. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 4.
  5. 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. 1–63.
  6. M. L. Gorodetsky, A. A. Savchenkov, V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
    [CrossRef] [PubMed]
  7. G. Schweiger, “Raman scattering on microparticles: size dependence,” J. Opt. Soc. Am. B 8, 1770–1778 (1991).
    [CrossRef]
  8. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957), p. 208.
  9. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343–352 (1993).
    [CrossRef]
  10. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, UK, 1992), Chap. 14.
    [CrossRef]
  11. 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]
  12. M. Abramowitz, I. E. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 10.
  13. J. B. Keller, “A geometrical theory of diffraction,” in Calculus of Variations and its Applications, Proceedings of Symposia in Applied Mathematics, L. M. Graves, ed. (McGraw-Hill, New York, 1958), Vol. 8.
    [CrossRef]
  14. P. Chyélek, “Resonance structure of Mie scattering: distance between resonances,” J. Opt. Soc. Am. A 7, 1609–1613 (1990).
    [CrossRef]
  15. J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.) 9, 24–75 (1960).
    [CrossRef]

1996 (1)

1993 (1)

1992 (1)

1991 (1)

1990 (1)

1980 (1)

1960 (1)

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

Barber, P. W.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 4.

Benner, R. E.

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. 1–63.

Bohren, C. F.

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

Chyélek, P.

Gorodetsky, M. L.

Hill, S. C.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 4.

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. 1–63.

Huffman, D. R.

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

Ilchenko, V. S.

Johnson, B. R.

Keller, J. B.

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

J. B. Keller, “A geometrical theory of diffraction,” in Calculus of Variations and its Applications, Proceedings of Symposia in Applied Mathematics, L. M. Graves, ed. (McGraw-Hill, New York, 1958), Vol. 8.
[CrossRef]

Khare, V.

V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1976).

Lam, C. C.

Leung, P. T.

Nussenzveig, H. M.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, UK, 1992), Chap. 14.
[CrossRef]

Rubinow, S. I.

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

Savchenkov, A. A.

Schweiger, G.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957), p. 208.

Wiscombe, W. J.

Young, K.

Ann. Phys. (N.Y.) (1)

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

Appl. Opt. (1)

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

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

Opt. Lett. (1)

Other (8)

M. Abramowitz, I. E. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 10.

J. B. Keller, “A geometrical theory of diffraction,” in Calculus of Variations and its Applications, Proceedings of Symposia in Applied Mathematics, L. M. Graves, ed. (McGraw-Hill, New York, 1958), Vol. 8.
[CrossRef]

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957), p. 208.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, UK, 1992), Chap. 14.
[CrossRef]

V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1976).

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

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), Chap. 4.

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. 1–63.

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Figures (5)

Fig. 1
Fig. 1

Positions of 131 TE resonances in (x, l) space for n = 1.5. The crosses mark the exact positions, whereas the l coordinates of the filled circles were calculated with Eq. (16). The solid lines are lines of constant linewidth according to approximation (16). The associated linewidth is given at the right edge of the respective line. This figure shows the accuracy of Eq. (16). Additionally the expansion order according to Wiscombe’s criterion is plotted as a dash–dot line.

Fig. 2
Fig. 2

Results of five Mie computations of the scattering cross section of a sphere of refractive index n = 1.5 with different expansion orders. The respective expansion order was determined with approximation (16), and the truncation linewidth is given at the right edge of the figure. The scale corresponds to the lowest curve; the other curves are vertically shifted by multiples of 0.08. This figure shows that it is possible to use approximation (16) as a filter; the smaller Δx trunc, the sharper the details.

Fig. 3
Fig. 3

Scattering cross section of a sphere with refractive index 1.5 for different expansion orders according to approximation (16). The lower curve corresponds to Δx trunc = 0.15 whereas the upper curve was computed with Δx trunc = 0.001. The scale corresponds to the lower curve; the upper curve is shifted by 0.05. This figure shows that approximation (16) may be used to a degree to extract the background contributions.

Fig. 4
Fig. 4

Positions of 131 TE resonances in (x, l) space for n = 1.5. The crosses mark the exact positions, whereas the l coordinates of the filled circles were calculated with approximation (24). The solid lines are lines of constant radial mode number according to approximation (24). The radial mode number is given at the right edge of the respective line. This figure shows the accuracy of approximation (24).

Fig. 5
Fig. 5

Contribution of all terms with radial mode number v = 7 to the scattering cross section in the interval 100 ≤ x ≤ 105 (solid curve). This curve was calculated by controlled start and truncation of the Mie series according to approximation (24) with v = 6.5 and v = 7.5, respectively. The envelope (dashed line) was directly computed from the relation for the scattering cross section and approximation (24) with v = 7.

Tables (2)

Tables Icon

Table 1 Linewidth and Order of Various Resonances in the Transition Region (x ≈ Λ = l + ½) with Comparison of Exact and Approximate Values

Tables Icon

Table 2 Values for the Parameter c [Compare Eq. (17)] for Different Combinations of the Truncation Linewidth and Refractive Index

Equations (27)

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N = x + 4 x 1 / 3 + 1 ,   0.02 x 8 x + 4.05 x 1 / 3 + 2 ,   8 < x < 4200 x + 4 x 1 / 3 + 2 ,   4200 x 20000 ,
Δ x = 2 n 2 - 1 η x 2 y l 2 x ,
η = 1 for   TE   modes Λ x 2 + Λ nx 2 - 1 for   TM   modes
Δ x = 2 n 2 - 1 η Λ x 2 - 1 exp 2 ψ r ,
ψ r = Λ 1 - x Λ 2 - arccosh Λ x .
Λ x = cosh   χ 1 + 1 2   χ 2 .
ψ r = Λ tanh   χ - χ - Λ 1 3 χ 3 .
Λ x + x 1 / 3 1 2 - 3 ψ r 2 / 3 ,
Λ x + x 1 / 3 1 2 3 2 ln F Δ x 2 / 3 ,
F = 2 n 2 - 1 η Λ x 2 - 1 .
Δ x Δ x trans   exp 2 ψ r .
Δ x Δ x trans 1 - 0.919 h - 0.333 h 3 + 0.096 h 4 2 ,
Δ x trans = 2.122 μ n 2 - 1 Λ 1 / 3 ,
μ = 1 for TE modes n 2 for TM modes ,
h = x - Λ Λ 1 / 3 .
l x + x 1 / 3 α   1 2 3 2 ln Δ x trans Δ x 2 / 3 - 1 2 ,
c = α   1 2 3 2 ln Δ x trans Δ x trunc 2 / 3 .
x l + 1 , v - x l , v = arctan n 2 - 1 n 2 - 1 ,
φ r - π 4 - δ B = v - 1 π ,
φ r = Λ nx Λ 2 - 1 - arccos Λ nx ,
Λ nx = cos   γ ,
φ r = Λ tan   γ - γ .
Λ nx - nx 1 / 3 1 2 3 φ r 2 / 3 .
l nx - nx 1 / 3 1 2 3 v - 1 4 π 2 / 3 + 1
v trans = x π n 2 - 1 - arccos 1 n + 3 4 ,
Q sca = 2 x l = 1 2 l + 1 | a l | 2 + | b l | 2 ,
Q env = 4 l + 2 x

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