Abstract

It is shown that, in the large-size-parameter region (xn2/2) the Re(an + bn) is independent of the summation index n. For real refractive indices m < 2.5 the dominating term of Re (an + bn) has an x dependence of the form sin2[(m − 1)x] leading to the periodicity of Re(an + bn) and of the extinction curve QEXT(x) given by Δx = π/(m − 1). The derived periodicity is the same as the periodicity of interference structure derived by using the anomalous diffraction approximation; however, our derivation is not limited by the anomalous diffraction condition m − 1 ≪ 1. At refractive indices m > 2.5 the extinction curve does not have a simple periodic structure, since several terms of approximately equal magnitudes and different x dependences contribute to the Re(an + bn).

© 1989 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  2. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
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    [CrossRef]
  4. P. Chýlek, J. T. Kiehl, M. K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978).
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    [CrossRef]
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    [CrossRef]
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1988

1985

1984

1983

1981

1980

H. M. Nussenzweig, W. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

1978

1977

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

1976

1973

1969

H. M. Nussenzweig, “High-frequency scattering by a transparent sphere,”J. Math. Phys. 10, 82–176 (1969).
[CrossRef]

1968

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985).

Ashkin, A.

Barber, P. W.

Benner, R. E.

Bennett, H. S.

Bohren, C. F.

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

Box, M. A.

Chýlek, P.

Conwell, P. R.

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

Dziedzic, J. M.

Fuchs, R.

Hill, S. C.

Huffman, D. R.

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

Hunter, B. A.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Kiefer, W.

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]

Kliewer, K. L.

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]

Lock, J. A.

Maier, B.

Messiah, A.

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1964), Vol. 1, p. 490.

Nussenzweig, H. M.

H. M. Nussenzweig, W. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

H. M. Nussenzweig, “High-frequency scattering by a transparent sphere,”J. Math. Phys. 10, 82–176 (1969).
[CrossRef]

Probert-Jones, J. R.

Ramaswamy, V.

Rosasco, G. J.

Rushforth, C. K.

Thurn, R.

van de Hulst, H. C.

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

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sec. 12.31.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Secs. 11.21 and 11.22.

Wiscombe, W.

H. M. Nussenzweig, W. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Appl. Opt.

J. Math. Phys.

H. M. Nussenzweig, “High-frequency scattering by a transparent sphere,”J. Math. Phys. 10, 82–176 (1969).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. Rev. A

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

Phys. Rev. Lett.

H. M. Nussenzweig, W. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

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

Other

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

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

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Secs. 11.21 and 11.22.

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1964), Vol. 1, p. 490.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sec. 12.31.

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Figures (7)

Fig. 1
Fig. 1

Normalized extinction cross section (efficiency for extinction) for a refractive index m = 1.33 exhibits slow oscillations (interference structure) with superimposed sharp resonance peaks. Numbers above the peaks denote the n index of the resonating partial wave.

Fig. 2
Fig. 2

Contributions of partial waves n = 17, 18, 19 to the QEXT(x) for m = 1.33. The partial wave n = 17 is the lowest one that shows separate peaks of an and bn resonances.

Fig. 3
Fig. 3

The periodic interference structure is shown clearly by the QEXT(x) curves for m = 1.5 and m = 2.0 with periods of oscillation Δx of 6.3 and 3.1, respectively.

Fig. 4
Fig. 4

At larger values of refractive index m no simple periodic interference structure is observed.

Fig. 5
Fig. 5

In the region of refractive indices 0.5 ≤ m ≤ 2.5 the c1 coefficient in Eq. (12) is much larger than coefficients c2 and c3 Consequently, the term containing c1 dominates the expression [relation (12)] for Re(an + bn).

Fig. 6
Fig. 6

All extinction curves QEXT for m < 2.5 show the same frequency of oscillation when plotted as a function of y = (m − 1)x.

Fig. 7
Fig. 7

Small imaginary part of the refractive index (|Im(m)| ≤ 0.01) reduces the amplitude; however, it does not change the periodicity of oscillation.

Equations (30)

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Q EXT ( x , m ) = 2 x 2 n = 1 ( 2 n + 1 ) Re [ a n ( x , m ) + b n ( x , m ) ]
a n ( x , m ) = Ψ n ( x ) Ψ n ( m x ) - m Ψ n ( m x ) Ψ n ( x ) ζ n ( x ) Ψ n ( m x ) - m Ψ n ( m x ) ζ n ( x ) ,
b n ( x , m ) = m Ψ n ( x ) Ψ n ( m x ) - Ψ n ( m x ) Ψ n ( x ) m ζ n ( x ) Ψ n ( m x ) - Ψ n ( m x ) ζ n ( x ) ,
Q EXT ( x , m ) 2 { - sin [ 2 x ( m - 1 ) ] x ( m - 1 ) } + 1 - cos [ 2 x ( m - 1 ) ] x 2 ( m - 1 ) 2 ,
Δ x = x m - 1 ,
Ψ n ( m x ) ~ sin ( m x - n π / 2 ) ,
ζ n ( x ) ~ sin ( x - n π / 2 ) - i cos ( x - n π / 2 ) .
a n ( x , m ) = A n ( x , m ) A n ( x , m ) - i C n ( x , m ) ,
b n ( x , m ) = B n ( x , m ) B n ( x , m ) - i D n ( x , m ) ,
A n ( x , m ) ~ C ( m x ) S ( x ) - m S ( m x ) C ( x ) ,
C n ( x , m ) ~ C ( m x ) C ( x ) - m S ( m x ) S ( x ) ,
B n ( x , m ) ~ m C ( m x ) S ( x ) - S ( m x ) C ( x ) ,
D n ( x , m ) ~ m C ( m x ) C ( x ) + S ( m x ) S ( x ) ,
A n ~ - ( 1 / 2 ) [ ( m + 1 ) sin ( m - 1 ) x + ( - 1 ) n ( m - 1 ) × sin ( m + 1 ) x ] ,
C n ~ ( 1 / 2 ) [ ( m + 1 ) cos ( m - 1 ) x - ( - 1 ) n ( m - 1 ) × cos ( m + 1 ) x ] ,
B n ~ ( 1 / 2 ) [ - ( m + 1 ) sin ( m - 1 ) x + ( - 1 ) n ( m - 1 ) × sin ( m + 1 ) x ] ,
D n ~ ( 1 / 2 ) [ ( m + 1 ) cos ( m - 1 ) x + ( - 1 ) n ( m - 1 ) × cos ( m + 1 ) x ] .
Re ( a n + b n ) = A n 2 ( B n 2 + D n 2 ) + B n 2 ( A n 2 + C n 2 ) ( A n 2 + C n 2 ) ( B n 2 + D n 2 ) ,
Re ( a n + b n ) ~ ( m + 1 ) 2 m 2 + 1 sin 2 [ ( m - 1 ) x ] + ( m - 1 ) 2 m 2 + 1 sin 2 [ ( m + 1 ) x ] + 2 ( m 2 - 1 2 m 2 + 1 ) 2 sin [ ( m - 1 ) x ] sin [ ( m + 1 ) x ] cos 2 m x 1 - ( m 2 - 1 2 m 2 + 1 ) 2 cos 2 2 m x ,
Re [ a n ( x , m ) ] ~ ( sin m x cos x - m cos m x sin x ) 2 sin 2 m x + m 2 cos 2 m x ,
Re [ b n ( x , m ) ] ~ ( m sin m x cos x - cos m x sin x ) 2 m 2 sin 2 m x + cos 2 m x
Re ( a n + b n ) ~ ( 1 + 2 m m 2 + 1 ) sin 2 [ ( m - 1 ) x ] ,
Δ x = x m - 1 .
c 1 = ( m + 1 ) 2 m 2 + 1 ,
c 2 = ( m - 1 ) 2 m 2 + 1 ,
c 3 = ( m 2 - 1 m 2 + 1 ) 2 .
0.5 m 2.5.
Re a n = sin 2 x ,
Re b n = cos 2 x ,
Re ( a n + b n ) = 1.

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