Abstract

We study the ripple structure in the scattered intensity predicted by Mie scattering theory in the angular behavior of the scattered intensity for homogeneous, dielectric spheres. We find that for small values of the phase shift parameter ρ=2kR|m-1|, where k=2π/λ, R is the sphere radius, and m is the relative refractive index, the ripples are periodic with spacing equal to π when plotted versus the dimensionless qR, where q=2k sin(θ/2) and θ is the scattering angle. However, as ρ increases, this outcome switches to nonuniform spacing of approximately π cos(θ/2). The latter spacing is equivalent to a uniform spacing of π/kR when plotted versus θ.

© 2002 Optical Society of America

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References

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  1. G. Mie, “Beitrage zur Optik trüber Medien speziel kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  2. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  3. M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic, New York, 1969).
  4. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  5. C. M. Sorensen, D. J. Fischbach, “Patterns in Mie scattering,” Opt. Commun. 173, 145–153 (2000).
    [CrossRef]
  6. A. Guinier, G. Gournet, C. B. Walker, K. L. Yudowitch, Small Angle Scattering of X-Rays (Wiley, New York, 1955).
  7. S. H. Maron, M. E. Elder, “Determination of latex particle size by light scattering I. Minimum intensity method,” J. Colloid Sci. 18, 107–118 (1963).
    [CrossRef]
  8. P. E. Pierce, S. H. Maron, “Prediction of minima and maxima in intensities of scattered light and of higher order Tyndall spectra,” J. Colloid Sci. 19, 658–672 (1964).
    [CrossRef]
  9. M. Kerker, W. A. Farone, L. B. Smith, E. Matijevic, “Determination of particle size by the minima and maxima in the angular dependence of the scattered light. Range of validity of the method,” J. Colloid Sci. 19, 193–200 (1964).
    [CrossRef]

2000

C. M. Sorensen, D. J. Fischbach, “Patterns in Mie scattering,” Opt. Commun. 173, 145–153 (2000).
[CrossRef]

1964

P. E. Pierce, S. H. Maron, “Prediction of minima and maxima in intensities of scattered light and of higher order Tyndall spectra,” J. Colloid Sci. 19, 658–672 (1964).
[CrossRef]

M. Kerker, W. A. Farone, L. B. Smith, E. Matijevic, “Determination of particle size by the minima and maxima in the angular dependence of the scattered light. Range of validity of the method,” J. Colloid Sci. 19, 193–200 (1964).
[CrossRef]

1963

S. H. Maron, M. E. Elder, “Determination of latex particle size by light scattering I. Minimum intensity method,” J. Colloid Sci. 18, 107–118 (1963).
[CrossRef]

1908

G. Mie, “Beitrage zur Optik trüber Medien speziel kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Bohren, C. F.

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

Elder, M. E.

S. H. Maron, M. E. Elder, “Determination of latex particle size by light scattering I. Minimum intensity method,” J. Colloid Sci. 18, 107–118 (1963).
[CrossRef]

Farone, W. A.

M. Kerker, W. A. Farone, L. B. Smith, E. Matijevic, “Determination of particle size by the minima and maxima in the angular dependence of the scattered light. Range of validity of the method,” J. Colloid Sci. 19, 193–200 (1964).
[CrossRef]

Fischbach, D. J.

C. M. Sorensen, D. J. Fischbach, “Patterns in Mie scattering,” Opt. Commun. 173, 145–153 (2000).
[CrossRef]

Gournet, G.

A. Guinier, G. Gournet, C. B. Walker, K. L. Yudowitch, Small Angle Scattering of X-Rays (Wiley, New York, 1955).

Guinier, A.

A. Guinier, G. Gournet, C. B. Walker, K. L. Yudowitch, Small Angle Scattering of X-Rays (Wiley, New York, 1955).

Huffman, D. R.

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

Kerker, M.

M. Kerker, W. A. Farone, L. B. Smith, E. Matijevic, “Determination of particle size by the minima and maxima in the angular dependence of the scattered light. Range of validity of the method,” J. Colloid Sci. 19, 193–200 (1964).
[CrossRef]

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

Maron, S. H.

P. E. Pierce, S. H. Maron, “Prediction of minima and maxima in intensities of scattered light and of higher order Tyndall spectra,” J. Colloid Sci. 19, 658–672 (1964).
[CrossRef]

S. H. Maron, M. E. Elder, “Determination of latex particle size by light scattering I. Minimum intensity method,” J. Colloid Sci. 18, 107–118 (1963).
[CrossRef]

Matijevic, E.

M. Kerker, W. A. Farone, L. B. Smith, E. Matijevic, “Determination of particle size by the minima and maxima in the angular dependence of the scattered light. Range of validity of the method,” J. Colloid Sci. 19, 193–200 (1964).
[CrossRef]

Mie, G.

G. Mie, “Beitrage zur Optik trüber Medien speziel kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Pierce, P. E.

P. E. Pierce, S. H. Maron, “Prediction of minima and maxima in intensities of scattered light and of higher order Tyndall spectra,” J. Colloid Sci. 19, 658–672 (1964).
[CrossRef]

Smith, L. B.

M. Kerker, W. A. Farone, L. B. Smith, E. Matijevic, “Determination of particle size by the minima and maxima in the angular dependence of the scattered light. Range of validity of the method,” J. Colloid Sci. 19, 193–200 (1964).
[CrossRef]

Sorensen, C. M.

C. M. Sorensen, D. J. Fischbach, “Patterns in Mie scattering,” Opt. Commun. 173, 145–153 (2000).
[CrossRef]

van de Hulst, H. C.

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

Walker, C. B.

A. Guinier, G. Gournet, C. B. Walker, K. L. Yudowitch, Small Angle Scattering of X-Rays (Wiley, New York, 1955).

Yudowitch, K. L.

A. Guinier, G. Gournet, C. B. Walker, K. L. Yudowitch, Small Angle Scattering of X-Rays (Wiley, New York, 1955).

Ann. Phys.

G. Mie, “Beitrage zur Optik trüber Medien speziel kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

J. Colloid Sci.

S. H. Maron, M. E. Elder, “Determination of latex particle size by light scattering I. Minimum intensity method,” J. Colloid Sci. 18, 107–118 (1963).
[CrossRef]

P. E. Pierce, S. H. Maron, “Prediction of minima and maxima in intensities of scattered light and of higher order Tyndall spectra,” J. Colloid Sci. 19, 658–672 (1964).
[CrossRef]

M. Kerker, W. A. Farone, L. B. Smith, E. Matijevic, “Determination of particle size by the minima and maxima in the angular dependence of the scattered light. Range of validity of the method,” J. Colloid Sci. 19, 193–200 (1964).
[CrossRef]

Opt. Commun.

C. M. Sorensen, D. J. Fischbach, “Patterns in Mie scattering,” Opt. Commun. 173, 145–153 (2000).
[CrossRef]

Other

A. Guinier, G. Gournet, C. B. Walker, K. L. Yudowitch, Small Angle Scattering of X-Rays (Wiley, New York, 1955).

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

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

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

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the envelopes of the Mie scattering curves for homogeneous, dielectric spheres (i.e., ignoring the ripple structure). Dashed lines, RDG limit at ρ=0 with slope -4 and the ρ limit with slope -2; solid line, envelope for an arbitrary ρ.

Fig. 2
Fig. 2

Spacing between successive maxima for the scattered light intensity for spheres with a size parameter of kR=50 but different indices of refractions m, hence different phase shift parameters ρ.

Fig. 3
Fig. 3

Ripple spacing plotted as either δqR or δqR/π cos(θ/2).

Fig. 4
Fig. 4

Normalized Mie scattered intensity for spheres with m=1.2 and kR=50, hence ρ=20 versus qR. Note how the ripple spacing becomes smaller at large qR.

Fig. 5
Fig. 5

Normalized Mie scattered intensity for spheres with m=1.2 and kR=50, hence ρ=20 versus θ. Note how the ripple spacing remains constant at large θ.

Fig. 6
Fig. 6

Ripple spacing in θ space, δθ, as a function of scattering angle.

Equations (11)

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q=4πλ-1sin(θ/2)
ρ=2kR|m-1|,
I(qR)0 whenqR<1
I(qR)-2 when1<qR<ρ
I(qR)-4 whenqR>ρ.
I(u)=[3(sin u-u cos u)/u3]2,
δqR=π,ρ<1,
δqR=π cos(θ/2),ρ5.
d(qR)dθ=kR cos(θ/2).
δθ=π/kR,ρ5.
2Rλ-1sin(θ/2)=ki orKi.

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