Abstract

For spheres of size a larger than light wavelength λ the Mie series converges slowly, and ordinary computations cannot provide enough (say, 100) significant digits. A digit array is proposed to overcome this inconvenience. This method transforms a number into an array of digits, and operations are performed with respect to individual digits. It implies that the number of significant digits could be as many as necessary. With this method, patterns of light scattering from moderate particles, whose size parameter q=(2π/λ)a ranges from 0.01 to 500, are computed for different relative refractive indices nd/n0 (nd and n0 are, respectively, refractive indices of the sphere and the surrounding medium) and are compared with previous results.

© 2002 Optical Society of America

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References

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  1. G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaller Metallösungen,” Ann. D. Physik 25, 377–445 (1908).
    [CrossRef]
  2. P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. D. Phys. 30, 57–136 (1909).
    [CrossRef]
  3. A. Clebsch, “Ueber die Reflexion an einer Kugelfläche,” Z. für Math. 61, 195–262 (1863).
  4. H. Lamb, “On the oscillations of a viscous spheroid,” Proc. Math. Soc. (London) 13, 51–66 (1881).
    [CrossRef]
  5. J. W. Strutt (Lord Rayleigh), “Investigation of the disturbance produced by a spherical obstacle on the waves of sound,” Proc. Math. Soc. (London) 4, 253–283 (1872).
  6. J. W. Strutt (Lord Rayleigh), “The incidence of light upon a transparent sphere of dimensions comparable with the wavelength,” Proc. R. Soc. London, Ser. A 84, 25–46 (1910).
    [CrossRef]
  7. L. V. Lorenz, “Sur la lumière réfléchie et réfractée par une sphère transparente,” Vidensk. Selsk. Skrifter 6, 1–62 (1890).
  8. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).
  9. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, London, 1999).
  10. W. T. Grandy, Jr., Scattering of Waves from Large Particles (Cambridge University, London, 2000).
  11. J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
    [CrossRef] [PubMed]
  12. H. Blumer, “Strahlungsdiagramme kleiner dielektrischer Kugeln,” Z. Phys. 32, 119–134 (1925).
    [CrossRef]
  13. H. Blumer, “Strahlungsdiagramme kleiner dielectrischer Kugeln. II,” Z. Phys. A 38, 304–328 (1926).
    [CrossRef]
  14. J. V. Dave, “Subroutines for computing the parameters of the electromagnetic radiation scattered by a sphere,” Report 320–3237 (IBM Science Center, Palo Alto, Calif., 1968).
  15. H. C. Bryant and A. J. Cox, “Mie theory and the glory,” J. Opt. Soc. Am. 56, 1529–1532 (1966).
    [CrossRef]
  16. T. S. Fahlen and H. C. Bryant, “Optical backscattering from single water drops,” J. Opt. Soc. Am. 58, 304–310 (1968).
    [CrossRef]
  17. S. T. Shipley and J. A. Weinman, “A numerical study of scattering by large dielectric spheres,” J. Opt. Soc. Am. 68, 130–134 (1978).
    [CrossRef]
  18. G. Grehan and G. Gouesbet, “Mie theory calculations: new progress, with emphasis on particle sizing,” Appl. Opt. 18, 3489–3493 (1979).
    [CrossRef] [PubMed]
  19. H. C. van de Hulst and R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991).
    [CrossRef] [PubMed]
  20. R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computation and the Airy approximation,” Appl. Opt. 30, 106–107 (1991).
    [CrossRef] [PubMed]

1991 (2)

1979 (1)

1978 (1)

1969 (1)

1968 (1)

1966 (1)

1926 (1)

H. Blumer, “Strahlungsdiagramme kleiner dielectrischer Kugeln. II,” Z. Phys. A 38, 304–328 (1926).
[CrossRef]

1925 (1)

H. Blumer, “Strahlungsdiagramme kleiner dielektrischer Kugeln,” Z. Phys. 32, 119–134 (1925).
[CrossRef]

1910 (1)

J. W. Strutt (Lord Rayleigh), “The incidence of light upon a transparent sphere of dimensions comparable with the wavelength,” Proc. R. Soc. London, Ser. A 84, 25–46 (1910).
[CrossRef]

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. D. Phys. 30, 57–136 (1909).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaller Metallösungen,” Ann. D. Physik 25, 377–445 (1908).
[CrossRef]

1890 (1)

L. V. Lorenz, “Sur la lumière réfléchie et réfractée par une sphère transparente,” Vidensk. Selsk. Skrifter 6, 1–62 (1890).

1881 (1)

H. Lamb, “On the oscillations of a viscous spheroid,” Proc. Math. Soc. (London) 13, 51–66 (1881).
[CrossRef]

1872 (1)

J. W. Strutt (Lord Rayleigh), “Investigation of the disturbance produced by a spherical obstacle on the waves of sound,” Proc. Math. Soc. (London) 4, 253–283 (1872).

1863 (1)

A. Clebsch, “Ueber die Reflexion an einer Kugelfläche,” Z. für Math. 61, 195–262 (1863).

Blumer, H.

H. Blumer, “Strahlungsdiagramme kleiner dielectrischer Kugeln. II,” Z. Phys. A 38, 304–328 (1926).
[CrossRef]

H. Blumer, “Strahlungsdiagramme kleiner dielektrischer Kugeln,” Z. Phys. 32, 119–134 (1925).
[CrossRef]

Bryant, H. C.

Clebsch, A.

A. Clebsch, “Ueber die Reflexion an einer Kugelfläche,” Z. für Math. 61, 195–262 (1863).

Cox, A. J.

Dave, J. V.

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. D. Phys. 30, 57–136 (1909).
[CrossRef]

Fahlen, T. S.

Gouesbet, G.

Grehan, G.

Lamb, H.

H. Lamb, “On the oscillations of a viscous spheroid,” Proc. Math. Soc. (London) 13, 51–66 (1881).
[CrossRef]

Lorenz, L. V.

L. V. Lorenz, “Sur la lumière réfléchie et réfractée par une sphère transparente,” Vidensk. Selsk. Skrifter 6, 1–62 (1890).

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaller Metallösungen,” Ann. D. Physik 25, 377–445 (1908).
[CrossRef]

Shipley, S. T.

Strutt, J. W.

J. W. Strutt (Lord Rayleigh), “The incidence of light upon a transparent sphere of dimensions comparable with the wavelength,” Proc. R. Soc. London, Ser. A 84, 25–46 (1910).
[CrossRef]

J. W. Strutt (Lord Rayleigh), “Investigation of the disturbance produced by a spherical obstacle on the waves of sound,” Proc. Math. Soc. (London) 4, 253–283 (1872).

van de Hulst, H. C.

Wang, R. T.

Weinman, J. A.

Ann. D. Phys. (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. D. Phys. 30, 57–136 (1909).
[CrossRef]

Ann. D. Physik (1)

G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaller Metallösungen,” Ann. D. Physik 25, 377–445 (1908).
[CrossRef]

Appl. Opt. (4)

J. Opt. Soc. Am. (3)

Proc. Math. Soc. (London) (2)

H. Lamb, “On the oscillations of a viscous spheroid,” Proc. Math. Soc. (London) 13, 51–66 (1881).
[CrossRef]

J. W. Strutt (Lord Rayleigh), “Investigation of the disturbance produced by a spherical obstacle on the waves of sound,” Proc. Math. Soc. (London) 4, 253–283 (1872).

Proc. R. Soc. London, Ser. A (1)

J. W. Strutt (Lord Rayleigh), “The incidence of light upon a transparent sphere of dimensions comparable with the wavelength,” Proc. R. Soc. London, Ser. A 84, 25–46 (1910).
[CrossRef]

Vidensk. Selsk. Skrifter (1)

L. V. Lorenz, “Sur la lumière réfléchie et réfractée par une sphère transparente,” Vidensk. Selsk. Skrifter 6, 1–62 (1890).

Z. für Math. (1)

A. Clebsch, “Ueber die Reflexion an einer Kugelfläche,” Z. für Math. 61, 195–262 (1863).

Z. Phys. (1)

H. Blumer, “Strahlungsdiagramme kleiner dielektrischer Kugeln,” Z. Phys. 32, 119–134 (1925).
[CrossRef]

Z. Phys. A (1)

H. Blumer, “Strahlungsdiagramme kleiner dielectrischer Kugeln. II,” Z. Phys. A 38, 304–328 (1926).
[CrossRef]

Other (4)

J. V. Dave, “Subroutines for computing the parameters of the electromagnetic radiation scattered by a sphere,” Report 320–3237 (IBM Science Center, Palo Alto, Calif., 1968).

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, London, 1999).

W. T. Grandy, Jr., Scattering of Waves from Large Particles (Cambridge University, London, 2000).

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Figures (14)

Fig. 1
Fig. 1

Intensities Iφ(θ) (solid) and Iθ(θ) (dashed) for q=0.1, m=1.5 [Iφ(0)=Iθ(0)=8.697×10-8], which is identical to previous results.9,10

Fig. 2
Fig. 2

Intensities Iφ(θ) (solid) and Iθ(θ) (dashed) for q=1.6, m=1.25 [Iφ(0)=Iθ(0)=1.809], which are identical to those in Ref. 12.

Fig. 3
Fig. 3

Intensities Iφ(θ) (solid) and Iθ(θ) (dashed) for q=1.6, m=1.5 [Iφ(0)=Iθ(0)=6.911]: (a) whole pattern; (b) enlarged central part.

Fig. 4
Fig. 4

Intensities Iφ(θ) (solid) and Iθ(θ) (dashed) for q=4, m=1.25 [Iφ(0)=Iθ(0)=152.0]: (a) whole pattern; (b) enlarged central part. Intensities are identical to those in Ref. 12.

Fig. 5
Fig. 5

Intensity Iθ(θ) for q=8, m=1.25 [Iθ(0)=3531]: (a) whole pattern; (b) enlarged central part. (Intensity is different from Fig. 4 of Ref. 12.)

Fig. 6
Fig. 6

Intensity Iφ(θ) for q=8, m=1.25 [Iφ(0)=3531]: (a) whole pattern; (b) enlarged central part. (Intensity is different from Fig. 4 of Ref. 12.)

Fig. 7
Fig. 7

Intensity Iθ(θ) for q=16, m=1.25 [Iθ(0)=3.468×104]: (a) whole pattern; (b) enlarged central part.

Fig. 8
Fig. 8

Intensity Iφ(θ) for q=16, m=1.25 [Iφ(0)=3.468×104]: (a) whole pattern; (b) enlarged central part.

Fig. 9
Fig. 9

Intensity Iθ(θ) for q=50, m=1.33 [Iθ(0)=1.929×106]: (a) whole pattern; (b) enlarged central part.

Fig. 10
Fig. 10

Intensity Iφ(θ) for q=50, m=1.33 [Iφ(0)=1.929×106]: (a) whole pattern; (b) enlarged central part.

Fig. 11
Fig. 11

Intensity Iθ(θ) for q=100, m=1.33 [Iθ(0)=1.658×108]: (a) whole pattern; (b) enlarged central part.

Fig. 12
Fig. 12

Intensity Iφ(θ) for q=100, m=1.33 [Iφ(0)=1.658×108]: (a) whole pattern; (b) enlarged central part.

Fig. 13
Fig. 13

Intensity Iθ(θ) for q=500, m=1.33 [Iθ(0)=1.530×1010]: (a) whole pattern; (b) enlarged central part.

Fig. 14
Fig. 14

Intensity Iφ(θ) for q=500, m=1.33 [Iφ(0)=1.530×1010]: (a) whole pattern; (b) enlarged central part.

Equations (49)

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Eθ(θ, ϕ)=-iE0 cos ϕ exp(iK0r)K0r×n=1 2n+1n(n+1)[anτn(cos θ)+bnπn(sin θ)],
Eϕ(θ, ϕ)=iE0 sin ϕ exp(iK0r)K0r×n=1 2n+1n(n+1)[anπn(cos θ)+bnτn(cos θ)],
πn(cos θ)Pn1(cos θ)sin θ,τn(cos θ)dPn1(cos θ)dθ,
Pn1(cos θ)=2-n sin θk=0[n/2](-1)k{n; k}{2n-2k;n}×(n-2k)(cos θ)n-2k-1,
{u; v}u!v!(u-v)!.
an=ψn(q)ψn(mq)-mψn(mq)ψn(q)ζn(q)ψn(mq)-mζn(q)ψn(mq),
bn=ψn(q)ψn(mq)-mψn(mq)ψn(q)ζn(q)ψn(mq)-mζn(q)ψn(mq),
q=K0a=2πnλa,m=ndn0.
ψn(z)=zjn(z)=sinz-nπ2k=0[π/2]Λn,k(z)+cosz-nπ2k=0[(n-1)/2]Ωn,k(z),
ζn(z)=zhn(z)=ψn(z)+i sinn+12π×cosz+nπ2k=0[n/2]Λn,k(z)-sinz+nπ2k=0[(n-1)/2]Ωn,k(z),
Λn,k(z)
=(-1)k(2k+1)(2k+2)(2k+n)(n-2k)!(2z)-2k,
Ωn,k(z)
=(-1)k(2k+2)(2k+3)(2k+n+1)(n-2k-1)!(2z)-2k-1,
dΛ(z)dz=-2kzΛ(z),dΩ(z)dz=-2k+1zΩ(z).
Eθ(θ)=-in=1 2n+1n(n+1)[anτn(cos θ)+bnπn(cos θ)];
Eφ(θ)=in=1 2n+1n(n+1)[anπn(cos θ)+bnτn(cos θ)],
r=ga1.a2aN-1aN×10M,
R:b1,b2,b3 ,, bN+2,
b1=1g=+-1g=-,
b2=M+3,
b3=a1,
bj=aj-2(j=3,4 ,, N+2).
A1:c1,c2,c3 ,, c8,c9,
A2:d1,d2,d3 ,, d8,d9.
c3=c4==c2+(d2-c2)=0,
c3+(d2-c2)=c3,
cj+(d2-c2)=cj
forj+(d2-c2)9,
A1:-1,1,9,8,7,6,5,4,3,
A2:-1,5,3,4,5,4,3,2,1,
A1:-1,5,0,0,0,0,9,8,7.
A:-1,5,3,4,5,4,12,10,8.
A:-1,5,3,4,5,5,3,0,8.
A1:-1,4,7,7,7,7,7,7,7,
A2:1,5,2,2,3,3,8,8,5.
A:1,5,1,4,5,6,1,0,8.
A1:-1,9,9,8,7,6,5,4,3,
A2:1,5,7,0,1,1,0,6,3.
A=-1,12,6,9,2,4,5,0,6.
A1=1,5,7,3,1,1,0,6,3,
A2=1,8,9,8,7,6,5,4,3.
7,3,1,1,0,6,3 7,3,1,1,0,6,3
9,8,7,6,5,4,3(×7)6,9,1,3,5,8,0,1.
3,9,7,4,8,2,93,9,7,4,8,2,9
 9,8,7,6,5,4,3(×4)3,9,5,0,6,1,7,2.
0,2,4,2,1,1,80,2,4,2,1,1,8
9,8,7,6,5,4,3(×0)0,0,0,0,0,0,0.
A:1,-1,7,4,0,2,4,5,1.

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