Abstract

The perturbation theory suggested by Shifrin is applied through the second order to the scattering of light from dielectric spheroids and finite cylinders. In the case of short dielectric cylinders, this technique provides an accurate prediction of the scattering pattern in its range of applicability, and this prediction is especially useful as no exact scattering solution exists. The validity of the perturbation theory is established by comparison with exact results for the spheroid, and excellent agreement is shown for ka(m − 1) ≈ 1, where k = 2π/λ, a is a representative target dimension, and m is the index of refraction. The results for the finite cylinder are refined from our previous work by a careful construction of the internal electrostatic solution. This allows the calculation of intensities for short cylinders. Comparisons are made between the spheroids and cylinders of equal volumes for aspect ratios ranging from ½ to 5, and significant differences are noted in some cases.

© 1984 Optical Society of America

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References

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  1. J. W. Strutt, Philos. Mag. 12, 81 (1881).
  2. G. Mie, Ann. Phys. Leipzig 25, 377 (1908).
    [CrossRef]
  3. J. R. Mentzner, Scattering and Diffraction of Radio Waves (Pergamon, Oxford, 1955).
  4. J. R. Wait, Electromagnetic Radiation by Small Particles (Wiley, New York, 1957).
  5. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  6. L. D. Cohen, R. D. Haracz, A. Cohen, C. Acquista, Appl. Opt. 22, 742 (1983).
    [CrossRef] [PubMed]
  7. K. S. Shifrin, Scattering of Light in a Turbid Medium (Moscow, 1951; NASA TTF-477, Washington, D.C., 1968).
  8. C. Acquista, Appl. Opt. 15, 2932 (1976).
    [CrossRef] [PubMed]
  9. S. Asano, G. Yamamoto, Appl. Opt. 14, 29 (1975).
    [PubMed]
  10. W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill, New York, 1968).
  11. See Ref. 8 and C. Acquista, “Shifrin’s Method Applied to Scattering by Tenuous Nonspherical Particles,” in Light Scattering by Irregularly Shaped Particles, D. Schuerman, Ed. (Plenum, New York, 1980).
    [CrossRef]
  12. J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964), pp. 68–77.

1983 (1)

1976 (1)

1975 (1)

1908 (1)

G. Mie, Ann. Phys. Leipzig 25, 377 (1908).
[CrossRef]

1881 (1)

J. W. Strutt, Philos. Mag. 12, 81 (1881).

Acquista, C.

L. D. Cohen, R. D. Haracz, A. Cohen, C. Acquista, Appl. Opt. 22, 742 (1983).
[CrossRef] [PubMed]

C. Acquista, Appl. Opt. 15, 2932 (1976).
[CrossRef] [PubMed]

See Ref. 8 and C. Acquista, “Shifrin’s Method Applied to Scattering by Tenuous Nonspherical Particles,” in Light Scattering by Irregularly Shaped Particles, D. Schuerman, Ed. (Plenum, New York, 1980).
[CrossRef]

Asano, S.

Cohen, A.

Cohen, L. D.

Haracz, R. D.

Kerker, M.

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

Mentzner, J. R.

J. R. Mentzner, Scattering and Diffraction of Radio Waves (Pergamon, Oxford, 1955).

Mie, G.

G. Mie, Ann. Phys. Leipzig 25, 377 (1908).
[CrossRef]

Shifrin, K. S.

K. S. Shifrin, Scattering of Light in a Turbid Medium (Moscow, 1951; NASA TTF-477, Washington, D.C., 1968).

Smythe, W. R.

W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill, New York, 1968).

Strutt, J. W.

J. W. Strutt, Philos. Mag. 12, 81 (1881).

Van Bladel, J.

J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964), pp. 68–77.

Wait, J. R.

J. R. Wait, Electromagnetic Radiation by Small Particles (Wiley, New York, 1957).

Yamamoto, G.

Ann. Phys. Leipzig (1)

G. Mie, Ann. Phys. Leipzig 25, 377 (1908).
[CrossRef]

Appl. Opt. (3)

Philos. Mag. (1)

J. W. Strutt, Philos. Mag. 12, 81 (1881).

Other (7)

K. S. Shifrin, Scattering of Light in a Turbid Medium (Moscow, 1951; NASA TTF-477, Washington, D.C., 1968).

W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill, New York, 1968).

See Ref. 8 and C. Acquista, “Shifrin’s Method Applied to Scattering by Tenuous Nonspherical Particles,” in Light Scattering by Irregularly Shaped Particles, D. Schuerman, Ed. (Plenum, New York, 1980).
[CrossRef]

J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964), pp. 68–77.

J. R. Mentzner, Scattering and Diffraction of Radio Waves (Pergamon, Oxford, 1955).

J. R. Wait, Electromagnetic Radiation by Small Particles (Wiley, New York, 1957).

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

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Figures (8)

Fig. 1
Fig. 1

Geometry of the scattering process. The axes x1,y1,z1 are the target axes with z1 the axis of symmetry. The axes x,y,z are the detector axes with z the direction toward the detector. The incident wave vector k has the spherical coordinates (k,β/2,ϕ1) in the target frame.

Fig. 2
Fig. 2

Scattering amplitudes I1 and I2 for light of wavelength λ = 2π μm incident along the axis of symmetry. The target has an index of refraction m = 1.33 and is an oblate spheroid of size parameter C = 1 (see text) and aspect ratio ½. The volume is 3.22 μm3. The solid lines represent the perturbative calculations through the second order. The open circles are the exact results of Ref. 9. The results for a cylinder of the same aspect ratio and volume are virtually the same as for the spheroid and, therefore, not shown.

Fig. 3
Fig. 3

Amplitudes I1 and I2 for light incident along the axis of symmetry. The target is a sphere of 1-μm radius. The solid lines represent the perturbative calculations through the second order, and the open circles represent the exact (Mie) theory. The dashed line represents perturbative results through the second order for a cylinder and aspect ratio 1. The volume of each target is 4.19 μm3.

Fig. 4
Fig. 4

Amplitude I1 for light incident along the axis of symmetry. The solid line represents the second-order perturbation theory for a spheroid of size parameter C = 3 and aspect ratio 5. The volume is 4.81 μm3. The open circles represent the exact results of Ref. 9. The dashed line represents the perturbative results for a cylinder of the same aspect ratio and volume.

Fig. 5
Fig. 5

Amplitude I2 for the same case as Fig. 4.

Fig. 6
Fig. 6

Geometry of the cylinder used to calculate the internal electrostatic field. Polar coordinates are used in evaluating the surface integrals (see text).

Fig. 7
Fig. 7

Polarization matrix elements for the finite dielectric cylinders: ×, aspect ratio of 5; ●, 1; ○, ½. The results are plotted along the axis of the cylinder. The top portion of the figure is for an ambient electric field directed parallel to the axis (TM), and the bottom portion is for an ambient field perpendicular to the axis (TE).

Fig. 8
Fig. 8

Amplitude I1 for light incident perpendicular to the axis of symmetry. The solid lines represent the perturbative spheroidal results and the dashed lines the cylindrical results. The top portion of the figure corresponds to targets of aspect ratio 2 and 1.61-μm3, while the bottom portion corresponds to targets of aspect ratio 5 and 4.81-μm3 volume.

Equations (26)

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E ( r ) = E 0 exp ( i k 0 r ) + × × d 3 r ( m 2 - 1 4 π ) × exp ( i k 0 r - r r - r E ( r ) - ( m 2 - 1 ) E ( r ) ,
E i ( r ) = A i j E 0 j exp ( i k 0 r ) ,
E i ( r ) = A ¯ i j E eff , j ( r ) ,
A ¯ = ( a TE 0 0 0 a TE 0 0 0 a TM )
E eff ( r ) = E 0 exp ( i k · r ) + n = 1 σ n E eff ( n ) ( r ) .
E eff , i ( r ) = A i j - 1 E j ( r ) = E i ( r ) + B i j E eff , j ( r ) ,
B = 1 - A .
E eff , i ( r ) = E 0 i exp ( i k 0 · r ) + ( i j + δ i j k 0 2 ) × d 3 r U ( r ) exp ( i k · r - r ) r - r α A ¯ j k E eff , k ( r ) + B i j E eff , j ( r ) U ( r ) ,
U ( r ) = { 1 , for r within the target , 0 , for r outside the target .
E eff ( 1 ) ( r ) = k 0 2 exp ( i k 0 r ) r u ( k 0 r ^ - k ^ 0 ) [ ( E 0 x 1 i ^ 1 + E 0 y 1 j ^ 1 ) a TE + E 0 z 1 k ^ 1 α TM ] ,
E eff ( 2 ) ( r ) = k 0 2 exp ( i k 0 r ) 2 π 2 r p d 3 p u ( p + k 0 r ^ ) u ( - p - k 0 ) p 2 - k 0 2 × { k 0 2 [ a TE 2 ( E 0 x 1 i ^ 1 + E 0 y 1 j ^ 1 ) + a TM 2 E 0 z 1 k ^ 1 ] - [ a TE 2 ( p · ( E 0 x 1 i ^ 1 + E 0 y 1 j ^ 1 ) + a TM 2 E o z 1 p · k ^ 1 ] p + p 2 - k 0 2 m 2 - 1 [ a TE ( 1 - a TE ) ( E 0 x 1 i ^ 1 + E 0 y 1 j ^ 1 ) + a TM ( 1 - a TM ) E 0 z 1 k ^ 1 ] } .
u ( x ) = d 3 r U ( r ) exp ( i x · r ) .
x r ^ x · r ^ - x ,
Θ = cos - 1 [ cos θ cos β / 2 - cos ϕ 1 sin θ sin β / 2 ] ,
E 0 = E 0 { [ ( cos θ cos β / 2 cos ϕ 1 - sin θ sin β / 2 ) sin ψ - cos θ sin ϕ 1 cos ψ ] i ^ + [ sin ϕ 1 sin ψ cos β / 2 + cos ϕ 1 cos ψ ] j ^ + [ sin θ sin ϕ 1 cos ψ - ( sin θ cos β / 2 cos ϕ 1 + cos θ sin β / 2 ) sin ψ ] k ^ } .
u ( p ) = V f [ a ( p 2 - 2 p ) ] , prolate , u ( p ) = V f [ a ( p 2 - 2 p ) ] , oblate ,
f ( x ) = 3 [ sin ( x ) - x cos ( x ) ] / x 3 .
a TM = 1 { m 2 - δ ( m 2 - 1 ) [ ( 1 - δ 2 ) coth - 1 δ + δ ] } , a TE = 2 { 2 + δ ( m 2 - 1 ) [ ( 1 - δ 2 ) coth - 1 δ + δ ] } .
a TM = 1 { m 2 - δ ( m 2 - 1 ) [ ( 1 + δ 2 ) cot - 1 δ - δ ] } ,
a TE = 2 { 2 - δ ( m 2 - 1 ) [ ( 1 + δ 2 ) cot - 1 δ - δ ] } ,
I 1 = k 0 2 r 2 E 0 2 E sc ψ = 0 2 ,             I 2 = k 0 2 r 2 E 0 2 E sc ψ = 90 ° 2 ,
E sc ( r ) = α E eff ( 1 ) ( r ) + α 2 E eff 2 ( r ) .
m 2 ϕ ( r 0 ) = ϕ 0 ( r 0 ) - m 2 - 1 4 π S ϕ ( r ) n | 1 r - r 0 | d S ,
m 2 E ( r 0 ) = E 0 + m 2 - 1 4 π S ϕ ( r ) 0 ( n | 1 r - r 0 | ) d S .
( m 2 a i - 1 ) E 0 i = m 2 - 1 4 π S ϕ ( r ) n [ ( r - r 0 ) i r - r 0 3 ] ,
ϕ ( r s ) = 2 m 2 + 1 ϕ 0 ( r s ) - m 2 - 1 m 2 + 1 1 2 π S ϕ ( r ) n | 1 r - r s | d S ,

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