Abstract

An iterative approach to the scattering of light from a finite dielectric cylinder first developed by Shifrin and extended by Acquista is applied to cases where the phase shift is <2, and the cylinder is arbitrarily oriented. It is found that the first 2 orders of the iteration converge to within 1% when the aspect ratio (length/diameter) of the cylinder is as small as 20. The results are compared to the exact theory for infinite cylinders, and the effects of finite size are calculated and discussed.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Strutt (Lord Rayleigh), 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); A. Blank, Trans. Chalmers Univ. Technol. Gothenburg 168 (1955); R. BurbergZ. Naturforsch. Teil A 11, 800 (1956); W. A. Farone, C. W. Querfeld, Rep. ERDA-281, U.S. Army Electron Research and Development Activity (1965). The procedure to be followed is found in H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1956).
  5. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  6. E. Weber, Electromagnetic Theory (Dover, New York, 1965), see Chap. 8.
  7. R. Gans, Ann. Phys. Leipzig 76, 29 (1925); I. Rocard, Rev. Opt. Theor. Instrum. 9, 97 (1930).
    [CrossRef]
  8. K. S. Shifrin, Scattering of Light in a Turbid Medium (Moscow, 1951), NASA TT F-477, Washington, D.C., 1968).
  9. C. Acquista, Appl. Opt. 15, 2932 (1976).
    [CrossRef] [PubMed]
  10. A. Cohen, Opt. Lett. 5, 150 (1980).
    [CrossRef] [PubMed]
  11. A. Cohen, C. Acquista, J. Opt. Soc. Am. 72, 531 (1982).
    [CrossRef]
  12. A numerical approach can be used such as that applied to a dielectric cube in T. W. Edwards, J. Van Bladel, Appl. Sci. Res. Sect. B 9, 151 (1961).
    [CrossRef]
  13. A. D. Yaghjian, Proc. IEEE 68, 248 (1980). This paper provides a thorough review of the integral equation approach to scattering problems and describes the matrix inversion techniques.
    [CrossRef]
  14. J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964), pp. 68–77.
  15. P. W. Barber, C. Yeh, Appl. Opt. 14, 2864 (1975).
    [CrossRef] [PubMed]

1982

1980

A. D. Yaghjian, Proc. IEEE 68, 248 (1980). This paper provides a thorough review of the integral equation approach to scattering problems and describes the matrix inversion techniques.
[CrossRef]

A. Cohen, Opt. Lett. 5, 150 (1980).
[CrossRef] [PubMed]

1976

1975

1961

A numerical approach can be used such as that applied to a dielectric cube in T. W. Edwards, J. Van Bladel, Appl. Sci. Res. Sect. B 9, 151 (1961).
[CrossRef]

1925

R. Gans, Ann. Phys. Leipzig 76, 29 (1925); I. Rocard, Rev. Opt. Theor. Instrum. 9, 97 (1930).
[CrossRef]

1908

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

1881

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

Acquista, C.

Barber, P. W.

Cohen, A.

Edwards, T. W.

A numerical approach can be used such as that applied to a dielectric cube in T. W. Edwards, J. Van Bladel, Appl. Sci. Res. Sect. B 9, 151 (1961).
[CrossRef]

Gans, R.

R. Gans, Ann. Phys. Leipzig 76, 29 (1925); I. Rocard, Rev. Opt. Theor. Instrum. 9, 97 (1930).
[CrossRef]

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 TT F-477, Washington, D.C., 1968).

Strutt, J. W.

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

Van Bladel, J.

A numerical approach can be used such as that applied to a dielectric cube in T. W. Edwards, J. Van Bladel, Appl. Sci. Res. Sect. B 9, 151 (1961).
[CrossRef]

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); A. Blank, Trans. Chalmers Univ. Technol. Gothenburg 168 (1955); R. BurbergZ. Naturforsch. Teil A 11, 800 (1956); W. A. Farone, C. W. Querfeld, Rep. ERDA-281, U.S. Army Electron Research and Development Activity (1965). The procedure to be followed is found in H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1956).

Weber, E.

E. Weber, Electromagnetic Theory (Dover, New York, 1965), see Chap. 8.

Yaghjian, A. D.

A. D. Yaghjian, Proc. IEEE 68, 248 (1980). This paper provides a thorough review of the integral equation approach to scattering problems and describes the matrix inversion techniques.
[CrossRef]

Yeh, C.

Ann. Phys. Leipzig

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

R. Gans, Ann. Phys. Leipzig 76, 29 (1925); I. Rocard, Rev. Opt. Theor. Instrum. 9, 97 (1930).
[CrossRef]

Appl. Opt.

Appl. Sci. Res. Sect. B

A numerical approach can be used such as that applied to a dielectric cube in T. W. Edwards, J. Van Bladel, Appl. Sci. Res. Sect. B 9, 151 (1961).
[CrossRef]

J. Opt. Soc. Am.

Opt. Lett.

Philos. Mag.

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

Proc. IEEE

A. D. Yaghjian, Proc. IEEE 68, 248 (1980). This paper provides a thorough review of the integral equation approach to scattering problems and describes the matrix inversion techniques.
[CrossRef]

Other

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

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

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

J. R. Wait, Electromagnetic Radiation by Small Particles (Wiley, New York, 1957); A. Blank, Trans. Chalmers Univ. Technol. Gothenburg 168 (1955); R. BurbergZ. Naturforsch. Teil A 11, 800 (1956); W. A. Farone, C. W. Querfeld, Rep. ERDA-281, U.S. Army Electron Research and Development Activity (1965). The procedure to be followed is found in H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1956).

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

E. Weber, Electromagnetic Theory (Dover, New York, 1965), see Chap. 8.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

General geometry for the scattering of light from a cylinder. The coordinates x1, y1, z1 refer to the cylinder reference frame, and x, y, z refer to the detector frame.

Fig. 2
Fig. 2

Scattering geometry for an incident TE wave. The unit vector k ˆ 0 is in the direction of incidence, which is shown in the x-z plane. The incident field direction is the vector E0 shown in the TE configuration.

Fig. 3
Fig. 3

Intensity I for the scattering of a plane wave from an infinite cylinder oriented at an angle β/2 = 30° with the propagation vector k0. The abscissa is the azimuthal angle ϕ shown in Fig. 1. The solid curve refers to the present work while the crosses × are the exact results of Ref. 5. The results are normalized at ϕ = 0, m = 1.5.

Fig. 4
Fig. 4

TE scattering from a cylinder of 0.5-μm radius and 100-μm length. The geometry is as shown in Fig. 2. m = 1.5. Aspect ratio = 100.

Fig. 5
Fig. 5

TE scattering from a cylinder of 0.05-μm radius and 10-μm length. The geometry is as shown in Fig. 2. m = 1.5. Aspect ratio = 100.

Fig. 6
Fig. 6

TE scattering from a cylinder of 0.5μm radius and 20-μm length. The geometry is as shown in Fig. 2. m = 1.5. Aspect ratio = 20.

Fig. 7
Fig. 7

TE scattering from a cylinder of 0.05-μm radius and 2.0-μm length. The geometry is as shown in Fig. 2. m = 1.5. Aspect ratio = 20.

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

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 ) ,
× [ exp ( i k 0 | r r | ) | r r | ] = grad div [ exp ( i k 0 | r r | ) | r r | ] + k 0 2 exp ( i k 0 | r r | ) | r r | + 4 π δ ( r r ) .
E ( r ) = E 0 exp ( i k 0 · r ) + ( grand div + k 0 2 ) × d 3 r exp ( i k 0 | r r | ) | r r | ( m 2 1 4 π ) E ( r ) .
E i ( r ) = A i j E e ff j ( r ) ,
A = ( a TE 0 0 0 a TE 0 0 0 a TM ) ,
E eff ( r ) = E 0 exp ( i k 0 · r ) + n = 1 ( m 2 1 4 π ) n E e ff ( n ) ( r ) .
E i ( r ) = E o i exp ( i k 0 · r ) + n = 1 ( m 2 1 4 π ) E eff , i ( n ) ( r ) = E o i exp ( i k 0 · r ) + j , k ( grad div + k 0 2 ) ij d 3 r exp ( i k 0 · | r r | ) | r r | × ( m 2 1 4 π ) A j k [ E o k exp ( i k 0 · r ) + n = 1 ( m 2 1 4 π ) n E eff , k ( n ) ( r ) ] .
( m 2 1 4 π ) E eff i ( 1 ) ( r ) = j k ( x i x j + k 0 2 δ i j ) × d 3 r exp ( i k 0 | r r | ) | r r | ( m 2 1 4 π ) A j k E o k × exp ( i k 0 · r )
( m 2 1 4 π ) 2 E eff i ( 2 ) ( r ) = j k ( x i x j + k 0 2 δ i j ) × d 3 r exp ( i k 0 | r r | ) | r r | ( m 2 1 4 π ) 2 A j k E eff k ( 1 ) ( r ) .
E eff ( 1 ) ( r ) = d 3 r ( grad div + k 0 2 ) exp ( i k 0 | r r | ) | r r | exp ( i k 0 · r ) × [ ( E 0 x 1 i ˆ 1 + E 0 y 1 j ˆ 1 ) a TE + E 0 z 1 k ˆ 1 a TM ] = exp ( i k 0 r ) r k 0 2 u ( k 0 r ˆ k ˆ 0 ) [ E 0 x 1 i ˆ 1 + E 0 y 1 j ˆ 1 ] a TE + E 0 z 1 k ˆ 1 a TM ] ,
U ( r ) = { 1 , inside the cylinder , 0 , outside the cylinder ,
u ( x ) = d 3 r U ( r ) exp ( i x · r ) = 2 π a 2 h sin ( x h / 2 ) J 1 ( x a ) ( x h / 2 ) ( x a ) ,
k ˆ 0 = ( cos ϕ 1 cos θ sin β / 2 + sin θ cos β / 2 ) i ˆ + sin ϕ 1 sin β / 2 j ˆ ( cos ϕ 1 sin θ sin β / 2 cos θ cos β / 2 ) k ˆ , x = k 0 [ ( sin θ + cos ϕ 1 sin β / 2 ) 2 + sin 2 ϕ 1 sin 2 β / 2 ] 1 / 2 , x = k 0 | [ cos θ cos β / 2 ] | .
[ a TE ( E 0 x 1 i ˆ 1 + E 0 y 1 j ˆ 1 ) + a TM E 0 z 1 k ˆ 1 ] / E 0 = [ ( cos θ cos ϕ 1 sin ψ cos β / 2 cos θ sin ϕ 1 cos ψ ) a TE sin θ sin β / 2 sin ψ a TM ] i ˆ + [ ( sin ϕ 1 sin ψ cos β / 2 + cos ϕ 1 cos ψ ) a TE ] j ˆ ,
E 0 = E 0 { [ ( cos 2 β / 2 cos ϕ 1 sin 2 β / 2 ) sin ψ cos β / 2 sin ϕ 1 cos ψ ] i ˆ + ( sin ϕ 1 sin ψ cos β / 2 + cos ϕ 1 cos ψ ) j ˆ + [ sin β / 2 sin ϕ 1 cos ψ ( sin β / 2 cos β / 2 cos ϕ 1 + cos β / 2 sin β / 2 ) sin ψ ] k ˆ } ,
E ( 2 ) ( r ) = d 3 r ( grad div + k 0 2 ) r exp ( i k 0 | r r | ) | r r | × d 3 r ( grad div + k 0 2 ) r exp ( i k 0 | r r | ) | r r | × exp ( i k 0 · r ) F ,
F i = j , k ( A i j A j k E 0 k )
F = a TE 2 ( E 0 x 1 i ˆ 1 + E 0 y 1 j ˆ 1 ) + a TM 2 E 0 z 1 k ˆ 1 = a TE 2 F TE + a TM 2 F TM .
E ( 2 ) ( r ) = 2 ( 2 π ) 2 d 3 r u ( p + k 0 r ˆ ) p 2 k 0 2 u ( p k 0 ) × [ ( p 2 + k 0 2 2 ) F ( p · F ) P ] ,
a TE = 1 ( m 2 1 ) g TE + 1 , a TM = 1 ( m 2 1 ) g TM + 1 ,
g TE = s ( s 2 1 ) 2 [ s s 2 1 1 2 ln ( s + 1 s 1 ) ] , g TM = ( s 2 1 ) [ 1 2 s ln ( s + 1 s 1 ) 1 ] , s = ( 1 4 a 2 / h 2 ) 1 / 2 .
a TE 2 m 2 + 1 , a TM 1 .
a TE = 2 0.9797 m 2 + 1.020 , a TM = 1 0.02029 m 2 + 0.9797 ,
E sc ( r ) E ( 1 ) ( r ) = exp ( i k 0 r ) r k 0 2 u ( k o r ˆ k 0 ) ( m 2 1 4 π ) ( A 1 ) ,
A 1 = 2 m 2 + 1 ( E 0 x 1 i ˆ 1 + E 0 y 1 j ˆ 1 ) + E 0 z 1 k ˆ 1 .
( A 1 ) = E 0 [ ( cos θ cos ϕ 1 sin ψ cos β / 2 cos θ sin ϕ 1 cos ψ ) 2 m 2 + 1 sin θ sin β / 2 sin ψ ] i ˆ + E 0 ( sin ϕ 1 sin ψ cos β / 2 + cos ϕ 1 cos ψ ) 2 m 2 + 1 j ˆ .
I = k 0 2 r 2 E 0 2 | E sc · i ˆ | 2 ,
I 22 = K | ( A 1 ) · j ˆ | ψ = 0 2 = K E 0 2 cos 2 ϕ 1 ,
I 21 = I 12 = K | ( A 1 ) · i ˆ | ψ = 0 2 = K E 0 2 cos 2 θ sin 2 ϕ 1 ,
I 11 = K | ( A 1 ) · i ˆ | ψ = 90 2 = K E 0 2 ( cos θ cos ϕ 1 cos β / 2 2 m 2 1 sin θ sin β / 2 ) 2 ,
K = ( m 2 1 ) 2 4 π 2 E 0 2 ( π a 2 h ) 2 sin 2 ( x h / 2 ) ( x h / 2 ) 2 J 1 2 ( x a ) ( x a ) 2 ,
I 2 = K | ( A 1 ) · ( A 1 ) | = K E 0 2 ( cos 2 θ sin 2 ϕ 1 + cos 2 ϕ 1 ) .
I 11 ( ϕ ) | b 0 + 2 n = 1 ( 1 ) n b n cos n ϕ | 2 ,
b n = n 2 p 2 α 2 J l H l J j 2 ( l 2 j 2 ) 2 + j 2 l 2 ( j H l 1 J j + l H l J l 1 ) ( m 2 l J l J j 1 j J l 1 J j ) n 2 p 2 α 2 H l 2 J j 2 ( l 2 j 2 ) + j 2 l 2 ( j H l 1 J j + l H l J l 1 ) ( m 2 l H l J l 1 j H l 1 J j )
m 2 l 2 j 2 = p 2 ( l m 2 ) = k 0 2 cos 2 θ ( l m 2 ) , l 2 = k 0 2 p 2 , j 2 = m 2 k 0 2 p 2 .
I 11 | b 0 2 b 1 cos ϕ | 2 | sin 2 θ 2 cos 2 θ cos ϕ / ( m 2 1 ) | 2 ,
E sc ( r ) E ( 1 ) ( r ) + E ( 2 ) ( r ) ,
I 2 = k 0 2 r 2 E 0 2 2 [ Re E sc ( r ) · Re E sc ( r ) ] ¯ = k 0 2 r 2 E 0 2 { | E ( 1 ) ( r ) | 2 + 2 Re ( E ( 1 ) ( r ) · E ( 2 ) ( r ) * ] } ,

Metrics