Abstract

An algorithm that extends the possible size-parameter range for the calculation of plane-wave light scattering from infinite homogeneous circular cylinders by use of Mie-type analysis has been developed. This algorithm is based on the calculation of the ratios of Bessel functions instead of the Bessel functions or their logarithmic derivatives directly. This algorithm agrees with existing methods (when those methods converge). It also converges for large size parameters when other algorithms often do not.

© 2006 Optical Society of America

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  1. R. K. Mueller, M. Kaveh, and G. Wade, "Reconstructive tomography and applications to ultrasonics," Proc. IEEE 67, 567-587 (1979).
    [CrossRef]
  2. R. K. Mueller, M. Kaveh, and R. D. Iverson, in Acoustical Imaging, A. F. Metherell, ed. (Plenum, 1980), Vol. 8, p. 615.
    [CrossRef]
  3. E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
    [CrossRef]
  4. O. R. Halse, K. K. Stamnes, and A. J. Devaney, "Three-dimensional diffraction tomography by two-dimensional sectioning," Opt. Commun. 224, 185-195 (2003).
    [CrossRef]
  5. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  6. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
    [CrossRef]
  7. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  8. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  9. M. Kerker, D. Cooke, W. A. Farone, and R. T. Jacobson, "Electromagnetic scattering from an infinite circular cylinder at oblique incidence. I. Radiance functions for m=1.46," J. Opt. Soc. Am. 56, 487-491 (1966).
    [CrossRef]
  10. L. Infeld, "The influence of the width of the gap upon the theory of antennas," Q. Appl. Math. 5, 113-132 (1947).
  11. J. V. Dave, "Scattering of electromagnetic radiation by a large, absorbing sphere," IBM J. Res. Dev. 13, 302-313 (1969).
    [CrossRef]
  12. W. J. Lentz, "Generating Bessel functions in Mie scattering calculations using continued fractions," Appl. Opt. 15, 668-671 (1976).
    [CrossRef] [PubMed]
  13. W. J. Wiscombe, "Improved Mie scattering algorithms," Appl. Opt. 19, 1505-1509 (1980).
    [CrossRef] [PubMed]
  14. H. Du, "Mie-scattering calculation," Appl. Opt. 43, 1951-1956 (2004).
    [CrossRef] [PubMed]
  15. D. Mackowski, http://atol.ucsd.edu/∼pflatau/scatlib/scatterlib.htm.
  16. P. W. Barber, J. F. Owen, and R. K. Chang, "Resonant scattering for characterization of axisymmetric dielectric objects," IEEE Trans. Antennas Propag. 30, 168-172 (1982).
    [CrossRef]
  17. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).
  18. R. T. Wang and H. C. van de Hulst, "Rainbows: Mie computations and the Airy approximation," Appl. Opt. 30, 106-117 (1991).
    [CrossRef] [PubMed]
  19. V. E. Cachorro and L. L. Salcedo, "New improvements for Mie scattering calculations," J. Electromagn. Waves Appl. 5, 913-926 (1991).
    [CrossRef]
  20. G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1966).

2004

2003

O. R. Halse, K. K. Stamnes, and A. J. Devaney, "Three-dimensional diffraction tomography by two-dimensional sectioning," Opt. Commun. 224, 185-195 (2003).
[CrossRef]

1991

R. T. Wang and H. C. van de Hulst, "Rainbows: Mie computations and the Airy approximation," Appl. Opt. 30, 106-117 (1991).
[CrossRef] [PubMed]

V. E. Cachorro and L. L. Salcedo, "New improvements for Mie scattering calculations," J. Electromagn. Waves Appl. 5, 913-926 (1991).
[CrossRef]

1982

P. W. Barber, J. F. Owen, and R. K. Chang, "Resonant scattering for characterization of axisymmetric dielectric objects," IEEE Trans. Antennas Propag. 30, 168-172 (1982).
[CrossRef]

1980

1979

R. K. Mueller, M. Kaveh, and G. Wade, "Reconstructive tomography and applications to ultrasonics," Proc. IEEE 67, 567-587 (1979).
[CrossRef]

1976

1969

J. V. Dave, "Scattering of electromagnetic radiation by a large, absorbing sphere," IBM J. Res. Dev. 13, 302-313 (1969).
[CrossRef]

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

1966

1947

L. Infeld, "The influence of the width of the gap upon the theory of antennas," Q. Appl. Math. 5, 113-132 (1947).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Barber, P. W.

P. W. Barber, J. F. Owen, and R. K. Chang, "Resonant scattering for characterization of axisymmetric dielectric objects," IEEE Trans. Antennas Propag. 30, 168-172 (1982).
[CrossRef]

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

Bohren, C. F.

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

Cachorro, V. E.

V. E. Cachorro and L. L. Salcedo, "New improvements for Mie scattering calculations," J. Electromagn. Waves Appl. 5, 913-926 (1991).
[CrossRef]

Chang, R. K.

P. W. Barber, J. F. Owen, and R. K. Chang, "Resonant scattering for characterization of axisymmetric dielectric objects," IEEE Trans. Antennas Propag. 30, 168-172 (1982).
[CrossRef]

Cooke, D.

Dave, J. V.

J. V. Dave, "Scattering of electromagnetic radiation by a large, absorbing sphere," IBM J. Res. Dev. 13, 302-313 (1969).
[CrossRef]

Devaney, A. J.

O. R. Halse, K. K. Stamnes, and A. J. Devaney, "Three-dimensional diffraction tomography by two-dimensional sectioning," Opt. Commun. 224, 185-195 (2003).
[CrossRef]

Du, H.

Farone, W. A.

Halse, O. R.

O. R. Halse, K. K. Stamnes, and A. J. Devaney, "Three-dimensional diffraction tomography by two-dimensional sectioning," Opt. Commun. 224, 185-195 (2003).
[CrossRef]

Hill, S. C.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

Huffman, D. R.

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

Infeld, L.

L. Infeld, "The influence of the width of the gap upon the theory of antennas," Q. Appl. Math. 5, 113-132 (1947).

Iverson, R. D.

R. K. Mueller, M. Kaveh, and R. D. Iverson, in Acoustical Imaging, A. F. Metherell, ed. (Plenum, 1980), Vol. 8, p. 615.
[CrossRef]

Jacobson, R. T.

Kaveh, M.

R. K. Mueller, M. Kaveh, and G. Wade, "Reconstructive tomography and applications to ultrasonics," Proc. IEEE 67, 567-587 (1979).
[CrossRef]

R. K. Mueller, M. Kaveh, and R. D. Iverson, in Acoustical Imaging, A. F. Metherell, ed. (Plenum, 1980), Vol. 8, p. 615.
[CrossRef]

Kerker, M.

Lentz, W. J.

Mackowski, D.

D. Mackowski, http://atol.ucsd.edu/∼pflatau/scatlib/scatterlib.htm.

Mueller, R. K.

R. K. Mueller, M. Kaveh, and G. Wade, "Reconstructive tomography and applications to ultrasonics," Proc. IEEE 67, 567-587 (1979).
[CrossRef]

R. K. Mueller, M. Kaveh, and R. D. Iverson, in Acoustical Imaging, A. F. Metherell, ed. (Plenum, 1980), Vol. 8, p. 615.
[CrossRef]

Owen, J. F.

P. W. Barber, J. F. Owen, and R. K. Chang, "Resonant scattering for characterization of axisymmetric dielectric objects," IEEE Trans. Antennas Propag. 30, 168-172 (1982).
[CrossRef]

Salcedo, L. L.

V. E. Cachorro and L. L. Salcedo, "New improvements for Mie scattering calculations," J. Electromagn. Waves Appl. 5, 913-926 (1991).
[CrossRef]

Stamnes, K. K.

O. R. Halse, K. K. Stamnes, and A. J. Devaney, "Three-dimensional diffraction tomography by two-dimensional sectioning," Opt. Commun. 224, 185-195 (2003).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

van de Hulst, H. C.

Wade, G.

R. K. Mueller, M. Kaveh, and G. Wade, "Reconstructive tomography and applications to ultrasonics," Proc. IEEE 67, 567-587 (1979).
[CrossRef]

Wang, R. T.

Watson, G. N.

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1966).

Wiscombe, W. J.

Wolf, E.

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Appl. Opt.

IBM J. Res. Dev.

J. V. Dave, "Scattering of electromagnetic radiation by a large, absorbing sphere," IBM J. Res. Dev. 13, 302-313 (1969).
[CrossRef]

IEEE Trans. Antennas Propag.

P. W. Barber, J. F. Owen, and R. K. Chang, "Resonant scattering for characterization of axisymmetric dielectric objects," IEEE Trans. Antennas Propag. 30, 168-172 (1982).
[CrossRef]

J. Electromagn. Waves Appl.

V. E. Cachorro and L. L. Salcedo, "New improvements for Mie scattering calculations," J. Electromagn. Waves Appl. 5, 913-926 (1991).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

O. R. Halse, K. K. Stamnes, and A. J. Devaney, "Three-dimensional diffraction tomography by two-dimensional sectioning," Opt. Commun. 224, 185-195 (2003).
[CrossRef]

Proc. IEEE

R. K. Mueller, M. Kaveh, and G. Wade, "Reconstructive tomography and applications to ultrasonics," Proc. IEEE 67, 567-587 (1979).
[CrossRef]

Q. Appl. Math.

L. Infeld, "The influence of the width of the gap upon the theory of antennas," Q. Appl. Math. 5, 113-132 (1947).

Other

R. K. Mueller, M. Kaveh, and R. D. Iverson, in Acoustical Imaging, A. F. Metherell, ed. (Plenum, 1980), Vol. 8, p. 615.
[CrossRef]

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

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

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

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

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1966).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

D. Mackowski, http://atol.ucsd.edu/∼pflatau/scatlib/scatterlib.htm.

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Figures (6)

Fig. 1
Fig. 1

Geometry of an infinite cylinder at oblique illumination by a plane wave.

Fig. 2
Fig. 2

Scattering efficiency Q sca , I for incident light with the electric field is parallel to the e ^ x e ^ z plane as a function of size parameter calculated with the algorithms (a) described in this paper, (b) of Bohren and Huffman, (c) of Mackowski, and (d) of Barber and Hill. The calculations used 1000 points.

Fig. 3
Fig. 3

Comparison of n c = n c ( 1 ) and n c = n c ( 2 ) . Scattering efficiency Q sca , I as a function of size parameter obtained with the algorithm described in this paper. The calculations used 1000 points.

Fig. 4
Fig. 4

Scattering efficiency Q sca , I as a function of size parameter calculated with the algorithms (a) described in this paper and (b) of Bohren and Huffman. The calculations used 10,000 points. For the results shown in (b), different summation cutoffs n c were used. The curves for n c = n c ( 1 ) and n c = n c ( 2 ) are identical. The peaks are due to morphology-dependent resonances of the cylinder.

Fig. 5
Fig. 5

Scattering efficiency Q sca , I as a function of size parameter calculated with the algorithms (i) described in this paper with n c = n c ( 2 ) and (ii) of Bohren and Huffman with n c = n c ( 1 ) . The calculations used 10,000 points.

Fig. 6
Fig. 6

Scattering efficiency Q sca , I as a function of size parameter calculated with the algorithms (a) described in this paper and (b) of Ref. [15] for an incident angle of ζ = 30 ° .

Tables (1)

Tables Icon

Table 1 Logarithmic Derivative D 0(1500) Calculated by the Algorithm Described by Bohren and Huffman a and Actual Value for Summation Cutoffs n c

Equations (31)

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

E s = - n = - E n [ b n I N n ( 3 ) + i a n I M n ( 3 ) ] ,
Q sca , I = 2 x [ b 0 I 2 + 2 n = 1 ( b n I 2 + a n I 2 ) ] ,
Q ext , I = 2 x Re ( b 0 I 2 n = 1 b n I ) ,
E s = - n = - E n [ b n II N n ( 3 ) + i a n II M n ( 3 ) ] .
Q sca , II = 2 x [ a 0 II 2 + 2 n = 1 ( a n II 2 + b n II 2 ) ] ,
Q ext , II = 2 x Re ( a 0 II 2 n = 1 a n II ) ,
n c ( 1 ) = Round ( x + 4.05 x 1 / 3 + 2 ) ,
n c ( 2 ) = Round { max [ ( x + 4.05 x 1 / 3 + 2 ) , m x ] } ,
J n ( z ) = 2 / ( π z ) { cos ( z - n π / 2 - π / 4 ) + exp [ Im ( z ) ] O ( z - 1 ) } ,
r n ( z ) = J n - 1 ( z ) J n ( z ) ,
s n ( z ) = Y n - 1 ( z ) Y n ( z ) ,
t n ( z ) = Y n ( z ) J n ( z ) .
r n ( z ) + r n + 1 - 1 ( z ) = 2 n z ,
s n ( z ) + s n + 1 - 1 ( z ) = 2 n z .
ln [ t n ( η ) ] = ln [ t n - 1 ( η ) ] + ln [ | r n ( η ) s n ( η ) | ] ,
sgn [ t n ( η ) ] = sgn [ t n - 1 ( η ) ] sgn [ r n ( η ) s n ( η ) ] ,
a n I = P n C n V n - B n D n W n V n + i D n 2 ,
a n II = - P n A n V n - i C n D n W n V n + i D n 2 ,
a n II = - P n A n V n - i C n D n W n V n + i D n 2 ,
b n II = - i P n C n W n + A n D n W n V n + i D n 2 ,
A n = i ξ { ξ [ r n ( η ) - n / η + η n ξ 2 ] - η r n ( ξ ) } ,
B n = ξ { m 2 ξ [ r n ( η ) - n / η + η n ( m ξ ) 2 ] - η r n ( ξ ) } ,
C n = n cos ( ζ ) η ( ξ 2 η 2 - 1 ) ,
D n = C n ,
V n = ξ ( m 2 ξ [ r n ( η ) - n / η + η n ( m ξ ) 2 ] - η s n ( ξ ) { [ r n ( ξ ) / s n ( ξ ) ] + i t n ( ξ ) } [ 1 + i t n ( ξ ) ] ) ,
W n = i ξ ( η s n ( ξ ) { [ r n ( ξ ) / s n ( ξ ) ] + i t n ( ξ ) } [ 1 + i t n ( ξ ) ] - ξ [ r n ( η ) - n / η + η n ξ 2 ] ) ,
P n = 1 [ 1 + i t n ( ξ ) ] .
l n ( z ) = ( Im ( z ) - ln ( 2 ) - n Re { ln ( z n ) + 1 - ( z n ) 2 - ln ( 1 + 1 - ( z n ) 2 ) } ) ln ( 10 ) .
V n ξ { m 2 ξ [ r n ( η ) - n / η + η n ( m ξ ) 2 ] - η s n ( ξ ) } ,
W n i ξ { η s n ( ξ ) - ξ [ r n ( η ) - n / η + η n ξ 2 ] } ,
n c ( 3 ) = Round { max [ ( x + 4.05 x 1 / 3 + 2 ) , m x , m ( x + 4.05 x 1 / 3 + 2 ) ] } .

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