Abstract

An efficient numerical procedure for computing the scattering coefficients of a multilayered sphere is discussed. The stability of the numerical scheme allows us to extend the feasible range of computations, both in size parameter and in number of layers for a given size, by several orders of magnitude with respect to previously published algorithms. Exemplifying results, such as scattering diagrams and cross-sectional curves, including the case of Gaussian beam illumination, are provided. Particular attention is paid to scattering at the rainbow angle for which approaches based on geometrical optics might fail to provide accurate enough results.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
    [CrossRef]
  2. G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
    [CrossRef]
  3. E. M. Khaled, S. C. Hill, P. W. Barber, “Light scattering by a coated sphere illuminated by a Gaussian beam,” Appl. Opt. 33, 3308–3314 (1994).
    [CrossRef] [PubMed]
  4. F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
    [CrossRef] [PubMed]
  5. Z. S. Wu, X. Q. Fu, “Scattering of fundamental Gaussian beams from a multilayered sphere,” Acta Electron. Sinica (China) 23, 32–36 (1995).
  6. G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase Doppler systems: study of a near-forward and a near-backward geometry,” Part. Part. Syst. Charact. 11, 133–145 (1994).
    [CrossRef]
  7. G. Gouesbet, G. Gréhan, “Last progress in generalized Lorenz–Mie theory with applications in multiphase flows,” in Advances in Multiphase Flow, A. Serizawa, T. Fukano, J. Bataille, eds., (Elsevier, Amsterdam, 1995)pp. 703–710.
  8. G. Martinot–Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams: II GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
    [CrossRef]
  9. G. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  10. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  11. O. B. Toon, T. P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. 20, 3657–3660 (1981).
    [CrossRef] [PubMed]
  12. A. L. Alden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  13. R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960–1967 (1985).
    [CrossRef] [PubMed]
  14. D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, “Internal absorption cross sections in a astratified sphere,” Appl. Opt. 29, 1551–1559 (1990).
    [CrossRef] [PubMed]
  15. Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
    [CrossRef]
  16. L. Kai, P. Massoli, “Scattering of electromagnetic plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
    [CrossRef] [PubMed]
  17. N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in micrometer range,” J. Laser Appl. 2, 37–42 (1990).
    [CrossRef]
  18. N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in Sixth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 20–23, ed. (University of Lisbon, Lisbon, Portugal, 1992), p. 15.5.
  19. M. Schneider, E. D. Hirleman, H. Saleheen, D. Q. Chowdhury, S. C. Hill, “Rainbows and radially-inhomogeneous droplets,” in Third International Congress on Optical Particle Sizing, Yokohama, Japan (Keio University, Yokohama, Japan, 1993), pp. 323–326.
  20. J. P. A. J. van Beeck, M. L. Riethmuller, “Nonintrusive measurements of temperature and size of raindrops,” Appl. Opt. 34, 1633–1639 (1995).
    [CrossRef] [PubMed]
  21. S. V. Sankar, K. M. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler/rainbow refractometer system for simultaneous measurement of droplet size, velocity, and refractive index,” in Third International Congress on Optical Particle Sizing, Yokohama, Japan (Keio University, Yokohama, Japan, 1993), pp. 275–284.
  22. J. P. A. J. van Beeck, M. L. Riethmuller, “Détermination non intrusive de la dimension et de la température des gouttes dans une pulvérisation,” in 4éme Congrés Francophone de V́locimétrie Laser, Poitiers, 26–29 Septembre 1994, (Université de Poitiers, Poitiers, France, 1994), article 2-2.
  23. K. Anders, N. Roth, A. Frohn, “Theoretical and experimental studies of the influence of internal temperature gradients on rainbow refractometry,” in PARTEC 95, Nuremberg, Germany (Nürnbergmesse GmbH, Nüremberg, Germany, 1995), 419–428.
  24. K. Anders, N. Roth, A. Frohn, “Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry,” Part. Part. Syst. Charact. 13, 125–129 (1996).
    [CrossRef]
  25. The sources of the fortran program are available from the authors, in particular from Z. S. Wu.
  26. L. Kai, P. Massoli, A. D’Alessio, “Some far-field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
    [CrossRef]
  27. K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of Generalized Lorenz–Mie Theory: faster algorithm for computations of beam shape coefficients gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
    [CrossRef]
  28. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere II. theory of the rainbow and the glory,” J. Math. Phys. 10, 125 (1969).
    [CrossRef]
  29. V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976 (1974).
    [CrossRef]
  30. J. A. Lock, “Theory of the observations made of high-order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
    [CrossRef] [PubMed]
  31. H. C. van de Hulst, Light Scattering by Small Particle (Dover, New York, 1981).
  32. P. L. Marston, “Rainbow phenomena and the detection of nonsphericity in drops,” Appl. Opt. 19, 680–685 (1980).
    [CrossRef] [PubMed]

1996 (1)

K. Anders, N. Roth, A. Frohn, “Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry,” Part. Part. Syst. Charact. 13, 125–129 (1996).
[CrossRef]

1995 (4)

Z. S. Wu, X. Q. Fu, “Scattering of fundamental Gaussian beams from a multilayered sphere,” Acta Electron. Sinica (China) 23, 32–36 (1995).

G. Martinot–Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams: II GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
[CrossRef] [PubMed]

J. P. A. J. van Beeck, M. L. Riethmuller, “Nonintrusive measurements of temperature and size of raindrops,” Appl. Opt. 34, 1633–1639 (1995).
[CrossRef] [PubMed]

1994 (4)

L. Kai, P. Massoli, “Scattering of electromagnetic plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
[CrossRef] [PubMed]

E. M. Khaled, S. C. Hill, P. W. Barber, “Light scattering by a coated sphere illuminated by a Gaussian beam,” Appl. Opt. 33, 3308–3314 (1994).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase Doppler systems: study of a near-forward and a near-backward geometry,” Part. Part. Syst. Charact. 11, 133–145 (1994).
[CrossRef]

L. Kai, P. Massoli, A. D’Alessio, “Some far-field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
[CrossRef]

1992 (1)

K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of Generalized Lorenz–Mie Theory: faster algorithm for computations of beam shape coefficients gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

1991 (1)

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

1990 (3)

1988 (1)

1987 (1)

1985 (1)

1981 (1)

1980 (1)

1974 (1)

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976 (1974).
[CrossRef]

1969 (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere II. theory of the rainbow and the glory,” J. Math. Phys. 10, 125 (1969).
[CrossRef]

1951 (1)

A. L. Alden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Ackerman, T. P.

Alden, A. L.

A. L. Alden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Altenkirch, R. A.

Anders, K.

K. Anders, N. Roth, A. Frohn, “Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry,” Part. Part. Syst. Charact. 13, 125–129 (1996).
[CrossRef]

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in micrometer range,” J. Laser Appl. 2, 37–42 (1990).
[CrossRef]

K. Anders, N. Roth, A. Frohn, “Theoretical and experimental studies of the influence of internal temperature gradients on rainbow refractometry,” in PARTEC 95, Nuremberg, Germany (Nürnbergmesse GmbH, Nüremberg, Germany, 1995), 419–428.

N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in Sixth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 20–23, ed. (University of Lisbon, Lisbon, Portugal, 1992), p. 15.5.

Angelova, M. I.

G. Martinot–Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams: II GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

Bachalo, W. D.

S. V. Sankar, K. M. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler/rainbow refractometer system for simultaneous measurement of droplet size, velocity, and refractive index,” in Third International Congress on Optical Particle Sizing, Yokohama, Japan (Keio University, Yokohama, Japan, 1993), pp. 275–284.

Barber, P. W.

Bhandari, R.

Bohren, G. F.

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

Buermann, D. H.

S. V. Sankar, K. M. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler/rainbow refractometer system for simultaneous measurement of droplet size, velocity, and refractive index,” in Third International Congress on Optical Particle Sizing, Yokohama, Japan (Keio University, Yokohama, Japan, 1993), pp. 275–284.

Chowdhury, D. Q.

M. Schneider, E. D. Hirleman, H. Saleheen, D. Q. Chowdhury, S. C. Hill, “Rainbows and radially-inhomogeneous droplets,” in Third International Congress on Optical Particle Sizing, Yokohama, Japan (Keio University, Yokohama, Japan, 1993), pp. 323–326.

D’Alessio, A.

L. Kai, P. Massoli, A. D’Alessio, “Some far-field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
[CrossRef]

Durst, F.

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase Doppler systems: study of a near-forward and a near-backward geometry,” Part. Part. Syst. Charact. 11, 133–145 (1994).
[CrossRef]

Fidrich, M. J.

S. V. Sankar, K. M. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler/rainbow refractometer system for simultaneous measurement of droplet size, velocity, and refractive index,” in Third International Congress on Optical Particle Sizing, Yokohama, Japan (Keio University, Yokohama, Japan, 1993), pp. 275–284.

Frohn, A.

K. Anders, N. Roth, A. Frohn, “Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry,” Part. Part. Syst. Charact. 13, 125–129 (1996).
[CrossRef]

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in micrometer range,” J. Laser Appl. 2, 37–42 (1990).
[CrossRef]

K. Anders, N. Roth, A. Frohn, “Theoretical and experimental studies of the influence of internal temperature gradients on rainbow refractometry,” in PARTEC 95, Nuremberg, Germany (Nürnbergmesse GmbH, Nüremberg, Germany, 1995), 419–428.

N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in Sixth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 20–23, ed. (University of Lisbon, Lisbon, Portugal, 1992), p. 15.5.

Fu, X. Q.

Z. S. Wu, X. Q. Fu, “Scattering of fundamental Gaussian beams from a multilayered sphere,” Acta Electron. Sinica (China) 23, 32–36 (1995).

Gouesbet, G.

G. Martinot–Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams: II GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase Doppler systems: study of a near-forward and a near-backward geometry,” Part. Part. Syst. Charact. 11, 133–145 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of Generalized Lorenz–Mie Theory: faster algorithm for computations of beam shape coefficients gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, “Last progress in generalized Lorenz–Mie theory with applications in multiphase flows,” in Advances in Multiphase Flow, A. Serizawa, T. Fukano, J. Bataille, eds., (Elsevier, Amsterdam, 1995)pp. 703–710.

Gréhan, G.

F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
[CrossRef] [PubMed]

G. Martinot–Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams: II GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase Doppler systems: study of a near-forward and a near-backward geometry,” Part. Part. Syst. Charact. 11, 133–145 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of Generalized Lorenz–Mie Theory: faster algorithm for computations of beam shape coefficients gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, “Last progress in generalized Lorenz–Mie theory with applications in multiphase flows,” in Advances in Multiphase Flow, A. Serizawa, T. Fukano, J. Bataille, eds., (Elsevier, Amsterdam, 1995)pp. 703–710.

Hill, S. C.

E. M. Khaled, S. C. Hill, P. W. Barber, “Light scattering by a coated sphere illuminated by a Gaussian beam,” Appl. Opt. 33, 3308–3314 (1994).
[CrossRef] [PubMed]

M. Schneider, E. D. Hirleman, H. Saleheen, D. Q. Chowdhury, S. C. Hill, “Rainbows and radially-inhomogeneous droplets,” in Third International Congress on Optical Particle Sizing, Yokohama, Japan (Keio University, Yokohama, Japan, 1993), pp. 323–326.

Hirleman, E. D.

M. Schneider, E. D. Hirleman, H. Saleheen, D. Q. Chowdhury, S. C. Hill, “Rainbows and radially-inhomogeneous droplets,” in Third International Congress on Optical Particle Sizing, Yokohama, Japan (Keio University, Yokohama, Japan, 1993), pp. 323–326.

Huffman, D. R.

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

Ibrahim, K. M.

S. V. Sankar, K. M. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler/rainbow refractometer system for simultaneous measurement of droplet size, velocity, and refractive index,” in Third International Congress on Optical Particle Sizing, Yokohama, Japan (Keio University, Yokohama, Japan, 1993), pp. 275–284.

Kai, L.

L. Kai, P. Massoli, “Scattering of electromagnetic plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
[CrossRef] [PubMed]

L. Kai, P. Massoli, A. D’Alessio, “Some far-field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
[CrossRef]

Kerker, M.

A. L. Alden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

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

Khaled, E. M.

Khare, V.

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976 (1974).
[CrossRef]

Lock, J. A.

Mackowski, D. W.

Maheu, B.

Marston, P. L.

Martinot–Lagarde, G.

G. Martinot–Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams: II GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

Massoli, P.

L. Kai, P. Massoli, A. D’Alessio, “Some far-field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
[CrossRef]

L. Kai, P. Massoli, “Scattering of electromagnetic plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
[CrossRef] [PubMed]

Menguc, M. P.

Naqwi, A.

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase Doppler systems: study of a near-forward and a near-backward geometry,” Part. Part. Syst. Charact. 11, 133–145 (1994).
[CrossRef]

Nussenzveig, H. M.

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976 (1974).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere II. theory of the rainbow and the glory,” J. Math. Phys. 10, 125 (1969).
[CrossRef]

Onofri, F.

Pouligny, B.

G. Martinot–Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams: II GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

Ren, K. F.

K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of Generalized Lorenz–Mie Theory: faster algorithm for computations of beam shape coefficients gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

Riethmuller, M. L.

J. P. A. J. van Beeck, M. L. Riethmuller, “Nonintrusive measurements of temperature and size of raindrops,” Appl. Opt. 34, 1633–1639 (1995).
[CrossRef] [PubMed]

J. P. A. J. van Beeck, M. L. Riethmuller, “Détermination non intrusive de la dimension et de la température des gouttes dans une pulvérisation,” in 4éme Congrés Francophone de V́locimétrie Laser, Poitiers, 26–29 Septembre 1994, (Université de Poitiers, Poitiers, France, 1994), article 2-2.

Roth, N.

K. Anders, N. Roth, A. Frohn, “Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry,” Part. Part. Syst. Charact. 13, 125–129 (1996).
[CrossRef]

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in micrometer range,” J. Laser Appl. 2, 37–42 (1990).
[CrossRef]

K. Anders, N. Roth, A. Frohn, “Theoretical and experimental studies of the influence of internal temperature gradients on rainbow refractometry,” in PARTEC 95, Nuremberg, Germany (Nürnbergmesse GmbH, Nüremberg, Germany, 1995), 419–428.

N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in Sixth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 20–23, ed. (University of Lisbon, Lisbon, Portugal, 1992), p. 15.5.

Saleheen, H.

M. Schneider, E. D. Hirleman, H. Saleheen, D. Q. Chowdhury, S. C. Hill, “Rainbows and radially-inhomogeneous droplets,” in Third International Congress on Optical Particle Sizing, Yokohama, Japan (Keio University, Yokohama, Japan, 1993), pp. 323–326.

Sankar, S. V.

S. V. Sankar, K. M. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler/rainbow refractometer system for simultaneous measurement of droplet size, velocity, and refractive index,” in Third International Congress on Optical Particle Sizing, Yokohama, Japan (Keio University, Yokohama, Japan, 1993), pp. 275–284.

Schneider, M.

M. Schneider, E. D. Hirleman, H. Saleheen, D. Q. Chowdhury, S. C. Hill, “Rainbows and radially-inhomogeneous droplets,” in Third International Congress on Optical Particle Sizing, Yokohama, Japan (Keio University, Yokohama, Japan, 1993), pp. 323–326.

Toon, O. B.

van Beeck, J. P. A. J.

J. P. A. J. van Beeck, M. L. Riethmuller, “Nonintrusive measurements of temperature and size of raindrops,” Appl. Opt. 34, 1633–1639 (1995).
[CrossRef] [PubMed]

J. P. A. J. van Beeck, M. L. Riethmuller, “Détermination non intrusive de la dimension et de la température des gouttes dans une pulvérisation,” in 4éme Congrés Francophone de V́locimétrie Laser, Poitiers, 26–29 Septembre 1994, (Université de Poitiers, Poitiers, France, 1994), article 2-2.

van de Hulst, H. C.

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

Wang, Y. P.

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Wu, Z. S.

Z. S. Wu, X. Q. Fu, “Scattering of fundamental Gaussian beams from a multilayered sphere,” Acta Electron. Sinica (China) 23, 32–36 (1995).

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Acta Electron. Sinica (China) (1)

Z. S. Wu, X. Q. Fu, “Scattering of fundamental Gaussian beams from a multilayered sphere,” Acta Electron. Sinica (China) 23, 32–36 (1995).

Appl. Opt. (9)

J. Appl. Phys. (1)

A. L. Alden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

J. Laser Appl. (1)

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in micrometer range,” J. Laser Appl. 2, 37–42 (1990).
[CrossRef]

J. Math. Phys. (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere II. theory of the rainbow and the glory,” J. Math. Phys. 10, 125 (1969).
[CrossRef]

J. Opt. Soc. Am. A (2)

Part. Part. Syst. Charact. (4)

L. Kai, P. Massoli, A. D’Alessio, “Some far-field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Localized approximation of Generalized Lorenz–Mie Theory: faster algorithm for computations of beam shape coefficients gnm,” Part. Part. Syst. Charact. 9, 144–150 (1992).
[CrossRef]

K. Anders, N. Roth, A. Frohn, “Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry,” Part. Part. Syst. Charact. 13, 125–129 (1996).
[CrossRef]

G. Gréhan, G. Gouesbet, A. Naqwi, F. Durst, “Trajectory ambiguities in phase Doppler systems: study of a near-forward and a near-backward geometry,” Part. Part. Syst. Charact. 11, 133–145 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976 (1974).
[CrossRef]

Pure Appl. Opt. (1)

G. Martinot–Lagarde, B. Pouligny, M. I. Angelova, G. Gréhan, G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams: II GLMT analysis,” Pure Appl. Opt. 4, 571–585 (1995).
[CrossRef]

Radio Sci. (1)

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Other (10)

The sources of the fortran program are available from the authors, in particular from Z. S. Wu.

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

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

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

G. Gouesbet, G. Gréhan, “Last progress in generalized Lorenz–Mie theory with applications in multiphase flows,” in Advances in Multiphase Flow, A. Serizawa, T. Fukano, J. Bataille, eds., (Elsevier, Amsterdam, 1995)pp. 703–710.

N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in Sixth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 20–23, ed. (University of Lisbon, Lisbon, Portugal, 1992), p. 15.5.

M. Schneider, E. D. Hirleman, H. Saleheen, D. Q. Chowdhury, S. C. Hill, “Rainbows and radially-inhomogeneous droplets,” in Third International Congress on Optical Particle Sizing, Yokohama, Japan (Keio University, Yokohama, Japan, 1993), pp. 323–326.

S. V. Sankar, K. M. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler/rainbow refractometer system for simultaneous measurement of droplet size, velocity, and refractive index,” in Third International Congress on Optical Particle Sizing, Yokohama, Japan (Keio University, Yokohama, Japan, 1993), pp. 275–284.

J. P. A. J. van Beeck, M. L. Riethmuller, “Détermination non intrusive de la dimension et de la température des gouttes dans une pulvérisation,” in 4éme Congrés Francophone de V́locimétrie Laser, Poitiers, 26–29 Septembre 1994, (Université de Poitiers, Poitiers, France, 1994), article 2-2.

K. Anders, N. Roth, A. Frohn, “Theoretical and experimental studies of the influence of internal temperature gradients on rainbow refractometry,” in PARTEC 95, Nuremberg, Germany (Nürnbergmesse GmbH, Nüremberg, Germany, 1995), 419–428.

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 (10)

Fig. 1
Fig. 1

Geometry of the problem under study.

Fig. 2
Fig. 2

Computational domain on (a) linear scale and (b) logarithmic scale.

Fig. 3
Fig. 3

Definition of a criterion for the number of layers required to describe a refractive index profile: (a) continuous profile index, (b) extinction and backward efficiency factors versus the number of layers for a plane wave.

Fig. 4
Fig. 4

Scattering properties of a blood cell with an off-axis location in a Gaussian beam: continuous curve, w 0 = 20 µm, filled circles, w 0 = 10 µm, filled squares, w 0 = 3 µm, filled triangles, w 0 = 1 µm. (a) Scattering intensity versus scattering angle, (b) polarization degree versus scattering angle.

Fig. 5
Fig. 5

Q ext and Q sca versus the distance to the beam waist. The particle is located at x 0 = 2 µm, y 0 = 2 µm. The parameter is the beam waist that takes on the values of 2, 3, 5, and 10 µm. The incident wavelength is 0.6328 µm.

Fig. 6
Fig. 6

Q ext and Q sca versus the beam-waist radius. The parameter is the particle location that takes on the values, in µm, of (x 0 = 0, y 0 = 0), (x 0 = 2, y 0 = 2), (x 0 = 5, y 0 = 5), and (x 0 = 2, y 0 = 10). z 0 is set to 100 µm. The wavelength is 0.6328 µm.

Fig. 7
Fig. 7

Scattering diagrams for a 50-layer particle. The parameter is the beam-waist size, expressed in number of wavelengths.

Fig. 8
Fig. 8

Scattering diagram in the rainbow angular range for a water droplet diameter of 250 µm.

Fig. 9
Fig. 9

Evolution of the scattering diagram in the rainbow angular range for a 250-µm-diameter droplet with a linear variation of the refractive index (1.33 at the core center and 1.36 at the outer surface). The droplet is divided by use of n layers of equal thickness.

Fig. 10
Fig. 10

Light scattered in the rainbow angle range for a nonlinear profile of the refractive index.

Tables (2)

Tables Icon

Table 1 Computation Time in minutes on a PC 486 DX at 33 MHz

Tables Icon

Table 2 Study of the influence of the Number of Layers on Rainbow Technique Measurementsa

Equations (31)

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

an=ΨnxLξnxL·HnamLxL-mLDn1xLHnamLxL-mLDn3xL,
bn=ΨnxLξnxL·mLHnbmLxL-Dn1xLmLHnbmLxL-Dn3xL,
Hnamj,xj=Ψnmjxj/χnmjxjDn1mjxj-AnjDn2mjxjΨnmjxj/χnmjxj-Anj,
Anj=Ψnmjxj-1χnmjxj-1·mjHnamj-1xj-1-mj-1Dn1mjxj-1mjHnamj-1xj-1-mj-1Dn2mjxj-1,
An1=0, Hnam1x1=Dn1m1x1;
Hnbmj, xj=Ψnmjxj/χnmjxjDn1mjxj-BnjDn2mjxjΨnmjxj/χnmjxj-Bnj,
Bnj=Ψnmjxj-1χnmjxj-1·mj-iHnbmj-1xj-1-mjDn1mjxj-1mj-1Hnbmj-1xj-1-mjDn2mjxj-1,
Bn1=0, Hnbm1x1=Dn1m1x1,
anm=gnmTMan, bnm=gnmTEbn,
Eθ=iE0 exp-ikrkr S2θ, φ,
Eϕ=-E0 exp-ikrkrS1θ, φ,
S1θ, φ=n=1m=-nn2n+1nn+1mgnmTManπnmcos θ+ignmTEbnτnmcos θexpimφ,
S2θ, φ=n=1m=-nn2n+1nn+1gnmTManτnmcos θ+imgnmTEbnπnmcos θexpimφ.
Is=S12+S22, P=S12-S22S12+S22.
Cext=λ2πn=1m=-nn2n+1nn+1n+1+m!n-m!×ReangnmTM2+bngnmTE2,
Csca=λ2πn=1m=-nn2n+1nn+1n+1+m!n-m!×an2gnmTM2+bn2gnmTE2,
σ=λ2/πn=1n+1/2-1nbn-an2.
Rnj=ψnmjxj-1χnmjxj-1χnmjxjψnmjxj.
Hnamj, xj=Dn1mjxj-AnjDn2mjxj1-Anj,
Anj=Rnj·mjHnamj-1xj-1-mj-1Dn1mjxj-1mjHnamj-1xj-1-mj-1Dn2mjxj-1,
Hnbmj, xj=Dn1mj-xj-BnjDn2mjxj1-Bnj,
Bnj=Rnj·mj-iHnbmj-1xj-1-mjDn1mjxj-1mj-1Hnbmj-1xj-1-mjDn2mjxj-1,
Ψnzχnz=Ψn-1zχn-1zDn2z+n/zDn1z+n/z,
ξnz1ξnz2=ξn-1z1ξn-1z2Dn-13z1-n/z1Dn-13z2-n/z2,
Ψnz1Ψnz2=Ψn-1z1Ψn-1z2Dn1z2+n/z2Dn1z1+n/z1,
Qext-QscaQext+Qsca<1%
m=c1+fr2/b2,
θ=-2* arcsin13m2-11/2+4* arcos1312-3m21/2m,
d=λ4cosρrgsin3ρrg1/2αi-αjθi-θj3/2,
sin ρrg=m2-1/31/2.
θ1-θ0=α1λ/4l,

Metrics