Abstract

The theory of light scattering by spheres possessing one or more spherical inhomogeneities is developed. The inhomogeneities, or subspheres, are of uniform but otherwise arbitrary composition and are restricted in size and number by only the volume of the host. Numerical results for single inclusions are considered in regard to questions that have arisen in the course of recent experimental research on morphology-dependent resonances in droplets. The modification of the light-absorbing properties of carbon by its entrainment in droplets is also studied. The predicted absorption by mass of soot particles in cloud droplets tends to be higher when the particles are centered in the droplets than when the more realistic eccentric inclusion model is used. Absorption by carbon that is internally mixed in sulfate hazes is more sensitive to the relative sizes of the sulfate and carbon particles, but absorption by eccentrically included grains tends not to differ greatly from that predicted by concentric models.

© 1995 Optical Society of America

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References

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  1. B. V. Bronk, M. J. Smith, S. Arnold, “Photon-correlation spectroscopy for small spherical inclusions in a micrometer-sized electrodynamically levitated droplet,” Opt. Lett. 18, 93–95 (1993).
    [CrossRef] [PubMed]
  2. J. Gu, T. E. Ruekgauer, J.-G. Xie, R. L. Armstrong, “Effect of particulate seeding on microdroplet angular scattering,” Opt. Lett. 18, 1293–1295 (1993).
    [CrossRef] [PubMed]
  3. P. Chýlek, D. Ngo, R. G. Pinnick, “Resonance structure of composite and slightly absorbing spheres,” J. Opt. Soc. Am. A 9, 775–780 (1992).
    [CrossRef]
  4. D. Ngo, R. G. Pinnick, “Suppression of scattering resonances in inhomogeneous microdroplets,” J. Opt. Soc. Am. A 11, 1352–1359 (1994).
    [CrossRef]
  5. H.-B. Lin, A. L. Huston, J. D. Eversole, A. J. Campillo, P. Chýlek, “Internal scattering effects on microdroplet resonant emission structure,” Opt. Lett. 17, 970–972 (1992).
    [CrossRef] [PubMed]
  6. R. L. Armstrong, J.-G. Xie, T. E. Ruekgauer, J. Gu, R. G. Pinnick, “Effects of submicrometer-sized particles on microdroplet lasing,” Opt. Lett. 18, 119–121 (1993).
    [CrossRef] [PubMed]
  7. J.-G. Xie, T. E. Ruekgauer, R. L. Armstrong, R. G. Pinnick, “Suppression of stimulated Raman scattering from microdroplets by seeding with nanometer-sized latex particles,” Opt. Lett. 18, 340–342 (1993).
    [CrossRef] [PubMed]
  8. Md. M. Mazumder, S. C. Hill, P. W. Barber, “Morphology-dependent resonances in inhomogeneous spheres: comparison of the layered T-matrix method and the time-independent perturbation method,” J. Opt. Soc. Am. A 9, 1844–1853 (1992).
    [CrossRef]
  9. J. A. Lock, “Interference enhancement of the internal fields at structural resonances of a coated sphere,” Appl. Opt. 29, 3180–3187 (1990).
    [CrossRef] [PubMed]
  10. M. Essien, R. L. Armstrong, R. G. Pinnick, “Lasing emission from an evaporating layered microdroplet,” Opt. Lett. 18, 762–764 (1993).
    [CrossRef] [PubMed]
  11. J. C. Knight, H. S. T. Driver, G. N. Robertson, “Interference modulation of Q values in a cladded-fiber whispering-gallery-mode laser,” Opt. Lett. 18, 1296–1298 (1993).
    [CrossRef] [PubMed]
  12. J. Podzimek, “Physical properties of coarse aerosol particles and haze elements in a polluted urban-marine environment,” J. Aerosol Sci. 21, 299–308 (1990).
    [CrossRef]
  13. F. Parungo, B. Kopcewicz, C. Nagamoto, R. Schnell, P. Sheridan, C. Zhu, J. Harris, “Aerosol properties in the Kuwait oil fire plumes: their morphology, size distribution, chemical composition, transport, and potential effects on climate,” J. Geophys. Res. 97D, 15867–15882 (1992).
    [CrossRef]
  14. P. Chýlek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,” J. Atmos. Sci. 41, 3076–3084 (1984).
    [CrossRef]
  15. G. L. Stephens, S.-C. Tsay, “On the cloud absorption anomaly,” Q. J. R. Meteorol. Soc. 116, 671–704 (1990).
    [CrossRef]
  16. J. G. Fikioris, N. K. Uzunoglu, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. 69, 1359–1366 (1979).
    [CrossRef]
  17. F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327–1335 (1992).
    [CrossRef]
  18. K. A. Fuller, “Morphology dependent resonances in eccentrically stratified spheres,” Opt. Lett. 19, 1272–1274 (1994).
    [CrossRef] [PubMed]
  19. K. A. Fuller, G. L. Stephens, B. D. Jersak, “Some advances in understanding light scattering by nonspherical particles,” in Proceedings of the 8th Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 319–321.
  20. K. A. Fuller, “Scattering and absorption by inhomogeneous spheres and sphere aggregates,” in Laser Applications in Combustion and Combustion DiagnosticsL. C. Liou, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1862, 249–257 (1993).
    [CrossRef]
  21. K. A. Fuller, “Scattering of light by coated spheres,” Opt. Lett. 18, 257–259 (1993).
    [CrossRef] [PubMed]
  22. K. A. Fuller, “Scattering and absorption by spheres containing arbitrarily located spherical inhomogeneities,” in Proceedings of the 1993 Scientific Conference on Obscuration and Aerosol Research, Battelle Edgewood Operations Rep. ERDEC-SP-019, pp. 443–474.Available from Battelle Edgewood Operations, 2113 Emmorton Park Road, Edgewood, Md. 21040, or through the National Technical Information Service.
  23. F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing several spherical inclusions,” Appl. Opt. 33, 484–493 (1994).
    [CrossRef] [PubMed]
  24. S. C. Hill, H. I. Saleheen, K. A. Fuller, “Volume current method for modeling light scattering by inhomogeneously perturbed spheres,” J. Opt. Soc. Am. A 12, 905–915 (1995).
    [CrossRef]
  25. J. H. Bruning, Y. T. Lo, “Multiple scattering of em waves by spheres. Parts I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
    [CrossRef]
  26. J. H. Bruning, Y. T. Lo, “Multiple scattering by spheres,” Antenna Laboratory Rep. 69-5 (Antenna Laboratory, Department of Electrical Engineering, Engineering Experiment Station, University of Illinois, Urbana, Ill.).
  27. K. A. Fuller, “Optical resonances and two-sphere systems,” Appl. Opt. 30, 4716–4731 (1991).
    [CrossRef] [PubMed]
  28. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  29. A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  30. F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327–1335 (1992).
    [CrossRef]
  31. K. A. Fuller, “Scattering and absorption cross sections of compounded spheres. I. Theory for external aggregation,” J. Opt. Soc. Am. A 11, 3251–3260 (1994).
    [CrossRef]
  32. K. A. Fuller, “Scattering and absorption cross sections of compounded spheres. II. Calculations for external aggregation,” J. Opt. Soc. Am. A 12, 881–892 (1995).
    [CrossRef]
  33. D. W. Mackowski, P. D. Jones, “Theoretical investigation of particles having a directionally dependent absorption cross section,” J. Thermophys. Heat Transfer (to be published).

1995 (2)

1994 (4)

1993 (7)

1992 (6)

1991 (1)

1990 (3)

G. L. Stephens, S.-C. Tsay, “On the cloud absorption anomaly,” Q. J. R. Meteorol. Soc. 116, 671–704 (1990).
[CrossRef]

J. Podzimek, “Physical properties of coarse aerosol particles and haze elements in a polluted urban-marine environment,” J. Aerosol Sci. 21, 299–308 (1990).
[CrossRef]

J. A. Lock, “Interference enhancement of the internal fields at structural resonances of a coated sphere,” Appl. Opt. 29, 3180–3187 (1990).
[CrossRef] [PubMed]

1984 (1)

P. Chýlek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,” J. Atmos. Sci. 41, 3076–3084 (1984).
[CrossRef]

1979 (1)

1971 (1)

J. H. Bruning, Y. T. Lo, “Multiple scattering of em waves by spheres. Parts I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
[CrossRef]

1951 (1)

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

Aden, A. L.

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

Armstrong, R. L.

Arnold, S.

Barber, P. W.

Borghese, F.

Bronk, B. V.

Bruning, J. H.

J. H. Bruning, Y. T. Lo, “Multiple scattering of em waves by spheres. Parts I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
[CrossRef]

J. H. Bruning, Y. T. Lo, “Multiple scattering by spheres,” Antenna Laboratory Rep. 69-5 (Antenna Laboratory, Department of Electrical Engineering, Engineering Experiment Station, University of Illinois, Urbana, Ill.).

Campillo, A. J.

Cheng, R. J.

P. Chýlek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,” J. Atmos. Sci. 41, 3076–3084 (1984).
[CrossRef]

Chýlek, P.

Denti, P.

Driver, H. S. T.

Essien, M.

Eversole, J. D.

Fikioris, J. G.

Fuller, K. A.

K. A. Fuller, “Scattering and absorption cross sections of compounded spheres. II. Calculations for external aggregation,” J. Opt. Soc. Am. A 12, 881–892 (1995).
[CrossRef]

S. C. Hill, H. I. Saleheen, K. A. Fuller, “Volume current method for modeling light scattering by inhomogeneously perturbed spheres,” J. Opt. Soc. Am. A 12, 905–915 (1995).
[CrossRef]

K. A. Fuller, “Scattering and absorption cross sections of compounded spheres. I. Theory for external aggregation,” J. Opt. Soc. Am. A 11, 3251–3260 (1994).
[CrossRef]

K. A. Fuller, “Morphology dependent resonances in eccentrically stratified spheres,” Opt. Lett. 19, 1272–1274 (1994).
[CrossRef] [PubMed]

K. A. Fuller, “Scattering of light by coated spheres,” Opt. Lett. 18, 257–259 (1993).
[CrossRef] [PubMed]

K. A. Fuller, “Optical resonances and two-sphere systems,” Appl. Opt. 30, 4716–4731 (1991).
[CrossRef] [PubMed]

K. A. Fuller, G. L. Stephens, B. D. Jersak, “Some advances in understanding light scattering by nonspherical particles,” in Proceedings of the 8th Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 319–321.

K. A. Fuller, “Scattering and absorption by inhomogeneous spheres and sphere aggregates,” in Laser Applications in Combustion and Combustion DiagnosticsL. C. Liou, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1862, 249–257 (1993).
[CrossRef]

K. A. Fuller, “Scattering and absorption by spheres containing arbitrarily located spherical inhomogeneities,” in Proceedings of the 1993 Scientific Conference on Obscuration and Aerosol Research, Battelle Edgewood Operations Rep. ERDEC-SP-019, pp. 443–474.Available from Battelle Edgewood Operations, 2113 Emmorton Park Road, Edgewood, Md. 21040, or through the National Technical Information Service.

Gu, J.

Harris, J.

F. Parungo, B. Kopcewicz, C. Nagamoto, R. Schnell, P. Sheridan, C. Zhu, J. Harris, “Aerosol properties in the Kuwait oil fire plumes: their morphology, size distribution, chemical composition, transport, and potential effects on climate,” J. Geophys. Res. 97D, 15867–15882 (1992).
[CrossRef]

Hill, S. C.

Huston, A. L.

Jersak, B. D.

K. A. Fuller, G. L. Stephens, B. D. Jersak, “Some advances in understanding light scattering by nonspherical particles,” in Proceedings of the 8th Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 319–321.

Jones, P. D.

D. W. Mackowski, P. D. Jones, “Theoretical investigation of particles having a directionally dependent absorption cross section,” J. Thermophys. Heat Transfer (to be published).

Kerker, M.

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

Knight, J. C.

Kopcewicz, B.

F. Parungo, B. Kopcewicz, C. Nagamoto, R. Schnell, P. Sheridan, C. Zhu, J. Harris, “Aerosol properties in the Kuwait oil fire plumes: their morphology, size distribution, chemical composition, transport, and potential effects on climate,” J. Geophys. Res. 97D, 15867–15882 (1992).
[CrossRef]

Lin, H.-B.

Lo, Y. T.

J. H. Bruning, Y. T. Lo, “Multiple scattering of em waves by spheres. Parts I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
[CrossRef]

J. H. Bruning, Y. T. Lo, “Multiple scattering by spheres,” Antenna Laboratory Rep. 69-5 (Antenna Laboratory, Department of Electrical Engineering, Engineering Experiment Station, University of Illinois, Urbana, Ill.).

Lock, J. A.

Mackowski, D. W.

D. W. Mackowski, P. D. Jones, “Theoretical investigation of particles having a directionally dependent absorption cross section,” J. Thermophys. Heat Transfer (to be published).

Mazumder, Md. M.

Nagamoto, C.

F. Parungo, B. Kopcewicz, C. Nagamoto, R. Schnell, P. Sheridan, C. Zhu, J. Harris, “Aerosol properties in the Kuwait oil fire plumes: their morphology, size distribution, chemical composition, transport, and potential effects on climate,” J. Geophys. Res. 97D, 15867–15882 (1992).
[CrossRef]

Ngo, D.

Parungo, F.

F. Parungo, B. Kopcewicz, C. Nagamoto, R. Schnell, P. Sheridan, C. Zhu, J. Harris, “Aerosol properties in the Kuwait oil fire plumes: their morphology, size distribution, chemical composition, transport, and potential effects on climate,” J. Geophys. Res. 97D, 15867–15882 (1992).
[CrossRef]

Pinnick, R. G.

Podzimek, J.

J. Podzimek, “Physical properties of coarse aerosol particles and haze elements in a polluted urban-marine environment,” J. Aerosol Sci. 21, 299–308 (1990).
[CrossRef]

Ramaswamy, V.

P. Chýlek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,” J. Atmos. Sci. 41, 3076–3084 (1984).
[CrossRef]

Robertson, G. N.

Ruekgauer, T. E.

Saija, R.

Saleheen, H. I.

Schnell, R.

F. Parungo, B. Kopcewicz, C. Nagamoto, R. Schnell, P. Sheridan, C. Zhu, J. Harris, “Aerosol properties in the Kuwait oil fire plumes: their morphology, size distribution, chemical composition, transport, and potential effects on climate,” J. Geophys. Res. 97D, 15867–15882 (1992).
[CrossRef]

Sheridan, P.

F. Parungo, B. Kopcewicz, C. Nagamoto, R. Schnell, P. Sheridan, C. Zhu, J. Harris, “Aerosol properties in the Kuwait oil fire plumes: their morphology, size distribution, chemical composition, transport, and potential effects on climate,” J. Geophys. Res. 97D, 15867–15882 (1992).
[CrossRef]

Sindoni, O. I.

Smith, M. J.

Stephens, G. L.

G. L. Stephens, S.-C. Tsay, “On the cloud absorption anomaly,” Q. J. R. Meteorol. Soc. 116, 671–704 (1990).
[CrossRef]

K. A. Fuller, G. L. Stephens, B. D. Jersak, “Some advances in understanding light scattering by nonspherical particles,” in Proceedings of the 8th Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 319–321.

Tsay, S.-C.

G. L. Stephens, S.-C. Tsay, “On the cloud absorption anomaly,” Q. J. R. Meteorol. Soc. 116, 671–704 (1990).
[CrossRef]

Uzunoglu, N. K.

van de Hulst, H. C.

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

Xie, J.-G.

Zhu, C.

F. Parungo, B. Kopcewicz, C. Nagamoto, R. Schnell, P. Sheridan, C. Zhu, J. Harris, “Aerosol properties in the Kuwait oil fire plumes: their morphology, size distribution, chemical composition, transport, and potential effects on climate,” J. Geophys. Res. 97D, 15867–15882 (1992).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Antennas Propag. (1)

J. H. Bruning, Y. T. Lo, “Multiple scattering of em waves by spheres. Parts I & II,” IEEE Trans. Antennas Propag. AP-19, 378–400 (1971).
[CrossRef]

J. Aerosol Sci. (1)

J. Podzimek, “Physical properties of coarse aerosol particles and haze elements in a polluted urban-marine environment,” J. Aerosol Sci. 21, 299–308 (1990).
[CrossRef]

J. Appl. Phys. (1)

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

J. Atmos. Sci. (1)

P. Chýlek, V. Ramaswamy, R. J. Cheng, “Effect of graphitic carbon on the albedo of clouds,” J. Atmos. Sci. 41, 3076–3084 (1984).
[CrossRef]

J. Geophys. Res. (1)

F. Parungo, B. Kopcewicz, C. Nagamoto, R. Schnell, P. Sheridan, C. Zhu, J. Harris, “Aerosol properties in the Kuwait oil fire plumes: their morphology, size distribution, chemical composition, transport, and potential effects on climate,” J. Geophys. Res. 97D, 15867–15882 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

Opt. Lett. (9)

R. L. Armstrong, J.-G. Xie, T. E. Ruekgauer, J. Gu, R. G. Pinnick, “Effects of submicrometer-sized particles on microdroplet lasing,” Opt. Lett. 18, 119–121 (1993).
[CrossRef] [PubMed]

B. V. Bronk, M. J. Smith, S. Arnold, “Photon-correlation spectroscopy for small spherical inclusions in a micrometer-sized electrodynamically levitated droplet,” Opt. Lett. 18, 93–95 (1993).
[CrossRef] [PubMed]

K. A. Fuller, “Scattering of light by coated spheres,” Opt. Lett. 18, 257–259 (1993).
[CrossRef] [PubMed]

J.-G. Xie, T. E. Ruekgauer, R. L. Armstrong, R. G. Pinnick, “Suppression of stimulated Raman scattering from microdroplets by seeding with nanometer-sized latex particles,” Opt. Lett. 18, 340–342 (1993).
[CrossRef] [PubMed]

M. Essien, R. L. Armstrong, R. G. Pinnick, “Lasing emission from an evaporating layered microdroplet,” Opt. Lett. 18, 762–764 (1993).
[CrossRef] [PubMed]

J. Gu, T. E. Ruekgauer, J.-G. Xie, R. L. Armstrong, “Effect of particulate seeding on microdroplet angular scattering,” Opt. Lett. 18, 1293–1295 (1993).
[CrossRef] [PubMed]

J. C. Knight, H. S. T. Driver, G. N. Robertson, “Interference modulation of Q values in a cladded-fiber whispering-gallery-mode laser,” Opt. Lett. 18, 1296–1298 (1993).
[CrossRef] [PubMed]

K. A. Fuller, “Morphology dependent resonances in eccentrically stratified spheres,” Opt. Lett. 19, 1272–1274 (1994).
[CrossRef] [PubMed]

H.-B. Lin, A. L. Huston, J. D. Eversole, A. J. Campillo, P. Chýlek, “Internal scattering effects on microdroplet resonant emission structure,” Opt. Lett. 17, 970–972 (1992).
[CrossRef] [PubMed]

Q. J. R. Meteorol. Soc. (1)

G. L. Stephens, S.-C. Tsay, “On the cloud absorption anomaly,” Q. J. R. Meteorol. Soc. 116, 671–704 (1990).
[CrossRef]

Other (6)

K. A. Fuller, G. L. Stephens, B. D. Jersak, “Some advances in understanding light scattering by nonspherical particles,” in Proceedings of the 8th Conference on Atmospheric Radiation (American Meteorological Society, Boston, Mass., 1994), pp. 319–321.

K. A. Fuller, “Scattering and absorption by inhomogeneous spheres and sphere aggregates,” in Laser Applications in Combustion and Combustion DiagnosticsL. C. Liou, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1862, 249–257 (1993).
[CrossRef]

K. A. Fuller, “Scattering and absorption by spheres containing arbitrarily located spherical inhomogeneities,” in Proceedings of the 1993 Scientific Conference on Obscuration and Aerosol Research, Battelle Edgewood Operations Rep. ERDEC-SP-019, pp. 443–474.Available from Battelle Edgewood Operations, 2113 Emmorton Park Road, Edgewood, Md. 21040, or through the National Technical Information Service.

J. H. Bruning, Y. T. Lo, “Multiple scattering by spheres,” Antenna Laboratory Rep. 69-5 (Antenna Laboratory, Department of Electrical Engineering, Engineering Experiment Station, University of Illinois, Urbana, Ill.).

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

D. W. Mackowski, P. D. Jones, “Theoretical investigation of particles having a directionally dependent absorption cross section,” J. Thermophys. Heat Transfer (to be published).

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Figures (17)

Fig. 1
Fig. 1

A wave front incident upon a sphere (a) is decomposed into components of VSH that are concentric with the sphere (b); i.e., the projections of the incident field onto these basis functions serve as the coefficients for the expansion of the incident wave in VSH.

Fig. 2
Fig. 2

Transmitted and reflected nth partial waves arising from scattering by a layered sphere. The complex amplitudes (1cn, 1dn) and (1an, 1bn) are those associated with, respectively, reflection and transmission of the nth incoming VSH component of the plane wave at a convex spherical surface. The coefficients (2an, 2bn) likewise correspond to reflections from the core. Transmission and reflection of these outgoing partial waves across the concave outer surface of the shell are represented, in order, by (1čn d ˇ 1 n) and ( a ˇ 1 n , b ˇ 1 n ).

Fig. 3
Fig. 3

Illustration of the multiple-scattering calculation of the scattering coefficients of a coated sphere. Shown are the first three TM contributions. Note that the incident field is illustrated in its VSH representation.

Fig. 4
Fig. 4

Scattering from an arbitrarily located spherical inclusion in an otherwise homogeneous spherical host. To find the coefficients of the partial fields scattered by the inclusion, we find the projections onto the normal modes of the inclusion for incoming partial waves concentric with the host. The sum of these projections is then multiplied by the Lorenz–Mie coefficient (reflection coefficient) of the inclusion.

Fig. 5
Fig. 5

Transmission and reflection, at the surface of the host, of the scattered field of an arbitrarily located spherical inclusion are determined from the projections of those scattered fields onto the basis functions of the host as a product of the sum of those projections and the concave Lorenz–Mie coefficients of the host.

Fig. 6
Fig. 6

Sample of the experimental data collected by Ngo and Pinnick.4 Lorenz–Mie theory for and experimental measurements of light scattering by an evaporating homogeneous glycerol droplet are compared with measurements made on glycerol hosts that had been seeded with latex spheres of various sizes and concentrations.

Fig. 7
Fig. 7

Dependence of scattered intensity, observed at 90° from the direction of incidence, on the axial position of two sizes of latex inclusion. The radius of the host is approximately 3.19 μm, and the wavelength of the incident radiation is 0.514 μm. Note that the inset is not drawn to scale—the inclusion would not be visible if it were.

Fig. 8
Fig. 8

Same as Fig. 7 but with the host tuned to a resonance and for three sizes of inclusion.

Fig. 9
Fig. 9

TE 46 2 resonance spectrum of a glycerol droplet for three sizes of latex and water inclusions. The inclusions are in the forward prominence of the resonating droplet.

Fig. 10
Fig. 10

Same as Fig. 9 but for five sizes of air bubbles in the glycerol.

Fig. 11
Fig. 11

Variations in the resonance spectrum of a glycerol droplet as a latex inclusion having a radius of 30 nm is located at several positions near the center of the forward MDR prominence. The thickest solid curve represents the spectrum of a homogeneous host.

Fig. 12
Fig. 12

Same as Fig. 11 but for a latex inclusion having a radius of 50 nm.

Fig. 13
Fig. 13

Variations in the resonance spectrum of a glycerol droplet with a 50-nm latex inclusion located near the forward surface of the droplet. (This is near the focal volume of the host.)

Fig. 14
Fig. 14

Orientation dependence of the gram-specific absorption cross sections of a carbon grain located at various radial distances from the center of a sulfate host. The refractive indices of the sulfate and carbon particles are taken to be 1.52 + 0.0i and 1.8 + 0.5i, respectively. In the example shown, an average over polarization has been taken.

Fig. 15
Fig. 15

Same as Fig. 14 but with the carbon entrained in a cloud droplet of radius 5 μm. The refractive index of water is taken here to be 1.33 + 0.0i.

Fig. 16
Fig. 16

Orientation-averaged absorption cross sections for carbon spheres at different radial distances from the center of sulfate haze elements.

Fig. 17
Fig. 17

Same as Fig. 16 but for carbon grains entrained in cloud droplets.

Equations (32)

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

[ č n 1 N m n ( 3 ) + d ˇ n 1 M m n ( 3 ) ] θ , ϕ = [ a ˇ n 1 N m n ( 1 ) + b ˇ n 1 M m n ( 1 ) + N m n ( 3 ) + M m n ( 3 ) ] θ , ϕ .
N 1 1 č n ξ n ( ρ 1 ) = ξ n ( η 1 ) + 1 a ˇ n ψ n ( η 1 ) ,
č 1 n ξ n ( ρ 1 ) = ξ n ( η 1 ) + a ˇ 1 n ψ n ( η 1 ) .
č n 1 = ψ n ( ρ 1 ) ξ n ( ρ 1 ) ξ n ( ρ 1 ) ψ n ( ρ 1 ) N 1 ξ n ( ρ 1 ) ψ n ( η 1 ) ξ n ( ρ 1 ) ψ n ( η 1 ) .
W { j n ( z ) , y n ( z ) } = z 2 i
D n M 1 = N 1 ξ n ( ρ 1 ) ψ n ( η 1 ) ξ n ( ρ 1 ) ψ n ( η 1 ) ,
č n 1 = i / D 1 n M .
a ˇ 1 n = [ N 1 ξ n ( ρ 1 ) ξ n ( η 1 ) ξ n ( ρ 1 ) ξ n ( η 1 ) ] / D n M 1 ,
d ˇ n 1 = ξ n ( η 1 ) ψ n ( η 1 ) ξ n ( η 1 ) ψ n ( η 1 ) ξ n ( ρ 1 ) ψ n ( η 1 ) N 1 ξ n ( ρ 1 ) ψ n ( η 1 )
= [ ξ n ( η 1 ) ψ n ( η 1 ) ψ n ( η 1 ) ξ n ( η 1 ) ] / D n E 1 = i / D n E 1 ,
b ˇ 1 n = [ N 1 ξ n ( ρ 1 ) ξ n ( η 1 ) ξ n ( ρ 1 ) ξ n ( η 1 ) ] / D n E 1
A E m n 1 = p m n [ a n 1 + c n 1 a n 2 č n 1 k = 0 ( a n 2 a ˇ n 1 ) k ] = p m n ( a n 1 + a n 2 c n 1 č n 1 1 a ˇ n 1 a n 2 ) ,
A H m n 1 = q m n [ b n 1 + d n 1 b n 2 d ˇ n 1 k = 0 ( b n 2 b ˇ n 1 ) k ] = q m n ( b n 1 + b n 2 d n 1 d ˇ n 1 1 b ˇ n 1 b n 2 ) ,
a m n ( 1 ) 2 = a n 2 ν μ ( p μ ν c ν 1 Ã m n μ ν + q μ ν d ν 1 B m n μ ν ) ,
A E m n 1 = j = 1 a m n ( j ) 1 , A H m n 1 = j = 1 b m n ( j ) 1 ,
a m n ( j ) 2 = a n 2 ν μ [ c μ ν ( j ) 1 Ã m n μ ν + d μ ν ( j ) 1 B m n μ ν ] ,
a m n ( j + 1 ) 1 = č n 1 ν μ [ a μ ν ( j ) 2 Ã m n μ ν + b μ ν ( j ) 2 B m n μ ν ] ,
c m n ( j + 1 ) 1 = a ˇ n 1 ν μ [ a μ ν ( j ) 2 Ã m n μ ν + b μ ν ( j ) 2 B m n μ ν ] .
E E = n = 1 m = n n [ p m n N m n ( 1 ) 1 + q m n M m n ( 1 ) 1 + A E m n 1 N m n ( 3 ) 1 + A H m n 1 M m n ( 3 ) 1 ] ,
E H = n = 1 m = n n { C E m n 1 N m n ( 1 ) 1 + C H m n 1 M m n ( 1 ) 1 + l > 1 [ A E μ ν l N m n ( 3 ) l + A H μ ν l M m n ( 3 ) l ] } ,
E I = n = 1 m = n n [ C E m n l N m n ( 1 ) l + C H m n l M m n ( 1 ) l ] ,
A E m n l = a n l ν μ [ C E μ ν 1 Ã m n μ ν + C H μ ν 1 B m n μ ν + l l ( A E μ ν l A m n μ ν + A H μ ν l B m n μ ν ) ] ,
A H m n l = b n l ν μ [ C H μ ν 1 Ã m n μ ν + C E μ ν 1 B m n μ ν + l l ( A H μ ν l A m n μ ν + A E μ ν l B m n μ ν ) ] ,
C E m n 1 = c n 1 p m n + a ˇ n 1 l 1 ν μ ( A E μ ν l à m n μ ν + A H μ ν l B m n μ ν ) ,
C H m n 1 = d n 1 q m n + b ˇ n 1 l 1 ν μ ( A H μ ν l à m n μ ν + A E μ ν l B m n μ ν ) ,
A E m n 1 = a n 1 p m n + č n 1 l 1 ν μ ( A E μ ν l à m n μ ν + A H μ ν l B m n μ ν ) ,
A H m n 1 = b n 1 q m n + d ˇ n 1 l 1 ν μ ( A H μ ν l à m n μ ν + A E μ ν l B m n μ ν ) ,
σ s = σ s 11 + σ s 22 + 2 Re ( σ s 12 ) ,
Re ( σ s 12 ) = 9 π | a 1 1 | 2 k 2 cos ( k d cos α ) × [ 2 a ( 1 , 1 , 1 , 1 , 0 ) j 0 ( k d ) + a ( 1 , 1 , 1 , 1 , 2 ) j 2 ( k d ) ] = 9 π | a 1 1 | 2 k 2 cos ( k d cos α ) k d × [ ( 1 k d 2 1 ) sin ( k d ) cos ( k d ) k d ] ,
Q s ( Q 1 + Q 2 ) = 1 3 2 cos ( k d cos α ) k d × [ ( 1 k d 2 1 ) sin ( k d ) cos ( k d ) k d ] .
A = σ a m = σ a ( specific gravity ) ( particle volume ) .
A = 1 2 0 π A ( α ) sin α d α .

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