Abstract

We have measured the differential scattering cross sections (phase functions I22) and the normalized extinction and scattering cross sections (efficiences) of composite spherical particles. The size parameter x = 2πr/λ was around 2π. Composite spheres consisted of nonabsorbing matrix containing a small amount (1.6 and 2.7% by volume) of highly absorbing inclusions. Such composite particles may represent a realistic model of fog or cloud droplets containing small amounts of carbon or a composite atmospheric aerosol particle. We have compared measured data with those calculated using seven different effective medium approximations. We have found that the approximations of Bruggeman and Maxwell Garnett, the generalization of dynamic effective medium approximation derived by Chylek and Srivastava, and the experimental waveguide method of determination of the effective refractive index lead to an acceptable agreement between calculated and measured values. The reduced χ2 values for these approximations ranged between 0.6 and 2.0. The remaining three approximations (volume averages of refractive indices or dielectric constants and the Maxwell Garnett relation with matrix and inclusion materials interchanged) lead to reduced χ2 values between 4.0 and 12.0 demonstrating a large disagreement between calculated and measured scattering characteristics.

© 1988 Optical Society of America

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References

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  1. H. Rosen, A. D. Hansen, L. Gundel, T. Novakov, “Identification of the Graphitic Carbon Component of Source and Ambient Particulates by Raman Spectroscopy and an Optical Attenuation Technique,” Appl. Opt. 17, 3859 (1978).
    [CrossRef] [PubMed]
  2. G. W. Grams, I. H. Blifford, D. A. Gillette, P. B. Russell, “Complex Index of Refraction of Airborne Soil Particle,” J. Appl. Meteorol. 13, 459 (1974).
    [CrossRef]
  3. T. A. Cahill, R. A. Ellred, “Elemental Composition of Arctic Particulate Matter,” Geophys. Res. Lett. 11, 413 (1984).
    [CrossRef]
  4. S. G. Warren, W. J. Wiscombe, “A Model for the Spectral Albedo of Snow: Snow Containing Atmospheric Aerosols,” J. Atmos. Sci. 37, 2734 (1980).
    [CrossRef]
  5. P. Chylek, V. Ramaswamy, V. Srivastava, “Albedo of Soot-Contaminated Snow,” J. Geophys. Res. 88, (C15), 10,837 (1983).
    [CrossRef]
  6. P. Chylek, V. Ramaswamy, R. Cheng, “Effect of Graphitic Carbon on the Albedo of Clouds,” J. Atmos Sci. 41, 3076 (1984).
    [CrossRef]
  7. P. Chylek, B. R. D. Gupta, N. C. Knight, C. A. Knight, “Distribution of Water in Hailstones,” J. Climate Appl. Meterol. 23, 1469 (1984).
    [CrossRef]
  8. D. Atlas, K. R. Hardy, J. Joss, “Radar Reflectivity of Storms Containing Spongy Hail,” J. Geophys. Res. 69, 1955 (1964).
    [CrossRef]
  9. J. Joss, A. N. Aufdermaur, “Experimental Determination of Radar Cross Sections of Artificial Hailstones Containing Water,” J. Appl. Meteorol. 4, 723 (1965).
    [CrossRef]
  10. A. L. Alden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
    [CrossRef]
  11. R. W. Fenn, H. Oser, “Scattering Properties of Concentric Soot–Water Spheres for Visible and Infrared Light,” Appl. Opt. 4, 1504 (1965).
    [CrossRef]
  12. R. G. Pinnick, S. G. Jennings, D. C. Boice, J. P. Cruncleton, “Attenuated Total Reflectance Measurements of the Complex Refractive Index of Laolinite Powder at CO2 Laser Wavelengths,” Appl. Opt. 24, 3274 (1985).
    [CrossRef] [PubMed]
  13. P. Chylek, V. Ramaswamy, R. Cheng, R. G. Pinnick, “Optical Properties and Mass Concentrations of Carbonaceous Smokes,” Appl. Opt. 20, 2980 (1981).
    [CrossRef] [PubMed]
  14. M. Sitarski, “Absorption of Infrared Radiation Inside an Explosively Boiling Fine Coal-Water Particle,” Particulate Sci. Technol. in press (1988).
  15. M. Sitarski, “On the Feasibility of Secondary Atomization of Small Slurry Droplets Exposed to Intense Thermal Radiation,” Combust. Sci. Technol. 54, 177 (1987).
    [CrossRef]
  16. G. A. Niklasson, C. G. Granqvist, O. Hunderi, “Effective Medium Models for the Optical Properties of Inhomogeneous Materials,” Appl. Opt. 20, 26 (1981).
    [CrossRef] [PubMed]
  17. J. B. Gillespie, S. G. Jennings, J. D. Lindberg, “Use of an Average Complex Refractive Index in Atmospheric Propagation Calculations,” Appl. Opt. 17, 989 (1978).
    [CrossRef] [PubMed]
  18. A. C. Lind, R. T. Wang, J. M. Greenberg, “Microwave Scattering by Nonspherical Particles,” Appl. Opt. 4, 1556 (1965).
    [CrossRef]
  19. P. Chylek, V. Srivastava, “Dielectric Constant of a Composite Inhomogeneous Medium,” Phys. Rev. B 27, 5098 (1983).
    [CrossRef]
  20. C. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Subsganzen,” Ann. Phys. Leipzig 24, 636 (1935).
    [CrossRef]
  21. J. C. Maxwell Garnett, “Colours in Metal Glasses and in Metallic Films,” Philos. Trans. R. Soc. London 203, 385 (1904).
    [CrossRef]
  22. D. Stroud, F. P. Pan, “Self-Consistent Approach to Electromagnetic Wave Propagation in Composite Media: Application to Model Granular Metals,” Phys. Rev. B 17, 1602 (1978).
    [CrossRef]
  23. S. Roberts, A. von Hippel, “A New Method of Measuring Dielectric Constant and Loss in the Range of Centimeter Waves,” J. Appl. Phys. 17, 610 (1946).
    [CrossRef]
  24. J. R. Taylor, An Introduction to Error Analysis (University Science Books, Mill Valley, CA, 1982), p. 221.
  25. P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969), p. 189.
  26. C. F. Bohren, “Applicability of Effective Medium Theories to Problems of Scattering and Absorption by Nonhomogeneous Atmospheric Particles,” J. Atmos. Sci. 43, 468 (1986).
    [CrossRef]

1987 (1)

M. Sitarski, “On the Feasibility of Secondary Atomization of Small Slurry Droplets Exposed to Intense Thermal Radiation,” Combust. Sci. Technol. 54, 177 (1987).
[CrossRef]

1986 (1)

C. F. Bohren, “Applicability of Effective Medium Theories to Problems of Scattering and Absorption by Nonhomogeneous Atmospheric Particles,” J. Atmos. Sci. 43, 468 (1986).
[CrossRef]

1985 (1)

1984 (3)

T. A. Cahill, R. A. Ellred, “Elemental Composition of Arctic Particulate Matter,” Geophys. Res. Lett. 11, 413 (1984).
[CrossRef]

P. Chylek, V. Ramaswamy, R. Cheng, “Effect of Graphitic Carbon on the Albedo of Clouds,” J. Atmos Sci. 41, 3076 (1984).
[CrossRef]

P. Chylek, B. R. D. Gupta, N. C. Knight, C. A. Knight, “Distribution of Water in Hailstones,” J. Climate Appl. Meterol. 23, 1469 (1984).
[CrossRef]

1983 (2)

P. Chylek, V. Ramaswamy, V. Srivastava, “Albedo of Soot-Contaminated Snow,” J. Geophys. Res. 88, (C15), 10,837 (1983).
[CrossRef]

P. Chylek, V. Srivastava, “Dielectric Constant of a Composite Inhomogeneous Medium,” Phys. Rev. B 27, 5098 (1983).
[CrossRef]

1981 (2)

1980 (1)

S. G. Warren, W. J. Wiscombe, “A Model for the Spectral Albedo of Snow: Snow Containing Atmospheric Aerosols,” J. Atmos. Sci. 37, 2734 (1980).
[CrossRef]

1978 (3)

1974 (1)

G. W. Grams, I. H. Blifford, D. A. Gillette, P. B. Russell, “Complex Index of Refraction of Airborne Soil Particle,” J. Appl. Meteorol. 13, 459 (1974).
[CrossRef]

1965 (3)

J. Joss, A. N. Aufdermaur, “Experimental Determination of Radar Cross Sections of Artificial Hailstones Containing Water,” J. Appl. Meteorol. 4, 723 (1965).
[CrossRef]

A. C. Lind, R. T. Wang, J. M. Greenberg, “Microwave Scattering by Nonspherical Particles,” Appl. Opt. 4, 1556 (1965).
[CrossRef]

R. W. Fenn, H. Oser, “Scattering Properties of Concentric Soot–Water Spheres for Visible and Infrared Light,” Appl. Opt. 4, 1504 (1965).
[CrossRef]

1964 (1)

D. Atlas, K. R. Hardy, J. Joss, “Radar Reflectivity of Storms Containing Spongy Hail,” J. Geophys. Res. 69, 1955 (1964).
[CrossRef]

1951 (1)

A. L. Alden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

1946 (1)

S. Roberts, A. von Hippel, “A New Method of Measuring Dielectric Constant and Loss in the Range of Centimeter Waves,” J. Appl. Phys. 17, 610 (1946).
[CrossRef]

1935 (1)

C. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Subsganzen,” Ann. Phys. Leipzig 24, 636 (1935).
[CrossRef]

1904 (1)

J. C. Maxwell Garnett, “Colours in Metal Glasses and in Metallic Films,” Philos. Trans. R. Soc. London 203, 385 (1904).
[CrossRef]

Alden, A. L.

A. L. Alden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

Atlas, D.

D. Atlas, K. R. Hardy, J. Joss, “Radar Reflectivity of Storms Containing Spongy Hail,” J. Geophys. Res. 69, 1955 (1964).
[CrossRef]

Aufdermaur, A. N.

J. Joss, A. N. Aufdermaur, “Experimental Determination of Radar Cross Sections of Artificial Hailstones Containing Water,” J. Appl. Meteorol. 4, 723 (1965).
[CrossRef]

Bevington, P. R.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969), p. 189.

Blifford, I. H.

G. W. Grams, I. H. Blifford, D. A. Gillette, P. B. Russell, “Complex Index of Refraction of Airborne Soil Particle,” J. Appl. Meteorol. 13, 459 (1974).
[CrossRef]

Bohren, C. F.

C. F. Bohren, “Applicability of Effective Medium Theories to Problems of Scattering and Absorption by Nonhomogeneous Atmospheric Particles,” J. Atmos. Sci. 43, 468 (1986).
[CrossRef]

Boice, D. C.

Bruggeman, C. A. G.

C. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Subsganzen,” Ann. Phys. Leipzig 24, 636 (1935).
[CrossRef]

Cahill, T. A.

T. A. Cahill, R. A. Ellred, “Elemental Composition of Arctic Particulate Matter,” Geophys. Res. Lett. 11, 413 (1984).
[CrossRef]

Cheng, R.

P. Chylek, V. Ramaswamy, R. Cheng, “Effect of Graphitic Carbon on the Albedo of Clouds,” J. Atmos Sci. 41, 3076 (1984).
[CrossRef]

P. Chylek, V. Ramaswamy, R. Cheng, R. G. Pinnick, “Optical Properties and Mass Concentrations of Carbonaceous Smokes,” Appl. Opt. 20, 2980 (1981).
[CrossRef] [PubMed]

Chylek, P.

P. Chylek, B. R. D. Gupta, N. C. Knight, C. A. Knight, “Distribution of Water in Hailstones,” J. Climate Appl. Meterol. 23, 1469 (1984).
[CrossRef]

P. Chylek, V. Ramaswamy, R. Cheng, “Effect of Graphitic Carbon on the Albedo of Clouds,” J. Atmos Sci. 41, 3076 (1984).
[CrossRef]

P. Chylek, V. Ramaswamy, V. Srivastava, “Albedo of Soot-Contaminated Snow,” J. Geophys. Res. 88, (C15), 10,837 (1983).
[CrossRef]

P. Chylek, V. Srivastava, “Dielectric Constant of a Composite Inhomogeneous Medium,” Phys. Rev. B 27, 5098 (1983).
[CrossRef]

P. Chylek, V. Ramaswamy, R. Cheng, R. G. Pinnick, “Optical Properties and Mass Concentrations of Carbonaceous Smokes,” Appl. Opt. 20, 2980 (1981).
[CrossRef] [PubMed]

Cruncleton, J. P.

Ellred, R. A.

T. A. Cahill, R. A. Ellred, “Elemental Composition of Arctic Particulate Matter,” Geophys. Res. Lett. 11, 413 (1984).
[CrossRef]

Fenn, R. W.

Gillespie, J. B.

Gillette, D. A.

G. W. Grams, I. H. Blifford, D. A. Gillette, P. B. Russell, “Complex Index of Refraction of Airborne Soil Particle,” J. Appl. Meteorol. 13, 459 (1974).
[CrossRef]

Grams, G. W.

G. W. Grams, I. H. Blifford, D. A. Gillette, P. B. Russell, “Complex Index of Refraction of Airborne Soil Particle,” J. Appl. Meteorol. 13, 459 (1974).
[CrossRef]

Granqvist, C. G.

Greenberg, J. M.

A. C. Lind, R. T. Wang, J. M. Greenberg, “Microwave Scattering by Nonspherical Particles,” Appl. Opt. 4, 1556 (1965).
[CrossRef]

Gundel, L.

Gupta, B. R. D.

P. Chylek, B. R. D. Gupta, N. C. Knight, C. A. Knight, “Distribution of Water in Hailstones,” J. Climate Appl. Meterol. 23, 1469 (1984).
[CrossRef]

Hansen, A. D.

Hardy, K. R.

D. Atlas, K. R. Hardy, J. Joss, “Radar Reflectivity of Storms Containing Spongy Hail,” J. Geophys. Res. 69, 1955 (1964).
[CrossRef]

Hunderi, O.

Jennings, S. G.

Joss, J.

J. Joss, A. N. Aufdermaur, “Experimental Determination of Radar Cross Sections of Artificial Hailstones Containing Water,” J. Appl. Meteorol. 4, 723 (1965).
[CrossRef]

D. Atlas, K. R. Hardy, J. Joss, “Radar Reflectivity of Storms Containing Spongy Hail,” J. Geophys. Res. 69, 1955 (1964).
[CrossRef]

Kerker, M.

A. L. Alden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

Knight, C. A.

P. Chylek, B. R. D. Gupta, N. C. Knight, C. A. Knight, “Distribution of Water in Hailstones,” J. Climate Appl. Meterol. 23, 1469 (1984).
[CrossRef]

Knight, N. C.

P. Chylek, B. R. D. Gupta, N. C. Knight, C. A. Knight, “Distribution of Water in Hailstones,” J. Climate Appl. Meterol. 23, 1469 (1984).
[CrossRef]

Lind, A. C.

A. C. Lind, R. T. Wang, J. M. Greenberg, “Microwave Scattering by Nonspherical Particles,” Appl. Opt. 4, 1556 (1965).
[CrossRef]

Lindberg, J. D.

Maxwell Garnett, J. C.

J. C. Maxwell Garnett, “Colours in Metal Glasses and in Metallic Films,” Philos. Trans. R. Soc. London 203, 385 (1904).
[CrossRef]

Niklasson, G. A.

Novakov, T.

Oser, H.

Pan, F. P.

D. Stroud, F. P. Pan, “Self-Consistent Approach to Electromagnetic Wave Propagation in Composite Media: Application to Model Granular Metals,” Phys. Rev. B 17, 1602 (1978).
[CrossRef]

Pinnick, R. G.

Ramaswamy, V.

P. Chylek, V. Ramaswamy, R. Cheng, “Effect of Graphitic Carbon on the Albedo of Clouds,” J. Atmos Sci. 41, 3076 (1984).
[CrossRef]

P. Chylek, V. Ramaswamy, V. Srivastava, “Albedo of Soot-Contaminated Snow,” J. Geophys. Res. 88, (C15), 10,837 (1983).
[CrossRef]

P. Chylek, V. Ramaswamy, R. Cheng, R. G. Pinnick, “Optical Properties and Mass Concentrations of Carbonaceous Smokes,” Appl. Opt. 20, 2980 (1981).
[CrossRef] [PubMed]

Roberts, S.

S. Roberts, A. von Hippel, “A New Method of Measuring Dielectric Constant and Loss in the Range of Centimeter Waves,” J. Appl. Phys. 17, 610 (1946).
[CrossRef]

Rosen, H.

Russell, P. B.

G. W. Grams, I. H. Blifford, D. A. Gillette, P. B. Russell, “Complex Index of Refraction of Airborne Soil Particle,” J. Appl. Meteorol. 13, 459 (1974).
[CrossRef]

Sitarski, M.

M. Sitarski, “On the Feasibility of Secondary Atomization of Small Slurry Droplets Exposed to Intense Thermal Radiation,” Combust. Sci. Technol. 54, 177 (1987).
[CrossRef]

M. Sitarski, “Absorption of Infrared Radiation Inside an Explosively Boiling Fine Coal-Water Particle,” Particulate Sci. Technol. in press (1988).

Srivastava, V.

P. Chylek, V. Srivastava, “Dielectric Constant of a Composite Inhomogeneous Medium,” Phys. Rev. B 27, 5098 (1983).
[CrossRef]

P. Chylek, V. Ramaswamy, V. Srivastava, “Albedo of Soot-Contaminated Snow,” J. Geophys. Res. 88, (C15), 10,837 (1983).
[CrossRef]

Stroud, D.

D. Stroud, F. P. Pan, “Self-Consistent Approach to Electromagnetic Wave Propagation in Composite Media: Application to Model Granular Metals,” Phys. Rev. B 17, 1602 (1978).
[CrossRef]

Taylor, J. R.

J. R. Taylor, An Introduction to Error Analysis (University Science Books, Mill Valley, CA, 1982), p. 221.

von Hippel, A.

S. Roberts, A. von Hippel, “A New Method of Measuring Dielectric Constant and Loss in the Range of Centimeter Waves,” J. Appl. Phys. 17, 610 (1946).
[CrossRef]

Wang, R. T.

A. C. Lind, R. T. Wang, J. M. Greenberg, “Microwave Scattering by Nonspherical Particles,” Appl. Opt. 4, 1556 (1965).
[CrossRef]

Warren, S. G.

S. G. Warren, W. J. Wiscombe, “A Model for the Spectral Albedo of Snow: Snow Containing Atmospheric Aerosols,” J. Atmos. Sci. 37, 2734 (1980).
[CrossRef]

Wiscombe, W. J.

S. G. Warren, W. J. Wiscombe, “A Model for the Spectral Albedo of Snow: Snow Containing Atmospheric Aerosols,” J. Atmos. Sci. 37, 2734 (1980).
[CrossRef]

Ann. Phys. Leipzig (1)

C. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Subsganzen,” Ann. Phys. Leipzig 24, 636 (1935).
[CrossRef]

Appl. Opt. (7)

Combust. Sci. Technol. (1)

M. Sitarski, “On the Feasibility of Secondary Atomization of Small Slurry Droplets Exposed to Intense Thermal Radiation,” Combust. Sci. Technol. 54, 177 (1987).
[CrossRef]

Geophys. Res. Lett. (1)

T. A. Cahill, R. A. Ellred, “Elemental Composition of Arctic Particulate Matter,” Geophys. Res. Lett. 11, 413 (1984).
[CrossRef]

J. Appl. Meteorol. (2)

G. W. Grams, I. H. Blifford, D. A. Gillette, P. B. Russell, “Complex Index of Refraction of Airborne Soil Particle,” J. Appl. Meteorol. 13, 459 (1974).
[CrossRef]

J. Joss, A. N. Aufdermaur, “Experimental Determination of Radar Cross Sections of Artificial Hailstones Containing Water,” J. Appl. Meteorol. 4, 723 (1965).
[CrossRef]

J. Appl. Phys. (2)

A. L. Alden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

S. Roberts, A. von Hippel, “A New Method of Measuring Dielectric Constant and Loss in the Range of Centimeter Waves,” J. Appl. Phys. 17, 610 (1946).
[CrossRef]

J. Atmos Sci. (1)

P. Chylek, V. Ramaswamy, R. Cheng, “Effect of Graphitic Carbon on the Albedo of Clouds,” J. Atmos Sci. 41, 3076 (1984).
[CrossRef]

J. Atmos. Sci. (2)

S. G. Warren, W. J. Wiscombe, “A Model for the Spectral Albedo of Snow: Snow Containing Atmospheric Aerosols,” J. Atmos. Sci. 37, 2734 (1980).
[CrossRef]

C. F. Bohren, “Applicability of Effective Medium Theories to Problems of Scattering and Absorption by Nonhomogeneous Atmospheric Particles,” J. Atmos. Sci. 43, 468 (1986).
[CrossRef]

J. Climate Appl. Meterol. (1)

P. Chylek, B. R. D. Gupta, N. C. Knight, C. A. Knight, “Distribution of Water in Hailstones,” J. Climate Appl. Meterol. 23, 1469 (1984).
[CrossRef]

J. Geophys. Res. (2)

D. Atlas, K. R. Hardy, J. Joss, “Radar Reflectivity of Storms Containing Spongy Hail,” J. Geophys. Res. 69, 1955 (1964).
[CrossRef]

P. Chylek, V. Ramaswamy, V. Srivastava, “Albedo of Soot-Contaminated Snow,” J. Geophys. Res. 88, (C15), 10,837 (1983).
[CrossRef]

Philos. Trans. R. Soc. London (1)

J. C. Maxwell Garnett, “Colours in Metal Glasses and in Metallic Films,” Philos. Trans. R. Soc. London 203, 385 (1904).
[CrossRef]

Phys. Rev. B (2)

D. Stroud, F. P. Pan, “Self-Consistent Approach to Electromagnetic Wave Propagation in Composite Media: Application to Model Granular Metals,” Phys. Rev. B 17, 1602 (1978).
[CrossRef]

P. Chylek, V. Srivastava, “Dielectric Constant of a Composite Inhomogeneous Medium,” Phys. Rev. B 27, 5098 (1983).
[CrossRef]

Other (3)

M. Sitarski, “Absorption of Infrared Radiation Inside an Explosively Boiling Fine Coal-Water Particle,” Particulate Sci. Technol. in press (1988).

J. R. Taylor, An Introduction to Error Analysis (University Science Books, Mill Valley, CA, 1982), p. 221.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969), p. 189.

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Figures (8)

Fig. 1
Fig. 1

Rectangular piece of composite material containing 1.6% water in an acrylic matrix. Water is dispersed in acrylic in a form of spherical polydispersion.

Fig. 2
Fig. 2

Schematic diagram of the experimental arrangement consisting of a microwave source 1, transmitting antenna 2, target 3, and receiving antenna 4 attached to a movable cart 5. The He–Ne laser 6 is used to position the target properly in the beam. The background signal scattered by the walls and the absorbing material 7 is canceled by the proper attenuation and phase adjustment 8 of the reference signal 9. The computer-controlled 12 target orientation mechanism 13 and a lock-in amplifier 10 are located in a control room 11. The second receiving antenna 14 and attenuator-phase shifter 15 are used for extinction measurements.

Fig. 3
Fig. 3

Comparison of measured and calculated (using Mie theory) phase function I22 for the case of the homogeneous acrylic sphere.

Fig. 4
Fig. 4

Measured phase function I22 of the pure acrylic sphere and spheres of water–acrylic mixtures containing 1.6 and 2.7% water.

Fig. 5
Fig. 5

Comparison of the measured phase function I22 of a composite water–acrylic sphere containing 1.6% water with Mie scattering calculations using the Bruggeman (solid line, χ2 = 0.80) and a Chylek-Srivastava iterative scheme (dashed line, χ2 = 0.62). Results using the Maxwell Garnett mixing rule for water inclusions in acrylic matrix are indistinguishable from those using the Bruggeman rule; and results using the waveguide measurement of the refractive index of an acrylic–water mixture are indistinguishable from those obtained using the Chylek-Srivastava iteration scheme.

Fig. 6
Fig. 6

Comparison of measured phase function I22 of a water–acrylic sphere (1.6% water) with Mie calculations using the volume average of refractive indices (long–short dash line, χ2 = 6.01), volume average of dielectric constants (short dash line, χ2 = 11.45) and Maxwell Garnett rule for acrylic inclusions in a water matrix (long dash line, χ2 = 10.66). Waveguide measurement of the refractive index of a water–acrylic mixture leads to χ2 = 0.62 (solid line).

Fig. 7
Fig. 7

Same as Fig. 5, however, for 2.7% water by volume.

Fig. 8
Fig. 8

Same as Fig. 6 but for 2.7% water.

Tables (7)

Tables Icon

Table I Spherical Targets of Pure Acrylic and Acrylic–Water Composite Used in Experimental Measurements

Tables Icon

Table II Phase Function Measurements of Homogeneous Acrylic Sphere and Acrylic-Water Composite Spheres for Linearly Polarized Radiation of Wavelength λ = 3.1835 cm with Electric Vector E in the Scattering Plane

Tables Icon

Table III Results of a Waveguide Method of Determination of the Refractive Index of Pure Acrylic and of the Effective Refractive Index m7 of the Acrylic–Water Composite with the Mode Radius of Spherical Water Inclusions rm = 0.02 cm

Tables Icon

Table IV Measured Normalized Extinction Cross Section Qext and Normalized Scattering Cross-Section Qsc Obtained by Numerical Integration of the Measured Phase Function and Calculated Normalized Absorption Cross-Section Qabs of Investigated Targets at the Wavelength λ = 3.1835 cm

Tables Icon

Table V Effective Refractive Indices of Acrylic–Water Composite Material Calculated Using the Eight Different Effective Medium Approximation Described in the Text

Tables Icon

Table VI Reduced χ2 Values Characterizing the Goodness of the Fit Between the Measured Phase Function and Phase Function Calculated Using Considered Effective Medium Approximations

Tables Icon

Table VII Normalized Extinction, Scattering, and Absorption Cross Section of Acrylic–Water Composite Spheres at λ = 3.1835 cm Calculated from the Mie Theory Using Effective Refractive Indices from Table V

Equations (12)

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

m 1 = v 0 m 0 + v A m A ,
m 2 2 = v 0 m 0 2 + v A m A 2 ,
v 0 m 0 2 - m 3 2 m 0 2 + 2 m 3 2 + v A m A 2 - m 3 2 m A 2 + 2 m 3 2 = 0.
m 4 2 = m 0 2 m A 2 + 2 m 0 2 + 2 v A ( m A 2 - m 0 2 ) m A 2 + 2 m 0 2 - v A ( m A 2 - m 0 2 ) .
m 5 2 = m A 2 m 0 2 + 2 m A 2 + 2 v 0 ( m 0 2 - m A 2 ) m 0 2 + 2 m A 2 - v 0 ( m 0 2 - m A 2 ) .
m eff 3 12 π 2 λ 3 i ( 1 - v A ) m 0 2 - m eff 2 m 0 2 + 2 m eff 2 + 3 v A 4 π k = 1 ( 2 k + 1 ) n ( r ) [ a k ( r , m ¯ A ) + b k ( r , m ¯ A ) ] d r = 0 ,
A ( m eff ) = i 12 π 2 λ 3 m eff 3 ,
B ( m eff ) = k = 1 ( 2 k + 1 ) n ( r ) [ a k ( r , m ¯ A ) + b k ( r , m ¯ A ) ] d r ,
( m eff 2 ) j + 1 = m 0 2 A j ( 1 - v A ) + B j A j ( 1 - v A ) - 2 B j ,
n ( r ) = a r α exp ( - α r / r m )
χ 2 = 1 N i = 1 N ( I i m - I i c ) 2 σ i 2
σ i 2 = 1 2 ( I i m + I i c ) ,

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