Abstract

A two-parameter phase function is described for highly anisotropic angular scattering distributions. The phase function has a convenient analytic representation since it is the generating function for the Gegenbauer (or ultraspherical) polynomials. Sample calculations illustrate the large variety of forward-to-backward scattering amplitudes and shapes that can be fit with the two parameters. Illustrative phase functions are given that approximate light scattering distributions predicted for several biological and atmospheric specimens from Rayleigh-Gans and Mie models. A suggested classification scheme for size and refractive index of microparticles in human blood is presented as a potential application of the phase function with two parameters.

© 1980 Optical Society of America

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  1. L. Henyey and J. Greenstein, "Diffuse Radiation in the Galaxy," Astrophys. J. 93, 70–83 (1941).
  2. G. Pedersen, N. McCormick, and L. Reynolds, "Transport Calculations of Light Scattering in Blood," Biophys. J. 16, 199–207, (1976).
  3. S. Pahor and M. Gros, "Optical Properties of Thick Fog Layers," Tellus 22, 321–327 (1970).
  4. V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, New York, 1975). (Translation of Russian 1972 edition.)
  5. H. G. Kaper, J. K. Shultis, and J. G. Veninga, "Numerical Evaluation of the Slab Albedo Problem Solution in One-Speed Anisotropic Transport Theory," J. Comp. Phys. 6, 288–313 (1970).
  6. R. Sanchez (private communication).
  7. B. Davision, Neutron Transport Theory (Oxford University, London, 1958).
  8. L. Reynolds, "Optical Diffuse Reflectance and Transmittance from an Anisotropically Scattering Finite Blood Medium," Ph.D. dissertation, Department of Electrical Engineering, University of Washington, Seattle, Washington, 1975 (unpublished).
  9. C. E. Siewert and M. M. R. Williams, "The Effect of Anisotropic Scattering on the Critical Slab Problem in Neutron Transport Theory Using a Synthetic Kernel," J. Phys. D: Appl. Phys. 10, 2031–2040 (1977).
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  13. J. Molcho and L. Reynolds, "A Note on the Computation of the Rayleigh-Gans Interference Factor for Particles with Rotational and Fore-Aft Symmetries," J. Colloid Interface Sci. 54, 149–150 (1976).
  14. H. van de Hulst, Light Scattering from Small Particles (Wiley, New York, 1957).
  15. W. Magnus, F. Oberhettinger, and R. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966).
  16. F. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956).

1979

C. Devaux, C. E. Siewert, and Y. Yener, "The Effect of Forward and Backward Scattering on the Spherical Albedo for a Finite Plane-Parallel Atmosphere," J. Quant. Spectrosc. Rad. Transf. 21, 505–509 (1979).

1977

A. Cohen, "Cloud-Base Water Content Measurements Using Single Wavelength Laser-Radar Data," App. Opt. 14, 2873–2877 (1977).

C. E. Siewert and M. M. R. Williams, "The Effect of Anisotropic Scattering on the Critical Slab Problem in Neutron Transport Theory Using a Synthetic Kernel," J. Phys. D: Appl. Phys. 10, 2031–2040 (1977).

1976

G. Pedersen, N. McCormick, and L. Reynolds, "Transport Calculations of Light Scattering in Blood," Biophys. J. 16, 199–207, (1976).

J. Molcho and L. Reynolds, "A Note on the Computation of the Rayleigh-Gans Interference Factor for Particles with Rotational and Fore-Aft Symmetries," J. Colloid Interface Sci. 54, 149–150 (1976).

1970

S. Pahor and M. Gros, "Optical Properties of Thick Fog Layers," Tellus 22, 321–327 (1970).

H. G. Kaper, J. K. Shultis, and J. G. Veninga, "Numerical Evaluation of the Slab Albedo Problem Solution in One-Speed Anisotropic Transport Theory," J. Comp. Phys. 6, 288–313 (1970).

1960

1941

L. Henyey and J. Greenstein, "Diffuse Radiation in the Galaxy," Astrophys. J. 93, 70–83 (1941).

Cohen, A.

A. Cohen, "Cloud-Base Water Content Measurements Using Single Wavelength Laser-Radar Data," App. Opt. 14, 2873–2877 (1977).

Davision, B.

B. Davision, Neutron Transport Theory (Oxford University, London, 1958).

Devaux, C.

C. Devaux, C. E. Siewert, and Y. Yener, "The Effect of Forward and Backward Scattering on the Spherical Albedo for a Finite Plane-Parallel Atmosphere," J. Quant. Spectrosc. Rad. Transf. 21, 505–509 (1979).

Greenstein, J.

L. Henyey and J. Greenstein, "Diffuse Radiation in the Galaxy," Astrophys. J. 93, 70–83 (1941).

Gros, M.

S. Pahor and M. Gros, "Optical Properties of Thick Fog Layers," Tellus 22, 321–327 (1970).

Henyey, L.

L. Henyey and J. Greenstein, "Diffuse Radiation in the Galaxy," Astrophys. J. 93, 70–83 (1941).

Hildebrand, F.

F. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956).

Kaper, H. G.

H. G. Kaper, J. K. Shultis, and J. G. Veninga, "Numerical Evaluation of the Slab Albedo Problem Solution in One-Speed Anisotropic Transport Theory," J. Comp. Phys. 6, 288–313 (1970).

Magnus, W.

W. Magnus, F. Oberhettinger, and R. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966).

McCormick, N.

G. Pedersen, N. McCormick, and L. Reynolds, "Transport Calculations of Light Scattering in Blood," Biophys. J. 16, 199–207, (1976).

Molcho, J.

J. Molcho and L. Reynolds, "A Note on the Computation of the Rayleigh-Gans Interference Factor for Particles with Rotational and Fore-Aft Symmetries," J. Colloid Interface Sci. 54, 149–150 (1976).

Oberhettinger, F.

W. Magnus, F. Oberhettinger, and R. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966).

Pahor, S.

S. Pahor and M. Gros, "Optical Properties of Thick Fog Layers," Tellus 22, 321–327 (1970).

Pedersen, G.

G. Pedersen, N. McCormick, and L. Reynolds, "Transport Calculations of Light Scattering in Blood," Biophys. J. 16, 199–207, (1976).

Reynolds, L.

G. Pedersen, N. McCormick, and L. Reynolds, "Transport Calculations of Light Scattering in Blood," Biophys. J. 16, 199–207, (1976).

J. Molcho and L. Reynolds, "A Note on the Computation of the Rayleigh-Gans Interference Factor for Particles with Rotational and Fore-Aft Symmetries," J. Colloid Interface Sci. 54, 149–150 (1976).

L. Reynolds, "Optical Diffuse Reflectance and Transmittance from an Anisotropically Scattering Finite Blood Medium," Ph.D. dissertation, Department of Electrical Engineering, University of Washington, Seattle, Washington, 1975 (unpublished).

Sanchez, R.

R. Sanchez (private communication).

Shultis, J. K.

H. G. Kaper, J. K. Shultis, and J. G. Veninga, "Numerical Evaluation of the Slab Albedo Problem Solution in One-Speed Anisotropic Transport Theory," J. Comp. Phys. 6, 288–313 (1970).

Siewert, C. E.

C. Devaux, C. E. Siewert, and Y. Yener, "The Effect of Forward and Backward Scattering on the Spherical Albedo for a Finite Plane-Parallel Atmosphere," J. Quant. Spectrosc. Rad. Transf. 21, 505–509 (1979).

C. E. Siewert and M. M. R. Williams, "The Effect of Anisotropic Scattering on the Critical Slab Problem in Neutron Transport Theory Using a Synthetic Kernel," J. Phys. D: Appl. Phys. 10, 2031–2040 (1977).

Sobolev, V. V.

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, New York, 1975). (Translation of Russian 1972 edition.)

Soni, R.

W. Magnus, F. Oberhettinger, and R. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966).

Spencer, D.

van de Hulst, H.

H. van de Hulst, Light Scattering from Small Particles (Wiley, New York, 1957).

Veninga, J. G.

H. G. Kaper, J. K. Shultis, and J. G. Veninga, "Numerical Evaluation of the Slab Albedo Problem Solution in One-Speed Anisotropic Transport Theory," J. Comp. Phys. 6, 288–313 (1970).

Williams, M. M. R.

C. E. Siewert and M. M. R. Williams, "The Effect of Anisotropic Scattering on the Critical Slab Problem in Neutron Transport Theory Using a Synthetic Kernel," J. Phys. D: Appl. Phys. 10, 2031–2040 (1977).

Yener, Y.

C. Devaux, C. E. Siewert, and Y. Yener, "The Effect of Forward and Backward Scattering on the Spherical Albedo for a Finite Plane-Parallel Atmosphere," J. Quant. Spectrosc. Rad. Transf. 21, 505–509 (1979).

App. Opt.

A. Cohen, "Cloud-Base Water Content Measurements Using Single Wavelength Laser-Radar Data," App. Opt. 14, 2873–2877 (1977).

Astrophys. J.

L. Henyey and J. Greenstein, "Diffuse Radiation in the Galaxy," Astrophys. J. 93, 70–83 (1941).

Biophys. J.

G. Pedersen, N. McCormick, and L. Reynolds, "Transport Calculations of Light Scattering in Blood," Biophys. J. 16, 199–207, (1976).

J. Colloid Interface Sci.

J. Molcho and L. Reynolds, "A Note on the Computation of the Rayleigh-Gans Interference Factor for Particles with Rotational and Fore-Aft Symmetries," J. Colloid Interface Sci. 54, 149–150 (1976).

J. Comp. Phys.

H. G. Kaper, J. K. Shultis, and J. G. Veninga, "Numerical Evaluation of the Slab Albedo Problem Solution in One-Speed Anisotropic Transport Theory," J. Comp. Phys. 6, 288–313 (1970).

J. Opt. Soc. Am.

J. Phys. D: Appl. Phys.

C. E. Siewert and M. M. R. Williams, "The Effect of Anisotropic Scattering on the Critical Slab Problem in Neutron Transport Theory Using a Synthetic Kernel," J. Phys. D: Appl. Phys. 10, 2031–2040 (1977).

J. Quant. Spectrosc. Rad. Transf.

C. Devaux, C. E. Siewert, and Y. Yener, "The Effect of Forward and Backward Scattering on the Spherical Albedo for a Finite Plane-Parallel Atmosphere," J. Quant. Spectrosc. Rad. Transf. 21, 505–509 (1979).

Tellus

S. Pahor and M. Gros, "Optical Properties of Thick Fog Layers," Tellus 22, 321–327 (1970).

Other

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, New York, 1975). (Translation of Russian 1972 edition.)

R. Sanchez (private communication).

B. Davision, Neutron Transport Theory (Oxford University, London, 1958).

L. Reynolds, "Optical Diffuse Reflectance and Transmittance from an Anisotropically Scattering Finite Blood Medium," Ph.D. dissertation, Department of Electrical Engineering, University of Washington, Seattle, Washington, 1975 (unpublished).

H. van de Hulst, Light Scattering from Small Particles (Wiley, New York, 1957).

W. Magnus, F. Oberhettinger, and R. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966).

F. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1956).

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