Abstract

Traditional methods of determining the optical constants of particulate materials by means of transmission, absorption, and reflectance measurements are known to be inherently inaccurate. The use of the Lorenz–Mie formalism to derive the optical constants from extinction data overcomes the problems associated with the traditional methods; but, as currently practiced, this method has severe limitations. In this paper we report an entirely new approach to determining the optical constants of aerosols from extinction data. This is an iterative method that uses the Lorenz–Mie formalism in conjunction with the Kramers–Kronig dispersion relations in order to derive the optical constants of the aerosol material. The theory of the method is developed in detail and is applied successfully to find the optical constants of an o-phosphoric-acid aerosol in the 7–14-μm infrared. The numerical procedure is shown to introduce an error of less than 1% in the determination of the o-phosphoric-acid optical constants. Limits on n, k and the particle size distribution for which the method is valid are indicated.

© 1981 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  3. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  4. J. V. Dave, Rep. no. 320-3237, IBM Scientific Center, Palo Alto, California (1968).
  5. G. Grehan and G. Gouesbet, “Mie theory calculations: new progress, with emphasis on particle sizing,” Appl. Opt. 18, 3489–3493 (1979).
    [CrossRef] [PubMed]
  6. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  7. W. C. Wells, G. Gal, and M. W. Munn, “Aerosol distributions in maritime air and predicted scattering coefficients in the infrared,” Appl. Opt. 16, 654–659 (1977).
    [CrossRef] [PubMed]
  8. E. M. Patterson, “Atmospheric extinction between 0.55 μ m and 10.6 μ m due to soil-derived aerosols,” Appl. Opt. 16, 2414–2418 (1977).
    [CrossRef] [PubMed]
  9. S. G. Jennings, R. G. Pinnick, and H. J. Auvermann, “Effects of particulate complex refractive index and particle size distribution variations on atmospheric extinction and absorption for visible through middle IR wavelengths,” Appl. Opt. 17, 3922–3929 (1978).
    [CrossRef] [PubMed]
  10. D. M. Roessler and F. R. Faxvog, “Opacity of black smoke: calculated variation with particulate size and refractive index,” Appl. Opt. 18, 1399–1403 (1979).
    [CrossRef] [PubMed]
  11. S. G. Jennings, R. G. Pinnick, and J. B. Gillespie, “Relation between absorption coefficient and imaginary index of atmospheric aerosol constituents,” Appl. Opt. 18, 1368–1371 (1979).
    [CrossRef] [PubMed]
  12. R. W. Bergstrom, “Extinction and absorption coefficients of the atmospheric aerosol as a function of particle size,” Beitr. Phys. Atmos. 46, 198–202 (1973).
  13. O. B. Toon, J. B. Pollack, and B. N. Khare, “The optical constants of several atmospheric aerosol species: ammonium sulfate, aluminum oxide, and sodium chloride,” J. Geophys. Res. 81, 5733–5748 (1976).
    [CrossRef]
  14. J. Janzen, “The refractive index of colloidal carbon,” J. Colloid Interface Sci. 69, 436–447 (1979).
    [CrossRef]
  15. P. J. Wyatt, “Some chemical, physical, and optical properties of fly ash particles,” Appl. Opt. 19, 975–983 (1980).
    [CrossRef] [PubMed]
  16. A. B. Pluchino, S. S. Goldberg, J. M. Dowling, and C. M. Randall, “Refractive-index measurements of single micron-sized carbon particles,” Appl. Opt. 19, 3371–3372 (1980).
    [CrossRef]
  17. M. Cardona, Optical Properties of Solids, S. Nudelman and S. S. Mitra, eds. (Plenum, New York, 1969).
  18. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).
  19. R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TiBr and TiCl,” Phys. Rev. B 1, 818–831 (1970).
    [CrossRef]
  20. R. K. Ahrenkiel, “Modified Kramers–Kronig analysis of optical spectra,” J. Opt. Soc. Am. 61, 1651–1655 (1971).
    [CrossRef]
  21. G. M. Hale and M. R. Querry, “Optical constants of water in the 200-nm to 200-μ m wavelength region,” Appl. Opt. 12, 555–563 (1973).
    [CrossRef] [PubMed]
  22. D. P. Gilra, “Collective excitations in small solid particles and astronomical applications,” Ph.D. thesis, University of Wisconsin, Madison, Wisconsin (1972).
  23. M. R. Querry, University of Missouri-Kansas City, Kansas City, Missouri 64110, personal communication (1978).
  24. M. R. Querry, “Molecular and crystalline electromagnetic properties of selected condensed materials in the infrared,” (1979).

1980 (3)

A. B. Pluchino, S. S. Goldberg, J. M. Dowling, and C. M. Randall, “Refractive-index measurements of single micron-sized carbon particles,” Appl. Opt. 19, 3371–3372 (1980).
[CrossRef]

P. J. Wyatt, “Some chemical, physical, and optical properties of fly ash particles,” Appl. Opt. 19, 975–983 (1980).
[CrossRef] [PubMed]

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

1979 (4)

1978 (1)

1977 (2)

1976 (1)

O. B. Toon, J. B. Pollack, and B. N. Khare, “The optical constants of several atmospheric aerosol species: ammonium sulfate, aluminum oxide, and sodium chloride,” J. Geophys. Res. 81, 5733–5748 (1976).
[CrossRef]

1973 (2)

R. W. Bergstrom, “Extinction and absorption coefficients of the atmospheric aerosol as a function of particle size,” Beitr. Phys. Atmos. 46, 198–202 (1973).

G. M. Hale and M. R. Querry, “Optical constants of water in the 200-nm to 200-μ m wavelength region,” Appl. Opt. 12, 555–563 (1973).
[CrossRef] [PubMed]

1971 (1)

1970 (1)

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TiBr and TiCl,” Phys. Rev. B 1, 818–831 (1970).
[CrossRef]

Ahrenkiel, R. K.

Auvermann, H. J.

Bachrach, R. Z.

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TiBr and TiCl,” Phys. Rev. B 1, 818–831 (1970).
[CrossRef]

Bergstrom, R. W.

R. W. Bergstrom, “Extinction and absorption coefficients of the atmospheric aerosol as a function of particle size,” Beitr. Phys. Atmos. 46, 198–202 (1973).

Brown, F. C.

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TiBr and TiCl,” Phys. Rev. B 1, 818–831 (1970).
[CrossRef]

Cardona, M.

M. Cardona, Optical Properties of Solids, S. Nudelman and S. S. Mitra, eds. (Plenum, New York, 1969).

Dave, J. V.

J. V. Dave, Rep. no. 320-3237, IBM Scientific Center, Palo Alto, California (1968).

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

Dowling, J. M.

A. B. Pluchino, S. S. Goldberg, J. M. Dowling, and C. M. Randall, “Refractive-index measurements of single micron-sized carbon particles,” Appl. Opt. 19, 3371–3372 (1980).
[CrossRef]

Faxvog, F. R.

Gal, G.

Gillespie, J. B.

Gilra, D. P.

D. P. Gilra, “Collective excitations in small solid particles and astronomical applications,” Ph.D. thesis, University of Wisconsin, Madison, Wisconsin (1972).

Goldberg, S. S.

A. B. Pluchino, S. S. Goldberg, J. M. Dowling, and C. M. Randall, “Refractive-index measurements of single micron-sized carbon particles,” Appl. Opt. 19, 3371–3372 (1980).
[CrossRef]

Gouesbet, G.

Grehan, G.

Hale, G. M.

Janzen, J.

J. Janzen, “The refractive index of colloidal carbon,” J. Colloid Interface Sci. 69, 436–447 (1979).
[CrossRef]

Jennings, S. G.

Kerker, M.

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

Khare, B. N.

O. B. Toon, J. B. Pollack, and B. N. Khare, “The optical constants of several atmospheric aerosol species: ammonium sulfate, aluminum oxide, and sodium chloride,” J. Geophys. Res. 81, 5733–5748 (1976).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).

Munn, M. W.

Patterson, E. M.

Pinnick, R. G.

Pluchino, A. B.

A. B. Pluchino, S. S. Goldberg, J. M. Dowling, and C. M. Randall, “Refractive-index measurements of single micron-sized carbon particles,” Appl. Opt. 19, 3371–3372 (1980).
[CrossRef]

Pollack, J. B.

O. B. Toon, J. B. Pollack, and B. N. Khare, “The optical constants of several atmospheric aerosol species: ammonium sulfate, aluminum oxide, and sodium chloride,” J. Geophys. Res. 81, 5733–5748 (1976).
[CrossRef]

Querry, M. R.

G. M. Hale and M. R. Querry, “Optical constants of water in the 200-nm to 200-μ m wavelength region,” Appl. Opt. 12, 555–563 (1973).
[CrossRef] [PubMed]

M. R. Querry, University of Missouri-Kansas City, Kansas City, Missouri 64110, personal communication (1978).

M. R. Querry, “Molecular and crystalline electromagnetic properties of selected condensed materials in the infrared,” (1979).

Randall, C. M.

A. B. Pluchino, S. S. Goldberg, J. M. Dowling, and C. M. Randall, “Refractive-index measurements of single micron-sized carbon particles,” Appl. Opt. 19, 3371–3372 (1980).
[CrossRef]

Roessler, D. M.

Toon, O. B.

O. B. Toon, J. B. Pollack, and B. N. Khare, “The optical constants of several atmospheric aerosol species: ammonium sulfate, aluminum oxide, and sodium chloride,” J. Geophys. Res. 81, 5733–5748 (1976).
[CrossRef]

van de Hulst, H. C.

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

Wells, W. C.

Wiscombe, W. J.

Wyatt, P. J.

Appl. Opt. (10)

A. B. Pluchino, S. S. Goldberg, J. M. Dowling, and C. M. Randall, “Refractive-index measurements of single micron-sized carbon particles,” Appl. Opt. 19, 3371–3372 (1980).
[CrossRef]

G. M. Hale and M. R. Querry, “Optical constants of water in the 200-nm to 200-μ m wavelength region,” Appl. Opt. 12, 555–563 (1973).
[CrossRef] [PubMed]

W. C. Wells, G. Gal, and M. W. Munn, “Aerosol distributions in maritime air and predicted scattering coefficients in the infrared,” Appl. Opt. 16, 654–659 (1977).
[CrossRef] [PubMed]

E. M. Patterson, “Atmospheric extinction between 0.55 μ m and 10.6 μ m due to soil-derived aerosols,” Appl. Opt. 16, 2414–2418 (1977).
[CrossRef] [PubMed]

S. G. Jennings, R. G. Pinnick, and H. J. Auvermann, “Effects of particulate complex refractive index and particle size distribution variations on atmospheric extinction and absorption for visible through middle IR wavelengths,” Appl. Opt. 17, 3922–3929 (1978).
[CrossRef] [PubMed]

S. G. Jennings, R. G. Pinnick, and J. B. Gillespie, “Relation between absorption coefficient and imaginary index of atmospheric aerosol constituents,” Appl. Opt. 18, 1368–1371 (1979).
[CrossRef] [PubMed]

D. M. Roessler and F. R. Faxvog, “Opacity of black smoke: calculated variation with particulate size and refractive index,” Appl. Opt. 18, 1399–1403 (1979).
[CrossRef] [PubMed]

G. Grehan and G. Gouesbet, “Mie theory calculations: new progress, with emphasis on particle sizing,” Appl. Opt. 18, 3489–3493 (1979).
[CrossRef] [PubMed]

P. J. Wyatt, “Some chemical, physical, and optical properties of fly ash particles,” Appl. Opt. 19, 975–983 (1980).
[CrossRef] [PubMed]

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

Beitr. Phys. Atmos. (1)

R. W. Bergstrom, “Extinction and absorption coefficients of the atmospheric aerosol as a function of particle size,” Beitr. Phys. Atmos. 46, 198–202 (1973).

J. Colloid Interface Sci. (1)

J. Janzen, “The refractive index of colloidal carbon,” J. Colloid Interface Sci. 69, 436–447 (1979).
[CrossRef]

J. Geophys. Res. (1)

O. B. Toon, J. B. Pollack, and B. N. Khare, “The optical constants of several atmospheric aerosol species: ammonium sulfate, aluminum oxide, and sodium chloride,” J. Geophys. Res. 81, 5733–5748 (1976).
[CrossRef]

J. Opt. Soc. Am. (1)

Phys. Rev. B (1)

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TiBr and TiCl,” Phys. Rev. B 1, 818–831 (1970).
[CrossRef]

Other (9)

D. P. Gilra, “Collective excitations in small solid particles and astronomical applications,” Ph.D. thesis, University of Wisconsin, Madison, Wisconsin (1972).

M. R. Querry, University of Missouri-Kansas City, Kansas City, Missouri 64110, personal communication (1978).

M. R. Querry, “Molecular and crystalline electromagnetic properties of selected condensed materials in the infrared,” (1979).

M. Cardona, Optical Properties of Solids, S. Nudelman and S. S. Mitra, eds. (Plenum, New York, 1969).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).

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

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

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

J. V. Dave, Rep. no. 320-3237, IBM Scientific Center, Palo Alto, California (1968).

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Figures (14)

Fig. 1
Fig. 1

Isopleths in n, k space of the dimensionless extinction coefficient for spheres with the indicated size parameter.

Fig. 2
Fig. 2

Dimensionless extinction coefficient αD for log-normally distributed spheres as a function of the mass median size parameter Xm. Individual curves are for k = 0, 0.2, 0.4, 0.6, 0.8, 1.0, and 2 and are easily identified since αD is monotonically increasing with respect to k at the left-hand side of the plots.

Fig. 3
Fig. 3

Variation of the dimensionless extinction coefficient with k as Xm increases. The curves for Xm = 1.38 and Xm = 1.44 bracket Xml.

Fig. 4
Fig. 4

Illustration of the numerical procedure for calculating k. αe is the measured extinction coefficient, and αc is the computed extinction coefficient.

Fig. 5
Fig. 5

Flow chart of the algorithm for determining the optical constants n, k from extinction measurements. n0 is an initial assumed value of the real part of the refractive index. n0 is typically taken to be 1.3.

Fig. 6
Fig. 6

Experimental arrangement used for o-phosphoric-acid aerosol measurements. WR, radiometer window (polyethylene), Ws, source window (polyethylene), L, path length (3.05 m), V, chamber volume (22 m3).

Fig. 7
Fig. 7

Comparison of the experimentally determined extinction spectrum (dot–dashed line) with a computed extinction spectrum (solid line) for 65% o-phosphoric acid.

Fig. 8
Fig. 8

Percent difference between the computed and measured extinction spectrum as a function of wavelength.

Fig. 9
Fig. 9

Comparison of n(λ) determined from extinction measurements (solid curve) with the measured values for 65% o-phosphoric acid (dashed curve).

Fig. 10
Fig. 10

Comparison of k(λ) determined from extinction measurements (solid curve) with the measured values for 65% o-phosphoric acid (dashed curve).

Fig. 11
Fig. 11

Percent difference between n(λ) determined from extinction measurements and the values of n(λ) for 65% o-phosphoric acid.

Fig. 12
Fig. 12

Percent difference between k(λ) determined from extinction measurements and the values of k(λ) for 65% o-phosphoric acid.

Fig. 13
Fig. 13

Comparison of n(λ) derived from a spectrum (circles) computed from Lorenz–Mie theory with the real parts used in the computation (solid curve).

Fig. 14
Fig. 14

Comparison of k(λ) derived from a spectrum (circles) computed from Lorenz–Mie theory with the imaginary parts used in the computation (solid curve).

Tables (1)

Tables Icon

Table 1 Approximate Values of the Upper Monotonicity Limit for the Mass Median Size Parameter

Equations (14)

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N ˜ ( λ ) = n ( λ ) - i k ( λ ) ,
α = i s i i m i ,
α = α ( z ) d M ,
α 0 = 3 Q e [ N ˜ ( λ ) , X ] 2 ρ D ,
α D = α λ ρ .
n ( λ 0 ) = 1 + 2 λ 0 2 π P 0 k ( λ ) d λ λ ( λ 0 2 - λ 2 ) ,
n ( λ 0 ) = n ( λ 1 ) + 2 ( λ 1 2 - λ 0 2 ) π P 0 λ k ( λ ) d λ ( λ 0 2 - λ 2 ) ( λ 1 2 - λ 2 ) .
α ( n , k ) = α 0 ( n , k , D ) d M .
d M = 1 2 π 1 ln σ g exp { - 1 / 2 [ ln ( D / D m ) / ln σ g ] 2 } d ln D ,
α ( k ) = α ( n 0 , k ) .
k 2 ( 1 + 6 / 5 X 2 ) 1 / 2
α D ( k ) = α ( k ) λ ρ ,
X m = π D m / λ ,
α e = - 1 C L ln T ,