Abstract

Explicit analytical expressions to represent the radiative properties of spherical particles in the small size-parameter range were obtained. These expressions were deduced following Penndorf’s approach of expanding the Mie coefficients in power series on the size parameter. However, in opposition to Penndorf’s original work—in which some errors were found and corrected—the Mie coefficients were expanded to the eighth power of the size parameter, which results in a five-term approximation to the extinction and scattering efficiencies. Also, expressions for the evaluation of the asymmetry factor and both polarized and unpolarized phase functions were deduced and presented. The results so obtained have proved to be very accurate, even for size parameters beyond the limit of validity of the approach utilized.

© 2001 Optical Society of America

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References

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  1. M. Caldas, V. Semião, “Modelling of optical properties for a polydispersion/gas mixture,” J. Quant. Spectrosc. Radiat. Transfer 62, 495–501 (1999).
    [CrossRef]
  2. M. Caldas, V. Semião, “A new approximate phase function for isolated particles and polydispersions,” J. Quant. Spectrosc. Radiat. Transfer 68, 521–542 (2001).
    [CrossRef]
  3. A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25, 1–53 (1999).
    [CrossRef]
  4. F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
    [CrossRef]
  5. B. T. N. Evans, G. R. Fournier, “Analytic approximation to randomly oriented spheroid extinction,” Appl. Opt. 33, 5796–5804 (1994).
    [CrossRef] [PubMed]
  6. M. I. Mishchenko, “Light scattering by size–shape distributions of randomly oriented axially symmetric particles of a size comparable to a wavelength,” Appl. Opt. 32, 4652–4666 (1993).
    [CrossRef] [PubMed]
  7. G. R. Fournier, B. T. N. Evans, “Bridging the gap between the Rayleigh and Thompson limits for spheres and spheroids,” Appl. Opt. 32, 6159–6166 (1993).
    [CrossRef] [PubMed]
  8. Y. Liu, W. P. Arnott, J. Hallett, “Anomalous diffraction theory for arbitrarily oriented finite circular cylinders and comparison with exact T-matrix results,” Appl. Opt. 37, 5019–5030 (1998).
    [CrossRef]
  9. M. Caldas, V. Semião, “Modelling of scattering and absorption coefficients for a polydispersion,” Int. J. Heat Mass Transf. 42, 4535–4548 (1999).
    [CrossRef]
  10. R. B. Penndorf, “Scattering and extinction coefficients for small absorbing and nonabsorbing aerosols,” J. Opt. Soc. Am. 52, 896–904 (1962).
    [CrossRef]
  11. A. Selamet, V. S. Arpaci, “Rayleigh limit—Penndorf extension,” Int. J. Heat Mass Transfer 32, 1809–1820 (1989).
    [CrossRef]
  12. J. C. Ku, J. D. Felske, “The range of validity of the Rayleigh limit for computing Mie scattering and extinction efficiencies,” J. Quant. Spectrosc. Radiat. Transfer 31, 569–574 (1984).
    [CrossRef]
  13. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  14. M. Kerker, The Scattering of Light (Academic, San Diego, Calif., 1969).
  15. W. A. Fiveland, W. J. Oberjohn, D. K. Cornelius, “COMO: A numerical model for predicting furnace performance in axisymmetric geometries,” Vol. 1, (U.S. Department of Energy, Washington, D.C., 1985).

2001 (1)

M. Caldas, V. Semião, “A new approximate phase function for isolated particles and polydispersions,” J. Quant. Spectrosc. Radiat. Transfer 68, 521–542 (2001).
[CrossRef]

1999 (3)

A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25, 1–53 (1999).
[CrossRef]

M. Caldas, V. Semião, “Modelling of scattering and absorption coefficients for a polydispersion,” Int. J. Heat Mass Transf. 42, 4535–4548 (1999).
[CrossRef]

M. Caldas, V. Semião, “Modelling of optical properties for a polydispersion/gas mixture,” J. Quant. Spectrosc. Radiat. Transfer 62, 495–501 (1999).
[CrossRef]

1998 (2)

1994 (1)

1993 (2)

1989 (1)

A. Selamet, V. S. Arpaci, “Rayleigh limit—Penndorf extension,” Int. J. Heat Mass Transfer 32, 1809–1820 (1989).
[CrossRef]

1984 (1)

J. C. Ku, J. D. Felske, “The range of validity of the Rayleigh limit for computing Mie scattering and extinction efficiencies,” J. Quant. Spectrosc. Radiat. Transfer 31, 569–574 (1984).
[CrossRef]

1962 (1)

Arnott, W. P.

Arpaci, V. S.

A. Selamet, V. S. Arpaci, “Rayleigh limit—Penndorf extension,” Int. J. Heat Mass Transfer 32, 1809–1820 (1989).
[CrossRef]

Caldas, M.

M. Caldas, V. Semião, “A new approximate phase function for isolated particles and polydispersions,” J. Quant. Spectrosc. Radiat. Transfer 68, 521–542 (2001).
[CrossRef]

M. Caldas, V. Semião, “Modelling of optical properties for a polydispersion/gas mixture,” J. Quant. Spectrosc. Radiat. Transfer 62, 495–501 (1999).
[CrossRef]

M. Caldas, V. Semião, “Modelling of scattering and absorption coefficients for a polydispersion,” Int. J. Heat Mass Transf. 42, 4535–4548 (1999).
[CrossRef]

Cornelius, D. K.

W. A. Fiveland, W. J. Oberjohn, D. K. Cornelius, “COMO: A numerical model for predicting furnace performance in axisymmetric geometries,” Vol. 1, (U.S. Department of Energy, Washington, D.C., 1985).

Evans, B. T. N.

Felske, J. D.

J. C. Ku, J. D. Felske, “The range of validity of the Rayleigh limit for computing Mie scattering and extinction efficiencies,” J. Quant. Spectrosc. Radiat. Transfer 31, 569–574 (1984).
[CrossRef]

Fiveland, W. A.

W. A. Fiveland, W. J. Oberjohn, D. K. Cornelius, “COMO: A numerical model for predicting furnace performance in axisymmetric geometries,” Vol. 1, (U.S. Department of Energy, Washington, D.C., 1985).

Fournier, G. R.

Hallett, J.

Jones, A. R.

A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25, 1–53 (1999).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light (Academic, San Diego, Calif., 1969).

Ku, J. C.

J. C. Ku, J. D. Felske, “The range of validity of the Rayleigh limit for computing Mie scattering and extinction efficiencies,” J. Quant. Spectrosc. Radiat. Transfer 31, 569–574 (1984).
[CrossRef]

Liu, Y.

Mishchenko, M. I.

Oberjohn, W. J.

W. A. Fiveland, W. J. Oberjohn, D. K. Cornelius, “COMO: A numerical model for predicting furnace performance in axisymmetric geometries,” Vol. 1, (U.S. Department of Energy, Washington, D.C., 1985).

Penndorf, R. B.

Schulz, F. M.

Selamet, A.

A. Selamet, V. S. Arpaci, “Rayleigh limit—Penndorf extension,” Int. J. Heat Mass Transfer 32, 1809–1820 (1989).
[CrossRef]

Semião, V.

M. Caldas, V. Semião, “A new approximate phase function for isolated particles and polydispersions,” J. Quant. Spectrosc. Radiat. Transfer 68, 521–542 (2001).
[CrossRef]

M. Caldas, V. Semião, “Modelling of optical properties for a polydispersion/gas mixture,” J. Quant. Spectrosc. Radiat. Transfer 62, 495–501 (1999).
[CrossRef]

M. Caldas, V. Semião, “Modelling of scattering and absorption coefficients for a polydispersion,” Int. J. Heat Mass Transf. 42, 4535–4548 (1999).
[CrossRef]

Stamnes, J. J.

Stamnes, K.

Van de Hulst, H. C.

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

Appl. Opt. (5)

Int. J. Heat Mass Transf. (1)

M. Caldas, V. Semião, “Modelling of scattering and absorption coefficients for a polydispersion,” Int. J. Heat Mass Transf. 42, 4535–4548 (1999).
[CrossRef]

Int. J. Heat Mass Transfer (1)

A. Selamet, V. S. Arpaci, “Rayleigh limit—Penndorf extension,” Int. J. Heat Mass Transfer 32, 1809–1820 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Quant. Spectrosc. Radiat. Transfer (3)

J. C. Ku, J. D. Felske, “The range of validity of the Rayleigh limit for computing Mie scattering and extinction efficiencies,” J. Quant. Spectrosc. Radiat. Transfer 31, 569–574 (1984).
[CrossRef]

M. Caldas, V. Semião, “Modelling of optical properties for a polydispersion/gas mixture,” J. Quant. Spectrosc. Radiat. Transfer 62, 495–501 (1999).
[CrossRef]

M. Caldas, V. Semião, “A new approximate phase function for isolated particles and polydispersions,” J. Quant. Spectrosc. Radiat. Transfer 68, 521–542 (2001).
[CrossRef]

Prog. Energy Combust. Sci. (1)

A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25, 1–53 (1999).
[CrossRef]

Other (3)

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

M. Kerker, The Scattering of Light (Academic, San Diego, Calif., 1969).

W. A. Fiveland, W. J. Oberjohn, D. K. Cornelius, “COMO: A numerical model for predicting furnace performance in axisymmetric geometries,” Vol. 1, (U.S. Department of Energy, Washington, D.C., 1985).

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Figures (5)

Fig. 1
Fig. 1

Relative error, in comparison with Mie theory results, of extinction efficiencies Qe from the Penndorf model and the present model.

Fig. 2
Fig. 2

Relative error, in comparison with Mie theory results, of scattering efficiencies Qs from the Penndorf model and the present model.

Fig. 3
Fig. 3

Relative error, in comparison with Mie theory results, of the product of the asymmetry factor and the scattering efficiency based on the Penndorf model and the present model.

Fig. 4
Fig. 4

Polarized and unpolarized phase functions for soot (m=2.2-i1.122, x=0.75).

Fig. 5
Fig. 5

Polarized and unpolarized phase functions for fly ash (m=1.5-i0.02, x=1.0).

Tables (4)

Tables Icon

Table 1 Coefficients Appearing in the Power Series Expansion to the Eighth Power of the First Five Mie Coefficients (a1, b1, a2, b2 and a3)

Tables Icon

Table 2 Functional Form of the Power Series Coefficients for the First Five Mie Coefficients Introduced in Table 1 (P, Q, R, S, T, U, V and W Are Parameters That Depend on m)

Tables Icon

Table 3 Root-Mean-Square Errors Relative to Mie Theory of Three-, Four-, and Five-Term Approximations in the Range 0x1, 0x|m|1a

Tables Icon

Table 4 Maximum Allowable Value of x|m| to Ensure a 1% Maximum Relative Error with Use of a Three-, Four-, or Five-Term Approximation

Equations (45)

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Qe=4x2Re[S(1)],
Qs=12-114i(μ)x2dμ.
i(μ)=i1(μ)+i2(μ)2,
i1(μ)=|S1(μ)|2,
i2(μ)=|S2(μ)|2.
g=1Qs12-114i(μ)x2μdμ,
P(μ)=1Qs4i(μ)x2,
P1(μ)=1Qs4i1(μ)x2,
P2(μ)=1Qs4i2(μ)x2,
P(μ)=P1(μ)+P2(μ)2.
S1(μ)=n=12n+1n(n+1)[anπn(μ)+bnτn(μ)],
S2(μ)=n=12n+1n(n+1)[bnπn(μ)+anτn(μ)].
S1=S13x3+S15x5+S16x6+S17x7+S18x8,
S2=S23x3+S25x5+S26x6+S27x7+S28x8.
S(1)=x32[3a13+(3a15+3b15+5a25)x2+3a16x3+(3a17+3b17+5a27+5b27+7a37)x4+3a18x5],
i1(μ)=|S13|2x6+2 Re(S13¯S15)x8+2 Re(S13¯S16)x9+{|S15|2+2 Re(S13¯S17)}x10+[2 Re(S16¯S15)+2 Re(S13¯S18)]x11,
i2(μ)=|S23|2x6+2 Re(S23¯S25)x8+2 Re(S23¯S26)x9+[|S25|2+2 Re(S23¯S27)]x10+[2 Re(S26¯S25)+2 Re(S23¯S28)]x11,
i(μ)=|S13|2+|S23|22x6+Re(S13¯S15+S23¯S25)x8+Re(S13¯S16+S23¯S26)x9+|S15|2+|S25|22+Re(S13¯S17+S23¯S27)x10+[Re(S16¯S15+S26¯S25)+Re(S13¯S18+S23¯S28)]x11.
P=m2-1m2+2,
Q=m2-2m2+2,
R=m6+20m4-200m2+200(m2+2)2,
S=m2-12m2+3,
T=m2-1(2m2+3)2,
U=m2-13m2+4,
V=m2-1,
W=(m2-1)(2m2-5).
Q=Q,R=18R,S=5S/P,
T=375T/P,U=28U/P,V=V/P,
W=5W/P.
4i1(μ)x2=4|P|2x41+115[Re(Q)+Re(V+S)μ]x2+43Im(P)x3+16300[7|Q|2+4 Re(R)+7(|V|2+|S|2)μ2+14 Re[(V+S)Q¯]μ+4 Re(W-T)μ+14 Re(SV¯)μ2+20 Re(V)(2μ2-1)+Re(U)(5μ2-1)]x4+245{Im[Q(P-P¯)]-Im[(V+S)P¯]μ}x5,
4i2(μ)x2=4|P|2x4μ2+115[Re(Q)μ2+Re(V-S)μ+2 Re(S)μ3]x2+43Im(P)μ2x3+16300{7|Q|2μ2+4 Re(R)μ2+7(|V|2+|S|2)+28|S|2(μ4-μ2)+14 Re[(V-S)Q¯]μ+4 Re(W+T)μ+28 Re(SQ¯)μ3-8 Re(T)μ3+14 Re(SV¯)×(2μ2-1)+20 Re(V)μ2+Re(U)(15μ4-11μ2)}x4+245{Im[Q(P-P¯)]μ2-Im[(V-S)P¯]μ-2 Im(SP¯)μ3}x5
4i(μ)x2=2|P|2x4(1+μ2)+115[Re(Q)(1+μ2)+2 Re(V)μ+2 Re(S)μ3]x2+43Im(P)(1+μ2)x3+16300[7|Q|2(1+μ2)+4 Re(R)(1+μ2)+7(|V|2+|S|2)×(1+μ2)+28|S|2(μ4-μ2)+28 Re(VQ¯)μ+8 Re(W)μ+28 Re(SQ¯)μ3-8 Re(T)μ3+14 Re(SV¯)×(3μ2-1)+20 Re(V)(3μ2-1)+Re(U)×(15μ4-6μ2-1)]x4+245{Im[Q(P-P¯)]×(1+μ2)-2 Im(VP¯)μ-2 Im(SP¯)μ3}x5.
Qe=x-4 Im(P)-215Im[P(Q+V+S)]x2+83Re(P2)x3-21575Im[P(R+W-T+5V+U)]x4+845Re(P2Q)x5,
Qs=83|P|2x41+115Re(Q)x2+43Im(P)x3+131500[35|Q|2+20 Re(R)+35|V|2+21|S|2]x4+245Im[Q(P-P¯)]x5.
Qa=x-4 Im(P)-215Im[P(Q+V+S)]x2-163[Im(P)]2x3-21575Im[P(R+W-T+5V+U)]x4-1645Im(P)Im(PQ)x5.
gQs=4225|P|2x6[5 Re(V)+3 Re(S)]+1210[35 Re(VQ¯)+21 Re(SQ¯)+10 Re(W)-6 Re(T)]x2-23[5 Im(VP¯)+3 Im(SP¯)]x3,
P1(μ)=32+415|P|2x6Qs[Re(V+S)μ]+12100{5 Re(U)(5μ2-1)+7|S|2(5μ2-3)+35|V|2(μ2-1)+100 Re(V)(2μ2-1)+70 Re(SV¯)μ2+70 Re[(V+S)Q¯]μ+20 Re(W-T)μ}x2-23{Im[(V+S)P¯]μ}x3,
P2(μ)=32μ2+415|P|2x6Qs[Re(V-S)μ+2 Re(S)μ3]+12100{5 Re(U)(15μ4-11μ2)+7|S|2(20μ4-23μ2+5)-35|V|2(μ2-1)+100 Re(V)μ2+70 Re(SV¯)(2μ2-1)+70 Re[(V-S)Q¯]μ+20 Re(W+T)μ+140 Re(SQ¯)μ3-40 Re(T)μ3}x2-23{Im[(V-S)P¯]μ+2 Im(SP¯)μ3}x3,
P(μ)=34(1+μ2)+215|P|2x6Qs2[Re(V)μ+Re(S)μ3]+12100[5 Re(U)(15μ4-6μ2-1)+14|S|2(10μ4-9μ2+1)+100 Re(V)×(3μ2-1)+70 Re(SV¯)(3μ2-1)+140 Re(VQ¯)μ+40 Re(W)μ+140 Re(SQ¯)μ3-40 Re(T)μ3]x2-43[Im(VP¯)μ+Im(SP¯)μ3]x3.
P(μ)=n=0NmaxAnPn(μ).
A0=1,
A1=12225|P|2x6Qs[5 Re(V)+3 Re(S)]+1210[35 Re(VQ¯)+21 Re(SQ¯)+10 Re(W)-6 Re(T)]x2-23[5 Im(VP¯)+3 Im(SP¯)]x3,
A2=12+255125|P|2x8Qs[350 Re(V)+245 Re(SV¯)+40 Re(U)-7|S|2],
A3=24225|P|2x6QsRe(S)+1210[7 Re(SQ¯)-2 Re(T)]x2-23Im(SP¯)x3,
A4=455125|P|2x8Qs[28|S|2+15 Re(U)].

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