Abstract

Comparisons are made between two solutions for light scattering and absorption by arbitrarily shaped or agglomerated particles. Both solutions have made improvements that result in better accuracy than that achieved by the original coupled-dipole solution of Purcell and Pennypacker [(PP); Astrophys. J. 186, 705 ( 1973)]. One solution, which has an additional self-interaction term, was derived by Iskander, Chen, and Penner [(ICP); Appl. Opt. 28, 3083 ( 1989)]. The other is Dungey and Bohren’s refinement of PP [(PP–DB); J. Opt. Soc. Am. 8, 81 ( 1991)], which calculates the dipole polarizability from the Doyle expression instead of from the Clausius-Mossotti (CM) relation. It is found that ICP has better overall accuracy than PP–DB, especially for absorption, regarding which PP–DB is even less accurate than PP. ICP is all-around theoretically sound, whereas PP–DB is not. For nonabsorbing materials, although corrected for the violation of zero extinction for single dipoles, PP–DB creates another violation of nonzero absorption. For the dipole refractive index, the Maxwell–Garnett (MG) relation is expected to be an accurate model, since it can be approximately deduced from ICP. MG can also be used as an accurate effective refractive-index model for inhomogeneous particles.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. E. Penner, C. R. Molenkamp, “Predicting the consequences of nuclear war: precipitation scavenging of smoke,” Aero. Sci. Tech. 10, 51–62 (1989).
    [CrossRef]
  2. S. G. O’Brien, G. H. Goedecke, “Scattering of millimeter waves by snow crystals and equivalent homogeneous symmetric particles,” Appl. Opt. 27, 2439–2444 (1988);“Propagation of polarized millimeter waves through falling snow,” Appl Opt. 27, 2445–2450 (1988).
    [CrossRef] [PubMed]
  3. M. Y. Lin, H. M. Lindsay, D. A. Weitz, R. C. Ball, R. Klein, P. Meakin, “Universal reaction-limited colloid aggregation,” Phys. Rev. A 41, 2005–2020 (1990).
    [CrossRef] [PubMed]
  4. J. D. Felske, P.-F. Hsu, J. C. Ku, “The effect of soot particle optical inhomogeneity and agglomeration on the analysis of light scattering measurements in flames,” J. Quant. Spec-trosc. Radiat. Transfer 35, 447–465 (1986).
    [CrossRef]
  5. J. C. Ku, K.-H. Shim, “Optical diagnostics and radiative properties of simulated soot agglomerates,” J. Heat Transfer 113, 953–958 (1991).
    [CrossRef]
  6. B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
    [CrossRef]
  7. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  8. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).
  9. G. H. Goedecke, S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2438 (1988).
    [CrossRef] [PubMed]
  10. A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. 79, 1–5 (1990).
    [CrossRef]
  11. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  12. M. F. Iskander, H. Y. Chen, J. E. Penner, “Optical scattering and absorption by branched chains of aerosols,” Appl. Opt. 28, 3083–3091 (1989).
    [CrossRef] [PubMed]
  13. D. S. Saxon, “Lectures on the scattering of light,” in The UCLA International Conference on Radiation and Remote Probing of the Atmosphere, J. G. Kuriyan, ed. (U. California Press, Los Angeles, Calif., 1974), pp. 227–308.
  14. J. C. Ku, K.-H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
    [CrossRef]
  15. A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London A 366, 111–127 (1979);Corrigendum, 375, 453–454 (1981).
    [CrossRef]
  16. A. R. Jones, “Scattering efficiency factors for agglomerates of small spheres,” J. Phys. D 12, 1661–1672 (1979).
    [CrossRef]
  17. C. E. Dungey, C. F. Bohren, “Light scattering by non-spherical particles: a refinement to the coupled-dipole method,” J. Opt. Soc. Am. A 8, 81–87 (1991).
    [CrossRef]
  18. W T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
    [CrossRef]
  19. M. F. Iskander, H. Y. Chen, J. E. Penner, “Resonance optical absorption by fractal agglomerates of smoke aerosols,” Atmos. Environ. 25A, 2563–2569 (1991).

1992 (1)

J. C. Ku, K.-H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

1991 (3)

C. E. Dungey, C. F. Bohren, “Light scattering by non-spherical particles: a refinement to the coupled-dipole method,” J. Opt. Soc. Am. A 8, 81–87 (1991).
[CrossRef]

M. F. Iskander, H. Y. Chen, J. E. Penner, “Resonance optical absorption by fractal agglomerates of smoke aerosols,” Atmos. Environ. 25A, 2563–2569 (1991).

J. C. Ku, K.-H. Shim, “Optical diagnostics and radiative properties of simulated soot agglomerates,” J. Heat Transfer 113, 953–958 (1991).
[CrossRef]

1990 (2)

A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. 79, 1–5 (1990).
[CrossRef]

M. Y. Lin, H. M. Lindsay, D. A. Weitz, R. C. Ball, R. Klein, P. Meakin, “Universal reaction-limited colloid aggregation,” Phys. Rev. A 41, 2005–2020 (1990).
[CrossRef] [PubMed]

1989 (3)

J. E. Penner, C. R. Molenkamp, “Predicting the consequences of nuclear war: precipitation scavenging of smoke,” Aero. Sci. Tech. 10, 51–62 (1989).
[CrossRef]

W T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

M. F. Iskander, H. Y. Chen, J. E. Penner, “Optical scattering and absorption by branched chains of aerosols,” Appl. Opt. 28, 3083–3091 (1989).
[CrossRef] [PubMed]

1988 (3)

1987 (1)

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[CrossRef]

1986 (1)

J. D. Felske, P.-F. Hsu, J. C. Ku, “The effect of soot particle optical inhomogeneity and agglomeration on the analysis of light scattering measurements in flames,” J. Quant. Spec-trosc. Radiat. Transfer 35, 447–465 (1986).
[CrossRef]

1979 (2)

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London A 366, 111–127 (1979);Corrigendum, 375, 453–454 (1981).
[CrossRef]

A. R. Jones, “Scattering efficiency factors for agglomerates of small spheres,” J. Phys. D 12, 1661–1672 (1979).
[CrossRef]

1973 (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Ball, R. C.

M. Y. Lin, H. M. Lindsay, D. A. Weitz, R. C. Ball, R. Klein, P. Meakin, “Universal reaction-limited colloid aggregation,” Phys. Rev. A 41, 2005–2020 (1990).
[CrossRef] [PubMed]

Bohren, C. F.

C. E. Dungey, C. F. Bohren, “Light scattering by non-spherical particles: a refinement to the coupled-dipole method,” J. Opt. Soc. Am. A 8, 81–87 (1991).
[CrossRef]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Chen, H. Y.

M. F. Iskander, H. Y. Chen, J. E. Penner, “Resonance optical absorption by fractal agglomerates of smoke aerosols,” Atmos. Environ. 25A, 2563–2569 (1991).

M. F. Iskander, H. Y. Chen, J. E. Penner, “Optical scattering and absorption by branched chains of aerosols,” Appl. Opt. 28, 3083–3091 (1989).
[CrossRef] [PubMed]

Doyle, W T.

W T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

Draine, B. T.

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Drolen, B. L.

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[CrossRef]

Dungey, C. E.

Felske, J. D.

J. D. Felske, P.-F. Hsu, J. C. Ku, “The effect of soot particle optical inhomogeneity and agglomeration on the analysis of light scattering measurements in flames,” J. Quant. Spec-trosc. Radiat. Transfer 35, 447–465 (1986).
[CrossRef]

Goedecke, G. H.

Hsu, P.-F.

J. D. Felske, P.-F. Hsu, J. C. Ku, “The effect of soot particle optical inhomogeneity and agglomeration on the analysis of light scattering measurements in flames,” J. Quant. Spec-trosc. Radiat. Transfer 35, 447–465 (1986).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Iskander, M. F.

M. F. Iskander, H. Y. Chen, J. E. Penner, “Resonance optical absorption by fractal agglomerates of smoke aerosols,” Atmos. Environ. 25A, 2563–2569 (1991).

M. F. Iskander, H. Y. Chen, J. E. Penner, “Optical scattering and absorption by branched chains of aerosols,” Appl. Opt. 28, 3083–3091 (1989).
[CrossRef] [PubMed]

Jones, A. R.

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London A 366, 111–127 (1979);Corrigendum, 375, 453–454 (1981).
[CrossRef]

A. R. Jones, “Scattering efficiency factors for agglomerates of small spheres,” J. Phys. D 12, 1661–1672 (1979).
[CrossRef]

Klein, R.

M. Y. Lin, H. M. Lindsay, D. A. Weitz, R. C. Ball, R. Klein, P. Meakin, “Universal reaction-limited colloid aggregation,” Phys. Rev. A 41, 2005–2020 (1990).
[CrossRef] [PubMed]

Ku, J. C.

J. C. Ku, K.-H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

J. C. Ku, K.-H. Shim, “Optical diagnostics and radiative properties of simulated soot agglomerates,” J. Heat Transfer 113, 953–958 (1991).
[CrossRef]

J. D. Felske, P.-F. Hsu, J. C. Ku, “The effect of soot particle optical inhomogeneity and agglomeration on the analysis of light scattering measurements in flames,” J. Quant. Spec-trosc. Radiat. Transfer 35, 447–465 (1986).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. 79, 1–5 (1990).
[CrossRef]

Lin, M. Y.

M. Y. Lin, H. M. Lindsay, D. A. Weitz, R. C. Ball, R. Klein, P. Meakin, “Universal reaction-limited colloid aggregation,” Phys. Rev. A 41, 2005–2020 (1990).
[CrossRef] [PubMed]

Lindsay, H. M.

M. Y. Lin, H. M. Lindsay, D. A. Weitz, R. C. Ball, R. Klein, P. Meakin, “Universal reaction-limited colloid aggregation,” Phys. Rev. A 41, 2005–2020 (1990).
[CrossRef] [PubMed]

Meakin, P.

M. Y. Lin, H. M. Lindsay, D. A. Weitz, R. C. Ball, R. Klein, P. Meakin, “Universal reaction-limited colloid aggregation,” Phys. Rev. A 41, 2005–2020 (1990).
[CrossRef] [PubMed]

Molenkamp, C. R.

J. E. Penner, C. R. Molenkamp, “Predicting the consequences of nuclear war: precipitation scavenging of smoke,” Aero. Sci. Tech. 10, 51–62 (1989).
[CrossRef]

O’Brien, S. G.

Penner, J. E.

M. F. Iskander, H. Y. Chen, J. E. Penner, “Resonance optical absorption by fractal agglomerates of smoke aerosols,” Atmos. Environ. 25A, 2563–2569 (1991).

J. E. Penner, C. R. Molenkamp, “Predicting the consequences of nuclear war: precipitation scavenging of smoke,” Aero. Sci. Tech. 10, 51–62 (1989).
[CrossRef]

M. F. Iskander, H. Y. Chen, J. E. Penner, “Optical scattering and absorption by branched chains of aerosols,” Appl. Opt. 28, 3083–3091 (1989).
[CrossRef] [PubMed]

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Saxon, D. S.

D. S. Saxon, “Lectures on the scattering of light,” in The UCLA International Conference on Radiation and Remote Probing of the Atmosphere, J. G. Kuriyan, ed. (U. California Press, Los Angeles, Calif., 1974), pp. 227–308.

Shim, K.-H.

J. C. Ku, K.-H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

J. C. Ku, K.-H. Shim, “Optical diagnostics and radiative properties of simulated soot agglomerates,” J. Heat Transfer 113, 953–958 (1991).
[CrossRef]

Tien, C. L.

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[CrossRef]

Weitz, D. A.

M. Y. Lin, H. M. Lindsay, D. A. Weitz, R. C. Ball, R. Klein, P. Meakin, “Universal reaction-limited colloid aggregation,” Phys. Rev. A 41, 2005–2020 (1990).
[CrossRef] [PubMed]

Aero. Sci. Tech. (1)

J. E. Penner, C. R. Molenkamp, “Predicting the consequences of nuclear war: precipitation scavenging of smoke,” Aero. Sci. Tech. 10, 51–62 (1989).
[CrossRef]

Appl. Opt. (3)

Astrophys. J. (2)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Atmos. Environ. (1)

M. F. Iskander, H. Y. Chen, J. E. Penner, “Resonance optical absorption by fractal agglomerates of smoke aerosols,” Atmos. Environ. 25A, 2563–2569 (1991).

J. Heat Transfer (1)

J. C. Ku, K.-H. Shim, “Optical diagnostics and radiative properties of simulated soot agglomerates,” J. Heat Transfer 113, 953–958 (1991).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

A. R. Jones, “Scattering efficiency factors for agglomerates of small spheres,” J. Phys. D 12, 1661–1672 (1979).
[CrossRef]

J. Quant. Spec-trosc. Radiat. Transfer (1)

J. D. Felske, P.-F. Hsu, J. C. Ku, “The effect of soot particle optical inhomogeneity and agglomeration on the analysis of light scattering measurements in flames,” J. Quant. Spec-trosc. Radiat. Transfer 35, 447–465 (1986).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

J. C. Ku, K.-H. Shim, “A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles,” J. Quant. Spectrosc. Radiat. Transfer 47, 201–220 (1992).
[CrossRef]

B. L. Drolen, C. L. Tien, “Absorption and scattering of agglomerated soot particles,” J. Quant. Spectrosc. Radiat. Transfer 37, 433–448 (1987).
[CrossRef]

Opt. Commun. (1)

A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. 79, 1–5 (1990).
[CrossRef]

Phys. Rev. A (1)

M. Y. Lin, H. M. Lindsay, D. A. Weitz, R. C. Ball, R. Klein, P. Meakin, “Universal reaction-limited colloid aggregation,” Phys. Rev. A 41, 2005–2020 (1990).
[CrossRef] [PubMed]

Phys. Rev. B (1)

W T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

Proc. R. Soc. London A (1)

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London A 366, 111–127 (1979);Corrigendum, 375, 453–454 (1981).
[CrossRef]

Other (2)

D. S. Saxon, “Lectures on the scattering of light,” in The UCLA International Conference on Radiation and Remote Probing of the Atmosphere, J. G. Kuriyan, ed. (U. California Press, Los Angeles, Calif., 1974), pp. 227–308.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Definition of variables for light scattering by an assembly of spherical dipolar subunits.

Fig. 2
Fig. 2

Eepresentation of a spherical particle by an array of 136 spherical dipolar subunits on a simple cubic lattice.

Fig. 3
Fig. 3

Comparison of efficiency factors Qe and Qs and forwardscattering cross section Cs,υυ(0°) for a 136-dipole sphere among PP, PP-DB, and ICP. Data are shown as the percentage differences from Mie results for an equivalent homogeneous sphere. xd = 0.15673xb. (a) mb = 1.33 + i0.0,(b) mb = 1.7 + i0.1,(c) mb = 1.38 + i0.275.

Equations (36)

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

E ( r ) [ 1 + 4 π 3 χ ( r ) ] = E inc ( r ) + χ ( r ) G ( r r ) E ( r ) d 3 r .
i = E inc , i + i 3 k 3 j = 1 j i N α j T i j j + s i i , i = 1 , 2 , 3 , , N .
α = ( 3 V c 4 π ) ( 1 + 2 ) = ( 3 V c + 2 ) χ ,
E inc , i = E 0 ê 0 exp ( i k z i ) , | E 0 ê 0 | 2 1 ,
T i j = [ a b c b a d c d e ] i j ,
a = 2 h 0 ( 1 ) ( γ ) h 2 ( 1 ) ( γ ) [ P 2 ( μ ) 1 / 2 cos ( 2 ψ ) P 2 2 ( μ ) ] , a = 2 h 0 ( 1 ) ( γ ) h 2 ( 1 ) ( γ ) [ P 2 ( μ ) + 1 / 2 cos ( 2 ψ ) P 2 2 ( μ ) ] , e = 2 h 0 ( 1 ) ( γ ) + 2 h 2 ( 1 ) ( γ ) P 2 ( μ ) , b = 1 / 2 h 2 ( 1 ) ( γ ) sin ( 2 ψ ) P 2 2 ( μ ) , c = h 2 ( 1 ) ( γ ) cos ( ψ ) P 2 1 ( μ ) , d = h 2 ( 1 ) ( γ ) sin ( ψ ) P 2 1 ( μ ) .
s i = 2 ( 4 π α i 3 V c ) [ exp ( i x i ) ( 1 i x i ) 1 ] k 3 α i ( 1 x i + 2 i 3 ) ,
E sca ( r ) = exp ( ikr ) ikr [ i k 3 ( I r ̂ r ̂ ) exp ( i k r ̂ r ) × α ( r ) V c ( r ) d 3 r ] = exp ( ikr ) ikr F ( r ̂ ) ,
F ( r ̂ ) = i k 3 i = 1 N α i exp ( i k r i cos β i ) ( Θ i θ ̂ + Φ i ϕ ̂ ) ,
C s , p p = k 2 | F p p ( r ̂ ) | 2 ,
C e = 4 π k 2 Re [ E 0 * ê 0 * F θ = ϕ = 0 ° ] ,
C a = 4 π k Im [ α ( r ) ] V c | ( r ) | 2 d 3 r ,
C s = k 2 | F ( r ) | 2 d Ω ,
C e = 4 π k Im [ i = 1 N α i exp ( i k z i ) E 0 * ê 0 * i ] ,
C a = 4 π k i = 1 N ( Im α i ) | i | 2 .
f ( d 1 d + 2 ) = ( b 1 b + 2 ) .
α = i 3 2 k 3 a 1 ,
a 1 = m ψ 1 ( m x ) ψ 1 ( x ) ψ 1 ( x ) ψ 1 ( m x ) m ψ 1 ( m x ) ξ 1 ( x ) ξ 1 ( x ) ψ 1 ( m x ) ,
i 6 π k 3 a 1 ( m d , x d ) = ( 3 V c 4 π ) ( b 1 b + 2 )
= ( 1 s ) 1 E inc ( 1 + s ) ( cos η î + sin η ĵ ) ,
Q e = 4 k R 2 Im [ α ( 1 + s ) ] ,
Q a = 4 k R 2 Im ( α ) | 1 + s | 2 ,
Q s = ( 8 / 3 ) k 4 R 2 | α | 2 | 1 + s | 2 ,
s [ x 2 A r ( 2 / 3 ) x 3 A i ] + i [ x 2 A i + ( 2 / 3 ) x 3 A r ]
Q e = 4 x A i + 8 x 3 A i A r + ( 8 / 3 ) x 4 ( A r 2 A i 2 ) ,
Q e = 4 x A i + 8 x 3 A i A r ( 16 / 3 ) x 4 A i 2 + 4 x 5 [ 1 + ( 4 / 9 ) x 2 ] A i | A | 2 ,
Q s = ( 8 / 3 ) x 4 | A | 2 { 1 + 2 x 2 A r ( 4 / 3 ) x 3 A i + x 4 [ 1 + ( 4 / 9 ) x 2 ] | A | 2 } ,
Q e Q a = ( 8 / 3 ) x 4 | A | 2 × [ 1 + ( 3 / 2 ) x A i + ( 2 / 3 ) x 3 A i ] Q s ,
Q e Q a Q s = ( 8 / 3 ) x 5 | A | 2 { ( 3 / 2 ) A i 2 x A r + 2 x 2 A i x 3 [ 1 + ( 4 / 9 ) x 2 ] | A | 2 } 0 .
Q e = Q a = 4 x A i ,
Q s = ( 8 / 3 ) x 4 | A | 2 ;
Q s = Q a = 0 and Q e Q s for nonabsorbing materials .
α = k 3 x 3 A [ 1 + 3 5 x 2 ( 2 + 2 ) + 2 i 3 x 3 A + ] .
Q e = Q a = 4 x { A i + 3 5 x 2 Im [ A ( 2 + 2 ) ] + 2 3 x 3 Re ( A 2 ) + } ,
Q e = 8 3 x 4 | A | 2 [ 1 + 6 5 x 2 | 2 + 2 | 2 + ] ;
Q e 0 , but Q e Q s and Q a 0 for nonabsorbing materials .

Metrics