Abstract

We derive formulas for rigorous transfer matrix calculations of absorption in multiple-coherent-scattering systems in which the scatterers are multiply coated spheres (not necessarily concentric). Any of the spherical coatings, cores, or host media may be composed of absorbing materials. For a nonabsorbing host media, the total absorption may be deduced from overall energy conservation. A more detailed description of the absorption is obtained through formulas yielding the absorption within individual scatterers and/or coatings. We present some illustrative applications of these formulas to the design of heterogeneous coated-sphere media exhibiting enhanced absorption.

© 2003 Optical Society of America

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References

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  1. B. Stout, J. C. Auger, J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
    [CrossRef]
  2. J. C. Auger, B. Stout, “A recursive centered T-matrix algorithm to solve the multiple scattering equation: numerical validation,” J. Quant. Spectrosc. Radiat. Transfer. (to be published).
  3. B. Stout, J. C. Auger, J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105–2128 (2001).
  4. J. C. Auger, B. Stout, R. G. Barrera, F. Curiel, “Scattering properties of rutile pigments located eccentrically within microvoids,” J. Quant. Spectrosc. Radiat. Transfer. 70, 675–695 (2001).
    [CrossRef]
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    [CrossRef]
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  9. T. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing, Wiley Series in Remote Sensing (Wiley, New York, 1985).
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  13. Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayed spheres: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
    [CrossRef]
  14. D. W. Mackowski, R. A. Altenkirch, M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551–1559 (1990).
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  15. O. B. Toon, T. P. Ackerman, “Algorithms for the calculations of scattering by stratified spheres,” Appl. Opt. 20, 3657–3660 (1981).
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  16. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–442 (1908).
    [CrossRef]
  17. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley-Interscience Publication (Wiley, New York, 1983).
  18. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  19. M. Sitarski, “Internal heating of multilayered aerosol particles by electromagnetic radiation,” Langmuir 3, 85–93 (1987).
    [CrossRef]
  20. P. R. Siqueira, K. Sarabandi, “T-matrix determination of effective permittivity for three-dimensional dense random media,” IEEE Trans. Antennas Propag. 48, 317–327 (2000).
    [CrossRef]
  21. U. Frisch, “Wave propagation in random media” in Probabilistic Methods in Applied Mathematics Vol. I, A. T. Bharucha-Reid, ed. (Academic, New York, 1968), pp. 76–198.
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  23. O. R. Cruzan, “Translation addition theorems for spherical vector wave functions,” Q. Appl. Math. 19, 15–24 (1962).

2002 (1)

B. Stout, J. C. Auger, J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

2001 (2)

B. Stout, J. C. Auger, J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105–2128 (2001).

J. C. Auger, B. Stout, R. G. Barrera, F. Curiel, “Scattering properties of rutile pigments located eccentrically within microvoids,” J. Quant. Spectrosc. Radiat. Transfer. 70, 675–695 (2001).
[CrossRef]

2000 (1)

P. R. Siqueira, K. Sarabandi, “T-matrix determination of effective permittivity for three-dimensional dense random media,” IEEE Trans. Antennas Propag. 48, 317–327 (2000).
[CrossRef]

1996 (2)

1995 (2)

1994 (2)

1991 (1)

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayed spheres: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

1990 (1)

1987 (1)

M. Sitarski, “Internal heating of multilayered aerosol particles by electromagnetic radiation,” Langmuir 3, 85–93 (1987).
[CrossRef]

1981 (1)

1962 (1)

O. R. Cruzan, “Translation addition theorems for spherical vector wave functions,” Q. Appl. Math. 19, 15–24 (1962).

1961 (1)

S. Stein, “Addition theorems for spherical wave function,” Quart. Appl. Math. 19, 15–24 (1961).

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–442 (1908).
[CrossRef]

Ackerman, T. P.

Altenkirch, R. A.

Auger, J. C.

B. Stout, J. C. Auger, J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

B. Stout, J. C. Auger, J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105–2128 (2001).

J. C. Auger, B. Stout, R. G. Barrera, F. Curiel, “Scattering properties of rutile pigments located eccentrically within microvoids,” J. Quant. Spectrosc. Radiat. Transfer. 70, 675–695 (2001).
[CrossRef]

J. C. Auger, B. Stout, “A recursive centered T-matrix algorithm to solve the multiple scattering equation: numerical validation,” J. Quant. Spectrosc. Radiat. Transfer. (to be published).

Barrera, R. G.

J. C. Auger, B. Stout, R. G. Barrera, F. Curiel, “Scattering properties of rutile pigments located eccentrically within microvoids,” J. Quant. Spectrosc. Radiat. Transfer. 70, 675–695 (2001).
[CrossRef]

Bohren, C. F.

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

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media, IEEE Press Series on Electromagnetic Waves (IEEE Press, New York, 1994).

Cruzan, O. R.

O. R. Cruzan, “Translation addition theorems for spherical vector wave functions,” Q. Appl. Math. 19, 15–24 (1962).

Curiel, F.

J. C. Auger, B. Stout, R. G. Barrera, F. Curiel, “Scattering properties of rutile pigments located eccentrically within microvoids,” J. Quant. Spectrosc. Radiat. Transfer. 70, 675–695 (2001).
[CrossRef]

Frisch, U.

U. Frisch, “Wave propagation in random media” in Probabilistic Methods in Applied Mathematics Vol. I, A. T. Bharucha-Reid, ed. (Academic, New York, 1968), pp. 76–198.

Gouesbet, G.

Gréhan, G.

Huffman, D. R.

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

Johnson, B. R.

Kai, L.

Kong, J. A.

T. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing, Wiley Series in Remote Sensing (Wiley, New York, 1985).

Lafait, J.

B. Stout, J. C. Auger, J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

B. Stout, J. C. Auger, J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105–2128 (2001).

Mackowski, D. W.

Massoli, P.

Menguc, M. P.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–442 (1908).
[CrossRef]

Mishchenko, M. I.

Onofri, F.

Sarabandi, K.

P. R. Siqueira, K. Sarabandi, “T-matrix determination of effective permittivity for three-dimensional dense random media,” IEEE Trans. Antennas Propag. 48, 317–327 (2000).
[CrossRef]

Shin, R. T.

T. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing, Wiley Series in Remote Sensing (Wiley, New York, 1985).

Siqueira, P. R.

P. R. Siqueira, K. Sarabandi, “T-matrix determination of effective permittivity for three-dimensional dense random media,” IEEE Trans. Antennas Propag. 48, 317–327 (2000).
[CrossRef]

Sitarski, M.

M. Sitarski, “Internal heating of multilayered aerosol particles by electromagnetic radiation,” Langmuir 3, 85–93 (1987).
[CrossRef]

Stein, S.

S. Stein, “Addition theorems for spherical wave function,” Quart. Appl. Math. 19, 15–24 (1961).

Stout, B.

B. Stout, J. C. Auger, J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

J. C. Auger, B. Stout, R. G. Barrera, F. Curiel, “Scattering properties of rutile pigments located eccentrically within microvoids,” J. Quant. Spectrosc. Radiat. Transfer. 70, 675–695 (2001).
[CrossRef]

B. Stout, J. C. Auger, J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105–2128 (2001).

J. C. Auger, B. Stout, “A recursive centered T-matrix algorithm to solve the multiple scattering equation: numerical validation,” J. Quant. Spectrosc. Radiat. Transfer. (to be published).

Toon, O. B.

Tsang, T. L.

T. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing, Wiley Series in Remote Sensing (Wiley, New York, 1985).

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Wang, Y. P.

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayed spheres: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Wu, Z. S.

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayed spheres: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Xu, Y. L.

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–442 (1908).
[CrossRef]

Appl. Opt. (6)

IEEE Trans. Antennas Propag. (1)

P. R. Siqueira, K. Sarabandi, “T-matrix determination of effective permittivity for three-dimensional dense random media,” IEEE Trans. Antennas Propag. 48, 317–327 (2000).
[CrossRef]

J. Mod. Opt. (2)

B. Stout, J. C. Auger, J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

B. Stout, J. C. Auger, J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105–2128 (2001).

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transfer. (1)

J. C. Auger, B. Stout, R. G. Barrera, F. Curiel, “Scattering properties of rutile pigments located eccentrically within microvoids,” J. Quant. Spectrosc. Radiat. Transfer. 70, 675–695 (2001).
[CrossRef]

Langmuir (1)

M. Sitarski, “Internal heating of multilayered aerosol particles by electromagnetic radiation,” Langmuir 3, 85–93 (1987).
[CrossRef]

Q. Appl. Math. (1)

O. R. Cruzan, “Translation addition theorems for spherical vector wave functions,” Q. Appl. Math. 19, 15–24 (1962).

Quart. Appl. Math. (1)

S. Stein, “Addition theorems for spherical wave function,” Quart. Appl. Math. 19, 15–24 (1961).

Radio Sci. (1)

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayed spheres: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Other (6)

U. Frisch, “Wave propagation in random media” in Probabilistic Methods in Applied Mathematics Vol. I, A. T. Bharucha-Reid, ed. (Academic, New York, 1968), pp. 76–198.

W. C. Chew, Waves and Fields in Inhomogeneous Media, IEEE Press Series on Electromagnetic Waves (IEEE Press, New York, 1994).

T. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing, Wiley Series in Remote Sensing (Wiley, New York, 1985).

J. C. Auger, B. Stout, “A recursive centered T-matrix algorithm to solve the multiple scattering equation: numerical validation,” J. Quant. Spectrosc. Radiat. Transfer. (to be published).

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

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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Figures (4)

Fig. 1
Fig. 1

Coated nonconcentric spherical scatterer.

Fig. 2
Fig. 2

Multiply coated spherical scatterer.

Fig. 3
Fig. 3

(a) Dimensionless effective absorption length plotted as a function of the size parameter, χ1=2πR1/λ0 for R1-R2=0.18R2. (b) Asymmetry parameter of the scatterings plotted as a function of χ1.

Fig. 4
Fig. 4

Twelve nearest-neighbor coherent-scattering calculations of laeff plotted as a function of the density 1/fs. The independent-scattering prediction is plotted as a dashed line for comparison.

Tables (2)

Tables Icon

Table 1 Individual Absorption Efficiencies Qabs(j) for a System of 13 Touching Coated Dielectric Spheres with χ1=2.2, n0=1.5, n1=1.5+0.01i, n2=2.5, R1-R2=0.18R2, and an Electric Field Polarized along the yˆ Axis

Tables Icon

Table 2 Orientation-Averaged Absorption Efficiencies and Isolated Sphere Efficiencies for the Same Aggregate and Parameters as in Table 1

Equations (95)

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Einc(r)=En=1m=-nn(Rg{Mnm(k0r)}[a]nmM+Rg{Nnm(k0r)}[a]nmN)=E(Rg{M(k0r)}, Rg{N(k0r)})aMaNERg{Ξt(k0r)}a.
Ξt(kr)(M1,-1(kr), M1,0(kr),,
N1,-1(kr), N1,0(kr),),
Mnm(kr)-hn(kr)Xnm(rˆ),
Nnm(kr)1kr {n(n+1)hn(kr)Ynm(rˆ)+[krhn(kr)]Znm(rˆ)},
E0=E0(j),e+E0(j),s,
E0=ERg{Ξt(k0r1(j))}e0(j)+EΞt(k0r1(j))f0(j),
E1(j)=ERg{Ξt(k1(j)r1(j))}e1(j)+EΞt(k1r2(j))f1(j)
=ERg{Ξt(k1(j)r1(j))}e1(j)+EΞt(k1(j)r1(j))βk1[1,2]f1(j)
=ERg{Ξt(k1(j)r2(j))}βk1[2,1]e1(j)+EΞt(k1(j)r2(j))f1(j),
E2(j)=ERg{Ξt(k2(j)r2(j))}e2(j),
iωμlμvHl(r)=×El(r),
k1ψn(k0R1)[e0]rmM+k1ξn(k0R1)[f0]nmM=k0ψn(k1R1)[e1]nmM+k0ξn(k1R1)[βk1[1,2]f1]nmM,
k1ψn(k0R1)[e0]nmN+k1ξn(k0R1)[f0]nmN=k0ψn(k1R1)[e1]nmN+k0ξn(k1R1)[βk1[1,2]f1]nmN,
k2ψn(k1R2)[βk1(2,1)e1]nmM+k2ξn(k1R2)[f1]nmM=k1ψn(k2R2)[e2]nmM,
k2ψn(k1R2)[βk1(2,1)e1]nmN+k2ξn(k1R2)[f1]nmN=k1ψn(k2R2)[e2]nmN,
μ1ψn(k0R1)[e0]nmM+μ1ξn(k0R1)[f0]nmM=μ0ψn(k1R1)[e1]nmM+μ0ξn(k1R1)[βk1[1,2]f1]nmM,
μ1ψn(k0R1)[e0]nmN+μ1ξn(k0R1)[f0]nmN=μ0ψn(k1R1)[e1]nmN+μ0ξn(k1R1)[βk1[1,2]f1]nmN,
μ2ψ2(k1R2)[βk1[2,1]e1]nmM+μ2ξn(k1R2)[f1]nmM=μ1ψn(k2R2)[e2]nmM,
μ2ψn(k1R2)[βk1[2,1]e1]nmN+μ2ξn(k1R2)[f1]nmN=μ1ψn(k2R2)[e2]nmN.
[f0]nmA=[T10]nA[e0]nmA+[Q01]nA[βk1[1,2]f1]nmA,
A=M,N,
[Tl+1,l]nM=ψn(klRl+1)ξn(klRl+1)×μl+1μl Φn(klRl+1)-ρl+1,lΦn(kl+1Rl+1)ρl+1,lΦn(kl+1Rl+1)-μl+1μl Ψn(klRl+1)ψn(klRl+1)ξn(klRl+1)×[T˜l+1,l]nM,
[Tl+1,l]nN=ψn(klRl+1)ξn(klRl+1)×μl+1μl Φn(kl+1Rl+1)-ρl+1,lΦn(klRl+1)ρl+1,lΨn(klRl+1)-μl+1μl Φn(kl+1Rl+1)ψn(klRl+1)ξn(klRl+1)×[T˜l+1,l]nN,
[Ql,l+1]nM=1ξn(klRl+1)ψn(kl+1Rl+1)×iμl+1μl Ψn(klRl+1)-ρl+1,lΦn(kl+1Rl+1)=1ξn(klRl+1)ψn(kl+1Rl+1) [Q˜l,l+1]nM,
[Ql,l+1]nN=1ξn(klRl+1)ψn(kl+1Rl+1)×iρl+1,lΨn(klRl+1)-μl+1μl Φn(kl+1Rl+1)1ξn(klRl+1)ψn(kl+1Rl+1) [Q˜l,l+1]nN.
O=OMMOMNONMONN,
[O]nm,νμAB=δA,Bδn,νδm,μ[O]nA.
f0=T10e0+Q01βk1[1,2]f1.
f1=T21βk1[2,1]e1,
e1=V10e0+U10f0,
[Vl+1,l]nM=iψn(klRl+1)ξn(kl+1Rl+1)×μl+1μl Φn(klRl+1)-ρl+1,lΨn(kl+1Rl+1)ψn(klRl+1)ξn(kl+1Rl+1)[V˜l+1,l]nM,
[Vl+1,l]nN=iψn(klRl+1)ξn(kl+1Rl+1)×ρl+1,lΦn(klRl+1)-μl+1μl Ψn(kl+1Rl+1)ψn(klRl+1)ξn(kl+1Rl+1)[V˜l+1,l]nN,
[Ul+1,l]nM=iξn(klRl+1)ξn(kl+1Rl+1)×μl+1μl Ψn(klRl+1)-ρl+1,lΨn(kl+1Rl+1)ξn(klRl+1)ξn(kl+1Rl+1)[U˜l+1,l]nM,
[Ul+1,l]nN=iξn(klRl+1)ξn(kl+1Rl+1)×ρl+1,lΨn(klRl+1)-μl+1μl Ψn(kl+1Rl+1)ξn(klRl+1)ξn(kl+1Rl+1)[U˜l+1,l]nN.
f0=T10+Q01βk1[1,2]T21βk1[2,1]V10I-Q01βk1[1,2]T21βk1[2,1]U10e0T10e0,
Tl+1,l=Tl+1,l+Ql,l+1βkl[l+1,l+2]Tl+2,l+1βkl[l+2,l+1]Vl+1,lI-Ql,l+1βkl[l+1,l+2]Tl+2,l+1βkl[l+2,l+1]Ul+1,l
fl+1=βkl+1[l+2,l+1]Pl,l+1(I-Tl+1,lTl+1,l-1)fl,
[Pl,l+1]nM[[Ql,l+1]nM]-1ξn(klRl+1)ψn(kl+1Rl+1)×[P˜l,l+1]nM,
[Pl,l+1]nN[[Ql,l+1]nN]-1ξn(klRl+1)ψn(kl+1Rl+1)×[P˜l,l+1]nN.
el=βkl[l,l+1]Tl+1,l-1fl,l=1,,L-1,
[eL]nmA=[ΛL,L-1]nA[fL-1]nmA,
[ΛL,L-1]nM=1ψn(kL-1RL)ψn(kLRL)×iρL,L-1μL-1μL ρL,L-1Φn(kLRL)-Φn(kL-1RL)1ψn(kL-1RL)ψn(kLRL) [Λ˜L,L-1]nM,
[ΛL,L-1]nN=1ψn(kL-1RL)ψn(kLRL)×iρL,L-1Φn(kLRL)-μL-1μL ρL,L-1Φn(kL-1RL)1ψn(kL-1RL)ψn(kLRL) [Λ˜L,L-1]nN.
[Tl+1,l]nA=[Tl+1,l]nA+[Ql,l+1]nA[Tl+2,l+1]nA[Vl+1,l]nA1-[Ql,l+1]nA[Tl+2,l+1]nA[Ul+1,l]nA,
A=M, N.
[f¯l]nmAξn(klRl+1)[fl]nmA(l=0,,L-1)
A=M,N,
[e¯0]nmAψn(k0R1)[e0]nmA,
[e¯l]nmAψn(klRl)[el]nmA(l=1,,L)A=M,N.
[f¯0]nmA=[T˜10]nA[e¯0]nmA,
[Tl+1,l]nAψn(klRl+1)ξn(klRl+1) [T˜l+1,l]nA,
[T˜l+1,l]nA=[T˜l+1,l]nA+ξn(kl+1Rl+1)ξn(kl+1Rl+2)ψn(kl+1Rl+2)ψn(kl+1Rl+1) [Q˜l,l+1]nA[T˜l+2,l+1]nA[V˜l+1,l]nA1-ξn(kl+1Rl+1)ξn(kl+1Rl+2)ψn(kl+1Rl+2)ψn(kl+1Rl+1) [Q˜l,l+1]nA[T˜l+2,l+1]nA[U˜l+1,l]nA,
[T˜L,L-1]nA=[T˜L,L-1]nA,A=M,N.
[f¯l+1]nmA=ψn(kl+1Rl+1)ξn(kl+1Rl+2)[P˜l,l+1]nA×1-[T˜l+1,l]nA[T˜l+1,l-1]nA[f¯l]nmA,
[e¯0]nmA=[T˜10-1]nmA[f¯0]nmA,
[e¯l]nmA=ψn(klRl)ψn(klRl+1) [T˜l+1,l-1]nA[f¯l]nmA,
(l=1,,L-1),
[e¯L]nmA=1ψn(kL-1RL)ξn(kL-1RL) [Λ˜L,L-1]nA[f¯L-1]nmA.
f0(j)=T10(j)e0(j)τ1(j)e0(j),
f0(j)=k=1NτN(j,k)β(k,0)a,
τN(N,N)=τ1(N)1-j,k=1N-1α(N,k)τN-1(k,j)α(j,N)τ1(N)-1,
τN(N,k)=τN(N,N)j=1N-1α(N,j)τN-1(j,k),kN,
τN(j,N)=k=1N-1τN-1(j,k)α(k,N)τN(N,N),jN.
τN(j,k)=τN-1(j,k)+i=1N-1τN-1(j,i)α(i,N)τN(N,k),jN,kN.
σecl=-1k2j,lNRe{exp[ik0(xl-xj)]pτN(j,l)p},
σscl=1k2Rej,k,l,iNexp[ik0(xi-xl)]×p[TN(j,l)]β(j,k)τN(k,i)p,
σeclo=-2πk2j,lNRe{Tr{τN(j,l)β(l,j)}},
σsclo=2πk2Rej,k,i,lNTr{[τN(j,l)]β(j,k)τN(k,i)β(i,l)}.
Pl(j)=-Arˆl(j)Sl(j)dA=-(Rl(j))22×dΩrˆl(j) Re{El(j)×Hl(j)*},
el(j)=(Vl,l-1(j)[Tl,l-1(j)]-1+Ul,l-1(j))fl-1(j),
σa,l(j)=1|kl(j)|2Reμ00 fl-1(j),Γl(j)fl-1(j),
Γl(j)=[Pl-1,l(j)(I-Tl,l-1(j)[Tl,l-1(j)]-1)]Cl(j)[Pl-1,l(j)(I-Tl,l-1(j)[Tl,l-1(j)]-1)]+[Vl,l-1(j)[Tl,l-1(j)]-1+Ul,l-1(j)]Dl(j)[Vl,l-1(j)[Tl,l-1(j)]-1+Ul,l-1(j)]+[Vl,l-1(j)[Tl,l-1(j)]-1+Ul,l-1(j)]Fl(j)[Pl-1,l(j)(I-Tl,l-1(j)[Tl,l-1(j)]-1)]+[Pl-1,l(j)(I-Tl,l-1(j)×[Tl,l-1(j)]-1)]Gl(j)[Vl,l-1(j)[Tl,l-1(j)]-1+Ul,l-1(j)].
[Cl(j)]nAi[η˜l(j)]Aξn*(kl(j)Rl(j))ξn(kl(j)Rl(j)),
[Dl(j)]nAi[η˜l(j)]Aψn*(kl(j)Rl(j))ψn(kl(j)Rl(j)),
[Fl(j)]nAi[η˜l(j)]Aψn*(kl(j)Rl(j))ξn(kl(j)Rl(j)),
[Gl(j)]nAi[η˜l(j)]Aψn(kl(j)Rl(j))ξn*(kl(j)Rl(j)),
A=M,N,
[η˜l(j)]M(l(j)/μl(j))1/2,[η˜l(j)]N(l(j),*/μl(j),*)1/2.
σa,1(j)o=2π|k1(j)|2i,l=1N×Reμ00×Tr{[τN(j,l)]Γ1(j)τN(j,i)β(i,l)}.
σa,l(j)=1|kl(j)|2A=M,Nn,m|ψn(j)(kl(j)Rl(j))×ξn(j)(kl(j)Rl(j))|2|[f ¯l-1(j)]nmA|2×Reμ00 (iΦn(kl(j)Rl(j))[η˜l(j)]A|[A˜l(j)]nA|2+iΨn(kl(j)Rl(j))[η˜l(j)]A|[B˜l(j)]nA|2+iΨn(kl(j)Rl(j))×[η˜l(j)]A[A˜l(j)]nA,*[B˜l(j)]nA+iΦn(kl(j)Rl(j))×[η˜l(j)]A[A˜l(j)]nA[B˜l(j)]nA,*),
[A˜l(j)]nA[V˜l,l-1(j)]nA[[T˜l,l-1(j)]nA]-1+[U˜l,l-1(j)]nA,
[B˜l(j)]nA[P˜l-1,l(j)]nA(1-[T˜l,l-1(j)]nA[[T˜l,l-1(j)]nA]-1).
σedispσadisp2n1ωc V1=4πn1λv V1,
la1N1σadisp=12f1kvn1,
kvla12f1n1=50f1,
kvlaeff=χ1{ν+2[g(ν-1)-νg2+gν+1]/(1-g)2}1/2n0fsQs,
Ynm(rˆ)=γnmn(n+1)Pnm(cos θ)exp(imϕ)rˆ=Ynm(θ, ϕ)rˆ,
Xnm(rˆ)=γnm-imsin θ Pnm(cos θ)exp(imϕ)θˆ+ddθ Pnm(cos θ)exp(imϕ)ϕˆ,
Znm(rˆ)=γnmddθ Pnm(cos θ)exp(imϕ)θˆ+imsin θ Pnm(cos θ)exp(imϕ)ϕˆ,
γnm=(2n+1)(n-m)!4πn(n+1)(n+m)!1/2.
[p]nmM=-in4πXnm*(k^i)e^i,
[p]nmN=-in+14πZnm*(k^i)e^i.
Ξt(kr)=Ξt(kr)β(kr0),r>r0, Ξt(kr)=Rg{Ξt(kr)}α(kr0),r<r0,Rg{Ξt(kr)}=Rg{Ξt(kr)}β(kr0),|rj|,
α(kr0)=A¯(kr0)B¯(kr0)B¯(kr0)A¯(kr0),β(kr0)Rg{α(kr0)},

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