Abstract

A formalism is developed for the calculation of the electromagnetic field scattered by a multilayered spheroidal particle. The suggested formalism utilizes the recursive approach with respect to passing from one layer to the next; thus it does not require an increase in the size of the equation matrices involved as the number of layers increases. The equations operate with matrices of the same size as for a homogeneous spheroid. The special cases of extremely prolate and weakly prolate spheroids are considered in more detail. It is shown that in such cases one can avoid the matrix calculations by instead using the iterative scalar calculations.

© 2000 Optical Society of America

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References

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  1. M. I. Mishchenko, L. D. Travis, “Light scattering by polydisperse rotationally symmetric nonspherical particles: linear polarization,” J. Quant. Spectrosc. Radiat. Transfer 51, 759–778 (1994).
    [CrossRef]
  2. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [CrossRef] [PubMed]
  3. N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
    [CrossRef]
  4. V. G. Farafonov, N. V. Voshchinnikov, V. V. Somsikov, “Light scattered by a core-mantled spheroidal particle,” Appl. Opt. 35, 5412–5426 (1996).
    [CrossRef] [PubMed]
  5. T. Onaka, “Light scattering by spheroidal grains,” Ann. Tokyo Astron. Observ. 18, 1–54 (1980).
  6. M. F. R. Cooray, I. R. Ciric, “Scattering of electromagnetic waves by a coated dielectric spheroid,” J. Electromagn. Waves Appl. 6, 1491–1507 (1992).
    [CrossRef]
  7. A. Cohen, M. Kleiman, “Mie scattering coefficients for multilayered magnetic particles,” J. Wave-Material Interaction 1, 252–256 (1986).
  8. C. F. Bohren, D. R. Huffman, Absorbing and Scattering of Light by Small Particles (Wiley, New York, 1983).
  9. Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multi-layered sphere: recursive algorithm,” Radio Sci. 26, 1393–1401 (1991).
    [CrossRef]
  10. I. Gurwich, N. Shiloah, M. Kleiman, “The recursive algorithm for electromagnetic scattering by tilted infinite circular multi-layered cylinder,” J. Quant. Spectrosc. Radiat. Transfer 63, 217–229 (1999).
    [CrossRef]
  11. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

1999 (1)

I. Gurwich, N. Shiloah, M. Kleiman, “The recursive algorithm for electromagnetic scattering by tilted infinite circular multi-layered cylinder,” J. Quant. Spectrosc. Radiat. Transfer 63, 217–229 (1999).
[CrossRef]

1996 (1)

1994 (1)

M. I. Mishchenko, L. D. Travis, “Light scattering by polydisperse rotationally symmetric nonspherical particles: linear polarization,” J. Quant. Spectrosc. Radiat. Transfer 51, 759–778 (1994).
[CrossRef]

1993 (1)

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

1992 (1)

M. F. R. Cooray, I. R. Ciric, “Scattering of electromagnetic waves by a coated dielectric spheroid,” J. Electromagn. Waves Appl. 6, 1491–1507 (1992).
[CrossRef]

1991 (1)

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multi-layered sphere: recursive algorithm,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

1986 (1)

A. Cohen, M. Kleiman, “Mie scattering coefficients for multilayered magnetic particles,” J. Wave-Material Interaction 1, 252–256 (1986).

1980 (1)

T. Onaka, “Light scattering by spheroidal grains,” Ann. Tokyo Astron. Observ. 18, 1–54 (1980).

1975 (1)

Asano, S.

Bohren, C. F.

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

Ciric, I. R.

M. F. R. Cooray, I. R. Ciric, “Scattering of electromagnetic waves by a coated dielectric spheroid,” J. Electromagn. Waves Appl. 6, 1491–1507 (1992).
[CrossRef]

Cohen, A.

A. Cohen, M. Kleiman, “Mie scattering coefficients for multilayered magnetic particles,” J. Wave-Material Interaction 1, 252–256 (1986).

Cooray, M. F. R.

M. F. R. Cooray, I. R. Ciric, “Scattering of electromagnetic waves by a coated dielectric spheroid,” J. Electromagn. Waves Appl. 6, 1491–1507 (1992).
[CrossRef]

Farafonov, V. G.

V. G. Farafonov, N. V. Voshchinnikov, V. V. Somsikov, “Light scattered by a core-mantled spheroidal particle,” Appl. Opt. 35, 5412–5426 (1996).
[CrossRef] [PubMed]

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

Gurwich, I.

I. Gurwich, N. Shiloah, M. Kleiman, “The recursive algorithm for electromagnetic scattering by tilted infinite circular multi-layered cylinder,” J. Quant. Spectrosc. Radiat. Transfer 63, 217–229 (1999).
[CrossRef]

Huffman, D. R.

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

Kleiman, M.

I. Gurwich, N. Shiloah, M. Kleiman, “The recursive algorithm for electromagnetic scattering by tilted infinite circular multi-layered cylinder,” J. Quant. Spectrosc. Radiat. Transfer 63, 217–229 (1999).
[CrossRef]

A. Cohen, M. Kleiman, “Mie scattering coefficients for multilayered magnetic particles,” J. Wave-Material Interaction 1, 252–256 (1986).

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, “Light scattering by polydisperse rotationally symmetric nonspherical particles: linear polarization,” J. Quant. Spectrosc. Radiat. Transfer 51, 759–778 (1994).
[CrossRef]

Onaka, T.

T. Onaka, “Light scattering by spheroidal grains,” Ann. Tokyo Astron. Observ. 18, 1–54 (1980).

Shiloah, N.

I. Gurwich, N. Shiloah, M. Kleiman, “The recursive algorithm for electromagnetic scattering by tilted infinite circular multi-layered cylinder,” J. Quant. Spectrosc. Radiat. Transfer 63, 217–229 (1999).
[CrossRef]

Somsikov, V. V.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, “Light scattering by polydisperse rotationally symmetric nonspherical particles: linear polarization,” J. Quant. Spectrosc. Radiat. Transfer 51, 759–778 (1994).
[CrossRef]

Voshchinnikov, N. V.

V. G. Farafonov, N. V. Voshchinnikov, V. V. Somsikov, “Light scattered by a core-mantled spheroidal particle,” Appl. Opt. 35, 5412–5426 (1996).
[CrossRef] [PubMed]

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

Wang, Y. P.

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multi-layered sphere: recursive algorithm,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Wu, Z. S.

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multi-layered sphere: recursive algorithm,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Yamamoto, G.

Ann. Tokyo Astron. Observ. (1)

T. Onaka, “Light scattering by spheroidal grains,” Ann. Tokyo Astron. Observ. 18, 1–54 (1980).

Appl. Opt. (2)

Astrophys. Space Sci. (1)

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

J. Electromagn. Waves Appl. (1)

M. F. R. Cooray, I. R. Ciric, “Scattering of electromagnetic waves by a coated dielectric spheroid,” J. Electromagn. Waves Appl. 6, 1491–1507 (1992).
[CrossRef]

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

M. I. Mishchenko, L. D. Travis, “Light scattering by polydisperse rotationally symmetric nonspherical particles: linear polarization,” J. Quant. Spectrosc. Radiat. Transfer 51, 759–778 (1994).
[CrossRef]

I. Gurwich, N. Shiloah, M. Kleiman, “The recursive algorithm for electromagnetic scattering by tilted infinite circular multi-layered cylinder,” J. Quant. Spectrosc. Radiat. Transfer 63, 217–229 (1999).
[CrossRef]

J. Wave-Material Interaction (1)

A. Cohen, M. Kleiman, “Mie scattering coefficients for multilayered magnetic particles,” J. Wave-Material Interaction 1, 252–256 (1986).

Radio Sci. (1)

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multi-layered sphere: recursive algorithm,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Other (2)

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

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

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Figures (1)

Fig. 1
Fig. 1

Geometry of the multilayered oblate spheroid illuminated by an electromagnetic plane wave at incident angle α relative to the z axis.

Equations (47)

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E0=yˆE0 expik0x sin α+z cos α
E0=xˆ cos α-zˆ sin αE0 expik0x sin α+z cos α
E=×Uzˆ+Vr, H=-i 1μk0 ××Uzˆ+Vr
E=i 1εk0 ××Uzˆ+Vr, H=×Uzˆ+Vr
x=f2ξ2-11-η21/2 cos φ, y=f2ξ2-11-η21/2 sin φ,  z=f2 ξη,
ηUj+f2 ξjVj=ηUj+1+f2 ξjVj+1, ξξUj+f2 ηVj=ξξUj+1+f2 ηVj+1|ξ=ξj, εjξjUj+f2 ηVj=εj+1ξjUj+1+f2 ηVj+11μjξηUj+f2 ξVj+1-η2ξ2-1ηξUj+f2 ηVj=1μj+1ξηUj+1+f2 ξVj+1+1-η2ξ2-1ηξUj+1+f2 ηVj+1|ξ=ξj.
Uj=m=1l=mamljRml1cj, ξ+cmljRml2cj, ξS¯mlcj, ηcos mφ, Vj=m=1l=mbmljRml1cj, ξ+dmljRml2cj, ξS¯mlcj, ηcos mφ,
l=m amljγnlj,j+1,mRml1cj, ξj+AmljRml2cj, ξj+ξjf2l=m bmljδnlj,j+1,mRml1cj, ξj+BmljRml2cj, ξj =l=m amlj+1γnlj+1,j+1,mRml1cj+1, ξj+Amlj+1Rml2cj+1, ξj+ξjf2l=m bmlj+1inlRml1cj+1, ξj+Bmlj+1Rml2cj+1, ξj,
εjξjl=m amljδnlj,j+1,mRml1cj, ξj+AmljRml2cj, ξj+εjf2l=m bmljγnlj,j+1,mRml1cj, ξj+BmljRml2cj, ξj=εj+1ξjl=m amlj+1inlRml1cj+1, ξj+Amlj+1Rml2cj+1, ξj+εj+1f2l=m bmlj+1γnlj+1,j+1,mRml1cj+1, ξj+Bmlj+1Rml2cj+1, ξj,
l=m amljδnlj,j+1,mRml1cj, ξj1+ξjDml1cj, ξj+AmljRml2cj, ξj1+ξjDml2cj, ξj+f2l=m bmljγnlj,j+1,mRml1cj, ξj+BmljRml2cj, ξj=l=m amlj+1inlRml1cj+1, ξj1+ξjDml1cj+1, ξj+Amlj+1Rml2cj+1, ξj1+ξjDml2cj+1, ξj+f2l=m bmlj+1γnlj+1,j+1,mRml1cj+1, ξj+Bmlj+1Rml2cj+1, ξj,
1μjl=m amljγnlj,j+1,mRml1cj, ξj+AmljRml2cj, ξj1μjf2l=m bmljδnlj,j+1,m×Rml1cj, ξj1+ξjDml1cj, ξj+BmljRml2cj, ξj1+ξjDml2cj, ξj+1μjξjξj2-1l=m amljκnlj,j+1,mRml1cj, ξj+AmljRml2cj, ξj+1μj1ξj2-1f2l=m bmljσnlj,j+1,m×Rml1cj, ξj+BmljRml2cj, ξj=1μj+1l=m amlj+1γnlj+1,j+1,mRml1cj+1, ξj+Amlj+1Rml2cj+1, ξj1μj+1f2l=m bmlj+1inlRml1cj+1, ξj1+ξjDml1cj+1, ξj+Bmlj+1Rml2cj+1, ξj1+ξjDml2cj+1, ξj+1μj+1ξjξj2-1l=m amlj+1κnlj+1,j+1,mRml1cj+1, ξj+Amlj+1Rml2cj+1, ξj+1μj+11ξj2-1f2l=m bmlj+1σnlj+1,j+1,mRml1cj+1, ξj+Bmlj+1Rml2cj+1, ξj,
δnlj,k,m=-11 S¯mncj, ηS¯mlck, ηdη, γnlj,k,m=-11 ηS¯mncj, ηS¯mlck, ηdη, κnlj,k,m=-11 S¯mncj, ηS¯mlck, η1-η2dη, snlj,k,m=-11 ηS¯mncj, η]S¯mlck, ηdη,
wa,klmj,j=iklRml1cj, ξj+AmljRml2cj, ξj,
amljRml1cj, ξj+AmljRml2cj, ξj=kνa,kmjwa,klmj,j,
wa,klmj,j+1=iklRml1cj+1, ξj+Amlj+1Rml2cj+1, ξj,
wb,klmj,j=iklRml1cj, ξj+BmljRml2cj, ξj,
Πamj,j,matrix, πa,klmj,j=iklRml1cj, ξj1+ξjDml1cj, ξj+AmljRml2cj, ξj1+ξjDml2cj, ξj,
ua,klmj,j=iklRml1cj,ξj+AmljRml2cj,ξj,
ub,klmj,j=iklRml1cj, ξj+AmljRml2cj, ξjf2
Γmj,j+1·Wamj,j·Vamj+ξjΔmj,j+1·Wbmj,j·Vbmj=Γmj+1,j+1·Wamj,j+1·Vamj+1+ξjWbmj,j+1·Vbmj+1, εjξjΔmj,j+1·Wamj,j·Vamj+εjΓmj,j+1·Wbmj,j·Vbmj=εj+1ξjWamj,j+1·Vamj+1+εj+1Γmj+1,j+1·Wbmj,j+1·Vbmj+1, Δmj,j+1·Πamj,j·Vamj+Γmj,j+1·Ubmj,j·Vbmj=Πamj,j+1·Vamj+1+Γmj+1,j+1·Ubmj,j+1·Vbmj+1, 1μjΓmj,j+1·Uamj,j+ξjξj2-1Kmj,j+1·Wamj,j·Vamj+1μjΔmj,j+1Πbmj,j+ξjξj2-1Smj,j+1·Wbmj,j·Vbmj=1μj+1Γmj+1,j+1·Uamj,j+1+ξjξj2-1Kmj+1,j+1·Wamj,j+1·Vamj+1+1μj+1Πbmj,j+1+ξjξj2-1Smj+1,j+1·Wbmj,j+1·Vbmj+1
κnlj,k,m=ξjξj2-1-11 S¯mncj, ηS¯mlck, η1-η2dη, σnlj,k,m=ξjξj2-1-11ηS¯mncj, ηS¯mlck, ηdη
Tj1ε+ma=Δmj,j+1-εjεj+1 Γmj+1,j+1-1·Γmj,j+1=Δmj,j+1-εjεj+1 Γmj+1,j+1-1·Δmj,j+1·Γmj+1,j+1, Tjm=1ξj Γmj+1,j+1-ξjΓmj+1,j+1-1,
Tj2ε-ma=1ξj Γmj,j+1-εjεj+1 ξjΓmj+1,j+1-1·Δmj,j+1 =1ξj Δmj,j+1·Γmj+1,j+1-εjεj+1 ξjΓmj+1,j+1-1·Δmj,j+1, Tj1ε-ma=Δmj+1,j-εj+1εj Γmj,j+1-1·Γmj+1,j+1=Δmj+1,j-εj+1εj Γmj+1,j+1-1·Δmj+1,j·Γmj+1,j+1, Tj2ε-ma=1ξj Δmj+1,j·Γmj+1,j+1-εj+1εj ξjΓmj,j+1-1=1ξj Δmj+1,j·Γmj+1,j+1-εj+1εj ξjΓmj+1,j+1-1·Δmj+1,j.
Tj1ε+m·Wbmj,j·Vbmj=Tjm·Wamj,j+1·Vamj+1-Tj2ε+m·Wamj,j·Vamj, Tj1ε-m·Wbmj,j+1·Vbmj+1=Tjm·Wamj,j·Vamj-Tj2ε-m·Wamj,j+1·Vamj+1,
Δmj,j+1·Πamj,j-Γmj,j+1·Ubmj,j·Wbmj,j-1·Tj1ε+ma-1·Tj2ε+ma·Wamj,j-Γmj+1,j+1·Ubmj,j+1·Wbmj,j+1-1·Tj1ε-ma-1·Tjm·Wamj,j·Vamj=Δmj+1,j+1·Πamj,j+1-Γmj,j+1·Ubmj,j·Wbmj,j-1·Tj1ε+ma-1·Tjm·Wamj,j+1·Γmj+1,j+1·Ubmj,j+1·Wbmj,j+1-1·Tj1ε-ma-1·Tj2ε-ma·Wamj,j+1·Vamj+1,
Γmj,j+1·Uamj,j+Kmj,j+1·Wamj,j-Smj,j+1·Tj1ε+ma-1·Tj2ε+ma·Wamj,j+1-Δmj,j+1·Πbmj,j·Wbmj,j-1·Tj1ε+ma-1·Tj2ε+ma·Wamj,j-μjμj+1Πbmj,j+1·Wbmj,j+1-1·Tj1ε-ma-1·Tjm·Wamj,j+Smj+1,j+1·Tj1ε-ma-1·Tjm·Wamj,j·Vamj=μjμj+1Γmj+1,j+1·Uamj,j+1+Kmj+1,j+1·Wamj,j+1+Smj+1,j+1·Tj1ε-ma-1·Tj2ε-ma·Wamj,j+1-Πbmj,j+1·Wbmj,j+1-1·Tj2ε-ma-1·Tj2ε-ma·Wamj,j+1-Δmj,j+1·Πbmj,j·Wbmj,j-1·Tj1ε+ma-1·Tjm·Wamj,j+1+Smj,j+1·Tj1ε+ma-1·Tjm·Wamj,j+1·Vamj+1,
Γmj,j+1=Γmj,j·Δmj,j+1, Γmj,j+1=Δmj,j+1·Γmj+1,j+1, Δmj,j+1=Δmj+1,j-1,
Hajm=1μjUamj,j·Wamj,j-1, Haj1m=1μj+1Uamj,j+1·Wamj,j+1-1,
Gajm=Πamj,j·Wamj,j-1, Gaj1m=Πamj,j+1·Wamj,j+1-1,
Gajm=ξjμjHajm+μjI, Gbjm=ξjμjHbjm+μjI
Δmj,j+1·Gajm-Γj1m·Hbjm·Tj12+ma,h-Γj1m·Hbj1m·Tj-ma,h-1 ·-Δmj,j+1·Γj1m·Hbjm·Tj+ma,h+Gaj1m-Γj1m·Hbj1m·Tj12ma,h =Δmj,j+1·Γj1m·Hajm-Gbjm·Tj12+ma,h+Ψj,εμma-Gbj1m·Tj-ma,h-1 ·-Δmj,j+1·Gbjm·Tj+ma,h+Ψj+1,εμma+Γj1m·Haj1m-Gbj1m·Tj12-ma,h, Δmj,j+1·Γj1m·Hbjm-Gajm·Tj12+mb,h-Gaj1m·Tj-mb,h-1 -Δmj,j+1·Gajm·Tj+mb,h+Γj1m·Hbj1m-Gaj1m·Tj12-mb,h=Δmj,j+1·Gbjm-Γj1m·Hajm·Tj12+mb,h+Ψj,εμmb-Γj1m·Haj1m·Tj-mb,h-1·-Δmj,j+1·Γj1m·Hajm·Tj+mb,h+Ψj+1,εμmb+Gbj1m-Γj1m·Haj1m·Tj12-mb,h
Tj1ε+mb=εjεj+1 Δmj,j+1-Γmj+1,j+1-1·Γmj,j+1, Tj2ε+mb=1ξjεjεj+1 Γmj,j+1-ξjΓmj+1,j+1-1·Δmj,j+1, Tj1ε-mb=εj+1εj Δmj+1,j-Γmj,j+1-1·Γmj+1,j+1, Tj2ε-mb=1ξjεj+1εj Δmj+1,j·Γmj+1,j+1-ξjΓmj,j+1-1;
Tj12-ma,h=Tj1ε-ma-1·Tj2ε-ma,  Tj12+ma,h=Tj1ε+ma-1·Tj2ε+maTj-ma,h=Tj1ε-ma-1·Tjm,  Tj+ma,h=Tj1ε+ma-1·Tjm, Ωj1-ma=Kmj+1,j+1+Smj+1,j+1·Tj1ε-ma-1·Tj2ε-ma, Ωj1+ma=Kmj+1,j+1+Smj+1,j+1·Tj1ε+ma-1·Tj2ε+ma, Sj1m=Smj+1,j+1;
Ψj,εμma=1μj Δmj,j+1·Ωj1+ma-1μj+1Sj1m·Tj-ma,h, Ψj+1,εμma=1μj+1 Ωj1-ma-1μj Δmj,j+1·Sj1m·Tj+ma,h;
Ψj,εμmb=1μj Δmj,j+1·Γj1mΩj1+mb-1μj+1 Kj1m·Tj-mb,h, Ψj+1,εμmb=1μj+1 Ωj1-mb-1μj Δmj,j+1·Γj1m·Kj1m·Tj+mb,h;
Δmj,j+1·ξjHajm-Γj1m·Hbjm·Tj12+ma,h+Δmj,j+1·Qajm-Γj1m-Γj1m·Hbj1m·Tj-ma,h-1·-Δmj,j+1·Γj1m·Hbjm·Tj+ma,h+Qaj1m+ξjHaj1m-Γj1m·Hbj1m·Tj12-ma,h=Δmj,j+1·Γj1m·Hajm-ξjHbjm·Tj12+ma,h+Λabj0-ξjHbj1m·Tj-ma,h-1·-Δmj,j+1·ξjHbjm·Tj+ma,h+Λabj1+Γj1m·Haj1m-ξjHbj1m·Tj12-ma,h,Δmj,j+1·Γj1m·Hbjm-ξjHajm·Tj12+mb,h-Θabj+-ξjHaj1m·Tj-mb,h-1·-ξjΔmj,j+1·Hajm·Tj+mb,h-Θabj(-)+Γj1m·Hbj1m-ξjHaj1m·Tj12-mb,h=Δmj,j+1·ξjHbjm-Γj1m·Hajm·Tj12+mb,h+Φbaj0-Γj1m·Haj1m·Tj-mb,h-1·-Δmj,j+1·Γj1m·Hajm·Tj+mb,h+Φbaj1+ξjHbj1m+-Γj1m·Haj1m·Tj12-mb,h,
Λabj0=-Δmj,j+1·Qbjm·Tj12+ma,h+Ψj,εμma-Qbj1m·Tj-ma,h, Λabj1=-Δmj,j+1·Qbjm·Tj+ma,h+Ψj+1,εμma-Qbj1m·Tj12-ma,h,
Θabj+=Δmj,j+1·QajmTj12+mb,h+Qaj1m·Tj-mb,h, Θabj(-)=Δmj,j+1·QajmTj+mb,h+Qaj1m·Tj12-mb,h,
Φbaj0=-Δmj,j+1·Qajm+Ψj,εμmb, Φbaj1=Ψj+1,εμmb-Qaj1m.
πa,klmj,j=iklRml1cj, ξj1+ξjDml1cj, ξj+AmljRml2cj, ξj1+ξjDml2cj, ξj=iklRml1cj, ξj+AmljRml2cj, ξj+ξjiklRml1cj, ξj+AmljRml2cj, ξj.
Πamj,j=Wamj,j+ξjUamj,j,
Δmj,j+1·Gajm-Δmj,j+1·Γj1m·Hbjm·Tj1ε+ma-1·Tj2ε+ma-Γj1m·Hbj1m·Tj1ε-ma-1·Tjm·Wamj,j·Vamj=Gaj1m-Γj1m·Hbj1m·Tj1ε-ma-1·Tj2ε-ma-Δmj,j+1·Γj1m·Hbjm·Tj1ε+ma-1·Tjm·Wamj,j+1·Vamj+1, Γmj,j+1·Hajm+Kmj,j+1-Smj,j+1·Tj1ε+ma-1·Tj2ε+ma-Δmj,j+1·Gbjm·Tj1ε+ma-1·Tj2ε+ma-μjμj+1Gbj1m·Tj1ε-ma-1·Tjm+Smj+1,j+1·Tj1ε-ma-1·Tjm·Wamj,j·Vamj=μjμj+1Γmj+1,j+1·Haj1m-Gbj1m·Tj1ε-ma-1·Tj2ε-ma+Kmj+1,j+1+Smj+1,j+1Tj1ε-ma-1Tj2ε-ma-Δmj,j+1·Gbjm·Tj1ε+ma-1·Tjm+Smj,j+1·Tj1ε+ma-1·Tjm·Wamj,j+1·Vamj+1.
Δmj,j+1·Gajm-Δmj,j+1·Γj1m·Hbjm·Tj12+ma,h-Γj1m·Hbj1m·Tj-ma,h·Wamj,j·Vamj=Gaj1m-Γj1m·Hbj1m·Tj12-ma,h-Δmj,j+1·Γj1m·Hbjm·Tj+ma,h·Wamj,j+1·Vamj+1, Δmj,j+1·Γj1m·Hajm+Ωj1ma-Gbjm·Tj12+ma,h-μjμj+1Gbj1m+Sj1mTj-ma,h·Wamj,j·Vamj=μjμj+1Γj1m·Haj1m-Gbj1m·Tj12-ma,h+Ωj1-ma-Δmj,j+1·Gbjm+Sj1m·Tj+ma,h·Wamj,j+1·Vamj+1.
Δmj,j+1·Gajm-Δmj,j+1·Γj1m·Hbjm·Tj12+ma,h+Γj1m·Hbj1m·Tj-ma,h·Wamj,j·Vamj=Gaj1m-Δmj,j+1·Γj1m·Hbjm·Tj+ma,h+Γj1m·Hbj1m·Tj12-ma,h·Wamj,j+1·Vamj+1, Δmj,j+1·Γj1m·Hajm-Δmj,j+1·Gbjm·Tj12+ma,h+μjμj+1Gbj1m·Tj-ma,h+Δmj,j+1·Ωj1+ma-μjμj+1Sj1m·Tj-ma,h·Wamj,j·Vamj=μjμj+1 Γj1m·Haj1m-Δmj,j+1·Gbjm·Tj+ma,h+μjμj+1Gbj1m·Tj12-ma,h+μjμj+1 Ωj1-ma-Δmj,j+1·Sj1m·Tj+ma,h·Wamj,j+1·Vamj+1,
Δmj,j+1·Γj1m·Hbjm-Δmj,j+1·Gajm·Tj12+mb,h+Gaj1m·Tj-mb,h·Wbmj,j·Vbmj=Γj1m·Hbj1m-Δmj,j+1·Gajm·Tj+mb,h+Gaj1m·Tj12-mb,h·Wbmj,j+1·Vbmj+1, Δmj,j+1·Gbjm-Δmj,j+1·Γj1m·Hajm·Tj12+mb,h+μjμj+1 Γj1m·Haj1m·Tj-mb,h+Δmj,j+1·Γj1mΩj1+mb-μjμj+1 Kj1m·Tj-mb,h·Wbmj,j·Vbmj=μjμj+1Gbj1m-Δmj,j+1·Γj1m·Hajm·Tj+mb,h+μjμj+1 Γj1m·Haj1m·Tj12-mb,h+μjμj+1 Ωj1-mb-Δmj,j+1·Γj1m·Kj1m·Tj+mb,h·Wbmj,j+1·Vbmj+1,
Tj1ε+ma·Tj1ε-ma=2I-εjεj+1 Γj1m-1Γjm-εj+1εj Γjm-1Γj1m, Tj2ε+ma·Tj2ε-ma=1ξj2 ΓjmΓj1m+ξj2Γj1m-1Γjm-1-εjεj+1+εj+1εjI, Tj1ε+mb·Tj1ε-mb=2I-εj+1εj Γj1m-1Γjm-εjεj+1 Γjm-1Γj1m, Tj2ε+mb·Tj2ε-mb=1ξj2 ΓjmΓj1m+ξj2Γj1m-1Γjm-1-εjεj+1+εj+1εjI.
Δmj,j+1·Gajm-Δmj,j+1·Γj1m·Hbjm·Tj12+ma,h+Γj1m·Hbj1m·Tj-ma,h-1·Gaj1m-Δmj,j+1·Γj1m·Hbjm·Tj+ma,h+Γj1m·Hbj1m·Tj12-ma,h=Δmj,j+1·Γj1m·Hajm-Δmj,j+1·Gbjm·Tj12+ma,h+μjμj+1Gbj1m·Tj-ma,h+Ψj,εμma-1μjμj+1 Γj1m·Haj1m-Δmj,j+1·Gbjm·Tj+ma,h+μjμj+1Gbj1m·Tj12-ma,h+Ψj+1,εμma, Δmj,j+1·Γj1m·Hbjm-Δmj,j+1·Gajm·Tj12+mb,h+Gaj1m·Tj-mb,h-1·Γj1m·Hbj1m-Δmj,j+1·Gajm·Tj+mb,h+Gaj1m·Tj12-mb,h=Δmj,j+1·Gbjm-Δmj,j+1·Γj1m·Hajm·Tj12+mb,h+μjμj+1 Γj1m·Haj1m·Tj-mb,h+Ψj,εμmb-1·μjμj+1Gbj1m-Δmj,j+1·Γj1m·Hajm·Tj+mb,h+μjμj+1 Γj1m·Haj1m·Tj12-mb,h+Ψj+1,εμmb.

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