Abstract

The scattering of electromagnetic waves by a dielectric structure that fills a slit aperture in a conducting screen is considered. The cavity consists of two zones of different materials separated by an arbitrarily shaped interface. A rigorous R-matrix multilayer modal method is applied, which gives numerical stability even for deep structures. Examples of the numerical results obtained are shown and discussed for both fundamental cases of polarization.

© 1998 Optical Society of America

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References

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  1. T. B. Hansen, A. D. Yaghjian, “Low-frequency scattering from two-dimensional perfect conductors,” IEEE Trans. Antennas Propag. 40, 1389–1402 (1992).
    [CrossRef]
  2. Y.-L. Kok, “Boundary value solution to electromagnetic scattering by a rectangular groove in a ground plane,” J. Opt. Soc. Am. A 9, 302–311 (1992).
    [CrossRef]
  3. T.-M. Wang, H. Ling, “A connection algorithm on the problem of EM scattering from arbitrary cavities,” J. Electromagn. Waves Appl. 5, 301–314 (1991).
  4. S.-K. Jeng, “Scattering from a cavity-backed slit in a ground plane—TE case,” IEEE Trans. Antennas Propag. 38, 1529–1532 (1990).
  5. R. F. Harrington, J. R. Mautz, “A generalized network formulation for aperture problems,” IEEE Trans. Antennas Propag. 24, 870–873 (1976).
    [CrossRef]
  6. J. F. Aguilar, E. R. Méndez, “Imaging optically thick objects in scanning microscopy: perfectly conducting surfaces,” J. Opt. Soc. Am. A 11, 155–167 (1994).
    [CrossRef]
  7. T. C. Rao, R. Barakat, “Plane wave scattering by a conducting cylinder partially buried in a ground plane. 1: TM case,” J. Opt. Soc. Am. A 6, 1270–1280 (1989).
    [CrossRef]
  8. T. C. Rao, R. Barakat, “Plane wave scattering by a conducting cylinder partially buried in a ground plane. 2: TE case,” J. Opt. Soc. Am. A 8, 1986–1990 (1991).
    [CrossRef]
  9. P. J. Valle, F. González, F. Moreno, “Electromagnetic wave scattering from conducting cylindrical structures on flat substrates: study by means of the extinction theorem,” Appl. Opt. 33, 512–523 (1994).
    [CrossRef] [PubMed]
  10. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  11. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  12. D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thickness,” J. Opt. Soc. Am. A 8, 755–762 (1991).
    [CrossRef]
  13. J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1977).
    [CrossRef]
  14. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
    [CrossRef]
  15. Y.-L. Kok, “General solution to the multiple-metallic-grooves scattering problem: the fast-polarization case,” Appl. Opt. 32, 2573–2581 (1993).
    [CrossRef] [PubMed]
  16. R. A. Depine, D. C. Skigin, “Scattering from metallic surfaces having a finite number of rectangular grooves,” J. Opt. Soc. Am. A 11, 2844–2850 (1994).
    [CrossRef]
  17. M. Kuittinen, J. Turunen, “Exact-eigenmode model for index-modulated apertures,” J. Opt. Soc. Am. A 13, 2014–2020 (1996).
    [CrossRef]
  18. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).
  19. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  20. D. C. Skigin, R. A. Depine, “R-matrix method for a surface with one groove of arbitrary profile,” Opt. Commun. 130, 307–316 (1996).
    [CrossRef]
  21. D. C. Skigin, R. A. Depine, “The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves,” J. Mod. Opt. 44, 1023–1036 (1997).
    [CrossRef]
  22. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
    [CrossRef]
  23. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
    [CrossRef]
  24. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]

1997

D. C. Skigin, R. A. Depine, “The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves,” J. Mod. Opt. 44, 1023–1036 (1997).
[CrossRef]

1996

1995

1994

1993

1992

T. B. Hansen, A. D. Yaghjian, “Low-frequency scattering from two-dimensional perfect conductors,” IEEE Trans. Antennas Propag. 40, 1389–1402 (1992).
[CrossRef]

Y.-L. Kok, “Boundary value solution to electromagnetic scattering by a rectangular groove in a ground plane,” J. Opt. Soc. Am. A 9, 302–311 (1992).
[CrossRef]

1991

1990

S.-K. Jeng, “Scattering from a cavity-backed slit in a ground plane—TE case,” IEEE Trans. Antennas Propag. 38, 1529–1532 (1990).

1989

1982

1981

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

1977

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1977).
[CrossRef]

1976

R. F. Harrington, J. R. Mautz, “A generalized network formulation for aperture problems,” IEEE Trans. Antennas Propag. 24, 870–873 (1976).
[CrossRef]

1975

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Aguilar, J. F.

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1977).
[CrossRef]

Awada, K. A.

Barakat, R.

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Depine, R. A.

D. C. Skigin, R. A. Depine, “The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves,” J. Mod. Opt. 44, 1023–1036 (1997).
[CrossRef]

D. C. Skigin, R. A. Depine, “R-matrix method for a surface with one groove of arbitrary profile,” Opt. Commun. 130, 307–316 (1996).
[CrossRef]

R. A. Depine, D. C. Skigin, “Scattering from metallic surfaces having a finite number of rectangular grooves,” J. Opt. Soc. Am. A 11, 2844–2850 (1994).
[CrossRef]

Fox, J. R.

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1977).
[CrossRef]

Gaylord, T. K.

González, F.

Hansen, T. B.

T. B. Hansen, A. D. Yaghjian, “Low-frequency scattering from two-dimensional perfect conductors,” IEEE Trans. Antennas Propag. 40, 1389–1402 (1992).
[CrossRef]

Harrington, R. F.

R. F. Harrington, J. R. Mautz, “A generalized network formulation for aperture problems,” IEEE Trans. Antennas Propag. 24, 870–873 (1976).
[CrossRef]

Jeng, S.-K.

S.-K. Jeng, “Scattering from a cavity-backed slit in a ground plane—TE case,” IEEE Trans. Antennas Propag. 38, 1529–1532 (1990).

Kok, Y.-L.

Kuittinen, M.

Li, L.

Ling, H.

T.-M. Wang, H. Ling, “A connection algorithm on the problem of EM scattering from arbitrary cavities,” J. Electromagn. Waves Appl. 5, 301–314 (1991).

Mautz, J. R.

R. F. Harrington, J. R. Mautz, “A generalized network formulation for aperture problems,” IEEE Trans. Antennas Propag. 24, 870–873 (1976).
[CrossRef]

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Méndez, E. R.

Moharam, M. G.

Moreno, F.

Morf, R. H.

Pai, D. M.

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Rao, T. C.

Skigin, D. C.

D. C. Skigin, R. A. Depine, “The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves,” J. Mod. Opt. 44, 1023–1036 (1997).
[CrossRef]

D. C. Skigin, R. A. Depine, “R-matrix method for a surface with one groove of arbitrary profile,” Opt. Commun. 130, 307–316 (1996).
[CrossRef]

R. A. Depine, D. C. Skigin, “Scattering from metallic surfaces having a finite number of rectangular grooves,” J. Opt. Soc. Am. A 11, 2844–2850 (1994).
[CrossRef]

Tamir, T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Turunen, J.

Valle, P. J.

Wang, T.-M.

T.-M. Wang, H. Ling, “A connection algorithm on the problem of EM scattering from arbitrary cavities,” J. Electromagn. Waves Appl. 5, 301–314 (1991).

Wilson, I. J.

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1977).
[CrossRef]

Yaghjian, A. D.

T. B. Hansen, A. D. Yaghjian, “Low-frequency scattering from two-dimensional perfect conductors,” IEEE Trans. Antennas Propag. 40, 1389–1402 (1992).
[CrossRef]

Appl. Opt.

IEEE Trans. Antennas Propag.

S.-K. Jeng, “Scattering from a cavity-backed slit in a ground plane—TE case,” IEEE Trans. Antennas Propag. 38, 1529–1532 (1990).

R. F. Harrington, J. R. Mautz, “A generalized network formulation for aperture problems,” IEEE Trans. Antennas Propag. 24, 870–873 (1976).
[CrossRef]

T. B. Hansen, A. D. Yaghjian, “Low-frequency scattering from two-dimensional perfect conductors,” IEEE Trans. Antennas Propag. 40, 1389–1402 (1992).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

J. Electromagn. Waves Appl.

T.-M. Wang, H. Ling, “A connection algorithm on the problem of EM scattering from arbitrary cavities,” J. Electromagn. Waves Appl. 5, 301–314 (1991).

J. Mod. Opt.

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

D. C. Skigin, R. A. Depine, “The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves,” J. Mod. Opt. 44, 1023–1036 (1997).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

M. Kuittinen, J. Turunen, “Exact-eigenmode model for index-modulated apertures,” J. Opt. Soc. Am. A 13, 2014–2020 (1996).
[CrossRef]

J. F. Aguilar, E. R. Méndez, “Imaging optically thick objects in scanning microscopy: perfectly conducting surfaces,” J. Opt. Soc. Am. A 11, 155–167 (1994).
[CrossRef]

R. A. Depine, D. C. Skigin, “Scattering from metallic surfaces having a finite number of rectangular grooves,” J. Opt. Soc. Am. A 11, 2844–2850 (1994).
[CrossRef]

T. C. Rao, R. Barakat, “Plane wave scattering by a conducting cylinder partially buried in a ground plane. 1: TM case,” J. Opt. Soc. Am. A 6, 1270–1280 (1989).
[CrossRef]

D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thickness,” J. Opt. Soc. Am. A 8, 755–762 (1991).
[CrossRef]

T. C. Rao, R. Barakat, “Plane wave scattering by a conducting cylinder partially buried in a ground plane. 2: TE case,” J. Opt. Soc. Am. A 8, 1986–1990 (1991).
[CrossRef]

Y.-L. Kok, “Boundary value solution to electromagnetic scattering by a rectangular groove in a ground plane,” J. Opt. Soc. Am. A 9, 302–311 (1992).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
[CrossRef]

Opt. Acta

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1977).
[CrossRef]

Opt. Commun.

D. C. Skigin, R. A. Depine, “R-matrix method for a surface with one groove of arbitrary profile,” Opt. Commun. 130, 307–316 (1996).
[CrossRef]

Other

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).

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Figures (7)

Fig. 1
Fig. 1

Configuration of the problem.

Fig. 2
Fig. 2

Multilayer approximation. This structure is considered for the plots in Figs. 3 and 4 below.

Fig. 3
Fig. 3

Intensity versus transmission angle for the structure in Fig. 2 approximated by 40 layers. The incident beam is p polarized and has w/λ=20 and θ0=0°. Other parameters are L/λ=2.4, h/λ=1, c/λ=0.8, ν1=1.3, ν2=1.5, and νr=νt=1.

Fig. 4
Fig. 4

Intensity versus transmission angle for the structure in Fig. 2 approximated by 40 layers for three values of c: c/λ=0, c/λ=0.4, and c/λ=0.8. The incident beam is s polarized and has w/λ=20 and θ0=0°. Other parameters are L/λ=2.4, h/λ=1, ν1=1.3, ν2=1.5, and νr=νt=1.

Fig. 5
Fig. 5

Exponential profiles used for the plots in Fig. 6 approximated by 40 layers.

Fig. 6
Fig. 6

Intensity versus transmission angle for the structures in Fig. 5. The incident beam is s polarized and has w/λ=20 and θ0=0°. Other parameters are L/λ=2.4, h/λ=1, c/λ=0.8, ν1=1.3, ν2=1.5, and νr=νt=1. (a) Profiles 1 and 2, (b) profiles 4 and 5.

Fig. 7
Fig. 7

Intensity versus transmission angle for the structure of Fig. 2 and different angles of incidence. The incident beam is s polarized and has w/λ=20 and θ0=0°. Other parameters are L/λ=2.4, h/λ=0.1, c/λ=0.4, ν1=1.3, ν2=1.5, and νr=νt=1.

Equations (69)

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

(2+kl2)f(x, y)=0,
finc(x, y)=-A(α)exp[i(αx-βry)]dα,
βr=kr2-α2ifkr2>α2iα2-kr2ifkr2<α2,
A(α)=1σπexp-(α-α0)2σ2,
frefl(x, y)=-Rq(α)exp[i(αx+βry)]dα,
ftrans(x, y)=-Tq(α)exp[i(αx-βty)]dα,
fjq(x, y)=m=1Um,jq(x)wm,jq(y)rectx-xminL,
wm,jq(y)=am,jq cos(vm,jqy)+bm,jq sin(vm,jqy),
Um,js(x)=sin[um,1,js(x-xmin)]forxmin<x<x1,jsin(um,1,jsΔ1j)cos[um,2,js(x-x1,j)]+um,1,jsum,2,jscos(um,1,jsΔ1j)sin[um,2,js(x-x1,j)],forx1,j<x<xmax,
Um,jp(x)=cos[um,1,jp(x-xmin)]forxmin<x<x1,jcos(um,1,jpΔ1j)cos[um,2,jp(x-x1,j)]-um,1,jpum,2,jp21sin(um,1,jpΔ1j)sin[um,2,jp(x-x1,j)]forx1,j<x<xmax,
um,l,jq=kl2-(vm,jq)2.
rect(x)=1if0<x<10otherwise.
1um,1,jscos(um,2,jsΔ2j)sin(um,1,jsΔ1j)
+1um,2,jssin(um,2,jsΔ2j)cos(um,1,jsΔ1j)=0,
um,2,jp2sin(um,2,jpΔ2j)cos(um,1,jpΔ1j)
+um,1,jp1cos(um,2,jpΔ2j)sin(um,1,jpΔ1j)=0
Ts(α)=12πm=1Ims(-α)am,1s,
bm,1s=k=1Sm,kak,1s;
am,1p=k=1Pm,kbk,1p,
Tp(α)=it2πβtm=1Imp(-α)vm,1pbm,1p;
Imq(α)=xminxmax 1ξ1q(x)Um,1q(x)exp(iαx)dx,
Cm,jq=xminxmax 1ξjq(x)[Um,jq(x)]2 dx,
Sm,k=-i2πCm,1svm,1s-βtIm(α)Ik(-α)dα,
Pm,k=it2πCm,1p-Im(α)Ik(-α) vk,1pβtdα,
ξjq(x)=1forq=sj(x)forq=p.
m=1Am,m(j)swm,js(yj)=wm,j+1s(yj),
m=1 Cm,jsCm,j+1s[Am,m(j)s]-1χm,js(yj)=χm,j+1s(yj),
m=1[Am,m(j)p]-1wm,jp(yj)=wm,j+1p(yj),
m=1 Cm,jpCm,j+1pAm,m(j)pχm,jp(yj)=χm,j+1p(yj),
χm,jq(y)=-d[wm,jq(y)]dy,
Am,m(j)s=1Cm,j+1sxminxmaxUm,js(x)Um,j+1s(x)dx,
Am,m(j)p=1Cm,jpxminxmax 1j(x)Um,jp(x)Um,j+1p(x)dx.
bm,Ms=m=1Mm,msam,1s,
am,Ms=m=1Nm,msam,1s,
Mm,ms=-cos(vm,MsyM-1){[(R12)m,m(M-1)]-1+Qm,ms}/vm,Ms+sin(vm,MsyM-1)×k=1((R22)m,k(M-1){[(R12)k,m(M-1)]-1+Qk,ms}-(R21)m,k(M-1)vk,1sSk,m),
Nm,ms=[(R12)m,m(M-1)]-1+Qm,msvm,Ms sin(vm,MsyM-1)+cos(vm,MsyM-1)Mm,ms,
Qm,ms=l=1k=1[(R12)m,l(M-1)]-1(R11)l,k(M-1)vk,1sSk,m,
bm,Mp=m=1Mm,mpbm,1p,
am,Mp=m=1Nm,mpbm,1p,
Mm,mp=-sin(vm,MpyM-1)×{[(R12)m,m(M-1)]-1vm,1p+Qm,mp}+cos(vm,MpyM-1)vm,Mp×k=1(-(R22)m,k(M-1)×{[(R12)k,m(M-1)]-1vm,1p+Qk,mp}+(R21)m,k(M-1)Pk,m),
Nm,mp=-[(R12)m,m(M-1)]-1vm,1p+Qm,mpcos(vm,MpyM-1)-sin(vm,MpyM-1)Mm,mp,
Qm,mp=l=1k=1[(R12)m,l(M-1)]-1(R11)l,k(M-1)Pk,m,
2π[A(α)exp(-iβryM)+Rs(α)exp(iβryM)]=m=1[am,Ms cos(vm,MsyM)+bm,Ms sin(vm,MsyM)]Jms(-α),
i-βr[-A(α)exp(-iβryM)
+Rs(α)exp(iβryM)]Jms(α)dα
=vm,MsCm,Ms[-am,Ms sin(vm,MsyM)+bm,Ms cos(vm,MsyM)],
-[A(α)exp(-iβryM)+Rp(α)exp(iβryM)]Jmp(α)dα=Cm,Mp[am,Mp cos(vm,MpyM)+bm,Mp sin(vm,MpyM)],
iβrr2π[-A(α)exp(-iβryM)+Rp(α)exp(iβryM)]=m=1[am,Mp sin(vm,MpyM)-bm,Mp cos(vm,MpyM)]vm,MpJmp(-α),
Jmq(α)=xminxmax 1ξMq(x)Um,Mq(x)exp(iαx)dx
2π[A(α)exp(-iβryM)+Rs(α)exp(iβryM)]
=m=1Jms(-α)k=1Fm,ksak,1s,
i-βr[-A(α)exp(-iβryM)
+Rs(α)exp(iβryM)]Jms(α)dα
=vm,MsCm,Msk=1Gm,ksak,1s.
-[A(α)exp(-iβryM)+Rp(α)exp(iβryM)]Jmp(α)dα
=Cm,Mpk=1Gm,kpbk,1p,
iβrr2π[-A(α)exp(-iβryM)+Rp(α)exp(iβryM)]
=m=1Jmp(-α)k=1Fm,kpbk,1p.
Ir(α)=|Rq(α)|2βr,
It(α)=|Tq(α)|2βt,
1Pinc-Ir(α)ξrq+It(α)ξtqdα=1,
Pinc=1ξrq-|A(α)|2βr dα,
ξγq=1forq=sγ,forq=p,
Fm,ks=-sin(vm,MshM)vm,Msl=1[(R12)m,l(M-1)]-1×δl,k+i=1(R11)l,i(M-1)vi,1sSi,k+cos(vm,MshM)l=1i=1j=1(R22)m,l(M-1)×[(R12)l,i(M-1)]-1[δk,iδi,j+(R11)i,j(M-1)vj,1sSj,k]-cos(vm,MshM)l=1(R21)m,l(M-1)vl,1sSl,k,
Fm,kp=vm,Mp sin(vm,MphM)l=1[(R12)m,l(M-1)]-1×vl,1pδl,k+i=1(R11)l,i(M-1)Pi,k+cos(vm,MphM)l=1i=1j=1(R22)m,l(M-1)×[(R12)l,i(M-1)]-1[vl,1pδk,iδi,j+(R11)i,j(M-1)Pj,k]-cos(vm,MphM)l=1(R21)m,l(M-1)Pl,k,
Gm,ks=sin(vm,MshM)l=1(R21)m,l(M-1)vl,1sSl,k-sin(vm,MshM)l=1(R22)m,l(M-1)i=1[(R12)l,i(M-1)]-1×δi,k+j=1(R11)i,j(M-1)vj,1sSj,k-cos(vm,MshM)vm,Msl=1[(R12)m,l(M-1)]-1×δl,k+i=1(R11)l,i(M-1)vi,1sSi,k,
Gm,kp=sin(vm,MphM)vm,Mpl=1(R21)m,l(M-1)Pl,k-sin(vm,MphM)vm,Mpl=1(R22)m,l(M-1)i=1[(R12)l,i(M-1)]-1
×[vi,1pδi,k+(R11)l,i(M-1)Pi,k]-cos(vm,MphM)l=1[(R12)m,l(M-1)]-1
×vl,1pδl,k+i=1(R11)l,i(M-1)Pi,k.

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