Abstract

An efficient method is developed for rigorously analyzing the scattering of light by a layered circular cylindrical object in a layered background, and it is applied to the study of the transmission of light through a subwavelength hole in a metallic film, where the hole may be filled by a dielectric material. The method relies on expanding the electromagnetic field (subtracted by one-dimensional solutions of the layered media) in one-dimensional modes, where the expansion “coefficients” are functions satisfying two-dimensional Helmholtz equations. A system of equations is established on the boundary of the circular cylinder to solve the expansion “coefficients.” The method effectively reduces the original three-dimensional scattering problem to a two-dimensional problem on the boundary of the cylinder.

© 2014 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  2. T. W. Ebbesen, H. J. Lezec, G. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2012 (3)

2011 (2)

2010 (2)

L. Yuan and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing hole arrays in a slab,” J. Opt. Soc. Am. B 27, 2568–2579 (2010).
[CrossRef]

F. J. García-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

2009 (1)

H. Xu, P. Zhu, H. G. Craighead, and W. W. Webb, “Resonantly enhanced transmission of light through subwavelength apertures with dielectric filling,” Opt. Commun. 282, 1467–1471 (2009).
[CrossRef]

2008 (1)

2007 (1)

2005 (5)

2004 (2)

F. Olyslager, “Discretization of continuous spectra based on perfectly matched layers,” SIAM J. Appl. Math. 64, 1408–1433 (2004).
[CrossRef]

S. Boscolo and M. Midrio, “Three-dimensional multiple-scattering technique for the analysis of photonic-crystal slabs,” J. Lightwave Technol. 22, 2778–2786 (2004).
[CrossRef]

2002 (1)

2001 (2)

1998 (2)

T. W. Ebbesen, H. J. Lezec, G. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
[CrossRef]

1997 (1)

R. P. Wang and M.-M. Dumitrescu, “Theory of optical modes in semiconductor microdisk lasers,” J. Appl. Phys. 81, 3391–3397 (1997).
[CrossRef]

1994 (2)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Bao, G.

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Bonod, N.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Boscolo, S.

Cao, Q.

Chaumet, P.

Chen, Z. M.

Chew, W. C.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves (Morgan & Claypool, 2009).

Craighead, H. G.

H. Xu, P. Zhu, H. G. Craighead, and W. W. Webb, “Resonantly enhanced transmission of light through subwavelength apertures with dielectric filling,” Opt. Commun. 282, 1467–1471 (2009).
[CrossRef]

De Zutter, D.

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
[CrossRef]

Derudder, H.

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
[CrossRef]

Dumitrescu, M.-M.

R. P. Wang and M.-M. Dumitrescu, “Theory of optical modes in semiconductor microdisk lasers,” J. Appl. Phys. 81, 3391–3397 (1997).
[CrossRef]

Ebbesen, T. W.

F. J. García-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, G. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Garcia de Abajo, F. J.

García-Vidal, F. J.

F. J. García-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Morento, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef]

Ghaemi, G. F.

T. W. Ebbesen, H. J. Lezec, G. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Ginste, D. V.

Granet, G.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Han, L.

W. P. Huang, L. Han, and J. Mu, “A rigorous circuit model for simulation of large-scale photonic integrated circuits,” IEEE Photon. J. 4, 1622–1638 (2012).
[CrossRef]

R. Wang, L. Han, J. Mu, and W. P. Huang, “Simulation of waveguide crossings and corners with complex mode-matching method,” J. Lightwave Technol. 30, 1795–1801 (2012).
[CrossRef]

Howe, D.

Hu, B.

W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves (Morgan & Claypool, 2009).

Huang, W. P.

Hugonin, J.-P.

Jin, J. M.

J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

Kataja, K.

Kuipers, L.

F. J. García-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

Lalanne, P.

Lenne, P.-F.

Lezec, H. J.

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, G. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Linke, R. A.

Lu, Y. Y.

Martin-Moreno, L.

F. J. García-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

Martín-Morento, L.

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Morento, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef]

Michielssen, E.

Midrio, M.

Moreno, E.

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Morento, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef]

Mu, J.

Nédélec, J.-C.

J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems (Springer, 2001).

Nevière, M.

Olkkonen, J.

Olyslager, F.

D. Pissoort, E. Michielssen, D. V. Ginste, and F. Olyslager, “Fast-multipole analysis of electromagnetic scattering by photonic crystal slabs,” J. Lightwave Technol. 25, 2847–2863 (2007).
[CrossRef]

F. Olyslager, “Discretization of continuous spectra based on perfectly matched layers,” SIAM J. Appl. Math. 64, 1408–1433 (2004).
[CrossRef]

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
[CrossRef]

Pellerin, K. M.

Pissoort, D.

Popov, E.

Porto, J. A.

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Morento, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef]

Rigneault, H.

Silberstein, E.

Song, D.

D. Song, L. Yuan, and Y. Y. Lu, “Fourier-matching pseudospectral modal method for diffraction gratings,” J. Opt. Soc. Am. A 28, 613–620 (2011).
[CrossRef]

D. Song and Y. Y. Lu, “Pseudospectral modal method for conical diffraction of gratings,” J. Mod. Opt., doi: 10.1080/09500340.2013.856484 (to be published).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Thio, T.

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, and T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, G. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Tong, M. S.

W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves (Morgan & Claypool, 2009).

Trefethen, L. N.

L. N. Trefethen, Spectral Methods in MATLAB (Society for Industrial and Applied Mathematics, 2000).

Wang, R.

Wang, R. P.

R. P. Wang and M.-M. Dumitrescu, “Theory of optical modes in semiconductor microdisk lasers,” J. Appl. Phys. 81, 3391–3397 (1997).
[CrossRef]

Webb, W. W.

H. Xu, P. Zhu, H. G. Craighead, and W. W. Webb, “Resonantly enhanced transmission of light through subwavelength apertures with dielectric filling,” Opt. Commun. 282, 1467–1471 (2009).
[CrossRef]

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, G. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Wu, H. J.

Xu, H.

H. Xu, P. Zhu, H. G. Craighead, and W. W. Webb, “Resonantly enhanced transmission of light through subwavelength apertures with dielectric filling,” Opt. Commun. 282, 1467–1471 (2009).
[CrossRef]

Yuan, L.

Zhu, P.

H. Xu, P. Zhu, H. G. Craighead, and W. W. Webb, “Resonantly enhanced transmission of light through subwavelength apertures with dielectric filling,” Opt. Commun. 282, 1467–1471 (2009).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34, 2138–2140 (1998).
[CrossRef]

IEEE Photon. J. (1)

W. P. Huang, L. Han, and J. Mu, “A rigorous circuit model for simulation of large-scale photonic integrated circuits,” IEEE Photon. J. 4, 1622–1638 (2012).
[CrossRef]

J. Appl. Phys. (1)

R. P. Wang and M.-M. Dumitrescu, “Theory of optical modes in semiconductor microdisk lasers,” J. Appl. Phys. 81, 3391–3397 (1997).
[CrossRef]

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. A (5)

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

Microw. Opt. Technol. Lett. (1)

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Nature (1)

T. W. Ebbesen, H. J. Lezec, G. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Opt. Commun. (1)

H. Xu, P. Zhu, H. G. Craighead, and W. W. Webb, “Resonantly enhanced transmission of light through subwavelength apertures with dielectric filling,” Opt. Commun. 282, 1467–1471 (2009).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Morento, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef]

Rev. Mod. Phys. (1)

F. J. García-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

SIAM J. Appl. Math. (1)

F. Olyslager, “Discretization of continuous spectra based on perfectly matched layers,” SIAM J. Appl. Math. 64, 1408–1433 (2004).
[CrossRef]

Other (7)

D. Song and Y. Y. Lu, “Pseudospectral modal method for conical diffraction of gratings,” J. Mod. Opt., doi: 10.1080/09500340.2013.856484 (to be published).
[CrossRef]

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves (Morgan & Claypool, 2009).

J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems (Springer, 2001).

L. N. Trefethen, Spectral Methods in MATLAB (Society for Industrial and Applied Mathematics, 2000).

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Figures (4)

Fig. 1.
Fig. 1.

Circular cylindrical object in a layered background. (a) Vertical cross section in the xz or yz plane, (b) horizontal cross section in the xy plane.

Fig. 2.
Fig. 2.

Contour plots of |Ez| (top panel) and |Hz| (bottom panel) at z=1.2236nm.

Fig. 3.
Fig. 3.

|Ez| along the x axis and |Ex| along both x and y axes for z=1.2236nm.

Fig. 4.
Fig. 4.

Normalized transmission T for a silver film with a circular hole filled with a dielectric medium of refractive index nc. (a) T versus film thickness D for hole radius R=50nm and nc=1.81, (b) T versus R for D=200nm and nc=1.81, (c) T versus nc for D=200nm and R=25nm.

Equations (47)

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

×E=ik0μH,×H=ik0εE,
E(0)(x,y,z)=E˜(0)(z)ei(αx+βy),H(0)(x,y,z)=H˜(0)(z)ei(αx+βy),
2Hzx2+2Hzy2+z[1μ(μHz)z]+k02εμHz=0,
2Ezx2+2Ezy2+z[1ε(εEz)z]+k02εμEz=0.
μHz(x,y,z)=[η(e)]2ϕ(e)(z)V(e)(x,y),
μddz(1μdϕ(e)dz)+k02εμϕ(e)=[η(e)]2ϕ(e),
2V(e)x2+2V(e)y2+[η(e)]2V(e)=0.
[EyEx]=ik0ϕ(e)[xV(e)yV(e)],[HxHy]=1μdϕ(e)dz[xV(e)yV(e)].
ν=(νx,νy),τ=(νy,νx),
Eτ=ik0ϕ(e)V(e)ν,Hτ=1μdϕ(e)dzV(e)τ,
Hz=0,εEz(x,y,z)=[η(h)]2ϕ(h)(z)V(h)(x,y),
εddz(1εdϕ(h)dz)+k02εμϕ(h)=[η(h)]2ϕ(h),
2V(h)x2+2V(h)y2+[η(h)]2V(h)=0.
Eτ=1εdϕ(h)dzV(h)τ,Hτ=ik0ϕ(h)V(h)ν.
ϕ(p)(zb)=ϕ(p)(zt)=0,p{e,h}.
ϕj(p)(z),ηj(p),p{e,h},j=1,2,3,
ϕj(l,p)(z),ηj(l,p),Vj(l,p)(x,y)
Hz=Hz(0)+1μ(0)j=1[ηj(0,e)]2ϕj(0,e)Vj(0,e),
Ez=Ez(0)+1ε(0)j=1[ηj(0,h)]2ϕj(0,h)Vj(0,h),
Hτ=Hτ(0)+1μ(0)j=1dϕj(0,e)dzVj(0,e)τ+ik0j=1ϕj(0,h)Vj(0,h)ν,
Eτ=Eτ(0)+1ε(0)j=1dϕj(0,h)dzVj(0,h)τik0j=1ϕj(0,e)Vj(0,e)ν.
Λj(l,p)Vj(l,p)=νVj(l,p)onΓ.
x2V(0)+y2V(0)+η2V(0)=0,r>R.
V(0)(r,θ)=m=amHm(1)(ηr)eimθ,r>R,
rV(0)(r,θ)=m=amηHm(1)(ηr)eimθ,r>R,
Λ(0)eimθ=λm(0)eimθ,m=0,±1,±2,,
λm(0)=ηHm(1)(ηR)Hm(1)(ηR).
rV(0)=Λ(0)V(0)atr=R.
x2V(1)+y2V(1)+η2V(1)=0,r<R,
V(1)(r,θ)=m=bmJm(ηr)eimθ,r<R,
Λ(1)eimθ=λm(1)eimθ,m=0,±1,±2,,
λm(1)=ηJm(ηR)Jm(ηR).
[A(11)0A(13)00A(22)0A(24)A(31)A(32)A(33)A(34)A(41)A(42)A(43)A(44)][x1x2x3x4]=[b1b2b3b4],
x1=[v1(0,e)v2(0,e)vN(0,e)],x2=[v1(0,h)v2(0,h)vN(0,h)],x3=[v1(1,e)v2(1,e)vN(1,e)],x2=[v1(1,h)v2(1,h)vN(1,h)],
vj(l,p)=[Vj(l,p)(r1)Vj(l,p)(r2)Vj(l,p)(rM)]
Aij(11)=1μ(0)(zi)[ηj(0,e)]2ϕj(0,e)(zi)I,Aij(13)=1μ(1)(zi)[ηj(1,e)]2ϕj(1,e)(zi)I,Aij(22)=1ε(0)(zi)[ηj(0,h)]2ϕj(0,h)(zi)I,Aij(24)=1ε(1)(zi)[ηj(1,h)]2ϕj(1,h)(zi)I,Aij(31)=1μ(0)(zi)zϕj(0,e)(zi)τ,Aij(32)=ik0ϕj(0,h)(zi)Λj(0,h),Aij(33)=1μ(1)(zi)zϕj(1,e)(zi)τ,Aij(34)=ik0ϕj(1,h)(zi)Λj(1,h),Aij(41)=ik0ϕj(s,1)(zi)Λj(0,e),Aij(42)=1ε(0)(zi)zϕj(0,h)(zi)τ,Aij(43)=ik0ϕj(h,1)(zi)Λj(1,e),Aij(44)=1ε(1)(zi)zϕj(1,h)(zi)τ,
b1=[b11b12b1N],
Hz(1)(r,z)Hz(0)(r,z)=Hz(1,r)(r,z)Hz(0,r)(r,z)
E=E(0)+E(s),H=H(0)+H(s),
Sz=12Re(ExH¯yEyH¯x)=Sz(0)+Sz(extra),
Sz(extra)=12Re[Ex(0)H¯y(s)Ey(0)H¯x(s)+Ex(s)H¯y(0)Ey(s)H¯x(0)+Ex(s)H¯y(s)Ey(s)H¯x(s)].
P(extra)=R2Sz(extra)dr.
T=P(extra)PΩ1(i),
PΩ1(i)=Ω1Sz(i)dr
Ijk(0,pp)=Ω0[xVj(0,p)xV¯k(0,p)+yVj(0,p)yV¯k(0,p)]dr,Ijk(0,pq)=Ω0[xVj(0,p)yV¯k(0,q)yVj(0,p)xV¯k(0,q)]dr,
Ijk(0,pp)=1η¯k2ηj2Γ[ηj2VjνVk¯η¯k2V¯kνVj]ds(r),Ijk(0,pq)=ΓVjτVk¯ds(r),
s(z)={1+St[(zz˜t)/(ztz˜t)]3,z>z˜t,1,z˜b<z<z˜t,1+Sb[(zz˜b)/(zbz˜b)]3,z<z˜b,

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