Abstract

A relatively simple and efficient numerical method is developed for analyzing the scattering of light by a layered cylindrical structure of arbitrary cross section surrounded by a layered background. The method significantly extends an existing vertical mode expansion method (VMEM) for circular or elliptic cylindrical structures. The original VMEM and its extension give rise to effective two-dimensional formulations for the three-dimensional scattering problems of layered cylindrical structures. The extended VMEM developed in this paper uses boundary integral equations to handle the two-dimensional Helmholtz equations that appear in the vertical mode expansion process. The method is applied to analyze the transmission of light through subwavelength apertures in metallic films and the scattering of light by metallic nanoparticles.

© 2015 Optical Society of America

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References

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2015 (2)

2014 (1)

2013 (1)

2012 (4)

2011 (3)

2010 (3)

2009 (2)

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

A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009).
[Crossref]

2008 (1)

2007 (1)

M. Hammer, “Hybrid analytical/numerical coupled-mode modeling of guided wave devices,” J. Lightw. Technol. 25(9), 2287–2298 (2007).
[Crossref]

2005 (2)

Y. Y. Lu and J. Zhu, “Propagating modes in optical waveguides terminated by perfectly matched layers,” IEEE Photon. Technol. Lett. 17, 2601–2603 (2005).
[Crossref]

J. Olkkonen, K. Kataja, and D. Howe, “Light transmission through a high index dielectric-filled sub-wavelength hole in a metal film,” Opt. Express 13, 6980–6989 (2005).
[Crossref] [PubMed]

2004 (3)

2002 (1)

G. Granet and J. P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A 4, S145–S149 (2002).
[Crossref]

2001 (2)

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microwave Theory Tech. 49(2), 349–354 (2001).
[Crossref]

E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[Crossref]

1999 (1)

B. K. Alpert, “Hybrid Gauss-trapezoidal quadrature rules,” SIAM J. Sci. Comput. 20(5), 1551–1584 (1999).
[Crossref]

1998 (3)

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]

A. D. Raki, A. B. Djuriic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271–5283 (1998).
[Crossref]

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34(22), 2138–2140 (1998).
[Crossref]

1997 (1)

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,” Microwave and Optical Technology Letters 7, 599–604 (1994).
[Crossref]

Alpert, B. K.

B. K. Alpert, “Hybrid Gauss-trapezoidal quadrature rules,” SIAM J. Sci. Comput. 20(5), 1551–1584 (1999).
[Crossref]

Araújo, M. G.

Baets, R.

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microwave Theory Tech. 49(2), 349–354 (2001).
[Crossref]

Berenger, J. P.

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

Bienstman, P.

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microwave Theory Tech. 49(2), 349–354 (2001).
[Crossref]

Boscolo, S.

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

Cao, Q.

Chew, W. C.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates,” Microwave and Optical Technology Letters 7, 599–604 (1994).
[Crossref]

W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves (Morgan & Claypool, 2008).

Colton, D.

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (John Wiley & Sons, 1983).

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed. (Springer-Verlag, 2013).
[Crossref]

Craighead, H. G.

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

Dal Negro, L.

De Zutter, D.

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microwave Theory Tech. 49(2), 349–354 (2001).
[Crossref]

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34(22), 2138–2140 (1998).
[Crossref]

Derudder, H.

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microwave Theory Tech. 49(2), 349–354 (2001).
[Crossref]

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34(22), 2138–2140 (1998).
[Crossref]

Djuriic, A. B.

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. 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]

Elazar, J. M.

Fan, S.

V. Liu and S. Fan, “S4: A free electromagnetic solver for layered periodic structures,” Computer Physics Communications 183, 2233–2244 (2012).
[Crossref]

Forestiere, C.

Gallinet, B.

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]

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]

Granet, G.

G. Granet and J. P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A 4, S145–S149 (2002).
[Crossref]

Hagness, S. C.

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

Hammer, M.

M. Hammer, “Hybrid analytical/numerical coupled-mode modeling of guided wave devices,” J. Lightw. Technol. 25(9), 2287–2298 (2007).
[Crossref]

Han, L.

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

Hesselink, L.

Howe, D.

Hu, B.

W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves (Morgan & Claypool, 2008).

Huang, W. P.

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

J. Mu and W. P. Huang, “Simulation of three-dimensional waveguide discontinuities by a full-vector mode-matching method based on finite-difference schemes,” Opt. Express 16, 18152–18163 (2008).
[Crossref] [PubMed]

Hugonin, J.-P.

Iadarola, G.

Jin, J. M.

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

Kataja, K.

Kern, A. M.

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed. (Springer-Verlag, 2013).
[Crossref]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (John Wiley & Sons, 1983).

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.

Landesa, L.

Le Ru, E. C.

Lezec, H. J.

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]

Li, L.

Liu, V.

V. Liu and S. Fan, “S4: A free electromagnetic solver for layered periodic structures,” Computer Physics Communications 183, 2233–2244 (2012).
[Crossref]

Lu, X.

Lu, Y. Y.

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Majewski, M. L.

Mansuripur, M.

Martin, O. J. F.

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]

Miano, G.

Midrio, M.

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

Moloney, J. V.

Mu, J.

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

J. Mu and W. P. Huang, “Simulation of three-dimensional waveguide discontinuities by a full-vector mode-matching method based on finite-difference schemes,” Opt. Express 16, 18152–18163 (2008).
[Crossref] [PubMed]

Obelleiro, F.

Olkkonen, J.

Olyslager, F.

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microwave Theory Tech. 49(2), 349–354 (2001).
[Crossref]

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34(22), 2138–2140 (1998).
[Crossref]

Plumey, J. P.

G. Granet and J. P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A 4, S145–S149 (2002).
[Crossref]

Raki, A. D.

Raziman, T. V.

Rivero, J.

Rubinacci, G.

Shi, H.

Shi, X.

Silberstein, E.

Solís, D. M.

Somerville, W. R. C.

Song, D.

Taboada, J. M.

Taflove, A.

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

Tamburrino, A.

Thio, T.

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, 2008).

Trefethen, L. N.

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

Wang, R.

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

Webb, W. W.

H. Xu, P. Zhu, H. G. Craighead, and W. W. Webb, “Resonantly enhanced transmission of light through subwave-length 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,” Microwave and Optical Technology Letters 7, 599–604 (1994).
[Crossref]

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]

Xu, H.

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

Yuan, L.

Zakharian, A. R.

Zhu, J.

J. Zhu and Y. Y. Lu, “Asymptotic solutions of eigenmodes in slab waveguides terminated by perfectly matched layers,” J. Opt. Soc. Am. A 30, 2090–2095 (2013).
[Crossref]

Y. Y. Lu and J. Zhu, “Propagating modes in optical waveguides terminated by perfectly matched layers,” IEEE Photon. Technol. Lett. 17, 2601–2603 (2005).
[Crossref]

Zhu, P.

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

Appl. Opt. (1)

Computer Physics Communications (1)

V. Liu and S. Fan, “S4: A free electromagnetic solver for layered periodic structures,” Computer Physics Communications 183, 2233–2244 (2012).
[Crossref]

Electron. Lett. (1)

H. Derudder, D. De Zutter, and F. Olyslager, “Analysis of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 34(22), 2138–2140 (1998).
[Crossref]

IEEE Photon. Technol. Lett. (1)

Y. Y. Lu and J. Zhu, “Propagating modes in optical waveguides terminated by perfectly matched layers,” IEEE Photon. Technol. Lett. 17, 2601–2603 (2005).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

P. Bienstman, H. Derudder, R. Baets, F. Olyslager, and D. De Zutter, “Analysis of cylindrical waveguide discontinuities using vectorial eigenmodes and perfectly matched layers,” IEEE Trans. Microwave Theory Tech. 49(2), 349–354 (2001).
[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. Lightw. Technol. (3)

M. Hammer, “Hybrid analytical/numerical coupled-mode modeling of guided wave devices,” J. Lightw. Technol. 25(9), 2287–2298 (2007).
[Crossref]

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

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

J. Opt. A (1)

G. Granet and J. P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A 4, S145–S149 (2002).
[Crossref]

J. Opt. Soc. Am. A (10)

E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[Crossref]

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]

J. M. Taboada, J. Rivero, F. Obelleiro, M. G. Araújo, and L. Landesa, “Method-of-moments formulation for the analysis of plasmonic nano-optical antennas,” J. Opt. Soc. Am. A 28, 1341–1348 (2011).
[Crossref]

C. Forestiere, G. Iadarola, G. Rubinacci, A. Tamburrino, L. Dal Negro, and G. Miano, “Surface integral formulations for the design of plasmonic nanostructures,” J. Opt. Soc. Am. A 29, 2314–2327 (2012).
[Crossref]

J. Zhu and Y. Y. Lu, “Asymptotic solutions of eigenmodes in slab waveguides terminated by perfectly matched layers,” J. Opt. Soc. Am. A 30, 2090–2095 (2013).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 (a): A cylindrical aperture in a metallic film; (b): A cylindrical particle on a substrate.
Fig. 2
Fig. 2 Top-left: a C-shaped aperture in a silver film; Top-right: transmission spectrum for a C-shaped hole; Bottom-left: transmission versus refractive index nc; Bottom-right: transmission versus horizontal angle of incident electric field.
Fig. 3
Fig. 3 Top-left: A “triangular” aperture in a metallic film; Top-right: the dependence of normalized transmission on wavelength; Bottom-left: the dependence on the refractive index nc of the material filling the aperture; Bottom-right: the dependence on the horizontal angle of the incident electric field.
Fig. 4
Fig. 4 Normalized scattering cross sections of a gold C-shaped cylindrical particle, for normal incident waves with different horizontal angles between the electric field and the y axis.
Fig. 5
Fig. 5 Normalized scattering cross sections of a gold cylindrical particle on a substrate with refractive index 1.5 for normal incident plane waves with different angles between the electric field and the y axis.

Tables (1)

Tables Icon

Table 1 Nodes δk and weights γk of the 10-th order Alpert’s hybrid Gauss-trapezoidal rule for functions with a logarithmic singularity.

Equations (30)

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H z = H z ( l ) + 1 μ ( l ) j = 1 [ η j ( l , e ) ] 2 ϕ j ( l , e ) V j ( l , e ) ,
E z = E z ( l ) + 1 ε ( l ) j = 1 [ η j ( l , h ) ] 2 ϕ j ( l , h ) V j ( l , h ) ,
2 V j ( l , p ) x 2 + 2 V j ( l , p ) y 2 + [ η j ( l , p ) ] 2 V j ( l , p ) = 0
Λ j ( l , p ) V j ( l , p ) = v V j ( l , p ) on Γ .
[ A ( 11 ) 0 A ( 13 ) 0 0 A ( 22 ) 0 A ( 24 ) A ( 31 ) A ( 32 ) A 33 A 34 A ( 41 ) A ( 42 ) A ( 43 ) A ( 44 ) ] [ x 1 x 2 x 3 x 4 ] = [ b 1 b 2 b 3 b 4 ] .
x q = [ v v 1 ( l , p ) v v 2 ( l , p ) v v N ( l , p ) ] , v v j l , p = [ v V j ( l , p ) ( r 1 ) v V j ( l , p ) ( r 2 ) v V j ( l , p ) ( r M ) ] ,
A i j ( 11 ) = 1 μ ( 0 ) ( z i ) [ η j ( 0 , e ) ] 2 ϕ j ( 0 , e ) ( z i ) N j ( 0 , e ) , A i j ( 13 ) = 1 μ ( 1 ) ( z i ) [ η j ( 1 , e ) ] 2 ϕ j ( 1 , e ) ( z i ) N j ( 1 , e ) , A i j ( 22 ) = 1 ε ( 0 ) ( z i ) [ η j ( 0 , h ) ] 2 ϕ j ( 0 , h ) ( z i ) N j ( 0 , h ) , A i j ( 24 ) = 1 ε ( 1 ) ( z i ) [ η j ( 1 , h ) ] 2 ϕ j ( 1 , h ) ( z i ) N j ( 1 , h ) , A i j ( 31 ) = 1 μ ( 0 ) ( z i ) z ϕ j ( 0 , e ) ( z i ) T N j ( 0 , e ) , A i j ( 32 ) = i k 0 ϕ j ( 0 , h ) ( z i ) I , A i j ( 33 ) = 1 μ ( 1 ) ( z i ) z ϕ j ( 1 , e ) ( z i ) T N j ( 1 , e ) , A i j ( 34 ) = i k 0 ϕ j ( 1 , h ) ( z i ) I , A i j ( 41 ) = i k 0 ϕ j ( 0 , e ) ( z i ) I , A i j ( 42 ) = 1 ε ( 0 ) ( z i ) z ϕ j ( 0 , h ) ( z i ) T N j ( 0 , h ) , A i j ( 43 ) = i k 0 ϕ j ( 1 , e ) ( z i ) I , A i j ( 44 ) = 1 ε ( 1 ) ( z i ) z ϕ j ( 1 , h ) ( z i ) T N j ( 1 , h ) ,
G ( r , r ) = i 4 H 0 ( 1 ) ( η | r r | ) , r r ,
V ( r ) = Γ [ G ( r , r ) V V ( r ) V ( r ) G ( r , r ) v ( r ) ] d s ( r )
V ( r * ) 2 = Γ [ G ( r * , r ) V V ( r ) V ( r ) G ( r * , r ) v ( r ) ] d s ( r )
( S ϕ ) ( r * ) = 2 Γ G ( r * , r ) ϕ ( r ) d s ( r ) ,
( K ϕ ) ( r * ) = 2 Γ G ( r * , r ) v ( r ) ϕ ( r ) d s ( r )
( I + K ) V = S v V on Γ ,
N = ( I + K ) 1 S .
V ( r ) = Γ [ V ( r ) G ( r , r ) v ( r ) G ( r , r ) V v ( r ) ] d s ( r )
V ( r * ) 2 = Γ [ V ( r ) G ( r * , r ) v ( r ) G ( r * , r ) V v ( r ) ] d s ( r )
( K I ) V = S v V on Γ ,
N = ( K I ) 1 S .
r = r ( t ) , 0 t 1 ,
ψ ( r * ) = Γ B ( r * , r ) ϕ ( r ) d s ( r ) , r * Γ ,
[ ψ ( r 1 ) ψ ( r 2 ) ψ ( r M ) ] B [ ϕ ( r 1 ) ϕ ( r 2 ) ϕ ( r M ) ] .
ψ ( r i ) = 0 1 B ( r i , r ( t ) ) ϕ ( r ( t ) ) w ( t ) d t ,
ϕ ( r ( t ) ) j = 1 M ϕ ( r j ) L ( t t j ) ,
B i j = 0 1 B ( r i , r ( t ) ) L ( t t j ) w ( t ) d t .
B i j = 0 1 B ( r i , r ( ζ + t i ) ) L ( ζ + t i t j ) w ( ζ + t i ) d ζ .
0 1 g ( ζ ) d ζ h k = 1 K 1 γ k [ g ( δ k h ) + g ( 1 δ k h ) ] + h k = K 2 M K 2 g ( t k ) ,
x ( t ) = 0.15 cos ( 2 π t ) + 0.05 sin ( 4 π t + 0.8 ) , y ( t ) = 0.1 sin ( 2 π t ) + 0.02 cos ( 4 π t ) ,
0 1 g ( ζ ) d ζ S left + S middle + S right ,
g ( ζ ) = g 0 ( ζ ) log ( ζ ) + g 1 ( ζ )
S left = k = 1 K 1 γ k g ( δ k h ) , S right = k = 1 k 1 γ k g ( 1 δ k h ) , S middle = k = K 2 M K 2 g ( k h ) ,

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