Abstract

We propose an efficient numerical method for the full-vectorial analysis of three-dimensional (3-D) optical waveguide discontinuities. In this method, the finite element method with higher adaptability and flexibility is employed to discretize the waveguide cross section. In order to calculate the square root of the characteristic matrix, the Denman-Beavers iterative scheme is used. Applying this method to 3-D strongly guiding waveguide discontinuity problems, the modal reflectivities of the fundamental TE-like and TM-like modes are calculated. These results show unique vector properties and significantly differ from those of scalar analysis because various mode couplings between the field components occur at the discontinuity facet and they cannot be ignored.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends

Kuniaki Kakihara, Naoya Kono, Kunimasa Saitoh, and Masanori Koshiba
Opt. Express 14(23) 11128-11141 (2006)

References

  • View by:
  • |
  • |
  • |

  1. M. Eguchi and Y. Tsuji, “Influence of reflected radiation waves caused by large mode field and large refractive index mismatches on splice loss evalution between elliptical-hole lattice core holey fibers and conventional fibers,” J. Opt. Soc. Am. B 30(2), 410–420 (2013).
    [Crossref]
  2. H.-P. Nolting and G. Sztefka, “Eigenmode matching and propagation theory of square meandertype couplers,” IEEE Photon. Technol. Lett. 4(12), 1386–1389 (1992).
    [Crossref]
  3. M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific/Kluwer Academic, 1992).
  4. Y. Tsuji and M. Koshiba, “Finite element method using port truncation by perfectlymatched layer boundary conditions for optical waveguide discontinuity problems,” J. Lightwave Technol. 20(3), 463–468 (2002).
    [Crossref]
  5. K. S. Ye, “Numerical solution of initial boundary value problems involving Maxwell’s equations,” IEEE Trans. Anlennas Propag. AF-14, 302–307 (1966).
  6. Y.-P. Chiou and H.-C. Chang, “Analysis of optical waveguide discontinuities using the Padé approximants,” IEEE Photon. Technol. Lett. 9(7), 964–966 (1997).
    [Crossref]
  7. H. El-Refaei, I. Betty, and D. Yevick, “The application of complex Padé approcimants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12(2), 158–160 (2000).
    [Crossref]
  8. H. A. Jamid and Md. Zahed M. Khan, “3-D Full Vectorial Analysis of Strong Optical Waveguide Discontinuities Using Pade Approximants,” IEEE J. Quantum Electron. 43(4), 343–349 (2007).
    [Crossref]
  9. S. Wu, J. Xiao, and X. Sun, “Full-vectorial analysis of optica waveguide discontinuities using denman-beavers iterative scheme,” J. Lightwave Technol. 33(2), 511–518 (2015).
    [Crossref]
  10. S. Inao, T. Sato, H. Hondo, M. Ogai, S. Sentsui, A. Otake, K. Yoshizaki, K. Ishihara, and N. Uchida, “High density multicore-fiber cable,” Proc. Int. Wire & Cable Symp., pp. 370–384, 1979.
  11. G. Le Noane, D. Boscher, P. Grosso, J. C. Bizeul, and C. Botton, “Ultra high density cables using a new concept of bunched multicore monomode fibers: A key for the future FTTH networks,” Proc. Int. Wire & Cable Symp., pp. 203–210, 1994.
  12. T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Design and fabrication of ultra-low crosstalk and low-loss multi-core fiber,” Opt. Express 19(17), 16576–16592 (2011).
    [Crossref]
  13. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single mode optical fiber with photonic crystal cladding,” Opt. Lett. 21(19), 1547–1549 (1996).
    [Crossref]
  14. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22(13), 961–963 (1997).
    [Crossref]
  15. J. Broeng, D. Mogilevstev, E. Barkou, and A. Bjarklev, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Technol. 5(3), 305–330 (1999).
    [Crossref]
  16. A. R. Bhagwat and A. L. Gaeta, “Nonlinear optics in hollow-core photonic bandgap fibers,” Opt. Express 16(7), 5035–5047 (2008).
    [Crossref]
  17. S. Kawai, A. Iguchi, and Y. Tsuji, “Study on high precision and stable finite element beam propagation method based on incomplete third order Hybrid Edge/Nodal Element,” J. Lightwave Technol. 36(11), 2278–2285 (2018).
    [Crossref]
  18. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
    [Crossref]
  19. M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guidedwave problems,” J. Lightwave Technol. 18(5), 737–743 (2000).
    [Crossref]
  20. T. Q. Tran and S. Kim, “Stability condition of finite-element beam propagation methods in lossy waveguide,” IEEE J. Quantum Electron. 50(10), 808–814 (2014).
    [Crossref]
  21. K. Morimoto and Y. Tsuji, “Analysis of butt coupling of optical waveguides using propagation operator method based on finite element method,” IEICE TRANSACTIONS on Electronics (Japanese Edition) J101-C(5), 210–216 (2018).

2018 (2)

K. Morimoto and Y. Tsuji, “Analysis of butt coupling of optical waveguides using propagation operator method based on finite element method,” IEICE TRANSACTIONS on Electronics (Japanese Edition) J101-C(5), 210–216 (2018).

S. Kawai, A. Iguchi, and Y. Tsuji, “Study on high precision and stable finite element beam propagation method based on incomplete third order Hybrid Edge/Nodal Element,” J. Lightwave Technol. 36(11), 2278–2285 (2018).
[Crossref]

2015 (1)

2014 (1)

T. Q. Tran and S. Kim, “Stability condition of finite-element beam propagation methods in lossy waveguide,” IEEE J. Quantum Electron. 50(10), 808–814 (2014).
[Crossref]

2013 (1)

2011 (1)

2008 (1)

2007 (1)

H. A. Jamid and Md. Zahed M. Khan, “3-D Full Vectorial Analysis of Strong Optical Waveguide Discontinuities Using Pade Approximants,” IEEE J. Quantum Electron. 43(4), 343–349 (2007).
[Crossref]

2002 (1)

2000 (2)

M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guidedwave problems,” J. Lightwave Technol. 18(5), 737–743 (2000).
[Crossref]

H. El-Refaei, I. Betty, and D. Yevick, “The application of complex Padé approcimants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12(2), 158–160 (2000).
[Crossref]

1999 (1)

J. Broeng, D. Mogilevstev, E. Barkou, and A. Bjarklev, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Technol. 5(3), 305–330 (1999).
[Crossref]

1997 (2)

Y.-P. Chiou and H.-C. Chang, “Analysis of optical waveguide discontinuities using the Padé approximants,” IEEE Photon. Technol. Lett. 9(7), 964–966 (1997).
[Crossref]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22(13), 961–963 (1997).
[Crossref]

1996 (1)

1994 (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[Crossref]

1992 (1)

H.-P. Nolting and G. Sztefka, “Eigenmode matching and propagation theory of square meandertype couplers,” IEEE Photon. Technol. Lett. 4(12), 1386–1389 (1992).
[Crossref]

1966 (1)

K. S. Ye, “Numerical solution of initial boundary value problems involving Maxwell’s equations,” IEEE Trans. Anlennas Propag. AF-14, 302–307 (1966).

Atkin, D. M.

Barkou, E.

J. Broeng, D. Mogilevstev, E. Barkou, and A. Bjarklev, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Technol. 5(3), 305–330 (1999).
[Crossref]

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[Crossref]

Betty, I.

H. El-Refaei, I. Betty, and D. Yevick, “The application of complex Padé approcimants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12(2), 158–160 (2000).
[Crossref]

Bhagwat, A. R.

Birks, T. A.

Bizeul, J. C.

G. Le Noane, D. Boscher, P. Grosso, J. C. Bizeul, and C. Botton, “Ultra high density cables using a new concept of bunched multicore monomode fibers: A key for the future FTTH networks,” Proc. Int. Wire & Cable Symp., pp. 203–210, 1994.

Bjarklev, A.

J. Broeng, D. Mogilevstev, E. Barkou, and A. Bjarklev, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Technol. 5(3), 305–330 (1999).
[Crossref]

Boscher, D.

G. Le Noane, D. Boscher, P. Grosso, J. C. Bizeul, and C. Botton, “Ultra high density cables using a new concept of bunched multicore monomode fibers: A key for the future FTTH networks,” Proc. Int. Wire & Cable Symp., pp. 203–210, 1994.

Botton, C.

G. Le Noane, D. Boscher, P. Grosso, J. C. Bizeul, and C. Botton, “Ultra high density cables using a new concept of bunched multicore monomode fibers: A key for the future FTTH networks,” Proc. Int. Wire & Cable Symp., pp. 203–210, 1994.

Broeng, J.

J. Broeng, D. Mogilevstev, E. Barkou, and A. Bjarklev, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Technol. 5(3), 305–330 (1999).
[Crossref]

Chang, H.-C.

Y.-P. Chiou and H.-C. Chang, “Analysis of optical waveguide discontinuities using the Padé approximants,” IEEE Photon. Technol. Lett. 9(7), 964–966 (1997).
[Crossref]

Chiou, Y.-P.

Y.-P. Chiou and H.-C. Chang, “Analysis of optical waveguide discontinuities using the Padé approximants,” IEEE Photon. Technol. Lett. 9(7), 964–966 (1997).
[Crossref]

Eguchi, M.

El-Refaei, H.

H. El-Refaei, I. Betty, and D. Yevick, “The application of complex Padé approcimants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12(2), 158–160 (2000).
[Crossref]

Gaeta, A. L.

Grosso, P.

G. Le Noane, D. Boscher, P. Grosso, J. C. Bizeul, and C. Botton, “Ultra high density cables using a new concept of bunched multicore monomode fibers: A key for the future FTTH networks,” Proc. Int. Wire & Cable Symp., pp. 203–210, 1994.

Hayashi, T.

Hondo, H.

S. Inao, T. Sato, H. Hondo, M. Ogai, S. Sentsui, A. Otake, K. Yoshizaki, K. Ishihara, and N. Uchida, “High density multicore-fiber cable,” Proc. Int. Wire & Cable Symp., pp. 370–384, 1979.

Iguchi, A.

Inao, S.

S. Inao, T. Sato, H. Hondo, M. Ogai, S. Sentsui, A. Otake, K. Yoshizaki, K. Ishihara, and N. Uchida, “High density multicore-fiber cable,” Proc. Int. Wire & Cable Symp., pp. 370–384, 1979.

Ishihara, K.

S. Inao, T. Sato, H. Hondo, M. Ogai, S. Sentsui, A. Otake, K. Yoshizaki, K. Ishihara, and N. Uchida, “High density multicore-fiber cable,” Proc. Int. Wire & Cable Symp., pp. 370–384, 1979.

Jamid, H. A.

H. A. Jamid and Md. Zahed M. Khan, “3-D Full Vectorial Analysis of Strong Optical Waveguide Discontinuities Using Pade Approximants,” IEEE J. Quantum Electron. 43(4), 343–349 (2007).
[Crossref]

Kawai, S.

Khan, Md. Zahed M.

H. A. Jamid and Md. Zahed M. Khan, “3-D Full Vectorial Analysis of Strong Optical Waveguide Discontinuities Using Pade Approximants,” IEEE J. Quantum Electron. 43(4), 343–349 (2007).
[Crossref]

Kim, S.

T. Q. Tran and S. Kim, “Stability condition of finite-element beam propagation methods in lossy waveguide,” IEEE J. Quantum Electron. 50(10), 808–814 (2014).
[Crossref]

Knight, J. C.

Koshiba, M.

Le Noane, G.

G. Le Noane, D. Boscher, P. Grosso, J. C. Bizeul, and C. Botton, “Ultra high density cables using a new concept of bunched multicore monomode fibers: A key for the future FTTH networks,” Proc. Int. Wire & Cable Symp., pp. 203–210, 1994.

Mogilevstev, D.

J. Broeng, D. Mogilevstev, E. Barkou, and A. Bjarklev, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Technol. 5(3), 305–330 (1999).
[Crossref]

Morimoto, K.

K. Morimoto and Y. Tsuji, “Analysis of butt coupling of optical waveguides using propagation operator method based on finite element method,” IEICE TRANSACTIONS on Electronics (Japanese Edition) J101-C(5), 210–216 (2018).

Nolting, H.-P.

H.-P. Nolting and G. Sztefka, “Eigenmode matching and propagation theory of square meandertype couplers,” IEEE Photon. Technol. Lett. 4(12), 1386–1389 (1992).
[Crossref]

Ogai, M.

S. Inao, T. Sato, H. Hondo, M. Ogai, S. Sentsui, A. Otake, K. Yoshizaki, K. Ishihara, and N. Uchida, “High density multicore-fiber cable,” Proc. Int. Wire & Cable Symp., pp. 370–384, 1979.

Otake, A.

S. Inao, T. Sato, H. Hondo, M. Ogai, S. Sentsui, A. Otake, K. Yoshizaki, K. Ishihara, and N. Uchida, “High density multicore-fiber cable,” Proc. Int. Wire & Cable Symp., pp. 370–384, 1979.

Sasaki, T.

Sasaoka, E.

Sato, T.

S. Inao, T. Sato, H. Hondo, M. Ogai, S. Sentsui, A. Otake, K. Yoshizaki, K. Ishihara, and N. Uchida, “High density multicore-fiber cable,” Proc. Int. Wire & Cable Symp., pp. 370–384, 1979.

Sentsui, S.

S. Inao, T. Sato, H. Hondo, M. Ogai, S. Sentsui, A. Otake, K. Yoshizaki, K. Ishihara, and N. Uchida, “High density multicore-fiber cable,” Proc. Int. Wire & Cable Symp., pp. 370–384, 1979.

Shimakawa, O.

St. J. Russell, P.

Sun, X.

Sztefka, G.

H.-P. Nolting and G. Sztefka, “Eigenmode matching and propagation theory of square meandertype couplers,” IEEE Photon. Technol. Lett. 4(12), 1386–1389 (1992).
[Crossref]

Taru, T.

Tran, T. Q.

T. Q. Tran and S. Kim, “Stability condition of finite-element beam propagation methods in lossy waveguide,” IEEE J. Quantum Electron. 50(10), 808–814 (2014).
[Crossref]

Tsuji, Y.

Uchida, N.

S. Inao, T. Sato, H. Hondo, M. Ogai, S. Sentsui, A. Otake, K. Yoshizaki, K. Ishihara, and N. Uchida, “High density multicore-fiber cable,” Proc. Int. Wire & Cable Symp., pp. 370–384, 1979.

Wu, S.

Xiao, J.

Ye, K. S.

K. S. Ye, “Numerical solution of initial boundary value problems involving Maxwell’s equations,” IEEE Trans. Anlennas Propag. AF-14, 302–307 (1966).

Yevick, D.

H. El-Refaei, I. Betty, and D. Yevick, “The application of complex Padé approcimants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12(2), 158–160 (2000).
[Crossref]

Yoshizaki, K.

S. Inao, T. Sato, H. Hondo, M. Ogai, S. Sentsui, A. Otake, K. Yoshizaki, K. Ishihara, and N. Uchida, “High density multicore-fiber cable,” Proc. Int. Wire & Cable Symp., pp. 370–384, 1979.

IEEE J. Quantum Electron. (2)

H. A. Jamid and Md. Zahed M. Khan, “3-D Full Vectorial Analysis of Strong Optical Waveguide Discontinuities Using Pade Approximants,” IEEE J. Quantum Electron. 43(4), 343–349 (2007).
[Crossref]

T. Q. Tran and S. Kim, “Stability condition of finite-element beam propagation methods in lossy waveguide,” IEEE J. Quantum Electron. 50(10), 808–814 (2014).
[Crossref]

IEEE Photon. Technol. Lett. (3)

H.-P. Nolting and G. Sztefka, “Eigenmode matching and propagation theory of square meandertype couplers,” IEEE Photon. Technol. Lett. 4(12), 1386–1389 (1992).
[Crossref]

Y.-P. Chiou and H.-C. Chang, “Analysis of optical waveguide discontinuities using the Padé approximants,” IEEE Photon. Technol. Lett. 9(7), 964–966 (1997).
[Crossref]

H. El-Refaei, I. Betty, and D. Yevick, “The application of complex Padé approcimants to reflection at optical waveguide facets,” IEEE Photon. Technol. Lett. 12(2), 158–160 (2000).
[Crossref]

IEEE Trans. Anlennas Propag. (1)

K. S. Ye, “Numerical solution of initial boundary value problems involving Maxwell’s equations,” IEEE Trans. Anlennas Propag. AF-14, 302–307 (1966).

IEICE TRANSACTIONS on Electronics (Japanese Edition) (1)

K. Morimoto and Y. Tsuji, “Analysis of butt coupling of optical waveguides using propagation operator method based on finite element method,” IEICE TRANSACTIONS on Electronics (Japanese Edition) J101-C(5), 210–216 (2018).

J. Comput. Phys. (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[Crossref]

J. Lightwave Technol. (4)

J. Opt. Soc. Am. B (1)

Opt. Express (2)

Opt. Fiber Technol. (1)

J. Broeng, D. Mogilevstev, E. Barkou, and A. Bjarklev, “Photonic crystal fibers: a new class of optical waveguides,” Opt. Fiber Technol. 5(3), 305–330 (1999).
[Crossref]

Opt. Lett. (2)

Other (3)

M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific/Kluwer Academic, 1992).

S. Inao, T. Sato, H. Hondo, M. Ogai, S. Sentsui, A. Otake, K. Yoshizaki, K. Ishihara, and N. Uchida, “High density multicore-fiber cable,” Proc. Int. Wire & Cable Symp., pp. 370–384, 1979.

G. Le Noane, D. Boscher, P. Grosso, J. C. Bizeul, and C. Botton, “Ultra high density cables using a new concept of bunched multicore monomode fibers: A key for the future FTTH networks,” Proc. Int. Wire & Cable Symp., pp. 203–210, 1994.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1.
Fig. 1. Optical waveguide with a butt-joint section.
Fig. 2.
Fig. 2. Calculation error of propagation operator by using DBI method and Pad$\rm \acute {e}$ approximation as a function of the iterative number of DBI method or Pad$\rm \acute {e}$ order.
Fig. 3.
Fig. 3. Buried rectangular waveguide: (a) 3-D waveguide geometry; (b) computational window.
Fig. 4.
Fig. 4. Modal reflectivity of the fundamental modes as a function of the core width corresponding to the waveguide structure shown in Fig. 3(a).
Fig. 5.
Fig. 5. Field distribution of the incident and reflected field components of the fundamental $E^x$ modes for waveguide structure which core width $w = 2.0~\mu \textrm {m}$ shown in Fig. 3(a). Incident field: (a)$E_x$ and (b)$E_y$; Reflected field: (c)$E_x$ and (d)$E_y$.
Fig. 6.
Fig. 6. Field distribution of the incident and reflected field components of the fundamental $E^x$ modes for waveguide structure which core width $w = 0.4~\mu \textrm {m}$ shown in Fig. 3(a). Incident field: (a)$E_x$ and (b)$E_y$; Reflected field: (c)$E_x$ and (d)$E_y$.
Fig. 7.
Fig. 7. Typrical rib waveguide: (a) 3-D waveguide geometry; (b) computational window.
Fig. 8.
Fig. 8. Modal reflectivity of the fundamental modes as a function of the core width corresponding to the waveguide structure shown in Fig. 7(a).
Fig. 9.
Fig. 9. Modal reflectivity of the fundamental modes and calculation time, as a function of the number of unknown variables in entire analysis domain.
Fig. 10.
Fig. 10. Modal reflectivity of the fundamental modes and matrix error calculated by Eq. (23), as a function of the number of iteration of DBI method.
Fig. 11.
Fig. 11. Field distribution of the incident, reflected and transmitted field components of the fundamental $E^x$ modes and $E^y$ modes for full-vectorial wave analysis with the waveguide structure which core width $w = 2.0~\mu \textrm {m}$ shown in Fig. 7(a). Incident field: (a)$E_x$ and (b)$H_x$; Reflected field: (c)$E_x$ and (d)$H_x$; Transmitted field: (e)$E_x$ and (f)$H_x$.
Fig. 12.
Fig. 12. Field distribution of the incident, reflected and transmitted field components of the fundamental $E^x$ modes and $E^y$ modes for scalar wave approximation analysis with the waveguide structure which core width $w = 2.0~\mu \textrm {m}$ shown in Fig. 7(a). Incident field: (a)$E_x$ and (b)$H_x$; Reflected field: (c)$E_x$ and (d)$H_x$; Transmitted field: (e)$E_x$ and (f)$H_x$.

Equations (24)

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

× ( p × Φ ) k 0 2 q Φ = 0
p = 1 ,             q   =   n 2       for   Φ   =   E
p = 1 / n 2 ,     q   =   1           for   Φ   =   H
Φ = ϕ ( x , y ) e j β z = [ { U } T { ϕ t } { V } T { ϕ t } j β { N } T { ϕ z } ] e j β z
( [ K ] β 2 [ M ] ) { ϕ } = { 0 }
{ ϕ } = [ { ϕ t } { ϕ z } ]
[ K ] = [ [ K t t ] 0 0 0 ]
[ M ] = [ [ M t t ] [ M t z ] [ M z t ] [ M z z ] ]
[ K t t ] = e e [ k 0 2 ( q { U } { U } T + q { V } { V } T ) p ( { V } T x { U } T y ) ( { V } T x { U } T y ) ] d x d y [ M t t ] = e e [ p ( { U } { U } T + { V } { V } T ) ] d x d y [ M t z ] = e e [ p ( { U } { N } T x + { V } { N } T y ) ] d x d y [ M z t ] = [ M t z ] T [ M z z ] = e e [ k 0 2 q { N } { N } T + p ( { N } x { N } T x + p { N } y { N } T y ) ] d x d y
d 2 { Φ } d z 2 + [ Q ] 2 { Φ } = 0
{ Φ } = { ϕ t ( + ) } e j [ Q ] z + { ϕ t ( ) } e j [ Q ] z
{ ϕ t , 1 ( + ) } + { ϕ t , 1 ( ) } = { ϕ t , 2 ( + ) }
Ψ = j c k 0 p ( × Φ )
Γ ( E i × H i ) i z d S = Γ j c k 0 [ Φ i × ( p × Φ i ) ] i z d S = c β i k 0 { ϕ t , i } T ( [ M t t ] { ϕ t , i } + [ M t z ] { ϕ z , i } ) .
{ ϕ z , i } = [ M t t ] [ M z t ] { ϕ t , i } .
Γ ( E i × H i ) i z d S           = c β i k 0 { ϕ t , i } T ( [ M t t ] [ M t z ] [ M z z ] 1 [ M z t ] ) { ϕ t , i } .
Γ ( E × H ) i z d S                 = c k 0 { ϕ t } T ( [ M t t ] [ M t z ] [ M z z ] 1 [ M z t ] ) [ Q ] { ϕ t }                 = c k 0 { ϕ t } T [ F ] { ϕ t }
[ F 1 ] { ϕ t , 1 ( + ) } [ F 1 ] { ϕ t , 1 ( ) } = [ F 2 ] { ϕ t , 2 ( + ) } .
{ ϕ t , 1 ( ) } = [ F 1 ] [ F 2 ] [ F 1 ] + [ F 2 ] { ϕ t , 1 ( + ) }
[ A ] k + 1 = [ A ] k + [ B ] k 1 2
[ B ] k + 1 = [ B ] k + [ A ] k 1 2
[ A ] 0 = [ Q ] 2
[ B ] 0 = [ I ]
Error = 1 N 2 i , j | a ~ i j a i j a i j , max |

Metrics