Abstract

In this work, a new implementation of the finite-difference (FD) modal method (FDMM) based on an iterative approach to calculate the eigenvalues and corresponding eigenfunctions of the Helmholtz equation is presented. Two relevant enhancements that significantly increase the speed and accuracy of the method are introduced. First of all, the solution of the complete eigenvalue problem is avoided in favor of finding only the meaningful part of eigenmodes by using iterative methods. Second, a multigrid algorithm and Richardson extrapolation are implemented. Simultaneous use of these techniques leads to an enhancement in terms of accuracy, which allows a simple method such as the FDMM with a typical three-point difference scheme to be significantly competitive with an analytical modal method.

© 2013 Optical Society of America

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References

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  1. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  2. 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]
  3. K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
    [CrossRef]
  4. G. Granet, A. L. Bakonirina, R. Karyl, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310 (2010).
    [CrossRef]
  5. A. M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Prog. Electromagn. Res. 106, 243–261 (2010).
    [CrossRef]
  6. 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]
  7. Y.-P. Chiou, W.-L. Yeh, and N.-Y. Shih, “Analysis of highly conducting lamellar gratings with multidomain pseudospectral method,” J. Lightwave Technol. 27, 5151–5159 (2009).
    [CrossRef]
  8. L. C. Botten, M. S. Craig, R. C. McPhedran, and J. L. Adams, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
    [CrossRef]
  9. G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
    [CrossRef]
  10. P. Lalanne and J.-P. Hugonin, “Numerical performance of finite-difference modal methods for the electromagnetic analysis of one-dimensional lamellar gratings,” J. Opt. Soc. Am. A 17, 1033–1042 (2000).
    [CrossRef]
  11. Y.-P. Chiou and C.-K. Shen, “Higher-order finite-difference modal method with interface conditions for the electromagnetic analysis of gratings,” J. Lightwave Technol. 30, 1393–1398 (2012).
    [CrossRef]
  12. D. Song and Y. Y. Lu, “High-order finite difference modal method for diffraction gratings,” J. Mod. Opt. 59, 800–808 (2012).
    [CrossRef]
  13. Y. Saad, Numerical Methods for Large Eigenvalue Problems, revised edition (SIAM, 2011).
  14. Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (SIAM, 2003).
  15. R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).
  16. I. Semenikhin, M. Zanuccoli, V. Vyurkov, E. Sangiorgi, and C. Fiegna, “Computational efficient solution of Maxwell’s equations for lamellar gratings,” in PIERS Proceedings, Moscow, Russia, August19–23, 2012, pp. 1521–1525.
  17. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  18. G. Bao, Z. Chen, and H. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106–1114 (2005).
    [CrossRef]
  19. E. Popov, B. Chernov, M. Nevière, and N. Bonod, “Differential theory: application to highly conducting gratings,” J. Opt. Soc. Am. A 21, 199–206 (2004).
    [CrossRef]
  20. K. Watanabe, “Study of the differential theory of lamellar gratings made of highly conducting materials,” J. Opt. Soc. Am. A 23, 69–72 (2006).
    [CrossRef]
  21. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  22. E. Popov and M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
    [CrossRef]
  23. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667–669 (1998).
    [CrossRef]
  24. Synopsys Sentaurus TCAD Rel. C-2009.06, Synopsys Inc. This is a purified database of silver refractive index that originally comes from: E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

2012

2011

2010

G. Granet, A. L. Bakonirina, R. Karyl, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310 (2010).
[CrossRef]

A. M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Prog. Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

2009

2006

2005

2004

2000

1999

1998

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

1996

1995

1981

L. C. Botten, M. S. Craig, R. C. McPhedran, and J. L. Adams, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, and J. L. Adams, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

Armeanu, A. M.

A. M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Prog. Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

G. Granet, A. L. Bakonirina, R. Karyl, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310 (2010).
[CrossRef]

Bakonirina, A. L.

Bao, G.

Bonod, N.

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, and J. L. Adams, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

Chen, Z.

Chernov, B.

Chiou, Y.-P.

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, and J. L. Adams, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

Ebbesen, T. W.

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

Edee, K.

Fiegna, C.

I. Semenikhin, M. Zanuccoli, V. Vyurkov, E. Sangiorgi, and C. Fiegna, “Computational efficient solution of Maxwell’s equations for lamellar gratings,” in PIERS Proceedings, Moscow, Russia, August19–23, 2012, pp. 1521–1525.

Gaylord, T. K.

Ghaemi, H. F.

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

Granet, G.

Grann, E. B.

Hugonin, J.-P.

Karyl, R.

Lalanne, P.

Lehoucq, R. B.

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).

Lezec, H. J.

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

Li, L.

Lu, Y. Y.

D. Song and Y. Y. Lu, “High-order finite difference modal method for diffraction gratings,” J. Mod. Opt. 59, 800–808 (2012).
[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]

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, and J. L. Adams, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

Moharam, M. G.

Morf, R. H.

Morris, G. M.

Nevière, M.

Palik, E. D.

Synopsys Sentaurus TCAD Rel. C-2009.06, Synopsys Inc. This is a purified database of silver refractive index that originally comes from: E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

Pommet, D. A.

Popov, E.

Saad, Y.

Y. Saad, Numerical Methods for Large Eigenvalue Problems, revised edition (SIAM, 2011).

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (SIAM, 2003).

Sangiorgi, E.

I. Semenikhin, M. Zanuccoli, V. Vyurkov, E. Sangiorgi, and C. Fiegna, “Computational efficient solution of Maxwell’s equations for lamellar gratings,” in PIERS Proceedings, Moscow, Russia, August19–23, 2012, pp. 1521–1525.

Schiavone, P.

A. M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Prog. Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

Semenikhin, I.

I. Semenikhin, M. Zanuccoli, V. Vyurkov, E. Sangiorgi, and C. Fiegna, “Computational efficient solution of Maxwell’s equations for lamellar gratings,” in PIERS Proceedings, Moscow, Russia, August19–23, 2012, pp. 1521–1525.

Shen, C.-K.

Shih, N.-Y.

Song, D.

D. Song and Y. Y. Lu, “High-order finite difference modal method for diffraction gratings,” J. Mod. Opt. 59, 800–808 (2012).
[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]

Sorensen, D. C.

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).

Thio, T.

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

Vyurkov, V.

I. Semenikhin, M. Zanuccoli, V. Vyurkov, E. Sangiorgi, and C. Fiegna, “Computational efficient solution of Maxwell’s equations for lamellar gratings,” in PIERS Proceedings, Moscow, Russia, August19–23, 2012, pp. 1521–1525.

Watanabe, K.

Wolff, P. A.

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

Wu, H.

Yang, C.

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).

Yeh, W.-L.

Yuan, L.

Zanuccoli, M.

I. Semenikhin, M. Zanuccoli, V. Vyurkov, E. Sangiorgi, and C. Fiegna, “Computational efficient solution of Maxwell’s equations for lamellar gratings,” in PIERS Proceedings, Moscow, Russia, August19–23, 2012, pp. 1521–1525.

J. Lightwave Technol.

J. Mod. Opt.

D. Song and Y. Y. Lu, “High-order finite difference modal method for diffraction gratings,” J. Mod. Opt. 59, 800–808 (2012).
[CrossRef]

J. Opt. Soc. Am. A

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]

G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
[CrossRef]

P. Lalanne and J.-P. Hugonin, “Numerical performance of finite-difference modal methods for the electromagnetic analysis of one-dimensional lamellar gratings,” J. Opt. Soc. Am. A 17, 1033–1042 (2000).
[CrossRef]

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[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]

K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
[CrossRef]

G. Granet, A. L. Bakonirina, R. Karyl, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310 (2010).
[CrossRef]

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

G. Bao, Z. Chen, and H. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106–1114 (2005).
[CrossRef]

E. Popov, B. Chernov, M. Nevière, and N. Bonod, “Differential theory: application to highly conducting gratings,” J. Opt. Soc. Am. A 21, 199–206 (2004).
[CrossRef]

K. Watanabe, “Study of the differential theory of lamellar gratings made of highly conducting materials,” J. Opt. Soc. Am. A 23, 69–72 (2006).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

E. Popov and M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

Nature (London)

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

Opt. Acta

L. C. Botten, M. S. Craig, R. C. McPhedran, and J. L. Adams, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

Prog. Electromagn. Res.

A. M. Armeanu, K. Edee, G. Granet, and P. Schiavone, “Modal method based on spline expansion for the electromagnetic analysis of the lamellar grating,” Prog. Electromagn. Res. 106, 243–261 (2010).
[CrossRef]

Other

Synopsys Sentaurus TCAD Rel. C-2009.06, Synopsys Inc. This is a purified database of silver refractive index that originally comes from: E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

Y. Saad, Numerical Methods for Large Eigenvalue Problems, revised edition (SIAM, 2011).

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (SIAM, 2003).

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM, 1998).

I. Semenikhin, M. Zanuccoli, V. Vyurkov, E. Sangiorgi, and C. Fiegna, “Computational efficient solution of Maxwell’s equations for lamellar gratings,” in PIERS Proceedings, Moscow, Russia, August19–23, 2012, pp. 1521–1525.

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Figures (11)

Fig. 1.
Fig. 1.

Sketch of the grating configuration. One period is depicted.

Fig. 2.
Fig. 2.

Relative error of eigenvalues versus their number n obtained from FD Eq. (11) compared with exact eigenvalues [Eq. (10)] in the case of ε=3, θ=0, N=200, and wavelength equal to the periodicity of the grating. Only the first few eigenvalues have good accuracy.

Fig. 3.
Fig. 3.

Relative error of eigenvalues versus their number n obtained from FD Eq. (11) compared with exact eigenvalues [Eq. (10)] in the case of ε=3, θ=0, and different numbers of mesh points N. Radiation wavelength is equal to periodicity of grating. It can be seen that Richardson extrapolation can significantly improve accuracy.

Fig. 4.
Fig. 4.

Relative error of the eigenvalue with maximum real part κ1 versus the number of expansion modes M for the metallic grating in Fig. 1 with ε1=ε2=(0.22+6.71i)2, λ=1μm, and θ=30° calculated for the TE polarization. Quantity q characterizes the rate of convergence.

Fig. 5.
Fig. 5.

Relative error of the eigenvalue with maximum real part κ1 versus the number of expansion modes M for the metallic grating in Fig. 1 with ε1=ε2=(0.22+6.71i)2, λ=1μm, and θ=30° calculated for the TM polarization. Quantity q characterizes the rate of convergence.

Fig. 6.
Fig. 6.

Relative error of the minus-first-order reflected efficiency R1 for the metallic grating in Fig. 1 with ε1=ε2=(0.22+6.71i)2, λ=1μm, and θ=30° calculated for the TE case by adopting the presented IFDMM at different numbers of mesh points (squares and diamonds) and by using the FMM (circles) as a function of (a) the number of expansion modes M and (b) calculation time. The AMM result (triangles) is presented as a reference.

Fig. 7.
Fig. 7.

Relative error of the zero-order reflected efficiency R0 for the metallic grating in Fig. 1 with ε1=ε2=(0.22+6.71i)2, λ=1μm, and θ=30° calculated for the TM case by adopting the presented IFDMM (diamonds) and by using the FMM (circles) as a function of (a) the number of expansion modes M and (b) calculation time. The AMM result (triangles) is presented as reference.

Fig. 8.
Fig. 8.

Relative error of the transmitted first-order T1 for the dielectric grating in Fig. 1 with parameters ε1=(2.3)2, ε2=(1.5)2, h=1μm, p=2μm, d=0.234p, λ=1μm, and θ=30° calculated for the TE case by adopting the presented IFDMM (diamonds) and by using the FMM (circles) as a function of (a) the number of expansion modes M and (b) calculation time. The AMM result (triangles) is presented as a reference.

Fig. 9.
Fig. 9.

Relative error of the transmitted first-order T1 for the dielectric grating in Fig. 1 with parameters ε1=(2.3)2, ε2=(1.5)2, h=1μm, p=2μm, d=0.234p, λ=1μm, and θ=30° calculatd for the TM case by adopting the presented IFDMM (diamonds) and by using the FMM (circles) as a function of the number of expansion modes M. The AMM result (triangles) is presented as a reference.

Fig. 10.
Fig. 10.

Minus first-order reflected efficiency R1 of the highly conducting lamellar grating in Fig. 1 [ε1=ε2=(0+10i)2, p=h=0.5μm, λ=0.6328μm, and θ=30°) calculated as a function of groove width g=pd. Calculations are performed for the TM case by adopting the presented IFDMM with M=31 (solid line). The AMM result with M=1000 (dots) is presented as a reference.

Fig. 11.
Fig. 11.

Zero-order transmission efficiency T0 of a silver rectangular rod grating lying on a glass substrate as a function of wavelength calculated by the IFDMM for the TM case with M=100 (solid line). The physical and geometrical parameters for the structure in Fig. 1 are as follows: ε1 corresponding to data from [24] for silver, ε2=(1.44)2, p=0.9μm, h=0.2μm, and groove width g=pd=0.02μm. Normal incidence is assumed (θ=0°). The AMM result with M=1000 (dots) is presented as a reference.

Equations (21)

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

2Eyz2+2Eyx2=k02εs(x)Ey,Ex=Ez=0(TE),
Ey(x+p,z)=exp(ik0n0psinθ)Ey(x,z),
Ey(x,z)=n[cs,n+exp(ik0κs,nz)+cs,nexp(ik0κs,nz)]ψs,n(x),
2ψs(x)k02x2+εs(x)ψs(x)=κsψs(x)
ψs(x+p)=exp(ik0n0psinθ)ψs(x).
1(k0Δx)2[ψs(xj1)2ψs(xj)+ψs(xj+1)]+εs(j)ψs(xj)=κsψs(xj),0jN1
ψs(xj+N)=exp(ik0n0psinθ)ψs(xj).
εs(j)=εs(x)[xjΔx/2,xj+Δx/2].
2ψ(x)x2=1Δx2[ψ(xΔx)2ψ(x)+ψ(x+Δx)]+Δx212ψ(4)(ζ),
ψn(x)=exp[ik0n0xsin(θ)+i2πnx/p],κn=ε(n0sin(θ)+nλ/p)2,n=0,±1,±2,
ψnFD(xj)=exp[ik0n0jΔxsin(θ)+i2πnj/N],κnFD=εsin2[(n0sin(θ)+nλ/p)k0Δx/2](k0Δx/2)2,n=N/2+1,,N/2.
m=1m=M[cs1,m+exp(ik0κs1,mzs)+cs1,mexp(ik0κs1,mzs)]ψs1,m(x)=m=1m=M[cs,m+exp(ik0κs,mzs)+cs,mexp(ik0κs,mzs)]ψs,m(x)m=1m=M[cs1,m+exp(ik0κs1,mzs)cs1,mexp(ik0κs1,mzs)]κs1,mψs1,m(x)=m=1m=M[cs,m+exp(ik0κs,mzs)cs,mexp(ik0κs,mzs)]κs,mψs,m(x).
(cs+cs1)=S(s1s)(cs1+cs).
2Hyz2=εs(x)x(1εs(x)Hyx)k02εs(x)Hy,Hx=Hz=0(TM)
Hy(x+p,z)=exp(ik0n0psinθ)Hy(x,z)
Hy(x,z)=n[cs,n+exp(ik0κs,nz)+cs,nexp(ik0κs,nz)]φs,n(x),
εs(x)k0x(1εs(x)φs(x)k0x)+εs(x)φs(x)=κsφs(x),
φs(x+p)=exp(ik0n0psinθ)φs(x).
εs,1(j)[φs(xj+1)φs(xj)(k0Δx)2εs,2(j+1/2)φs(xj)φs(xj1)(k0Δx)2εs,2(j1/2)+φs(xj)]=κsφs(xj),
εs,1(j)=1/1/εs(x)[xjΔx/2,xj+Δx/2],εs,2(j1/2)=εs(x)[xjΔx,xj],εs,2(j+1/2)=εs(x)[xj,xj+Δx],
φs(xj+N)=exp(ik0n0psinθ)φs(xj).

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