Abstract

A Fourier-matching pseudospectral modal method [PSMM(f)] is developed for analyzing lamellar diffraction gratings or grating stacks. A Chebyshev pseudospectral method is first used to accurately calculate the eigenmodes of the grating layers, and then the Fourier coefficients are matched at the interfaces between the layers. Compared with an existing pseudospectral modal method based on point matching, the PSMM(f) is more robust and accurate. The method performs better than the standard Fourier modal method for gratings involving metals.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
    [CrossRef]
  2. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
    [CrossRef]
  3. L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106(1981).
    [CrossRef]
  4. P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square wave gratings—application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916(1982).
    [CrossRef]
  5. A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511–538 (1987).
    [CrossRef]
  6. L. Li, “A modal analysis of lamellar diffraction gratings in cornical mountings,” J. Mod. Opt. 40, 553–573 (1993).
    [CrossRef]
  7. L. Li, “Multilayer modal method for diffration gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  8. J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
    [CrossRef]
  9. M. Foresti, L. Menez, and A. V. Tishchenko, “Modal method in deep metal-dielectric gratings: the decisive role of hidden modes,” J. Opt. Soc. Am. A 23, 2501–2509 (2006).
    [CrossRef]
  10. S. Campbell, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Modal method for classical diffraction by slanted lamellar gratings,” J. Opt. Soc. Am. A 25, 2415–2426 (2008).
    [CrossRef]
  11. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509(1966).
    [CrossRef]
  12. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  13. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  14. 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]
  15. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  16. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  17. 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]
  18. N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A 24, 3781–3788(2007).
    [CrossRef]
  19. I. Gushchin and A. V. Tishchenko, “Fourier modal method for relief gratings with oblique boundary conditions,” J. Opt. Soc. Am. A 27, 1575–1583 (2010).
    [CrossRef]
  20. K. M. Gundu and A. Mafi, “Reliable computation of scattering from metallic binary gratings using Fourier-based modal methods,” J. Opt. Soc. Am. A 27, 1694–1700 (2010).
    [CrossRef]
  21. K. M. Gundu and A. Mafi, “Constrained least squares Fourier modal method for computing scattering from metallic binary gratings,” J. Opt. Soc. Am. A 27, 2375–2380 (2010).
    [CrossRef]
  22. 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]
  23. 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]
  24. K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462(2005).
    [CrossRef]
  25. G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, 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]
  26. 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]
  27. L. N. Trefethen, Spectral Methods in MATLAB (Society for Industrial and Applied Mathematics, 2000).
    [CrossRef]
  28. L. N. Trefethen, “Is Gauss quadrature better than Clenshaw–Curtis?” SIAM Rev. 50, 67–87 (2008).
    [CrossRef]

2010 (4)

2009 (1)

2008 (2)

2007 (1)

2006 (1)

2005 (1)

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462(2005).
[CrossRef]

2000 (1)

1999 (1)

1996 (3)

1995 (1)

1994 (1)

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

1993 (2)

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

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

1987 (1)

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511–538 (1987).
[CrossRef]

1982 (2)

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square wave gratings—application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916(1982).
[CrossRef]

M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

1981 (3)

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

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

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

1978 (1)

1966 (1)

Adams, J. L.

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

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

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

Andriamanampisoa, L. B.

Armeanu, A. M.

Botten, L. C.

S. Campbell, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Modal method for classical diffraction by slanted lamellar gratings,” J. Opt. Soc. Am. A 25, 2415–2426 (2008).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

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

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

Burckhardt, C. B.

Campbell, S.

Chiou, Y.-P.

Craig, M. S.

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

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

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

de Sterke, C. M.

Edee, K.

G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, 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]

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462(2005).
[CrossRef]

Foresti, M.

Gaylord, T. K.

Granet, G.

Guizal, B.

Gundu, K. M.

Gushchin, I.

Hugonin, J. P.

Knop, K.

Lalanne, P.

Li, L.

Lyndin, N. M.

Mafi, A.

McPhedran, R. C.

S. Campbell, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Modal method for classical diffraction by slanted lamellar gratings,” J. Opt. Soc. Am. A 25, 2415–2426 (2008).
[CrossRef]

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511–538 (1987).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

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

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

Menez, L.

Miller, J. M.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Moharam, M. G.

Morf, R. H.

Morris, G. M.

Noponen, E.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Parriaux, O.

Raniriharinosy, K.

Roberts, A.

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511–538 (1987).
[CrossRef]

Sanda, P. N.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square wave gratings—application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916(1982).
[CrossRef]

Schiavone, P.

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462(2005).
[CrossRef]

Sheng, P.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square wave gratings—application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916(1982).
[CrossRef]

Shih, N.-Y.

Stepleman, R. S.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square wave gratings—application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916(1982).
[CrossRef]

Taghizadeh, M. R.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Tishchenko, A. V.

Trefethen, L. N.

L. N. Trefethen, “Is Gauss quadrature better than Clenshaw–Curtis?” SIAM Rev. 50, 67–87 (2008).
[CrossRef]

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

Turunen, J.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Vasara, A.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Yeh, W.-L.

J. Lightwave Technol. (1)

J. Mod. Opt. (2)

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511–538 (1987).
[CrossRef]

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

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (14)

M. Foresti, L. Menez, and A. V. Tishchenko, “Modal method in deep metal-dielectric gratings: the decisive role of hidden modes,” J. Opt. Soc. Am. A 23, 2501–2509 (2006).
[CrossRef]

N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A 24, 3781–3788(2007).
[CrossRef]

S. Campbell, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Modal method for classical diffraction by slanted lamellar gratings,” J. Opt. Soc. Am. A 25, 2415–2426 (2008).
[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]

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]

L. Li, “Multilayer modal method for diffration gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[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. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

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

G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, 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]

I. Gushchin and A. V. Tishchenko, “Fourier modal method for relief gratings with oblique boundary conditions,” J. Opt. Soc. Am. A 27, 1575–1583 (2010).
[CrossRef]

K. M. Gundu and A. Mafi, “Reliable computation of scattering from metallic binary gratings using Fourier-based modal methods,” J. Opt. Soc. Am. A 27, 1694–1700 (2010).
[CrossRef]

K. M. Gundu and A. Mafi, “Constrained least squares Fourier modal method for computing scattering from metallic binary gratings,” J. Opt. Soc. Am. A 27, 2375–2380 (2010).
[CrossRef]

Jpn. J. Appl. Phys. (1)

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462(2005).
[CrossRef]

Opt. Acta (3)

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

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

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

Opt. Commun. (1)

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Phys. Rev. B (1)

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square wave gratings—application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916(1982).
[CrossRef]

SIAM Rev. (1)

L. N. Trefethen, “Is Gauss quadrature better than Clenshaw–Curtis?” SIAM Rev. 50, 67–87 (2008).
[CrossRef]

Other (1)

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

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 (2)

Fig. 1
Fig. 1

Example 3 (TM case): diffraction efficiency of the zeroth reflected order calculated by the FMM (circles) and the PSMM(f) (pluses).

Fig. 2
Fig. 2

Example 4 (TM case): diffraction efficiency of the zeroth reflected order calculated by (a) the FMM, (b) the PSMM(f) with N 1 = M , (c) the PSMM(f) with N 1 = ( M + 1 ) / 2 .

Tables (7)

Tables Icon

Table 1 Example 1 (TE Case): Diffraction Efficiency of the Minus-First Reflected Order ( RE 1 ) Calculated by the FMM, the PSMM(p), and the PSMM(f)

Tables Icon

Table 2 Example 1 (TM Case): Diffraction Efficiency of the Zeroth Reflected Order ( RE 0 ) Calculated by the FMM, the PSMM(p), and the PSMM(f)

Tables Icon

Table 3 Example 2 (TE Case): Diffraction Efficiency of the Zeroth Transmitted Order ( TE 0 ) Calculated by the FMM and the PSMM(f)

Tables Icon

Table 4 Example 2 (TM Case): Diffraction Efficiency of the First Transmitted Order ( TE 1 ) Calculated by the FMM and the PSMM(f)

Tables Icon

Table 5 Example 3 (TE Case): Diffraction Efficiency of the Minus-First Reflected Order ( RE 1 ) Calculated by the FMM and the PSMM(f)

Tables Icon

Table 6 Example 3 (TM Case): Diffraction Efficiency of the Zeroth Reflected Order ( RE 0 ) Calculated by the FMM and the PSMM(f)

Tables Icon

Table 7 Example 4 (TM Case): Diffraction Efficiencies of the Zeroth Reflected Order Calculated by the PSMM(f)

Equations (50)

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

x ( 1 ε u x ) + y ( 1 ε u y ) + k 0 2 μ u = 0 ,
ε ( x , y ) = ε ( j ) ( x ) , μ ( x , y ) = μ ( j ) ( x ) , y j 1 < y < y j .
u ( i ) = exp { i [ α 0 x β 0 ( * ) ( y D ) ] } , y > D .
α 0 = k 0 ε ( * ) μ ( * ) sin θ , β 0 ( * ) = k 0 ε ( * ) μ ( * ) cos θ .
u ( r ) ( x , y ) = m = R m exp { i [ α m x + β m ( * ) ( y D ) ] } , y > D ,
u ( t ) ( x , y ) = m = T m exp [ i ( α m x β m ( 0 ) y ) ] , y < 0 ,
α m = α 0 + 2 π m L , β m ( 0 ) = k 0 2 ε ( 0 ) μ ( 0 ) α m 2 , β m ( * ) = k 0 2 ε ( * ) μ ( * ) α m 2 ,
u ( x , y ) = l = 1 { a l ( j ) exp [ i β l ( j ) ( y y j 1 ) ] + b l ( j ) exp [ i β l ( j ) ( y j y ) ] } g l ( j ) ( x ) ,
ε d d x ( 1 ε d g d x ) + k 0 2 ε μ g = β 2 g , 0 < x < L ,
g ( L ) = e i α 0 L g ( 0 ) ,
1 ε ( L ) d g d x ( L ) = e i α 0 L ε ( 0 + ) d g d x ( 0 + ) .
u ( x , y j ) = u ( x , y j + ) , 1 ε ( x , y j ) u y ( x , y j ) = 1 ε ( x , y j + ) u y ( x , y j + ) ,
ξ p , k = x p 1 + x p x p 1 2 [ 1 cos ( k π q p ) ] , 0 k q p .
[ g ( x p 1 + ) g p g ( x p ) ] C ( p ) [ g ( x p 1 ) g p g ( x p ) ] ,
g p = [ g ( ξ p , 1 ) , g ( ξ p , 2 ) , , g ( ξ p , q p 1 ) ] T ,
C k l ( p ) = 2 x p x p 1 { ( 2 q p 2 + 1 ) / 6 if     k = l = 0 , ( 2 q p 2 + 1 ) / 6 if     k = l = q p , 0.5 γ k / ( 1 γ k 2 ) if     0 < k = l < q p , ( 1 ) k + l σ k σ l 1 / ( γ k γ l ) otherwise ,
C ( p ) = [ c 1 ( p ) c 2 ( p ) c 3 ( p ) c 4 ( p ) C 5 ( p ) c 6 ( p ) c 7 ( p ) c 8 ( p ) c 9 ( p ) ] ,
g ( x p 1 + ) c 1 ( p ) g ( x p 1 ) + c 2 ( p ) g p + c 3 ( p ) g ( x p ) ,
g p c 4 ( p ) g ( x p 1 ) + C 5 ( p ) g p + c 6 ( p ) g ( x p ) ,
g ( x p ) c 7 ( p ) g ( x p 1 ) + c 8 ( p ) g p + c 9 ( p ) g ( x p ) .
1 ε ( x p ) [ c 7 ( p ) g ( x p 1 ) + c 8 ( p ) g p + c 9 ( p ) g ( x p ) ] = 1 ε ( x p + ) [ c 1 ( p + 1 ) g ( x p ) + c 2 ( p + 1 ) g p + 1 + c 3 ( p + 1 ) g ( x p + 1 ) ] ,
e i α 0 L ε ( x 0 + ) [ c 1 ( 1 ) g ( x 0 ) + c 2 ( 1 ) g 1 + c 3 ( 1 ) g ( x 1 ) ] = 1 ε ( x P ) [ c 7 ( P ) g ( x P 1 ) + c 8 ( P ) g P + c 9 ( P ) g ( x P ) ] .
[ g ( x 0 ) g ( x 1 ) g ( x P ) ] = A 0 [ g 1 g 2 g P ] .
[ f ( x p 1 + ) f p f ( x p ) ] = B ( p ) [ g ( x p 1 + ) g p g ( x p ) ] , B ( p ) = E ( p ) C ( p ) [ E ( p ) ] 1 C ( p ) ,
B ( p ) = [ b 1 ( p ) b 2 ( p ) b 3 ( p ) b 4 ( p ) B 5 ( p ) b 6 ( p ) b 7 ( p ) b 8 ( p ) b 9 ( p ) ] .
[ B 5 ( p ) + k 0 2 E s ( p ) U s ( p ) ] g p + b 4 ( p ) g ( x p 1 ) + b 6 ( p ) g ( x p ) = β 2 g p ,
A 1 [ g 1 g 2 g P ] + A 2 [ g ( x 0 ) g ( x 1 ) g ( x P ) ] = β 2 [ g 1 g 2 g P ] ,
A 2 = [ b 4 ( 1 ) b 6 ( 1 ) b 4 ( 2 ) b 6 ( 2 ) b 4 ( P ) b 6 ( P ) ] .
A [ g 1 g 2 g P ] = β 2 [ g 1 g 2 g P ] ,
u ( x , y ) l = 1 N j { a l ( j ) exp [ i β l ( j ) ( y y j 1 ) ] + b l ( j ) exp [ i β l ( j ) ( y j y ) ] } g l ( j ) ( x ) ,
0 L [ u ( x , y j + ) u ( x , y j ) ] exp ( i α m x ) d x = 0 ,
0 L [ 1 ε ( x , y j ) u y ( x , y j ) 1 ε ( x , y j + ) u y ( x , y j + ) ] exp ( i α m x ) d x = 0 ,
0 L u ( x , y 0 ) exp ( i α m x ) = L T m ,
0 L 1 ε ( x , y 0 ) u y ( x , y 0 ) exp ( i α m x ) = i β m ( 0 ) ε ( 0 ) L T m .
0 L u ( x , y J + ) exp ( i α m x ) = L ( R m + δ m 0 ) ,
0 L 1 ε ( x , y J + ) u y ( x , y J + ) exp ( i α m x ) = i β m ( * ) ε ( * ) L ( R m δ m 0 ) ,
u y ( x , y ) l = 1 N j i β l ( j ) { a l ( j ) exp [ i β l ( j ) ( y y j 1 ) ] b l ( j ) exp [ i β l ( j ) ( y j y ) ] } g l ( j ) ( x ) .
0 L u ( x , y j ) exp ( i α m x ) d x l = 1 N j I l m ( j ) [ a l ( j ) γ l ( j ) + b l ( j ) ] ,
0 L u ( x , y j 1 + ) exp ( i α m x ) d x l = 1 N j I l m ( j ) [ a l ( j ) + b l ( j ) γ l ( j ) ] ,
0 L y u ( x , y j ) ε ( x , y j ) exp ( i α m x ) d x l = 1 N j J l m ( j ) [ a l ( j ) γ l ( j ) b l ( j ) ] ,
0 L y u ( x , y j 1 + ) ε ( x , y j 1 + ) exp ( i α m x ) d x l = 1 N j J l m ( j ) [ a l ( j ) b l ( j ) γ l ( j ) ] ,
γ l ( j ) = exp [ i β l ( j ) ( y j y j 1 ) ] ,
I l m ( j ) = 0 L g l ( j ) ( x ) exp ( i α m x ) d x ,
J l m ( j ) = i β l ( j ) 0 L g l ( j ) ( x ) ε ( j ) ( x ) exp ( i α m x ) d x .
Fz = h ,
F = [ F 00 F 01 F 11 F 12 F J J F J , J + 1 ] , z = [ z 0 z 1 z J z J + 1 ] , h = [ 0 0 h J ] .
min z | | Fz h | | .
ε ( x ) = { ε ( * ) , if     0 < x < 0.5 L , ε ( 0 ) , if     0.5 L < x < L .
ε ( x ) = { 5.29 , if     0 < x < 0.234 L , 1 , if     0.234 L < x < L .
ε ( x ) = { 2.5 , if     0 < x < 0.5 L , 1.69 , if     0.5 L < x < L .

Metrics