Abstract

The coupled wave theory dealing with optics of discontinuous two-dimensional (2D) periodic structures is reformulated by using Fourier factorization with complex polarization bases, which is a generalized implementation of the fast Fourier factorization rules. The modified approach yields considerably improved convergence properties, as shown on an example of a 2D quartz grating. The method can also be applied to the calculation of 2D photonic band structures or nonperiodic cylindrical devices, and can be generalized to elements with arbitrary cross-sections.

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References

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  1. R. Petit (ed.), Electromagnetic Theory of Gratings (Springer, 1980).
  2. M. Neviere and E. Popov, Light Propagation in Periodic Media: Diffraction Theory and Design (Marcel Dekker, New York, 2003).
  3. D. Maystre, "Rigorous vector theories of diffraction gratings," Prog. Opt. 21, 1-67 (1984).
    [CrossRef]
  4. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ., 1995)
  5. K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
    [CrossRef] [PubMed]
  6. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870- 1876 (1996).
    [CrossRef]
  7. L. Li, "Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials," J. Mod. Opt. 45, 1313-1334 (1998).
    [CrossRef]
  8. B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
    [CrossRef]
  9. E. Popov and M. Neviere, "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]
  10. L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
    [CrossRef]
  11. K. Watanabe, R. Petit, and M. Neviere, "Differential theory of gratings made of anisotropic materials," J. Opt. Soc. Am. A 19, 325-334 (2002).
    [CrossRef]
  12. K. Watanabe, "Numerical integration schemes used on the differential theory for anisotropic gratings," J. Opt. Soc. Am. A 19, 2245-2252 (2002).
    [CrossRef]
  13. L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).
    [CrossRef]
  14. P. Boyer, E. Popov, M. Neviere, and G. Tayeb, "Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section," J. Opt. Soc. Am. A 21, 2146- 2153 (2004).
    [CrossRef]
  15. N. Bonod, E. Popov, M. Neviere, "Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications," Opt. Commun. 245, 355-361 (2005).
    [CrossRef]
  16. N. Bonod, E. Popov, M. Neviere, "Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect," Opt. Commun. 244, 389-398 (2005).
    [CrossRef]
  17. E. Noponen and J. Turunen, "Eigenmode method for electromagnetic synthesis of diffractive elements with three dimensional profiles," J. Opt. Soc. Am. A 11, 2494-2502 (1994).
    [CrossRef]
  18. S. Visnovsky and K. Yasumoto, "Multilayer anisotropic bi-periodic diffraction gratings," Czech. J. Phys. 51, 229-247 (2001).
    [CrossRef]
  19. R. Antos, S. Visnovsky, J. Mistrik, and T. Yamaguchi, "Magneto-optical polar-Kerr-effect spectroscopy on 2Dperiodic subwavelength arrays of magnetic dots," International Journal of Microwave and Optical Technology 1, 905-909 (2006).
  20. A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for twodimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
    [CrossRef]
  21. T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, andW. Osten, "Normal vector method for convergence improvement using the RCWA for crossed gratings," J. Opt. Soc. Am. A 24, 2880-2890 (2007).
    [CrossRef]
  22. P. Gotz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Express 16, 17295-17301 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17295.
    [CrossRef] [PubMed]
  23. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1997).
  24. N. P. K. Cotter, T. W. Preist, and J. R. Sambles, "Scattering matrix approach to multilayer diffraction," J. Opt. Soc. Am. A 12, 1097-1103 (1995).
    [CrossRef]
  25. L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996).
    [CrossRef]
  26. S. Kaushik, "Vector Fresnel equations and Airy formula for one-dimensional multilayer and surface-relief gratings," J. Opt. Soc. Am. A 14, 596-609 (1997).
    [CrossRef]
  27. R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, "Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials," J. Appl. Phys. 100, 054906 (2006).
    [CrossRef]
  28. E. D. Palik (ed.), Handbook of Optical Constants of Solids (Academic, 1998).

2008

2007

2006

R. Antos, S. Visnovsky, J. Mistrik, and T. Yamaguchi, "Magneto-optical polar-Kerr-effect spectroscopy on 2Dperiodic subwavelength arrays of magnetic dots," International Journal of Microwave and Optical Technology 1, 905-909 (2006).

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for twodimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[CrossRef]

R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, "Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials," J. Appl. Phys. 100, 054906 (2006).
[CrossRef]

2005

N. Bonod, E. Popov, M. Neviere, "Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications," Opt. Commun. 245, 355-361 (2005).
[CrossRef]

N. Bonod, E. Popov, M. Neviere, "Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect," Opt. Commun. 244, 389-398 (2005).
[CrossRef]

2004

2003

L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).
[CrossRef]

2002

2001

S. Visnovsky and K. Yasumoto, "Multilayer anisotropic bi-periodic diffraction gratings," Czech. J. Phys. 51, 229-247 (2001).
[CrossRef]

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

2000

1998

L. Li, "Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials," J. Mod. Opt. 45, 1313-1334 (1998).
[CrossRef]

1997

1996

1995

1994

1990

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

1984

D. Maystre, "Rigorous vector theories of diffraction gratings," Prog. Opt. 21, 1-67 (1984).
[CrossRef]

Antos, R.

R. Antos, S. Visnovsky, J. Mistrik, and T. Yamaguchi, "Magneto-optical polar-Kerr-effect spectroscopy on 2Dperiodic subwavelength arrays of magnetic dots," International Journal of Microwave and Optical Technology 1, 905-909 (2006).

R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, "Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials," J. Appl. Phys. 100, 054906 (2006).
[CrossRef]

Benisty, H.

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for twodimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[CrossRef]

Bonod, N.

N. Bonod, E. Popov, M. Neviere, "Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications," Opt. Commun. 245, 355-361 (2005).
[CrossRef]

N. Bonod, E. Popov, M. Neviere, "Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect," Opt. Commun. 244, 389-398 (2005).
[CrossRef]

Boyer, P.

Chan, C. T.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Chernov, B.

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

Cotter, N. P. K.

David, A.

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for twodimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[CrossRef]

Frenner, K.

Gotz, P.

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Horie, M.

R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, "Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials," J. Appl. Phys. 100, 054906 (2006).
[CrossRef]

Kaushik, S.

Kerwien, N.

Li, L.

Maystre, D.

D. Maystre, "Rigorous vector theories of diffraction gratings," Prog. Opt. 21, 1-67 (1984).
[CrossRef]

Mistrik, J.

R. Antos, S. Visnovsky, J. Mistrik, and T. Yamaguchi, "Magneto-optical polar-Kerr-effect spectroscopy on 2Dperiodic subwavelength arrays of magnetic dots," International Journal of Microwave and Optical Technology 1, 905-909 (2006).

R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, "Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials," J. Appl. Phys. 100, 054906 (2006).
[CrossRef]

Neviere, M.

N. Bonod, E. Popov, M. Neviere, "Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications," Opt. Commun. 245, 355-361 (2005).
[CrossRef]

N. Bonod, E. Popov, M. Neviere, "Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect," Opt. Commun. 244, 389-398 (2005).
[CrossRef]

P. Boyer, E. Popov, M. Neviere, and G. Tayeb, "Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section," J. Opt. Soc. Am. A 21, 2146- 2153 (2004).
[CrossRef]

K. Watanabe, R. Petit, and M. Neviere, "Differential theory of gratings made of anisotropic materials," J. Opt. Soc. Am. A 19, 325-334 (2002).
[CrossRef]

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

E. Popov and M. Neviere, "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]

Noponen, E.

Osten, W.

Otani, Y.

R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, "Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials," J. Appl. Phys. 100, 054906 (2006).
[CrossRef]

Petit, R.

Pistora, J.

R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, "Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials," J. Appl. Phys. 100, 054906 (2006).
[CrossRef]

Popov, E.

N. Bonod, E. Popov, M. Neviere, "Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect," Opt. Commun. 244, 389-398 (2005).
[CrossRef]

N. Bonod, E. Popov, M. Neviere, "Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications," Opt. Commun. 245, 355-361 (2005).
[CrossRef]

P. Boyer, E. Popov, M. Neviere, and G. Tayeb, "Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section," J. Opt. Soc. Am. A 21, 2146- 2153 (2004).
[CrossRef]

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

E. Popov and M. Neviere, "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]

Preist, T. W.

Rafler, S.

Ruoff, J.

Sambles, J. R.

Schuster, T.

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Tayeb, G.

Turunen, J.

Visnovsky, S.

R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, "Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials," J. Appl. Phys. 100, 054906 (2006).
[CrossRef]

R. Antos, S. Visnovsky, J. Mistrik, and T. Yamaguchi, "Magneto-optical polar-Kerr-effect spectroscopy on 2Dperiodic subwavelength arrays of magnetic dots," International Journal of Microwave and Optical Technology 1, 905-909 (2006).

S. Visnovsky and K. Yasumoto, "Multilayer anisotropic bi-periodic diffraction gratings," Czech. J. Phys. 51, 229-247 (2001).
[CrossRef]

Watanabe, K.

Weisbuch, C.

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for twodimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[CrossRef]

Yamaguchi, S.

R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, "Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials," J. Appl. Phys. 100, 054906 (2006).
[CrossRef]

Yamaguchi, T.

R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, "Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials," J. Appl. Phys. 100, 054906 (2006).
[CrossRef]

R. Antos, S. Visnovsky, J. Mistrik, and T. Yamaguchi, "Magneto-optical polar-Kerr-effect spectroscopy on 2Dperiodic subwavelength arrays of magnetic dots," International Journal of Microwave and Optical Technology 1, 905-909 (2006).

Yasumoto, K.

S. Visnovsky and K. Yasumoto, "Multilayer anisotropic bi-periodic diffraction gratings," Czech. J. Phys. 51, 229-247 (2001).
[CrossRef]

Czech. J. Phys.

S. Visnovsky and K. Yasumoto, "Multilayer anisotropic bi-periodic diffraction gratings," Czech. J. Phys. 51, 229-247 (2001).
[CrossRef]

International Journal of Microwave and Optical Technology

R. Antos, S. Visnovsky, J. Mistrik, and T. Yamaguchi, "Magneto-optical polar-Kerr-effect spectroscopy on 2Dperiodic subwavelength arrays of magnetic dots," International Journal of Microwave and Optical Technology 1, 905-909 (2006).

J. Appl. Phys.

R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, "Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials," J. Appl. Phys. 100, 054906 (2006).
[CrossRef]

J. Mod. Opt.

L. Li, "Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials," J. Mod. Opt. 45, 1313-1334 (1998).
[CrossRef]

J. Opt. A

L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).
[CrossRef]

J. Opt. Soc. Am. A

E. Noponen and J. Turunen, "Eigenmode method for electromagnetic synthesis of diffractive elements with three dimensional profiles," J. Opt. Soc. Am. A 11, 2494-2502 (1994).
[CrossRef]

S. Kaushik, "Vector Fresnel equations and Airy formula for one-dimensional multilayer and surface-relief gratings," J. Opt. Soc. Am. A 14, 596-609 (1997).
[CrossRef]

L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
[CrossRef]

L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (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]

N. P. K. Cotter, T. W. Preist, and J. R. Sambles, "Scattering matrix approach to multilayer diffraction," J. Opt. Soc. Am. A 12, 1097-1103 (1995).
[CrossRef]

E. Popov and M. Neviere, "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]

K. Watanabe, R. Petit, and M. Neviere, "Differential theory of gratings made of anisotropic materials," J. Opt. Soc. Am. A 19, 325-334 (2002).
[CrossRef]

K. Watanabe, "Numerical integration schemes used on the differential theory for anisotropic gratings," J. Opt. Soc. Am. A 19, 2245-2252 (2002).
[CrossRef]

P. Boyer, E. Popov, M. Neviere, and G. Tayeb, "Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section," J. Opt. Soc. Am. A 21, 2146- 2153 (2004).
[CrossRef]

T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, andW. Osten, "Normal vector method for convergence improvement using the RCWA for crossed gratings," J. Opt. Soc. Am. A 24, 2880-2890 (2007).
[CrossRef]

Opt. Commun.

N. Bonod, E. Popov, M. Neviere, "Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications," Opt. Commun. 245, 355-361 (2005).
[CrossRef]

N. Bonod, E. Popov, M. Neviere, "Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect," Opt. Commun. 244, 389-398 (2005).
[CrossRef]

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

Opt. Express

Phys. Rev. B

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for twodimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[CrossRef]

Phys. Rev. Lett.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Prog. Opt.

D. Maystre, "Rigorous vector theories of diffraction gratings," Prog. Opt. 21, 1-67 (1984).
[CrossRef]

Other

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ., 1995)

R. Petit (ed.), Electromagnetic Theory of Gratings (Springer, 1980).

M. Neviere and E. Popov, Light Propagation in Periodic Media: Diffraction Theory and Design (Marcel Dekker, New York, 2003).

E. D. Palik (ed.), Handbook of Optical Constants of Solids (Academic, 1998).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1997).

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Figures (3)

Fig. 1.
Fig. 1.

Configuration of the optical problem (a) and propagation algorithm (b).

Fig. 2.
Fig. 2.

Distribution of u in the periodic cell, corresponding to Models B (a) and C (b), and the radial dependence of the ellipticity along two lines ϕ = 0 and ϕ = 30° (c).

Fig. 3.
Fig. 3.

Convergence properties of diffraction efficiencies of periodically arranged cylindrical holes in the [0,0] (a, b) and [-1,0] (c, d) orders of diffraction for the s (a, c) and p (b, d) polarizations.

Equations (9)

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

( ε + [ y 2 x y x y x 2 ] + t 1 ε t ε ) [ E x E y ] = z 2 [ E x E y ] ,
z [ H y H x ] = i ( ε + [ y 2 x y x y x 2 ] ) [ E x E y ]
ε ( x , y ) = m , n ε mn exp [ i ( mpx + nqy ) ] ,
E x ( y ) ( x , y , z ) = m , n f x ( y ) , mn exp [ i ( p m x + q n y + sz ) ]
[ [ E x ] [ E y ] ] = [ [ F ] ] [ [ E u ] [ E v ] ] , [ [ F ] ] = [ [ [ ξ ] ] [ [ ζ * ] ] [ [ ζ ] ] [ [ ξ * ] ] ] ,
[ [ D x ] [ D y ] ] = [ [ F ] ] [ [ [ 1 ε ] ] 1 [ [ 0 ] ] [ [ 0 ] ] [ [ ε ] ] ] [ [ F ] ] 1 [ [ E x ] [ E y ] ] .
u = [ ξ ζ ] = e [ cos θ sin θ sin θ cos θ ] [ cos η i sin η ] ,
θ ( r , ϕ ) = ϕ ,
η ( r , ϕ ) = { π 8 ( 1 + cos πr R ) ( r R ) π 8 { 1 + cos π [ r + D ( ϕ ) 2 R ] D ( ϕ ) R } ( r > R )

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