Abstract

Anisotropic lamellar sub-wavelength gratings (nanogratings) are described by Effective Medium Approximation (EMA). Analytical formulas for effective medium optical parameters of nanogratings from arbitrary anisotropic materials are derived using approximation of zero-order diffraction mode. The method is based on Rigorous Coupled Wave Analysis (RCWA) combined with proper Fourier factorization method. Good agreement between EMA and the rigorous model is observed, where slight differences are explained by the influence of evanescent higher Fourier harmonics in the nanograting.

© 2006 Optical Society of America

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2005 (2)

2004 (2)

K. Watanabe and K. Yasumoto, "Fourier Modal Theory of Rectangular Dot Gratings Made of Anisotropic and Conducting Materials," Proc. of SPIE 5445, 218-221 (2004).
[CrossRef]

G. E. Jellison, "Generalized ellipsometry for materials characterization," Thin Solid Films 450, 42-50 (2004).
[CrossRef]

2003 (1)

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

2002 (1)

2001 (1)

2000 (2)

J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Selingson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. of SPIE 3998, 125-134 (2000).
[CrossRef]

C. Zhang, B. Yang, X. Wu, T. Lu, Y. Zheng, and W. Su, "Calculation of the effective dielectric function of composites with periodic geometry," Physica B 293, 16-32 (2000).
[CrossRef]

1999 (1)

1998 (1)

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 (4)

F. Garc´ýa-Vidal, J. M. Pitarke, and J. B. Pendry, "Effective medium theory of the optical properties of aligned carbon nanotubes," Phys. Rev. B 78, 4289 (1997).
[CrossRef]

C.-Y. You, S.-C. Shin, and S.-Y. Kim, "Modified effective-medium theory for magneto-optical spectra of magnetic materials," Phys. Rev. B 55, 5953-5958 (1997).
[CrossRef]

J. Turunen, M. Kuittinen, and P. Vahimaa, "Form-birefringence limits of Fourier-expansion methods in grating theory: arbitrary angle of incidence," J. Opt. Soc. Am. A 14, 2314-2316 (1997).
[CrossRef]

P. Lalanne and D. Lemercier-Lalanne, "Depth dependence of the effective properties of subwavelength gratings," J. Opt. Soc. Am. A 14, 450-458 (1997).
[CrossRef]

1996 (6)

1995 (2)

H. Kikuta, H. Yoshida, and K. Iwata, "Ability and limitation of effective medium theory for subwavelength gratings," Opt. Rev. 2, 92-99 (1995).
[CrossRef]

G. Campbell and R. Kostuk, "Effective-medium theory of sinusoidally modulated volume holograms," J. Opt. Soc. Am. A 12, 1113-1117 (1995).
[CrossRef]

1991 (1)

1983 (1)

1979 (1)

D. E. Aspnes and J. B. Theeten, "Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry," Phys. Rev. B 20, 3292-3302 (1979).
[CrossRef]

1974 (1)

P. Johnson and R. W. Christy, "Optical constants of transition metals: Ti and V and Cr and Mn and Fe and Co and Ni and Pd," Phys. Rev. B 9, 5056-5070 (1974).
[CrossRef]

1956 (1)

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Abdulhalim, I.

J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Selingson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. of SPIE 3998, 125-134 (2000).
[CrossRef]

Allgair, J.

J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Selingson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. of SPIE 3998, 125-134 (2000).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes and J. B. Theeten, "Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry," Phys. Rev. B 20, 3292-3302 (1979).
[CrossRef]

Benoit, D.

J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Selingson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. of SPIE 3998, 125-134 (2000).
[CrossRef]

Braymer, B.

J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Selingson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. of SPIE 3998, 125-134 (2000).
[CrossRef]

Campbell, G.

Cao, Q.

Christy, R. W.

P. Johnson and R. W. Christy, "Optical constants of transition metals: Ti and V and Cr and Mn and Fe and Co and Ni and Pd," Phys. Rev. B 9, 5056-5070 (1974).
[CrossRef]

Collin, S.

Deguzman, P. C.

Faeyrman, M.

J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Selingson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. of SPIE 3998, 125-134 (2000).
[CrossRef]

Garc´ýa-Vidal, F.

F. Garc´ýa-Vidal, J. M. Pitarke, and J. B. Pendry, "Effective medium theory of the optical properties of aligned carbon nanotubes," Phys. Rev. B 78, 4289 (1997).
[CrossRef]

Granet, G.

Guérineau, N

Guizal, B.

Haidner, H.

Haïýdar, R

Hershey, R.

J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Selingson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. of SPIE 3998, 125-134 (2000).
[CrossRef]

Hugonin, J.-P.

Iwata, K.

H. Kikuta, H. Yoshida, and K. Iwata, "Ability and limitation of effective medium theory for subwavelength gratings," Opt. Rev. 2, 92-99 (1995).
[CrossRef]

Jellison, G. E.

G. E. Jellison, "Generalized ellipsometry for materials characterization," Thin Solid Films 450, 42-50 (2004).
[CrossRef]

Johnson, P.

P. Johnson and R. W. Christy, "Optical constants of transition metals: Ti and V and Cr and Mn and Fe and Co and Ni and Pd," Phys. Rev. B 9, 5056-5070 (1974).
[CrossRef]

Kazaryan, A.

D. Stroud and A. Kazaryan, "Optical sum rules and effective-medium theories for a polycrystalline material: Application to a model for polypyrrole," Phys. Rev. B 53, 7076-7084 (1996).
[CrossRef]

Kikuta, H.

H. Kikuta, H. Yoshida, and K. Iwata, "Ability and limitation of effective medium theory for subwavelength gratings," Opt. Rev. 2, 92-99 (1995).
[CrossRef]

Kim, S.-Y.

C.-Y. You, S.-C. Shin, and S.-Y. Kim, "Modified effective-medium theory for magneto-optical spectra of magnetic materials," Phys. Rev. B 55, 5953-5958 (1997).
[CrossRef]

Kipfer, P.

Kostuk, R.

Kuittinen, M.

Lajunen, H.

Lalanne, P.

Lemercier-Lalanne, D.

Li, L.

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

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

Litt, L. C.

J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Selingson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. of SPIE 3998, 125-134 (2000).
[CrossRef]

Lu, T.

C. Zhang, B. Yang, X. Wu, T. Lu, Y. Zheng, and W. Su, "Calculation of the effective dielectric function of composites with periodic geometry," Physica B 293, 16-32 (2000).
[CrossRef]

Morris, G.

Nevi`ere, M.

Nordin, G. P.

Pendry, J. B.

F. Garc´ýa-Vidal, J. M. Pitarke, and J. B. Pendry, "Effective medium theory of the optical properties of aligned carbon nanotubes," Phys. Rev. B 78, 4289 (1997).
[CrossRef]

Petit, R.

Pitarke, J. M.

F. Garc´ýa-Vidal, J. M. Pitarke, and J. B. Pendry, "Effective medium theory of the optical properties of aligned carbon nanotubes," Phys. Rev. B 78, 4289 (1997).
[CrossRef]

Primot, J.

Robinson, J. C.

J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Selingson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. of SPIE 3998, 125-134 (2000).
[CrossRef]

Rokushima, K.

Rytov, S. M.

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Selingson, J.

J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Selingson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. of SPIE 3998, 125-134 (2000).
[CrossRef]

Shin, S.-C.

C.-Y. You, S.-C. Shin, and S.-Y. Kim, "Modified effective-medium theory for magneto-optical spectra of magnetic materials," Phys. Rev. B 55, 5953-5958 (1997).
[CrossRef]

Silberstein, E.

Stork, W.

Streibl, N.

Stroud, D.

D. Stroud and A. Kazaryan, "Optical sum rules and effective-medium theories for a polycrystalline material: Application to a model for polypyrrole," Phys. Rev. B 53, 7076-7084 (1996).
[CrossRef]

Su, W.

C. Zhang, B. Yang, X. Wu, T. Lu, Y. Zheng, and W. Su, "Calculation of the effective dielectric function of composites with periodic geometry," Physica B 293, 16-32 (2000).
[CrossRef]

Tervo, J.

Theeten, J. B.

D. E. Aspnes and J. B. Theeten, "Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry," Phys. Rev. B 20, 3292-3302 (1979).
[CrossRef]

Turunen, J.

Vahimaa, P.

Velghe, S.

Vincent, G

Watanabe, K.

K. Watanabe and K. Yasumoto, "Fourier Modal Theory of Rectangular Dot Gratings Made of Anisotropic and Conducting Materials," Proc. of SPIE 5445, 218-221 (2004).
[CrossRef]

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

Whitney, U.

J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Selingson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. of SPIE 3998, 125-134 (2000).
[CrossRef]

Wu, X.

C. Zhang, B. Yang, X. Wu, T. Lu, Y. Zheng, and W. Su, "Calculation of the effective dielectric function of composites with periodic geometry," Physica B 293, 16-32 (2000).
[CrossRef]

Xu, Y.

J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Selingson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. of SPIE 3998, 125-134 (2000).
[CrossRef]

Yamakita, J.

Yang, B.

C. Zhang, B. Yang, X. Wu, T. Lu, Y. Zheng, and W. Su, "Calculation of the effective dielectric function of composites with periodic geometry," Physica B 293, 16-32 (2000).
[CrossRef]

Yasumoto, K.

K. Watanabe and K. Yasumoto, "Fourier Modal Theory of Rectangular Dot Gratings Made of Anisotropic and Conducting Materials," Proc. of SPIE 5445, 218-221 (2004).
[CrossRef]

Yoshida, H.

H. Kikuta, H. Yoshida, and K. Iwata, "Ability and limitation of effective medium theory for subwavelength gratings," Opt. Rev. 2, 92-99 (1995).
[CrossRef]

You, C.-Y.

C.-Y. You, S.-C. Shin, and S.-Y. Kim, "Modified effective-medium theory for magneto-optical spectra of magnetic materials," Phys. Rev. B 55, 5953-5958 (1997).
[CrossRef]

Zalicki, P.

J. Allgair, D. Benoit, R. Hershey, L. C. Litt, I. Abdulhalim, B. Braymer, M. Faeyrman, J. C. Robinson, U. Whitney, Y. Xu, P. Zalicki, and J. Selingson, "Manufacturing considerations for implementattion of scatterometry for process monitoring," Proc. of SPIE 3998, 125-134 (2000).
[CrossRef]

Zhang, C.

C. Zhang, B. Yang, X. Wu, T. Lu, Y. Zheng, and W. Su, "Calculation of the effective dielectric function of composites with periodic geometry," Physica B 293, 16-32 (2000).
[CrossRef]

Zheng, Y.

C. Zhang, B. Yang, X. Wu, T. Lu, Y. Zheng, and W. Su, "Calculation of the effective dielectric function of composites with periodic geometry," Physica B 293, 16-32 (2000).
[CrossRef]

J. Mod. Opt. (1)

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: Pure Appl. Opt. (1)

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

J. Opt. Soc. Am. (1)

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

P. Lalanne and D. Lemercier-Lalanne, "Depth dependence of the effective properties of subwavelength gratings," J. Opt. Soc. Am. A 14, 450-458 (1997).
[CrossRef]

E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, "Use of diffraction theories in integrated optics," J. Opt. Soc. Am. A 18, 2865-2875 (2001).
[CrossRef]

G. Campbell and R. Kostuk, "Effective-medium theory of sinusoidally modulated volume holograms," J. Opt. Soc. Am. A 12, 1113-1117 (1995).
[CrossRef]

J. Turunen, "Form-birefringence limits of Fourier-expansion methods in grating theory," J. Opt. Soc. Am. A 13, 1013-1018 (1996).
[CrossRef]

J. Turunen, M. Kuittinen, and P. Vahimaa, "Form-birefringence limits of Fourier-expansion methods in grating theory: arbitrary angle of incidence," J. Opt. Soc. Am. A 14, 2314-2316 (1997).
[CrossRef]

P. Lalanne and G. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-783 (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]

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]

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

Opt. Express (3)

Opt. Lett. (1)

Opt. Rev. (1)

H. Kikuta, H. Yoshida, and K. Iwata, "Ability and limitation of effective medium theory for subwavelength gratings," Opt. Rev. 2, 92-99 (1995).
[CrossRef]

Phys. Rev. B (5)

D. E. Aspnes and J. B. Theeten, "Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry," Phys. Rev. B 20, 3292-3302 (1979).
[CrossRef]

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Figures (12)

Fig. 1.
Fig. 1.

Structure with one-dimensional lamellar grating layer. In terms of the grating period Λ, the fill factor is defined as f = a/Λ.

Fig. 2.
Fig. 2.

Piecewise constant function of the permittivity in y for the one-dimensional lamellar grating. εH and εL are the permittivities of alternating materials, a and b are widths of the stripes and the inter-space, respectively. The space period Λ can be related to the fill factor as f = a/Λ. Two periods Λ are shown.

Fig. 3.
Fig. 3.

Convergence of the RCWA code applied for numerical modeling in this article. Logarithms of diffraction efficiency differences are shown as a dependence on number of positive modes in truncated Fourier series. Values are obtained as a decadic logarithms of differences between value for given number of modes and precise value (value taken for much larger number of modes). Chosen period of grating with fill factor f = 0.5 is Λ = 100 nm and incidence angle is ϕ= 45 degrees.

Fig. 4.
Fig. 4.

Dependence of the effective parameters (components of ε in Eq. (12)) on the az-imuthal angle ϕ. Fill factor of the grating is f = 0.5 and period is Λ = 5 nm. Curves show real and imaginary parts of the ordinary and extraordinary effective refractive indices fitted from rigorous data for the polar angles ranging from 0 to 90 degrees. Effective parameters are plotted with respect to the coordinate system fixed with grating stripes.

Fig. 5.
Fig. 5.

Dependence of the diagonal effective parameters on the fill factor f for period Λ = 5 nm in complex plane. Left figure shows ordinary εo and extraordinary εe relative permittivity tensor elements, where dash and dash-dotted lines represent values obtained from EMA, while solid lines are exact fitted values. Right figure shows the same dependence for the corresponding ordinary no and extraordinary ne refractive index. Small steps in fill factor are used and steps of 0.1 are denoted by small green and yellow circles. The point f = 0 (ε= 1) represents air and the point f = 1 (ε= -13.53 + i19.57) represents cobalt.

Fig. 6.
Fig. 6.

Detail of the dependence of the imaginary part of extraordinary effective parameters on fill factor. Left figure shows extraordinary ℑ(εe ) effective permittivity and right figure introduces extraordinary effective refractive index ℑ(ne ). Dashed and dash-dotted values correspond to EMA, while solid lines are values from rigorous fit.

Fig. 7.
Fig. 7.

Dependence of the real and imaginary part of ordinary refractive index on the period of the grating, for the choice of the fill factor f = 0.5. The curve shows difference between the zero-order approximation N = 0 (constant black solid line) and the rigorous model fit. Influence of higher Fourier harmonics is illustrated by fits of rigorous data for truncation order N = 1 and exact fit (here N = 10).

Fig. 8.
Fig. 8.

Real and imaginary part of extraordinary refractive index dependent on the period of the grating, for the choice of the fill factor f = 0.5. The curve shows difference between the zero-order approximation N = 0 (constant black solid line) and the rigorous model fit. Influence of higher Fourier harmonics is illustrated by fits of rigorous data for truncation order N = 1 and exact fit (here N = 10).

Fig. 9.
Fig. 9.

Figure shows orientations of optical axes of uniaxial ZnO with respect to the geometry of lamellar grating.

Fig. 10.
Fig. 10.

Effective parameters of ZnO anisotropic grating dependent on filling factor f are plotted in the complex plane. Values of the effective parameters are plotted in coordinate system aligned with the optical axes of the ZnO (rotated by 45 degs from lamellas) for the period Λ = 5 nm. On the left figure diagonal effective parameters are shown, while on the right figure is off-diagonal parameter. Fitted values are compared with analytical formulas in (11) (dashed and dash-dotted lines). Steps of 0.1 in fill factor are denoted by small green and yellow circles. Point f = 0 represents air with ε= 1, while points f = 1 represent in the left figure ordinary and extraordinary permittivity of ZnO: εxx = εzz = 6.43 + i3.00, εyy = 7.19 + i0.70. In the right figure point f = 1 is also zero.

Fig. 11.
Fig. 11.

Real and imaginary part of effective permittivities as a function of the period of ZnO grating with fill factor f = 0.5. Dashed and dash-dotted lines shows EMA values in static limit (Λ → 0).

Fig. 12.
Fig. 12.

Dependence of the real and imaginary part of the off-diagonal element of the effective permittivity tensor εxy . Dashed and dash-dotted lines shows EMA values in static limit (Λ → 0).

Equations (24)

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[ D x D y D z ] = [ ε xx ε xy ε xz ε yx ε yy ε yz ε zx ε zy ε zz ] [ E x E y E z ] .
[ D x D z E y ] = [ ε xx ε xy ε yy 1 ε yx ε xz ε xy ε yy 1 ε yz ε xy ε yy 1 ε zx ε zy ε yy 1 ε yx ε zz ε zy ε yy 1 ε yz ε zy ε yy 1 ε yy 1 ε yx ε yy 1 ε yz ε yy 1 ] [ E x E z D y ] .
[ D x D z E y ] = [ ε xx ε xy ε yy 1 ε yx ε xz ε xy ε yy 1 ε yz ε xy ε yy 1 ε zx ε zy ε yy 1 ε yx ε zz ε zy ε yy 1 ε yz ε zy ε yy 1 ε yy 1 ε yx ε yy 1 ε yz ε yy 1 ] [ E x E z D y ] .
[ D x D y D z ] = Q [ E x E y E z ] ,
Q = [ ε xx ε xy ε yy 1 ε yx + ε xy ε yy 1 ε yy 1 1 ε yy 1 ε yx ε xy ε yy 1 ε yy 1 1 ε xz ε xy ε yy 1 ε yz + ε xy ε yy 1 ε yy 1 1 ε yy 1 ε yz ε yy 1 1 ε yy 1 ε yx ε yy 1 1 ε yy 1 1 ε yy 1 ε yz ε zx ε zy ε yy 1 ε yx + ε zy ε yy 1 ε yy 1 1 ε yy 1 ε yx ε zy ε yy 1 ε yy 1 1 ε zz ε zy ε yy 1 ε yz + ε zy ε yy 1 ε yy 1 1 ε yy 1 ε yz ] .
Q isotropic = [ ε 0 0 0 ε 1 1 0 0 0 ε ] .
ε ( y ) = f ε ( H ) + ( 1 f ) ε ( L ) + n 0 ( ε ( H ) ε ( L ) ) sin ( nπf ) n π exp ( i n 2 π Λ y ) .
ε ( y ) ( N = 0 ) = f ε ( H ) + ( 1 f ) ε ( L ) .
ε eff = [ ε xx eff 0 0 0 ε yy eff 0 0 0 ε xx eff ] .
ε xx eff = Q 0 , xx = f ε ( H ) + ( 1 f ) ε ( L ) ,
ε yy eff = Q 0 , yy = ε ( H ) ε ( L ) f ε ( L ) + ( 1 f ) ε ( H ) ,
ε rs eff = Q 0 , rs = f ( ε rs ( H ) ε ry ( H ) ε ys ( H ) ε yy ( H ) ) + ( 1 f ) ( ε rs ( L ) ε ry ( L ) ε ys ( L ) ε yy ( L ) ) +
+ f ε ry ( H ) ε yy ( L ) + ( 1 f ) ε ry ( L ) ε yy ( H ) ε yy ( H ) ε yy ( L ) [ f ε yy ( L ) + ( 1 f ) ε yy ( H ) ] [ f ε ys ( H ) ε yy ( L ) + ( 1 f ) ε ys ( L ) ε yy ( H ) ] ,
ε yy eff = Q 0 , yy = ε yy ( H ) ε yy ( L ) f ε yy ( L ) + ( 1 f ) ε yy ( H )
ε tu eff = Q 0 , tu = tu ( H ) ε yy ( L ) + ( 1 f ) ε tu ( L ) ε yy ( H ) f ε yy ( L ) + ( 1 f ) ε yy ( H ) ,
ε ϕ = R ϕ 1 ε R ϕ ,
R ϕ = [ cos ϕ sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 ] .
( ε yy ) = f ε L 2 ( ε H ) [ f ε L + ( 1 f ) ( ε H ) ] 2 + [ ( 1 f ) ( ε H ) ] 2 .
( ε yy ) f ε L 2 ( ε H ) ( 1 f ) 2 ε H 2 ,
F = [ F N F 0 F N ] ,
f ( y ) = n = N N F n exp ( i 2 πny Λ ) .
F n = 1 Λ 0 Λ f ( y ) exp ( i 2 πny Λ ) d y .
T = [ F 0 F 1 F 2 F 3 F 2 N F 1 F 0 F 1 F 2 F 2 F 1 F 0 F 1 F 3 F 3 F 2 F 1 F 0 F 2 F 1 F 2 N F 3 F 2 F 1 F 0 ]
T ij = F i j .

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