## Effective medium approximation of anisotropic lamellar nanogratings based on Fourier factorization

Optics Express, Vol. 14, Issue 8, pp. 3114-3128 (2006)

http://dx.doi.org/10.1364/OE.14.003114

Acrobat PDF (298 KB)

### 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

## 1. Introduction

1. 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]

2. H.-T. Huang and F. L. Terry Jr., “Erratum to ‘Spectroscopic ellipsometry and reflectometry from gratings (Scatterometry) for critical dimension measurement and in situ and real-time process monitoring’,” Thin Solid Films **468**, 339–346 (2004). [CrossRef]

3. G. P. Nordin and P. C. Deguzman, “Broadband form birefringent quarter-wave plate for the mid-infrared wavelength region,” Opt. Express **5**, 163–168 (1999). [CrossRef] [PubMed]

4. R. Häidar, G. Vincent, N. Guérineau, S. Collin, S. Velghe, and J. Primot, “Wollaston prism-like devices based on blazed dielectric subwavelength gratings,” Opt. Express **13**, 9941–9953 (2005). [CrossRef] [PubMed]

5. H. Lajunen, J. Turunen, and J. Tervo, “Design of polarization gratings for broadband illumination,” Opt. Express **13**, 3055–3067 (2005). [CrossRef] [PubMed]

6. 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]

7. 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]

8. F. García-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]

9. 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]

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

11. 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]

12. 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]

## 2. Theory

### 2.1. Diffraction grating theory and Fourier factorization

*y*-axis and the plane of incidence is rotated by the angle

*ϕ*from

*yz*plane.

13. K. Rokushima and J. Yamakita, “Analysis of anisotropic dielectric gratings,” J. Opt. Soc. Am. **73**, 901–908 (1983). [CrossRef]

15. 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]

16. 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]

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

18. 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]

*ε*

_{ij}╖ is not satisfactory because of two reasons, which are connected to each other:

- First, standard factorization decreases precision of modeling and there is much more terms in truncated series needed.
- Second, it does not obeys boundary conditions inside the periodic structure.

19. K. Watanabe, R. Petit, and M. Nevière, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A **19**, 325–334 (2002). [CrossRef]

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

**E**and the displacement

**D**can be written in the form

*ε*

_{ij}in the form of Fourier series and after that to use the Toeplitz matrices representing each element in (1). But this traditional approach is not correct and for truncated series leads to numerical errors in any used numeric implementation.

20. 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]

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

*yz*plane into two groups: continuous and discontinuous. The continuous field components are

*E*

_{x},

*E*

_{z}, and

*D*

_{y}, while discontinuous are

*D*

_{x},

*D*

_{z}, and

*E*

_{y}(for our choice of coordinate system, see Fig. 1).

*as follows:*

**Q***is crucial for the zero-order approximation which is introduced in the next section.*

**Q**22. 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]

### 2.2. Effective medium approximation

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

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

25. 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]

*N*= 0, see Appendix), Toeplitz matrix ╓∙╖ reduces to its central element as follows:

_{(N=0)}(central element of Toeplitz matrix) from Eq. (8) and explicit evaluation of the elements of matrix

*in (5) leads to the simple analytical formulas for arbitrary anisotropic materials. Similarly, application to the elements of the matrix*

**Q**

*Q*_{isotropic}in Eq. (6) gives analytical formulas for isotropic grating.

### 2.3. Grating from isotropic materials

*Q*_{isotropic}(6) and identifying

**µ**^{eff}with

**Q**_{isotropic}:

*H*) and (

*L*) denote materials of the stripes and the space in between stripes. The analytical formulas (10) for the diagonal elements of

**ε**^{eff}can be found e.g. in Refs. [26

26. W. Stork, N. Streibl, H. Haidner, and P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. **16**, 1921–1923 (1991). [CrossRef] [PubMed]

28. 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]

### 2.4. General anisotropic grating

*r*,

*s*∈ {

*x*,

*z*}. The element

*tu*) ∈ {

*xy*,

*yx*,

*yz*,

*zy*}.

29. M. Foldyna, K. Postava, D. Ciprian, and J. Pištora, “Modeling of magneto-optical properties of periodic nanos-tructures,” J. Magn. Magn. Mater. **290–291**, 120–123 (2005). [CrossRef]

## 3. Rigorous modeling of effective parameters

*ψ*and Δ) or polarimetric quantities (Mueller matrix components [31]) are computed using RCWA model. In this paper we calculate the Mueller matrix components, which include information about ellipsometric phases and have sensitivity to the off-diagonal elements of permittivity tensor. (ii) Then the calculated quantities are fitted in frame of the model of uniform layer of effective medium with the same thickness as grating, which enables us to obtain the effective permittivity tensor. We apply standard Levenberg–Marquardt optimization procedure to minimize

*χ*

^{2}error function [32]. For each modeled quantity we calculate also standard error, which enables us to estimate standard errors of the fitted effective parameters and adjust correct relative weights between different quantities (Mueller matrix components). We always fit simultaneously all Mueller matrix components for several angles of incidence (typically from 0 to 85 degrees with 2.5 degree steps).

## 3.1. Isotropic grating

*d*of the grating is always 200 nm. The period Λ is chosen to be far from sub-wavelength limit (Λ < λ/2). Real and imaginary parts of silicon refractive index are taken from Ref. [33]:

*n*

_{Si}= 3.8812,

*k*

_{Si}= 0.0196. Used optical parameters of cobalt for this wavelength

*n*

_{Co}= 2.265 and

*k*

_{Co}= 4.32 can be found in Ref. [34

34. 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]

*N*= 20 modes, which is clearly sufficient even for gratings with the period Λ = 100 nm (numerical error of diffraction efficiencies is smaller than 10

^{-6}for TE and 10

^{-4}for TM mode).

*n*

_{o}= √

*ε*

_{xx}and

*n*

_{e}= √

*ε*

_{yy}, respectively. Note that reasonable fit can not be obtained with only isotropic medium.

*ϕ*= 0 the plane of incidence is perpendicular to the grating lamellas (planar diffraction geometry). Effects of grating rotation (

*ϕ*≠ 0 – conical diffraction geometry) can be simply described by rotation of the effective permittivity tensor

*by angle*

**ε***ϕ*. In the case of coordinate system in Fig. 1 the rotated permittivity tensor

*can be obtained as*

**ε**_{ϕ}*is in the form*

**R**_{ϕ}*on the azimuthal angle*

**ε***ϕ*. Consequently, main results of this paper are discussed for planar geometry,

*i. e*., for the plane of incidence perpendicular to the grating stripes (

*ϕ*= 0).

*n*

_{o}= √

*ε*

_{xx}and the extraordinary refractive index

*n*

_{e}= √

*ε*

_{yy}can differ from analytical formulas depending on the fill factor and period of the grating, but for the small periods they are in very good agreement even for large interval of fill factors. Figure was obtained by fitting all Mueller matrix components obtained from rigorous modeling of the diffraction grating in planar configuration.

*ε*

_{L}) = 0:

*f*≪ 0.5 and large norm of relative permittivity of second material |

*ε*

_{H}| ≫ |

*ε*

_{L}| lead to desired small values. Discussed dichroic property of metallic gratings is used for design of polarizers for infrared spectral range.

## 3.2. Anisotropic grating

*λ*= 374 nm is

*ε*

_{0,ZnO}= 6.43 + i3.00 and extraordinary permittivity has value

*ε*

_{e,ZnO}= 7.19 + i0.70 [35

35. G. E. Jellison, “Generalized ellipsometry for materials characterization,” Thin Solid Films **450**, 42–50 (2004). [CrossRef]

*d*= 200 nm with period of grating Λ = 5 nm.

## 4. Conclusions

## Appendix

*N*positive and

*N*negative terms of Fourier series, where total number of coefficients is 2

*N*+ 1. Schematically Fourier coefficients of periodic function

*f*(

*y*) with period Λ can be written as column vector

**F**in form:

*F*

_{n}can be obtained using the integral

**F**of function

*f*(

*y*) is denoted simply as

**F**= ⌈

*f*⌉ or

**F**=⌈

*f*(

*y*)⌉.

*of dimension (2*

**T***N*+ 1) × (2

*N*+ 1) can be produced from vector of Fourier coefficients

**F**with 4

*N*+ 1 elements as follows:

**F**and Toeplitz matrix

*produced from this vector is as follows:*

**T***is obtained from function*

**T***f*(

*y*) denoted simply as

*= ╓*

**T***f*╖ or

*= ╓*

**T***f*(

*y*)╖. Central element of Topelitz matrix

*is denoted in this work as ╓*

**T***T*╖

_{(N=0)}.

## Acknowledgments

## References and links

1. | 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 |

2. | H.-T. Huang and F. L. Terry Jr., “Erratum to ‘Spectroscopic ellipsometry and reflectometry from gratings (Scatterometry) for critical dimension measurement and in situ and real-time process monitoring’,” Thin Solid Films |

3. | G. P. Nordin and P. C. Deguzman, “Broadband form birefringent quarter-wave plate for the mid-infrared wavelength region,” Opt. Express |

4. | R. Häidar, G. Vincent, N. Guérineau, S. Collin, S. Velghe, and J. Primot, “Wollaston prism-like devices based on blazed dielectric subwavelength gratings,” Opt. Express |

5. | H. Lajunen, J. Turunen, and J. Tervo, “Design of polarization gratings for broadband illumination,” Opt. Express |

6. | D. E. Aspnes and J. B. Theeten, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B |

7. | 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 |

8. | F. García-Vidal, J. M. Pitarke, and J. B. Pendry, “Effective medium theory of the optical properties of aligned carbon nanotubes,” Phys. Rev. B |

9. | C.-Y. You, S.-C. Shin, and S.-Y. Kim, “Modified effective-medium theory for magneto-optical spectra of magnetic materials,” Phys. Rev. B |

10. | H. Kikuta, H. Yoshida, and K. Iwata, “Ability and limitation of effective medium theory for subwavelength gratings,” Opt. Rev. |

11. | 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 |

12. | E. Silberstein, P. Lalanne, J.-P. Hugonin, and Q. Cao, “Use of diffraction theories in integrated optics,” J. Opt. Soc. Am. A |

13. | K. Rokushima and J. Yamakita, “Analysis of anisotropic dielectric gratings,” J. Opt. Soc. Am. |

14. | M. Nevière and E. Popov, |

15. | P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A |

16. | G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A |

17. | L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A |

18. | L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A |

19. | K. Watanabe, R. Petit, and M. Nevière, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A |

20. | K. Watanabe and K. Yasumoto, “Fourier Modal Theory of Rectangular Dot Gratings Made of Anisotropic and Conducting Materials,” Proc. of SPIE |

21. | L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. |

22. | L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A: Pure Appl. Opt. |

23. | G. Campbell and R. Kostuk, “Effective-medium theory of sinusoidally modulated volume holograms,” J. Opt. Soc. Am. A |

24. | J. Turunen, “Form-birefringence limits of Fourier-expansion methods in grating theory,” J. Opt. Soc. Am. A |

25. | 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 |

26. | W. Stork, N. Streibl, H. Haidner, and P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. |

27. | S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP |

28. | P. Lalanne and D. Lemercier-Lalanne, “Depth dependence of the effective properties of subwavelength gratings,” J. Opt. Soc. Am. A |

29. | M. Foldyna, K. Postava, D. Ciprian, and J. Pištora, “Modeling of magneto-optical properties of periodic nanos-tructures,” J. Magn. Magn. Mater. |

30. | M. Foldyna, K. Postava, D. Ciprian, and J. Pištora, “Modeling of magneto-optical properties of lamellar nano-gratings,” J. Alloy. Compd. (to be published) (2006). |

31. | R. M. A. Azzam and N. M. Bashara, |

32. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

33. | H. Piller, “Silicon (Amorphous) (a-Si),” in |

34. | 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 |

35. | G. E. Jellison, “Generalized ellipsometry for materials characterization,” Thin Solid Films |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(050.2770) Diffraction and gratings : Gratings

(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry

(160.1190) Materials : Anisotropic optical materials

(310.3840) Thin films : Materials and process characterization

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: January 30, 2006

Revised Manuscript: April 2, 2006

Manuscript Accepted: April 11, 2006

Published: April 17, 2006

**Citation**

Martin Foldyna, Razvigor Ossikovski, Antonello De Martino, Bernard Drevillon, Kamil Postava, Dalibor Ciprian, Jaromír Pištora, and Koki Watanabe, "Effective medium approximation of anisotropic lamellar nanogratings based on Fourier factorization," Opt. Express **14**, 3114-3128 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3114

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### References

- 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]
- H.-T. Huang and F. L. Terry and Jr., "Erratum to ’Spectroscopic ellipsometry and reflectometry from gratings (Scatterometry) for critical dimension measurement and in situ and real-time process monitoring’," Thin Solid Films 468, 339-346 (2004). [CrossRef]
- G. P. Nordin and P. C. Deguzman, "Broadband form birefringent quarter-wave plate for the mid-infrared wavelength region," Opt. Express 5, 163-168 (1999). [CrossRef] [PubMed]
- R . Haïýdar, G . Vincent, N . Guérineau, S. Collin, S. Velghe, and J. Primot, "Wollaston prism-like devices based on blazed dielectric subwavelength gratings," Opt. Express 13, 9941-9953 (2005). [CrossRef] [PubMed]
- H. Lajunen, J. Turunen, and J. Tervo, "Design of polarization gratings for broadband illumination," Opt. Express 13, 3055-3067 (2005). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- H. Kikuta, H. Yoshida, and K. Iwata, "Ability and limitation of effective medium theory for subwavelength gratings," Opt. Rev. 2, 92-99 (1995). [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]
- 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]
- K. Rokushima and J. Yamakita, "Analysis of anisotropic dielectric gratings," J. Opt. Soc. Am. 73, 901-908 (1983). [CrossRef]
- M. Nevi`ere and E. Popov, Light Propagation in periodic media: Differential theory and design (Marcel Dekker, 2002).
- 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]
- 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]
- 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, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A: Pure Appl. Opt. 5, 345-355 (2003). [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]
- W. Stork, N. Streibl, H. Haidner, and P. Kipfer, "Artificial distributed-index media fabricated by zero-order gratings," Opt. Lett. 16, 1921-1923 (1991). [CrossRef] [PubMed]
- S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).
- 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]
- M. Foldyna, K. Postava, D. Ciprian, and J. Pi¡stora, "Modeling of magneto-optical properties of periodic nanostructures," J. Magn. Magn. Mater. 290-291, 120-123 (2005). [CrossRef]
- M. Foldyna, K. Postava, D. Ciprian, and J. Pi¡stora, "Modeling of magneto-optical properties of lamellar nanogratings," J. Alloy. Compd. (to be published) (2006).
- R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, p. 149, 3rd ed. (North-Holland, Amsterdam, 1989).
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The art of scientifique computing, 2nd ed. (Cambridge, 2002).
- H. Piller, "Silicon (Amorphous) (a-Si)," in Handbook of Optical Constants of Solids, E. D. Palik, ed., p. 571 (Academic Press, 1991).
- 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]
- G. E. Jellison, "Generalized ellipsometry for materials characterization," Thin Solid Films 450, 42-50 (2004). [CrossRef]

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