1. Introduction

Traditional optical characterization methods (reflection and transmission spectroscopy, ellip-sometry) propose considerable instrumental advantages, for example, non-destructive and non-invasive character, fast response for monitoring of real-time processes, and relatively simple and cheap experimental setups. In recent years, the optical methods become standards of process monitoring and quality control in semiconductor technology. New trends in ultra large scale integrated circuits result in the realization of higher-density semiconductor devices. There is a need for adaptation of the optical methods for characterization and control of lithographic and nanostructurization processes and determination of optical functions of nanosize objects (for instance quantum wires and dots). Straightforward method of increasing sensitivity to sub-wavelength and nanosize objects is to produce a periodic system from such objects, usually lamellar gratings [1, 2].

On the other hand, the sub-wavelength gratings behave as effective media with artificial optical properties, which have no analogy among natural materials. Symmetry reduction by nanostructurization results in a strong form anisotropy and gives possibility to apply nanograting as polarizing devices. Recently, a birefringent quarter wave plate [3], polarizing beam-splitters [4], and other quasi-achromatic polarizing devices designed from dielectric gratings [5] have been reported.

The above applications require careful modeling of electromagnetic field reflection and transmission by the grating structure. There are several rigorous methods that differ by solution approach, numerical implementation, applicability, and computation time. The most spread one is the Rigorous Coupled Wave Analysis (RCWA), which is also applied in this paper. However, many applications demand significant reduction of computation time. Effective Medium Approximation (EMA) provide efficient description of sub-wavelength structures as surface roughness [6], nanocomposite and polycrystalline materials [7, 8, 9], and periodic structures [10, 11, 12].

This article deals with effective permittivity tensor of arbitrary anisotropic sub-wavelength lamellar gratings. In Section 2 the theory necessary for modeling based on the RCWA is presented with the proper Fourier factorization rule applied to the Fourier series expansions of permittivity tensor. In Section 3 analytical formulas for effective dielectric tensor of lamellar grating with arbitrary anisotropy are derived as approximation by zero-order diffraction mode. The approximate formulas are compared with the RCWA model in Section 4. We show that the sub-wavelength gratings can be described by an effective medium with parameters slightly different from those obtained from the simple EMA. The differences are explained as effects of the higher evanescent Fourier harmonics in the nanograting.

2. Theory

2.1. Diffraction grating theory and Fourier factorization

Figure 1 schematically shows the one-dimensional lamellar grating and the coordinate system used in this paper. The grating lamellas are perpendicular to y-axis and the plane of incidence is rotated by the angle ϕ from yz plane.

Monochromatic plane-wave propagation is assumed for modeling of the electromagnetic field in the grating. The RCWA is used for the field description [13, 14]. This method is based on Fourier series representation of the field and material parameters. Slower convergence for TM polarized incident light in the highly conducting gratings was improved by representing the permittivity tensor according to the Fourier factorization rules [15, 16, 17]. Boundary conditions, stating that the tangential components of the field vectors are continuous at the interface, are realized using the scattering matrix algorithm [18], which allows numerically stable computations for deep gratings.

As the appropriate Fourier factorization plays a crucial role in this paper it is discussed in more details in following text. The truncated Fourier expansion of the permittivity tensor strongly influences convergence properties of the used numerical algorithms. Standard factorization of permittivity tensor by using Toeplitz matrices (see Appendix for details) for each permittivity tensor element ╓ε_ij ╖ is not satisfactory because of two reasons, which are connected to each other:

 

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.

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In the following general factorization rules are derived for anisotropic gratings based on the idea that the boundary conditions should be respected also at this stage of modeling [19] together with proper use of the Li factorization rule [17].

Relation between the electric intensity E and the displacement D can be written in the form

At this point, it is possible to express elements of permittivity tensor 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.

To decrease computation time together with keeping accuracy of result, there is general tendency to have the numerical realization, which embodies fast convergence for lower number of terms of the Fourier series. Improvement of the convergence speed can be achieved by appropriate application of boundary conditions also for the permittivity tensor elements representations. This improvement for the lamellar gratings is denoted as the Fourier factorization [14].

There is also another reason, why the appropriate representation of permittivity tensor elements is necessary. Bloch mode propagating in the sub-wavelength lamellar structures is a plane wave in the first approximation. Therefore, if one can find appropriate factorization rules for permittivity functions, zero eigenmode used in Fourier series representation corresponds directly to the zero Bloch’s mode of the structure. This allows direct study of behavior of physical propagating modes by analysis of behavior of Fourier series plane waves.

In this paper we introduce derivation of the factorized permittivity tensor matrix based on physical foundation [20, 21]. For the application of this method, it is necessary to separate the field vector components considering incidence in the 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).

Rewriting (1) with respect to the regrouped field components leads to

At this stage the left side of (2) is composed of discontinuous functions, while on the right side we have multiplication of the continuous and discontinuous functions. Therefore, Toeplitz matrices can be used as representation of the components of the matrix (2) in Fourier basis.

Use of Toeplitz matrix notation (see Appendix for details) leads to the following system of equations:

After reordering equations to original state, the permittivity tensor can be represented by matrix Q as follows:

For isotropic medium with permittivity ε the matrix Q in Eq. (5) reduces to the form:

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

Presented approach is suitable only for one-dimensional (1D) gratings. For two-dimensional (2D) dot gratings problems appear with Fourier factorization rules, which in present form have to be applied separately in two directions. Results will be different even for zero-order approximation, if Fourier factorization in two directions is applied in different order. This is caused by non-uniqueness of Fourier representation of 2D periodic medium [22].

2.2. Effective medium approximation

In this subsection the zero-mode approximation algorithm is used to get analytic EMA formulas describing sub-wavelength grating [23, 24, 25]. The idea is based on simple assumption that for the sub-wavelength gratings the only propagating mode is the zero diffraction order (this holds for the most cases if the period is small enough). Therefore taking into account only zero diffraction order while using appropriate Fourier factorization is sufficient to find effective optical parameters of the diffraction grating. This idea allows to obtain analytical formulas for any anisotropic configuration of several mixed materials (grating lamellas and inter-space materials).

In the case of binary lamellar grating the permittivity profile can be described by function at Fig. 2. Therefore, the Fourier series expansion for this profile takes the following form:

In the zero-mode approximation (denoted as N = 0, see Appendix), Toeplitz matrix ╓∙╖ reduces to its central element as follows:

So that change of the operator ╓∙╖ (complete Toeplitz matrix) in Eq. (5) into operator ╓∙╖_(N=0) (central element of Toeplitz matrix) from Eq. (8) and explicit evaluation of the elements of matrix Q in (5) leads to the simple analytical formulas for arbitrary anisotropic materials. Similarly, application to the elements of the matrix Q _isotropic in Eq. (6) gives analytical formulas for isotropic grating.

2.3. Grating from isotropic materials

In the case of sub-wavelength diffraction gratings, the description of the grating by a simple EMA anisotropic layer can be used. According to the symmetry of lamellar grating composed from isotropic materials, the effective layer can be described by the effective uniaxial permittivity tensor

Analytical formulas for ${\mathit{\epsilon }}_{\mathit{\text{xx}}}^{\text{eff}}$ and ${\mathit{\epsilon }}_{\mathit{\text{yy}}}^{\text{eff}}$ can be obtained by application of the operator from Eq. (8) to the matrix Q _isotropic (6) and identifying µ ^eff with Q _isotropic:

 

Fig. 2. Piecewise constant function of the permittivity in y for the one-dimensional lamellar grating. ε_H and ε_L 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.

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where indices (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, 27, 28] and represent simple idea of weighted average of permittivities and reciprocal permittivities.

2.4. General anisotropic grating

Described algorithm can be analogically applied to arbitrary anisotropic materials of the lamellar grating. Formulas obtained for such configurations are more complicated, but can be used for approximation of effective parameters without need of use of the complex rigorous models. The following formulas obtained for a general permittivity tensors in analogy with 2.3 are the main results of this section. The elements ${\epsilon }_{\mathit{\text{xx}}}^{\text{eff}}$, ${\epsilon }_{\mathit{\text{zz}}}^{\text{eff}}$, ${\epsilon }_{\mathit{\text{xz}}}^{\text{eff}}$, and ${\epsilon }_{\mathit{\text{zx}}}^{\text{eff}}$ are in the form:

where r,s ∈ {x,z}. The element ${\epsilon }_{\mathit{\text{yy}}}^{\text{eff}}$ is in the form:

and the elements ${\epsilon }_{\mathit{\text{xy}}}^{\text{eff}}$, ${\epsilon }_{\mathit{\text{yx}}}^{\text{eff}}$, ${\epsilon }_{\mathit{\text{zy}}}^{\text{eff}}$, and ${\epsilon }_{\mathit{\text{yz}}}^{\text{eff}}$ are

where (tu) ∈ {xy,yx,yz,zy}.

Formulas (11) can be used to estimate effective dielectric tensor components of lamellar grating consisting of materials with arbitrary anisotropy. Important applications are optically active materials and materials with anisotropy induced by electro-optic and magneto-optic effects [29, 30].

3. Rigorous modeling of effective parameters

In this subsection simple analytical EMA is compared with the rigorous model based on RCWA. We show that the sub-wavelength gratings can be well described by an effective medium with parameters derived in previous section from simple zero-mode approximation. Slight difference originates from effects of evanescent higher-order diffraction waves.

Fitted effective parameters are obtained using two-step procedure: (i) Ellipsometric (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).

Two different structures are used for numerical modeling: (i) isotropic case represented by absorbing (metallic) Cobalt grating on Silicon substrate and (ii) general anisotropic case represented by ZnO uniaxial grating rotated by 45 degrees from lamel direction.

3.1. Isotropic grating

Properties of isotropic sub-wavelength gratings are shown on the example of cobalt grating on silicon substrate at wavelength 633 nm. In following calculations the deepness 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].

Figure 3 shows, in what manner the precision of modeled polarimetric quantity depends on number of positive and negative modes in truncated Fourier series. In further analysis we employ 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).

 

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.

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From the fit of Mueller matrix components calculated for different angles of incidence we obtain the effective complex ordinary and extraordinary refractive indices: n_o = √ε_xx and n_e = √ε_yy , respectively. Note that reasonable fit can not be obtained with only isotropic medium.

For the azimuthal angle 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 ε by angle ϕ. In the case of coordinate system in Fig. 1 the rotated permittivity tensor ε_ϕ can be obtained as

where the rotation matrix R_ϕ is in the form

This assumption can be confirmed by fitting data from range of polar angles for each particular azimuthal angle separately. Figure 4 shows independence of the fitted effective tensor ε on the azimuthal angle 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).

 

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.

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In Fig. 5 the effective parameters dependent on fill factor of the cobalt grating are compared with the simple analytical formulas (10) in complex plane. The period of Λ = 5 nm was chosen, which is much smaller that the sub-wavelength limit. The ordinary refractive index 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.

It is shown in Fig. 5 and in details in Fig. 6 that for rather large interval of filling factors the imaginary parts of the extraordinary effective parameters are very small. Therefore, sub-wavelength grating consisting of highly absorbing metal and small permittivity lossless material demonstrates for small fill factors natural dichroism resulting from the fact that absorption in the extraordinary direction is very small. On contrary, imaginary part of the effective parameter in ordinary direction grows faster with increasing fill factor. This is general property of these types of sub-wavelength gratings. It can be shown by taking imaginary part of EMA value of extraordinary permittivity from Eq. (10) and assuming that imaginary part permittivity of inter-space material is zero ℑ(ε_L ) = 0:

 

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 ε_o 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 n_o and extraordinary n_e 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.

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Fig. 6. Detail of the dependence of the imaginary part of extraordinary effective parameters on fill factor. Left figure shows extraordinary ℑ(ε_e ) effective permittivity and right figure introduces extraordinary effective refractive index ℑ(n_e ). Dashed and dash-dotted values correspond to EMA, while solid lines are values from rigorous fit.

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Omitting smaller positive values in denominator of (14) leads to the formula

where assumptions of small fill factor 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.

Dependence of the effective parameters on the period of the grating Λ is illustrated in Figures 7 and 8 for the fill factor f = 0.5. The simple analytical EMA model based on the assumption of existence of only zero-diffraction order is represented by constant line in the figure. However, this assumption is not fulfilled for the real grating structures. The origin of the differences between the simple analytical formulas and the fitted values is related to higher-order Fourier harmonics. For the one-dimensional lamellar grating these Fourier series terms correspond to the physical propagation modes, if the Fourier factorization rule is applied properly. Despite the fact, that only the zero-order (specular) beam diffracts from a sub-wavelength grating, the higher-order Fourier harmonics representing the evanescent waves influence the effective parameters. Different curves in Figs. 7 and 8 correspond to the different number of retained orders in truncated Fourier series in the rigorous modeling. The first- and minus-first-order Fourier harmonics gives dominant trends in dependencies shown in Figs. 7 and 8. Moreover, we can conclude that only small number of Fourier harmonics is necessary for rigorous nanograting modeling, where in our case this number is ten (ten positive and ten negative Fourier harmonics have to be included). Also errors of the fit of the rigorous RCWA data by effective values are shown by error bars in Figs. 7 and 8. The increasing value of the errors is reasoned by decreasing the validity of EMA modeling of diffraction gratings for large periods.

3.2. Anisotropic grating

In this subsection analytical EMA formulas are compared with rigorous RCWA model for the case of anisotropic grating. We show that effective parameters of the grating can be in the zero period limit very well described by the analytical formulas (11).

Anisotropic grating is from uniaxial ZnO with ordinary permittivity for chosen wavelength λ = 374 nm is ε _0,ZnO = 6.43 + i3.00 and extraordinary permittivity has value ε _e,ZnO = 7.19 + i0.70 [35]. The wavelength near absorption bandgap was selected for high difference between ordinary and extraordinary refractive indices and also motivated by potential application in the area of ultraviolet lasers. The grating lamellas are rotated by the angle of 45 degrees as illustrated in Fig. 9. Deepness of grating is d = 200 nm with period of grating Λ = 5 nm.

Figure 10 shows comparison of EMA and rigorous model. Values obtained from Eqs. (11) are plotted with dashed and dash-dotted lines for the period Λ = 5 nm. Very good correspondence confirms validity of Eqs. (11) in limit Λ → 0. Right panel of Fig. 10 shows detail of off-diagonal element of anisotropic effective dielectric tensor. The anisotropy originates from symmetry reduction along direction perpendicular to stripes.

Figures 11 and 12 show dependency of the fitted effective parameters on the period of grating Λ. Values correspond very well to the proposed EMA values in (11) for Λ → 0. Similarly as for the isotropic case we propose explanation of differences between the rigorous model and EMA as influence of higher Fourier harmonics.

 

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).

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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).

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Fig. 9. Figure shows orientations of optical axes of uniaxial ZnO with respect to the geometry of lamellar grating.

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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: ε_xx = ε_zz = 6.43 + i3.00, ε_yy = 7.19 + i0.70. In the right figure point f = 1 is also zero.

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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).

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4. Conclusions

The main result of this article is the proposition of the general method of obtaining analytical EMA formulas for arbitrary permittivity tensor element (called zero-order approximation). It is applied for the calculation of effective permittivity tensor of anisotropic lamellar nano-gratings with general anisotropy. The method is illustrated on the case of binary gratings, how-ever, generalization to more complex lamellar gratings consisting of three or more materials is straightforward. Introduction of the higher Fourier harmonics into the rigorous model shows their influence on the effective parameters of nanograting when the period of the grating is increasing. Effects of grating period, fill factor, and azimuthal rotation has been discussed.

 

Fig. 12. Dependence of the real and imaginary part of the off-diagonal element of the effective permittivity tensor ε_xy . Dashed and dash-dotted lines shows EMA values in static limit (Λ → 0).

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We propose possible applications of the EMA approach described in this article: (i) design of new artificial anisotropic materials for applications as polarizing devices; (ii) using progressive fitting of experimental spectroscopic data for advanced characterization of nanogratings; (iii) applications in electro-optics and magneto-optics of periodic grating systems; (iv) important for deeper understanding of behavior of anisotropic diffraction gratings and nanogratings.