Influence of cross-correlation effects on the optical quantities of rough films

Daniel Franta; Ivan Ohlídal; David Nečas

doi:10.1364/OE.16.007789

1. Introduction

In the optical characterization of thin films the influence of boundary roughness on the optical quantities of these films is encountered. There are several approaches for including this roughness into formulae for the optical quantities of rough films. This paper deals with the influence of boundary roughness on the optical quantities describing light coherently reflected from thin films with rough boundaries. In practice it is interesting if the total scattered flux detected by a detector is negligible compared to the flux of coherently reflected light. This case corresponds to practical problems frequently taking place in ellipsometry and spectrophotometry. If boundary roughness is sufficiently small in comparison with the wavelength of incident light it is possible to use the Rayleigh–Rice theory (RRT) [1–8]. The RRT is a vector perturbation approach of the solution of the Maxwell equations on slightly rough surfaces and thin films. Within this approach the electromagnetic field is calculated up to the second order of the perturbation parameter σk₀ in the specular direction (where σ and k ₀ denote the rms value of the heights of roughness irregularities and vacuum wave number, respectively). When the perturbation parameter is sufficiently smaller than unity the RRT utilized up to the second order represents a sufficiently precise solution of the Maxwell equations on the rough surfaces. In our foregoing paper [8] it was shown that the RRT was usable for $σ \underset{\approx}{<} 10$ nm when the optical quantities were calculated in near-UV, visible and near-IR regions.

The theoretical results presented in our paper [6] imply that there is an influence of crosscorrelation between both the rough boundaries of the single thin layer in expression of their optical quantities. This cross-correlation makes the formulae relatively complex in comparison with the formulae corresponding the rough surfaces presented in our papers [7, 8]. Therefore, it is reasonable to deal with conditions permitting to consider the boundaries isolated. Then the Fresnel coefficients derived for the single rough surface can be used to express the Fresnel coefficients describing the Fresnel coefficients of both the individual boundaries of the rough layer. The aim of this paper is to advance the discussion of the conditions allowing for isolation of the boundaries, i. e. it will deal with the simplification of the formulae within the 2×2 matrix formalism [9].

The discussion will be performed on the basis of comparing the simulated data for the examples selected. This means that the ellipsometric quantities and reflectances will be studied.

2. Theoretical background

2.1. Smooth system

For the discussion the 2×2 matrix formalism presented in Refs. [9] will be used. This formalism describes the solution of the Maxwell’s equations in isotropic stratified system when the plane wave falls on this system. This solution can be separated into the plane eigenwaves corresponding to the p and s polarizations. For each polarization the electromagnetic field inside the j-th layer is represented by two plane waves, i. e. by the right-going and left-going waves described with amplitudes Â+_qj and Â ^- _qj, respectively (see Fig. 1). Below index q will represent both the polarization, i. e. q=p or s.

The boundary conditions for the electromagnetic fields give the following matrix equation for the j-th boundary of the system [9]:

{\vec{A}}_{q, j - 1} = B_{qj} {\vec{A}}_{qj},

Fig. 1. Schematic diagram of the j-th layer of the isotropic thin film system.

Download Full Size | PDF

Fig. 2. Schematic diagram representing thin film system containing only one layer.

Download Full Size | PDF

where A⃗_qj denotes the vector whose components are formed by the complex amplitudes Â ⁺ _qj and Â ^- _qj, i.e.

{\vec{A}}_{qj} = (\frac{{\hat{A}}_{qj}^{+}}{{\hat{A}}_{qj}^{-}}) .

The boundary matrix B _qj is defined for the smooth boundary as follows:

B_{qj} = \frac{1}{{\hat{t}}_{qj}} (\begin{matrix} 1 & {\hat{r}}_{qj} \\ {\hat{r}}_{qj} & 1 \end{matrix}) .

This matrix expresses the binding conditions for the amplitudes mentioned at the j-th boundary. The symbols r̂_qj and t̂_qj represent the Fresnel reflection and transmission coefficients, respectively, for the wave incident onto the smooth boundary from the left side [10, 11].

The amplitudes of the waves inside of the j-th layer corresponding to the j-th and (j+1)-th boundaries are connected with the elements of the phase matrix T _j, i.e.

T_{j} = (\begin{matrix} \exp (i {\hat{X}}_{j}) & 0 \\ 0 & \exp (- i {\hat{X}}_{j}) \end{matrix}),

where

{\hat{X}}_{j} = \frac{2 π}{λ} d_{j} \sqrt{{\hat{n}}_{j}^{2} - \sin^{2} θ_{0}} .

In the foregoing equation the symbols d_j and n̂_j represent the thickness and the complex refractive index of the j-th layer, respectively. The symbols λ and θ ₀ denote the wavelength of incident light and angle of incidence, respectively.

A multilayer system consisting of N layers is described by means of the overall matrix M _q as follows:

M_{q} = B_{q 1} T_{1} B_{q 2} T_{2} B_{q 3} . . . B_{q, N} T_{N} B_{q, N + 1} .

The reflection and transmission Fresnel coefficients of the system are expressed as follows

{\hat{r}}_{q} = \frac{M_{q, 21}}{M_{q, 11}} and {\hat{t}}_{q} = \frac{1}{M_{q, 11}} .

For the single layer system it is necessary to use the following equation for expressing the overall matrix

M_{q} = B_{q 1} T_{1} B_{q 2} = B_{q}^{(SB)} ({\hat{n}}_{a}, {\hat{n}}_{f}) T (d, {\hat{n}}_{f}) B_{q}^{(SB)} ({\hat{n}}_{f}, {\hat{n}}_{s}),

where symbol B _q ^(SB) denotes the boundary matrix of the smooth boundary. Each boundary matrix B _q ^(SB) depends only on the refractive indices of the adjacent media and phase matrix T depends only on thickness and refractive index of the corresponding layer.

2.2. Rough system

For the rough boundary the boundary matrix B _qj is expressed in a more general way:

B_{qj} = \frac{1}{{\hat{t}}_{qj}^{+}} (\begin{matrix} 1 & - {\hat{r}}_{qj}^{-} \\ {\hat{r}}_{qj}^{+} & {\hat{t}}_{qj}^{+} {\hat{t}}_{qj}^{-} - {\hat{r}}_{qj}^{+} {\hat{r}}_{qj}^{-} \end{matrix}),

where symbols r̂ ⁺ _qj, r̂ ^- _qj, t̂ ⁺ _qj and t̂ ^- _qj denote the Fresnel reflection and transmission coefficients of the rough boundary. Superscripts + and - correspond to the waves incident on the boundary from the left and right sides, respectively (see Fig. 1). Note that for the smooth boundaries it is fulfilled:

{\hat{r}}_{qj}^{+} = - {\hat{r}}_{qj}^{-} and {\hat{t}}_{qj}^{+} {\hat{t}}_{qj}^{-} - {\hat{r}}_{qj}^{+} {\hat{r}}_{qj}^{_} = 1

so that (9) is transformed into (3) for the smooth boundary. Of course, (10) are not true for the rough boundary. The reflection Fresnel coefficient for the isolated rough boundary derived using the RRT [7, 8] is as follows

{\hat{r}}_{qj} = {\hat{r}}_{qj}^{(0)} + σ_{j}^{2} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} {\hat{f}}_{qj} (K_{x}, K_{y}) w_{j} (K_{x} - n_{0} k_{0} \sin θ_{0}, K_{y}) d K_{x} d K_{y},

where r̂ ⁽⁰⁾ _qj, σ_j, f̂_qj and w_j represent the Fresnel coefficient of the corresponding smooth boundary, the rms value of heights, complex function presented in [7] and normalized power spectral density function (NPSDF) of the rough boundary, respectively. A similar equation is valid for the transmission Fresnel coefficient of the rough boundary [8].

Within the RRT the reflection Fresnel coefficient of the rough layer is given in the following more complex form [6]

{\hat{r}}_{q} = {\hat{r}}_{q}^{(0)} + σ_{1}^{2} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} {\hat{f}}_{1 q} (K_{x}, K_{y}) w_{1} (K_{x} - n_{0} k_{0} \sin θ_{0}, K_{y}) d K_{x} d K_{y}

where w ₁₂ is the normalized cross-correlation power spectral density function

w_{12} (K_{x}, K_{y}) = c_{12} (K_{x}, K_{y}) \sqrt{w_{1} (K_{x}, K_{y}) w_{2} (K_{x}, K_{y})},

Fig. 3. Schematic diagrams of the different types of the rough layers.

Download Full Size | PDF

where c ₁₂ is the cross-correlation spectral function (CCSF) satisfying the following relation

∣ c_{12} (K_{x}, K_{y}) ∣ \leq 1 .

Of course, a similar equation as in Eq. (12) is fulfilled for the transmission Fresnel coefficient. The overall matrix M _q of the rough single layer system is generally expressed in the same way as that of rough single boundary in Eq. (9)

M_{q} = \frac{1}{{\hat{t}}_{q}^{+}} (\begin{matrix} 1 & - {\hat{r}}_{q}^{-} \\ {\hat{r}}_{q}^{+} & {\hat{t}}_{q}^{+} {\hat{t}}_{q}^{-} - {\hat{r}}_{q}^{+} {\hat{r}}_{q}^{-} \end{matrix}) .

The difference is only in the fact that the Fresnel coefficients are those of the rough single layer. It is then possible to formally write the foregoing equation in the following way

M_{q} = B_{q}^{(RL)} (σ_{1}, w_{1}, σ_{2}, w_{2}, c_{12}, d, {\hat{n}}_{a}, {\hat{n}}_{f}, {\hat{n}}_{s}),

where matrix B q ^(RL) represents the effective boundary matrix of the rough layer. Symbols d, n̂ _a, n̂ _f and n̂ _s represent the mean thickness and complex refractive indices of the ambient, layer and substrate, respectively.

If each of the two boundaries of the rough layer is seen as isolated Eq. (16) can be simplified as follows

M_{q} = B_{q 1} T_{1} B_{q 2} = B_{q}^{(RB)} (σ_{1}, w_{1}, {\hat{n}}_{a}, {\hat{n}}_{f}) T (d, {\hat{n}}_{f}) B_{q}^{(RB)} (σ_{2}, w_{2}, {\hat{n}}_{f}, {\hat{n}}_{s}),

where matrix B _q ^(RB) denotes boundary matrix defined in Eq. (9). The right side of Eq. (17) evidently represents a isolated roughness approximation (IRA) of true expression of B _q ^(RL) occurring in Eq. (16). The justification of this approximation will be studied in the forthcoming sections.

3. Numerical analysis

It is necessary to select concrete forms of NPSDF and CCSF occurring in Eqs. (16) and (17) if the concrete calculations are performed. In the following the same isotropic Gaussian form of the autocorrelation functions will be assumed. Then the NPSDF of the rough boundaries is given by the Gaussian function, i. e.

w (K_{x}, K_{y}) = w (K) = \frac{τ^{2}}{4 π} e^{- \frac{K^{2} τ^{2}}{4}},

where K ²=K ² x+K ² _y and τ denotes the autocorrelation length. The following selected four models of the rough layer will be investigated:

Identical rough layer (IRL): σ₁=σ₂=σ, w ₁(K)=w ₂(K)=w(K), c ₁₂(K)=1
Uncorrelated rough layer (URL): σ₁=σ₂=σ, w₁(K)=w ₂(K)=w(K), c ₁₂(K)=0
Upper boundary rough layer (UBRL): σ ₁=σ, σ₂=0, w ₁(K)=w(K)
Lower boundary rough layer (LBRL): σ₁=0, σ₂=σ, w ₂(K)=w(K).

Schematic diagrams of these models are introduced in Fig. 3. First we calculated the ellipsometric quantities P⃗ ^(RRT) and reflectance R ^(RRT) using the true formula Eq. (16) within the spectral region 1.2–6.5 eV (190–1000 nm) for rough SiO₂ layers placed onto silicon single crystal substrates. Within this spectral region the ellipsometric quantities were calculated for five incidence angles, i. e. 55°, 60°, 65°, 70° and 75° while the reflectance was calculated for 6°. Symbol P⃗ represents Poincaré vector whose components are expressed in the following manner [12]:

\vec{P} = (I_{s}, I_{cII}, I_{cIII}) = (\sin 2 Ψ \sin Δ, \sin 2 Ψ \cos Δ, \cos 2 Ψ) .

As the ellipsometric quantities the components of the Poincaré vector P⃗ were selected because problems connected with the discontinuities in spectral dependencies occurring for the standard ellipsometric quantities such as Ψ and Δ or complex ellipsometric ratio $\hat{ρ}$ =tan Ψ exp(iΔ) are for them avoided. This is so because the close polarization states are actually close on the Poincaré sphere [12]. These data calculated were fitted using formulae corresponding to the IRA given by Eq. (17). For fitting the following merit function Q was employed:

Q (σ, τ, d) = \sum_{i = 1}^{N_{i}} ∣ {\vec{P}}^{(RRT)} - {\vec{P}}^{(IRA)} (σ, τ, d) ∣ W_{i}

where the summation is performed over the values of the quantities calculated. Symbols W_i and W_j represents the weights estimated on the basis of the experimental accuracy usually achieved using the experimental equipment (W_i=2·10⁵ and W_j=2·10⁶). The degree of agreement between the values of the optical quantities calculated using the RRT and IRA is described by quantity χ defined thus:

χ = \sqrt{\frac{Q}{3 N_{i} + N_{j}}},

where N_i and N_j denote the number of the ellipsometric and reflectance values, respectively.

In Fig. 4 the comparison of the values of the optical quantities calculated using the RRT and IRA is carried out for the selected rough layer. In this comparison model IRL was utilized for d=25 nm, σ=5 nm and τ=50 nm when the RRT was applied. The results of the fit of the calculated data by means of IRA are as follows: d′=24.8 nm, σ′=3.43 nm and τ′=29.1 nm. It can be seen that these results differ considerably from the values of corresponding quantities employed for data calculated using the RRT. Moreover, it is evident that the quality of the fit is relatively good. This is demonstrated by the value of χ=3.50. For comparison the fit of the RRT data calculated was performed using the formulae corresponding to the smooth layer. This fit is poor because χ=17.6 (see Fig. 4). In this case the thickness was found to have a value d=25.0 nm.

The results characterizing the remaining fits are presented using quantities (σ′-σ)/σ, (τ′-τ)/τ, (d′-d)/d and χ in the graphical form in Figs. 5 and 6. In Fig. 5 the quantities characterizing the fits corresponding to the different models of the rough layer are introduced as functions of thickness d. The values of the roughness parameters employed in the RRT calculation were selected in values of σ=5 nm and τ=50 nm. From this figure it can be seen that the quality of the fits represented by χ is dependent on whether one or two boundaries are rough. If the lower boundary is rough the quality of the fits is inferior. This is caused by the greater contrast of the refractive indices of the adjacent media. Furthermore, it can be seen that except the IRL the IRA is more suitable for the greater thicknesses from the point of view of the quality of the fit. As for the correctness of determining the parameters using the IRA the dependence on the thickness is pronounced. It is evident that for thicknesses greater than 250 nm the IRA gives practically true values of the parameters sought. From the point of view of the correctness of rms of the heights σ the IRA gives inferior results for the IRL model compared with the URL model. This is apparently caused by the correlation of the boundaries in the IRL model.

Of course, the usability of the IRA also depends on the value of the autocorrelation length of the boundaries. This is illustrated using Fig. 6. In this figure the dependencies of (σ′-σ)/σ, (τ′-τ)/τ, (d′-d)/d and χ on the thickness are introduced for the IRL model characterized with the different values of the autocorrelation length τ. From this figure it can be seen that with the increasing autocorrelation length the IRA gives inferior results and fits. This conclusion is evident for the rms value of heights σ in particular. It is necessary to emphasize that the instability of results of the fits corresponding to τ=100 nm is caused by the fact that the ellipsometric quantities and reflectance depend only weakly on the value of the autocorrelation length for such large values [6–8].

Finally, it can be formulated that the following inequalities existing between the thicknesses of the rough films and autocorrelation length of their rough boundaries that justify the use of the IRA. If the boundaries are statistically independent it holds that d≳τ, while for the identical rough layer it is fulfilled that d≳3τ. These inequalities follow from the numerical analysis and our experiences.

4. Other approximations

In practice the effective medium approximation (EMA) [13] is most frequently employed for interpreting the experimental data belonging to slightly rough layers. In the previous related paper [8] we demonstrated that the EMA was not satisfactory if the scattering of light was nonnegligible in interaction of light with the selected rough surface. This is illustrated in Fig. 7 where one can see that there are relatively great differences between the ellipsometric quantities and reflectance corresponding to RRT and EMA. Note that within the EMA the study obtained the thickness of the effective layers d _ef=1.59 nm for the same simulated data as in the foregoing section. It is necessary to point out that within the EMA approach it was assumed that the upper and lower rough boundaries exhibited the same effective thickness. In the previous paper [8] it was shown that the EMA only describes roughness components corresponding to high spatial frequencies with the following rms value of heights σ_H=0.5d _ef=0.79 nm. On the other hand the scalar diffraction theory (SDT) [14,15] expresses the influence of the roughness components corresponding to low spatial frequencies. The SDT can be combine with the RRT or EMA in a easy way. Within these combinations the local Fresnel coefficients taking place in the SDT are substituted by the Fresnel coefficients express using the RRT or EMA. Therefore we combined the EMA and SDT for the description of the interaction of light with the rough surfaces with low and high frequencies in our previous paper [8]. When the combined approach EMA&SDT was employed for the treatment of the RRT data calculated here the values obtained were: σ_H=0.61 nm and σ_L=6.24 nm. Symbol σ_L denotes the rms value of the components with the low frequencies. The total rms value of the heights is σ_T=(σ² _H+σ² _L)^1/2=6.27 nm. Of course, the SDT can also be employed individually. The results of this individual application of the SDT are represented in Table 1. From the foregoing it can be implied that the fit of the data simulated is the best for the IRA discussed in the foregoing section. However, the roughness parameters determined using all the approximations are relatively far from those selected for the data simulated using the RRT. This is caused by the fact that within the EMA and SDT the influence of spatial frequencies of roughness comparable with the wavelength of incident light are not included [8] and within the IRA the cross-correlation effects are neglected. In Table 2 it can be seen that the IRA yields the relatively close fit and values of the roughness parameters for the rough films with larger thicknesses which was emphasized in the foregoing discussion of Figs. 5 and 6. From this table it is also evident that the other approximations, i. e. EMA, SDT and EMA&SDT, give similarly unsatisfactory results as for the thinner rough films (see also Fig. 8).

Fig. 4. Spectral dependencies of the components of the Poincaré vector P⃗ for the incidence angle of 65° and reflectance R corresponding to rough layer calculated by the RRT for the IRL model (d=25 nm, σ=5 nm and τ=50 nm). The fits of the RRT data were performed by formulae belonging to the IRA and smooth layer.

Download Full Size | PDF

Fig. 5. The thickness dependencies of the relative differences (σ′-σ)/σ, (τ′-τ)/τ and (d′-d)/d between the fitted parameters and parameters selected for the RRT simulation and quantity χ expressing the quality of the fits for the different models of rough layer (σ=5 nm and τ=50 nm).

Download Full Size | PDF

Fig. 6. The thickness dependencies of the relative differences (σ′-σ)/σ, (τ′-τ)/τ and (d′-d)/d between the fitted parameters and parameters selected for the RRT simulation and quantity χ expressing the quality of the fits for the IRL model of rough layer corresponding to the different values of the autocorrelation length τ (σ=2 nm).

Download Full Size | PDF

Fig. 7. Spectral dependencies of the differences between the values of the optical quantities P⃗ and R calculated using the RRT and using the approximations: EMA, SDT, EMA&SDT and smooth layer. The IRL model was used for the RRT calculation with the following parameters: d=25 nm, σ=5 nm and τ=50 nm.

Download Full Size | PDF

Table 1. The values of the parameters found using the individual approximations by fitting the RRT simulated data calculated for the IRL model with the following parameters: d=25 nm, σ=5 nm and τ=50 nm.

View Table | View all tables in this article

Table 2. The values of the parameters found using the individual approximations by fitting the RRT simulated data calculated for the IRL model with the following parameters: d=500 nm, σ=5 nm and τ=50 nm.

View Table | View all tables in this article

The research also attempted to combine the IRA with SDT in order to include the influence of the cross-correlation effects. We found that for the thinner rough layers this combination represents a certain improvement of the treatment of the simulated data, illustrated in Table 1 and Fig. 9. The total rms value of the heights σ_T=(σ² _H+σ²)^1/2=4.75 nm was determined in an accordance with the rms value selected for the simulated data in this case. On the other hand, for the thicker rough films the combination of IRA and SDT did not improve the treatment of the simulated data as is apparent from Table 2. The best fit corresponds to σ_L=0, this means that approximation IRA&SDT degenerated to IRA.

Fig. 8. Spectral dependencies of the components of the Poincaré vector P⃗ for the incidence angle of 65° and reflectance R corresponding to rough layer calculated by the RRT for the IRL model (d=500 nm, σ=5 nm and τ=50 nm). The fits of the RRT data were performed by formulae belonging to the IRA, EMA&SDT and smooth layer.

Download Full Size | PDF

Fig. 9. Spectral dependencies of the differences between the values of the optical quantities P→ and R calculated using the RRT and using the approximations: IRA, IRA&SDT and smooth layer. The IRL model was used for the RRT calculation with the following parameters: d=25 nm, σ=5 nm and τ=50 nm.

Download Full Size | PDF

5. Conclusion

On the basis of the numerical analysis the influence of neglecting the cross-correlation effects on the optical quantities corresponding to coherently reflected light, i. e. ellipsometric parameters and reflectance, of rough layers has been investigated within the framework of the Rayleigh-Rice theory (RRT) that is sufficiently accurate for the rough films considered. The neglecting of the cross-correlation effects has been performed by means of the isolated roughness approximation (IRA), based on the assumption that the individual rough boundaries are considered to be isolated. Within this approximation the Fresnel coefficients of the individual boundaries of the rough layers are expressed as for the single rough surface. It has been shown that the IRA approach is applicable to rough layers with lateral dimensions of the irregularities smaller than the distance of the boundaries, i. e. the mean layer thickness. The minimum mean thickness from which the IRA is applicable increases with increasing correlation of the boundaries: for uncorrelated boundaries it must hold that d≳τ whereas for completely correlated boundaries d≳3τ. In addition, the IRA has been compared with the other approximation, i. e. effective medium approximation (EMA), scalar diffraction theory (SDT) and their combination. It has been found that all these approximations are less suitable for the treatment of the data in comparison with the IRA. It follows that the approximations mentioned do not yield true values of parameters sought and they yield inferior fits. Furthermore, the approximation based on combining IRA and SDT has been tested. It has been shown that this approximation improves the characterization results if the mean thickness of the rough layer is smaller than the lateral dimensions of the irregularities. For relatively thick layers no improvement has been indicated. The results achieved in this paper are significant for the treatment of the experimental data obtained for rough layers occurring in practice (see e. g. [16]).

Acknowledgments

This work was supported by Ministry of Education of the Czech Republic under contract MSM 0021622411 and by Ministry of Industry and Trade of the Czech Republic under contract TANDEM, FT–TA3/142.

References and links

1. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951). [CrossRef]

2. G. R. Valenzuela, “Depolarization of EM Waves by Slightly Rough Surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967). [CrossRef]

3. K. Krishen, “Scattering of Electromagnetic Waves from a layer with rough front and plane back (Small Perturbation Method by Rice),” IEEE Trans. Antennas Propag. AP-18, 573–576 (1970). [CrossRef]

4. R. Schiffer, “Reflectivity of a slightly rough surface,” Appl. Opt. 26, 704–712 (1987). [CrossRef] [PubMed]

5. J. I. Larruquert, J. A. Mendez, and J. A. Aznarez, “Far-ultraviolet reflectance measurements and optical-constants of unoxidized aluminium films,” Appl. Opt. 34, 4892–4899 (1995). [CrossRef] [PubMed]

6. D. Franta and I. Ohlídal, “Ellipsometric Parameters and Reflectances of Thin Films with Slightly Rough Boundaries,” J. Mod. Opt. 45, 903–934 (1998). [CrossRef]

7. D. Franta and I. Ohlídal, “Comparison of Effective Medium Approximation and Rayleigh-Rice Theory Concerning Ellipsometric Characterization of Rough Surfaces,” Opt. Commun. 248, 459–467 (2005). [CrossRef]

8. D. Franta and I. Ohlídal, “Influence of Lateral Dimensions of the Irregularities on the Optical Quantities of Rough Surfaces,” J. Opt. A-Pure Appl. Opt. 8, 763–774 (2006). [CrossRef]

9. I. Ohlídal and D. Franta, “Ellipsometry of Thin Film Systems,” in Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 2000), Vol. 41, pp. 181–282.

10. A. Vašíček, Optics of Thin Films (North-Holland, Amsterdam, 1960).

11. Z. Knittl, Optics of Thin Films (Wiley, London, 1976).

12. M. Born and E. Wolf, Principles of Optics, 3 ed. (Pergamon Press, Oxford, 1965).

13. D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of Microscopic Surface Roughness by Spectroscopic Ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979). [CrossRef]

14. I. Ohlídal and F. Lukeš, “Ellipsometric parameters of rough surfaces and of a System Substrate-Thin Film with rough boundaries,” Opt. Acta 19, 817–843 (1972). [CrossRef]

15. I. Ohlídal, F. Vižd’a, and M. Ohlídal, “Optical Analysis by means of Spectroscopic Reflectometry of single and double layers with correlated randomly rough boundaries,” Opt. Eng. 34, 1761–1768 (1995). [CrossRef]

16. D. Franta, I. Ohlídal, and D. Nečas, “Optical quantities of rough films calculated by Rayleigh-Rice theory,” Phys. Status Solidi C 5, 1395–1398 (2008). [CrossRef]

approximation	d [nm]	σ_H [nm]	σ_L [nm]	σ [nm]	τ [nm]	σ_T	χ
smooth	25.0	—	—	—	—	0	17.6
EMA	24.6	0.79	—	—	—	0.79	12.5
SDT	24.9	—	6.98	—	—	6.98	8.5
EMA&SDT	24.7	0.61	6.24	—	—	6.27	6.0
IRA	24.8	—	—	3.43	29.1	3.43	3.5
IRA&SDT	24.8	—	3.60	3.11	25.6	4.75	2.7

approximation	d [nm]	σ_H [nm]	σ_L [nm]	σ [nm]	τ [nm]	σ_T	χ
smooth	500.2	—	—	—	—	0	33.0
EMA	499.3	0.83	—	—	—	0.83	28.7
SDT	500.2	—	8.09	—	—	8.09	20.1
EMA&SDT	499.6	0.63	7.66	—	—	7.69	18.2
IRA	500.0	—	—	5.10	50.8	5.10	3.5
IRA&SDT	500.0	—	0	5.10	50.8	5.10	3.5

approximation	d [nm]	σ_H [nm]	σ_L [nm]	σ [nm]	τ [nm]	σ_T	χ
smooth	25.0	—	—	—	—	0	17.6
EMA	24.6	0.79	—	—	—	0.79	12.5
SDT	24.9	—	6.98	—	—	6.98	8.5
EMA&SDT	24.7	0.61	6.24	—	—	6.27	6.0
IRA	24.8	—	—	3.43	29.1	3.43	3.5
IRA&SDT	24.8	—	3.60	3.11	25.6	4.75	2.7

approximation	d [nm]	σ_H [nm]	σ_L [nm]	σ [nm]	τ [nm]	σ_T	χ
smooth	500.2	—	—	—	—	0	33.0
EMA	499.3	0.83	—	—	—	0.83	28.7
SDT	500.2	—	8.09	—	—	8.09	20.1
EMA&SDT	499.6	0.63	7.66	—	—	7.69	18.2
IRA	500.0	—	—	5.10	50.8	5.10	3.5
IRA&SDT	500.0	—	0	5.10	50.8	5.10	3.5

Influence of cross-correlation effects on the optical quantities of rough films

Abstract

1. Introduction

2. Theoretical background

2.1. Smooth system

2.2. Rough system

3. Numerical analysis

4. Other approximations

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Tables (2)

Equations (24)

Optics Express