## Abstract

A decomposition model with depolarization matrix is proposed to characterize the optical and physical properties of anisotropic thin films with rough surfaces. In the proposed approach, the refractive index and thickness of the thin film are inversely extracted using a pure polarization matrix, and the surface roughness of the film is characterized using a depolarization matrix. The validity of the proposed method is demonstrated by comparing the experimental results for the refractive index and thickness of a thin film with the analytical results obtained using the effective ellipsometric parameters of the film. The results show that the proposed method provides a reliable means of obtaining the optical and physical properties of thin films with fine or coarse rough surfaces. Importantly, the proposed method not only enables the coarse surface roughness of thin-film samples to be determined in a non-contact optical manner, but also provides a more versatile approach than the well-known effective medium approximation (EMA) model, which is restricted to the characterization of samples with low surface roughness.

© 2016 Optical Society of America

## 1. Introduction

Ellipsometry is a well-known method for characterizing the optical properties of thin films by means of the amplitude ratio *tan(Ψ)* and phase difference *Δ* of the p- and s- polarized light waves [1–3]. However, the conventional ellipsometry is suitable only for specimens with surface roughness sufficiently fine that the Fresnel model and the effective medium approximation (EMA) method [4, 5] can be applied. Fujiwara et al. [6] confirmed that the EMA model gives the best overall fitting results for the thin film with surface roughness of 10-20 nm. The EMA method also cannot be used for semiconductor alloys or transparent thin films with small refractive indices [7, 8] and is inapplicable to samples comprising two-dimensional island structures on a substrate [9, 10]. When the specimen surface is coarse, the polarization properties cannot be adequately characterized by Ψ and Δ alone and the effects of depolarization must be considered. Thus, various models such as the rigorous coupled wave analysis (RCWA) [11] or the dispersion model [12] are for taking the surface roughness into account, and the Mueller matrix ellipsometry [13, 14] is one of the most widely applied methods.

The Mueller matrix ellipsometry provides the means to determine all 16 elements of the Mueller matrix of an optical sample [15]. In addition, it also provides a potential technique for dealing with the depolarization effects caused by scattering from the rough surface of a substrate [16] or grating structure [17, 18]. Nee [19] used a null ellipsometry method to determine the principal Mueller matrix of a sample with surface roughness based on the depolarization effects observed at different incident polarization angles. Letnes *et al.* [20] calculated the Mueller matrix for the scattering of light from two-dimensional rough surfaces and showed that the ability to model polarization effects in light scattering media enables a better interpretation of experimental data and facilitates the design of surfaces with useful polarization effects. However, both studies [19, 20] only characterized the effects of depolarization within the principal Mueller matrix. That is, a complete model of the depolarization matrix itself was not derived. Cloude [21] proposed a new technique called depolarization synthesis for calculating the Mueller matrix depolarization. However, this technique was quite complicated and Mueller matrix depolarization was obtained numerically only. Williams *et al.* [22] proposed a method for extracting the Mueller matrix of rough specimens utilizing a depolarization matrix and cross polarization matrix correlated with the specimen roughness and specimen reflectance. It was shown that the analytical results for the Mueller matrix elements were in good agreement with the experimental values. Therefore, the depolarization matrix proposed in [22] is also adopted in this study.

In a previous study of the present group [23], a Stokes-Mueller ellipsometry was employed to extract the optical/physical properties of thin film without considering depolarization effects due to surface roughness. In the present study, the Stokes-Mueller ellipsometry is extended to determine the depolarization matrix and the optical/physical properties (i.e., thickness, refractive indices and surface roughness) of anisotropic thin films with either fine or coarse rough surfaces based on a decomposition model. In the proposed approach, a genetic algorithm (GA) fitting technique is used to inversely extract the refractive index and thickness of the thin-film sample based on the pure polarization matrix and the surface roughness *Ra* based on the depolarization matrix. The feasibility of the proposed method is confirmed by comparing the experimental results for the effective ellipsometric parameters of thin-film samples with varying degrees of surface roughness with the analytical results.

## 2. Decomposition model for characterization of thin-film samples with fine/coarse rough surfaces

The Stokes parameters completely describe un-polarized light and partially-polarized light in accordance with

*S*represents the total intensity; S

_{0}_{1}is the difference in intensity of the horizontal linearly polarized components and vertical linearly polarized components;

*S*is the difference in intensity of the linearly polarized components oriented at + 45° and −45°, respectively; and S

_{2}_{3}is the difference in intensity of the right and left circularly polarized components, respectively.

Every optical thin film sample can be described by a 4x4 Mueller matrix as S = [M]_{thinfilm} S′, where *S* is the Stokes vector of the output light and *S′* is the Stokes vector of the input light. The general form is given as

In this study, the proposed decomposition model for a thin-film optical sample with a rough surface is shown in Fig. 1. As shown, the film is modelled as a two-layer structure, in which the optical properties are described by *M _{D}* (i.e., the depolarization Mueller matrix) and

*M*(i.e., the reflectance Mueller matrix), respectively. It is noted that

_{R}*d*is the thickness of surface roughness layer and

_{s}*d*is the thickness of thin film, respectively. The light passes the rough surface layer, thin film-substrate and reflects back to the rough surface layer. Thus, the new model to describe the Mueller matrix of the two-layer thin film structure as illustrated in Fig. 1 can be expressed as

The reflectance Mueller matrix *M _{R}* can be obtained based upon the reflectance Jones matrix

*J*calculated by Berreman 4x4 matrix [24, 25]. Figure 2 shows the illustration of the electric fields of a two-layer thin film structure. The matrix

_{R}*T*presents the electric fields of the incident and reflected waves onto the surface of a thin film and the matrix

_{i}*T*presents the electric fields of the transmitted waves at the thin film/substrate interface. The matrix

_{t}*P(-z)*transfers the field components of multiple reflected waves through the thin film and

*T*converts the tangential components into the incident and reflected waves. As described by Schubert in [25], the electric fields can be expressed as

_{i}^{−1}*T*is the transfer matrix and is defined as $T={T}_{i}{}^{-1}P(-z){T}_{t}$.

*E*are the electric field of incident and reflectance waves of s-polarization, respectively.

_{is}and E_{rs}*E*are the electric field of incident and reflectance waves of p-polarization, respectively. While

_{ip}and E_{rp}*E*and

_{ts}*E*are the electric field of transmitted waves of s- and p-polarization, respectively.

_{tp}Thus, the reflectance Jones matrix for a two-layer thin film structure without considering depolarization effects can be expressed as [26]

where*r*are the amplitude reflectance coefficients and defined as the ratio of reflected electric field to incident electric field.

_{sp}, r_{ss}, r_{ps}, r_{pp}Subsequently, the formula for the conversion of Jones matrix in Eq. (5) to Mueller matrix by calculating the light intensity (the Stokes vector) from the electric fields (the Jones vectors) is applied [27] and thus *M _{R}* represents the reflectance Mueller matrix and has the form

It is noted that the elements in the reflectance matrix *M _{R}* are a function of the incident angle, the refractive index of the surrounded environment, the refractive indices and thickness of the thin film, and the refractive index of the substrate. In this study, the influence of interface layer formed at (anisotropic thin film)/(substrate) interface is neglected when the thickness of the thin film is larger than 300

**Å**[28]. The multiple reflections at the interface of air-thin film and thin film-substrate can be simulated by reflectance matrix

*M*with Berreman model [24].

_{R}Meanwhile, the depolarization matrix *M _{D}* caused by scattering effects from the rough surface has the form [22]

*P*and

_{1}, P_{2}*P*are the average lengths of the transformed axes due to depolarization. In addition, the top row elements

_{3}*A*and

_{1,}A_{2}*A*are asymmetric in the amounts of depolarization of the oppositely-directed input vectors.

_{3}## 3. Inverse extraction of depolarization matrix based on the output Stokes vectors

Given the use of the light with four different input polarization states, namely three linear polarization states (0°_{,} 45°, and 90°) and one circular polarization state (right hand (R-)), the input Stokes vectors are as follows: S′_{0°} = [1,1,0,0]^{T}, S′_{45°} = [1,0,1,0]^{T}, S′_{90°} = [1,-1,0,0]^{T}, and S′_{R} = [1,0,0,1]^{T}. Thus, the output Stokes vectors are given by

The elements of Mueller matrix M_{thinfilm} in Eq. (2) are obtained using Eqs. (8)-(11). Importantly, the four output Stokes vectors provide a sufficient number of constraints to inversely extract the depolarization matrix *M _{D}*, refractive index and thickness of a thin film using a GA fitting technique for which the detailed procedure can be found in [29]. The aim of the GA process is to minimize the following fitness function

*S’*and

_{i,theory}*S’*are the simulated output Stokes vector calculated from Eqs. (8)-(11) and the measured output Stokes vector by experiment in a P′-S′ coordinate system rotated at an angle of θ to the X-Y coordinate system described in detail in [23];

_{i,experiment}*k*is the index of the Stokes vectors (

*S*) while S

_{1}, S_{2}, S_{3}_{0}are normalized to 1; and

*i*is the state of polarization of the incident light (i.e., 0°, 45°, 90°, and R-). The GA process commences by generating an initial random population of candidate solutions for the depolarization matrix elements (i.e.,

*A*and

_{1}, A_{2}, A_{3}, P_{1}, P_{2}*P*), refractive indices and thickness of the thin film. The proposed model described as Eq. (3) is then applied to calculate the output Stokes vector of the thin film for each candidate solution. Finally, the Stokes vector is substituted into Eq. (12) to determine the corresponding fitness (i.e., error). If the error between the theoretical output Stokes vector and the experimental output Stokes vector is less than a pre-defined tolerance, the GA is terminated and the optimal values of the depolarization matrix elements, refractive indices and thickness of the thin film are recovered from the corresponding candidate solution. As described in the following section, the extracted values of the refractive indices and thickness of the thin film can also be obtained by performing a GA fitting procedure based on the effective ellipsometric parameters derived from the pure polarization matrix.

_{3}## 4. Inverse extraction of refractive index and thickness properties based on effective ellipsometric parameters

Having obtained the depolarization matrix *M _{D}* via the GA fitting technique described in Section 2, the so-called pure polarization matrix of the sample can be obtained by multiplying both sides of Eq. (3) by

*M*in order to eliminate the depolarization effects. In other words, the pure polarization Mueller matrix, which relates only to the reflectance Mueller matrix

_{D}^{−1}*M*of the thin film, can be expressed as

_{R}A previous study by the present group [23] introduced a polarization scanning ellipsometry based on the concept of “effective ellipsometric parameters”, in which the p- and s- waves considered in the traditional ellipsometry technique were redefined as p′- and s′- waves in a P′-S′ coordinate system rotated at an arbitrary angle of *θ* relative to the original X-Y coordinate system. It was shown that the effective ellipsometric parameters fall within the ranges of 0° ≤ ψ′_{pp} ≤ 90°, 0° ≤ ψ′_{ps} ≤ 90°, 0° ≤ ψ′_{sp} ≤ 90°, 0° ≤ Δ′_{pp} ≤ 360, 0° ≤ Δ′_{ps} ≤ 360° and 0° ≤ Δ′_{sp} ≤ 360°, and can be expressed in terms of the Mueller elements as follows:

*M*are the elements of the pure polarization matrix obtained from Eq. (13) and they are a function of scanning angle

_{ij}*θ*. In extracting the refractive index and thickness of the thin film using the effective ellipsometric parameters, the second GA fitness function is formulated as

*E*are the effective ellipsometric parameters calculated from Eqs. (14)-(19) with

_{Experiment}*M*obtained from experiment,

_{pure}*E*are the effective ellipsometric parameters calculated from Eqs. (14)-(19) with

_{Theory}*M*obtained in Eq. (13), and

_{pure}*j*= 1 to 6

*(ψ′*and

_{pp}, ψ′_{ps}, ψ′_{sp}, Δ′_{pp},Δ′_{ps}*Δ′*, respectively) and

_{sp}*abs*is the absolute value.

A series of simulations were demonstrated to investigate the effects of experimental errors in the measured output Stokes vectors on the accuracy of the extracted effective ellipsometric parameters. The theoretical values of *Ψ′ _{pp}, Ψ′_{ps}, Ψ′_{sp}, Δ′_{pp}, Δ′_{ps}, Δ′_{sp}* were calculated using Eqs. (14)-(19) for an anisotropic film sample with two sets of random assumed input parameters of (Ψ′

_{pp}= 50°, Ψ′

_{ps}= 20°, Ψ′

_{sp}= 20°, Δ′

_{pp}= 70°, Δ′

_{ps}= 80°, and Δ′

_{sp}= 80°) and (Ψ′

_{pp}= 70°, Ψ′

_{ps}= 50°, Ψ′

_{sp}= 10°, Δ′

_{pp}= 50°, Δ′

_{ps}= 110°, and Δ′

_{sp}= 260°), and they were compared with the extracted values. The corresponding extracted results are presented in Fig. 3 for output Stokes vectors with commercial Stokes polarimeter given accuracies of ± 0.5% [PAX5710, Thorlabs Co.]. For the first set of assumed values of effective ellipsometric parameters

*Ψ′*and

_{pp}, Ψ′_{ps}, Ψ′_{sp}, Δ′_{pp}, Δ′_{ps}*Δ′*, the corresponding extracted values are found to deviate from the input values by ± 0.0015°, ± 0.006°, ± 0.006°, ± 0.014°, ± 0.009°, and ± 0.03°, respectively. For the second set of assumed values of effective ellipsometric parameters

_{sp}*Ψ′*and

_{pp}, Ψ′_{ps}, Ψ′_{sp}, Δ′_{pp}, Δ′_{ps}*Δ′*, the corresponding extracted values are found to deviate from the input values by ± 0.004°, ± 0.007°, ± 0.005°, ± 0.009°, ± 0.003°, and ± 0.02°. It is found that the error is small in extracting the effective ellipsometric parameters in the P′-S′ coordinate system.

_{sp}## 5. Simulation results

Simulations were performed to investigate the depolarization matrix of a thin-film sample consisting of anisotropic hafnium oxide-HfO_{2} thin films with surface roughness values of 6 Å and 200 Å, respectively, deposited on a silicon substrate. In conducting the simulations, the Bruggeman EMA model [6] was used to calculate the Mueller depolarization matrix *M _{D}* caused by the rough surface of the thin film. Moreover, according to [6] the fractional area of the EMA layer was specified as

*f*= 0.5, and thus the refractive index of the effective layer was obtained as

_{s}*n*= 1.51. It is noted that the

_{s}*M*Mueller matrix for the anisotropic HfO

_{R}_{2}thin film was extracted from the Jones vector following the procedure described in Section 2. In performing the simulations, the refractive index of the silicon substrate Si was given as

*n*= 3.88; the ordinary refractive index and extraordinary refractive index of HfO

_{t}_{2}were given as

*n*= 2.1 and

_{o}*n*= 2.05, respectively; and the thin-film thickness was given as 215 nm. Finally, the Euler angles were set as

_{e}*θ*= 15°,

_{e}*φ*= 90° and

_{e}*ψ*= 15°, the incident angle

*ϕ*of the input light was given as 50° and the wavelength of the light beam was 632.8 nm.

_{i}Tables 1 and 2 show the results obtained for the depolarization matrix *M _{D}*, refractive indices and thin-film thickness of the two samples using the GA fitting method with the fitness function given in Eq. (12). Note that in implementing the GA process for determining an efficient region of the search space value for unknown values of the refractive index, thickness, and depolarization matrix elements, the search space was firstly chosen in a large scale to define an initial value of these parameters. When the deviation of initial value is large, the search space will be redefined in the new range associated with the new value of unknown parameters with the smaller deviation. Repeat the same process and the search space can be narrowed down until the deviation of the extracted parameters is the smallest. Finally, the optimized search spaces for the refractive indices, thickness and depolarization matrix elements were set as 1 ≤ n

_{e}≤ 3, 1 ≤ n

_{o}≤ 3, 100 nm ≤ d ≤ 300 nm, and −1 ≤ (A

_{1}, A

_{2}, A

_{3}, P

_{1}, P

_{2}and P

_{3}) ≤ 1, respectively. Figure 3 presents the simulation results of the GA curve fitting method and the decomposition model for normalized Stokes vectors (in four input polarized states of 0°, 90°, 45° and R-) and the corresponding maximum error in Eq. (12) was 10

^{−2}. Note that the blue solid line represents the simulation results by the GA curve fitting method, while the dashed red line represents the simulation results by the decomposition model.

The GA fitting technique based on the fitness function given in Eq. (20) was also used to extract the refractive indices and thickness of the two thin-film samples. The optimized search spaces for the refractive indices and thickness were again set as 1 ≤ n_{e} ≤ 3, 1 ≤ n_{o} ≤ 3, and 100 nm ≤ d ≤ 300 nm, respectively. The results are also presented in Tables 1 and 2 and the corresponding maximum error in Eq. (20) was 10^{−2}.

As shown, for the sample with low surface roughness, the top row elements of the depolarization matrix approach 0 and the diagonal matrix elements approach 1. Consequently, the depolarization matrix *M _{D}* approximates a unit matrix, as shown in Table 1. For both samples, a good agreement is again observed between the extracted values of

*n*,

_{e}*n*and

_{o}*d*and the known values. In addition, the accuracy of the results obtained using the effective ellipsometric parameters (

*ψ′*and

*Δ’*) is slightly better than that of the results obtained using the output Stokes vectors. Furthermore, the validity of the proposed model is confirmed by a good agreement in Fig. 4 between the simulation results using the decomposition model and GA fitting method for normalized Stokes vectors.

## 6. Experimental setup and results

Figure 5 presents a schematic illustration of the experimental setup for extracting the output Stokes vectors and effective ellipsometric parameters on the P’-S’ coordinate system. As shown, the major items of equipment include a stable frequency HeNe-laser (SL 02/2, SIOS Co., central wavelength 632.8nm), a polarizer (GTH5M, Thorlab Co.) to produce linear polarized light, a quarter-wave plate (QWP0-633-04-4-R10, CVI Co.) to generate circular polarized light, and a second polarizer (GTH5M, Thorlab Co.) set to an appropriate angle in the range of *θ* = 0° ~180°. In performing the experiments, the optical intensity of the input linear polarization beam was calibrated using a neutral density filter (NDC-100C-2, Oneset Co.) and power meter detector (8842A, OPHIR Co.). Finally, the output Stoke vectors were measured using a commercial Stokes polarimeter (PAX5710, Thorlabs Co.) with an accuracy of ± 0.5%. It is noted that the changing of polarization states of light after passing though the neutral density filter is small and it can be neglected. Thus after calibration, the power meter in Fig. 5 is removed for the subsequent tests. To ensure a precise alignment of the optical components in the experimental setup, a pin hole was placed in front of each component in turn, and the position of the component was adjusted such that the reflected laser beam passed through the pin hole. The rotational position of the second polarizer was adjusted using a mechanical stage (SG-60YAW, Sigma Koki Co.). In performing the experiments, the Stokes polarimeter was rotated to an angle of 80° in order to establish the incident angle of 50°, as shown in Fig. 4. It is noted that the chosen incident angle was based on the best fitting results of Stokes vectors and effective ellipsometric parameters as showed in Fig. 6 and Fig. 7, respectively. It is also noted that for a small roughness sample, the reflected beam is the specular reflectance. However, for a high roughness sample, the sample is rotated through a small angle so that the diffuse beam still falls to the fixed Stokes polarimeter. Thus, the incident angle is slightly changed by the diffuse reflectance. It is found that this small incident angle error ( ± 2°) only slightly affects the inverse results extracted by the proposed method and can be neglected. Meanwhile, the scanning polarized angle was adjusted incrementally from 0° to 180° in steps of 15° by rotating the second polarizer. Finally, the R-hand circular polarization light beam was produced by removing the second polarizer from the experimental configuration.

The experiments were performed using an HfO_{2} thin film deposited on an isotropic silicon substrate. According to the manufacturer’s specification, the sample properties were as follows: refractive indices of HfO_{2} thin film: *n _{o}* = 2.10 and

*n*= 2.05 with the uncertainties of ± 0.05; thickness of thin film:

_{e}*d*= 215 nm with the uncertainty of ± 6%; Euler angles of thin film:

*θ*= 15°,

_{e}*φ*= 90° and

_{e}*ψ*= 15°; and refractive index of silicon substrate:

*n*= 3.88. It is noted that the absorption of the silicon substrate and anisotropic thin film HfO

_{2}was ignored. Thin-film samples with three different surface roughness were prepared, namely

*R*= 6 Å, 197 Å, and 2000 Å. Note that the thin film is coated on the substrate by Co-sputtering system (ULVAC, ACS-4000-C3) and the surface roughness value

_{a}*Ra*is average roughness measured by an atomic force microscope (Dimension Icon, Bruker Corp.) with the high resolution non-contact golden silicon probes (NSG10, NT-MDT). It is also noted that the high surface roughness sample was specially made by coating the thin film at the back side of the wafer with

*R*= 2000 Å. Thus, in this case the roughness of wafer backside can be used to define the roughness of coated thin film on the substrate.

_{a}#### 6.1 Inverse extraction of refractive indices, thickness and depolarization matrix

To determine the output Stokes vectors of each sample, scanning polarized angles of 0° to 180° were obtained in steps of 15° by rotating the second polarizer in Fig. 5. In implementing the GA fitting method based on the output Stokes vectors (i.e., the fitness function given in Eq. (12)), the optimized search spaces for the refractive indices, thickness and depolarization matrix elements were set as 1 ≤ n_{e} ≤ 3, 1 ≤ n_{o} ≤ 3, 100 nm ≤ d ≤ 300 nm, and −1≤ (A_{1}, A_{2}, A_{3}, P_{1}, P_{2}, P_{3}) ≤1, respectively, and the corresponding maximum error in Eq. (12) for three samples was 10^{−2}, respectively (note that the optimal search space was chosen following the process described in section 5). As a result, Tables 3, 4 and 5 present the extracted results for the thin-film samples with surface roughness values of *R _{a}* = 6 Å, 197 Å, and 2000 Å, respectively. Note that the results for each sample represent the average values obtained over four repeated measurements.

To determine the effective ellipsometric parameters of each sample, scanning polarized angles of 0° to 180° were obtained in steps of 15° by rotating the second polarizer in Fig. 5. As described in [23], the axes of the four polarization input lights beams (i.e., 0°, 45°, 90°, and R-) in the P′-S′ coordinate system must be rotated through an additional scanning angle of θ in order to convert them from the X-Y coordinate frame. Therefore, the output Stokes vector can be expressed with respect to the P′-S′ coordinate system as

*R(θ)*is a rotation angle matrix with angle

*θ*and

*M*is the pure polarization matrix calculated in Eq. (13).

_{Pure}Overall, Eq. (21) provides sufficient information to extract the effective ellipsometric parameters of the thin film at any value of *θ*. In using the GA fitting method with the fitness function given in Eq. (20), the optimized search spaces for the refractive indices and thickness were again defined as 1 ≤ n_{e}≤ 3, 1 ≤ n_{o} ≤ 3, and 100 nm ≤ d ≤ 300 nm, respectively, and the corresponding maximum error in Eq. (20) was 10^{−2}. The extraction results for the thin film samples with surface roughness values of *R _{a}* = 6 Å, 197 Å, and 2000 Å are also illustrated in Tables 3, 4 and 5, respectively. (Note that the results again show the mean values obtained over four repeated measurements.)

It is seen in Table 3 that for the sample with low surface roughness (*R _{a}* = 6 Å), a good agreement exists between the extracted values of the refractive indices and thickness of the thin film and the known values. Moreover, the depolarization matrix approximates a unit matrix. However, for a higher surface roughness of

*R*= 197 Å, the elements of the depolarization matrix change in accordance with the depolarization effect. It is noted that the diagonal elements in

_{a}*M*are more sensitive to the changing of the surface roughness than the top row elements. While the surface roughness increases, the value of the diagonal elements,

_{D}*P*, decrease from 1 to −1 and the value of

_{1}, P_{2}, P_{3}*A*increase from 0 to 1. A small deviation exists between the extracted values of the refractive indices and thickness and the known values. For the sample with a surface roughness of

_{1}, A_{2}, A_{3}*R*= 6 Å, the extracted values of the ordinary/extraordinary refractive indices and thickness of thin film deviate from the known values by 0.03, 0.01 and 0.26 nm, respectively. Similarly, for the sample with a surface roughness of

_{a}*R*= 197 Å, the corresponding deviations are 0.02, 0.01 and 0.46 nm, respectively. For the sample with a surface roughness of

_{a}*R*= 2000 Å, the corresponding deviations are 0.05, 0.02 and 0.54 nm, respectively. In practice, the deviations of the extracted values from the known values for each sample are a result most likely of imperfect optical components and system misalignments. Comparing the results obtained for the ordinary/extraordinary refractive indices and thickness based on the output Stokes vectors with those obtained based on the effective ellipsometric parameters, the latter results are found to be more accurate.

_{a}Figure 6 illustrates the GA-fitted results for the normalized Stokes vectors (in four input polarized states of 0°, 90°, 45° and R-) of the thin film samples with those obtained from experimental results. Figure 7 illustrates the GA-fitted results for the six effective ellipsometric parameters of the thin film sample with those obtained experimentally using the pure polarization matrix given in Eq. (13). As shown, the good agreement between the two sets of results confirms the feasibility of the proposed GA-fitting method. It is noted that if the HfO_{2} thin film was treated as an isotropic material and a similar calculation process was performed to extract its thickness and refractive index, the results were not reasonable as compared to the results shown in Tables 3, 4, and 5. Furthermore, the hafnium oxide film is made of many randomly oriented grains, whose size is small compared to the size of the laser spot. Thus, the result of the measurement was an average over many grains, which would effectively form an anisotropic property for the respectively hafnium oxide film. Thus, the hafnium oxide film was reasonably treated as an anisotropic material in this proposed technique.

If the conventional EMA theory is applied directly, the ordinary/extraordinary refractive indices and thickness of the thin-film sample with a surface roughness of *R _{a} = 6 Å* are obtained as 2.12, 2.06 and 215.4 nm, respectively. Moreover, the surface roughness is obtained as 8 Å based upon the assumption of the fractional area

*f*as 0.5. It is seen that these values are in good agreement with the known values since the depolarization matrix approaches a unit matrix, and hence the conventional EMA theory can still be applied successfully. Similar results are obtained for the thin-film sample with a surface roughness of

_{s}*R*= 197 Å. The corresponding values of

_{a}*n*and

_{o}, n_{e}*d*are 2.15, 2.07 and 217.5 nm, respectively. The surface roughness is obtained as 213 Å with the fractional area

*f*as 0.5. However, for the sample with a high surface roughness of

_{s}*R*= 2000 Å, the computed values of

_{a}*n*and

_{o}, n_{e}*d*are found to be 2.4, 2.15 and 240 nm, respectively. Moreover, the surface roughness is found to be 2180 Å. In other words, the conventional EMA theory fails to take adequate account of the scattering effect induced by the large surface roughness and therefore yields the poor measurement result.

#### 6.2 Inverse Extraction of Surface Roughness Ra

The surface roughness of the thin-film sample was measured using a hybrid method comprising the depolarization matrix and the EMA model. In the EMA method, the rough surface layer is modeled as an effective layer with refractive index *n _{s}*, thickness

*d*and fraction area

_{s}*f*and its Mueller matrix is determined via the Fresnel equation. Thus, in the proposed hybrid method, the GA fitting method was employed to extract

_{s}*n*and

_{s}*d*of the rough surface layer by minimizing the function

_{s}*M*are the elements of the experimental depolarization matrix of sample,

_{D}(n)*M*are the elements of the Mueller matrix calculated by the EMA model with the structure in an EMA film, and

_{EMA}(n)*n*(1~16) is the index of the Mueller matrix elements.

In implementing the GA fitting process for the sample with a surface roughness of *R _{a}* = 2000 Å, the optimized search spaces for the refractive index and thickness were set as 1.50 ≤ n

_{s}≤ 1.6 and 1900 Å ≤ d

_{s}≤ 2200 Å, respectively (note that the optimal search space was chosen following the process described in Section 5). As a result, the extracted values of

*n*and

_{s}*d*were found to be 1.51 and 2010 Å and the corresponding standard deviations were ± 0.04 and ± 0.18 nm, respectively. Note that four repeated measurements were performed. Given the extracted value of

_{s}*n*, the fractional area

_{s}*f*was obtained as 0.51 and was thus deemed reasonable. For the sample with a surface roughness of

_{s}*R*= 197 Å, the optimized search spaces were defined as 1.50 ≤ n

_{a}_{s}≤ 1.60 and 180 Å ≤ d

_{s}≤ 220 Å, respectively. The extracted values of

*n*and

_{s}*d*were found to be 1.54 and 201 Å, respectively, with standard deviations of ± 0.05 and ± 0.14 nm. The fractional area

_{s}*f*was obtained as 0.48 and was thus also deemed reasonable. For the sample with a surface roughness of

_{s}*R*= 6 Å, the optimized search spaces were defined as 1.50≤ n

_{a}_{s}≤ 1.60 and 1 Å ≤ d

_{s}≤ 10 Å, respectively. The extracted values of

*n*and

_{s}*d*were found to be 1.56 and 7.8 Å, respectively, with standard deviations of ± 0.03 and ± 0.04 nm. The fractional area

_{s}*f*was obtained as 0.46 and was thus also deemed reasonable. The small deviation between the extracted values and known values of the refractive index

_{s}*n*and thickness

_{s}*d*can be attributed to imperfect optical components, system misalignments, and human error. It is noted that the EMA method [4–6] was unable to provide any specified definition for

_{s}*d*. However, in our proposed technique, a combination method between EMA and decomposition model was developed to extract

_{s}*d*by using the depolarization Mueller matrix. The extracted values for

_{s}*d*were compared to the AFM measurement results. The AFM results show

_{s}*R*(average roughness),

_{a}*R*(root mean square), and

_{p}*R*(skewness), respectively. It was found that our extracted values

_{sk}*d*were close to the

_{s}*R*value. Thus,

_{a}*R*is defined as the average roughness. Furthermore, the extracted value of

_{a}*R*obtained by our proposed approach was only the surface roughness at the illuminated spot. The differences in position of the laser spot from repeated experiment would cause deviations due to the non-uniformity of the rough surface. The roughness of the non-uniform surface would be the average value of

_{a}*R*at different spots.

_{a}## 7. Conclusions and discussions

This study has proposed a decomposition method based on the Mueller matrix for inversely extracting the thickness and refractive indices of isotropic/anisotropic thin films together with the depolarization matrix caused by surface roughness. In addition, a method has been proposed for extracting the surface roughness *R _{a}* of a thin-film sample using the depolarization matrix and EMA model. The validity of the proposed approach has been confirmed by comparing the simulated values of the effective ellipsometric parameters of samples with various degrees of surface roughness with the experimental values. It has been demonstrated that a good agreement exists between the extracted values of the refractive index and thickness of rough thin film samples with the known values. For example, given a fine surface roughness of

*R*= 6 Å, the extracted values of the ordinary/extraordinary refractive indices and thickness of thin film deviate from the known values by just 0.03, 0.01 and 0.26 nm, respectively. Similarly, for a coarse surface roughness of

_{a}*R*= 2000 Å, the extracted values of the ordinary/extraordinary refractive indices and thickness of thin film deviate from the known values by 0.05, 0.02 and 0.54 nm, respectively.

_{a}The method proposed in this study provides a straightforward and reliable approach for characterizing the optical properties of thin films with depolarization effects. Notably, the method described here represents the first time in the literature that the coarse surface roughness (*R _{a}* = 2000 Å) of thin films has been extracted successfully in a non-contact optical manner using a Stokes-Mueller ellipsometry. Furthermore, the method is applicable to both fine and coarse surface roughness samples, and therefore overcomes the limitation of the conventional EMA method, which is applicable only to samples with a fine surface roughness.

## Acknowledgment

The authors gratefully acknowledge the financial support provided to this study by the Ministry of Science and Technology of Taiwan under Grant No. MOST 104-2221-E-006-125-MY2, and MOST 105-3113-E-006-002. This research was, in part, supported by the Ministry of Education, Taiwan, “The Aim for the Top University Project” to the National Cheng Kung University (NCKU).

## References and links

**1. **K. Vedam, “Spectroscopy ellipsometry: a historical overview,” Thin Solid Films **313–314**, 1–9 (1998). [CrossRef]

**2. **H. Fujiwara, *Spectroscopic Ellipsometry Principle and Application* (John Wiley & Sons, 2007).

**3. **D. D. Engelsen, “Ellipsometry of anisotropic film,” J. Opt. Soc. Am. **61**(11), 1461–1466 (1971).

**4. **A. Malsi, R. Kalyanaraman, and H. Garcia, “From Mie to Fresnel through effective medium approximation with multipole contributions,” J. Opt. **16**(6), 065001 (2014). [CrossRef]

**5. **T. Yang, S. Goto, M. Kawata, K. Uchida, A. Niwa, and J. Gotoh, “Optical properties of Gan thin films on sapphire substrates characterized by variable angle spectroscopic ellipsometry,” Jpn. J. Appl. Phys. **37**(2), 1105–1108 (1998). [CrossRef]

**6. **H. Fujiwara, J. Koh, P. I. Rovira, and R. W. Collins, “Assessments of effective medium theories in the analysis of nucleation and microscopic surface roughness evolution for semiconductor thin films,” Phys. Rev. B **61**(16), 10832–10843 (2000). [CrossRef]

**7. **M. Erman, J. B. Theeten, P. Chambon, S. M. Kelso, and D. E. Aspnes, “Optical properties and damage analysis of GaAs single crystals partly amorphized by ion implantation,” J. Appl. Phys. **56**(10), 2664–2671 (1984). [CrossRef]

**8. **P. G. Snyder, J. A. Woollam, S. A. Alterovitz, and B. Johs, “Modeling Al_{x} Ga_{1-x} As optical constant as function of composition,” J. Appl. Phys. **6**(11), 5925–5926 (1990). [CrossRef]

**9. **R. H. Muller and J. C. Farmer, “Macroscopic optical model for the ellipsometry of an underpotential deposit: lead on copper and silver,” Surf. Sci. **135**(1-3), 521–531 (1983). [CrossRef]

**10. **J. C. Farmer and R. H. Muller, “Effect of Rhodamine-B on the electrodeposition of lead on copper,” J. Electrochem. Soc. **132**(2), 313–319 (1985). [CrossRef]

**11. **J. Qiu, W. J. Zhang, L. H. Liu, P. F. Hsu, and L. J. Liu, “Reflective properties of randomly rough surfaces under large incidence angles,” J. Opt. Soc. Am. A **31**(6), 1251–1258 (2014). [CrossRef] [PubMed]

**12. **D. Nečas, I. Ohlídal, D. Franta, M. Ohlídal, V. Čudek, and J. Vodák, “Measurement of thickness distribution, optical constants, and roughness parameters of rough nonuniform ZnSe thin films,” Appl. Opt. **53**(25), 5606–5614 (2014). [CrossRef] [PubMed]

**13. **L. M. S. Aas, P. G. Ellingsen, B. E. Fladmark, P. A. Letnes, and M. Kildemo, “Overdetermined broadband spectroscopic Mueller matrix polarimeter designed by genetic algorithms,” Opt. Express **21**(7), 8753–8762 (2013). [CrossRef] [PubMed]

**14. **O. Svensen, M. Kildemo, J. Maria, J. J. Stamnes, and Ø. Frette, “Mueller matrix measurements and modeling pertaining to Spectralon white reflectance standards,” Opt. Express **20**(14), 15045–15053 (2012). [CrossRef] [PubMed]

**15. **S. A. Hall, M. A. Hoyle, J. S. Post, and D. K. Hore, “Combined Stokes vector and Mueller matrix polarimetry for materials characterization,” Anal. Chem. **85**(15), 7613–7619 (2013). [CrossRef] [PubMed]

**16. **S. C. Siah, B. Hoex, and A. G. Aberle, “Accurate characterization of thin films on rough surfaces by spectroscopic ellipsometry,” Thin Solid Films **545**, 451–457 (2013). [CrossRef]

**17. **X. Chen, S. Liu, C. Zhang, H. Jiang, Z. Ma, T. Sun, and Z. Xu, “Accurate characterization of nanoimprinted resist patterns using Mueller matrix ellipsometry,” Opt. Express **22**(12), 15165–15177 (2014). [CrossRef] [PubMed]

**18. **H. T. Huang, W. Kong, and F. L. Terry Jr., “Normal incidence spectroscopic ellipsometry for critical dimension monitoring,” Appl. Phys. Lett. **78**(25), 3983–3985 (2001). [CrossRef]

**19. **S. M. F. Nee and T. W. Nee, “Principle Mueller matrix of reflection and scattering measured for a one dimensional rough surface,” Opt. Eng. **41**(5), 994–1001 (2002). [CrossRef]

**20. **P. A. Letnes, A. A. Maradudin, T. Nordam, and I. Simonsen, “Calculation of the Mueller matrix for scattering of light from two dimensional rough surfaces,” Phys. Rev. A **86**(3), 031803 (2012). [CrossRef]

**21. **S. R. Cloude, “Depolarization synthesis: understanding the optics of Mueller matrix depolarization,” J. Opt. Soc. Am. A **30**(4), 691–700 (2013). [CrossRef] [PubMed]

**22. **M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. **25**(20), 3616–3622 (1986). [CrossRef] [PubMed]

**23. **Y. L. Lo, Y. F. Chung, and H. H. Lin, “Polarization scanning ellipsometry method for measuring effective ellipsometric parameters of isotropic and anisotropic thin film,” J. of Light. Tech. **31**(14), 2361–2369 (2013). [CrossRef]

**24. **D. W. Berreman, “Optics in stratified and anisotropic media 4x4 matrix formulation,” J. Opt. Soc. Am. **62**(4), 502–510 (1972). [CrossRef]

**25. **M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B Condens. Matter **53**(8), 4265–4274 (1996). [CrossRef] [PubMed]

**26. **M. Schubert, B. Rheinlander, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from unaxial TiO_{2},” J. Opt. Soc. Am. A **13**(4), 875–883 (1996). [CrossRef]

**27. **R. Barakat, “Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization thoery,” Opt. Commun. **38**(3), 159–161 (1981). [CrossRef]

**28. **C. M. Herzinger, B. Johs, W. A. Mcgahan, J. A. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. **83**(6), 3323–3336 (1998). [CrossRef]

**29. **N. Nguyen-Huu, Y. L. Lo, Y. B. Chen, and T. Y. Yang, “Realization of integrated polarizer and color filters based on subwavelength metallic gratings using a hybrid numerical scheme,” Appl. Opt. **50**(4), 415–426 (2011). [CrossRef] [PubMed]