**2. Materials and Methods**

Figure 1 schematically illustrates the configuration of the RPAE system. A monochromator containing a rotatable grating for the wavelength scan was employed to disperse the light from the source. The monochromatic light from the exit slit passed through a collimator lens, a fixed polarizer, and a rotating polarizer in sequence before incidence on the sample. Subsequently, the reflected light went through a rotating analyzer and entered a detector for data acquisition. The acquired analog signal was converted to a digital one for data processing. The initial azimuths of polarizing elements were set along the direction perpendicular to the incident plane. The angular velocity of the rotating analyzer was controlled to be twice that of the rotating polarizer. The optical system was aligned and calibrated precisely by a low-power He-Ne laser to realize a continuously variable incident angle in a range of 45◦–90◦ , with a computer-controlled resolution of 0.001◦ or a visual resolution of 0.005◦ [18]. The spectroscopic measurement was performed routinely through a wavelength scan in a spectral range of 300–800 nm with an interval of 10 nm. Au was selected as the test material for the low penetration depth in the visible range with a great optical stability in the atmospheric environment.

**Figure 1.** Schematic configuration of the RPAE optical system. (1) The continuous light source; (2) the monochromator consisting of two spherical mirrors and a rotatable plane grating; (3) lightcollimating lens; (4) rotatable filters; (5) fixed polarizer; (6) rotating polarizer; (7) stepping motors; (8) rotating stage; (9) sample rotator; (10) sample; (11) rotating analyzer; (12) photomultiplier; (13) computer to control the monochromator, stepping motors, filters, rotating table, sample stage, and the photomultiplier; (14) laser used for alignment; and (15) mirrors to guide the laser beam for alignment. **Figure 1.** Schematic configuration of the RPAE optical system. (1) The continuous light source; (2) the monochromator consisting of two spherical mirrors and a rotatable plane grating; (3) lightcollimating lens; (4) rotatable filters; (5) fixed polarizer; (6) rotating polarizer; (7) stepping motors; (8) rotating stage; (9) sample rotator; (10) sample; (11) rotating analyzer; (12) photomultiplier; (13) computer to control the monochromator, stepping motors, filters, rotating table, sample stage, and the photomultiplier; (14) laser used for alignment; and (15) mirrors to guide the laser beam for alignment.

where *A* represents the azimuth of analyzer and *I*0−*I*4 are coefficients of one direct and four harmonic components, which are obtained by applying the discrete Fourier analysis as

where *Ai* is the *i*th analyzer azimuth in the measurement period. Accordingly, the ellipso-

13 2

*II I I I*

+ − <sup>+</sup>

2( 2 ) tan = ,

<sup>−</sup> Δ =

1 3

cos , 2( )( 2 )

13 2

*II I II I*

+ − + +

9( 2 ) tan = , 2(2 4 )

12 4

3( ) 4( 4 ) cos . 8( )( 2 )

+− + Δ =

<sup>2</sup> ( )cos( ) 1,2,3,4

[ ]

[ ]

+ +−

13 2 4 2 1/2 1 31 3 2

*II I I I II I I*

+ +−

1 3 1 1/2 1 31 3 2

*I I I II I I*

3

1/2

1/2

= = , (2)

(3)

(4)

1

*i*

1

2

ψ

ψ

*n* <sup>=</sup>

metric parameters are determined by [16−18]:

and

*k ii*

*I I A kA k*

*n*

The two sets of solutions are self-consistent to quantitatively verify the reliability of The light intensity at the detector for RPAE is expressed as:

$$I(A) = I\_0 + I\_1 \cos A + I\_2 \cos 2A + I\_3 \cos 3A + I\_4 \cos 4A,\tag{1}$$

since the value of *I*4 is the smallest in Equation (1). For bulk material measured at an incident angle of *θ* in the atmosphere, the dielectric function is determined with the wellknown equation: where *A* represents the azimuth of analyzer and *I*0−*I*<sup>4</sup> are coefficients of one direct and four harmonic components, which are obtained by applying the discrete Fourier analysis as

$$I\_k = \frac{2}{n} \sum\_{i=1}^{n} I(A\_i) \cos(kA\_i) \quad k = 1, 2, 3, 4,\tag{2}$$

The accuracy and stability are evaluated by the MARE. The value of the MARE is given by: where *A<sup>i</sup>* is the *i*th analyzer azimuth in the measurement period. Accordingly, the ellipsometric parameters are determined by [16–18]:

$$\begin{aligned} \tan \psi\_1 &= \left[ \frac{2(I\_1 + I\_3 - 2I\_2)}{I\_1 + I\_3} \right]^{1/2}, \\ \cos \Delta\_1 &= \frac{I\_1 - 3I\_3}{\left[ 2(I\_1 + I\_3)(I\_1 + I\_3 - 2I\_2) \right]^{1/2}}, \end{aligned} \tag{3}$$

and

$$\begin{aligned} \tan \psi\_2 &= \left[ \frac{9(I\_1 + I\_3 - 2I\_2)}{2(2I\_1 + I\_2 + 4I\_4)} \right]^{1/2}, \\ \cos \Delta\_2 &= \frac{3(I\_1 + I\_3) - 4(I\_2 + 4I\_4)}{\left[8(I\_1 + I\_3)(I\_1 + I\_3 - 2I\_2)\right]^{1/2}}. \end{aligned} \tag{4}$$

The two sets of solutions are self-consistent to quantitatively verify the reliability of the results without other instruments. We prefer to use Equation (3) in the experiment, since the value of *I*<sup>4</sup> is the smallest in Equation (1). For bulk material measured at an incident angle of *θ* in the atmosphere, the dielectric function is determined with the well-known equation:

$$\widetilde{\varepsilon} = \sin^2 \theta \left[ 1 + \tan^2 \theta \left( \frac{1 - \tan \psi \cdot e^{i\Delta}}{1 + \tan \psi \cdot e^{i\Delta}} \right)^2 \right]. \tag{5}$$

The accuracy and stability are evaluated by the MARE. The value of the MARE is given by:

$$\text{MARE} = \frac{1}{n} \cdot \sum\_{i=1}^{n} \left| \frac{\mathbf{x}\_{\text{i}}^{\text{measured}} - \mathbf{x}\_{\text{i}}^{\text{reference}}}{\mathbf{x}\_{\text{i}}^{\text{reference}}} \right| \times 100\% \,\tag{6}$$

where *n* represents the amount of data. where *n* represents the amount of data.

*Crystals* **2021**, *11*, x FOR PEER REVIEW 4 of 12

#### **3. Results 3. Results**

*3.1. Incident Angle and Principal Angle 3.1. Incident Angle and Principal Angle* 

The incident angle satisfying the condition of ∆ = 90◦ is defined as the principal angle [45]. The error was proved theoretically to be reduced to obtain the highest precision in determining the optical constants when measured at the principal angle [17]. The spectrum of the principal angle for the Au sample was both theoretically and experimentally investigated in our previous work [46]. In this section, the ellipsometric measurements were performed by the RPAE at a series of incident angles in a range of 55◦ to 80◦ , with an interval of 5◦ , to evaluate the accuracy and stability. The incident angle satisfying the condition of Δ = 90° is defined as the principal angle [45]. The error was proved theoretically to be reduced to obtain the highest precision in determining the optical constants when measured at the principal angle [17]. The spectrum of the principal angle for the Au sample was both theoretically and experimentally investigated in our previous work [46]. In this section, the ellipsometric measurements were performed by the RPAE at a series of incident angles in a range of 55° to 80°, with an interval of 5°, to evaluate the accuracy and stability.

measured reference i i

referen

ce i <sup>1</sup> MARE 100% *<sup>n</sup> x x*

<sup>−</sup> =⋅ × , (6)

i=

1

*n x*

#### 3.1.1. Spectroscopic Measurement 3.1.1. Spectroscopic Measurement

The dielectric function spectra of the Au sample at various incident angles (Figure 2) were determined from the measured ellipsometric parameters with Equation (5). The spectra showed great agreement in most of the wavelength range. On the other hand, discrepancies were observed obviously in some regions, especially in the long-wavelength range. The reference dielectric function was obtained by applying the Model dielectric function [47] and Drude model [48] to the spectra of various incident angles. Accordingly, the spectrum of the principal angle was calculated with the method presented in [46], as shown in Figure 3. The dielectric function spectra of the Au sample at various incident angles (Figure 2) were determined from the measured ellipsometric parameters with Equation (5). The spectra showed great agreement in most of the wavelength range. On the other hand, discrepancies were observed obviously in some regions, especially in the long-wavelength range. The reference dielectric function was obtained by applying the Model dielectric function [47] and Drude model [48] to the spectra of various incident angles. Accordingly, the spectrum of the principal angle was calculated with the method presented in [46], as shown in Figure 3.

**Figure 2.** Dielectric functions of bulk Au determined at six incident angles. **Figure 2.** Dielectric functions of bulk Au determined at six incident angles.

**Figure 3.** Spectrum of the principal angle for bulk Au. **Figure 3.** Spectrum of the principal angle for bulk Au. **Figure 3.** Spectrum of the principal angle for bulk Au.

The RPAE gave two solutions to determine the values of *ψ* and Δ with Equations (3) and (4). Theoretically, the results extracted by the two solutions were expected to be equal. The differences between the two results, defined as δ*ψ* = *ψ*1 – *ψ*2 and δΔ = Δ1 – Δ2, are used generally to evaluate the reliability of measurement. The values of δ*ψ* and δΔ in the spectral range are exhibited in Figure 4. For the incident angles in 65°–80°, the differences between the two sets varied around 0 in the spectral range, which implied good credibility for measurement. Meanwhile, the differences of 55° and 60° were relatively large, especially in the long wavelength range. As indicated in Figure 3, the principal angle increased significantly in the long wavelength range, reaching approximately 80°. Consequently, larger measurement errors occurred at incident angles of 55° and 60° away from the principal angle, leading to the significant discrepancy between the two solutions. The RPAE gave two solutions to determine the values of *ψ* and ∆ with Equations (3) and (4). Theoretically, the results extracted by the two solutions were expected to be equal. The differences between the two results, defined as δ*ψ* = *ψ*<sup>1</sup> – *ψ*<sup>2</sup> and δ∆ = ∆<sup>1</sup> – ∆2, are used generally to evaluate the reliability of measurement. The values of δ*ψ* and δ∆ in the spectral range are exhibited in Figure 4. For the incident angles in 65◦–80◦ , the differences between the two sets varied around 0 in the spectral range, which implied good credibility for measurement. Meanwhile, the differences of 55◦ and 60◦ were relatively large, especially in the long wavelength range. As indicated in Figure 3, the principal angle increased significantly in the long wavelength range, reaching approximately 80◦ . Consequently, larger measurement errors occurred at incident angles of 55◦ and 60◦ away from the principal angle, leading to the significant discrepancy between the two solutions. The RPAE gave two solutions to determine the values of *ψ* and Δ with Equations (3) and (4). Theoretically, the results extracted by the two solutions were expected to be equal. The differences between the two results, defined as δ*ψ*= *ψ*1 – *ψ*2 and δΔ = Δ1 – Δ2, are used generally to evaluate the reliability of measurement. The values of δ*ψ* and δΔ in the spectral range are exhibited in Figure 4. For the incident angles in 65°–80°, the differences between the two sets varied around 0 in the spectral range, which implied good credibility for measurement. Meanwhile, the differences of 55° and 60° were relatively large, especially in the long wavelength range. As indicated in Figure 3, the principal angle increased significantly in the long wavelength range, reaching approximately 80°. Consequently, larger measurement errors occurred at incident angles of 55° and 60° away from the principal angle, leading to the significant discrepancy between the two solutions.

**Figure 4.** Discrepancies of the two sets of ellipsometric parameters at six incident angles in a spectral range. **Figure 4.** Discrepancies of the two sets of ellipsometric parameters at six incident angles in a spectral range. **Figure 4.** Discrepancies of the two sets of ellipsometric parameters at six incident angles in a spectral range.

The MARE values of the dielectric function at different incident angles are given in Table 1. The results demonstrate that the accuracy was dependent on the wavelength and corresponding principal angle. The measured results at 65°, 70°, and 75° turned out to be more accurate than those measured at 55°, 60°, and 80° in a wavelength range of 300−800 nm, which was consistent with the analysis based on the spectrum of the principal angle. The MARE values of the dielectric function at different incident angles are given in Table 1. The results demonstrate that the accuracy was dependent on the wavelength and corresponding principal angle. The measured results at 65°, 70°, and 75° turned out to be more accurate than those measured at 55°, 60°, and 80° in a wavelength range of 300−800 nm, which was consistent with the analysis based on the spectrum of the principal angle. The MARE values of the dielectric function at different incident angles are given in Table 1. The results demonstrate that the accuracy was dependent on the wavelength and corresponding principal angle. The measured results at 65◦ , 70◦ , and 75◦ turned out to be more accurate than those measured at 55◦ , 60◦ , and 80◦ in a wavelength range of 300–800 nm, which was consistent with the analysis based on the spectrum of the principal angle.


**Table 1.** MARE values of the dielectric functions at different incident angles. **Table 1.** MARE values of the dielectric functions at different incident angles.

#### 3.1.2. Monochromatic Measurement 3.1.2. Monochromatic Measurement

The monochromatic measurements at different incident angles were tested by performing 100 repeated measurements at a single wavelength of 350 nm. The real and imaginary parts of the reference dielectric constant at 350 nm were determined as *ε*<sup>1</sup> = −1.09 and *ε*<sup>2</sup> = 5.2, respectively. The principal angle of the Au sample at 350 nm was calculated to be 69.43◦ . Figure 5a,b display the distribution of the measured ellipsometric parameters and determined dielectric constants, respectively, at six incident angles. The data amount at each incident angle in an accurate region (relative error of ±2%, illustrated in Figure 5) is counted and listed in Table 2. The statistical values demonstrated that the results at 70◦ , 65◦ , and 75◦ had more accurate data compared with the others, which was consistent with the theoretical analysis of the principal angle. The monochromatic measurements at different incident angles were tested by performing 100 repeated measurements at a single wavelength of 350 nm. The real and imaginary parts of the reference dielectric constant at 350 nm were determined as *ε*1 = −1.09 and *ε*2 = 5.2, respectively. The principal angle of the Au sample at 350 nm was calculated to be 69.43°. Figures 5a and 5b display the distribution of the measured ellipsometric parameters and determined dielectric constants, respectively, at six incident angles. The data amount at each incident angle in an accurate region (relative error of ± 2%, illustrated in Figure 5) is counted and listed in Table 2. The statistical values demonstrated that the results at 70°, 65°, and 75° had more accurate data compared with the others, which was consistent with the theoretical analysis of the principal angle.

**Figure 5.** Distribution of the 100 (**a**) repeatedly-measured ellipsometric parameters, and (**b**) determined dielectric constants for six incident angles at a wavelength of 350 nm. **Figure 5.** Distribution of the 100 (**a**) repeatedly-measured ellipsometric parameters, and (**b**) determined dielectric constants for six incident angles at a wavelength of 350 nm.


**Table 2.** Amounts of data points in an accurate region shown in Figure 5 in the 100 repeated measurements for different incident angles at a wavelength of 350 nm. **Table 2.** Amounts of data points in an accurate region shown in Figure 5 in the 100 repeated measurements for different incident angles at a wavelength of 350 nm.

The repeated measurement procedure was performed subsequently on two other wavelengths of 550 and 750 nm at different incident angles. The corresponding MAREs of the real part of the dielectric constant with varying incident angles were calculated at three wavelength points, as indicated in Figure 6. The principal angles of the Au sample at 350, 550, and 750 nm were determined to be 69.43◦ , 70.53◦ , and 78.01◦ , respectively. The MARE versus incident angle implies that the measurement exhibited higher accuracy and smaller error with the incident angle close to the principal angle. For example, the MARE at 750 nm decreased significantly with the increasing incident angle, which was attributed to the corresponding principal angle of 78.01◦ . At incident angles of 65◦ , 70◦ , and 75◦ , the MAREs turned out to be relatively small at all three wavelengths in Figure 6, representing the short, middle, and long wavelength parts in the spectral range. Accordingly, the results indicated that these three incident angles enabled accurate measurement for the Au sample and some other typical metals. The repeated measurement procedure was performed subsequently on two other wavelengths of 550 and 750 nm at different incident angles. The corresponding MAREs of the real part of the dielectric constant with varying incident angles were calculated at three wavelength points, as indicated in Figure 6. The principal angles of the Au sample at 350, 550, and 750 nm were determined to be 69.43°, 70.53°, and 78.01°, respectively. The MARE versus incident angle implies that the measurement exhibited higher accuracy and smaller error with the incident angle close to the principal angle. For example, the MARE at 750 nm decreased significantly with the increasing incident angle, which was attributed to the corresponding principal angle of 78.01°. At incident angles of 65°, 70°, and 75°, the MAREs turned out to be relatively small at all three wavelengths in Figure 6, representing the short, middle, and long wavelength parts in the spectral range. Accordingly, the results indicated that these three incident angles enabled accurate measurement for the Au sample and some other typical metals.

**Figure 6.** Values of MARE versus the incident angle at three wavelengths. **Figure 6.** Values of MARE versus the incident angle at three wavelengths.

#### *3.2. Azimuthal Error 3.2. Azimuthal Error*

In the rotating element ellipsometers, the measurement is fundamentally based on the detection of different polarization states, which is usually realized with the rotating elements. Consequently, the azimuthal error of the polarizing element significantly affects the performance of the instrument. In this subsection, we experimentally investigated the effect of the azimuthal error on the accuracy and stability of the results. The initial azimuths of the polarizer and analyzer were adjusted rotationally by a certain angle from the s-axis to act as the azimuthal errors, represented as δ*θ*P and δ*θ*A, respectively. In the rotating element ellipsometers, the measurement is fundamentally based on the detection of different polarization states, which is usually realized with the rotating elements. Consequently, the azimuthal error of the polarizing element significantly affects the performance of the instrument. In this subsection, we experimentally investigated the effect of the azimuthal error on the accuracy and stability of the results. The initial azimuths of the polarizer and analyzer were adjusted rotationally by a certain angle from the s-axis to act as the azimuthal errors, represented as δ*θ*<sup>P</sup> and δ*θ*A, respectively.

#### 3.2.1. Theoretical Analysis 3.2.1. Theoretical Analysis

For the measurement with the azimuthal error *δ*, assuming that the condition *A* = 2*P* is still satisfied, the expression of light intensity in Equation (1) is modified as: For the measurement with the azimuthal error *δ*, assuming that the condition *A* = 2*P* is still satisfied, the expression of light intensity in Equation (1) is modified as:

$$I\_{\delta}(A) = I\_0 + I\_1 \cos(A + \delta) + I\_2 \cos 2(A + \delta) + I\_3 \cos 3(A + \delta) + I\_4 \cos 4(A + \delta). \tag{7}$$

Compared with Equation (2), the four harmonic components are determined as:

Compared with Equation (2), the four harmonic components are determined as:

$$I\_{k\delta} = \frac{2}{\cos k\delta \cdot n} \sum\_{i=1}^{n} I\_{\delta}(A\_i) \cos(kA\_i) = \sec k\delta \cdot I\_k \quad k = 1, 2, 3, 4. \tag{8}$$

Consequently, the ellipsometric parameters obtained in experiment are given as:

$$\begin{aligned} \tan\psi\_{\delta 1} &= \left[\frac{2(I\_1 \cdot \sec\delta + I\_3 \cdot \sec 3\delta - 2I\_2 \cdot \sec 2\delta)}{I\_1 \cdot \sec\delta + I\_3 \cdot \sec 3\delta}\right]^{1/2}, \\ \cos\Delta\_{\delta 1} &= \frac{I\_1 \cdot \sec\delta - 3I\_3 \cdot \sec 3\delta}{\left[2(I\_1 \cdot \sec\delta + I\_3 \cdot \sec 3\delta)(I\_1 \cdot \sec \delta + I\_3 \cdot \sec 3\delta - 2I\_2 \cdot \sec 2\delta)\right]^{1/2}} \end{aligned} \tag{9}$$

and

$$\begin{split} \tan \psi\_{\delta 2} &= \left[ \frac{9(I\_1 \cdot \sec \delta + I\_3 \cdot \sec 3\delta - 2I\_2 \cdot \sec 2\delta)}{2(2I\_1 \cdot \sec \delta + I\_2 \cdot \sec 2\delta + 4I\_4 \cdot \sec 4\delta)} \right]^{1/2} \\ \cos \Delta\_{\delta 2} &= \frac{3(I\_1 \cdot \sec \delta + I\_3 \cdot \sec 3\delta) - 4(I\_2 \cdot \sec 2\delta + 4I\_4 \cdot \sec 4\delta)}{\left[8(I\_1 \cdot \sec \delta + I\_3 \cdot \sec 3\delta)(I\_1 \cdot \sec \delta + I\_3 \cdot \sec 3\delta - 2I\_2 \cdot \sec 2\delta)\right]^{1/2}} .\end{split} \tag{10}$$
