Fast and full range measurements of ellipsometric parameters using a 45&#x00B0; dual-drive symmetric photoelastic modulator

Li Kewu; Zhang Rui; Jing Ning; Chen Youhua; Zhang Minjuan; Wang Liming; Wang Zhibin

doi:10.1364/OE.25.005725

1. Introduction

The ellipsometric parameters $ψ$ $(0^{\circ} \leq ψ \leq 90^{\circ})$ and $Δ$ $(- 180^{\circ} \leq Δ \leq 180^{\circ})$ express the amplitude ratio and phase difference between p- and s-polarization, respectively, when light is reflected from or transmitted through the interface between different media [1,2]. The simultaneous measurements of the ellipsometric parameters $ψ$ and $Δ$ provide a wealthy of information about the sample under investigation. In a standard ellipsometry, the physical properties of a thin film sample, such as the refractive index, the extinction coefficient, and the film thickness, can be determined from $ψ$ and $Δ$ by using the Fresnel equations [3]. Extensive applications of the thin film thickness monitoring in optics (windows, mirrors, beam splitters, filters, etc.), solar panels, display devices, semiconductor devices, and optoelectronic devices have been implemented [4–6]. Furthermore, the ellipsometric parameters measurements are also used as effective detection methods in biosensing, surface magneto-optical effect detection, and nanostructure characterization [7–9].

Modern Ellipsometry configurations include rotating-analyzer ellipsometry (RAE), rotating-compensator ellipsometry (RCE), and phase-modulation ellipsometry (PME) [1,10]. The RAE optical configuration is simple, but the $Δ$ range limited to only 0°–180° can be measured, and the measurement errors increase at near $Δ = 0^{\circ}$ and $180^{\circ}$ . Using the RCE configuration, $ψ$ and $Δ$ can be measured over the full range, and the measurement sensitivity is uniform, so this ellipsometry configuration have been successfully commercialized. However, the frequency of the mechanical rotating analyzer or compensator (typically tens of hertz) is slow, which limits the ellipsometry measurement rate. Also, the artifacts such as the system instability and light beam drift caused by the mechanical rotation are difficult to eliminated. With the PME configuration, a photoelastic modulator (PEM) with typical working frequency of 50 kHz or multiple PEMs with different working frequencies are employed, which can achieve fast or real-time measurements.

For a single PEM, two optical configurations have been reported, one that has the fast axis of PEM parallel to the x direction and the analyzer oriented at −45°, and the other with the fast axis of PEM and the analyzer are oriented at 45° and 0°, respectively [11,12]. In the first one, $Δ$ can be measured over the full range, but the range of $ψ$ limited only 0°–45° can be measured, and the measurement errors increase at near $ψ = 0^{\circ}$ and 45°. In the second one, the measurement range of $Δ$ is half the full range $(0^{\circ} \leq Δ \leq 180^{\circ})$ , and the measurement errors increase at near $Δ = 0^{\circ}$ and 180°. Full range and uniformly sensitive measurements of $ψ$ and $Δ$ can be obtained by performing the two measurements. However, this would be ineffective in real-time monitoring. For multiple PEMs configurations, two or four PEMs are used, and the entire Mueller matrix of the sample can be measured [13,14]. Thus $ψ$ and $Δ$ can be fast measured over the full range, but the control process of multiple PEMs is complex, and the instrument cost is increased.

Recently, we demonstrated a 45° dual-drive symmetric PEM device and showed that a pure standing wave and pure traveling wave modulation mode were achieved [15,16]. For the pure traveling wave modulation mode, the retardation magnitude is a constant, and the modulation axis performs circular motion at half of the PEM working frequency. This is very similar to a rotating wave plate or compensator, but the modulation frequency is much higher, and there is no need for any mechanical rotating parts, thus the artifacts of rotating a wave plate or compensator, such as low working frequency, the detection light beam drift, and the system instability, can be avoided. We apply the modulation mode in an ellipsometry system, in which the PEM operation control and data processing are based on a field programmable gate array (FPGA). The in situ, fast, and full range measurements of $ψ$ and $Δ$ can be realized.

2. Design and principle

2.1 Optical scheme

The ellipsometric parameter measurement system is schematically shown in Fig. 1. The incident light is collimated, and the polarization of the light is oriented at 45° after passing through a polarizer oriented at 45°. Then, the light is reflected from a sample and modulated by a 45° dual-drive symmetric PEM, finally passing through an analyzer oriented at 0° to be detected.

Fig. 1 Schematic diagram of the ellipsometry system based on a 45° dual-drive symmetric PEM working on the pure traveling modulation mode. A and B are the piezoelectric actuators, C is the symmetric photoelastic crystal, LC, FPGA, AD and PC represents a LC (inductor-capacitor) oscillator circuit, a field programmable gate array, an analog to digital converter, and notebook PC, respectively. A, B, and C constitute the PEM’s optical head. FPGA and LC constitute the PEM’s controller.

Download Full Size | PDF

The Stokes vector and Mueller matrix approach are used to analyze the measurement system. The Stokes vector of the incident light oriented at 45° can be written as [17]

S_{i n} = I_{0} {[\begin{matrix} 1 & 0 & 1 & 0 \end{matrix}]}^{T},

where _{$I_{0}$} is the light intensity after the polarizer. For an isotropic sample, the Muller matrix is given by [10,18]

M_{s} = \frac{1}{2} [\begin{matrix} 1 & - N & 0 & 0 \\ - N & 1 & 0 & 0 \\ 0 & 0 & C & S \\ 0 & 0 & - S & C \end{matrix}],

Where

N = \cos 2 ψ

,

S = \sin 2 ψ \sin Δ

, and

C = \sin 2 ψ \cos Δ

._$ψ$ and _$Δ$ are the two ellipsometric parameters, which are defined by the ratio _$ρ$ of reflection coefficients _{$r_{p}$} and _{$r_{s}$} for p- and s-polarized light, and

ρ = \frac{r_{p}}{r_{s}} = \tan ψ e^{i Δ}

[1].

For the 45° dual-drive symmetric PEM controller, a FPGA provides two channel square wave signals whose frequencies and phases can be regulated, and a LC oscillator circuit translates the square waves signals into sinusoidal high voltages to drive the PEM to work. The amplitudes of the two driving voltages can be changed by adjusting the power supply. When the amplitudes of the stress waves driven by the actuators A and B are equal and their phase difference is_{$\frac{π}{2}$}, the 45° dual-drive symmetric PEM works on the pure traveling wave mode, in which the modulation axis performs a circular motion, and the retardation magnitude is a constant [15,16]. This polarization characteristic can be described using the Muller matrix as

M_{P E M} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos^{2} 2 θ + \sin^{2} 2 θ \cos δ & \cos 2 θ \sin 2 θ (1 - \cos δ) & - \sin 2 θ \sin δ \\ 0 & \cos 2 θ \sin 2 θ (1 - \cos δ) & \sin^{2} 2 θ + \cos^{2} 2 θ \cos δ & \cos 2 θ \sin δ \\ 0 & \sin 2 θ \sin δ & - \cos 2 θ \sin δ & \cos δ \end{matrix}],

The azimuth angle of the modulation axis _{$θ = \frac{2 π f_{0} t}{2}$}, _{$f_{0}$} is the PEM resonance working frequency. _$δ$ is the retardation magnitude that is proportional to the driving voltage amplitude.

The Stokes vector of the light detected by the detector is expressed as

S_{o u t} = M_{A} M_{P E M} M_{s} S_{i n},

where _{$M_{A}$} is the Muller matrix of the analyzer oriented at 0° [19]. Substituting Eqs. (1)–(3) and _{$θ = \frac{2 π f_{0} t}{2}$} into Eq. (4), we obtain the first element of the Stokes vector _{$S_{o u t}$}, which is the time-dependent intensity of the light that the detector can detect [20]

\begin{array}{l} I_{(t)} = \frac{I_{0}}{4} [2 - N (1 + \cos δ) + 2 S \sin δ \sin (2 π f_{0} t) - N (1 - \cos δ) \cos (4 π f_{0} t) + \\ C (1 - \cos δ) \sin (4 π f_{0} t)], \end{array}

2.2 Data processing

In our system, the FPGA not only provides the driving signal for the PEM but also performs to the analog to digital converter (ADC) sampling control. The DC term, the first harmonic term, and the second harmonic term of the light intensity signals are also extracted in the FPGA based on the digital phase lock technology. As shown in Fig. 2, the FPGA provides the PEM driving signals, controls the ADC sampling, and generates sine and cosine reference sequences of harmonic terms.

Fig. 2 Drive signals (a), reference signals generated by the FPGA and the digital signal converted by the ADC (b). A and B represent drive signals for the piezoelectric actuators A and B, respectively, _{$f_{s}$} is the ADC’s sampling frequency.

Download Full Size | PDF

As shown in Fig. 2, the ADC sampling frequency is set as _{$f_{s}$}. Thus, sampling points for the first harmonic term in a full period are _{$Q = f_{s} / f_{0}$} (generally,_{$Q \geq 3$}). For q integer periods, the total signal sampling points are _{$M = q \times Q$}, and the input signals_{$I_{(t)}$}are converted to digital signals.

\begin{matrix} \begin{array}{l} I_{(k)} = K \frac{I_{0}}{4} [2 - N (1 + \cos δ) + 2 S \sin δ \sin \frac{2 π k}{Q} - \\ N (1 - \cos δ) \cos \frac{4 π k}{Q} + C (1 - \cos δ) \sin \frac{4 π k}{Q}] \end{array} & (k = 0, 1, ..., M - 1), \end{matrix}

where K is a photoelectric conversion coefficient of the detector. For the digital signal sequences, the DC term can be obtained from Eq. (6) by direct summation.

V_{D C} = \sum_{k = 0}^{M - 1} I_{(k)} = M K \frac{I_{0}}{4} [2 - N (1 + \cos δ)],

Multiplying the digital signal sequences in Eq. (6) by the first harmonic sine and cosine reference sequences _{$R_{1 f} s_{(k)} = \sin (\frac{2 π k}{Q})$} and _{$R_{1 f} c_{(k)} = \cos (\frac{2 π k}{Q})$}, respectively. Adding all the terms, the AC signals are eliminated.

{\begin{matrix} I_{1 s} = Σ_{k = 0}^{M - 1} I_{(k)} R_{1 f} s_{(k)} = M K \frac{I_{0}}{4} S \sin δ \cos φ \\ I_{1 c} = Σ_{k = 0}^{M - 1} I_{(k)} R_{1 f} c_{(k)} = M K \frac{I_{0}}{4} S \sin δ \sin φ \end{matrix},

where _$φ$ is the phase difference between the first harmonic signal and the first harmonic reference, which represents the phase delay between the modulation signal and the driving voltage signal. The phase delay depends on the PEM and drive circuits, and its value can be precisely determined by calibration. Similarly, multiplying the digital signal sequences in Eq. (6) by the second harmonic sine and cosine reference sequences _{$R_{2 f} s_{(k)} = \sin (\frac{4 π k}{Q})$} and _{$R_{2 f} c_{(k)} = \cos (\frac{4 π k}{Q})$}, and adding all terms, we obtain

{\begin{matrix} I_{2 s} = Σ_{k = 0}^{M - 1} I_{(k)} R_{2 f} s_{(k)} = M K \frac{I_{0}}{8} [N (1 - \cos δ) \sin 2 φ - C (1 - \cos δ) \cos 2 φ] \\ I_{2 c} = Σ_{k = 0}^{M - 1} I_{(k)} R_{2 f} c_{(k)} = M K \frac{I_{0}}{8} [N (1 - \cos δ) \cos 2 φ + C (1 - \cos δ) \sin 2 φ] \end{matrix},

As shown in Fig. 2, the first and second harmonic references are synchronous, so the phase delay of the second harmonic signal is two times to that of the first harmonic signal, which is _{$2 φ$} in Eq. (9). Combining Eqs. (7)–(9), the optimum choice for _$δ$ is _{$\frac{π}{2}$}, so that these important trigonometric functions of ellipsometric parameters _$ψ$and _$Δ$such as as _{$S = \sin 2 ψ \sin Δ$}, _{$C = \sin 2 ψ \cos Δ$}, and _{$N = \cos 2 ψ$} can be achieved simultaneously, which provide accurate measurements of _$ψ$and _$Δ$over the full range from 0° to 90° and from −180° to 180°, respectively. The two ellipsometric parameters are solved out

{\begin{matrix} \begin{array}{l} Δ = \arctan [\frac{(1 - \cos δ) I_{1 S}}{2 \sin δ \cos φ (I_{2 C} \cos 2 φ - I_{2 S} \sin 2 φ)}] \begin{matrix}  \end{matrix} \\ or \begin{matrix}  \end{matrix} \arctan [\frac{(1 - \cos δ) I_{1 C}}{2 \sin δ \sin φ (I_{2 C} \cos 2 φ - I_{2 S} \sin 2 φ)}] \end{array} \\ ψ = \frac{1}{2} \arctan [\frac{I_{2 C} \cos 2 φ - I_{2 S} \sin 2 φ}{\cos Δ (I_{2 C} \cos 2 φ + I_{2 S} \sin 2 φ)}] \end{matrix} .

Since the data processing in the FPGA is run in parallel, Eqs. (7)–(9) can be processed at the same time. Then, the data are sent to the PC to complete the calculation of the ellipsometric parameters, which guarantees the data processing speed and precision.

3. Experiment

The experimental setup is built based on a 45° dual-drive symmetric PEM, as illustrated in Fig. 1. A low noise laser diode operating at the wavelength of 650 nm is employed as the light source. Both the polarizer and analyzer are Glan—Taylor polarizers with an extinction ratio greater than 10⁵:1. An Altera EP3C FPGA is used to provide the PEM driving signal, control a fast and precise 12 bit ADC clock frequency, and then complete the digital signal processing.

3.1 Calibrations of the retardation magnitude _$δ$ and the phase delay _$φ$

The pure traveling wave modulation mode of the 45°dual-drive symmetric PEM is applied in the ellipsometry system. For this modulation mode, the retardation is a constant, and it is preferable to choose it as _{$\frac{π}{2}$} in this system. Using the methods reported in references [16], without any sample, the laser beam passes through the polarizer, then directly passes through the 45° dual-drive symmetric PEM and the analyzer to be detected. Thus, the pure traveling wave mode can be achieved by regulating the driven voltage signals of the actuators A and B. With the FPGA digital signal processing, the parameters _$δ$ and _$φ$ can be precisely calibrated.

The resonant working frequency of the 45° dual-drive symmetric PEM is 49.956kHz. The sampling frequency of the ADC is set to 3.2 MHz, and 50 integer periods of the first harmonic and 100 integer periods of the second harmonic are chosen for one single data digital processing output. The calibration results (Fig. 3) show that the pure traveling modulated mode are achieved and the retardation magnitude δ is about _{$\frac{π}{2}$}, when the amplitudes of the driving voltage signals of the actuators A and B are 159 V and 150 V, respectively, and the phase difference of the two driving voltages is 87.49° (channel _{$V_{A}$} to channel _{$V_{B}$}).

Fig. 3 The pure traveling modulated mode calibration, (a) Waveform observation via an oscilloscope, channels _{$V_{A}$} and _{$V_{B}$} represent the driving voltage signal of the actuators A and B, respectively. Channel I is the modulation signal. (b) The retardation magnitude and phase delay calibration.

Download Full Size | PDF

As _{$V_{A}$} goes ahead of _{$V_{B}$} by 87.49°, the modulation axis performs circular motion in the counter clockwise direction [16]. The average value of the retardation magnitude is _{$\bar{δ} = 1 .5712 \begin{matrix} \end{matrix} rad$}, and the standard deviation is _{$σ_{δ} = 3 .0e-4 \begin{matrix} \end{matrix} rad$}. The average value of phase difference is _{$2 φ = 76 {.267}^{\circ}$}, and the standard deviation is _{$σ_{2 φ} = {0.006}^{\circ}$}.

3.2 Experimental results and discussion

A 115.4 ± 0.4 nm thickness SiO₂ thin film sample (provided by Ellitop Scientific Co.,Ltd. China) was tested to identify the properties of the experimental setup. The sample is shown in Fig. 4(a). The SiO₂ thin film was deposited on the Si wafer. The incident angle was chosen as 70°, which is close to the Brewster’s angle of the Si wafer [3]. The waveform observation via an oscilloscope is also shown in the Fig. 4(b).

Fig. 4 SiO₂/Si thin film sample (a) and waveform observation via an oscilloscope (b). _{$V_{A}$}, _{$V_{B}$}, and I have the same meaning as in Fig. 3.

Download Full Size | PDF

Comparing the waveform shown in Fig. 4(b) to that in Fig. 3(a), we can observe that _{$V_{A}$} and _{$V_{B}$} are kept very well, and the ellipsometric parameters of the SiO₂/Si thin film sample are loaded in the modulated signal. For the digital processing, the consecutive measurements of the ellipsometric parameters _$ψ$ and _$Δ$ are shown in Fig. 5.

Fig. 5 Consecutive measurements of the ellipsometric parameters, _$Δ$(a) and _$ψ$(b).

Download Full Size | PDF

According to Fig. 5, the acquisition rate of the ellipsometric parameter is about 1ms/point. The measurement values are presented in Table 1. A commercial spectroscopic ellipsometry (model ESS01, Ellitop, China) with the RCE configuration was also used to measure the sample. The incidence angle was also set as 70°, and the measurement results at 650 nm are also shown in Table 1.

Table 1. Measured ellipsometric parameters

View Table

The values of the ellipsometric parameters measured by our system are close to those obtained by the commercial ellipsometry. For the 115.4 ± 0.4 nm thickness SiO₂/Si thin film sample, the ellipsometric parameters ( $ψ$ , $Δ$ ) are calculated as (49.18 ± 0.32°, 81.36 ± 0.10°) by using the Fresnel equations with substituting into the refractive indexes and extinction coefficients of SiO₂ and Si [3,21,22], when the light source is operated at the wavelength of 650 nm, and the incidence angle is set as 70°. The mean values of $ψ$ and $Δ$ given by our system are closer to the theoretical calculations than those given by the commercial ellipsometry. The value of $ψ$ given by our system is acceptable, but $Δ$ has a deviation of 81.36°-80.741° = 0.619°, which corresponds to a relative error of 0.619°/81.36° = 0.76%. According to Eq. (10), it can be judged that the deviation is mainly caused by the deviation of the PEM’s retardation magnitude. Besides, the deviation of the azimuth angles of the polarizers and the incident angle will also introduce measurement errors. But all the deviations are expected to reduce as much as possible after carefully and accurately calibrating.

Furthermore, the corresponding standard deviations in Table 1 indicate that our system has repeatability and sensitivity nearly one order higher, which is a substantial improvement over the commercial system. In addition, the measurement time interval of our system is in milliseconds. Therefore, our system realizes sensitive and fast measurements of the ellipsometric parameters. As there are no mechanical adjustments in the measurement processing, our system is also appropriate for the in situ measurement application.

Similar to other general PEMs, the photoelastic crystal of the 45° dual-drive symmetric PEM is also made of isotropic optical materials, such as fused silica, CaF₂, ZnSe and Si, etc. The PEM can operate in a wide spectral range from vacuum UV to terahertz so that our system holds a great promise for the realization of wide spectroscopic ellipsometry measurements, especially in the infrared and terahertz regions. However, the high measurement speed may be limited by the spectral measurement technique. This is where we're planning to take this research next.

4. Summary

We demonstrate an ellipsometric parameter measuring system based on a 45° dual-drive symmetric PEM that works on a pure traveling modulation mode. The PEM control and data processing are performed in a FPGA. The trigonometric functions of ellipsometric parameters _$ψ$and _$Δ$ such as_{$\sin 2 ψ \sin Δ$}, _{$\sin 2 ψ \cos Δ$}, and _{$\cos 2 ψ$} can be measured simultaneously, which provide accurate measurements of _$ψ$ and _$Δ$ over full range of 0°–90° and (−180°)–180°, respectively. The experimental results for the tested SiO₂/Si thin film sample show that the standard deviation related the repeatability and sensitivity of this system are at 10⁻³°, and the acquisition rate is 1ms/point. There is no need for any mechanical adjustment of the operation in the measurement process. Thus, This method overcomes the shortages of the traditional RAE, RCE and PME configurations, and realizes the in-stiu, fast and full range measurements of the ellipsometric parameters. Furthermore, the system is promising for the automation integration.

Funding

International Science and Technology Cooperation Special (ISTC) (2013DFR10150); National Science Foundation (NSF) (11647089, 61471325, 61505179, 61505180).

References and links

1. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (John Wiley & Sons, 2007).

2. R. M. A. Azzam, “Stokes-vector and Mueller-matrix polarimetry,” J. Opt. Soc. Am. A 33(7), 1396–1408 (2016). [CrossRef] [PubMed]

3. H. Tompkins, and E. A. Irene, Handbook of Ellipsometry (William Andrew, 2005).

4. K. B. J. Rodenhausen, D. Schmidt, T. Kasputis, A. K. Pannier, E. Schubert, and M. Schubert, “Generalized ellipsometry in-situ quantification of organic adsorbate attachment within slanted columnar thin films,” Opt. Express 20(5), 5419–5428 (2012). [CrossRef] [PubMed]

5. R. Petkovšek, J. Petelin, J. Možina, and F. Bammer, “Fast ellipsometric measurements based on a single crystal photo-elastic modulator,” Opt. Express 18(20), 21410–21418 (2010). [CrossRef] [PubMed]

6. Y. Battie, L. Broch, A. En Naciri, J.-S. Lauret, M. Guézo, and A. Loiseau, “Diameter dependence of the optoelectronic properties of single walled carbon nanotubes determined by ellipsometry,” Carbon 83, 32–39 (2015). [CrossRef]

7. R. S. Moirangthem, Y. C. Chang, and P. K. Wei, “Ellipsometry study on gold-nanoparticle-coated gold thin film for biosensing application,” Biomed. Opt. Express 2(9), 2569–2576 (2011). [CrossRef] [PubMed]

8. N. Maccaferri, A. Berger, S. Bonetti, V. Bonanni, M. Kataja, Q. H. Qin, S. van Dijken, Z. Pirzadeh, A. Dmitriev, J. Nogués, J. Åkerman, and P. Vavassori, “Tuning the magneto-optical response of nanosize ferromagnetic Ni disks using the phase of localized plasmons,” Phys. Rev. Lett. 111(16), 167401 (2013). [CrossRef] [PubMed]

9. K. Thyagarajan, C. Santschi, P. Langlet, and O. J. Martin, “Highly improved fabrication of Ag and Al nanostructures for UV and nonlinear plasmonics,” Adv. Optical Mater. 4(6), 871–876 (2016). [CrossRef]

10. E. Garcia-Caurel, A. De Martino, J. P. Gaston, and L. Yan, “Application of spectroscopic ellipsometry and Mueller ellipsometry to optical characterization,” Appl. Spectrosc. 67(1), 1–21 (2013). [CrossRef] [PubMed]

11. S. N. Jasperson and S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40(6), 761–767 (1969). [CrossRef]

12. O. Acher, E. Bigan, and B. Drevillon, “Improvements of phase‐modulated ellipsometry,” Rev. Sci. Instrum. 60(1), 65–77 (1989). [CrossRef]

13. G. E. Jellison and F. A. Modine, “Two-modulator generalized ellipsometry: experiment and calibration,” Appl. Opt. 36(31), 8184–8189 (1997). [CrossRef] [PubMed]

14. O. Arteaga, J. Freudenthal, B. Wang, and B. Kahr, “Mueller matrix polarimetry with four photoelastic modulators: theory and calibration,” Appl. Opt. 51(28), 6805–6817 (2012). [CrossRef] [PubMed]

15. K. W. Li, Z. B. Wang, L. M. Wang, and R. Zhang, “45° double-drive photoelastic modulation,” J. Opt. Soc. Am. A 33(10), 2041–2046 (2016). [CrossRef] [PubMed]

16. K. W. Li, L. M. Wang, R. Zhang, and Z. B. Wang, “Modulation axis performs circular motion in a 45° dual-drive symmetric photoelastic modulator,” Rev. Sci. Instrum. 87(12), 123103 (2016). [CrossRef] [PubMed]

17. A. Zeng, F. Li, L. Zhu, and H. Huang, “Simultaneous measurement of retardance and fast axis angle of a quarter-wave plate using one photoelastic modulator,” Appl. Opt. 50(22), 4347–4352 (2011). [CrossRef] [PubMed]

18. A. Laskarakis, S. Logothetidis, E. Pavlopoulou, and M. Gioti, “Mueller matrix spectroscopic ellipsometry: formulation and application,” Thin Solid Films 455, 43–49 (2004). [CrossRef]

19. Y. W. Liu, G. A. Jones, Y. Peng, and T. H. Shen, “Generalized theory and application of Stokes parameter measurements made with a single photoelastic modulator,” J. Appl. Phys. 100(6), 063537 (2006). [CrossRef]

20. H. Satozono, “Elimination of artifacts derived from the residual birefringence of a phase modulator for circular dichroism by retardation domain analysis,” Opt. Lett. 40(7), 1161–1164 (2015). [CrossRef] [PubMed]

21. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55(10), 1205–1208 (1965). [CrossRef]

22. D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of si, ge, gap, gaas, gasb, inp, inas, and insb from 1.5 to 6.0 ev,” Phys. Rev. B 27(2), 985–1009 (1983). [CrossRef]

Ellipsometric parameter	Our system			Commercial spectroscopic ellipsometry (@650 nm)
Ellipsometric parameter	Mean value (°)	Standard deviation (°)	Acquisition rate (s/point)	Mean value (°)	Standard deviation (°)	Acquisition rate (s/point)
$ψ$	49.028	0.0027	0.001	49.64	0.02	1.6
$Δ$	80.741	0.0070	0.001	79.86	0.04	1.6

Fast and full range measurements of ellipsometric parameters using a 45° dual-drive symmetric photoelastic modulator

Abstract

1. Introduction