## Abstract

An accurate optical coherent ellipsometer (OCE) is proposed and setup in which a two-frequency paired linear polarized laser beam is integrated with a common-path heterodyne interferometer. This OCE is able to precisely measure the optical properties of scattering specimen by measuring ellipsometric parameters (*ψ*, *Δ*). In the mean time the degree of polarization *P*, and degree of coherence *χ* of incident two-frequency linear polarized laser beam are measured too. In the experiment, both smooth and ground BK7 glass plates were tested in which the optical parameters (*ψ*, *Δ*, *P*, *χ*) were obtained precisely. Comparing with conventional ellipsometers, OCE can characterize scattering specimen precisely and excludes the scattering effect.

© 2008 Optical Society of America

## 1. Introduction

Ellipsometer enables to characterize optical properties of materials and thin films quantitatively via the measurement of the ellipsometric parameters, *ψ* and *Δ*, of the reflected polarized light by specimen [1]. Theoretically, a linear polarized light which can be treated as the combination of one p-polarized and one s-polarized light waves is reflected by the specimen. The complex amplitudes of the reflection coefficients *r*
_{p}, *r*
_{s} of the p- and s-waves respectively are expressed by

Thus the ratio of *r*
_{p} to *r*
_{s}, *ρ*, can be expressed by

(*ψ*, *Δ*) represents the elliptical polarization of the reflected light quantitatively whereas *ψ* is the arctangent of the ratio of the p-amplitude to the s-amplitude and *Δ* is the phase retardation between the two waves. Thus, the optical properties of tested thin film including the refractive index and thickness can be obtained in terms of (*ψ*, *Δ*) simultaneously.

Different setups of the ellipsometers were developed in order to measure (*ψ*, *Δ*) [1–3] precisely. Among these methods, the nulling ellipsometer (NE) is based on PCSA configuration, the sequential arrangement of a polarizer (P), a compensator (C), a specimen (S) and an analyzer (A), in which P and A are rotated in order to find the minimum outgoing intensity for the measurement of (*ψ*, *Δ*). However, the accuracy of NE is limited by the rotating stage. Meanwhile, the speed of measurement is slow because of the nulling condition is required [4]. In contrast, the rotating ellipsometer (RE) [5–7] in which either the polarizer or the analyzer is rotated is setup with the configuration of PSA. Thus, (*ψ*, *Δ*) is determined by means of measuring intensities at different rotation angles. The simple geometry of RE presents the advantage compared with conventional ellipsometers. However, a limited dynamic range on *Δ* (0°<*Δ*<180°) and a limited rotating speed (<100 Hz) are resulted in RE. In a phase modulation ellipsometer (PME) [8–11], a polarizer, a photo elastic modulator (PEM), a specimen and an analyzer are arranged sequentially. The PME is able to be modulated at a high frequency so that the measured light intensity emergent from the PME includes a dc component and a serious of harmonics ac components. Thus, (*ψ*, *Δ*) can be obtained in terms of the amplitudes of the measured dc and the fundamental and second harmonic ac components of the detected signal. Because PME can operate at high speed (≈50 KHz), it is capable of real time measurement. However, a calibration on residual linear birefringence in PEM is required in order to measure (*ψ*, *Δ*) at high accuracy [11]. Similarly, the ambiguity on *Δ* limits the dynamic range in PME too.

Fundamentally, to non-scattering sample, (*ψ*, *Δ*) is able to fully describe the optical properties of a specimen [12] that does not depolarize and decorrelate the incident laser beam. In contrast, Nee *et al.* [12–14] studied the optical properties of rough specimen by NE and has defined the Mueller matrix **M** for a depolarizing tested specimen by

*T*, *P*, *D*
_{v} and *D*
_{u} are the transmittance, the degree of polarization, the cross- and co-polarized depolarizations of the specimen respectively. The depolarization *D* is related to the different depolarization components by

In order to characterize a scattering specimen that may depolarize and decorrele the incident laser beam, an accurate optical coherent ellipsometer (OCE) is proposed and setup. In this OCE, a two-frequency paired linearly polarized laser beam (TPL), a tested specimen and an analyzer are arranged sequentially to construct a TPLSA configuration. The OCE inherently presents the advantages of common-path heterodyne interferometer and is able to provide high signal-to-noise ratio (SNR) and wide dynamic range of the heterodyne signal [15–17]. These result in high precision on (*ψ*, *Δ*) measurements apparently. In the mean time, OCE can measure the depolarization and decorrelation of a laser beam as well as characterize a scattering specimen without being affected by the scattering effect of the specimen.

In Section 2, the working principle of OCE based on Mueller matrix formulation is derived. The experimental verification of OCE being able to measure degree of polarization (DOP) and the degree of coherence (DOC) of a TPL and specimen characteristics are demonstrated and discussed in Section 3 and 4. Finally, the features of OCE and its advantages compared with conventional ellipsometers are analyzed and summarized in Section 5.

## 2. Principle

#### 2.1 The Stokes vector of TPL

A TPL can be produced by using a single frequency laser in conjunction with an electro-optic modulator (EOM) driven by a saw tooth signal at frequency *δω* [18]. When a linear polarized light whose polarization is 45° to the x-axis is transmitting through an EOM, the phase retardation Δ*φ* of the linearly polarized p- and s-waves is generated as Δ*φ*=δ*ωt*. Then the Stokes vector of the two-frequency and highly correlated paired linearly polarized laser beam can be calculated by

*I*
_{0} is the intensity of the single frequency laser and *δω*=*ω*
_{p}-*ω*
_{s}, where *ω*
_{p} and *ω*
_{s} are the temporal frequencies of the p- and s- waves respectively.

#### 2.2 Theory of optical coherent ellipsometer (OCE)

The setup of OCE is shown in Fig. 1. A TPL whose Stokes vector **S _{in}** is given by Eq. (6) is incident onto a tested specimen whose Mueller matrix

**M**is given by Eq. (4). For the analyzer whose angle is adjusted at

*A*degrees from the x-axis, the output intensity sensed by a photo detector can be expressed by

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}={\left(\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right)}^{\mathrm{t}}{T}_{a}\left(\begin{array}{cccc}1& \mathrm{cos}2A& \mathrm{sin}2A& 0\\ \mathrm{cos}2A& {\mathrm{cos}}^{2}2A& \mathrm{sin}2A\mathrm{cos}2A& 0\\ \mathrm{sin}2A& \mathrm{sin}2A\mathrm{cos}2A& {\mathrm{sin}}^{2}2A& 0\\ 0& 0& 0& 0\end{array}\right)$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\xb7T\left(\begin{array}{cccc}1& -P\mathrm{cos}2\psi & 0& 0\\ -P\mathrm{cos}2\psi & 1-2{D}_{v}& 0& 0\\ 0& 0& P\mathrm{sin}2\psi \mathrm{cos}\Delta & P\mathrm{sin}2\psi \mathrm{sin}\Delta \\ 0& 0& -P\mathrm{sin}2\psi \mathrm{sin}\Delta & P\mathrm{sin}2\psi \mathrm{cos}\Delta \end{array}\phantom{\rule{.2em}{0ex}}\right){I}_{0}\left(\begin{array}{c}1\\ 0\\ \mathrm{cos}\delta \omega t\\ -\mathrm{sin}\delta \omega t\end{array}\right)$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}={I}_{\mathrm{dc}}+{I}_{\mathrm{ac}}\mathrm{cos}\left(\delta \omega t+\Delta \right),$$

*T*
_{a} and **A** are the transmittance and the Mueller matrix of the analyzer respectively. **D** is a 4×1 row Stokes vector of the photo detector and the superscript t means transpose.

We define *γ* as the ratio of the measured *I*
_{ac} to *I*
_{dc}. From Eqs. (7b) and (7c), *γ* is given by

Rearrange Eq. (8a), we have

The ellipsometric parameter *Δ* can be obtained by measuring the phase difference in Eq. (7a) directly via a lock-in amplifier (LIA). *γ*(*A*) can be measured from Eq. (8a). The second ellipsometric parameter *ψ* and the degree of polarization *P* can be obtained by least square fitting between the measured *γ*(*A*) and Eq. (8b).

#### 2.3 Degree of coherence (DOC)

DOC is a characteristic of a modulated laser, and is defined as the ratio of the modulated intensity to the average intensity. For an ideal TPL such as **S**
_{in} of Eq. (6), DOC=1. If an ideal TPL passes through a depolarizing sample, the DOC *χ* of the output laser will be affected by the sample. We analyze the polarization and coherence of this output laser by scanning an analyzer after the sample as shown in Fig. 1. The DOC of the laser after the analyzer is given by the *γ* of Eq. (8a). *χ* is the maximum value of *γ* when

is satisfied. Thus, *χ* is given by

*χ* is a function of *P* and *ψ*. Fig. 2 shows the computer simulation of Eq. (10) for different *P* and *ψ*. We include also *ψ*>45° in the figure for the transmission through a sample. If we have a non-depolarizing sample, *P*=1, then the reflected laser beam is still highly coherent with *χ*=1. In contrast, if the sample is depolarizing (*P*<1), then *χ*<*P*<1 and the output laser loses some coherency as well as some polarization. *χ* is seriously degraded when *ψ* is close to 0° or 90° as shown in Fig. 2.

## 3. Experimental setup

To demonstrate the capabilities of the OCE as described previously, three different experiments are conducted: (1) measurement of the reflection by a glass plate (2) phase retardation measurement and (3) measurement of depolarization by rough glass plates. As shown in Fig. 1, the beat frequency of the TPL is chosen as *δω*=2 KHz. The analyzer A selects the components of the reflected beam along its polarization direction and the optical heterodyned signal is generated at photo detector D by Eq.(7).

#### 3.1 Reflection of incident TPL by thick BK7 glass plate

The sample is a thick and smooth BK7 glass plate whose refractive index is 1.515 (at 632.8 nm) and the thickness is 8 mm. Reflection from a smooth surface has *P*=1 and *Δ*=0 or π. The reflected TPL by the air/glass interface at different incident angles *θ* results in different values of *ψ* accordingly. For a fixed incident angle *θ*, the *γ* associated with different azimuth angle of *A* are obtained by measuring the intensity of *I*
_{dc} and *I*
_{ac} of the detected signal via a digital voltmeter simultaneously. *ψ* is obtained by least square fitting between the measured *γ*(*A*) and Eq. (8b).

#### 3.2 Phase retardation of Soliel Babinet compensator (SBC)

The Mueller matrix of a perfect SBC, **M _{c}**, can be expressed by

Γ is the phase retardation between the p- and s-waves. Since the SBC is assumed perfect, *P*=1, *ψ*=45° and *Δ*=Γ. Figure 3 shows the optical setup of the experiment in which the TPL transmits through the SBC normally and then through the analyzer A_{1} whose azimuth angle is fixed at 45°. In the mean time, a reference beam is generated by using a beam-splitter (BS) and a secondary analyzer A_{2} whose azimuth angle is also set at 45°. The signals from both beams are fed into a lock-in amplifier to measure the phase between them. The intensities *I*
_{sig} from the signal beam and *I*
_{ref} from the reference beam are

Where *T*
_{BS} and *R*
_{BS} are respectively the transmittance and reflectance of the BS as shown in Fig. 3. Thus, the phase difference between *I*
_{sig} and *I*
_{ref} is equal to Γ. Theoretically, there is one extra phase difference equal to Δ*Φ*=*nδωl*/*c* between the p- and s-waves in OCE due to their slightly different temporal frequencies of the p- and s-waves. However, Δ*Φ*≈0 because of *δω*=2 KHz, *c*=3×10^{-10} cm/sec and *l*≈20 cm in this experiment. Similarly the phase difference *Φ*
_{BS} induced by the BS can be calibrated when no specimen is tested in Fig. 3.

#### 3.C Degree of polarization (DOP) by a scattered glass plate

Let **M _{sca}** represent the Mueller matrix of a scattering specimen. For normal transmission through an isotropic sample,

*ψ*=45 and

*Δ*=0, and the Mueller matrix is diagonalized. Thus, the depolarized Stokes vector of the emerging laser beam after transmitting through a scattering specimen can be expressed by

Where *T*
_{sca}, *P*
_{sca}, are the transmittance and the polarization of the scattering specimen, **S _{in}** and

**S**are the Stokes vectors of the incident and emerging TPL respectively. If the specimen consists of two scattering glass plates stacked together, then the effective polarization for the normal transmission through the set is

_{sca}The DOP of the emerging TPL by the scattering specimen is also *P*
_{T}.

Figure 4 shows the experiment for measuring the DOP for normal transmission through a scattering sample. The azimuth angle of the analyzer A is set at 45°. Two ground glass plates (G1 and G2) of different roughnesses are prepared for the experiment. Each ground glass plate is produced by sanding a smooth glass plate with emery paper only on one side of the glass plate. The ground glass plate reduces the DOP of the output TPL by the same amount as the polarization of the sample.. According to Eq. (8a), when the analyzer is adjusted to 45° and TPL is normal incidence onto the specimen (*ψ*=45°), then

To consider a scattering specimen which is combination of two ground glasses G1 and G2 stacked together, *P*
_{T}=*P*
_{1}
*P*
_{2} according to Eq. (14).

## 4. The experimental results

#### 4.1 Thick glass plate

In Fig. 1, TPL is obliquely incident onto a smooth BK7 glass plate that does not induce depolarization to the output laser beam. Then, *ψ* can be obtained separately by least square fitting of Eq. (8b) with the measured *γ* at different orientation of the analyzer. In Fig. 5 the discrete dots are the measured data and the solid lines are the best-fit curves according to Eq. (8). The best-fit *ψ* are obtained as 10.63°, 16.38°, 25.68° and 36° with the confidence R^{2}≥0.999 for Figs. 5(a), 5(b), 5(c) and 5(d) at different incident angles respectively. These experimental results verify the principle of the OCE properly.

#### 4.2 Phase retardation of SBC

As described in section 3.2 together with Fig. 3, the result of the measured phase retardation of a SBC for different thicknesses is shown in Fig. 6. The consistence between the measured and calibrated data from the manufacture (Special Optics, 8-400-UNCTD) is shown obviously. The error percentage in this measurement is less than 1% (Table 1). The dynamic range of phase retardation in this experiment is 0° to 360° experimentally.

## 5. DOP of TPL by ground glass plate

As shown in Fig. 5, a perfect TPL is normally incident onto a glass plate (*ψ*=45°) and then through an analyzer with the azimuth angle *A* adjusted at 45°. The measured *P* is the ratio of *I*
_{ac} to *I*
_{dc} as given by Eq. (15). Table 2 shows the depolarization effect of three ground glass plates (G1, G2 and G3) where G3 is produced by stacking G1 and G2 together. Then, the depolarization *D* are obtained by the relation of *D*=1 - *P*. In Table 2, the direct measurement *P** for the DOP of G3 agrees well with the prediction *P* by multiplication of the DOP of G1 and G2. The small difference of 0.0008 might be caused by the misalignment during the measurement or by the limited solid angle of photo detector in the interferometer.

## 6. Conclusions

In this research, an accurate optical coherent ellipsometer (OCE) is proposed and setup and the experimental verifications of OCE are demonstrated. Fundamentally, an OCE integrates an optical heterodyne interferometer with a two-frequency paired linear polarized laser beam that provides excellent modulation in the TPL-SA configuration. The simple operation principles for measuring (*ψ*, *Δ*) by OCE have been devised and three experiments were performed. One experiment measures *ψ* for the reflection from a smooth glass plate by measuring the DOC (*γ*) of light passing through the analyzer. The measured *γ* at different analyzer angles fit very well to the prediction by Eq. (8) to obtain *ψ*. The second experiment measures *Δ* for a Soleil-Babinet compensator by measuring the phase difference between the sample beam and the reference beam using a lock-in amplifier. The measured *Δ* agree excellently with the calibrated *Δ* provided by the manufacture. The third experiment measures the depolarizations for two single scattering samples and also for the two samples stacked together. The measured depolarization for the stacked sample agrees well with the prediction from the measured depolarizations for the two separate samples. In conclusion, all three experiments agree well with the expectations. The devised OCE provides a fast and clean modulation in phase and very simple algebra for extracting the ellipsometric parameters. This system is capable of real time measurement of (*ψ*, *Δ*) by using the Zeeman laser as the TPL.

## Acknowlegment

This research was partially supported by Nation Science Council of Taiwan through grant # NSC 96-2221-E-010-002-MY2

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