## Abstract

A circularly polarized ellipsometer was developed to enable real-time measurements of the optical properties of materials. Using a four photo-detector quadrature configuration, a phase modulated ellipsometer was substantially miniaturized which has the ability to achieve a high precision detection limit. With a proven angular resolution of 0.0001 deg achieved by controlling the relative positions of a triangular prism, a paraboloidal and a spherical mirror pair, this new ellipsometer possesses a higher resolution than traditional complex mechanically controlled configurations. Moreover, the addition of an algorithm, *FTA* (fault tolerance algorithm) was adopted to compensate for the imperfections of the opto-mechanical system which can decrease system measurement reliability. This newly developed system requires only one millisecond or less to complete the measurement task without having to adopt any other modulation approach. The resolution achieved can be as high as 4x10^{−7} RIU (refractive index unit) which is highly competitive when compared with other commercially available instruments. Our experimental results agreed well with the simulation data which confirms that our quadrature-based circularly polarized ellipsometer with *FTA* is an effective tool for precise detection of the optical properties of thin films. It also has the potential to be used to monitor the refractive index change of molecules in liquids.

© 2011 OSA

## 1. Introduction

*Ellipsometry* is a useful significant optical measurement technique which can be used especially to detect the optical attributes of samples non-invasively [1,2]. In 1887, the first ellipsometric system was developed by Drude *et al*. For many years, the famous mathematical “Drude model” was proposed to verify the relationship between several optical constants of materials. A large number of ellipsometry related studies were undertaken which included protein detection and protein binding assay in solids or liquids [3,4], two-dimensional imaging measurements [5–7], and *in situ* spectral ellipsometry [8,9]. Over the past decades, many commercially available ellipsometers such as nulling ellipsometers, photometric ellipsometers, and interferometric ellipsometers, have been developed. Photometric ellipsometry can be further categorized into either a rotating optical element type or a phase modulation configuration type [1,10]. The first type ellipsometry includes such as *rotating analyzer ellipsometry*, *rotating analyzer ellipsometry with compensator*, and *rotating compensator ellipsometry*. The minimum measuring time is only limited by the rotating speed of the optical elements (e.g. analyzers, compensators). The second type ellipsometry includes *phase modulation ellipsometry* which has the advantage of providing an accurate phase modulation for ellipsometric measurements. Usually, the configuration utilizes liquid crystal molecules or a photo-elastic modulator. A significant disadvantage of this type of system is that it is hard to convert into a portable measurement tool for use outside of the laboratory.

Quadrature interferometry can distinguish the phase difference between *p*- and *s*- polarization. Previous research showed that it can be used in an AVID^{TM} system (Advanced Vibrometer/Interferometer Device [11]), a heterodyne surface plasmon device, and an interferometer enabled phase interrogation technique tool [12,13]. To evaluate the two ellipsometry parameters Ψ and Δ, a four photo-detector quadrature configuration was adopted to effectively avoid the disadvantages associated with typical PSA (polarizer-sample-analyzer) ellipsometers and phase-modulated ellipsometers. In summary, starting with the goal to develop a portable ellipsometer for point-of-care (** PoC**) applications, we developed a circularly polarized ellipsometer which has the advantages of having an easy to implement optical configuration and a full range of ellipsometric measurements.

In working to develop a ** PoC** (point-of-care) ellipsometer, we integrated ellipsometry with quadrature interferometry. In our previous work, we developed a multifunctional optical metrology system called OBMorph

^{TM}which consisted of an interferometer, an ellipsometer, and a surface plasmon resonance sensor enabled angular interrogations and phase interrogations [14,15]. Our simulations showed that we can precisely control the incidence angles to estimate the effective refractive index of samples. It can facilitate the ellipsometric inverse calculation by utilizing obtained effective refractive index of samples. Moreover, a

*FTA*(fault tolerance algorithm) was adopted to suppress the errors associated with imperfections [16,17]. Both theoretical and experimental results verified the advantages of our circularly polarized ellipsometer with a

*FTA*addition. The resolution of our circularly polarized ellipsometer was identified to be 4x10

^{−7}RIU (refractive index unit) which is precise enough to measure specific interactions of human CRP (C-Reactive Protein) and anti-CRP. In addition, it is also precise enough to measure different concentrations of biomarkers, such as the tuberculosis inhibitor

*Dihydrofolate reductase*(DHFR) which was successfully demonstrated in a previous research [17]. In summary, our circularly polarized ellipsometer with its simple optical arrangement, can improve real-time measurements while offering high sensitivity. This new ellipsometer configuration has the potential for broader applications in the future than traditional ellipsometric configurations.

## 2. Design and principle

#### 2.1. Optical arrangement configuration

Our circularly polarized ellipsometer used a 0.3mW laser diode at 635nm wavelength as the light source. A varying incidence angle system and a quadrature configuration were adopted (Fig. 1
). A polarizer 1 oriented at 45 deg was used to make sure the *p-* and *s-* polarizations had equal intensity levels and equal initial phases. The polarization directions were defined as shown in Fig. 1. A 50:50 ratio non-polarized beamsplitter was used to make the *p*- and *s*- polarized light beams impinge onto the sample simultaneously. The varying incidence angle configuration was composed of a triangular prism, a paraboloidal mirror and a spherical mirror. The paraboloidal mirror changed the incident angle with a positioning of the triangular prism when the paraboloidal mirror focal point was set to the center of the sensing region (Fig. 1). To precisely position the triangular prism, a linearly motorized stage with encoder possessing a 0.12406μm per count was controlled using a LabVIEW program. Our set-up easily approached a 0.0001 deg accuracy while maintaining a wide incidence ranging between 18 deg to 78 deg. The returning light from the spherical mirror passed through a quadrature configuration which includs a quarter-wavelength plate oriented on a fast axis at 45 deg and along four polarizers at 0 deg, 45 deg, 90 deg, and 135 deg orientations. The arranged incident polarization at 45 deg can be treated as the summation of a *p-* polarization and a *s-* polarization light beam with no phase difference between them. Considering that a common path configuration carries both *p-* and *s-* polarization responses, a set of orthogonal signals was obtained in the circularly polarized ellipsometer. The mathematical details are discussed further in Eq. (4).

#### 2.2. Circularly polarized ellipsometer with fault tolerance algorithm

In the past, homodyne interferometry adopted a three-step phase-shifting method to measure phase difference [18]. In 1987, a five-step digital phase-shifting algorithm was developed by P. Hariharan that used the five known phase shifting terms to determine unknown phase data [19]. More specifically, the relationship of a phase difference $\Delta \varphi $ and modulation term β solved from the intensity of the light beam detected as **I _{1}~I_{5}** leads to

A rotating analyzer method developed by R. Naraoka in 2005 focused on the phase detection of surface plasmon resonance utilizing modulation of the four different polarized states. Its high repeatability and high precision characteristics made it easy to achieve a 10^{−7} RIU (refractive index unit) accuracy [20]. Our circularly polarized ellipsometer is more suitable than a rotating analyzer configuration since there is no need to rotate the analyzers. Quadrature configurations have been applied before as a phase shifting tool in SPR (surface plasmon resonance) devices for many years [12,13]. The Jones transformation matrix for a quarter-wavelength plate of our system (in Fig. 2
) when ${\text{M=45}}^{\text{o}}$ and $\text{\delta =}\text{\pi}/\text{4}$can be expressed as

*p-*and

*s-*polarizations have the same amplitude as represented by ${\text{E}}_{\text{x}}\propto \text{exp}\left(\text{i}{\varphi}_{\text{0p}}\right)$ and ${\text{E}}_{\text{y}}\propto \text{exp}\left(\text{i}{\varphi}_{\text{0s}}\right)\text{,}$ where ${\varphi}_{\text{0p}}$ and ${\varphi}_{\text{0s}}$ denote the initial phase difference of the

*p-*and

*s-*polarizations. The ${\varphi}_{\text{p}}$ and ${\varphi}_{\text{s}}$ denote the phase difference caused by the samples. We denote the parameter $\Delta =\left({\varphi}_{\text{0p}}\text{+}{\varphi}_{\text{p}}\right)\text{-}\left({\varphi}_{\text{0s}}\text{+}{\varphi}_{\text{s}}\right),$ to represent the relative phase difference between the

*p-*and

*s-*polarizations. In our double-pass ellipsometer, Δ is one of the two ellipsometry parameters. Furthermore, r

_{p}and r

_{s}denote the reflection coefficient of the

*p-*and

*s-*polarizations respectively. For the configuration shown in Fig. 2, we can obtain the calculated electric field of each detector by substituting the azimuthal angle θ at 0 deg, 45 deg, 90 deg, and 135 deg into Eq. (3)

_{1}~I

_{4}are:

Compared with a single-pass configuration ellipsometer, the experimental data undertook the following changes: (1) tanΨ became (tanΨ)^{2}; and (2) Δ became 2Δ in our double-pass configuration. Considering the signs of the two terms of the intensity difference, a sine term $\left({\text{I}}_{\text{1}}{\text{-I}}_{\text{3}}\right)\text{,}$ and a cosine term $\left({\text{I}}_{\text{2}}{\text{-I}}_{\text{4}}\right)$, we can extend the full dynamic range of Δ from -π to π as shown by

For either$\left({\text{I}}_{\text{1}}{\text{-I}}_{\text{3}}\right)\left({\text{I}}_{\text{2}}{\text{-I}}_{\text{4}}\right)\text{>0,}$and$\left({\text{I}}_{\text{1}}{\text{-I}}_{\text{3}}\right)\text{<0,}$ or $\left({\text{I}}_{\text{1}}{\text{-I}}_{\text{3}}\right)\left({\text{I}}_{\text{2}}{\text{-I}}_{\text{4}}\right)\text{<0,}$and$\left({\text{I}}_{\text{1}}{\text{-I}}_{\text{3}}\right)\text{>0,}$ Δ calculated from Eq. (5) must add a π term to make sure the range of Δ is correct. Assuming that $\left|{\text{r}}_{\text{p}}\right|/\left|{\text{r}}_{\text{s}}\right|=\text{tan\Psi ,}$ Eq. (6) contains another ellipsometry parameter Ψ where the sign of the tan Ψ is defined to be positive.

A circularly polarized ellipsometer which has the important advantage of having no modulation during measurement will always be superior to a traditional ellipsometer. We actually obtained the ellipsometry parameters by the four intensity levels according to Eqs. (5) and (6) for static measurements. According to Eq. (4), a sine term $\left({\text{I}}_{\text{1}}{\text{-I}}_{\text{3}}\right)\text{,}$ and a cosine term $\left({\text{I}}_{\text{2}}{\text{-I}}_{\text{4}}\right)$removing the unnecessary DC signals, take the form of Lissajous PQ signals. The way to produce the Lissajous signals needed before the measurement was to rotate the HWP (half-wavelength plate) from the initial 0 deg polarization to π. This HWP rotation changes the phase difference of the sine and the cosine terms from 0 to 2π such that complete Lissajous signals are obtained. We demonstrated that some imperfections may affect the circularly polarized ellipsometer and result in the appearance of elliptical PQ signals which can only be caused by mis-adjustment or mis-alignment. More specifically, according to Eq. (4), less than perfect circular type Lissajous signals can only be caused by optical components located behind the second NPBS (Fig. 1). In other words, less than perfect circular Lissajous signals are the result of effects not appearing in the common path. That is, different reflection coefficients of the *p*- and *s*- polarizations while passing the optical components in the system will not cause the Lissajous signals deviating from the circle [17]. The *FTA* (fault tolerance algorithm) can be viewed as a way to make sure various experimental tests are compared using an equal baseline. More specifically, the sensitivity over different parts of the Lissajous PQ signals will not be identical if the Lissajous figure is not a circle. The higher the ratio between the long and the short axes of a Lissajous figure, the higher the sensitivity difference. In other words, there is lower sensitivity for the data obtained along the sharp curve of an ellipse and higher sensitivity along the smooth side of an ellipse. For calibrating the eccentric ellipse cases, the following derivation shows the results of the *FTA* method. Equation (7) defined each position of a general ellipse with a $2\times 2$ rotational matrix including a major radius${\text{R}}_{\text{A}}$, a minor radius${\text{R}}_{\text{B}}$, eccentric center$\left({\text{B}}_{\text{p}}{\text{,B}}_{\text{q}}\right)$, and starting phase ${\varphi}_{\text{0}}$. Considering the $\widehat{p}$ and $\widehat{q}$ coordinates, ${\text{A}}_{\text{p}}$ and ${\text{A}}_{\text{q}}$ were individual amplitudes of the PQ signals with each phase retardation, ${\varphi}_{\text{p}}$ and ${\varphi}_{\text{q}}$ (Fig. 3
). Comparing Eqs. (7) and (8), the relationship between $\left({\text{A}}_{\text{p}}{\text{,A}}_{\text{q}}\text{,\theta}\right)$ and $\left({\text{R}}_{\text{A}}{\text{,R}}_{\text{B}}\text{,}{\varphi}^{\text{*}}\right)$ are independent of ${\varphi}_{\text{0}}$ and can be denoted by Eq. (9).

Given that ${\varphi}_{\text{q}}\text{=0}$, Eq. (10) shows a linearization form with the five parameters during an inverse matrix operation, and assuming ${\text{C}}_{\text{p}}{\text{=A}}_{\text{p}}\text{sin}\left({\varphi}_{\text{p}}\right)$ and ${\text{D}}_{\text{p}}{\text{=A}}_{\text{p}}\text{cos}\left({\varphi}_{\text{p}}\right)$, we obtained

The steps of the fault tolerance algorithm can be summarized as follows:

- 1. The resulting intensity values were obtained using a LabVIEW program. We had the measured PQ signals $\left({\text{p}}_{\text{i}}{\text{,q}}_{\text{i}}\right)$. The initial phase${\varphi}_{\text{i}}$ can be retrieved as shown by
- 2. We can use $\left({p}_{i}\text{,}{q}_{i}\text{,}{\varphi}_{i}\right)$directly solve the five parameters $\left({\overline{\text{B}}}_{\text{p}},{\overline{\text{B}}}_{\text{q}},{\overline{\text{C}}}_{\text{p}},{\overline{\text{C}}}_{\text{q}},{\overline{\text{D}}}_{\text{q}}\right)$ using Eq. (11). During the iterative calculation, the final characterized matrix can be obtained. Herein, the calculated square sum of $\text{cos}\varphi $ and $\text{sin}\varphi $ is necessary to be one as a result of the normalization,$$\left[\begin{array}{c}\text{cos}\varphi \\ \text{sin}\varphi \end{array}\right]\text{=}\frac{\left[\begin{array}{ccc}{\overline{\text{C}}}_{\text{q}}& {\overline{\text{D}}}_{\text{q}}& {\overline{\text{B}}}_{\text{q}}\\ {\overline{\text{C}}}_{\text{p}}& \text{0}& {\overline{\text{B}}}_{\text{p}}\end{array}\right]\left[\begin{array}{c}\text{q}\\ \text{p}\\ \text{1}\end{array}\right]}{\left|\left[\begin{array}{ccc}{\overline{\text{C}}}_{\text{q}}& {\overline{\text{D}}}_{\text{q}}& {\overline{\text{B}}}_{\text{q}}\\ {\overline{\text{C}}}_{\text{p}}& \text{0}& {\overline{\text{B}}}_{\text{p}}\end{array}\right]\left[\begin{array}{c}\text{q}\\ \text{p}\\ \text{1}\end{array}\right]\right|}\text{.}$$
- 3. The resulting five parameters can correct the measured intensity values by an optimal re-mapping conversed into a circle in Lissajous space. Figure 4 shows the successful adjustment of our experimental data into a modified unit circle calibrated by
*FTA.*The following results, seen after*FTA*modification, allowed our imperfect system to retrieve the exact ellipsometry parameters without an over precise optical alignment.

## 3. Results and discussions

To verify the proposed circularly polarized ellipsometer, the experimental results included a solid phase and a liquid phase measurements. It is known that conventional ellipsometers are easily subjected to great fluctuations caused by incidences from air passing through layers of a liquid sample. A prism-coupled scheme can be used to approach liquid phase measurements precisely. We obtained ellipsometry parameters in liquids by utilizing a half-spherical prism made by a high index material, SF2 (n = 1.64) instead of an original flat sensing platform as seen in Fig. 1. More specifically, adding a half-spherical prism converted the original ellipsometer for dry samples (Fig. 1) into an ellipsometer for liquid samples. Real-time measurements of our ellipsometer enable us to monitor fast reactions in the range of a few milliseconds and to detect the temporal optical properties of film layers. More specifically, the influences of the inaccurate positions of the optical devices (e.g. rotated waveplates, polarizers, and compensators) can easily result in measurement errors [1,21–23]. Compared with traditional ellipsometers, our circularly polarized ellipsometer can minimize the detection time, limited only by the capability of the data acquisition facility.

To continue measuring the response of a refractive index, a serial gradient concentration of an alcohol solution was prepared by evaporation. The refractive index extended from 1.332 (pure water) to 1.3556 (pure alcohol). We used a commercially available refractive index meter (Kyoto Electronics Manufacturing Co., RA-130) to quantify the real-time index change. Based on a specific angular set-up (e.g. Brewster angle for non-liquid phase measurements and surface plasmon resonance angle for liquid samples) the obvious variation of the ellipsometry parameters versus the angular spectrum was facilitated to determine the optical properties when an inverse calculation was processed. For example, the surface plasmon enhanced ellipsometric measurement is one such case where a substantial change in Δ when the surface plasmon (SP) exited [24]. In our design, we developed a double-pass configuration sensing platform which enhanced the change of Δ twice with more sensitivity. Figure 5
demonstrates the measurement of Δ in liquids. We obtained two different trends from the raw data (dark triangle) and from the data after calibration with the *FTA* (circle). The solid curve was simulated by a commercial program (Film Wizard Software^{TM}, v.9.0.4) with optimal surface plasmon angle at 65 deg and with a chip layer composed of 50 nm Au and 1 nm Cr. The results after calculation with *FTA* shown in Fig. 5 matched the simulations well. In other words, it demonstrates that the *FTA* can correct each intensity level to reduce the imperfections while adjusting the opto-mechanical system. Considering the sensitivity of our circularly polarized ellipsometer, we can see the sharpest linear region of the response located near$n=1.3523~1.3556$. The sensitivity is denoted by a linear region of the maximum response change with a proper noise level at 0.005 deg. The phase difference was multiplied by a triple noise level, and then divided by the response change. The calculated sensitivity of our ellipsometer was identified to be as high as 4x10^{−7} RIU. Compared with a general phase surface plasmon device, our circularly polarized ellipsometer based on its ability to detect small refractive index changes in sample solutions, demonstrates high potential for use in monitoring interactions in liquid phase environments [25].

The sensitivity of the instruments depends on the accurate resolution of the angular control. Previous research measured several angular positions including the maximum and minimum reflectance of air and water layers based on a 50nm Au chip (Fig. 6
). Results showed that the minimum calculated count value was 64498 for the maximum reflectance of air layer (38.4 deg), and the maximum count was 166901 for the minimum reflectance of water layer (63.8 deg). According to the design of a paraboloidal mirror for a quadratic function, the count values can be converted into an incident angle and expressed as $\text{\theta}=$
$\text{0 .000184908282\eta}$
$\text{+26 .235498261818}$, with range from 38 deg to 63 deg where η denotes the encoded count value. We confirmed the angular resolution of 0.0001 deg [17]. Figure 6 shows the experimental set-up of the prism type when the *p-* polarized light beam impinged from a prism (SF2, $\text{n}=\text{1 .64}$) through a 1 nm Cr layer and a 50 nm Au layer to the interface of the metal and dielectric sample solution. The surface plasmon (SP) dispersion relationship is defined as follows: $\mathrm{sin}\zeta =\mathrm{Re}\sqrt{{\epsilon}_{2}{\epsilon}_{3}/\left({\epsilon}_{2}+{\epsilon}_{3}\right){\epsilon}_{1}},$ where ${\epsilon}_{1}$ denotes the relative permittivity of the prism, ${\epsilon}_{2}$ denotes the relative permittivity of the metal layer including a golden layer and a chromium layer, ${\epsilon}_{3}$ denotes the relative permittivity of the sample layer, and ζ denotes the specific angle with the minimum reflectance. A perfect linear relationship focusing on $\mathrm{sin}\zeta $ versus the effective refractive index of the sample layer (N) (Fig. 6) can be found. In addition, the angular control of our system is accurate when the experimental results and simulation are fitted together as shown in Fig. 6. We successfully observed that the various surface plasmon exiting tests with the refractive index of the outer sample layer (glucose solution) set at 1.332 (50 ug/mL), 1.3359 (200 ug/mL), 1.3386 (400 ug/mL), 1.3404 (600 ug/mL), and 1.3505 (1000 ug/mL). According to the above results, the observation of the effective refractive index of sample layers can directly be estimated by detecting the angle ζ of the minimum reflectance on the basis of the clear linear relationship between sinζ and effective refractive index of samples (N.) It is useful to pursue the following analysis of an inverse calculation. This approach provided us with a good way to monitor the real-time index change of the sample.

Similarly, we prepared two types of samples which were deposited on a 50 nm Au/1 nm Cr and 46 nm Au/4 nm Ti/40 nm ITO (Indium tin oxide) upon the base of the SF2 ($\text{n}=\text{1 .64}$) individually. Figure 7
shows the ellipsometric measurements Ψ and Δ, with its incidence from 40 deg to 65 deg at each step of a 5 deg increase in a non-liquid phase environments. Figure 7(a) indicates the results of the Sample #1 (50 nm Au/1 nm Cr), where the dark circle and diamond represent the Δ and Ψ obtained. The measurement results of the Sample #2 (46 nm Au/4 nm Ti/40 nm ITO) is shown in Fig. 7(b). For comparison of the experimental results, a commercially available nulling-type ellipsometer EP^{3} (Nanofilm Co.) was used to provide the reference data, whereas a four-zone method was adopted to measure the ellipsometry parameters. The circle and diamond represent the experimental data measured by EP^{3} as shown in Figs. 7(a) and 7(b). According to the variations of the ellipsometry parameters, Δ typically possesses more sensitivity than Ψ. Figure 7 shows the results measured by our system which matched very well with the simulated results. In summary, our newly developed circularly polarized ellipsometer offers a reliable measurement method for practical applications.

## 4. Conclusions

We designed a new circularly polarized ellipsometer based on quadrature configuration which combines a useful *FTA* algorithm which together can enhance system reliability. Our system possesses a precise incident angle controlled scheme and a double-pass configuration which offers more sensitivity. To overcome imperfections of an opto-mechanical configuration, the *FTA* was successfully incorporated which can correct misalignment induced measurement errors. Therefore, our circularly polarized ellipsometer offers more advantages than other conventional ellipsometer configurations. The minimum measurement time of our system is only limited by the data acquisition facility which allows for more real-time ellipsometric measurements. Our new design has high potential to be used as a portable instrument. In addition, our experimental results show that the obtained sensitivity can be as high as 4x10^{−7} RIU. We showed that our ellipsometric configuration can be used to measure Ψ and Δ in liquids or non-liquids.

## Acknowledgments

This work was supported by the Taiwan National Science Council (NSC) through Project NSC 98-2627-E-002-003.

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