## Abstract

For quality control in high volume manufacturing of thin layers and for tracking of physical and chemical processes, ellipsometry is a common measurement technology. For such kinds of applications we present a novel approach of fast ellipsometric measurements. Instead of a conventional setup that uses a standard photo-elastic modulator, we use a 92 kHz Single Crystal Photo-Elastic Modulator (SCPEM), which is a LiTaO3 crystal with a size of 28 × 9 × 4 mm. This small, simple, and cost-effective solution also offers the advantage of direct control of the retardation via the current amplitude, which is important for repeatability of the measurements. Instead of a Lock-In Amplifier, an automated digital processing based on a fast analog to digital converter controlled by a highly flexible Field Programmable Gate Array is used. This and the extremely compact and efficient polarization modulation allow fast ellipsometric testing where the upper limit of measurement rates is mainly limited by the desired accuracy and repeatability of the measurements. The standard deviation that is related to the repeatability +/–0.002° for dielectric layers can be easily reached.

©2010 Optical Society of America

## 1. Introduction

Ellipsometry plays an important role in many applications like thin coatings, photovoltaic devices, semiconductor devices, flat panel displays, optoelectronic devices, and biological and chemical engineering. Ellipsometry measures the polarizing effect of a sample via the ellipsometric angles ψ and Δ in order to deduce basic physical parameters such as the thickness and refractive index of optical layers [1]. For this the polarization of the probe light beam needs to be changed in a defined manner during the measurement. How this is done divides ellipsometers into three categories, namely with slow (based on rotating wave-plates [1,10,11]), medium (with liquid crystals retarders [3]), and fast polarization modulation (based on a PEM – Photo-Elastic Modulator [2,4,5,8,12]). The first method produces extremely accurate results +/–0.001° at very low sampling rates (several seconds for one sample). The second method is faster with less accuracy, and the third, to which our setup belongs, currently achieves sampling rates of up to 1 kHz with accuracy +/–0.01°. The modulator is in this case composed of one piece of glass that is excited on a resonance frequency with one or two actuators [6,7]. The resulting mechanical oscillation induces a modulated birefringence in the glass, via the photo-elastic effect, and hence a modulated polarizing effect. Combinations of different ellipsometric methods are also possible [14].

The ellipsometric parameters *ψ* and Δ can be calculated from the 0th, 1st, and 2nd harmonics of the signal of the photo diode (Section 2). These are usually measured with a Lock-In Amplifier.

The main advantage of our setup is the usage of a single piezo-electric crystal as a modulator, which in comparison to a conventional PEM, which is composed of at least two glued components, is more favorable in terms of size, price, control, and handling [9,13]. The crystal is made of LiTaO_{3}, with dimensions of 28 × 9.5 × 4 mm in *x*-, *y*-, and *z*-coordinates. The light travels along the z-axis, which is the optical axis (this crystal from the crystal symmetry group 3m is uniaxial). The electrodes are on the *y*-surfaces and a longitudinal *x*-oscillation is excited. The voltage amplitude for half wave retardation amplitude for 633 nm is ~2V.

This solution approach enables easier and more direct control of optical retardation as there is no transmission of the mechanical oscillation from an actuator to the optical modulator.

The evaluation is digital, as first demonstrated in [2], where a spectroscopic ellipsometer based on a classical PEM is presented. Due to the limitations of electronics and due to the use of a lamp (as necessary for spectroscopic ellipsometry) the accuracy and measurement velocity is much lower than in the system presented here. Further we refer to recent work [5], where a detailed error analysis for a digital (Fourier) evaluation is presented and applied to a birefringence measurement, based again on a classical PEM, and similar to an ellipsometric measurement.

Our setup uses custom-made electronics, which is also very important for driving the SCPEM as well as for data acquisition and processing. It is based on a Field Programmable Gate Array (FPGA), which replaces the conventional Lock-In Amplifier usually used in PEM-based ellipsometers and enables high sampling rates with high flexibility.

This approach realizes high sampling rates with sufficient repeatability with a compact design. Here one must distinguish between absolute or relative accuracy and repeatability. The latter, which makes an evaluation based on the constancy of the signals for a continuous measurement on a constant sample, is decisive for monitoring applications for which our system is especially suited due to the high sampling rate. As the system is mainly intended for the measurement of fast temporal or spatial changes in layer thickness or its optical properties, it is therefore more important to have high repeatability than high absolute accuracy.

We mention that it is straightforward to extend this method to spectroscopic ellipsometry [2] and, with a synchronized pulsed light source, to imaging ellipsometry, as done with a conventional PEM in [4]. In the latter case the angle-dependent natural birefringence of the crystal must be taken into account [9].

## 2. Theoretical background of fast ellipsometry

Figure 1 shows the optical setup of a typical ellipsometric measurement with a photo-elastic modulator. The polarization of the light from the low noise laser source is oriented at 45° with respect to the axis perpendicular to the figure plane. The optical axis (z-axis) of the modulator is parallel to the laser beam in the case of ideal adjustment. The x-axis of the crystal is parallel to the paper plane.

The modulator changes the phase of the p-polarization with respect to the s-polarization by an angle (or retardation) *δ*, which varies according to:

*δ*

_{0}is the retardation dc-value or off-set (depending on pre-stresses and on the parallelism between the light beam and the optical axis of the crystal),

*δ*

_{1}is the amplitude of the 1st harmonic of the retardation course, and Ω is the modulator frequency

The ellipsometric angles *ψ* and Δ are defined for a reflection as

*R*

_{p}and

*R*

_{s}are the complex reflection coefficients for p- and s-polarized light, respectively. Their measurement within the setup of Fig. 1 uses a polarization-modulated light beam. The reflected beam passes an analyzer and hits a detector that records the temporal dependency of the resulting intensity. For a proper sample without any depolarizing effect the temporal transmission course

*T*through the analyzer is now given by

*= Δ +*

_{S}*δ*

_{0}. If now the dc-value component

*I*

_{0}and the first two harmonics components

*I*

_{1,2}of this function are calculated, one obtains the following relations: (

*J*

_{0,1,2}are the 0th, 1st, and 2nd order Bessel functions):

By adjusting the retardation amplitude to *δ*
_{1} = *δ _{B}* = 2.4048…rad, which corresponds to the first zero of

*J*

_{0}, the ratios of the first harmonic to the DC component and the second harmonic to the DC component are given by:

In order to calculate the ellipsometric angles from the ratio of the particular components it is useful to introduce the following intensity ratio:

*ψ*holds only for

*δ*

_{1}=

*δ*, whereas that for Δ

_{B}_{S}holds for all

*δ*

_{1}.

To judge now the sensitivity of the measurement on errors of *I*
_{0}, *I*
_{1}, *I*
_{2}, and *δ,* the partial derivatives of Δ, and *ψ* as functions of *δ*, *I*
_{0,1,2} can be calculated. Figures 2
and 3
show these as functions of Δ and *ψ*. Obviously *ψ*-values near 45° are difficult to measure, since all partial derivatives of *ψ* show a singularity at *ψ* = 45°. Small inaccuracies in the measurement of the harmonics *I*
_{0,1,2} or a small deviation of *δ*
_{1} from the desired value then cause large errors.

## 3. Experimental setup

The laser source is an ultra low noise laser diode (RMS noise 0.06% in the bandwidth from 10 Hz to 10 MHz, wavelength 635 nm, 4 mW). The extinction rates of the polarizers are approximately 1:10^{5}. We used several samples: glass, sapphire, polished aluminum, copper, steel, and zinc.

#### 3.1 Calibration of retardation

To directly measure the retardation amplitude and to calibrate the SCPEM one has to check the light intensity *I* going through parallel polarizers with the SCPEM in between, tilted by 45° around the optical path. It is sufficient to track the dc-value *I*
_{0} in dependence on *δ*
_{1} [6]:

*δ*

_{1}is exactly the first zero of the 0th Bessel function (

*J*

_{0}(

*δ*

_{1}) = 0 →

*δ*

_{1}=

*δ*= 2.4048…),

_{B}*I*

_{0}takes exactly 50% of the value measured at

*δ*

_{1}= 0.

#### 3.2 Control of retardation

A piezo-electric crystal is a linear system generating an alternating current when excited with a harmonic electrical field. For a constant excitation frequency the amplitude of deformation and electrical current *i*
_{1} is directly proportional to the voltage amplitude *u*
_{1} of excitation. Since the retardation amplitude *δ*
_{1} is linearly dependent on the strain amplitude, which of course is linearly dependent on the deformation amplitude, the amplitude of retardation and of current must be linearly related for a fixed excitation frequency (Fig. 4
).

This linear relation allows control of the retardation via control of the current. When the excitation frequency *f* is tuned to any resonance frequency *f*
_{R}, where a sharp resonance peak is found, current and retardation are extremely sensitive to small changes in frequency, which must therefore be controlled with high precision. However the linear relation between the current and retardation amplitude is constant only within the resonance bandwidth *f*
_{FWHM} (typically *f*
_{R}/*f*
_{FWHM} = ~5000):

*i*

_{1}are of utmost importance to keep the retardation amplitude

*δ*

_{1}at the desired value.

#### 3.3 SCPEM driver and measuring unit

A schematic diagram of the driver and the measuring units is presented in Fig. 5 .

The role of the driver unit is to keep the retardation amplitude of the modulator fixed. For the dynamic ellipsometric measurements it is very important to control the retardation amplitude of the modulator very precisely. How the retardation amplitude affects the final results can be seen from Figs. 2(a) and 3(a), which show the influence of variation in initial retardation on ellipsometric angles. The retardation amplitude of the SCPEM is directly proportional to the electrical current. Within the resonance bandwidth the factor between both values is constant (Eq. (9)). The resonance peak of the SCPEM used in our experiments is quite narrow (relative FWHM is in the range of 10^{−4}). Therefore the electronics must first precisely control the frequency and second stabilize the current amplitude. This is realized by a digital measurement of current amplitude and phase. The information on amplitude is used for current stabilization and retardation adjustment. The phase information is used to lock the electronic clock frequency to the crystal resonance frequency with high relative accuracy (better than 10^{−6}).

In the measuring unit the photodiode signal is converted to a digital signal by using a fast and precise 14 bit analog to digital converter (ADC). Signal components that correspond to the DC, first harmonic, and second harmonic contribution are separated from the initial signal by digital signal processing. The time that is needed for a single measurement directly depends on the acquisition time. It is determined by the ADC clock frequency and the number of captured samples per single measurement. Therefore in order to achieve a lower acquisition time, the number of sampling points has to be decreased and/or the sampling frequency has to be increased. However due to the high frequency noise of the system that comes from the photodiode, amplifiers, and ADC, there is a limit to the sampling frequency. Above this limit the repeatability of the system deteriorates. Furthermore the number of sampling points of course directly influences the accuracy and repeatability. Therefore there is a trade-off between the repeatability of the measurements and acquisition time.

## 4. Results and discussion

Typical intensity courses for different dielectric and metallic surfaces are shown in Fig. 6
clearly showing the first and second harmonic components of the photodiode signal. From the upper left graph to the bottom right the share of the first harmonic component *I*
_{1} is increasing. As clearly shown by Eq. (7), the ellipsometric angle Δ depends on *I*
_{1}/*I*
_{2}. A higher share of the first harmonic *I*
_{1} means a lower Δ (for 90° < Δ < 180°). Therefore for the dielectric material Δ is close to 180°, and for the metals Δ is much smaller.

Examples of series of consecutive measurements of Δ angle for dielectric surfaces are presented in Fig. 7 . The lower and upper two graphs represent measurements taken at acquisition times of 0.3 ms and 20 ms respectively. Obviously a longer acquisition time leads to better repeatability, as expected. From the graphs it is also clear that there is no significant drift in measurements and therefore the repeatability can be well represented by the standard deviation of the measurement.

The effect of acquisition time on standard deviation is summarized in Table 1 . Clearly the standard deviation increases with decreasing acquisition time, which can in principle be reduced to the period time of the SCPEM-oscillation, i.e. to nearly 10µs. A further reduction is possible by the usage of an SCPEM with higher frequency; up to 1 MHz can be realized rather easy, corresponding to acquisition times of 1µs.

Measured Δ- and *ψ*-values and the corresponding standard deviations are shown in Table 2
. Regarding standard deviation there is a significant difference between metal and dielectric surfaces. The standard deviation of angle Δ is smaller in the case of dielectric surfaces. This is directly related to the absolute value of these angles. From Fig. 2(a) it is clear that the influence of initial retardation variation *δ*
_{1} is larger for angles around 150° (typical for metals) in comparison to the angles close to 180°, which are typical for reflection from dielectric surfaces. The influence of variations of the ratio *I*
_{1}/*I*
_{2} do not differ significantly for metal or dielectric surfaces – see Fig. 2(b).

The difference in the standard deviation of angle *ψ* corresponding to dielectric and metal surfaces is much higher. For the metal *ψ* is close to 45°. From Fig. 3(a) it is clear that in that case *ψ* is very sensitive to any variation in the initial retardation *δ*
_{1} and therefore to variation in the electrical current of the modulator. Furthermore, any variation in the measurement that is reflected in the variation of the ratio of *I*
_{1} and *I*
_{0} as well as *I*
_{2} and *I*
_{0} also strongly influences the standard deviation of the final results and therefore the repeatability of the final value of *ψ*, when this angle is close to 45° – see Figs. 3(a) and 3(b).

## 5. Conclusion

The proposed ellipsometer based on a 92 kHz-SCPEM shows a simple, compact, and cost-effective design while offering the highest flexibility in operation. We consider two main effects that influence the repeatability of the results: variations in the ratio between the frequency components of the measured photodiode signal and variations in the initial retardation of the modulator. The former depends mainly on data acquisition and processing and therefore on the quality of the electronics and is limited by its noise. The latter is related to the stability of the modulator operation. Using a single crystal modulator instead of a standard approach potentially offers very good possibilities for direct control of the optical response. Therefore by controlling the electrical current of the modulator we directly control the mechanical oscillation which is directly related to the optical behavior of the crystal. This is an important advantage over standard photo-elastic modulators, where the control of the mechanical oscillation may be influenced by the properties of the connection between actuator and glass.

In general, as expected the repeatability strongly depends on acquisition time, which was tested with this prototype in the range 0.03–20 ms. Even shorter acquisition times are possible. A further development is planned to increase the working frequency by using higher harmonics of the SCPEM or by using a different crystal with a higher fundamental frequency. The proposed measurement device is ideally suited for monitoring applications, quality control for large volume production (for example thin film control in photovoltaic panel production), and tracking of fast physical and chemical reactions.

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