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Parallel spectroscopic ellipsometry for ultra-fast thin film characterization

Andrey Nazarov, Michael Ney, and Ibrahim Abdulhalim

Open Access Open Access

Abstract

Spectroscopic ellipsometer (SE) is an essential optical metrology tool commonly used to characterize thin films and monitor fabrication processes. However, it relies on mechanical rotation of a polarizer or a photo-elastic phase modulator which are limited in speed and prone to errors when handling dynamic processes. The constant trend of micro-electronics dimensions shrinkage and increase of the wafer area necessitates faster and more accurate tools. A fast SE design based on parallel snapshot detection of three signals at different polarizations is proposed and demonstrated. Not relying on mechanical rotation nor serial phase modulation, it is more accurate and can reach acquisition rates of hundreds of measurements per second.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Ellipsometry is an old thin films and surfaces characterization technique that dates back to Paul Drude’s work [1,2] who applied it to the surfaces of crystals coated with over-layers. The extracted ellipsometric parameters $\Psi $ and $\Delta $, originally used by Drude to characterize polarized light, describe the sample’s induced change of the probe beam’s state of polarization (and intensity) and can be used to characterize the sample’s structure and composition in a very accurate manner using some a-priori model of the sample. The $\Psi $ and $\Delta $ parameters are defined by the ratio between the p- and s- polarized light components reflected from the sample, ${{{r_p}} \mathord{\left/ {\vphantom {{{r_p}} {{r_s}}}} \right.} {{r_s}}} = \tan (\Psi )\exp ({i\Delta } )$, where ${r_p}$ and ${r_s}$ are the Fresnel complex reflection coefficients of the p- and s- polarized light components respectively. The physical meaning of $\Psi $ is the ratio between the absolute value of ${r_p}$ and ${r_s}$, while $\Delta $ is the phase difference between the two reflected polarization components, parallel (p) and perpendicular (s) to the plane of incidence. When the ellipsometric measurement is performed at many wavelengths it is termed as spectroscopic ellipsometry (SE) which became recently, with the rise of parallel spectrometers, the main non-contact thin films characterization technique. SE is now of great use in the semiconductors industry [3,4], display panels [5], solar cells [6,7], bio-sensing [8] and more. One remarkable application developed during the last two decades for SE is the measurement of the transistor gate width in the nano-electronics industry in what is called critical dimension (CD) to overcome the diffraction limit of optical microscopy when the CD became less than 200nm [9,10]. It was found that $\Psi $ and $\Delta $ are highly sensitive to the linewidth of one dimensional subwavelength grating down to linewidths of the order of few tens of nm or less [11]. The methodology also known historically as optical scatterometry it is basically SE, has helped the mass production of nanoelectronic circuits in an easy and fast way. However, when the Si wafer size becomes 450mm there is a need for faster systems and yet maintaining the same precision or even better. Later on SE started to be used also for monitoring the overlay misregistration between layers, another important parameter which represents the positioning error of the stepper [12]. Fast SE can also be useful for biosensing as it can monitor the refractive index variations with high precision and hence the concentration of analytes in liquids such as water and blood [13].

Ellipsometric measurements are usually made in one of several configurations: rotating analyzer ellipsometry (RAE) [14], rotating polarizer-analyzer ellipsometry (RPAE) [15–20], rotating compensator ellipsometry (RCE) [14], phase modulation ellipsometry (PME) [14], and Muller matrix ellipsometry [14]. In RAE and RCE the optical system includes a rotating element which is either set to discrete orientations or set to perform a continuous rotation for a single measurement point, i.e. several sequential data samples are required for a single measurement point. The rotating element limits the operating speed of the system and may result in long acquisition times, thus the utilization of the ellipsometry system in high throughput systems is limited. Another important limitation that arises from the rotation process is the inability to measure dynamic processes, since one cannot stop the process itself, the dynamic nature of the sample introduces un-desired changes between sequential data acquisitions performed for the same measurement point. In addition, mechanical rotation is prone to errors which affect the precision.

Real-time SE measurement has been reported for thin film deposition using a commercial system [6], the reported acquisition time for the full spectra (690 wavelengths in the range of 193-1690 nm) is 0.25s. In another ellipsometry system used for production [5], relying on a rotating polarizer, a maximum speed of 40 measurements of 128 wavelengths per second has been achieved. Fast electro-optical devices such as photo-elastic modulators [8] (PEM) can reach much faster operation speeds (1ms acquisition speed), but since the PEM is a dispersive element spectroscopic (ellipsometry) analysis is very difficult.

In this work a new approach is presented for parallel spectroscopic ellipsometry based on polarization diversity configuration. The presented configuration consists of three channels, where each channel has a fixed analyzer at different orientation followed by a spectrometer. There are no rotating elements in this system, and only a single snapshot is needed to acquire a full spectroscopic data set per measurement point, hence the system is inherently more accurate as errors originating from the mechanical rotation and exact determination of the polarizer/analyzer positions are excluded. The speed of such system is limited mainly by the acquisition rate of the spectrometer and may potentially reach acquisition rates of hundreds/thousands of measurements per second or more limited by the speed of the spectrometers alone.

2. Methods

In RAE a single measurement point is taken by rotating the analyzer continuously for a finite duration of time or by sequentially rotating it to several predetermined discrete orientations and taking a snapshot at each angle. The need for the rotation of elements means non instantaneous and possibly long data acquisition, which requires the sample to be still during the acquisition of data for each measurement point in order to avoid parasitic measurement errors. The long data acquisition time limits the applicability of the ellipsometric system and which makes these methods limited at best, and possibly not applicable, for high throughput production/inspection tools in nano-fabs.

A suitable configuration for parallel SE (based on the classic RAE type configuration) is achieved by splitting the light beam reflected from the sample into, for example, (the minimal number of) three independent measurement channels. The theoretical justification for this method is summarized in the following section (see supplementary for full derivation):

2.1 Theoretical justification

Fast ellipsometry may be achieved once the need for rotation of an optical element (or performing some other ‘scanning’ operation using an active element to achieve a similar effect to rotation that is time consuming) is eliminated. We propose to solve this limitation by using a limited, yet sufficient, number of ellipsometric measurements in parallel. In this case, each channel measures the following signal

$${I_{CHi}} = {a_i}\left\{ \begin{array}{l} {|{{q_{i1}}r{ _{S,ss}}} |^2}{\cos^2}{A_i}{\cos^2}({{\alpha_{PM}} - {\theta_P}} )+ {|{{q_{i2}}r{ _{S,pp}}} |^2}{\sin^2}{A_i}{\sin^2}({{\alpha_{PM}} - {\theta_P}} )\\ + \sin ({2{A_i}} )\sin ({2({{\alpha_{PM}} - {\theta_P}} )} ){\mathop{\textrm {Re}}\nolimits} ({{q_{i1}}q_{i2}^\ast r{ _{S,ss}}r_{S,pp}^\ast } )\end{array} \right\},$$
where $i = 1,2,3,$ ${a_i}$ is the transmission of the lens that couples the light into the spectrometer's fiber, ${A_i}$ is the orientation of the analyzer with respect to the p-polarization upon the sample, ${\alpha _{PM}}( = 90 - \theta )$ is the angle between the surface of the sample and the optical axis ($\theta$ is the incidence angle on the sample, see Fig. 1), ${\theta _P}$ is the orientation of the polarizer with respect to the p-polarization upon the sample, $r{ _{S,ss}}$ and $r{ _{S,pp}}$ are the complex reflection coefficients of the sample for s- and p- polarization respectively, and ${q_{i1}}$, ${q_{i2}}$ are (for our setup configuration):
$$\begin{array}{l} \textrm{for i = 1}:\textrm{ }{q_{11}} = {t_{BS2,pp}}{t_{BS1,pp}}{r_{PM2,pp}}{r_{PM1,pp}}\textrm{ ; }{q_{12}} = {t_{BS2,ss}}{t_{BS1,ss}}{r_{PM2,ss}}{r_{PM1,ss}}\\ \textrm{for i = }2:\textrm{ }{q_{21}} = {r_{BS2,pp}}{t_{BS1,pp}}{r_{PM2,pp}}{r_{PM1,pp}}\textrm{ ; }{q_{22}} = {r_{BS2,ss}}{t_{BS1,ss}}{r_{PM2,ss}}{r_{PM1,ss}}\\ \textrm{for i = }3:\textrm{ }{q_{31}} = {t_{BS2,pp}}{r_{BS1,pp}}{r_{PM2,pp}}{r_{PM1,pp}}\textrm{ ; }{q_{32}} = {t_{BS2,ss}}{r_{BS1,ss}}{r_{PM2,ss}}{r_{PM1,ss}} \end{array},$$
where $t/{r_{BS/PM}}$ are the transmission/reflection coefficient of the beam splitter (BS) or the parabolic mirror (PM), and the subscript pp/ss refers to the p- or s- polarization respectively.

 figure: Fig. 1.

Fig. 1. Schematic drawing of parallel spectroscopic ellipsometer. L1-L3 – light collection lenses, A1-A3 – Analyzers, BS – beam splitter, M1-M2 – mirrors, P – polarizer, L4-L5 – beam shrinker, L6 – light source collimation lens, θ – incidence angle of illumination upon the sample.

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The difference between the three channels is the orientation of the analyzer. Two of the channels are set to be orthogonal one to the other and parallel to the p- and s- polarization axes, wherein the third channel is set to an angle exactly between them at 45°$({{A_1} = 0;\textrm{ }{A_2} = 90;\textrm{ }{A_3} = 45} )$. This configuration allows the system to acquire all data needed to measure the ellipsometric parameters by a single measurement, without the use of rotation or some other active elements.

The measurement process is as described hereafter; the sample placed on a stage between the two parabolic mirrors (see Fig. 1), and the signal is recorded by three detectors (spectrometers for parallel SE) in parallel. The three signals are:

$$\begin{aligned} {I_{CH1}} &= {a_1}\{{{{|{{q_{11}}r{ _{S,ss}}} |}^2}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )} \},\\ {I_{CH2}} &= {a_2}\{{{{|{{q_{22}}r{ _{S,pp}}} |}^2}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} )} \},\\ {I_{CH3}} &= \frac{{{a_3}}}{2}\left\{ \begin{array}{l} {|{{q_{31}}r{ _{S,ss}}} |^2}{\cos^2}({{\alpha_{PM}} - {\theta_P}} )+ {|{{q_{32}}r{ _{S,pp}}} |^2}{\sin^2}({{\alpha_{PM}} - {\theta_P}} )\\ + \sin ({2({{\alpha_{PM}} - {\theta_P}} )} ){\mathop{\textrm {Re}}\nolimits} ({{q_{i1}}q_{i2}^\ast r{ _{S,ss}}r_{S,pp}^\ast } )\end{array} \right\} \end{aligned}.$$
From the 1st and the 2nd channel ${|{r{ _{S,ss}}} |^2}$ and ${|{r{ _{S,pp}}} |^2}$, are obtained (and using the calibration as described in section 2.3) it is possible to extract $\Psi $,
$$\tan (\Psi )= \left|{\frac{{{r_{S,pp}}}}{{{r_{S,ss}}}}} \right|= \frac{{|{{r_{ref,pp}}} |}}{{|{{r_{ref,ss}}} |}}\sqrt {\frac{{{I_{CH2}}}}{{{I_{CH2,ref}}}}\frac{{{I_{CH1.ref}}}}{{{I_{CH1}}}}} .$$
In order to find $\Delta $, the real term that includes the multiplication of $r{ _{S,ss}}$ by $r_{S,pp}^\ast $ needs to be extracted from the expression of the 3rd channel.
$${a_3}{\mathop{\textrm {Re}}\nolimits} ({{q_{31}}q_{32}^\ast r{ _{S,ss}}r_{S,pp}^\ast } )= \frac{{2{I_{CH3}} - {a_3}{{|{{q_{31}}r{ _{S,ss}}} |}^2}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )- {a_3}{{|{{q_{32}}r{ _{S,pp}}} |}^2}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} )}}{{\sin ({2({{\alpha_{PM}} - {\theta_P}} )} )}},$$
And using the calibration procedure we receive:
$${a_3}{\mathop{\textrm {Re}}\nolimits} ({{q_{31}}q_{32}^\ast r{ _{S,ss}}r_{S,pp}^\ast } )= \frac{{2{I_{CH3}} - \frac{{{I_{CH3,ref,{A_3} = 0}}{I_{CH1}}}}{{{I_{CH1,ref}}}} - \frac{{{I_{CH3,ref,{A_3} = 90}}{I_{CH2}}}}{{{I_{CH2,ref}}}}}}{{\sin ({2({{\alpha_{PM}} - {\theta_P}} )} )}}.$$
Once this expression is divided by ${q_{31}}q_{32}^\ast r{ _{S,ss}}r_{S,pp}^\ast $, we can find $\Delta $ using:
$$\frac{{{a_3}{\mathop{\textrm {Re}}\nolimits} ({{q_{31}}q_{32}^\ast r{ _{S,ss}}r_{S,pp}^\ast } )}}{{{a_3}|{{q_{31}}q_{32}^\ast {r_{S,ss}}r_{S,pp}^\ast } |}} = \cos ({q + \Delta } ).$$
The value of q is known from the calibration procedure, therefore, $\Delta $ is found using the following relation:
$$\begin{array}{l} q = {\cos ^{ - 1}}({\cos ({q + {\Delta _{ref}}} )} )- {\Delta _{ref}}\\ \Delta = {\cos ^{ - 1}}({\cos ({\Delta + q} )} )- q \end{array}.$$
In spectroscopic ellipsometry (SE), the ellipsometric parameters $\Psi $ and $\Delta $ are to be measured over a range of wavelengths by taking a sequence of discrete monochromatic measurements or by using a white light source and a spectrometer capturing a range of wavelengths at a single-shot. The denser and wider the spectrum of acquired wavelengths, the sensitivity and accuracy of the characterization of the sample improves – especially for complex samples. However, the increase in the spectral sampling data may prolong the acquisition duration, especially if the wavelengths are captured sequentially. The derived expressions above maybe used for both single a wavelength and a spectroscopic measurement. By using wide spectrum source and spectrometers instead of the regular detectors, a spectroscopic measurement can be taken – here resulting in a parallel acquisition of the ellipsometric parameters $\Psi $ and $\Delta $ over the entire spectrum in parallel (thus it is PSE – parallel SE) and in a single shot.

In the case of PSE the limiting factor is only the acquisition speed of the spectrometers (as long as the light source is powerful enough to allow reaching the minimum available integration time). Fast, compact and relatively cheap spectrometers are available and can reach as fast as 0.2 msec (rate of 5 kHz) per scan.

2.2 Parallel SE setup configuration

An experimental setup used to test and verify the suggested parallel spectroscopic ellipsometry concept is depicted in Fig. 1. The light source is a wide spectrum Xenon arc lamp (200-1200nm, yet at this stage we utilize only the visible portion of the spectrum) coupled to an optical fiber. The output of the fiber is connected to an enhanced aluminum parabolic mirror based fiber collimator (L6), then the collimated light beam is passed through lenses L5 and L4 combined together to reduce the size of the beam diameter. To set the polarization of the light illuminating the sample, the collimated light beam is passed through a liner polarizer (P). After passing the polarizer, the light reaches a silver mirror (M1) that reflects the light on the sample at a desired incidence angle θ. The light reflected from the sample is reflected again by a second silver mirror (M2) whose role is to direct the light towards the assembly of components comprising the ‘detection head’. The collimated light beam from M2 is divided into four beams by three 50:50 beam splitters (BS), three are utilized. Each one of the three beams are directed to one of three parallel analysis channels where they are passed through an analyser (A1, A2 and A3) that is oriented at a different angle and is focused by a lens (L1, L2 and L3) into the input fiber of a fast spectrometer (AvaSpec-ULS2048CL-EVO). All three spectrometers are connected using a trigger wire, and synchronously acquire the signals simultaneously. These three signals that are measured simultaneously are the core of the parallel spectroscopic ellipsometry concept.

It is important to note that the light throughput is a slightly limiting factor in the current system configuration. The light source, a Xenon arc lamp, is coupled into an optical fiber whose output is collimated using a parabolic mirror. Since the fiber core diameter is 100µm, the collimation is not ideal, and a portion of the light is lost. Currently our measurement consists acquisition at 250µs of 2048 wavelengths (1100 of them are used for the analysis) in the 196-764nm range, where a batch of 4 sequential measurements are averaged together to increase SNR and provide better accuracy. By improving the light collimation and in general the illumination head efficiency, thus reducing light loss (increasing SNR), higher acquisition rates can be achieved resulting in a better or even full utilization of the maximum acquisition rate of the used spectrometers (2500Hz).

2.3 Calibration

Like most measurement systems, this system requires a preliminary calibration process which is performed once per system configuration (angle of incidence setting). The calibration process is very simple and requires three measurements of a well-known reference sample (a piece of Si). In each measurement, the orientation of the analyzer in channel 3 is changed. The flow of the calibration process is as described below:

  • 1. First, the analyzers are set to be: ${A_1} = 0;\textrm{ }{A_2} = 90;\textrm{ }{A_3} = 0$. This reduces the expressions for the three channels to:
    $$\begin{array}{l} {I_{CH1,ref}} = {a_1}\{{{{|{{q_{11}}r{ _{ref,ss}}} |}^2}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )} \}\\ {I_{CH2,ref}} = {a_2}\{{{{|{{q_{22}}r{ _{ref,pp}}} |}^2}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} )} \}\\ {I_{CH3,ref,{A_3} = 0}} = {a_3}\{{{{|{{q_{31}}r{ _{ref,ss}}} |}^2}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )} \} \end{array}.$$
    where ${I_{CHi,ref}}$ is the signal measured in the channels from the reference sample, $r{ _{ref,ss}}$ and $r{ _{ref,pp}}$ are the reflection coefficients of the reference sample for s- and p- polarizations respectively. From these measurements the following are extracted:
    $$\begin{array}{l} {a_1}{|{{q_{11}}} |^2} = \frac{{{I_{CH1,ref}}}}{{{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{r{ _{ref,ss}}} |}^2}}}\\ {a_2}{|{{q_{22}}} |^2} = \frac{{{I_{CH2,ref}}}}{{{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{r{ _{ref,pp}}} |}^2}}}\\ {a_3}{|{{q_{31}}} |^2} = \frac{{{I_{CH3,ref,{A_3} = 0}}}}{{{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{r{ _{ref,ss}}} |}^2}}} \end{array}.$$
  • 2. Second, the analyzers are set to be:${A_1} = 0;\textrm{ }{A_2} = 90;\textrm{ }{A_3} = 90$. Note that the analyzers in the 1st and 2nd channels haven't changed, so only the change in the 3rd channel is presented. This reduces the expression to become:
    $${I_{CH3,ref,{A_3} = 90}} = {a_3}\{{{{|{{q_{32}}r{ _{ref,ss}}} |}^2}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )} \}.$$

    From this measurement we extract the following:

    $${a_3}{|{{q_{32}}} |^2} = \frac{{{I_{CH3,ref,{A_3} = 90}}}}{{{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{r{ _{ref,pp}}} |}^2}}}.$$

  • 3. Third, the analyzers are set to be:${A_1} = 0;\textrm{ }{A_2} = 90;\textrm{ }{A_3} = 45$. Again, only the change in the 3rd channel is presented.
    $${I_{CH3,ref,{A_3} = 45}} = \frac{{{a_3}}}{2}\left\{ \begin{array}{l} {|{{q_{31}}r{ _{ref,ss}}} |^2}{\cos^2}({{\alpha_{PM}} - {\theta_P}} )+ {|{{q_{32}}r{ _{ref,pp}}} |^2}{\sin^2}({{\alpha_{PM}} - {\theta_P}} )\\ + \sin ({2({{\alpha_{PM}} - {\theta_P}} )} )2{\mathop{\textrm {Re}}\nolimits} ({{q_{31}}q_{32}^\ast r{ _{ref,ss}}r_{ref,pp}^\ast } )\end{array} \right\},$$

From this measurement we extract the following:

$$\begin{array}{l} {a_3}{\mathop{\textrm {Re}}\nolimits} ({{q_{31}}q_{32}^\ast {r_{ref,ss}}{r^\ast }_{ref,pp}} )\\ = \frac{{2{I_{CH3,ref,{A_3} = 45}} - {a_3}{{|{{q_{31}}} |}^2}{{|{{r_{ref,ss}}} |}^2}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )- {a_3}{{|{{q_{32}}} |}^2}{{|{{r_{ref,pp}}} |}^2}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} )}}{{\sin ({2({{\alpha_{PM}} - {\theta_P}} )} )}}\\ = \frac{{2{I_{CH3,ref,{A_3} = 45}} - {I_{CH3,ref,{A_3} = 0}} - {I_{CH3,ref,{A_3} = 90}}}}{{\sin ({2({{\alpha_{PM}} - {\theta_P}} )} )}} \end{array}.$$

In order to extract $\Delta $ for the sample the accumulated (parasitic ellipsometric) effect, noted by the complex quantity q, of the optical elements in the system need to be found and eliminated. For that we need to use the following expression:

$${a_3}|{{q_{31}}{q_{32}}r_{ref,ss}^\ast r_{ref,pp}^\ast } |= \frac{{\sqrt {{I_{CH3,ref,{A_3} = 0}} \cdot {I_{CH3,ref,{A_3} = 90}}} }}{{|{\cos ({{\alpha_{PM}} - {\theta_P}} )} ||{\sin ({{\alpha_{PM}} - {\theta_P}} )} |}}.$$
Once we divide Eqs. (14) and (15) we receive:
$$\frac{{{a_3}{\mathop{\textrm {Re}}\nolimits} ({{q_{31}}q_{32}^\ast r{ _{ref,ss}}r_{ref,pp}^\ast } )}}{{{a_3}|{{q_{31}}{q_{32}}r_{ref,ss}^\ast r_{ref,pp}^\ast } |}} = \cos ({q + {\Delta _{ref}}} ).$$
Since the reference sample is known, ${\Delta _{ref}}$ is known precisely and can be used to find q:
$$q = {\cos ^{ - 1}}({\cos ({q + {\Delta _{ref}}} )} )- {\Delta _{ref}}.$$

2.4 Data processing

In order to extract the ellipsometric parameters from the parallel measurements, first the system must be calibrated. The calibration is done using three measurements of a well-known sample. There are three measurements in the calibration process, each measurement is stored and used to extract ${a_i}$, ${q_{i1}}$, ${q_{i2}}$ and q.

Once a measurement is performed and the data from all three channels is collected, the ellipsometric parameters $\Psi $ and $\Delta $ are extracted. After extracting these parameters, a process of fitting between the measurement and an a priori calculated library of the ellipsometric parameters based on possible models of the possible samples is performed. The fitting process can be accelerated by any a priori knowledge on the sample under test, thus limiting the span of parameters through which the fitting process needs to scan (which may include thickness of layers, materials, complex geometries of structures etc.). For example, in the case of thin film thickness estimation, a rough estimate of the expected thickness is used combined with some model of the structure – number of layers and layer materials. In order to find the best fit the root mean square error (RMSE) between measured and theoretical curves is calculated and required to be minimized. The thickness that gives the least RMSE is considered as the measured thickness. We will continue with the case study of thin films, but remember that the method is applicable for any structure, as long as a matching library of sample models with the calculated ellipsometric parameters are available for the process of fitting.

The accuracy and speed of the fitting process can be controlled by changing different parameters in the software: 1) the resolution of layer thicknesses and the number of layers considered that are used for calculating the theoretical curves - determining the number of candidate curves to be tested for fitting, 2) the amount of wavelengths at which the data is measured and fitted, 3) using smoothing functions for the measured and theoretical curves (not used in this study), 4) weighting parameters for the RMSE calculation.

In this study two methods for defining the fitting resolution were used: 1) The resolution and fitting points spread on a fixed span (e.g. 2000 points evenly spaced in the range 10nm-10µm), 2) the resolution changed dynamically with the a priori estimated/expected value (e.g. for a given thickness d, 2000 points evenly spaced in the range ${d \mathord{\left/ {\vphantom {d 2}} \right.} 2}$-${{3d} \mathord{\left/ {\vphantom {{3d} 2}} \right.} 2}$). Both approaches worked well and are easy to implement. Also, pre-written libraries were used to try and reduce the fitting time of the software, but this did not reduce the speed dramatically.

The RMSE calculation is done separately for the two ellipsometric parameters, and the final value is the square root of the sum of the squares of the individual RMSE’s. It is possible to favor the RMSE of one parameter over the other after visual examination of the results; in this study the same weighting factor (0.5) was used for both ellipsometric parameters. It is also possible to give different weights to different data sets depending on the signal to noise ration of each spectral region, however here we used the same weight.

The resolution of thickness extraction is 5-10nm for the 5µm sample, and 5nm for the other samples – ranging from 50 to 318nm. The resolution can be reduced on the expense of processing time.

Since all layers are on top of a silicon substrate, a native silicon substrate with a 2.2nm oxide layer was used as the reference sample. The following dispersion models for SiO2 [21,22], Si3N4 [23,24] and (C8H8)n [25] were used in order to extract the thickness of the layers:

$$\begin{array}{l} {n^2}({Si{O_2}} )= 1 + \frac{{0.6961663{\lambda ^2}}}{{{\lambda ^2} - {{0.0684043}^2}}} + \frac{{0.4079426{\lambda ^2}}}{{{\lambda ^2} - {{0.1162414}^2}}} + \frac{{0.8974794{\lambda ^2}}}{{{\lambda ^2} - {{9.896161}^2}}}\\ {n^2}({S{i_3}{N_4}} )= 1 + \frac{{2.8939{\lambda ^2}}}{{{\lambda ^2} - {{0.13967}^2}}}\\ {n^2}({{{({{C_8}{H_8}} )}_n}} )= 1 + \frac{{1.4435{\lambda ^2}}}{{{\lambda ^2} - 0.020216}} \end{array}.$$

3. Results and discussion

Our experiments concentrated on three types of samples: 1) Silicon oxide (SiO2), 2) Silicon nitride (Si3N4) and 3) Polystyrene (C8H8)n, on top of a silicon substrate. The thicknesses of all samples are known.

3.1 Single measurement of static samples

The first sample measured was a 5µm SiO2 layer on top of a silicon substrate. First the sample's thickness was verified using a commercial ellipsometer ‘alpha-SE’ (by J.A. Woollam Co.). Three measurements were performed of the sample and found that the mean value of the thickness is 5.027µm (see results in Table 1). The measurement was performed at various angles to better understand the capabilities of the PSE system. Also, to check the repeatability and the ease of alignment, the measurements were made at different times (days apart) after re-alignment of the system. Figure 2 presents the results for a 5µm SiO2 sample at different incidence angles.

 figure: Fig. 2.

Fig. 2. Measured Ψ and Δ for silicon oxide sample of 5µm. (a) - incidence angle of 64deg, extreacted thickness is 5.02µm with RMSE of 31.37deg, (b) - incidence angle of 67deg, extreacted thickness is 5.02µm with RMSE of 8.08deg, (c) - incidence angle of 70deg, extreacted thickness is 5.03µm with RMSE of 12.6deg, (d) - incidence angle of 64deg, extreacted thickness is 5.02µm with RMSE of 16.47deg.

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Tables Icon

Table 1. Measurement results of a 5µm SiO2 sample using the PSE system, and comparison to the results from Woollam’s alpha-SE ellipsometer results.

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The Brewster angle of silicon is ∼73°; at this incidence angle the system is most sensitive to intensity changes of the different polarizations. However, the alignment of the system must be very accurate when aligning the polarizer. Therefore, the system was tested at 3 more angles in order to see if more accurate results can be obtained at neighboring incidence angles. Four angles 64°, 67°, 70° and 73° were used. It is evident from Fig. 2 that the measurement at 73° is noisier than other angles although the measured thickness is still close to the expected value. Also, from all performed measurements, it is seen that at 67° the same values are received, and the restrictions on the alignment of the polarizers were less strict. After reviewing all measurements, it was decided to use 67° as the incidence angle, although all other tested angles are also suitable. The noisier signal at Brewster angle is understood as the p-polarized reflectivity vanishes, hence small deviation from this angle improved the signal to noise ratio.

Table 1 summarizes the performed measurements. The mean value for the thickness for the SiO2 achieved was 5.023µm. It can be seen that most results are consistent and in the vicinity of the expected value. Note that near the same incidence angle the variability of the commercial system is 32nm while our system gives 0nm variability at the nominal angles of 67° or 70°. This is not surprising as our system does not use any mechanical motion and all signals are grabbed simultaneously. On the other hand commercial systems based on mechanical rotation of polarizers are prone to errors due to vibrations and errors in the determination of the polarizer/analyzer angles. Similarly dynamically changing phase modulators such as PEMs are source of errors as the phase determination with applied electrical signal may vary as well as be temperature dependent.

Second, a series of C8H8 samples were measured, their thicknesses varied from 51nm up to 318nm. The thicknesses were measured using a commercial ellipsometry system ‘SE800’ (by SENTECH Instruments Gmbh) for comparison. Figures 3 and 4 present the measurements. All measurements were performed at an incidence angle of 67°. The average deviation from the expected values is 0.65%, when the maximum deviation of 4.4% is for the 230nm layer.

 figure: Fig. 3.

Fig. 3. Measured Ψ and Δ for polystyrene sample of 51nm, the extreacted thickness is 49.98nm with RMSE of 11.7471deg (a), sample of 60nm, the extreacted thickness is 60.00nm with RMSE of 12.4266deg (b), sample of 66nm, the extreacted thickness is 64.97nm with RMSE of 46.4949deg (c), sample of 70nm, the extreacted thickness is 69.97nm with RMSE of 50.2605deg (d).

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 figure: Fig. 4.

Fig. 4. Measured Ψ and Δ for polystyrene sample of 172 nm, the extreacted thickness is 174.92 nm with RMSE of 10.8772deg (a), sample of 180 nm, the extreacted thickness is 174.92 nm with RMSE of 8.0036deg (b), sample of 201 nm, the extreacted thickness is 199.90 nm with RMSE of 8.2324deg (c), sample of 219 nm, the extreacted thickness is 224.89 nm with RMSE of 10.208deg (d), sample of 230 nm, the extreacted thickness is 219.89 nm with RMSE of 9.2275deg (e), sample of 318 nm, the extreacted thickness is 319.84 nm with RMSE of 16.8384deg (f).

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The third sample measured was a 100nm Si3N4 layer on top of a silicon substrate. The thickness was measured using a commercial ellipsometry system ‘SE800’ (by SENTECH Instruments Gmbh) for comparison. The thickness varied between 100.2 to 102.95nm depending on the measurement settings (measured values: 100.7, 102.5, 100.2, and 102.95nm). Figure 5 presents the measurement using the parallel SE system. The measurement was taken at an incidence angle of 67°. The minimal RMSE corresponds to Si3N4 layer of 99.95nm, which is a 0.05% deviation from the expected value of ∼100nm.

 figure: Fig. 5.

Fig. 5. Measured Ψ and Δ for silicon nitride sample of 100 nm. The extracted thickness (d) and the corresponding minimal RMSE are stated in the title of each graph.

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The fourth sample measured was a 106nm SiO2 layer on top of a silicon substrate. The thickness was measured using a commercial ellipsometry system SE800 (by SENTECH Instruments Gmbh) for comparison. The thickness varied between 106.01 to 106.06nm depending on the measurement settings (measured values: 106.06, 106.01, and 106.02nm). Figure 6 presents the measurement using the parallel SE system. The measurement was made at an incidence angle of 67°. The minimal RMSE is found for a SiO2 layer of 107.35nm, which is a 1.27% deviation from the expected value.

 figure: Fig. 6.

Fig. 6. Measured Ψ and Δ for silicon oxide sample of 100 nm. The extracted thickness (d) and the corresponding minimal RMSE are stated in the title of each graph.

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It can be seen in Figs. 3–6 that the measured values of $\Psi $ are not as smooth as the theoretical values showing bumps in the curves (no smoothing was used on the measured curves) especially around 570nm and 650nm. The effect is better seen in the thinner layers (thickness less than 200nm). This is not due to sample non-uniformity since the spot size falling on the sample is large enough for averaging it. Also, since the lights in the room during the measurements were off, the effect was not caused by stray light that entered the system. It might be an error in the dispersion model used for the theoretical calculations of the reference sample.

3.2 Multiple measurements of a static sample

In order to further base our claim of the good repeatability of the system we have taken 50 subsequent measurements of a 5µm SiO2 layer on top of a silicon substrate. The thicknesses and the corresponding RMSE values are presented in Fig. 7. The measurement is done at an incidence angle of 67°. The fitting process is done for intervals of 5nm in thickness, the fitted thickness value varied between two discrete values and the average value is found to be 5.0241µm. Same measurements were analyzed with a 1nm interval, and the average value was found to be close to the previous one 5.0246µm. Therefore, it is evident that even at 5nm fitting resolution the results are accurate and computation time might be saved, hence, a faster system.

 figure: Fig. 7.

Fig. 7. Thickness and RMSE for a burst of 50 measurements for a 5µm SiO2 sample.

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3.3 Lateral scan of a variable thickness sample

In order to support our claim that the proposed technique is suitable for scanning a sample, a 5µm SiO2 sample mounted on a motorized stage was further tested. The sample was moved at 3 different speeds and bursts of 10 measurements were taken. Before the scan began, a reference measurement was taken with a still stage; the extracted thickness was 5.0425µm (which lays in the range measured by alpha-SE). All measurements gave the same thickness as the reference test as can be seen in Fig. 8. At 0.5mm/sec scan the last 3 measurements had relatively higher RMSE values; it is caused as a result of the limitation of the range available for the stage and the size of the sample.

 figure: Fig. 8.

Fig. 8. Thickness and RMSE for a lateral scan of a 5µm SiO2 sample at various scan rates.

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To further test the abilities of the system two adjacent samples of different thicknesses were tested (to simulate variable thickness step sample). The previously measured 5µm and 100nm SiO2 samples were used. Figure 9 presents the result of the scan. It can be seen that the different samples are distinguishable and the transition zone between them is clear. The scan speed for this measurement is 0.1mm/sec.

 figure: Fig. 9.

Fig. 9. Thickness and RMSE for a lateral scan of two adjacent samples of 100 nm and 5µm SiO2 samples.

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4. Conclusion

In this work a new fast and a more accurate approach for spectroscopic ellipsometry is proposed and demonstrated experimentally. Our main goal was to find a new implementation for spectroscopic ellipsometry, similar to RAE method, but without its’ shortcomings that will result in an accurate and fast system to satisfy the needs of nano-electronics industry where the Si wafer size becomes 450mm without sacrificing the precision. The proposed system key design feature is its analysis (detection) head composed of 3 parallel channels, where each channel includes the elements used in a simple RAE configuration but with an analyzer which static (as well as the polarizer in the illumination path). The use of 3 parallel spectroscopic channels is the reason for the system ability to receive all needed data in one snapshot. A commercial ellipsometer was used to first verify the thickness of our tested samples, and we have also demonstrated that the results were maintained and with lower variability when the samples were measured over a period of time and at faster rates. Different materials and different thicknesses were tested varying from 50nm up to 5µm and verified the repeatability of the results by measurement on different days and after realigning and re-calibrating the system. We have also performed dynamic measurements by laterally moving samples under the PSE setup using a motorized stage, and measured them at various scan rates as well as measuring a sample with varying thickness - thus confirming the ability of the proposed system to monitor thickness variations at high scan speed. The proposed system is limited mainly by the acquisition speed of the spectrometers. Nowadays, faster spectrometers are becoming more available in the market and this system can potentially reach hundreds and even thousands of measurements per second and eventually lead to real-time spectroscopic ellipsometry. This might trigger new applications of spectroscopic ellipsometry for example in monitoring dynamic processes at surfaces, biological and chemical fast processes.

Appendix: System's light path description and the full derivation of equations used for extraction of the ellipsometric parameters

For the derivation of the equations we follow the light path as illustrated in the Fig. 1. The light source is a wide spectrum Xenon arc lamp (200-1200nm) coupled to an optical fiber. The output of the fiber is connected to a parabolic fiber collimator (L6); then the collimated light beam is passed through lenses L5 and L4 for a smaller beam diameter. To set the polarization of the light illuminating the sample, the collimated light beam is passed through a linear polarizer (P). After passing the polarizer, the light reaches a silver mirror (M1) that reflects the light on the sample at an incidence angle θ. The light reflected from the sample is reflected again by a second silver mirror (M2). The collimated light beam is splitted into three beams by three 50:50 beam splitters (BS). Each one of these three beams is passed through an analyser (A1, A2 and A3) at different orientation at each channel; then the light is focused by a lens (L1, L2 and L3) into the input fiber of a fast spectrometer. All three spectrometers are connected using a trigger wire, and are acquiring the signals simultaneously

The direction of light emitted by the source is in the positive z-axis,

$${k_{in}} = \left( \begin{array}{l} 0\\ 0\\ 1 \end{array} \right),$$
and the electric field at the input of the system is,
$${E_{in}} = \left( \begin{array}{c} {E_x}\\ {E_y}\\ 0 \end{array} \right) = \left( \begin{array}{c} {E_p}\\ {E_s}\\ 0 \end{array} \right) = E\left( \begin{array}{c} \cos {\theta_p}\\ \sin {\theta_p}\\ 0 \end{array} \right).$$
The direction of the electric field after reflection of the first mirror is at an angle ${\alpha _{PM}}( = 90 - \theta )$– the angle between the surface of the sample and the optical axis ($\theta$ is the incidence angle on the sample). And the light interacts with the sample at an angle of $\theta = {90^ \circ } - {\alpha _{PM}}$ relative to the normal of the sample plane.

Since the p- and s- polarization switch roles at different interaction points of the system, close attention should be payed to that during the derivation of the equations.

$$\left( \begin{array}{l} {E_{p,PM1\textrm{ out}}}\\ {E_{s,PM1\textrm{ out}}} \end{array} \right) = \left( {\begin{array}{{cc}} {{r_{PM1,pp}}}&0\\ 0&{{r_{PM1,ss}}} \end{array}} \right)\left( \begin{array}{l} {E_{p,PM1\textrm{ in}}}\\ {E_{s,PM1\textrm{ in}}} \end{array} \right) = \left( \begin{array}{l} {r_{PM1,pp}} \cdot {E_{p,PM1\textrm{ in}}}\\ {r_{PM1,ss}} \cdot {E_{s,PM1\textrm{ in}}} \end{array} \right).$$
In order to align the linear polarization of the source at 45 degrees in respect to the p- and s- polarizations at the mirror; the polarizer P is set at an angle of ${45^ \circ } + {\alpha _{PM}}$. Therefore, the input field is only multiplied by the p- and s- reflection coefficient of the mirror. A projection matrix may be used instead of rotation, but rotation leads to much simpler expressions.

At the interaction point with the sample, the sign of the p- polarization is changed and sign of the s- polarization remains the same. Also, it is important to note that p- polarization at the mirror becomes the s- polarization of the sample, and s- polarization at the mirror becomes the p- polarization at the sample.

After light reflection only the sign of the electric field is changed:

$$\begin{aligned} \left( \begin{array}{l} {E_{p,S\textrm{ out}}}\\ {E_{s,S\textrm{ out}}} \end{array} \right) &= \left( {\begin{array}{{cc}} {{r_{S,pp}}}&0\\ 0&{{r_{S,ss}}} \end{array}} \right)\left( \begin{array}{l} {E_{p,S\textrm{ in}}}\\ {E_{s,S\textrm{ in}}} \end{array} \right) = \left( {\begin{array}{{cc}} {{r_{S,pp}}}&0\\ 0&{{r_{S,ss}}} \end{array}} \right)\left( \begin{array}{c} {E_{s,PM1\textrm{ out}}}\\ - {E_{p,PM1\textrm{ out}}} \end{array} \right)\\ &= \left( \begin{array}{c} {r_{S,pp}} \cdot {E_{s,PM1\textrm{ out}}}\\ - {r_{S,ss}} \cdot {E_{p,PM1\textrm{ out}}} \end{array} \right) = \left( \begin{array}{c} {r_{S,pp}}{r_{PM1,ss}} \cdot {E_{s,PM1\textrm{ in}}}\\ - {r_{S,ss}}{r_{PM1,pp}} \cdot {E_{p,PM1\textrm{ in}}} \end{array} \right) \end{aligned}.$$
At the interaction with the second mirror the sign of the p- polarization is changed and the sign of s- polarization is unchanged. Again, the p- polarization of the sample becomes the s- polarization for the second mirror, and the s- polarization of the sample becomes the p- polarization for the mirror second mirror:
$$\begin{aligned} \left( \begin{array}{l} {E_{p,PM2\textrm{ out}}}\\ {E_{s,PM2\textrm{ out}}} \end{array} \right) &= \left( {\begin{array}{{cc}} {{r_{PM2,pp}}}&0\\ 0&{{r_{PM2,ss}}} \end{array}} \right)\left( \begin{array}{l} {E_{p,PM2\textrm{ in}}}\\ {E_{s,PM2\textrm{ in}}} \end{array} \right) = \left( {\begin{array}{{cc}} {{r_{PM2,pp}}}&0\\ 0&{{r_{PM2,ss}}} \end{array}} \right)\left( \begin{array}{c} - {E_{s,S\textrm{ out}}}\\ {E_{p,S\textrm{ out}}} \end{array} \right)\\ &= \left( \begin{array}{c} - {r_{PM2,pp}} \cdot {E_{s,S\textrm{ out}}}\\ {r_{PM2,ss}} \cdot {E_{p,S\textrm{ out}}} \end{array} \right) = \left( \begin{array}{c} {r_{PM2,pp}}{r_{S,ss}}{r_{PM1,pp}} \cdot {E_{p,PM1\textrm{ in}}}\\ {r_{PM2,ss}}{r_{S,pp}}{r_{PM1,ss}} \cdot {E_{s,PM1\textrm{ in}}} \end{array} \right) \end{aligned}.$$
After reflection off the second mirror, the light is split into 3 beams by three beam splitters. Since the polarization axes are rotated, the detection head is rotated by ${\alpha _{PM}}$ (for the same reasons the polarizer P is set to ${45^ \circ } + {\alpha _{PM}}$). Each detection channel goes through different path in the beam splitters: CH1 is reached after double transmission, CH2 is reached after transmission followed by a reflection, and CH3 is reached after reflection followed by a transmission.

For CH1 the expression is:

$$\begin{aligned} E_{BS,CH1}^T &= \left( {\begin{array}{{cc}} {{t_{BS2,pp}}}&0\\ 0&{{t_{BS2,ss}}} \end{array}} \right)\left( {\begin{array}{{cc}} {{t_{BS1,pp}}}&0\\ 0&{{t_{BS1,ss}}} \end{array}} \right)\left( \begin{array}{l} {E_{p,PM2\textrm{ out}}}\\ {E_{s,PM2\textrm{ out}}} \end{array} \right)\\ &= \left( \begin{array}{c} {t_{BS2,pp}}{t_{BS1,pp}} \cdot {E_{p,PM2\textrm{ out}}}\\ {t_{BS2,ss}}{t_{BS1,ss}} \cdot {E_{s,PM2\textrm{ out}}} \end{array} \right) = \left( \begin{array}{c} {t_{BS2,pp}}{t_{BS1,pp}}{r_{PM2,pp}}{r_{S,ss}}{r_{PM1,pp}} \cdot {E_{p,PM1\textrm{ in}}}\\ {t_{BS2,ss}}{t_{BS1,ss}}{r_{PM2,ss}}{r_{S,pp}}{r_{PM1,ss}} \cdot {E_{s,PM1\textrm{ in}}} \end{array} \right). \end{aligned}$$
Let's substitute the following expressions with:
$$\begin{array}{l} {q_{11}} = {t_{BS2,pp}}{t_{BS1,pp}}{r_{PM2,pp}}{r_{PM1,pp}}\\ {q_{12}} = {t_{BS2,ss}}{t_{BS1,ss}}{r_{PM2,ss}}{r_{PM1,ss}}. \end{array}$$
Before reaching the spectrometer, which is the detection device, the light is passed through an analyzer A1. Therefore:
$$\begin{aligned} E_{CH1}^T &= \left( {\begin{array}{{cc}} 1&0\\ 0&0 \end{array}} \right)\left( {\begin{array}{{cc}} {\cos {A_1}}&{\sin {A_1}}\\ { - \sin {A_1}}&{\cos {A_1}} \end{array}} \right)E_{BS,CH1}^T = \\ &= \left( {\begin{array}{{cc}} {\cos {A_1}}&{\sin {A_1}}\\ 0&0 \end{array}} \right)\left( \begin{array}{c} {q_{11}}{r_{S,ss}} \cdot {E_{p,PM1\textrm{ in}}}\\ {q_{12}}{r_{S,pp}} \cdot {E_{s,PM1\textrm{ in}}} \end{array} \right)\\ &= \left( \begin{array}{c} \cos {A_1} \cdot {q_{11}}{r_{S,ss}} \cdot {E_{p,PM1\textrm{ in}}} + \sin {A_1} \cdot {q_{12}}{r_{S,pp}} \cdot {E_{s,PM1\textrm{ in}}}\\ 0 \end{array} \right) \end{aligned},$$
and the intensity is therefore the multiplication of the field by its complex conjugate:
$$\begin{aligned} {I_{CH1}} &= {|{\cos {A_1} \cdot {q_{11}}{r_{S,ss}} \cdot {E_{p,PM1\textrm{ in}}} + \sin {A_1} \cdot {q_{12}}{r_{S,pp}} \cdot {E_{s,PM1\textrm{ in}}}} |^2}\\ &= {|{{q_{11}}{r_{S,ss}}} |^2}{\cos ^2}{A_1}{\cos ^2}({{\alpha_{PM}} - {\theta_P}} )+ {|{{q_{12}}{r_{S,pp}}} |^2}{\sin ^2}{A_1}{\sin ^2}({{\alpha_{PM}} - {\theta_P}} )\\ &\textrm{ } + 0.5\sin ({2{A_1}} )\sin ({2({{\alpha_{PM}} - {\theta_P}} )} ){\mathop{\textrm {Re}}\nolimits} ({{q_{11}}{r_{S,ss}} \cdot {{({{q_{12}}{r_{S,pp}}} )}^\ast }} )\\ &= {|{{q_{11}}{r_{S,ss}}} |^2}{\cos ^2}{A_1}{\cos ^2}({{\alpha_{PM}} - {\theta_P}} )+ {|{{q_{12}}{r_{S,pp}}} |^2}{\sin ^2}{A_1}{\sin ^2}({{\alpha_{PM}} - {\theta_P}} )\\ &\textrm{ } + 0.5\sin ({2{A_1}} )\sin ({2({{\alpha_{PM}} - {\theta_P}} )} ){\mathop{\textrm {Re}}\nolimits} ({{q_{11}}q_{12}^\ast {r_{S,ss}}{r^\ast }_{S,pp}} )\end{aligned},$$
For CH2 the expression is:
$$\begin{aligned} E_{BS,CH1}^T &= \left( {\begin{array}{{cc}} {{r_{BS2,pp}}}&0\\ 0&{{r_{BS2,ss}}} \end{array}} \right)\left( {\begin{array}{{cc}} {{t_{BS1,pp}}}&0\\ 0&{{t_{BS1,ss}}} \end{array}} \right)\left( \begin{array}{l} {E_{p,PM2\textrm{ out}}}\\ {E_{s,PM2\textrm{ out}}} \end{array} \right)\\ &= \left( \begin{array}{c} {r_{BS2,pp}}{t_{BS1,pp}} \cdot {E_{p,PM2\textrm{ out}}}\\ {r_{BS2,ss}}{t_{BS1,ss}} \cdot {E_{s,PM2\textrm{ out}}} \end{array} \right) = \left( \begin{array}{c} {r_{BS2,pp}}{t_{BS1,pp}}{r_{PM2,pp}}{r_{S,ss}}{r_{PM1,pp}} \cdot {E_{p,PM1\textrm{ in}}}\\ {r_{BS2,ss}}{t_{BS1,ss}}{r_{PM2,ss}}{r_{S,pp}}{r_{PM1,ss}} \cdot {E_{s,PM1\textrm{ in}}} \end{array} \right) \end{aligned}.$$
Let's substitute the following expressions with:
$$\begin{array}{l} {q_{21}} = {r_{BS2,pp}}{t_{BS1,pp}}{r_{PM2,pp}}{r_{PM1,pp}}\\ {q_{22}} = {r_{BS2,ss}}{t_{BS1,ss}}{r_{PM2,ss}}{r_{PM1,ss}} \end{array}.$$
In CH2 we have an analyzer A2:
$$\begin{aligned} E_{CH2}^T &= \left( {\begin{array}{{cc}} 1&0\\ 0&0 \end{array}} \right)\left( {\begin{array}{{cc}} {\cos {A_2}}&{\sin {A_2}}\\ { - \sin {A_2}}&{\cos {A_2}} \end{array}} \right)E_{BS,CH2}^T = \\ &= \left( {\begin{array}{{cc}} {\cos {A_2}}&{\sin {A_2}}\\ 0&0 \end{array}} \right)\left( \begin{array}{c} {q_{21}}{r_{S,ss}} \cdot {E_{p,PM1\textrm{ in}}}\\ {q_{22}}{r_{S,pp}} \cdot {E_{s,PM1\textrm{ in}}} \end{array} \right)\\ &= \left( \begin{array}{c} \cos {A_2} \cdot {q_{21}}{r_{S,ss}} \cdot {E_{p,PM1\textrm{ in}}} + \sin {A_2} \cdot {q_{22}}{r_{S,pp}} \cdot {E_{s,PM1\textrm{ in}}}\\ 0 \end{array} \right) \end{aligned},$$
and the intensity is:
$$\begin{aligned} {I_{CH2}} &= {|{\cos {A_2} \cdot {q_{21}}{r_{S,ss}} \cdot {E_{p,PM1\textrm{ in}}} + \sin {A_2} \cdot {q_{22}}{r_{S,pp}} \cdot {E_{s,PM1\textrm{ in}}}} |^2}\\ &= {|{r{q_{21}}{r_{S,ss}}} |^2}{\cos ^2}{A_2}{\cos ^2}({{\alpha_{PM}} - {\theta_P}} )+ {|{{q_{22}}{r_{S,pp}}} |^2}{\sin ^2}{A_2}{\sin ^2}({{\alpha_{PM}} - {\theta_P}} )\\ &\textrm{ } + 0.5\sin ({2{A_2}} )\sin ({2({{\alpha_{PM}} - {\theta_P}} )} ){\mathop{\textrm {Re}}\nolimits} ({{q_{21}}{r_{S,ss}} \cdot {{({{q_{22}}{r_{S,pp}}} )}^\ast }} )\\ &= {|{{q_{21}}{r_{S,ss}}} |^2}{\cos ^2}{A_2}{\cos ^2}({{\alpha_{PM}} - {\theta_P}} )+ {|{{q_{22}}{r_{S,pp}}} |^2}{\sin ^2}{A_2}{\sin ^2}({{\alpha_{PM}} - {\theta_P}} )\\ &\textrm{ } + 0.5\sin ({2{A_2}} )\sin ({2({{\alpha_{PM}} - {\theta_P}} )} ){\mathop{\textrm {Re}}\nolimits} ({{q_{21}}q_{22}^\ast {r_{S,ss}}{r^\ast }_{S,pp}} )\end{aligned},$$
For CH3 the expression is:
$$\begin{aligned} E_{BS,CH3}^T &= \left( {\begin{array}{{cc}} {{t_{BS2,pp}}}&0\\ 0&{{t_{BS2,ss}}} \end{array}} \right)\left( {\begin{array}{{cc}} {{r_{BS1,pp}}}&0\\ 0&{{r_{BS1,ss}}} \end{array}} \right)\left( \begin{array}{l} {E_{p,PM2\textrm{ out}}}\\ {E_{s,PM2\textrm{ out}}} \end{array} \right)\\& = \left( \begin{array}{c} {t_{BS2,pp}}{r_{BS1,pp}} \cdot {E_{p,PM2\textrm{ out}}}\\ {t_{BS2,ss}}{r_{BS1,ss}} \cdot {E_{s,PM2\textrm{ out}}} \end{array} \right) = \left( \begin{array}{c} {t_{BS2,pp}}{r_{BS1,pp}}{r_{PM2,pp}}{r_{S,ss}}{r_{PM1,pp}} \cdot {E_{p,PM1\textrm{ in}}}\\ {t_{BS2,ss}}{r_{BS1,ss}}{r_{PM2,ss}}{r_{S,pp}}{r_{PM1,ss}} \cdot {E_{s,PM1\textrm{ in}}} \end{array} \right) .\end{aligned}$$
Let's substitute the following expressions with:
$$\begin{aligned} {q_{31}} &= {t_{BS2,pp}}{r_{BS1,pp}}{r_{PM2,pp}}{r_{PM1,pp}}\\ {q_{32}} &= {t_{BS2,ss}}{r_{BS1,ss}}{r_{PM2,ss}}{r_{PM1,ss}}. \end{aligned}$$
In CH2 we have an analyzer A3:
$$\begin{aligned} E_{CH3}^T &= \left( {\begin{array}{{cc}} 1&0\\ 0&0 \end{array}} \right)\left( {\begin{array}{{cc}} {\cos {A_3}}&{\sin {A_3}}\\ { - \sin {A_3}}&{\cos {A_3}} \end{array}} \right)E_{BS,CH3}^T\\ &= \left( {\begin{array}{{cc}} {\cos {A_3}}&{\sin {A_3}}\\ 0&0 \end{array}} \right)\left( \begin{array}{c} {q_{31}}{r_{S,ss}} \cdot {E_{p,PM1\textrm{ in}}}\\ {q_{32}}{r_{S,pp}} \cdot {E_{s,PM1\textrm{ in}}} \end{array} \right)\\ &= \left( \begin{array}{c} \cos {A_3} \cdot {q_{31}}{r_{S,ss}} \cdot {E_{p,PM1\textrm{ in}}} + \sin {A_3} \cdot {q_{32}}{r_{S,pp}} \cdot {E_{s,PM1\textrm{ in}}}\\ 0 \end{array} \right) \end{aligned},$$
and the intensity is:
$$\begin{aligned} {I_{CH3}} &= {|{\cos {A_3} \cdot {q_{31}}{r_{S,ss}} \cdot {E_{p,PM1\textrm{ in}}} + \sin {A_3} \cdot {q_{32}}{r_{S,pp}} \cdot {E_{s,PM1\textrm{ in}}}} |^2}\\ &= {|{{q_{31}}{r_{S,ss}}} |^2}{\cos ^2}{A_3}{\cos ^2}({{\alpha_{PM}} - {\theta_P}} )+ {|{{q_{32}}{r_{S,pp}}} |^2}{\sin ^2}{A_3}{\sin ^2}({{\alpha_{PM}} - {\theta_P}} )\\ &\quad + 0.5\sin ({2{A_3}} )\sin ({2({{\alpha_{PM}} - {\theta_P}} )} ){\mathop{\textrm {Re}}\nolimits} ({{q_{31}}{r_{S,ss}} \cdot {{({{q_{32}}{r_{S,pp}}} )}^\ast }} )\\ &= {|{{q_{31}}{r_{S,ss}}} |^2}{\cos ^2}{A_3}{\cos ^2}({{\alpha_{PM}} - {\theta_P}} )+ {|{{q_{32}}{r_{S,pp}}} |^2}{\sin ^2}{A_3}{\sin ^2}({{\alpha_{PM}} - {\theta_P}} )\\ & \quad + 0.5\sin ({2{A_3}} )\sin ({2({{\alpha_{PM}} - {\theta_P}} )} ){\mathop{\textrm {Re}}\nolimits} ({{q_{31}}q_{32}^\ast {r_{S,ss}}{r^\ast }_{S,pp}} )\end{aligned},$$
In each channel the light is collected by a focusing lens, this is taken into account by a1, a2 and a3 in the following expressions:
$$\begin{aligned} {I_{CH1}} &= {a_1}\{{{{|{{q_{11}}{r_{S,ss}}} |}^2}{{\cos }^2}{A_1}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )+ {{|{{q_{12}}{r_{S,pp}}} |}^2}{{\sin }^2}{A_1}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} )} \\ & \quad\textrm{ } {\textrm{ } + 0.5\sin ({2{A_1}} )\sin ({2({{\alpha_{PM}} - {\theta_P}} )} ){\mathop{\textrm {Re}}\nolimits} ({{q_{11}}q_{12}^\ast {r_{S,ss}}{r^\ast }_{S,pp}} )} \}\\ {I_{CH2}} &= {a_2}\{{{{|{{q_{21}}{r_{S,ss}}} |}^2}{{\cos }^2}{A_2}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )+ {{|{{q_{22}}{r_{S,pp}}} |}^2}{{\sin }^2}{A_2}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} )} \\ & \quad\textrm{ } { + 0.5\sin ({2{A_2}} )\sin ({2({{\alpha_{PM}} - {\theta_P}} )} ){\mathop{\textrm {Re}}\nolimits} ({{q_{21}}q_{22}^\ast {r_{S,ss}}{r^\ast }_{S,pp}} )} \}\\ {I_{CH3}} &= {a_3}\{{{{|{{q_{31}}{r_{S,ss}}} |}^2}{{\cos }^2}{A_3}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )+ {{|{{q_{32}}{r_{S,pp}}} |}^2}{{\sin }^2}{A_3}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} )} \\ & \quad\textrm{ } {\textrm{ } + 0.5\sin ({2{A_3}} )\sin ({2({{\alpha_{PM}} - {\theta_P}} )} ){\mathop{\textrm {Re}}\nolimits} ({{q_{31}}q_{32}^\ast {r_{S,ss}}{r^\ast }_{S,pp}} )} \}\end{aligned}.$$
Calibration process

For the system calibration we take a known sample, for example a silicon wafer with a ∼2nm native oxide layer.

Let us choose the following angles (degrees) for the analyzers: ${A_1} = 0;\textrm{ }{A_2} = 90;\textrm{ }{A_3} = 0$.

For these values, the intensities are simplified to the following expressions:

$$\begin{aligned} {I_{CH1,ref}} &= {a_1}\{{{{|{{q_{11}}{r_{ref,ss}}} |}^2}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )} \}\\ {I_{CH2,ref}} &= {a_2}\{{{{|{{q_{22}}{r_{ref,pp}}} |}^2}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} )} \}\\ {I_{CH3,ref,{A_3} = 0}} &= {a_3}\{{{{|{{q_{31}}{r_{ref,ss}}} |}^2}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )} \}\end{aligned}.$$
From Eq. (37) we can extract:
$$\begin{aligned} {a_1}{|{{q_{11}}} |^2} &= \frac{{{I_{CH1,ref}}}}{{{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{{r_{ref,ss}}} |}^2}}}\\ {a_2}{|{{q_{22}}} |^2} &= \frac{{{I_{CH2,ref}}}}{{{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{{r_{ref,pp}}} |}^2}}}\\ {a_3}{|{{q_{31}}} |^2} &= \frac{{{I_{CH3,ref,{A_3} = 0}}}}{{{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{{r_{ref,ss}}} |}^2}}} \end{aligned}.$$
Now let us define the analyzers to be at (note that only A3 is changed):${A_1} = 0;\textrm{ }{A_2} = 90;\textrm{ }{A_3} = 90$. In this case, only the expression for CH3 is changed:
$$\begin{aligned} {I_{CH1,ref}} &= {a_1}\{{{{|{{q_{11}}{r_{ref,ss}}} |}^2}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )} \}\\ {I_{CH2,ref}} &= {a_2}\{{{{|{{q_{22}}{r_{ref,pp}}} |}^2}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} )} \}\\ {I_{CH3,ref,{A_3} = 90}} &= {a_3}\{{{{|{{q_{32}}{r_{ref,pp}}} |}^2}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} )} \}\end{aligned}.$$
From Eq. (39) we can extract:
$$\begin{aligned} {a_1}{|{{q_{11}}} |^2} &= \frac{{{I_{CH1,ref}}}}{{{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{{r_{ref,ss}}} |}^2}}}\\ {a_2}{|{{q_{22}}} |^2} &= \frac{{{I_{CH2,ref}}}}{{{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{{r_{ref,pp}}} |}^2}}}\\ {a_3}{|{{q_{32}}} |^2} &= \frac{{{I_{CH3,ref,{A_3} = 90}}}}{{{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{{r_{ref,pp}}} |}^2}}} \end{aligned}.$$
Now let us define the analyzers to be at (note that again only A3 is changed):${A_1} = 0;\textrm{ }{A_2} = 90;\textrm{ }{A_3} = 45$. In this case, again, only the expression for the third channel changed:
$$\begin{aligned} {I_{CH1,ref}} &= {a_1}\{{{{|{{q_{11}}{r_{ref,ss}}} |}^2}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )} \}\\ {I_{CH2,ref}} &= {a_2}\{{{{|{{q_{22}}{r_{ref,pp}}} |}^2}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} )} \}\\ {I_{CH3,ref,{A_3} = 45}} &= \frac{{{a_3}}}{2}\left\{ \begin{array}{l} {|{{q_{31}}{r_{ref,ss}}} |^2}{\cos^2}({{\alpha_{PM}} - {\theta_P}} )+ {|{{q_{32}}{r_{ref,pp}}} |^2}{\sin^2}({{\alpha_{PM}} - {\theta_P}} )\\ + \sin ({2({_{PM} - {\theta_P}} )} ){\mathop{\textrm {Re}}\nolimits} ({{q_{31}}q_{32}^\ast {r_{ref,ss}}{r^\ast }_{ref,pp}} )\end{array} \right\} \end{aligned}.$$
Since ${\alpha _{PM}} - {\theta _P}$ is known, the cos/sin of this angle is a number.

From the expression for CH3 in Eq. (41) together with Eqs. (38) and (40) we receive:

$$\begin{array}{l} {a_3}{\mathop{\textrm {Re}}\nolimits} ({{q_{31}}q_{32}^\ast {r_{ref,ss}}{r^\ast }_{ref,pp}} )\\ = \frac{{2{I_{CH3,ref,{A_3} = 45}} - {a_3}{{|{{q_{31}}} |}^2}{{|{{r_{ref,ss}}} |}^2}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )- {a_3}{{|{{q_{32}}} |}^2}{{|{{r_{ref,pp}}} |}^2}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} )}}{{\sin ({2({{\alpha_{PM}} - {\theta_P}} )} )}}\\ = \frac{{2{I_{CH3,ref,{A_3} = 45}} - {I_{CH3,ref,{A_3} = 0}} - {I_{CH3,ref,{A_3} = 90}}}}{{\sin ({2({{\alpha_{PM}} - {\theta_P}} )} )}} \end{array}.$$
To extract q, the effect of the optical elements accumulated in the system, we need the following:
$$\frac{{{a_3}{\mathop{\textrm {Re}}\nolimits} ({{q_{31}}q_{32}^\ast {r_{ref,ss}}r_{ref,pp}^\ast } )}}{{{a_3}|{{q_{31}}q_{32}^\ast {r_{ref,ss}}r_{ref,pp}^\ast } |}} = \frac{{{\mathop{\textrm {Re}}\nolimits} ({Q \cdot {e^{iq}}{r_{ref,ss}}r_{ref,pp}^\ast } )}}{{Q|{{r_{ref,ss}}r_{ref,pp}^\ast } |}} = \frac{{{\mathop{\textrm {Re}}\nolimits} ({{e^{iq}}{r_{ref,ss}}r_{ref,pp}^\ast } )}}{{|{{r_{ref,ss}}r_{ref,pp}^\ast } |}} = \cos ({{\Delta _{ref}} + q} ),$$
$$q = {\cos ^{ - 1}}({\cos ({{\Delta _{ref}} + q} )} )- {\Delta _{ref}}.$$
Note: for the previous transition Eq. (43) we need to remember that $\cos (w )= {\mathop{\textrm {Re}}\nolimits} ({{e^{iw}}} )\textrm{ },\textrm{ }{r_x} \cdot r_y^\ast{=} |{{r_x}r_y^\ast } |{e^{i({\varphi _x} - {\varphi _y})}}$.

Therefore, we need to express ${a_3}|{{q_{31}}q_{32}^\ast {r_{ref,ss}}r_{ref,pp}^\ast } |$ (same as ${a_3}|{{q_{31}}{q_{32}}{r_{ref,ss}}{r_{ref,pp}}} |$):

$$\begin{array}{l} {a_3}|{{q_{31}}{q_{32}}} |= \sqrt {{a_3}{{|{{q_{31}}} |}^2} \cdot {a_3}{{|{{q_{32}}} |}^2}} = \sqrt {\frac{{{I_{CH3,ref,{A_3} = 0}}}}{{{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{{r_{ref,ss}}} |}^2}}} \cdot \frac{{{I_{CH3,ref,{A_3} = 90}}}}{{{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{{r_{ref,pp}}} |}^2}}}} \\ {a_3}|{{q_{31}}{q_{32}}{r_{ref,ss}}{r_{ref,pp}}} |= \frac{{\sqrt {{I_{CH3,ref,{A_3} = 0}} \cdot {I_{CH3,ref,{A_3} = 90}}} }}{{|{\cos ({{\alpha_{PM}} - {\theta_P}} )} ||{\sin ({{\alpha_{PM}} - {\theta_P}} )} |}}\frac{1}{{|{{r_{ref,ss}}} ||{{r_{ref,pp}}} |}}\sqrt {{{|{{r_{ref,ss}}} |}^2}{{|{{r_{ref,pp}}} |}^2}} \\ {a_3}|{{q_{31}}{q_{32}}{r_{ref,ss}}{r_{ref,pp}}} |= \frac{{\sqrt {{I_{CH3,ref,{A_3} = 0}} \cdot {I_{CH3,ref,{A_3} = 90}}} }}{{|{\cos ({{\alpha_{PM}} - {\theta_P}} )} ||{\sin ({{\alpha_{PM}} - {\theta_P}} )} |}} \end{array}.$$
Now we can find the added value q to the measurement of Δ.

Measurement

For the measurement, the analyzers remained fixed at the same orientation as in the third measurement of the calibration:${A_1} = 0;\textrm{ }{A_2} = 90;\textrm{ }{A_3} = 45$. The sample of interest is placed instead of the reference, and now we have:

$$\begin{array}{l} {I_{CH1}} = {a_1}\{{{{|{{q_{11}}{r_{S,ss}}} |}^2}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )} \}\\ {I_{CH2}} = {a_2}\{{{{|{{q_{22}}{r_{S,pp}}} |}^2}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} )} \}\\ {I_{CH3}} = \frac{{{a_3}}}{2}\left\{ \begin{array}{l} {|{{q_{31}}{r_{S,ss}}} |^2}{\cos^2}({{\alpha_{PM}} - {\theta_P}} )+ {|{{q_{32}}{r_{S,pp}}} |^2}{\sin^2}({{\alpha_{PM}} - {\theta_P}} )\\ + \sin ({2({_{PM} - {\theta_P}} )} ){\mathop{\textrm {Re}}\nolimits} ({{q_{31}}q_{32}^\ast {r_{S,ss}}r_{S,pp}^\ast } )\end{array} \right\} \end{array}.$$
From the first two expressions in Eq. (46) we receive:
$$\begin{array}{l} {|{{r_{S,ss}}} |^2} = \frac{{{I_{CH1}}}}{{{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} ){a_1}{{|{{q_{11}}} |}^2}}} = \frac{{{I_{CH1}}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{{r_{ref,ss}}} |}^2}}}{{{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} ){I_{CH1,ref}}}} = \frac{{{I_{CH1}}}}{{{I_{CH1,ref}}}}{|{{r_{ref,ss}}} |^2}\\ {|{{r_{S,pp}}} |^2} = \frac{{{I_{CH2}}}}{{{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} ){a_2}{{|{{q_{22}}} |}^2}}} = \frac{{{I_{CH2}}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{{r_{ref,pp}}} |}^2}}}{{{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} ){I_{CH2,ref}}}} = \frac{{{I_{CH2}}}}{{{I_{CH2,ref}}}}{|{{r_{ref,pp}}} |^2} \end{array}.$$
From Eq. (47) we receive the ellipsometric parameter Ψ is:
$$\tan (\Psi )= \left|{\frac{{{r_{S,pp}}}}{{{r_{S,ss}}}}} \right|= \sqrt {\frac{{\frac{{{I_{CH2}}}}{{{I_{CH2,ref}}}}{{|{{r_{ref,pp}}} |}^2}}}{{\frac{{{I_{CH1}}}}{{{I_{CH1,ref}}}}{{|{{r_{ref,ss}}} |}^2}}}} = \frac{{|{{r_{ref,pp}}} |}}{{|{{r_{ref,ss}}} |}}\sqrt {\frac{{{I_{CH2}}}}{{{I_{CH2,ref}}}}\frac{{{I_{CH1,ref}}}}{{{I_{CH1}}}}} .$$
From the expression for CH3 in Eq. (46) together with Eqs. (38) and (40) we receive:
$$\begin{array}{l} {a_3}{\mathop{\textrm {Re}}\nolimits} ({{q_{31}}q_{32}^\ast {r_{S,ss}}{r^\ast }_{S,pp}} )\\ = \frac{{2{I_{CH3,S}} - {a_3}{{|{{q_{31}}} |}^2}{{|{{r_{S,ss}}} |}^2}{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} )- {a_3}{{|{{q_{32}}} |}^2}{{|{{r_{S,pp}}} |}^2}{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} )}}{{\sin ({2({{\alpha_{PM}} - {\theta_P}} )} )}}\\ = \frac{{2{I_{CH3,S}} - \frac{{{I_{CH3,ref,{A_3} = 0}}}}{{{{|{{r_{ref,ss}}} |}^2}}}\frac{{{I_{CH1}}}}{{{I_{CH1,ref}}}}{{|{{r_{ref,ss}}} |}^2} - \frac{{{I_{CH3,ref,{A_3} = 90}}}}{{{{|{{r_{ref,pp}}} |}^2}}}\frac{{{I_{CH2}}}}{{{I_{CH2,ref}}}}{{|{{r_{ref,pp}}} |}^2}}}{{\sin ({2({{\alpha_{PM}} - {\theta_P}} )} )}}\\ = \frac{{2{I_{CH3,S}} - \frac{{{I_{CH3,ref,{A_3} = 0}} \cdot {I_{CH1}}}}{{{I_{CH1,ref}}}} - \frac{{{I_{CH3,ref,{A_3} = 90}} \cdot {I_{CH2}}}}{{{I_{CH2ref}}}}}}{{\sin ({2({{\alpha_{PM}} - {\theta_P}} )} )}} \end{array}.$$
Now let's divide this by ${a_3}|{{q_{31}}q_{32}^\ast {r_{S,ss}}r_{S,pp}^\ast } |$ (same as ${a_3}|{{q_{31}}{q_{32}}{r_{S,ss}}{r_{S,pp}}} |$):
$${a_3}|{{q_{31}}{q_{32}}} |= \sqrt {{a_3}{{|{{q_{31}}} |}^2} \cdot {a_3}{{|{{q_{32}}} |}^2}} = \sqrt {\frac{{{I_{CH3,ref,{A_3} = 0}}}}{{{{\cos }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{{r_{ref,ss}}} |}^2}}} \cdot \frac{{{I_{CH3,ref,{A_3} = 90}}}}{{{{\sin }^2}({{\alpha_{PM}} - {\theta_P}} ){{|{{r_{ref,pp}}} |}^2}}}} .$$
Using Eq. (47) we receive:
$$\begin{array}{l} {a_3}|{{q_{31}}{q_{32}}{r_{S,ss}}{r_{S,pp}}} |\\ = \frac{{\sqrt {{I_{CH3,ref,{A_3} = 0}} \cdot {I_{CH3,ref,{A_3} = 90}}} }}{{|{\cos ({{\alpha_{PM}} - {\theta_P}} )} ||{\sin ({{\alpha_{PM}} - {\theta_P}} )} |}}\frac{1}{{|{{r_{ref,ss}}} ||{{r_{ref,pp}}} |}}\sqrt {\frac{{{I_{CH1}}}}{{{I_{CH1,ref}}}}{{|{{r_{ref,ss}}} |}^2}\frac{{{I_{CH2}}}}{{{I_{CH2,ref}}}}{{|{{r_{ref,pp}}} |}^2}} \\ = \frac{{\sqrt {{I_{CH3,ref,{A_3} = 0}} \cdot {I_{CH3,ref,{A_3} = 90}}} }}{{|{\cos ({{\alpha_{PM}} - {\theta_P}} )} ||{\sin ({{\alpha_{PM}} - {\theta_P}} )} |}}\sqrt {\frac{{{I_{CH1}}}}{{{I_{CH1,ref}}}}\frac{{{I_{CH2}}}}{{{I_{CH2,ref}}}}} \end{array}.$$
Now we can use the expression for q from Eq. (44) from the calibration and extract Δ:
$$\frac{{{a_3}{\mathop{\textrm {Re}}\nolimits} ({{q_{31}}q_{32}^\ast {r_{S,ss}}r_{S,pp}^\ast } )}}{{{a_3}|{{q_{31}}q_{32}^\ast {r_{S,ss}}r_{S,pp}^\ast } |}} = \frac{{{\mathop{\textrm {Re}}\nolimits} ({Q \cdot {e^{iq}}{r_{S,ss}}r_{S,pp}^\ast } )}}{{Q|{{r_{S,ss}}r_{S,pp}^\ast } |}} = \frac{{{\mathop{\textrm {Re}}\nolimits} ({{e^{iq}}{r_{S,ss}}r_{S,pp}^\ast } )}}{{|{{r_{S,ss}}r_{S,pp}^\ast } |}} = \cos ({\Delta + q} ),$$
$$\Delta = {\cos ^{ - 1}}({\cos ({\Delta + q} )} )- q.$$
Therefore, using Eqs. (48) and (53) the ellipsometric parameters are extracted. The values extracted by Eqs. (48) and (53) are compared, for example, to a data base until the best match is found.

Funding

Ministry of Science, Technology and Space (3-15200).

Acknowledgments

This research is supported partially by the Israel Ministry of Science, Technology and Space through the binational Israel-Slovania program.

Disclosures

The authors declare no conflicts of interest.

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Figures (9)
Fig. 1.
Fig. 1. Schematic drawing of parallel spectroscopic ellipsometer. L1-L3 – light collection lenses, A1-A3 – Analyzers, BS – beam splitter, M1-M2 – mirrors, P – polarizer, L4-L5 – beam shrinker, L6 – light source collimation lens, θ – incidence angle of illumination upon the sample.

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Fig. 2.
Fig. 2. Measured Ψ and Δ for silicon oxide sample of 5µm. (a) - incidence angle of 64deg, extreacted thickness is 5.02µm with RMSE of 31.37deg, (b) - incidence angle of 67deg, extreacted thickness is 5.02µm with RMSE of 8.08deg, (c) - incidence angle of 70deg, extreacted thickness is 5.03µm with RMSE of 12.6deg, (d) - incidence angle of 64deg, extreacted thickness is 5.02µm with RMSE of 16.47deg.

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Fig. 3.
Fig. 3. Measured Ψ and Δ for polystyrene sample of 51nm, the extreacted thickness is 49.98nm with RMSE of 11.7471deg (a), sample of 60nm, the extreacted thickness is 60.00nm with RMSE of 12.4266deg (b), sample of 66nm, the extreacted thickness is 64.97nm with RMSE of 46.4949deg (c), sample of 70nm, the extreacted thickness is 69.97nm with RMSE of 50.2605deg (d).

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Fig. 4.
Fig. 4. Measured Ψ and Δ for polystyrene sample of 172 nm, the extreacted thickness is 174.92 nm with RMSE of 10.8772deg (a), sample of 180 nm, the extreacted thickness is 174.92 nm with RMSE of 8.0036deg (b), sample of 201 nm, the extreacted thickness is 199.90 nm with RMSE of 8.2324deg (c), sample of 219 nm, the extreacted thickness is 224.89 nm with RMSE of 10.208deg (d), sample of 230 nm, the extreacted thickness is 219.89 nm with RMSE of 9.2275deg (e), sample of 318 nm, the extreacted thickness is 319.84 nm with RMSE of 16.8384deg (f).

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Fig. 5.
Fig. 5. Measured Ψ and Δ for silicon nitride sample of 100 nm. The extracted thickness (d) and the corresponding minimal RMSE are stated in the title of each graph.

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Fig. 6.
Fig. 6. Measured Ψ and Δ for silicon oxide sample of 100 nm. The extracted thickness (d) and the corresponding minimal RMSE are stated in the title of each graph.

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Fig. 7.
Fig. 7. Thickness and RMSE for a burst of 50 measurements for a 5µm SiO2 sample.

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Fig. 8.
Fig. 8. Thickness and RMSE for a lateral scan of a 5µm SiO2 sample at various scan rates.

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Fig. 9.
Fig. 9. Thickness and RMSE for a lateral scan of two adjacent samples of 100 nm and 5µm SiO2 samples.

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Tables (1)

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Table 1. Measurement results of a 5µm SiO2 sample using the PSE system, and comparison to the results from Woollam’s alpha-SE ellipsometer results.

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Equations (53)

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I C H i = a i { | q i 1 r S , s s | 2 cos 2 A i cos 2 ( α P M − θ P ) + | q i 2 r S , p p | 2 sin 2 A i sin 2 ( α P M − θ P ) + sin ⁡ ( 2 A i ) sin ⁡ ( 2 ( α P M − θ P ) ) Re ( q i 1 q i 2 ∗ r S , s s r S , p p ∗ ) } ,
for i = 1 :   q 11 = t B S 2 , p p t B S 1 , p p r P M 2 , p p r P M 1 , p p  ;  q 12 = t B S 2 , s s t B S 1 , s s r P M 2 , s s r P M 1 , s s for i =  2 :   q 21 = r B S 2 , p p t B S 1 , p p r P M 2 , p p r P M 1 , p p  ;  q 22 = r B S 2 , s s t B S 1 , s s r P M 2 , s s r P M 1 , s s for i =  3 :   q 31 = t B S 2 , p p r B S 1 , p p r P M 2 , p p r P M 1 , p p  ;  q 32 = t B S 2 , s s r B S 1 , s s r P M 2 , s s r P M 1 , s s ,
I C H 1 = a 1 { | q 11 r S , s s | 2 cos 2 ( α P M − θ P ) } , I C H 2 = a 2 { | q 22 r S , p p | 2 sin 2 ( α P M − θ P ) } , I C H 3 = a 3 2 { | q 31 r S , s s | 2 cos 2 ( α P M − θ P ) + | q 32 r S , p p | 2 sin 2 ( α P M − θ P ) + sin ⁡ ( 2 ( α P M − θ P ) ) Re ( q i 1 q i 2 ∗ r S , s s r S , p p ∗ ) } .
tan ⁡ ( Ψ ) = | r S , p p r S , s s | = | r r e f , p p | | r r e f , s s | I C H 2 I C H 2 , r e f I C H 1. r e f I C H 1 .
a 3 Re ( q 31 q 32 ∗ r S , s s r S , p p ∗ ) = 2 I C H 3 − a 3 | q 31 r S , s s | 2 cos 2 ( α P M − θ P ) − a 3 | q 32 r S , p p | 2 sin 2 ( α P M − θ P ) sin ⁡ ( 2 ( α P M − θ P ) ) ,
a 3 Re ( q 31 q 32 ∗ r S , s s r S , p p ∗ ) = 2 I C H 3 − I C H 3 , r e f , A 3 = 0 I C H 1 I C H 1 , r e f − I C H 3 , r e f , A 3 = 90 I C H 2 I C H 2 , r e f sin ⁡ ( 2 ( α P M − θ P ) ) .
a 3 Re ( q 31 q 32 ∗ r S , s s r S , p p ∗ ) a 3 | q 31 q 32 ∗ r S , s s r S , p p ∗ | = cos ⁡ ( q + Δ ) .
q = cos − 1 ( cos ⁡ ( q + Δ r e f ) ) − Δ r e f Δ = cos − 1 ( cos ⁡ ( Δ + q ) ) − q .
I C H 1 , r e f = a 1 { | q 11 r r e f , s s | 2 cos 2 ( α P M − θ P ) } I C H 2 , r e f = a 2 { | q 22 r r e f , p p | 2 sin 2 ( α P M − θ P ) } I C H 3 , r e f , A 3 = 0 = a 3 { | q 31 r r e f , s s | 2 cos 2 ( α P M − θ P ) } .
a 1 | q 11 | 2 = I C H 1 , r e f cos 2 ( α P M − θ P ) | r r e f , s s | 2 a 2 | q 22 | 2 = I C H 2 , r e f sin 2 ( α P M − θ P ) | r r e f , p p | 2 a 3 | q 31 | 2 = I C H 3 , r e f , A 3 = 0 cos 2 ( α P M − θ P ) | r r e f , s s | 2 .
I C H 3 , r e f , A 3 = 90 = a 3 { | q 32 r r e f , s s | 2 cos 2 ( α P M − θ P ) } .
a 3 | q 32 | 2 = I C H 3 , r e f , A 3 = 90 sin 2 ( α P M − θ P ) | r r e f , p p | 2 .
I C H 3 , r e f , A 3 = 45 = a 3 2 { | q 31 r r e f , s s | 2 cos 2 ( α P M − θ P ) + | q 32 r r e f , p p | 2 sin 2 ( α P M − θ P ) + sin ⁡ ( 2 ( α P M − θ P ) ) 2 Re ( q 31 q 32 ∗ r r e f , s s r r e f , p p ∗ ) } ,
a 3 Re ( q 31 q 32 ∗ r r e f , s s r ∗ r e f , p p ) = 2 I C H 3 , r e f , A 3 = 45 − a 3 | q 31 | 2 | r r e f , s s | 2 cos 2 ( α P M − θ P ) − a 3 | q 32 | 2 | r r e f , p p | 2 sin 2 ( α P M − θ P ) sin ⁡ ( 2 ( α P M − θ P ) ) = 2 I C H 3 , r e f , A 3 = 45 − I C H 3 , r e f , A 3 = 0 − I C H 3 , r e f , A 3 = 90 sin ⁡ ( 2 ( α P M − θ P ) ) .
a 3 | q 31 q 32 r r e f , s s ∗ r r e f , p p ∗ | = I C H 3 , r e f , A 3 = 0 ⋅ I C H 3 , r e f , A 3 = 90 | cos ⁡ ( α P M − θ P ) | | sin ⁡ ( α P M − θ P ) | .
a 3 Re ( q 31 q 32 ∗ r r e f , s s r r e f , p p ∗ ) a 3 | q 31 q 32 r r e f , s s ∗ r r e f , p p ∗ | = cos ⁡ ( q + Δ r e f ) .
q = cos − 1 ( cos ⁡ ( q + Δ r e f ) ) − Δ r e f .
n 2 ( S i O 2 ) = 1 + 0.6961663 λ 2 λ 2 − 0.0684043 2 + 0.4079426 λ 2 λ 2 − 0.1162414 2 + 0.8974794 λ 2 λ 2 − 9.896161 2 n 2 ( S i 3 N 4 ) = 1 + 2.8939 λ 2 λ 2 − 0.13967 2 n 2 ( ( C 8 H 8 ) n ) = 1 + 1.4435 λ 2 λ 2 − 0.020216 .
k i n = ( 0 0 1 ) ,
E i n = ( E x E y 0 ) = ( E p E s 0 ) = E ( cos ⁡ θ p sin ⁡ θ p 0 ) .
( E p , P M 1  out E s , P M 1  out ) = ( r P M 1 , p p 0 0 r P M 1 , s s ) ( E p , P M 1  in E s , P M 1  in ) = ( r P M 1 , p p ⋅ E p , P M 1  in r P M 1 , s s ⋅ E s , P M 1  in ) .
( E p , S  out E s , S  out ) = ( r S , p p 0 0 r S , s s ) ( E p , S  in E s , S  in ) = ( r S , p p 0 0 r S , s s ) ( E s , P M 1  out − E p , P M 1  out ) = ( r S , p p ⋅ E s , P M 1  out − r S , s s ⋅ E p , P M 1  out ) = ( r S , p p r P M 1 , s s ⋅ E s , P M 1  in − r S , s s r P M 1 , p p ⋅ E p , P M 1  in ) .
( E p , P M 2  out E s , P M 2  out ) = ( r P M 2 , p p 0 0 r P M 2 , s s ) ( E p , P M 2  in E s , P M 2  in ) = ( r P M 2 , p p 0 0 r P M 2 , s s ) ( − E s , S  out E p , S  out ) = ( − r P M 2 , p p ⋅ E s , S  out r P M 2 , s s ⋅ E p , S  out ) = ( r P M 2 , p p r S , s s r P M 1 , p p ⋅ E p , P M 1  in r P M 2 , s s r S , p p r P M 1 , s s ⋅ E s , P M 1  in ) .
E B S , C H 1 T = ( t B S 2 , p p 0 0 t B S 2 , s s ) ( t B S 1 , p p 0 0 t B S 1 , s s ) ( E p , P M 2  out E s , P M 2  out ) = ( t B S 2 , p p t B S 1 , p p ⋅ E p , P M 2  out t B S 2 , s s t B S 1 , s s ⋅ E s , P M 2  out ) = ( t B S 2 , p p t B S 1 , p p r P M 2 , p p r S , s s r P M 1 , p p ⋅ E p , P M 1  in t B S 2 , s s t B S 1 , s s r P M 2 , s s r S , p p r P M 1 , s s ⋅ E s , P M 1  in ) .
q 11 = t B S 2 , p p t B S 1 , p p r P M 2 , p p r P M 1 , p p q 12 = t B S 2 , s s t B S 1 , s s r P M 2 , s s r P M 1 , s s .
E C H 1 T = ( 1 0 0 0 ) ( cos ⁡ A 1 sin ⁡ A 1 − sin ⁡ A 1 cos ⁡ A 1 ) E B S , C H 1 T = = ( cos ⁡ A 1 sin ⁡ A 1 0 0 ) ( q 11 r S , s s ⋅ E p , P M 1  in q 12 r S , p p ⋅ E s , P M 1  in ) = ( cos ⁡ A 1 ⋅ q 11 r S , s s ⋅ E p , P M 1  in + sin ⁡ A 1 ⋅ q 12 r S , p p ⋅ E s , P M 1  in 0 ) ,
I C H 1 = | cos ⁡ A 1 ⋅ q 11 r S , s s ⋅ E p , P M 1  in + sin ⁡ A 1 ⋅ q 12 r S , p p ⋅ E s , P M 1  in | 2 = | q 11 r S , s s | 2 cos 2 A 1 cos 2 ( α P M − θ P ) + | q 12 r S , p p | 2 sin 2 A 1 sin 2 ( α P M − θ P )   + 0.5 sin ⁡ ( 2 A 1 ) sin ⁡ ( 2 ( α P M − θ P ) ) Re ( q 11 r S , s s ⋅ ( q 12 r S , p p ) ∗ ) = | q 11 r S , s s | 2 cos 2 A 1 cos 2 ( α P M − θ P ) + | q 12 r S , p p | 2 sin 2 A 1 sin 2 ( α P M − θ P )   + 0.5 sin ⁡ ( 2 A 1 ) sin ⁡ ( 2 ( α P M − θ P ) ) Re ( q 11 q 12 ∗ r S , s s r ∗ S , p p ) ,
E B S , C H 1 T = ( r B S 2 , p p 0 0 r B S 2 , s s ) ( t B S 1 , p p 0 0 t B S 1 , s s ) ( E p , P M 2  out E s , P M 2  out ) = ( r B S 2 , p p t B S 1 , p p ⋅ E p , P M 2  out r B S 2 , s s t B S 1 , s s ⋅ E s , P M 2  out ) = ( r B S 2 , p p t B S 1 , p p r P M 2 , p p r S , s s r P M 1 , p p ⋅ E p , P M 1  in r B S 2 , s s t B S 1 , s s r P M 2 , s s r S , p p r P M 1 , s s ⋅ E s , P M 1  in ) .
q 21 = r B S 2 , p p t B S 1 , p p r P M 2 , p p r P M 1 , p p q 22 = r B S 2 , s s t B S 1 , s s r P M 2 , s s r P M 1 , s s .
E C H 2 T = ( 1 0 0 0 ) ( cos ⁡ A 2 sin ⁡ A 2 − sin ⁡ A 2 cos ⁡ A 2 ) E B S , C H 2 T = = ( cos ⁡ A 2 sin ⁡ A 2 0 0 ) ( q 21 r S , s s ⋅ E p , P M 1  in q 22 r S , p p ⋅ E s , P M 1  in ) = ( cos ⁡ A 2 ⋅ q 21 r S , s s ⋅ E p , P M 1  in + sin ⁡ A 2 ⋅ q 22 r S , p p ⋅ E s , P M 1  in 0 ) ,
I C H 2 = | cos ⁡ A 2 ⋅ q 21 r S , s s ⋅ E p , P M 1  in + sin ⁡ A 2 ⋅ q 22 r S , p p ⋅ E s , P M 1  in | 2 = | r q 21 r S , s s | 2 cos 2 A 2 cos 2 ( α P M − θ P ) + | q 22 r S , p p | 2 sin 2 A 2 sin 2 ( α P M − θ P )   + 0.5 sin ⁡ ( 2 A 2 ) sin ⁡ ( 2 ( α P M − θ P ) ) Re ( q 21 r S , s s ⋅ ( q 22 r S , p p ) ∗ ) = | q 21 r S , s s | 2 cos 2 A 2 cos 2 ( α P M − θ P ) + | q 22 r S , p p | 2 sin 2 A 2 sin 2 ( α P M − θ P )   + 0.5 sin ⁡ ( 2 A 2 ) sin ⁡ ( 2 ( α P M − θ P ) ) Re ( q 21 q 22 ∗ r S , s s r ∗ S , p p ) ,
E B S , C H 3 T = ( t B S 2 , p p 0 0 t B S 2 , s s ) ( r B S 1 , p p 0 0 r B S 1 , s s ) ( E p , P M 2  out E s , P M 2  out ) = ( t B S 2 , p p r B S 1 , p p ⋅ E p , P M 2  out t B S 2 , s s r B S 1 , s s ⋅ E s , P M 2  out ) = ( t B S 2 , p p r B S 1 , p p r P M 2 , p p r S , s s r P M 1 , p p ⋅ E p , P M 1  in t B S 2 , s s r B S 1 , s s r P M 2 , s s r S , p p r P M 1 , s s ⋅ E s , P M 1  in ) .
q 31 = t B S 2 , p p r B S 1 , p p r P M 2 , p p r P M 1 , p p q 32 = t B S 2 , s s r B S 1 , s s r P M 2 , s s r P M 1 , s s .
E C H 3 T = ( 1 0 0 0 ) ( cos ⁡ A 3 sin ⁡ A 3 − sin ⁡ A 3 cos ⁡ A 3 ) E B S , C H 3 T = ( cos ⁡ A 3 sin ⁡ A 3 0 0 ) ( q 31 r S , s s ⋅ E p , P M 1  in q 32 r S , p p ⋅ E s , P M 1  in ) = ( cos ⁡ A 3 ⋅ q 31 r S , s s ⋅ E p , P M 1  in + sin ⁡ A 3 ⋅ q 32 r S , p p ⋅ E s , P M 1  in 0 ) ,
I C H 3 = | cos ⁡ A 3 ⋅ q 31 r S , s s ⋅ E p , P M 1  in + sin ⁡ A 3 ⋅ q 32 r S , p p ⋅ E s , P M 1  in | 2 = | q 31 r S , s s | 2 cos 2 A 3 cos 2 ( α P M − θ P ) + | q 32 r S , p p | 2 sin 2 A 3 sin 2 ( α P M − θ P ) + 0.5 sin ⁡ ( 2 A 3 ) sin ⁡ ( 2 ( α P M − θ P ) ) Re ( q 31 r S , s s ⋅ ( q 32 r S , p p ) ∗ ) = | q 31 r S , s s | 2 cos 2 A 3 cos 2 ( α P M − θ P ) + | q 32 r S , p p | 2 sin 2 A 3 sin 2 ( α P M − θ P ) + 0.5 sin ⁡ ( 2 A 3 ) sin ⁡ ( 2 ( α P M − θ P ) ) Re ( q 31 q 32 ∗ r S , s s r ∗ S , p p ) ,
I C H 1 = a 1 { | q 11 r S , s s | 2 cos 2 A 1 cos 2 ( α P M − θ P ) + | q 12 r S , p p | 2 sin 2 A 1 sin 2 ( α P M − θ P )     + 0.5 sin ⁡ ( 2 A 1 ) sin ⁡ ( 2 ( α P M − θ P ) ) Re ( q 11 q 12 ∗ r S , s s r ∗ S , p p ) } I C H 2 = a 2 { | q 21 r S , s s | 2 cos 2 A 2 cos 2 ( α P M − θ P ) + | q 22 r S , p p | 2 sin 2 A 2 sin 2 ( α P M − θ P )   + 0.5 sin ⁡ ( 2 A 2 ) sin ⁡ ( 2 ( α P M − θ P ) ) Re ( q 21 q 22 ∗ r S , s s r ∗ S , p p ) } I C H 3 = a 3 { | q 31 r S , s s | 2 cos 2 A 3 cos 2 ( α P M − θ P ) + | q 32 r S , p p | 2 sin 2 A 3 sin 2 ( α P M − θ P )     + 0.5 sin ⁡ ( 2 A 3 ) sin ⁡ ( 2 ( α P M − θ P ) ) Re ( q 31 q 32 ∗ r S , s s r ∗ S , p p ) } .
I C H 1 , r e f = a 1 { | q 11 r r e f , s s | 2 cos 2 ( α P M − θ P ) } I C H 2 , r e f = a 2 { | q 22 r r e f , p p | 2 sin 2 ( α P M − θ P ) } I C H 3 , r e f , A 3 = 0 = a 3 { | q 31 r r e f , s s | 2 cos 2 ( α P M − θ P ) } .
a 1 | q 11 | 2 = I C H 1 , r e f cos 2 ( α P M − θ P ) | r r e f , s s | 2 a 2 | q 22 | 2 = I C H 2 , r e f sin 2 ( α P M − θ P ) | r r e f , p p | 2 a 3 | q 31 | 2 = I C H 3 , r e f , A 3 = 0 cos 2 ( α P M − θ P ) | r r e f , s s | 2 .
I C H 1 , r e f = a 1 { | q 11 r r e f , s s | 2 cos 2 ( α P M − θ P ) } I C H 2 , r e f = a 2 { | q 22 r r e f , p p | 2 sin 2 ( α P M − θ P ) } I C H 3 , r e f , A 3 = 90 = a 3 { | q 32 r r e f , p p | 2 sin 2 ( α P M − θ P ) } .
a 1 | q 11 | 2 = I C H 1 , r e f cos 2 ( α P M − θ P ) | r r e f , s s | 2 a 2 | q 22 | 2 = I C H 2 , r e f sin 2 ( α P M − θ P ) | r r e f , p p | 2 a 3 | q 32 | 2 = I C H 3 , r e f , A 3 = 90 sin 2 ( α P M − θ P ) | r r e f , p p | 2 .
I C H 1 , r e f = a 1 { | q 11 r r e f , s s | 2 cos 2 ( α P M − θ P ) } I C H 2 , r e f = a 2 { | q 22 r r e f , p p | 2 sin 2 ( α P M − θ P ) } I C H 3 , r e f , A 3 = 45 = a 3 2 { | q 31 r r e f , s s | 2 cos 2 ( α P M − θ P ) + | q 32 r r e f , p p | 2 sin 2 ( α P M − θ P ) + sin ⁡ ( 2 ( P M − θ P ) ) Re ( q 31 q 32 ∗ r r e f , s s r ∗ r e f , p p ) } .
a 3 Re ( q 31 q 32 ∗ r r e f , s s r ∗ r e f , p p ) = 2 I C H 3 , r e f , A 3 = 45 − a 3 | q 31 | 2 | r r e f , s s | 2 cos 2 ( α P M − θ P ) − a 3 | q 32 | 2 | r r e f , p p | 2 sin 2 ( α P M − θ P ) sin ⁡ ( 2 ( α P M − θ P ) ) = 2 I C H 3 , r e f , A 3 = 45 − I C H 3 , r e f , A 3 = 0 − I C H 3 , r e f , A 3 = 90 sin ⁡ ( 2 ( α P M − θ P ) ) .
a 3 Re ( q 31 q 32 ∗ r r e f , s s r r e f , p p ∗ ) a 3 | q 31 q 32 ∗ r r e f , s s r r e f , p p ∗ | = Re ( Q ⋅ e i q r r e f , s s r r e f , p p ∗ ) Q | r r e f , s s r r e f , p p ∗ | = Re ( e i q r r e f , s s r r e f , p p ∗ ) | r r e f , s s r r e f , p p ∗ | = cos ⁡ ( Δ r e f + q ) ,
q = cos − 1 ( cos ⁡ ( Δ r e f + q ) ) − Δ r e f .
a 3 | q 31 q 32 | = a 3 | q 31 | 2 ⋅ a 3 | q 32 | 2 = I C H 3 , r e f , A 3 = 0 cos 2 ( α P M − θ P ) | r r e f , s s | 2 ⋅ I C H 3 , r e f , A 3 = 90 sin 2 ( α P M − θ P ) | r r e f , p p | 2 a 3 | q 31 q 32 r r e f , s s r r e f , p p | = I C H 3 , r e f , A 3 = 0 ⋅ I C H 3 , r e f , A 3 = 90 | cos ⁡ ( α P M − θ P ) | | sin ⁡ ( α P M − θ P ) | 1 | r r e f , s s | | r r e f , p p | | r r e f , s s | 2 | r r e f , p p | 2 a 3 | q 31 q 32 r r e f , s s r r e f , p p | = I C H 3 , r e f , A 3 = 0 ⋅ I C H 3 , r e f , A 3 = 90 | cos ⁡ ( α P M − θ P ) | | sin ⁡ ( α P M − θ P ) | .
I C H 1 = a 1 { | q 11 r S , s s | 2 cos 2 ( α P M − θ P ) } I C H 2 = a 2 { | q 22 r S , p p | 2 sin 2 ( α P M − θ P ) } I C H 3 = a 3 2 { | q 31 r S , s s | 2 cos 2 ( α P M − θ P ) + | q 32 r S , p p | 2 sin 2 ( α P M − θ P ) + sin ⁡ ( 2 ( P M − θ P ) ) Re ( q 31 q 32 ∗ r S , s s r S , p p ∗ ) } .
| r S , s s | 2 = I C H 1 cos 2 ( α P M − θ P ) a 1 | q 11 | 2 = I C H 1 cos 2 ( α P M − θ P ) | r r e f , s s | 2 cos 2 ( α P M − θ P ) I C H 1 , r e f = I C H 1 I C H 1 , r e f | r r e f , s s | 2 | r S , p p | 2 = I C H 2 sin 2 ( α P M − θ P ) a 2 | q 22 | 2 = I C H 2 sin 2 ( α P M − θ P ) | r r e f , p p | 2 sin 2 ( α P M − θ P ) I C H 2 , r e f = I C H 2 I C H 2 , r e f | r r e f , p p | 2 .
tan ⁡ ( Ψ ) = | r S , p p r S , s s | = I C H 2 I C H 2 , r e f | r r e f , p p | 2 I C H 1 I C H 1 , r e f | r r e f , s s | 2 = | r r e f , p p | | r r e f , s s | I C H 2 I C H 2 , r e f I C H 1 , r e f I C H 1 .
a 3 Re ( q 31 q 32 ∗ r S , s s r ∗ S , p p ) = 2 I C H 3 , S − a 3 | q 31 | 2 | r S , s s | 2 cos 2 ( α P M − θ P ) − a 3 | q 32 | 2 | r S , p p | 2 sin 2 ( α P M − θ P ) sin ⁡ ( 2 ( α P M − θ P ) ) = 2 I C H 3 , S − I C H 3 , r e f , A 3 = 0 | r r e f , s s | 2 I C H 1 I C H 1 , r e f | r r e f , s s | 2 − I C H 3 , r e f , A 3 = 90 | r r e f , p p | 2 I C H 2 I C H 2 , r e f | r r e f , p p | 2 sin ⁡ ( 2 ( α P M − θ P ) ) = 2 I C H 3 , S − I C H 3 , r e f , A 3 = 0 ⋅ I C H 1 I C H 1 , r e f − I C H 3 , r e f , A 3 = 90 ⋅ I C H 2 I C H 2 r e f sin ⁡ ( 2 ( α P M − θ P ) ) .
a 3 | q 31 q 32 | = a 3 | q 31 | 2 ⋅ a 3 | q 32 | 2 = I C H 3 , r e f , A 3 = 0 cos 2 ( α P M − θ P ) | r r e f , s s | 2 ⋅ I C H 3 , r e f , A 3 = 90 sin 2 ( α P M − θ P ) | r r e f , p p | 2 .
a 3 | q 31 q 32 r S , s s r S , p p | = I C H 3 , r e f , A 3 = 0 ⋅ I C H 3 , r e f , A 3 = 90 | cos ⁡ ( α P M − θ P ) | | sin ⁡ ( α P M − θ P ) | 1 | r r e f , s s | | r r e f , p p | I C H 1 I C H 1 , r e f | r r e f , s s | 2 I C H 2 I C H 2 , r e f | r r e f , p p | 2 = I C H 3 , r e f , A 3 = 0 ⋅ I C H 3 , r e f , A 3 = 90 | cos ⁡ ( α P M − θ P ) | | sin ⁡ ( α P M − θ P ) | I C H 1 I C H 1 , r e f I C H 2 I C H 2 , r e f .
a 3 Re ( q 31 q 32 ∗ r S , s s r S , p p ∗ ) a 3 | q 31 q 32 ∗ r S , s s r S , p p ∗ | = Re ( Q ⋅ e i q r S , s s r S , p p ∗ ) Q | r S , s s r S , p p ∗ | = Re ( e i q r S , s s r S , p p ∗ ) | r S , s s r S , p p ∗ | = cos ⁡ ( Δ + q ) ,
Δ = cos − 1 ( cos ⁡ ( Δ + q ) ) − q .
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