Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2023; 7(4): 428-434
Published online August 25, 2023 https://doi.org/10.3807/COPP.2023.7.4.428
Copyright © Optical Society of Korea.
Byeong-Kwan Yang^{1,2} , Jin Seung Kim^{2}
Corresponding author: ^{*}bkyang@jbnu.ac.kr, ORCID 0000-0003-4914-9626
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The compensators used in spectroscopic ellipsometers are usually assumed to be ideal linear wave-plates. In reality, however, they are elliptical waveplates, because they are usually made by bonding two or more linear waveplates of different materials with slight misalignment. This induces systematic error when they are modeled as linear waveplates. We propose an improved calibration method based on an optical model that regards an elliptical waveplate as a combination of a circular waveplate (rotator) and a linear waveplate. The method allows elimination of the systematic error, and the residual error of optic axis measurement is reduced to 0.025 degrees in the spectral range of 450–800 nm.
Keywords: Dual-rotating compensator, Elliptical waveplate, Mueller matrix decomposition, Spectroscopic ellipsometry
OCIS codes: (120.2130) Ellipsometry and polarimetry; (240.2130) Ellipsometry and polarimetry; (260.1440) Birefringence
The most general polarization characteristics of an anisotropic object are usually described by using a real 4 × 4 Mueller matrix [1]. The Mueller calculus, based on Mueller matrices, can describe the interaction of an object even with partially polarized light, and has applications now extending to the measurement of critical dimensions and cancer detection [2, 3].
When measuring the Mueller matrix of an object using a dual-rotating compensator type spectroscopic ellipsometer, two waveplates are placed in parallel and the object is inserted between them. Then the compensators are rotated with different angular speeds, and the intensity of the light transmitted through (or reflected from) them is measured. The Mueller matrix of the object is determined from the Fourier coefficients of the measured signal, using the system matrix of the ellipsometer [4, 5].
The accuracy of the measurement is maximized when the condition number of the system matrix is minimized. The condition number is strongly correlated to the retardance of the waveplates, the optimal value of which is 127 degrees [6, 7]. For wider spectral range of the measurement, composite achromatic waveplates are commonly used. A composite waveplate is made by bonding two or more waveplates of different materials with their optic axes aligned in orthogonal directions [8]. However, if the alignment of the optic axes is imperfect, the resulting composite waveplate becomes an elliptical waveplate instead of a linear one [9, 10]. In such a situation, an unwanted systematic error can be included in the final result. To improve accuracy, the systematic error should be reduced or eliminated.
Among various methods of calibration developed to reduce the systematic error, the most popular are the eigenvalue calibration method (ECM) [11, 12] and the self-calibration method (SCM) [13, 14]. ECM does not rely on any assumption about the properties of the optical components in the ellipsometer, but it demands at least three times measurements for calibration and necessitates the use of a linear waveplate as a calibration sample. Although SCM requires only one measurement for calibration, the waveplates are assumed to be linear; thus when composite waveplates are used, it is necessary to characterize them separately [14–17].
We propose an improved self-consistent calibration method (ISCM) with which one can significantly reduce the systematic errors. This method, like ECM, does not make any assumptions about the properties of the optical components. As a result, it does not require individual measurements for composite waveplates. However, unlike ECM, it does not require the use of a linear waveplate. The Mueller matrix of an elliptical waveplate can be decomposed into the product of the rotator and linear-waveplate Mueller matrices, or the product of the linear-waveplate and rotator Mueller matrices with the order reversed. In this paper, the Mueller matrices of the two compensators are decomposed in such a way that each Mueller matrix of the linear waveplate faces the sample. As a result, we can calibrate an ellipsometer with elliptical phase plates by replacing it with an equivalent ellipsometer with ideal linear waveplates and optical rotators combined.
In part II we explain the layout of the spectroscopic ellipsometer and the SCM. In part III we explain the method of calibrating the instrument, using an optical model of an elliptical waveplate as a combination of a linear waveplate and a polarization rotator.
Figure 1 shows a schematic of a transmission-type spectroscopic ellipsometer. The polarizer and analyzer are fixed, and thus the measured signal is not influenced by the polarization dependence of the detector or the variation in polarization of the light source. The two compensators C_{1} and C_{2} are achromatic waveplates, and the sample is placed in between them. They rotate about the common system axis with different angular speeds in the ratio
where
In SCM we take the system axis as the
where
where
Figure 2 shows the variations of the calibration parameters with wavelength that are obtained using the SCM. The measurement is repeated at every rotation of 8 degrees, to yield 45 data points in one cycle. The compensators used are achromatic composite waveplates, consisting of a quartz plate and a MgF_{2} plate bonded together with their optic axes orthogonal. The angles
The Mueller matrix
or
These expressions imply that an elliptical waveplate can be made by superposing a rotator and a linear waveplate.
Adopting this model, we replace compensator C_{1} with a superposition of a rotator C_{1C} and a linear waveplate C_{1L} as in Eq. (4), and replace compensator C_{2} with a superposition of a linear waveplate C_{2L} and a rotator C_{2C} as in Eq. (5) (Fig. 3). In this system we measure the spectrum by rotating C_{1} (= C_{1C}C_{1L}) and C_{2} (= C_{2L}C_{2C}), keeping the ratio of their rotation speeds 1:5.
The Mueller matrix of a rotator with optical rotatory power (ORP)
This matrix is invariant with respect to the rotation of the coordinate system about the
So, we can assume that all rotators are fixed and only C_{1L} and C_{2L} rotate. Linearly polarized light passing through a rotator remains linearly polarized, but its polarization direction is rotated. Now the combination of the polarizer and the rotator C_{1C} works as an effective polarizer, whose transmission angle is the effective polarizer angle
Now the ellipsometer built using composite waveplates as compensators can be modeled as one with ideal linear waveplates, as shown in Fig. 3(b). However, in this model, the polarizer and analyzer are replaced with the effective polarizer and effective analyzer respectively. When we apply the SCM to this model, the reference for the azimuthal angle should be the effective polarizer angle
As the ORP
The Fourier coefficients with respect to the new reference can be obtained from Eq. (2) by changing variables in the following way:
where
We attach a prime to the Fourier coefficients in Eq. (2) after changing their variables according to Eq. (8), to distinguish them from those before the change. Each retardance of the linear waveplates C_{1L} and C_{2L} has the same form and value as in Eq. (3a):
If we set
The nonzero Fourier coefficients obtained from the signal measured using the polarizing prism (which is assumed to be ideal as a sample for calibration) are
We can determine the remaining calibration parameters from Eqs. (12)–(13):
where
Figure 4 shows the results of the new calibration method. As described above, the retardances of the linear waveplates are the same as those shown in Fig. 2. The reason is that the main difference in the optical models of the two calibration methods (the SCM and the analytic method) is the rotation of the reference, which changes only the phase of the Fourier coefficients.
Because the ORP
Figure 5 shows the results of measurements obtained using a Glan-Tompson polarizer as a sample, for comparison of the two calibration methods. It displays azimuthal angle
In the SCM, the azimuthal angle oscillates with an amplitude of 0.5 degrees, but if the elliptical-waveplate model with analytic method is applied, the angle is almost constant with just small ripples; The standard deviation is 0.025 degrees over the whole wavelength range. Because the transmission axis of a Glan-Tompson polarizer is independent of wavelength, we can see that the new calibration method of the analytic approach works well. Figure 5(b) shows that the ellipticity is nearly zero, as expected, and we see that the polarization state of maximum transmittance is linear polarization.
In the SCM result of Fig. 5(a), the optic axis oscillates because we set the transmission axis
Therefore, the systematic error can be significantly reduced if we compensate for the rotation of the reference due to the dispersion in the SCM. One way of doing this is to rotate the reference to make the azimuthal angle in Fig. 5(a) be equal to zero. That can be done by transforming the measured Mueller matrix
We call this the ISCM, because we can use the conventional SCM and do not need to know the analytic solution [Eqs. (13)] of a calibration sample.
Figure 6 is a comparison of the results obtained using the analytic method and ISCM. The sample is a zeroth-order composite waveplate. The eigenpolarization of the waveplate is extracted from its Mueller matrix using the polar-decomposition technique [22]. Figure 6(a) shows the retardance, and Figs. 6(b) and 6(c) show the azimuthal angle of the slow axis and the ellipticity of the eigenpolarization ellipse respectively.
Both the azimuthal angle of the slow axis and the ellipticity oscillate with wavelength. Nonzero ellipticity indicates that the waveplate is elliptical, of course. The two methods yield almost the same results; The differences in the retardances and the azimuthal angles are less than ±0.08 degrees and ±0.09 degrees respectively. The two ellipticities are well matched within the instrumental error. The problem that arises if we regard the elliptical waveplate as a linear one is simply the rotation of the reference with wavelength, and a solution is to fix the reference using a polarizing prism as a calibrating sample. Then the two calibration methods, the analytic method and the ISCM, become equivalent. However, the ISCM does not require the analytic solution of Eqs. (13). The error coming from the measurement for calibration remains as background error in the subsequent real measurement.
It has been shown that systematic error can be significantly reduced by applying a new, improved calibration method to a dual-rotating compensator type spectroscopic ellipsometer. The key ideas are modeling the compensators as elliptical waveplates instead of linear ones, and considering an elliptical waveplate as a combination of a rotator and a linear-waveplate. We can place the rotators of the two compensators to face the polarizer and the analyzer respectively. Then the ellipsometer system becomes equivalent to one built using ideal linear waveplates, but with different orientations of the effective polarizer and the effective analyzer.
In the SCM the systematic error shows oscillation with wavelength, which comes from the dispersion of the model’s circular waveplates. To set up a fixed reference against the dispersion, it is necessary to characterize the system using a sample with fixed optic axis, such as a polarizing prism, for additional calibration.
We have proposed two calibration methods to eliminate the systematic error. One is based on the analytic theory for a polarizing prism as a calibration sample. The other is an improvement upon the SCM, by adding a correction for optic axis rotation. The two methods yield well matched results in the test measurement of a zeroth-order composite waveplate within the instrumental error of ±0.08 and ±0.09 degrees for retardance and optic axis respectively.
The authors declare no conflicts of interest.
Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.
The authors received no financial support for the research, authorship, and/or publication of this article.
Curr. Opt. Photon. 2023; 7(4): 428-434
Published online August 25, 2023 https://doi.org/10.3807/COPP.2023.7.4.428
Copyright © Optical Society of Korea.
Byeong-Kwan Yang^{1,2} , Jin Seung Kim^{2}
^{1}Jiny Photonics Inc., Jeonju 55124, Korea
^{2}Institute of Photonics and Information Technology, Department of Physics, Jeonbuk National University, Jeonju 54896, Korea
Correspondence to:^{*}bkyang@jbnu.ac.kr, ORCID 0000-0003-4914-9626
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The compensators used in spectroscopic ellipsometers are usually assumed to be ideal linear wave-plates. In reality, however, they are elliptical waveplates, because they are usually made by bonding two or more linear waveplates of different materials with slight misalignment. This induces systematic error when they are modeled as linear waveplates. We propose an improved calibration method based on an optical model that regards an elliptical waveplate as a combination of a circular waveplate (rotator) and a linear waveplate. The method allows elimination of the systematic error, and the residual error of optic axis measurement is reduced to 0.025 degrees in the spectral range of 450–800 nm.
Keywords: Dual-rotating compensator, Elliptical waveplate, Mueller matrix decomposition, Spectroscopic ellipsometry
The most general polarization characteristics of an anisotropic object are usually described by using a real 4 × 4 Mueller matrix [1]. The Mueller calculus, based on Mueller matrices, can describe the interaction of an object even with partially polarized light, and has applications now extending to the measurement of critical dimensions and cancer detection [2, 3].
When measuring the Mueller matrix of an object using a dual-rotating compensator type spectroscopic ellipsometer, two waveplates are placed in parallel and the object is inserted between them. Then the compensators are rotated with different angular speeds, and the intensity of the light transmitted through (or reflected from) them is measured. The Mueller matrix of the object is determined from the Fourier coefficients of the measured signal, using the system matrix of the ellipsometer [4, 5].
The accuracy of the measurement is maximized when the condition number of the system matrix is minimized. The condition number is strongly correlated to the retardance of the waveplates, the optimal value of which is 127 degrees [6, 7]. For wider spectral range of the measurement, composite achromatic waveplates are commonly used. A composite waveplate is made by bonding two or more waveplates of different materials with their optic axes aligned in orthogonal directions [8]. However, if the alignment of the optic axes is imperfect, the resulting composite waveplate becomes an elliptical waveplate instead of a linear one [9, 10]. In such a situation, an unwanted systematic error can be included in the final result. To improve accuracy, the systematic error should be reduced or eliminated.
Among various methods of calibration developed to reduce the systematic error, the most popular are the eigenvalue calibration method (ECM) [11, 12] and the self-calibration method (SCM) [13, 14]. ECM does not rely on any assumption about the properties of the optical components in the ellipsometer, but it demands at least three times measurements for calibration and necessitates the use of a linear waveplate as a calibration sample. Although SCM requires only one measurement for calibration, the waveplates are assumed to be linear; thus when composite waveplates are used, it is necessary to characterize them separately [14–17].
We propose an improved self-consistent calibration method (ISCM) with which one can significantly reduce the systematic errors. This method, like ECM, does not make any assumptions about the properties of the optical components. As a result, it does not require individual measurements for composite waveplates. However, unlike ECM, it does not require the use of a linear waveplate. The Mueller matrix of an elliptical waveplate can be decomposed into the product of the rotator and linear-waveplate Mueller matrices, or the product of the linear-waveplate and rotator Mueller matrices with the order reversed. In this paper, the Mueller matrices of the two compensators are decomposed in such a way that each Mueller matrix of the linear waveplate faces the sample. As a result, we can calibrate an ellipsometer with elliptical phase plates by replacing it with an equivalent ellipsometer with ideal linear waveplates and optical rotators combined.
In part II we explain the layout of the spectroscopic ellipsometer and the SCM. In part III we explain the method of calibrating the instrument, using an optical model of an elliptical waveplate as a combination of a linear waveplate and a polarization rotator.
Figure 1 shows a schematic of a transmission-type spectroscopic ellipsometer. The polarizer and analyzer are fixed, and thus the measured signal is not influenced by the polarization dependence of the detector or the variation in polarization of the light source. The two compensators C_{1} and C_{2} are achromatic waveplates, and the sample is placed in between them. They rotate about the common system axis with different angular speeds in the ratio
where
In SCM we take the system axis as the
where
where
Figure 2 shows the variations of the calibration parameters with wavelength that are obtained using the SCM. The measurement is repeated at every rotation of 8 degrees, to yield 45 data points in one cycle. The compensators used are achromatic composite waveplates, consisting of a quartz plate and a MgF_{2} plate bonded together with their optic axes orthogonal. The angles
The Mueller matrix
or
These expressions imply that an elliptical waveplate can be made by superposing a rotator and a linear waveplate.
Adopting this model, we replace compensator C_{1} with a superposition of a rotator C_{1C} and a linear waveplate C_{1L} as in Eq. (4), and replace compensator C_{2} with a superposition of a linear waveplate C_{2L} and a rotator C_{2C} as in Eq. (5) (Fig. 3). In this system we measure the spectrum by rotating C_{1} (= C_{1C}C_{1L}) and C_{2} (= C_{2L}C_{2C}), keeping the ratio of their rotation speeds 1:5.
The Mueller matrix of a rotator with optical rotatory power (ORP)
This matrix is invariant with respect to the rotation of the coordinate system about the
So, we can assume that all rotators are fixed and only C_{1L} and C_{2L} rotate. Linearly polarized light passing through a rotator remains linearly polarized, but its polarization direction is rotated. Now the combination of the polarizer and the rotator C_{1C} works as an effective polarizer, whose transmission angle is the effective polarizer angle
Now the ellipsometer built using composite waveplates as compensators can be modeled as one with ideal linear waveplates, as shown in Fig. 3(b). However, in this model, the polarizer and analyzer are replaced with the effective polarizer and effective analyzer respectively. When we apply the SCM to this model, the reference for the azimuthal angle should be the effective polarizer angle
As the ORP
The Fourier coefficients with respect to the new reference can be obtained from Eq. (2) by changing variables in the following way:
where
We attach a prime to the Fourier coefficients in Eq. (2) after changing their variables according to Eq. (8), to distinguish them from those before the change. Each retardance of the linear waveplates C_{1L} and C_{2L} has the same form and value as in Eq. (3a):
If we set
The nonzero Fourier coefficients obtained from the signal measured using the polarizing prism (which is assumed to be ideal as a sample for calibration) are
We can determine the remaining calibration parameters from Eqs. (12)–(13):
where
Figure 4 shows the results of the new calibration method. As described above, the retardances of the linear waveplates are the same as those shown in Fig. 2. The reason is that the main difference in the optical models of the two calibration methods (the SCM and the analytic method) is the rotation of the reference, which changes only the phase of the Fourier coefficients.
Because the ORP
Figure 5 shows the results of measurements obtained using a Glan-Tompson polarizer as a sample, for comparison of the two calibration methods. It displays azimuthal angle
In the SCM, the azimuthal angle oscillates with an amplitude of 0.5 degrees, but if the elliptical-waveplate model with analytic method is applied, the angle is almost constant with just small ripples; The standard deviation is 0.025 degrees over the whole wavelength range. Because the transmission axis of a Glan-Tompson polarizer is independent of wavelength, we can see that the new calibration method of the analytic approach works well. Figure 5(b) shows that the ellipticity is nearly zero, as expected, and we see that the polarization state of maximum transmittance is linear polarization.
In the SCM result of Fig. 5(a), the optic axis oscillates because we set the transmission axis
Therefore, the systematic error can be significantly reduced if we compensate for the rotation of the reference due to the dispersion in the SCM. One way of doing this is to rotate the reference to make the azimuthal angle in Fig. 5(a) be equal to zero. That can be done by transforming the measured Mueller matrix
We call this the ISCM, because we can use the conventional SCM and do not need to know the analytic solution [Eqs. (13)] of a calibration sample.
Figure 6 is a comparison of the results obtained using the analytic method and ISCM. The sample is a zeroth-order composite waveplate. The eigenpolarization of the waveplate is extracted from its Mueller matrix using the polar-decomposition technique [22]. Figure 6(a) shows the retardance, and Figs. 6(b) and 6(c) show the azimuthal angle of the slow axis and the ellipticity of the eigenpolarization ellipse respectively.
Both the azimuthal angle of the slow axis and the ellipticity oscillate with wavelength. Nonzero ellipticity indicates that the waveplate is elliptical, of course. The two methods yield almost the same results; The differences in the retardances and the azimuthal angles are less than ±0.08 degrees and ±0.09 degrees respectively. The two ellipticities are well matched within the instrumental error. The problem that arises if we regard the elliptical waveplate as a linear one is simply the rotation of the reference with wavelength, and a solution is to fix the reference using a polarizing prism as a calibrating sample. Then the two calibration methods, the analytic method and the ISCM, become equivalent. However, the ISCM does not require the analytic solution of Eqs. (13). The error coming from the measurement for calibration remains as background error in the subsequent real measurement.
It has been shown that systematic error can be significantly reduced by applying a new, improved calibration method to a dual-rotating compensator type spectroscopic ellipsometer. The key ideas are modeling the compensators as elliptical waveplates instead of linear ones, and considering an elliptical waveplate as a combination of a rotator and a linear-waveplate. We can place the rotators of the two compensators to face the polarizer and the analyzer respectively. Then the ellipsometer system becomes equivalent to one built using ideal linear waveplates, but with different orientations of the effective polarizer and the effective analyzer.
In the SCM the systematic error shows oscillation with wavelength, which comes from the dispersion of the model’s circular waveplates. To set up a fixed reference against the dispersion, it is necessary to characterize the system using a sample with fixed optic axis, such as a polarizing prism, for additional calibration.
We have proposed two calibration methods to eliminate the systematic error. One is based on the analytic theory for a polarizing prism as a calibration sample. The other is an improvement upon the SCM, by adding a correction for optic axis rotation. The two methods yield well matched results in the test measurement of a zeroth-order composite waveplate within the instrumental error of ±0.08 and ±0.09 degrees for retardance and optic axis respectively.
The authors declare no conflicts of interest.
Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.
The authors received no financial support for the research, authorship, and/or publication of this article.