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A liquid crystal variable retarder-based reflectance difference spectrometer for fast, high precision spectroscopic measurements
TSF-33021; No of Pages 5 Thin Solid Films xxx (2013) xxx–xxx
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A liquid crystal variable retarder-based reflectance difference spectrometer for fast, high precision spectroscopic measurements Chunguang Hu ⁎, Pengfei Xie, Shuchun Huo, Yanning Li ⁎, Xiaotang Hu State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin, 300072, China
a r t i c l e
i n f o
Available online xxxx Keywords: Reflectance difference spectrometer Liquid crystal variable retarder Systematic errors Measurement quality
a b s t r a c t We present a liquid crystal variable retarder (LCVR)-based design for reflectance difference (RD) spectrometry, which offers one high quality RD spectrum in the visible range within several seconds. The measurement principle and the instrument development of this design are provided. As the LCVR is a key component, investigations are focused on its wavelength-dependent optical characteristics, the systematic errors induced by its imperfections, and the temperature effect on the measurement results. With careful calibration and data correction, the qualities of corrected RD spectra, defined as standard deviations of the RD signals as a function of wavelength and time, are better than 3 × 10−4 and 1.3 × 10−3, respectively. © 2013 Published by Elsevier B.V.
1. Introduction Reflectance difference spectroscopy (RDS), also called reflectance anisotropy spectroscopy, has high surface sensitivity and is widely used in the fields of semiconductor, thin film growth, biological detection, and so on [1]. A common instrument of the RD spectrometry was developed in the 1980s [2]. It is based on a photoelastic modulator (PEM) with a working frequency of several tens of kHz. Thus, the measurement time at one wavelength goes down to millisecond scale. This type of instrument can also acquire RD spectra in a broad wavelength range by using a wavelength-scanning monochromator, but it is time consuming [3]. In order to realize fast spectroscopic measurements, a rotating-compensator (RC)-based RD spectrometer was developed, which offers an RD spectrum from 1.5 to 4.5 eV within 10 s [4]. However, the optical imperfections of a super-achromatic compensator and the mechanical rotation-induced measurement errors complicate the procedures for signal calculation and system calibration and correction. Precise synchronization between the detector and the rotation action of the compensator is needed by using a sophisticated control mechanism. A liquid crystal variable retarder (LCVR) allows a controllable phase retardation [5], which depends on the applied electric field, but is compact in size with respect to the PEM. The LCVR has been an alternative to the polarization modulation in various designs of the ellipsometry/ polarimetry [6–8]. Many of those designs are of single-wavelength measurements as the retardation value of the LCVR is a nonlinear function of the applied voltage. Here, we propose an LCVR-based spectroscopic RD
⁎ Corresponding authors. Tel.: +86 22 5801 8712; fax: +86 22 2740 6941. E-mail addresses: [email protected] (C. Hu), [email protected] (Y. Li).
spectrometer with the following reasons: First, there is no optical element mechanically rotated during measurement when the LCVR is applied as phase retardation modulator. Errors related to the mechanical rotation are avoided. At the same time, the structure of the instrument becomes simple thanks to the compact size of the LCVR. Second, the retardation of the LCVR can either be modulated at a low frequency or be fixed. This permits the LCVR to work with a linear array detector for fast spectroscopic measurements. Third, the control protocol is easy because only two steps are needed sequentially and repeatedly: set the LCVR to a desired retardation value and read the intensity spectrum from the detector. In this paper, the mathematical modeling as well as the data collection and reduction are presented in Section 2. A detailed instrumentation is given in Section 3. The electro-optical characteristics of the LCVR as a function of wavelength and applied voltage are measured in Section 4. At the end, the defects of the LCVR-induced systematic errors, the temperature influence on the measurement precision, and the measurement repeatability are explored in Section 5.
2. Measurement principle An LCVR-based RD spectrometer (LC-RDS) discussed below follows a configuration of polarizer-LCVR sample analyzer. Similar to the PEMbased design, one LCVR device is used as phase retardation modulator. It is noted that this design cannot measure all polarization properties of the sample, e.g., depolarization. Using Muller matrix approach, the intensity imposing on one pixel of the array detector, i.e., one wavelength channel, can be expressed as Eq. (1) [3]. Here, the azimuthal orientations of the polarizer, the LCVR, the sample, and the analyzer are set at 0, 45°, 0, and 45°, respectively, and it is supposed that the sample has no depolarization effect.
0040-6090/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.tsf.2013.12.034
Please cite this article as: C. Hu, et al., Thin Solid Films (2013), http://dx.doi.org/10.1016/j.tsf.2013.12.034
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C. Hu et al. / Thin Solid Films xxx (2013) xxx–xxx
I ¼ I 0 ð1 þ N cosδ þ S sinδÞ;
ð1Þ
where N = − cos2ψ, S = sin2ψsinΔ, and ψ and Δ are ellipsometric parameters of the sample. I0 is the base intensity and should be constant during measurement. δ is the retardation of the LCVR. Eq. (1) is the same as the one achieved from the one-PEM-based RD spectrometer [3]. However, a nonlinear relationship between the retardation and the applied voltage exists in the LCVR device. A sinusoidal modulation of the applied voltage does not lead to a sinusoidal change of the retardation. The way for data reduction in the PEM-based RD spectrometer is not fit for the LVCR-based design. As an alternative solution, scanning the retardation step by step is carried out in our design. In principle, only three intensity signals of I at different retardation values of the LCVR are needed to solve Eq. (1). In order to optimize the measurement quality, the least square method is applied. The best approximations of N and S are deduced using Eq. (2) according to a set of I acquired at n different values of δ. 0
1 0 I1 1 B I2 C B 1 B C¼B @ ⋮ A @⋮ In 1
cosδ1 cosδ2 ⋮ cosδn
1 1 sinδ1 0 I0 sinδ2 C C@ I N A; ⋮ A 0 I0 S sinδn
ðnN N3Þ:
ð2Þ
One gets the RD signal of Δr/r as below [4] Δr N þ iS pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼2 r 1 þ 1−N 2 −S2
ð3Þ
Here, Δr is the difference in reflectance of normal incidence plane-polarized light between two orthogonal directions in the surface plane. r is the mean reflectance. Both N and S are normally much less than 1, and the real and imaginary parts of the RD signal approximately equal to N and S, respectively. Since the intensity spectrum is measured in parallel using a linear array detector, the RD spectra are gained by a parallel calculation for all wavelengths. 3. Instrumentation The schematic diagram of LC-RDS is shown in Fig. 1. A white light source (Ocean optics, DH-2000-BAL) emits light, which passes through a linear polarizer (Edmund optics, Glan-Thompson type) and an LCVR (Thorlabs, LCC1113-A) and then reaches to the sample. The reflected
beam goes through an analyzer, which is the second linear polarizer and is fed into a spectrometer (Ocean optics, QE65Pro) via a multimode fiber. Intensity spectrum is recorded by a detector array in the spectrometer. Different to the ellipsometer, the angle between the incident beam and the reflected beam is less than 5° for RD spectrometer. It is a basic requirement to obtain the in-plane optical anisotropy of the sample. The control procedure for one measurement is performed as follows: 1) change the retardation of the LCVR, start an integration of the detector afterwards, and read out the intensity spectrum when the integration is completed. 2) Repeat the first step at n different retardations of the LCVR. 3) According to the collected intensity spectra and the corresponding retardations of the LCVR, the RD spectra are deduced using Eqs. (2) and (3). 4. Calibration The retardation δ of the LCVR in Eq. (2) has to be known before data reduction. Since the commercial LCVR used in our setup is only calibrated at one wavelength by the manufacturer, it is necessary to determine the retardation value of the LCVR as a function of applied voltage and wavelength [9,10]. However, it is reported that the fast axis orientation of the LCVR might be also dependent on the voltage [11]. We follow references [6] and [12] and do a calibration with the steps below: 1) The fast axis of the polarizer is placed to be orthogonal to the one of the analyzer, in which case there is no light passing through the twoelement structure. 2) The LCVR is placed between the polarizer and the analyzer. By rotating the LCVR, its azimuthal axis is determined when no light or a minimum intensity is measured from the three-element constructed setup. The mathematical expression is 2
I ¼ I0 ð1−cosδÞsin ð2LÞ
ð4Þ
where L is the fast axis orientation of the LCVR with respect to the active axis of the polarizer. Note that this step does not work well when the retardation is close to a value of 2π or an integral multiple of 2π. 3) The analyzer is fixed at a place where its fast axis is parallel to the polarizer. The retardation of the LCVR is determined by fitting the intensities as a function of the rotation angle of the LCVR according to Eq. (5). h i 2 I ¼ I 0 2−ð1−cosδÞsin ð2LÞ
ð5Þ
89.90 89.70
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Wavelength (nm) Fig. 1. (Color online) Scheme of the liquid crystal variable retarder-based reflectance difference spectrometer. See Ref. [5] for details of the LCVR.
Fig. 2. (Color online) Orientation of the fast axis of the LCVR vs. wavelength and voltage. The value of the fast axis is the indication number of the rotation stage in unit of degree.
Please cite this article as: C. Hu, et al., Thin Solid Films (2013), http://dx.doi.org/10.1016/j.tsf.2013.12.034
C. Hu et al. / Thin Solid Films xxx (2013) xxx–xxx
As shown in Fig. 2, the orientation of the fast axis of the LCVR is constant in a wavelength range from 400 to 800 nm and a voltage range from 2.5 to 10 V. It means that the LCVR device is suitable for spectroscopic measurements. The contour in Fig. 3(a) is the measured wrapped retardation of the LCVR as a function of voltage and wavelength. The threshold voltage starting to drive the LCVR is around 0.9 V. In the range from 0.9 to 3 V, the retardation of the LCVR decreases quickly when increasing the voltage. However, the change rate of the retardation becomes slow since the voltage is over than 3 V. The retardation curves at the selected voltages are plotted as a function of wavelength in Fig. 3(b). When the voltage is less than 3 V, the retardation curve is folded because its real value is beyond the analytical range, i.e., [0, π], of an inverse cosine function. In ideal conditions, the curve is folded at both edges of the analytical range. However, it is not the case for the measurement data, especially at short wavelength. This problem might be caused by the imperfections of the LCVR, for example, the liquid crystal material scatters the light and introduces slight depolarization [13]. For safety, we do RD measurement in the range from 2.5 to 10 V.
5. Experiments 5.1. Systematic errors The systematic errors are explored in a straight through configuration, where the polarizer, the LCVR, the sample, and the analyzer are placed in a line. Therefore, the sample is air and the ideal values of N
a
3
and S at all wavelengths are zero. As a result, the intensity in Eq. (1) should be independent of the voltage. An experiment was done in the following conditions. One 16-time averaged intensity spectrum was obtained with a time of 128 ms. Sixty spectra were sequentially acquired by changing the voltage in the range from 2.5 to 9.9 V. Then, one RD spectrum was calculated from these intensity spectra according to Eqs. (2) and (3). The total time for one measurement is about 9 s, counting the switch time of the liquid crystal as the applied voltage is changed. The measurement time is mainly restricted by the effective power of the light on the detector and the minimum integration time and the signal-to-noise ratio of the detector. Figs. 4(a) and 4(b) show 16-time averaged real and imaginary spectra of the RD signal in dash line, respectively. Both of them are diverged from the desired value of zero and have oscillations along wavelength. The oscillations might be introduced by the interference occurring in the two-glass-plate formed cavity of the LCVR [7,14]. The divergence of the curve could come from another imperfection of the LCVR, a various transmission of the LC cell along with the applied voltage [12,13]. For example, the intensity curve at the wavelength of 450 nm as a function of applied voltage is shown in the inset at the up-left corner of Fig. 4 (a). An unwanted change of the intensity occurs in the range from 2.5 to 3.5 V. However, the intensity curve at the wavelength of 750 nm, shown in the inset at the bottom-right corner of Fig. 4(b), is more constant along the voltage. Consequently, the RD signals at 750 nm have smaller deviation than the RD signals at 450 nm. We hold both 16-averaged fluctuation curves measured above as systematic errors and correct the raw measurement data by subtracting these systematic errors. The corrected spectra turn to be flat as plotted in Fig. 4 in solid line. Their standard deviations as a function of wavelength are better than 3 × 10−4.
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Wavelength (nm) Fig. 3. (Color online) (a) Wrapped retardation of the LCVR vs. wavelength and voltage in unit of radian. (b) Curves of the retardation along with wavelength at the selected voltages.
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Wavelength (nm) Fig. 4. (Color online) Uncorrected and corrected spectra of the Re(Δr/r) and Im(Δr/r) signals of air. The inset graphs are the intensity curves along with voltage at the wavelengths of 450 nm and 750 nm.
Please cite this article as: C. Hu, et al., Thin Solid Films (2013), http://dx.doi.org/10.1016/j.tsf.2013.12.034
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C. Hu et al. / Thin Solid Films xxx (2013) xxx–xxx
Now, the question is whether the fluctuation curves are constant in time and are really systematic errors. In order to answer this question and make clear the stability of the design, an experiment continuously measuring the RD spectra of air was examined over 3 hours and at a temperature of 23 °C. The 82-time measurements were obtained with a temporal interval of 2 minutes. The curves of the real and imaginary RD signals versus time at three selected wavelengths are plotted in Fig. 5. The standard deviations of the RD signals versus time vary with wavelength. At short wavelength, for example, at 400 nm, the standard deviations of both N and S are better than 1.4 × 10−4 and 5.5 × 10−4, respectively. However, the standard deviations of both N and S at 800 nm are increased to 5.3 × 10−4 and 1.4 × 10−3, respectively. Fortunately, there are no obvious temporal effects changing the RD signals. As a conclusion, the fluctuation curves in Fig. 4 are systematic errors and can be directly eliminated by subtracting them from the measurement data. The fact that the standard deviation of the RD signal at long wavelength is larger than the standard deviation at short wavelength might be introduced by a reason below. For the same voltage range, the retardation variation of the LCVR at short wavelength is larger than the variation at long wavelength. Larger variation in retardation means a better ability to overcome the noise-induced error when a least square method is used. Accordingly, the measurement indicates a less sensitivity to the intensity noise at short wavelength than at long wavelength. A solution for the above problem is to keep the same retardation range and the same sampling number for all wavelengths. 5.2. Temperature influence It is known that the retardation of the LCVR is temperature dependence [9,14]. We also checked the thermal effect on the measurement
quality. With the same straight through configuration discussed in Section 5.1, the RD spectra of air were measured at three different temperatures, 21 °C, 23 °C, and 25 °C. All spectra were corrected with the aforementioned systematic errors, which were measured at 23 °C. In Fig. 6(a), the oscillation comes back to the corrected spectra of N at 21 °C and 25 °C. And three spectra have positions in a sequence corresponding to the temperatures. It implies that there is offset imposed on the real spectra of the RD signal by a change of the temperature. In Fig. 6(b), the oscillation effect on the spectra of S at 21 °C and 25 °C is even large. However, there is no offset happened with the changes of temperature. General speaking, thermal effect is a big problem to high precision LC-RDS measurement, but this design is also available for a measurement with a precision requirement less than 3 × 10− 3. This type of temperature-induced error might be caused by both the length change of the cavity in the LCVR and the change of the intrinsic optical property of the liquid crystal material. A reasonable way to eliminate this problem is to equip a temperature controller for the LCVR, which is already available in commercial products. 6. Conclusions An LCVR-based RD spectrometer has been developed for fast spectroscopic measurements. This design has a simpler optical structure and easier operation procedure than an RCRD spectrometer. The achieved RD spectrum covers a wavelength range from 400 to 800 nm and one measurement takes several seconds. However, the defects of the LCVR degrade the measurement quality and attentions have to be paid on them. One is the interference occurring in the cavity of the LCVR, which brings a systematic error fluctuating along with wavelength to RD spectra. Another is the temperature influence on the LCVR, which introduces even more complicate errors to the measurement results. Fortunately, with a data processing and a good control of
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Fig. 6. (Color online) Temperature influence on the Re(Δr/r) (a) and Im(Δr/r) (b) signals.
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C. Hu et al. / Thin Solid Films xxx (2013) xxx–xxx
temperature, the standard deviation of the corrected RD spectrum as a function of wavelength is better than 3 × 10−4. The stability in the whole spectral range is proven as good as 1.3 × 10−3 over 3 hours. Acknowledgements The authors gratefully acknowledge the support from the National Natural Science Foundation of China (grant no. 61008028), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (grant no. 201140), and the Natural Science Foundation of Tianjin, China (grant no. 11JCJBJC25700). References [1] P. Weightman, D.S. Martin, R.J. Cole, T. Farrell, Rep. Prog. Phys. 68 (2005) 1251.
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[2] D.E. Aspnes, J.P. Harbison, A.A. Studna, L.T. Florez, J. Vac. Sci. Technol. A 6 (1988) 1327. [3] O. Acher, B. Drèvillon, Rev. Sci. Instrum. 63 (1992) 5332. [4] C.G. Hu, L.D. Sun, J.M. Flores-Camacho, M. Hohage, C.Y. Liu, X.T. Hu, P. Zeppenfeld, Rev. Sci. Instrum. 81 (2010) 043108. [5] http://www.meadowlark.com/store/LC_Digital_Controller_manual_30.pdf2011. [6] J.M. Bueno, J. Opt. A Pure Appl. Opt. 2 (2000) 216. [7] A. De Martino, Y.K. Kim, E. Garcia-Caurel, B. Laude, B. Drèvillon, Opt. Lett. 28 (2003) 616. [8] N.J. Pust, J.A. Shaw, Appl. Opt. 45 (2006) 5470. [9] J.S. Baba, P.R. Boudreaux, Appl. Opt. 46 (2007) 5539. [10] J. Ladstein, M. Kildemo, G.K. Svendsen, I.S. NerbØ, F. Stabo-Eeg, in: M. Glogarova, P. Palffy-Muhoray, M. Copic (Eds.), Liquid Crystal and Applications in optics, Proc. SPIE, 6587, 2007, p. 65870D, (Prague, Czech Republic, April 16, 2007). [11] P. Terrier, J.M. Charbois, V. Devlaminck, Appl. Opt. 49 (2010) 4278. [12] S.T. Wu, U. Efron, L.D. Hess, Appl. Opt. 23 (1984) 3911. [13] R.A. Chipman, in: M. Bass (Ed.), Handbook of Optics, vol. 1, McGraw Hill, New York, 2010, (chap. 15.31). [14] R.L. Heredero, N. Uribe-Patarroyo, T. Belenguer, G. Ramos, A. sánchez, M. Reina, V. Martínez Pillet, A. Álvarez-Herrero, Appl. Opt. 46 (2007) 689.
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Contents lists available at ScienceDirect
Thin Solid Films journal homepage: www.elsevier.com/locate/tsf
A liquid crystal variable retarder-based reflectance difference spectrometer for fast, high precision spectroscopic measurements Chunguang Hu ⁎, Pengfei Xie, Shuchun Huo, Yanning Li ⁎, Xiaotang Hu State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin, 300072, China
a r t i c l e
i n f o
Available online xxxx Keywords: Reflectance difference spectrometer Liquid crystal variable retarder Systematic errors Measurement quality
a b s t r a c t We present a liquid crystal variable retarder (LCVR)-based design for reflectance difference (RD) spectrometry, which offers one high quality RD spectrum in the visible range within several seconds. The measurement principle and the instrument development of this design are provided. As the LCVR is a key component, investigations are focused on its wavelength-dependent optical characteristics, the systematic errors induced by its imperfections, and the temperature effect on the measurement results. With careful calibration and data correction, the qualities of corrected RD spectra, defined as standard deviations of the RD signals as a function of wavelength and time, are better than 3 × 10−4 and 1.3 × 10−3, respectively. © 2013 Published by Elsevier B.V.
1. Introduction Reflectance difference spectroscopy (RDS), also called reflectance anisotropy spectroscopy, has high surface sensitivity and is widely used in the fields of semiconductor, thin film growth, biological detection, and so on [1]. A common instrument of the RD spectrometry was developed in the 1980s [2]. It is based on a photoelastic modulator (PEM) with a working frequency of several tens of kHz. Thus, the measurement time at one wavelength goes down to millisecond scale. This type of instrument can also acquire RD spectra in a broad wavelength range by using a wavelength-scanning monochromator, but it is time consuming [3]. In order to realize fast spectroscopic measurements, a rotating-compensator (RC)-based RD spectrometer was developed, which offers an RD spectrum from 1.5 to 4.5 eV within 10 s [4]. However, the optical imperfections of a super-achromatic compensator and the mechanical rotation-induced measurement errors complicate the procedures for signal calculation and system calibration and correction. Precise synchronization between the detector and the rotation action of the compensator is needed by using a sophisticated control mechanism. A liquid crystal variable retarder (LCVR) allows a controllable phase retardation [5], which depends on the applied electric field, but is compact in size with respect to the PEM. The LCVR has been an alternative to the polarization modulation in various designs of the ellipsometry/ polarimetry [6–8]. Many of those designs are of single-wavelength measurements as the retardation value of the LCVR is a nonlinear function of the applied voltage. Here, we propose an LCVR-based spectroscopic RD
⁎ Corresponding authors. Tel.: +86 22 5801 8712; fax: +86 22 2740 6941. E-mail addresses: [email protected] (C. Hu), [email protected] (Y. Li).
spectrometer with the following reasons: First, there is no optical element mechanically rotated during measurement when the LCVR is applied as phase retardation modulator. Errors related to the mechanical rotation are avoided. At the same time, the structure of the instrument becomes simple thanks to the compact size of the LCVR. Second, the retardation of the LCVR can either be modulated at a low frequency or be fixed. This permits the LCVR to work with a linear array detector for fast spectroscopic measurements. Third, the control protocol is easy because only two steps are needed sequentially and repeatedly: set the LCVR to a desired retardation value and read the intensity spectrum from the detector. In this paper, the mathematical modeling as well as the data collection and reduction are presented in Section 2. A detailed instrumentation is given in Section 3. The electro-optical characteristics of the LCVR as a function of wavelength and applied voltage are measured in Section 4. At the end, the defects of the LCVR-induced systematic errors, the temperature influence on the measurement precision, and the measurement repeatability are explored in Section 5.
2. Measurement principle An LCVR-based RD spectrometer (LC-RDS) discussed below follows a configuration of polarizer-LCVR sample analyzer. Similar to the PEMbased design, one LCVR device is used as phase retardation modulator. It is noted that this design cannot measure all polarization properties of the sample, e.g., depolarization. Using Muller matrix approach, the intensity imposing on one pixel of the array detector, i.e., one wavelength channel, can be expressed as Eq. (1) [3]. Here, the azimuthal orientations of the polarizer, the LCVR, the sample, and the analyzer are set at 0, 45°, 0, and 45°, respectively, and it is supposed that the sample has no depolarization effect.
0040-6090/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.tsf.2013.12.034
Please cite this article as: C. Hu, et al., Thin Solid Films (2013), http://dx.doi.org/10.1016/j.tsf.2013.12.034
2
C. Hu et al. / Thin Solid Films xxx (2013) xxx–xxx
I ¼ I 0 ð1 þ N cosδ þ S sinδÞ;
ð1Þ
where N = − cos2ψ, S = sin2ψsinΔ, and ψ and Δ are ellipsometric parameters of the sample. I0 is the base intensity and should be constant during measurement. δ is the retardation of the LCVR. Eq. (1) is the same as the one achieved from the one-PEM-based RD spectrometer [3]. However, a nonlinear relationship between the retardation and the applied voltage exists in the LCVR device. A sinusoidal modulation of the applied voltage does not lead to a sinusoidal change of the retardation. The way for data reduction in the PEM-based RD spectrometer is not fit for the LVCR-based design. As an alternative solution, scanning the retardation step by step is carried out in our design. In principle, only three intensity signals of I at different retardation values of the LCVR are needed to solve Eq. (1). In order to optimize the measurement quality, the least square method is applied. The best approximations of N and S are deduced using Eq. (2) according to a set of I acquired at n different values of δ. 0
1 0 I1 1 B I2 C B 1 B C¼B @ ⋮ A @⋮ In 1
cosδ1 cosδ2 ⋮ cosδn
1 1 sinδ1 0 I0 sinδ2 C C@ I N A; ⋮ A 0 I0 S sinδn
ðnN N3Þ:
ð2Þ
One gets the RD signal of Δr/r as below [4] Δr N þ iS pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼2 r 1 þ 1−N 2 −S2
ð3Þ
Here, Δr is the difference in reflectance of normal incidence plane-polarized light between two orthogonal directions in the surface plane. r is the mean reflectance. Both N and S are normally much less than 1, and the real and imaginary parts of the RD signal approximately equal to N and S, respectively. Since the intensity spectrum is measured in parallel using a linear array detector, the RD spectra are gained by a parallel calculation for all wavelengths. 3. Instrumentation The schematic diagram of LC-RDS is shown in Fig. 1. A white light source (Ocean optics, DH-2000-BAL) emits light, which passes through a linear polarizer (Edmund optics, Glan-Thompson type) and an LCVR (Thorlabs, LCC1113-A) and then reaches to the sample. The reflected
beam goes through an analyzer, which is the second linear polarizer and is fed into a spectrometer (Ocean optics, QE65Pro) via a multimode fiber. Intensity spectrum is recorded by a detector array in the spectrometer. Different to the ellipsometer, the angle between the incident beam and the reflected beam is less than 5° for RD spectrometer. It is a basic requirement to obtain the in-plane optical anisotropy of the sample. The control procedure for one measurement is performed as follows: 1) change the retardation of the LCVR, start an integration of the detector afterwards, and read out the intensity spectrum when the integration is completed. 2) Repeat the first step at n different retardations of the LCVR. 3) According to the collected intensity spectra and the corresponding retardations of the LCVR, the RD spectra are deduced using Eqs. (2) and (3). 4. Calibration The retardation δ of the LCVR in Eq. (2) has to be known before data reduction. Since the commercial LCVR used in our setup is only calibrated at one wavelength by the manufacturer, it is necessary to determine the retardation value of the LCVR as a function of applied voltage and wavelength [9,10]. However, it is reported that the fast axis orientation of the LCVR might be also dependent on the voltage [11]. We follow references [6] and [12] and do a calibration with the steps below: 1) The fast axis of the polarizer is placed to be orthogonal to the one of the analyzer, in which case there is no light passing through the twoelement structure. 2) The LCVR is placed between the polarizer and the analyzer. By rotating the LCVR, its azimuthal axis is determined when no light or a minimum intensity is measured from the three-element constructed setup. The mathematical expression is 2
I ¼ I0 ð1−cosδÞsin ð2LÞ
ð4Þ
where L is the fast axis orientation of the LCVR with respect to the active axis of the polarizer. Note that this step does not work well when the retardation is close to a value of 2π or an integral multiple of 2π. 3) The analyzer is fixed at a place where its fast axis is parallel to the polarizer. The retardation of the LCVR is determined by fitting the intensities as a function of the rotation angle of the LCVR according to Eq. (5). h i 2 I ¼ I 0 2−ð1−cosδÞsin ð2LÞ
ð5Þ
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Wavelength (nm) Fig. 1. (Color online) Scheme of the liquid crystal variable retarder-based reflectance difference spectrometer. See Ref. [5] for details of the LCVR.
Fig. 2. (Color online) Orientation of the fast axis of the LCVR vs. wavelength and voltage. The value of the fast axis is the indication number of the rotation stage in unit of degree.
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C. Hu et al. / Thin Solid Films xxx (2013) xxx–xxx
As shown in Fig. 2, the orientation of the fast axis of the LCVR is constant in a wavelength range from 400 to 800 nm and a voltage range from 2.5 to 10 V. It means that the LCVR device is suitable for spectroscopic measurements. The contour in Fig. 3(a) is the measured wrapped retardation of the LCVR as a function of voltage and wavelength. The threshold voltage starting to drive the LCVR is around 0.9 V. In the range from 0.9 to 3 V, the retardation of the LCVR decreases quickly when increasing the voltage. However, the change rate of the retardation becomes slow since the voltage is over than 3 V. The retardation curves at the selected voltages are plotted as a function of wavelength in Fig. 3(b). When the voltage is less than 3 V, the retardation curve is folded because its real value is beyond the analytical range, i.e., [0, π], of an inverse cosine function. In ideal conditions, the curve is folded at both edges of the analytical range. However, it is not the case for the measurement data, especially at short wavelength. This problem might be caused by the imperfections of the LCVR, for example, the liquid crystal material scatters the light and introduces slight depolarization [13]. For safety, we do RD measurement in the range from 2.5 to 10 V.
5. Experiments 5.1. Systematic errors The systematic errors are explored in a straight through configuration, where the polarizer, the LCVR, the sample, and the analyzer are placed in a line. Therefore, the sample is air and the ideal values of N
a
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and S at all wavelengths are zero. As a result, the intensity in Eq. (1) should be independent of the voltage. An experiment was done in the following conditions. One 16-time averaged intensity spectrum was obtained with a time of 128 ms. Sixty spectra were sequentially acquired by changing the voltage in the range from 2.5 to 9.9 V. Then, one RD spectrum was calculated from these intensity spectra according to Eqs. (2) and (3). The total time for one measurement is about 9 s, counting the switch time of the liquid crystal as the applied voltage is changed. The measurement time is mainly restricted by the effective power of the light on the detector and the minimum integration time and the signal-to-noise ratio of the detector. Figs. 4(a) and 4(b) show 16-time averaged real and imaginary spectra of the RD signal in dash line, respectively. Both of them are diverged from the desired value of zero and have oscillations along wavelength. The oscillations might be introduced by the interference occurring in the two-glass-plate formed cavity of the LCVR [7,14]. The divergence of the curve could come from another imperfection of the LCVR, a various transmission of the LC cell along with the applied voltage [12,13]. For example, the intensity curve at the wavelength of 450 nm as a function of applied voltage is shown in the inset at the up-left corner of Fig. 4 (a). An unwanted change of the intensity occurs in the range from 2.5 to 3.5 V. However, the intensity curve at the wavelength of 750 nm, shown in the inset at the bottom-right corner of Fig. 4(b), is more constant along the voltage. Consequently, the RD signals at 750 nm have smaller deviation than the RD signals at 450 nm. We hold both 16-averaged fluctuation curves measured above as systematic errors and correct the raw measurement data by subtracting these systematic errors. The corrected spectra turn to be flat as plotted in Fig. 4 in solid line. Their standard deviations as a function of wavelength are better than 3 × 10−4.
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Wavelength (nm) Fig. 3. (Color online) (a) Wrapped retardation of the LCVR vs. wavelength and voltage in unit of radian. (b) Curves of the retardation along with wavelength at the selected voltages.
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Wavelength (nm) Fig. 4. (Color online) Uncorrected and corrected spectra of the Re(Δr/r) and Im(Δr/r) signals of air. The inset graphs are the intensity curves along with voltage at the wavelengths of 450 nm and 750 nm.
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C. Hu et al. / Thin Solid Films xxx (2013) xxx–xxx
Now, the question is whether the fluctuation curves are constant in time and are really systematic errors. In order to answer this question and make clear the stability of the design, an experiment continuously measuring the RD spectra of air was examined over 3 hours and at a temperature of 23 °C. The 82-time measurements were obtained with a temporal interval of 2 minutes. The curves of the real and imaginary RD signals versus time at three selected wavelengths are plotted in Fig. 5. The standard deviations of the RD signals versus time vary with wavelength. At short wavelength, for example, at 400 nm, the standard deviations of both N and S are better than 1.4 × 10−4 and 5.5 × 10−4, respectively. However, the standard deviations of both N and S at 800 nm are increased to 5.3 × 10−4 and 1.4 × 10−3, respectively. Fortunately, there are no obvious temporal effects changing the RD signals. As a conclusion, the fluctuation curves in Fig. 4 are systematic errors and can be directly eliminated by subtracting them from the measurement data. The fact that the standard deviation of the RD signal at long wavelength is larger than the standard deviation at short wavelength might be introduced by a reason below. For the same voltage range, the retardation variation of the LCVR at short wavelength is larger than the variation at long wavelength. Larger variation in retardation means a better ability to overcome the noise-induced error when a least square method is used. Accordingly, the measurement indicates a less sensitivity to the intensity noise at short wavelength than at long wavelength. A solution for the above problem is to keep the same retardation range and the same sampling number for all wavelengths. 5.2. Temperature influence It is known that the retardation of the LCVR is temperature dependence [9,14]. We also checked the thermal effect on the measurement
quality. With the same straight through configuration discussed in Section 5.1, the RD spectra of air were measured at three different temperatures, 21 °C, 23 °C, and 25 °C. All spectra were corrected with the aforementioned systematic errors, which were measured at 23 °C. In Fig. 6(a), the oscillation comes back to the corrected spectra of N at 21 °C and 25 °C. And three spectra have positions in a sequence corresponding to the temperatures. It implies that there is offset imposed on the real spectra of the RD signal by a change of the temperature. In Fig. 6(b), the oscillation effect on the spectra of S at 21 °C and 25 °C is even large. However, there is no offset happened with the changes of temperature. General speaking, thermal effect is a big problem to high precision LC-RDS measurement, but this design is also available for a measurement with a precision requirement less than 3 × 10− 3. This type of temperature-induced error might be caused by both the length change of the cavity in the LCVR and the change of the intrinsic optical property of the liquid crystal material. A reasonable way to eliminate this problem is to equip a temperature controller for the LCVR, which is already available in commercial products. 6. Conclusions An LCVR-based RD spectrometer has been developed for fast spectroscopic measurements. This design has a simpler optical structure and easier operation procedure than an RCRD spectrometer. The achieved RD spectrum covers a wavelength range from 400 to 800 nm and one measurement takes several seconds. However, the defects of the LCVR degrade the measurement quality and attentions have to be paid on them. One is the interference occurring in the cavity of the LCVR, which brings a systematic error fluctuating along with wavelength to RD spectra. Another is the temperature influence on the LCVR, which introduces even more complicate errors to the measurement results. Fortunately, with a data processing and a good control of
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Fig. 6. (Color online) Temperature influence on the Re(Δr/r) (a) and Im(Δr/r) (b) signals.
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C. Hu et al. / Thin Solid Films xxx (2013) xxx–xxx
temperature, the standard deviation of the corrected RD spectrum as a function of wavelength is better than 3 × 10−4. The stability in the whole spectral range is proven as good as 1.3 × 10−3 over 3 hours. Acknowledgements The authors gratefully acknowledge the support from the National Natural Science Foundation of China (grant no. 61008028), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (grant no. 201140), and the Natural Science Foundation of Tianjin, China (grant no. 11JCJBJC25700). References [1] P. Weightman, D.S. Martin, R.J. Cole, T. Farrell, Rep. Prog. Phys. 68 (2005) 1251.
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Please cite this article as: C. Hu, et al., Thin Solid Films (2013), http://dx.doi.org/10.1016/j.tsf.2013.12.034