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Measuring the optical rotation based on the Fast Fourier Transform
Optik 123 (2012) 1404–1406
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Measuring the optical rotation based on the Fast Fourier Transform Bochun Wu a , Hongzhi Jia a,∗ , Guizhen Xia b a Shanghai Key Laboratory of Modern Optical System, School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China b Laboratory Management Centre, CAPF Shanghai Institute of Politics, Shanghai 200435, China
a r t i c l e
i n f o
Article history: Received 11 March 2011 Accepted 25 July 2011
Keywords: Optical rotation Faraday magneto-optic effect Fast Fourier Transform
a b s t r a c t Based on the structure of polarizer–Faraday modulator–analyzer, a method of measuring the optical rotation is proposed by using Fast Fourier Transform to analyze the output signal of optical material. The output signal detected by a photodetector comprises two AC components, which angular frequencies are ω and 2ω, respectively. So, the amplitude ratio of different frequency signals can be obtained by using the Fast Fourier Transform. According to the relationship between the amplitude ratio and optical rotation, the optical rotation of material can be detected. Furthermore, the experiment results show that the correlation coefficient between the measured optical rotation and the sugar solution concentration is found to be 0.99979. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction It is well known that the polarization plane of linearly polarized light may rotate, when it passes through certain optical material. This is because these materials contain asymmetric structure of the chiral material, such as quartz crystal or sugar solution. The research shows that optical rotation of the sugar solution depends upon the thickness of the medium, the concentration of the sugar solution, and the wavelength of the measurement light source [1,2]. By measuring the optical rotation of optical solution, we can detect its concentration. So the optical rotation is a very important parameter of the chiral material. In some food, it is required to control the sugar concentration and the other optically active substances [3]. Also the glucose is a key diagnostic parameter for many diseases, there is a valuable application in monitoring blood sugar in patients [4–6]. According to the optical heterodyne technique, some researchers proposed different methods to measure the optical rotation. Feng et al. [7] used an optical polarized heterodyne polarimeter and lock-in technique to detect the concentration of a glucose solution obtained from the detection of a phase sensitive heterodyning signal. Based on the principle of Feng et al., Lin et al. [8] proposed an improved optical method by using a MachZehnder interferometer to enhance the measurement solution. But Lin’s optical configuration and the associated algorithm were more complicated. Lin et al. [9] based on an electro-optic (EO)
∗ Corresponding author. Tel.: +86 21 55274144; fax: +86 21 55271877. E-mail address: [email protected] (H. Jia). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.08.020
modulated circular heterodyne interferometer and phase-lock technique to measure the optical rotation angle. Additionally, in 2006, Lo et al. [10] adopted a liquid-crystal (LC) rotator which was driven with a sinusoidal voltage to instead the Faraday modulator. But the capability of glucose sensing would be influenced by the electro-optic characteristics of the LC rotator. In 2007, Flores et al. [11] proposed a polarimeter to use an analyzer mounted on rotator 360 stage instead of employing a Faraday modulator. However, the rotation angle was 0.95 when the glucose concentration was zero. In this paper, we use a photodetector to convert the optical signal which through the magneto-optic modulation of Faraday coil and material to electrical signal. We find that the amplitude ratio of different frequency signals can be obtained by using the Fast Fourier Transform. According to the relationship between the amplitude ratio and optical rotation, the optical rotation of material can be detected. 2. Theoretical background Fig. 1 presents a schematic illustration of the optical configuration to measure the optical rotation. As shown, the system comprises a He–Ne laser, a polarizer, a Faraday modulator, a sample, an analyzer, a photodetector. The He–Ne laser with wavelength 632.8 nm is employed as the light source. The polarizer and the analyzer are adjusted orthogonally. In the system the Faraday modulator is modulated with a sinusoidal electrical current. The light beam from the He–Ne laser turns into polarized light when it passes through the polarizer. Then the plane of the polarized light periodically changes with the current change in Faraday modulator. After this, the analyzer detects optical signal
B. Wu et al. / Optik 123 (2012) 1404–1406
1405
Fig. 1. Schematic diagram of the experimental setup for measuring the optical rotation.
Fig. 3. Fast Fourier Transform of the simulation output waveform.
Based on Fast Fourier Transform, we could convert the time domain of output signal to frequency domain. Therefore, the amplitude ratio A is easily obtained. Consequently, the optical rotation ˛ can be determined from: Fig. 2. The simulation output waveform with optical material.
˛=
ˇ A 4
(7)
from sample solution. The photodetector changes the optical signal into electrical signal. According to the Malus’ law, when is the angle between the axis of polarizer and analyzer which deviates from the orthogonal position, the light intensity after the analyzer is:
Fig. 3 shows the result of simulation output waveform after Fast Fourier Transform. We can find that the y-axis amplitude in frequency ω and 2ω have apparent values. So, if we get the amplitude ratio A, the optical rotation can be estimated from Eq. (7).
I = I0 sin2
3. Experimental setup and results
(1)
So, after the polarized light passes through the Faraday modulator and sample solution with the optical rotation ˛, could be expressed as: =˛+V ·B·L
(2)
where V is the Verdet constant of the material inside the Faraday modulator, L is the sample path-length, B = B0 sin(ωt) is the magnetic induction generated by the sinusoidal current and the angular frequency is ω. Since the maximum modulated angle of the Faraday modulator ˇ = V · L · B0 , Eq. (2) can be written as: = ˛ + ˇ sin(ωt)
(3)
Using Eq. (1) and (3), the light intensity after the analyzer can be expressed as: I = I0 sin2 [˛ + ˇ sin(ωt)]
Fig. 1 shows the experimental setup. The frequency of the current in which through the coil of the Faraday modulator was 50 Hz. After testing, the maximum modulated angle of the Faraday modulator ˇ was 2.27◦ . In the experiment, the rotation tube length was 100 mm. The sample sugar concentrations with different weights were dissolved in water separately and stirred fully, then stopped for a little while. Fig. 4 shows the waveform with sugar solution 0.015 g/ml, which was detected by photodetector. Although Fig. 4 was similar to Fig. 2, the waveform was influenced seriously by the noise. So, we used Fast Fourier Transform to analyze the output signal (see Fig. 5). The Faraday modulator was operating at a frequency of 50 Hz, so the x-axis frequency in 50 Hz and 100 Hz had the amplitude values. When we got the amplitude ratio A, the optical rotation ˛ can be
(4)
Assuming ˛ and ˇ are small, we obtain I
2
= I0 × [˛ + ˇ sin(ωt)] = I0 × [˛2 + 2˛ˇ sin(ωt) + ˇ2 sin2 (ωt)] 1 − cos(2ωt) = I0 × ˛2 + 2˛ˇ sin(ωt) + ˇ2 (5) 2 1 2 ˇ2 2 = I0 × ˛ + ˇ + 2˛ˇ sin(ωt) − cos(2ωt) 2 2
The expression in Eq. (5) shows that the output optical signal includes DC component, AC components of angular frequency ω and 2ω. Fig. 2 shows the simulated output waveform with sample active solution. As a result, the amplitude ratio of angular frequency ω to 2ω is A, which could be expressed as: A=
Aω 2˛ˇ 4˛ = = A2ω ˇ (1/2)ˇ2
(6)
Fig. 4. The output waveform of photodetector (sugar solution with concentration of 0.015 g/ml).
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B. Wu et al. / Optik 123 (2012) 1404–1406
commercial electricity. The frequency of 50 Hz for the modulator increases the noise coupled from the coil into the photodetector [12]. If the maximum Faraday modulation angle ˇ can be measured accurately and the frequency of the Faraday coil is more stable, the measurement accuracy can be further improved. Besides, the temperature and purity of the sugar solution cannot be ignored [10]. Therefore, the result confirms the ability of this method to obtain precise measurements of the optical rotation. 4. Conclusions
Fig. 5. Fast Fourier Transform of photodetector output waveform (sugar solution with concentration of 0.015 g/ml). Table 1 The data of experiment (27 ◦ C). No.
Sugar solution concentration (g/ml)
Amplitude ratio
Optical rotation (◦ )
1 2 3 4 5 6 7 8 9
0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040
0 0.535 0.996 1.533 2.054 2.548 3.012 3.543 4.039
0 0.304 0.565 0.870 1.166 1.446 1.709 2.011 2.292
In this study, a measurement based on Fast Fourier Transform is successfully developed to measure the optical rotation. The optical structure was composed of a polarizer, sample, a Faraday modulator and an analyzer. We analyzed the output optical intensity after the analyzer and found that the waveform included two AC components if there was optical active substance in the optical path. The optical rotation could be measuring by using Fast Fourier Transform to get the amplitude ratio of the two AC components. When compared with previous study, the structure of this method is simple and its correlation coefficient is higher than that in Ref. [11]. Moreover, the main characteristic of our presented method is using Fast Fourier Transform to process the output signal, which results not only in a good linear output but also take advantage of the improvement in signal to noise ratio. Acknowledgements This work was supported by the Innovation Fund Project For Graduate Student of Shanghai (JWCXSL1022), the Innovation Funds from Shanghai Committee of Science & Technology (09YZ211) and partly supported by the Shanghai Leading Academic Discipline Project (No. S30502) and the Programs (08DZ2272800) from Shanghai Committee of Science & Technology. References
Fig. 6. The relationship between the sugar solution concentration and optical rotation.
estimated from Eq. (5). Using this method, we measured the different sugar concentration’s optical rotation listed in Table 1. Fig. 6 was determined by the experimental data collected for each one solute. Fig. 6 presents the experimental relationship between the optical rotation and the sugar concentration. The correlation coefficient for the linear regression for sugar concentration was 0.99979. From Eq. (7), it can be seen that optical rotation ˛ is proportional to the maximum modulated angle of the Faraday modulator ˇ. In the experiment, ˇ cannot be measured exactly, and then it would have an important impact on the measurement results. Also, the current through the coil of Faraday modulator is driven by
[1] L.D. Barron, Molecular Light Scattering and Optical Activity, Cambridge Press, London, 1982. [2] I. Alex Vitkin, D. Richard, Claire Laszlo, L. Whyman, Effects of molecular asymmetry of optically active molecules on the polarization properties of multiply scattered light, Opt. Express 10 (4) (2002) 222–229. [3] H.J. King, C. Chou, S.T. Lu, Optical heterodyne polarimeter for measuring the chiral parameter and the circular refraction indices of optical activity, Opt. Lett. 18 (22) (1993) 1970–1972. [4] C. Chou, C.Y. Han, W.C. Kuo, Y.C. Huang, C.M. Feng, J.C. Shyu, Noninvasive glucose monitoring in vivo with an optical heterodyne polarimeter, Appl. Opt. 37 (1998) 3553–3557. [5] J.S. Baba, B.D. Cameron, S. Theru, G.L. Cote, Effect of temperature, pH, and corneal birefringence on polarimetric glucose monitoring in the eye, J. Biomed. Opt. 7 (3) (2002) 321–328. [6] C.E. Ferrante do Amaral, B. Wolf, Current development in non-invasive glucose monitoring, Med. Eng. Phys. 30 (5) (2007) 541–549. [7] C.-M. Feng, Y.-C. Huang, J.-G. Chang, M. Chang, C. Chou, A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement, Opt. Commun. 141 (1997) 314–321. [8] J.-Y. Lin, K.-H. Chen, D.-C. Su, Improved method for measuring small optical rotation angle of chiral medium, Opt. Commun. 238 (2004) 113–118. [9] J.-F. Lin, C.-C. Chang, C.-D. Syu, Y.-L. Lo, S.-Y. Lee, A new electro-optic modulated circular heterodyne interferometer for measuring the rotation angle in a chiral medium, Opt. Laser Eng. 47 (2009) 39–44. [10] Y.-L. Lo, T.-C. Yu, A polarimetric glucose sensor using a liquid-crystal polarization modulator driven by a sinusoidal signal, Opt. Commun. 259 (2006) 40–48. [11] J.L. Flores, G. García-Torales, C.R. Juárez, M.E. Cota, C.P. Ávila, Novel optical polarimeter using an analyzer mounted on rotator 360 stage, Proc. SPIE 6422 (2007) 642211. [12] A. Arnaud, F. Silveira, E.M. Frins, A. Dubra, C.D. Perciante, J.A. Ferrari, Precision synchronous polarimeter with linear response for the measurement of small rotation angles, Appl. Opt. 39 (16) (2000) 2601–2604.
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Measuring the optical rotation based on the Fast Fourier Transform Bochun Wu a , Hongzhi Jia a,∗ , Guizhen Xia b a Shanghai Key Laboratory of Modern Optical System, School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China b Laboratory Management Centre, CAPF Shanghai Institute of Politics, Shanghai 200435, China
a r t i c l e
i n f o
Article history: Received 11 March 2011 Accepted 25 July 2011
Keywords: Optical rotation Faraday magneto-optic effect Fast Fourier Transform
a b s t r a c t Based on the structure of polarizer–Faraday modulator–analyzer, a method of measuring the optical rotation is proposed by using Fast Fourier Transform to analyze the output signal of optical material. The output signal detected by a photodetector comprises two AC components, which angular frequencies are ω and 2ω, respectively. So, the amplitude ratio of different frequency signals can be obtained by using the Fast Fourier Transform. According to the relationship between the amplitude ratio and optical rotation, the optical rotation of material can be detected. Furthermore, the experiment results show that the correlation coefficient between the measured optical rotation and the sugar solution concentration is found to be 0.99979. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction It is well known that the polarization plane of linearly polarized light may rotate, when it passes through certain optical material. This is because these materials contain asymmetric structure of the chiral material, such as quartz crystal or sugar solution. The research shows that optical rotation of the sugar solution depends upon the thickness of the medium, the concentration of the sugar solution, and the wavelength of the measurement light source [1,2]. By measuring the optical rotation of optical solution, we can detect its concentration. So the optical rotation is a very important parameter of the chiral material. In some food, it is required to control the sugar concentration and the other optically active substances [3]. Also the glucose is a key diagnostic parameter for many diseases, there is a valuable application in monitoring blood sugar in patients [4–6]. According to the optical heterodyne technique, some researchers proposed different methods to measure the optical rotation. Feng et al. [7] used an optical polarized heterodyne polarimeter and lock-in technique to detect the concentration of a glucose solution obtained from the detection of a phase sensitive heterodyning signal. Based on the principle of Feng et al., Lin et al. [8] proposed an improved optical method by using a MachZehnder interferometer to enhance the measurement solution. But Lin’s optical configuration and the associated algorithm were more complicated. Lin et al. [9] based on an electro-optic (EO)
∗ Corresponding author. Tel.: +86 21 55274144; fax: +86 21 55271877. E-mail address: [email protected] (H. Jia). 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.08.020
modulated circular heterodyne interferometer and phase-lock technique to measure the optical rotation angle. Additionally, in 2006, Lo et al. [10] adopted a liquid-crystal (LC) rotator which was driven with a sinusoidal voltage to instead the Faraday modulator. But the capability of glucose sensing would be influenced by the electro-optic characteristics of the LC rotator. In 2007, Flores et al. [11] proposed a polarimeter to use an analyzer mounted on rotator 360 stage instead of employing a Faraday modulator. However, the rotation angle was 0.95 when the glucose concentration was zero. In this paper, we use a photodetector to convert the optical signal which through the magneto-optic modulation of Faraday coil and material to electrical signal. We find that the amplitude ratio of different frequency signals can be obtained by using the Fast Fourier Transform. According to the relationship between the amplitude ratio and optical rotation, the optical rotation of material can be detected. 2. Theoretical background Fig. 1 presents a schematic illustration of the optical configuration to measure the optical rotation. As shown, the system comprises a He–Ne laser, a polarizer, a Faraday modulator, a sample, an analyzer, a photodetector. The He–Ne laser with wavelength 632.8 nm is employed as the light source. The polarizer and the analyzer are adjusted orthogonally. In the system the Faraday modulator is modulated with a sinusoidal electrical current. The light beam from the He–Ne laser turns into polarized light when it passes through the polarizer. Then the plane of the polarized light periodically changes with the current change in Faraday modulator. After this, the analyzer detects optical signal
B. Wu et al. / Optik 123 (2012) 1404–1406
1405
Fig. 1. Schematic diagram of the experimental setup for measuring the optical rotation.
Fig. 3. Fast Fourier Transform of the simulation output waveform.
Based on Fast Fourier Transform, we could convert the time domain of output signal to frequency domain. Therefore, the amplitude ratio A is easily obtained. Consequently, the optical rotation ˛ can be determined from: Fig. 2. The simulation output waveform with optical material.
˛=
ˇ A 4
(7)
from sample solution. The photodetector changes the optical signal into electrical signal. According to the Malus’ law, when is the angle between the axis of polarizer and analyzer which deviates from the orthogonal position, the light intensity after the analyzer is:
Fig. 3 shows the result of simulation output waveform after Fast Fourier Transform. We can find that the y-axis amplitude in frequency ω and 2ω have apparent values. So, if we get the amplitude ratio A, the optical rotation can be estimated from Eq. (7).
I = I0 sin2
3. Experimental setup and results
(1)
So, after the polarized light passes through the Faraday modulator and sample solution with the optical rotation ˛, could be expressed as: =˛+V ·B·L
(2)
where V is the Verdet constant of the material inside the Faraday modulator, L is the sample path-length, B = B0 sin(ωt) is the magnetic induction generated by the sinusoidal current and the angular frequency is ω. Since the maximum modulated angle of the Faraday modulator ˇ = V · L · B0 , Eq. (2) can be written as: = ˛ + ˇ sin(ωt)
(3)
Using Eq. (1) and (3), the light intensity after the analyzer can be expressed as: I = I0 sin2 [˛ + ˇ sin(ωt)]
Fig. 1 shows the experimental setup. The frequency of the current in which through the coil of the Faraday modulator was 50 Hz. After testing, the maximum modulated angle of the Faraday modulator ˇ was 2.27◦ . In the experiment, the rotation tube length was 100 mm. The sample sugar concentrations with different weights were dissolved in water separately and stirred fully, then stopped for a little while. Fig. 4 shows the waveform with sugar solution 0.015 g/ml, which was detected by photodetector. Although Fig. 4 was similar to Fig. 2, the waveform was influenced seriously by the noise. So, we used Fast Fourier Transform to analyze the output signal (see Fig. 5). The Faraday modulator was operating at a frequency of 50 Hz, so the x-axis frequency in 50 Hz and 100 Hz had the amplitude values. When we got the amplitude ratio A, the optical rotation ˛ can be
(4)
Assuming ˛ and ˇ are small, we obtain I
2
= I0 × [˛ + ˇ sin(ωt)] = I0 × [˛2 + 2˛ˇ sin(ωt) + ˇ2 sin2 (ωt)] 1 − cos(2ωt) = I0 × ˛2 + 2˛ˇ sin(ωt) + ˇ2 (5) 2 1 2 ˇ2 2 = I0 × ˛ + ˇ + 2˛ˇ sin(ωt) − cos(2ωt) 2 2
The expression in Eq. (5) shows that the output optical signal includes DC component, AC components of angular frequency ω and 2ω. Fig. 2 shows the simulated output waveform with sample active solution. As a result, the amplitude ratio of angular frequency ω to 2ω is A, which could be expressed as: A=
Aω 2˛ˇ 4˛ = = A2ω ˇ (1/2)ˇ2
(6)
Fig. 4. The output waveform of photodetector (sugar solution with concentration of 0.015 g/ml).
1406
B. Wu et al. / Optik 123 (2012) 1404–1406
commercial electricity. The frequency of 50 Hz for the modulator increases the noise coupled from the coil into the photodetector [12]. If the maximum Faraday modulation angle ˇ can be measured accurately and the frequency of the Faraday coil is more stable, the measurement accuracy can be further improved. Besides, the temperature and purity of the sugar solution cannot be ignored [10]. Therefore, the result confirms the ability of this method to obtain precise measurements of the optical rotation. 4. Conclusions
Fig. 5. Fast Fourier Transform of photodetector output waveform (sugar solution with concentration of 0.015 g/ml). Table 1 The data of experiment (27 ◦ C). No.
Sugar solution concentration (g/ml)
Amplitude ratio
Optical rotation (◦ )
1 2 3 4 5 6 7 8 9
0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040
0 0.535 0.996 1.533 2.054 2.548 3.012 3.543 4.039
0 0.304 0.565 0.870 1.166 1.446 1.709 2.011 2.292
In this study, a measurement based on Fast Fourier Transform is successfully developed to measure the optical rotation. The optical structure was composed of a polarizer, sample, a Faraday modulator and an analyzer. We analyzed the output optical intensity after the analyzer and found that the waveform included two AC components if there was optical active substance in the optical path. The optical rotation could be measuring by using Fast Fourier Transform to get the amplitude ratio of the two AC components. When compared with previous study, the structure of this method is simple and its correlation coefficient is higher than that in Ref. [11]. Moreover, the main characteristic of our presented method is using Fast Fourier Transform to process the output signal, which results not only in a good linear output but also take advantage of the improvement in signal to noise ratio. Acknowledgements This work was supported by the Innovation Fund Project For Graduate Student of Shanghai (JWCXSL1022), the Innovation Funds from Shanghai Committee of Science & Technology (09YZ211) and partly supported by the Shanghai Leading Academic Discipline Project (No. S30502) and the Programs (08DZ2272800) from Shanghai Committee of Science & Technology. References
Fig. 6. The relationship between the sugar solution concentration and optical rotation.
estimated from Eq. (5). Using this method, we measured the different sugar concentration’s optical rotation listed in Table 1. Fig. 6 was determined by the experimental data collected for each one solute. Fig. 6 presents the experimental relationship between the optical rotation and the sugar concentration. The correlation coefficient for the linear regression for sugar concentration was 0.99979. From Eq. (7), it can be seen that optical rotation ˛ is proportional to the maximum modulated angle of the Faraday modulator ˇ. In the experiment, ˇ cannot be measured exactly, and then it would have an important impact on the measurement results. Also, the current through the coil of Faraday modulator is driven by
[1] L.D. Barron, Molecular Light Scattering and Optical Activity, Cambridge Press, London, 1982. [2] I. Alex Vitkin, D. Richard, Claire Laszlo, L. Whyman, Effects of molecular asymmetry of optically active molecules on the polarization properties of multiply scattered light, Opt. Express 10 (4) (2002) 222–229. [3] H.J. King, C. Chou, S.T. Lu, Optical heterodyne polarimeter for measuring the chiral parameter and the circular refraction indices of optical activity, Opt. Lett. 18 (22) (1993) 1970–1972. [4] C. Chou, C.Y. Han, W.C. Kuo, Y.C. Huang, C.M. Feng, J.C. Shyu, Noninvasive glucose monitoring in vivo with an optical heterodyne polarimeter, Appl. Opt. 37 (1998) 3553–3557. [5] J.S. Baba, B.D. Cameron, S. Theru, G.L. Cote, Effect of temperature, pH, and corneal birefringence on polarimetric glucose monitoring in the eye, J. Biomed. Opt. 7 (3) (2002) 321–328. [6] C.E. Ferrante do Amaral, B. Wolf, Current development in non-invasive glucose monitoring, Med. Eng. Phys. 30 (5) (2007) 541–549. [7] C.-M. Feng, Y.-C. Huang, J.-G. Chang, M. Chang, C. Chou, A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement, Opt. Commun. 141 (1997) 314–321. [8] J.-Y. Lin, K.-H. Chen, D.-C. Su, Improved method for measuring small optical rotation angle of chiral medium, Opt. Commun. 238 (2004) 113–118. [9] J.-F. Lin, C.-C. Chang, C.-D. Syu, Y.-L. Lo, S.-Y. Lee, A new electro-optic modulated circular heterodyne interferometer for measuring the rotation angle in a chiral medium, Opt. Laser Eng. 47 (2009) 39–44. [10] Y.-L. Lo, T.-C. Yu, A polarimetric glucose sensor using a liquid-crystal polarization modulator driven by a sinusoidal signal, Opt. Commun. 259 (2006) 40–48. [11] J.L. Flores, G. García-Torales, C.R. Juárez, M.E. Cota, C.P. Ávila, Novel optical polarimeter using an analyzer mounted on rotator 360 stage, Proc. SPIE 6422 (2007) 642211. [12] A. Arnaud, F. Silveira, E.M. Frins, A. Dubra, C.D. Perciante, J.A. Ferrari, Precision synchronous polarimeter with linear response for the measurement of small rotation angles, Appl. Opt. 39 (16) (2000) 2601–2604.