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An instrument for measuring low optical rotation angle
Optik 122 (2011) 733–738
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Optik journal homepage: www.elsevier.de/ijleo
An instrument for measuring low optical rotation angle Jing-Fung Lin a,∗ , Jyh-Shyang Wu b , Cong-Hui Huang c , Jiann-Shing Jeng d a Department of Computer Application Engineering, Far East University, No. 49, Jhonghua Road, Sinshih Township, Tainan County 74448, Taiwan b Department of Energy Application Engineering, Far East University, No. 49, Jhonghua Road, Sinshih Township, Tainan County 74448, Taiwan c Department of Automation and Control Engineering, Far East University, No. 49, Jhonghua Road, Sinshih Township, Tainan County 74448, Taiwan d Department of Material Science and Engineering, Far East University, No. 49, Jhonghua Road, Sinshih Township, Tainan County 74448, Taiwan
a r t i c l e
i n f o
Article history: Received 1 November 2009 Accepted 24 May 2010
Keywords: Circular birefringence Optical rotation Phase-locked
a b s t r a c t This paper develops a linear heterodyne interferometer based on phase-locked extraction for measuring low optical rotation angle. The validity of the proposed design is demonstrated by measuring a half-wave plate. The average relative error in the measured rotation angle of the half-wave plate is determined as just 0.74%. When applied to the measurement of glucose solutions with concentrations ranging from 0 to 1.2 g/dl, the average relative error in the measured rotation angle of glucose solutions is determined to be 1.46%. The correlation coefficient between the measured rotation angle and the glucose concentration is determined to be 0.999991, while the standard deviation is just 0.00051◦ . The current system is capable of measuring glucose concentration as low as 0.01 g/dl with an error of 6.67% in the rotation angle measurement. Overall, the experimental results demonstrate the ability of the proposed system to obtain highly accurate measurement of the optical rotation angle. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction Certain optical materials such as glucose which contain chiral organic molecules in solution are said to be optically active. when the linearly polarized light passes through such materials, its plane of polarization may rotate. This phenomenon is referred to as optical rotation or circular birefringence [1] and is a result of the asymmetric structure of the chiral materials. Obtaining accurate measurements of the rotation angles of optically active materials is a key concern in the optics field. The current study focuses on the particular case of the rotation angle measurement of glucose samples with different concentrations. This measurement process is an essential task in many research and drug discovery fields since glucose is known to be a key diagnostic parameter for many metabolic disorders. Several researchers [2–8] proposed different modulated mechanisms in their interferometers for measuring the optical rotation. For example, in 1997, Cameron and Cóte [3] designed the glucose sensing digital closed-loop processing system. This measurement system was based on the common-path heterodyne interference technique. It utilized Faraday rotators (FR) for both modulation and
∗ Corresponding author. E-mail address: [email protected] (J.-F. Lin). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.05.015
compensation. This method used a closed-loop processing system and a lock-in amplifier to obtain the rotation angle of the glucose. This system required use of a feedback mechanism in order to reduce the system instability of FR modulator. For strong absorbing materials, a reflection method can be adapted. Using a photo-elastic modulator (PEM) and a phasesensitive detection technique, the differential circular reflections of incident left-circularly and right-circularly polarized lights; the optical activity of absorbing materials can be obtained successfully. However, a static birefringence in a standard PEM, which could not be neglected, would degrade the accuracy [4]. It was also noted that the modulated interference effects induced by multiple laser light beam reflections at the PEM optical element surfaces were potential sources of spurious signals at the fundamental and harmonic PEM frequencies. In 2004, Lin et al. [5] proposed a heterodyne Mach–Zehnder interferometer to enhance the measurement resolution about to 6 × 10−5◦ . However, Lin’s optical configuration and the associated algorithm were more complicated. In 2006, Lo and Yu [6] adopted a liquid crystal (LC) modulator to modulate the azimuth of the linearly polarized light in a sinusoidal signal and developed a new signal-processing algorithm for the measurement of glucose concentrations. The standard deviation in rotation angle level of 0.00551◦ has been obtained, with a 0.998773 correlation coefficient between the reference and the measured values. However,
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the LC modulator has the drawback of slow frequency response [6]. In 2009, Lin et al. [7,8] proposed both circular heterodyne interferometers based on electro-optic (EO) modulation to measure the glucose concentrations, with one phase-locked technique, respectively. The resolution of rotation angle is half that of the phase resolution obtained using the lock-in amplifier. These two optical systems are capable of measuring glucose samples with concentration as low as 0.2 g/dl with a standard deviation of 0.0053◦ and 0.0065◦ , respectively. Additionally, in 2007, Chou et al. [9] adopted a Zeeman laser and proposed a polarized photon-pairs heterodyne polarimetry where a balanced detector scheme for optical rotation angle detection is setup. Optical rotation angle is conversed into the amplitude of an amplitude-modulated heterodyne signal, and the sensitivity for optical rotation detection by this polarimeter is determined as 5.5 × 10−5◦ /cm. However, the cost of the Zeeman laser is prohibitive. Further, Lin [10] proposed a technique for the concurrent measurement of the principal axis angle, the phase retardation, and the rotation angle of optical samples. In the proposed approach, a rotating-wave-plate Stokes polarimeter is used to extract the 2 × 2 central elements of the Mueller matrix of the sample of interest by two probe lights linearly polarized at angles of 0◦ and 45◦ , respectively. The measurement uncertainty of optical rotation angle induced by the principal axis angle, the phase retardation, and the deviation in the direction of probe lights is analyzed. However, the minimum measurable glucose concentration is limited to be 0.1 g/dl, which is influenced by the accuracy of the adopted Stokes polarimeter. In this study, a polarimeter capable of measuring low optical rotation angle in a chiral medium is developed successfully. The polarimeter is based on an electro-optic modulated linear heterodyne interferometer, and the proposed algorithm based on the phase-locked technique is feasible to measure the optical rotation angle precisely. The current system is capable of measuring glucose concentrations as low as 0.01 g/dl with an error of 6.67% in the corresponding measured optical rotation angle.
Fig. 1. Schematic diagram of a measurement system for the rotation angle by an electro-optic modulated linear heterodyne interferometer (a) first step, (b) second step.
wavelength 632.8 nm propagates initially through a polarizer and then through an EO modulator regulated by a saw-tooth waveform signal supplied by a function generator. The light beam then passes successively through the chiral sample and the linearly birefringent medium, and finally an analyzer. The output light intensity is then detected by a photodetector. Subsequently, the final signals I1 and I2 in Fig. 1 are processed by a lock-in amplifier in phase-locked, respectively, for extracting the rotation angle of the chiral medium. According to the Jones matrix formalism, the vectors of electric field emerging from the first and second measurements in the common-path linear heterodyne interferometer as shown in Fig. 1 can be expressed, respectively, as
2. Methodology Fig. 1 presents a schematic illustration of the proposed linear heterodyne interferometer, which has the ability to measure the rotation angle of a chiral medium by means of a simple two phase-locked extractions technique. The configurations in Fig. 1(a) and (b) are based on the common-path linear heterodyne interferometer with electro-optic (EO) modulation, and after the first measurement in Fig. 1(a) when the principal axis of the linearly birefringent material is set to 45◦ and the analyzer is set to 0◦ , then both the linearly birefringent material and analyzer are E1 = A1 (0◦ ) · LB1 (45◦ , ˇ) · S() · EO(90◦ , ωt) · P(45◦ ) · Ein
=
1 0
0 0
⎡ ⎣
cos
ˇ 2
i sin
ˇ 2
⎤
i sin
ˇ 2
⎦ ˇ 2
cos
cos
sin
− sin
cos
e−i(ωt)/2
0
1
ei(ωt)/2
0
2 1 2
1 2 1 2
0 E0
(1)
eiω0 t ,
and E2 = A2 (−45◦ ) · LB2 (0◦ , ˇ) · S() · EO(90◦ , ωt) · P(45◦ ) · Ein
=
1 2 1 − 2
1 2 1 2
−
⎡ ⎣
cos
ˇ 2
+ i sin 0
ˇ 2
cos
ˇ 2
⎤ 0
⎦
− i sin
rotated to add −45◦ increments to their principal axis and transmission axis, respectively, shown as in Fig. 1(b), and the second measurement is executed. In Fig. 1, a He–Ne laser light beam of
ˇ 2
cos − sin
sin cos
e−i(ωt)/2 0
0 e
i(ωt)/2
1 2 1 2
1 2 1 2
0 E0
eiω0 t ,
(2)
where E0 is the amplitude of the incident electric field which is parallel to the y-axis, Ein , A1 (0◦ ) is the Jones matrix of the analyzer A1 parallel to the x-axis. Meanwhile, LB1 (45◦ , ˇ) is the Jones matrix
J.-F. Lin et al. / Optik 122 (2011) 733–738
735
of the linearly birefringent medium LB1 with unknown retardance ˇ, whose principal axis is set at 45◦ to the x-axis, and LB2 (0◦ , ˇ) is the Jones matrix of the linearly birefringent medium LB2 with unknown retardance ˇ, whose principal axis is set to the x-axis, and is considered as rotating the principal axis angle of LB1 from 45◦ to 0◦ . S() represents the Jones matrix of the chiral medium, and is its rotation angle. Furthermore, A2 (−45◦ ) is the Jones matrix of the analyzer A2 , whose transmission axis is set at −45◦ to the xaxis, and is considered as rotating the transmission axis of A1 from 0◦ to −45◦ . EO (90◦ , ωt) is the Jones matrix of the EO modulator positioned with its fast axis parallel to the y-axis and driven by a saw-tooth waveform voltage at a frequency of ω, and P (45◦ ) is the Jones matrix of the polarizer set at 45◦ to the x-axis. As a result, the intensities expression for the transmitted lights, therefore, could be expressed, respectively, as I1 = Idc1 (1 + cos(ˇ) sin(2) cos(ωt) − sin(ˇ) sin(ωt)) = Idc1 + R1 cos(ωt + 1 ),
(3)
and
Fig. 2. Simulated optical rotation angle induced by EO deviation angle.
expressed as
I2 = Idc2 (1 − cos(ˇ) cos(2) cos(ωt) − sin(ˇ) sin(ωt)) = Idc2 − R2 cos(ωt − 2 ),
(4)
where Idc1 = Idc2 = E02 /4 is the DC component of the output light intensities, respectively, I1 and I2 , and E02 is the intensity of the input light. The notation R1 represents Idc1
2
Idc2 R2 represents tan−1
2
(cos(ˇ) sin(2)) + (sin(ˇ)) , 1 represents tan−1
tan(ˇ) cos(2)
2
tan(ˇ) sin(2)
,
2
(cos(ˇ) cos(2)) + (sin(ˇ)) , and 2 represents
From Eqs. (3) and (4), a lock-in amplifier can be employed to lock the AC component of the output intensities, I1 and I2 , respectively, at the reference frequency ω in order to measure the phase terms: i.e., 1 and 2 . Consequently, the optical rotation angle, , of the chiral medium can be determined from 1 = tan−1 2
tan(2 ) tan(1 )
EO(90 + d, ωt) =
.
(5)
From Eq. (5), the optical rotation angle can be estimated by two phase-lock extractions and is independent of the retardance of the linearly birefringent material. The phase-locked technique, although has been proposed in previous studies for the measurement of a chiral medium [7,8] or the linearly birefringent medium [11,12], its application to optical rotation measurement in this study is new. Above all, from the inspection of experimental results of glucose solution in front of a half-wave plate, the current system is capable of measuring low rotation angle corresponding to glucose concentration as low as 0.01 g/dl. When compared with Refs. [7,8,10], the phase-locked technique and intensity signalprocessing, respectively, the proposed algorithm in this study is feasible to measure low rotation angle. Accordingly, there are various sources of potential measurement error in the proposed system of rotation angle measurement, including misorientation of the EO modulator, variations of the retardance or orientation of the principal axis in the linearly birefringent medium, and misorientations of the polarizer and analyzer. The misalignment in the polarizer and analyzer could be easily controlled. Consequently, the present study for rotation angle measurement restricts its attention to the principal source of measurement errors, namely, misorientation of the EO modulator. If the principal axis angle of the EO modulator is to be set at 90◦ to the y-axis and the deviation angle is given by d, the Jones matrix of the EO modulator driven by a saw-tooth wave of frequency ω is
ωt ωt − i cos 2d sin 2 2 ωt −i sin 2d sin 2
−i sin 2d sin cos
ωt 2
ωt ωt + i cos 2d sin 2 2
.
(6)
Following a process of calculation, it can be shown that the phases 1d and 2d of the output light intensities of optical configuration in Fig. 1(a) and (b), respectively, are given by
1d
= tan
−1
and
.
cos
◦
2d = tan−1
2 cos(2d) sin(ˇ) ((1 + cos(4d)) sin(2) − sin(4d) cos(2)) cos(ˇ)
2 cos(2d) sin(ˇ) (sin(4d) sin(2) + (1 + cos(4d)) cos(2)) cos(ˇ)
(7)
. (8)
Furthermore, substituting 1d and 2d into the phase algorithm given in Eq. (5) yields the rotation angle d . Subsequently, the simulated rotation angles and relative errors of rotation angle measurements can be expressed as d =
1 tan−1 2
(1 + cos(4d)) sin(2) − sin(4d) cos(2) sin(4d) sin(2) + (1 + cos(4d)) cos(2)
,
(9)
and d = d − d=0 = −d.
(10)
Subsequently, the simulated rotation angles are illustrated in Fig. 2, for the variation of rotation angle in chiral medium is from 0◦ to 2◦ and the variation of deviation angle in EO modulator is from −1◦ to 1◦ . It is noted the simulated rotation angles are independent of the retardance of the linearly birefringent medium. Fig. 2 indicates that the simulated rotation angles are obtained from subtract of the EO modulator deviation angles from the theoretical rotation angles, i.e., the differences between the simulated rotation angles and the theoretical rotation angles are relative to the magnitude and direction of deviation angles in the EO modulator. Further, the simulated relative errors of the rotation angle are linear, which are different from the nonlinearity errors in Refs. [7,8]. Hence, the accurate alignment of EO modulator plays an important role in the rotation angle measurement. 3. Calibration and experimental results To enhance the precision of the rotation angle measurement obtained using the phase-lock amplifier, it is first necessary to calibrate the measurement system. In the current study, this calibration process involved two stages, namely (1) aligning the various
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optical components within the system; and (2) determining optimal experimental parameter settings for the EO modulator [7,8]. The first stage commenced by ensuring that the EO modulator was precisely positioned such that its fast axis was parallel to the y-axis. Accordingly, the input polarizer was rotated to the y-axis and the analyzer adjusted to the x-axis to form a crossed-polarizer pair. The EO modulator was then inserted into the system and the analyzer was rotated to the y-axis. In this configuration, the AC intensity component is completely suppressed if the EO modulator is precisely aligned with the y-axis. Accordingly, the position of the EO modulator was progressively adjusted until the amplitude of the AC output intensity detected in the lock-in amplifier fell to zero. The high resolution of the amplifier ensured that a precise alignment of the EO modulator could be obtained. Having aligned the EO modulator, the input polarizer and the analyzer were both rotated such that their transmission axes were orientated at 45◦ and 0◦ to the x-axis, respectively. As described earlier, the EO modulator in the current optical setup is driven using a saw-tooth waveform. To determine an appropriate value of the DC-biased voltage for the EO modulator, the voltage was increased incrementally and the rotation angle of a calibrated half-wave plate sample was computed at each step. The value of the DC-biased voltage at which the computed value of the rotation angle coincided with the known angle of the calibrated sample was then selected as an appropriate setting for the current experimental measurements. The full set of tuning parameters for the EO modulator was found to be a 80 V DC-biased voltage (provided by a controller supplied by Conoptics Inc., Model 370) and a 1 kHz saw-tooth signal with a DC-offset voltage of 0.49 V and an AC amplitude of 1.68 V (provided by a function generator supplied by Tektronix AFG 310). The time constant of the lock-in amplifier was 10 s and its voltage sensitivity was 100 mV. System noise was reduced by boosting the power delivered to the photodetector to the maximum tolerable limit in order to improve the signalto-noise ratio and using a high-sensitivity setting on the lock-in amplifier to fully utilize its dynamic range [13]. The validity of the proposed metrology system and analytical scheme was investigated using a sample as a half-wave plate (Casix Inc., Model WPF1225-633-/2) positioned in front of a quarterwave plate (CVI Inc., Model QWP0-633-04-4-R10). Note that a half-wave plate was deliberately chosen as the sample here since such plate is an ideal optical rotator and is readily available. In general, a half-wave plate rotates the polarization of linearly polarized light through twice the angle between the principal (i.e., fast) axis and the plane of polarization. In other words, placing the principal axis of the half-wave plate at an angle of 45◦ to the polarization plane results in a polarization rotation of 90◦ . In the verification experiment, the rotation angle was measured as the principal axis angle of the half-wave plate was rotated incrementally through a full 45◦ (from −22.5◦ to 22.5◦ in increments of 2.5◦ ) while the principal axis angle of the quarter-plate was fixed at 45◦ in the first step of measurement and fixed at 0◦ in the second step of measurement, respectively. Fig. 3 presents the circular birefringence property of the halfwave plate. From inspection, the rotation angle measurements are found to have an average absolute error (Pmeasured − Pactual ) of just 0.125◦ , with a standard deviation of 0.132◦ . Meanwhile, the average normalized error (see Eq. (11)) is determined to be 0.74%.
Pmeasured − Pactual × 100%. ε= Pactual
Fig. 3. Variation of measured rotation angle (sample: half-wave plate).
of the rotation angle and the actual values is determined to be 0.999979. Thus, it is evident that the proposed metrology system has a highly linear response. Overall, the results presented in Fig. 3 confirm the ability of the proposed system to obtain highlyaccurate measurement of the rotation angle property of the chiral sample. To demonstrate the repeatability of the measurement system, the rotational angle was measured 13 times with the principal axis of the half-wave plate set to an angle of 2.5◦ . The measurement results are presented in Fig. 4. From inspection, the average value of the rotational angle is found to be 5.012◦ and the standard deviation to be 0.017◦ , i.e., equivalent to a 1 value of ∼0.34% of the average value. Therefore, the results confirm the ability of the optical system to obtain highly precise measurement of the optical rotation angle. The feasibility of the proposed metrology system was further verified using a sample as a glucose solution of a specified concentration positioned in front of a half-wave plate with its principal axis angle set at 45◦ in the first measurement. Note that in this arrangement, the half-wave plate behaves as a linearly birefringent material. The glucose samples were prepared from a 60 mg/ml stock glucose solution produced by dissolving 6 g of d-glucose (Merck Ltd.) in a total volume of 100 ml of de-ionized water. Individual glucose samples with concentrations ranging from 0 to 1.2 g/dl in 0.2 g/dl increments (0.1 g/dl, 0.05 g/dl, and 0.01 g/dl are included) were prepared by diluting the stock solution with an appropriate volume of de-ionized water. The glucose samples were injected into sample cells with a length of 50 mm and were then
(11)
From Eq. (11), it is clear that when Pactual = 0, the normalized error tends to infinity [10]. Therefore, this particular error value is excluded when calculating the average value of the normalized error. The correlation coefficient between the measured values
Fig. 4. Measurement repeatability test (sample: a half-wave plate with rotation angle of 5◦ ).
J.-F. Lin et al. / Optik 122 (2011) 733–738
Fig. 5. Variation of measured rotation angle (sample: glucose solution).
Fig. 6. Measurement repeatability test (sample: glucose solution with concentration of 0.01 g/dl).
introduced individually into the optical measurement system in order to measure their respective rotation angles. Further, the equation which describes the rotation of the polarization plane of light corresponding to the concentration of glucose is expressed as C=
100 , L[]
737
(12)
where the glucose concentration C in a liquid solution, expressed in grams per deciliter of the solution, is the observed rotation in degrees, L is the sample path in decimeters, and in general [] is the specific rotation depending on temperature, wavelength and pH level. Thus, optical path length, temperature, wavelength, and pH level have an uncertainty associated with their value and contribute to the total uncertainty of glucose concentration. Fig. 5 presents the experimental relationship between the rotation angle and the glucose concentration for a driving signal with a frequency of 1 kHz. A correlation coefficient value of 0.999991 indicates a good linear response, with a standard deviation of 0.00051 ◦ , i.e., with a measurement deviation of 0.002 g/dl, which is lower than that of 0.075 g/dl in Ref. [14]. According to the 52.7◦ of the specific rotation angle in glucose solution at 633 nm and 20 ◦ C in Ref. [15], the average relative error in the measured rotation angles of glucose solutions is just 1.46%, which is lower that that of 1.82% [10]. Specifically, the measured rotation angle is 0.00246◦ for the glucose concentration of 0.01 g/dl and the relative error of rotation angle is determined to be 6.67%. To verify the repeatability of the measurement system, a glucose solution with a concentration of 0.01 g/dl was measured 13 times. The corresponding results are presented in Fig. 6. The average value of the measured rotation angle for glucose sample with a concentration of 0.01 g/dl is found to be 0.00254◦ , and the standard deviation to be 0.0001◦ , i.e., equivalent to a 1 value of ∼3.94% of the average value. The sensitivity for optical rotation detection is calculated at 2 × 10−5◦ /cm from the standard deviation and the length of sample cell. Furthermore, according to Ref. [15], the average relative error is determined to be just 3.70%, which is lower than that of 6.67% in Fig. 5. From the experimental results, the phase instability of electro-optic modulator due to the DC-biased voltage drift and thermal effect may mainly cause the variation of the measured rotation angle [12,16]. The temperature and perturbations of the glucose solution cannot be ignored [6]. Therefore, the results confirm the ability of the optical system to obtain highly precise measurements and excellent repeatability characteristics in the optical rotation angle. The 1 h stability of the proposed measurement system for a glucose solution with a concentration of 0.01 g/dl was measured as
illustrated in Fig. 7. From Fig. 7, the average stability value and the average absolute error for the rotation angle of glucose solution are 0.00244◦ and 0.0002◦ (7.54%), respectively. The proposed measuring system has good stability within at least 15–20 min short terms, illustrated as in Fig. 7. From the experimental results of stability, as shown in Fig. 7, as described before, the phase instability of EO modulator due to the DC-biased voltage drift and thermal effect may mainly cause the variation of the measured rotation angle [12,16]. A solution needed to assure enough accuracy of measured rotation angle by the stabilization of EO modulator DC-biased voltage drift can be found in Refs. [8,17], which is called the proportional–integral-derivative (PID) locking system. To stabilize the bias drift using PID or PI controller, a beam is split off after the EO modulator, and the light beam then passes through an analyzer set at an angle of 45◦ and is measured by a photodetector. The measured sinusoidal waveform phase is compared to the phase of saw-tooth waveform driven by a function generator to find the phase error, which is sent to the PI controller. The output from the PI controller is used to adjust the DC-biased voltage to the EO modulator via a general-purpose interface bus (GPIB) power supply. The limitation of this method has been found to be the voltage adjustment resolution of the GPIB power supply [17]. In this study, it is noted that ten measurements were intentionally taken for each different data point in Fig. 3 to assure the accuracy of measured rotation angle in half-wave plate, while
Fig. 7. Measurement stability test (sample: glucose solution with concentration of 0.01 g/dl).
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twelve measurements were taken for each different data point in Fig. 5 to assure the accuracy of measured rotation angle in glucose solution. 4. Discussions and conclusions In this study, an optical polarimeter capable of measuring low optical rotation angle in a chiral medium is developed successfully. The system is based on an electro-optic modulated linear heterodyne interferometer and the phase-locked technique is used to measure the optical rotation angle precisely. From the inspection of a sample as a half-wave plate, the average relative error in the rotation angle of a half-wave plate has been obtained as 0.74%. Further, a standard deviation in rotation angle level of 0.00051◦ has been obtained for glucose solutions with concentrations ranging from 0 to 1.2 g/dl, with a correlation coefficient value of 0.999991 between the reference values and the measured values. Moreover, the relative error of the rotation angle corresponding to 0.01 g/dl glucose concentration has been determined to be 6.67%. As compared to the others [6–8,10], the standard deviation of the developed system is superior to those previous studies. The sensitivity is comparable to that in Ref. [9]. The measurable low concentration of 0.01 g/dl is one order lower than 0.2 g/dl in Refs. [6–8] or than 0.1 g/dl in [10]. As the glucose level of a normal human being is around 0.1 g/dl and for clinical use a relative concentration error of less than 15% is required [14], therefore the developed system has its potential in low glucose concentration measurement. It is noted that aligning the EO modulator precisely is crucial for measurement accuracy in the rotation angle measurement. Consequently, the proposed system has the advantages as a compact common-path configuration, easy calibration, high linearity, and is not influenced by the intensity variation. Finally, it is noted that the optical rotation angle can be determined without priority knowledge of phase retardance in the linearly birefringent medium. When using a beam splitter to combine Fig. 1(a) and (b) into a real time measurement system, a phase compensator is needed to suppress the transmission/reflection phase-retardation effect of the beam splitter, thereby enhancing the precision of the measuring performance.
Acknowledgement The financial support provided to this study by the National Science Council of Taiwan under Grant NSC-97-2221-E-269-010 is gratefully acknowledged. References [1] A. Yariv, P. Yeh, Optical Waves in Crystal, John Wiley and Sons, 1984. [2] T.W. King, G.L. Cóte, R. McNichols, M.J. Goetz, Multispectral polarimetric glucose detection using a single Pockels cell, Opt. Eng. 33 (1994) 2746–2753. [3] B.D. Cameron, G.L. Cóte, Noninvasive glucose sensing utilizing a digital closedloop polarimetric approach, IEEE Trans. Biomed. Eng. 44 (1997) 1221–1227. [4] C. Chou, Y.C. Huang, M. Chang, Precise optical activity measurement of quartz plate by using a true phase-sensitive technique, Appl. Opt. 36 (1997) 3604–3609. [5] 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. [6] 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. [7] 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. Las. Eng. 47 (2009) 39–44. [8] J.F. Lin, Glucose concentration measurement using an electro-optically modulated circular polariscope, J. Mod. Opt. 56 (2009) 992–999. [9] C. Chou, K.H. Chiang, K.Y. Liao, Y.F. Chang, C.E. Lin, Polarized photon-pairs heterodyne polarimetry for ultrasensitive optical activity detection of a chiral medium, J. Phys. Chem. B 111 (2007) 9919–9922. [10] J.F. Lin, Concurrent measurement of linear and circular birefringence using rotating-wave-plate Stokes polarimeter, Appl. Opt. 47 (2008) 4529–4539. [11] S.Y. Lee, J.F. Lin, Y.L. Lo, Measurements of phase retardation and principal axis angle using an electro-optic modulated Mach–Zehnder interferometer, Opt. Las. Eng. 43 (2005) 704–720. [12] J.F. Lin, Y.L. Lo, The new circular heterodyne interferometer with electro-optic modulation for measurement of the optical linear birefringence, Opt. Commun. 260 (2006) 486–492. [13] S. Ohkubo, N. Umeda, Near-field scanning optical microscope based on fast birefringence measurements, Sens. Mater. 13 (2001) 433–443. [14] C. Pu, Z. Zhu, Y.H. Lo, A surface-micromachined optical self-homodyne polarimetric sensor for noninvasive glucose monitoring, IEEE Photon. Technol. Lett. 12 (2000) 190–192. [15] B. Wang, Measurement of circular and linear birefringence in chiral media and optical materials using the photoelastic modulator, Proc. SPIE 355 (1999) 294–302. [16] E. Compain, B. Drevillon, High-frequency modulation of the four states of polarization of light with a single phase modulator, Rev. Sci. Instrum. 69 (1998) 1574–1580. [17] J. Snoddy, Y. Li, F. Ravet, X. Bao, Stabilization of electro-optic modulator bias voltage drift using a lock-in amplifier and a proportional–integral-derivative controller in a distributed brillouin sensor system, Appl. Opt. 46 (2007) 1482–1485.
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Optik journal homepage: www.elsevier.de/ijleo
An instrument for measuring low optical rotation angle Jing-Fung Lin a,∗ , Jyh-Shyang Wu b , Cong-Hui Huang c , Jiann-Shing Jeng d a Department of Computer Application Engineering, Far East University, No. 49, Jhonghua Road, Sinshih Township, Tainan County 74448, Taiwan b Department of Energy Application Engineering, Far East University, No. 49, Jhonghua Road, Sinshih Township, Tainan County 74448, Taiwan c Department of Automation and Control Engineering, Far East University, No. 49, Jhonghua Road, Sinshih Township, Tainan County 74448, Taiwan d Department of Material Science and Engineering, Far East University, No. 49, Jhonghua Road, Sinshih Township, Tainan County 74448, Taiwan
a r t i c l e
i n f o
Article history: Received 1 November 2009 Accepted 24 May 2010
Keywords: Circular birefringence Optical rotation Phase-locked
a b s t r a c t This paper develops a linear heterodyne interferometer based on phase-locked extraction for measuring low optical rotation angle. The validity of the proposed design is demonstrated by measuring a half-wave plate. The average relative error in the measured rotation angle of the half-wave plate is determined as just 0.74%. When applied to the measurement of glucose solutions with concentrations ranging from 0 to 1.2 g/dl, the average relative error in the measured rotation angle of glucose solutions is determined to be 1.46%. The correlation coefficient between the measured rotation angle and the glucose concentration is determined to be 0.999991, while the standard deviation is just 0.00051◦ . The current system is capable of measuring glucose concentration as low as 0.01 g/dl with an error of 6.67% in the rotation angle measurement. Overall, the experimental results demonstrate the ability of the proposed system to obtain highly accurate measurement of the optical rotation angle. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction Certain optical materials such as glucose which contain chiral organic molecules in solution are said to be optically active. when the linearly polarized light passes through such materials, its plane of polarization may rotate. This phenomenon is referred to as optical rotation or circular birefringence [1] and is a result of the asymmetric structure of the chiral materials. Obtaining accurate measurements of the rotation angles of optically active materials is a key concern in the optics field. The current study focuses on the particular case of the rotation angle measurement of glucose samples with different concentrations. This measurement process is an essential task in many research and drug discovery fields since glucose is known to be a key diagnostic parameter for many metabolic disorders. Several researchers [2–8] proposed different modulated mechanisms in their interferometers for measuring the optical rotation. For example, in 1997, Cameron and Cóte [3] designed the glucose sensing digital closed-loop processing system. This measurement system was based on the common-path heterodyne interference technique. It utilized Faraday rotators (FR) for both modulation and
∗ Corresponding author. E-mail address: [email protected] (J.-F. Lin). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.05.015
compensation. This method used a closed-loop processing system and a lock-in amplifier to obtain the rotation angle of the glucose. This system required use of a feedback mechanism in order to reduce the system instability of FR modulator. For strong absorbing materials, a reflection method can be adapted. Using a photo-elastic modulator (PEM) and a phasesensitive detection technique, the differential circular reflections of incident left-circularly and right-circularly polarized lights; the optical activity of absorbing materials can be obtained successfully. However, a static birefringence in a standard PEM, which could not be neglected, would degrade the accuracy [4]. It was also noted that the modulated interference effects induced by multiple laser light beam reflections at the PEM optical element surfaces were potential sources of spurious signals at the fundamental and harmonic PEM frequencies. In 2004, Lin et al. [5] proposed a heterodyne Mach–Zehnder interferometer to enhance the measurement resolution about to 6 × 10−5◦ . However, Lin’s optical configuration and the associated algorithm were more complicated. In 2006, Lo and Yu [6] adopted a liquid crystal (LC) modulator to modulate the azimuth of the linearly polarized light in a sinusoidal signal and developed a new signal-processing algorithm for the measurement of glucose concentrations. The standard deviation in rotation angle level of 0.00551◦ has been obtained, with a 0.998773 correlation coefficient between the reference and the measured values. However,
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the LC modulator has the drawback of slow frequency response [6]. In 2009, Lin et al. [7,8] proposed both circular heterodyne interferometers based on electro-optic (EO) modulation to measure the glucose concentrations, with one phase-locked technique, respectively. The resolution of rotation angle is half that of the phase resolution obtained using the lock-in amplifier. These two optical systems are capable of measuring glucose samples with concentration as low as 0.2 g/dl with a standard deviation of 0.0053◦ and 0.0065◦ , respectively. Additionally, in 2007, Chou et al. [9] adopted a Zeeman laser and proposed a polarized photon-pairs heterodyne polarimetry where a balanced detector scheme for optical rotation angle detection is setup. Optical rotation angle is conversed into the amplitude of an amplitude-modulated heterodyne signal, and the sensitivity for optical rotation detection by this polarimeter is determined as 5.5 × 10−5◦ /cm. However, the cost of the Zeeman laser is prohibitive. Further, Lin [10] proposed a technique for the concurrent measurement of the principal axis angle, the phase retardation, and the rotation angle of optical samples. In the proposed approach, a rotating-wave-plate Stokes polarimeter is used to extract the 2 × 2 central elements of the Mueller matrix of the sample of interest by two probe lights linearly polarized at angles of 0◦ and 45◦ , respectively. The measurement uncertainty of optical rotation angle induced by the principal axis angle, the phase retardation, and the deviation in the direction of probe lights is analyzed. However, the minimum measurable glucose concentration is limited to be 0.1 g/dl, which is influenced by the accuracy of the adopted Stokes polarimeter. In this study, a polarimeter capable of measuring low optical rotation angle in a chiral medium is developed successfully. The polarimeter is based on an electro-optic modulated linear heterodyne interferometer, and the proposed algorithm based on the phase-locked technique is feasible to measure the optical rotation angle precisely. The current system is capable of measuring glucose concentrations as low as 0.01 g/dl with an error of 6.67% in the corresponding measured optical rotation angle.
Fig. 1. Schematic diagram of a measurement system for the rotation angle by an electro-optic modulated linear heterodyne interferometer (a) first step, (b) second step.
wavelength 632.8 nm propagates initially through a polarizer and then through an EO modulator regulated by a saw-tooth waveform signal supplied by a function generator. The light beam then passes successively through the chiral sample and the linearly birefringent medium, and finally an analyzer. The output light intensity is then detected by a photodetector. Subsequently, the final signals I1 and I2 in Fig. 1 are processed by a lock-in amplifier in phase-locked, respectively, for extracting the rotation angle of the chiral medium. According to the Jones matrix formalism, the vectors of electric field emerging from the first and second measurements in the common-path linear heterodyne interferometer as shown in Fig. 1 can be expressed, respectively, as
2. Methodology Fig. 1 presents a schematic illustration of the proposed linear heterodyne interferometer, which has the ability to measure the rotation angle of a chiral medium by means of a simple two phase-locked extractions technique. The configurations in Fig. 1(a) and (b) are based on the common-path linear heterodyne interferometer with electro-optic (EO) modulation, and after the first measurement in Fig. 1(a) when the principal axis of the linearly birefringent material is set to 45◦ and the analyzer is set to 0◦ , then both the linearly birefringent material and analyzer are E1 = A1 (0◦ ) · LB1 (45◦ , ˇ) · S() · EO(90◦ , ωt) · P(45◦ ) · Ein
=
1 0
0 0
⎡ ⎣
cos
ˇ 2
i sin
ˇ 2
⎤
i sin
ˇ 2
⎦ ˇ 2
cos
cos
sin
− sin
cos
e−i(ωt)/2
0
1
ei(ωt)/2
0
2 1 2
1 2 1 2
0 E0
(1)
eiω0 t ,
and E2 = A2 (−45◦ ) · LB2 (0◦ , ˇ) · S() · EO(90◦ , ωt) · P(45◦ ) · Ein
=
1 2 1 − 2
1 2 1 2
−
⎡ ⎣
cos
ˇ 2
+ i sin 0
ˇ 2
cos
ˇ 2
⎤ 0
⎦
− i sin
rotated to add −45◦ increments to their principal axis and transmission axis, respectively, shown as in Fig. 1(b), and the second measurement is executed. In Fig. 1, a He–Ne laser light beam of
ˇ 2
cos − sin
sin cos
e−i(ωt)/2 0
0 e
i(ωt)/2
1 2 1 2
1 2 1 2
0 E0
eiω0 t ,
(2)
where E0 is the amplitude of the incident electric field which is parallel to the y-axis, Ein , A1 (0◦ ) is the Jones matrix of the analyzer A1 parallel to the x-axis. Meanwhile, LB1 (45◦ , ˇ) is the Jones matrix
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735
of the linearly birefringent medium LB1 with unknown retardance ˇ, whose principal axis is set at 45◦ to the x-axis, and LB2 (0◦ , ˇ) is the Jones matrix of the linearly birefringent medium LB2 with unknown retardance ˇ, whose principal axis is set to the x-axis, and is considered as rotating the principal axis angle of LB1 from 45◦ to 0◦ . S() represents the Jones matrix of the chiral medium, and is its rotation angle. Furthermore, A2 (−45◦ ) is the Jones matrix of the analyzer A2 , whose transmission axis is set at −45◦ to the xaxis, and is considered as rotating the transmission axis of A1 from 0◦ to −45◦ . EO (90◦ , ωt) is the Jones matrix of the EO modulator positioned with its fast axis parallel to the y-axis and driven by a saw-tooth waveform voltage at a frequency of ω, and P (45◦ ) is the Jones matrix of the polarizer set at 45◦ to the x-axis. As a result, the intensities expression for the transmitted lights, therefore, could be expressed, respectively, as I1 = Idc1 (1 + cos(ˇ) sin(2) cos(ωt) − sin(ˇ) sin(ωt)) = Idc1 + R1 cos(ωt + 1 ),
(3)
and
Fig. 2. Simulated optical rotation angle induced by EO deviation angle.
expressed as
I2 = Idc2 (1 − cos(ˇ) cos(2) cos(ωt) − sin(ˇ) sin(ωt)) = Idc2 − R2 cos(ωt − 2 ),
(4)
where Idc1 = Idc2 = E02 /4 is the DC component of the output light intensities, respectively, I1 and I2 , and E02 is the intensity of the input light. The notation R1 represents Idc1
2
Idc2 R2 represents tan−1
2
(cos(ˇ) sin(2)) + (sin(ˇ)) , 1 represents tan−1
tan(ˇ) cos(2)
2
tan(ˇ) sin(2)
,
2
(cos(ˇ) cos(2)) + (sin(ˇ)) , and 2 represents
From Eqs. (3) and (4), a lock-in amplifier can be employed to lock the AC component of the output intensities, I1 and I2 , respectively, at the reference frequency ω in order to measure the phase terms: i.e., 1 and 2 . Consequently, the optical rotation angle, , of the chiral medium can be determined from 1 = tan−1 2
tan(2 ) tan(1 )
EO(90 + d, ωt) =
.
(5)
From Eq. (5), the optical rotation angle can be estimated by two phase-lock extractions and is independent of the retardance of the linearly birefringent material. The phase-locked technique, although has been proposed in previous studies for the measurement of a chiral medium [7,8] or the linearly birefringent medium [11,12], its application to optical rotation measurement in this study is new. Above all, from the inspection of experimental results of glucose solution in front of a half-wave plate, the current system is capable of measuring low rotation angle corresponding to glucose concentration as low as 0.01 g/dl. When compared with Refs. [7,8,10], the phase-locked technique and intensity signalprocessing, respectively, the proposed algorithm in this study is feasible to measure low rotation angle. Accordingly, there are various sources of potential measurement error in the proposed system of rotation angle measurement, including misorientation of the EO modulator, variations of the retardance or orientation of the principal axis in the linearly birefringent medium, and misorientations of the polarizer and analyzer. The misalignment in the polarizer and analyzer could be easily controlled. Consequently, the present study for rotation angle measurement restricts its attention to the principal source of measurement errors, namely, misorientation of the EO modulator. If the principal axis angle of the EO modulator is to be set at 90◦ to the y-axis and the deviation angle is given by d, the Jones matrix of the EO modulator driven by a saw-tooth wave of frequency ω is
ωt ωt − i cos 2d sin 2 2 ωt −i sin 2d sin 2
−i sin 2d sin cos
ωt 2
ωt ωt + i cos 2d sin 2 2
.
(6)
Following a process of calculation, it can be shown that the phases 1d and 2d of the output light intensities of optical configuration in Fig. 1(a) and (b), respectively, are given by
1d
= tan
−1
and
.
cos
◦
2d = tan−1
2 cos(2d) sin(ˇ) ((1 + cos(4d)) sin(2) − sin(4d) cos(2)) cos(ˇ)
2 cos(2d) sin(ˇ) (sin(4d) sin(2) + (1 + cos(4d)) cos(2)) cos(ˇ)
(7)
. (8)
Furthermore, substituting 1d and 2d into the phase algorithm given in Eq. (5) yields the rotation angle d . Subsequently, the simulated rotation angles and relative errors of rotation angle measurements can be expressed as d =
1 tan−1 2
(1 + cos(4d)) sin(2) − sin(4d) cos(2) sin(4d) sin(2) + (1 + cos(4d)) cos(2)
,
(9)
and d = d − d=0 = −d.
(10)
Subsequently, the simulated rotation angles are illustrated in Fig. 2, for the variation of rotation angle in chiral medium is from 0◦ to 2◦ and the variation of deviation angle in EO modulator is from −1◦ to 1◦ . It is noted the simulated rotation angles are independent of the retardance of the linearly birefringent medium. Fig. 2 indicates that the simulated rotation angles are obtained from subtract of the EO modulator deviation angles from the theoretical rotation angles, i.e., the differences between the simulated rotation angles and the theoretical rotation angles are relative to the magnitude and direction of deviation angles in the EO modulator. Further, the simulated relative errors of the rotation angle are linear, which are different from the nonlinearity errors in Refs. [7,8]. Hence, the accurate alignment of EO modulator plays an important role in the rotation angle measurement. 3. Calibration and experimental results To enhance the precision of the rotation angle measurement obtained using the phase-lock amplifier, it is first necessary to calibrate the measurement system. In the current study, this calibration process involved two stages, namely (1) aligning the various
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optical components within the system; and (2) determining optimal experimental parameter settings for the EO modulator [7,8]. The first stage commenced by ensuring that the EO modulator was precisely positioned such that its fast axis was parallel to the y-axis. Accordingly, the input polarizer was rotated to the y-axis and the analyzer adjusted to the x-axis to form a crossed-polarizer pair. The EO modulator was then inserted into the system and the analyzer was rotated to the y-axis. In this configuration, the AC intensity component is completely suppressed if the EO modulator is precisely aligned with the y-axis. Accordingly, the position of the EO modulator was progressively adjusted until the amplitude of the AC output intensity detected in the lock-in amplifier fell to zero. The high resolution of the amplifier ensured that a precise alignment of the EO modulator could be obtained. Having aligned the EO modulator, the input polarizer and the analyzer were both rotated such that their transmission axes were orientated at 45◦ and 0◦ to the x-axis, respectively. As described earlier, the EO modulator in the current optical setup is driven using a saw-tooth waveform. To determine an appropriate value of the DC-biased voltage for the EO modulator, the voltage was increased incrementally and the rotation angle of a calibrated half-wave plate sample was computed at each step. The value of the DC-biased voltage at which the computed value of the rotation angle coincided with the known angle of the calibrated sample was then selected as an appropriate setting for the current experimental measurements. The full set of tuning parameters for the EO modulator was found to be a 80 V DC-biased voltage (provided by a controller supplied by Conoptics Inc., Model 370) and a 1 kHz saw-tooth signal with a DC-offset voltage of 0.49 V and an AC amplitude of 1.68 V (provided by a function generator supplied by Tektronix AFG 310). The time constant of the lock-in amplifier was 10 s and its voltage sensitivity was 100 mV. System noise was reduced by boosting the power delivered to the photodetector to the maximum tolerable limit in order to improve the signalto-noise ratio and using a high-sensitivity setting on the lock-in amplifier to fully utilize its dynamic range [13]. The validity of the proposed metrology system and analytical scheme was investigated using a sample as a half-wave plate (Casix Inc., Model WPF1225-633-/2) positioned in front of a quarterwave plate (CVI Inc., Model QWP0-633-04-4-R10). Note that a half-wave plate was deliberately chosen as the sample here since such plate is an ideal optical rotator and is readily available. In general, a half-wave plate rotates the polarization of linearly polarized light through twice the angle between the principal (i.e., fast) axis and the plane of polarization. In other words, placing the principal axis of the half-wave plate at an angle of 45◦ to the polarization plane results in a polarization rotation of 90◦ . In the verification experiment, the rotation angle was measured as the principal axis angle of the half-wave plate was rotated incrementally through a full 45◦ (from −22.5◦ to 22.5◦ in increments of 2.5◦ ) while the principal axis angle of the quarter-plate was fixed at 45◦ in the first step of measurement and fixed at 0◦ in the second step of measurement, respectively. Fig. 3 presents the circular birefringence property of the halfwave plate. From inspection, the rotation angle measurements are found to have an average absolute error (Pmeasured − Pactual ) of just 0.125◦ , with a standard deviation of 0.132◦ . Meanwhile, the average normalized error (see Eq. (11)) is determined to be 0.74%.
Pmeasured − Pactual × 100%. ε= Pactual
Fig. 3. Variation of measured rotation angle (sample: half-wave plate).
of the rotation angle and the actual values is determined to be 0.999979. Thus, it is evident that the proposed metrology system has a highly linear response. Overall, the results presented in Fig. 3 confirm the ability of the proposed system to obtain highlyaccurate measurement of the rotation angle property of the chiral sample. To demonstrate the repeatability of the measurement system, the rotational angle was measured 13 times with the principal axis of the half-wave plate set to an angle of 2.5◦ . The measurement results are presented in Fig. 4. From inspection, the average value of the rotational angle is found to be 5.012◦ and the standard deviation to be 0.017◦ , i.e., equivalent to a 1 value of ∼0.34% of the average value. Therefore, the results confirm the ability of the optical system to obtain highly precise measurement of the optical rotation angle. The feasibility of the proposed metrology system was further verified using a sample as a glucose solution of a specified concentration positioned in front of a half-wave plate with its principal axis angle set at 45◦ in the first measurement. Note that in this arrangement, the half-wave plate behaves as a linearly birefringent material. The glucose samples were prepared from a 60 mg/ml stock glucose solution produced by dissolving 6 g of d-glucose (Merck Ltd.) in a total volume of 100 ml of de-ionized water. Individual glucose samples with concentrations ranging from 0 to 1.2 g/dl in 0.2 g/dl increments (0.1 g/dl, 0.05 g/dl, and 0.01 g/dl are included) were prepared by diluting the stock solution with an appropriate volume of de-ionized water. The glucose samples were injected into sample cells with a length of 50 mm and were then
(11)
From Eq. (11), it is clear that when Pactual = 0, the normalized error tends to infinity [10]. Therefore, this particular error value is excluded when calculating the average value of the normalized error. The correlation coefficient between the measured values
Fig. 4. Measurement repeatability test (sample: a half-wave plate with rotation angle of 5◦ ).
J.-F. Lin et al. / Optik 122 (2011) 733–738
Fig. 5. Variation of measured rotation angle (sample: glucose solution).
Fig. 6. Measurement repeatability test (sample: glucose solution with concentration of 0.01 g/dl).
introduced individually into the optical measurement system in order to measure their respective rotation angles. Further, the equation which describes the rotation of the polarization plane of light corresponding to the concentration of glucose is expressed as C=
100 , L[]
737
(12)
where the glucose concentration C in a liquid solution, expressed in grams per deciliter of the solution, is the observed rotation in degrees, L is the sample path in decimeters, and in general [] is the specific rotation depending on temperature, wavelength and pH level. Thus, optical path length, temperature, wavelength, and pH level have an uncertainty associated with their value and contribute to the total uncertainty of glucose concentration. Fig. 5 presents the experimental relationship between the rotation angle and the glucose concentration for a driving signal with a frequency of 1 kHz. A correlation coefficient value of 0.999991 indicates a good linear response, with a standard deviation of 0.00051 ◦ , i.e., with a measurement deviation of 0.002 g/dl, which is lower than that of 0.075 g/dl in Ref. [14]. According to the 52.7◦ of the specific rotation angle in glucose solution at 633 nm and 20 ◦ C in Ref. [15], the average relative error in the measured rotation angles of glucose solutions is just 1.46%, which is lower that that of 1.82% [10]. Specifically, the measured rotation angle is 0.00246◦ for the glucose concentration of 0.01 g/dl and the relative error of rotation angle is determined to be 6.67%. To verify the repeatability of the measurement system, a glucose solution with a concentration of 0.01 g/dl was measured 13 times. The corresponding results are presented in Fig. 6. The average value of the measured rotation angle for glucose sample with a concentration of 0.01 g/dl is found to be 0.00254◦ , and the standard deviation to be 0.0001◦ , i.e., equivalent to a 1 value of ∼3.94% of the average value. The sensitivity for optical rotation detection is calculated at 2 × 10−5◦ /cm from the standard deviation and the length of sample cell. Furthermore, according to Ref. [15], the average relative error is determined to be just 3.70%, which is lower than that of 6.67% in Fig. 5. From the experimental results, the phase instability of electro-optic modulator due to the DC-biased voltage drift and thermal effect may mainly cause the variation of the measured rotation angle [12,16]. The temperature and perturbations of the glucose solution cannot be ignored [6]. Therefore, the results confirm the ability of the optical system to obtain highly precise measurements and excellent repeatability characteristics in the optical rotation angle. The 1 h stability of the proposed measurement system for a glucose solution with a concentration of 0.01 g/dl was measured as
illustrated in Fig. 7. From Fig. 7, the average stability value and the average absolute error for the rotation angle of glucose solution are 0.00244◦ and 0.0002◦ (7.54%), respectively. The proposed measuring system has good stability within at least 15–20 min short terms, illustrated as in Fig. 7. From the experimental results of stability, as shown in Fig. 7, as described before, the phase instability of EO modulator due to the DC-biased voltage drift and thermal effect may mainly cause the variation of the measured rotation angle [12,16]. A solution needed to assure enough accuracy of measured rotation angle by the stabilization of EO modulator DC-biased voltage drift can be found in Refs. [8,17], which is called the proportional–integral-derivative (PID) locking system. To stabilize the bias drift using PID or PI controller, a beam is split off after the EO modulator, and the light beam then passes through an analyzer set at an angle of 45◦ and is measured by a photodetector. The measured sinusoidal waveform phase is compared to the phase of saw-tooth waveform driven by a function generator to find the phase error, which is sent to the PI controller. The output from the PI controller is used to adjust the DC-biased voltage to the EO modulator via a general-purpose interface bus (GPIB) power supply. The limitation of this method has been found to be the voltage adjustment resolution of the GPIB power supply [17]. In this study, it is noted that ten measurements were intentionally taken for each different data point in Fig. 3 to assure the accuracy of measured rotation angle in half-wave plate, while
Fig. 7. Measurement stability test (sample: glucose solution with concentration of 0.01 g/dl).
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twelve measurements were taken for each different data point in Fig. 5 to assure the accuracy of measured rotation angle in glucose solution. 4. Discussions and conclusions In this study, an optical polarimeter capable of measuring low optical rotation angle in a chiral medium is developed successfully. The system is based on an electro-optic modulated linear heterodyne interferometer and the phase-locked technique is used to measure the optical rotation angle precisely. From the inspection of a sample as a half-wave plate, the average relative error in the rotation angle of a half-wave plate has been obtained as 0.74%. Further, a standard deviation in rotation angle level of 0.00051◦ has been obtained for glucose solutions with concentrations ranging from 0 to 1.2 g/dl, with a correlation coefficient value of 0.999991 between the reference values and the measured values. Moreover, the relative error of the rotation angle corresponding to 0.01 g/dl glucose concentration has been determined to be 6.67%. As compared to the others [6–8,10], the standard deviation of the developed system is superior to those previous studies. The sensitivity is comparable to that in Ref. [9]. The measurable low concentration of 0.01 g/dl is one order lower than 0.2 g/dl in Refs. [6–8] or than 0.1 g/dl in [10]. As the glucose level of a normal human being is around 0.1 g/dl and for clinical use a relative concentration error of less than 15% is required [14], therefore the developed system has its potential in low glucose concentration measurement. It is noted that aligning the EO modulator precisely is crucial for measurement accuracy in the rotation angle measurement. Consequently, the proposed system has the advantages as a compact common-path configuration, easy calibration, high linearity, and is not influenced by the intensity variation. Finally, it is noted that the optical rotation angle can be determined without priority knowledge of phase retardance in the linearly birefringent medium. When using a beam splitter to combine Fig. 1(a) and (b) into a real time measurement system, a phase compensator is needed to suppress the transmission/reflection phase-retardation effect of the beam splitter, thereby enhancing the precision of the measuring performance.
Acknowledgement The financial support provided to this study by the National Science Council of Taiwan under Grant NSC-97-2221-E-269-010 is gratefully acknowledged. References [1] A. Yariv, P. Yeh, Optical Waves in Crystal, John Wiley and Sons, 1984. [2] T.W. King, G.L. Cóte, R. McNichols, M.J. Goetz, Multispectral polarimetric glucose detection using a single Pockels cell, Opt. Eng. 33 (1994) 2746–2753. [3] B.D. Cameron, G.L. Cóte, Noninvasive glucose sensing utilizing a digital closedloop polarimetric approach, IEEE Trans. Biomed. Eng. 44 (1997) 1221–1227. [4] C. Chou, Y.C. Huang, M. Chang, Precise optical activity measurement of quartz plate by using a true phase-sensitive technique, Appl. Opt. 36 (1997) 3604–3609. [5] 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. [6] 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. [7] 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. Las. Eng. 47 (2009) 39–44. [8] J.F. Lin, Glucose concentration measurement using an electro-optically modulated circular polariscope, J. Mod. Opt. 56 (2009) 992–999. [9] C. Chou, K.H. Chiang, K.Y. Liao, Y.F. Chang, C.E. Lin, Polarized photon-pairs heterodyne polarimetry for ultrasensitive optical activity detection of a chiral medium, J. Phys. Chem. B 111 (2007) 9919–9922. [10] J.F. Lin, Concurrent measurement of linear and circular birefringence using rotating-wave-plate Stokes polarimeter, Appl. Opt. 47 (2008) 4529–4539. [11] S.Y. Lee, J.F. Lin, Y.L. Lo, Measurements of phase retardation and principal axis angle using an electro-optic modulated Mach–Zehnder interferometer, Opt. Las. Eng. 43 (2005) 704–720. [12] J.F. Lin, Y.L. Lo, The new circular heterodyne interferometer with electro-optic modulation for measurement of the optical linear birefringence, Opt. Commun. 260 (2006) 486–492. [13] S. Ohkubo, N. Umeda, Near-field scanning optical microscope based on fast birefringence measurements, Sens. Mater. 13 (2001) 433–443. [14] C. Pu, Z. Zhu, Y.H. Lo, A surface-micromachined optical self-homodyne polarimetric sensor for noninvasive glucose monitoring, IEEE Photon. Technol. Lett. 12 (2000) 190–192. [15] B. Wang, Measurement of circular and linear birefringence in chiral media and optical materials using the photoelastic modulator, Proc. SPIE 355 (1999) 294–302. [16] E. Compain, B. Drevillon, High-frequency modulation of the four states of polarization of light with a single phase modulator, Rev. Sci. Instrum. 69 (1998) 1574–1580. [17] J. Snoddy, Y. Li, F. Ravet, X. Bao, Stabilization of electro-optic modulator bias voltage drift using a lock-in amplifier and a proportional–integral-derivative controller in a distributed brillouin sensor system, Appl. Opt. 46 (2007) 1482–1485.