Improved common-path optical heterodyne interferometer
for measuring small optical rotation angle of chiral medium
Ju-Yi Lee
a,*, Der-Chin Su
b aDepartment of Mechanical Engineering, National Central University, No. 300 Jhongda Road, Jhongli City 32001, Taiwan, ROC
b
Institute of Electro-Optical Engineering, National Chiao Tung University, 1001 Ta-Hsueh Rd., Hsin-Chu 30050, Taiwan, ROC Received 26 April 2005; received in revised form 22 June 2005; accepted 24 June 2005
Abstract
An improved common-path optical heterodyne interferometry for measuring small optical rotation angle of chiral liquid is presented. The optical rotation angle can be determined by the phase difference between the test and reference signals. By means of inserting a phase retarder behind the test sample, the measuring resolution of optical rotation angle is greatly enhanced. The theoretical prediction indicates the resolution is better than 3.5· 105degree.
2005 Elsevier B.V. All rights reserved.
PACS: 07.60.j; 33.55.A; 78.20.Ek
Keywords: Heterodyne interferometry; Optical rotation angle; Optical activity; Chiral medium
1. Introduction
The chiral or optical activity medium has the ability to rotate the polarization plane of light. The measurement of the rotation angle with an optical polarimeter has existed for several years [1–9]. There are several successful methods such as high accuracy universal polarimetry (HAUP)
[4,5] and heterodyne interferometry [6–9] to
improve the measurement resolution of the opti-cal rotation angle. The optiopti-cal configuration of HAUP is the conventional polarimetry setup, and its method is based on a least-squares refine-ment of the transmitted light intensity as a func-tion of the azimuthal angles of the polarizer and the analyzer. The optical rotation angle is ob-tained by the fitting parameters provided that the systematic errors have been removed. In het-erodyne interferometry, the rotation angle is esti-mated by means of the measurement of the phase difference between two interference signals. Chou et al. [8] proposed the circular polarized 0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.optcom.2005.06.056
*
Corresponding author. Tel.: +886 3 426 7307; fax: +886 3 425 4501.
E-mail address:[email protected](J.-Y. Lee).
optical heterodyne interferometer to effectively measure the optical rotation of a scattered chiral medium. In their works, the measurement resolu-tion is about 103degree, and a large amount of denser chiral medium is needed to make the optical rotation angle measurable. On the other
hand, Lin et al. [9] presented a heterodyne
Mach-Zehnder interferometer to enhance the measurement resolution, and their measurement resolution is about 6· 105degree. However, LinÕs optical configuration and the associated algorithm are more complicated.
In this paper, we present an improved
com-mon-path optical heterodyne interferometer
[10], which is similar to ChouÕs configuration [8] to measure the optical rotation angle. In our method, a phase retarder is inserted behind the test sample and the phase difference variation between the test signal and the reference signal is greatly enhanced. Benefiting from the common-path optical configuration and the heterodyne interferometric phase measurement, this method has the advantages of high stability against sur-rounding vibration, high measurement resolu-tion, and relatively straightforward operation. The theoretical prediction indicates its resolution is better than 3.5· 105degree, which is almost 30 times better than that obtained with ChouÕs method. In addition, not only is our measure-ment resolution higher than that of LinÕ s method, but also the optical configuration is more com-pact. The feasibility is demonstrated.
2. Principle
The schematic diagram of this improved com-mon-path optical heterodyne interferometer is shown in Fig. 1. The light passing through an electro-optic modulator EOM is divided by a beam-splitter BS into two parts-the reflected light and the transmitted light. The reflected light
passes through an analyzer ANr and enters a
photodetector Dr. If the amplitude of the light
detected by Dr is Er, then the intensity measured
by Dr is Ir= |Er|2. It is regarded as the reference
signal. On the other hand, the transmitted light passes through a half-wave plate H, a test
sam-ple S, a phase retarder R, an analyzer ANt,
and finally enters a photodetector Dt. If the
amplitudes of the test light is Et, then the
inten-sity measured by Dtis It= |Et|2. It is regarded as
the test signal.
For convenience, the z-axis is chosen along the propagation direction and the y-axis is along the vertical direction. Let the light coming from a laser be linearly polarized at 45 with respect to the x-axis, then its Jones vector can be written as
Ein¼ 1 ffiffiffi 2 p 1 1 . ð1Þ
If the fast axis of EOM is along the x-axis, and an external sawtooth voltage signal with angular frequency x and amplitude Vk/2 (the half-wave
voltage of EOM) is applied to EOM, then the retardation produced by EOM can be expressed as xt [9]. If the transmission axis of ANr is 45
with respect to the x-axis, then we have Er¼ ANr EOMðxtÞ Ein ¼1 2 1 1 1 1 eixt=2 0 0 eixt=2 ! 1 ffiffiffi 2 p 1 1 ¼ 1ffiffiffi 2 p cos xt 2 1 1 . ð2Þ
Hence, the intensity of the reference signal is
Ir¼ Ej jr2¼12ð1þ cos xtÞ. ð3Þ z x He-Ne laser ANt(45°) ANr(45°) Dt Dr BS R(0°) S(α) EOM Lock-in Amp. V H(22.5°) t
Fig. 1. Schematic diagram for the improved common-path optical heterodyne interferometer. EOM, electro-optic modu-lator; BS, beam splitter; H, half-wave plate; S, tested sample; R, phase retarder; AN, analyzer; D, photodetector.
On the other hand, if the fast axes of H and R, and the transmission axis of ANtare set to 22.5, 0,
and 45 with respect to the x-axis, respectively, then the Jones vector of the test beam is
Et¼ ANt R SðaÞ H EOMðxtÞ Ein ¼1 2 1 1 1 1 eid=2 0 0 eid=2 ! cos a sin a sin a cos a 1ffiffiffi 2 p 1 1 1 1 eixt=2 0 0 eixt=2 ! 1 ffiffiffi 2 p 1 1 ¼1 2 cos a cos xt 2 þ i sin a sin xt 2 eid2 h sin a cosxt 2 i cos a sin xt 2 eid2 i 1 1 ; ð4Þ where S(a) is the Jones matrix of the chiral
med-ium[6], and a is its optical rotation angle, and d
is the phase retardation of R. The intensity of the test signal is given by
It¼ Ej jt 2 ¼1 2½1 A cosðxt þ /Þ; ð5Þ where A¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2dþ sin 2a cos dð Þ2 q ð6Þ and /¼ tan1 tan d sin 2a . ð7Þ
If these two sinusoidal signals It and Ir are sent
to a lock-in amplifier, then their phase difference /can be obtained. The relationships between the phase difference / and the optical rotation angle a for three different phase retardations (i.e., d= 170, 175, and 178) are shown in Fig. 2. It is clear that the slope of the curve becomes steeper as the phase retardation increases. That is, a phase retarder with large phase retardation will be used to get high measurement resolution.
From Fig. 2 or Eq. (7), it can be seen that /
= 90 as a = 0. Therefore, we define the phase difference variation w as
w¼ /ðaÞ / 0 ¼ tan1 tan d
sin 2a
90. ð8Þ
Consequently, Eq.(8)can be rewritten as
a¼ 1 2sin
1
tan d tan w
ð Þ. ð9Þ
It is obvious from Eq.(9)that a can be calculated with the measurement of w under experimental conditions in which d is specified.
3. Experiments and results
To show the validity of this method, a He–Ne laser with wavelength 632.8 nm modulated by an
electro-optic modulator (New Focus, Model
4002) was used as a heterodyne light source. The frequency difference between p- and s-polarized
components [10] was 1 kHz. The experimental
condition d = 178 was chosen for high measure-ment resolution. A half-wave plate is traditionally used as an optical rotator like a Faraday rotator. Therefore, we inserted another half-wave plate to replace the chiral medium S to simulate the optical rotation. The experimental results of w versus a are shown inFig. 3, where the symbol ‘‘’’ represents the measured data and the solid curve represents the theoretical curve (curve A) calculated from
Eq. (8). It is obvious that they match well. For
comparison, the theoretical curves of w versus it a calculated by LinÕs method (curve B) [9] and ChouÕs method (curve C) [8] are also shown in Fig. 3. It is clear that the slope of curve A is almost double that of curve B and 30· that of curve C. It
= 170° δ δ = 175° δ= 178° -45 -50 -55 -60 -65 -70 -75 -80 -85 0 -90
Phase difference (degree)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Optical rotation angle α (degree)
φ
Fig. 2. The relationships between / and a for three different phase retarders.
is thus seen that this method can enhance the mea-surement resolution.
Next, the optical rotation angles of six d-glu-cose solutions in different weight percent, namely, 0.1%, 0.3%, 0.5%, 0.7%, 0.9%, and 1% g/cc, were measured. Each solution was held in a quartz cell of 4 cm length. The measured data of w and their associated a values are listed inTable 1, where the reference optical rotation angles aref are also
in-cluded for comparison. The reference optical rota-tion can be calculated with the equarota-tion[2,10,11]
aref ¼ asCgL; ð10Þ
where as is the specific rotation in degree/dm/(g/
cc), Cgis the concentration in g/cc, and L is the cell
length in decimeters, dm. The value of as can be
obtained from [9] and it is 44.8/dm/g/cc at
633 nm wavelength.
4. Discussion
From Eqs. (5) and (6), it can be seen that the amplitude of the test signal depends on a and d. In our experiments the condition d = 178 was chosen, and the minimum contrast of the test signal approximated to 0.035 as a = 0. It is still enough for a lock-in amplifier (Stanford, Model SR850) to measure the phase difference in our experiments. If d was chosen to approach to 179 or 180, then the contrast of the test signal would be too low to be measured.
From Eq.(9), we can get Da¼ tan d sec
2w
ð Þ
2 cos 2a Dw; ð11Þ
where Da and Dw are the errors in a and w, respec-tively. As a approaches zero, Eq. (11) can be rewritten as
Daffi tan d
2 Dw. ð12Þ
Considering the angular resolution of the lock-in amplifier, the second harmonic error and the polarization-mixing error, the total errors Dw decrease to 0.002 in our experiments [9]. Substi-tuting d = 178 and Dw = 0.002 into Eq. (12), we get Da = 3.5· 105degree.
If a scattered chiral medium is tested, an extra phase difference /s occurs. According to [8], it
can be expressed as /s¼ nx c 3l0 s 4la 1=2 L; ð13Þ
where n is the refractive index, c is the speed of light, l0
s and la are the reduced scattering and
absorption coefficients, respectively, and L is the thickness of the scattering medium. If a poly-styrene suspension with 1% vol. concentration, l0
s¼ 0.3 cm1, la= 0.001 cm1, and n = 1.33 is
tested under our experimental conditions, we get ws 104degree. This value is far smaller
than Dw (0.002), so the extra phase difference /s due to scattering can be ignored. Hence,
Table 1
Experimental results and the reference data
Concentration (g/cc) w() a() aref() 0.1% 1.081 0.019 0.018 0.3% 3.022 0.053 0.054 0.5% 5.356 0.094 0.090 0.7% 6.823 0.120 0.125 0.9% 9.238 0.163 0.161 1.0% 10.051 0.177 0.179
w, phase difference variation; a, optical rotation angle; aref,
reference optical rotation angle calculated from Eq.(10)and[9].
0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 25 30 C: Chou’s method B: Lin’s method A: this method
Phase difference variation
Ψ
(degree)
Optical rotation angle α (degree)
Fig. 3. Measurement results and theoretical curves of w versus a.
our experimental conditions may be suitable for measuring a scattered chiral medium.
5. Conclusion
An improved common-path optical heterodyne interferometer for the measuring the small optical rotation angle of a chiral medium is presented. By means of inserting a phase retarder behind the test sample and the heterodyne interferometric phase detection technique, it demonstrates a measure-ment resolution better than 3.5· 105degree. In addition, this method has some merits such as sim-ple optical configuration, real-time measurement, and high stability against surrounding vibration and air turbulence.
Acknowledgments
The authors cordially thank Dr. Cheng-Chih Hsu and Dr. Chyan-Chyi Wu for their useful help
on the manuscript. This study was supported in part by the National Science Council, Taiwan, under contract NSC 93-2218-E-008-036.
References
[1] G.L. Co´te, M.D. Fox, R.B. Northrop, IEEE Trans. Biomed. Eng. 39 (1992) 752.
[2] B.D. Cameron, G.L. Co´te, IEEE Trans. Biomed. Eng. 44 (1997) 1127.
[3] C. Chou, Y.C. Huang, C.M. Feng, M. Chang, Jpn. J. Appl. Phys. 36 (1997) 356.
[4] H. Meekes, A. Janner, Phys. Rev. B 38 (1988) 8075. [5] C.L. Folcia, J. Ortega, J. Etxebarria, J. Phys. D: Appl.
Phys. 32 (1999) 2266.
[6] T. Mitsui, K. Sakurai, Appl. Opt. 35 (1996) 2253. [7] C.M. Wu, Y.T. Chuang, Sens. Actuators A 116 (2004) 145. [8] C. Chou, W.C. Kuo, T.S. Hsieh, H.K. Teng, Opt.
Commun. 230 (2004) 259.
[9] J.Y. Lin, K.H. Chen, D.C. Su, Opt. Commun. 238 (2004) 113.
[10] J.Y. Lee, D.C. Su, Opt. Commun. 162 (1999) 710. [11] N. Berova, K. Nakanishi, R.W. Woody (Eds.), Circular
Dichroism: Principles and Applications, second ed., Wiley-VCH, 2000 (Chapter 1).
