## Abstract

We proposed a new method for small optical rotation measurement. The method is based on the use of a half-wave plate and the high-stability common-path heterodyne interferometry. When the azimuth angle is at 22.5^{°} of a half-wave plate, the phase has a distinct change caused by the small polarization rotation of the test beam. The optical rotation can be obtained from the relationship between the phase and the azimuth angle of the wave plate. The resolution of polarization rotation measurement can achieve 1.6 × 10^{−5} degree/mm and the detection sensitivity on circular birefringence of glucose-water solution can be up to δ|n_{r} − n_{l} | = 5.6 × 10^{−11}.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In the early 19th century, Biot & Fresnel discovered that when linearly polarized light passed through an optically active medium, the light would maintain the initial polarization state and start to rotate. Optically active medium can be divided into left-handed and right-handed mediums [1]. The polarization would gradually rotate to the left as the thickness of the left-handed medium increases when the linearly polarized light enters a left-handed medium. On the other hand, the polarization would rotate to the right as linearly polarized light enters a right-handed medium. This kind of medium is ubiquitous in nature, especially in biology, where there are many molecules with optical activity. For example, the protein composed of amino acid is a kind of biological molecule with optical activity. Feng et al. [2] used the Mach-Zehnder interferometer to measure the optical rotation of the glucose concentration. The rotation angle was directly converted to the phase between the two test signals. Because the rotation angle was small for most solutions, the resolution was not high. The stability of the light intensity and the non-common optical path structure could reduce the system accuracy. Lin et al. [3] proposed an optical heterodyne polarimeter, using the linearly polarized light to reflect on the interface of a chiral liquid. When its incident angle was close to the critical angle, the coupling coefficient of the reflected light would be amplified which could be used to measure up to an accuracy of $2 \times {10^{ - 8}}$ for the chiral parameter and $1 \times {10^{ - 5}}$ for the average refractive index of a chiral liquid, respectively. Chou [4] presented an optical heterodyne interferometer using a Zeeman laser to measure the optical rotation in a scattered chiral medium solution whose scattering particle size was 1.072 $\mu m$ in diameter and the volume concentration was 4.6%. The use of a circularly polarized photon pair (CPPP) from the two orthogonal circularly polarized light waves was able to reduce the scattering effect and improve the sensitivity on optical rotation measurement.

Chou et al. [5] reused the Zeeman laser as the light source of the polarized light common-path optical heterodyne interferometer to measure the quartz Cornu depolarizer (QCD) optically active object. Its balanced detection circuit plus a differential amplifier could greatly reduce most of the noise from laser system and environment, and then obtain accurate and highly sensitive optical rotation measurement results. The detection sensitivity of the optical activity of a quartz crystal was measured as $8.5 \times {10^{ - 10}}$. Steinbacher et al. [6] presented a setup combining common-path optical heterodyne interferometry and an accumulative technique for femtosecond laser spectroscopy to measure small changes in optical rotatory power. It could be used for the photoreaction of chiral sulfoxides, and simultaneously recorded linear absorption spectra. The presented polarimeter had an experimentally determined angular resolution of 0.10 mdeg, a measurement time of only one second, an interaction length of only 250 µm, and a probe beam size (aperture) of only 30 µm. Kumagai et al. [7] proposed a Sagnac interferometer for the glucose concentration measurement. The angle of optical rotation was measured by detecting the phase difference between clockwise and counterclockwise circular polarized light that propagated in a sensing loop with a polarization-maintaining optical fiber. The angular resolution of optical rotation is about $5.4 \times {10^{ - 4}}$ degree. Liu et al. [8] proposed a low-coherence interferometer for the optical rotation measurement in phase-sensitive detection. They used the polarization maintaining fiber (PMF) and polarization optic system to create the left- and right-handed circular states. These two states experienced the same optical path in the reference arm and test arm of the interferometer. A Wollaston prism separated two interferences, the optical rotation can be achieved by the phase difference between these two signals.

Cao et al. [9] proposed an optical rotation measurement system based on the centroid algorithm. The beam of the semiconductor laser was converted into linearly polarized passes through the polarizer (Glan-Taylor prism). Then, the polarized light was received by a photodiode after transmitting through the sample tube and analyzer (Glan-Taylor prism) fixed on a step-motor rotating stage. With the photodiode, the light signals were transformed into electrical signals, taking advantage of processing light signals with or without sample to obtain the optical rotation angle through the step difference between two centroids. After calibrating with several standard quartz tubes and measuring glucose solution with different concentration, the relative error and precision of the system were determined to be 0.4% and 0.004$^\circ $. Although this method was simple, the accuracy was not high. All the above methods did not use water as the test specimen to obtain further confirmation. Goldberg et al. [10] proposed a curve-fitting polarimetric method based on a self-referenced optical rotation polarimeter. They used two high-contrast (1 : ${10^5}$) Glan–Thompson polarizers as the polarizer and analyzer and used an Electro-Optic modulator for phase modulation. The calculated accuracy is $8.1 \times {10^{ - 5}}{\; }$degree, and the calculated precision is $2.22 \times {10^{ - 4}}$ degree.

In this paper we propose a new method for measuring tiny polarization, which is based on the common-path heterodyne interferometry combining with a half-wave plate (HWP) for measurement. When the HWP is rotated to a specific azimuth angle at 22.5$^\circ $, the phase difference changes greatly, and can be used to measure the optically active medium. The analytic polarization rotation is $1.6 \times {10^{ - 5}}\textrm{degree}/\textrm{mm}$, and the detection sensitivity of $\mathrm{\delta }|{{\; }{\textrm{n}_\textrm{r}} - {\textrm{n}_\textrm{i}}} |$ is as high as $5.6 \times {10^{ - 11}}$. Such a high angle resolution is a result that is difficult to achieve by ordinary measuring instruments. In the case of high resolution of polarization rotation measurement, in addition to measuring the concentration of glucose, the optical rotation of different water qualities is also measured to further demonstrate the feasibility of this method.

## 2. Principles

#### 2.1 Principles and experimental setup

The reason why we use the common-path heterodyne interference as the experimental light source is that the reference light (s-polarized) and the test light (p-polarized) are overlapping in the same path and cancel the phase change with each other so as not to be affected by environmental interference. This can improve the accuracy and reduce the error of the experiment. The structure is simple and can accurately resolve the phase. Not only can it be used to measure the optical rotation of optically active objects, it can also be used to measure long wavelength changes [11,12], small angle measurement [13–15], and small displacement [16–18]. The experimental structure is shown in Fig. 1, using the common-path heterodyne interferometry combining with a HWP and the test sample uses a vessel with an inner length of 5 cm to hold the solution. The experimental setup of this study is divided into two parts. Part 1 is a heterodyne light source, including a He-Ne laser (λ=632.8 nm), two polarization beam splitters (PBS1, PBS2), two acousto-optic modulators (AOM1, AOM2), two mirrors (M1, M2), and two apertures (Iris1, Iris2). Initially, in order to generate a heterodyne light source, we use PBS1 to divide the laser light into the p- and s- polarizations. The modulation frequencies of AOM1 and AOM2 are 80.00 and 80.01 MHz, respectively. After the s-polarized light is reflected by the mirror (M2) and then incidents on AOM2, the +1 order beam is reflected by PBS2. Since the modulated light will have the -1, 0 and +1 order lights, we use the aperture (Iris1or Iris2) to block the -1 and 0 order lights which are not used in this experiment, and we only take out the +1 order light. The p-polarized light from PBS1 passing through the AOM1 and reflected by the mirror (M1) is combined with the s-polarized light by PBS2. A heterodyne light source with a frequency difference of 10kHz can be obtained. Part 2 is the optical measurement architecture, which includes a beam splitter (BS), two analyzers ($\textrm{A}{\textrm{N}_\textrm{r}},{\; A}{\textrm{N}_\textrm{t}}$) with the same transmission axis at azimuth angle of 45$^\circ $, two photodetectors (${\textrm{D}_\textrm{r}},{\textrm{D}_\textrm{t}}$), a HWP (${\textrm{W}_{\mathrm{\lambda }/2}}$) and the test medium (Sample). The heterodyne light source is divided into the transmitted light and reflected light by BS. The reflected light passes through an analyzer ($\textrm{A}{\textrm{N}_\textrm{r}}$) and received by a photodetector $({\textrm{D}_\textrm{r}}$). The detected interference signal is used as the reference signal ${\textrm{I}_\textrm{r}}$. The transmitted light passes through the test medium (Sample) and the HWP (${\textrm{W}_{\mathrm{\lambda }/2}}$), and then passes through an analyzer ($\textrm{A}{\textrm{N}_\textrm{t}}$). The interference signal detected by the photodetector ${\textrm{D}_\textrm{t}}$ is regarded as the test signal ${\textrm{I}_\textrm{t}}$. Finally, a lock-in amplifier (the Stanford Research Systems, SR830) is used to capture the phase difference between the test signal ${\textrm{I}_\textrm{t}}$ and the reference signal ${\textrm{I}_\textrm{r}}$.

Because each interference is from combining the partial components of the p- and s-polarizations on the transmission axis of the analyzer and two polarization lights are always on the same optical path, the signal is the common-path heterodyne interference signal. It is stable enough without concerning the environment disturbance. In addition, the test signal ${\textrm{I}_\textrm{t}}\; $and reference signal ${\textrm{I}_\textrm{r}}\; $are from the same heterodyne light source, the same air disturbance in the light source (Part 1) can cancel each other.

#### 2.2 Formula derivation

This section mainly describes the derived formula. The optical measurement structure is shown in Fig. 2. In the process of derivation, the best measurement conditions are found in the relationship curve between the phase difference and the azimuth angle of the HWP. From Fig. 2, ${\textrm{E}_1}$ represents the optical field intensity reflected by BS and the $\textrm{Jon}{\textrm{e}^{{\prime}}}\textrm{s}$ Matrix of the reference signal can be written as

**S** represents the test specimen, and ${\textrm{E}_2}$ is the intensity of electric field of the transmitted light from the BS. Assuming that the phase retardation of the HWP is Γ, the $\textrm{Jon}{\textrm{e}^{{\prime}}}\textrm{s}$ Matrix of the test light can be written as

#### 2.3 MATLAB simulation

According to the formula presented in Section 2-2, we can compare the phase difference between ${\textrm{I}_\textrm{t}}$ and ${\textrm{I}_\textrm{r}}$. The value is between 0 and 2π. For the convenience of explanation, we will convert 0 ∼ 2π to -180°∼180°. And the function of phase difference $\phi$ can be written as

In Eqs. (13) to (16), the range of the arctangent function is from -π/2 to π/2. In order to calculate the phase difference conveniently, we define the phase retardation nearly equal to ${\pm} \mathrm{\pi }$ when the arctangent function has two variables. From the above relationship, we use MATLAB to draw the relationship between the phase difference $\phi $ and the azimuth angle θ. If the wave plate used is a nearly half-wave plate, we let the phase retardation Γ be a positive (175°∼179°), the result of the simulated phase difference is shown in Fig. 3(a) The largest sensitive angles are at $22.5^\circ ,{\; }67.5^\circ ,{\; }112.5^\circ ,{\; }157.5^\circ ,{\; \; }202.5^\circ ,{\; }247.5^\circ ,{\; }292.5^\circ ,{\; }$and $337.5^\circ $. Finally, using the result of Γ as a positive value in Fig. 3(a), when θ=22.5$^\circ $, we take out the phase $\phi $ versus the Γ and draw it as shown in Fig. 3(b). It reveals that the phase shift caused by the slight change of phase retardation is smaller than the change of the azimuth angle.

Taking out the relationship diagram between $\mathrm{\Gamma } = 175^\circ{\sim} 179^\circ $ and its azimuth angle during 22.5°±0.2 degrees, the phase difference as shown in Fig. 4(a), and the slope diagram (angular sensitivity) is shown in Fig. 4(b). The better linearity range is when the phase retardation is less than $177^\circ $.

For the same reason, if Γ is negative (-179°∼-175°), the relationship between the simulated phase difference $\phi $ and the azimuth angle θ is plotted as shown in Fig. 5(a). The largest sensitive angles are the same at $22.5^\circ ,{\; }67.5^\circ ,{\; }112.5^\circ ,{\; }157.5^\circ ,{\; }202.5^\circ ,{\; }247.5^\circ $, $292.5^\circ {\; }$, and $337.5^\circ $. Then, we use the result of Γ as a negative value in Fig. 5(a), when θ=22.5$^\circ $, taking out the phase $\phi $ versus Γ, and draw it as Fig. 5(b). It also reveals that the phase change caused by the retardation change is very small at this azimuth angle. It is almost negligible if compared with the phase change caused by the change of $\mathrm{\theta }$. It can be found from Fig. 3(b) and Fig. 5(b) that at the azimuth angle θ=22.5$^\circ $, the maximum phase change is only about 1.4$^\circ $.

Taking out the relationship diagram between $\mathrm{\Gamma} = -175^\circ \sim -179^\circ$ and its azimuth angle during $22.5^\circ{\pm} 0.2^\circ \; $, the phase difference is shown in Fig. 6(a), and the slope diagram is shown in Fig. 6(b). The better linearity range is when the absolute phase retardation is less than $177^\circ $.

## 3. Experimental results and discussion

#### 3.1 Phase retardation measurement results of a half-wave plate

For ensuring the phase retardation of the HWP in this experiment. The experimental results without any test specimen are shown in Fig. 7(a). The solid line is the MATLAB simulation result of Γ=+176.78°. The dotted line is the result of the experiment. From Eqs. (10) ∼ (12), if $\theta = 0^\circ ,$ then $\phi = \mathrm{\Gamma }$, we can prove that the phase retardation Γ of the half-wave plate is +176.78$^\circ $. When the azimuth angle of the half wave plate is equal to 22.5$^\circ $, we assume ±0.2$^\circ $ for the azimuth angle tolerance to find the average slope of this line segment, as shown in Fig. 7(b). The slope of the phase difference is -98.2 (degree/degree) in the range of $\mathrm{\theta } = 22.3^\circ {\; }\sim {\; }22.7^\circ $.

This phase change caused by the azimuth rotation is much larger than that of the phase retardation variation. Even if the medium has a small phase retardation variation, the phase shift caused by the phase retardation is very small and can be ignored. The polarization rotation and azimuth rotation are relative motions. In the absence of azimuth rotation, the phenomenon of phase shift appears, which means that the phase shift is caused by the polarization rotation and the polarization rotation is the negative azimuth rotation ($\alpha ={-} \Delta \theta $). So we directly calculate the polarization rotation by using the equation $\alpha = \frac{{\Delta \phi }}{{98.2}}$.

#### 3.2 *When the azimuth angle of the half-wave plate is equal to 0*$^\circ $*, the effect of phase retardation of the glucose*

We use 50mm-long glucose solution as the test specimen in this experiment. The concentration is from 100 mg/dL to 1000 mg/dL, and the measurement is performed with an increase of 100 mg/dL each time. When the azimuth of the HWP is equal to 0$^\circ $, the measurement result is shown in Fig. 8. The abscissa of Fig. 8 is the concentration and its unit is milligrams per deciliter (mg/dL), which is the unit of measurement used in the United States for blood glucose concentration. The ordinate is the total phase retardation including the retardations of the HWP and glucose solutions and its unit is degree.

The up and down jump of the phase difference indicates the change of the phase retardation of the solution, and there is only a vibration in the phase retardation of ±0.2$^\circ $. The average total phase retardation is $176.78^\circ .{\; }$Because its distribution is a jump up and down, there is no correlation with the concentration of glucose. Therefore, the decreasing (or increasing) phase retardation is only related to the fast and slow axes of the birefringence of the glucose solution. In addition, because the fast or slow axis is random, the total phase retardation will slightly jump up and down regardless of the concentration. From Fig. 8, we can confirm once again that the phase retardation is equal to $176.78^\circ $. Because its phase slope near at $\theta = 22.5^\circ $ is $ - 98.2^\circ $ (degree/degree) in Fig. 7(b), the phase change caused by the slight variation of phase retardation is much smaller than the phase change caused by the polarization rotation. However, that does not affect the measurement results of polarization rotation.

#### 3.3 When the half-wave plate rotates to θ=22.5°, the specific rotation number of different concentrations of glucose and fructose

When the HWP is rotated to θ=22.5°, the temperature $T = 25\circ{C}$, the phase shift ($\Delta \phi = \phi - {\phi _0}$, where ${\phi _0} = 91.7^\circ $ is the initial phase) measurement results of high and low concentration glucose solutions are shown in Figs. 9(a) and 9(b), and the results of fructose solutions are shown in Fig. 9(c) and Fig. 9(d), respectively. When the concentration increases, the phase changes will increase, too. This phase variable is only caused by the optical activity of the solutions. The greater polarization rotation angle, the greater phase change will be measured.

Because of the polarization rotation and the azimuth rotation are relative motions ($\mathrm{\alpha }\textrm{=} - \Delta \mathrm{\theta }$), and from Figs. 9(a), 9(b) and Fig. 3(b), the phase shift of glucose solution is positive $(\Delta \phi > 0)$. The meaning is that $\Delta \theta = ({\theta - 22.5^\circ } )< 0$, the wave plate is equivalent to clockwise rotation, so the polarization rotation ($\mathrm{\alpha } > 0)$ is counterclockwise and right-handed. In other words, from Figs. 9(c), 9(d), and Fig. 5(a), the phase shift of fructose solution is negative ($\Delta \phi < 0$) and the polarization rotation is left-handed ($\alpha < 0$).

The phase shift of Figs. 9(a)∼9(d) have been subtracted from the initial phase difference by $91.7^\circ $ (without any solution) and then are divided by value of 98.2 (degree/degree) of the phase retardation $\mathrm{\Gamma } = 176.78^\circ $, the results of optical rotation (or polarization rotation) measurements for the 50 mm long high- and low-concentration glucose solutions are shown in Figs. 10(a) and 10(b), respectively. And the optical rotation measurement results of high- and low-concentration fructose solutions are shown in Figs. 10(c) and 10(d), respectively.

The solid line is the optical rotation $\alpha $ per millimeter and the dotted line is the specific rotation number $[\alpha ]$. Based on the specific rotation definition [19],

*l*is the solution length (the unit is $dm = 0.1m$),

*c*is the concentration (the unit is gram per milliliter). The specific rotation of the glucose and fructose solutions are near to 47${\pm} 1.5$ and -83.5${\pm} 2.5$ ($deg \times mL\ast {g^{ - 1}}d{m^{ - 1}}$), respectively for the wavelength of 632.8 nm. Thus, the experimental error is about ${\pm} $3%. Although we use the average of all values as the specific rotation, the determined error is all values minus the average. We know the specific rotation within this concentration range and also the associated error. In addition, the specific rotation may not be a fixed value as it may change with the change of concentration.

In this paper, we used the average slope of phase curve to evaluate the optical rotation for convenience. In fact, the phase difference diagram $\phi \; $is a curve and not a straight line. Thus the nonlinear error exists and affects the calculations of azimuth angle $\theta $ and optical rotation $\alpha \; $. $\; $The theoretical calculation of the phase error is shown in Fig. 11. The maximum phase error is ${\pm} 0.26^\circ $ within the measurement range of $ - 0.2^\circ \le \alpha \le 0.2^\circ $. It is larger than the resolution of phase measurement. Thus, the accuracy of the polarization rotation is only dependent on the nonlinear error. Therefore, $\Delta {\alpha _{max}}/\textrm{mm} = \frac{{0.26^\circ }}{{98.2 \times 50}} = 5.3 \times {10^{ - 5}}{\; }$degree/mm. For the resolution of phase measurement, the standard deviation is $0.08^\circ \; $from our past experience [17], so the standard deviation of $\alpha $ per millimeter is ${\pm} 1.6 \times {10^{ - 5}}$ degree/mm and the detection sensitivity on circular birefringence of glucose-water solution up to $\delta |{{n_r} - {n_l}} |= 5.6 \times {10^{ - 11}}$.

#### 3.4 Optical rotation of different water qualities

When the HWP is rotated to θ=22.5°, $T = 25\circ{C},\; $the container length is 50 mm, the phase shift and optical rotation measurement results of different water are shown in Figs. 12(a) and 12(b), respectively. From Fig. 12(a), the phase shifts after the phase differences subtracting the initial phase in air by 91.7$^\circ $ are negative and then dividing the results by the average slope of -98.2 degree/degree of Γ=17$6.78^\circ $ is $\Delta \theta $, and $\alpha ={-} \Delta \theta .$ The optical rotations per millimeter of different water qualities are obtained as shown in Fig. 12(b). These optical rotations are left-handed and the average rotation is $ - 2.0 \times {10^{ - 4}}{\; }degree/mm$. The distilled and slightly alkaline water have larger phase shifts and optical rotations, and the tap water has the smallest optical rotation. It might be revealed that larger optical rotation has a higher water quality.

## 4. Conclusions

The small optical rotation measurement based on the common-path heterodyne interferometry and the use of a half-wave plate was proposed in this paper. The measurable range of the polarization rotation is from 0$^\circ $ to ±0.$2^\circ $. Because the phase error can reach ${\pm} 0.26^\circ $, the accuracy of the optical rotation can be up to $5.3 \times {10^{ - 5}}\; \textrm{degree}/\textrm{mm}$. For the phase standard deviation is $0.08^\circ {\; },$ the deviation of $\alpha $ is ${\pm} 1.6 \times {10^{ - 5}}$ degree/mm and the optical rotations of different water qualities are negative (left-handed) and have been successfully measured. From the experimental results, the error of the specific rotation measurement is about ${\pm} 3\%.$ The feasibility of this method is demonstrated. In this paper, we emphasize that the proposed method can measure very tiny optical rotation even for water. The optical rotation may be another important parameter for water.

## Funding

National Formosa University (EN2020-09170456074223).

## Acknowledgment

M.H. Chiu thanks the MOST in Taiwan for supporting this work.

## Disclosures

The authors declare no conflicts of interest.

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