1. Introduction

Chirality is a property of asymmetry that means an object cannot be superposed on its mirror image. It is significant for people working in the fields of life sciences and pharmaceuticals. Because many biomolecules and drugs are chiral they induce optical rotations (OR) [1–2], measuring the OR value can help determine the purity and the type of enantiomer (levorotary or dextrorotatory) of these chiral substances. In the past, there were many methods of OR measurement, most of which lacked precision and real-time measurement. Because they need a very precise rotator and rotation time to confirm the OR value, it has been difficult to determine the purity of chiral materials immediately. In addition, when two enantiomer substances coming from different directions are mixed together, the optical rotation measurement before and after mixing is crucial for the determination of the concentration and specific rotation of one of the solutions. Thus, it is foremost to have convenient, immediate, efficient and accurate measurement. The specific rotation measurement is beneficial for determining the composition and purity of a drug. Therefore, OR measurement has many applications, for example, in hospital clinical diagnosis, chemical research, sugar production, and the pharmaceutical industry.

If the measurement method of OR is distinguished according to the optical path structure, it can be divided into three types of methods [3]: 1. polarizer-analyzer, 2. polarizer-Faraday modulator-analyzer, and 3. laser-phase modulator-analyzer. The first type of method is the classical method which measures the intensity from the extinction point to analyze the OR value based on the sinusoidal curve-fitting of light intensity [4–8]. It can achieve an accuracy of about 0.001$^\circ $. In the second type of method, adjust the current of the Faraday modulator to rotate the polarization direction to the extinction direction; the negative value of rotation angle is the OR value of the sample. It can achieve a high accuracy ($1.3 \times {10^{ - 4}}^\circ $), but the rotation angle of Faraday modulator is limited within a small range [9–10]. The third type of method, for rotating the polarization direction or measuring the phase difference to calculate the OR may use the EOM (electro-optic modulator), MOM (magneto-optic modulator), AOM (acousto-optic modulator), or a rotating electromechanical component as a phase shifter or a frequency modulator in the heterodyne interferometry. The resolution of OR measurement is increased from $1 \times {10^{ - 4}}^\circ \; \; $ to $3.63 \times {10^{ - 5}}^\circ $. [11–17]. For example, J.Y. Lee et al. [13] proposed “Improved common-path optical heterodyne interferometer for measuring small optical rotation angle of chiral medium”. They used the EOM based common-path heterodyne interferometer, a half-wave plate and a phase retarder to measure the OR with an optimal resolution of $3.5 \times {10^{ - 5}}^\circ $.

In 2019, Ma et al. [3] proposed “pixelated-polarization-camera-based polarimetry” for wide real-time OR measurement. The algorithm is based on the Stokes parameters and used the Mueller matrix to calculate the four intensity values on each pixel which is one of four directions of polarizations to achieve the OR value. The accuracy is about $1 \times {10^{ - 4}}^\circ $.

In 2020, Harvie et al. proposed a “high-resolution polarimeter formed from inexpensive optical parts” [18]. This method is based on the sliding discrete Fourier transform, analyzing the frequency and waveform of the object light and the reference light and the phase difference between the two lights. The object light first passes through a polarizer, then passes through the sample, and finally passes through the reference light through the same point of the rotating analyzer, where the frequency of the rotation is f and the resulting frequency of intensity signal is $2f$. The reference light also passes through a polarizer, bypasses the sample, and then passes through the same rotating analyzer. Its intensity signal frequency is the same as the object light. The two light signals are presented in a sinusoidal waveform, the resolution of OR is ${\pm} 1.3 \times {10^{ - 3}}^\circ $, and the accuracy is ${\pm} 2.8 \times {10^{ - 3}}^\circ .$

Although the resolution of the OR measurement can reach $1 \times {10^{ - 3}}^\circ $ in the above methods, there are some methods that are inconvenient and inefficient to use. Therefore, we proposed a common-path heterodyne polarimeter with sufficient resolution, very convenient operation and an easy working theory, which can obtain the OR value and specific optical rotation in real-time. In our method, due to the OR effect of the sample, the two orthogonal polarizations of the test beam are rotated the same angle, and when passing through a half-wave plate at the azimuth angle of $22.5^\circ $, will have the largest phase difference between the two polarizations [14]. If the half-wave plate is close to the real half-wave plate, the angular sensitivity of OR will be higher. Although the phase change can be seen in the derivation of the formula, there are still some errors that are not discussed. In this paper, the derivation formula will be explained in detail and all possible sources of error will be discussed.

In order to prove the feasibility and convenience of this method, we must investigate the actual systematic error and resolution. These may include phase resolution, the slope of the simulation curve, and the accuracy of OR.

2. Principle

In 1992, Yoshino et al. proposed the common-path heterodyne interferometry (CPHI) used in the fiber sensor system [19]. In 1996, Chiu et al. proposed a method based on the CPHI for identifying the fast axis and phase retardation of a wave plate [20]. From Fig. 1, the phase retardation from $175^\circ $ to $179^\circ ,\textrm{all}$ the phase simulation curves have the largest phase slopes at the azimuths of $22.5^\circ ,\textrm{ }67.5^\circ ,\textrm{}112.5^\circ ,\textrm{}157.5^\circ ,\textrm{}202.5^\circ ,\textrm{ }247.5^\circ ,\textrm{}292.5^\circ ,\textrm{and}337.5^\circ $ of the half-wave plates. In other words, these azimuth angles have the best angular sensitivity.

 

Fig. 1. Simulation curves of phase difference between the s- and p-polarizations versus the fast-axis orientation of the half-wave plates.

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Since the polarization rotation and wave plate rotation are in relative motion to each other, using the characteristic of high angular sensitivity of a half-wave plate at a specific azimuth of $22.5^\circ $ in phase measurement can obtain polarization rotation or OR precisely.

The basic working principle is as follows:

From Fig. 2, a heterodyne light source with two orthogonal polarizations (this will be described in detail in section 3, Fig. 4) is divided by a beam splitter (BS) into two beams, ${E_1}$ and ${E_2}.\textrm{}$ Assume that the optical frequencies of the horizontal and vertical polarizations are $\textrm{}{f_1}$ and $\textrm{}{f_2}$, respectively.

 

Fig. 2. The basic setup of measurement. BS, beam splitter; ${W_{\lambda /2}}(\theta ),\; $ half-wave plate; $\textrm{AN}$, analyzer; $\textrm{D},\; $ photodetector; Sample($\mathrm{\alpha }$), sample with optical rotation $\mathrm{\alpha }$.

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Therefore, the beat frequency is $\textrm{}f = {f_2} - {f_1}$. The reference light (${E_{1\textrm{}}})\textrm{}$ passes through an analyzer ($A{N_r}$) which transmission axis is at $45^\circ $, and then that causes the components of two polarized lights to interfere with each other. The optical interference signal received and converted into an electrical signal by a photodetector ${D_r}$, is the reference signal ${I_r}$.

The object light (${E_2}\; )$ passes through a sample ($\textrm{S}$) and a half-wave plate (${W_{\lambda /2}}(\theta ),\; $) with an azimuth of $\mathrm{\theta \;\ }$ degrees, and then passes through another analyzer ($A{N_t}\; $) which transmission axis is also at $45^\circ $. The test sample is an optically active solution. Assume that its OR is $\mathrm{\alpha \;\ }$ degrees. The Jones matrix is written as

Thus, the AC signal of interfering intensity of the object light received by a photodetector (${D_t}$) can be written as

For example, assuming the phase retardation $\mathrm{\Gamma }$ is in the range of $175^\circ{\sim} 179^\circ $ and the azimuth $\theta = 22.5^\circ $, $\phi \textrm{}$ can be simulated versus $\alpha \textrm{}$ for $\alpha \; $ in the range of ${\pm} 1.5^\circ $ by use of Eq. (11). Since our approach is measuring the value of phase $\phi $ and then finding the OR$\textrm{}(\alpha )$, we let the coordinates of this simulation curve be reversed, as shown in Fig. 3, and the value $\textrm{}\alpha $ is calculated by substituting the measured phase $\phi $ into the fitted equation of the curve, where the OR$\textrm{}(\alpha )$ can be written as

Some simulation curves of $\alpha \textrm{}$ versus $\phi \textrm{ }$ for $\mathrm{\Gamma }$ values in a range of $175^\circ{\sim} 179^\circ \textrm{ }$ at $\mathrm{\theta } = 22.5^\circ \textrm{ }$ are shown in Fig. 3.

 

Fig. 3. The simulated curves of $\mathrm{\alpha \;\ }$ versus $\phi \textrm{ }$ for different values of $\mathrm{\Gamma } = 175^\circ{\sim} 179^\circ $ and $\mathrm{\theta } = 22.5^\circ $

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From Fig. 3, a larger value of $\mathrm{\Gamma }$ will have a larger value of ${\raise0.7ex\hbox{${d\phi }$} \!\mathord{/ {\vphantom {{d\phi } {d\alpha }}} }\!\lower0.7ex\hbox{${d\alpha }$}}$ thus a better angular resolution of $\alpha $. When α approaches zero, $\phi $ approaches $90^\circ $. If $\phi > 90^\circ $, then $\alpha < 0$, and the sample is left-handed matter (levorotary); if $\textrm{}\phi < 90^\circ $, then $\alpha > 0$, and the sample is right-handed matter (dextrorotatory). Through the judgment of the positive and negative values of $\mathrm{\alpha }$, the optical rotation or concentration of the analyte can be determined. It can also be expressed in specific optical rotation $[\alpha ]$ and defined as

3. Measurement results and discussions

3.1 Experimental setup

The sketch of the OR experiment is shown in Fig. 4, this polarimeter is based on CPHI combined with a half-wave plate, and we use a quartz box with an inner length of 5 cm as a container for the analyte. We split the experimental architecture into two parts. The first part is to generate a heterodyne light source and the second part is the measurement setup.

 

Fig. 4. Experimental setup. Laser: Helium-neon laser (λ=632.8 nm), or solid-state laser (λ=532 nm); ${\textrm{W}_{{\raise0.7ex\hbox{$\lambda $} \!\mathord{/ {\vphantom {\lambda 2}} }\!\lower0.7ex\hbox{$2$}}}}$: half-wave plate; M1, M2, M3, M4, M5: Mirror; PBS1, PBS2: Polarization Beam Splitter; P1, P2: Polarizer; AOM1, AOM2: Acousto-Optic Modulator; BS: Beam Splitter; $A{N_r},\textrm{}A{N_t}$: Analyzer; ${D_r},\textrm{}{D_t}:\textrm{}$ Photodetector; Lock-in Amplifier (Stanford Research Systems, SR830).

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Firstly, we use a half-wave plate (${\textrm{W}_{\mathrm{\lambda }/2}}({45^\circ } )$) to adjust the intensities of the p-polarized (transmitted from PBS1) and s-polarized (reflected from PBS1) lights to be equal. Because the polarization beam splitter (PBS1) has a limited extinction ratio, each polarized light will still be slightly mixed with other polarized light, so the polarizer (P1, P2) is used to make the polarized light purer, and then the p- and s-polarized light beams are incident into the acousto-optic modulators AOM1 (the modulating frequency $\textrm{ }{f_1}$ is 80.00 MHz) and AOM2 (the modulating frequency $\textrm{}{f_2}$ is 80.01 MHz), respectively.

Then the lights are reflected into the second polarization beam splitter (PBS2) by mirrors (M3, M4). Because the two output lights of AOMs have -1, 0, + 1 orders, we want to totally overlap these two +1 orders together, so we must use the aperture (Iris) to filter out other orders. Then, completely overlapping these two +1 order lights becomes a heterodyne light source with a beat frequency of 10 kHz ($f = {f_2} - {f_1}$).

The second part is the optical measurement system, using a beam splitter (BS) to divide the heterodyne light into reference light and object light. The reference light passes through an analyzer ($A{N_r}$) with an azimuth of $45^\circ $, and then is received by the photodetector (${D_r}$). The interference signal is the reference signal (${I_r}$); the object light passes through the 5-centimeter-long sample and the half-wave plate (${W_{\lambda /2\; }}({22.5^\circ } )$) with azimuth angle of $22.5^\circ $, and then passes through another analyzer ($A{N_t}$) at $45^\circ $, and finally enters the photodetector (${D_t}$) to obtain an interfering signal which is the test signal (${I_t}$). The phase difference between the test signal (${I_t}$) and reference signal (${I_r}$) can be measured by using the lock-in amplifier (SR830). Use the software MATLAB to analyze, substitute this phase value into Eq. (12) to find the OR value of the sample, and the specific rotation is obtained by Eq. (13).

3.2 To determine the real value of the half-wave plate

The ideal phase retardation of a half-wave plate is $\mathrm{\Gamma } = 180^\circ $. In order to measure the OR more accurately, the phase retardation of the half-wave plate must be determined before the experiment begins. First, in Fig. 4, the container is removed, the half-wave plate is rotated from $0^\circ $ to $360^\circ $, and the phase value is measured in real-time. The experimental results are shown in Figs. 5(a) and 5(b). The solid blue line is the simulation result of MATLAB, and the red dashed line is the measurement result of the experiment. From Figs. 5(a) and 5(b), the experimental results are consistent with the simulation results, and the phase retardations of these two half-wave plates are $177.2^\circ $ (for λ=632.8 nm) and $178.08^\circ $ (for λ = 532 nm), respectively.

 

Fig. 5. Phase retardation measurement results of the half-wave plates (a) for $\mathrm{\lambda } = 632.8\textrm{nm};(\textrm{b} )\textrm{}$ for $\mathrm{\lambda } = 532\textrm{nm};$

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The phase retardations of the two half-wave plates used in the OR experiment have been measured to be $177.2^\circ $ and 178.08°, respectively. We set the azimuth of the half-wave plates at $22.5^\circ $ ($\mathrm{\theta } = 22.5^\circ $), and assume -1.5°≤$\mathrm{\alpha }$≤1.5°. Substituting these values into Eq. (11) to simulate the curves of $\textrm{}\phi \textrm{}$ versus $\alpha $.

We swap the $\phi $ coordinate with the $\alpha $ coordinate, and then the polynomial equation $\mathrm{\alpha }(\phi )$ can be found by curve fitting. From Figs. 6(a) and 6(b), the abscissa is the phase value measured by the experiment, and the ordinate is the OR that we want to obtain. Therefore, the value of OR can be obtained by substituting the phase value into the polynomial equation.

 

Fig. 6. The simulation curve results of optical rotation $\mathrm{\alpha }$ versus the phase $\phi $: (a)for $\mathrm{\;\ \Gamma } = 177.2^\circ ;\textrm{}$(b)for $\mathrm{\Gamma } = 178.08^\circ $.

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The curve fitting equations for $\mathrm{\Gamma } = 177.2^\circ $ and $\mathrm{\Gamma } = 178.08^\circ $ are shown below

3.3 Optical rotation measurement results

Once we have determined the phase retardation of the half-wave plate and set its azimuth (θ) at 22.5°, we can place the sample as shown in Fig. 4. First, in the case of an empty quartz box, measure out the phase value and substitute it into Eq. (14) or Eq. (15) to calculate the OR value and call it as ${\alpha _1}$. And then pour the test solution into the quartz box, measure out the phase value and substitute it into the same formula to obtain the second OR value ${\alpha _2}$. Subtract ${\alpha _1}$ from ${\alpha _2}$, and the OR of the solution is obtained. Then we use red light with a wavelength of 632.8 nm, as a light source.

The samples are glucose, fructose, and vitamin C, and their concentrations are in the range from 0.5 [g/dl] to 2 [g/dl], which are poured into the quartz box one by one. The OR measurement values are shown in Figs. 7 (a), (c), and (e), respectively. The calculated specific rotation $[\alpha ]$ are shown in Figs. 7 (b), (d), and (f), respectively. From Figs. 7(a), (c), and (e), the magnitude of OR increases linearly as the concentration increases. However, from Figs. 7(b), (d), and (f), we know that there is still a little up and down fluctuation in the specific rotation, which is reasonable, because there will be some errors in the experiment. This error may come from the concentration blending error or from the change in temperature.

 

Fig. 7. (a), (c), and (e) are the optical rotation results; (b), (d), and (f) are the specific rotation results of glucose, fructose, and vitamin C in red light ($\mathrm{\lambda } = 632.8\textrm{nm}$) measurements for different concentrations, respectively.

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In the same solutions and concentration range, we switched to a green laser with a wavelength of 532 nm, and repeated the above experimental process. The experimental results of OR values of glucose, fructose, and vitamin C are shown in Figs. 8(a), (c), and (e), respectively. The calculation results of specific rotation $[\alpha ]\textrm{}$ are shown in Figs. 8(b), (d), and (f), respectively.

 

Fig. 8. (a), (c), and (e) are the optical rotation results; (b), (d), and (f) are the specific rotation results of glucose, fructose, and vitamin C in green light ($\mathrm{\lambda } = \textrm{}532\textrm{nm}$) measurements for different concentrations, respectively.

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Each point in Fig. 7 and Fig. 8 is the average of 100 consecutive data. The trend in Fig. 8 is similar to Fig. 7, except that the OR values at different wavelengths are slightly shifted.

4. System analysis and discussions

From Figs. 6(a) and (b), the curves of optical rotation $\alpha $ versus the phase $\phi $. The resolution of OR can be defined as

 

Fig. 9. The differential curves of $\mathrm{\alpha \;\ }$ versus $\phi $: (a) for $\mathrm{\Gamma } = 177.2^\circ $; (b) for $\mathrm{\Gamma } = 178.08^\circ $.

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When $\mathrm{\;\ \Gamma } = 177.2^\circ $, the minimum value of $|{{\raise0.7ex\hbox{${d\alpha }$} \!\mathord{/ {\vphantom {{d\alpha } {d\phi }}} }\!\lower0.7ex\hbox{${d\phi }$}}} |$ is $1.73 \times {10^{ - 2}}$ [degree/degree] as shown in Fig. 9(a), and when $\mathrm{\Gamma } = 178.08^\circ $, the minimum value of $|{{\raise0.7ex\hbox{${d\alpha }$} \!\mathord{/ {\vphantom {{d\alpha } {d\phi }}} }\!\lower0.7ex\hbox{${d\phi }$}}} |$ is $1.18 \times {10^{ - 2}}$ [degree/degree] as shown in Fig. 9(b). Because the standard deviation of phase measurement is $0.06^\circ ,$ and then for Γ=177.2° and Γ=178.08°, the best accuracies of OR are $1.0 \times {10^{ - 3}}^\circ $ and $7 \times {10^{ - 4}}^\circ $, respectively, for the range of $- 1.5^\circ \le \alpha \le 1.5^\circ $, and the maximum error of $\Delta {\alpha _{max}} = 4.1 \times {10^{ - 3}}^\circ $ for Γ=177.2°; $\Delta {\alpha _{max}} = 3.3 \times {10^{ - 3}}^\circ $ for Γ=178.08°. Therefore, the larger $\mathrm{\Gamma }$ has better OR resolution. The average of specific rotations of test mediums and their standard deviations are shown in Table 1.

Table 1. The standard deviation of specific rotation^a,b,c

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The percentage of standard deviation is in the range of 1.32%∼7.6%. As the wavelength increases, the magnitude of specific rotation value also increases. From the literature [21–22], the specific rotation measurements of glucose, fructose and vitamin C at a wavelength of 589.3 nm were 52.7, -92.25, and 21 [dl${\times} $degree/gdm], respectively. If calculated by the interpolation of wavelength, the relative error percentages of glucose, fructose and vitamin C relative to the reference value are 0.037%, 0.279% and 0.727%, respectively. The results we obtained are quite consistent with these literature values, so the feasibility of this method can be proved. The optimal concentration resolution can be written as

We can see that the slope of Figs. 8(a), (c), (e) are 0.265, -0.45, 0.1 [degree/g/dl], respectively. From Fig. 9(b), and Eq. (17), the best resolutions of the concentration measurements of glucose, fructose, and vitamin C are $2.7 \times {10^{ - 3}},\; \; 1.6 \times {10^{ - 3}}$, $7.1 \times {10^{ - 3}}\; $[g/dl], respectively, in $\mathrm{\Gamma } = 178.08^\circ $ for phase standard deviation $\Delta {\phi _{min}} = 0.06^\circ $.

The error that can occur in the azimuth of the half-wave plate under human operation, for example, is $\lambda = 632.8\textrm{nm},\mathrm{\Gamma } = 177.2^\circ $. The phase error caused by human operation error in actual measurement is shown in Fig. 10.

 

Fig. 10. The phase error due to human operation of $\mathrm{\theta }$.

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Assume that the operational error of azimuth angle deviated from $22.5^\circ $ is ${\pm} 0.025^\circ ,$ and the phase error is about ${\pm} 0.05^\circ .$ Since the phase error of CPHI is $0.06^\circ $. Thus, the root mean square value of total phase error is $\sqrt {\frac{{{{0.05}^2} + {{0.06}^2}}}{2}} = 0.055^\circ{\approx} 0.06^\circ $. Therefore, we can think of the resolution or accuracy of the phase as $0.06^\circ $.

Now considering the increase in phase retardation caused by the solution to be measured, what is the amount of change in OR measurement caused by it? We actually measure the concentration range of fructose solution 0.5∼2.5 [g/dl], the azimuth of the half-wave plate is set at $0^\circ $, and the overall phase retardation can be measured, that is, including the phase retardation of the solution, quartz box and the half-wave plate. The experimental results are shown in Fig. 11(a). The average phase change is about 0.1°, indicating that the average increase in phase retardation is 0.1°. If we assume that Γ=178.08° as an example, ΔΓ=0.1°, the calculated OR error is $8 \times {10^{ - 4}}^\circ $, as shown in Fig. 11(b). Therefore, the overall OR resolution is equal to $8 \times {10^{ - 4}}^\circ $. The magnitude of the phase error of CPHI depends on the stability of the optical architecture and whether the two lights overlap exactly. If the phase error can be reduced to $0.01^\circ $, the resolution of the OR can be increased to $1.3 \times {10^{ - 4}}^\circ $. Such results are almost comparable to the result of J.Y. Lee et al. [13]. However, our architecture lacks one retarder component.

 

Fig. 11. (a)Total phase retardation measurement at $\mathrm{\theta } = 0^\circ $; (b) The measurement error of optical rotation due to the increase of phase retardation.

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5. Conclusions

We take three common OR materials glucose, fructose, and vitamin C in measurement. The percentage standard deviation of the specific rotation is 1.32%∼7.6% in the measurable range of $- 1.5^\circ \le \alpha \le 1.5^\circ $. Because a higher phase retardation has a higher resolution of OR, in this case, Γ=178.08° has higher measurement resolution. The optimum accuracy of the OR value is $8 \times {10^{ - 4}}^\circ{\approx} 1.0 \times {10^{ - 3}}^\circ $, and the best concentration resolution is $\textrm{}1.6 \times {10^{ - 3}}\; $[g/dl] for fructose solution. If the phase error can be improved to $0.01^\circ $, the optimal resolution of OR can reach $1.3 \times {10^{ - 4}}^\circ $ and the best resolution of fructose concentration is $\textrm{}2.7 \times {10^{ - 4\; }}[{g/dl} ]$.

The principle of this method is simple and easy to operate. Because it is a common-path heterodyne structure, it can avoid environmental disturbance and measure very small OR in real-time. The focus of this paper is to revise past derivation formulas [14], explore the sources of various errors, find the best OR resolution, study the influence of light source wavelengths, and confirm the feasibility of this method.