## 2 Experimental Section  
### Experimental Setup  
For the optical measurement, a He-Ne laser with a wavelength of 632.8 nm was employed as the light source. When the light was passing through the first half wave-plate (HWP1) (NTHU, Hsinchu, Taiwan), the polarization state was adjusted parallel to the incident plane, then divided by a beam splitter (BS1) and fed into a pair of acousto-opto-modulators (AOM1 and AOM2) (NTHU, Hsinchu, Taiwan). A beam combiner (PBS) (NTHU, Hsinchu, Taiwan) was used to combine the output beams from each AOM, and then the heterodyne light source could be obtained. The heterodyne light source consisted of two component beams with frequencies \(\omega_{1}\) and \(\omega_{2}\), and the polarization of each beam was orthogonal to the other. The one perpendicular to the incidence plane is called the TE mode, while the other is in TM mode. The prototype of an angular displacement-enhanced heterodyne polarimeter has been built and the theoretical analysis has been discussed in [13], in which two angles represent the optical rotation and the phase difference. The relationship between the phase difference \(\varphi\) and the intensity of the beam of the laser onto the HWP1 can be expressed as following [13]:  
\[I_{m}=k_{a1}^{2}+k_{a2}^{2}+2k_{a1}k_{a2}\cos[(\omega_{1}-\omega_{2})t+\varphi], \tag{1}\]where  
\[k_{a1}= \left(1-\cos^{2}\frac{\delta}{2}\sin 2\beta+\sin^{2}\frac{\delta}{2} \sin(4\theta+2\beta)\right)^{\frac{1}{2}} \tag{2}\] \[k_{a2}= \left(1+\cos^{2}\frac{\delta}{2}\sin 2\beta-\sin^{2}\frac{\delta}{2} \sin(4\theta+2\beta)\right)^{\frac{1}{2}} \tag{3}\]  
and phase difference \(\varphi\) can be further obtained from  
\[\varphi=\tan^{-1}\left[\frac{\sin\delta\cos 2\theta}{\cos^{2}\frac{\delta}{2} \cos 2\beta-\sin^{2}\frac{\delta}{2}\cos(4\theta+2\beta)}\right] \tag{4}\]  
The double-distilled water (ddH\({}_{2}\)O) was injected into the sample tube and used as control. Due to the nature of optical inactivity, the change of the angular displacement is zero (_i.e._, \(\beta=0\)). The phase difference \(\varphi\) can be simplified as the following [13]:  
\[\varphi_{\beta=0}=\tan^{-1}\left[\frac{\sin\delta\cos 2\theta}{\cos^{2}\frac{ \delta}{2}-\sin^{2}\frac{\delta}{2}\cos(4\theta)}\right] \tag{5}\]  
For all optical measurement, the phase retardation \(\delta\) of \(178.5^{\circ}\) and azimuth \(\theta=23.2^{\circ}\) was set to maintain the larger amplification with great signal-to-noise (SNR). Equation (4) was employed to calculate \(\varphi\), between the reference signal and measurement signal, and with adjustment, \(\beta\) varied from \(0^{\circ}\) to \(0.01^{\circ}\) by increasing \(0.001^{\circ}\) for each measurement. The amplification factor defined by the ratio of \(\varphi\) over \(\beta\) is about 42.1. The thermoelectric cooler (TEC) (NTHU, Hsinchu, Taiwan) was used to heat the sample tube to a temperature from 25 \({}^{\circ}\)C to 55 \({}^{\circ}\)C. The temperature deviation during the measurements was less than \(\pm 0.3\)\({}^{\circ}\)C. A Soleil-Babinet compensator (SBC) (NTHU, Hsinchu, Taiwan) was employed to enhance the optical displacement. This SBC was controlled to rotate \(0.05^{\circ}\) azimuth degrees per second (_i.e._, rotate \(180^{\circ}\) within 60 min), and the sampling frequency is 0.5 Hz. Finally, the phase difference \(\varphi\) as a function of phase retardation \(\delta\), azimuth \(\theta\) of the \(\delta\)-WP, and angular displacement \(\beta\) can be simply read by the lock-in amplifier (LIA). Figure 1 shows a schematic drawing of the applied setup.  
Figure 1: _Cont._