Optimal weak measurement scheme for chiral molecular detection based on photonic spin Hall effect

Tingting Tang; Ke Shen; Jie Li; Xiao Liang; Yujie Tang; Chaoyang Li; Yu He; Yu He

doi:10.1364/OE.500812

1. Introduction

Chiral molecules, also known as chiral compounds, are pairs of molecules with the same molecular weight and structure but in opposite left-right order, which are called "enantiomers" [1,2]. Chiral molecules are widely found in amino acids, sugars, DNA, and medicines [3–7], identification and quantitative analysis of chiral conformations are of great importance in fields such as bioassays and pharmacology [8–14]. Current methods for the direct detection of chiral molecules include polarimeter OR measurements, circular dichroism (CD), gas chromatography (GC), capillary electrophoresis (CE), and nuclear magnetic resonance (NMR) [15–17]. Among them, optical rotatory (OR), which refers to the rotation of the polarization plane that occurs when polarized light passes through a substance, is an effective tool for determining chiral conformations [18]. However, chiral optical signals are usually very weak, although OR is fast-responding and non-contact. A polarimeter can measure OR by mechanically rotating the polarizer angle, but the measurement precision is still limited by the extinction ratio of the device itself and the quantum noise from the coherent light source [19]. Recently, a photonic nanostructure that preserves helicity to enhance optical rotation was proposed [20]. It is suitable for most molecular solutions but requires very complex structures and cannot separate OR and CD signals.

In recent years, quantum weak measurement techniques have been proposed to significantly amplify the weak signal, providing a new approach in the field of precision detection [21–26]. In 2008, the photonic spin Hall effect at the air-glass interface was observed for the first time using quantum weak measurements with a precision of 10$^{-10}$ $m$ [27]. In 2009, weak deflection measurements of light beams were made using weak measurements with an precision of 10$^{-13}$ rad [28]. In 2016, an innovative precision method based on a nonlinear weak measurements model was proposed by Qiu et al., estimating the OR angle of glucose and fructose with a precision of 0.2 $mdeg$ [29]. In 2018, a high sensitivity scheme for OR estimation based on weak measurements in the frequency domain was demonstrated by Li et al., the resolution for Proline detection is about 10$^{-9}$ $mol/ ml$, which is 2 orders of magnitude higher than that of common methods [30]. Meanwhile, the weak measurement system with PSHE as a probe discriminates and simultaneously measures different signals. In 2017, PSHE, as a probe was used to simultaneously measure the OR angle and refractive index change (RIC) of a mixed solution of glucose and fructose [31]. However, the above results reflective interfaces are mostly prisms or single-layer material, and reflective surface material with turnable parameter is not utilized to further increase the PSHE. Furthermore, weak measurement schemes optimized for parameters such as beam waist, transmission distance, and angle of incidence have not been discussed, which could also improve the detection sensitivity.

In this paper, we investigate the weak measurement system of chiral molecules using PSHE as a probe. The detection sensitivity is increased by enhancing the refractive index gradient of the reflective surface film medium. Unlike the glass or monolayer metal interface reflections in the above literature, Ce:YIG-YIG-SiO$_{2}$ with a large refractive index gradient was chosen as the reflective surface material to increase the sensitivity of the weak measurement system. In addition, Ce:YIG-YIG-SiO$_{2}$ as a magnetic multilayer material can also adjust the PSHE by applying a magnetic field for the purpose of optimization for different substances to be measured. The parameters such as the incident angle, transmission distance $z$ and waist $\omega _0$ are further calculated and adjusted to optimize the weak measurement system. Our results show that the PSHE is highly sensitive to the OR signal under a parameter-optimized multilayer film. The magnitude of the PSHE shift is proportional to the OR signal (namely, the concentration of chiral solutions). Besides, the direction of beam shift is determined by the direction of chirality. The amplified PSHE is positive for the right-handed rotation, while it is negative for the left-handed rotation. The experimental results are in good agreement with the theoretical model. Optimizing weak measurement systems offers a high precision tool for chiral drug detection.

The experimental setup is given in Fig. 1. The incidence plane is the $xoz$ plane. The wavelength of incident light used in the experiment is 632.8 $nm$. The beam passes through L1 and P1 to adjust the polarization state and beam waist. Consider the optical axis of P1 is horizontal, and it passes through a chiral molecular solution, the polarization plane of the light undergoes a rotation with an angle of $\alpha$, the wave function can be given by

(1)$$\left|\Psi_{i}\right\rangle=\int d k_{x} d k_{y} \varphi\left(k_{x}, k_{y}\right)\left|k_{x}\right\rangle\left|k_{y}\right\rangle\left|\psi_{i}\right\rangle .$$

Where $\left |\psi _{i}\right \rangle =\cos \alpha |H\rangle +\sin \alpha |V\rangle$ is the pre-selection. $\varphi \left (k_{i x}, k_{i y}\right )=\frac {1}{\sqrt {2 \pi }} \omega _{0} \exp \left [-\frac {(R+i z)\left (\boldsymbol {k}_{ix}^{2}+\boldsymbol {k}_{iy}^{2}\right )}{2 \boldsymbol {k}}\right ]$ denotes the transverse distribution of the wave packet. $k_{ix}$, $k_{iy}$ are the $x$- and $y$- components of the wave vector. $z$ represents the transmission distance of the beam in free space. $R=k \omega _{0}^{2} / 2$ is the Rayleigh length. Then the light beam is reflected on a prism coated with a Ce: YIG-YIG-SiO$_2$ film. Where YIG is a typical garnet-type ferrite, Among them, YIG is a typical garnet-type ferrite, and Ce:YIG as its adulterant material is widely utilized in the microwave field as YIG. The multilayer film structure is obtained by sequential deposition of YIG and Ce:YIG on a silica substrate. The transverse wavevector components propagate in slightly different directions deviating from the central wave-vector, the reflection matrix is

(2)$$\mathrm{M}_{\mathrm{R}}=\left[\begin{array}{cc} r_{p p}-\frac{k_{r y}}{k_{0}}\left(r_{p s}-r_{s p}\right) \cot \theta_{i} & r_{p s}+\frac{k_{r y}}{k_{0}}\left(r_{p p}-r_{s s}\right) \cot \theta_{i} \\ r_{s p}+\frac{k_{r y}}{k_{0}}\left(r_{p p}-r_{s s}\right) \cot \theta_{i} & r_{s s}-\frac{k_{r y}}{k_{0}}\left(r_{p s}-r_{s p}\right) \cot \theta_{i} \end{array}\right]$$

Where $r_{ss}$, $r_{sp}$, $r_{ps}$, $r_{pp}$ denotes several reflection coefficients of Ce:YIG-YIG-SiO$_2$ film, which can be calculated by the transfer matrix method. To find the reflection coefficient of a multilayer film, let the plane wave propagating in layer $n$ be $\mathbf {E}^{(n)}=\mathbf {E}_{0}^{(n)} \exp [i(\omega t-\mathbf {k} \cdot \mathbf {r})]$, and the wave equation be

(3)$$\mathbf{k}^{(n) 2} \mathbf{E}_{0}^{(n)}-\mathbf{k}^{(n)}\left(\mathbf{k}^{(n)} \cdot \mathbf{E}_{0}^{(n)}\right)=\frac{\omega^{2}}{c^{2}} \overleftrightarrow{\varepsilon}^{(n)} \mathbf{E}_{0}^{(n)} .$$

Fig. 1. Schematic diagram of the experimental setup for weak measurements where from left to right are laser ($\lambda$=632.8$nm$); diaphragm; half-wave plate (HWP); lens 1 (L1, $f$=50$mm$); Glan polarizer 1 (P1, horizontal optical axis); experimental sample and prism with thin film(Ce:YIG-YIG-SiO$_{2}$ ); Glan polarizer 2 (P2, vertical optical axis ); lens 2 (L2, $f$=250$mm$); charge-coupled device(CCD).

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$\mathbf {E}_{0}^{(n)}$ is the amplitude of the electric field, and $\mathbf {k}^{(n)}$ is the wave vector. According to the boundary condition, the electric field amplitude matrix of each layer in the structure can be expressed as

(4)$$\mathbf{Q}=\mathbf{D}^{(0)^{{-}1}} \prod_{n=1}^{3} \mathbf{D}^{(n)} \mathbf{P}^{(n)} \mathbf{D}^{(n)^{{-}1}} \mathbf{D}^{(4)} .$$

The matrices $\mathbf {D}^{(n)}$($n$=1,2,3) and $\mathbf {P}^{(n)}$($n$=1,2,3) represent the Dynamic and Propagation matrices, respectively, for each layer. By solving banding Eq. (4), we get reflection coefficient.

(5)$$r_{p p}=\left(\frac{E_{0 p}^{(r)}}{E_{0 p}^{(i)}}\right)_{E_{E_{s}^{(i)}}=0}=\frac{Q_{11} Q_{43}-Q_{41} Q_{13}}{Q_{11} Q_{33}-Q_{13} Q_{31}}, \quad r_{p s}=\left(\frac{E_{0 p}^{(r)}}{E_{0 s}^{(i)}}\right)_{E_{0 p}^{(i)}=0}=\frac{Q_{41} Q_{33}-Q_{43} Q_{31}}{Q_{11} Q_{33}-Q_{13} Q_{31}} \\$$

(6)$$r_{s p}=\left(\frac{E_{0 s}^{(r)}}{E_{0 p}^{(i)}}\right)_{E_{0 s}^{(i)}=0}=\frac{Q_{11} Q_{23}-Q_{21} Q_{13}}{Q_{11} Q_{33}-Q_{13} Q_{31}}, \quad r_{s s}=\left(\frac{E_{0 s}^{(r)}}{E_{0 s}^{(i)}}\right)_{E_{0 p}^{(i)}=0}=\frac{Q_{21} Q_{33}-Q_{23} Q_{31}}{Q_{11} Q_{33}-Q_{13} Q_{31}}$$

Considering Eqs. (2)–(6), the polarization state of reflected light from the Ce:YIG-YIG-SiO$_2$ film can be written as

(7)$$\begin{aligned} \left|\psi_{r}\right\rangle= & \left\{\left[r_{p p}+\cot \theta_{i} \frac{k_{y}}{k_{0}}\left(r_{s p}-r_{p s}\right)\right] \cos \alpha+\left[r_{p s}+\cot \theta_{i} \frac{k_{y}}{k_{0}}\left(r_{p p}+r_{s s}\right)\right] \sin \alpha\right\}|H\rangle \\ & +\left\{\left[r_{s p}-\cot \theta_{i} \frac{k_{y}}{k_{0}}\left(r_{p p}+r_{s s}\right)\right] \cos \alpha+\left[r_{s s}+\cot \theta_{i} \frac{k_{y}}{k_{0}}\left(r_{s p}+r_{p s}\right)\right] \sin \alpha\right\}|V\rangle \end{aligned}$$

For PSHE calculation, the polarization state of the reflected light in the spin basis set is

(8)$$\left|\psi_{r}^{ {\pm}}\right\rangle \approx \frac{1}{\sqrt{2}}\left\{\begin{array}{l} r_{p p}\left[\exp \left({\pm} i k_{y} \delta_{H}\right) \mp i \frac{r_{s p}}{r_{p p}} \exp \left({\pm} i k_{y} \delta_{H}^{\prime}\right)\right] \cos \alpha \\ \mp i r_{s s}\left[\exp \left({\pm} i k_{y} \delta_{V}\right) \mp i \frac{r_{p s}}{r_{s s}} \exp \left({\pm} i k_{y} \delta_{V}^{\prime}\right)\right] \cos \alpha \end{array}\right\}| \pm\rangle$$

Where $| \pm \rangle =(|H\rangle \pm i|V\rangle ) / \sqrt {2}$ denotes the right- and the left-handed circularly polarized state. In this case, $\delta _{H}=\left (1+r_{s s} / r_{p p}\right ) \cot \theta _{i} / k_{0}$ and $\delta _{H}=\left (1+r_{p p} / r_{s s}\right ) \cot \theta _{i} / k_{0}$ represent the PSHE generated by the reflection of horizontally and vertically polarized light at the tri-layers thin film of a nonmagnetic field. $\delta _{H}^{\prime }=\left (1-r_{p s} / r_{s p}\right ) \cot \theta _{i} / k_{0}$ and $\delta _{H}^{\prime }=\left (1-r_{s p} / r_{p s}\right ) \cot \theta _{i} / k_{0}$ denotes the additional PSHE resulting from the reflection of horizontally and vertically polarized light on the tri-layers sample with a magnetic field, respectively. It is worth noting that the PSHE is very small. In order to measure this small displacement, post-selection is introduced by P2 in the measurement scheme, and $\left |\psi _{f}\right \rangle =|V\rangle$. The final state of the whole system is given by

(9)$$\left|\Psi_{f}\right\rangle=\int d k_{x} d k_{y} \varphi\left(k_{x}, k_{y}\right)\left|k_{x}\right\rangle\left|k_{y}\right\rangle\left|\psi_{f}\right\rangle\left\langle\psi_{f} \mid \psi_{r}\right\rangle$$

It is easy to calculate the PSHE after weak measurement by using the method of geometric center of gravity integration, as shown in Eq. (10).

(10)$$|y\rangle=\frac{\left\langle\Psi_{f}\left|i \partial_{k_{{\perp}}}\right| \Psi_{f}\right\rangle}{\left\langle\Psi_{f} \mid \Psi_{f}\right\rangle}$$

In order to obtain an optimal weak measurement scheme, we compare the PSHE models in Ce:YIG-YIG-SiO$_2$ multilayer film and Si film. For different reflective surfaces, the relationship between PSHE and OR angle are shown in Fig. 2. Among them, the multilayer membrane structure consists of SiO$_2$ ($\varepsilon _{1} = 4.0$, $d _{1} = 1mm$), YIG ($\varepsilon _{2} = 14$, $d _{2} = 600nm$), Ce:YIG ($\varepsilon _{3} = 17.34+0.1966i$, $d _{3} = 500nm$). The red curve denotes amplified PSHE of Ce:YIG-YIG-SiO$_2$ film versus the OR angle and black curve stand for amplified PSHE of SiO$_2$ film upon different OR angle. The gray grid area is inversion region in which the amplified beam displacement are extremely sensitive to the OR angle, and that is why we can detect the OR signal in high precision with weak measurement. We can see that the use of multilayer films as reflective surface can increase the range of spin shift by about 1.3 times compared to that of Si film. Additionally, it can improve OR measurement sensitivity by approximately 1.8 times. The illustration shows the distribution of light intensity under two types of reflective materials at different OR angle. A bigger OR angle causes a larger beam center shift, which means the two spots is more asymmetrical and the SHEL is greater. From it, we can also see the intensity center of the beam for Ce:YIG-YIG-SiO$_2$ thin film is shifting more obvious, while the changes for Si film are significantly weaken. It indirectly indicates that the sensitivity of selecting a Ce:YIG-YIG-SiO$_2$ thin film to detect optical rotation angle is higher. The reflective materials plays a very important role in modulating the transverse shift of PSHE.

Fig. 2. Relationship between PSHE and OR angle and actual spot map for reflection from multilayer film and Si.

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Furthermore, the effect of different parameters on OR angle measurement is explored. The angle of incidence, transmission distance, applied magnetic field and the beam waist were simulated in the simulation to find the most suitable measurement parameters. Fig. 3(a) shows the curve of PSHE versus incident angle in the range of 50${\circ }$ to 74${\circ }$. As the angle of incidence increases, the slope of the curve becomes larger, and the sensitivity of the measurement becomes higher. From Fig. 3(b), the colorful lines correspond to changes of PSHE with the increase of transmission distance. $\textit {z}$ is adjusted by changing the focus of L2 in a confocal system composed of L1 and L2. Increasing the transmission distance enlarges the amplified shift of PSHE, which improves the OR angle detection system’s sensitivity. According to Fig. 3(c), the influence of different magnetic field directions on amplified shift is also different. Through numerical calculations above, it can be known that changing the direction of the magnetic field can change the relative value of $r_{ps}$ and $r_{sp}$, thereby affecting the original spin splitting of PSHE on Ce:YIG-YIG-SiO$_2$ thin film. The detection range of the system increases with polar magnetic field and transverse magnetic field. The detection sensitivity is expected to expand under a longitudinal magnetic field applied. The curve of PSHE versus beam waist in the range of 30um to 70um are reported in Fig. 3(d). When the beam waist is 30${\mu }m$, the PSHE reaches nearly 800 um, which is almost 2.5 times higher than that with the beam waist being 70${\mu }m$. It is obvious that the sensitivity and precision of the system decreases as the spot becomes smaller due to the increase the beam waist. Note that the $\omega _0$ can be adjusted by changing the focus of L1 and the distance between the lens and the reflecting surface [32]. Under the optimal weak measurement conditions described above, the PSHE is as high as 600 ${\mu }m$, much higher than our previous work [33]. And the precision can be improved to 10$^{-7} rad$, two orders of magnitude lower than previous work. Comparing the results of qiu [29] and li [30] mentioned above, the precision is improved by about an order of magnitude.

Fig. 3. Relationship between PSHE and the parameters of the weak measurement system. (a) Variation of PSHE with OR angle at different incidence angles. (b) Variation of PSHE with OR angle at different transmission distance.(c) Variation of PSHE with OR angle at different directions of applied magnetic field. (d) Variation of PSHE with OR angle at different beam waists.

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To visualize the modulation of PSHE by chiral solutions of different rotations, we theoretically analyzed the intensity profiles of the reflected beams when the chiral solutions were left-handed (L-tryptophan) and right-handed (D-tryptophan). The left columns of Fig. 4(a) and Fig. 4(b) show the simulation results for L-tryptophan and D-tryptophan molecules at a concentration of 0.5 $mg/ml$. Meanwhile, the 3D light intensity distribution and contour mapping are given in Fig. 4, where the light intensity is normalized. From Fig. 4(a), it can be seen that when the sample is a solution of L- chiral molecule, the intensity distribution of the spot moves from the center to the upper left. Similarly, when the sample is a solution of D- chiral molecule, the intensity distribution of the light spot moves from the center to the lower right. Thus, chiral solutions with opposite chirality also amplify PSHE in the opposite direction. The black solid line represents the theoretical curve of PSHE displacement and the red dots represent the corresponding experimental results as shown in the right column of Fig. 4(a) and Fig. 4(b). The PSHE of both L-tryptophan and D-tryptophan are linear functions with respect to the OR angle. The amplified displacements of D-tryptophan range from 0${\mu }m$ to - 453${\mu }m$, and the amplified displacement of L-tryptophan ranges from 0${\mu }m$ to 428${\mu }m$. Combining Figs. 4(a) and 4(b), it can be obtained that the concentration of chiral molecule solution is directly proportional to PSHE, and that D-tryptophan and L-tryptophan chiral molecules affect PSHE in opposite directions.

Fig. 4. (a) Simulated light intensity distribution at 0.5 mg/ml dextrotryptophan molecular solution as a sample and PSHE variation with concentration of dextrotryptophan molecular solution (b) Simulated light intensity distribution at 0.5 mg/ml levotryptophan molecular solution as a sample and PSHE variation with concentration of levotryptophan molecular solution. The black line in the figure shows the plot of 0-1 $mg/ml$ chiral molecule solution concentration versus PSHE simulation, and the red dots are experimental data.

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Artemisinin is still the most effective drug against malaria. It exists in the flowers and leaves of Chinese herbal medicine Artemisia annua with a very low content. Thus, chirality and concentration detection of arteannuin is of great importance in research and development of arteannuin properties. Based on the above analysis, we measured the OR angles of artemisinin solutions at concentrations ranging from 0.05 $mg/ml$ to 0.25 $mg/ml$ using a parameter-optimized weak measurement system with PSHE as the probe at room temperature. The experiment is carried out at room temperature. Artemisinin is almost insoluble in water but dissolved in ethanol and the solutions at different concentrations are configured separately. The incidence angle in the experiment is chosen to be 74${\circ }$. The focal length of L1 is determined to be $f$=50$mm$ to maximize $\omega _0$, $\omega _0$=53.7${\mu }m$. The transmission distance $z$ is 250$mm$. Pure ethanol is used as a blank group and adjusting P2 slightly to obtain a symmetrical double spot, as shown in the insert map of Fig. 5. Then, placing configured solutions at different concentrations in the samples successively and the center of gravity of the double spot will change.

Fig. 5. Simulated and experimental curves of PSHE versus concentration changes of artemisinin solution.

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Generally, the specific rotation for solutions is defined as

(11)$$\left [ \alpha \right ] = \frac{\alpha }{l\times c}$$

where $l$ and $c$ are the optical path length (i.e., thickness of cell, $dm$) and the concentration of chiral sample solution ($g/ml$). Therefore, we obtain the specific rotations of the artemisinin solutions (+74.08${\circ }$) from the numerical fitting results. According to the Chinese Pharmacopoeia, this value is +75-78${\circ }$ [34]. This result confirms the rationality and feasibility of our method to estimate the OR angle of Artemisinin solution with weak measurements. According to the experimental results, the beam shift sensitivity to the OR angle is approximately $6.38\times 10^{6}{\mu }m/mrad$. The standard deviation of the CCD (Duma Optronics, beamon U3-E) for positiondetection in our experiment is about 1.5 ${\mu }m$, so the precision in our experiment reaches $2.35\times 10^{-7}$ $rad$.

In conclusion, we have demonstrated the weak measurement of chirality and concentration detection for artemisinin solution by using PSHE as a probe. The addition of a film with a large refractive index gradient as a reflecting surface and the tuning of the parameters of the weak measurement system led to an increase in measurement sensitivity and precision. Rotations of the preselected polarization states caused by the weak OR signal caused incredible changes in the weak values. The amplified PSHE corresponding to the weak values was measured to estimate the OR angle of the sample. These initiatives improve detection sensitivity to the $10^{6}{\mu }m/mrad$ level and, at the same time, increase measurement precision to the $10^{-7} rad$ level. This method have extensive applications in life sciences, stereochemistry, and pharmacology.

Funding

National Natural Science Foundation of China (62175021); Chengdu Technology Innovation and Research and Development Project (2021-YF05-02420-GX, 2021-YF05-02422-GX, 2021-YF08-00159-GX); Scientific Research Foundation of Chengdu University of Information Technology (KYTZ202244).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Optimal weak measurement scheme for chiral molecular detection based on photonic spin Hall effect

Abstract

1. Introduction

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (11)

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