1. Introduction

The method of quantum weak measurement was first proposed by Aharonov, Albert, and Vaidman in 1988 [1]. By properly adjusting the pre-selection and post-selection states and, at the same time, maintaining a small interaction strength, we can obtain values that are several times larger than the eigenvalue. The measured system and the pointer system are weakly coupled. After the interaction, the weak value is contained in the pointer state. This phenomenon was first experimentally realized in 1991 by Ritchie et al. [2]. The process that amplifies many parameters is what we actually need and it is called weak value amplification (WVA). This theory not only provides a deeper explanation for quantum physics [3, 4], but also shows the potential of precision measurement [5, 6].

Nowadays, the measurement method based on weak value amplification (WVA) has become the focus of public attention. Its amplification mechanism has high measurement accuracy and potential, so it can be used for both the observation of physical phenomena and the detection of physical parameters. This is usually achieved by amplifying the small signal from the difference of each eigenstate. Based on different application demand, four kinds of distribution are widely used to realize weak measurement techniques. These are time-domain, frequency-domain, spatial-domain, and polarization angle distribution. Those methods have shown promising applicability and achieved reasonably high precision, in various application aspect. Representative works are listed below:

Recently, the frequency-domain weak value amplification technology has also been applied to biosensors, such as: combining weak measurement with Mach-Zehnder interferometer to measure blood glucose concentration in mice [15], combining weak measurement techniques with total internal reflection (TIR) sensors to realize real-time monitoring of biological macromolecules interaction [16] and molecularly imprinted polymers (MIP) sensor based on weak measuring technology [17]. These greatly improve the sensitivity of such sensors. However, due to the single working area selected by these kinds of sensors, the light intensity of the selected work area is very weak. In spite of the relatively high sensitivity, the system is greatly influenced by the noise, which leads to a certain extent of precision reduction.

Previous work in the field of frequency-domain weak measurement, polarization state or spin state is often selected as the eigenstate of interaction A. Based on the theory of [18, 19], the interacting Hamiltonian is expressed as:

Here g(t) represents the coupling strength, which satisfies${\displaystyle \int g(t)dt=k}$, and P is the photon momentum. We can obtain a measurement result with higher precision by using the imaginary part of the weak value of A. The transversal shift is created by the total phase difference we introduce with optical instruments in the light path. In a frequency-domain weak measurement system, the interacting Hamitonian is determined by the total phase difference, and pre- and post-selection states. Within a certain adjustable range, total phase difference between polarized lights can be used to generate the interaction of weak measurement, corresponding with certain pre- and post-selection states. The sets of total phase difference and selection states are known here as working areas of weak measurement.

In this paper, for the first time we propose the theory of multi-operating areas of frequency domain weak measurement sensor. Furthermore, we complete the experimental verification of four working areas under the total internal reflection non-marking weak measurement sensor system. We put sodium chloride solution with different concentration gradients in the total internal reflection sensor based on the frequency domain weak measurement technology. We then measure it under different working areas, and obtain the sensitivity and resolution of the sensor under different working areas. The sensitivity of the A working area is 1.85 um/RIU and the resolution is 1.944 × 10⁻⁶ RIU. The sensitivity of the B working area is 1.42 um/RIU and the resolution is 8.150 × 10⁻⁷ RIU. The sensitivity of the C working area is 1.18 um/RIU, and the resolution is 7.564 × 10⁻⁷ RIU. The sensitivity of the D working area is 0.98 um/RIU and the resolution is 6.808 × 10⁻⁷ RIU. It is thus clear that we can choose different operation areas based on weak measurement technology according to the needs of different levels of sensitivity and resolution.

2. Theory

We measure the observable operator A of the system with eigenvalues 1 and −1 for the two orthogonal polarizations, and consider the phase difference in the propagation direction of light.

The weak value amplification (WVA) is mainly composed of three parts: system preselection, weak interaction between the system (observable) and the measurement device, and post-selection of the system. The experimental device used in our experiment is shown in Fig. 1. In the experiment, we use front and back polarizers in order to make the pre-selection and post-selection. When the beam passes through Soleil-Babinet compensator (SBC), the weak interaction is coupled between the polarization operator and the photon longitudinal momentum. As is researched in [11, 18], the imaginary part of the weak value also has a real dramatic advantage, which corresponds to the spectral shift of the photon. It should be noticed that weak value amplification will not provide more information of the measurement itself, as Ferrie and Combes [23] have illustrated. However, WVA gains a higher sensitivity advantage at the cost of lower detection probability.

 

Fig. 1 The schematic diagram of the weak measurement system.

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This paper adopts reflective phase-sensitive weak measurement method, the schematic diagram of the system is shown in Fig. 1. The light source used in this experiment is a super-luminescent diode centered on 830nm (SLD, Thorlabs Inc., IPSDD0803, 830 nm, 5 mW, Inphenix). After passing through a collimating lens, it is filtered through a Gaussian filter (center wavelength 830 nm, FWHM 10 nm), The pre-selection is then prepared through a half-wave plate (HWP), the pre-selection is conducted through the first linear polarizer (Thorlabs LPVIS050-MP, extinction ratio is 100000:1), and the beam is divided into two orthogonal polarized light (H and V). The initial state can be expressed as$|{\psi }_i〉=\sin \alpha |H〉+\cos \alpha |V〉$, α is the angle between the polarization axis of the pre-selected polarizer and the horizontal direction. The light is then completely reflected at an incident angle of 74° (greater than the critical angle, calculated as 59.4°) by the K9 substrate on the K9 prism with a refractive index of 1.51 without changing the light intensity, but only increasing the phase difference Δ related to refractive index between p and s polarization. There is a flow path on the prism surface, which contains the liquid that needs to be measured. The prism then uses the Soleil-Babinet compensator (SBC, Thorlabs Inc., SBC-IR) to introduce the necessary initial phase difference between the horizontal polarization component and the vertical polarization component at the appropriate working area. After that, second linear polarizers are used for post selection and placed at the angle of β. The light path then enters the spectrometer through the collimating lens.

In our optical path, the total phase difference between H and V lights has two main sources: the optical path difference ΔS caused by the phase compensator and the phase difference Δ caused by the total internal reflection effect. The phase difference between P light and S light is determined by the difference between the inner and outer refractive index of the total internal reflection prism, and the light intensity does not change after TIR. Based on our experiment setup, we use vertically polarized light (V light) and horizontally polarized light (H light) to replace P light and S light, respectively. According to Fresnel’s Formula, Δ can be expressed as

θ is the sum of incident angles n = n₁/n₂, n₁ and n₂ are the refractive index of the prism and the analyses, respectively. The light then passes through the SBC modulator and the phase of the direct H and V polarization is continuously increased from zero by adjusting SBC. The weak interaction between polarization operator and longitudinal momentum leads to the spectral shift of the imaginary part of the weak value.

For incident light of wavelength λ, the phase difference caused by SBC can be expressed as [20]

where n_e and n_o denote the refractive indices for the extraordinary and ordinary components, h indicates the thickness of the wave plate. Because of $\lambda \approx {\lambda }_0$, we can express the total phase difference in the form of${\delta }_0+\epsilon (\lambda )-\Delta \approx 0$, where${\delta }_0=\pm \frac{\pi \left(n_e-n_0\right)h{\chi }^2}{{\lambda }_0}$, $\epsilon (\lambda )={\delta }_0\cdot \frac{{\lambda }_0-\lambda }{\lambda }$ represent the tiny amount of influence of different wavelengths on the total phase difference. As the SBC increases the phase difference, the influence of ε(λ) also increases. We can draw the strength of the interaction as $k=({\delta }_0+\epsilon (\lambda )-\Delta )/P_0$, which is related to the coupling strength in (1). As shown in Fig. 1, we use the polarizer to construct the pre-selection state$|{\psi }_{pre}〉=\sin \alpha |H〉+\cos \alpha |V〉$, where α represents the angle between the polarizer direction and the vertical direction. After the total internal reflection surface and the SBC, a post-selection polarizer with an angle β between the polarization direction and the vertical direction is-assembled in the light path, so that the system's post-selection state can be expressed as$|{\psi }_{post}〉=\sin \beta |H〉+\cos \beta e^{i({\delta }_0(\lambda )+\epsilon (\lambda )-{\Delta }_0)}|V〉$.

The weak value of operator$A=|H〉〈H|-|V〉〈V|$ can be expressed as

The imaginary part of the weak value is

where $\gamma =\cot \alpha \cot \beta$. We usually choose to measure the state of $\alpha \approx \frac{\pi }{4}$, $\beta \approx -\frac{\pi }{4}$, ${\delta }_0+\epsilon (\lambda )-\Delta \approx 0$. At this time, $\gamma \approx 1$, $\epsilon (\lambda )\approx 0$, so Im(A_ω) can be expressed as

which is independent from λ. We have already made calculations for this case in our previous work [16]. At this time, the momentum shift should be $\delta P=2k{(\Delta P)}^2\mathrm{Im}(A_{\omega })$ [2], where ΔP is the uncertainty of the photons corresponding to the spectrum width of the SLD, and k is the interaction strength [11], which is relatively small. We can rewrite δP using the relation P = 2π∕λ, thus giving the spectrum shift

In this case, the result of the calculation will be consistent with the conclusion of [11, 15]. However, this is not the only limit that satisfies the weak measurement condition of the system. According to the theory given by AAV in [1], $\Delta P\ll \underset{n}{\max }\frac{\left|〈{\psi }_{post}|{\psi }_{pre}〉\right|}{k{\left|〈{\psi }_{post}|{\widehat{A}}^n|{\psi }_{pre}〉\right|}^{\frac{1}{n}}}$, weak measurement state is valid in the condition that ΔP is sufficiently small. We can calculate whether the system is still under weak measurement state according to $\Delta P\ll \underset{n}{\max }\frac{\left|〈{\psi }_{post}|{\psi }_{pre}〉\right|}{k{\left|〈{\psi }_{post}|{\widehat{A}}^n|{\psi }_{pre}〉\right|}^{\frac{1}{n}}}$. Based on the relation $\lambda =\frac{2\pi }{P}$, the condition can be expressed as

We can see here if $|{\delta }_0+\epsilon (\lambda )-\Delta |$ is relatively small (The maximum total phase difference we choose is 4π, satisfying the condition of (8)), appropriate α and β values can be chosen, corresponding to the selected state before and after the angle, to make the system a sensitive statue. In our previous work, we chose$\alpha \approx \frac{\pi }{4}$, $\beta \approx -\frac{\pi }{4}$ as the pre- and post-selection states angular respectively. But other α, β value can also be used in weak measurement, if the corresponding phase difference ${\delta }_0+\epsilon (\lambda )-\Delta$ satisfies (8).

Supposing we choose a different phase difference to make the value of δ₀ increases by 2π. In the same α and β angles, we can still establish weak measurement conditions. The value of A_ω is not affected by the increase in δ₀, but the final light intensity is affected by the increase in δ₀.

According to [1], the last eigenstate (the measurement state) should be expressed as

The state of measuring device will turn out to be [21]

where $e^{-\frac{{(\lambda -{\lambda }_0-\delta \lambda )}^2}{4\Delta {\lambda }^2}}$denotes the shift of the center wavelength after interaction of system state ${\varphi }_i{}^2(\lambda )=e^{-\frac{{(\lambda -{\lambda }_0)}^2}{2\Delta {\lambda }^2}}$ with incident Gaussian waves to δλ. Without considering ε(λ), $|〈{\psi }_{post}|{\psi }_{pre}{〉|}^2$denotes the reduction of light intensity caused by the pre-selection and post-selection states, which can be regarded as constants. But after considering influence from ε(λ), we can express${\varphi }_0{}^2(\lambda )$ as:

which varies with λ. The form of the final state of the system ${\varphi }_0{}^2(\lambda )$ should be the result of a trigonometric function and a center-shifted Gaussian function. In the formula, ${\delta }_0+\epsilon (\lambda )$ denotes that the phase difference of the phase compensator changes is not the same as the one caused by different wavelengths of H and V lights. When δ₀ is small, we have$\epsilon (\lambda )\ll 1$. In this case, (11) indicates that the influence of the pre and post selected states on the center wavelength shift amount and peak intensity is almost negligible. When the phase difference is small, the center wavelength shift amount is only determined by Im(A_ω), and $|〈{\psi }_{post}|{\psi }_{pre}{〉|}^2$ only affects the light intensity in the weak measurement state and does not affect the center wavelength. This is consistent with our previous derivation. However, with the increase of δ₀, its influence in $|〈{\psi }_{post}|{\psi }_{pre}{〉|}^2$becomes non-negligible. The central wavelength shift amount and peak intensity are determined by Im(A_ω) and $|〈{\psi }_{post}|{\psi }_{pre}{〉|}^2$ together.

It can be seen from the above that when a phase modulator is used in the optical path and a large total phase difference is caused, the peak wavelength and the shift of the center wavelength are affected simultaneously by both Im(A_ω) and $|〈{\psi }_f|{\psi }_i{〉|}^2$ when a WVA phenomenon occurs.

In this paper, we choose four working areas, which are $\alpha \approx \frac{\pi }{4}$,$\beta \approx \frac{\pi }{4}$, ${\delta }_0+\epsilon (\lambda )-\Delta \approx \pi$; $\alpha \approx \frac{\pi }{4}$, $\beta \approx -\frac{\pi }{4}$, ${\delta }_0+\epsilon (\lambda )-\Delta \approx 2\pi$; $\alpha \approx \frac{\pi }{4}$, $\beta \approx \frac{\pi }{4}$, ${\delta }_0+\epsilon (\lambda )-\Delta \approx 3\pi$; $\alpha \approx \frac{\pi }{4}$, $\beta \approx -\frac{\pi }{4}$, ${\delta }_0+\epsilon (\lambda )-\Delta \approx 4\pi$,according to (8), to conduct experiments on four working areas with WVA phenomena. These four working areas are recorded as A, B, C and D. We found that when Δ is relatively small, $e^{2g\lambda \mathrm{Im}(A_{\omega })}$ has a relatively extensive influence on the system, with high and low light intensity. When Δ is large, ${|〈{\phi }_{out}|{\phi }_{in}〉|}^2$ has more influence on the system, the system sensitivity is low, and the light intensity is high and stable. The experimental results are in good agreement with the calculated curve.

3. Experimental observation

3.1 Application range reasonableness test

In order to detect the rationality of the four working areas of the total internal reflection unmarked sensor based on the weak measurement technology, we measured the sensitivity and resolution of phase difference in these four working areas. The phase difference between V and H is changed by adjusting SBC. The relationship between the shift of the center wavelength and the value of SBC rotation (the phase difference) is shown in Fig. 2. We also theoretically predict the equation for center wavelength shift and total phase difference.

 

Fig. 2 Figure (a) shows the simulation curves and experimental results of the weak measurement operating area obtained by adjusting the SBC under the condition that the front and rear selection states are parallel to each other, and denoted as A and C, respectively. The theoretical curves and experimental data of the pink and black dashed boxes in the upper part of figure (a) are shown in the lower left and right figures respectively. Figure (b) shows the simulation curve and experimental results of the weak measurement working area obtained by adjusting the SBC under the condition that the front and rear selection states are perpendicular to each other, and denoted as B and D, respectively. The theoretical curves and experimental data of the pink and black dashed boxes in the upper part of figure (b) are shown in the lower left and right figures respectively.

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The four working areas of the sensor based on the weak measurement technology are shown in Fig. 2. The center wavelength shifts slowly to a minimum value as the total phase difference increases. It then drastically increases to a maximum, and then slowly drops again. It is obvious that each working area has relatively high sensitivity of the total phase difference.

3.2 A series of concentration gradient sodium chloride solution experiments

We first adjust the pre-selection and post-selection states of the total internal reflection sensor to be parallel to each other based on the weak measurement technique. We then adjust the SBC so that the phase difference is near π. Then we pass the deionized water into the flow cell on the prism surface then accordingly adjust working area of the SBC system to be placed in the area of extremely high sensitivity mentioned above. This working area is regarded to as the-working area A.

Because during the process of introducing different concentrations of sodium chloride solution, the pressure change caused by the liquid flow on the prism surface also affects the refractive index of the prism surface. In order to control the variables, we only collect the data of the central wavelength shift in the case where the solution in the flow channel is stationary each time the sodium chloride solution is completed.

The data is recorded by the OCEAN VIEW HR4000 spectrometer with its own program. The four concentrations of sodium chloride solution with concentrations of 0.45%, 0.90%, 1.35%, and 1.80% are then recorded and the data is taken in sequence after the solution in the flow channel is stationary. Then we pass the deionized water again and record the data, in order to verify the repeatability of the system.

We adjust the SBC so that the phase difference is near 3π. We then pass the deionized water into the flow cell on the prism surface, and accordingly adjust the working point of the SBC system to be placed in the area of extremely high sensitivity mentioned above. This working area is regarded as the working area C. The experiment is then repeated with different concentrations of sodium chloride solution, and the data is once again recorded.

Secondly we adjust the pre-selection and post-selection states of the total internal reflection sensor to be vertical to each other based on the weak measurement technique. We then adjust the SBC so that phase difference is near 2π, and then we pass the deionized water into the flow cell on the prism surface. The working area of the SBC system is then accordingly adjusted to be placed in the area of extremely high sensitivity mentioned above. This-working area is regarded as the working area B.

Similarly, we adjust the SBC so that phase difference is near 4π. We then pass the deionized water into the flow cell on the prism surface, and accordingly adjust the working area of the SBC system to be placed in the area of extremely high sensitivity mentioned above. This working area is regarded as the-working area D. The experiment is then repeated with different concentrations of sodium chloride solution, and the data is once again recorded.

In the above work, the integration time at the four working areas of A, B, C and D is uniformly set to 5 ms. We use the smooth program (set to 4 in the experiment) in the OCEAN VIEW spectrometer to smooth the spectrum, so as to reduce the influence of high frequency noise corresponding to the noise from the light source and detector on the system. We have obtained the bimodal peak image corresponding to each concentration gradient under the four working areas, as shown in Fig. 3.

 

Fig. 3 The bimodal spectra of different concentrations of sodium chloride solution at four operating areas A, B, C, and D.

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3.2.1 Comparison of overall light intensity at four working areas

Better system signal-to-noise ratio is attained through stronger light intensity for weak measurement systems, and smaller noise impact on the system. We define the enclosed area of the four peaks of the four working areas based on the weak measurement system above as the weak measurement sensor light intensity (WMSLI).

We calculated the WMSLI of 0.9% sodium chloride solution for four operating areas as a comparison. The results are shown in Fig. 4, WMSLI of A, B, C and D increase in turn.

 

Fig. 4 Intensity integration contrast of 0.9% sodium chloride solution at four-working areas A, B, C, D (The error bar indicates the standard deviation for three measurements).

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3.2.2 Sensitivity and resolution at four operating areas

We process the data of the central wavelength shift caused by a series of concentration gradients of sodium chloride solutions obtained from the above four experimental areas. The refractive index concentration dependence of the sodium chloride solution is 1.47 × 10⁻³ C [22], where C is the mass percentage. Thus the 1.8% sodium chloride solution compared with deionized water can cause 2.646 × 10⁻³ RIU refractive index change.

In order to compare the refractive index sensitivity and refractive index resolution of different working areas, we perform the same sodium chloride solution experiments in the four working areas A, B, C, and D, respectively, and obtain the following results.

For the A working area, in Fig. 5(a), the 1.8% sodium chloride solution causes center wavelength shift of 4.9 nm compared to deionized water.The corresponding sensitivity is known as 1.85 um/RIU by the formula δλ/λn. The standard deviation of the center wavelength deviation is calculated by using average 100 different successive data. The insert in the Fig. 5(a) shows the shifts measuring 1.8% sodium chloride solution with the standard deviation σ_s of 0.00120 nm. So by the formula$\sigma =3{\sigma }_s/(\delta \lambda /\delta n)$, the calculator resolution is 1.944 × 10⁻⁶ RIU.

 

Fig. 5 Figures (a), (b), (c), and (d) show the results of the sodium chloride solution at four operating areas A, B, C, and D, respectively, and a linear plot. In each figure, the left picture shows the experimental results of the central wavelength shift of NaCl, 0.45%, 0.9%, 1.35%, and 1.80% of five concentration gradient sodium chloride solutions; the right picture shows the linearly-fitted spectral shift of each operating area (The error bar (red) indicates the standard deviation for three measurements).

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After the system-reaching equilibrium, standard deviations of wavelength shifts were calculated by averaging 100 successive 100 data. The inset of Fig. 3(b) shows the shifts measuring 1.8% sodium chloride solution. The standard deviation σ_s is 0.00197 nm.

For the B working area, in Fig. 5(b), the 1.8% sodium chloride solution causes center wavelength shift of 3.76 nm compared to deionized water. The corresponding sensitivity is known as 1.42 um/RIU by the formula δλ/λn. The standard deviation of the center wavelength deviation is calculated by using average 100 different successive data. The insert in the Fig. 5(b) shows the shifts measuring 1.8% sodium chloride solution with the standard deviation σ_s of 0.000395nm. So by the formula$\sigma =3{\sigma }_s/(\delta \lambda /\delta n)$, the calculator resolution is 8.150 × 10⁻⁷ RIU.

For the C working area, in Fig. 5(c), the 1.8% sodium chloride solution causes center wavelength shift of 3.13 nm compared to deionized water. The corresponding sensitivity is known as 1.18 um/RIU by the formula δλ/λn. The standard deviation of the center wavelength deviation is calculated by using average 100 different successive data. The insert in the Fig. 5(b) shows the shifts measuring 1.8% sodium chloride solution with the standard deviation σ_s of 0.000298 nm. So by the formula$\sigma =3{\sigma }_s/(\delta \lambda /\delta n)$, the calculator resolution is 7.564 × 10⁻⁷ RIU.

For the D working area, in Fig. 5(d), the 1.8% sodium chloride solution causes center wavelength shift of 2.60 nm compared to deionized water. The corresponding sensitivity is known as 0.98 um/RIU by the formula δλ/λn. The standard deviation of the center wavelength deviation is calculated by using average 100 different successive data. The insert in the Fig. 5(a) shows the shifts measuring 1.8% sodium chloride solution with the standard deviation σ_s of 0.000228 nm. So by the formula$\sigma =3{\sigma }_s/(\delta \lambda /\delta n)$, the calculator resolution is 6.808 × 10⁻⁷ RIU.

It should be noted that the working areas satisfying the weak measurement conditions may be more than four. A larger δ₀-Δ limit may also obtain weak value amplification effect, but it may also transit to a classical state because the Eq. (8) is not satisfied. Therefore, only the smallest four limiting factors were studied as our working areas (A, B, C, D).

The advantages and disadvantages of each working areas are summarized in Table 1.

Table 1. Advantages and disadvantages of each working areas

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Among the four working areas, A has the highest sensitivity, but the resolution is the lowest, while D has the highest resolution and the sensitivity is the lowest. In practical measurement applications, we can select different working areas to satisfy different requirement. If higher measurement stability is required, the D area with a high signal-to-noise ratio should be selected. If a higher measurement sensitivity is required and the signal-to-noise ratio is not so important, the A area can be selected for measurement. Working area B and C are the alternatives moderate choices to satisfy both requirements.

Since the limits of the four working areas are mutually exclusive, it is difficult to apply two working areas to the same optical path. Multiplexing can be a possible way to simultaneously increase sensitivity and resolution.

4. Conclusion

In summary, this paper proposes a theoretical scheme of multiple working areas for weak measurement in frequency domain and discuss the limits of the working areas. The experimental verification is carried out in the total internal reflection non-marking weak measurement system, where four limits are selected as the working areas. In all of the four working areas, we can detect the phase difference information with a relatively high sensitivity. We also summarize the advantages and disadvantages of different working areas, which can help us to apply weak value amplification method to more practical applications.

Our highest resolution reaches 6.81 × 10⁻⁷ RIU in working area D, higher than our past work of 3.6 × 10⁻⁶ RIU [16] and existing weak measurement work of a phase difference of ~10⁻⁵ rad [23], corresponding to a ~10⁻⁶ RIU index difference according to Eq. (3). Our highest sensitivity reaches 1.85 um/RIU in working area A, remaining a close level to our previous work of 1.64 um/RIU [16], but not the best comparing to other works such as SPR. Multiple working area method can be combined with some other methods to achieve better results, as well as multiplexing to obtain advantages of different working areas. Our research suggests that this frequency domain quantum weak measurement optimization method has the potential of strong and promising practicability, and has great potential and merits in high-precision measurement such as biomedical sensing.