High-precision temperature measurement with adjustable operating range based on weak measurement

High-precision temperature measurement is widely used in industrial production [1], biomechanics [2], rehabilitation applications [3, 4], manufacture of distributed feedback lasers [5, 6] and physical properties of materials [7]. Traditional thermometry, such as thermocouples [8–10] and thermal resistance [11–13], has been studied and required in all kinds of fields. Although these thermometries are well-known for temperature measurement, the precision of submillikelvin ranges are subject to drift, resistive heating and a considerable investment [14]. How to realize high precision is a crucial problem for temperature measurement in practical applications. Many researchers have focused on optical methods for high-precision temperature measurements. By using dual-wavelength pseudo-heterodyne phase detection, a miniature extrinsic fiber-optic-based Fizeau temperature sensor with a typical cavity length of several hundred microns has been demonstrated, and a resolution of 0.006 °C over a range of 35 °C has been obtained with a bandwidth of 30 Hz [15]. Using the temperature dependence of reflection group-delay in a linearly chirped fiber Bragg grating [16], the thermometry can realize a resolution of 8.9 × 10⁻³ °C. A liquid-in-glass thermometer has been developed in which the meniscus level is monitored by an interferometer, allowing temperature changes of below 2 μK to be resolved [17].

As an outstanding metrological technique, weak measurement, as proposed by Aharonov et al [18], has been used to observe physical phenomena and precisely measure small parameters such as velocities [19], the spin Hall effect [20], optical beam deflections [21], frequency variation [22], time delay [23], phase shifts [24–26] and temperature measurement [14, 27, 28]. Weak measurement can suppress the technical noises to increase the signal-to-noise ratio by choosing appropriate pre- and post-selections [29, 30]. Recently, Egan and Stone reported the temperature precision of 0.2 mK at room temperature using a fluid-filled prism and sensing the deviation of a laser beam based on weak measurement [14]. Salazar-Serrano et al [31] increased the sensitivity of temperature sensors based on fiber Bragg gratings via the real part of weak measurement, which performed a spectral shift of the reflected signal of about 0.035 nm/°C. Li et al [27] demonstrated a precise temperature measurement method by using nematic liquid crystals (NLCs) and a Sagnac interferometer based on weak measurement, which can achieve a precision of 3 × 10⁻⁶ °C.

Both high precision and large operating range are to be considered for temperature measurement in practical application. In our previous work [28], we have realized a temperature measurement with a large operating range and the accuracy of 2.4 × 10⁻⁶ °C based on weak measurement. The extra phase generated by Soleil–Babinet compensators (SBC) in the experiment has been employed to change the operating range. However, the accuracy of the extra phase delay is crucial to the adjustment of operating range for high-precision temperature measurement. In this paper, we demonstrate a high-precision temperature measurement with adjustable operating range based on weak measurement. By introducing a dynamic extra time delay to the post-selection, the operating range of measurement can be precisely modulated by coarse and fine adjustment. The higher the resolution of the extra time delay is, the more precise modulation the system can achieve when realizing temperature measurement with adjustable operating area. The optimized extra time delay pumped by a femtosecond laser can achieve the accuracy of 3.09 × 10⁻² as, which is about two orders of magnitude higher than that of SBC. In the experiment, the temperature variation of the NLC leads to a weak interaction between the system and the pointer. In a certain operating range, the resolution of the temperature measurement could be 8.03 × 10⁻⁷ °C with a currently available spectrometer, and the average sensitivity is 38 nm/°C at 28 °C.

In the proposed method, weak measurement is used to measure the time delay resulting from temperature variation. 'System' and 'pointer' of the physical quantum state are considered as the polarization and frequency of a light beam, respectively. Consider a light beam is preselected in a linear polarization state $\vert {\varphi }_{i}\rangle =\frac{1}{\sqrt{2}}\left(\vert H\rangle +\vert V\rangle \right)$, where |H⟩ and |V⟩ stand for the horizontal and vertical polarization states, respectively. The pointer is a continuous degree of freedom in state $\vert \phi \rangle =\int \mathrm{d}\omega \enspace f\left(\omega \right)\vert \omega \rangle$, where the frequency-domain wave function of the pointer $f\left(\omega \right)={\left(\pi {\sigma }^{2}\right)}^{-1/4}\enspace \mathrm{exp}\left[-{\left(\omega -{\omega }_{0}\right)}^{2}/2{\sigma }^{2}\right]$. Then, a weak interaction is introduced into the system due to the temperature change of NLC. NLC is an outstanding optical material, which possesses a large thermo-optic coefficient (dn/dT) of about 10⁻³/°C and a nonlinear refractive index coefficient (n₂) up to 10⁻³ cm² W⁻¹ [32]. The varied time delay between two orthogonal optical polarizations resulted from the temperature variation is given by [28]

$$\begin{equation}{\Delta}\tau =\frac{d}{c}\cdot {\Delta}\left({\Delta}n\right)=\frac{d}{c}\cdot \frac{\partial \left({\Delta}n\right)}{\partial T}{\Delta}T,\end{equation} \tag{ 1 }$$

where d is the thickness of NLCs and c is the velocity of light. The variation of birefringence can be expressed as ${\Delta}\left({\Delta}n\right)=\frac{\partial \left({\Delta}n\right)}{\partial T}{\Delta}T$, where ΔT is the small varied temperature. And ${\Delta}n\left(T\right)={\left({\Delta}n\right)}_{0}{\left(1-\frac{T}{{T}_{\mathrm{c}}}\right)}^{\beta }$, where (Δn)₀ is the birefringence of NLCs in the crystalline state (T = 0 K), T = T₀ + ΔT and T₀ is the initial temperature variable, T_c is the clearing temperature of the NLCs and β is a material constant [33]. The interaction can be described by a unitary operator [18]

$$\begin{equation}\hat{U}={\mathrm{e}}^{-\mathrm{i}\frac{{\Delta}\tau }{2}\hat{A}\hat{\omega }},\end{equation} \tag{ 2 }$$

where the system observable $\hat{A}=\vert H\rangle \langle H\vert -\vert V\rangle \langle V\vert$. After the interaction, the evolved state takes the form $\vert {\Psi}\rangle =\hat{U}\vert {\varphi }_{i}\rangle \vert \phi \rangle$. Then, the system is post-selected in the state

$$\begin{equation}\vert {\varphi }_{f}\rangle =\frac{1}{\sqrt{2}}\left({\mathrm{e}}^{-\mathrm{i}\left(\frac{\omega {\tau }_{f}}{2}-\varepsilon \right)}\vert H\rangle -{\mathrm{e}}^{\mathrm{i}\left(\frac{\omega {\tau }_{f}}{2}-\varepsilon \right)}\vert V\rangle \right),\end{equation} \tag{ 3 }$$

where ɛ is the tiny tilt of the post-selection polarizer and τ_f is the introduced extra time delay of the post-selection. In order to obtain adjustable operating range, τ_f can be generated by the SBC and another NLC sample pumped by a laser which can induce the ultra-small extra time delay ${\tau }_{{f}_{1}}$. NLCs show excellent nonlinear optical properties, which are associated with laser-induced director axis re-orientation effects. Based on the third-order nonlinear optical effect [36], the refractive index change is linearly related to the light intensity by Δn = n₂ I_p , where n₂ is the nonlinear coefficient and I_p is the intensity of the pump beam. The corresponding ultra-small extra time delay can be expressed as

$$\begin{equation}{\tau }_{{f}_{1}}=\frac{d}{c}\cdot {n}_{2}{I}_{p}=\frac{d}{c}\cdot \frac{{n}_{2}P}{S},\end{equation} \tag{ 4 }$$

where P and S are the power and area of the pump beam, respectively. ${\tau }_{{f}_{1}}$ has higher accuracy than that of SBC. Here, τ_f can be expressed as ${\tau }_{f}={\tau }_{{f}_{1}}+{\tau }_{f2}-{\tau }_{{T}_{0}}$, where τ_f2 is generated by the SBC, ${\tau }_{{T}_{0}}$ is the initial time delay resulted from NLCs when at the initial temperature T₀. Consequently, the extra time delay of the post-selection is generated by NLC2, SBC and the initial temperature of NLC1. And the accuracy of the optimized extra time delay can be as high as ${\tau }_{{f}_{1}}$.

The system is projected to the state |φ_f ⟩. The global state can be decomposed as

$$\begin{equation}\vert {\phi }_{f}\rangle =\langle {\varphi }_{f}\vert {\Psi}\rangle =\langle {\varphi }_{f}\vert \left[\mathrm{cos}\left(\frac{{\Delta}\tau }{2}\hat{\omega }\right)\vert {\varphi }_{i}\rangle -i\enspace \mathrm{sin}\left(\frac{{\Delta}\tau }{2}\hat{\omega }\right)\hat{A}\vert {\varphi }_{i}\rangle \right]\vert \phi \rangle .\end{equation} \tag{ 5 }$$

Consequently, the probability distribution of ω takes the form

$$\begin{equation}{P}_{f}\left(\omega \right)=\vert \langle \omega \vert {\phi }_{f}\rangle {\vert }^{2}={\mathrm{sin}}^{2}\left(\omega \frac{{\Delta}\tau -{\tau }_{f}}{2}+\varepsilon \right){f}^{2}\left(\omega \right).\end{equation} \tag{ 6 }$$

The frequency shift is calculated as

$$\begin{align}\hfill {\Delta}\omega & =\frac{\int \omega {P}_{f}\left(\omega \right)\mathrm{d}\omega }{\int {P}_{f}\left(\omega \right)\mathrm{d}\omega }-{\omega }_{0}\hfill \\ \hfill & =\frac{\left({\Delta}\tau -{\tau }_{f}\right){\sigma }^{2}\enspace {\mathrm{e}}^{-{\left({\Delta}\tau -{\tau }_{f}\right)}^{2}{\sigma }^{2}/4}\enspace \mathrm{sin}\left({\omega }_{0}\left({\Delta}\tau -{\tau }_{f}\right)+2\varepsilon \right)}{2\left[1-{\mathrm{e}}^{-{\left({\Delta}\tau -{\tau }_{f}\right)}^{2}{\sigma }^{2}/4}\enspace \mathrm{cos}\left({\omega }_{0}\left({\Delta}\tau -{\tau }_{f}\right)+2\varepsilon \right)\right]}.\hfill \end{align} \tag{ 7 }$$

Due to the instability of the light source, low extinction ratio of the polarizers, the influence of the indium-tin oxides glasses which sandwich the NLCs, and the imperfections of the fibers, there will be several kinds of the stray light entering the detector [28, 30]. Here, we consider these typical noises are superimposed on the signal. The probability distribution at the receiving end can be expressed as P'(ω) = P_f (ω) + Σb_m F_m (ω), where F_m (ω) is the noise function with average frequency ω_m ', and b_m is the intensity of the normalized noise. The frequency shift influenced by noise can be written as

$$\begin{align}\hfill {\Delta}\omega & =\frac{\int \omega {P}^{\prime }\left(\omega \right)\mathrm{d}\omega }{\int {P}^{\prime }\left(\omega \right)\mathrm{d}\omega }-{\omega }_{0}\hfill \\ \hfill & =\frac{\tau {\sigma }^{2}\enspace {\mathrm{e}}^{-{\tau }^{2}{\sigma }^{2}/4}\enspace \mathrm{sin}\left({\omega }_{0}\tau +2\varepsilon \right)+4{\Sigma}{b}_{m}\left({\omega }_{m}^{\prime }-{\omega }_{0}\right)}{2\left[1-{\mathrm{e}}^{-{\tau }^{2}{\sigma }^{2}/4}\enspace \mathrm{cos}\left({\omega }_{0}\tau +2\varepsilon \right)+2{\Sigma}{b}_{m}\right]},\hfill \end{align} \tag{ 8 }$$

where τ = Δτ − τ_f . It is noted that by changing τ_f continuously to satisfy ω₀ τ_f − 2ɛ ≈ 2nπ (n = 0, 1, 2, 3...), different linear areas could be achieved with different n, where the frequency shift Δω changes dramatically and linearly with Δτ.

With the limitation of Δτ ≪ ɛ/ω₀, Δτ ≪ τ_f and ω₀ τ_f − 2ɛ ≈ 2nπ, the spectral shift can be calculated as

$$\begin{equation}{\Delta}\omega \approx -\frac{{\omega }_{0}^{2}/\left(n\pi +\varepsilon \right)}{1+\chi {\Sigma}{b}_{m}}{\Delta}\tau +\frac{2{\Sigma}{b}_{m}\left({\omega }_{m}^{\prime }-{\omega }_{0}\right)}{2{\Sigma}{b}_{m}+{\left(n\pi +\varepsilon \right)}^{2}{\sigma }^{2}/{\omega }_{0}^{2}},\end{equation} \tag{ 9 }$$

where $\chi =2{\omega }_{0}^{2}/\left[{\left(n\pi +\varepsilon \right)}^{2}{\sigma }^{2}\right]$. Here, the factor χ of Σb_m indicates the robustness of the noise with different n and the slope between Δω and Δτ represents the sensitivity of the linear areas. As n = 0, the factor χ increases dramatically for ɛ ≪ nπ. In our experiment, ${\omega }_{0}^{2}/\left({\varepsilon }^{2}{\sigma }^{2}\right)$ is of the order of 10⁸ and the sensitivity deteriorates sharply when Σb_m > 10⁻⁵. In this case, the received light intensity is so weak that it is covered by the noise. On the other hand, when n increases, χ becomes smaller, which means the robustness against noise is better. But the slope goes down and the sensitivity gradually decreases. Hence, to choose a proper linear area n, the trade-off between robustness and sensitivity is worth serious consideration in practice. When the robustness against the noise is satisfactory, the most sensitive linear area should be selected in the experiment. In this case, the condition ω₀ τ_f − 2ɛ ≈ 2nπ indicates that τ_f should be smaller when n decreases. In order to get higher sensitivity, the measurement should be in a linear area where n is smaller, which must lead to τ_f being smaller.

Figure 1 shows the experimental setup. As illustrated, a Gaussian light beam was emitted from a commercial superluminescent diode (SLD, Thorlabs, S5FC1005S) with the central wavelength λ₀ of 1550 nm and the full width at half maximum (FWHM) of 50 nm. After passing through a collimating lens, the light was pre-selected by the first Glan polarizer (P1) with the optical axis at an angle of π/4. Then, the first NLC (NLC1) sample was put into the light path to couple the system and pointer. It was attached to the transparent glass whose temperature was controlled by a temperature controller (HCS302, Instec Co.) with an accuracy of 0.01 °C. Next, the light entered into the post-selection composed of NLC2, SBCs, QWP and P2. The extra time delay of the post-selection can be controlled by NLC2 and SBCs and the initial temperature of NLC1. The second NLC (NLC2) to generate ${\tau }_{{f}_{1}}$ was pumped by a femtosecond erbium laser (MenloSystems C-Fiber 780) which delivers a pulse of 70 fs and a power P_m = 100 mW. The power was attenuated by a neutral density (ND) filter (Thorlabs, NDC). The area of the light spot is about 8.6 mm². These two NLCs were respectively sandwiched between two indium-tin oxide glass substrates with 20 μm thick spacers. Followed by the NLC2, two Soleil-Babinet compensators (Thorlabs Inc. SBC-IR) were mounted in the light path, whose phase adjustment range could be about 4π. After that, the light was postselected by a quarter-wave plate (QWP) rotated −π/4 to the fast axis and the second Glan polarizer (P2) with its optical axis at an angle −π/4 −ɛ, where ɛ is about 0.017 rad in the experiment. Finally, the light was coupled into the fiber by a lens and monitored by a spectrometer (Yokogawa, AQ6370D).

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To verify the impact of noise and choose a proper linear area, figure 2 shows the comparison between the ideal case in figure 2(a) and the noise-influenced case in figure 2(a). The difference between figures 2(a) and (b) is merely due to the influence of the noise and there is no additional phase. Figure 2(a) shows the wavelength shift as the time delay τ increases without considering the influence of noise. In practice, however, the noise cannot be ignored. Comparing figures 2(a) and (b) shows the numerical simulation for the wavelength shift by considering the influence of the noise. The blue dots are experimental results and every two adjacent blue dots have an abscissa interval of 29 as. In this way, the greater the ordinate interval of the two adjacent blue dots in the linear areas is, the higher the sensitivity can be. The noise has a significant influence on the linear areas especially on the area n = 0. In the ideal case without the influence of the noise, the linear area of n = 0 has the highest sensitivity but with lowest signal intensity and the sensitivity decreases as n gradually increase. In practice, with the influence of the noise, the sensitivity of the linear area at n = 0 can deteriorate sharply which is four orders of magnitude lower than that in ideal case. This parameter can not be measured here because the noise intensity covers the weak signal light with the lowest signal-to-noise ratio (SNR). As n increases, the sensitivity of the linear areas becomes worse but the influence of the noise gets smaller which means the robustness against the noise can be better. Therefore, there is a trade-off between sensitivity and robustness which constrains the size of n. Hence, the trade-off between sensitivity and robustness must be weighted carefully. In our experiment, as the robustness against the noise in the linear areas is satisfactory except in the linear area n = 0, we keep the measurement in the linear area n = 1 by making the extra time delay τ_f satisfy ω₀ τ_f − 2ɛ ≈ 2π in order to get higher sensitivity.

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To generate the high resolution and large dynamic range of the extra time delay, we used both the pumped NLC2 and SBCs. In the experiment, SBCs were firstly used for coarse adjustment to roughly find the linear area (n = 1). Secondly, we changed the power of the pump laser from NLC2 for fine adjustment in order to keep the system correctly in the linear area. Therefore, the large operating range is mainly determined by the SBCs and the resolution of the extra time delay is improved by the NLC2. In our experiment, the NLC2, a kind of optical nonlinear material, was used to generate the ultra-small extra time delay ${\tau }_{{f}_{1}}$, which was pumped by a laser. The power of the pump beam was controlled by the ND filter which comes with angular graduations and is on a rotating axle with the angular resolution of Δa = 1°. As shown in figure 3, the blue spots show the wavelength shift as the power of the pump laser increases in the experiment. The results indicate that in linear area n = 1, the wavelength shift of 3.2 nm is achieved with the power change of only 5 mW and the time delay generated by the power of the pump beam could be 1.87 as/mW. The accuracy of the pump beam power is about 16.5 μW. Based on equation (4), the accuracy of the time delay is about 3.09 × 10⁻², compared with the SBC's resolution of the order of attoseconds. Consequently, the pumped NLC2 can realize a high-resolution time delay about two orders of magnitude higher than that of SBC. The two SBCs can realize a large phase range of about 4π. Hence, the optimized extra time delay of the post-selection can make the high-precision temperature measurement achieve a large dynamic operating range. We have improved the system by two orders of magnitude over that of a traditional phase delay compensator, which makes it easier to modulate the system and is more suitable for higher-precision measurements with adjustable operating range. Additionally, the accuracy of the extra time delay can be further improved by mounting the ND filter on a higher-precision rotating axle, or replacing the NLC2 with other optical nonlinear materials such as transparent liquid crystals [34, 35] and GaAs [36] which have smaller refractive index coefficients.

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In the experiment, we firstly use two SBCs to keep the measurement in the linear area (n = 1) at an initial temperature T₀. Secondly, we changed the power of the pump laser from NLC2 for fine adjustment in order to keep the system in the right linear area. Then, the temperature is changed to generate Δτ and the parameter Δτ can be deduced from the frequency shift based on equation (9). The temperature change ΔT can be obtained according to equation (1). The measured temperature can be calculated as T = T₀ + ΔT. When the initial temperature of NLC1 changes to realize temperature measurement with adjustable operating range, the original extra time delay also changes because the temperature variation can cause changes in its birefringence. The system may then exceed the original linear operating area. Therefore, changing the extra time delay to satisfy ω₀ τ_f ≈ 2nπ + 2ɛ(n = 1) can keep the system in the linear operating area. The higher the resolution of the extra time delay is, the more precise modulation the system can achieve when realizing temperature measurement with an adjustable operating area. In this way, we can achieve a high-precision temperature measurement with an adjustable operating range.

The experiment results are shown in figure 4. In figure 4(a), we realized the temperature measurement between 18 °C and 21 °C. The initial temperature T₀ is 18 °C. The linear operating range can reach to about 0.7 °C at a fixed pump power. By rotating the axle of the ND filter, the power of the pump laser can be attenuated to different values. When the measured temperature gradually increases, the operating range can be achieved by increasing the pumping light power from 0 mW to 60 mW. Compared with the operating range at the pump power of 0 mW, the operating range has been moved right to about 1.7 °C at 60 mW. By adjusting the SBCs, the measurement at the initial temperature of about 28 °C can be roughly in the operating area in figure 4(b). Because the variation of birefringence for NLC1 is associated with the temperature, the linear operating range has changed into about 0.5 °C. The four parts top to bottom show that the linear operating range gradually moves to the right due to the slowly increased temperature, which is achieved by increasing the pump power. As the pump power increases from 0 mW to 60 mW, the operating range can move right to about 0.8 °C which is smaller than that in figure 4(a). It indicates that the operating range becomes smaller as the temperature measured is larger, which is consistent with equation (1). Additionally, the experimental results also indicate that when the operating range moves on the order 10⁻¹ °C of magnitude, the power of the pump laser should change to about 10 mW. When realizing a higher-precision temperature measurement, the change power of the pump laser will be smaller which is easy to realize by using the filter with a higher-precision attenuation or replacing NLC2 with other materials that have smaller refractive index coefficients. Consequently, our scheme has promising applications in the high-precision temperature measurement field. The average sensitivity in figure 4(b) can reach 38 nm/°C. The dynamic temperature range depends on the thermo-optic material of the NLCs sample and the resolution of the temperature measurement could be 8.03 × 10⁻⁷ °C with currently available spectrometer of a spectral resolution Δλ = 0.04 pm. Here, we focus on the theoretical estimation of accuracy, which is the ideal accuracy under extreme conditions of a spectral resolution of the currently available spectrometer [24]. In future work, we will consider the impact of the readout noise from the spectrometer on the precision of temperature measurement.

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We have investigated a high-precision temperature measurement with adjustable operating range based on weak measurement. By conducting a detailed analysis of the noise influence caused by the imperfection of the elements, the trade-off between sensitivity and robustness against the noise should be considered when choosing the proper linear area. When the robustness against noise is satisfactory, the linear area with higher sensitivity can be achieved by introducing a higher-accuracy extra time delay to the postselection. As shown in the proof-of-principle experiments, the optimized extra time delay introduced by the pumped NLC and two SBCs has both large dynamic range and high accuracy of about 3.09 × 10^−2, which is about two orders of magnitude higher than that of SBC. Consequently, the temperature measurement can achieve high resolution and adjustable operating range synchronously based on weak measurement with the optimized extra time delay of the post-selection. The average sensitivity can reach 38 nm/°C at the initial temperature of 28 °C. The dynamic temperature range depends on the thermo-optic material of the NLCs sample and the resolution of the measurement could be 8.03 × 10⁻⁷ °C with a currently available spectrometer.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61631014, 61701302 and 61901258), and the fund of State Key Laboratory of Advanced Optical Communication Systems and Networks.