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Abstract
The noise floor is a vital specification that determines the minimum detectable signal in the phase measurement. However, the noise floor in optical phase measurement conducted via conventional optical interferometry tends to approach the intrinsic limit. In this study, a low noise phase measurement of a fiber optic sensor conducted via weak value amplification is experimentally demonstrated. The system has a flat, wideband frequency response from 0.1 Hz to 10 kHz, as well as adequate linearity. The operating band is wider than the present sensor using the same mechanism. In particular, the system noise floor is measured to be -98 dB at 1 Hz and -155 dB at 1 kHz. The results indicate that the minimum detectable signal can reach as low as 5.6 × 10−6 rad at 1 Hz and 8 × 10−9 rad at 1 kHz. In addition, it is demonstrated that the noise result of the proposed system is two-order of magnitude lower than that of the typical interferometric fiber optic sensors through the comparison experiment. With the characteristic of low-noise, the system is promising in the field of weak signal detection such as underwater acoustic signal detection, seismic wave detection, and mineral resource exploration.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical phase measurements play an important role in detecting physical parameters of interest, such as pressure, temperature, current, and strain [1–10]. In optical phase measurement systems, the noise floor determines the minimum optical phase that can be detected. Hence, it is important to minimize the noise floor. In general, low-noise measurements of the optical phase are carried out via optical interferometry, particularly interferometric fiber optic sensors (IFOS). However, the noise floor in IFOS systems tends to approach the intrinsic limit [11–15]. A recently proposed method for measuring the optical phase, called weak value measurement, has garnered much interest. This method is capable of magnifying the signal and suppressing the noise simultaneously [16–18]. Due to its unique characteristics, weak value measurement can overcome the intrinsic noise floor of optical interferometry. As a result, it has been applied to the measurement of a wide range of quantities, including longitudinal shifts, phases, polarization rotation, optical Faraday differential refraction, birefringence, and the Poynting vector [19–27].
Recently, the weak value measurement technique has been applied to obtain a high level of precision and sensitivity for an extended range of phase estimations in a free-space system, reaching a precision of $10^{-5}$ rad [28]. In addition, weak value amplification (WVA) has been applied to a stress-induced birefringence detection system, which achieved a high resolution of $5 \times 10^{-10}$ [29]. As demonstrated, the resolution based on the WVA technique is higher by two orders of magnitude compared to the conventional method of optical phase interferometry [29]. Luo et al. conducted a hydro pressure detection experiment using a short segment of fiber under water [23]. The system rivaled similar free-space schemes that used bulk optical components, and it had a resolution of $10^{-5}$ rad. Luo et al. also introduced WVA into a fiber optic sensor system to achieve a high-sensitivity and low-frequency fiber optic hydrophone [24]. In the system, a segment of polarization-maintaining (PM) fiber with a length of 0.8 m was wound on a polycarbonate (PC) tube to form a hydrophone probe. The phase sensitivity of this hydrophone was −173 dB (re 1 rad/$\mu$Pa) within the 0.1-50 Hz band, and the noise-equivalent pressure was measured to be 1.3 $\times 10^{-6}$ Pa/$\sqrt {\text {Hz}}$ at 10 Hz. In this report, the operating band was limited to 0.1-50 Hz, as the frequency of sound under water is generally low. Additionally, the noise result was given only in the form of the noise-equivalent pressure, which is more suitable for hydrophone applications. However, the operating band of the sensing system utilizing WVA was far broader than the demonstrated frequency band, indicating that the system can be used for a wider range of applications. In addition, the natural noise floor and noise sources of the system, which significantly influence the system performance and play an important role in practical applications, require further investigation. The influence of the WVA ratio should also be analyzed further.
In this study, we built a polarization-based WVA fiber optic sensor system and demonstrate that the operating band can be extended to as high as 10 kHz. Apart from its high-performance detection capability, the noise floor of the system is extremely low, and it penetrates the intensity noise limit via a large WVA ratio. We propose that the noise floor of the fiber optic sensor utilizing WVA is primarily determined by the polarization-induced intensity noise and the intensity noise of the laser.
The remainder of this paper is organized as follows. In Section 2, the mechanism of the polarization-based WVA fiber optic sensor is introduced, and the linear region of the system’s operation is determined and simulated. In Section 3, an all-fiber polarization-based WVA fiber optic sensor is introduced. In Section 4, the experimental results of the time-domain response, frequency response, and linearity are presented, and the scheme designed in this study is verified to have the properties of a high-performance fiber optic sensor. Furthermore, the noise floor of the system is measured to be low. There are two reasons for this: the low level of the noise sources and the magnifying role of the WVA. In Section 5, we conclude that the system is feasible for applications requiring low-noise and wideband detection of phases with weak signals.
2. Methods
A system and a continuous pointer, which are the initial states of $\left |\psi _{i}\right \rangle$ and $|\phi _i\rangle =\int d x \phi _i(x)|x\rangle$ (where $|\phi _i\rangle$ is the input measurement meter state and $|\phi _i(x)|^{2}$ is a probability density function for variable $x$), are generally included in the WVA scheme. The system and the continuous pointer are coupled by a weak interaction $\hat {U}(g)=\exp [-i g \hat {A} \hat {x}]$, where $\hat {A}$ is the weak value of the observable system and the parameter $g$ denotes the coupling strength [30]. The weak interaction is fundamental to the weak measurement between the measurement device and the system. There are three stages in the weak measurement: 1) the pre-selection is prepared in an initial state $\left |\psi _{i}\right \rangle$, 2) an observable weak coupling $\hat {A}$ is introduced by a discrete detector, and 3) the post-selection is prepared in the final state $\left |\psi _{f}\right \rangle$. The observable weak value evolves significantly, whereas the initial state $\left |\psi _{f}\right \rangle$ is almost orthogonal to the final state $\left |\psi _{i}\right \rangle$. When the initial and the final states deviate from the nearly orthogonal states, the linear WVA is still valid, but the observable weak value $\hat {A}$ decreases sharply and the noise of the system increases.
The polarization-based WVA fiber optic sensor is shown in Fig. 1. The horizontal and vertical components ($|H\rangle$ and $|V\rangle$, respectively) of the optical polarization state are regarded as eigenstates.
Fig. 1. Basic mechanism of the polarization-based WVA fiber optic sensor. (A) The three stages in the weak signal measurement: pre-selection, weak interaction, and post-selection. (B) Evolution of the polarization state along the optical path depicted in (A). The input polarization state is along the slow axis of the leading PM fiber. The in-line polarizers work along the slow axis and block with the fast axis. The x-axis is the fast axis of the PM fiber, and the y-axis is the slow axis of the PM fiber with panda eyes, which are represented by two dark circles. (a) Polarization state of the leading fiber after the first polarizer. (b) Polarization state after $45^{\circ }$ is spliced at the pre-selection stage. (c) Polarization state after $90^{\circ }$ is spliced at the weak interaction stage. (d) Polarization state after $45^{\circ }+\phi$ is spliced at the post-selection stage.
The pre-selection state can be written as
The pre-selection state represents linearly polarized light at $45^{\circ }$. The frequency of light $\omega$ was chosen as the input meter variable, the input measurement meter state of which is in the form The probability density function of the laser frequency can be regarded as a Gaussian distribution with $f(\omega ) = \left (\pi \sigma ^{2}\right )^{-1 / 2} \exp \left [-\left (\omega -\omega _{0}\right )^{2} / 2 \sigma ^{2}\right ]$. The Stokes polarization operator $\hat {A}=|H\rangle \langle H|-| \mathrm {V}\rangle \langle \mathrm {V}|$ is a weak coupling observable in the system. The Hamiltonian interaction $H=g(t) \hat {A} \otimes \hat {\omega }$, where $g(t) \ll 1$ is the instantaneous coupling strength in the WVA system, is coupled to the system and input meter variable. A small optical delay $\tau$ is introduced by the weak interaction between the two orthogonal polarization states. Thus, the initial joint state for the system and the pointer variable evolves intoFig. 2. ICR as a function of phase shift $\alpha$ for different post angles $\phi$. (a) ICR of the WVA scheme with symmetric post-selected angles; (b) ICR of the WVA scheme of optical fiber. The linear region is limited to $|\alpha | / 5 \ll |\phi |$ and is valid for different post-selected angles $\phi$. A smaller $\phi$ leads to a higher WVA ratio (the slope of the linear region) and a smaller width for the linear region.
3. Experimental setup
The experimental setup of the polarization-based WVA fiber optic sensor is shown in Fig. 3. A PM semiconductor laser with a central wavelength of 1550 nm and a bandwidth of 1.6 kHz was selected as the light source. The light from the laser, which had an output power of $\simeq$ 14 mW, passed through the first integrated inline polarizer and was converted into linearly polarized light. The pre-selected intensity light $I_{0}$ was measured through the tap output (380 $\mu W$) of a PM fiber optic coupler with a coupling ratio of $95\%$. The other output of the coupler went through the $45^{\circ }$ spliced point, which was regarded as the pre-selection state of the system. Then the light propagated as $45^{\circ }$ linearly polarized light (pre-selection) into the sensing fiber PMF2 at an angle of $45^{\circ }$ relative to the slow axis. Because the propagation constants $\beta _x$ and $\beta _y$ were different for the two orthogonal mode components corresponding to the fast and slow axes, the phase difference between the two orthogonal mode components was linearly proportional to the propagation length. Subsequently, the polarized light transmitted along the slow axis of the PM fiber PMF2 propagated along the fast axis of the PM fiber PMF3 if PMF2 and PMF3 were orthogonally spliced. In the system, PMF2 and PMF3 had the same length of 1.6 m. The phase difference should be zero if no ambient signal was detected, which was regarded as a balanced state. In the balanced state, the light should be right $45^{\circ }$ linearly polarized just before the $45^{\circ }+\phi$ spliced point, where $\phi$ is a small post angle and can be measured using Eq. (9). This progression is referred to as the weak interaction. The polarization direction of the light beam was nearly along the fast axis of the PM fiber PMF4 after the third spliced point. Because the polarization analyzer was blocked with the fast axis, the output of the polarization analyzer was weak and was received by the photodiode. Post-selection was achieved by the second $45^{\circ }+\phi$ spliced point and by the polarization analyzer. The amplification coefficients could be adjusted by changing the post angle $\phi$.
When an ambient signal was applied to the sensing fiber PMF2, it was converted into a phase shift between the horizontal polarization $|H\rangle$ and the vertical polarization $|V\rangle$. In this study, the ambient signal was detected by a piezoelectric transducer (PZT), on which the sensing fiber was wound with glue. The phase shift in the sensing fiber due to the change in radius of the PZT is given by
Fig. 3. Experimental setup for all-fiber WVA system. LD: 1550-nm single-frequency laser with a bandwidth of 1.6 kHz (supplier: RIO Inc., S/N: 801818); Polarizer and analyzer: inline polarizer (supplier: Accelink Inc., PN: ILP15500001128S-0702, extinction ratio: 28 dB); PMF1-PMF4: PM fiber (supplier: YOFC, PN: PM1550-125-18/250); OC: PM fiber optic couplers (supplier: YOFC, PN: FPMC-55-1-A-P-1-G-0-2-1-2, coupling ratio: 3 dB); Meter: optical power meter (supplier: Accelink Inc., PN:PMSII-B); PZT: piezoelectric transducer (radius: 20 mm); PD: photodiode (supplier: FEMTO, PN:OE-200-IN2); DAQ: data acquisition system (supplier: National Instruments, PN: 151944B-04L); the green cross dots represent the spliced point: PMF1 and PMF2 were $45^{\circ }$ spliced together; PMF2 and PMF3 were $90^{\circ }$ spliced and their lengths were equal; PMF3 and PMF4 were $45^{\circ }+\phi$ spliced. The PM fiber wound on a PZT in the green background represents the sensing section. The input signal could be loaded using a signal generator.
Suppose that the PZT is driven by a sinusoidal signal $V= V_0 \sin (\omega _s t)$, where $V_0$ is the amplitude of the driving voltage, $\omega _s$ is the frequency, and $t$ is the time. Then the change in the radius of the PZT is given by:
where $\delta _{R}\left (\omega _{s}\right )$ is the amplitude response of $\omega _{s}$, and it is about 8 nm/V across the band ranging from 0.1 Hz to 10 kHz for the PZT that is used. Then the phase shift introduced by the change in length of the fiber turns around the PZT, and the change in the fiber’s refractive index is given by [31]To avoid the effect of twisting, PMF2 needed to be wound on the PZT along a certain principle axis to ensure that the light remained almost linearly polarized after passing through PMF2 and PMF3. In this way, the detection of the phase shift, which was introduced by the variation in the length of the sensing fiber, was converted into the detection of the variation in the post-selected light intensity according to the mechanism of WVA.
4. Results and discussion
4.1 Performance of weak signal detection
First, the time-domain response of the system was tested. In this experiment, we recorded the output intensity of the system to determine the phase variation within the fiber that was induced by the PZT. Standard sinusoidal signals with various frequencies were used to test the time-domain response of the system. The sampling rate was 1 MS/s, the conversion gain of the photodiode was $10^5$ V/W at 1550 nm, and the bandwidth was 400 kHz. The light intensity of the pre-selection was $I_0$ = 7.22 mW, and the light intensity of post-selection without excitation was $I_{1}$ = 23.1 $\mu$W. Then, the post angle was determined to be $\phi = 0.0566$ rad and the WVA ratio was $\operatorname {Im}\left (A_{\omega }\right ) = 17.63$ according to Eqs. (9) and (6), respectively. The experimental results of the time-domain response are presented in Fig. 4. Meanwhile, the theoretical phase amplitude is calculated according to Eq. (13), which is $3.2\times 10^{-3}$ rad. It is observed that the experimental results are almost consistent with the theoretical result. The phase signal can be retrieved through the system with low distortion at the selected frequencies, indicating that the upper limit of the operating band can be extended to as high as 10 kHz. In addition, in the ultra-low band, the performance of this system was better than that of the IFOS. This superior performance was a result of the low noise of the system, which is discussed in the following section. In addition to the above mentioned advantages over the IFOS system, the WVA system also requires no complicated demodulation method.
Fig. 4. Time-domain response of the WVA system at the following frequencies: (a) 0.1 Hz, (b) 1 Hz, (c) 100 Hz, and (d) 10 kHz.
Second, the frequency response of the system was tested. The driving voltage of the signal detected by the PZT was set to 4 Vpp, and the frequency was scanned from 0.1 Hz to 10 kHz. The test was repeated 100 times to obtain the response at a single signal frequency, and the experimental response was an average of the 100 tests. The amplitude of the phase variation was confined within the linear region obtained from the linear condition $|\alpha | / 5 \ll |\phi |$, where $\phi$ was measured to be 0.0566 rad in the experiment. The experimental results are shown in Fig. 5(a). The phase shift was about −49.5 dB re 1 rad across the testing band, and the variation was 1.35 dB. In this sense, the frequency response was reasonably flat over a wide band, which is applicable in a wide range of fields.
Fig. 5. (a) Frequency response for the WVA system from 0.1 Hz to 10 kHz at a fixed voltage of 4 Vpp for the PZT. (b) Variation in the phase shift (dB re 1 rad) in the fiber optic as a function of the driving voltage (dB re 1 Vpp) for the PZT. The red points represent experimental data. The solid line in (b) shows the expected phase shift based on a linear extrapolation of accurate measurements of the phase shift at the voltages higher than −10 dB. WVA allows a small phase shift to be measured; the WVA ratio $\operatorname {Im}\left (A_{\omega }\right )$ was 17.63.
The linearity of the system was also tested. In the test, the signal frequency was fixed at 1 Hz, while the driving voltage was changed from 20 to 10 Vpp. The results are presented in Fig. 5(b), which indicates a linearity that is consistent with Eq. (13). The weakest phase signal that could be detected reached as low as −96.73 dB re 1 rad, which corresponded to a driving voltage of 20 mVpp. This indicates that the minimum detectable phase of the system was on the order of $10^{-5}$ rad at 1 Hz, accounting for the requirement of a signal-to-noise ratio of at least 10 dB for a detection.
4.2 Noise characterization
The minimum detectable phase signal was limited by the noise floor of the polarization-based WVA fiber optic sensor. By amplifying a small phase shift in the sensing fiber with a large amplification factor, the system demonstrated that it possessed an ultra-low noise floor.
To confirm the minimum phase signal that could be detected by the system, we decreased the driving voltage as much as possible until its amplitude merged into the noise floor. The frequency of interest was set at 1 Hz. As is shown in Fig. 6, the power spectrum density (PSD) of the weakest signal reached as low as −88.89 dB re 1 rad/$\sqrt {\text {Hz}}$ when the driving voltage was 20 mVpp, which was $1.6 \times 10^{-5}$ rad in the frequency spectrum. In addition, the PSD of the system’s noise floor, which determines the minimum detectable signal of the system, was derived.
Furthermore, to highlight the ultra-low noise characteristic of the system, we constructed an IFOS that was tested by the $3\times 3$ demodulation method. The noise floor of the IFOS, which is presented in Fig. 6, was used as the contrast. The result was the average of 500 tests. According to the result, the noise floor of the IFOS system was approximately −55 dB at 1 Hz (re 1 $\text {rad}/ \sqrt {\text {Hz}}$) and −106 dB at 1 kHz. In contrast, the noise of the WVA system was approximately −98 dB at 1 Hz and −155 dB at 1 kHz. This indicates that the minimum detectable signal of the polarization-based WVA fiber optic sensor was $5.6 \times 10^{-6}$ rad at 1 Hz and $8 \times 10^{-9}$ rad at 1kHz. Compared to the IFOS system, the noise floor of the WVA system was approximately 43 dB lower at 1 Hz and 50 dB lower at 1 kHz. Therefore, the WVA system could detect a weak phase signal with an amplitude 100 times lower than the IFOS system could.
Fig. 6. PSD of the minimum detectable phase shift (blue curve), noise floor of WVA system (red curve), and noise floor of the interferometric fiber optic sensor (yellow curve) with a $3\times 3$ demodulation algorithm as a function of frequency. The blue curve in the violet dashed circle is the minimum signal at 1 Hz that is detected in the experiment.
4.3 Influence of noise source and WVA ratio on ultra-low noise
The polarization-induced intensity noise and intensity noise of the laser were found to be the main components contributing to the noise floor. To verify this, we built another experimental setup to measure the polarization-induced intensity noise. In the setup, the polarizer and analyzer were almost perpendicularly spliced once the balanced polarization interferometry was removed. In this way, the polarization-induced noise was measured; the result is shown in Fig. 7. In addition, the laser intensity noise was measured by directly detecting the output of the light source with the photodiode. The PSD of the noise source is presented in Fig. 7. All curves are the averages of 10 tests.
Fig. 7. Polarization-induced noise (blue curve), intensity noise of the laser (yellow curve), and noise floor of the WVA system (red curve) as a function of frequency. The units of the noise floor of the system, shown on the left axis, are dB re 1 rad/$\sqrt {\text {Hz}}$. The units of the polarization-induced noise and laser intensity noise, shown on the right axis, are dB re 1 V/$\sqrt {\text {Hz}}$.
The result of the polarization-induced intensity noise floor included the intensity noise of the light source. This was because the polarization-induced noise curve was acquired through the direct detection of the optical intensity. A comparison of the results of the polarization-induced intensity noise and the intensity noise of the laser is shown in Fig. 7. The polarization-induced noise, which had the typical characteristics of 1/f noise, was far higher than the intensity noise of the light source within the band from 0.1 Hz to approximately 1 kHz. This indicates that polarization-induced noise dominated within the low-frequency band. However, at frequencies higher than 1 kHz, the level of polarization-induced intensity noise tended to be lower than that of the laser intensity noise. Therefore, the laser intensity noise played a leading role in the high-frequency band.
Next, the relationship between the polarization-induced noise floor and the noise of the WVA system was considered. Figure 7 indicates that the curve of the polarization-induced noise was similar to that of the WVA system, which suggests that the source of the system noise was the polarization-induced intensity noise. The relationship between the polarization-induced intensity noise and the noise floor of the system can be expressed by the Fourier transform of Eq. (10) as
The WVA ratio, which determines the decline of the noise floor from the total intensity noise, is crucial for the system. To verify the influence of the WVA ratio on the noise floor, we established another polarization-based WVA fiber optic sensor with a post-angle of $\phi = 0.3582$ rad. The measured noise floors of the two systems are shown in Fig. 8. According to Eq. (6), the WVA ratio was 2.67 for the $\phi = 0.3582$ rad case, while it was 17.63 for the $\phi = 0.0566$ rad case. According to Eq. (16), the difference between the noise floors in the two WVA ratio cases could be predicted to be $10 \lg (2.6712/16.73)^2=-16.39$ dB. In Fig. 8, the noise floor with the high WVA ratio was approximately 16 dB lower than that with the low WVA ratio, which agrees well with the theoretical prediction. Therefore, the post-angle should be set as small as possible. However, considering the linear region of the WVA, the maximum signal that can be detected is limited by the post-angle. Therefore, the post-angle should be assigned by considering both requirements in the design.
The WVA system possesses a low-noise floor for two main reasons. First, the WVA system utilizes polarization interference to obtain phase information through one-path balanced polarization interferometry. Compared to the noise floor of the traditional IFOS, which is primarily limited by the phase and frequency noise of the laser, the noise floor of the WVA system is primarily determined by the polarization-induced intensity noise and the intensity noise of the laser. The contribution of the two noise sources is far lower than the phase and frequency noise of the laser, thereby enabling the WVA system to obtain a noise floor that is far lower than that of the IFOS. Second, the WVA method converts the detection of phase variations into the measurement of the ICR with a large WVA ratio. Therefore, the noise floor of the WVA system can penetrate the intensity noise limit to achieve an ultra-low noise floor for weak signal detection.
5. Conclusions
In this study, a polarization-based WVA fiber optic sensor was experimentally demonstrated to have a flat and wideband frequency response from 0.1 Hz to 10 kHz. The system also exhibited adequate linearity in this range. The system noise level was measured to be −98 dB at 1 Hz and −155 dB at 1 kHz, which indicates that the minimum detectable signal amplitude was as low as $5.6 \times 10^{-6}$ rad at 1 Hz and $8 \times 10^{-9}$ rad at 1 kHz. By comparison, the system noise floor was approximately two orders of magnitude lower than that of the IFOS. Theoretical and experimental investigations of the noise sources revealed that the polarization-induced intensity noise was the primary source of low-frequency noise output, whereas the laser intensity noise contributed significantly to the high-frequency noise floor component. The low levels of the two key noise sources were demonstrated to be the basis for the low-noise floor of the WVA system. The system mechanism determined that the noise floor was converted from the intensity noise sources, and that the WVA ratio was the conversion coefficient. Therefore, the noise floor can be further decreased owing the intensity noise via a large WVA ratio. Because of the WVA ratio, the WVA system can measure the small phase shifts caused by the physical parameters of interest. However, the upper measurement range should be limited to the linear range to avoid the non-linearity, which should be noted in the application. The technique is suited to numerous applications such as underwater acoustic signal detection, seismic wave detection, and mineral resource exploration.
Funding
National Natural Science Foundation of China (52101391, 61904204); The Research Plan Program of the National University of Defense Technology (ZK19-36).
Disclosures
The authors declare no conflicts of interest.
Data Availability
The 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.
References
1. C. K. Kirkendall and A. Dandridge, “Overview of high performance fibre-optic sensing,” J. Phys. D: Appl. Phys. 37(18), R197–R216 (2004). [CrossRef]
2. D. Xu, F. Yang, D. Chen, F. Wei, H. Cai, Z. Fang, and R. Qu, “Laser phase and frequency noise measurement by michelson interferometer composed of a 3× 3 optical fiber coupler,” Opt. Express 23(17), 22386–22393 (2015). [CrossRef]
3. Z. Qu, S. Guo, C. Hou, J. Yang, and L. Yuan, “Real-time self-calibration pgc-arctan demodulation algorithm in fiber-optic interferometric sensors,” Opt. Express 27(16), 23593–23609 (2019). [CrossRef]
4. L. Wang, J. He, F. Li, and Y. Liu, “Ultra low frequency phase generated carrier demodulation technique for fiber sensors,” Chin. J. Laser 38(4), 0405001 (2011). [CrossRef]
5. F. Shang, Y. Yu, L. Ma, Y. Qi, and J. Song, “Analysis on leading-fiber-induced doppler noise in interferometric fbg sensor arrays using polarization switching and pgc hybrid processing method,” Opt. Express 29(11), 16118–16134 (2021). [CrossRef]
6. M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, “Adaptive homodyne measurement of optical phase,” Phys. Rev. Lett. 89(13), 133602 (2002). [CrossRef]
7. Y. Dong, P. Hu, M. Ran, Z. Le, H. Fu, H. Yang, and R. Yang, “Phase modulation depth setting technique of a phase-generated-carrier under aoim in fiber-optic interferometer with laser frequency modulation,” Opt. Express 28(21), 31700–31713 (2020). [CrossRef]
8. T. Becker, O. Ziemann, R. Engelbrecht, and B. Schmauss, “Optical strain measurement with step-index polymer optical fiber based on the phase measurement of an intensity-modulated signal,” Sensors 18(7), 2319 (2018). [CrossRef]
9. A. J. Rogers, “Distributed optical-fibre sensors for the measurement of pressure, strain and temperature,” Phys. Rep. 169(2), 99–143 (1988). [CrossRef]
10. A. Gosteva, M. Haiml, and U. Keller, “Measurement of the spectrally-resolved absolute phase difference between orthogonal optical modes using a nonlinear beat signal,” Opt. Express 13(22), 8913–8920 (2005). [CrossRef]
11. C. Li, X. Xu, J. Tang, J. Xu, and G. Guo, “Ultrasensitive phase estimation with white light,” Phys. Rev. A 83(4), 044102 (2011). [CrossRef]
12. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96(1), 010401 (2006). [CrossRef]
13. W. Ni, “Quantum-mechanical noise in an interferometer: intrinsic uncertainty vs measurement uncertainty,” in International Quantum Electronics Conference, (Optica Publishing Group, 1986) p. THGG6.
14. B. Yurke, S. L. McCall, and J. R. Klauder, “Su (2) and su (1, 1) interferometers,” Phys. Rev. A 33(6), 4033–4054 (1986). [CrossRef]
15. Y. Yang, Z. Luo, Z. Wang, Y. Zhang, T. Chang, M. Yu, L. Cheng, C. Wu, and H. Cui, “High-performance fiber optic interferometric hydrophone based on push–pull structure,” IEEE Trans. Instrum. Meas. 70, 1–13 (2021). [CrossRef]
16. Y. Kedem, “Using technical noise to increase the signal-to-noise ratio of measurements via imaginary weak values,” Phys. Rev. A 85(6), 060102 (2012). [CrossRef]
17. A. N. Jordan, J. Martínez-Rincón, and J. C. Howell, “Technical advantages for weak-value amplification: When less is more,” Phys. Rev. X 4(1), 011031 (2014). [CrossRef]
18. A. Nishizawa, K. Nakamura, and M.-K. Fujimoto, “Weak-value amplification in a shot-noise-limited interferometer,” Phys. Rev. A 85(6), 062108 (2012). [CrossRef]
19. N. Brunner and C. Simon, “Measuring small longitudinal phase shifts: weak measurements or standard interferometry?” Phys. Rev. Lett. 105(1), 010405 (2010). [CrossRef]
20. B. de Lima Bernardo, S. Azevedo, and A. Rosas, “Ultrasmall polarization rotation measurements via weak value amplification,” Phys. Lett. A 378(30-31), 2029–2033 (2014). [CrossRef]
21. A. Parks and S. Spence, “Weak value amplification of an optical faraday differential refraction effect,” Appl. Opt. 51(16), 3364–3369 (2012). [CrossRef]
22. J. Dressel, K. Y. Bliokh, and F. Nori, “Classical field approach to quantum weak measurements,” Phys. Rev. Lett. 112(11), 110407 (2014). [CrossRef]
23. Z. Luo, H. Wu, L. Xu, T. Chang, P. Wu, C. Du, and H. Cui, “Phase measurement of optical fiber via weak-value amplification,” IEEE Sens. J. 19(16), 6742–6747 (2019). [CrossRef]
24. Z. Luo, Y. Yang, Z. Wang, M. Yu, C. Wu, T. Chang, P. Wu, and H. L. Cui, “Low-frequency fiber optic hydrophone based on weak value amplification,” Opt. Express 28(18), 25935–25948 (2020). [CrossRef]
25. Y. Xu, C. Zhou, L. Shi, X. Zhang, T. Guan, C. Guo, Z. Li, X. Xing, Y. Ji, L. Liu, and Y. He, “Imaging sensor for the detection of the flow battery via weak value amplification,” Anal. Chem. 93(38), 12914–12920 (2021). [CrossRef]
26. L. J. Salazar-Serrano, D. Barrera, W. Amaya, S. Sales, V. Pruneri, J. Capmany, and J. P. Torres, “Enhancement of the sensitivity of a temperature sensor based on fiber bragg gratings via weak value amplification,” Opt. Lett. 40(17), 3962–3965 (2015). [CrossRef]
27. L. Luo, X. Qiu, L. Xie, X. Liu, Z. Li, Z. Zhang, and J. Du, “Precision improvement of surface plasmon resonance sensors based on weak-value amplification,” Opt. Express 25(18), 21107–21114 (2017). [CrossRef]
28. Z. Li, J. Qiu, L. Xie, L. Luo, X. Liu, Z. Zhang, C. Ren, and J. Du, “Retaining high precision and sensitivity for an extended range of phase estimation via modulated weak measurement,” Appl. Phys. Lett. 113(19), 191103 (2018). [CrossRef]
29. Y. Wang, S. Chen, S. Wen, and H. Luo, “Realization of ultra-small stress birefringence detection with weak-value amplification technique,” Appl. Phys. Lett. 118(16), 161104 (2021). [CrossRef]
30. F. Li, J. Huang, and G. Zeng, “Adaptive weak-value amplification with adjustable postselection,” Phys. Rev. A 96(3), 032112 (2017). [CrossRef]
31. W. Ha, S. Lee, Y. Jung, J. K. Kim, and K. Oh, “Acousto-optic control of speckle contrast in multimode fibers with a cylindrical piezoelectric transducer oscillating in the radial direction,” Opt. Express 17(20), 17536–17546 (2009). [CrossRef]
