1. Introduction

Exceptional points (EPs), the spectral singularities within non-Hermitian systems [1], distinguish themselves from Hermitian degenerate points (DPs). At EPs, both eigenvalues and eigenvectors merge into degeneracy simultaneously [2], resulting in a reduction of the overall dimensionality [3]. EPs have garnered substantial research attention in the fields of optics, acoustics, quantum mechanics, and beyond, leading to a myriad of intriguing phenomena. These include loss-induced transparency [4], coherent full absorption [5], power oscillation [6], topological chirality [7], and laser control [8,9]. Additionally, the pivotal eigenvalue bifurcation characteristic of EPs can significantly amplify the non-intrinsic perturbation effect [10,11]. In theory, the sensitivity of the frequency response approaches infinity as perturbation diminishes, holding paramount significance in sensing applications [12–14]. L. Yang et al. experimentally verified a square root dependence between the frequency splitting induced by nano-scale objects and the disturbance intensity, stemming from the square root topology around the second-order EP [15,16]. Kerry Vahala et al. achieved continuous control near the EP using an optical microcavity, resulting in a four-fold enhancement of the Sagnac effect [17]. Meanwhile, CW Qiu et al. observed sensitivity enhancement surpassing traditional limits using a wireless PT (Parity-time)-symmetric LC (Inductor-capacitor) microsensor implanted in the abdomen of rats [18]. These experiments have convincingly demonstrated the potential for sensing enhancement in the vicinity of EPs.

However, it is often challenging to apply this simple and efficient method to practical sensing because any unavoidable disturbance, defects in all manufacturing processes, and drift of the operating point will keep the sensor system away from the EP, thereby reducing high sensitivity [19–24]. Numerical simulations by Christian Wolff et al. [25,26] demonstrate that the eigenstates exhibit exponential divergence due to noise. To detect small resonance splits and maintain a sufficiently long operating time near the exceptional point, it is necessary to meticulously design the feedback system. WJ Wan et al. proposed an approach for achieving precise WGM frequency tuning using probes [27]. Furthermore, R. El-Ganainy et al. introduced the notion of hypersurfaces of exceptional points, referred to as ‘exceptional surfaces’ (ES) [28], this method further enhances robustness while maintaining the high sensitivity of the EPs system. GQ Qin et al. first reported the experimental observation of an ES-based sensor, which provides an important solution to this problem [29]. How to effectively introduce a high-speed feedback controller to design the control system to compensate for the stability of the operating point and keep the system running long enough to detect weak signals at room temperature is a major problem in the practical application of EP-based sensors [30–33]. However, due to the coupling strength of the sensing target, the practical versatility of this architecture remains to be developed.

Inspired by these remarkable developments and the growing demand for applications harnessing the power of EP enhancement, this work introduces a novel sensing system based on the EP-enhanced effect referred to EPTDS. The system employs a cascaded frequency tracking loop with two different response speeds to individually compensate for frequency errors under varying bandwidths, ensuring that the second-order PT-symmetric non-Hermitian photonic structure remains consistently in proximity to the EP. The system achieves a lock-in time of 10 ms, with a frequency error margin narrower than 10 kHz and an error range of less than 3% before locking. Experimental tests under rotational conditions confirm a substantial increase in the system's non-linear response, with a sensitivity improvement of 121-fold at a rotational speed of 0.05 ^°/s, accompanied by a wide-ranging frequency detection capability spanning three orders of magnitude. In an ambient temperature environment, the system exhibits an Allan deviation-based Bias Instability (BI) of 0.036 ^°/h. Crucially, EPTDS not only enables enhanced angular velocity detection at the EP but also accommodates various other sensing applications capable of inducing frequency shifts [34–37]. The outcomes of this study are poised to contribute significantly to the development of sensing applications in the vicinity of EPs and offer valuable insights into the state control of non-Hermitian systems.

2. System scheme design and analysis

Due to the nonlinear effects of EP in non-Hermitian systems, in this work, we first designed a non-Hermitian optical sensor module as a sensitive unit based on a high-Q fiber ring resonator (FRR). Two FRRs with identical parameters, including cavity length (L) and radius (R), are coupled together and connected to different fiber couplers. The first resonator (FRR₁) is an active resonator containing Er³⁺ doped fiber, capable of gaining through 980 nm laser pumping. The second resonator (FRR₂) is a passive (without gain medium) resonator wrapped around a PZT (lead zirconate titanate) exhibiting loss, and its loss is balanced to the gain in FRR₁ to achieve PT symmetry conditions. To achieve controllable coupling between the resonators, we employ a high-precision tunable coupler, adjusting the distance precisely to control the coupling strength. Optical gain in FRR₁ is provided by the emission of Er³⁺ ions within the 1550 nm band, pumped by the 980 nm laser. As shown in Fig. 1, the transmission spectrum of FRR₂ near the 1550 nm wavelength is obtained with a Lorentzian fit, yielding a half-width at half-maximum (FWHM) of approximately 0.95 MHz. By adjusting the pump field power, FRR₁ with an FWHM of about 0.98 MHz is achieved while maintaining nearly identical cavity quality factors, approximately 2 × 10⁸. Figure 1(b) illustrates the evolution of the transmission spectrum of FRR₁ under different optical gain. The transmission spectrum of FRR₁ transforms from a downward resonance valley to an upward resonance peak. The system reaches the EP when the resonant frequencies of the two ring cavities are equal.

 

Fig. 1. The transmission spectra schematic of two optical FRRs. (a) The transmission spectra of the active resonator FRR₁ and the passive resonator FRR₂. (b) The evolution of the transmission spectrum of FRR1 under different optical gain.

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As shown in Fig. 2(a), we configured the EPTDS and experimentally tested the effective sensing of nonlinear enhancement. The light emitted by a 1550 nm narrow linewidth tunable external-cavity semiconductor laser passes through an isolator before entering the intensity modulator (IM). After intensity modulation, it is input into an integrated optical phase modulator. The optical field is synchronously modulated by a high-frequency sinusoidal signal with a frequency of ω_m, and then coupled into the PT-symmetric system. After the sensing signal is enhanced and output in the non-Hermitian optical system, it is converted into an electrical signal at the photodetector and then detected by the electronic readout device. The photodetector simultaneously monitors changes in the input optical power and provides continuous stabilization of the optical power level by providing feedback to the IM. The two resonant frequencies of the optical sensor's resonators are equal, and the coupling strength κ_c and gain/loss γ_1,2 satisfy the PT symmetry-breaking critical condition. This controls the system near the EP. At this EP, the sensing signal simultaneously affects the frequency difference and linewidth of both supermodes, which is reflected in the linewidth variation of the transmission peak. By utilizing the synchronous characteristics of multiple synchronized demodulation curves and the transmission waveform, the information on the spectral linewidth is extracted and, ultimately, the sensing signal is detected.

 

Fig. 2. Schematic diagram of the EPTDS scheme and the dual-loop frequency tracking scheme based on coherent demodulation. (a) illustrates the optical circuit principle of the experiment, where the output light from the external cavity diode laser (ECDL) enters the resonator after intensity and phase modulation. The PT symmetric module serves as a non-Hermitian optical sensor, and the closed-loop frequency error signal near EP is introduced to compensate the frequency tracking loop in the cascaded control system. Here, the abbreviations are defined as follows: ISO: Optical Isolator, IM: Intensity Modulator, IOPM: Integrated Optical Phase Modulator, WDM: Wave Division Multiplexer, EDF: Erbium-Doped Fiber, PD: Photodetector, FPGA: Field Programmable Gate Array, PZT: Lead Zirconate Titanate, DDS: Direct Digital Synthesizer, LA: Linear Amplifier. (b) and (c) outline the control logic for system stabilization near the EP. Two distinct frequency tracking control loops are utilized to precisely control both the laser frequency and the resonator frequency variation.

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However, unlike traditional methods of achieving EP, we do not slowly modulate the temperature to control the resonant frequencies of the cavities, nor do we need to continuously adjust the gain, loss, or coupling points of the non-Hermitian optical system in a constant-temperature environment, as these approaches can reduce the flexibility of the sensing system. Instead, we control the EP by tuning the resonant frequencies. As the sensing signals can cause variations in the resonant frequencies, moving the system away from the EP, we continuously monitor and calibrate the system to ensure that the resonant frequency difference remains adjusted in real-time, staying close to zero for higher frequency response. Other parameters are fixed in the control loop after the initial EP tuning calibration and are periodically recalibrated. The calibration methods include adjusting the 980 nm pump power intensity and coupling strength. Due to thermal drift in the resonant cavity, the system's gain typically changes by a few Hz within an hour, which is negligible in our experiments.

To achieve long-term control of EP, we have introduced the CDCC (combinatorial demodulation-based dual-loop cascaded control) technique to track fluctuations in real-time. The system's frequency tracking primarily consists of high and low-frequency control loops. The laser frequency control loop compensates for low-frequency errors over a wide range, while the resonator frequency control loop rapidly establishes stability in the optical system's proximity to the EP, concurrently detecting real-time frequency shifts caused by sensing signals. Firstly, it is essential to ensure that the output frequency of the laser remains close to the resonant frequencies of the two cavities. This is achieved by adjusting the laser's output frequency to track the low-frequency and wide-range variations of the resonant frequency, ensuring significant frequency tracking even in the presence of environmental noise. Secondly, we need to ensure the synchronization of the resonant frequencies between the two cavities. For rapid frequency tuning between the cavities, we utilize the resonators wound around PZT elements. The PZT frequency controller has a smaller tunable range compared to the laser, allowing real-time tracking of rapid resonant frequency fluctuations. Its tuning speed can exceed 100 MHz/s and is dynamically adjusted by the control circuit. In the CDCC technique, a dual-tracking system is established to track the two closed-loop error signals simultaneously.

As shown in Fig. 2(b), within the optical output loop, we generated the light signal ${E_0}{e^{i{\omega _L}t}}$ using an ECDL with a natural bandwidth of 50 kHz and a central wavelength of 1550 nm. The ECDL was driven with a triangular wave signal at a frequency of ω_t= 50 Hz, with periodic fluctuations in frequency not exceeding one free spectral range (FSR). The laser frequency control loop continuously monitored the transmission spectrum of the non-Hermitian optical system in real-time, determining the relative variation between the transmission peak frequency and the center frequency of the spectral domain fluctuations. To retain low-frequency error information while removing high-frequency random errors, a low-pass filter (LPF) was employed. The LPF had a bandwidth set at 1 kHz. With the assistance of a first-order proportional-integral (PI) controller, the feedback signal was fed into a digital-to-analog converter (DAC) and the driving circuitry. Ultimately, this feedback loop regulated the central frequency of the output light, enabling real-time tracking of the slow drift in the resonant frequency. The laser frequency control loop tuning range is consistent with the laser's output wavelength range, exceeding 15 GHz. Simultaneously, this frequency tracking was also incorporated into the system's output signal to compensate for errors introduced by low-frequency noise.

In the high-frequency resonant frequency tracking loop (Fig. 2(c)), a high-frequency sinusoidal wave is modulated onto the ECDL's output optical signal using IOPM for carrier suppression (∼50 dB). The modulation amplitude is set as a fixed multiple of the half-wave voltage. To improve CDCC method accuracy, optimize demodulation gain, the sinusoidal wave frequency is modulated around the optimal frequency (<100 kHz), and optical frequency is adjusted in real-time by the DDS module based on transmission spectrum linewidth, enhancing the output signal-to-noise ratio. The transmission spectrum with non-Hermitian optical sensing information is converted by PD and analog-to-digital converter (ADC) to a digital signal, synchronously demodulated by sinusoidal waves of different frequencies. After tuning by the lock-in amplifier, the frequency difference of the two ring cavities within the harmonic demodulation spectrum is determined. The high-frequency error signal is extracted as input to the PI controller, fed back through DAC and linear amplifier to the PZT frequency controller.

The PZT frequency controller adjusts the input voltage of the solidified ring PZT using inverse piezoelectric effect, with a response bandwidth >1 MHz and frequency tuning accuracy of 20 Hz, enabling real-time adjustment of relative frequency error between resonant cavities. The tuning coefficient of the resonant frequency control loop is 1.25 MHz/V, with a tuning range of approximately 25 MHz. The tuning range is set not to exceed 1 FSR of the resonant cavity, allowing for precise adjustment of the resonance frequency for high-frequency fluctuations. Both frequency control loops have closed-loop periods as integer multiples, ensuring synchronous operation and frequency tracking error of 0 during the control process. By analyzing frequency tracking's closed-loop error model and designing adaptive Proportional-Integral-Derivative (PID) based controller, the system achieves long-term, rapid tracking, locking near EP.

Finally, EPTDS extracts the feedback signal from the frequency control loop as the sensing output signal, where the signal intensity is proportional to the spectral linewidth. Compared to traditional methods, EPTDS enables long-term EP control and signal detection at room temperature, enhancing the system's robustness. It is crucial to emphasize that if the optical system is implemented on a single-chip platform, the need for such complex frequency control loops and environmental noise feedback might become unnecessary. This represents the future development prospect of the system. The shorter optical path and on-chip integration will ensure relative stability against frequency fluctuations caused by the environment, leading to increased signal-to-noise ratio and output responsiveness, thus endowing the system with superior performance.

3. Experimental results and discussion

3.1 Non-Hermitian optical sensing module

Prior to testing the EPTDS system, we conducted tests on the non-Hermitian optical sensing module. This module consists of two optical fiber ring resonators, FRR₁ and FRR₂, with identical parameters, characterized by gains or losses represented as γ₁ and γ₂, respectively. We monitored the output power of a tunable laser at 980 nm in real-time and used a power feedback loop to maintain a constant output power, ensuring the pump power inside the cavity remained unchanged. To compare the phase transition processes between the non-Hermitian and Hermitian systems, we adjusted the output of the 980 nm laser to set two different output states. One state involved increasing the pump power of the 980 nm laser to achieve |γ₁|=|γ₂| and γ₁+ γ₂= 0, at which point the optical system precisely reached the PT-symmetric condition, classifying it as a non-Hermitian system. In the other state, we reduced the intensity of the 980 nm signal until γ₁= γ₂. In this case, the system contained two lossy cavities with identical Q-values and no gain, ensuring energy conservation within the system. The characteristic eigenvalues of the two orthogonal modes became real, transforming the system into a Hermitian system. We adjusted the coupling strength, κ_c, of the two cavity modes separately under both the PT-symmetric and coupled-dissipative conditions, observing the changes in the transmission spectra. The results are depicted in Fig. 3.

 

Fig. 3. The real (a) and imaginary (b) parts of the eigenfrequencies of the two supermodes. The variation of eigenvalues is divided into two parts, with the EP as the boundary. The deep red dashed line represents the theoretical transmission model for the non-Hermitian system, while the light red solid line represents the theoretical transmission model for the Hermitian system. The deep blue spherical dots and light blue star-shaped dots represent the respective experimental test results.

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The real and imaginary parts of the eigenfrequencies represent their frequencies and linewidths, respectively. The coupling of the resonators generates two supermodes, with Δω being the eigenfrequency difference between them. In theory, the coupling strength at the EP is defined as κ_EP=|γ₁-γ₂|/2. For Hermitian systems, as κ_c increases, the frequency difference between the two supermodes linearly increases, while the linewidth remains constant during this process. In contrast to Hermitian systems, for non-Hermitian systems, as the coupling strength increases from 0, three distinct phases are observed: PT-symmetry breaking phase, critical state (EP), and PT-symmetric phase. Using κ_EP as the boundary, the orthogonal modes exhibit two distinct behaviors with increasing coupling strength, as shown in Fig. 3. The light yellow background represents the PT-symmetry breaking phase. When the coupling strength κ_c is relatively low, the eigenfrequencies of the two coupled modes remain unchanged, but there are two different linewidths. As κ_c increases and approaches κ_EP, the linewidth of the orthogonal modes sharply decreases. Beyond the critical value of κ_EP, the system enters the symmetric phase, where the linewidths of the cavity modes degenerate, and the eigenfrequencies split, with the splitting rate slowing down as κ_c increases. The theoretical results exhibit slight deviations from the experimental results due to the imperfect equality of gain and loss in the two resonators. However, the system still exhibits the phase transition characteristics of PT-symmetric systems, displaying pronounced nonlinear features, which provide the basis for us to enhance nonlinear sensing using non-Hermitian optical modules.

3.2 Frequency response of EPTDS

To validate the frequency-locking performance and output characteristics of the EPTDS scheme, we documented the system parameters’ fluctuations and comparative results in ambient temperature conditions, as depicted in Fig. 4(a). Over a sampling period of 100 seconds, we employed a temperature sensor to measure temperature variations of approximately 0.15 ^°C and their corresponding effects on laser output frequency. Given that both the detection error signal and the feedback signal within the laser frequency control loop fall under the category of digital signals in the signal processing module, we subjected them to normalization. Their magnitudes were equal, but their directions were opposite, and their trends corresponded to changes in temperature. The low-frequency fluctuations induced by temperature variations were effectively tracked by the frequency controller, resulting in a synchronous shift of the laser center frequency of approximately 200 MHz during this time period.

 

Fig. 4. The frequency output characteristics of the EPTDS scheme. (a) A comparison of the system's detection error signal (red), feedback signal from the proportional-integral controller (green), and the range of laser frequency output (yellow) under a 100 s variation at room temperature (blue); (b) Experimental results of the system's dynamic frequency control characteristics, with the inset in the bottom left corner showing the output response during the locking moment, and the inset in the top right corner displaying the pulse response during the locked state; (c) A comparison of the long-term frequency output results before (red) and after (green) locking the EPTDS scheme.

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In the pulse response experiment, we conducted further testing of the dynamic performance of the resonance frequency control loop. As illustrated in Fig. 4(b), the system exhibits a steady-state time of approximately 10 ms during the transition from open-loop output to closed-loop locking. Following full system locking, we applied pulse signals generated by the digital signal processing unit within the FPGA (field programmable gate array) to the PZT frequency controller, effectively serving as input to the closed-loop system. The pulse signals had a digital intensity of 100, equivalent to inducing a resonance frequency variation of 1.5 kHz. Experimental results indicate that the steady-state time for the system's pulse response is 45 milliseconds, showcasing EPTDS's capability to achieve rapid response and effective control of resonance frequency fluctuations.

Finally, Fig. 4(c) compares the test results of the system's output frequency response during a 4000 s static experiment with a temperature variation rate of 0.5 ^°C/h at room temperature. In contrast to the significant random fluctuations of the open-loop system output at ±100 kHz, the closed-loop system using the CDCC method maintains frequency tracking near the EP within 1 hour, with a noise width less than 10 kHz and noise error less than 3% of the pre-locking level. Further enhancement in stability can be achieved by optimizing the system parameters. This demonstrates that the system can achieve long-term locking of the EP in non-Hermitian optical systems.

3.3 Sensing experiment of EPTDS

To further test the sensing enhancement effect of the EPTDS scheme, we simulated the system's input signal using the Sagnac effect during the rotation process and designed experiments to validate the system's output performance. Firstly, we utilized a high-resolution rotational displacement platform with a nominal rotational resolution and range of 0.01 ^°/s and 1000 ^°/s, respectively, and an output data sampling rate of 200 Hz to test the system's output response at different rotational speeds.

Define the response enhancement factor (R.E.) as the ratio of the system's output response to the disturbance intensity, denoted as $\Delta {\omega _{\varepsilon - PT}}/\Delta {\omega _\varepsilon }$. Assuming a rotational displacement platform speed of Ω and a resonant cavity area perpendicular to the axis of rotation as A, the change in output spectrum FWHM under the Sagnac effect is given by:

The theoretical value of $R.E.$ under the EP can be obtained using the following formula:

According to the above equation, we derived the theoretical enhancement coefficient near the EP. By fitting the output results at different rotational speeds, we reconstructed the system's enhancement effect regarding signal intensity. We then compared this with the theoretical values in ideal conditions, as shown in Fig. 5. Figures 5(a)-(b) respectively depict the output responses of EP-enhancement testing methods and traditional testing methods at rotational speeds of ±0.1 ^°/s, ± 1 ^°/s, ± 2 ^°/s and ±3 ^°/s. The step signals with different frequency responses vary with changes in rotational speed. Despite the presence of noise near the EP and system locking errors, the output responses exhibit clear nonlinear enhancement characteristics, although they are smaller in comparison to theoretical values. Figure 5(c) illustrates the relationship between rotational speed and frequency response, with each data point representing the average result after a 5s sampling period. The system conducted rotation tests ranging from 0.05 ^°/s to 40 ^°/s, corresponding to a frequency detection range of 50 Hz to 30 kHz, achieving a span of three orders of magnitude. The intensity variation of the curved lines is proportional to the square root of the rotational speed, showing excellent agreement with theory. We then calculated the response enhancement factor under the Sagnac effect, as displayed in Fig. 5(d). The system's ultimate sensitivity lies around 0.05 ^°/s, and the experimental reaches 121-fold, whereas the theoretical enhancement value at this point is 322-fold. To further understand this, the insets in Fig. 5(c) and (d) illustrate amplified contrast results in the region of weak signals. The slope of the curves in the logarithmic-logarithmic plot is 0.5, validating the non-Hermitian degeneracy characteristics of the system near the EP.

 

Fig. 5. Comparison of Experimental and Theoretical Results. (a) and (b): Using the same resonant cavity, EPTDS is compared to traditional testing methods at ±0.1 ^°/s, ± 1 ^°/s, ± 2 ^°/s and ±3 ^°/s, based on output step signals. EPTDS exhibits nonlinear variations consistent with theoretical predictions. (c) Comparison of frequency response of EPTDS under varying rotational speeds. The arched curved lines represent the nonlinear enhancement of sensing response near the EP. (d) The comparison graph of the fitting results for the responsiveness enhancement factor shows that the system's measured enhancement limit is approximately 121 times, while the theoretical enhancement is 322 times at this point.

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It is noteworthy that there is a difference between the theoretically enhanced factor near EP and the experimentally measured actual enhancement factor. This situation is actually caused by two factors in the experiment: frequency locking errors and incomplete balancing of system parameters, along with residual noise interference. Although we have effectively improved the output signal-to-noise ratio through methods such as combined modulation and cascaded control, a small amount of noise error still weakens the system's response enhancement. With further optimization of experimental parameters and signal processing methods, the discrepancy in response enhancement compared to theoretical values is expected to decrease.

To gain deeper insights into the performance of the EPTDS, we subjected it to testing and analysis for sinusoidal rotation and static stability in a room temperature environment. We utilized a rotating displacement platform with a frequency of 0.1 Hz to measure the tracking performance of the system at varying angular velocities. The experimental results are shown in Fig. 6(a). Throughout the entire measurement period, we initiated equivalent sinusoidal rotations from a stationary state and sampled the data for 50 s. The sinusoidal fluctuation had an amplitude of 0.1π and a period of 10 s. Its frequency response exhibited an approximate sinusoidal signal with a peak of 11.4 kHz. The relative phase measurement error was almost negligible, clearly demonstrating the system's nonlinear response to sinusoidal modulation near the EP. This finding indicates that even with dynamic fluctuations in the input signal, the system can still achieve precise and stable tracking near the EP.

 

Fig. 6. Test Results of EPTDS. (a) Comparison of system output (yellow), frequency detection error (green), feedback parameters (red), and rotational speed (blue) under sinusoidal rotation. (b) Allan deviation results for rotational speed and output noise under static testing

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Finally, for the long-term BI test related to frequency detection results, we calculated the static output response's Allan deviation and normalized it through fitting with the R.E.. This ultimately provided the Allan deviation in units of rotational speed. The final results, as shown in Fig. 6(b), indicate that over an extended period, the system consistently maintains stability near the EP and exhibits excellent frequency stability. Near a 100 s averaging time, a BI of 0.036 ^°/h can be observed, meeting the performance requirements for navigation-grade gyroscopes. However, the signal-to-noise ratio near EP decreases, random errors increase, and they decrease with increasing averaging time. As the sensitivity limit approaches, the response enhancement near the EP deteriorates. To further validate this result, the inset in Fig. 6(b) illustrates the non-normalized frequency noise results, similarly indicating noise enhancement near the EP. These results demonstrate that our system not only achieves steady-state control near the EP but also exhibits low noise levels and excellent dynamic performance.

4. Conclusion

In summary, we demonstrate a precise and controllable novel EP tracking and detection system, EPTDS, and provide a detailed explanation of its sensing principles and detection logic. The system is ensured to maintain long-term stability near the EP through the use of laser frequency comb tuning and CDCC technology. By employing sinusoidal phase modulation and dual-synchronized demodulation to analyze non-Hermitian spectral sensing information, we identify the directional output results. Compared to open-loop systems, EPTDS operates without being constrained by constant temperature conditions, reducing the output error to less than 3% and achieving a steady-state lock within approximately 10 ms. Leveraging the Sagnac effect, we test the feasibility of the system's enhanced sensing for long-term responses. The resonance frequencies resulting from the backward propagation due to rotation exhibit a slight splitting. At a rotation speed of 0.05 ^°/s near the EP, the output responsiveness is amplified by 121 times. The frequency response detection range extends from 50 Hz to 30 kHz, surpassing three orders of magnitude. Furthermore, we also test the system's dynamic response under sinusoidal rotation. Finally, we analyze the stability results of the static test at a bias rate of 0.036 ^°/h.

Near the EP, the system exhibits remarkable sensitivity to sensing signals. However, even tiny levels of noise can manifest as divergent responses. Our work is dedicated to mitigating the influence of traditional noise on output responsiveness and enhancing the signal-to-noise ratio. In fact, many input signal that causes relative frequency fluctuations between the two resonant cavities can be utilized for sensing tests using EPTDS. Angular velocity sensing is just one of the most easily implemented and detected forms among various non-Hermitian sensing applications. While these efforts may not improve the ultimate sensitivity near the EP, they demonstrate the testing potential of non-Hermitian optical sensing, which offers higher responsiveness and precision compared to traditional approaches. Additionally, our experiments provide a valuable avenue for non-Hermitian optical sensing applications. If parameters in the feedback system can be balanced more skillfully or if an optical platform based on chip and waveguide designs can be utilized, it may lead to more precise implementations with simpler logic and better frequency stability than our experiment achieved. This would ultimately realize the long-pursued goal of high-resolution nonlinear-enhanced optical sensors.