Fiber-optic sensing technology with its intrinsic advantages of low loss, light weight, large bandwidth, immunity to electromagnetic interference, and sensitivity to diverse measurands has made rapid development. The ability to realize a high-precision and reconfigurable-sensitivity fiber-optic sensor will enable significant advances in practical applications including radio telescope arrays [1–3], biomedical diagnosis [4], seismic observation [5,6], and gravitational wave detection [7,8]. In general, fiber-optic sensors make use of an optical fiber as a sensing element to transmit light from the source to the receiver through the point to be measured, where the light parameter (amplitude, wavelength, phase, polarization, etc.) will be modulated as a function of the measurand (temperature [9], strain [10], displacement [11], refractive index [12], angular velocity [13], etc.). Many high-precision measurement tasks tend to utilize the optical phase to convey sensing information, and a typical structure is the laser interferometer. A single-frequency laser interferometer essentially encodes the optical phase introduced by the sensing element into the interference fringe pattern of the acquired spectrum, but it is seriously affected by the laser frequency noise [14–16]. It is worth mentioning that the original measurand in some specific applications is extremely small, so it would be very advantageous to improve the measurement sensitivity if the optical phase can be amplified or enhanced [10,17,18]. An efficient phase amplification method has been proposed with the assistance of a harmonic generation effect in a laser frequency-shifted feedback interferometer, but for a higher-order harmonic, its signal-to-noise ratio (SNR) shows a gradual decline compared with the fundamental wave, which has certain limitations on the recovery of the amplified phase [18]. On the other hand, the weakness of the signal means that noise sources associated with the detector readout should be reduced to an extremely low level [19]. Although some digital phase demodulation techniques have been widely known, such as phase generation carrier or using a $3\times 3$ coupler [6], the phase measurement still faces great challenges in detection limit due to the limited instrument noise floor.

In recent years, much effort has been concentrated on achieving microwave photonic interrogation [20,21], and the emerging attention on optoelectronic oscillators (OEOs) has brought new approaches to constructing fiber-optic sensors for measurement. An OEO [22,23] is a typical microwave photonic system with a closed feedback loop consisting of optical paths and circuits. Owing to the high quality (Q)-factor energy storage element in the cavity, it is easy to produce self-sustained microwave signals with high frequency stability, so it is promising to achieve high precision when applied to sensing. Earlier, the measurand was suggested to be inserted into the cavity [24–26]. The phase perturbation caused by cavity delay jitter is cumulatively enhanced and converted to the oscillating frequency shift. However, in a classical OEO, a long optical fiber with low loss always serves as a portion of its cavity in order to obtain high spectral purity, which is susceptible to environmental perturbations and difficult to distinguish from the measurand. The other common approach takes advantage of a single-passband microwave photonic filter (MPF) [27–30], which acts as both a frequency-selective device and a sensing unit. The MPF can be implemented based on phase-modulation to intensity-modulation (PM-IM) conversion, constructed by an optical notch filter that is sensitive to the measurand. As the measurand induces a change of notch wavelength, the central frequency of the MPF and thus the oscillating frequency of the OEO will be changed. Also, the optical phase variation can be mapped to that of the oscillating signal [31]. Unfortunately, the essential optical phase is not amplified, and the limited sensing sensitivity makes it unsuitable for diverse needs. It is worth noting that a high-sensitivity angular velocity measurement based on an OEO embedding a Sagnac interferometer is demonstrated in Ref. [32]. The phase detected by the interferometer will produce an oscillating frequency shift of the OEO, but the sensitivity is less reconfigurable. More importantly, the random drift of the laser frequency will greatly deteriorate the accuracy of the sensing results, which is avoided in our proposal.

In this paper, a novel optoelectronic hybrid oscillating interrogation technique is proposed, and the sensing performance characterization is experimentally demonstrated by fiber-optic delay jitter detection. Two heterodyne signals carrying the amplified sensing information are injected into a frequency conversion pair and the valuable phase difference is converted to the easily measured oscillating frequency shift benefiting from the cumulative enhancement in the oscillating cavity. In contrast to the phase measurement without an oscillating cavity, the detection limit can reach an improvement of up to 42 dB at a 10 Hz frequency offset with a cavity delay of 1 μs. The reconfigurable sensitivity can be flexibly constructed by using different frequency intervals of the optical source or changing the cavity delay, which is mainly derived from an electrical narrowband filter. By a motorized variable delay line (MDL), comparison experiments are performed with a linear relationship between frequency shift and delay jitter obtained. Thanks to the all-electric oscillating cavity, the proposed sensor is outstanding in its insensitivity to environmental perturbations. Crucially, the sensing precision is not limited by the instrument and cavity delay but is determined by the cavity noise limit. The frequency stability of the sensor is measured, and a minimum Allan deviation of 2.7 attoseconds at an averaging time of 0.2 s is achieved with a frequency interval of 150 GHz, which shows the prospect for precision metrology.

Figure 1 illustrates the schematic diagram of the proposed optoelectronic hybrid oscillating fiber-optic sensor. A continuous wave from a laser is modulated by a stable RF signal in an electro-optic phase modulator (PM) to generate an optical frequency comb (OFC) source. The frequency of each comb line can be expressed as $f_0+k{f}_{\mathrm{RF}}$, where $k$ is an integer representing the $k$th comb line from the optical carrier, and $f_0$ and $f_{\mathrm{RF}}$ are the central frequencies of the laser and the RF signal, respectively. The OFC is coupled into a Mach–Zehnder interferometer (MZI) consisting of two optical couplers (OCs) and two interference arms. One arm is used as a reference arm, and the other arm is the sensing arm, on which the sensing fiber is placed. It is assumed that the fiber-optic delay difference between the two arms can be denoted as ${\tau }_0+\tau (t)$, where $\tau (t)$ is the fiber-optic delay jitter introduced by the measurand in contrast to the fixed delay ${\tau }_0$. A polarization controller (PC) is necessary for adjustment to the appropriate interference state and an acousto-optic modulator (AOM) is exploited to realize heterodyne detection. At the output of the interferometer, two optical spectral regions of the frequency comb are extracted and split into two paths. Then the photoelectric detection with cascaded electrical amplification and filtering is carried out, respectively. At each branch, an electrical signal with the fiber-optic delay difference weighted by the corresponding optical comb line frequency is generated, which is given by $$V_{1,2}(t)\propto \cos (2\pi f_{m}t+2\pi (f_0+k{f}_{\mathrm{RF}})({\tau }_0+\tau (t))),$$(1)where $f_m$ is the driving frequency of the AOM. The BPF1 centered at $f_m$ is designed to filter out harmonics generated by the amplifier (EA1). When the $m$th and $n$th comb lines are selected, the phase difference associated with the sensing information between two branch signals can be expressed as ${\phi }_s(t)=2\pi (m-n)f_{\mathrm{RF}}\tau (t)$. It should be mentioned that coupling two lasers is more direct and purer to obtain a larger frequency interval; however, the high noise of typical free-running lasers means that the frequency stability of the heterodyne source is several orders of magnitude worse than those achieved with existing synthesizers, particularly at low (GHz) frequencies [33], which may lead to increased measurement deviation.

For cumulative enhancement, an all-electric oscillating cavity is introduced, where the oscillating signal is first up-converted, bandpass-filtered by an intermediate frequency (IF) filter (BPF2), and then down-converted. The IF filter with a central frequency of $f_m+f_{\mathrm{osc}}$ works to block the image frequency of the mixer output $f_m-f_{\mathrm{osc}}$, which also carries the sensing information and may cause the same frequency crosstalk when down-conversion. Considering that the group delay of the IF filter is expressed as ${\tau }_{\mathrm{IF}}$, the phase noise of the laser ${\theta }_0(t)$ will be transferred to the oscillating cavity as ${\theta }_0(t-{\tau }_{\mathrm{IF}})-{\theta }_0(t)$, thus worsening the cavity noise limit. To solve this problem, an IF filter with a negligible group delay or a laser with a narrow linewidth is preferred. The resulting signal after twice frequency conversion is amplified, filtered, and fed back to the injection of the up-conversion. The main cavity delay and single-mode operation are both provided by a narrowband filter (BPF3) centered at the oscillating frequency. It is well known that the essence of an oscillator is a positive feedback loop, where the additive noise in the oscillating cavity is accumulated in each amplification and filtering until the gain is saturated, thus producing a stable oscillating output. The closed-loop configuration allows self-reproduction once the amplitude and phase conditions of oscillation are met. In addition, a phase shifter (PS) is inserted in one of the branches to tune the oscillating frequency as needed. After the two branch signals are injected through the frequency conversion pair, the phase difference containing the sensing information will be cumulatively enhanced, thus causing the frequency shift of the oscillating signal. Following the regenerative feedback approach, the oscillating frequency can be given by $$f_{\mathrm{osc}}(t)=\frac{(m-n)f_{\mathrm{RF}}}{{\tau }_{\text{cavity}}}\tau (t)+\frac{N}{{\tau }_{\text{cavity}}}+\frac{{\varphi }_0}{2\pi {\tau }_{\text{cavity}}},$$(2)where ${\tau }_{\text{cavity}}$ is the cavity delay and $N(=0,1,2,\dots )$ is the order of the oscillating mode. ${\varphi }_0$ is a constant that includes the phase introduced by the fixed delay as well as the PS. Mathematically, in the range of an FSR, the change in oscillating frequency with sensing delay jitter is given by $$\mathrm{\Delta }{f}_{\mathrm{osc}}=\frac{(m-n)f_{\mathrm{RF}}}{{\tau }_{\text{cavity}}}\mathrm{\Delta }\tau .$$(3)

It can be seen from Eq. (3) that when the fiber-optic delay of the sensing arm changes, the oscillating frequency changes in proportion, which can be measured by a high-resolution and high-speed frequency counter. The sensing sensitivity can be improved by enlarging the frequency interval of the two comb lines $(m-n)f_{\mathrm{RF}}$ or reducing the cavity delay ${\tau }_{\text{cavity}}$. The sensing range is determined by the FSR, because an excessive delay change would cause a mode jump, resulting in incorrect measurements. By setting a reasonable frequency interval and cavity delay, a specific sensor can be constructed to meet actual needs in different situations.

The theoretical detection limit depends on the sensing sensitivity and instrument resolution. Supposing that the phase noise floor of the instrument is expressed as $S_{\phi }(f)$, the detection limit with respect to the frequency offset can be given by ${\tau }_w(f)=\frac{{\tau }_{\text{cavity}}\sqrt{f^2{S}_{\phi }(f)}}{(m-n)f_{\mathrm{RF}}}$; while in phase measurement without an oscillating cavity, the phase difference is directly detected, and the detection limit is given by ${\tau }_{\mathrm{wo}}(f)=\frac{\sqrt{S_{\phi }(f)}}{2\pi (m-n)f_{\mathrm{RF}}}$. Thus, the transfer function between detection limit improvement and frequency offset is calculated as $$\begin{gathered} H(f)=-10{\log }_{10}({\tau }_w(f)/{\tau }_{wo}(f)) \\ =-10{\log }_{10}(2\pi f{\tau }_{\text{cavity}}). \end{gathered}$$(4)

One can see that for a given instrument resolution, a great improvement occurs when the frequency offset is low, and it gets less at a rate of $-10\mathrm{dB}/\mathrm{decade}$ as the frequency offset is increased. Note that the introduction of the oscillating cavity with a cavity delay of 1 μs can result in a detection limit improvement of up to 42 dB at a 10 Hz frequency offset. In other words, if a target detection limit is set, then this optoelectronic hybrid oscillating sensor will greatly reduce the instrument resolution requirement, and a shorter cavity delay does mean a more remarkable effect.

In the proposed sensor, we obtain the amplified sensing information by heterodyne detection of two optical spectral regions and cumulatively enhance the phase difference by an oscillating cavity with a frequency conversion pair. However, in addition to the valuable phase difference ${\phi }_s(t)$, the phase noise inherent in the oscillating cavity ${\phi }_n(t)$ is also involved in the cumulative enhancement. Fortunately, provided that the phase noise is sufficiently suppressed below the phase difference, the sensing results can be considered credible. Even if the instrument has a poor noise floor, we can always achieve the target detection limit by reducing the cavity delay, which indicates that the sensing precision is limited by the cavity noise rather than the instrument. Commonly, Allan deviation is used in the field of metrology for time-domain characterization. The sensing precision in the unit of delay along with averaging time in this work can be calculated as $\sigma (\tau )=\frac{{\tau }_{\text{cavity}}{\sigma }_f(\tau )}{(m-n)f_{\mathrm{RF}}}$, where ${\sigma }_f(\tau )$ represents the frequency instability of the oscillating cavity, equivalent to a transfer function applied to the SSB phase noise power spectral density [34]: ${\sigma }_f(\tau )=2\sqrt{{\int }_0^{\infty }{f}^2{S}_{\mathrm{osc}}(f)\frac{{\sin }^4\pi f\tau }{{(\pi f\tau )}^2}\mathrm{d}f}$, where $S_{\mathrm{osc}}(f)=\frac{{|{\int }_{-\infty }^{\infty }{\phi }_n(t)e^{i2\pi ft}\mathrm{d}t|}^2}{{(2\pi f{\tau }_{\text{cavity}})}^2}$. As a result, it can be confirmed that the sensing precision is independent of the cavity delay.

A proof-of-concept experiment based on the setup shown in Fig. 1 is performed. A single-frequency optical carrier is generated by a narrow-linewidth laser with a central frequency of 193.41 THz and an output power of 16 dBm. Before injecting into the PM, an erbium-doped fiber amplifier (EDFA) is employed to amplify the optical power because of the considerable power loss in the subsequent electro-optic and photoelectric conversion. The RF signal is provided by a frequency-adjustable microwave signal generator. In MZI, a motorized variable optical delay line (General Photonics, MDL-002) with a range of 560 ps and a resolution of 1 fs is placed on one of the arms to simulate the sensing information of the change in fiber-optic delay. The driving frequency of the AOM is 80 MHz. A single input multiple output waveshaper (WS) is used to extract two optical spectral regions, including a selected comb line and its optical frequency shift, respectively. Since the heterodyne detection of each optical spectral region generates an electrical signal containing the sensing information, the WS is required to be a reconfigurable optical filter to meet different frequency intervals, where the central frequency needs to be aligned with the selected comb line frequency and the bandwidth is not greater than the comb line interval.

We generate two different OFCs by using RF signals with 10 GHz and 25 GHz, respectively, and the power of the RF signal needs to be fine-tuned to make the required comb line power larger. The optical spectra at the output of the WS can be observed by an optical spectrum analyzer (OSA). Due to the coarse resolution, the optical frequency shift is not visible. As shown in Fig. 2(a), a frequency interval of 20 GHz is composed of $\pm 1$st comb lines of the 10 GHz OFC, and the residual sideband is suppressed by more than 20 dB. Figure 2(b) shows that a larger frequency interval of 150 GHz is composed of $\pm 3$rd comb lines of the 25 GHz OFC, and the residual sideband rejection ratio is more than 40 dB. The photoelectric conversion is performed at two branches with the same components, that is a photodetector (PD), a low-noise amplifier (LNA), and a BPF with a passband range of 75–85 MHz.

In the oscillating cavity, two mixers sandwiched an IF filter is utilized to complete twice frequency conversion, where the IF filter has a central frequency of 105 MHz and a 3 dB bandwidth of 25 MHz. For comparison, we apply two custom-made narrowband filters with a close central frequency of about 24 MHz but different bandwidths and group delays, which can be seen in Fig. 2(c). Note that the bandwidth of the narrower filter is about 400 kHz, and the group delay within its bandwidth has a fluctuation between 2.15 μs and 2.57 μs. The other filter has a bandwidth of about 2 MHz, and the group delay within its bandwidth is symmetric relative to the center (0.69 μs) with a fluctuation of 0.36 μs. The FSR is about 400 kHz and 1.1 MHz, correspondingly. Moreover, two LNAs with a small signal gain of 23 dB are required to ensure sufficient cavity gain for self-sustaining. Using an oscillating cavity with a frequency conversion pair, the fiber-optic delay jitter on the sensing arm of the MZI can be interrogated in the oscillating frequency shift. Figures 2(d) and 2(e) show the electrical spectrum of the generated signal for the filter with a bandwidth of 400 kHz and 2 MHz under static measurement by an electrical spectrum analyzer (ESA). When the MDL is fixed at different fiber-optic delays, the oscillating signal will work at different frequencies. The SNR of the generated signal is as high as 90 dB. Adjacent modes are displayed when using the 2 MHz filter, and the spurs suppression reaches 50 dB. It can also be seen that the FSR is not equal at different oscillating frequencies, which is caused by the dispersion of group delay. Fortunately, it will hardly affect the sensing results.

To investigate the dynamic characteristics of the proposed sensor, we set the scanning speed of the MDL to 1 ps/s and use a frequency counter (Keysight 53230A) with a gate time of 1 ms to collect the instantaneous frequencies of the generated signal. Sensing beyond the range is analyzed, and the frequency shift is calculated as the difference from the center of oscillating frequency change. Note that in practical applications, the initial frequency should be set at this central frequency with the help of the PS. As shown in Figs. 3(a) and 3(b), with the unidirectional increase of fiber-optic delay jitter, the frequency shift presents a linear decline with periodic changes. The period is the reciprocal of the frequency interval, and the frequency shift range is equal to the FSR. Given that the mode jump can lead to a fuzzy judgment, the period is indeed the sensing range. When the curve is linearly fitted, we can utilize the slope to approximately represent the sensing sensitivity. Thus, for the 400 kHz filter in Fig. 3(a), a sensor with a sensitivity of 8.3 kHz/ps and a range of 50 ps is realized when the frequency interval is 20 GHz, while when the frequency interval is 150 GHz, the sensing sensitivity is raised to 62 kHz/ps and the sensing range is scaled down to 6.7 ps. For the 2 MHz filter in Fig. 3(b), the shorter group delay results in a larger FSR, and the sensitivity of the sensor is further improved at the same frequency interval, reaching 22.3 kHz/ps and 162 kHz/ps, respectively. Since the group delay of the narrowband filter is not flat, it indicates that the sensitivity of the sensor operating at different oscillating frequencies is not exactly the same. In Fig. 3(c), within the frequency shift range of $\pm 200\mathrm{kHz}$, the sensing delay jitter of several discrete points is estimated in proportion to the change in oscillating frequency according to the measured filter group delay, which is in good agreement with the actual measured result. A greater sensing sensitivity means less delay jitter for the same frequency shift. The slight deviation of delay jitter may come from the error of cavity delay measurement, where the fractional delay error is directly delivered.

To better characterize the measurement noise floor, a power divider is employed to produce two identical signals from the driving source with a frequency of 80 MHz, replacing the original branch signals injected into mixers. The phase difference is fixed, so the generated oscillating signal contains no sensing information. The phase noise of the oscillating cavity with a narrowband filter of 400 kHz and 2 MHz is measured by a phase noise analyzer and the result is shown in Fig. 4. As can be seen, the shape of the two curves is approximately the same with a gap of 9.5 dB, which matches the group delay relationship between the two filters. The phase noise reaches $-108.4\mathrm{dBc}/\mathrm{Hz}$ at 1 kHz, $-130.5\mathrm{dBc}/\mathrm{Hz}$ at 10 kHz when using the 400 kHz filter, and $-96.5\mathrm{dBc}/\mathrm{Hz}$ at 1 kHz, $-120.9\mathrm{dBc}/\mathrm{Hz}$ at 10 kHz when using the 2 MHz filter, respectively. At the same time, the corresponding phase noise of the phase measurement with the same detection limit is also calculated, with a result as low as $-144\mathrm{dBc}/\mathrm{Hz}$ at 1 kHz, $-147\mathrm{dBc}/\mathrm{Hz}$ at 10 kHz. We clearly see that the introduction of the oscillating cavity makes the instrument resolution requirement relaxed. Specifically, when the frequency offset is 1 kHz, the phase noise floor of the instrument is relaxed by 36.4 dB for the 400 kHz filter and 46 dB for the 2 MHz filter, which is consistent with the theory. To produce an oscillator with very low cavity noise, design rules are described where both thermal noise and flicker noise coming from the amplifier are considered [35,36]. It is reported that any components with a nonlinear phase response will give rise to amplitude-to-phase (AM-PM) conversion, particularly when the components are operated in saturation [37]. Therefore, in such an all-electric oscillating cavity with a frequency conversion pair, the mixer will be the key nonlinear component in addition to the amplifier, which indicates that the amplitude noise in two branches, such as from the shot noise of the PD and the relative intensity noise (RIN) of the laser, is also likely to be converted into the phase noise of the cavity.

Then, over 8 s instantaneous frequencies with a gate time of 10 μs for two filters are collected, respectively. Through the Allan deviation evaluation, we prove that the sensing precision of the proposed sensor can outperform the existing advanced distance measurement techniques [38,39], which can be seen from Fig. 5. The Allan deviation is unified into the unit of delay. Overall, it shows a trend of first decreasing and then increasing. The minimum value occurs when the averaging time is about 0.2 s, reaching 2.7 attoseconds with a frequency interval of 150 GHz. We believe that different responses of the two filters lead to the working state difference of the amplifiers in the oscillating cavity, resulting in a non-covering between two curves of experimental results. Considering that a larger frequency interval widened to above 1 THz can be implemented [40], the sensing precision will be further optimized, and a level as low as sub-attosecond can be expected, which is well suited to sensing applications that require very high precision but have very small variable ranges.

This paper illustrated an optoelectronic hybrid oscillating fiber-optic sensor and successfully applied it to fiber-optic delay jitter detection with the advantages of reconfigurable sensitivity, attosecond precision, and simple configuration. The optical phase of two comb line frequencies storing the sensing information is photoelectrically converted by heterodyne interference. The phase difference is cumulatively enhanced and reflected in the frequency shift assisted by an all-electric oscillating cavity with a frequency conversion pair, which greatly improves the sensitivity and detection limit of the sensor. The group delay of the narrowband filter constitutes the main cavity delay, where the flatness and accuracy will be directly delivered to the sensing linearity and measurement deviation. Without long optical fiber in the sensor, the insensitivity to environmental perturbations as well as the stability of the generated microwave signal can be outstanding, and the sensing precision only depends on the cavity noise limit. In addition, a larger frequency interval of the optical source can be achieved through cascaded modulators to further improve the sensing precision, while the corresponding sensing range will be reduced. The proposal could also be applied to temperature, strain, and other practical sensing scenarios that can be equivalent to the change in fiber-optic delay and would show great application potential in precision and flexible sensing interrogation systems.