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Abstract
The multiplexing of fiber ring resonators (FRRs) for no crosstalk loss sensing is proposed and demonstrated experimentally. The difference between the parallel and series FRRs is theoretically elaborated to determine the multiplexing scheme. The frequency response properties of the cascaded FRRs at distinct radio frequency (RF) working points are compared and analyzed. The optical carrier-based microwave interferometry system is implemented to verify the numerical investigation and exhibit the multiplexing of phase-shift based demodulation at diverse RF working points. Enhanced by the phase-shift amplification and the series configuration, each FRR can be independently demodulated by recording the phase of frequency response at the specific RF working point. The experimental results indicate that the sensitivity of transmittance reaches −0.341 rad with the advantage of robustness and immunity to power fluctuation. Owing to the prominent contribution of insensitive points and the series strategy, the crosstalk of multiplexing for loss sensing between two FRRs is eliminated virtually, which matches well with the theory. The proposed scheme provides an innovative approach for multiplexing the phase-based FRRs sensors without additional expenditure.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Fiber ring resonators (FRRs) are the essential wavelength-selective and energy-stored devices with high quality factor, which have been widely researched in the application of Kerr frequency combs [1], fiber-optic gyro [2], temporal cavity solitons [3], microscopic dynamics [4], fiber laser [5,6] and metrology [7,8]. With the advantages of small size, light weight, immunity to electromagnetic interference and etc [9,10], the FRR-based sensors have attracted significant attention. As a common approach, the intensity-based sensors with advantages of simple structure and low cost are widely implemented by transforming the change of sensing parameters to the variation of loss, which is restricted by the fluctuation of light source. Fiber-loop ring-down technology is proposed to overcome this weakness and the loss-based sensing is employed to indirectly measure the temperature [11], strain [12], refractive index [13] and other parameters [14,15]. However, the small duty cycle of the time domain ring-down technique leads to a relatively long measuring time [16,17]. To further enhance measurement efficiency, the conventional microwave-assisted schemes are achieved by recording the magnitude and phase spectrum to reconstruct the time-domain signal, whose principle is intrinsically same to the pulse-based demodulation [17]. But the distinctive characters of the magnitude and phase spectrum are ignored [18].
Distinguishing from previous microwave-assisted scheme, optical carrier-based microwave interferometry (OCMI) technology is reported and demonstrated by researching the magnitude and phase response spectrum [19,20]. The OCMI technology combined with various interferometers has been implemented to achieve diverse sensing applications with several fantastic advantages, including low dependence on the polarization state, multimode interference and etc [21,22]. Apart from the intensity-based demodulation, the OCMI-based sensors can be classified to the frequency drift and phase shift scheme according to their demodulation methods [23–25]. The frequency-based sensors demodulate the frequency drift of resonance in the magnitude spectrum and the sensitivity can be enhanced by the vernier effect, where an expensive microwave source (GHz) and an electrical spectrum analyzer is essential. As for phase-shift-based OCMI sensors with the dependence on vector network analyzer (VNA), the sensitivity is improved by phase-shift amplification and the sensing of single parameter is conducted. Consisting of the low-cost RF source (MHz) and lock-in amplifier, the traditional phase-shift sensors obtain the advantage of low-cost, rapid response and high sensitivity, but it is challenging for the phase-based schemes to multiplex without the increasing of cost and complexity [26,27].
In this paper, to the best of our knowledge, we firstly propose and demonstrate a creative phase multiplexing of RF-assisted FRRs based on phase-shift amplification. The numerical derivation of FRRs is systemically illustrated. To determine the multiplexing scheme, the difference between the parallel and series FRRs is demonstrated by simulation. Inspired by the phase-shift amplification, the insensitive points of phase shift are applied in the multiplexing of double FRRs sensors without crosstalk. Due to the simplicity and efficiency, the OCMI system is implemented to verify the numerical investigation of proposed scheme and demonstrate the multiplexing of phase shift at diverse RF working points. The transmittances of FRRs are varied by the attenuators to simulate the loss-based sensors. The sensitivity of transmittance in each FRR is relatively enhanced by the phase shift amplification and selecting the proper configuration of FRRs. And the crosstalk of multiplexing is eliminated virtually owing to the unique signature of insensitive points and the series cascading. Additionally, the scheme can be performed by switching RF working point of the conventional phase-shift sensing system, consisting of low-cost RF source and lock-in amplifier.
2. Principle and numerical investigation
2.1 Fiber ring resonator
The schematic diagram of the FRRs is exhibited in Fig. 1. Two FRRs with different lengths and transmittances are cascaded in series and work as two filters. In many cases, the mathematical model combining optical domain and microwave domain is developed to completely describe the theory of FRR [28]. Here, we mainly focus on the interference in microwave domain. The steady-state analysis of microwave is used to more simply investigate the basic mechanism in the proposed structure and describe the frequency transmission response function.
The transmission of FRR is intrinsically the interference in RF domain, which can be derived from the steady-state interference of OCMI. Supposing that the loss and phase shift of optical coupler (OC) are ignored, the interaction of single FRR can be derived by the matrix relations:
The 3 dB coupler is used in this paper, for which $t$ and $k$ are equal to 0.5. $\varphi _m$ ($m$ is 1 and 2) can be simplified to:
Observed from Eq. (6), the total phase shift of FRR$_m$ varies periodically with the increasing of the radio frequency (RF) $f$ and the period is determined by ${c}/{L_m}$, called as free spectral range (FSR). To estimate the effect of phase shift on the transmittance sensitivity, the phase amplification sensitivity $G_m$ of the transmittance $\alpha _m$ for single FRR can be expressed as:
Based on the phase shift amplification technology, the sensitivity of transmittance can be flexibly tuned by sweeping radio frequency. As revealed in Eq. (7), the derivatives ${\rm {d}}\varphi _m /{\rm {d}}{\alpha _m}$ are equal to non-zero constants with $f$ at FSR$\cdot (2k+1)/4$ and $\theta _m$ at $(2k+1)\pi /2$ ($k$ is the natural number), called as the linear working points. And the derivative of the loss insensitive points are 0 with $f$ at FSR$\cdot k/2$ and $\theta _m$ at $k\pi$. In Fig. 2, the numerical investigations of magnitude and phase response spectra of two FRRs with different FSRs and transmittances are demonstrated. The phase insensitive points of two FRRs are located at different frequencies ($f_1$ and $f_2$) as illustrated in Fig. 2(b). In previous researches, the unique signature of the insensitive points is overlooked due to overemphasis on the improvement of sensitivity. The insensitive points provide a novel approach for multiplexing of phase-shift sensors without crosstalk by switching RF.
Fig. 2. Numerical investigations of (a) magnitude and (b) phase spectra for FRR$_1$ with FSR$_1$ of 16.67 MHz and FRR$_2$ with FSR$_2$ of 25 MHz.
2.2 Multiplexing scheme
The multiplexing schemes can be classified to parallel and series FRRs and the performance of both are investigated to determine proper multiplexing one. The frequency response functions $H_{series}$ and $H_{parallel}$ of series and parallel FRRs can be derived as:
The simulated frequency response spectra are depicted in Fig. 3. The transmittances of FRR$_1$ and FRR$_2$ are set to 1 and 0.5 in the simulation. According to Eq. (8), the phase shift of total system for series FRRs is simply equal to $\varphi _1+\varphi _2$ without the impact of the magnitude as shown in Fig. 3(b). On the contrary, the magnitude will inflect the total phase shift of the parallel FRRs, which enhances the impact of crosstalk and the complexity of demodulation method. Moreover, the series FRRs obtain larger contrast in both magnitude and phase spectra compared with the parallel FRRs.
Fig. 3. The simulated frequency response of (a) magnitude and (b) phase spectra for each single FRR and two cascaded schemes of FRRs.
To further demonstrate the features of the different cascaded schemes, the simulation on the phase shift of two cascaded systems is executed with the transmittance of FRR$_1$ ranging from 0 to 1. As shown in Fig. 4, the crosstalk suppression point of parallel scheme appears at 9.3 MHz and the no crosstalk point of series scheme occurs at 8.33 MHz. With zooming up these areas, it can be noticed that the curves of different transmittances have diverse intersections, leading to crosstalk in multiplexing in Fig. 4(a). But for the series multiplexing, all curves have the same intersection at 8.33 MHz in Fig. 4(b), where the phase shift of FRR$_1$ is 0 and does not contribute to the total phase. Additionally, the series multiplexing has higher sensitivity than parallel. Hence, the series multiplexing has a much better performance and the insensitive points of FRRs are the key to achieve the dual-parameters sensing without crosstalk.
Fig. 4. The simulated phase spectrum of (a) parallel and (b) series multiplexing. The insertions are the enlarged view of the crosstalk suppression point and the no crosstalk point.
The loss insensitive points (also called as no crosstalk points) of one FRR can be the multiplexing points for another FRR and the sensitivity is tunable at each RF according to the phase-shift amplification technology. Therefore, it is essential to survey the sensitivity of multiplexing points and the crosstalk between two sensing elements. The insensitive points of series and parallel multiplexing for FRR$_1$ at 8.33 and 9.3 MHz are selected to work as multiplexing points for FRR$_2$. The numerical relationship between the phase and the transmittance of the FRR$_2$ is depicted in Fig. 5(a). The sensitivity of series scheme at the multiplexing point of FRR$_2$ is much higher than the parallel one. And it is close to the linear point of series scheme, where the sensitivity is enhanced by the phase shift amplification. Moreover, the fluctuation of transmittance in FRR$_1$ leads to phase shift of all curves except the series scheme at multiplexing point. The RF at 12.5 MHz is the insensitive point for FRR$_2$ and the multiplexing point for FRR$_1$. By demodulating the phase at 8.33 and 12.5 MHz, the transmittances of FRR$_1$ and FRR$_2$ can be obtained independently. As revealed in Fig. 5(b), there is no crosstalk between two FRRs for transmittance sensing in theory. Therefore, the series scheme provides an effective way to achieve multiplexed dual-FRRs sensing.
Fig. 5. (a) The numerical calculation of the relationship between the phase and transmittance of FRR$_2$ with different cascaded schemes at various RF points. (b) The transmittance crosstalk between two FRRs demodulated by the phase change at the corresponding multiplexing points.
Considering the loss and delay of the system apart from FRRs, the frequency response function of total system $H\left ( f \right )$ can be expressed as:
3. Results and discussions
The experimental arrangement of the RF-assisted interference system with sweeping frequency is illustrated in Fig. 6. In order to satisfy the requirement of rated detection power for the photodetector (PD) and avoid the optical interference, an erbium-doped optical fiber amplifier (EDFA, Amonics AEDFA-C-DWDM-23-B-FA) is exploited as the optical source to provide enough power and sufficiently short coherence length. The sweeping RF signal generated by the vector network analyzer (VNA, Rohde & Schwarz ZVL6) is modulated onto the input light by an electro-optic modulator (EOM, AVANEX). Such modulated RF signal travels through the two cascaded FRRs formed by two optical OCs along with the propagation of input light and is subsequently received by PD. The PD converts the optical signal to electrical signal and sent it to VNA for obtaining the frequency response spectra of both magnitude and phase. As elaborated in previous section, the multiplexing of two FRRs is established on the basis of selecting proper frequencies. Two variable optical attenuators (VOA) are embedded in the two FRRs respectively to adjust the loss of FRRs, which can also be substituted by other loss-based sensors.
Fig. 6. Experimental setup of the RF-assisted interference system for the multiplexed fiber ring resonators with sweeping frequency interrogation.
After subtracting the phase offset, the magnitude and phase spectra of each FRR and two cascaded FRRs acquired by VNA are presented in Figs. 7(a) and (b) respectively. Consistent with the numerical investigation, the magnitude contrast of FRR$_{2}$ is smaller than that of FRR$_{1}$ due to the larger insertion loss induced by the VOA$_{2}$, which is similar in the phase spectrum. The magnitude spectrum of the cascaded FRRs in series is the product of each singe FRR’s spectrum. Such series configuration is manifested in the phase spectrum as the sum of FRR$_{1}$ and FRR$_{2}$, as proved by the difference curve in purple between the phase spectrum of FRR$_{1}$+FRR$_{2}$ and the sum of that of each single FRR. The small dip in this difference curve corresponds to the distortion as denoted in Fig. 7(a). The FSRs of the two FRRs are 12.7 MHz and 18.9 MHz, which are inversely proportional to the optical path of fiber ring. Influenced by the intrinsic responsivity and background noise of the instrument, a waveform distortion appears in the frequency response especially the magnitude spectrum at around 14 MHz, which is inevitable with our existing instruments. Also, the power fluctuation would affect the magnitude spectrum whose contrast, i.e. the extinction ratio, is related to the loss. For the loss sensing demodulated by the extinction ratio of magnitude response, a frequency-sweeping RF source is necessary and it’s difficult to achieve multiplexing. Additionally, the period of frequency sweeping will lead to a relatively long measuring time. On the contrary, the phase spectrum is less susceptible to these disturbances, and the phase-shift amplification demodulation method only requires a proper fixed frequency.
Fig. 7. (a) The magnitude and (b) phase spectra of frequency response for each FRR and the cascaded FRRs.
For the sake of clarifying the multiplexing mechanism of the cascaded FRRs, the loss sensing properties at variable RF points are investigated as depicted in Fig. 8. The enlarged view of the magnitude spectra in Fig. 7(a) is marked with each characteristic RF point and presented in Fig. 8(a). The no crosstalk points in yellow and red refer to the loss independent RF points, corresponding to the cumulative phase $\theta$ of a round trip in the fiber ring being equal to $k\pi$ ($k$ is the natural number). The linear working points in blue for both FRRs which hold only under the condition of $\theta = k\pi + \pi /2$. At such points, the phase change is linearly related to the transmittance of fiber ring link. The loss sensitivities at the RF points of $\theta = 2k\pi + \pi /2$ are opposite to that of $\theta = (2k+1)\pi + \pi /2$. For consistency and comparison, the RF points on the left side of the no crosstalk points are mainly studied and demonstrated.
Fig. 8. Enlarged view of (a) the magnitude interferogram for each FRR centered at 6.35 MHz and (c) the illustration for waveform distortion of FRR$_{2}$. The relationships between the phase change and the transmittances of (b) VOA$_{1}$ and (d) VOA$_{2}$ in the corresponding FRRs at different RF.
The relationships between the phase change and the transmittance of VOA$_{1}$ at the linear working point of 3.175 MHz, the no crosstalk point of 6.35 MHz and the nonlinear point of 4.975 MHz are presented in Fig. 8(b). The relationship curve at nonlinear point is fitted according to Eq. (6), while other curves are linearly fitted. Obviously, these three curves show the linear, constant and nonlinear trends which match well with the principle. The phase change has little offset with the varying of loss in FRR$_{1}$, indicating a well insensitivity to the loss. In order to realize the multiplexing of two FRRs, the multiplexing RF working point of FRR$_{1}$ should be the no crosstalk point of FRR$_{2}$, i.e. 9.45 MHz as shown in Fig. 8(a). However, due to the partial waveform distortion at some frequencies caused by the instruments, the sensing performance is not always gratifying at each multiplexing RF working point. The enlarged view of FRR$_{2}$’s distorted magnitude spectrum for instance is depicted in Fig. 8(c). As a result, there is a intercept in the yellow relationship curve of FRR$_{2}$ at the multiplexing RF working point, implying that an additional phase is introduced. A nonlinear point at 43.75 MHz is harnessed as an instance to exhibit a smaller intercept in Fig. 8(d), where the waveform has less conspicuous distortion. Furthermore, the multiplexing RF working point of 28.35 MHz is adopted to conduct the loss sensing of FRR$_{1}$ as shown in Fig. 8(b), which is the no crosstalk point of FRR$_{2}$. Such frequency enables the quasi-linear demodulation of FRR$_{1}$ and insensitivity to loss of FRR$_{2}$ simultaneously.
To clarify the anti-crosstalk capability of the proposed multiplexing scheme with the two FRRs, the transmittance crosstalk between them tested by individually varying the VOA$_{1}$ and VOA $_{2}$ at the corresponding multiplexing RF working points is shown in Figs. 9(a) and (b). As the loss tuned by VOA$_{1}$ or VOA $_{2}$ changes, the transmittance of FRR$_{2}$ or FRR $_{1}$ remains nearly constant with the maximum change range of 0.0174 and 0.0177 respectively. Apart from the crosstalk, the influences of the power fluctuation caused by the optical source, PD or other instruments on the frequency responses of magnitude and phase are also researched. Figure 9 presents the stability test against power fluctuation for relative magnitude and phase of the system at the multiplexing RF working point of 6.35 MHz. This power fluctuation is introduced by randomly adjusting the optical source power. Apparently, the phase hardly changes with the power fluctuation compared with the relative magnitude, proving that the phase-shift amplification based demodulation has a great stability for anti power fluctuation.
Fig. 9. The transmittance crosstalk between two FRRs tested by (a) VOA$_{1}$ and (b) VOA$_{2}$. (c) The evolution of relative magnitude and phase at different source power.
Overall, the OCMI system is implemented with two FRRs to verify the proposed multiplexing scheme. The VNA is used to acquire the frequency response of the system by sweeping RF, thereby obtain the characteristic RF working points, especially the loss insensitive points. Such insensitive points of both FRRs can be harnessed as the multiplexed RF working points of each other. For a certain FRR, its working RF at the loss insensitive points can be calculated with the condition of $\theta = \pi$ according to Eq. (3). Supposing the FSRs of several cascaded FRRs in series are $\Delta f_{n}$ ($n$ is the positive integer from 1 to the quantity of FRRs) which are different and have no common divisor (besides 1) with each other, the corresponding multiplexing RF points for each FRR are $p_{n}\cdot {q/2}$, ($q$ is the positive integer), where $p_{n}$ is the least common multiple of all $\Delta f_{n}$ except itself. Once the working frequency is determined, the VNA in Fig. 6 can be replaced by a RF source and a lock-in amplifier (LIA) as shown in Fig. 10, which could further improve the demodulation accuracy and efficiency. With the coupler splitting ratio of 50:50 adopted in this work, the total loss increases as the FRR quantity increases. Hence, the quantity of multiplexed FRRs is determined by the responsivity of PD and the rated working frequency range of the employed instruments. In addition, the loss sensing in proposed scheme can be utilized to measuring loss-related parameters such as fiber bending, fiber aging and radiation-induced fiber loss or other parameters such as temperature, strain, refractive index, gas and magnetic field by accessing the corresponding loss-dependent sensors into the FRR. On the basis of measuring the parameters mentioned above, the multiplexed loss sensing can be applied to the real potential applications including simultaneous sensing of magnetic field and temperature in the power grid system, fiber aging and radiation in the spacecraft, refractive index and temperature of solution in the battery, and so on.
4. Conclusion
The multiplexing of RF-assisted FRRs based on phase-shift amplification is proposed and experimentally demonstrated with OCMI. As far as we know, this is the first time to realize multiplexing of FRRs for phase-based sensing by switching proper RF working point. For the phase-shift-amplified system, the theoretical calculation reveals that the multiplexing of two FRRs in series is better than that in parallel. Consistent with the numerical simulation, the experimental results indicate that the insensitive points of both FRRs contribute critically to the multiplexing. Due to the relationship between sensitivity and frequency, the loss sensitivity of each FRR can be flexibly adjusted by tuning RF working point. As a result, with the contribution of phase-shift amplification and series strategy, the sensitivity is enhanced and the crosstalk of multiplexing is eliminated virtually. Distinguished from the highly dependence of the conventional OCMI-based sensors on VNA, the OCMI in this work is only utilized to verify the proposed scheme. The multiplexed RF working points can actually be directly calculated without obtaining complete frequency response. Therefore the VNA can be replaced by low-cost RF source and LIA for more accurate and effective sensing in practice. In addition to the loss sensing presented in this work, the proposed scheme can also be extended to measuring other physical parameters which can be linked with loss-based sensors.
Funding
National Natural Science Foundation of China (62175073).
Disclosures
The authors declare no conflicts of interest.
Data availability
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.
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