Sensitivity-improved fiber optic current sensor based on an optoelectronic oscillator utilizing a dispersion induced microwave photonic filter

Naihan Zhang; Muguang Wang; Beilei Wu; Mengyao Han; Jing Zhang; Yan Liu; Guofang Fan

doi:10.1364/OE.440551

1. Introduction

Fiber optic current sensors (FOCSs) based on the Faraday effect have become a potential alternative to traditional inductive current sensors in electric power transmission system [1] and electro-winning industry [2]. Traditional inductive current sensors have drawbacks of high power consumption, large footprint and complex configurations, which limit their application in some harsh environments. In contrast, FOCS not only has the advantages of small size, corrosion free and electromagnetic resistance, but also can measure current with high fidelity due to a large bandwidth and the absence of magnetic saturation. Besides, FOCS avoids the use of extensive copper wiring, and its operation is safer and more flexible, which facilitates the transformation from conventional substations to digital substation automation.

When the current changes, a magneto-optic phase shift is introduced in the FOCS between the left and right circularly polarized lights transmitted in the fiber. The original FOCS was inspired by mature fiber-optic gyroscope technology, whose sensing unit was a Sagnac loop interferometer [3]. Nowadays, most of the commercial FOCSs work in reflection mode, and the interrogation technology is based on irreversible phase modulation [4]. The reason why the reflective FOCS is preferred is that the reflective interferometer is not affected by the Sagnac effect, and the magneto-optic phase shift it produces is twice that of the Sagnac interferometer. These advantages endow the reflective FOCS with excellent performance for both alternating current and direct current (DC) measurements. They can be roughly classified into two types: closed-loop [5] and open-loop [6]. In general, the measurement range of closed-loop sensor is larger than that of open-loop sensor. However, both of them need to monitor the intensity of the interference signal to demodulate the magneto-optic phase shift, which makes them susceptible to light source power fluctuations and system noise. In order to overcome the influence of light source power fluctuations, the polarimetric FOCS [7,8] has been proposed by using two independent photodetectors (PDs) to detect the intensity of two orthogonal polarized lights, and obtaining the magneto-optic phase shift through signal processing. However, the two separate fiber links are affected differently by the environment, which may also cause additional measurement errors. The scheme, which uses integrated-optic polarization splitter and normalization algorithm, can partly solve this problem [9]. In addition, traditional FOCS usually increase measurement sensitivity by using long fiber or special fibers [8,10]. The former will aggravate the impact of environmental factors on the measurement, while the latter is expensive and complex to fabricate.

The optoelectronic oscillator (OEO), a key device of microwave photonics, has been researched since 1990s [11,12]. In the last several decades, it has been explored for its application potential in various fields, such as communication system [13], high speed signal processing [14] and sensing [15]. The OEO is a hybrid oscillating loop, which can form a self-sustained oscillation and generate a radio frequency signal based on feedback from the electrical and optical domains [16]. The low loss and low dispersion of the optical fiber in RF band provide the OEO with high energy storage capabilities. And the optical components have large bandwidth and frequency-independent loss. These characteristics enable OEO to generate oscillating signal with high spectral purity, low phase noise, and wide frequency tuning range. Generally, OEO-based fiber optic sensors rely on microwave photonic filters (MPFs) for mapping the measurand to microwave frequency [17]. Compared with traditional fiber optic sensors, OEO can use mature and high-precision microwave frequency interrogation technology for frequency monitoring, such as digital signal processor or an electric spectrum analyzer (ESA), which makes it possible to achieve high-speed and high-resolution sensing. In recent years, different physical parameter measurements, such as magnetic field [18], transverse load [19], temperature [20] and angular velocity [21], have been demonstrated based on OEO technology to achieve high-performance sensing. The OEO-based FOCS [22] can map the current changes to the frequency domain. Therefore, the influence of system noise, loss, and light source power fluctuation on the measurement resolution and error is weakened to a large extent. However, the frequency shift induced by the change of the OEO phase-matching relationship is limited within a free spectral range (FSR), so the sensitivity is relatively low and the influence of OEO frequency instability on the measurement is more obvious.

In this paper, we propose a novel OEO-based fiber-optic current measurement scheme with improved sensitivity using cascaded dispersion induced MPF [23,24] and two-tap MPF. The center frequency of the cascaded MPF is determined by the phase difference between the optical carrier and ±1st-order sidebands, which is designed to be transformed from magneto-optic phase shift induced by Faraday effect. A small change in optical phase difference will cause a larger shift in the center frequency of the dispersion induced MPF and the frequency shift is no longer limited within an FSR due to the introduction of the MPF. In the experiment, the frequency shift reaches as high as 407.9 MHz as the current increases by 1 A. However, under the same conditions, the frequency only changes 173.1 kHz when the current increases by 1 A in our previous work [22]. Consequently, the current sensitivity is increased by about three orders of magnitude. Since the current variation can be obtained by monitoring the oscillating frequency change in microwave domain, the proposed FOCS also possess a fast demodulation capability. Moreover, the influence of the deviation of the polarization direction after the three PCs on the measurement is also simulated and analyzed.

2. Configuration and principle

Figure 1 illustrates the configuration of the proposed OEO-based FOCS. The linearly polarized light emitted from a tunable laser source (TLS) enters a quarter-wave retarder (QWR) after passing through a polarization controller (PC₁) and an optical circulator (OC₁). The PC₁ makes the polarization direction of the incident light at an angle of 45° to the principal axis of the polarization-maintaining fiber (PMF) before QWR. Therefore, two orthogonal x, y polarized light waves along the PMF principal axes are converted into left and right circularly polarized lights via the QWR. The circularly polarized lights are transmitted along the spun fiber coil and reflected by a reflector, and then they retrace the optical path with the exchanged polarization state. After that, the QWR changes the circularly polarized lights back to linearly polarized lights, and a magneto-optic phase shift Δφ caused by the Faraday effect is added between x and y polarized lights. A PC₂ aligns the x polarized light to the principal axis of the Mach–Zehnder modulator (MZM). As a result, the LiNbO₃ MZM, which has polarization-dependent properties, only modulates the x polarized light. The light wave is sent to the linearly chirped fiber Bragg grating (LCFBG) via an OC₂ and the reflected light is amplified by an erbium-doped fiber amplifier (EDFA). Then the light wave is coupled into a polarization multiplexed composite cavity, which consists of a PC₃, a polarization beam splitter (PBS), two optical fiber paths with different length and a polarization beam combiner (PBC). Thereinto, the PC₃ makes the light polarization direction at a 45° angle to the principal axis of the PBS, then the x, y polarized lights interfere on two principal axes of the PBS (PBS_d and PBS_e). Thus, the interfered optical carrier and sidebands at points d and e are orthogonal. After the photoelectric conversion by a PD, the microwave signal passes through an electrical bandpass filter (EBPF) and is divided into two parts by a power divider (Div). One path is amplified by an electrical amplifier (EA) and sent back to the MZM to form the OEO loop, the other is monitored by an ESA.

Fig. 1. Configuration of the OEO-based FOCS and the schematic of the light polarization state and optical spectrum evolution. (MZM_P and MZM_O: polarization directions in parallel and orthogonal to the principal axis of the MZM, PBS_d and PBS_e: polarization directions aligned with one and the other principal axis of the PBS corresponding to points d and e.).

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The key component in the OEO is a dispersion-induced MPF formed by the TLS, MZM, LCFBG and PD, the function of the MPF is equivalent to that of an electrical filter used to select the oscillating frequency of the OEO. Mathematically, the transfer function of the MPF can be expressed by [25]

(1)$${H_1}({\omega _\textrm{m}}) = \cos ({\omega _\textrm{m}}^2\pi Dc/{\omega _\textrm{c}}^\textrm{2} - \varphi )$$

where ω_m and ω_c are the angular frequency of the microwave signal and optical carrier, D (ps/nm) is the dispersion of the LCFBG, c is the light velocity in vacuum and φ is the phase difference between the optical carrier and ±1st-order sidebands. The 3-dB bandwidth of dispersion-induced MPF is relatively wide and not narrow enough to meet the condition of single-mode oscillation of OEO. Thus, a two-tap MPF formed by a polarization multiplexed composite cavity is employed to finely select the oscillating frequency, due to that it can narrow the equivalent bandwidth of MPF. The transfer function of the two-tap MPF is

(2)$${H_2}({\omega _\textrm{m}}) = |{\sin ({\omega_\textrm{m}}\Delta \tau /2)} |$$

where Δτ is the time delay difference between two optical paths. The FSR of the two-tap MPF can be calculated as 1/Δτ. Therefore, the final MPF response can be written as H(ω_m)=H₁(ω_m)•H₂(ω_m). In theory, only the frequency component, which obtains the maximum gain and meets the phase-matching relationship in the OEO loop, will oscillate. Therefore, the oscillating frequency can be expressed as

(3)$${f_\textrm{m}} = \frac{{{k_1}}}{{{\tau _{\textrm{OEO}}}}} = \frac{{{k_2}}}{{\Delta \tau }} = {f_\textrm{c}}\sqrt {\frac{\varphi }{{\pi Dc}}}$$

where k₁ and k₂ are integers, τ_OEO is the time delay of the OEO loop, and f_c is the frequency of the optical carrier. In the proposed scheme, the DC bias of the MZM is set at the minimum transmission point for carrier suppression modulation. At the output of the MZM, an optical carrier and ±1st-order sidebands with orthogonal polarization states are generated. The phase difference φ between optical carrier and sidebands is equal to the sum of the initial phase difference φ₀ caused by three PCs and the magneto-optic phase shift Δφ caused by the current change, which can be written as φ=φ₀+Δφ. The magneto-optic phase shift is

(4)$$\Delta \varphi = \textrm{4}V{N_\textrm{f}}\int H dz = 4V{N_\textrm{f}}{N_\textrm{s}}I$$

where V is the Verdet constant of the spun fiber (according to experiment results in [22], V≈1 µrad/A). N_f and N_s are the number of spun fiber coils and solenoid turns, respectively. I is the applied current in the solenoid. In this way, Eq. (3) can be rewritten as

(5)$${f_m} = \frac{{{k_1}}}{{{\tau _{\textrm{OEO}}}}} = \frac{{{k_2}}}{{\Delta \tau }} = {f_c}\sqrt {\frac{{{\varphi _0} + 4V{N_\textrm{f}}{N_\textrm{s}}I}}{{\pi Dc}}}$$

When the current increases from 0 A to 1 A, the oscillating frequency shift can be represented as

(6)$$\Delta {f_m} = {f_c}\left( {\sqrt {\frac{{{\varphi_0} + 4V{N_\textrm{f}}{N_\textrm{s}}}}{{\pi Dc}}} - \sqrt {\frac{{{\varphi_0}}}{{\pi Dc}}} } \right)$$

The current measurement resolution R_I is determined by the equivalent FSR of the OEO (i.e., the FSR of the two tap MPF) and the current sensitivity, which can be written as

(7)$${R_I} = {1 / {\left[ {\Delta \tau \cdot {f_c}\left( {\sqrt {\frac{{{\varphi_0} + 4V{N_\textrm{f}}{N_\textrm{s}}}}{{\pi Dc}}} - \sqrt {\frac{{{\varphi_0}}}{{\pi Dc}}} } \right)} \right]}}$$

To date, the mapping relationship between current and microwave frequency has been established. Therefore, the current variation can be measured by simply monitoring the oscillating frequency of the OEO.

3. Experiment results

In order to demonstrate the proposed scheme, an experiment is performed based on the setup shown in Fig. 1. Both the bandwidth and central frequency of the dispersion-induced MPF is related to the dispersion of the LCFBG. Thus, the dispersion needs to be carefully designed. The LCFBG used in our experiment is directly written in a hydrogen-loaded fiber using a phase mask exposed under a 248 nm KrF excimer laser. The reflection spectrum and group delay line of the LCFBG measured by a commercial chromatic dispersion measurement system (CD400, PE. Fiberoptics Ltd) is shown in Fig. 2(a). The central wavelength, bandwidth and the dispersion of the LCFBG are 1549.3 nm, 0.35nm and 2800 ps/nm, respectively. Thus, the wavelength and the output power of the TLS (Emcore Corporation) are set to be 1549.3 nm and 7 dBm to ensure that the optical carrier can be highly reflected. The 3-dB bandwidth of the MZM (JDS Uniphase) is 10 GHz. In order to measure the carrier suppression effect of the MZM, a commercial microwave source (Anritsu, MG3692C) is employed to drive the MZM, and the optical spectrum at the output of the MZM is obtained by an optical spectrum analyzer (YOKOGAWA, AQ6370D). It can be seen from Fig. 2(b) that the carrier suppression ratio can reach as high as 16.96 dB. The PD (Multiplex, MTRX192L) has a 3-dB bandwidth of 10 GHz and a responsivity of 0.8 A/W. In order to eliminate the influence of low-frequency signal on the quality of oscillating signal, an EBPF with a bandwidth of 3.8 GHz (5 GHz-8.8 GHz) is added to select the first passband of the dispersion-induced MPF. An EA (JDS Uniphase, H301-1210) with a 3-dB bandwidth of 10 GHz is used to compensate the loop loss.

Fig. 2. (a) The measured reflection spectrum and group delay line of the LCFBG. (b) Optical spectrum of the carrier suppression modulation.

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Then the frequency response of the cascaded MPF is measured by a vector network analyzer (VNA, Keysight, E5063A), whose sweep bandwidth is set to 4 GHz from 5 GHz to 9 GHz. Figure 3 illustrates the measurement and simulation results. As shown in Fig. 3(a), the final MPF response retains the envelope of the dispersion-induced MPF and superimposes many narrow peaks realized by the two-tap MPF. The FSR of the two-tap MPF is 72 MHz, corresponding to a 2.8 m length difference between two optical paths. Since the optical field amplitudes of the two polarization-orthogonal optical paths are almost the same, the two-tap MPF has a deep notch. The central frequency and bandwidth of the dispersion-induced MPF are about 7.05 GHz and 3.3 GHz from 5.34 GHz to 8.64 GHz, which are basically consistent with the simulation results shown in Fig. 3(b).

Fig. 3. The (a) measured and (b) simulated frequency response of the final MPF (D=2800 ps/nm, f_c=193.502 THz, φ=0.14π, Δτ=13.89 ns).

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A QWR, spun fiber coils and a reflector jointly form a reflective current sensing unit, as shown in the dashed box in Fig. 1. After closing the OEO loop, the current sensing is conducted. In order to fabricate QWR, an elliptical core fiber (Beat length: ∼13.48 mm) is cut with a length of an odd multiple of a quarter beat length and 45° spliced with the PMF. The highly-birefringent spun fiber (IVG Fiber, LBE1300) can resist significant bending and twisting. The reflectivity of the reflector can reach more than 90%. The coil number of the spun fiber is 21, the solenoid has a length of 19 cm and a turn number of 3700. A DC regulated power supply (5A100V, accuracy: 0.1%) is used to adjust the current in the solenoid. Because the current sensitivity is only related to the number of spun fiber coils but not the fiber length. A too long fiber will increase transmitted loss and cost, but a too small fiber coil diameter may slightly introduce linear birefringence and bending loss. Therefore, the above factors need to be comprehensively considered to select the spun fiber length. In the practical application, the length selection of spun fiber coils needs to follow the guidelines of short length, large number of fiber coils and not too small diameter. In the experiment, in order to directly compare the current sensitivity of the two methods under the same conditions, we choose the same number of spun fiber coils as our previous work. When current increases from 0 A to 4A, the oscillating frequency variation is measured by an ESA (Agilent N9010A). The current measurement results are shown in Fig. 4. The black curve is the theoretical result derived from Eq. (5). It can be seen from Fig. 4(a) that the maximum error between the experiment and theoretical result is 0.075 A in this experiment. In the case of same solenoid turns (3700) and same fiber coils (21), it is worth noting that the current sensitivity is improved by three orders of magnitude compared with our previously proposed OEO-based FOCS, in which the oscillating frequency only changes 173.1 kHz when the current increases by 1 A [22]. The reason is that the frequency shift is limited within an FSR of the single-loop OEO (∼MHz) in [22] due to the fact that the magneto-optic phase shift changes the oscillating frequency to meet the OEO phase-matching relationship in principle. While the oscillating frequency varies 407.9 MHz as the current increases by 1 A in this paper. The operating mechanism is that the dispersion-induced MPF can directly convert small optical phase shifts into large microwave frequency variations (more than a 72 MHz equivalent FSR). It can also be seen from Eq. (5) that the current is proportional to the square of the MPF center frequency.

Fig. 4. (a) Oscillating frequency and the corresponding measurement error versus the current. (b) Electrical spectra of the oscillating signals as the current increases.

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4. Discussion

4.1 Resolution and the measurement range

According to the experimental results and Eq. (6), the current resolution can be calculated to be about 0.18 A. The theoretical resolution of the OEO-based FOCS in [22] is better than that in this paper. However, the ESA needs to work at 1 Hz resolution to achieve the theoretical resolution in [22], the better the resolution of the ESA means the slower the scanning speed. Therefore, the OEO needs to be in a very stable state to achieve such an excellent resolution, which is difficult in practical application. The resolution of the proposed scheme in this paper can be further improved by increasing the fiber length difference between the two optical paths in the two-tap MPF. However, a smaller equivalent FSR means a greater possibility of mode hopping. Therefore, it is necessary to make an appropriate trade-off between frequency stability and resolution. Besides, it can be seen from Eq. (6) that the sensitivity can be increased by decreasing the dispersion of the LCFBG or increasing the spun fiber coil number, which can also improve the resolution. Since the frequency response of dispersion-induced MPF is a cosine function, the theoretical measurement range of magneto-optic phase shift is 0-π, corresponding to a current range of 0–9.88 A. In practical applications, an EBPF with a wider bandwidth can be used to achieve an ideal measurement range.

4.2 Causes of measurement errors

Firstly, the frequency stability will affect the measurement results. The ESA is set at “maxhold” mode to evaluate the frequency stability. It can be seen from Fig. 5(a) the frequency shift of the oscillating signal is 63 kHz in 5 mins, which may cause a current measurement error of 0.00015 A. As for our previous work [22], the frequency stability measurement results show that the frequency shift is about 30 kHz in 5 mins, which may cause a current measurement error of 0.197 A. It can be known that when the frequency shifts caused by environment parameters are in the same order of magnitude, the influence of the frequency shift on the measurement error in this paper is about three orders of magnitude smaller than that in our previous work. Environmental factors may affect the equivalent loop delay of OEO, which will change FSR of the OEO. Since the group delay curve of LCFBG is not very smooth, the dispersion of LCFBG may change slightly as the reflection spectrum of LCFBG shifts. And the random variation of the optical polarization state in the single-mode fiber link will cause the change of the phase difference between orthogonally polarized lights. As a result, the center frequency of the MPF will changed. All the above factors will lead to an oscillating frequency fluctuation. In the practical application, the PMFs can be used to construct the fiber links, and LCFBG can be specially packaged to resist environmental interference. Secondly, the excessively high side mode will aggravate the mode hopping, this phenomenon may cause a measurement error of at least 0.18 A. Figure 5(b) shows the equivalent FSR of the OEO is 72 MHz and the side mode suppression ratio (SMSR) is as high as 54.16 dB, which can verify the effective filtering performance of the cascaded MPF. From the experiment results in Fig. 4(a), it can also be known that there is no measurement error caused by the mode hopping. The phase noise level is also measured to evaluate the performance of our proposed OEO-based FOCS more comprehensively. In the sensing experiment, the loop length of the OEO is 48 m, and the phase noise is −74.09 dBc/Hz@10 kHz. The phase noise is decreased to −90.53 dBc/Hz by increasing the loop length to 552 m. However, a longer loop means a greater possibility of mode hopping. Note that the calculated phase noise from the frequency spectrum measured by ESA superimposes the amplitude noise and the noise of the ESA itself, which cannot truly reflect the actual phase noise of the proposed OEO sensing system. As for our proposed OEO-based FOCS, compared to the phase noise, the frequency stability of the oscillating signal has a more important effect on the sensing performance. Therefore, in order to ensure a good sensing accuracy, the OEO need to has a shorter loop length. Besides, the measurement uncertainty determined by the system resolution will also introduce errors to the experimental results. For OEO-based sensing, this uncertainty corresponds to a frequency range of ±1/2FSR (i.e., ±0.085A). Another source of measurement error is the change of the Verdet constant caused by slight temperature variation in the experiment, which can be compensated by designing a retarder with an appropriate deviation degree from perfect quarter wave retardation [26]. In the future, we will consider to further perform the relevant experiments and use the estimation method [27] to evaluate the uncertainty of the current measurement results, which can more accurately analyse the measurement error.

Fig. 5. (a) Frequency stability measurement for 5 mins. (b) The electrical spectrum of an oscillating signal with a 6 GHz frequency.

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5.3 Influence of optical polarization direction deviation

The optical polarization state is an important factor affecting the performance of the FOCS [28]. In our proposed scheme, three PCs need to be set to keep the output light at a specific optical polarization direction: 1. the light polarization direction after the PC₁ should be at an angle of 45° to the principal axis of the PMF before QWR. 2. The x polarized light after the PC₂ should be aligned with the principal axis of the MZM. 3. The light polarization direction after the PC₃ should to be at an angle of 45° to the principal axis of the PBS. If the PCs do not make the light polarization direction in an ideal state, as shown in Fig. 6. then the optical signal after the OC₁ can be expressed as

(8)$${E_{\textrm{OC}1}}(t) = \left[ {\begin{array}{{c}} {{E_x}(t)}\\ {{E_y}(t)} \end{array}} \right] = {E_{\textrm{in}}}(t) \cdot \left[ {\begin{array}{{c}} {\cos {\alpha_1}{e^{j(\Delta \varphi \textrm{ + }{\varphi_\textrm{1}})}}}\\ {\sin {\alpha_1}} \end{array}} \right]$$

where E_in(t)=E₀e^jω_c^t is the incident light before the PMF, ω_c is the angular frequency of the optical carrier. α₁ is the polarization direction angle between the incident light and the principal axis of the PMF. Δφ is the magneto-optic phase shift. φ_n is the phase difference between two orthogonally polarized lights caused by PC_n. Assuming that the x polarized light is at an angle of α₂ to the principal axis of the MZM, then the input signal of MZM can be expressed as

(9)$${E_{\textrm{MZMin}}}(t) = \left[ \begin{array}{l} {E_{\textrm{MZMp}}}(t)\\ {E_{\textrm{MZMo}}}(t) \end{array} \right] = \left[ {\begin{array}{{cc}} {\cos {\alpha_2}{e^{j{\varphi_2}}}}&{ - \sin {\alpha_2}}\\ {\sin {\alpha_2}{e^{j{\varphi_2}}}}&{\cos {\alpha_2}} \end{array}} \right]{E_{\textrm{OC1}}}(t)$$

Fig. 6. The schematic diagram of the optical polarization direction after three PCs. PMF_F and PMF_S: polarization direction aligned with fast and slow axis of the PMF.

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Due to the polarization-dependent properties of the MZM, the light incident along the principal axis of the MZM will be modulated, while the light incident orthogonal to the principal axis will not. Thus, the output signal from the MZM can be written as

(10)$${E_{\textrm{MZMout}}}(t) = \left[ {\begin{array}{{cc}} \begin{array}{l} {J_1}(\beta )({e^{j{\omega_\textrm{m}}t}} + {e^{ - j{\omega_\textrm{m}}t}})\\ \textrm{0} \end{array}&\begin{array}{l} \textrm{0}\\ \textrm{1} \end{array} \end{array}} \right]{E_{\textrm{MZMin}}}(t)$$

where J_n denotes the nth-order Bessel function of the first kind, β is the modulation index. ω_m is the angular frequency of the microwave signal. Assuming that the polarization direction of the output light of PC₃ is at an angle of α₃ to the principal axis of the PBS, the interfered signal after PBS is given by

(11)$$\begin{aligned} {E_{\textrm{PBSout}}}(t) &= \left[ \begin{array}{l} {E_{\textrm{PBSd}}}(t)\\ {E_{\textrm{PBSe}}}(t) \end{array} \right] = \left[ {\begin{array}{{cc}} {\cos {\alpha_3}{e^{j{\varphi_3}}}}&{ - \sin {\alpha_3}}\\ {\sin {\alpha_3}{e^{j{\varphi_3}}}}&{\cos {\alpha_3}} \end{array}} \right]{E_{\textrm{MZMout}}}(t)\\ &= {E_{\textrm{in}}}(t) \cdot \left[ {\begin{array}{{c}} {{E_{\textrm{sd}}}({e^{j{\omega_\textrm{m}}t}} + {e^{ - j{\omega_\textrm{m}}t}}){e^{j{\theta_{\textrm{sd}}}}} + {E_{cd}}{e^{j{\theta_{\textrm{cd}}}}}}\\ {{E_{\textrm{se}}}({e^{j{\omega_\textrm{m}}t}} + {e^{ - j{\omega_\textrm{m}}t}}){e^{j{\theta_{\textrm{se}}}}} + {E_{ce}}{e^{j{\theta_{\textrm{ce}}}}}} \end{array}} \right] \end{aligned}$$

where E_sd, E_se, E_cd and E_ce are the amplitudes of the interfered sidebands and optical carrier at point d and e. θ_sd, θ_se, θ_cd and θ_ce, are the phases of the interfered sidebands and optical carrier at point d and e. Since the center frequency of the cascaded MPF is only determined by the phase difference between the optical carrier and sidebands, here we only discuss the influence of the optical polarization direction on the θ_sd, θ_se, θ_cd and θ_ce. According to the vector operation principle, these phases can be calculated as

(12)$$\left\{ \begin{array}{l} {\theta_{cd}} = \pi + \arccos \left( {\frac{{A + B\cos (\Delta \varphi + {\varphi_1}\textrm{ + }{\varphi_\textrm{2}})}}{{\sqrt {{B^2} + {A^2} + 2A \cdot B\cos (\Delta \varphi + {\varphi_1}\textrm{ + }{\varphi_\textrm{2}})} }}} \right)\\ {\theta_{sd}} = \pi - \arccos \left( {\frac{{C - E\cos (\Delta \varphi + {\varphi_1}\textrm{ + }{\varphi_\textrm{2}})}}{{\sqrt {{E^2} + {C^2} - 2C \cdot E\cos (\Delta \varphi + {\varphi_1}\textrm{ + }{\varphi_\textrm{2}})} }}} \right)\textrm{ + }{\varphi_\textrm{3}}\\ {\theta_{ce}} = \arccos \left( {\frac{{A + B\cos (\Delta \varphi + {\varphi_1}\textrm{ + }{\varphi_\textrm{2}})}}{{\sqrt {{B^2} + {A^2} + 2A \cdot B\cos (\Delta \varphi + {\varphi_1}\textrm{ + }{\varphi_\textrm{2}})} }}} \right)\\ {\theta_{se}} = \pi - \arccos \left( {\frac{{C - E\cos (\Delta \varphi + {\varphi_1}\textrm{ + }{\varphi_\textrm{2}})}}{{\sqrt {{E^2} + {C^2} - 2C \cdot E\cos (\Delta \varphi + {\varphi_1}\textrm{ + }{\varphi_\textrm{2}})} }}} \right)\textrm{ + }{\varphi_\textrm{3}} \end{array} \right.$$

where A = sinα₁cosα₂, B = cosα₁sinα₂, C = sinα₁sinα₂ and E = cosα₁cosα₂. It can be seen from Eq. (11) that θ_se-θ_ce=θ_sd-θ_cd+π, this means that the coefficient of the two-tap MPF is [1−1]. It can also be seen from Eq. (11) that the phase difference between two orthogonally polarized lights φ_n caused by PCs only affects the initial phase difference between the optical carrier and the sidebands (i.e., the initial oscillating frequency), and it will not cause measurement errors after the initial frequency is calibrated. Therefore, we take θ_se-θ_ce as an example and assume that φ_n=0 to discuss the relationship between Δφ and the final phase difference between the optical carrier and the sidebands. It can be known from Eq. (11) that α₃ has no effect on the phases.

We simulated the influence of α₁ and α₂ on θ_se-θ_ce, and the simulation results are shown in Fig. 7. As shown in Fig. 7 (a) and (b), no matter whether α₁ is at the ideal angle (45°), the deviation of α₂ from the ideal angle (0°) will cause θ_se-θ_ce to be unequal to Δφ, and the relationship between θ_se-θ_ce and Δφ is no longer linear. The greater the deviation, the greater the difference between θ_se-θ_ce and Δφ. However, when α₂=0°, no matter whether α₁ deviates from 45°, θ_se-θ_ce is equal to Δφ. When α₂≠0°, the greater the deviation of α₁ from 45°, the greater the difference between θ_se-θ_ce and Δφ, especially when Δφ is closer to π. If θ_se-θ_ce≠Δφ, the relationship between the oscillating frequency shift and the current variation in Eq. (5) is invalid. Therefore, in the experiment, it is necessary to make the PC₂ in an ideal state as much as possible, so that the influence of the polarization direction on the measurement can be minimized. If the PMFs can be used to replace the SMF, the PC₂ can be removed, and the function of the PC₁ and PC₃ can be replaced by 45°-splice. Consequently, this influence can be eliminated.

Fig. 7. The simulated relationship between θ_se-θ_ce and Δφ, when (a) α₁ is at an ideal angle (α₁=45°), (b) α₁ is not at an ideal angle (α₁=27°), (c) α₂ is at an ideal angle (α₂=0°) and (d) α₂ is not at an ideal angle (α₂=18°).

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5. Conclusion

In conclusion, we have proposed and experimentally demonstrated an OEO-based FOCS. Since the dispersion-induced MPF is extremely sensitive to optical phase changes, the current sensitivity is significantly increased by three orders of magnitude compared with our previous work. Due to the improved sensitivity, the influence of frequency instability of the OEO on measurement error is weakened, which reduces the requirements for the OEO working environment. In addition, the long fiber or special fibers in sensing unit and narrow-band electrical filter in OEO loop can be avoided, which makes the proposed scheme more cost-effective. The cost can be further reduced by using a digital signal processor module to replace the ESA and the system complexity can be reduced by using the 45°-splice to replace the PCs or applying the integration technology. Due to the mapping principle of current to frequency, the negative influence of system noise and light source power fluctuation on measurement is greatly weakened. This high sensitivity and high speed optical current sensing scheme can further promote the practical application of FOCSs based on Faraday effect in electric power transmission grids. Since fiber optic sensing can achieve remote measurement, the OEO system can be placed on the remote platform and would not be affected by the magnetic field. So, the shielding and complicated mechanical stabilization can be avoided in the proposed system. The OEO system has a low total power consumption, the power consumption of the MZM is smaller than 10 W, the laser and EA is smaller than 5 W, the PD is only 750 mW and other devices are passive. And integrated technology can be used to further reduce power consumption. In the future, we will integrate and package the OEO system in a box and a simple stabilization scheme may be used to make it more suitable for practical applications, just like the current commercial OEO [29]. Therefore, the OEO-based FOCS can achieve lower power consumption, better frequency stability and phase noise performance.

Funding

National Natural Science Foundation of China (61775015, 61801017, U2006217); Fundamental Research Funds for the Central Universities (2020YJS003, 2021JBZ103).

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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Sensitivity-improved fiber optic current sensor based on an optoelectronic oscillator utilizing a dispersion induced microwave photonic filter

Abstract

1. Introduction

2. Configuration and principle

3. Experiment results

4. Discussion

4.1 Resolution and the measurement range

4.2 Causes of measurement errors

5.3 Influence of optical polarization direction deviation

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (12)

Optics Express