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Abstract
For an interferometric fiber-optic gyroscope (IFOG), the angle random walk, which represents the sensitivity of rotation detection, is mainly limited by the relative intensity noise (RIN) of a broadband source. Using a single-mode fiber ring resonator (SM-FRR) to filter the spectrum of a broadband light source is a common strategy for reducing the RIN at the proper IFOG frequency. However, this method depends on the polarization cross-coupling within the SM-FRR. We model the effect of polarization cross-coupling on the SM-FRR. Then, to further reduce the RIN, we introduce a polarization-maintaining fiber ring resonator (PM-FRR), which mitigates the effect of polarization cross-coupling on the SM-FRR. Using the PM-FRR as a spectrum filter, the RIN is reduced to –143 dB/Hz, with a reduction ratio of 25 dB, and the angle random walk in the IFOG is improved by over five times from 1.17 to 0.223 mdeg/h1/2 using a 2.1 km sensing coil.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Broadband sources, especially those with amplification of spontaneous emission (ASE), are widely used in interferometric fiber-optic gyroscopes (IFOGs) [1,2]. There are two essential parameters in IFOGs, namely, the angular random walk (ARW) and bias instability (BI) [1,3,4]. The ARW is generally limited by the noise of optical sources, while the BI mainly depends on reciprocity, which is generally related to the length of the fiber coil. When using broadband sources, the ARW is inversely proportional to the coil length, and the BI is highly correlated with length. Moreover, the ARW is limited by the source relative intensity noise (RIN), accounting for at least 90% of the total noise power [5]. Hence, the RIN should be suppressed and the ARW improved while using a short fiber coil.
Various optical methods have been proposed to reduce the RIN [6]. Hakimi et al. [5,7] and Suo et al. [8] used a semiconductor optical amplifier to reshape the spectrum to increase the 3 dB linewidth. Two RIN subtraction architectures were established by iXblue [9]. Zheng et al. [10] added a Faraday rotator mirror after the coupler to reduce the RIN at the proper frequency. Electronic methods have also been implemented to reduce the RIN. Overmodulation is commonly adopted to mitigate noise by using a modulation phase close to π instead of π/2 in an IFOG [11]. Killian et al. [12] used two photodetectors to detect the optical power and determined the time difference of light intensity fluctuations by cross-correlation to offset the RIN. Zhang et al. [13] used a Wiener filter for intensity noise reduction.
However, the abovementioned approaches show a limited RIN suppression of approximately 10 dB, and the RIN still dominates optical noise. To further suppress the RIN and improve the ARW, a single-mode fiber ring resonator (SM-FRR) can be added after the broadband source as a spectral filter. This can be done because the RIN spectrum corresponds to the autocorrelation of the optical spectrum [14,15]. Various depolarized IFOGs have been implemented using this method [16,17]. Nonetheless, the polarization cross-coupling in the SM-FRR limits RIN suppression.
We propose a model to analyze the effects of polarization cross-coupling on the spectrum and a method for RIN suppression. We implemented a polarization-maintaining fiber ring resonator (PM-FRR) and IFOG to evaluate noise suppression and ARW improvement experimentally. By adding the PM-FRR, the proposed IFOG achieves the same ARW as that of an IFOG with a much longer fiber coil. The suppression of RIN only depends on the performance of PM-FRR. If the same ARW improvement is achieved in a traditional IFOG and an IFOG with PM-FRR, the length of sensing coil is inversely proportional to the ARW improvement ratio.
2. Principles
2.1 Noise of broadband sources
Noise in an IFOG mainly comprises the photon shot noise, RIN, and thermal noise, whose power values are denoted as σ2sh, σ2RIN, and σ2th, respectively. As these noise sources are uncorrelated, the total noise power is given by σ2total = σ2sh + σ2RIN + σ2th. Let Rf be the detector resistance and V0 be the detector bias voltage. The output voltage of the detector V is given by
In current IFOGs, the typical detection light power is in the order of tens of microwatts. Thus, σ2sh and σ2th are much smaller than σ2RIN, and further reducing the RIN power can reduce the total noise power.
The RIN is closely related to the spectrum shape. Let S(υ) be the power spectral density of the optical spectrum. The power spectral density of RIN in broadband sources can be expressed as [14,15]
where $P = \int_0^\infty {S(\upsilon )d\upsilon }$ is the average power of light interference.The setup of a RIN analysis system is shown in Fig. 1. The power spectral density of the detector voltage of RIN, SV(f), is analyzed by an electronic spectrum analyzer. In addition, a multimeter measures the DC signal of detected voltage V. In this case, the RIN is given by
where $\Delta V = V - {V_0}$ and $\overline {\Delta V}$ is the $\Delta V$ average over time.2.2. ARW calculation
The ARW of IFOG is affected by the light source power and modulation depth. Let φm be the modulation depth and φn be the phase noise. The output power of an IFOG at rest is given by
where P0 is the detection light power at φm = 0. As ${\varphi _\textrm{n}}$ is nearly zero, the phase noise can be calculated as where $\Delta P$ is the power fluctuation of the light source and $\Delta P = P - {P_0}$. According to the definition of ARW, it can be expressed as [18]3. Experiments and results
3.1 SM-FRR
A diagram of the proposed SM-FRR is shown in Fig. 2. The SM-FRR modulates the optical spectrum periodically and reduces the RIN at proper frequency fp of the IFOG, where the resonator length equals the IFOG coil length. To increase the resonator finesse, an erbium-doped fiber amplifier is inserted in the resonator, forming an active SM-FRR.
The spectral response of the resonator is calculated as [16]
3.2 Polarization cross-coupling
In the SM-FRR, polarization changes along with light transmission in the resonator because of the residual birefringence in single-mode fiber, constraining the RIN reduction to FSR/2. We propose a model to numerically analyze the effects of polarization cross-coupling. Polarization cross-coupling can be equivalently modeled as the polarization rotator shown in Fig. 3.
See Supplement 1 for details on the spectral response calculation in the SM-FRR with polarization cross-coupling, where the equivalent rotation angle of the polarization rotator is θp and the corresponding coupling ratio is Kp = 1 – cos2θp. Kp = 0 indicates no polarization cross-coupling in the resonator, corresponding to an ideal SM-FRR.
The effects of polarization cross-coupling ratio Kp on spectral transmission ${T_{FRR}}(\nu )$ and RIN suppression are illustrated in Figs. 4(a) and (b), respectively. With increasing polarization coupling, RIN suppression shows a downward trend. When the polarization cross-coupling ratio increases from 0 to 0.1, RIN suppression by the SM-FRR worsens by 3.2 dB, from 15.58 to 12.38 dB. Because polarization cross-coupling is pervasive in single-mode fibers and cannot be eliminated in the corresponding systems, RIN suppression is limited.
Fig. 4. (a) Spectral transmission of SM-FRR with polarization cross-coupling. (b) RIN frequency response of SM-FRR with polarization cross-coupling. The inset shows the RIN frequency response according to coupling ratio Kp at FSR/2. The negative gain corresponds to suppression. For these simulation results, LFRR = 2.1 km, G = 6.8 dB, K1 = K2 = 0.5, αc = 0.3 dB, and αf = 0.2 dB/km. Kp ranged from 0 to 1 in intervals of 0.2.
3.3 PM-FRR
We propose a PM-FRR RIN suppression architecture, whose diagram is shown in Fig. 5. The architecture handles the polarization cross-coupling of the SM-FRR. Two polarizers are added to the optical fiber path in the PM-FRR. The polarizers allow light to pass from the slow axis and block light from the fast axis. One polarizer is added after ASE with central wavelength at 1540 nm, spectral width of 6 nm, and light power of 10 mW. Another polarizer is placed before the fiber segment in the resonator to eliminate the effect of polarization cross-coupling and maintain Kp close to 0.
Fig. 5. Diagram of PM-FRR setup; (MIOC, multifunctional integrated optical circuit). The RIN analysis system is connected to switch A for testing the RIN frequency transfer and an IFOG is connected to switch B for testing the improvement in ARW. The diagram of the RIN analysis system is shown in Fig. 1.
With the PM-FRR, polarization cross-coupling is eliminated, and Kp is nearly 0. MFLI and KEITHLEY 2010 is used in RIN analysis system to measure the noise and DC signal, respectively. The experimental results with and without the PM-FRR are shown in Fig. 6. The PM-FRR improves RIN suppression at 25 dB from −118 to −143 dB/Hz around f = FSR/2 with a 6 nm-linewidth ASE. There is an 8 dB improvement compared with a –17 dB suppression of the SM-FRR [17]. The corresponding phase RIN power spectral density is reduced from 1220 to 68.6 nrad/Hz1/2 in an optical interferometer with a π/2 modulation depth.
Fig. 6. RIN spectrum of PM-FRR output. The blue solid line shows the RIN of the ASE output. The red solid line shows the experimental output with the PM-FRR, and the black dashed line shows that from simulations. The output power of the EDFA is 70 mW. The other device parameters for simulation are the same as those indicated in Fig. 4.
3.4 ARW improvement
The RIN at proper frequency fp degrades the ARW of the IFOG, as shown in Section 2.2. We use a PM-FRR as a spectral filter added after the source for coupling in an IFOG, whose modulation and demodulation is performed at fp with a π/2 modulation depth and a square wave. Because the RIN is considerably suppressed in the resonator at fp, the ARW of the IFOG is improved accordingly.
We constructed an IFOG as illustrated in Fig. 5 (switch B). The length of both the optical fiber in the PM-FRR and fiber sensing coil was 2.1 km, and the diameter of the sensing coil was 11 cm. Thus, the proper frequency of the IFOG and FSR/2 was approximately 48 kHz. The output power of the EDFA is 70 mW. The other parameters in the resonator were set as in the simulations (Section 3.2).
By acquiring the IFOG output at 840 Hz and 3 dB measurement bandwidth of 100 Hz, we obtained the angular velocity in the time domain. Then, the Allan deviations of the outputs with and without the PM-FRR were calculated. The results are shown in Fig. 7.
Fig. 7. (a) Time-domain outputs of test for IFOG at 840 Hz. (b) Allan deviation of IFOG with (orange lines) and without (blue lines) PM-FRR.
The ARW can be obtained from the Allan deviation. The ARW of the IFOG improves by over five times when using the PM-FRR, from 1.17 to 0.223 mdeg/h1/2, which corresponds to the ARW of an IFOG with a much longer fiber coil length and equal parameters as those evaluated, compared with only three times ARW improvement in SM-FRR [17]. In addition, the RIN is no longer dominant in the IFOG operation. In fact, the shot noise accounts for approximately half of the total noise, indicating that the ARW can be further improved by increasing the power of the light source.
4. Conclusion
We improve the ARW of an IFOG using a PM-FRR to reduce the RIN near the proper frequency of the IFOG and demonstrate our method theoretically and experimentally. We introduce a model to analyze polarization cross-coupling on an SM-FRR and use a PM-FRR to eliminate this effect. Experimental results shows that the RIN affecting the ARW is suppressed by 25 dB. By using the PM-FRR, the ARW in the designed IFOG is improved by more than five times, from 1.17 to 0.223 mdeg/h1/2, indicating that noise is suppressed by approximately 80.9%, and the reduced power of RIN is comparable to that of shot noise. Our method can improve the design of high-precision IFOGs without increasing the sensing coil length and diameter.
Funding
National Natural Science Foundation of China (62075193); Natural Science Foundation of Zhejiang Province (LD22F050002); Major Scientific Project of Zhejiang Laboratory (2019MB0AD01); Ten Thousand Talent Program of Zhejiang Province (2017r51010).
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.
Supplemental document
See Supplement 1 for supporting content.
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