1. Introduction

The interferometric fiber optic gyroscopes (IFOGs) are a kind of angular rate sensor based on Sagnac effect [1,2]. They are started to be investigated in the mid 1970s, opening the way for a fully solid-state rotation sensor [3]. In the past four decades, IFOG has attracted much attention and experienced rapid development. Since the advantages of long life, high reliability and sensitivity, IFOGs have been widely utilized for civilian and military applications [4–6].

In general, the long-term drift and short-term noise of FOG are the two most important performance characteristics, which can be quantified by the bias instability (BI) and the angle random walk (ARW), respectively [1,7]. BI is the long-term output drift from the real rotation rate in gyro. Typically, the environmental variation and polarization non-reciprocity (PN) error are recognized as the major sources [8–10]. ARW characterizes FOG’s white noise component [1], which affects the short-term performance [11] and sensitivity. The white noise is introduced mainly by the relative intensity noise in broadband light source, the thermal phase noise as well as the backscattering of the optical beam inside the fiber optic coil, and the shot noise in photodetector [12–14].

Of the above two performance characteristics, different applications have different concerns. Inertial navigation is the original and major application of IFOGs. When calculating the attitude of the carrier, the integration operation similar to the average is adopted, so the short-term noise is filtered out and only the effect of long-term drift is retained [9,15]. Therefore, for inertial navigation, whether the gyro can achieve small long-term drift is very important. However, in rotational seismology, the short-term noise is of greater concern instead, as the major seismic signals are directly measured without any filters and distribute within a frequency range of 0.01 Hz to 100 Hz [16–18]. Hence, the short-term noise directly limits the detection sensitivity. However, there is no conflict between BI and ARW in essence and in more special applications, the reference-grade IFOGs are required to guarantee both low BI and low ARW meanwhile.

Dual-polarization configuration is proposed recently, in which two orthogonal polarized light beams co-exist in the fiber coil and sense the rotation simultaneously for a better signal-to-noise ratio (SNR). In dual-polarization IFOG, BI and ARW are still the most significant characteristics we concerned. Coherent phase error and dual-polarization-intensity-type error are two kinds of concomitant errors that pose threat to the long-term perfrmnace of dual-polarization IFOG. The coherent phase error comes from the interference between a primary wave and a dual-polarization coupling wave. Therefore, the coherent phase error can be suppressed by reducing the coherence of the primary wave and dual-polarization coupling wave, such as adding a fiber delay line in one polarization or using two independent light sources to generate two polarizations [19]. And the dual-polarization-intensity-type error is the parasitic interference between two dual-polarization coupling waves, and reverse phase modulation is proposed as an effective reduction approach [20]. In addition to the BI, the ARW of dual-polarization IFOG has also been studied. The excess relative intensity noise (RIN), originates from the random beat of different frequencies of the light, is the primary cause of short-term noise in most IFOGs [21–23]. Many efforts have been made to suppress excess RIN, and the main idea is to introduce a reference arm [24,25]. For dual-polarization IFOG, in the Ref. [26], the two orthogonal polarized light beams have strong correlation and become reference arm to each other, and thus the excess RIN is effectively compensated owing to the opposite parities. The thermal phase noise, which also degrades the short-term performance especially when the length of fiber coil is several kilometers long. Fortunately, high order eigen frequency modulation is proved to be an effective method to suppress the walk-off component of thermal phase noise [27].

Although the above four major noise or error have been studied and suppressed with their corresponding methods, the long-term drift and short-term noise have still not been suppressed meanwhile in dual-polarization IFOG before. The reason is that there are some conflicts among them if we want to reduce the four major noises meanwhile. For example, the suppression of the coherent phase error requires two light sources or a delay fiber line, whereas the reduction of the excess relative intensity noise demands only one light source and the delay fiber line is undesired. It means that these two kinds of noise cannot be suppressed in the meantime as their configurations are mutually exclusive.

In this paper, we propose a newly designed dual-polarization IFOG configuration based on a four-port circulator to improve the BI and ARW performance meanwhile. First, we introduce the new dual-polarization IFOG scheme and analyze the two kinds of polarization non-reciprocity (PN) errors and excess RIN by using Jones matrix method. With the combination of four-port circulator and fiber delay line, the coherence between primary and the coupling waves is reduced greatly whereas the RIN correlation of the two primary waves is significantly guaranteed. Then the effectiveness and comparison of the scheme based on a four-port circulator is investigated experimentally. By using a 2-km polarization maintaining coil with open-loop configuration, an ARW of $5.0\times 10^{-5\circ }/\sqrt {\mathrm {h}}$ and a BI of $9.0\times 10^{-5\circ }/\mathrm {h}$ are achieved in detecting the rotation rate of Earth. The results confirm that the proposed dual-polarization IFOG scheme can effectively improves the short-term sensitivity and the long-term stability meanwhile.

2. Theory and analytical model

2.1 Propagation process

The dual-polarization IFOG based on a four-port circulator is schematically illustrated in Fig. 1(a). First, a polarizer is placed after the light source to guarantee the highly correlative RIN in two output beams of the polarization maintain (PM) coupler. A fiber delay line is set before the Port 3 to generate an optical path difference and reduce the coherence of two light beams temporarily. Then the two beams are polarized by two multi-functional integrated optical chips (MIOCs) again and each beam is split and exists from the two ports of MIOCs, which crresponding to clockwise (CW) and counter-clockwise (CCW) paths, respectively. The two output ports of MIOC 1 are connected to the two $o$-ports (ordinary ports) shown in Fig. 1(c) of the two polarization beam splitters/combiners (PBS/Cs), while the ports of MIOC 2 are connected to the two $e$-ports (extraordinary ports). The PBS/Cs couple the orthogonal polarized waves into a PM fiber coil by the $com.$-ports (combined-ports) and ensure that light beams of two polarizations return to different MIOCs after transmitting along the coil. Afterwards two signal waves return to the Port 2 and Port 4. Note that the waves at Port 2 have no delay line path yet until guided to Port 3 and then experience the same fiber delay line. The waves at Port 4 have one time delay line path already, and will be guided to Port 1 in which no delay line is set. Finally, the two waves are detected by two photodetectors (PD 1 and PD 2), respectively.

 

Fig. 1. (a) Scheme of the dual-polarization IFOG; (b) transmission diagram of four port circulator; (c) the inside structure and ports’ definition of PBS/Cs.

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The four-port circulator plays two vital roles: first, it places an optical path difference between the dual-polarization coupling waves and primary waves before entering at PDs, which means the residual coherent phase error can be avoid. Secondly, both the two signal waves carrying Sagnac phase pass through the delay coil only once, and hence there is no optical path difference at photodetectors and the excess RIN is high related. The detail noise analysis will be expressed in the next subsection.

2.2 Analysis model of the dual-polarization IFOG based on the four-port circulator

The Jones matrix method is utilized to analyze the light porpogation in this configuration. As shown in Fig. 1(a), the incident light beam $\mathbf {E}_0$ is polarized with a high polarization extinction ratio (PER) and then split by a 50:50 coupler. The Jones matrices of the incident light, the polarizer and coupler can be expressed as:

An MIOC is regarded as a combination of a polarizer, a coupler and two modulators with opposite modulation phase shown in Fig. 2. The input polarized light beams will be polarized again with the extinction of $\varepsilon$ and the $y$-polarization component becomes $\varepsilon ^2 E_y$. Since the extinction coefficient $\varepsilon$ is quite small, the term $\varepsilon ^2 E_y$ can be neglected. Shown as Fig. 3(a), the electrical field of the four output light leaving MIOCs can be expressed as:

 

Fig. 2. An equivalent model of MIOCs.

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Fig. 3. The schematic diagram of (a): PBC; (b): PBS.

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The PBS/C is based on the birefringent crystal shown in Fig. 1(c), and the crystal will polarize the light again. For the polarized light incident to the $o$-port, it can pass through directly and become the $x$ polarized light of the fiber which still propagates in fast axis. However, for the polarized light incident to the $e$-port, the propagation axis will change from fast to slow axis, so called $y$ polarized light, which just like a $90^{\circ }$ fiber splicing point. Hence, the transfer matrices of the PBS/Cs can be written as:

The transfer matrices of the fiber coil in the CW and CCW directions can be written as [30–32]:

Note that $\theta _o\approx 0^{\circ }$ and $\theta _e\approx 90^{\circ }$, all the $y$-polarization of $\mathbf {E}_\mathrm {rx1}, \mathbf {E}_\mathrm {rx2}, \mathbf {E}_\mathrm {ry1}, \mathbf {E}_\mathrm {ry2}$ will be suppressed again by the MIOCs. Therefore, the $y$-polarization of the four beams can be overlooked and only the $x$-polarization is taken into consideration. The four beams then will pass through the MIOCs, the four-port circulator and return to photodetectors (PDs) finally. Note that there is a time delay from port 2 to port 3, and thus the primary $\hat {\mathbf {x}}$ component electrical field of light on both PDs can be expressed as follows:

2.3 Theory of PN error suppression

In PD1, the primary reciprocal waves are $\mathbf {E}_\mathrm {xx}^\mathrm {CW}$ and $\mathbf {E}_\mathrm {xx}^\mathrm {CCW}$, the Sagnac phase can be derived from the interference of them without PN phase error. The main polarization coupling waves are $\mathbf {E}_\mathrm {yx}^\mathrm {CW}$ and $\mathbf {E}_\mathrm {yx}^\mathrm {CCW}$, which couple from $y$ polarization to $x$ polarization while passing through the fiber coil. Although the coupling waves can be reduced by increasing the refractive index difference of fast and slow axis, they can not be avoid absolutely. Moreover, these coupling waves are not extincted by any polarizer as they both pass through the two MIOCs in non-extinct axis and pose a potential threat to the purity of Sagnac phase extraction. Specifically, if the delay fiber is removed and $\phi _D = 0$ in Eq. (16), the coupling waves can interfere with the primary waves and itself, causing the prime PN phase error.

There are two kinds of PN phase error called coherent phase error and dual-polarization-intensity-type error in this scheme. With the help of the fiber delay line whose length $L_D$ is much larger than the cohence length of light source, the primary waves $\mathbf {E}_\mathrm {xx}^\mathrm {CW}$ and $\mathbf {E}_\mathrm {xx}^\mathrm {CCW}$ will not interfer with the coupling waves $\mathbf {E}_\mathrm {yx}^\mathrm {CW}$ and $\mathbf {E}_\mathrm {yx}^\mathrm {CCW}$, thus the coherence phase error can be well-handled. However, the interference between coupling waves will sill exist and interfere without reverse modulation, causing the dual-polarization-intensity-type error. The light intensity of the two PDs according to the Eqs. (16) and (17) can be expressed as :

2.4 Theory of RIN suppression

RIN is the prime noise that degrades the ARW, and thus it is necessary to elaborate the principle of RIN suppression. For the excess RIN of the light source, it is usually modeled as $I_0=\overline {I}_0+I_N(t)$ [22], where $\overline {I}_0$ is a non-time-varying constant, $I_N(t)$ is the random light intensity fluctuation noise that exhibits white noise characteristics within the limited bandwidth of PDs. In dual-polarizaiton IFOG, the RIN in the two orthogonal polarized light has opposite parities and strong correlation, which means the most direct method is use them to suppression each other. Firstly, it is not troublesome to get rid of the DC part of $I_\mathrm {PD1}$ and $I_\mathrm {PD2}$ by substracting the average value. Then based on the RIN suppression method in Ref. [26], the signal preprocessing can be applied to the IFOG outputs of PD 1 and PD 2 as follows:

By adjusting the length of the delay fiber coil to meet with $\tau _d = K/f_m, (K=1,2,3\dots )$, $\sin (n\omega _m\tau _d)=0, \sin [(2n-1)\omega _m\tau _d/2] = 0$ and the first four harmonics reach the highest amplitudes. The odd and even harmonics can be recombined as Eqs. (22) and (23). However, since the length of delay fiber can be calculated by $L_D=\tau _dc/n=2KL$, the shortest ideal length is about 4090 m, which is absolutely unacceptable. Fortunately, high-order eigen frequency modulation is proposed [27] and this can shorten the fiber delay length greatly according to the above relationship. For instance, if $k$-th eigen frequency is adopted, the shortest ideal length of the fiber delay line becomes

By seperating the odd and even harmonics, the amplitude of each harmonics becomes twice the original. When the IFOG is at rest or under a small rotation, $\phi _s$ is close to zero, the value of $I_{\mathrm {sub}}(t)$ approaches zero whereas the $I_{\mathrm {add}}(t)$ is still comparable to the original outputs. It means the excess RIN in Eqs. (18) and (19) main contibuted by even harmonics, thus the background RIN of odd harmonics in Eq. (22) is greatly reduced.

Futhermore, a numerical simulation is carried out to verify the statement where a series of typical parameter values are all consistent with the following practical experiments. The coil is set with the length of 2045 m and diameter of 14 cm. The modulation frequency $f_m$ is set as the eigen frequency of the fiber coil which is about 50 kHz. The RIN is added on the original light power $\overline {I}_0$ with zero-mean Gaussian white noise and in the two PDs , the $I_N(t)$ are totally the same. The $\overline {I}_0$ is set as 1 mW. The RIN level is determined by the SNR of source intensity as $20\mathrm {log}_{10}({\overline {I}_0}/I_N(t))$, and is set to 60 dB. The simulation results are shown as Fig. 4 and it is suggested that the amplitude of each harmonics has been doubled and the noise floor of odd harmonics declines significantly. With the RIN-suppression and sensitivity-enhanced processing, the short-term performance of the dual-polarization IFOG can be elevated effectively.

 

Fig. 4. A simulation of the power spectrum analysis by fast Fourier transform: (a) the original signal of PD 1 or PD 2; (b) the comparison of the added signal and subtracted signal.

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To sum up, by adopting the fiber delay line with ideal length and the proposed RIN-suppression method, the compensated outputs of two PDs get the odd and even harmonics seperated and the SNR of harmonics are also get enormously enhanced in the mean time. Therefore, the ARW of the dual-polarization IFOG will be improved effectively in theory.

3. Experimental results

We further present an experimental verification and comparison of the proposed configuration. In the setup shown in Fig. 1, all devices are pretest to make sure their performance are stable even in complicated environmental condition. An amplified spontaneous emission (ASE) light source (average light intensity $I_0$ = 15 dBm, central wavelength $\lambda _c$ = 1550 nm, spectral width $\Delta \lambda$ = 40 nm, rectangular spectrum) with degree of polarization (DOP) below 1.0${\%}$ is adopted. The PM fiber coil is wound using the quadrupole symmetric method with a fiber length of 2045 m and an average diameter of 14 cm, which gives an enclosed area of 71.6 $\mathrm {m}^2$ and an eigen frequency of 50 kHz. The 11th eigen frequency is applied to reduce thermal phase noise and hence the shortest available length of fiber delay line is about 371.8 m. The modulation depth is set to 2.7 rad to obtain the optimal SNR of the sinusoidal modulation [21]. The detected signals of PD 1 and PD 2 are acquired using a 24-bit ADC PXI-5922 to reduce the artifacts caused by the quantization. The sampling rate of 6 million samples per second (MSPS) and the sampling length of 30000, which means the rotation rate output rate is 200 Hz.

Besides the scheme based on the four-port circulator named new-proposed scheme, another two dual-polarization schemes shown in Fig. 5 named ARW-comparison scheme [26] and BI-comparison scheme [19] are built up for the comparison as well. In the ARW-comparison scheme, the fiber delay line is removed to guarantee the strong correlation of the two orthogonal polarized light, hence the RIN can be effectively suppressed and the ARW is impressive, whereas the coherent phase error is still remained and will degrades the BI performance; The BI-comparison scheme is based on two independent light sources, hence the coherence phase error is avoided and the BI is better, whereas the RIN cannot be suppressed resulting in a poor ARW. Furthermore, these three kinds of configurations are all utilized the same one fiber coil and placed in the same environment to avoid the performance difference comes from coils and external noise.

 

Fig. 5. The configurations of (a) the ARW-comparison scheme and (b) the BI-comparison scheme.

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First, we need to calculate the correlation of the RIN carried by the two orthogonal polarizations in the above three schemes, which is an important prerequisite for an effective suppression [25]. The signals of PD 1 and PD 2 are acquired without applying any modulation on the MIOCs, respectively. The degree of correlation is calculated and presented in Fig. 6, and the highest correlation of the two polarizations in the new-proposed scheme reaches $99.5\%$, which is as high as the ARW-comparison scheme. On the contrary, since the RIN in two light source is independent totally, the highest correlation in BI-comparison scheme is as low as $2\%$ apparently.

 

Fig. 6. The normalized correlation coefficient of the three kinds of dual-polarization configuration.

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We carried out long-term stability tests, targeting the local equivalent rotation velocity of earth, which is $9.667^\circ /h$ at our laboratory’s latitude (39.99$^\circ$N). Figure 7(a) shows a 9-h test rotation rates demodulated from the signals of PD 1, PD 2, and the compensated output of the new-proposed scheme. It is clear that, compared with the two raw output, the compensated output shows remarkably less random noise, which validates the effectiveness of the RIN suppression in this new scheme. Besides, Fig. 7(c) shows the compensated output comparison of the above three schemes. Since the RIN in two polarizations of the new-proposed scheme and ARW-comparison scheme maintain high correlation and is well-suppressed, they have the similar random noise which are both much lower than the BI-comparison scheme without RIN suppression.

 

Fig. 7. (a) The time-domain outputs of a 9-h test of the new-proposed scheme, and (b) detail 200-ms output of the test in (a); (c) The comparison of the time domain outputs of three schemes.

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In addition, the Allan deviation curves and the root power spectral density (PSD) method are adopted to quantitatively evaluate the performance of the above configurations. It is found that the ARW and BI are significantly improved in this dual-polarization configuration based on the four-port circulator, which are $5.0\times 10^{-5\,\circ }/\mathrm {\sqrt {h}}$ and $9.0\times 10^{-5\,\circ }/\mathrm {h}$, respectively. The RIN suppression method and high-order eigen frequency modulation enhance the short-term sensitivity greatly. Meanwhile, the combination of the fiber delay line and four-port circulator avoid the PN error and coherence error, which makes the long-term drift significantly improved. The PSD results are shown in Fig. 8(b), in which the root PSD floor reaches $20\,\mathrm {nrad/s/\sqrt {Hz}}$ and is flat in the frequency range of 0.001 Hz to 30 Hz. As for the frequencies higher than 30 Hz, the root PSD estimates may be disturbed by ambient vibrations, which cannot be completely excluded in our laboratory environment. Compared with the ARW-comparison scheme and BI-comparison scheme, the ARW and root PSD of this new-proposed scheme are much lower, and the BI is also the lowest because of the less coherence phase error.

 

Fig. 8. (a) The Allan deviation curves and (b) the root power spectral density of the above three kinds of schemes’ 9-h outputs.

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Table 1 shows the detail comparison, it is no doubt that a significant performance improvement of the new-proposed scheme is achieved. The ARW and root PSD are reduced by 8-fold, and the bias instability is reduced by almost 9-fold compared with the raw output. The test results are in good agreement with our previous analysis and indicate the new dual-polarization scheme proposed in this paper can effectively guarantee both ARW and BI. In addition, SRS-5000 and iXblue prototype FOG are also presented in Table 1, which are considered as the highest performance FOG. Compared with them, our laboratory test achieves similar performance with smaller enclosed area of the fiber coil, providing a more scheme choice for building the highest performance FOG.

Table 1. Performance comparison of the high-performance IFOG.

View Table

4. Conclusion

In this paper, a newly designed dual-polarization IFOG based on a four-port circulator is proposed and the propagation process is detail introduced. Four kinds of primary noise related to the long-term drift and short-term sensitivity, which include coherence phase error, dual-polarization-intensity-type error, excess RIN and thermal phase noise, are well handled meanwhile. Laboratory tests to confirm the performance and stability of dual-polarization IFOGs are conducted, and three schemes are placed in the same environment to detect the Earth’s rotation rate with a 2-km PM coil. The long-term and short-term performance of our newly designed dual-polarization IFOG is well optimized meanwhile with a ARW of $5.0\times 10^{-5\,\circ }/\mathrm {\sqrt {h}}$ and a BI of $9.0\times 10^{-5\,\circ }/\mathrm {h}$. And the power spectrum density of $20\,\mathrm {nrad/s/\sqrt {Hz}}$ is almost flat from 0.001 Hz to 30 Hz. The above test results indicate that this dual-polarization IFOG is of great significance for promoting the reference-grade performance IFOG and the practical implementation in sensing feild is in great feasibility.

A. Four-port Circulator

The schematic diagram of the four-port circulator used in this work is shown in Fig. 9, in which the four ports are marked as $\mathrm {Port}\,i, (i=1, 2, 3, 4)$. This device consists of two polarization beam splitters, two prisms, one $45^{\circ }$ Faraday rotator, and one half-wave plate. As shown in Fig. 9(a), the light waves incident to the Port 1 will firstly be split to the ordinary and extraordinary waves through transmission and reflection, respectively, at the interface of the prims that constitute the cube [36]. Then the ordinary waves will pass through and arrive at the Faraday rotator directly, while the extraordinary waves will reflected by the prism and reach the Faraday rotator as well. For a linearly polarized left-to-right direction wave, the net polarization rotation angle through the combination of the Faraday rotator and the half-wave plate is $90^{\circ }$, but for a right-to-left direction wave, the rotation angle is zero. In the left-to-right direction, the Faraday rotator imparts a polarization rotation of $45^{\circ }$ (clockwise) and the phase plate rotates the light another $45^{\circ }$ (also clockwise), hence we get a net $90^{\circ }$ clockwise (CW) rotation.

 

Fig. 9. The inside configuration of the four-port optical circulator and the optical wave transmission path: (a) from Port 1 to Port 2; (b) from Port 2 to Port 3. PBS indicates a polarizing beam splitter

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In the right-to-left direction, the phase plate rotates the light in the same direction (in relation to the propagation direction) as before, that is, counter-clockwise (CCW) at $45^{\circ }$. The Faraday rotator however rotates the polarization in the opposite direction (in relation to the propagation direction) as it did before, that is, clockwise by the same $45^{\circ }$. That means the two polarizations are rotated in the opposite direction and there is no net change in polarization. Therefore, for the light transmit from Port 1 to Port 2, the two orthogonal linearly polarized waves will be rotated by $90^{\circ }$, and the ordinary and extraordinary waves are exchanged meanwhile. Finally, with the combination of the prism and PBS, the two waves are combined together and exist from the Port 2. Based on this onfiguration, the circulator loops in the sequence Port 1 $\rightarrow$ 2 $\rightarrow$ 3 $\rightarrow$ 4 $\rightarrow$ 1 among the four ports as shown in Fig. 1(b).

Funding

National Key Research and Development Program of China (2019YFC1509501); National Natural Science Foundation of China (62201016, U1939207).

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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