1. Introduction

At present, interferometric fiber optic gyroscopes (IFOGs) have been widely utilized in numerous studies and applications owing to their high precision and portability [1–3]. In general, the performance of a gyroscope is evaluated in terms of long-term drift and short-term noise [4]. For inertial navigation, which is the original application of a gyroscope, low long-term drift is an important attribute to reduce accumulative errors [5,6]. By contrast, short-term noise, which directly limits the detection sensitivity, is of greater concern in rotational seismology, another rapidly growing gyroscope field in which all aspects of rotational motions induced by earthquakes, explosions, and ambient vibrations are studied [7,8]. Such application requires the sensors to be able to measure the amplitudes in order of $10^{-7}\,\mathrm {rad/s}$ within a frequency range of $0.01\,\mathrm {Hz}$ to $100\,\mathrm {Hz}$ [9,10]. Large-frame ring laser gyroscopes (RLGs) are first adopted for this purpose owing to their extremely high sensitivity (better than $\mathrm {nrad/s/\sqrt {Hz}}$) [11–14]. However, they are large and have an extremely delicate operation; in comparison, IFOGs usually lack the required high sensitivity, but are small and robust [15]. As a result, a series of studies have been conducted to develop a portable IFOG with high sensitivity [16–19].

An effective way to promote the sensitivity of an IFOG, is to enhance the Sagnac effect by utilizing a fiber coil with a large enclosed area [20]. Based on this, many large and even giant sized IFOGs have been designed. Clivati et al. [21] demonstrated an IFOG that utilizes a multiplexed fiber network with an enclosed area of $20\,\mathrm {km^{2}}$ and achieved a sensitivity of $10\,\mathrm {nrad/s/\sqrt {Hz}}$. Li et al. [22] designed an IFOG with an enclosed area of $2910\,\mathrm {m^{2}}$ achieving a sensitivity of $3.5\,\mathrm {nrad/s/\sqrt {Hz}}$. iXblue [20] designed a series of IFOG products and prototypes with an enclosed area range of $225$ to $1250\,\mathrm {m^{2}}$, corresponding to a sensitivity of $20$ to $4.5\,\mathrm {nrad/s/\sqrt {Hz}}$. Nevertheless, the benefits of the enclosed area enlargement are somehow limited, because a longer fiber not only causes more thermal phase noise [22], it also increases the shot noise owing to a stronger light attenuation [23]. In addition, their sizes significantly weaken their portability. Therefore, an alternative approach to reducing the amount of noise is indispensable.

A bottleneck to IFOG noise suppression is excess relative intensity noise (RIN). RIN comes from the random beating between various frequency components of light [24–26]. To suppress the RIN, a common scheme applying a conventional “minimum configuration” [27] is to draw a separate optical path from the light source that operates as a reference arm. The light from the reference arm can either be converted into an electronic signal, or after passing through a fiber delay line [28], and thereby compensate for the RIN from the measurement arm by signal processing algorithms [29,30]. Alternatively, the reference light can also be directly injected back into the output light with an appropriate attenuation to optically compensate for the RIN [25,31].

Recently, a dual-polarization IFOG has been extensively investigated and has shown a good performance under various ambient conditions [32–37]. Note that the “dual-polarization configuration” is different than the conventional “minimum configuration,” which operates using a single polarization. For the dual-polarization configuration, 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 this case, an extra benefit in RIN suppression is offered, i.e., the two polarizations experience almost identical optical paths, and can therefore be used as mutual reference arms for each other.

In this study, we propose a modified dual-polarization configuration to promote the sensitivity of IFOGs for rotational seismology applications. In this design, it was found that the RINs in two polarizations are identical and synchronized, and can thus be effectively suppressed through compensation with the help of a high-order eigenfrequency modulation under an open-loop configuration. At the same time, through compensation, the coherent phase errors upon harmonic demodulation are also eliminated. We experimentally verify the effectiveness of the proposed design by using a polarization-maintaining (PM) fiber coil with an enclosed area of $68\,\mathrm {m^2}$, and achieve a reference-grade [3] angular random walk (ARW) of $50\,\mathrm {\mu ^\circ /\sqrt {h}}$, and a flat self-noise passband of $20\,\mathrm {nrad/s/\sqrt {Hz}}$ within a frequency range of $0.01\,\mathrm {Hz}$ to $30\,\mathrm {Hz}$. The sensitivity of this IFOG is comparable to a widely used commercial IFOG product BlueSeis3A (iXblue), the enclosed area ($225\,\mathrm {m^2}$) of which is more than 3-times that of our own [20]. Therefore, the proposed IFOG scheme shows the advantages of a high sensitivity and high portability at the same time.

2. Theory and principles

2.1 Analysis model of the dual-polarization IFOG

The proposed dual-polarization IFOG is schematically illustrated in Fig. 1. A polarizer is placed between the amplified spontaneous emission (ASE) light source (LS) and a $50/50$ polarization-maintaining (PM) coupler, ensuring that the two output beams have equal intensity accompanied by a highly correlated RIN [30,31], which is the key point of effective RIN suppression in such a two-arm configuration. Two multi-functional integrated optical chips (MIOCs) are used to split the light into clockwise (CW) and counter-clockwise (CCW) paths, respectively, and a reverse phase modulation [37] is further applied simultaneously, to eliminate the non-reciprocity error from polarization cross-coupling. Two polarization beam splitters/combiners (PBS/Cs) are used to inject the light from MIOC 1 and MIOC 2 (both in the same polarization corresponding to the polarizer) into the two orthogonal (x and y) polarizations of the PM fiber coil. After a round trip, the light beams in x and y polarizations are output by two PM circulators, and are then detected by two photodetectors (PD 1 and PD 2), respectively.

 

Fig. 1. Schematic setup of a dual-polarization IFOG. The intensity of polarized light input into the coupler is presented as $I(t)$. In addition, $\pm \phi _m(t)$ are a pair of sinusoidal phase modulation signals with opposite signs and x and y present the two orthogonal polarizations of the fiber coil.

Download Full Size | PDF

The IFOG response is derived by using the Jones matrix method (explicitly given in Appendix A1). According to the output expressions of the PDs, as shown in Eqs. (20) and (21), we set the modulated phase difference $\Delta \phi _{m}(t)$ to a sinusoidal shift of $\phi _{b} \cos (\omega _m t) /2$, where $\phi _b$ is the modulation depth and the modulation frequency $f_m = \omega _m/2\pi$. The intensities detected by PD 1 and PD 2 under an open-loop configuration can be written as follows:

2.2 Dual-polarization compensation

To compensate for the noises, the signal processing can be applied to the IFOG outputs of PD 1 and PD 2, as follows:

It is noteworthy that, for the polarization-modulation terms, the compensation process does not affect the Sagnac phase shift as proved in Appendix A2. In addition, the summing operation under $I_{\textrm {even}}(t)$ still improves the SNR as we elaborated on in our previous study on dual-polarization IFOGs [37]. Not only does the subtraction operation under $I_{\textrm {odd}}(t)$ eliminate most of the unmodulated terms, it also suppresses the RIN carried by the polarization-modulation terms, the details of which are presented in the later section.

At the same time, the coherent phase errors raised from the coherent-modulation term can also be suppressed through Eqs. (3) and (4). In this case, the odd harmonics are $I^{x,y}_c \textstyle \sum \nolimits _{n=1}^{\infty }C^n_{\textrm {odd}}$, and the even harmonics are $I^{x,y}_c \textstyle \sum \nolimits _{n=1}^{\infty }C^n_{\textrm {even}}$, as detailed in Appendix A3. These harmonics are the noise components and cause coherent phase errors in the harmonic demodulation. However, by applying the compensation, the residual error terms can be expressed as follows:

2.3 Theory of RIN suppression

Because the compensation process reasonably handles the above-mentioned noises, we further elaborate the principle of suppressing the RIN appearing in the polarization-modulation terms, which can be decomposed into two parts as indicated in [25]:

As noted from Eqs. (10) and (11), when the IFOG is at rest or is under a small rotation (namely, $\phi _s$ is close to zero), the value of $I_{\textrm {odd}}^{n}(t)$ approaches zero as compared to $I_{\textrm {PD\,1,2}}^{n}(t)$ in Eqs. (8) and (9). Such fact indicates that the RIN errors in such odd harmonics are effectively suppressed in the dual-polarization configuration though the compensation process. By contrast, the magnitude of $I_{\textrm {even}}^{n}(t)$ is still comparable to the original outputs of $I_{\textrm {PD\,1,2}}^{n}(t)$. Therefore, for rotational seismology, the proposed dual-polarization IFOG is capable of suppressing the RIN by mutually adopting the outputs as the reference and signal arms.

3. Experimental results

We further present an experimental verification of the proposed configuration. As illustrated in Fig. 1, an ASE light source with a spectral width of $40\,\mathrm {nm}$ and a center wavelength of $1550\,\mathrm {nm}$ is adopted. The PM fiber coil is wound using the quadrupole symmetric method with a fiber length of $2000\,\mathrm {m}$ and an average diameter of $13.6\,\mathrm {cm}$, which gives an enclosed area of $68\,\mathrm {m^2}$. The detected signals of PD 1 and PD 2 are acquired using an NI-5922, with a sampling rate of 5 million samples per second (MSPS) and a sampling length of $25000$.

First, to verify the correlation of the RIN carried by the two orthogonal polarizations, we acquired the signals from PD 1 and PD 2, respectively, without applying any modulation under the MIOCs. Further, we calculated the degree of correlation between them. As the result presented in Fig. 2 indicates, the correlation of the two polarizations reaches $99.03\,\%$, and the correlation is quite stable with respect to the short and long time scales, which is an important prerequisite for an effective suppression of the RIN.

 

Fig. 2. The degree of correlation between the two orthogonal polarizations when no modulation is applied.

Download Full Size | PDF

To evaluate the performance of the proposed dual-polarization IFOG, we carried out an 8-h test in which the fiber coil was placed flat on a horizontal surface, targeting the component of the Earth’s rate of rotation in the local vertical direction ($9.67\,\mathrm {^\circ /h}$, corresponding to the latitude of our laboratory). All devices of the IFOG are placed in a room with a relatively stable temperature. To obtain the optimal SNR of the sinusoidal modulation, the modulation depth $\phi _b$ is set to $2.7$ [24], and a modulation frequency at 5-times the eigenfrequency ($f_{e} = 1/2\tau _g$) is adopted to reduce the thermal phase noise from the fiber coil [22].

Figure 3(a) shows the rotation rates ($\Omega$) of the 8-h test demodulated from the signals of PD 1 and PD 2, and the compensated output (Comp), simultaneously, with an output rate of $200\,\mathrm {Hz}$. Moreover, the output is zoomed to a smaller time scale of $200\,\mathrm {ms}$, as presented in Fig. 3(b). It was determined that the output with compensation processing shows remarkably less random noise compared with the raw output, which validates the effectiveness of the proposal RIN suppression method. At the same time, the local rate of the Earth’s rotation is accurately extracted, which also proves the correctness of the compensation method.

 

Fig. 3. (a) The time domain outputs of an 8-h test for the dual-polarization IFOG at a rate of 200 Hz, and (b) the first 200-ms output of the test in (a).

Download Full Size | PDF

In addition, to quantitatively evaluate the performance of the proposed IFOG, the Allan deviation curves and the root power spectral density (PSD) of the three sets of data are illustrated in Fig. 4(a) and (b) and combined with the indicator analysis presented in Table 1. It was found that a significant improvement in the IFOG performance is achieved through the dual-polarization compensation method. The ARW and self-noise is reduced by 8-fold, and the bias instability (BI) is reduced by more than 6-fold.

 

Fig. 4. The quantitative indicator analysis corresponding to the 8-h test shown in Fig. 3. (a) Allan deviation curves and (b) the root PSDs.

Download Full Size | PDF

Table 1. Comparison of ARW, BI, and self-noise.

View Table | View all tables in this article

In particular, the compensation result exhibits an ultra-low short-term noise, which is represented by an ARW of $50\,\mathrm {\mu ^\circ /\sqrt {h}}$ within the Allan deviation curve (calculated at a correlation time of $\mathrm {1\,s}$), and a flat self-noise floor of $20\,\mathrm {n rad/s/\sqrt {Hz}}$ over a frequency range of $\mathrm {0.01\,Hz}$ to $\mathrm {30\,Hz}$ in the root PSD. At frequencies of higher than $30\,\mathrm {Hz}$, the self-noise estimates may be disturbed by ambient vibrations, which cannot be completely excluded in our laboratory environment. The long-term drift is represented by a BI of $4.3 \times 10^{-4}\,\mathrm {^\circ /h}$ with a correlation time of $200\,\mathrm {s}$. At longer time periods that correspond to frequencies of lower than $0.01\,\mathrm {Hz}$, the self-noise gradually deteriorates until $40\,\mathrm {n rad/s/\sqrt {Hz}}$. This is mainly because the residual coherent phase error varies under the long time scale owing to the time-varying polarization cross-coupling in the fiber coil, which is not completely eliminated by the compensation.

To summarize, the sensitivity of the proposed IFOG is sufficient for many rotational seismology applications. Its self-noise is below $10^{-7}\,\mathrm {rad/s/\sqrt {Hz}}$ and reaches a level of $20\,\mathrm {nrad/s/\sqrt {Hz}}$ within the usually considered frequency range of $\mathrm {0.01\,Hz}$ to $\mathrm {100\,Hz}$. In addition, the laboratory test results [39] of a widely used three-component IFOG seismometer, BlueSeis3A (iXblue), are presented in Table 1 for comparison, of which the fiber coil has a length of $5\,\mathrm {km}$ and a diameter of $18\,\mathrm {cm}$, giving an enclosed area of $225\,\mathrm {m^2}$ [20]. Compared with BlueSeis3A, our IFOG achieves the same short-term noise and sensitivity levels, although the long-term drift and flat self-noise passband performance cannot be fully matched. Nevertheless, in our IFOG, the enclosed area of the fiber coil is less than one-third that of BlueSeis3A. This provides the advantage of a prominent portability for rotational seismology applications.

4. Discussion of other noises

The short-term noise is the most important attribute of an IFOG for rotational seismology applications, because it is related directly to the sensitivity during a fast response. Among the many different short-term noise origins, the RIN is dominant and is thus the highest priority and the main target of our proposal. However, in addition to the RIN, there are other noises that require attention.

The coherent-modulation terms accompanying the two polarizations also result in noise in the IFOG. The coherent phase errors fluctuate in both the short and long time scales and inter-play with a correlation of the light. In our previous studies on the dual-polarization IFOGs, the coherent phase errors were almost completely suppressed by introducing a sufficient long fiber delay line that makes the two polarizations incoherent. As a result, the long-term stability of the IFOGs was guaranteed [36]. In this study, to ensure the correlation of two polarizations for RIN suppression, the fiber delay line was removed, resulting in a decrease in the long-term drift performance because the coherent phase errors only approach zero under a small rotation approximation. The self-noise of our IFOG gradually increases under the long time scale. Nevertheless, the long-term stability is still acceptable in terms of rotation seismology applications, but a much lower short-term noise is obtained, even reaching the ARW level of the reference-grade IFOG [3] of $50\,\mathrm {\mu ^\circ /\sqrt {h}}$ with a fiber coil of $68\,\mathrm {m^2}$.

Another usually overlooked noise is the thermal phase noise. Such noise is induced by the random thermal fluctuations of the fiber and increases with the fiber length. In our previous study [22], we found that the thermal phase noise is more significant in the giant sized IFOG (fiber length $L\geqslant 10\,\mathrm {km}$), and can be effectively reduced through the modulation with a high-order eigenfrequency, and thus the thermal phase noise is less important in such conventional sized IFOGs ($L\,=\,0.3 - 2\,\mathrm {km}$). In this study, because a significant portion of the RIN has been suppressed, we can note the impact of thermal phase noise in our setup even when the fiber-coil length is only $2\,\mathrm {km}$. We applied two 1-h tests to detect the rotation rate of the Earth, but with different modulation frequencies, namely, the first-order eigenfrequency (${f_e}$) of $52\,\mathrm {kHz}$ and the fifth-order eigenfrequency ($5{f_e}$) of $260\,\mathrm {kHz}$. The test results and related root PSDs are presented in Fig. 5 and Table 2.

 

Fig. 5. The root PSDs of the 1-h tests with different modulation frequencies. Dotted lines indicate the outputs under the first-order eigenfrequency ($f_e$), and solid lines indicate the outputs under the fifth-order eigenfrequency ($5f_e$).

Download Full Size | PDF

Table 2. Comparison of self-noise.

View Table | View all tables in this article

As the outputs of PD 1 and PD 2 in Fig. 5 show, the thermal phase noise is immersed in the RIN and is thus too small to be noticed from the self-noise plots. This fact is true for both $f_e$ and $5f_e$ modulations (corresponding to without or with thermal phase noise suppression), suggesting that the RIN is indeed dominant. By contrast, the difference between $f_e$ and $5f_e$ modulations stand out when the compensation is applied. The self-noise of $5f_e$ modulation is more than two times lower than that of $f_e$ modulation. Such improvement is due to the high-order eigenfrequency modulation, and proves that the RIN in our setup is even lower than the thermal phase noise.

5. Conclusions

In this study, we propose a high-sensitivity IFOG based on the dual-polarization configuration for rotational seismology applications. By adopting a compensation processing technique, the proposed IFOG is capable of effectively suppressing the RIN from the light source and the coherent phase error caused by the polarization cross-coupling. We demonstrate experimentally that, by using a reverse sinusoidal modulation at high-order eigenfrequency, an ultra-low ARW of $50\,\mathrm {\mu ^\circ /\sqrt {h}}$ is achieved in detecting the rotation rate of the Earth. In addition, the self-noise reached a level of $20\,\mathrm {n rad/s/\sqrt {Hz}}$ over a frequency range of $\mathrm {0.01\,Hz}$ to $\mathrm {30\,Hz}$. Such a high sensitivity is obtained on a fiber coil with an enclosed area of only $68\,\mathrm {m^2}$, and thus makes the dual-polarization IFOG a promising candidate for portable rotational seismology sensors.