1. Introduction

Interferometric fiber optic gyroscope (IFOG) plays an important role in navigation, space exploration and military applications due to its high precision and good stability in angular velocity measurement, large dynamic range and small volume. Physical fields such as magnetic field, temperature and vibration, can lead to measurement errors and hold back IFOG from meeting higher precision requirements [1–4]. The related theories and solutions for the issues from the individual fields have been thoroughly studied [5–14], however, we found experimentally that under the combination of the magnetic field and varying temperature, the drift of IFOG showed a tendency towards low precision and poor stability. This tendency cannot be explained with current linear theories. Moreover, the magnetic field and temperature cross-coupling drift (MTCD) significantly decreased the precision of IFOG, which led us to the research described in this paper.

Previous studies about the drift in IFOG were researched from magnetic field to temperature separately. Due to the magneto-optical Faraday effect, magnetic field can change the polarization state of light propagating in fibers, leading to a nonreciprocal phase error (NPE) in IFOG. The fiber sensing coil is the main source of the magnetic NPE (MNPE) which is related to the magnetic field, the linear birefringence and the twist of the fiber [10–14]. The polarization-maintaining (PM) optical fiber, having high linear birefringence (>10³ rad/m), can suppress the MNPE while single-mode fiber used in D-IFOG cannot. Because of the tension force, twist and the ellipticity of the fiber core, the MNPE in D-IFOG is much larger than that in PM-IFOG [14].

Due to the Shupe Effect, the temperature-induced nonreciprocal phase error (TNPE) is related to the rate of temperature change [6,7]. Even if there is no change in temperature, the profile of thermal stress in the fiber coil still varies because of the different materials used in the skeleton and the coil. This variance results in a birefringence distribution, which is another part of TNPE, across the fiber coil which is temperature dependent [8,15–17]. MTCD is the term we use to describe the combination of MNPE and TNPE. In PM-IFOG, the birefringence caused by thermal stress is less significant than the inherent one, so the MTCD is quite small [16]. However, in D-IFOG, the birefringence induced by thermal stress cannot be ignored.

In section 2, we establish a theoretical model to analyze and develop a calculating equation of MTCD. Simulation results of the thermal stress, as well as the related birefringence distributions and MTCD, are given in section 3. After that, in section 4, we describe the experiments. The experimental results verify the theoretical model and the simulations. And we discuss the difference between the experiments and the simulations. Finally, we present our conclusion in section 5.

2. Theoretical model

The D-IFOG system is shown in Fig. 1 $B$ stands for the radial magnetic field. $l_1$~$l_4$ are four sections of polarization maintaining fiber (PMF), constituting a pair of Lyot depolarizers. The birefringence of the PMF in Lyot depolarizers equals $\Delta {\beta }_b$. The coupling angle between $l_1$ and $l_3$ is ${\theta }_1$, and between $l_2$ and $l_4$ is ${\theta }_2$. The fiber coil is based on standard single mode fiber. The fiber’s diameter is 165 μm and the coil’s average radius is $R$. The total length of the light path from port 1# to port 2# is $L$. The coil can be divided into $m$ sections with each section having a length of $dz$, thus $L=dz\cdot m$. IOC is an integrated optical chip which includes a polarizer, a Y-branch waveguide and an electro-optical modulator.

 

Fig. 1 The schematic of D-IFOG. SLD: Super Luminescent Diode; PIN: Photo Detector; IOC: Integrated Optical Chip; l₁ ~l₄: Polarization-Maintaining Fibers.

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We first deduce the equations applied to the single wavelength. For the clockwise (CW) light, the residual birefringence of the $i$ section is $\Delta \beta \left(i\right)$, the twist rate is $\phi \left(i\right)$, and the birefringence related to the thermal stress is $\Delta {\beta }_T\left(T,i\right)$ at temperature $T$. $\zeta \left(i\right)$ is the circular birefringence resulting from $B$:

We assume that the polarizer in IOC is ideal and the magnitudes of the two wave vectors are both 1 after port 0#. The attenuation factor of fiber is ignored since the distance travelled by the two lights are the same. Hence, the total transmission matrix of CW light propagating through port 0#, 1#, 2# and then port 0#:

The symbol * means conjugation. The wave vector is represented as:

In the case of the counter clockwise (CCW) light, the transmission matrix of the $i$ section is:

where

Similarly, wave vector turns to:

The interference intensity of light is:

This last step in Eq. (15) shows the interfering results a single wavelength. Because broadband light sources are used in practical IFOGs, we then derive the equations across broadband wavelength. Parameters, including $\Delta \beta$, $\phi$ and $V$, need to be expressed as a function of light frequency $v$. Hence, the parameters$a$, $b$, $c$ and $d$ are also functions of $v$.

The total interfering intensity is:

In D-IFOG, $I_0=0.5$. When temperature equals $T$, the NPE can be written as:

Taking $T=20°C$ as a reference temperature and defining a cross-coupling degree $k_{B,T}$ which symbolize of MTCD:

_{$k_{B,T}$} indicates the degree of MTCD in D-IFOG.

3. Numerical simulation

The distribution of $\Delta {\beta }_T\left(T,i\right)$ is the key element in the equations in section 3 above. In this paper, a finite element method is used to simulate the thermal stress in the fiber coil and calculate the birefringence distribution. We present a finite element simulation model in ANSYS software (Fig. 2(a)). The skeleton bracing the fiber coil is made of aluminum alloy with the thickness of 5 mm. The average radius of the fiber coil is 35 mm and the height is 17 mm. The heating source lies under the skeleton. The convection condition in closed air box is applied on the top of the skeleton and the side of the coil. The thermal stress analyses are performed across temperatures ranging from −20 °C to 60 °C.

 

Fig. 2 Numerical model. (a): Finite element model in ANSYS software; (b): Cross-section of the fiber coil fabricated using symmetric quadrupolar winding method. There are 7.5 total quadrupolar elements and two are plotted.

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Figure 2(b) shows a cross-section of the fiber coil. The coil is wound using a symmetric quadrupolar winding method. The fiber is the quartz single-mode fiber. The diameter of the fiber’s cross-section is 250 μm and the total length is about 450 m. The dotted lines indicate the light-propagation path across the coil. Material parameters in simulation are listed in Table 1.

Table 1. Material parameters for ANSYS simulation

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Figure 3 presents the thermal stress results in the coil at −20 and 60 °C. It can be observed that the thermal stresses are different between the radial and axial directions. This difference between the normal stress in the radial and axial directions contributes to the birefringence. From the elasto-optical effect, we can get the value of the birefringence at each point.

 

Fig. 3 Thermal stress in the coil at −20 and 60 °C. (a) Normal stress in radial direction at −20 °C, (b) normal stress in axial direction at −20 °C, (c) normal stress in radial direction at 60 °C, and (d) normal stress in axial direction at 60 °C. At each of these two temperatures, the maximum normal stress reaches approximately 50 MPa and the maximum difference between the two directions is about 60 MPa.

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Figure 4 shows the birefringence distribution induced by thermal stress along the CW light path. Different temperatures are represented by curves with different colors. The maximum variation of the refractive index difference is about ± 10⁻⁴, at both −20 °C and 60 °C. The corresponding birefringence is ± 800 rad/m which is much larger than the inherent residual birefringence. The minimal variation of the refractive index difference appears at 20 °C simply because the environmental temperature (zero thermal stress in ANSYS) is set to 20 °C.

 

Fig. 4 Birefringence distributions at different temperatures across the CW light path. The maximum variation of the refractive index difference is about ± 10⁻⁴, and the corresponding birefringence is ± 800 rad/m.

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The light source in numerical calculations has perfect Gaussian distribution with a center wavelength of 1310 nm and a bandwidth of 40 nm (FWHM). The two coupling angles of the Lyot depolarizers are set to 44.9° and 45.2°. The twist rate is 0.1 rad/m, which is the typical value in ordinary single mode fibers [13]. The inherent residual birefringence is 20 rad/m in bent fiber under tension [21]. The dispersion coefficient is approximated as a constant in the wavelength range. Substituting the thermal stress induced birefringence into Eq. (3), we obtained MTCD in D-IFOG under 1 mT magnetic field at different temperatures, which is the red curve shown in Fig. 5. For comparison, drifts without the magnetic field is shown by the blue curve in Fig. 5. At temperatures near 20 °C, MTCD is quite stable. While at low and high temperatures, MTCD oscillates irregularly. A tiny variation in temperature may cause a considerable difference in MTCD. In addition, the direction of the drift changes at certain temperatures. According to Eq. (17), the cross-coupling degree $k_{B,T}$ varies from −220% to 70%.

 

Fig. 5 Simulation and experimental results of MTCD at different temperatures (Earth’s rotation has been subtracted). The simulation drift is the red curve and the drift under 1 mT magnetic field is the blue curve.

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4. Experimental results and discussion

In order to verify the theoretical model as well as the numerical simulations, we set up a D-IFOG experimental platform. Parameters including light wavelength and size of the coil are kept the same as those used in numerical simulations. All components are put separately on a platform which is installed in the middle of a Helmholtz coil, as shown in Fig. 6, where the fiber coils were put inside an aluminum chamber with a temperature control device at the bottom. The temperature in the chamber can be adjusted by a programmable voltage source. The blue curve in Fig. 5 shows the experimental results of MTCD at different temperatures under 1 mT magnetic field, through the temperature range from −20 °C to 60 °C.

 

Fig. 6 Experimental platform for measuring MTCD at different temperatures.

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As shown in Fig. 5, under 1 mT magnetic field, the drift varies from −26.5 to 19.5 °/h. The cross-coupling degree varies from −170% to 38% in the full temperature range. Compared with the simulation result, MTCDs are in the same order of magnitudes. The experimental result shows a tendency which is similar with the simulation. However, the absolute values of the errors of different temperatures are quite different, with a maximum of 19.5 °/h at 50°C. There are two main issues to be considered in this case: one is that in the simulation (see section 3), we assumed ideal cylindrical symmetry in the thermal stress and induced birefringence, whereas in a practical IFOG, this condition is barely satisfied. For example, the outermost fiber of the coil isn’t a complete circle and there may be small differences in the number of loops in each layer. Another issue is that, the twist rate and the residual birefringence are set to typical values because these parameters cannot be measured accurately in practice.

5. Conclusions

The MNPE of a D-IFOG varies with the temperature. The magnitude is not equal to the simple sum of MNPE and TNPE. A temperature dependent cross-coupling drift exists. The mechanism of this drift is the interaction of the nonreciprocal circular birefringence caused by the magnetic field and the linear birefringence resulting from thermal stress in the fiber. In this paper, we focused on the theoretical study and the model of numerical calculation, and achieved equations for MTCD.

MTCD deteriorates the environmental adaptability of D-IFOGs. Except for adding a magnetic shield or establishing a precise compensating model for varying temperature, novel methods are urged to solve this problem. Based on the studies/experiments mentioned in this paper, we are now engaged in a low MNPE D-IFOG across the full temperature range. Preliminary experimental results have confirmed the feasibility of our new plan.