A New Optical Method for Suppressing Radial Magnetic Error in a Depolarized Interference Fiber Optic Gyroscope

Based on the theory of the radial magnetic error (RME) in depolarized interference fiber optic gyroscopes (D-IFOGs) under magnetic field, a new optical method is proposed to decrease the RME by adding a suppressing section fiber (SSF) in D-IFOGs. A related theoretical model is established, and the solutions of the parameters of the SSF are obtained with numerical calculations. Then the results of the suppressed RME are simulated. An experimental system is set up to verify the theory and simulation, and the experimental results prove that the RME can be suppressed effectively with a SSF added in the D-IFOG. The magnitude of the RME can be reduced to one-tenth of the original.


The theoretical model of the optical method for suppressing RME
The RME is closely related to the linear birefringence, the distribution of the twist on the fiber coil, the intensity and orientation of the magnetic field 6 . In a certain fiber coil, there must be linear birefringence and twist, which will produce dramatic RME in D-IFOG under magnetic field. The reduction of RME can be realized by adding a section of SSF. By adjusting the twist and linear birefringence of the added fiber, the two beams along the clockwise (CW) orientation and the counter clockwise (CCW) orientation in the fiber produce approximately equivalent phase variety when there is only radial magnetic field and no rotation. In other words, the magnetic sensitivity of the D-IFOG will be greatly decreased. Then the destination of reducing the RME can be realized. Figure 2 is the optical scheme of the RME suppression in D-IFOG. The light source and detector are omitted. As shown in Fig. 1, the integrated optical component (IOC) consists of polarizer, Y wave guide and electrodes used for phase modulation and feedback. The IOC can be equivalent to an ideal polarizer. l 1~l4 are four sections of polarization maintaining fiber (PMF) with the linear birefringence of Δβ b . l 1 and l 3 constitute a depolarizer, as well as l 2 and l 4 . The 1# and 2# are two welding points with designed angle of 45°, while the actual angles are θ 1 and θ 2 respectively. The 45° errors of 1# and 2# are Δθ 1 and Δθ 2 . B R represents the radial magnetic field. The original fiber coil is between point A and B. The SSF is located between point B and C. The length, linear birefringence and twist of the SSF will be determined after calculating and simulating.
According to the ref. 18 , the transmission matrixes of the light in fiber are calculated based on an infinitesimal method. Assuming that the length of the SSF is L s and the SSF is divided into m s pieces, the length of the i-th piece is dz = L s /m s . When the beam along CW orientation goes through the SSF, the sequence numbers of the little pieces are 0, 1, …, i, …, m s − 2, m s − 1, orderly. Then the transmission matrix of CW light for the whole SSF is, cs i s s , 2 2 Equation (2) is the transmission matrix of the CW light in the i-th piece. Δβ s (i) is the linear birefringence of the SSF. ϕ s (i) and ζ(i) is the circular birefringence of the SSF induced by twist and Faraday Effect respectively. ζ(i) = V*B(i), where V is the Verdet constant of the material, B(i) is the intensity of the magnetic field in the i-th piece. When the CW light passes through the whole nonreciprocal light path, the transmission matrix is given by is the transmission matrix in PMF, and θ θ θ ccs i s s , 2 2 Equation (6) is the transmission matrix of the CCW light in the i-th piece. When the CCW light passes through the whole nonreciprocal light path, the transmission matrix is given by , except that u ccs , Δβ s , ϕ s , η ccs are substituted by u cc , Δβ, ϕ, η cc of the original fiber coil.
There is a π 2 phase modulation in IFOG. Therefore the interference intensity of the light along the CW and CCW orientation can be shown as Where Δφ Bs is the phase difference of CW light and CCW light after adding SSF.
Assuming that After simplifying, the interference light intensity with the broad spectrum is given by where ν G( ) is the normalized power spectral density (NPSD) of the light source. After adding SSF, the RME of D-IFOG is given by

Bs
Where k is the scale factor. According to equation (9 Bs s Introduce the suppression effect δ, which is the ratio of the RMEs of D-IFOG after and before adding the SSF Where Δφ B is the phase difference of CW light and CCW light without SSF. As shown in the equation (15), for δ < 1, as δ decreases, the Δφ Bs becomes small too. It means a stronger suppression for the error. While in situation that δ > 1, it means the Δφ Bs is enlarged after adding a SSF. For a better suppression effect, it is necessary to get the situation in which δ→0. As shown in equation (14), there must be I s →2. In equation (12), the integrity within the spectrum is substituted by the calculation of average wavelength, then the equation (12) can be further simplified as In order to obtain the parameters of the SSF, we need to solve the equation bellow, Where Δθ 1 and Δθ 2 are constant. This is a very complicated transcendental equation. Every variable is the product of several matrixes in equation (2) and (6), and is closely related to the parameter Δβ s and ϕ s , as well as the length of the i-th piece dz. So the solutions are not explicit and can be solved only through numerical calculations. The solution procedure and results will be given in section 3.

Solution and simulation of the optical model for suppressing the RME
For one certain type of D-IFOG, we simulate and calculate the optical model for suppressing the RME. Firstly, set the length of the SSF to zero (namely there is no SSF), and set the orientation of the radial magnetic field arranging from 0° to 360°. Then the RME of the original D-IFOG is obtained. Secondly, set the range of the length, the linear birefringence, and the twist of the SSF, and scan these three parameters in scope respectively. If the maximal RME is zero or less than ten percent of the original D-IFOG (that is δ < 10%) with one set of the parameters, this set of parameters is one of the solutions satisfied with the suppression effect. Finally, the solution space is gotten as shown in Fig. 3 through scanning these parameters. The solution space contains all of the parameter combinations. They are not only satisfied with the suppression effect, but also constrained by the scanning range and scanning step of three parameters (length, linear birefringence, twist). As shown in Fig. 3, three axes represent length, linear birefringence and twist of the SSF. Figure 4 is the contour map of the RME after adding the SSF. It adopts several two-dimensional mappings to show the relationship between the parameters of the SSF and the RME of D-IFOG. The lengths of the SSF in four subgraphs are 0.4 m, 0.6 m, 0.8 m and 1.0 m respectively. In each subgraph, the horizontal axis and vertical axis represent linear birefringence and twist. The label value on the curve shows the RME after adopting the optical model and the unit is (°/h/Gs).
As shown in Fig. 4, the RME with SSF inside the system has obvious relationship with each parameter of SSF. Comparing the curves of zero RME in four subgraphs, we know that as the length of the SSF increases, the needed linear birefringence and twist become small. For the curve of zero RME in any subgraph, it is centrosymmetric about the origin. In addition, we can also see in Fig. 4 that, for a certain suppression result, a smaller (or larger) twist is needed if the linear birefringence becomes larger (or smaller). These conclusions will provide guideline for practical operation in experiment.
The RME is about 1°/h/Gs for one certain type of D-IFOG. Three solutions of SSF are used to simulate the suppression results. The intensity of magnetic field varies from 1 to 10 G and the orientation of magnetic field changes from 0 to 360°. The results are shown in Fig. 5.  As shown in Fig. 5, for one solution (a set of parameters), after adding SSF, the RME is still linear with the intensity of the magnetic field and varies as a cosine function of the orientation of the magnetic field. That is, the SSF is equal to a zoom system and doesn't change the orientation of the magnetic sensitive axis (the orientation of the maximum RME). The maximum RMEs for the three solutions are all no larger than 0.06°/h/Gs, which is satisfied with the requirement of the suppression effect.

Experimental verification of the optical method
The schematic diagram of the experimental facilities is shown in Fig. 6(a) and the objects are shown in Fig. 6(b). The Helmholtz coils can produce a uniform magnetic field with Φ250 mm × 250 mm area in the center. These Helmholtz coils are powered by an adjustable current source and the intensity of the magnetic field can be varied from 0 to 20 G. Testing platform and Helmholtz coils are placed on the horizontal isolate foundation. D-IFOG, SSF and other devices are placed on the experiment board which is put on the testing platform. During   Fig. 6(b) is put over the experiment board to reduce noises caused by the air disturbance. The D-IFOG is powered at least 1 h before each experiment to avoid the influence of the starting drift. The experiment is divided into two steps. Firstly, the radial magnetic character of the original D-IFOG is tested, including the intensity and orientation of the maximum RME. Secondly, the SSF is added in the D-IFOG, and appropriate operation such as twisting and pressing is applied to this fiber. Then the radial magnetic character of the D-IFOG with SSF is gotten to analyze the suppression effect.
In step 1, the RME of the original D-IFOG is measured in radial magnetic fields with a same intensity but different orientations. Then the sensitive axis and the maximum value of the RME can be obtained. The orientation of the magnetic field is indicated by the scale line on the testing platform. There are eight scale lines representing sixteen orientations with an angle interval of 22.5°. The result of the experiment is shown in Fig. 7. The RME shows a proximately sinusoidal relation with the orientation of the magnetic field. We get a maximum RME of about 1.3°/h/Gs and a magnetic sensitive axis at 60° and 240°. In Fig. 7, the vertical line at each testing point represents the test error, which is twice of the standard deviation σ. So the test value can be expressed as the average with a deviation of ±σ. σ is about 0.1 °/h/Gs.
Then the SSF is added in the D-IFOG and is looped and pressed properly. The SSF is fixed once it is adjusted to the appropriate position. Then the magnetic character is tested again. The result is shown in Fig. 8.
Comparing Fig. 7 wih Fig. 8, we get the suppression effect in experiment as δ = .
. × ≈ 0 09 1 35 100% 7% (18) This indicates that with proper parameters of SSF, a suppression effect less than 10% can be obtained. After suppression, the orientation of the magnetic sensitive axis remains the same. The RME still remains a sinusoidal relationship with the orientation of the magnetic field. The experimental results agree with the simulation in section 3.

Conclusion
In this paper we put forward an optical method for suppressing RME in a D-IFOG. The RME is reduced by adding a SSF in the D-IFOG system. The theoretical model of the RME in D-IFOG with SSF is shown. By numerical  calculations, the solutions for the parameters of the added SSF are obtained. Three solutions are chosen to simulate the suppression results. Then an experimental system is set up and the experimental results shows excellent agreement with the simulation. Experiments prove that, with this method, the RME can be reduced to one-tenth of the original. The work shows great possibility to improve the environmental adaptability for medium-to-high precision D-IFOG. Data availability. The datasets generated during the current study are available from the corresponding author on reasonable request.