1. Introduction

In recent decades, interferometric fiber-optic gyroscopes (IFOG) have become the most important category of rotation sensors for high accuracy applications. An IFOG detects the phase difference (Sagnac effect) between two counter-propagating light waves to measure inertial rotation of a fiber coil [1, 2]. The “minimal scheme” is well-known in the field of designing IFOGs, which has defined several reciprocity requirements [3]. Among the requirements, the polarizer is always suggested to be used to ensure polarization reciprocity. When the polarizer is used, only one polarization is maintained for rotation sensing, while the other polarization is eliminated to minimize nonreciprocal light beams. For better performance in these IFOGs, a polarizer with higher polarization extinction ratio (PER) is required. Otherwise, the bias stability of a IFOG will deteriorate because of polarization nonreciprocity (PN) erorrs. Moreover, high performance polarization maintaining elements are required in high precision IFOGs, which usually enhance cost and abandon the all-fiber structure. This PER requirement is a limitation for both the polarization maintaining IFOG (PM-IFOG) [4–8] and the depolarized IFOG [9–13].

In these IFOGs based on the conventional minimal scheme, two couplers are necessarily used to ensure reciprocity. One or two of the couplers may be possibly replaced by Y-junction waveguide, circulators, or any other elements with similar function, but the number of such elements can not be reduced to one. The two-coupler scheme is required by both polarization reciprocity and coupler reciprocity. In this case, the free port of the coil coupler can not been used for signal detection as it is nonreciprocal, and this is the reason why it is usually referred to as the “nonreciprocal port”. The nonreciprocal port receives errors induced by coupler non-reciprocity (CN) and PN [2]. CN appears as a comparatively stable bias in the IFOG output, which can be easily aligned. The PN errors are large and unstable at this port, because light has not be filtered by the polarizer for the second time when traveling back from the fiber coil. In IFOG applications, such unstable PN errors are not allowed as it deteriorates the bias stability dramatically.

Recently, optical compensation was proposed as a new approach for error reduction in PM-IFOGs [14,15]. Different from the conventional way of eliminating one polarization, the optical compensated PM-IFOG utilized two orthogonal polarizations simultaneously. PN errors in two polarizations had opposite signs, which were mutually canceled while summing up their optical intensity. PN errors could be totaly eliminated if the two polarizations have same intensity and no coherence. Furthermore, we proposed the all-depolarized IFOG for realizing optical compensation in depolarized IFOGs [16, 17]. The all-depolarized IFOGs had particular advantages of low cost, all-fiber components, and low structural complexity. Moreover, its performance was much better than old IFOG designs which had depolarized light source but did not ensure optical compensation [18]. In the all-depolarized IFOG, high performance was guaranteed by the low degree of polarization (DOP) instead of a high PER. However, as PN error reduction mechanism was totally different from conventional IFOGs, optically compensated IFOGs still had unique properties undiscovered.

In this paper, we study optically compensated polarization reciprocity based on the all-depolarized IFOG. A new discovery is that optical compensation is also effective for PN error reduction at the conventional nonreciprocal port. In this case, the nonreciprocal port of an IFOG can be used as a reciprocal port for rotation sensing. Not only the polarizer, but also the first coupler in the minimal scheme can be omitted in an optically compensated IFOG. In other words, the conventional “minimal scheme” is not minimal, and an even simpler scheme is proposed as the new minimal scheme for optically compensated IFOGs.

The construction of this paper is as follows. In Section 2, we describe basic structural principles of the all-depolarized IFOG, and analyze theoretically how PN noise is reduced by optical compensation at both the reciprocal port and the nonreciprocal port. In Section 3, we present simulation and experimental results to show the difference between optically compensated IFOGs and conventional IFOGs. Finally, Section 4 presents a summary of our work.

2. Structural principle and theory analysis

Figure 1 shows all the optical configurations we are going to discuss in this paper. The conventional minimal scheme for IFOGs is shown by Fig. 1(a). It uses at least two couplers and one polarizer to ensure reciprocity. In applicable IFOGs, more structural principles should be followed for sensitive and stable operation. For example, the PM-IFOG requires polarization maintaining components and a polarization maintaining fiber (PMF) coil. An applicable structure of the depolarized IFOG is showed in Fig. 1(b) [12]. The structure has several Lyot depolarizers constructed by PMF, an ordinary single mode fiber (SMF) coil, and still a polarizer. Here DP 1 is for depolarizing the light source, so that light intensity has a stable loss (3 dB) when passing through the polarizer. DP 2 and DP 3 are used to avoid the polarization errors induced by random birefringence along the SMF coil. The PZT works as a phase modulator. The interference signal is detected by PD 1, where the rotation signal can be demodulated. Usually, there is not PD 2 in IFOG products, as PD 2 connects to the nonreciprocal port of the coil coupler. The nonreciprocal port contains large detection errors due to CN and PN [2]. CN induces a stable bias to the output of the IFOG, which can be calibrated and does no harm to the IFOG’s bias stability. On the other hand, PN induces errors which fluctuate over time and degrade IFOG performance crucially.

 

Fig. 1 Optical constructions for discussion. S, PD, DC, P, DP, and PZT stand for light sources, photodetectors, directional couplers, polarizers, depolarizers, and piezoelectric transducers respectively. (a) The minimal scheme for conventional IFOGs. (b) The conventional depolarized IFOG. (c) The all-depolarized IFOG based on optical compensation. (d) The minimal scheme for optically compensated IFOGs.

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We use a common block diagram to show the polarization flow in IFOGs, as given by Fig. 2. We analyze PN errors in IFOGs by Jones Matrices (see Appendix for detail). The DOP before the PF is noted as d₀. The DOP at the entrance of coil is noted as d. PN errors are induced by unpredictable coupling between two polarizations inside the coil, and we notice d (or PER) at the coil entrance is the key influencing factor for restricting these PN errors.

For conventional IFOGs, the PF is a polarizer. At the reciprocal port, the PN error is derived as (see Appendix I)

 

Fig. 2 Polarization flow in IFOGs. Light enters the Sagac coil after a polarization filter (PF). The PF can be a polarizer or a depolarizer depending on the type of IFOGs. When light travels back out from the coil, there are two ports for detecting the interference signal. The reciprocal port (RP) is detected by PD 1, where light goes through the PF again. The nonreciprocal port (NRP) is detected by PD 2, where light does not go back through the PF.

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However, the nonreciprocal port contains a large PN error even though the polarizer is perfect. The error is derived as (see Appendix I)

Our analysis shows that going back trough the polarizer is a basic requirement for eliminating PN errors in conventional IFOGs. The nonreciprocal port does not meet this requirement and thus large PN errors always exist. PN errors can not be eliminated in the nonreciprocal port even if the polarizer is ideal. Due to these reasons, the “one polarizer between two couplers” principle is necessary for conventional IFOGs, and only the reciprocal port can be used for rotation sensing.

Fortunately, the “one polarizer between two couplers” scheme is not the only way for PN error reduction. Differently in optically compensated IFOGs, two polarizations are simultaneously used and PN errors are suppressed by compensation [14–17]. In these designs, PN errors in two polarizations have opposite polarities, and thus errors are mutually canceled when summing up the intensity of two polarizations. The all-depolarized IFOG is a setup applying optical compensation in depolarized IFOGs [16,17], as shown in Fig. 1(c). Here DP 4 is used instead of the polarizer. DP 4 is the key component for generating the light beam with two orthogonal polarizations. Theory analysis shows that PN errors can be totally reduced when two polarizations have same intensity and no coherence.

Equations in Appendix II show that PN errors in two polarizations have opposite signs, and they cancel each other at both the reciprocal port and the reciprocal port. PN errors after compensation at the reciprocal port and the nonreciprocal port are given by (see Appendix II)

For clear comparison between the two kinds of IFOGs, we use signal to noise ratio (SNR) to measure the influence of PN errors as shown in Fig. 3. A high SNR stands for a low level of PN errors. At the reciprocal port shown by Fig. 3(a), both kinds of IFOGs can reduce PN errors by their own polarization filtering technique. In the all-depolarized IFOG, PN errors is reduced by optical compensation at the depolarized point (|d| = 0). For the conventional IFOG with a polarizer, single-polarization operation (|d| = 1) also eliminates PN errors. However, the situation is different at the nonreciprocal port, as shown in Fig. 3(b). The optically compensated IFOG still works well at this port at the depolarized sate (|d| = 0). But conventional IFOGs has a limited SNR at their working point (|d| = 1). The remaining error amplitude depends on the level of polarization coupling in the fiber coil. In depolarized fiber coil shown in Fig. 1(b) and 1(c), polarization coupling is always large and hence PN errors are large correspondingly. Even in a PMF coil, polarization coupling is inevitable as real PMF and PM components have limited PERs.

 

Fig. 3 PN errors in the optically compensated IFOG and the conventional IFOG.

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To avoid misunderstanding, we should notice the state |d| = 0 has different meanings for the optically compensated IFOG and the conventional IFOG. In our deviation, |d| = 0 stands for equal intensity on the x polarization and the y polarization. In the optically compensated IFOG, |d| = 0 means depolarized polarization state, as the x polarization and the y polarization are incoherent. But for the conventional IFOG, the two polarizations are coherent, thus |d| = 0 could be the circular polarization state or some particular elliptic polarization states.

In this section, we have proved that both two ports in the optically compensated IFOG have reduced PN errors due to optical compensation. In contrast to conventional IFOGs, the nonreciprocal port is made effective for rotation sensing. In addition, remaining bias at the nonreciprocal port is induced by CN which is stable and can be aligned. Hence, the nonreciprocal port turns to be “reciprocal” in optically compensated IFOGs. This advantage is also verified by simulation and experimental results in the following section. From this point of view, two couplers are now redundant. Only one coupler is necessary for the minimal scheme of optically compensated IFOGs, as shown in Fig. 1(d). Based on this structural principle, stable IFOG can be achieved with only one coupler and without any polarizer or any polarization maintaining devices.

3. Simulation and experiments

In practice, polarization coupling in the Sagnac coil is not stable because of environmental instability such as acoustic vibration and temperature fluctuation. As a result, large PN errors make detection results wave up and down. Before experiments, we simulate the performance of the optically compensated all-depolarized IFOG against unstable environment. Target rotation rate is set as 9.67°/h, in accordance with our experiments. Unstable polarization coupling is induced in the IFOG by time-varying coupling coefficients. Simulation results at the reciprocal port are shown in Fig. 4. We can see that PN errors on both two polarizations are large but with opposite signs. After compensation, PNs errors are effectively reduced. This simulation describes how optical compensation reduces PN errors. Even though polarization coupling is large, the induced PN errors can be compressed to a low level as long as |d| is small.

 

Fig. 4 Simulation of the optically compensated all-depolarized IFOG against unstable polarization coupling. PN errors on two polarizations (x and y) are large but with opposite polarities. After summing up two results (sum), PN errors are notably reduced. As d is close to 0, the compensated result is much more stable than uncompensated results.

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Figure 5 illustrates the performance comparison between the conventional depolarized IFOG and the all-depolarized IFOG. Figure 5(a) and 5(b) show a group of simulated performance comparison. We simulate IFOG output at both ports. It is clear that PN errors are low at the reciprocal port for both IFOG configurations. However, their performances are quite different at the nonreciprocal port. The conventional IFOG has large errors at the nonreciprocal port, while the all-depolarized IFOG has also stable results at this port. These results are consistent with our theory prediction.

 

Fig. 5 Simulation and experimental results for comparison between the conventional depolarized IFOG and the all-depolarized IFOG. (a)(b) are simulation results. (c)(d) are experimental results. (e)(f) are experimental results analyzed by Allan variance. Blue lines are results for the reciprocal port. Red lines are results for the nonreciprocal port.

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We performed two experiments on the conventional depolarized IFOG and the all-depolarized IFOG respectively. The environmental conditions were the same for the two experiments, while two IFOGs both detecting the Earth’s rotation rate (9.67°/h projected at our laboratory latitude 40 °N). An amplified spontaneous emission (ASE) source was used, which had a center frequency at 1550 nm and a band width of 70 nm. Its decoherence length was calculated as L_dc = 34 μm. The Lyot depolarizers are constructed with PMF, in which the refractive index difference between two axes was Δn = 5 × 10⁻⁴. Hence $\overline{AB}$ in D₁ should be longer than L_dp = L_dc/Δn = 0.068 m. Differently in D₂, D₃, and D₄, a much longer depolarizing length L′_dp was required because of the possible birefringence in the SMF coil [19]. We used the same coil for both two IFOGs. It had a 2097 m length and a 0.14 m diameter, and the fiber core diameter was 125 μm. Regarding this coil, L′_dp = 1.25 m was required. For robust performance, we used redundant depolarizing lengths L₀ = 0.15 m and L′₀ = 2.0 m instead of L_dp = 0.068 m and L′_dp = 1.25 m. Accordingly, $\overline{AB}$ parts in D1, D2, D3, and D4 were chosen 0.15 m, 2 m, 8 m, and 32m respectively. Every $\overline{BC}$ was twice of its connected $\overline{AB}$. For practical PMF, its length and Δn were unstable mainly because of temperature changes [7]. In our design, the redundant lengths ensured the depolarizers valid for depolarizing over hundreds of °C. Two polarizations in PMF pieces were always incoherent in our experiments under room temperature range (15°C–25°C). Hence the IFOG avoided the influence of temperature related PMF instability.

For all the following experiments, we used the same signal processing system as shown in Fig. 6. Sinusoidal phase modulation was applied on the IFOGs, and traditional harmonic demodulation was used [8]. We detected the 1^st harmonic, the 2^nd harmonic, and the 4^th harmonic simultaneously for monitoring the modulation depth while determining the rotation rate. The signals received by two PDs (Thorlabs) were digitized by a dual-channel digitizer (NI PXI-5922) and conveyed to a computer. The digitizer had a resolution of 22 bits at the sampling rate 2 MS/s. The computer finished all the demodulation and data collection procures by digital signal processing.

 

Fig. 6 Signal processing system for IFOGs. Fast Fourier transform (FFT) is used for getting harmonic amplitudes. Ω₁ and Ω₂ are output rotation rates for the reciprocal port and the nonreciprocal port respectively.

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Figure 5(c) shows testing results by two ports in the conventional depolarized IFOG. By contrast, the results got in the all-depolarized IFOG are shown in Fig. 5(d). For both groups of data, the integrate time was 0.35s and the test length was 2 hours. Allan variance analysis is used for noise evaluation [20], as shown in Fig. 5(e) and 5(f). For the conventional IFOG, the results at the reciprocal port were stable, which had low bias drift values about 0.01 °/h. However, the result at the nonreciprocal port had large fluctuations, and the corresponding bias drift was as large as 0.11 °/h. For the all-depolarized IFOG, the results at both PDs were stable and had similar low bias drifts about 0.01 °/h. The only difference between two ports was that the nonreciprocal port had a offset about 40 °/h, which was stable and did not influence the IFOG performance. In applications, the stable bias can be calibrated by simply subtracting the offset value at the output.

By calculating the Allan variances, we also get the detailed noise indices as given in Table 1. Besides the bias drift, long-term noise indices such as rate random walk and rate ramp were also large at the nonreciprocal port in the conventional IFOG. On the other hand, these indices are similar between the reciprocal port and the nonreciprocal port in the all-depolarized IFOG. All those long term noises are suppressed while PN errors are compensated at the nonreciprocal port.

Table 1. Allan variance indices of the IFOGs

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In addition, we carried out a simplified inclination survey on the all-depolarized IFOG. The platform carrying the IFOG was tilted in different angles so that the target rotation rate was verified (i.e. changing the projection of Earth’ rotation rate). The test range was from 40° tilted south to 40° tilted north, and the interval between two tilt angles was 10°. The test result for each tilt angle was an average of 100 consecutive data points, where the integrate time was still 0.35 s. Two port outputs versus theoretical rotation input are shown in Fig. 7. Just as expected, the constant bias at the nonreciprocal port does not influence its response. In the testing range, linearity calculation values are 0.99973 and 0.99978 for the reciprocal port and the nonreciprocal port respectively. Two ports have similar linearity, which is another evidence for that the nonreciprocal port performs as well as the reciprocal port in optically compensated IFOGs.

 

Fig. 7 Inclination survey using the all-depolarized IFOG.

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The experimental phenomena are consistent with our simulation. In the conventional IFOG, the nonreciprocal port has large errors. This is the reason why this port is usually unused. On the other hand, the optically compensated all-depolarized IFOG has two low-error ports for rotation sensing. Due to this advantage, it is promising to further reduce IFOG noise by processing two detection results with estimation algorithms [21]. In a few IFOG applications where one detection port can already satisfy the users, we can reduce one coupler for less structural complexity.

4. Conclusion

In conclusion, we have analyzed the polarization reciprocity in optically compensated IFOGs. It is proved that optical compensation is valid for PN error reduction at both the reciprocal port and the nonreciprocal port. In the optically compensated all-depolarized IFOG, two ports are polarization reciprocal, and they obtain similar detection stability. This is a significant difference from the conventional IFOG where there is a nonreciprocal port with large PN errors. As advantages of optically compensated IFOGs, we can either get two detection results with one single fiber coil or reduce the structural complexity by omitting one coupler. Furthermore, we propose a minimal scheme for optically compensated IFOGs, which has advantages of low cost, low structural complexity, and all-fiber structure. This prototype is a good candidate for next generation of industrial IFOGs, which requires not only detection stability but also economical efficiency.

Appendix I: PN error analysis in conventional IFOGs

We use Jones Matrices to analyze the PN errors in IFOGs. Common loss on two polarizations out of the coil is neglected, as they do not contribute to PN errors. In the conventional IFOG, We assume the polarizer polarizes light on x direction. The matrices for the polarizer and normalized light fields after the polarizer are given as [22]

(5)$${\mathbf{P}}_x=\left[\begin{array}{cc}1& 0\\ 0& \epsilon \end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathbf{E}}_C=\left[\begin{array}{c}\sqrt{(1+d)/2}\\ \sqrt{(1-d)/2}\end{array}\right]e^{j{\omega }_{0}t}.$$

where ω₀ is the optical frequency, and d is the DOP after the polarizer (d ≈ 1).

Light splits at DC 2 and forms two counter-propagating beams in the coil. When two beams meet again at the reciprocal port, the transmission matrices for counter-propagating beams have reciprocal forms as [3, 23]

(6)$${\mathbf{M}}_r^{+}=\left[\begin{array}{cc}{C}_{r1}& C_{r2}\\ C_{r3}& C_{r4}\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathbf{M}}_r^{-}=\left[\begin{array}{cc}{C}_{r1}& C_{r3}\\ C_{r2}& C_{r4}\end{array}\right].$$

The superscripts ‘+’ and ‘−’ stands for clockwise and counterclockwise respectively. C_r1, C_r2, C_r3, and C_r4 are complex coefficients, which are derived by multiplying transmission matrices for DC 2, DP 2, DP 3, and the fiber coil. Similarly, there are also symmetric transmission matrices for light arriving at the nonreciprocal port as

(7)$${\mathbf{M}}_{nr}^{+}=\left[\begin{array}{cc}{C}_{nr1}& C_{nr2}\\ C_{nr3}& C_{nr4}\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathbf{M}}_{nr}^{-}=\left[\begin{array}{cc}{C}_{nr1}& C_{nr3}\\ C_{nr2}& C_{nr4}\end{array}\right].$$

Light fields and light intensity at the reciprocal port are given by

(8)$${\mathbf{E}}_r^{+}={\mathbf{P}}_x{\mathbf{M}}_r^{+}{\mathbf{E}}_C{e}^{j{\varphi }_r},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathbf{E}}_r^{-}={\mathbf{P}}_x{\mathbf{M}}_r^{-}{\mathbf{E}}_C,$$

(9)$$I_r=<{\left|{\mathbf{E}}_r^{+}+{\mathbf{E}}_r^{-}\right|}^2>=I_{r0}+\sqrt{p_r^2+q_r^2}\text{cos}(\varphi -\mathrm{\Delta }{\varphi }_r),$$

Here ϕ_r = ϕ_S + Δϕ(t) includes Sagnac phase shift ϕ_S and modulation induced phase Δϕ(t). I_r0 is a direct-current component which does not relate to the rotation rate. Δϕ_r = arctan(p_r/q_r) is PN induced phase error at the reciprocal port.

Differently at the nonreciprocal port, light is detected by PD 2 without going back through the polarizer. The fields and intensity are changed into

(10)$${\mathbf{E}}_{nr}^{+}={\mathbf{M}}_{nr}^{+}{\mathbf{E}}_C{e}^{j{\varphi }_{nr}},{\mathbf{E}}_{nr}^{-}={\mathbf{M}}_{nr}^{-}{\mathbf{E}}_C,$$

(11)$$I_{nr}=<{\left|{\mathbf{E}}_{nr}^{+}+{\mathbf{E}}_{nr}^{-}\right|}^2>=I_{nr0}+\sqrt{p_{nr}^2+q_{nr}^2}\text{cos}(\varphi -\mathrm{\Delta }{\varphi }_{nr}),$$

where ϕ_nr = ϕ_S + Δϕ(t) +ϕ_C. The additional phase shift ϕ_C is due to nonreciprocal coupling in DC 2. Δϕ_nr = arctan(p_nr/q_nr) are PN induced phase error at the nonreciprocal port. For an ideal coupler, ϕ_C = π.

The parameters for both ports are shown in Table 2. Here p stands for p_r or p_nr. q stands for q_r or q_nr. We have defined A_rij = |C_riC_rj|Γ(z_rij) and A_nrij = |C_nriC_nrj|Γ(z_nrij), with i, j ∈ {1, 2, 3, 4}. Here Γ(z) is the source’s degree of coherence [13]. z_rij and z_nrij are birefringent delays induced by $C_{ri}{C}_{rj}^{*}$ and $C_{nri}{C}_{nrj}^{*}$ respectively. ϕ_rij and ϕ_nrij are the phases of $C_{ri}{C}_{rj}^{*}$ and $C_{nri}{C}_{nrj}^{*}$ respectively. In the derivation, we have ignored two-order small terms, and used several empirical formulas as

(12)$$\left|C_{r1}{C}_{r2}\right|=\left|C_{r3}{C}_{r4}\right|\ne \left|C_{r1}{C}_{r3}\right|=\left|C_{r2}{C}_{r4}\right|,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\varphi }_{r12}=\pi +{\varphi }_{r34}\ne {\varphi }_{r13}=\pi +{\varphi }_{r24}.$$

Table 2. Derivation results for conventional IFOGs

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At the reciprocal port, when d = 1, we get p_r = 0 and Δϕ_r = arctan(p_r/q_r) = 0. The closer d is to 1, the smaller PN error we get. By a similar derivation, we can also prove d = −1 (ideally polarized along y axis) eliminates PN errors at the reciprocal port. At the nonreciprocal port, however, Δϕ_nr = arctan(p_nr/q_nr) ≠ 0 when d = 1. Therefore, there are always large PN errors even if an ideal polarizer is used.

Appendix II: PN error analysis in the optically compensated IFOG

In the all-depolarized IFOG, the coil entrance is connected to Point C (the end of DP 4), where normalized fields are written as

(13)$${\mathbf{E}}_C=\left[\begin{array}{c}\sqrt{(1+d)/2}{e}^{-j\mathrm{\Delta }\beta L}\\ \sqrt{(1-d)/2}\end{array}\right]e^{j{\omega }_{0}t}.$$

where d ≈ 1 for depolarized light, and the decoherence length L is ensured by DP 4.

Light fields arriving at the reciprocal port are derived by

(14)$${\mathbf{E}}_r^{+}={\mathbf{M}}_r^{+}{\mathbf{E}}_C{e}^{j{\varphi }_r},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathbf{E}}_r^{-}={\mathbf{M}}_r^{-}{\mathbf{E}}_C$$

The interference signals for two polarizations at the reciprocal port are then derived as

(15)$$I_{rx}=<{\left|{\mathbf{E}}_{rx}^{+}+{\mathbf{E}}_{rx}^{-}\right|}^2>=I_{rx0}+\sqrt{p_{rx}^2+q_{rx}^2}\text{cos}(\varphi -\mathrm{\Delta }{\varphi }_{rx}),$$

(16)$$I_{ry}=<{\left|{\mathbf{E}}_{ry}^{+}+{\mathbf{E}}_{ry}^{-}\right|}^2>=I_{ry0}+\sqrt{p_{ry}^2+q_{ry}^2}\text{cos}(\varphi -\mathrm{\Delta }{\varphi }_{ry}),$$

where Δϕ_rx = arctan(p_rx/q_rx) and Δϕ_ry = arctan(p_ry/q_ry) are PN induced phase errors in x and y polarizations respectively.

Light intensity on two polarizations is incoherently added up at PD 1 as

(17)$$I_{rsum}=I_{r0}+\sqrt{{\left(p_{rx}+p_{ry}\right)}^2+{\left(q_{rx}+q_{ry}\right)}^2}\text{cos}(\varphi -\mathrm{\Delta }{\varphi }_r),$$

where $\mathrm{\Delta }{\varphi }_r=\text{arctan}\frac{p_{rx}+p_{ry}}{q_{rx}+q_{ry}}$ is the compensated phase error at the reciprocal port.

At the nonreciprocal port. Light fields reaching PD 2 are derived by

(18)$${\mathbf{E}}_{nr}^{+}={\mathbf{M}}_{nr}^{+}{\mathbf{E}}_C{e}^{j{\varphi }_{nr}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathbf{E}}_{nr}^{-}={\mathbf{M}}_{nr}^{-}{\mathbf{E}}_C,$$

Similarly, we can derive the compensated PN error $\mathrm{\Delta }{\varphi }_{nr}=\text{arctan}\frac{p_{nrx}+p_{nry}}{q_{nrx}+q_{nry}}$.

Parameters and the final results for both ports are shown in Table 3. For both ports, q_x and q_y have the same sign, but p_x and p_y have opposite signs. This leads to opposite signs of Δϕ_x and Δϕ_y, which means PN errors in two polarizations have opposite polarities. After compensation, the errors are canceled by each other. The final PN errors Δϕ_r and Δϕ_nr both decrease with |d|.

Table 3. Derivation results for optically compensated IFOGs

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Acknowledgments

This work was supported by 973 Program of China No. 2013CB329205, 973 Program of China No. 2010CB328203, and the National Natural Science Foundation of China (NSFC) under grant No. 61307089.

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