Quadrature demodulation with synchronous difference for interferometric fiber-optic gyroscopes

Zinan Wang; Yi Yang; Yongxiao Li; Xiaoqi Yu; Zhenrong Zhang; Zhengbin Li

doi:10.1364/OE.20.025421

1. Introduction

Optical gyroscopes are rotation sensors detecting the phase shift induced by the Sagnac effect [1, 2]. They are widely used in industrial and military applications such as aircraft, satellites, and remote control devices. With superiority over other types of gyroscopes in precision and cost, interferometric fiber-optic gyroscopes (IFOGs) have been intensively studied over the past few decades, and commercial products are available [3, 4]. In applications, however, the performance of the IFOG is limited by both short term noise and long term instability [5, 6]. Suppressing noise is thus important for IFOG designing.

Phase modulation is introduced into IFOGs for a main purpose of maximizing its sensitivity [3, 4, 6]. Sinusoidal phase modulation, square wave modulation, triangular modulation and other ways of modulation were suggested for different situations or different configurations of IFOGs [4, 7–10]. Meanwhile, tricks in modulation and signal processing are also important for suppressing noise in IFOGs. Both short term noise and long term instability were proved possible to be reduced by related methods [9–13]. This kind of noise reduction methods adds almost no complexity into the optical structure of the IFOG, and thus they are especially convenient in application.

Among short term noises, the optical intensity noise produced by broadband sources becomes the major component at the output when optical power is relatively high [13]. A noise reference is mostly used for intensity noise subtraction. Moeller et al. first applied intensity noise substraction in the IFOG in 1991, by utilizing the unused port of source coupler as a noise reference [12]. Similar approaches are studies by Rabelo et al. and Polynkin et al. [13,14]. From another point of view, Blake et al. showed that intensity noise could be relegated to the quadrature channel by adjusting the modulation depth in sine wave modulated IFOGs, thus noise in the in-phase signal was reduced [9].

The bias drift is an important parameter representing long term instability of IFOGs. Methods based on polarization controlling were unusually used for bias drift reduction [15–19]. In addition, it is also possible for reducing bias drift without changing optical components of the IFOG. A method for bias drift reduction was proposed by Wang et al. in 2011, by subtracting two signals with different polarities at the output [10]. The two signals are obtained in a time division manner by switching the polarity of the modulation function periodically.

Benefited from all these ideas, we see the potential for suppressing simultaneously short term noise and long term instability, by utilizing simple phase modulation and signal processing methods. Specifically, we find the quadrature channel mentioned in Ref. [9] can also be used as a noise reference. Differently from the noise references used in Ref. [12–14], the quadrature channel contained not only the short term intensity noise from the light source but also the long term instability from the fiber coil. Therefore, the quadrature channel and the in-phase channel can be used together for reducing both short term noise and long term instability, by subtracting two signals in a similar way with Ref. [10].

In this paper we propose a novel method of quadrature demodulation with synchronous difference for suppressing both short term noise and long term instability in IFOGs. Two channels of detected results are simultaneously generated with the same expectation of amplitude but different signs. These two channels are also regarded as each other’s noise references. Synchronous difference between the two results is used as the output, where signals are added up and common noise is eliminated. Theory and experimental results show that both short term noise and long term instability of the IFOG are notably reduced by this method.

The construction of this paper is as follows. In Section 2, we describe the requirement and the process of quadrature demodulation, and present mathematical forms of relevant signals. In section 3, we analyze theoretically how noise is reduced during the synchronous difference procedure. Section 4 reports experimental results on an IFOG for performance comparison between traditional demodulation and the quadrature demodulation with synchronous difference, and Section 5 presents a summary of the work.

2. Theory of quadrature demodulation for the synchronous difference

The polarization maintaining IFOG is one of the typical IFOG structures meeting the requirement of reciprocity [3, 20, 21]. Figure 1 shows the open-loop polarization maintaining IFOG used in our experiment, and implementation details of modulation and demodulation.

Fig. 1 Quadrature demodulation with synchronous difference for an open-loop polarization maintaining IFOG. The experiment setup includes a C-Band ASE light source with a spectrum of 40 nm and output power of 13 dBmW, a circulator, a photodetector (PD) with a response of 10 μA/mW and load resistant of 100 KΩ, a Y-junction waveguide with polarizer and phase modulation function, a 1982 m long coil of polarization maintaining fiber (PMF), a dual-channel signal generator, a dual-channel digitizer with flexible resolution of 16–24 bits, and a computer.

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In our novel method, the phase modulation function is given by

ϕ_{m} (t) = ϕ_{0} [sin (ω_{m} t) - cos (ω_{m} t)] = \sqrt{2} ϕ_{0} sin (ω_{m} t - \frac{π}{4}),

which has a phase delay of 45° with the reference signal V_I(t) = V_f sin(ω_mt). When eigenfrequency modulation is used [3, 13], we have ω_m = π/τ, and the modulation introduced phase difference as

Δ ϕ (t) = ϕ_{m} (t) - ϕ_{m} (t - τ) = 2 \sqrt{2} ϕ_{0} cos (ω_{m} t - \frac{π}{4}),

This function ensures that the in-phase signal and the quadrature signal have the same expectation of amplitude but with opposite signs. In this case, the 1^st harmonic of detected signal is changed from Eq. (26) in Appendix to

\begin{array}{l} I_{1 H} (t) & = & I_{0} η J_{1} (ϕ_{b}) sin (ϕ_{s}) sin (ω_{m} t + \frac{3 π}{4}) \\ = & - \frac{\sqrt{2}}{2} I_{0} η J_{1} (ϕ_{b}) sin (ϕ_{s}) sin (ω_{m} t) + \frac{\sqrt{2}}{2} I_{0} η J_{1} (ϕ_{b}) sin (ϕ_{s}) cos (ω_{m} t), \end{array}

where I₀ is the source intensity coupled to the optical circuit, η is the response of the PD, J_n is the n^th Bessel function of the first kind, and

ϕ_{b} = 2 \sqrt{2} ϕ_{0}

.

We apply coherent detection to get amplitudes and signs of the in-phase signal and the quadrature signal. The reference sine wave V_I(t) = V_f sin(ω_mt) is used as the local oscillator (LO). The quadrature LO is a cosine wave V_Q(t) = V_f cos(ω_mt), which has a π/2 phase difference with V_I(t). The detected signal is multiplied by two LOs respectively, with amplitude scaling as

\frac{1}{V_{f}} I (t) V_{I} (t) = - \frac{\sqrt{2}}{2} I_{0} η J_{1} (ϕ_{b}) sin (ϕ_{s}) + h . c .,

\frac{1}{V_{f}} I (t) V_{Q} (t) = \frac{\sqrt{2}}{2} I_{0} η J_{1} (ϕ_{b}) sin (ϕ_{s}) + h . c . .

Here h.c. stands for high-frequency components, which do not affect final results. After low-pass filtering (LPF), we get useful DC components

I_{I} = - \frac{\sqrt{2}}{2} I_{0} η J_{1} (ϕ_{b}) sin (ϕ_{s}),

I_{Q} = \frac{\sqrt{2}}{2} I_{0} η J_{1} (ϕ_{b}) sin (ϕ_{s}) .

Differently with the traditional demodulation in Appendix, the rotation induced phase shift is simultaneously detected by two signals as

ϕ_{I} = arctan \frac{\sqrt{2} I_{I} J_{2} (ϕ_{b})}{I_{2 H} J_{1} (ϕ_{b})} = - ϕ_{s},

ϕ_{Q} = arctan \frac{\sqrt{2} I_{Q} J_{2} (ϕ_{b})}{I_{2 H} J_{1} (ϕ_{b})} = ϕ_{s},

where the 2^nd harmonic I_2H and ϕ_b are derived in a traditional way (see Appendix). By substituting ϕ_I and ϕ_Q for ϕ_s in Eq. (21) in Appendix respectively, we get two detection results of the rotation rate with different signs as Ω_I = −Ω, and Ω_Q = Ω. As shown by Eq. (8) and Eq. (9), the factor

\sqrt{2}

and the arc-tangent function ensure that scale factors of the two final outputs are linear and have theoretical values of −1 and 1, respectively.

3. Analysis of noise reduction in the synchronous difference

We have described the signal processing in demodulation from Eq. (3) to Eq. (9), where signals are in their ideal forms. In practice, detected results have error terms induced by noise, which are changing over time [10, 21]

Ω_{I} (t) = - Ω + N_{+} (t) - N_{-} (t) + N_{1} (t),

Ω_{Q} (t) = Ω + N_{+} (t) + N_{-} (t) + N_{2} (t),

where N₊(t) is common noise between Ω_I and Ω_Q, and N₋(t) is differential noise between them. N₁(t) and N₂(t) are independent random noise parts. It should be noticed here that common noise and differential noise are defined between two outputs Ω_I and Ω_Q, in spite of different signs with Ω. As Ω_Q(t) and Ω_I(t) are obtained simultaneously, synchronous difference is conveniently carried out in the novel method to eliminate common noise as

Ω_{out} (t) = \frac{Ω_{Q} (t) - Ω_{I} (t)}{2} = Ω + N_{-} (t) + \frac{N_{2} (t) - N_{1} (t)}{2} .

Equation (10) and Eq. (11) show that Ω_I(t) and Ω_Q(t) are orthogonal detection results of the same target Ω. Two signals modulated on the light wave travel through the same circuit simultaneously, thus both short term noise and long term instability in two results have a considerable common part in N₊(t).

First, we study short term noise reduction with the elimination of N₊(t). Short term noise mainly affecting the IFOG performance includes thermal noise, shot noise, and light intensity noise as [11, 13]

< i_{N}^{2} > = < i_{T}^{2} > + < i_{S}^{2} > + < i_{I}^{2} > = (\frac{4 k T}{R_{L}} + 2 e < i > + \frac{< i >^{2}}{Δ ν}) B,

where subscripts T, S, and I stands for thermal, shot, and intensity, respectively. k is the Boltzmann constant, T is absolute temperature, e is electron charge, Δν is the source bandwidth, R_L is detection load resistance, <i> is mean electric current at the PD, and B is the detection bandwidth. As optical intensity noise is proportional to the square of the light intensity, it becomes the major component of short term noise at the output when the optical power reaching the detector is more than a few tens of microwatts [13].

The optical intensity noise at the IFOG output can be expressed by [9, 13]

n_{P D} (t) = η n_{S} (t) g_{Q} (t),

where n_s(t) is the inherent intensity noise of the light source, and g_Q(t) = g[t − (π/4ω)] is the IFOG transfer function when Eq. (1) is used as phase modulation. Here g(t) is the IFOG transfer function when traditional sine wave modulation is used (see Appendix). The light intensity received by the PD is given by

I_{P D} = I_{D} + n_{P D} (t) = η (I_{0} + n_{S} (t)) g_{Q} (t),

where I_D is the detection signal without intensity noise (see Appendix). The noise part around the 1^st harmonic of sin(ωt) in the beat signal n_S(t)g_Q(t) should be considered, which finally affects detection results. If the 45° phase shift is not set between the modulation signal and reference signal, the distribution of this part of noise in the in-phase signal and the quadrature signal will not be equal in most cases [9]. In our modulation, a phase shift of 45° is set between the modulation signal and the reference signal, and the eigenfrequency modulation is used, so that two channels have the same magnitude of signal theoretically. According to the symmetry of two channels in this case, the power of intensity noise should be commonly distributed in the in-phase signal and the quadrature signal as

< {[n_{I} (t)]}^{2} > \approx < {[n_{Q} (t)]}^{2} > \approx \frac{1}{2} < {[n_{1 H} (t)]}^{2} >,

where, n_I(t), n_Q(t) are effective intensity noise in I_I(t) and I_Q(t), and n₁_H (t) is the noise part around the 1^st harmonic of sin(ωt) in n_PD(t). From this point of view, two channels have a considerable common part of short term noise in N₊(t), although signals have opposite signs.

Numerical simulation for IFOG outputs with intensity noise is carried out according to Eq. (15). White Gaussian noise is added to the source intensity for modeling the influence of intensity noise. Simulation for a source intensity SNR of 35dB is shown by Fig. 2. Ω_I(t), Ω_Q(t), Ω_out (t), and Ω_T (t) have mean values of −9.287°/h, 9.161°/h, 9.224°/h, and 9.358°/h, respectively. Noise amplitudes quantified by standard deviation are 1.667°/h, 1.726°/h, 0.604°/h, and 0.623°/h for Ω_I(t), Ω_Q(t), Ω_out (t), and Ω_T (t), respectively. Here Ω_I(t) is negative in accordance with Eq. (10). The noise in Ω_out (t) is obviously reduced by the difference procedure, as its deviation is much lower than single channel results Ω_I(t) and Ω_Q(t). The noise amplitude in Ω_T (t) is also lower than Ω_I(t) and Ω_Q(t), for the difference noise part between $I_{I}^{2}$ and $I_{Q}^{2}$ is reduced by the procedure given by Eq. (28) in Appendix. The noise in Ω_out (t) is lower than Ω_T (t), indicating that intensity noise reduction by the linear difference procedure is more effective.

Fig. 2 Numerical simulation for IFOG outputs with optical intensity noise. The signal-to-noise ratio (SNR) of the source intensity is 35 dB. Theoretical value of the rotation rate is 9.667°/h in accordance with the following experiment. 5000 samples are given, where the sampling time is 0.15 s. Single channel results Ω_I and Ω_Q, differential result Ω_out, and the traditional demodulation result Ω_T are put forward for comparison.

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For further comparison of four outputs in intensity noise limited IFOGs, simulation of noise amplitudes at different SNRs of source intensity are given in Fig. 3(a). For each output at each SNR, the noise amplitude is represented by the standard deviation of 5000 samples, where simulation parameters are the same as in Fig. 2. It can be seen that noise amplitudes in Ω_I(t) and Ω_Q(t) are close to each other in accordance with Eq. (16). Ω_out (t) has the lowest noise amplitude among the four outputs in all considered SNRs, and its curve is much lower than single channel results Ω_I(t) and Ω_Q(t). This is a clear evidence for the reduction of noise due to the synchronous difference procedure. Furthermore, the elimination of N₊(t) in Eq. (12) is proved more effective than the difference noise reduction in Eq. (28), as the curve for Ω_out (t) is lower than the curve for Ω_T (t). When the SNR is higher than 40 dB, the intensity noise reduction is not so obvious as low SNRs, which in practice may be submerged by other kind of noise such as phase noise induced by vibration of light circuit. In other words, performance of different demodulation methods is similar to each other when noise amplitude is very low.

Fig. 3 (a) Numerical simulation for output noise amplitude versus SNR of source intensity. Noise amplitudes are quantified as standard deviations for Ω_I, Ω_Q, Ω_out, and Ω_T. (b) Numerical simulation for output bias versus SNR of source intensity. Biases are calculated by mean values of Ω_I, Ω_Q, Ω_out, and Ω_T.

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The bias performance against intensity noise for the four outputs is also studied in the simulation, as show in Fig. 3(b). The traditional demodulation result Ω_T (t) perform badly in low SNRs, where large values of bias appear. Unfortunately, the amplitude of intensity noise may fluctuate over time in applications. As a result, the bias of Ω_T (t) will wave up and done, i.e., the bias stability of Ω_T (t) will get worse due to unstable intensity noise. On the other hand, biases of Ω_out (t), Ω_I(t), and Ω_Q(t) are scarcely affected by the intensity noise. As the mathematical expectation of white Gaussian noise is zero, it contribute no bias or bias instability into the output if the system is linear. In the low rotation rate case, the Sagnac phase shift ϕ_S is very small. For instance, in the experiment detecting the Earth’s rotation rate in next section, ϕ_S = 1.774 × 10⁻⁴ rad. In quadrature demodulation procedure described from Eq. (3) to Eq. (9), the only nonlinear functions are Eq. (8) and Eq. (9), which are almost linear when ϕ_S is small and thus arctan(ϕ_S) ≈ ϕ_S. However, Ω_T (t) is obtained by a nonlinear function as Eq. (28), even short term noise will affect its bias and bias stability.

As a conclusion of short term noise analysis, the quadrature demodulation with synchronous difference is better than either the single channel demodulation or the traditional demodulation in intensity noise limited IFOGs. Ω_out (t) has the lowest noise amplitude, and its bias is robust against short term noise.

In addition, long term instability can also be reduced during the synchronous difference. The bias drift of an IFOG is caused by temperature fluctuation, the Faraday effect, electric device instability, and other low-frequency noise sources [4, 10, 16, 19, 22]. Polarization controlling methods are mostly used for bias drift reduction, which mainly focus on the drift induced by polarization nonreciprocity effects. Meanwhile, the bias drift can also be suppressed by methods related to modulation and signal processing, as some drift components have distinguishable features with the real rotation rate.

The unstable biases in Ω_I(t) and Ω_Q(t) can be written in terms of common part B₊(t), differential part B₋(t), and independent parts B₁(t) and B₂(t). Considering only long term instability, detection results can be written as

Ω_{I} (t) = - Ω + B_{I} (t) = - Ω + B_{+} (t) - B_{-} (t) + B_{1} (t),

Ω_{Q} (t) = Ω + B_{Q} (t) = Ω + B_{+} (t) + B_{-} (t) + B_{2} (t),

Ω_{out} (t) = Ω + B_{-} (t) + \frac{B_{2} (t) - B_{1} (t)}{2} .

Here the eliminated common part B₊(t) also stands for the bias component independent of the polarity of the rotation rate. In many cases of practical IFOGs, B₊(t) is the main part of total bias [10]. The elimination of B₊(t) is thus effective for bias drift reduction in these IFOGs, where the main part of bias drift is intrinsically independent of the polarity of the rotation rate.

Both short term noise and long term instability are predicted to be reduced in Ω_out, due to the elimination of the common noise term N₊(t). In this way, gyroscope precision is enhanced. These are proved by the following experimental results.

4. Experimental results

The open-loop IFOG shown in Fig. 1 was horizontally placed on a stable optical bench, sensing only the Earth’s rotation rate. The theoretical value is 9.666°/h, as the laboratory latitude is 39.99° north and the rotation rate of the earth is 15.041°/h [23]. The sampling rate and the resolution of the digitizer were 2 MHz and 22 bits respectively, and the theoretical resolution of the IFOG was 0.007°/h accordingly. The source intensity was 13 dBmW, so that intensity noise was a big part of short term noise at the output. A phase modulation frequency of 52.35 KHz was used, which was an accurately measured value of the coil eigenfrequency. The traditional sinusoidal demodulation and the quadrature demodulation were both carried out for comparison. The test was carried out in a laboratory on the 4^th floor of a building, where human activities might introduce additional noise.

IFOG output data comparison is shown in Fig. 4. We can see short term noise in Ω_out is reduced notably by the synchronous difference in comparison with Ω_I and Ω_Q, as the output data is more concentrated. In the time domain results, we should also notice that there are some high spikes due to environmental acoustic vibrations. These spikes are mainly caused by human activities near our laboratory and appear only occasionally, thus they do not contribute much to long term statistical properties. Besides the occasional spikes, there are also contentious acoustic vibrations which have similar influence with other short term noise such as intensity noise and thermal noise. All these kinds of short term noise can be treated as a whole from a mathematical perspective, and be discussed together in the comparison.

Fig. 4 Experimental results for IFOG outputs. The test was during a period of 50 min, where the sample time was 0.148 s. Ω_I, Ω_Q, Ω_out, and Ω_T are put forward for comparison. The test was carried out in a laboratory on the 4^th floor of a building, where human activities might introduce additional noise due to acoustic vibration of light paths. This additional noise did not affect our comparison.

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Distribution of Ω_I and Ω_Q is illustrated in Fig. 5 as a scatter diagram, where solid curves represent fitting curves of their PDFs with a Gaussian distribution model. The scatter center is on the 135° line in accordance with Eq. (3). To quantify the short term noise reduction, we use the standard deviation of each output data to calculate the SNR improvement as [13]

R_{d B} = 20 log [\frac{min (σ_{I}, σ_{Q})}{σ_{out}}],

where σ_I = 1.048°/h, σ_Q = 1.045°/h, and σ_out = 0.732°/h are standard deviation values for Ω_I, Ω_Q, and Ω_out, respectively. The enhancement has a value of at least R_dB=3.09 dB from a single channel result to the final output, as an evidence of short term reduction in the synchronous difference procedure.

Fig. 5 Distribution of Ω_I and Ω_Q detected in experiment. The solid curves represent probability density functions (PDFs) of Ω_I, Ω_Q, and Ω_out. Ω_I has a mean value of −6.80°/h and standard deviation of 1.048°/h. Ω_Q has a mean value of 6.43°/h and standard deviation of 1.045°/h. Ω_out has a mean value of 6.61°/h and standard deviation of 0.732°/h. They all have negative biases, as their absolute mean values are lower than the theoretical value of 9.666°/h. The stable bias is not included in IFOG noise for it can be calibrated.

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The short term noise performance of Ω_out is similar to Ω_T, which has a standard deviation of σ_T = 0.720°/h ≈ σ_out. This result is consistent with theoretical analysis at high SNRs of source intensity in Fig. 3, where intensity noise difference between Ω_out and Ω_T is small and easily emerged by other kinds of noise. To further evaluate the performance difference between Ω_out and Ω_T, Allan variance method is used [24]. As shown in Fig. 6, it is the square root of the Allan variance σ(τ) versus cluster time τ. A significant phenomenon is that long term noise is notably suppressed, for the curve goes down rapidly when τ becomes larger. The numerical value of bias drift is obtained by reading the bottom of the curve. The bias drift of the traditional demodulated result Ω_T is 0.040°/h, and the value is reduced to 0.004°/h by the quadrature demodulation method for the same IFOG.

Fig. 6 Allan variance analysis of the IFOG. The bias drift of the traditional demodulated result is 0.040°/h, and the value is reduced to 0.004°/h by the quadrature demodulation with synchronous difference.

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Experiments are carried out on an open-loop configuration of the IFOG. With adjusted design, this method is also promising to be applied in closed-loop IFOGs [25].

5. Conclusions

In conclusion, we have shown a novel method of quadrature demodulation for IFOGs, where the synchronous difference of two detected results is used as a low-noise output. Both short term noise and long term instability are reduced by the synchronous difference procedure. Experimental results on an open-loop IFOG are consistent with theory analysis. In comparison with the traditional way of demodulation, the bias drift of the same IFOG is reduced by one order of magnitude using this method.

The most important advantage of the method is that it need not change the optical structure of the IFOG, so that it is possible to be applied in most sine wave modulated IFOGs for noise reduction without increasing their cost or complexity. In addition, it is a method that simultaneously suppresses short term noise and long term instability, which are both significant limitations of IFOGs. Therefore, this quadrature demodulation method is very profitable for designing high precision IFOGs.

Appendix: Traditional sinusoidal phase modulation and demodulation for IFOGs

In the IFOG, the Sagnac phase shift is given by [4]

ϕ_{s} = 2 π \frac{L D}{λ c} Ω,

where L and D are the length and the diameter of the fiber coil, c is the speed of light in vacuum, λ is the effective wave length of the light source, and Ω is the rotation rate. The interference signal received by the PD can be written as

I_{D} = η I_{0} g (t) = \frac{1}{2} η I_{0} {1 + cos [ϕ_{s} + Δ ϕ (t)]}

where I₀ is the source intensity, η is the response of the PD, and g(t) is the is the transfer function of the IFOG. For sinusoidal modulation ϕ_m(t) = ϕ₀ sin(ω_mt), we have

Δ ϕ (t) = ϕ_{m} (t) - ϕ_{m} (t - τ) = 2 ϕ_{0} sin (ω_{m} τ / 2) cos (ω_{m} t),

In this case, g(t) can be expanded by Bessel functions as

\begin{array}{l} g (t) & = & \frac{1}{2} {1 + J_{0} (ϕ_{b}) cos (ϕ_{s}) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} J_{2 n} (ϕ_{b}) cos [2 n ω_{m} (t - \frac{τ}{2})] cos (ϕ_{s}) \\ + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} J_{2 n - 1} (ϕ_{b}) cos [(2 n - 1) ω_{m} (t - \frac{τ}{2})] sin (ϕ_{s})} \end{array}

where J_n is the n^th Bessel function of the first kind, and ϕ_b = 2ϕ₀ sin(ω_mτ/2). The eigenfrequency ω_m = π/τ is suggested for optimal detection, so that ϕ_b = 2ϕ₀ [3,13]. In demodulation, ϕ_b is determined by J₄(ϕ_b)/J₂(ϕ_b) = I_4H/I_2H, and ϕ_s is obtained by

ϕ_{s} = arctan \frac{I_{1 H} J_{2} (ϕ_{b})}{I_{2 H} J_{1} (ϕ_{b})} .

Here I_nH is the amplitude of the n^th harmonic. It is usually assumed that the light intensity and the modulation depth are stable, i.e., I₀ and ϕ_b are both constants. Then only the 1^st harmonic is used for detection as

I_{1 H} (t) = - I_{0} η J_{1} (ϕ_{b}) sin (ϕ_{s}) cos (ω_{m} t - \frac{π}{2}) = I_{0} η J_{1} (ϕ_{b}) sin (ϕ_{s}) sin (ω_{m} t + π),

whose amplitude is denoted by I_1H, and the Sagnac phase shift is determined as

ϕ_{s} = arcsin [\frac{I_{1 H}}{I_{0} η J_{1} (ϕ_{b})}] .

In traditional demodulation, the amplitude of the first harmonic is determined by a single channel result, or obtained as

I_{1 H} = \sqrt{I_{I}^{2} + I_{Q}^{2}},

disregarding the phase of I_1H (t). Here I_I is the amplitude of the in-phase signal of the 1^st harmonic, and I_Q is the amplitude of the quadrature signal of the 1^st harmonic. Then the detection result is derived by Eq. (25) or Eq. (27).

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Quadrature demodulation with synchronous difference for interferometric fiber-optic gyroscopes

Abstract

1. Introduction

2. Theory of quadrature demodulation for the synchronous difference

3. Analysis of noise reduction in the synchronous difference

4. Experimental results

5. Conclusions

Appendix: Traditional sinusoidal phase modulation and demodulation for IFOGs

References and links

Cited By

Figures (6)

Equations (28)

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