Amplitude Control of Phase Modulation for Dithered Closed-loop Fiber Optic Gyroscope
- Author: Chong Kyoung-Ho, Chong Kil-To, Kim Young-Chul
- Organization: Chong Kyoung-Ho; Chong Kil-To; Kim Young-Chul
- Publish: Journal of the Optical Society of Korea Volume 16, Issue4, p401~408, 25 Dec 2012
The amplitude error of phase modulator used in closed-loop fiber optic gyroscope has occurred by the temperature dependency of the electro-optic coefficient, and also can be due to the square-wave dither signal which is generally applied for eliminating the deadzone. This error can cause bias drift and scale factor error. This paper analyzes the temperature dependency of the modulation amplitude and the relationship with the scale factor of the gyroscope, and deals with an amplitude control method. The error calculation logic considering the dither signal is implemented on the signal processing module. The result of experiments from a prototype gyroscope shows the effect of the modulation amplitude control and a considerable improvement on performances.
Phase modulation , Integrated optic circuits , Fiber optic gyroscope
The introduction of the Fiber Optic Gyroscope(FOG) occurred in the late 1970s after the invention of optical fibers. With help received from the researchers in the field of interferometry and telecommunications, The FOG has proven to be a leading technology in various applications for imaging systems, antennas, precise aiming of telescopes, and navigation systems [1-5].
The FOG is a mature technology and was originally designed as a low-cost alternative to the Ring Laser Gyroscope [2-3]. Due to the solid-state optical configuration and fiber technology, the FOG’s performance could lead to 0.001 deg/hr stability in miniature design [4-5]. The main optical components of FOG are composed of fiber, integrated optic chip(IOC), coupler, light source, and detector.
One of the key elements in determining the bias and scale factor of FOG is the modulator’s characteristics in the IOC [6,11-12]. The main function of the IOC is to split the light and to modulate the phases which are combined in a π/2-biased phase for maximum sensitivity, Sagnac phase shift by input rate, and feedback phase for compensating the rate. The important parameters of the IOC for improving the performance of gyroscope are polarization crosstalk, polarization extinction ratio, and modulation amplitude efficiency. Among them, the required level of polarization crosstalk and extinction ratio can be achieved by the manufacturing process.
However, as the modulation amplitude efficiency is intrinsically induced by temperature variation, controlling of the modulation amplitude should be needed because the bias and scale factor are strongly dependent on the feedback element which is directly proportional to the modulation amplitude coefficient of the IOC [7-13]. In addition, the modulation amplitude error can also occur due to the square-wave dither signal which is composed of plus and minus virtual rate. This error can cause bias drift and scale factor error in a closed-loop FOG.
This paper analyzes the temperature dependency of phase modulation amplitude from the effective electro-optic coefficient. In order to investigate how the modulation amplitude error affects the gyroscope, the relationship between temperature dependency of the phase modulation amplitude and the scale factor of the gyroscope is investigated. The error calculation method is improved for the gyroscope with the square-wave dither signal, and implemented on the signal processing module. Some of experimental results with and without the amplitude control loop are also discussed.
The FOG is based on the Sagnac effect, which produces a phase difference proportional to the rotation rate . Light from the broadband source is directed through the first coupler and a single-mode filter to a second coupler included on the IOC. The second coupler divides the light into two parts: one that will propagate clockwise through the sensing coil, and the other that will propagate counter-clockwise. They have opposite phase shift sensitivities to rotation about the coil axis. The Sagnac phase difference between clockwise and counter-clockwise becomes :
λis the central wavelength of the light source ; Lis the length of fiber ; Dis the average loop diameter of the fiber ; Ω is the input rate ; Δ ？Ris the Sagnac phase shift ; KOSFis the optical scale factor of gyroscope.
Figure 1 shows the general configuration of optics and IOC in FOG.
The phase difference induced by input rate is expressed as a cosine relationship for optical power. Once the broadband source generates the power, the photodector output through Sagnac loop in fiber spool
Pd is expressed as :
P0 is the peak power of the photodetector output.
To maximize the sensitivity of the gyroscope, the square-wave modulation-demodulation method is commonly used [12-14]. When the square-wave signal with Δ
？m amplitude is generated, the photodetector output during a period is expressed as :
？R is small ; Δ ？m is π/2.
When the square-wave modulation is applied and the input rate occurs, the photodetector output signal is shown in Fig. 2.
P0 is 1, Prateis simply equal to Δ ？R. The equation (5) is only available if both P0 and Δ ？m are stable. When the phase modulator has a small amplitude error, the equation (5) can be replaced Δ ？m with Δ ？m + δ:
？R is small ; Δ ？m is π/2.
From the equation (6), photodetector output has parasitic error due to a nonlinear term cos
δ, and can fluctuate due to environmental change. Fig. 3 shows the photodetector output when the phase modulation amplitude error occurs.
In order to reduce the error, the temperature characteristics
of the phase modulator should be analyzed first. The structure of the phase modulator is shown in Fig. 1. The modulator has two electrodes which can double the modulation efficiency. When one electrode of LiNbO3 is modulated, the phase modulation amplitude becomes  :
γis the optical confinement factor, defined as the portion of optical mode that is confined in the active layer ; Lm is the modulation length ; Δ nis the optical index change in the waveguide active layer.
Assume that the optical wavelength is constantly stabilized by a thermo-electric cooler. The temperature dependency of phase modulation amplitude
d(Δ ？m)/ dTis proportional to the temperature dependency of the optical index d(Δ n)/ dTbecause γand Lmare relatively small [7-8].
Here, we should look into the optical index change in detail. When the voltage
Vis applied to the electrodes on the modulator, the optical index change can be expressed as :
neffis the effective optical index of the active layer ; reffis the effective electro-optic coefficient; Vis the applied voltage ; dis the spatial gap between the electrodes across which the voltage is applied.
From equation (8), the temperature dependency of optical index is composed of
neffand reff. The reports shows that the temperature dependency of reffis approximately 4.9×10-4/℃
Because the temperature dependency of
reffis typically ten times higher than the temperature dependency of neff, combining equation (7) and (8), the temperature dependency of phase modulation amplitude can be written as :
In this section, the error calculation method from the photodetector output will be derived and an improved method will be suggested. A control loop modeling and the relationship between temperature dependence of phase modulation amplitude and scale factor of a gyroscope will be discussed next.
It is impossible to find the phase modulation error from the ±π/2 biasing modulation alone because we don’t know the reference power level of the photodetector output, as shown in Fig. 3. In order to find the phase modulation error, it is necessary to generate additional modulation, ±3 π/2. Fig. 4 shows the photodetector output with both ±π/2 and ±3π/2 modulation.
The modulation steps(pulses) come from the D/A converter through the digital steps determined by the digital register. The closed loop signal processing uses both ±π/2 biasing modulation and ramp modulation. The ramp modulation integrated by input rate digitally is used as feedback phase, which compensate the Sagnac phase proportional to the input rate. Typically, the maximum total steps, combined bias and ramp modulation, are limited to 2π because the maximum output voltage of the D/A converter is set to 2π.
When total steps reach the 2π-limit, the step register is reset, and ±π/2 biasing modulation is changed into ±3π/2 automatically. Considering the modulation amplitude error
δ, the photodetector output before and after 2π-reset can be expressed as :
The modulation amplitude error can be expressed by subtracting
P1 from P2.
In practice, the modulation amplitude error is calculated by the difference between average of ±π/2 steps and average of ±3π/2 steps in order to minimize the noise. The steps can increase or decrease with the input rate value. For a low rate, the steps increase due to a long modulation period that takes long for a small ramp bit to accumulate until 2π. Likewise, the steps decrease in a high rate and vary quickly from ±π/2 steps to ±3π/2 steps. Applying averaging method, the modulation amplitude error can be expressed as :
n : number of photodetector outputs for ±π/2 modulation step
m : number of photodetector outputs during ±3π/2 step
Ai : photodetector output(P1i) for ±π/2 modulation step
Bj : photodetector output(P2j) for ±3π/2 modulation step
This equation is commonly used because it can be simply calculated [13-14]. But in a case of adding a dither signal, equation (13) doesn’t match for exact calculation. Fig. 5 shows the photodetector output with dither signal.
Dithering is a widely used method in gyroscopes to overcome the deadzone where it can’t detect any rate [12-14]. The square-wave dither signal used in FOG changes the photodetector output power because it generates the additional phase shift in feedback modulation. In order to calculate the error considering dither amplitude Δ
？dith, we should apply Δ ？m+ Δ ？dithto equation (12), instead of Δ ？m:
This equation has a nonlinear term cosΔ
？diththat should be cancelled or compensated. The modulation amplitude error without dithering (Fig. 5(b)) can be successfully calculated by equation (13) without additional error. But the modulation amplitude error with dithering can include a residual error induced by dither amplitude, Δ ？dith.
The average of photodetector output for even steps
Bmresults in the same as T2 because the total average of Δ ？dithis cancelled. But if steps have odd number like An, the additional error occurs. Assume that the number of photodetector output Anis 5 during ±π/2 step as shown in Fig. 5(c). The average is calculated as :
Kis T1 is the average of An; T2 is the average of Bm
In order to remove the above additional error term
P0cosΔ ？dith/10, let’s define SΔ ？mas the sum of photodetector outputs and DΔ ？mas the difference from the first output to sum of following outputs. By subtracting DΔ ？mfrom SΔ ？m, we can remove the additional error as below :
nis the odd number ; Δ ？m= π/2.
n, SΔ ？m, and DΔ ？mare known values which can be extracted from the digital logic. The average value T1 is calculated by simply dividing (2 n-2), as shown in Table 1. The same rule for ±3π/2 can be applied.
The modulation amplitude error calculated from Table 1 can be compensated by the signal control loop shown in Fig. 6. The modulation amplitude controlled by PWM generator which transforms the digital value to pulse can adjust the range of D/A converter. A control loop model expressed in Laplace transform is shown in Fig. 7.
Kopis the open loop gain ; Kiis the integrator gain for stabilizing the loop ; Kampis the gain of the D/A converter ; Kmod is the modulation coefficient ; G( s) is the controller for closing the loop ; Kmis the gain for amplitude compensation.
When the gyroscope operates closed-loop mode in rate, the scale factor is not related to the open loop gain any more if the open loop gain is sufficiently large. The relationship between the control loop transfer function and scale factor can be expressed as :
The scale factor of the gyroscope depends on three factors,
KOSF, Kamp, and Kmod. Among them, the temperature dependency of KOSFand Kampcan be kept below a few ppm by wavelength stabilization and precision gain reference i: phase modulation sequences
circuits. So, the temperature dependency of modulation amplitude coefficient
dKmod/ dTis dominantly proportional to dneff/ dTfrom equation (9), and directly influences the scale factor of the gyroscope.
G( s) has a single-pole integrator. The input-output transfer function for the amplitude control loop is as below :
τis the time constant for the loop ; Kis the DC gain.
When the loop is closed, the output
Cm( s) has a desired value depending on temperature change and the feedback compensation α, shown in Fig. 7, is performed as follows :
The optical head and signal processing modules to verify the effect of amplitude control were implemented on prototype FOG. The optical head was made with Polarization Maintaining (PM) fiber, 1550 nm broadband source, IOC, 2×2 coupler, and photodetector. A 900 m PM fiber with a low birefringence and a small cladding diameter was chosen for good bias performance, wound in fiber spool with quadrupolar winding for reducing the temperature and vibration errors. A broadband source was made with 980 nm-pump laser diode, WDM, and coupler. This configuration promises good stability in power and wavelength. A push-pull modulator with long electrodes on the IOC was designed for high modulation efficiency. A photodetector with low dark noise and low capacitance was chosen for a low noise, high efficiency amplifier design. Special care should be taken in designing the photoamplifier as a small current below a few nano-ampere(nA) corresponding to small input rate can flow on circuits. The signal processing module includes constant current driving circuits for supplying constant power to the laser diode, digital modulation-demodulation logic, control logics for rate and modulation amplitude control, and DSP unit for compensation and digital communication. The optical head and signal processing module are shown in Fig. 8.
A Figure 9 shows a simulation result which compares a simple averaging method with the suggested method. In an amplitude open-loop mode, the amplitude error was increased up to 0.6% as dither amplitude was changed from 9 deg/sec to 15 deg/sec, while the suggested method was unchanged. This error can lead to a bias drift or scale factor error, even though the error goes to zero in an amplitude closed-loop mode.
The error without amplitude control shown in Fig. 10 is approximately 26,000 ppm(peak-to-peak) over full temperature range, which corresponds to 325 ppm/℃, a reasonable value for the typical temperature dependency of
reffas stated earlier in equation (9). Relatively the error with amplitude control is significantly reduced.
Figure 11 shows the relationship between amplitude control and scale factor. The scale factor is not only related to modulation amplitude
Kmod but can be affected by optical scale factor KOSF, according to equation (17). The variation of Kampand Kmod can be compensated by amplitude control loop but the variation of KOSFcannot be controlled. In order to find how much the modulation amplitude error can affect the scale factor, the amplitude control loop was tested without rotation (upper graph) and next, the scale factor was tested in rotation (lower graph). The upper graph residual error when the amplitude is controlled, existing unexpected temperature drift by modulator
or electronics. The lower graph has a similar curve to upper’s and the peak-to-peak values of both errors are almost same, 489 ppm and 444 ppm, respectively, and likewise, the residual errors by third order fitting are 64 ppm and 60 ppm, respectively. This result shows the scale factor is closely related to modulation amplitude. The scale factor residual error also corresponds to a sufficiently low value, acceptable for a navigation system.
The bias stability, one of the important parameters in determining gyroscope performance, was tested at room temperature. Even in room environment, the temperature is varied approximately 10℃ because the electronics mounted on the inner side of the gyro assembly heats up optical components. In Fig. 12, the bias instability without amplitude control is 0.32 deg/hr (1 sigma), changed along the temperature variation, while it is reduced to 0.1 deg/hr in an amplitude control. The Allan variance in Fig. 13 shows the difference of long-term bias instability more clearly. The Allan variance in an amplitude control is about 0.03 deg/hr, improved three times than without control (The bottom line of datum indicates the Allan variance value)
In this paper, we have discussed the control method of phase modulation amplitude in dithered closed-loop FOG. The origins induced by the modulation amplitude error have been investigated in terms of the effective electro-optic coefficient, pointing out the predominant factor of the scale factor error in gyroscope. The solution to suppressing the modulation amplitude error was proposed, enabling to the exact calculation in a square-wave dithering. We have shown that the control loop effectively eliminated the modulation amplitude error and was successfully implemented on prototype FOG, resulting in improved performances.
[FIG. 1.] Configuration of FOG optics and the structure of the IOC.
[FIG. 2.] Phase modulation and photodetector output when input rate occurs.
[FIG. 3.] Photodetector output with ±π/2 modulation when the modulation amplitude error occurs.
[FIG. 4.] Photodetector output with both ±π/2 and ±3π/2 modulation when the modulation amplitude error occurs.
[FIG. 5.] (a) Phase modulation steps (b) photodetector output without dithering (c) photodetector output with dithering.
[TABLE 1.] The average of steps during ±π/2
[FIG. 6.] Signal processing block diagram for phase modulation amplitude control.
[FIG. 7.] Modeling of phase modulation amplitude control loop.
[FIG. 8.] Optical head and signal processing module for prototype FOG.
[FIG. 9.] Modulation amplitude calculation error vs dither amplitude (simulation).
[FIG. 10.] Modulation amplitude error (measurement).
[FIG. 11.] Scale factor error (measurement).
[FIG. 12.] Bias instability (measurement).
[FIG. 13.] Allan variance (measurement).