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Resonator micro optic gyro (RMOG) is a promising candidate for applications requiring small, light and robust gyros. In optical passive ring resonator gyros, clockwise and counter clockwise lightwaves are modulated at different frequencies to reduce the backscattering induced noise. The effectiveness of this technique, however, is determined by the carrier suppression level. Accurate modulation index and high environmental temperature stability is required for achieving high total carrier suppression for the traditional single phase modulation technique (SPMT). In this paper, we propose an RMOG based on the double phase modulation technique (DPMT). Compared with the traditional SPMT, two additional phase modulations are added to provide additional carrier suppression. It is found that the control accuracy of the modulation index and temperature stability is relaxed more than 30 times. It is easily performed for reducing the backscattering error below the shot noise limited sensitivity. The modulation parameters in the DPMT are analyzed and optimized. Based on the optimum parameters of the DPMT, a bias stability of 1.85 × 10−4 rad/s is successfully demonstrated in the polarization maintaining silica waveguide resonator with the length of 7.9 cm. This is the best result reported to date, to the best of our knowledge, for a waveguide type passive ring resonator gyro.
©2011 Optical Society of America
1. Introduction
Optic gyros with no moving parts based on the Sagnac effect [1] are the perfect candidate for the inertial navigation systems. There are two very successful types of optic gyros: He-Ne ring laser gyro (RLG) [2] and interferometric fiber optic gyro (IFOG) [3]. With the development over past several decades, these two types of optical gyros show a good measurement performance, although there are some other potential alternatives [4]. However, the conventional RLG and IFOG are not easy to apply in the areas required for cheap and micro gyros, for the reason that they are complicated in fabrication process with a lot of individual components. With the development of the monolithically integrated optical components on waveguide circuits and batch micro fabrication techniques, integrated optical gyroscope can solve difficulty in gyro technology, such as the requirement of light weight and small size, low costs, low power consumption and so on [5].
Integrated optical gyroscope can be divided into two kinds of active and passive. For the active ones, the use of miniaturized ring laser for integrated RLG was proposed and researched in recent years [6-8]. A semiconductor ring laser (SRL) with a single mode fiber loop as an optical inertial rotation sensor was reported [9]. The experimental results verify that SRL’s can be operated as an active ring laser gyroscopes based on the Sagnac effect [9].The lock-in problem already observed in the RLG can be circumvented by the optoelectronic integrated circuits design and the SRL shows a potential to become a monolithic rotation sensor [10]. However, the performance of a monolithic SRL as an integrated optical gyro has yet to be reported. For the passive ones, resonator micro optic gyro (RMOG) [11] is proposed early in 1983 to reduce size, weight, and cost compared to the IFOG. The first ever fully integrated RMOG was made on glass substrate by Northrop Corporation in 1990 [12]. The RMOG with silica based waveguides on a silicon substrate has been researched for several decades aiming at the fully integration onto a single chip [13-19]. For example, the rate response of the RMOG has been tested over the range of 1 to 200 °/s [15]. Of course, some other integrated optic ring resonators based on the other materials are proposed for the improvement of the RMOG, such as a 6 cm diameter ring resonator with a finesse of 10 in Ti: LiNbO3 waveguide [20] and low loss InP-based ring resonators [21]. Although these two materials show promising prospects of passive and active integration for integrated optical gyroscope, the experimental results of the RMOG are expected to be improved [20]. On the other hand, the methods of compensation in propagation loss are proposed to obtain a very large finesse, such as a 1.6 cm diameter active ring resonator with a finesse of 250 in a neodymium-doped glass waveguide [22] and a silica-on-silicon ring resonator with the integration of two semiconductor optical amplifiers (SOAs) [23]. The minimum rotation rate for the latter one is predicted as low as 0.5 °/h with a ring radius of 2 mm and an input power of 24 dBm [23]. However, another pump laser is required [22,23] which will bring in more complexity in the RMOG and the experiment results with these active ring resonators have not been reported yet. In a word, the RMOG with silica based waveguides on a silicon substrate have been developed for several decades [13~19], which is very competitive compared with the other integrated optical gyroscopes based on the other materials and technologies described above up to now. The main achievements on RMOG with silica-on-silicon technology are based on the two facts. Firstly, silica-on-silicon technology can achieve very low propagation loss (around 0.0085dB/cm) and very high finesse (~132) of waveguide ring resonator (WRR) to enhance the gyro sensitivity [24]. Secondly, it is a relatively mature technology commonly used in the optical telecommunication [25]. The final goal of the RMOG is to combine many active/passive waveguide-based devices with different functions on a single chip. A possible solution is based on a silicon platform. Some silicon active components have been demonstrated successfully, such as silicon optical modulators [26], SiGe photo-detectors [27], and silicon Raman lasers with narrow linewidth [28]. These wonderful achievements open the promising way for the RMOG in silicon platform.
In the research of the resonator optic gyro (ROG), the resonator fiber optic gyro (RFOG) utilizing the fiber ring resonator has reported the resolution of 0.36 °/h with the YAG laser source by T. Imai et al, early in 1996 [29]. However, unlike the development of the RFOG, the realization of the low-noise RMOG comparable to the RFOG has not been reported yet. Up to now, a short-term bias stability has been improved to 0.46°/s, which is the best ever demonstrated in silica waveguide ring resonator with the ring length as short as 7.9cm [17]. The RMOG mainly suffers from the backscattering induced noise, which is theoretically estimated to be about 4500 °/s for a traditional silica waveguide ring resonator [17]. High carrier suppression level is crucial to reduce the backscattering induced noise [30]. K. Takiguchi and K. Hotate et al. demonstrated that a carrier suppression as high as 80 dB can be achieved when a binary phase shift keying (BPSK) scheme is applied with an acousto-optic modulator (AOM) in the RFOG [31]. For the development of an integrated RMOG, however, AOM is not a practical device, because it is hard to be integrated. It was also proposed to apply the BPSK or ternary phase shift keying modulation (TPSK) with a thermo-optic (TO) phase modulator in a waveguide-type ring resonator gyro [14,32]. However, their effectiveness is severely reduced when they are applied with the phase modulator. The limitation comes from not only the narrow bandwidth of the TO modulator, but also the implementation applied with the phase modulator. The latter influence is inherent and crucial. When the BPSK is applied with an AOM (a frequency shifter), high carrier suppression level is easily obtained. However, carrier suppression is limited at 15dB or ~35 dB for the BPSK or TPSK applied with a phase modulator theoretically.
While aiming at the development of an integrated RMOG with phase modulators having better performance, we adopt the lithium niobate (LN) integrated phase modulators. Recently, the backscattering induced noise is largely reduced by the carrier suppression with the phase modulation technique applied with the LiNbO3 phase modulators [17]. LiNbO3 phase modulators are beneficial to the miniaturization and integration of the RMOG. A total carrier suppression of 120 dB is required to reduce this backscattering error below the shot noise limited sensitivity of the RMOG with this traditional single phase modulation technique (SPMT). Applied with SPMT, the corresponding modulation amplitude accuracy is about 1.8 mV with a half-wave voltage of 3 V. The half-wave voltage is a temperature dependent parameter. Temperature drift is restrained to 1.6 °C required for a carrier suppression of 120 dB when the temperature coefficient of the half-wave voltage of the LiNbO3 phase modulator is about 500 ppm/°C [33]. These severe conditions limit the RMOG for practical applications.
In this paper, we propose an RMOG using the double phase modulation technique (DPMT) on the basis of the SPMT [16]. A part of experimental results of this work was presented by our group in the Conference on Lasers and Electro-Optics 2010 (CLEO 2010) [19], while this paper provides more detailed introductions, analyses, and descriptions. Compared with the traditional SPMT, two additional phase modulations are added to provide the additional carrier suppression. It is found that the control accuracy of the modulation index and temperature stability is relaxed dramatically. The modulation amplitude accuracy is down to 60 mV required for the 120 dB carrier suppression. With a temperature fluctuation of 51 °C, a carrier suppression of 120 dB can be maintained. The influences of the additional carrier suppression on the gyro performances are further investigated. Simulation shows that the influence of the additional carrier suppression modulation on gyro signal detection can be neglected when its frequency is 103 times smaller than the modulation frequency for signal detection. An RMOG system is constructed based on this method. The silica WRR is 7.9 cm long with only one turn, which is designed for polarization maintaining. The free spectral range (FSR), the full width at half maximum (FWHM) of the resonance curve, the finesse (F) of the WRR and the resonance depth are 2.61 GHz, 56.3 MHz, 46.3 and 94%, respectively. In the experiment, a single eigen-state of polarization (ESOP) is excited with polarized lightwave along with the polarization-axis of the polarization maintaining fiber and polarization maintaining waveguide. With the optimum parameters of the DPMT, a bias stability of 1.85 × 10−4 rad/s is successfully demonstrated.
2. Principle and Simulation
2.1 Principle of the DPMT
Figure 1 shows the setup of the RMOG based on the DPMT. All the fibers in the system are polarization maintaining. Lightwave from a FL (linewidth less than 50 kHz) is equally divided by a coupler C1 and injected into the silica WRR in CW and CCW lightwaves. PM1, PM2, PM3, and PM4 are driven by the sinusoidal waveforms from the signal generators SG1, SG2, SG3, and SG4 with the modulation frequencies f1, f2, f3, and f4, respectively. The CCW and the CW lightwaves from the silica WRR are detected by the InGaAs PIN photodetectors, PD1 and PD2, respectively. The output of the PD2 is fed back through the lock-in amplifier LIA2 to the FBC to compensate the fluctuations in the resonant frequency and/or the central frequency of the FL. A proportional integrator is adopted in the FBC to eliminate the residual error at the lock-in frequency [17]. The demodulated signal of the CCW lightwave from LIA1 is used as the open-loop readout of the rotation rate.
Fig. 1 Experimental setup of the RMOG based on the DPMT. FL: fiber laser; ISO: isolator; C1, C2: couplers; PM1, PM2, PM3, PM4: phase modulators; SG1, SG2, SG3, SG4: signal generators; CIR1, CIR2: circulators; Sync: synchronized signal; PLC: planar lightwave circuit; WRR: waveguide ring resonator; PD1, PD2: photodetectors; LIA1, LIA2: lock-in amplifiers; FBC: feedback circuit.
Figure 2 shows the schematic configuration of the DPMT. The LiNbO3 phase modulators are used to modulate the phase of the lightwaves from the laser. In Fig. 2(a), two phase modulators are used. PM1 and PM2 are driven by the sinusoidal waveforms with the modulation frequencies f1 and f2, respectively. f1 = ω1/2π, f2 = ω2/2π, here ω1 and ω2 are the angular frequencies. One of the phase modulators, for example, PM1 is used for gyro signal detection. The amplitude of the sinusoidal voltage wave V1 is carefully optimized to reduce the carrier [17]. After the lightwave is phase modulated, the carrier is reduced to the factor of J0(M1). J0(M1) denotes the zero-order Bessel function. M1 is the modulation index, M1 = πV1/Vπ1, here Vπ1 is the half-wave voltage of PM1. By setting M1 at the value satisfying J0(M1) = 0, the carrier is completely suppressed. The deviation from the optimum modulation index causes a residual carrier. Phase modulator PM2 is added for the additional carrier suppression which is shown in Fig. 2(a). Using the series of two phase modulators, the residual carrier caused by the first phase modulator PM1 is expected to be effectively reduced by the additional phase modulator PM2. In Fig. 2 (b), two sinusoidal phase modulations are realized in one optical phase modulator through an adder. Since the modulation signals are applied with the phase of the lightwaves, these two models are the same in theoretical analysis. The following analysis is based on the model of Fig. 2(a). Figure 2(c) shows the analysis model of the WRR with the DPMT.
Fig. 2 (a) Basic model of the DPMT with two phase modulators (PM), (b) Basic model of the DPMT with one PM and an adder, (c) Analysis model of the WRR with the DPMT.
As shown in Fig. 1, in the CCW direction, after being modulated by two phase modulators PM1 and PM2, the optical field at the entrance of the WRR is written as:
where Jn(M1) and Jm(M2) are the first kind of Bessel functions with n-th and m-th orders, respectively. E0 is the amplitude of the input electrical field. ω0 is the center angular frequency of the highly coherent laser. M1 and M2 are the phase modulation indexes. M1 = πV1/Vπ1, M2 = πV2/Vπ2, here Vπ1 and Vπ2 are the half-wave voltages of PM1 and PM2, respectively. V1 and V2 are the amplitudes of the sinusoidal wave.X. Zhang and H. Ma et al. analyzed the RFOG based on the single phase modulation technique (SPMT) applied with the LiNbO3 phase modulators, which can be applied for the RMOG [34]. Combined with the Bessel function expansion and the overlapping field methods [34], the output field of the WRR is written as:
where Hω(n,m) and Φω(n,m) are the amplitude and phase transfer functions of the WRR, respectively. Hω(n,m), Φω(n,m), and ωn,m are the functions of nω1 + mω2 in the DPMT instead of the item of nω1 in the SPMT.The modulated lightwave output from the WRR is then converted to the electric signal by the photodetector (PD). The output voltage VPD-out(t) from the PD is expressed as
where c is the light velocity in vacuum, ε0 is the permittivity in vacuum. Rv is the response factor of the photodetector in the unit of V/mW. m, n, m′, n′ are integers. Equation (3) contains the spectrum at (n-n′)f1 ± (m-m′)f2. The spectrum at f1 deduced from Eq. (3) is expressed asThe voltage components at frequency f1 are demodulated by the LIA with the pass band width less than f2. The demodulated voltage from the LIA with the unit gain is expressed as
where φ1 is reference phase from the LIA.2.2 Analysis and simulation of the DPMT
The ultimate sensitivity of the RMOG is determined by the shot noise of the photodetector. The minimum rotation rate Ωmin is given by [35]:
where λ0 is the vacuum wavelength; f0 is the central frequency of the laser; c is the velocity of light in vacuum; h is Planck's constant; A is the area of the WRR; F is the finesse of the WRR; Pi is the light power at detector; ηD is the quantum efficiency of the photodetector; τ is the integration time. Using the parameters of the RMOG (λ0 = 1550 nm; A = 5 × 10−4 m2; F = 46.3; Pi = 200 µW; ηD = 0.5; τ = 10 s), the minimum rotation rate Ωmin is 8.1 × 10−5 rad/s. The backscattering induced error is theoretically estimated to be about ~79 rad/s without any countermeasure in the RMOG [17]. A method for reducing this error is to suppress the carrier in each direction and can be calculated by [30]where D is the diameter of the WRR; L is the total length of the WRR; N is the number of suppressed carriers; σR is the backscattering coefficient in the WRR. ΔV/V is the suppressed carrier voltage error, which is proportional to the zero-order Bessel function of the modulation index M, J0(M), with the sinusoidal phase modulation applied [36]. Carrier suppression is applied in both the CW and CCW lightwaves at the same time to achieve higher total suppression. The total suppression is proportional to the product of J0(M1) and J0(M2) in the SPMT. The total carrier suppression is as high as 120 dB required to reduce the backscattering error below the shot noise limited sensitivity. This high carrier suppression requires the high accuracy of the amplitude of the modulated signal and the high precision of half-wave voltage of the phase modulator. The half-wave voltage is a temperature dependent parameter. The typical temperature coefficient of the half-wave voltage of the LiNbO3 phase modulator is about 500 ppm/°C. The high carrier suppression requires a high temperature stable environment. These severe conditions limit the RMOG to practical applications.These severe requirements are largely relaxed when the DPMT is used. The total carrier suppression is proportional to the product of J0(M1), J0(M2), J0(M3), and J0(M4). As shown in Fig. 3(a) , the accuracy of the modulation amplitude is relaxed from 1.8 mV to 60 mV. When the modulation amplitude is stable, the temperature stability requirement relaxes from 1.6 °C to 51 °C after applying the DPMT as shown in Fig. 3 (b). That is to say, more than 30 times relaxation is achieved compared with the SPMT for the backscattering error suppression.
Fig. 3 Calculated backscattering error in the RMOG. (a) relationship between the modulation amplitude accuracy and the backscattering noise with the DPMT (bottom curve) and the SPMT (upper curve), respectively. (b) relationship between the temperature drift and the backscattering error in the RMOG with the DPMT (bottom curve) and the SPMT (upper curve) applied, respectively. The half-wave voltage of the phase modulator is 3.0 V and its temperature coefficient is 500 ppm/°C.
The above analysis shows that the two phase modulations being added for the additional carrier suppressions in the DPMT reduce the backscattering error below the shot noise limited of the RMOG effectively. The influences of the additional carrier suppression on the gyro performances are further investigated. Figure 4(a) shows the relationship between the slope of the demodulation curve with the unit gain and the modulation frequencies. The maximum slope is 5.8 mV/MHz when the modulation frequency f1 is near 10 MHz. The slope of the demodulation curve decreases as f2 increases. The deterioration is less than 10−7 when f2 is smaller than f1 by three orders. As the principle of the DPMT described above, the influence of f2 on the transfer function Hω(n,m)exp[jΦω(n,m)] is neglected when f2 is much smaller than f1.
Fig. 4 Simulation results of the RMOG with the DPMT. (a) influence of modulation frequencies on the slope of the demodulation curve with the unit gain of the LIA when the phase modulation indices for both M1 and M2 are 2.38. The incident light power of the photodetector is about 200 µW. The responsivity of the photodetector Rv is 0.848 V/mW; (b) normalized demodulation curve with the modulation frequencies f1 of 9 MHz, f2 of 1 kHz, and the modulation indices for both M1 and M2 of 2.38.
Figure 4(b) shows the normalized demodulation curve. The modulation frequencies f1 and f2 are 9 MHz and 1 kHz applied to the phase modulators PM1 and PM2, respectively. The modulation indices are 2.38 for both phase modulators PM1 and PM2, which is about 1% deviation from 2.405 and easily be done [30]. As seen from Fig. 4(b), near the resonant point, there is a good linear region in the demodulation curve with the DPMT similar to the SPMT, which is used to keep resonance and obtain the gyro signal. As Fig. 4(b) shows, the maximum range of resonant frequency difference corresponding to the linear region is ± 23.49 MHz. The WRR is composed of a waveguide ring with 7.9 cm in length and 2.5 cm in diameter. As a result, the theoretical scale-factor of RMOG is 11.1 kHz/(rad/s). Its maximum detection rotation rate range is ± 2.11 × 103 rad/s theoretically. It indicates that RMOG has a large dynamic range for tactical application.
3. Experiments and Results
The experimental setup of the RMOG based on the DPMT is shown in Fig. 1. According to the simulation results as shown in Fig. 4, the modulation frequencies f1 and f3 should be set around 10MHz to maximize the slope of the demodulation curve. The modulation frequencies f2 and f4 should be smaller than f1 and f3 by three orders, respectively. To remove the influence of the backscattering intensity, not only the modulation frequencies f1 and f3 should be different but also their higher harmonic should not overlap each other. This can be satisfied if f1 and f3 are 9.0 MHz and 9.3 MHz in our experiments, respectively. Correspondingly, the modulation frequencies f2 and f4 are chosen 1.0 kHz and 1.1 kHz, respectively. The amplitudes of the sinusoidal voltage wave V1, V2, V3, and V4 determine the total carrier suppression. The proportional integrator is adopted in the FBC to eliminate the residual error at the lock-in frequency [17]. The demodulation signal of the CCW lightwave from LIA1 is used as the open-loop readout of the rotation rate. Furthermore, the whole system is mounted on a vibration isolation platform.
Polarization-maintaining silica PLC is fabricated and adopted instead of the single-mode chip that we used previously [16] to reduce the polarization-induced noise. The length of the silica WRR is 7.9 cm. As a result, the free spectral range (FSR) of the silica WRR is 2.61GHz. A sawtooth-wave signal from a signal generator (SG) tunes the central frequency of the FL linearly with respect of time. The resonant curves for the CW and CCW lightwaves of the silica WRR are detected by photodetector PD2 and PD1, respectively. At the output of the LIA1 and LIA2, the demodulation curves are detected correspondingly. Figure 5(a) and 5(b) shows the measured results for the CCW lightwave. The measured full width at half maximum is about 56.3 MHz. So the finesse of the fabricated silica WRR is about 46.3. Using Eq. (6), the theoretical sensitivity is calculated at 8.1 × 10−5 rad/s with the launching power at the photodetector of 0.2 mW and the integration time of 10 s.
Fig. 5 Measured resonance curve and the demodulation curve for the CCW lightwave. (a) resonance curve for the CCW lightwave. (b) demodulation curve for the CCW lightwave with a gain of 80. The modulation frequencies f1 and f2 are 9.0 MHz and 1.0 KHz, respectively. The modulation indices M1 and M2 are 2.38. The optical power at the photodetector is about 200 µW. The responsivity of the photodetector Rv is 0.848 V/mW.
Figure 5(b) shows the measured demodulation curve for the CCW lightwave. The gain of LIA1 in the practical RMOG is about 80. Computed from the CCW demodulation curves, the slope of the demodulation curve for the linear part with unit gain is 5.4 mV/MHz. The maximum range of resonant frequency difference corresponding to the linear region is ± 23.67 MHz. These experimental results have a good agreement with the simulation described in Fig. 4. With the theoretical scale-factor of RMOG is 194 Hz/(°/s) and the LIA1 gain of 8000, the practical scale factor of the RMOG is 8.4 mV/(°/s).
Figure 6 shows the experimental results of the RMOG compared with the DPMT and SPMT. As Fig. 6(a) shows, the bias stability of the RMOG with the SPMT deteriorates when the modulation amplitude V3 deviates from the optimum value. This is because the backscattering induced noise becomes dominant when the total carrier suppression is not high enough. With the same deviation of V3 from the optimum value, the bias stability of the RMOG is 1.85 × 10−4 rad/s with the DPMT as the lower curve of Fig. 6(a) shows. It shows that the modulation amplitude accuracy is largely relaxed when the DPMT is used. In the deviation ± 1% of optimum value, the bias stability with DPMT application is at least two orders magnitude less than the stability with SPMT application. All these show the significant effectiveness of the DPMT. With the DPMT, the relatively stable result of the RMOG shows that backscattering induced noise is not a major noise source.
Fig. 6 Experimental results of the RMOG compared with the DPMT and the SPMT. (a) Influence of the modulation amplitude applied with PM3 on the bias stability of the RMOG with the SPMT (f1 = 9.0 MHz, f3 = 9.3 MHz, M1 = 2.38) and the DPMT (f1 = 9.0 MHz, f2 = 1 KHz, f3 = 9.3 MHz, f4 = 1.1 KHz, M1 = M2 = M4 = 2.38). The half-wave voltage of the phase modulator is 3.0 V. (b) Typical outputs of the RMOG over 60 seconds with an integration time of 10 s. M1 = M2 = M3 = M4 = 2.38; f1 = 9.0 MHz, f2 = 1 KHz, f3 = 9.3 MHz, f4 = 1.1 KHz.
Figure 6 (b) shows the gyro output signal from LIA1 when the system is at rest. The noise level of 1.85 × 10−4 rad/s for 60 seconds duration is obtained. According to the theoretical calculation as shown in Fig. 3(a), 1% deviation from the optimum amplitude (M = 2.38), the backscattering error is lower than 8.1 × 10−5 rad/s with the DPMT. The experimental result is about twice of the theoretical data. The residual noise may be from the other noise sources such as Kerr effect and polarization fluctuations. And the reduction of these two noises will also benefit for the improvement of the long term stability.
The output of gyro is roughly proportional to τ-1/2 (τ is an integration time), when the residual noise is mainly limited by the random noise [37]. This relationship is also can be theoretically obtained by the Eq. (6) [35], where the shot noise is one of the random noise. If the residual noise of RMOG was mainly limited by random noise, the bias stability with SPMT will be reduced from 0.46°/s [17] to 0.15°/s for the same integration time of 10s. The bias stability of 1.85 × 10−4 rad/s (~0.01°/s) obtained by the DPMT is still 15 times lower than 0.15°/s. It shows the effectiveness of the DPMT. For the reason that the carrier suppression of DPMT can easily reach the 160 dB, while the carrier suppression of SPMT is 100 dB [17]. As described above, the carrier suppression of 120dB is required for the ultimate sensitivity of the RMOG. However, the suppression of one sinusoidal modulation can easily be done to one or two percent [30], which indicates that a total carrier suppression ~80 dB of SPMT can easily be obtained. So, the carrier suppression 100dB of SPMT is obtained by the careful adjustment of the modulation amplitude. On the other hand, the accuracy of the modulation amplitude and the temperature stability requirement are relaxed after the DPMT is applied. As shown in Fig. 3, more than 30 times relaxation is achieved compared with the SPMT for the backscattering error suppression. All these show the improvement with DPMT compared with the SPMT applied to the RMOG.
For the practical application, the integration time exceeding 0.1 s is incompatible with several gyro applications. Besides the integration of optical components, the next step we do is that the reduction of integration time while keeping high sensitivity. The Saganc effect is very weak in this small silica WRR with a diameter of 2.5cm and a ring length of 7.9 cm. The minimum rotation rate Ωmin is 8.1 × 10−4 rad/s ( = 167°/h) for the integration time of 0.1s, which is calculated by Eq. (6) with the finesse of 46.3 and incident light power in PD of 0.2 mW ( = −7dBm). Obviously, it is far from the requirement in the range of 1~10°/h for tactical application. So, we firstly realize the higher sensitivity with high integration time of 10s. And shown in Fig. 6 (b), the experiment results show that the bias stability is close to the shot noise limited sensitivity based on the DPMT. Then we will improve the shot noise limited sensitivity of the RMOG with the other methods, such as the increment of incident light power, the finesse of the WRR [22] and the totoal length of ring resonator [38]. Finally, the integration time will be reduced to be compatible with several gyro applications.
When the experimental setup swings from CCW to CW direction on a rotary table, the gyro signal of the RMOG is obtained successfully as shown in Fig. 7 . In the experiment, the result is sampled when the rotary table swings from CW to CCW directions with a rotation rate of 1.75 × 10−3 rad/s. The experimental results are basically consistent with the theoretical curve ignoring the time delay. It is the best result ever reported in the rotation test of the RMOG, which is benefit from the thorough reduction of the backscattering induced noise with the DPMT.
Fig. 7 Experimental result of the RMOG based on the DPMT. Gyro output when the rotary table swings from CW to CCW directions with a rotation rate of 1.75 × 10−3 rad/s.
4. Conclusions
In conclusion, we first propose an RMOG based on the DPMT. And we have demonstrated theoretically and experimentally the effectiveness of the DPMT applied to the RMOG. The DPMT is easily performed with the integrated optical phase modulators to reduce the backscattering error to the order of 10−6 rad/s, which is below the shot noise limited sensitivity of the RMOG. The control accuracy of the modulation index and the environmental temperature stability is largely relaxed compared with the traditional SPMT. Based on the optimum parameters of the DPMT, a bias stability of 1.85 × 10−4 rad/s is successfully demonstrated in the silica waveguide resonator with the length of 7.9 cm. To the best of our knowledge, this result is the best ever demonstrated in a waveguide type passive ring resonator gyro. For the application of RMOG, the shot noise limited sensitivity has to be improved and the integration time has to be reduced. Other noise sources are expected to be further reduced to improve the long term stability, such as the Kerr effect and polarization fluctuation induced noise in the system.
Acknowledgements
The authors thank the Specialized Research Fund for the Doctoral Program of High Education, PR China (20060335064) for financial support. And thanks to the reviewers from OE for giving the meaningful comments.
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