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A transmissive resonator optic gyro (TROG) based on silica waveguide ring resonator with improved long-term bias stability is reported in this paper. The modeling of a transmissive resonator used in optic gyro is carried out. The polarization dependence of resonator and the influences of phase modulator’s residual intensity modulation on the gyro output are analyzed. The resonator is simulated, designed, fabricated, tested and used to build up a TROG prototype. A bias stability of 0.22°/s over one hour test with an integration time of 10s is successfully demonstrated. No obvious drift has been found from the Allan variance analysis result of a 10000s test data, which means that the TROG prototype has an improved long-term drift characteristic.
© 2014 Optical Society of America
1. Introduction
Optic gyros based on the Sagnac Effect have high reliability and electromagnetic perturbation immunity because there is no movable component in the rotation sensing unit [1]. Consequently, they have been widely used in the fields of inertial navigation and attitude determination, and have gradually dominated the high-mid grade gyro market instead of the mechanical gyros [2,3]. Benefitted from the integrated circuit and planar lightwave circuit, the realization of all integrated optic gyro on a single chip had been proposed and are developing more and more quickly [4,5]. At present, the long-term bias stability of the resonator integrated optic gyro is of rad/s order, the one-hour bias stability is of °/s order, and the short-term bias stability (less than 3min) is below 0.1°/s. A bias stability of 0.67°/s [6] and 0.41°/s [7] over one hour test have been reported respectively in 2013. The best short-term bias stability result reported is 0.01°/s with test time of 60s and integration time of 10s [8], which is much better than the one-hour bias stability. Therefore, the low-frequency drift is the main barrier that hinders the improvement of the resonator integrated optic gyro’s performance. The factors that influence the gyro’s low-frequency drift are complex. Backreflection noise, polarization fluctuation noise, temperature induced variation of the resonator’s transfer function, phase modulator’s residual intensity modulation (RIM) and etc., will cause the gyro bias drift, which means the reciprocity of the sensing unit is worse than needed, and the nonreciprocal error is too large.
There are two basic kinds of ring resonator, transmissive resonator and reflective resonator. Compared to the reflective resonator, the transmissive resonator introduces an extra coupler to couple the light outward, resulting in increase of the total loss of a round-trip and decrease of the finesse of the resonator [9], and consequently the sensitivity of the gyro becomes lower. The unit loss of planar light waveguide is much larger than that of the fiber, so it is essential for the waveguide ring resonator (WRR) to decrease the total loss as much as possible, and this may be an important reason why the reflective resonator has been attracting more attention in the field of resonator integrated optic gyro for a long time. However, an optic gyro based on a reflective resonator is not so reciprocal. The phase difference between the through port and the cross port of an optical directional coupler is a little away from π/2 because of differential normal mode loss at the coupling area [10], which leads to a fact that the transfer function of a transmissive resonator is more symmetrical than that of a reflective resonator [11]. Moreover, a transmissive resonator may be more conducive to the inhibition of the polarization-induced error compared with a reflective resonator [12]. In this paper, it is also supported that a resonator optic gyro based on a transmissive resonator will perform better than one based on a reflective resonator according to the analyses of polarization dependence and the influence of phase modulator’s RIM on the gyro output.
To make the finesse of the transmissive WRR as high as possible, the waveguide material and the optimization of the resonator design and fabrication are important. Many kinds of materials can be used to fabricate the waveguide of integrated optic gyro, including silica, silicon nitride [13], PMMA [14], InP [15] and etc. The silica waveguide has the lowest unit loss among them. The silica-on-silicon technology has achieved very low propagation loss (around 0.85dB/m) of WRR early in 1994 [16]. A loss of 0.3dB/m had been realized for a silica-based waveguide in 2004 [17]. And the unit loss as low as 0.037dB/m has been successfully demonstrated in 2011, which is close to that of optical fiber when first considered a viable technology [18]. The propagation loss of the ring resonator fundamentally limits the performance of the resonator integrated optic gyro; therefore silica is a more competitive material than others up to now.
In this paper, a transmissive resonator optic gyro (TROG) based on buried silica-on-silicon waveguide is reported. Firstly the transmissive resonator used in optic gyro is modeled. The polarization dependence of resonator and the influences of phase modulator’s RIM on the gyro output are analyzed. The transmissive WRR based on buried silica waveguide is then designed, fabricated and tested. An effective polarization alignment method is applied in fiber splicing. In the end, a TROG prototype is built up and tested. A bias stability of 0.22°/s over one hour test with an integration time of 10s is successfully demonstrated, and there is no obvious drift found in a 10000s test via Allan variance analysis.
2. Principle
2.1 Sensitivity of TROG
The sketch map of silica-based TROG is shown in Fig. 1. A beam of narrow linewidth laser is filtered by a WRR after being polarized, split and phase modulated. The counterclockwise (CCW) and clockwise (CW) light propagated in the WRR couple out via the output coupler Cout and then are detected by photodetectors PDccw and PDcw, respectively. In order to suppress the WRR’s backscattering noise and the frequency noise of the laser source, and then to enhance the sensitivity and signal-to-noise ratio of the gyro system, the CCW and CW light are applied antiphase triangle modulation before entering the WRR, producing antiphase quasi square signals in PDccw and PDcw. The amplitude of the quasi square signal is proportional to the frequency difference between the laser and the central resonant frequency of the WRR in a certain range. By a frequency servo control the central frequency of the laser is locked to the resonant frequency of the CCW path of the WRR, and then the demodulation of the PDcw signal is taken to be the open-loop output of the gyro. According to the Sagnac effect, the resonant frequencies of CCW path and CW path are not equal, and the difference between them is proportional to the rotation speed.
The performance of the TROG is related to the accuracy of the frequency discrimination; hence the transfer function of the WRR is the fundamental factor when we design the TROG. The influence of the laser spectrum linewidth on the finesse of the WRR can be ignored if it is much less than the linewidth of the WRR. The normalized transfer function of the transmissive resonator can be represented as [19–21]:
where, k1 and k2, α1 and α2 are the coupling ratios and extra losses of Cin and Cout; αL/2 is the propagation loss of half round-trip of the WRR. Herein an approximation is used in discussion. I0 refers to the light intensity, f is the laser frequency, FSR is the free spectral range of the WRR, φ is the phase bias. The finesse of the WRR can be derived from Eq. (1), expressed as:We define the ratio of the maximum value over the minimum value of the transfer function as the extinction ratio (ER) of the WRR, and it can be calculated by
It can be seen that the minimum value of the normalized transfer function would be much closer to zero even if the finesse is not very high. Hence the expression of sensitivity limit δΩ of the TROG is a little different from the reflective resonator optic gyro’s, which has something to do with the resonance depth. Appendix A gives the complete derivation process.
where, t is integration time, e refers to the electron charge, Imax is the peak photocurrent of the photodetector without saturation, c is the light velocity in vacuum, λ is the operation wavelength in vacuum, A is the effective area of the WRR, and N is the number of the resonator loops. For a single loop WRR without cross waveguide, N = 1. The production of the three variables F, A and N can be named equivalent sensing area (ESA). The relation between the sensitivity limit and the ESA of a gyro resonator is shown in Fig. 2, from which it can be seen that if the diameter of the single loop WRR is 35mm, the finesse should be bigger than 59 in order to reach a sensitivity of 5°/h.
Fig. 2 Relation between the sensitivity limit and the ESA according to Eq. (4). The parameters are as follows: c = 3 × 108m/s, λ = 1.55 × 10−6m, Rf = 40KΩ, VPP = 2V, t = 10s, e = 1.6 × 10−19C.
2.2 Polarization dependence
In the following discussion, we assume that the WRR is operated at TE mode. The TE and TM mode of a WRR cannot be at resonance simultaneously due to the weak birefringence caused by stress and geometrical asymmetry of the waveguide. When the laser frequency is locked to the TE resonant frequency of a WRR, a reflective WRR plays the role of TE mode notching filter. The TE component is attenuated to the operating point while the TM component is passing through with little attenuation, and they are detected by a PD after split by a 3dB coupler. On the contrary, a transmissive WRR plays the role of TE polarizer. The TM component is greatly attenuated in the transmission port because the laser frequency cannot satisfy the TM resonant condition.
We define the ratio of TE transmittance over TM transmittance as the polarization extinction ratio (PER) of WRR. For simplification, the splitting ratios and extra losses of the two couplers of WRR are assumed to be polarization independent, and the unit propagation losses of TE and TM mode of the waveguide are the same, which means that the transfer functions of TE and TM mode are almost the same except their effective index and initial phase. According to Eq. (1a), if the TE mode is at resonance and φTE = 0, then the resonant frequency of TE mode can be expressed as:
where, FSRTE is the TE mode free spectral range of the ring waveguide, and m is an integer. Plug Eq. (5) into Eq. (1a), we can getThen the PER can be calculated as:
The PER varies as the raised-cosine function of φTE/TM according to Eq. (7), and the PER reaches its maximum when φTE/TM = π, written as:
The variation of the environment temperature causes fluctuations of the birefringence difference of the waveguide, φTE/TM and the PER of the WRR. Figure 3 shows the contour map of PER. It can be seen that in order to maintain the PER at a relatively high level, φTE/TM must be controlled in a certain range, and the higher the finesse is, the more tolerance the TE/TM mode phase difference has, and the larger the maximum PER can reach. This provides theoretical guidance for WRR sifting and temperature control.
2.3 RIM reduction by transmissive resonator
According to the analysis above, a transmissive resonator performs like a narrow bandwidth polarizer, while a reflective resonator does not. There are always polarization alignment errors or polarization crosstalk in the light paths. For a reflective resonator, the errors along the TM polarization direction, for example the RIM of the phase modulator, are directly detected by photodetector; for a transmissive resonator, these errors are attenuated by PER times. The RIM of the phase modulator used in our experiment is about −27dB, and the polarization fluctuation in the pigtail between the modulator and the resonator is assumed to be −30dB, then the nonreciprocal amplitude error of the quasi square signal due to the RIM and the independent random polarization fluctuation can reach −57dB. According to Eq. (13), the maximum slope of the experimental setup shown in Fig. 1 can be expressed as follows:
If the diameter and the finesse of the WRR are 35mm and 59, respectively, the maximum slope is about 2.2 × 10−5 (°/s)−1, hence a nonreciprocal error of −57dB can generate a bias fluctuation of 0.09°/s at least in the reflection port, which is much bigger than 5°/h. According to Fig. 3, the PER of the transmission port can reach 30dB if φTE/TM∈(3π/4, 5π/4), then the nonreciprocal error would drop down to −87dB, and the corresponding bias fluctuation would be 0.3°/h, which is below the sensitivity limit.
However, the RIM along the TE polarization direction does cause distortion of the quasi square waveform for both reflective and transmissive resonator. If the resonator and the Y modulator are coupled directly or integrated together, the nonreciprocal error induced by polarization fluctuation can be greatly reduced.
3. Design, fabrication and characterization
As mentioned above, the gyro sensitivity limit and the ESA of the WRR are in inverse proportion. Figure 4 shows the relation between the bend radius of a rounded square structure and the corresponding ESA. The structure is shown in inset of Fig. 4(a). In this simulation, we suppose that the bend loss is approximately inversely proportional to the bend radius squared in millimeter in the range from 5mm to 17.5mm, and the bend loss is 0.1dB when the bend radius is 5mm. The loss due to mode mismatch in the joint area of the straight waveguide and the curved waveguide is not taken into consideration. The simulation results show that the rounded rectangle/square structure has larger effective area in given wafer size, however, the bend loss would increase if the bend radius decreased; therefore, the rounded rectangle structure is not beneficial to increase the ESA. Although the area is smaller than the rounded rectangle/square structure, the circle structure can reach the lowest level of total loss, so it has the highest finesse and may have larger ESA. In general, if the unit propagation loss is low enough, circle is the optimal structure in given wafer size.
Fig. 4 Relation between the bend radius r and the corresponding ESA value (N = 1). a) simulation structure and the relation between r and the area; b) total loss of a round-trip of the WRR; c) finesse of the WRR if k = 0.01; d) ESA. The max ESA point is left-shift with the unit propagation loss increasing, but the ESA remains at a high level when the structure is more circle-like.
According to Eq. (1b), the smaller the coupling ratio k, the extra loss α and the unit propagation loss αL are, the closer to 1 the intermediate q is and the higher the finesse is. The extra loss equals zero if there is no coupling at all between the bus waveguide and ring waveguide; hence the weaker the coupling interaction between the bus waveguide and ring waveguide is, the smaller the extra loss is, which means that the extra loss and coupling ratio of a coupler are not independent. In addition, in order to reduce the propagation loss as much as possible, large cross section and low index contrast waveguide is designed. These measures are targeted for high finesse. However, higher finesse means lower transmittance of the WRR in given unit propagation loss. Figure 5 shows the relations among coupling ratio, finesse and transmittance of a transmissive WRR with a diameter of 35mm under different unit propagation losses. If the transmittance of the WRR is too small, the actual extinction ratio of the TROG system would drop down due to the weak signal light and strong background noise. Therefore a balance between the finesse and transmittance must be considered according to the laser power and the total loss of the whole optical circuit.
Fig. 5 Relation among coupling ratio, finesse and transmittance of a transmissive WRR with a diameter of 35mm. We suppose that the two couplers are the same, and the extra loss of the coupler is fixed at 0.013dB.
The WRR is fabricated on silicon substrate. The bottom cladding layer is thermally-grown SiO2. A 6μm-thick doped SiO2 is then deposited using PECVD (plasma enhanced chemical vapor deposition). The waveguide cores are defined with UV lithography and dry etch process. The top cladding layer is then completed with PECVD SiO2. The wafer is carefully annealed after every step of deposition. In the end, a glass is stick up on the top cladding layer as a shield and for the convenience of coupling to the fiber.
The diameter of the ring resonator fabricated based on buried silica waveguide is 35mm. The results of the transfer function tests are shown in Fig. 6. According to the fitted curve, the finesse of the WRR is 59.6, the quality factor is 6.13 × 106, and the resonance depth of the reflection port is 0.85. The sensitivity limit of the TROG prototype based on this WRR can reach about 5°/h according to Eq. (4).
Fig. 6 Transfer function tests of the WRR. a) fiber-coupled transmissive WRR (the red dash line outlines the ring resonator); b) output and fit of the reflection port; c) output and fit of the transmission port. The data is obtained by sweeping the central frequency of a tunable narrow linewidth laser via PZT tuning.
4. Experiments and results
The silica-based TROG prototype is built up based on Fig. 1. The central wavelength of the tunable laser is 1550nm, and its linewidth is less than 1kHz. A Y-type multifunctional modulator made of proton-exchange lithium niobate is used to polarize, split and modulate the laser. A polarization maintaining isolator is placed between the modulator and the laser source in order to avoid the laser source from being influenced by the echo light. Two single-mode isolators are placed ahead of the photodetectors, which are made of InGaAs and integrated with low-noise pre-amplifier and fixed transimpedance, to isolate the photodetectors’ backreflection light. The data acquisition, signal processing and automatic control of the frequency-lock loop and the output loop are realized with Labview and FPGA
The fiber splicing is monitored on-line and real-time in order to make the TM peak of the WRR as small as possible, and then the polarization alignment error between the incident light and the TE mode is controllable and can reach the limit of the splicer. The incident laser power is increased properly in order to increase the TM peak power, and then the judgment error can be decreased. Meanwhile the TE peak detected by photodetector is saturated, shown in Fig. 7. The fitting results show that the TE and TM peak values are 39.4V and 20mV, respectively, which means that, neither the polarization axis alignment error between the incident light and the TE mode of the WRR, nor the polarization cross talk of the resonator itself, is larger than 33dB; and the TE/TM mode phase difference is about 0.99π at room temperature, therefore the polarization fluctuation-induced noise can be greatly suppressed.
Fig. 7 Curve fitting of the PD signal. The TE peak is saturated, while the TM peak is almost submerged in noise. The TM peak value can be obtained more precisely after proper smoothing.
The scale factor of the TROG prototype is tested and calculated based on equivalent input method [22]. The TROG prototype is then tested in room temperature on a static table. A bias stability of 0.22°/s over one hour test with an integration time of 10s is successfully demonstrated. No obvious drift can be found from the Allan variance analysis result of a 10000s test data, which means that the TROG prototype has an improved long-term drift characteristic (Fig. 8).
Fig. 8 Test results of the TROG prototype. a) scale factor test based on equivalent input; b) 1h static test ; c) allan variance of the rotation rate data of a 10000s test, from which it can be seen that there is no obvious long-term drift.
5. Discussion
We designed and fabricated a silica-based transmissive WRR whose finesse and quality factor are close to that of the reflective WRR with nearly the same size [7], which does lay a sound experimental foundation for the improvement of the integrated optic gyro. From the test results shown above we can see that the gyro has no obvious long-term drift, which is attributed to the improved reciprocity of the transmissive resonator and the suppression of polarization fluctuation by fiber splicing optimization.
However, the short-term performance is in the same level with the long-term one, and the bias is much bigger than the bias stability numerically, which may be both caused by the backreflection from the inner or outer defects of the WRR. Although the backreflection noise has been reduced by the hybrid phase modulation and integer period sampling method, the difference of the initial phase between the backreflection light and the signal light will change with the temperature and stress, resulting in fluctuation of the gyro output within a certain range, and this short-term fluctuation is much bigger than the white noise level within the corresponding bandwidth. Consequently, some new evaluation indices must be added in in WRR sifting besides testing its finesse and transmittance, and this is what we will do next.
6. Conclusion
A transmissive resonator optic gyro based on silica waveguide ring resonator is reported in this paper. Analyses show that transmissive resonator performs like a narrow bandwidth polarizer, therefore the errors caused by polarization alignment errors or polarization crosstalk can be greatly suppressed. The finesse and the quality factor of the transmissive WRR we designed and fabricated are 59.6 and 6.13 × 106, respectively. The polarization suppression ratio of the TE mode over TM mode is reduced down to 33dB via fiber splicing optimization. A bias stability of 0.22°/s over one hour test with an integration time of 10s is successfully demonstrated in the TROG prototype. This work lays a sound experimental foundation for the improvement of the integrated optic gyro by using a transmissive waveguide ring resonator in the future.
Appendix
Sensitivity limit of TROG
The normalized transfer function of a transmissive resonator used in a TROG system, whose linewidth is much less than its free spectrum range (FSR), can be approximated by a Lorentz Function in which no trigonometric function items are included in the range between f0-FSR/2 and f0+FSR/2, given by:
The demodulation curve can be expressed as:
where, δf refers to the difference between the laser frequency and the resonant frequency of the ring resonator, Δf is the linewidth of the ring resonator, fb is the equivalent modulation frequency bias. The slope of demodulation curve at δf=0 (which means that the laser is locked to the ring resonator) can be calculated by:The slope reaches its maximum when:
The corresponding slopes and the value of the normalized transfer function can be calculated by:
We assume that the photon shot noises of PDcw and PDccw are independent; then the frequency fluctuation of the output loop caused by photon shot noise can be represented as:
where, SNR is the signal-to-noise ratio at f=fbias and only the photon shot noise is considered, expressed as:where, Ibias is the photocurrent at f=fbias. One thing to note is that the max sensitivity point does not coincide with the max SNR point in general. According to Eq. (14) and Eq. (17), Eq. (16) can be simplified as:Take the Sagnac effect into consideration, the sensitivity limit of the TROG can be written as:
Acknowledgments
The authors would like to acknowledge financial support from the National Natural Science Foundation of China (No. 61171004) and the Institute of Opto-Electronic Technology of Beihang University.
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