# Design and Optimization of a Resonant Micro-Optic Gyroscope Based on a Transmissive Silica Waveguide Resonator

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{3}substrate with a Q factor of 2.4 × 10

^{6}[14].

^{5}and an effective area of 10 mm

^{2}; the resolution of the gyroscope was 150 °/h [15]. Feng et al. fabricated a silica waveguide ring resonator with a Q factor of 1.4 × 10

^{7}, for which a long-term bias stability of 0.013 °/s was reported [16].

^{6}and the long-term bias stability was 0.22°/s [19].

## 2. Principle and Simulation

_{in}, E

_{through}, and E

_{drop}are the input port, through port, and drop port of the light fields, respectively. The light is divided into two parts when it arrives at coupling region 1: part of it travels to the through port, while the other couples into the circular waveguide. In the same way, E

_{1}will be spilt into two parts when arriving at coupling region 2: one part couples with a straight waveguide and then travels to the drop port, while the other continues propagating around the circular waveguide. When it meets the resonance conditions, the light of the ring resonator will reach a dynamic balance.

_{1}, k

_{2}, t

_{1}, and t

_{2}are defined as the coupling coefficients and transmission coefficients of coupling region 1 and coupling region 2, respectively. In this paper, we assume that the coupling is lossless, in which case the parameters satisfy the following equation:

_{1}and g

_{2}are the gaps between the straight waveguides and the circular waveguide, respectively. The transmission loss of the light during one cycle in the circular waveguide can be described using the round-trip loss factor a:

_{t}represents the resonant spectrum of the straight through port and the red curve T

_{d}represents the resonant spectrum of the drop port. The full width at half maximum (FWMH), Δf, and the quality factor, Q, can be expressed as follows:

_{tmax}and T

_{tmin}are the maximum and minimum of the transfer function of through port, respectively.

_{1}and t

_{2}.

_{2}, can be regarded as a whole factor to simplify the analysis.

_{in}, I

_{out}represent the input and output light intensity of the photodetector, respectively. FSR represents the free spectrum width of the resonance curve, as shown in Figure 1b.

_{1}= 0.9798 and t

_{2}= 0.9759 when the slope of the demodulation curve is at its maximum at the resonant frequency point, and the corresponding gyroscope sensitivity is at its maximum at this time.

_{1}= 6.9 μm and g

_{2}= 6.7 μm can be obtained through the beam propagation method (BPM) simulation.

## 3. Design and Fabrication

_{1}= 1.456 and n

_{2}= 1.445, respectively. First, the SiO

_{2}was thermally grown as the bottom cladding layer. Second, a 6 μm thick SiO

_{2}doped with Ge, which can increase the refractive index of the waveguide core, was deposited by plasma-enhanced chemical vapor deposition (PECVD). A 6 μm wide core was processed by lithography and the dry etch technique; this size can support single-mode transmission. Then, the top cladding layer was covered with borophosphosilicate glass (BPSG), which can be used to make the refractive index of the top cladding equal to that of the bottom cladding. The thickness of the top and bottom layer was 15 μm, which can reduce the leakage loss of the cladding layer. The wafer was annealed after each process to realize stress compensation and reduce polarization-dependent loss caused by birefringence. Finally, a layer of glass was covered on the top cladding for protecting the connection and packaging between the waveguides and the optical fiber. In order to carry out the comparison test, three groups of resonators with different gaps were fabricated (g

_{1}= 6.9 μm, g

_{2}= 6.7 μm; g

_{1}= g

_{2}= 6.9 μm; and g

_{1}= g

_{2}= 6.7 μm).

## 4. Experiment

## 5. Results

_{1}= 6.9 μm and g

_{2}= 6.7 μm) after Lorentz fitting is shown in Figure 4. The black curve refers to the resonant spectrum and the red curve indicates the output voltage of the triangle wave sweep signal. The corresponding scan voltage difference was 0.615 V. The frequency modulation coefficient of the laser was 15 MHz/V; therefore, we determined that the FWHM of the resonator was 9.22 MHz. Furthermore, we calculated that the Q was 2.1 × 10

^{7}and the finesse was 119.

_{1}= 6.9 μm and g

_{2}= 6.9 μm) and 203.1 °/h (g

_{1}= 6.7 μm and g

_{2}= 6.7 μm). The test results show that the gyro index of the resonator (g

_{1}= 6.9 μm and g

_{2}= 6.7 μm) is best, and the test results were consistent with the simulation results.

## 6. Conclusions

^{7}, which is the highest quality among the reported transmissive resonators. A bias stability of 183.7 °/h over a one-hour test was successfully demonstrated, which is the best index of the optical gyroscope, based on the silicon dioxide transmissive resonators. The results show that our design method is feasible and provides ideas for the design of high quality transmissive optical waveguide resonators. In addition, the quality factor, finesse, and zero-bias stability of the gyroscope were close to those of the reflective resonator with nearly the same size. This provides a sound foundation for the improvement of the micro-optic gyroscope.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ciminelli, C.; Dell’Olio, F.; Campanella, C.; Armenise, M.N. Photonic technologies for angular velocity sensing. Adv. Opt. Photon.
**2012**, 2, 370–404. [Google Scholar] [CrossRef] - Dell’Olio, F.; Indiveri, F.; Innone, F.; Russo, P.D.; Ciminelli, C.; Armenise, M. System test of an optoelectronic gyroscope based on a high Q-factor InP ring resonator. Opt. Eng.
**2014**, 53, 127104. [Google Scholar] [CrossRef] - Guillén-Torres, M.; Cretu, E.; Jaeger, N.A.F.; Chrostowski, L. Ring resonator optical gyroscopes-parameter optimization and robustness analysis. J. Light. Technol.
**2012**, 30, 1802–1817. [Google Scholar] [CrossRef] - Kalantarov, D.; Search, C.P. Effect of input–output coupling on the sensitivity of coupled resonator optical waveguide gyroscopes. J. Opt. Soc. Am. B
**2013**, 30, 377–381. [Google Scholar] [CrossRef] - Ma, H.; Zhang, J.; Wang, L.; Lu, Y.; Ying, D.; Jin, Z. Resonant micro-optic gyro using a short and high-finesse fiber ring resonator. Opt. Lett.
**2015**, 40, 5862–5865. [Google Scholar] [CrossRef] [PubMed] - Jin, Z.; Zhang, G.; Mao, H.; Ma, H. Resonator micro optic gyro with double phase modulation techniqueusing an FPGA-based digital processor. Opt. Commun.
**2012**, 285, 645–649. [Google Scholar] [CrossRef] - An, P.; Zheng, Y.; Yan, S.; Xue, C.; Wang, W.; Liu, J. High-Q microsphere resonators for angular velocity sensing in gyroscopes. Appl. Phys. Lett.
**2015**, 106, 327. [Google Scholar] [CrossRef] - Wang, J.; Feng, L.; Wang, Q. Reduction of angle random walk by in-phase triangular phase modulation technique for resonator integrated optic gyro. Opt. Express
**2016**, 24, 5463–5468. [Google Scholar] [CrossRef] [PubMed] - Mao, H.; Ma, H.; Jin, Z. Polarization maintaining silica waveguide resonator optic gyro using double phase modulation technique. Opt. Express
**2011**, 19, 4632–4643. [Google Scholar] [CrossRef] [PubMed] - Dell’Olio, F.; Conteduca, D.; Ciminelli, C.; Armenise, M.N. New ultrasensitive resonant photonic platform for label-free biosensing. Opt. Express
**2015**, 23, 28593–28604. [Google Scholar] [CrossRef] [PubMed] - Chang, X.; Ma, H.; Jin, Z. Resonance asymmetry phenomenon in waveguide-type optical ring resonator gyro. Opt. Commun.
**2012**, 285, 1134–1139. [Google Scholar] [CrossRef] - Spencer, D.T.; Bauters, J.F.; Heck, M.J.R.; Bowers, J.E. Integrated waveguide coupled Si
_{3}N_{4}resonators in the ultrahigh-Q regime. Optica**2014**, 1, 153–157. [Google Scholar] [CrossRef] - Qian, K.; Tang, J.; Guo, H.; Liu, W.; Liu, J.; Xue, C.; Zheng, Y.; Zhang, C. Under-coupling whispering gallery mode resonator applied to resonant micro-optic gyroscope. Sensors
**2017**, 17, 100. [Google Scholar] [CrossRef] [PubMed] - Vannahme, C.; Suche, H.; Reza, S. Integrated optical Ti:LiNbO
_{3}ring resonator for rotation rate sensing. In Proceedings of the 13th European Conference on Integrated Optics (ECIO), Copenhagen, Denmark, 25–27 April 2007. [Google Scholar] - Ciminell, C.; D’Agostino, D.; Carnicella, G.; Dell’Olio, F.; Conteduca, D.; Ambrosius, H.P.M.M.; Smit, M.K.; Armenise, M.N. A high-Q InP resonant angular velocity sensor for a monolithically integrated optical gyroscope. IEEE Photon. J.
**2015**, 8, 6800418. [Google Scholar] [CrossRef] - Wang, J.; Feng, L.; Wang, Q.; Jiao, H.; Wang, X. Suppression of backreflection error in resonator integrated optic gyro by the phase difference traversal method. Opt. Lett.
**2016**, 41, 1586–1589. [Google Scholar] [CrossRef] [PubMed] - Haavisto, J.; Pajer, G.A. Resonance effects in low-loss ring waveguides. Opt. Lett.
**1980**, 5, 510–512. [Google Scholar] [CrossRef] [PubMed] - Ma, H.; Chen, Z.; Yang, Z.; Yu, X.; Jin, Z. Polarization-induced noise in resonator fiber optic gyro. Appl. Opt.
**2012**, 51, 6708–6717. [Google Scholar] [CrossRef] [PubMed] - Feng, L.; Wang, J.; Zhi, Y.; Tang, Y.; Wang, Q.; Li, H.; Wang, W. Transmissive resonator optic gyro based on silica waveguide ring resonator. Opt. Express
**2014**, 22, 27565–27575. [Google Scholar] [CrossRef] [PubMed] - Poon, J.K.S.; Scheuer, J.; Mookherjea, S.; Paloczi, G.T.; Huang, Y.; Yariv, A. Matrix analysis of microring coupled-resonator optical waveguides. Opt. Express
**2004**, 12, 90–103. [Google Scholar] [CrossRef] [PubMed] - Ying, D.; Wang, Z.; Mao, J.; Jin, Z. An open-loop RFOG based on harmonic division technique to suppress LD’s intensity modulation noise. Opt. Commun.
**2016**, 378, 10–15. [Google Scholar] [CrossRef] - Matejček, M.; Šostronek, M. New experience with Allan variance: Noise analysis of accelerometers. In Proceedings of the Communication and Information Technologies (KIT), Vysoke Tatry, Slovakia, 4–6 October 2017; pp. 1–4. [Google Scholar]

**Figure 1.**(

**a**) Structural diagram of the transmissive ring resonator; (

**b**) resonant spectrum of the resonator.

**Figure 2.**(

**a**) Schematic of key fabrication and process steps of the resonator: (i) SiO

_{2}of the bottom cladding layer by being thermally grown; (ii) core layer by PECVD; (iii) ultraviolet lithography; (iv) dry etch; and (v) SiO

_{2}of the bottom cladding layer by PECVD. (

**b**) SEM image of a coupling region cross-section.

**Figure 3.**(

**a**) Schematic diagram of the test system for the resonator; (

**b**) schematic diagram of the gyroscope test system.

**Figure 6.**(

**a**) One-hour static test result; (

**b**) Allan standard deviation of the rotation data (g

_{1}= 6.9 μm and g

_{2}= 6.7 μm).

**Table 1.**Quality factors and scale factors of resonators with different parameters tested by the experiment.

g_{1} (μm) | g_{2} (μm) | Quality Factor | Scale Factor (mV/°/s) |
---|---|---|---|

6.9 | 6.7 | 2.1 × 10^{7} | 1.34 |

6.9 | 6.9 | 1.5 × 10^{7} | 0.79 |

6.7 | 6.7 | 1.2 × 10^{7} | 0.69 |

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**MDPI and ACS Style**

Zhang, W.; Liu, W.; Guo, H.; Tang, J.; Liu, J. Design and Optimization of a Resonant Micro-Optic Gyroscope Based on a Transmissive Silica Waveguide Resonator. *Electronics* **2022**, *11*, 3355.
https://doi.org/10.3390/electronics11203355

**AMA Style**

Zhang W, Liu W, Guo H, Tang J, Liu J. Design and Optimization of a Resonant Micro-Optic Gyroscope Based on a Transmissive Silica Waveguide Resonator. *Electronics*. 2022; 11(20):3355.
https://doi.org/10.3390/electronics11203355

**Chicago/Turabian Style**

Zhang, Wei, Wenyao Liu, Huiting Guo, Jun Tang, and Jun Liu. 2022. "Design and Optimization of a Resonant Micro-Optic Gyroscope Based on a Transmissive Silica Waveguide Resonator" *Electronics* 11, no. 20: 3355.
https://doi.org/10.3390/electronics11203355