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Article

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

by 1,2,*, 1,2, 1,2, 1,2 and 1,2,*
1
Shanxi Province Key Laboratory of Quantum Sensing and Precision Measurement, North University of China, Taiyuan 030051, China
2
Key Laboratory of Electronic Testing Technology, School of Instrument and Electronics, North University of China, Taiyuan 030051, China
*
Authors to whom correspondence should be addressed.
Electronics 2022, 11(20), 3355; https://doi.org/10.3390/electronics11203355
Received: 22 September 2022 / Revised: 13 October 2022 / Accepted: 14 October 2022 / Published: 18 October 2022
(This article belongs to the Special Issue Recent Advances in Intelligent Transportation Systems)

Abstract

:
In order to optimize the performance of a resonant micro-optic gyroscope, a well-designed transmissive structure, which was obtained by optimizing the transmission coefficient of a coupler was used as the core component of a planar waveguide optical gyroscope. By analyzing the relationship between the resonator’s transmission coefficient and the gyroscope’s scale factor, the optical waveguide resonator sensing element with optimal parameters for the resonant micro-optic gyroscope was obtained. A scale factor of 1.34 mV/°/s was achieved using an open-loop system, and a bias stability of 183.7 °/h over a one-hour test was successfully demonstrated.

1. Introduction

A resonant micro-optic gyroscope is a novel optoelectronic hybrid integrated sensor with great potential to realize miniaturized all-solid-state devices and monolithic device integration [1,2]. Benefitting from the mature process of integrated circuits and planar light waveguide circuits, the realization of all integrated resonant micro-optic gyroscopes on a single chip has been proposed for years and is developing rapidly [3,4]. Because a waveguide ring resonator is the core sensing element of a gyroscope, its performance directly determines the precision of the gyroscope. The design and fabrication of a high quality resonator has become an important research topic for optical waveguide gyroscopes.
A number of parameters are directly related to a gyroscope’s performance, including the quality factor and effective area of a resonator. In view of the development trend of gyroscope miniaturization and integration, the question of how to optimize the parameters of a resonator of a finite size to improve the performance of a gyroscope is urgent. In recent years, research on resonators for resonant optical gyroscopes has developed rapidly [5,6,7,8,9,10,11,12,13]. Vannahme et al. fabricated a 6 cm diameter ring resonator on a LiNbO3 substrate with a Q factor of 2.4 × 106 [14].
Ciminelli et al. fabricated an InP-based spiral resonator with a quality factor (Q factor) of 6 × 105 and an effective area of 10 mm2; 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 × 107, for which a long-term bias stability of 0.013 °/s was reported [16].
Planar optical resonators include transmissive structures and reflective structures. Compared with a reflective resonator, a transmissive resonator has one more coupling region, which introduces an additional coupling loss [17]; however, it seems to be more advantageous, not only in the sense that the structure is more symmetrical and reciprocal, but also in regard to the better suppression of the polarization fluctuation [18]. Moreover, circulators are not required when constructing the gyroscope system. Feng et al. designed a transmissive resonator optic gyroscope based on a silica waveguide ring resonator, for which the quality factor was 6.13 × 106 and the long-term bias stability was 0.22°/s [19].
In this paper, a resonant micro-optic gyroscope based on a transmissive silica waveguide ring resonator is reported. First, we established a physical model of the transmissive resonator, deduced the transfer function, analyzed the relationship between the resonator’s transmission coefficient and the gyroscope’s scale factor, and obtained the transmission coefficient at the maximum sensitivity. Then, micro-electro-mechanical-system (MEMS) processing was used to process the resonator. Finally, a gyroscope test system was built to carry out experiments. The experimental results agreed with the simulation results.

2. Principle and Simulation

A transmissive resonator consists of two straight waveguides and a circular waveguide. The structure diagram is shown in Figure 1a: Ein, Ethrough, and Edrop 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, E1 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.
In Figure 1, k1, k2, t1, and t2 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:
k 1 2 + t 1 2 = 1 k 2 2 + t 2 2 = 1
In Figure 1, g1 and g2 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:
a = 10 α L / 10
where α is the c-band transmission loss and ϕ is the round-trip phase of the resonator.
The transmission matrix method is used to analyze the transmission of light between the straight waveguides and the circular waveguide [20]; the relationship between the parameters can be expressed as follows:
( E t h r o u g h E 1 ) = ( t 1 i k 1 i k 1 t 1 ) ( E i n E 2 )
E d r o p = k 2 a 1 / 2 e i π / 2 e i ϕ / 2 E 1
According to the above formulae, the normalized transfer function of the transmissive resonator can be represented as:
T t h r o u g h ( ϕ ) = | E t h r o u g h E i n | 2 = t 1 2 + a 2 t 2 2 2 a t 1 t 2 cos ϕ 1 + a 2 t 1 2 t 2 2 2 a t 1 t 2 cos ϕ
T d r o p ( ϕ ) = | E d r o p E i n | 2 = a k 1 2 k 2 2 1 + a 2 t 1 2 t 2 2 2 a t 1 t 2 cos ϕ
The resonance curve is plotted by the transfer function, as shown in Figure 1b. The blue curve Tt represents the resonant spectrum of the straight through port and the red curve Td 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:
Δ f = c n π L arccos 2 a t 1 t 2 1 + a 2 t 1 2 t 2 2
Q = f Δ f F W H M = n π L λ arccos 2 a t 1 t 2 1 + a 2 t 1 2 t 2 2
where c is the speed of light in a vacuum, n is the refractive index of the waveguide, L is the perimeter of the circular waveguide, f is the resonant frequency, and λ is the operation wavelength in a vacuum.
The resonant depth, h, of the through port spectrum can be derived as follows:
h = T t max T t min T t max = 1 ( t 1 a t 2 1 a t 1 t 2 ) 2
where Ttmax and Ttmin are the maximum and minimum of the transfer function of through port, respectively.
Equations (7) and (8) show that when the length of the circular resonator is fixed, the Q will be determined by the transmission coefficients t1 and t2.
In a reflective resonant optical gyroscope, the best performance is achieved when the resonator is undercoupling and the resonant depth is 0.75 [13]. Therefore, in a transmissive resonator, the coupling loss from coupling region 2 can be approximately considered as a part of the loss in the reflective resonant. In this way, the round-trip loss factor, a, and the transmission coefficient, t2, can be regarded as a whole factor to simplify the analysis.
The slope, l, of the demodulation curve in the linear region is defined as a scale factor, which can be considered as the sensitivity of the gyroscope. The specific formula [21] is as follows:
l = d I o u t d Δ f | Δ f = 0
I o u t = I i n { t 1 2 + a 2 t 2 2 2 a t 1 t 2 cos 2 π ( Δ f / 2 + f ) F S R 1 + a 2 t 1 2 t 2 2 2 a t 1 t 2 cos 2 π ( Δ f / 2 + f ) F S R t 1 2 + a 2 t 2 2 2 a t 1 t 2 cos 2 π ( Δ f / 2 f ) F S R 1 + a 2 t 1 2 t 2 2 2 a t 1 t 2 cos 2 π ( Δ f / 2 f ) F S R }
where Iin, Iout 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.
In this article, the diameter of the circular waveguide is 6 cm. The experimental test results show that the c-band transmission loss, α, of the silica optical waveguide is 0.017 dB/cm.
In combination with Equations (9) to (11), we can calculate that t1 = 0.9798 and t2 = 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.
In this case, we only need to design the gaps between the straight waveguides and the circular waveguide to match the above transmission coefficient. Then, g1 = 6.9 μm and g2 = 6.7 μm can be obtained through the beam propagation method (BPM) simulation.

3. Design and Fabrication

A transmissive ring resonator was fabricated on a silicon substrate (Figure 2). The refractive indices of the core and the overlay were n1 = 1.456 and n2 = 1.445, respectively. First, the SiO2 was thermally grown as the bottom cladding layer. Second, a 6 μm thick SiO2 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 (g1 = 6.9 μm, g2 = 6.7 μm; g1 = g2 = 6.9 μm; and g1 = g2 = 6.7 μm).

4. Experiment

The experimental test system was built up to measure the resonance spectrum of the transmissive silica waveguide ring resonator (Figure 3a). A tunable laser with a central wavelength of 1550 nm and a spectral linewidth of 300 KHz was used as an incident light source. A triangular voltage signal was applied to the laser for the linear scanning of the laser frequency. An isolator was placed between the laser and the resonator in order to avoid the laser from being influenced by the echo light. The light was then coupled into the resonator. A photodetector was used to convert the light output from the resonator into an electrical signal. The resonance spectrum could be observed on the oscilloscope.
Based on the above system, an experimental testing system of a resonant micro-optic gyroscope based on a transmissive silica waveguide ring resonator in an output loop state was established (Figure 3b). A Y-branch multifunctional phase modulator made of proton exchange lithium niobate was used to modulate the optical signals. The modulated signals entered into the resonator from the two input ports; the two light waves were transmitted around the circular waveguide in opposite directions and then outputted to the photodetectors, PD1 and PD2, from the two drop ports of the resonator. Because of the structural advantages of the transmissive resonator compared with the reflective resonator, there was no need to use the circulators. The photodetectors converted the light intensity signals into current signals, which were then converted into voltage signals by transimpedance amplifiers. The demodulated signal from a lock-in amplifier (LIA1) was used to supply feedback to the frequency locking module to lock the laser’s central frequency to the resonance point of the resonator through a PI controller, and the other demodulated signal from LIA2 was used as the gyroscope output signal. The data acquisition and signal processing of the frequency-locked loop and the output loop were realized with Field Programmable Gate Array (FPGA).

5. Results

The resonant curve of the drop port of the transmissive resonator (g1 = 6.9 μm and g2 = 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 × 107 and the finesse was 119.
The transmissive resonators with different parameters were connected to the gyroscope system for rotation tests. Step signals from the gyroscope system could be obtained by adjusting the rotating speed of the rotary table with ±20 °/s, ±40 °/s, ±60 °/s, and ±80 °/s, respectively, as shown in Figure 5a. Then, the scale factor of the resonant optical gyroscope system based on the transmissive resonator was calculated by the least square method, as shown in Figure 5b.
According to the above test steps, the quality factor, finesse, and scale factors of the gyroscope system of the three transmissive optical waveguide resonators with different parameters were tested, respectively. The results are shown in Table 1.
The resonant micro-optic gyroscope based on a transmissive silica waveguide ring resonator with optimal parameters was tested at room temperature on a static table. A minimum of Allan deviation with 122 °/h over a one-hour test was successfully demonstrated (see Figure 6). Then, a bias stability of 183.7 °/h was calculated by dividing the minimum Allan deviation by 0.664 [22]. The bias stability of the other two groups of resonators were 191.7 °/h (g1 = 6.9 μm and g2 = 6.9 μm) and 203.1 °/h (g1 = 6.7 μm and g2 = 6.7 μm). The test results show that the gyro index of the resonator (g1 = 6.9 μm and g2 = 6.7 μm) is best, and the test results were consistent with the simulation results.

6. Conclusions

We designed and fabricated a transmissive silica waveguide ring resonator with various gaps. The modeling of a transmissive resonator used in a micro-optic gyroscope was carried out, and the relationship between the resonator’s transmission coefficient and the gyroscope’s sensitivity was identified. A scale factor of 1.34 mV/°/s was achieved using an open-loop system, which was the highest value attained among all the resonators that we fabricated; this result was consistent with those of the simulations. The quality of the resonator was up to 2.1 × 107, 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

Conceptualization, W.Z. and W.L.; methodology, H.G.; software, W.Z.; validation, J.T., W.L. and W.Z.; writing—original draft preparation, W.Z.; writing—review and editing, W.Z.; supervision, J.T.; funding acquisition, J.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China (No. 51727808, 51821003 and 51922009).

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. 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]
  2. 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]
  3. 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]
  4. 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]
  5. 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]
  6. 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]
  7. 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]
  8. 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]
  9. 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]
  10. 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]
  11. 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]
  12. Spencer, D.T.; Bauters, J.F.; Heck, M.J.R.; Bowers, J.E. Integrated waveguide coupled Si3N4 resonators in the ultrahigh-Q regime. Optica 2014, 1, 153–157. [Google Scholar] [CrossRef]
  13. 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]
  14. Vannahme, C.; Suche, H.; Reza, S. Integrated optical Ti:LiNbO3 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]
  15. 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]
  16. 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]
  17. Haavisto, J.; Pajer, G.A. Resonance effects in low-loss ring waveguides. Opt. Lett. 1980, 5, 510–512. [Google Scholar] [CrossRef] [PubMed]
  18. 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]
  19. 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]
  20. 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]
  21. 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]
  22. 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 1. (a) Structural diagram of the transmissive ring resonator; (b) resonant spectrum of the resonator.
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Figure 2. (a) Schematic of key fabrication and process steps of the resonator: (i) SiO2 of the bottom cladding layer by being thermally grown; (ii) core layer by PECVD; (iii) ultraviolet lithography; (iv) dry etch; and (v) SiO2 of the bottom cladding layer by PECVD. (b) SEM image of a coupling region cross-section.
Figure 2. (a) Schematic of key fabrication and process steps of the resonator: (i) SiO2 of the bottom cladding layer by being thermally grown; (ii) core layer by PECVD; (iii) ultraviolet lithography; (iv) dry etch; and (v) SiO2 of the bottom cladding layer by PECVD. (b) SEM image of a coupling region cross-section.
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Figure 3. (a) Schematic diagram of the test system for the resonator; (b) schematic diagram of the gyroscope test system.
Figure 3. (a) Schematic diagram of the test system for the resonator; (b) schematic diagram of the gyroscope test system.
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Figure 4. Curve fitting of the resonant spectrum (g1 = 6.9 μm and g2 = 6.7 μm).
Figure 4. Curve fitting of the resonant spectrum (g1 = 6.9 μm and g2 = 6.7 μm).
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Figure 5. (a). Rotation step signals; (b) least square fitting (g1 = 6.9 μm and g2 = 6.7 μm).
Figure 5. (a). Rotation step signals; (b) least square fitting (g1 = 6.9 μm and g2 = 6.7 μm).
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Figure 6. (a) One-hour static test result; (b) Allan standard deviation of the rotation data (g1 = 6.9 μm and g2 = 6.7 μm).
Figure 6. (a) One-hour static test result; (b) Allan standard deviation of the rotation data (g1 = 6.9 μm and g2 = 6.7 μm).
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Table 1. Quality factors and scale factors of resonators with different parameters tested by the experiment.
Table 1. Quality factors and scale factors of resonators with different parameters tested by the experiment.
g1 (μm)g2 (μm)Quality FactorScale Factor (mV/°/s)
6.96.72.1 × 1071.34
6.96.91.5 × 1070.79
6.76.71.2 × 1070.69
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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

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