It has been known for four decades that high finesse millimeter-scale optical cavities can produce high-performance, compact optical gyroscopes. Yet, a practical implementation of such a device has been hindered by multiple technical challenges, including Rayleigh scattering and optical nonlinearity of the cavity material. In this Letter we report on the implementation of an integrated passive gyroscope using a monolithic cavity characterized by 7 mm diameter, finesse of , and Rayleigh backscattering less than 10 ppm. The device is characterized with quantum noise limited angle random walk of , and bias drift of , corresponding to detection of rotation-originated optical path change of .
© 2017 Optical Society of America
Interferometric optical measurement techniques enable measurement of extremely small distances. For instance, the Advanced Laser Interferometric Gravitational Observatory (aLIGO) allows measurements of optical path change of the order of , which is 3 orders of magnitude smaller than the classical electron radius (). Naturally, this suggests the use of optical interferometers for detection of mechanical rotation that changes the distance traveled by light propagating clockwise (CW) and counterclockwise (CCW) with respect to the rotation axis, due to the Sagnac effect [1–4]. In a passive ring cavity, the nonreciprocity of the counterpropagating waves results in a phase difference between them and as a consequence, a spatial shift of the interference fringe pattern created by the fields inside the cavity. The phase difference can be directly measured, and the signal in such a measurement is proportional to the finesse of the cavity. In this work we report an integrated high finesse optical microcavity capable of measuring rotation-induced optical path change at the level of at 1 s integration time.
Laser [5–7], fiber optic [3,8,9], and micro-electro-mechanical system (MEMS) gyroscopes [10–13] constitute three types of mature rotation sensors used outside the laboratory. Each type has its own challenges, so the ongoing research is devoted to improvement of the existing technology and also to the development of novel gyroscopes free from the inherent disadvantages of existing approaches. One of the outstanding goals is to reduce the size of the optical rotation sensors without sensitivity degradation . A variety of microphotonic gyroscopes, including ring microlaser, photonic crystal microfiber , and waveguide microring resonator [16–21], have been recently demonstrated. While significant improvement in the performance of these devices has been achieved, the ultimate goal of creating a high-sensitivity and low-drift optical micro-gyroscope has not been previously realized.
We focus on research and development of miniature passive optical gyroscopes based on the Sagnac effect. The initial development of this type of sensor was driven by the development of high-quality optical fibers. The length of the optical fiber Sagnac cavity can be increased significantly with limited actual increase in the size of the interferometric device . It was realized that the gyroscope can have a practically useful modular structure . The device reached maturity in the middle of the 1980s, with the development of integrated fiber-optic components , as well as polarization-maintaining fibers . Further reduction of the size of fiber-based gyroscopes is difficult because of the limitation in the reduction of volume of the fiber coil, as well as bending loss of sub-centimeter coils. There are several hurdles preventing fiber-optic gyroscopes from reaching the fundamental limit of their performance. To achieve a sensitivity that approaches the quantum limit, all sources of nonreciprocity, other than that induced by the Sagnac effect, must be eliminated. Zero-point errors and fluctuations mask the Sagnac phase shift when counterpropagating waves accumulate a nonreciprocal phase shift due to optical components or environmentally induced disturbances.
Resonant passive optical gyroscopes were introduced as an alternative to fiber optical gyroscopes [25–28]. Resonant devices have a smaller size compared with fiber optic gyros, and they do not have a lock-in range for small rotation rates, compared with laser gyroscopes. Unlike interferometric fiber optical gyroscopes, passive resonant gyroscopes have not been developed beyond the proof-of-principle experiments, because of the technical complexity of their implementation. The activity in this area dropped significantly until recently, when the interest in small gyros surged to fill the need for accurate and compact devices suitable for application on small moving platforms.
In the work reported here, we utilized ultra-high- crystalline whispering gallery mode resonators (WGMRs) [29,30] to create a gyroscope. The WGMR gyroscope has numerous advantages over fiber-optic devices  since its sources of nonreciprocity are limited in number. For example, in a fiber gyroscope significant measurement errors emerge if the counterpropagating waves do not travel along the fiber with the same state of polarization and the same modal distribution . Spatial, as well as optical, modal filters and polarization controllers must be employed to reduce these effects. The WGMR gyroscope is not impacted by this limitation because the monolithic resonator itself does not change mode polarization. Moreover, because crystalline materials generally have significant birefringence, there is no cross talk among modes having TE and TM polarizations. Our gyroscope does not need polarization control elements, while fiber-optic devices do.
Another source of error in fiber-optic devices is backscattering, which induces a cross talk between the clockwise and the counterclockwise waves. In fiber-optic gyroscopes, scattering usually includes Rayleigh backscattering as well as backreflections at the interfaces. As we show below, the problem is significantly less severe in the WGMR-based gyroscope. The magneto-optical Faraday effect due, for example, to electromagnetic interference is a nonreciprocal effect that can potentially add to the Sagnac phase . This effect is not important for crystalline resonators.
We found the performance of the WGMR gyroscope is limited because of optical nonlinearity of the resonator host material and the residual resonant Rayleigh scattering. The self- and cross-phase modulation effects and nonlinearity of the material limit the accessible level of angle random walk (ARW) of the gyroscope, while Rayleigh scattering, together with misalignment due to mechanical stress release, lead to bias drift of the gyroscope signal. With optimization of physical parameters, we achieved quantum noise limited ARW of and bias drift of in a gyroscope based on a WGMR having 7 mm in diameter. This is at least 1 order of magnitude better when compared with previous results [14–21]. This performance was achieved with 80 μW of total optical power interrogating the resonator mode.
The schematic diagram of the gyroscope setup is shown in Fig. 1. The setup includes a distributed feedback (DFB) semiconductor laser, stabilized with using a piezo-tunable WGMR. The laser radiation is frequency modulated at 50 kHz and directed to a 50/50 beam splitter, reflected from 50/50 service beam splitters, and then sent to an input evanescent prism coupler attached to a high- WGMR. The resonator has 200 and 12 kHz loaded and intrinsic bandwidth, respectively. The linear mode frequency splitting resulting from Rayleigh scattering is less than 1 kHz. The DFB laser emits 20 mW of power, 99.5% of which is intentionally attenuated so that the total optimal power at the resonator input is 140 μW. The power of the optical signals reflected from the resonator input coupling prism is about 40 μW in each direction (power at photodiodes PD1 and PD2). There is another coupling prism attached to the resonator (not shown in the picture) and used for tracking the field inside the resonator. The total transmitted power through that service prism is 60 μW. The total power attenuated in the resonator is less than 1%.
The laser frequency is locked to a mode of the resonator in the CCW direction using the Pound–Drever–Hall (PDH) technique. Photodiode PD1 is used for this purpose. The signal from the photodiode is sent to the lock-in amplifier and a proportional–integral–derivative (PID) controller, as shown in Fig. 1. The rotation signal is inferred from phase measurement of the of the frequency position of the CW mode using detector PD2. The measurement is performed using two lock-in amplifiers.
The gyroscope is integrated on an optical micro-bench, as shown in Fig. 2. The volume of the physics package is . All parts, including the semiconductor DFB laser chip, a modulatable service WGMR, the gyroscope resonator, and the photodiodes are assembled on the micro-bench.
The gyroscope was calibrated using a high-precision rotation stage. To measure the sensitivity of the device, it was left at rest and the rotation signal was measured (Fig. 3). The sensitivity increases with the increase of the averaging time at a short time scale, saturates at about 1 to 10 s, and then decreases due to drift. The short term sensitivity is limited by the noise of the pump light, while the drift results from the drift of the Sagnac loop parameters and imbalance of the power of the pump light. Thermal gradients and interaction among various resonator modes are responsible for the interferometer instability. We found that warping of the interferometer plane and change of the position of the beam spots on the photodiodes is the major reason for the observed long-term drift.
There are other mechanisms limiting the sensitivity; for instance, linear interaction among the resonator modes. The WGMR supports many nearly nondegenerate modes. The pump light is selectively coupled to one of these modes. The residual coupling with other modes changes the measured position of the mode center and, hence, results in a change of the measured signal. This effect results in bias drift if the WGM spectrum changes due to stress relief, thermal gradients, and aging of the gyroscope package. Solutions of this problem include spatial matching of the input beam profile to a selected WGM to reduce the coupling to other modes, as well as making the alignment and coupling to the CW and CCW modes as symmetric as possible to minimize the differential coupling drift.
The variation in differential power of the pump light occurs due to several major causes. Fluctuation of the laser power before entering the gyroscope interferometer is one of them. Change of the ratio of CW to CCW circulating power is the second one. This could result from a change in the beam split ratio, or if there is a residual interference (fringing) effect in each arm of the interferometer. Any change in the CW and CCW coupling coefficients that depends on the alignment of the pump light at both the input and output of the interferometer also impacts the performance.
The major challenge of making a sensitive micro-gyroscope is the small scale factor resulting from the small area of the resonator. The interferometer must have exceptionally good stability because of the small signal. We estimated the theoretical values of the bias drift of the gyroscope and found the achieved interferometer path differential drift was at the sub-nanometer level. The optical power difference was stabilized to better than a nanowatt.
In the gyro, the frequency splitting between WGMs corresponding to CW and CCW electromagnetic waves propagating in a resonator rotating with angular velocity is measured. Let us assume that the resonator rotates CW. In this case, the frequencies of the CW () and CCW () modes shift .
To understand the importance of the resonator size for the measurement sensitivity note that the frequency shift in Eq. (1) can be obtained by changing the resonator radius by a small value:4). Such a high sensitivity is achievable because the measurement is performed for the nearly degenerate CW and CCW modes, and is not practically impacted by the environmental fluctuations.
Let us assume a linear coupling of the CW and CCW waves in the resonator. To describe this coupling, we introduce a set of nonlinear equations:32] defined as , where is an optical cubic nonlinearity coefficient and is the effective mode volume; and and stand for the external coherent optical pumping (, where is an average optical pump power for each direction.
For the field amplitude of the output light, we have the following relationship:
Assuming that , , and , the nonlinear frequency shift becomes . This is a large number that can exceed the value of the coefficient, which is less than 1 kHz in our device. However, since the alignment can be relatively good, , the value of the nonlinear frequency shift can be reduced significantly.
The ARW of the gyroscope is limited by the shot noise and by quantum backaction noise:
In conclusion, a high-performance gyroscope based on a high- WGMR is demonstrated. The results confirm the feasibility of the technology and pave the way for achieving ARW and bias drift of the order of typical values of optical fiber gyroscopes. We have evaluated the limitation of the device sensitivity and conclude that further improvements of the performance of the device are readily feasible.
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