On-chip modulation for rotating sensing of gyroscope based on ring resonator coupled with Mach-Zehnder interferometer

An improving structure for resonance optical gyro inserting a Mach-Zehnder Interferomete (MZI) into coupler region between ring resonator and straight waveguide was proposed. The different reference phase shift parameters in the MZI arms are tunable by thermo-optic effect and can be optimized at every rotation angular rate point without additional phase bias. Four optimum paths are formed to make the gyroscope to work always at the highest sensitivity.

Silicon optical modulator is typical integrating modulator, which could have been realized in both monolithic and hybrid forms. To the present, silicon-based modulators which operate via carrier depletion have been demonstrated at data rates up to 50 Gb/s 28 . Applying an electric field to a material may change the refractive indices (real and imaginary). Unfortunately, the centrosymmetric crystal structure of silicon dose not exhibit the Pockels effect (a linear electro-optical effect) and Kerr effect and Franz-Keldish effect are weak at 1.55 μ m 29 . Thus making electro-optic silicon modulators is difficult to realize 30 . Alternative methods are required to achieve modulation in silicon. One option is plasma dispersion effect, in which the concentration of free charges in silicon contributes to the loss via absorption 31 . Generally, in electrorefraction modulators, a balance should be struck between electrorefraction and eletroabsorption 32 . The effect of free carrier losses due to current injection is judged by observing the change in Q-factor during tuning, where any losses will lower the Q-factor. But there is not any change in Q-factor for thermo-optic effect 33 , whereas electro-optic modulators have large insertion loss usually more than 4 dB 34 . And the other practical option is thermal modulation owing to the large thermo-optic coefficient (TO) of silicon [35][36][37][38] . Early in 2000 years K. Suzuki and K. Hotate 39 had proposed and experimented the countermeasure for Backscattering induced noise via Thermo-optic phase modulation in a compact micro-optic gyro. Moreover, the modulator speed was increased via the application of a thermal bias holding the modulator at a higher average temperature with respect to the substrate heat sink 40 . In 2010, Morichetti. F et al. showed that resonator optical waveguide (CROW) delay lines fabricated on a silicon on insulator (SOI) platform at 100 Gbit/s. Heaters for SOI technology reveal to be more than 120 times faster than their counterpart in silica technology 41 . Though thermal modulation is comparatively low speed for high frequencies required by telecommunications applications, for sensing applications only relative moderate modulation speeds are need.
In this paper, an improving resonance structure for gyroscope was proposed, which inserts a Mach-Zehnder Interferometer (MZI) as a coupler between ring resonator and straight waveguide and each arm is introduced a different reference phase shift Δ Φ 1 , Δ Φ 2 , called MZI-coupler resonant waveguide optical gyroscope (MZIC-RWOG) (Fig. 1). When the phase shift in the ring resonator is changed by Sagnac effect, the proposed schematic for integrated optical gyroscope can keep the minimum detectable rotation rate (resolution) value through modulating phase shift using thermo-optic effect in coupled region of MZIC-RWOG structure and do not need to add external phase bias in MZIC-RWOG, which is more suitable for modulating on a chip.

Results
In this work, we only consider the thermo-optic effects to control reference phase shift Δ Φ 1,2 in the arms of MZI. As current injected into the MZI electrode, heating changes the waveguide's refractive index, and introduces a phase shift 35,36 :  (a typical value for silicon at λ = 1550 nm 38 ), a temperature change of 8.3 K is needed to achieve a π -phase shift. The sign and magnitude of the TO coefficient are primarily determined by the material density and polarizability. The change of TO coefficient is due to the difference of the propagation constant and optical confinement, an analytical expression for description of channel waveguides TO coefficient is 42 where n 0 is the channel waveguide refractive index and n 1 is the cladding refractive index and Γ is the confinement factor which defines the fraction of optical mode inside the core region. If ( / ) = dn dT 0, that means to achieve temperature-insensitivity in the effective index of a waveguide device by balancing the overall TO coefficient using with a negative TO coefficient of cladding to compensating positive TO coefficient of the core. However, our aims are temperature-sensitivity and controlling the phase shift through thermo-optic effect. We define the slope S (K −1 ) between phase shift and temperature change is φ = ∆ /∆ S T . Figure 2 show the slope S (K −1 ) versus the material TO coefficients of core with different confinement factors Γ = 50%, 70%, 90% when the material TO coefficients of cladding is set to − 0.0003 K −1 , 0 K −1 and 0.0003 K −1 .
Here confinement factor Γ depend only on the n 0,1 and waveguide cross section geometry at room temperature. Thermal expansion coefficient is ignored since much smaller than TO coefficient in silicon. Moreover, a smaller confinement factor Γ can get the larger S (K −1 ) if material TO coefficients of cladding is negative (see Fig. 2(a)) and a larger confinement factor Γ can get the larger S (K −1 ) if material TO coefficients of cladding is positive (see Fig. 2(c)). Next, the transmission characteristic of MZIC-RWOG in Fig. 1 can be obtained by using transfer matrix approach 43 , where k 1 , k 2 are the coupling coefficient and t 1 , t 2 are the through coupling coefficient of directional couplers respectively. When the coupling loss is neglected, + = k t 1 are satified. The one round-trip amplitude propagation attenuation is = α π − a e R 2 , in which α is the propagation loss with the units of m −1 , and φ is the phase shift a light experiences one round-trip, which is composed by two terms: where w, n, c and Ω is the angular frequency of the light, effective index of the ring, the speed of light in vacuum and rotation angular rate, respectively. The second term φ s in equation (4) is the phase shift induced by the Sagnac effect when the sensor rotate about an axis perpendicular to the plane of optical resonator at an angular velocity Ω .
By comparing the transfer function of MZIC-RWOG with all-pass resonant waveguide optical gyroscope (RWOG), which is composed by one straight waveguide and a single ring resonator, the effective through coupling coefficient t eff can be found According to 22,[44][45][46] , the resolution δΩ min (which is defined as the minimum detectable angular rate) is expressed if the resonator is circular: The amplitude noise term δ = ( and i D are the standard deviation of the photocurrent and the maximum photodiode current (specific parameters are listed in Table 1).

Discussion
In order to indicate the enhancement effect on sensitivity of MZIC-RWOG and compare the sensitivity of MZIC-RWOG with RWOG, we must firstly optimize the parameters of RWOG. The necessary global optimizing parameters include the area of ring resonator A, the coupled coefficient k between ring resonator and straight waveguide, the round-trip propagation loss α if the coupled loss is neglected and the phase shift φ which ensures the gyroscope obtain maximum sensitivity when rotation rate is null (so-called bias phase shift φ bias ). According to transfer matrix approach, the analytical expression of the all-pass structure RWOG for the minimum detectable angular rate δΩ min can be calculated which is a function of A, k, α and φ . Although it is possible to optimize δΩ min using the exact formula, numerical calculation for obtaining minimum value of δΩ min is more convenient. More specifically, we can scan parameters (A, k, α , φ ) and calculate every value of resolution δΩ in a setting range, then find out both the minimum value δ Ω min and optimizing parameters pair (A opt , k opt , α opt , φ bias ). For the optimizing parameters pair (A opt , k opt , α opt , φ bias ), except φ bias the other three parameters are all determinated by processing technology and cannot be changed after fabricating.
For the level of current processing technology the propagation loss of silicon based waveguide is about 1-5 dB/cm and every value of propagation loss is according to an optimum perimeter for a RWOG. All the parameter pair (L, k, φ) are scanned to find out the minimum value of δΩ min after setting a fixing value for propagation loss. From Fig. 3 we can see that there is always an optimum L corresponding to δΩ min for every loss value and the optimum parameters are shown as Table 2.
Note that the optimizing parameters pair (A opt , k opt , α opt , φ bias ) only ensure maximum sensitivity at zero rotation-rate point since the parameters have been fixed after fabrication, thus the parameters (A opt , k opt , α opt , φ bias ) pair is not optimal when the gyroscope is rotating. That is to say, the gyroscope is only designed by static optimization and not dynamic optimization because the coupled coefficient k cannot be controlled after fabrication. If it assumes that coupled coefficient k is tunable for every Sagnac phase shift φ s caused by rotation angular rate Ω, then we can find out a path of dynamic optimization of k to obtain minimum Ω. In Fig. 4(a), it shows that the two situations: k is fixed at static optimization point 0.8046 (black line) and k is tunable (red line). Although the resolution δΩ min is the same at φ bias for the two situations, there is small enhancement for δΩ with the Sagnac phase shift φ s increasing when k is tunable and the dynamic optimized path for k is shown in the Fig. 4(b) which is one-to-one corresponding to Fig. 4(a). From Fig. 4(b) the value of coupled coefficient k ranges from 0.7763 to 0.8440 which is corresponding to a range of gap between ring resonator and straight waveguide. Next we will analyze the structure of MZI-coupler resonant waveguide optical gyroscope (MZIC-RWOG) showed in Fig. 1 and show that MZIC-RWOG has higher sensitivity than RWOG via dynamic optimization. For a MZIC-RWOG there is need to optimize parameters pair (L m , k m , α m , φ m , Δ Φ 1 , Δ Φ 2 ), and the subscript m here indicates the parameters of MZIC-RWOG. Note that we assume k m = k 1 = k 2 here, because k 1 , k 2 are symmetrical in the equation (3), so the optimum value of k 1 and k 2 must be the same. In addition, the propagation loss α m is assumed to be 3 dB/cm and C IL = 1 from now on, unless stated otherwise.
As in the case of RWOG, all the parameters pairs (L m , k m , α m , φ m , Δ Φ 1 , Δ Φ 2 ) could be scanned and find out optimum parameters pairs for minimum resolution δΩ min . It is worth noting that there is not only one pair Φ (φ m , Δ Φ 1 , Δ Φ 2 ) when L m and k m are fixed at optimum values. That is to say, there are many pairs (L m-opt , k m-opt , Φ (φ m , Δ Φ 1 , Δ Φ 2 )) for minimum resolution δΩ min . Then if bias phase shift φ m-bias is set to zero, one of optimum parameters pairs could be found (L m-opt = 0.0554, k m-opt = 0.461, φ m-bias = 0, Δ Φ 1-opt = 1.3635, Δ Φ 2-opt = 1.7279), which we will discuss specifically later in this paper. Furthermore, the minimum resolution of MZIC-RWOG is the same   with RWOG (0.1247 deg/s) when they have the same perimeter. It is well understood that the optimum perimeters (L opt , L m-opt ) are equal for both MZIC-RWOG and RWOG due to the same propagation loss, so the Sagnac effect is not enhanced in a MZIC-RWOG. Indeed, the MZIC-RWOG discussed in this paper dose not enhance the absolute sensitivity for gyro, because the Sagnac effect is relate to the enclosed area which is not changed for MZIC-RWOG. The optimum perimeter L m-opt and coupled coefficient k m-opt of MZIC-RWOG have been obtained and these two parameters cannot be changed once the resonator on the chip is fabricated. Hence the only tunable parameter is Φ (φ m , Δ Φ 1 , Δ Φ 2 ) after fabricated. According to equation (4), φ m is composed by bias phase shift φ m-bias and Sagnac phase shift φ m-Sagnac (φ φ φ = + − − m m bias m Sagnac ). Once bias phase shift φ m-bias is determined and to be a constant, φ m is only increased as the Sagnac phase shift φ m-Sagnac . Along with an increasing Sagnac phase shift φ m-Sagnac , thus, we have three methods to optimize the resolution δΩ: a) Δ Φ 1 is tunable, Δ Φ 2 is fixed at optimum value; b) Δ Φ 1 is fixed at optimum value, Δ Φ 2 is tunable; c) Δ Φ 1 and Δ Φ 2 are both tunable.
In the following sections, we will discuss specifically the three methods and the control effect for rotation rate resolution δΩ respectively. The red line in Fig. 5(a) shows the optimum dynamic path of Δ Φ 1 versus the Sagnac phase shift φ m-Sagnac when Δ Φ 2 is fixed at optimum value Δ Φ 2-opt . In the middle of the contour the optimum path of Δ Φ 1 goes through the black point, which is the global optimized point (Δ Φ 1-opt , Δ Φ 2-opt ) when bias phase shift φ m-bias is set to zero. The maximum optimized range for rotation rate resolution is approximately 2 deg/s resulted from the colorbar in Fig. 5(a). Meanwhile, the red line in Fig. 5(b) shows the optimum dynamic path of Δ Φ 2 versus the Sagnac phase shift φ m-Sagnac when Δ Φ 1 is fixed at optimum value Δ Φ 1-opt . The optimum path of Δ Φ 2 goes through the black point, which is the global optimized point (Δ Φ 1-opt , Δ Φ 2-opt ) when bias phase shift  φ m-bias is set to zero. The maximum optimized range for rotation rate resolution is approximately 1.5 deg/s resulted from the colorbar in Fig. 5(b). The preliminary result is, hence, that optimum trajectory tracking control for Δ Φ 1 can obtain higher sensitivity than Δ Φ 2 , and the tuning range is less 1.28 times than tuning Δ Φ 2 . By optimizing parameters Φ 1 and Φ 2 from 0 to 2π , there are four optimum trajectory lines to make the sensitivity keeping the minimum value with the change of φ ( Fig. 6(a)). That means when the phase shift φ is fixed, there are four extreme points at the Φ 1 -Φ 2 plane from 0 to 2π . Each of optimum line is corresponding to the phase shift φ from − 0.5 to 0.5. Figure 6(b) shows the change in the temperature (Δ T 1 − Δ T 2 ) corresponding to (Δ Φ 1 − Δ Φ 2 ) in Fig. 6(a) with the different confinement coefficient Γ . High confinement coefficient Γ needs less change of the temperature.
Next, we compared the resolution of dynamic optimized MZIC-RWOG on optimum trajectory with the optimized RWOG in Fig. 7. It is found that the resolution for MZIC-RWOG keeps the minimum value through tuning together Φ 1 and Φ 2 follow with the trajectories in Fig. 6 when the phase shift in the ring resonator is changed by Sagnac effect. Dynamic optimization for MZIC-RWOG can play a better performance. This can be explained that we reset the bias phase shift through tuning together Δ Φ 1 and Δ Φ 2 all the time, so the resolution keeps the minimum value. Hence, MZIC-RWOG can play a better dynamic performance than RWOG.
Finally, we induce the relation between rotation rate Ω and thermal modulation temperature. It is found that the resolution for MZIC-RWOG keeps the minimum value through tuning together Φ 1 and Φ 2 following with the trajectories in Figs 6 and 7 when the Sagnac phase shift φ s in the ring resonator is changed from − 0.5 to 0.5. The variation phase difference is 0.88 (rad) for the modulation phase shift Φ 1 and Φ 2 from Therefore, we can obtain Δ T as a function of Sagnac phase shift φ s via combining equation (1)   In Fig. 8 shows the modulation temperature is as a function of rotation rate Ω with different heater lengths. The shorter heater length needs larger modulation temperature. The parameters using for simulation are confinement factor Γ is 90%, perimeter of ring resonator L = 0.0554 m, T-O coefficient for silicon is 1.86 × 10 −4 (K −1 ), thermal expansion coefficemnt is 2.6 × 10 −4 (K −1 ). Based on the structure of MZIC-RWOG the optimum dynamic trajectory of reference phase shift Δ Φ 1 and Δ Φ 2 is showed when the other parameters were fixed at optimum value. The proposed schematic for integrated optical gyroscope can keep the minimum resolution value through phase-tunable method using thermo-optic effect in coupled region of MZIC-RWOG structure, which is potential for on-chip modulating.

Methods
An improving resonant gyro called MZIC-RWOG is proposed and the optical field distribution and wave propagation are calculated according to transfer matrix approach. Numerical calculation, which is carried out by Matlab code, for obtaining minimum value of resolution of gyro is used for parameters dynamically optimizing.