Introduction

Chip-scale optical inertial rotation sensors1,2,3,4,5,6 based on Sagnac effect7 have distinct advantages in aspects of physical size, weight, power consumption, cost, compatibility with technology for producing microelectronics and have the most potential to become the next generation optical gyroscopes after ring laser gyro (RLG) and fiber optic gyro (FOG). Nevertheless, on-chip optical gyroscope with tiny area is bound to accumulate less phase shift and less sensitive than RLG and FOG. In order to make on-chip optical gyroscope more sensitive, various coupled-cavities structures such as coupled resonator optical waveguides (CROWs) and side-coupled integrated spaced sequence of resonators (SCISSOR) were proposed8,9,10 and slow-light effect were introduced into those structures. However, loss ultimately limits the achievable sensitivity in CROW structure. Physically, slow light is a resultant property of the highly dispersive structure and has no direct link with the enhancement of gyro’s sensitivity9. It is beneficial to introduce enhanced effect to increase accumulated frequency or phase shift, such as fast light11,12,13,14,15 or nonlinear Kerr effect16,17. Another effective way is utilizing ultralow propagation loss of optical waveguide18 or introducing active gain into waveguide for compensating loss19,20,21. In addition to enhanced effect and ultralow propagation loss, parameter optimization is an essential part to make the chip size gyroscope work at the most sensitive point22. All gyroscopes must be optimized with respect to their experimentally adjustable parameters. In the case of resonant gyroscopes, this means that the sensitivity must be calculated using optimum values for coupling coefficients and phase biases23,24,25. While, the parameters of on-chip rotation sensors are all influenced by processing technology and cannot be changed after fabricating. That is to say, the gyroscope is only designed by static optimization and not dynamic optimization.

An external phase modulator always provides necessary phase bias in conventional interferometric fiber optic gyro (IFOG). This modulator is usually made by using Piezoelectric ceramic (PZT) or electro-optic crystal materials lithium niobate (LiNbO3) and phase modulation is produced by driven voltage. Due to fiber winding on Piezoelectric ceramic (PZT), PZT phase modulator is not suitable for integrating on a chip, in addition, the frequency using PZT phase modulation is low which is not satisfied the requirement of feedback control components. On the other hand, Y-branch waveguide made by LiNbO3 can produce high modulation frequency and integrate beam splitter, polarizer and modulator on a chip. However, LiNbO3 phase modulators were adopted to realize the frequency modulation, which made it incompatible with silicon process technology and standard CMOS electronics and thus it is harder to integrate into complete systems on a chip. LiNbO3 modulator is still high cost with respect to silicon optical modulator. Silicon has the higher refractive index than LiNbO3, which is advantaged to size reduction. To the best our knowledge, LiNbO3 waveguide propagation loss (0.05 dB/cm26) is still higher than silicon based waveguide (0.01 dB/cm27) for the transmission wavelength of 1550–1580 nm.

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/s28. 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 μm29. Thus making electro-optic silicon modulators is difficult to realize30. 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 absorption31. Generally, in electrorefraction modulators, a balance should be struck between electrorefraction and eletroabsorption32. 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 effect33, whereas electro-optic modulators have large insertion loss usually more than 4 dB34. And the other practical option is thermal modulation owing to the large thermo-optic coefficient (TO) of silicon35,36,37,38. Early in 2000 years K. Suzuki and K. Hotate39 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 sink40. 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 technology41. 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 ΔΦ1, ΔΦ2, 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.

Figure 1
figure 1

MZI-coupler resonant waveguide optical gyroscope (MZIC-RWOG).

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Results

In this work, we only consider the thermo-optic effects to control reference phase shift ΔΦ1,2 in the arms of MZI. As current injected into the MZI electrode, heating changes the waveguide’s refractive index and introduces a phase shift35,36:

where λ is the wavelength, is the thermo-optic coefficient (TO) and represents thermalexpansion coefficient. ΔT is the change in the temperature (Kelvin) and Lheater is the heater length. For example, if the Lheater is 500 μm and (a typical value for silicon at λ = 1550 nm38), 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 is42

where n0 is the channel waveguide refractive index and n1 is the cladding refractive index and Γ is the confinement factor which defines the fraction of optical mode inside the core region. If , 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 .

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 n0,1 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)).

Figure 2
figure 2

The slope S between phase shift and temperature change versus the material TO coefficients of core for different confinement factor Γ = 50%, 70%, 90% when λ = 1550 nm, Lheater is 500 μm.

The material TO coefficients of cladding are (a) −0.0003 K−1, (b) 0 K−1 and (c) 0.0003 K−1.

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Next, the transmission characteristic of MZIC-RWOG in Fig. 1 can be obtained by

using transfer matrix approach43,

where k1, k2 are the coupling coefficient and t1, t2 are the through coupling coefficient of directional couplers respectively. When the coupling loss is neglected, and are satified. The one round-trip amplitude propagation attenuation is , 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 ϕs 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 teff can be found

According to22,44,45,46, the resolution (which is defined as the minimum detectable angular rate) is expressed if the resonator is circular:

The amplitude noise term and iD are the standard deviation of the photocurrent and the maximum photodiode current (specific parameters are listed in Table 1).

Table 1 List of parameters.
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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 ϕbias). According to transfer matrix approach, the analytical expression of the all-pass structure RWOG for the minimum detectable angular rate can be calculated which is a function of A, k, α and ϕ. Although it is possible to optimize using the exact formula, numerical calculation for obtaining minimum value of is more convenient. More specifically, we can scan parameters (A, k, α, ϕ) and calculate every value of resolution in a setting range, then find out both the minimum value δΩmin and optimizing parameters pair (Aopt, kopt, αopt, ϕbias). For the optimizing parameters pair (Aopt, kopt, αopt, ϕbias), except ϕbias 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 after setting a fixing value for propagation loss. From Fig. 3 we can see that there is always an optimum L corresponding to for every loss value and the optimum parameters are shown as Table 2.

Table 2 List of optimum parameters for different propagation loss.
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Figure 3
figure 3

Rotation rate resolution versus resonator perimeter for the RWOG.

The parameters ϕ and k are all fixed at optimum value (CIL = 1).

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Note that the optimizing parameters pair (Aopt, kopt, αopt, ϕbias) only ensure maximum sensitivity at zero rotation-rate point since the parameters have been fixed after fabrication, thus the parameters (Aopt, kopt, αopt, ϕbias) 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 ϕs 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 is the same at ϕbias for the two situations, there is small enhancement for with the Sagnac phase shift ϕs 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.

Figure 4
figure 4

(a) Rotation rate resolution versus phase shift in the resonator when k is fixed at static optimization point 0.8046 (black line) Rotation rate resolution versus phase shift in the resonator when k is tunable (red line) (b) the optimized path of k is one-to-one corresponding to (a). All the case are assumed α = 3 dB/cm, Lopt = 0.0554 m, ϕbias = 1.2353.

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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 (Lm, km, αm, ϕm, ΔΦ1, ΔΦ2) and the subscript m here indicates the parameters of MZIC-RWOG. Note that we assume km = k1 = k2 here, because k1 , k2 are symmetrical in the equation (3), so the optimum value of k1 and k2 must be the same. In addition, the propagation loss αm is assumed to be 3 dB/cm and CIL = 1 from now on, unless stated otherwise.

As in the case of RWOG, all the parameters pairs (Lm, km, αm, ϕm, ΔΦ1, ΔΦ2) could be scanned and find out optimum parameters pairs for minimum resolution . It is worth noting that there is not only one pair Φ (ϕm, ΔΦ1, ΔΦ2) when Lm and km are fixed at optimum values. That is to say, there are many pairs (Lm-opt, km-opt, Φ(ϕm, ΔΦ1, ΔΦ2)) for minimum resolution . Then if bias phase shift ϕm-bias is set to zero, one of optimum parameters pairs could be found (Lm-opt = 0.0554, km-opt = 0.461, ϕm-bias = 0, ΔΦ1-opt = 1.3635, ΔΦ2-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 (Lopt, Lm-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 Lm-opt and coupled coefficient km-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(ϕm, ΔΦ1, ΔΦ2) after fabricated. According to equation (4), ϕm is composed by bias phase shift ϕm-bias and Sagnac phase shift ϕm-Sagnac (). Once bias phase shift ϕm-bias is determined and to be a constant, ϕm is only increased as the Sagnac phase shift ϕm-Sagnac. Along with an increasing Sagnac phase shift ϕm-Sagnac, thus, we have three methods to optimize the resolution : a) ΔΦ1 is tunable, ΔΦ2 is fixed at optimum value; b) ΔΦ1 is fixed at optimum value, ΔΦ2 is tunable; c) ΔΦ1 and ΔΦ2 are both tunable.

In the following sections, we will discuss specifically the three methods and the control effect for rotation rate resolution respectively. The red line in Fig. 5(a) shows the optimum dynamic path of ΔΦ1 versus the Sagnac phase shift ϕm-Sagnac when ΔΦ2 is fixed at optimum value ΔΦ2-opt. In the middle of the contour the optimum path of ΔΦ1 goes through the black point, which is the global optimized point (ΔΦ1-opt, ΔΦ2-opt) when bias phase shift ϕm-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 ΔΦ2 versus the Sagnac phase shift ϕm-Sagnac when ΔΦ1 is fixed at optimum value ΔΦ1-opt. The optimum path of ΔΦ2 goes through the black point, which is the global optimized point (ΔΦ1-opt, ΔΦ2-opt) when bias phase shift ϕm-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 ΔΦ1 can obtain higher sensitivity than ΔΦ2 and the tuning range is less 1.28 times than tuning ΔΦ2.

Figure 5
figure 5

(a) The optimum path of ΔΦ1 when ΔΦ2 is fixed at optimum value 1.7279. (b) The optimum path of ΔΦ2 when ΔΦ1 is fixed at optimum value 1.3635. All the case are assumed α = 3 dB/cm, Lm-opt = 0.0554 m, km-opt = 0.461.

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By optimizing parameters Φ1 and Φ2 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 Φ1-Φ2 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 (ΔT1 − ΔT2) corresponding to (ΔΦ1 − ΔΦ2) in Fig. 6(a) with the different confinement coefficient Γ. High confinement coefficient Γ needs less change of the temperature.

Figure 6
figure 6

(a) The four optimum trajectory lines through tuning ΔΦ1 and ΔΦ2 together when α = 3 dB/cm, Lm-opt = 0.0554 m, km-opt = 0.461. (b) The change in the temperature (ΔT1 − ΔT2) is corresponding to (ΔΦ1 −Δ Φ2) in (a) with the different confinement coefficient Γ when Lheater = 500 μm, , λ = 1550 nm.

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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 Φ1 and Φ2 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 ΔΦ1 and ΔΦ2 all the time, so the resolution keeps the minimum value. Hence, MZIC-RWOG can play a better dynamic performance than RWOG.

Figure 7
figure 7

Comparing the resolution of dynamic optimized MZIC-RWOG with the optimized RWOG.

All the case are assumed α = 3 dB/cm, Lopt = 0.0554 m.

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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 Φ1 and Φ2 following with the trajectories in Figs 6 and 7 when the Sagnac phase shift ϕs 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 Φ1 and Φ2 from Fig. 6. Since the modulation phase shift Φ1 and Φ2 are a linear change with the increasing of Sagnac phase shift, the linear relation is

Therefore, we can obtain ΔT as a function of Sagnac phase shift ϕs via combining equation (1) and (7)

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).

Figure 8
figure 8

ΔT is as a function of rotation rate Ω with different heater length Lheater = 100 μm, 300 μm and 500 μm.

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Based on the structure of MZIC-RWOG the optimum dynamic trajectory of reference phase shift ΔΦ1 and ΔΦ2 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.

Additional Information

How to cite this article: Zhang, H. et al. On-chip modulation for rotating sensing of gyroscope based on ring resonator coupled with Mach-Zehnder interferometer. Sci. Rep. 6, 19024; doi: 10.1038/srep19024 (2016).