## Abstract

A side-coupled spaced sequence of resonators (SCSSOR) displays strong dispersion with a magnitude much larger than that of conventional waveguides. We investigate the Sagnac effect in a SCSSOR structure. An explicit expression of the Sagnac phase difference of the structure is derived and discussed. The results show the sensitivity is proportional to the number and dispersion intensity of resonators. Compared with other resonator structures, one advantage is that it is not necessary to preserve the same circumference of each resonator, which is difficult to realize in practice. The results also predict a SCSSOR structure can be used for the realization of highly sensitive and compact integrated rotation sensors and gyroscopes.

© 2011 OSA

## 1. Introduction

Rotation sensors are widely used in satellites, remote control devices, positioning systems, and in many other industrial and military affairs. Although rotation sensors based on both waveguide and fiber have become commercially available, people have been looking for new manners to develop more highly sensitive and compact rotation sensors. The developments of the dispersion structure (the structure in which the dispersion is induced by the physical configuration of elements constituting the structure, not by the medium of it) [1–9] may open a new way to enhance the sensitivity and compactness of rotation sensors, and in the past few years much attention has been devoted to developing rotation sensors based on dispersion structures, such as photonic-crystal-based coupled-cavity waveguides [10,11], coupled-resonator optical waveguides [12], coupled-resonator-induced transparency structures [13–15], optical microcoil resonators [16], and ring resonators [17,18]. As indicated in Refs [10,11,16], the rotation of dispersion structures modifies the dispersion relations of the counterpropagating electromagnetic waves, thus generating a rotation-dependent phase shift for each wave. As a result, the dispersion relations can increase the accumulated phase difference between the counterpropagating waves, which was firstly experimentally demonstrated by Zhang *et al*. using a single fiber resonator [18]. Zhang *et al*. also pointed out rotation sensitivity was proportional to the dispersion intensity.

A side-coupled spaced sequence of resonators (SCSSOR) is illustrated in Fig. 1 [8,9]. The structure is totally transmissive at all frequencies, and displays strong dispersion with a magnitude much larger than that of conventional waveguides because of the critical dependence of the time propagating through the structure on the detuning of the optical wave from the resonance frequency [8]. With the development of manufacturing techniques, the circumference of a microring resonator can be hundreds and even tens of optical wavelengths such that one can realize very compact integrated SCSSORs. Especially in a SCSSOR, it is not necessary to keep the same circumference of each resonator, which is difficult to realize in practice. However, in other dispersion structures consisting of many resonators, asynchronous resonance of each resonator can greatly decrease the transmission and destroy the Sagnac effect. In this Letter we investigate the Sagnac effect in a SCSSOR with the application for a highly sensitive and compact rotation sensor.

## 2. Sagnac effect in a SCSSOR

As shown in Fig. 1, a SCSSOR consists of an input/output waveguide and *M* resonators. In the structure the resonators are independent and there is no coupling between each other, so we first consider the fields of a single resonator. The coupling between the waveguide and a resonator can be described by a transfer matrix [8,19]:

*t*and

*k*are the transmission and coupling coefficients, and the matrix is unitary such that ${t}^{2}+{k}^{2}=1$. The round-trip phase shift $\varphi ={n}_{e}\omega L/c$, where ${n}_{e}$ is the effective index,

*ω*is the angular frequency,

*c*is the speed of light in vacuum, and

*L*is the total length of the resonator. Then we can obtainwhere

*Φ*is the whole phase shift induced by the resonator.

In an arbitrary closed path, the phase difference between the counterpropagating waves introduced by rotation is given by the well-known Sagnac effect [20]

where*A*,

*R*and

*N*are the area, radius and loop number of the resonator respectively, and

*Ω*is the rotation angular velocity. For the two waves, the round-trip phase shifts are ${n}_{e}\omega L/c\pm 2N\omega A\Omega /{c}^{2}$.

Apparently the round-trip phase shifts are perturbed by the Sagnac effect, and the perturbation modifies the dispersion relation. Therefore, it is expected the Sagnac effect can be enhanced in the resonator with strong dispersion. As ${n}_{g}=(c/L)\partial \Phi /\partial \omega $ is the group index and $\Omega R\ll c$, we can obtain

Then the phase difference between the counterpropagating waves, introduced by the resonator, in an arbitrary frequency is

In a SCSSOR, the resonators are independent with each other, so the total phase difference of the structure should be accumulated.

Figure 2 shows the phase difference of a resonator at different rotary velocities and frequencies. It can be seen that the phase difference is equal to zero when the resonator is stationary ($\Omega =0$), which can be easily deduced from Eq. (6). Apparently the phase difference increases as rotary velocity increases, owing to the modification of dispersion relation induced by rotation. Because the dispersion is strongest at the resonance frequency and gets weaker as the frequency detuning increases, the phase difference distributes symmetrically about the resonance frequency, and gets the maximum at the resonance frequency. In fact from Eq. (6) we can obtain ${\partial \Delta \Phi /\partial \omega |}_{\text{resonance \hspace{0.17em} frequency}}=0$.

The total phase difference of a SCSSOR structure is the summation of the phase difference of each resonator. Here we can call coincident resonance condition that the incident light frequency is within the operation dispersion bandwidth of each resonator. When all the resonators resonate at the incident light frequency, we obtain the perfect coincident resonance and the maximal total phase difference. And if the resonance frequency of a resonator gets away from the incident light frequency or even gets out of the dispersion band, the resonator has small or no contribution to the total phase difference. Therefore, we should keep the same circumference of each resonator to obtain the maximal phase difference, i.e., the maximal sensitivity. However, in practice it is impossible to preserve the same circumference. Fortunately the resonators are independent with each other, so that the detuning of any resonators does not influence others and cannot destroy the dispersion and Sagnac effect of the whole structure.

## 3. Experiment

Under our experimental condition, the SCSSOR structure consists of two fiber resonators: ${n}_{e}=1.5$, $R=0.12$m, $N=20$, and ${t}^{2}=0.6$. The experiment arrangement for measurement of the phase difference in the structure is shown in Fig. 3 . The wavelength and spectral linewidth of the tunable laser are 1550nm and 10 KHz. The function generator is used to linearly tune the laser frequency by setting a triangular wave voltage signal to the laser, so the interference spectrum can be obtained. The incident light (${I}_{in}$) equally divided by the 3dB coupler, forming counterpropagating waves (marked by the blue and red arrows). Finally, the counterpropagating waves are combined by the 3dB coupler where the detected signal (${I}_{out}$) depends on the phase difference:

All devices shown in Fig. 3 are set on a rotary table. The rotary speed of the table is from 0 to $2\pi \text{\hspace{0.17em}}rad/s$, and the absolute accuracy is $0.05\text{\hspace{0.17em}}rad/s$. In fact, in order to obtain the coincident resonance, the circumference difference of the two resonators must be within one hundred nm. In the experiment, the two fiber resonators cannot satisfy the coincident resonance, so only one resonator can take effect at its resonance frequency and the interference spectrum peaks of the resonators are separate as shown in Fig. 4(a) . When we tune the laser frequency near the resonance frequency of any resonator, the interference spectra at different rotary velocities can be obtained, shown in Fig. 4(b). The change of the interference light intensity is maximal at the resonance frequency, resulting from dependence of phase difference on rotary velocity and frequency discussed in Fig. 2. Therefore, in a SCSSOR rotation sensor, we should measure the interference light intensity at the resonance frequency for high sensitivity. It also can be predicted that as the rotary velocity increases unceasingly, the interference light intensity at the resonance frequency should firstly reach the maximum value and then decrease. The result in Fig. 4 also demonstrates that the detuning of a resonator does not destroy the Sagnac effect of the whole structure.

Figure 5 shows the theoretical and experimental phase difference at the resonance frequency against rotary velocity. The theoretical and experimental results do not coincide completely, because the light is not totally transmitted due to the loss in the couplers. Moreover, there are noise resulting from the large electric current of the rotary table, and small deviations of the transmission and coupling coefficients in the experiment.

With the development of micro-fabrication technique, miniature rotation sensors constructed by integrated SCSSOR structures may be realized. Compared with fiber resonator, it is easier to obtain the coincident resonance in an integrated SCSSOR structure by using high precision fabrication technique or tuning the resonators (using the electro-optic or thermo-optic effects). Figure 6
shows a compact SCSSOR structure which can be feasibly used in a rotation sensor. The structure can be fabricated on silicon-on-insulator, which allows strong confinement of light due to its high index contrast between the silicon core and the silica cladding. It can be seen that the structure sufficiently utilizes the area of the wafer, and can reduce the footprint of the rotation sensor. Especially, the Sagnac effect can be enhanced, because each counterpropagating wave keeps propagating in the same direction (counterclockwise and clockwise marked by the blue and red arrows respectively) in all resonators. It should be noted that one shortcoming of the rotation sensor constructed by integrated SCSSOR structure is the small areas of the microresonators, which can decrease the Sagnac effect. Fortunately we can increase the number of resonators to offset the decreasing Sagnac effect, which can be seen from Eq. (7). Another more effective way to enhance the Sagnac effect is to increase the dispersion intensities of microresonators by increasing the transmission coefficient. Figure 7
shows *n _{g}*/

*n*in a microresonator versus transmission coefficient.

_{e}## 4. Conclusions

In this letter, we investigate the Sagnac effect in a SCSSOR structure with the application for a highly sensitive and compact rotation sensor. An explicit expression of the Sagnac phase difference of the structure is derived and discussed. The sensitivity is proportional to the number and dispersion intensity of resonators. It should be noted that the enhancement of the sensitivity can be obtained not only by positive dispersion but also by negative dispersion.

And compared with other resonator structures, one advantage is that preserving the same circumference of each resonator is not necessary, because the resonators are independent with each other and the detuning of any resonators does not destroy the dispersion and Sagnac effect of the whole structure, which is demonstrated in the experiment. Finally, we propose a compact SCSSOR structure which can be feasibly used for the realization of highly sensitive and compact integrated rotation sensors and gyroscopes.

## Acknowledgements

The research is supported by the National Natural Science Foundation of China under Grant Nos. 61078006 and 60878006.

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