## Abstract

Rotation sensitivity of optical gyroscopes with ring resonators and two input/output waveguides in a coplanar add-drop filter configuration is studied. First, the gyroscope with a single resonator is analyzed, which is shown to have slightly higher sensitivity than the one with one waveguide. Next, the sensor with two identical resonators coupled through waveguides is investigated, which turns out to have half the sensitivity of the one with a single resonator when compared for the same footprints. The last point is valid when the resonators have the same coupling coefficients to the waveguides in the sensor with two resonators.

© 2010 OSA

## 1. Introduction

Optical gyroscopes have a number of advantages over their mechanical counterpart based on either conservation of angular momentum or the Coriolis force. They operate on the principle of the Sagnac effect, i.e. rotation of the frame of reference in which light is propagating induces relative phase shift [1]. Consequently, they do not require any moving part, making them more robust and free from friction. It also implies that they do not suffer from inherent drift term. In addition, they are not affected by rotation in the orthogonal direction. Optical gyroscopes can be generally divided into two groups: non-resonant and resonant. Most in the first group use interference of two lights after traveling in opposite directions to experience the Sagnac effect in opposite ways. On the other hand, resonant gyroscopes utilize loops such as ring resonators to accumulate phase shifts. At first glance, it appears that such accumulation can enhance the rotation sensitivity dramatically. However, accumulated loss during the resonance in the loop undermines such enhancement so that, on the whole, it produces just mild improvement in sensitivity. It is common to the both of the groups that the rotation sensitivity is proportional to the ratio of the area enclosed by the optical path to the length of that path.

Recently, there have been resurged interests in optical gyroscopes with various novel physical structures, such as photonic crystals, micro-ring resonators, coupled resonator optical waveguides (CROWs), Bragg gratings, Sagnac loops, and so on [2–10]. Another important setup worthy of investigation, which utilizes ring resonators is one with two input/output waveguides [11], which is often used as an add-drop filter. This configuration is particularly interesting because it bears features of the both groups of optical gyroscopes mentioned earlier. On one hand, it utilizes resonators, but on the other hand, its rotation sensitivity can be improved by combining its two outputs as in the non-resonant group. In this paper, we will report analysis on this gyroscope configuration with two input/output waveguides, and compare the results to the previous works. Section 2 will provide the analysis on the gyroscope with a single resonator. Study on the sensor with two resonators will be presented in section 3. The discussion will be mainly focused on the sensitivity of the gyroscopes to an angular rate.

## 2. Single resonator

A schematic diagram of an optical gyroscope with a single resonator with two input/output waveguides is presented in Fig. 1(a)
. Since the configuration is generally used as an add-drop filter, the name convention of the ports is adopted from that of the filter: *in*, *out*, *add*, and *drop*. For the analysis, a coupling matrix formalism introduced by Poon et al [12] is used. With it, relationship among the mode amplitudes can be expressed as,

*κ*and

*t*are coupling and transmission coefficients, respectively, which satisfy |

*κ*|

^{2}+ |

*t*|

^{2}= 1.

*ϕ*is the phase shift a signal experiences during the half of the roundtrip, which is composed of three terms as the following equation.

_{R}*R*,

*α*,

*n*,

_{ring}*ω*, and

*c*are radius of the ring, loss coefficient, effective index of the ring, angular frequency of the light, and the speed of light in vacuum, respectively. The second term in Eq. (2),

*ϕ*is the phase shift induced by the Sagnac effect when the sensor experiences a rotation at an angular rate of

_{SR}*Ω*. For an arbitrarily shaped optical path, the Sagnac phase shift

*ϕ*can be computed by [13],where

_{Sagnac}**A**is an area vector enclosed by the light path. Equation (3) implies that the phase shift depends on neither the location of the center of rotation nor the shape of the surface area

*A*, well-known facts for the Sagnac effect [1]. If

*a*= 1 and

_{in}*a*= 0,

_{add}*b*and

_{out}*b*become,

_{drop}There exists a specific value of *ϕ _{ring}* that leads to the highest rotation sensitivity when other parameters are fixed. This is marked as

*ϕ*in Fig. 1(b), which coincides with the maximum slope in the curve. The sensitivity

_{ring,bias}*S*is defined as,

*P*is an input power.

_{0}*P*is an output power which can be |

*b*|

_{out}^{2}, |

*b*|

_{drop}^{2}, or combination of those two, |

*b*|

_{comb}^{2}. Figure 2(a) shows the calculated maximum sensitivity

*S*

_{ma}_{x}, found at an optimized

*ϕ*for different values of

_{ring,bias}*κ*. The curve1 in Fig. 2(a) is when the output is obtained at the drop port and the curve2 at the out port. Those two curves are somewhat below the theoretical limit for a resonant fiber optic gyroscope (RFOG) suggested in [9],which is plotted as a dashed line in Fig. 2(a). It is because at

*ϕ*, the signal is separated into two ports so that only a part of the signal can contribute to the sensing when the output is obtained at only one port.

_{ring,bias}The overall maximum sensitivity (at the optimum *κ*) can be significantly increased by combining the light outputs from the two ports (see Fig. 2(a), curve3). It can be improved further by introducing additional phase difference *ϕ _{comb}* between the two outputs (see Fig. 1(a) and curve4 of Fig. 2(a)). It is interesting to find that the overall maximum sensitivity calculated at zero

*ϕ*is close to the limit for the RFOG, as given by Eq. (6). It is noteworthy that the overall maximum sensitivity increases significantly (by about a factor of 1.4) by introducing additional phase difference between the two ports. However, it should be noted that an extra area will be required to combine the two output signals, which may cancel out the benefit of such an increase in sensitivity. The optimum phase difference

_{comb}*ϕ*varies over different

_{comb,opt}*κ*values as depicted in Fig. 2(b), and is found to be π/2 for large |

*κ*|.

Next, effects of the loss coefficient *α* and the radius of the ring *R* on the maximum sensitivity were examined. Figure 3(a)
shows the effect of *α* when *R* is fixed at 5 cm. Device parameters such as *κ*, *ϕ _{ring}*, and

*ϕ*were adjusted until the maximum sensitivity was resulted in for a given set of

_{comb}*α*and

*R*. The optimum value of |

*κ*| is approximately (π

*Rα*)

^{0.5}for small

*Rα*as in the case of a single resonator with one input/output waveguide [9]. Figure 3(a) shows that the maximum sensitivity is inversely proportional to

*α*until it starts to deviate from the trend at large

*α*value. Figure 3(b) shows that the maximum sensitivity is directly proportional to

*R*for small

*R*value. A modified figure of merit

*S*is defined for the maximum rotation sensitivity of a single resonator with two input/output waveguides (when

_{max},_{2IO,SR}*Rα*is small), as below.

## 3. Two resonators in parallel

For an add/drop filter application, the configuration of Fig. 1(a) often includes multiple resonators to improve the pass band characteristics. In this section, a resonant optical gyroscope with two resonators as illustrated in Fig. 4 is analyzed (notice the name changes of the mode amplitudes). The analysis is limited to the case of two identical resonators without direct coupling between them, and the same coupling coefficients to the waveguides between them. Then, coupling between the ring resonators and the waveguides can be expressed as,

The coupling waveguide sections with the lengths of *L _{wg}* can be described as,

*n*is the effective index of the waveguide. The Sagnac phase shift for the lights circling inside the ring resonators is a little bit different from the single resonator case because a portion of light rotates around a bigger loop indicated by a dashed line in Fig. 4.

_{wg}*ϕ*is as defined in Eq. (2). Combining Eqs. (8)-(10), two outputs are found as,

_{R}*ϕ*,

_{ring}*ϕ*, and

_{wg}*ϕ*. Figure 5(a) plots the calculated maximum sensitivity versus |

_{comb}*κ*|. It can be seen that for low |

*κ*| (in this example, when |

*κ*| < 0.2), the maximum sensitivity with the out port (curve2) is higher than that with the drop port (curve1). It is the opposite case for high |

*κ*|. Once again, the maximum sensitivity is increased by combining the two outputs (curve3), and further increased by introducing additional phase difference

*ϕ*between the two outputs (curve4). However, unlike the case of the single resonator, the maximum sensitivity with

_{comb}*ϕ*is not heavily dependent on the value of

_{comb,opt}*κ*, and the overall maximum sensitivity is found when |

*κ*| value is almost close to 1. Figure 5(b) shows the effect of

*L*on the maximum sensitivity. As expected, the overall maximum sensitivity of a sensor with two resonators and two input/output waveguides

_{wg}*S*is a linear function of

_{max,2IO,TR}*L*, which can be expressed as the following formula.

_{wg}Compared to the one with a single resonator of section 2, the current sensor with two resonators has half the maximum sensitivity for the same footprint as shown below.

This is not surprising because, after all, a circle is the optimum shape for the gyroscope (in other words, it has the highest area-to-perimeter ratio among all two dimensional shapes). It should be noted that this result is valid for the specific configuration considered in this work.

## 4. Conclusion

Resonant optical gyroscopes with two input/output waveguides were investigated with a special focus on the rotation sensitivity. It was shown that a sensor with a single resonator and two waveguides can have higher sensitivity than the one with a single waveguide. It was also shown that in the configuration considered, coupling two identical ring resonators through waveguides reduces the sensitivity to half compared to the one with a single resonator of the same footprints. This result was obtained in the case of the same coupling coefficients between the resonators and the waveguides in the sensor with two resonators.

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