## Analysis of resonant optical gyroscopes with two input/output waveguides |

Optics Express, Vol. 18, Issue 17, pp. 18200-18205 (2010)

http://dx.doi.org/10.1364/OE.18.018200

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### 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

1. E. J. Post, “Sagnac effect,” Rev. Mod. Phys. **39**(2), 475–493 (1967). [CrossRef]

2. U. Leonhardt and P. Piwnicki, “Ultrahigh sensitivity of slow-light gyroscope,” Phys. Rev. A **62**(5), 055801 (2000). [CrossRef]

10. Y. Zhang, H. Tian, X. Zhang, N. Wang, J. Zhang, H. Wu, and P. Yuan, “Experimental evidence of enhanced rotation sensing in a slow-light structure,” Opt. Lett. **35**(5), 691–693 (2010). [CrossRef] [PubMed]

11. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. **15**(6), 998–1005 (1997). [CrossRef]

## 2. Single resonator

*in*,

*out*,

*add*, and

*drop*. For the analysis, a coupling matrix formalism introduced by Poon et al [12

12. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express **12**(1), 90–103 (2004). [CrossRef] [PubMed]

*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.where

_{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

_{Sagnac}13. R. B. Hurst, J.-P. R. Wells, and G. E. Stedman, “An elementary proof of the geometrical dependence of the Sagnac effect,” J. Opt. A, Pure Appl. Opt. **9**(10), 838–841 (2007). [CrossRef]

**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

1. E. J. Post, “Sagnac effect,” Rev. Mod. Phys. **39**(2), 475–493 (1967). [CrossRef]

*a*= 1 and

_{in}*a*= 0,

_{add}*b*and

_{out}*b*become,

_{drop}*ϕ*that leads to the highest rotation sensitivity when other parameters are fixed. This is marked as

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

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

*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

9. M. Terrel, M. J. F. Digonnet, and S. Fan, “Performance comparison of slow-light coupled-resonator optical gyroscopes,” Laser Photonics. Rev. **3**(5), 452–465 (2009). [CrossRef]

*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}*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*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

_{comb}*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*|.

*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*, and

_{ring}*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

9. M. Terrel, M. J. F. Digonnet, and S. Fan, “Performance comparison of slow-light coupled-resonator optical gyroscopes,” Laser Photonics. Rev. **3**(5), 452–465 (2009). [CrossRef]

*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

*L*can be described as,

_{wg}*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. where

_{wg}*ϕ*is as defined in Eq. (2). Combining Eqs. (8)-(10), two outputs are found as, Using Eq. (11), maximum sensitivity was calculated with varying parameters. In this case, there are three phase biases that can be independently adjusted for the maximum sensitivity, namely

_{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}## 4. Conclusion

## References and links

1. | E. J. Post, “Sagnac effect,” Rev. Mod. Phys. |

2. | U. Leonhardt and P. Piwnicki, “Ultrahigh sensitivity of slow-light gyroscope,” Phys. Rev. A |

3. | A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. |

4. | B. Z. Steinberg, “Rotating photonic crystals: a medium for compact optical gyroscopes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

5. | J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. |

6. | M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A |

7. | B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation induced superstructure in slow-light waveguides with mode degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B |

8. | C. Peng, Z. B. Li, and A. S. Xu, “Optical gyroscope based on a coupled resonator with the all-optical analogous property of electromagnetically induced transparency,” Opt. Express |

9. | M. Terrel, M. J. F. Digonnet, and S. Fan, “Performance comparison of slow-light coupled-resonator optical gyroscopes,” Laser Photonics. Rev. |

10. | Y. Zhang, H. Tian, X. Zhang, N. Wang, J. Zhang, H. Wu, and P. Yuan, “Experimental evidence of enhanced rotation sensing in a slow-light structure,” Opt. Lett. |

11. | B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. |

12. | J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express |

13. | R. B. Hurst, J.-P. R. Wells, and G. E. Stedman, “An elementary proof of the geometrical dependence of the Sagnac effect,” J. Opt. A, Pure Appl. Opt. |

**OCIS Codes**

(060.2800) Fiber optics and optical communications : Gyroscopes

(130.6010) Integrated optics : Sensors

(230.4555) Optical devices : Coupled resonators

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: June 21, 2010

Revised Manuscript: July 22, 2010

Manuscript Accepted: July 23, 2010

Published: August 9, 2010

**Citation**

Dooyoung Hah and Dan Zhang, "Analysis of resonant optical gyroscopes with two input/output waveguides," Opt. Express **18**, 18200-18205 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-18200

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### References

- E. J. Post, “Sagnac effect,” Rev. Mod. Phys. 39(2), 475–493 (1967). [CrossRef]
- U. Leonhardt and P. Piwnicki, “Ultrahigh sensitivity of slow-light gyroscope,” Phys. Rev. A 62(5), 055801 (2000). [CrossRef]
- A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233(1-3), 107–112 (2004). [CrossRef]
- B. Z. Steinberg, “Rotating photonic crystals: a medium for compact optical gyroscopes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056621 (2005). [CrossRef] [PubMed]
- J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 96(5), 053901 (2006). [CrossRef] [PubMed]
- M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A 75(5), 053807 (2007). [CrossRef]
- B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation induced superstructure in slow-light waveguides with mode degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B 24(5), 1216 (2007). [CrossRef]
- C. Peng, Z. B. Li, and A. S. Xu, “Optical gyroscope based on a coupled resonator with the all-optical analogous property of electromagnetically induced transparency,” Opt. Express 15(7), 3864–3875 (2007). [CrossRef] [PubMed]
- M. Terrel, M. J. F. Digonnet, and S. Fan, “Performance comparison of slow-light coupled-resonator optical gyroscopes,” Laser Photonics. Rev. 3(5), 452–465 (2009). [CrossRef]
- Y. Zhang, H. Tian, X. Zhang, N. Wang, J. Zhang, H. Wu, and P. Yuan, “Experimental evidence of enhanced rotation sensing in a slow-light structure,” Opt. Lett. 35(5), 691–693 (2010). [CrossRef] [PubMed]
- B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]
- J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12(1), 90–103 (2004). [CrossRef] [PubMed]
- R. B. Hurst, J.-P. R. Wells, and G. E. Stedman, “An elementary proof of the geometrical dependence of the Sagnac effect,” J. Opt. A, Pure Appl. Opt. 9(10), 838–841 (2007). [CrossRef]

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