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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 17 — Aug. 16, 2010
  • pp: 18200–18205
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Analysis of resonant optical gyroscopes with two input/output waveguides

Dooyoung Hah and Dan Zhang  »View Author Affiliations


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

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

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

]. 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.

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)
Fig. 1 (a) Schematic diagram of an optical gyroscope with a single resonator with two input/output waveguides, which is subject to rotation at an angular rate of Ω. (b) Power at the out port |bout|2 versus ϕring, phase shift during half the roundtrip inside the ring resonator. Ω = 0.
. 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

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]

] is used. With it, relationship among the mode amplitudes can be expressed as,
[aaddbdrop]=1κ*[t*11t][0exp(iϕR)exp(iϕR)0](1κ)[t11t*][ainbout],
(1)
where κ and t are coupling and transmission coefficients, respectively, which satisfy |κ|2 + |t|2 = 1. ϕR is the phase shift a signal experiences during the half of the roundtrip, which is composed of three terms as the following equation.
ϕR=ϕring+ϕSRiπRα2=πRnringωc+πR2Ωωc2iπRα2,
(2)
where R, α, nring, 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), ϕSR is the phase shift induced by the Sagnac effect when the sensor experiences a rotation at an angular rate of Ω. For an arbitrarily shaped optical path, the Sagnac phase shift ϕSagnac can be computed by [13

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]

],
ϕSagnac=2ω(AΩ)/c2,
(3)
where 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]

]. If ain = 1 and aadd = 0, bout and bdrop become,

bout=tt*exp(i2ϕR)1(t*)2exp(i2ϕR)    and   bdrop=|κ|2exp(iϕR)1(t*)2exp(i2ϕR).
(4)

There exists a specific value of ϕring that leads to the highest rotation sensitivity when other parameters are fixed. This is marked as ϕring,bias in Fig. 1(b), which coincides with the maximum slope in the curve. The sensitivity S is defined as,
S=1P0dPdΩ,
(5)
where P0 is an input power. P is an output power which can be |bout|2, |bdrop|2, or combination of those two, |bcomb|2. Figure 2(a)
Fig. 2 (a) Calculated maximum sensitivity Smax versus |κ| for a gyroscope with a single resonator and two waveguides. Curve1: drop port. Curve2: out port. Curve3: combined output with ϕcomb = 0. Curve4: combined output with optimized ϕcomb (or ϕcomb,opt). Dashed horizontal line: Smax,RFOG of Eq. (6). (b) ϕcomb,opt versus |κ|. R and α values used for the calculation are 5 cm and 0.06 m−1, respectively. Wavelength λ 0 of 700 nm is used throughout the analysis.
shows the calculated maximum sensitivity Sma x, found at an optimized ϕring,bias for different values of 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]

],
Smax,RFOG=4Rω/(33αc2),
(6)
which is plotted as a dashed line in Fig. 2(a). It is because at ϕring,bias, 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.

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 ϕcomb 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 ϕcomb 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 ϕcomb,opt varies over different 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)
Fig. 3 Maximum sensitivity Smax of a gyroscope with a single resonator and two waveguides at optimized κ, ϕring, and ϕcomb (a) versus α for R = 5 cm, and (b) versus R for α = 0.06 m−1.
shows the effect of α when R is fixed at 5 cm. Device parameters such as κ, ϕring, and ϕcomb were adjusted until the maximum sensitivity was resulted in for a given set of α and R. The optimum value of |κ| is approximately (π)0.5 for small 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]

]. 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 Smax,2IO,SR is defined for the maximum rotation sensitivity of a single resonator with two input/output waveguides (when is small), as below.

Smax,2IO,SR=1.1Rω/(αc2)1.4Smax,RFOG.
(7)

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
Fig. 4 Schematic diagram of an optical gyroscope with two resonators in a coplanar add-drop filter configuration.
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,

[binb1]=P[aina1],  [bdropb1]=P[adropa1],[boutb2]=P[aouta2],  and  [baddb2]=P[aadda2],
(8a)
where P=[tκκ*t*].
(8b)

The coupling waveguide sections with the lengths of Lwg can be described as,
adrop=baddexp(iϕW) and  aout=binexp(iϕW),
(9a)
where ϕW=ϕwg+ϕSWiLwgα2=nwgLwgωc+RLwgΩωc2iLwgα2.
(9b)
nwg 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.
a1=b1exp(iϕR), a2=b2exp(iϕR),
(10a)
a1=t*a1exp(iϕR)κ*adropexp[i(ϕR+ϕSW)],
(10b)
and  a2=t*a2exp(iϕR)κ*aoutexp[i(ϕR+ϕSW)],
(10c)
where ϕR is as defined in Eq. (2). Combining Eqs. (8)-(10), two outputs are found as,
bout={BCexp(iϕW)Dexp(iϕR)[|t|2(t*)2exp(i2ϕR)]}/(B2D2)
(11a)
and bdrop={CDexp(iϕW)+[D2B|κ|2]exp(iϕR)}/(B2D2),
(11b)
where B=1(t*)2exp(i2ϕR), C=t2|t|2exp(i2ϕR),
(11c)
and D=|κ|2exp[i(ϕR+ϕW+ϕSW)].
(11d)
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 ϕring, ϕwg, and ϕcomb. Figure 5(a)
Fig. 5 (a) Calculated maximum sensitivity Smax versus |κ| for a gyroscope with two resonators and two waveguides when Lwg = 11 cm. Curve1: drop port. Curve2: out port. Curve3: combined output with ϕcomb = 0. Curve4: combined output with ϕcomb = ϕcomb,opt. (b) Calculated Smax versus Lwg. R and α values used for the calculation are 5 cm and 0.06 m−1, respectively.
plots the calculated maximum sensitivity versus |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 ϕcomb between the two outputs (curve4). However, unlike the case of the single resonator, the maximum sensitivity with ϕcomb,opt is not heavily dependent on the value of κ, and the overall maximum sensitivity is found when |κ| value is almost close to 1. Figure 5(b) shows the effect of Lwg on the maximum sensitivity. As expected, the overall maximum sensitivity of a sensor with two resonators and two input/output waveguides Smax,2IO,TR is a linear function of Lwg, which can be expressed as the following formula.

Smax,2IO,TR=1.1(R2+Lwgπ)ωαc2.
(12)

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.

Smax,2IO,TR per areaSmax,2IO,SR per area=[1.1(R2+Lwgπ)ωαc2]/(πR2+2RLwg)(1.1Rωαc2)/(πR2)=12.
(13)

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.

References and links

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]

3.

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]

4.

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]

5.

J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 96(5), 053901 (2006). [CrossRef] [PubMed]

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 75(5), 053807 (2007). [CrossRef]

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 24(5), 1216 (2007). [CrossRef]

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 15(7), 3864–3875 (2007). [CrossRef] [PubMed]

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]

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]

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]

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]

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

  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]
  3. 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]
  4. 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]
  5. J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 96(5), 053901 (2006). [CrossRef] [PubMed]
  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 75(5), 053807 (2007). [CrossRef]
  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 24(5), 1216 (2007). [CrossRef]
  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 15(7), 3864–3875 (2007). [CrossRef] [PubMed]
  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]
  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]
  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]
  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]

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