Abstract

We analyze how the evanescent coupling, κe, between the outermost resonators and input/output waveguides of an N-resonator coupled resonator optical waveguide (CROW) gyroscope affects the transmission and sensitivity to rotations. For constant coupling between resonators κ, rotation sensitivities increase as both κe0 and κe1 while between these two limits the sensitivity has a minimum. In the weak coupling regime, κe0, the sensitivity is enhanced by Fabry–Perot oscillations due to the impedance mismatch at the input/output waveguide–CROW interface, while in the strong coupling regime, κe1, the sensitivity is reduced because the outermost resonators no longer contribute to the CROW transmission. For small N the sensitivity is proportional to N2, while at larger N the sensitivity begins to decrease due to resonator losses.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2012

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photon. Rev. 6, 74–96 (2012).
[CrossRef]

C. Sorrentino, J. Toland, and C. P. Search, “Ultra-sensitive chip scale Sagnac gyroscope based on periodically modulated coupling of a coupled resonator optical waveguide,” Opt. Express 20, 354–363 (2012).
[CrossRef]

R. Novitski, B. Z. Steinberg, and J. Scheuer, “Losses in rotating degenerate cavities and a coupled-resonator optical-waveguide rotation sensor,” Phys. Rev. A 85, 023813 (2012).
[CrossRef]

2011

2010

2009

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

M. A. Terrel, M. J. F. Digonnet, and S. Fan, “Performance limitation of a coupled resonant optical waveguide gyroscope,” J. Lightwave Technol. 27, 47–54 (2009).
[CrossRef]

C. P. Search, J. R. E. Toland, and M. Zivkovic, “Sagnac effect in a chain of mesoscopic quantum rings,” Phys. Rev. A 79, 053607 (2009).
[CrossRef]

2007

2006

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

P. Chak and J. E. Sipe, “Minimizing finite-size effects in artificial resonance tunneling structures,” Opt. Lett. 31, 2568–2570 (2006).
[CrossRef]

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” Appl. Phys. Lett. 89, 041122 (2006).
[CrossRef]

2004

1999

Armenise, M. N.

Boag, A.

Campanella, C. E.

Canciamilla, A.

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photon. Rev. 6, 74–96 (2012).
[CrossRef]

Capmany, J.

Cardenas, J.

Chak, P.

Ciminelli, C.

Dell’Olio, F.

Digonnet, M. J. F.

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

M. A. Terrel, M. J. F. Digonnet, and S. Fan, “Performance limitation of a coupled resonant optical waveguide gyroscope,” J. Lightwave Technol. 27, 47–54 (2009).
[CrossRef]

Ding, J.

Y.-H. Ye, J. Ding, D.-Y. Jeong, I. C. Khoo, and Q. M. Zhang, “Finite-size effect on one-dimensional coupled-resonator optical waveguides,” Phys. Rev. E 69, 056604 (2004).
[CrossRef]

Domenech, J. D.

Fan, S.

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

M. A. Terrel, M. J. F. Digonnet, and S. Fan, “Performance limitation of a coupled resonant optical waveguide gyroscope,” J. Lightwave Technol. 27, 47–54 (2009).
[CrossRef]

Ferrari, C.

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photon. Rev. 6, 74–96 (2012).
[CrossRef]

Hah, D.

Huang, Y.

Jeong, D.-Y.

Y.-H. Ye, J. Ding, D.-Y. Jeong, I. C. Khoo, and Q. M. Zhang, “Finite-size effect on one-dimensional coupled-resonator optical waveguides,” Phys. Rev. E 69, 056604 (2004).
[CrossRef]

Kaston, Z. A.

Khoo, I. C.

Y.-H. Ye, J. Ding, D.-Y. Jeong, I. C. Khoo, and Q. M. Zhang, “Finite-size effect on one-dimensional coupled-resonator optical waveguides,” Phys. Rev. E 69, 056604 (2004).
[CrossRef]

Lee, R. K.

Lipson, M.

Liu, H.-C.

Luo, L.-W.

Melloni, A.

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photon. Rev. 6, 74–96 (2012).
[CrossRef]

Mookherjea, S.

Morichetti, F.

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photon. Rev. 6, 74–96 (2012).
[CrossRef]

Munoz, P.

Muriel, M. A.

Novitski, R.

R. Novitski, B. Z. Steinberg, and J. Scheuer, “Losses in rotating degenerate cavities and a coupled-resonator optical-waveguide rotation sensor,” Phys. Rev. A 85, 023813 (2012).
[CrossRef]

O’Boyle, M.

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” Appl. Phys. Lett. 89, 041122 (2006).
[CrossRef]

Paloczi, G. T.

Poitras, C.

Poon, J. K. S.

Scherer, A.

Scheuer, J.

Search, C. P.

C. Sorrentino, J. Toland, and C. P. Search, “Ultra-sensitive chip scale Sagnac gyroscope based on periodically modulated coupling of a coupled resonator optical waveguide,” Opt. Express 20, 354–363 (2012).
[CrossRef]

J. R. E. Toland, Z. A. Kaston, C. Sorrentino, and C. P. Search, “Chirped area coupled resonator optical waveguide gyroscope,” Opt. Lett. 36, 1221–1223 (2011).
[CrossRef]

C. P. Search, J. R. E. Toland, and M. Zivkovic, “Sagnac effect in a chain of mesoscopic quantum rings,” Phys. Rev. A 79, 053607 (2009).
[CrossRef]

J. R. E. Toland and C. P. Search have prepared a manuscript titled, “Sagnac gyroscope using a two dimensional array of coupled optical microresonators,” Phys. Rev. A (to be published).

Sekaric, L.

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” Appl. Phys. Lett. 89, 041122 (2006).
[CrossRef]

Sipe, J. E.

Sorrentino, C.

Steinberg, B. Z.

R. Novitski, B. Z. Steinberg, and J. Scheuer, “Losses in rotating degenerate cavities and a coupled-resonator optical-waveguide rotation sensor,” Phys. Rev. A 85, 023813 (2012).
[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, 1216–1224 (2007).
[CrossRef]

Terrel, M.

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

Terrel, M. A.

Toland, J.

Toland, J. R. E.

J. R. E. Toland, Z. A. Kaston, C. Sorrentino, and C. P. Search, “Chirped area coupled resonator optical waveguide gyroscope,” Opt. Lett. 36, 1221–1223 (2011).
[CrossRef]

C. P. Search, J. R. E. Toland, and M. Zivkovic, “Sagnac effect in a chain of mesoscopic quantum rings,” Phys. Rev. A 79, 053607 (2009).
[CrossRef]

J. R. E. Toland and C. P. Search have prepared a manuscript titled, “Sagnac gyroscope using a two dimensional array of coupled optical microresonators,” Phys. Rev. A (to be published).

Vlasov, Y.

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” Appl. Phys. Lett. 89, 041122 (2006).
[CrossRef]

Wiederhecker, G. S.

Xia, F.

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” Appl. Phys. Lett. 89, 041122 (2006).
[CrossRef]

Xu, Y.

Yariv, A.

Ye, Y.-H.

Y.-H. Ye, J. Ding, D.-Y. Jeong, I. C. Khoo, and Q. M. Zhang, “Finite-size effect on one-dimensional coupled-resonator optical waveguides,” Phys. Rev. E 69, 056604 (2004).
[CrossRef]

Zhang, D.

Zhang, Q. M.

Y.-H. Ye, J. Ding, D.-Y. Jeong, I. C. Khoo, and Q. M. Zhang, “Finite-size effect on one-dimensional coupled-resonator optical waveguides,” Phys. Rev. E 69, 056604 (2004).
[CrossRef]

Zivkovic, M.

C. P. Search, J. R. E. Toland, and M. Zivkovic, “Sagnac effect in a chain of mesoscopic quantum rings,” Phys. Rev. A 79, 053607 (2009).
[CrossRef]

Adv. Opt. Photon.

Appl. Phys. Lett.

F. Xia, L. Sekaric, M. O’Boyle, and Y. Vlasov, “Coupled resonator optical waveguides based on silicon-on-insulator photonic wires,” Appl. Phys. Lett. 89, 041122 (2006).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Laser Photon. Rev.

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

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photon. Rev. 6, 74–96 (2012).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

R. Novitski, B. Z. Steinberg, and J. Scheuer, “Losses in rotating degenerate cavities and a coupled-resonator optical-waveguide rotation sensor,” Phys. Rev. A 85, 023813 (2012).
[CrossRef]

C. P. Search, J. R. E. Toland, and M. Zivkovic, “Sagnac effect in a chain of mesoscopic quantum rings,” Phys. Rev. A 79, 053607 (2009).
[CrossRef]

Phys. Rev. E

Y.-H. Ye, J. Ding, D.-Y. Jeong, I. C. Khoo, and Q. M. Zhang, “Finite-size effect on one-dimensional coupled-resonator optical waveguides,” Phys. Rev. E 69, 056604 (2004).
[CrossRef]

Phys. Rev. Lett.

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

Other

J. R. E. Toland and C. P. Search have prepared a manuscript titled, “Sagnac gyroscope using a two dimensional array of coupled optical microresonators,” Phys. Rev. A (to be published).

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Figures (6)

Fig. 1.
Fig. 1.

Microring CROW gyroscope showing the relationship between the resonators, transfer matrices, and repetitive unit cell defined in the text. sin and sout are the input signal and transmitted field, respectively, such that T(ϕS)=|sout/sin|2. Based on the propagation direction of sout in the figure, the number of resonators must be odd due to phase matching.

Fig. 2.
Fig. 2.

Maximum transmission slope (dT/dϕS)max for N=11 (blue curves), 15 (green curves), and 23 (red curves) resonators as a function of κe for (a) κ=0.05 and (b) κ=0.1. Solid curves represent lossless resonators (Qint1=0), dashed curves are Qint=106, and dotted curves are Qint=105. The resonator mode number in all cases is m=162 for R=11.5μm Si resonators. For each of the three line styles, red curves are the topmost, green curves are in the middle, and blue are on the bottom.

Fig. 3.
Fig. 3.

(a) Transmission spectra of an N=7 gyro for κe=0.1 (blue dashed curve) and κe=0.61 (green solid curve). In both cases κ=0.01, and they have identical maximum slopes (dT/dϕS)max=417.6. (b) Transmission spectra of an N=13 CROW gyro with κe=0.107 (blue dashed curve) and N=15 with κe=0.843 (green solid curve). Both have κ=0.05 and (dT/dϕS)max=1719. These results are for lossless resonators.

Fig. 4.
Fig. 4.

(a) (dT/dϕS)max versus N in the weak coupling and strong coupling (inset) regime. Data markers are numerically calculated slopes, while curves are quadratic fits, (dT/dϕS)max=α+βN2. Blue curves with crosses are κe=0.05, while green curves with squares are κe=0.1. Inset shows κe=0.95. In all cases κ=0.1 and the microresonator mode number is m=162, assuming R=11.5μm Si resonators. (b) log((dT/dϕS)max) comparing an N-ring CROW to a single resonator RFOG with equal enclosed geometric area Aeff=NπR2. Green curves (top three curves) are for the CROW with κ=κe=0.1, and black curves (bottom three curves) are the RFOG with waveguide–resonator coupling κe=0.1. In both (a) and (b), solid curves represent lossless resonators, dashed curves are Qint=106, and dotted curves are Qint=105.

Fig. 5.
Fig. 5.

(a) (dT/dϕS)max versus N for Qint=4.75×104, showing a decrease in sensitivity for large N. For all curves κ=0.1 and the mode number is m=162, while the waveguide couplings of the curves from topmost to bottommost curve in descending order are κe=0.05 (blue curve), κe=0.1 (green curve), κe=0.2 (red curve), κe=0.7 (purple curve), and κe=0.95 (black curve). (b) log((dT/dϕS)max) versus Qint1 for N=7 and 15, showing a linear decrease in sensitivity with increasing Qint1. Blue curves are κe=0.05, red curves are κe=0.1, and green curves are κe=0.2. Solid curves are N=7, while dashed curves are N=15. Note that at Qint1=0 the curves in descending order from topmost to bottommost are κe=0.05, N=15, κe=0.1, N=15, κe=0.2, N=15, κe=0.05, N=7, κe=0.1, N=7, and κe=0.2, N=7.

Fig. 6.
Fig. 6.

κe value at which the sensitivity (dT/dϕS)max is a minimum versus resonator number N for κ=0.1 (circles) and κ=0.2 (x’s). The red dashed and green solid curves are the fit (κe)min=κ+1N+1. This was calculated for lossless resonators, but the results are unchanged when losses are included.

Equations (10)

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ϕS=2πωΩR2/c2.
Uin=iκe(ei(ϕp+ϕS/2)1κeei(ϕp+ϕS/2)1κeei(ϕp+ϕS/2)ei(ϕp+ϕS/2)),
Uout=1κe(ei(ϕp+ϕS/2)1κeei(ϕp+ϕS/2)1κeei(ϕp+ϕS/2)ei(ϕp+ϕS/2))×(1κκ1κ1κ1κκ),
URHM=iκ(1κei(ϕp+ϕS/2)ei(ϕp+ϕS/2)ei(ϕp+ϕS/2)1κei(ϕp+ϕS/2)),
ULHM=iκ(1κei(ϕpϕS/2)ei(ϕpϕS/2)ei(ϕpϕS/2)1κei(ϕpϕS/2)),
TCCW=Uout(ULHMURHM)MULHMUin=(T11T12T21T22),
TM=(AuM1(x)uM2(x)BuM1(x)B*uM1(x)A*uM1(x)uM2(x)).
S=dT/dΩ=(2πωR2/c2)(dT/dϕS).
Ωmin=π2SmaxhcΔfληPopt,
(dTdϕS)max(N+1)2κeeωnLeff/cQint=(N+1)2κee(N1κ+2κe)mπQint.

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