Abstract

We present a theoretical investigation on the interplay between the intra-cavity backscattering and the losses out of the cavity when dual contra-propagating inputs are launched into a traveling wave microresonator, which has random surface defects (or backscatters) that are intentionally introduced. By adjusting the amplitude and/or phase differences between the dual inputs, the interaction of the cavity modes with the backscatters can be controlled. Consequently, the transmission and dispersion of the output light can be easily manipulated. This feature makes the dual input scheme highly attractive for continuously tunable fast and slow light applications, particularly if active tuning elements, such as p-i-n diode and heater, are absent in the cavity. Continuous tunability in the group delay of the fast and slow light is also demonstrated at the C-band wavelength of 1.55μm.

© 2010 Optical Society of America

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References

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2010 (4)

X. Luo and A. W. Poon, “Electro-optical tunable time delay and advance in silicon microring resonator-based notch filters,” in Conference on Lasers and Electro-Optics, 2010 OSA Technical Digest (Optical Society of America, 2010), paper CTuHH1.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

A. H. Atabaki, B. Momeni, A. A. Eftekhar, E. S. Hosseini, S. Yegnanarayanan, and A. Adibi, “Tuning of resonance-spacing in a traveling-wave resonator device,” Opt. Express 18, 9447–9455 (2010).
[CrossRef] [PubMed]

Y. H. Wang, C. S. Ma, X. Yan, and D. M. Zhang, “Analysis for amplifying characteristics of Er3+−Yb3+-co-doped microring resonators,” Opt. Laser Technol. 42, 336–340 (2010).
[CrossRef]

2009 (2)

2008 (3)

2007 (3)

2006 (2)

F. Morichetti, A. Melloni, C. Canavesi, F. Persia, M. Martinelli, and M. Sorel, “Tunable slow-wave optical delay-lines,” in Slow and Fast Light, 2006 OSA Technical Digest (Optical Society of America, 2006), paper MB2.

D. Leuenberger, J. Yao, M. M. Lee, and M. C. Wu, “Experimental demonstration of MEMS-tunable slow light in silicon microdisk resonators,” in Slow and Fast Light, 2006 OSA Technical Digest (Optical Society of America, 2006), paper TuC6.

2005 (1)

2004 (2)

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[CrossRef] [PubMed]

C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry–Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16, 506–508 (2004).
[CrossRef]

2002 (1)

R. W. Boyd, “Slow light now and then,” Nat. Photonics 2, 497–530 (2002).

2000 (1)

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

1998 (2)

S. A. Backes and J. R. A. Cleaver, “Microdisk laser structures for mode control and directional emission,” J. Vac. Sci. Technol. B 16, 3817–3820 (1998).
[CrossRef]

B. E. Little, S. T. Chu, and H. A. Haus, “Second-order filtering and sensing with partially coupled traveling waves in a single resonator,” Opt. Lett. 23, 1570–1572 (1998).
[CrossRef]

1997 (1)

Adibi, A.

Almeida, V. R.

C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry–Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16, 506–508 (2004).
[CrossRef]

Atabaki, A. H.

Backes, S. A.

S. A. Backes and J. R. A. Cleaver, “Microdisk laser structures for mode control and directional emission,” J. Vac. Sci. Technol. B 16, 3817–3820 (1998).
[CrossRef]

Barrios, C. A.

C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry–Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16, 506–508 (2004).
[CrossRef]

Bian, Z.

D. G. Rabus, Z. Bian, and A. Shakouri, “Ring resonator lasers using passive waveguides and integrated semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13, 1249–1256 (2007).
[CrossRef]

Borselli, M.

Boyd, R. W.

R. W. Boyd, “Slow light now and then,” Nat. Photonics 2, 497–530 (2002).

Canavesi, C.

F. Morichetti, A. Melloni, C. Canavesi, F. Persia, M. Martinelli, and M. Sorel, “Tunable slow-wave optical delay-lines,” in Slow and Fast Light, 2006 OSA Technical Digest (Optical Society of America, 2006), paper MB2.

Chen, D. R.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Chu, C. H.

Chu, S. T.

Cleaver, J. R. A.

S. A. Backes and J. R. A. Cleaver, “Microdisk laser structures for mode control and directional emission,” J. Vac. Sci. Technol. B 16, 3817–3820 (1998).
[CrossRef]

Cohen, O.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[CrossRef] [PubMed]

Dogariu, A.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

Dong, P.

Eftekhar, A. A.

Hane, K.

Haus, H. A.

He, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Hosseini, E. S.

Johnson, T.

Jones, R.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[CrossRef] [PubMed]

Kanamori, Y.

Kikuchi, T.

Kokubun, Y.

Kuzmich, A.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

Laine, J. P.

Lee, M. M.

D. Leuenberger, J. Yao, M. M. Lee, and M. C. Wu, “Experimental demonstration of MEMS-tunable slow light in silicon microdisk resonators,” in Slow and Fast Light, 2006 OSA Technical Digest (Optical Society of America, 2006), paper TuC6.

Leuenberger, D.

D. Leuenberger, J. Yao, M. M. Lee, and M. C. Wu, “Experimental demonstration of MEMS-tunable slow light in silicon microdisk resonators,” in Slow and Fast Light, 2006 OSA Technical Digest (Optical Society of America, 2006), paper TuC6.

Li, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Li, Q.

Liao, L.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[CrossRef] [PubMed]

Lin, C. Y.

Lipson, M.

S. Manipatruni, P. Dong, Q. Xu, and M. Lipson, “Tunable superluminal propagation on a silicon microchip,” Opt. Lett. 33, 2928–2930 (2008).
[CrossRef] [PubMed]

C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry–Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16, 506–508 (2004).
[CrossRef]

Little, B. E.

Liu, A.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[CrossRef] [PubMed]

Liu, F.

Luo, X.

X. Luo and A. W. Poon, “Electro-optical tunable time delay and advance in silicon microring resonator-based notch filters,” in Conference on Lasers and Electro-Optics, 2010 OSA Technical Digest (Optical Society of America, 2010), paper CTuHH1.

Ma, C. S.

Y. H. Wang, C. S. Ma, X. Yan, and D. M. Zhang, “Analysis for amplifying characteristics of Er3+−Yb3+-co-doped microring resonators,” Opt. Laser Technol. 42, 336–340 (2010).
[CrossRef]

Manipatruni, S.

Martinelli, M.

F. Morichetti, A. Melloni, C. Canavesi, F. Persia, M. Martinelli, and M. Sorel, “Tunable slow-wave optical delay-lines,” in Slow and Fast Light, 2006 OSA Technical Digest (Optical Society of America, 2006), paper MB2.

Melloni, A.

F. Morichetti, A. Melloni, C. Canavesi, F. Persia, M. Martinelli, and M. Sorel, “Tunable slow-wave optical delay-lines,” in Slow and Fast Light, 2006 OSA Technical Digest (Optical Society of America, 2006), paper MB2.

Momeni, B.

Morichetti, F.

F. Morichetti, A. Melloni, C. Canavesi, F. Persia, M. Martinelli, and M. Sorel, “Tunable slow-wave optical delay-lines,” in Slow and Fast Light, 2006 OSA Technical Digest (Optical Society of America, 2006), paper MB2.

Nakamura, S.

Nicolaescu, R.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[CrossRef] [PubMed]

Ozdemir, S. K.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Painter, O.

Panepucci, R. R.

C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry–Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16, 506–508 (2004).
[CrossRef]

Paniccia, M.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[CrossRef] [PubMed]

Persia, F.

F. Morichetti, A. Melloni, C. Canavesi, F. Persia, M. Martinelli, and M. Sorel, “Tunable slow-wave optical delay-lines,” in Slow and Fast Light, 2006 OSA Technical Digest (Optical Society of America, 2006), paper MB2.

Poon, A. W.

X. Luo and A. W. Poon, “Electro-optical tunable time delay and advance in silicon microring resonator-based notch filters,” in Conference on Lasers and Electro-Optics, 2010 OSA Technical Digest (Optical Society of America, 2010), paper CTuHH1.

Qiu, M.

Rabus, D. G.

D. G. Rabus, Z. Bian, and A. Shakouri, “Ring resonator lasers using passive waveguides and integrated semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13, 1249–1256 (2007).
[CrossRef]

Rubin, D.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[CrossRef] [PubMed]

Rubio, D. S.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[CrossRef] [PubMed]

Schmidt, B. S.

C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry–Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16, 506–508 (2004).
[CrossRef]

Shakouri, A.

D. G. Rabus, Z. Bian, and A. Shakouri, “Ring resonator lasers using passive waveguides and integrated semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13, 1249–1256 (2007).
[CrossRef]

Sorel, M.

F. Morichetti, A. Melloni, C. Canavesi, F. Persia, M. Martinelli, and M. Sorel, “Tunable slow-wave optical delay-lines,” in Slow and Fast Light, 2006 OSA Technical Digest (Optical Society of America, 2006), paper MB2.

Su, Y.

Takahashi, K.

Thapliya, R.

Thévenaz, L.

L. Thévenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008).
[CrossRef]

Tian, Y.

Wang, J.

Wang, L. J.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

Wang, T.

Wang, T. J.

Wang, Y. H.

Y. H. Wang, C. S. Ma, X. Yan, and D. M. Zhang, “Analysis for amplifying characteristics of Er3+−Yb3+-co-doped microring resonators,” Opt. Laser Technol. 42, 336–340 (2010).
[CrossRef]

Wu, M. C.

D. Leuenberger, J. Yao, M. M. Lee, and M. C. Wu, “Experimental demonstration of MEMS-tunable slow light in silicon microdisk resonators,” in Slow and Fast Light, 2006 OSA Technical Digest (Optical Society of America, 2006), paper TuC6.

Xiao, Y. F.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Xu, Q.

Yan, X.

Y. H. Wang, C. S. Ma, X. Yan, and D. M. Zhang, “Analysis for amplifying characteristics of Er3+−Yb3+-co-doped microring resonators,” Opt. Laser Technol. 42, 336–340 (2010).
[CrossRef]

Yang, L.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Yao, J.

D. Leuenberger, J. Yao, M. M. Lee, and M. C. Wu, “Experimental demonstration of MEMS-tunable slow light in silicon microdisk resonators,” in Slow and Fast Light, 2006 OSA Technical Digest (Optical Society of America, 2006), paper TuC6.

Ye, T.

Yegnanarayanan, S.

Zhang, D. M.

Y. H. Wang, C. S. Ma, X. Yan, and D. M. Zhang, “Analysis for amplifying characteristics of Er3+−Yb3+-co-doped microring resonators,” Opt. Laser Technol. 42, 336–340 (2010).
[CrossRef]

Zhang, Z.

Zhu, J.

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

D. G. Rabus, Z. Bian, and A. Shakouri, “Ring resonator lasers using passive waveguides and integrated semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13, 1249–1256 (2007).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

C. A. Barrios, V. R. Almeida, R. R. Panepucci, B. S. Schmidt, and M. Lipson, “Compact silicon tunable Fabry–Perot resonator with low power consumption,” IEEE Photon. Technol. Lett. 16, 506–508 (2004).
[CrossRef]

J. Lightwave Technol. (1)

J. Vac. Sci. Technol. B (1)

S. A. Backes and J. R. A. Cleaver, “Microdisk laser structures for mode control and directional emission,” J. Vac. Sci. Technol. B 16, 3817–3820 (1998).
[CrossRef]

Nat. Photonics (3)

J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D. R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[CrossRef]

R. W. Boyd, “Slow light now and then,” Nat. Photonics 2, 497–530 (2002).

L. Thévenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008).
[CrossRef]

Nature (2)

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406, 277–279 (2000).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615–618 (2004).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Laser Technol. (1)

Y. H. Wang, C. S. Ma, X. Yan, and D. M. Zhang, “Analysis for amplifying characteristics of Er3+−Yb3+-co-doped microring resonators,” Opt. Laser Technol. 42, 336–340 (2010).
[CrossRef]

Opt. Lett. (4)

Other (3)

X. Luo and A. W. Poon, “Electro-optical tunable time delay and advance in silicon microring resonator-based notch filters,” in Conference on Lasers and Electro-Optics, 2010 OSA Technical Digest (Optical Society of America, 2010), paper CTuHH1.

F. Morichetti, A. Melloni, C. Canavesi, F. Persia, M. Martinelli, and M. Sorel, “Tunable slow-wave optical delay-lines,” in Slow and Fast Light, 2006 OSA Technical Digest (Optical Society of America, 2006), paper MB2.

D. Leuenberger, J. Yao, M. M. Lee, and M. C. Wu, “Experimental demonstration of MEMS-tunable slow light in silicon microdisk resonators,” in Slow and Fast Light, 2006 OSA Technical Digest (Optical Society of America, 2006), paper TuC6.

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Figures (9)

Fig. 1
Fig. 1

TWMR with dual contra-propagating inputs. The intra-cavity backscattering is enhanced by intentionally introducing random surface defects on the cavity.

Fig. 2
Fig. 2

Variation of | Q cri , T | and | Q cri , R | with δ of the proposed TWMR for Q int < Q ext ( Q int = 2.1 × 10 4 , Q ext = 5.1 × 10 4 ) and Q int > Q ext ( Q int = 5.1 × 10 4 , Q ext = 5.1 × 10 4 ). Only Q mut = 1.5 × 10 4 and A = 1 apply to Figs. 3, 4, while Q mut = 8 × 10 3 and A = 0.1 are used to show the effects of tuning Q mut and A. Mode splitting occurs if | Q cri | Q mut .

Fig. 3
Fig. 3

Evolution in the (i) transmission, (ii) effective phase shift, and (iii) group delay responses for the output light at the through ports (a)–(c) and reflection ports (d)–(f) of the proposed TWMR in the coupling state of Q int < Q ext ( Q int = 2.1 × 10 4 , Q ext = 5.1 × 10 4 ) and Q mut = 1.5 × 10 4 as the phase modulation δ between the dual inputs is varied (with A = 1 ).

Fig. 4
Fig. 4

Evolution in the (i) transmission, (ii) effective phase shift, and (iii) group delay responses at the through ports (a)–(c) and reflection ports (d)–(f) of the proposed TWMR in the coupling state of Q int > Q ext ( Q int = 5.1 × 10 4 , Q ext = 2.1 × 10 4 ) and Q mut = 1.5 × 10 4 as the phase modulation δ between the dual inputs is varied (with A = 1 ).

Fig. 5
Fig. 5

Demonstration of the continuous tunability of the transmission and group delay at λ = λ 0 = 1.55 μ m for (a) Q int < Q ext ( Q int = 2.1 × 10 4 , Q ext = 5.1 × 10 4 ) and (b) Q int > Q ext ( Q int = 5.1 × 10 4 , Q ext = 2.1 × 10 4 ) as the phase modulation δ between the dual inputs of the proposed TWMR is adjusted (with A = 1 ). The plots with thicker (thinner) lines have Q mut = 8 × 10 3 ( Q mut = 1.5 × 10 4 ) .

Fig. 6
Fig. 6

(i) FOMs of group delay and (ii) the FOMs of transmission as functions of Q int for the TWMR with Q ext = 5.1 × 10 4 and λ = λ 0 = 1.55 μ m , with no critical coupling. The values of Q mut are 8.0 × 10 3 for (a), 1.5 × 10 4 for (b), and 2.0 × 10 4 for (c). Region to the left (right) of the vertical line in the plot is the coupling state of Q int < Q ext ( Q int > Q ext ) .

Fig. 7
Fig. 7

(i) FOMs of group delay and (ii) the FOMs of transmission as functions of Q int for the TWMR with Q ext = 5.1 × 10 4 , λ 0 = 1.55 μ m when critical coupling occurs. The values of Q mut are set to 2.2 × 10 4 for (a) and Q mut for (b). The asymptote labeled as CC marks the critical coupling point.

Fig. 8
Fig. 8

FOMs of bandwidth as a function of Q int for the proposed TWMR with Q ext = 5.1 × 10 4 , Q mut = 2.0 × 10 4 , and λ = λ 0 = 1.55 μ m .

Fig. 9
Fig. 9

Demonstration of using the variation in the phase and amplitude modulation (δ and A) between the dual inputs of the proposed TWMR to achieve continuous tunability in the transmission and group delay at a fixed wavelength of 1.55 μ m for the through port with (a) Q int < Q ext ( Q int = 3.2 × 10 4 , Q ext = 6.4 × 10 4 ) and (b) Q int > Q ext ( Q int = 6.4 × 10 4 , Q ext = 3.2 × 10 4 ) and Q mut of 2.2 × 10 4 .

Tables (1)

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Table 1 Comparisons Between the Commonly Used Active Tuning Schemes in TWMR with Our Proposed Dual Inputs Scheme

Equations (4)

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d E ccw d t = ( j Δ ω γ net ) E ccw j μ E cw j κ | S + 1 | ,     d E cw d t = ( j Δ ω γ net ) E cw j μ E ccw j κ | S + 2 | ,
τ T ( Δ ω n ) = S 1 S + 1 = 1 j [ Δ ω n M 2 Q mut ] + 1 2 Q ext + 1 2 Q int Q ext ( j [ Δ ω n + 1 2 Q mut ] + 1 2 Q ext + 1 2 Q int ) ( j [ Δ ω n 1 2 Q mut ] + 1 2 Q ext + 1 2 Q int ) ,
τ R ( Δ ω n ) = S 2 S + 1 = U j [ M Δ ω n 1 2 Q mut ] + M [ 1 2 Q ext + 1 2 Q int ] Q ext ( j [ Δ ω n + 1 2 Q mut ] + 1 2 Q ext + 1 2 Q int ) ( j [ Δ ω n 1 2 Q mut ] + 1 2 Q ext + 1 2 Q int ) ,
Q cri , T = j M Q int ± M 2 Q int 2 Q ext 2 + Q int 2 ( Q ext 2 Q int 2 ) ( Q ext Q int ) 1 ,     Q cri , R = j Q int ± M 2 ( Q int 2 Q ext 2 ) Q int 2 M ( Q ext 2 Q int 2 ) ( Q ext Q int ) 1 .

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