Abstract

The temporal coupled-mode theory (TCMT) for a ring–bus–ring Mach–Zehnder interferometer device is developed by taking energy conservation into account. The intercavity interaction in the device is facilitated via a tricoupler, which makes the decay of modes quantitatively different from that in other existing resonator schemes. The TCMT is related to the transfer matrix formalism with energy conservation and the Q factor, and it predicts results in good agreement with the experimental results. The mode analysis from the TCMT is quite illustrative because it can mimic the transparency as an electromagnetically induced transparency expression. The analysis of the tricoupler is applicable for analyzing the transparent resonance in two other similar configurations.

© 2012 Optical Society of America

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References

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2011

2010

Q. Li, T. Wang, Y. Su, M. Yan, and M. Qiu, “Coupled mode theory analysis of mode-splitting in coupled cavity system,” Opt. Express 18, 8367–8382 (2010).
[CrossRef]

S. Kocaman, X. Yang, J. F. McMillan, M. B. Yu, D. L. Kwong, and C. W. Wong, “Observations of temporal group delays in slow-light multiple coupled photonic crystal cavities,” Appl. Phys. Lett. 96, 221111 (2010).
[CrossRef]

2009

2008

K. Preston, P. Dong, B. Schmidt, and M. Lipson, “High-speed all-optical modulation using polycrystalline silicon microring resonators,” Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

2006

2004

L. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, “Tunable delay line with interacting whispering-gallery-mode resonators,” Opt. Lett. 29, 626–628 (2004).
[CrossRef]

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[CrossRef]

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[CrossRef]

2003

1997

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

1995

Y. Li and M. Xiao, “Observation of quantum interference between dressed states in an electromagnetically induced transparency,” Phys. Rev. A 51, 4959–4962 (1995).
[CrossRef]

1972

Boyd, R. W.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Bozhevolnyi, S. I.

Chang, H.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Chu, S. T.

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

Darmawan, S.

Dong, P.

K. Preston, P. Dong, B. Schmidt, and M. Lipson, “High-speed all-optical modulation using polycrystalline silicon microring resonators,” Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

Fan, S.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[CrossRef]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[CrossRef]

Foresi, J.

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

Fuller, K. A.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Han, Z.

Haus, H. A.

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

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

Ilchenko, V. S.

Joannopoulos, J. D.

Kocaman, S.

S. Kocaman, X. Yang, J. F. McMillan, M. B. Yu, D. L. Kwong, and C. W. Wong, “Observations of temporal group delays in slow-light multiple coupled photonic crystal cavities,” Appl. Phys. Lett. 96, 221111 (2010).
[CrossRef]

Kwong, D. L.

S. Kocaman, X. Yang, J. F. McMillan, M. B. Yu, D. L. Kwong, and C. W. Wong, “Observations of temporal group delays in slow-light multiple coupled photonic crystal cavities,” Appl. Phys. Lett. 96, 221111 (2010).
[CrossRef]

Laine, J.-P.

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

Li, Q.

Li, Y.

Y. Li and M. Xiao, “Observation of quantum interference between dressed states in an electromagnetically induced transparency,” Phys. Rev. A 51, 4959–4962 (1995).
[CrossRef]

Lipson, M.

K. Preston, P. Dong, B. Schmidt, and M. Lipson, “High-speed all-optical modulation using polycrystalline silicon microring resonators,” Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. 24, 1433–1439 (2006).
[CrossRef]

Little, B. E.

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

Maleki, L.

Manolatou, C.

Matsko, A. B.

McMillan, J. F.

S. Kocaman, X. Yang, J. F. McMillan, M. B. Yu, D. L. Kwong, and C. W. Wong, “Observations of temporal group delays in slow-light multiple coupled photonic crystal cavities,” Appl. Phys. Lett. 96, 221111 (2010).
[CrossRef]

Mei, T.

Preston, K.

K. Preston, P. Dong, B. Schmidt, and M. Lipson, “High-speed all-optical modulation using polycrystalline silicon microring resonators,” Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

Qiu, M.

Rosenberger, A. T.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Savchenkov, A. A.

Schmidt, B.

K. Preston, P. Dong, B. Schmidt, and M. Lipson, “High-speed all-optical modulation using polycrystalline silicon microring resonators,” Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

Smith, D. D.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Snyder, A. W.

Su, Y.

Suh, W.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[CrossRef]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[CrossRef]

Tobing, L. Y. M.

Wang, T.

Wang, Z.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[CrossRef]

Wong, C. W.

S. Kocaman, X. Yang, J. F. McMillan, M. B. Yu, D. L. Kwong, and C. W. Wong, “Observations of temporal group delays in slow-light multiple coupled photonic crystal cavities,” Appl. Phys. Lett. 96, 221111 (2010).
[CrossRef]

Xiao, M.

Y. Li and M. Xiao, “Observation of quantum interference between dressed states in an electromagnetically induced transparency,” Phys. Rev. A 51, 4959–4962 (1995).
[CrossRef]

Xu, Q.

Yan, M.

Yang, X.

S. Kocaman, X. Yang, J. F. McMillan, M. B. Yu, D. L. Kwong, and C. W. Wong, “Observations of temporal group delays in slow-light multiple coupled photonic crystal cavities,” Appl. Phys. Lett. 96, 221111 (2010).
[CrossRef]

Yu, M. B.

S. Kocaman, X. Yang, J. F. McMillan, M. B. Yu, D. L. Kwong, and C. W. Wong, “Observations of temporal group delays in slow-light multiple coupled photonic crystal cavities,” Appl. Phys. Lett. 96, 221111 (2010).
[CrossRef]

Zhang, D. H.

Zhang, Y.

Appl. Phys. Lett.

S. Kocaman, X. Yang, J. F. McMillan, M. B. Yu, D. L. Kwong, and C. W. Wong, “Observations of temporal group delays in slow-light multiple coupled photonic crystal cavities,” Appl. Phys. Lett. 96, 221111 (2010).
[CrossRef]

K. Preston, P. Dong, B. Schmidt, and M. Lipson, “High-speed all-optical modulation using polycrystalline silicon microring resonators,” Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

IEEE J. Quantum Electron.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[CrossRef]

J. Lightwave Technol.

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

C. Manolatou and M. Lipson, “All-optical silicon modulators based on carrier injection by two-photon absorption,” J. Lightwave Technol. 24, 1433–1439 (2006).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Rev. A

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Y. Li and M. Xiao, “Observation of quantum interference between dressed states in an electromagnetically induced transparency,” Phys. Rev. A 51, 4959–4962 (1995).
[CrossRef]

Other

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

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Figures (3)

Fig. 1.
Fig. 1.

Schematic of the RBR-MZI. The insets present the two characteristics of mode coupling in the RBR. The dashed box indicates the tricoupler.

Fig. 2.
Fig. 2.

Transmissions of an RBR-MZI system. The transparent window occurs at ω=(ω1+ω2)/2 between ω1 and ω2. For the solid curve ω2ω1=1.2GHz and for the dashed curve ω2ω1=2.4GHz, where Qi/Qe50 is applied.

Fig. 3.
Fig. 3.

(a) Image of a fabricated RBR-MZI. (b) Measured (thick-faded curve), TMF (solid curve) and TCMT (dashed curve) transmissions of balanced arm length RBR-MZI. (c) CRIT with different linewidths.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

du/dt=(iΩΓiΓ)u+iHTsin,sout=sin+iHu,
u=[u1u2],Ω=[ω100ω2],Γi=[τi1100τi21],H=[η1η2]T,Γ=12[η12η1η2η1η2η22],
tMZI=i(iΔω1+τi11)(iΔω2+τi21)(iΔω1+τi11+η12/2)(iΔω2+τi21+η22/2)η12η22/4,
Tmin|tMZI|24τi12η14,Tmax[(ω2ω1)/2]4[η12τi11(ω2ω1)2/4]2.
TMZI=1(2η12)(2η12)(Δω¯)2(2η12)2(Δω¯)2+4[(Δω¯)2(ω2ω1)2/4]2,
Π(ω2ω1)2+4τi122η12+(ω2ω1).
s+sin=(iΔω1+τi11)(iΔω2+τi21)(iΔω1+τi11+η12)(iΔω2+τi21+η22)η12η22,

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