Abstract

Coupled resonators are commonly used to achieve tailored spectral responses and allow novel functionalities in a broad range of applications. The Temporal Coupled-Mode Theory (TCMT) provides a simple and general tool that is widely used to model these devices. Relying on TCMT to model coupled resonators might however be misleading in some circumstances due to the lumped-element nature of the model. In this article, we report an important limitation of TCMT related to the prediction of dark states. Studying a coupled system composed of three microring resonators, we demonstrate that TCMT predicts the existence of a dark state that is in disagreement with experimental observations and with the more general results obtained with the Transfer Matrix Method (TMM) and the Finite-Difference Time-Domain (FDTD) simulations. We identify the limitation in the TCMT model to be related to the mechanism of excitation/decay of the supermodes and we propose a correction that effectively reconciles the model with expected results. Our discussion based on coupled microring resonators can be useful for other electromagnetic resonant systems due to the generality and far-reach of the TCMT formalism.

© 2016 Optical Society of America

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2015 (7)

A. Karalis and J. D. Joannopoulos, “Temporal coupled-mode theory model for resonant near-field thermophotovoltaics,” Appl. Phys. Lett. 107, 141108 (2015).
[Crossref]

Y. Jia, M. Qiu, H. Wu, Y. Cui, S. Fan, and Z. Ruan, “Theory of Half-Space Light Absorption Enhancement for Leaky Mode Resonant Nanowires,” Nano Lett. 15, 5513–5518 (2015).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

M. Zhang, S. Shah, J. Cardenas, and M. Lipson, “Synchronization and Phase Noise Reduction in Micromechanical Oscillator Arrays Coupled through Light,” Phys. Rev. Lett. 115, 163902 (2015).
[Crossref] [PubMed]

J. Wu, P. Cao, T. Pan, Y. Yang, C. Qiu, C. Tremblay, and Y. Su, “Compact on-chip 1×2 wavelength selective switch based on silicon microring resonator with nested pairs of subrings,” Photon. Res. 3, 9–14 (2015).
[Crossref]

Q. Vinckier, F. Duport, A. Smerieri, K. Vandoorne, P. Bienstman, M. Haelterman, and S. Massar, “High-performance photonic reservoir computer based on a coherently driven passive cavity,” Optica 2, 438–446 (2015).
[Crossref]

M. C. M. M. Souza, G. F. M. Rezende, L. A. M. Barea, A. A. G. von Zuben, G. S. Wiederhecker, and N. C. Frateschi, “Spectral engineering with coupled microcavities: active control of resonant mode-splitting,” Opt. Lett. 40, 3332–3335 (2015).
[Crossref] [PubMed]

2014 (5)

M. Mancinelli, M. Borghi, F. Ramiro-Manzano, J. M. Fedeli, and L. Pavesi, “Chaotic dynamics in coupled resonator sequences,” Opt. Express 22, 14505–14516 (2014).
[Crossref] [PubMed]

C. M. Gentry and M. A. Popovic, “Dark state lasers,” Opt. Lett. 39, 4136 (2014).
[Crossref] [PubMed]

H. Yu, M. Pantouvaki, P. Verheyen, G. Lepage, P. Absil, W. Bogaerts, and J. Van Campenhout, “Silicon dual-ring modulator driven by differential signal,” Opt. Lett. 39, 6379–6382 (2014).
[Crossref] [PubMed]

J. Müller, F. Merget, S. Sharif Azadeh, J. Hauck, S. R. García, B. Shen, and J. Witzens, “Optical peaking enhancement in high-speed ring modulators,” Sci. Rep. 4, 6310 (2014).
[Crossref] [PubMed]

M. C. M. M. Souza, L. A. M. Barea, F. Vallini, G. F. M. Rezende, G. S. Wiederhecker, and N. C. Frateschi, “Embedded coupled microrings with high-finesse and close-spaced resonances for optical signal processing,” Opt. Express 22, 20179 (2014).
[Crossref]

2013 (4)

L. A. M. Barea, F. Vallini, P. F. Jarschel, and N. C. Frateschi, “Silicon technology compatible photonic molecules for compact optical signal processing,” Appl. Phys. Lett. 103, 201102 (2013).
[Crossref]

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested Configuration of Silicon Microring Resonator With Multiple Coupling Regimes,” IEEE Phot. Tech. Lett. 25, 580–583 (2013).
[Crossref]

C. Mesaritakis, V. Papataxiarhis, and D. Syvridis, “Micro ring resonators as building blocks for an all-optical high-speed reservoir-computing bit-pattern-recognition system,” J. Opt. Soc. Am. B 30, 3048–3055 (2013).
[Crossref]

R. Haldar, S. Das, and S. K. Varshney, “Theory and Design of Off-Axis Microring Resonators for High-Density On-Chip Photonic Applications,” J. Lightwave Technol. 31, 3976–3985 (2013).
[Crossref]

2012 (4)

S. Sandhu and S. Fan, “Lossless intensity modulation in integrated photonics,” Opt. Express 20, 4280 (2012).
[Crossref] [PubMed]

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From Electromagnetically Induced Transparency to Superscattering with a Single Structure : A Coupled-Mode Theory for Doubly Resonant Structures,” Phys. Rev. Lett. 108, 083902 (2012).
[Crossref]

C. Huang, J. Fan, and L. Zhu, “Dynamic nonlinear thermal optical effects in coupled ring resonators,” AIP Adv. 2, 032131 (2012).
[Crossref]

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[Crossref]

2010 (1)

2009 (3)

H. Benisty, “Dark modes, slow modes, and coupling in multimode systems,” J. Opt. Soc. Am. B 26, 718–724 (2009).
[Crossref]

Q. Xu, “Silicon dual-ring modulator,” Opt. Express 17, 20783–20793 (2009).
[Crossref] [PubMed]

R. E. Hamam, A. Karalis, J. Joannopoulos, and M. Soljačić, “Efficient weakly-radiative wireless energy transfer: An EIT-like approach,” Ann. Phys. 324, 1783–1795 (2009).
[Crossref]

2008 (1)

2007 (1)

2006 (1)

2004 (1)

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]

2000 (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
[Crossref]

1997 (1)

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]

1991 (1)

H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[Crossref]

Absil, P.

Barea, L. A. M.

M. C. M. M. Souza, G. F. M. Rezende, L. A. M. Barea, A. A. G. von Zuben, G. S. Wiederhecker, and N. C. Frateschi, “Spectral engineering with coupled microcavities: active control of resonant mode-splitting,” Opt. Lett. 40, 3332–3335 (2015).
[Crossref] [PubMed]

M. C. M. M. Souza, L. A. M. Barea, F. Vallini, G. F. M. Rezende, G. S. Wiederhecker, and N. C. Frateschi, “Embedded coupled microrings with high-finesse and close-spaced resonances for optical signal processing,” Opt. Express 22, 20179 (2014).
[Crossref]

L. A. M. Barea, F. Vallini, P. F. Jarschel, and N. C. Frateschi, “Silicon technology compatible photonic molecules for compact optical signal processing,” Appl. Phys. Lett. 103, 201102 (2013).
[Crossref]

Benisty, H.

Bienstman, P.

Bogaerts, W.

Borghi, M.

Cao, P.

J. Wu, P. Cao, T. Pan, Y. Yang, C. Qiu, C. Tremblay, and Y. Su, “Compact on-chip 1×2 wavelength selective switch based on silicon microring resonator with nested pairs of subrings,” Photon. Res. 3, 9–14 (2015).
[Crossref]

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested Configuration of Silicon Microring Resonator With Multiple Coupling Regimes,” IEEE Phot. Tech. Lett. 25, 580–583 (2013).
[Crossref]

Cardenas, J.

M. Zhang, S. Shah, J. Cardenas, and M. Lipson, “Synchronization and Phase Noise Reduction in Micromechanical Oscillator Arrays Coupled through Light,” Phys. Rev. Lett. 115, 163902 (2015).
[Crossref] [PubMed]

Catrysse, P. B.

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From Electromagnetically Induced Transparency to Superscattering with a Single Structure : A Coupled-Mode Theory for Doubly Resonant Structures,” Phys. Rev. Lett. 108, 083902 (2012).
[Crossref]

Chak, P.

Chipouline, A.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[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]

Chua, S.-L.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Cui, Y.

Y. Jia, M. Qiu, H. Wu, Y. Cui, S. Fan, and Z. Ruan, “Theory of Half-Space Light Absorption Enhancement for Leaky Mode Resonant Nanowires,” Nano Lett. 15, 5513–5518 (2015).
[Crossref] [PubMed]

Das, S.

Duport, F.

Etrich, C.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[Crossref]

Fan, J.

C. Huang, J. Fan, and L. Zhu, “Dynamic nonlinear thermal optical effects in coupled ring resonators,” AIP Adv. 2, 032131 (2012).
[Crossref]

Fan, S.

Y. Jia, M. Qiu, H. Wu, Y. Cui, S. Fan, and Z. Ruan, “Theory of Half-Space Light Absorption Enhancement for Leaky Mode Resonant Nanowires,” Nano Lett. 15, 5513–5518 (2015).
[Crossref] [PubMed]

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From Electromagnetically Induced Transparency to Superscattering with a Single Structure : A Coupled-Mode Theory for Doubly Resonant Structures,” Phys. Rev. Lett. 108, 083902 (2012).
[Crossref]

S. Sandhu and S. Fan, “Lossless intensity modulation in integrated photonics,” Opt. Express 20, 4280 (2012).
[Crossref] [PubMed]

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]

Fedeli, J. M.

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]

Frateschi, N. C.

M. C. M. M. Souza, G. F. M. Rezende, L. A. M. Barea, A. A. G. von Zuben, G. S. Wiederhecker, and N. C. Frateschi, “Spectral engineering with coupled microcavities: active control of resonant mode-splitting,” Opt. Lett. 40, 3332–3335 (2015).
[Crossref] [PubMed]

M. C. M. M. Souza, L. A. M. Barea, F. Vallini, G. F. M. Rezende, G. S. Wiederhecker, and N. C. Frateschi, “Embedded coupled microrings with high-finesse and close-spaced resonances for optical signal processing,” Opt. Express 22, 20179 (2014).
[Crossref]

L. A. M. Barea, F. Vallini, P. F. Jarschel, and N. C. Frateschi, “Silicon technology compatible photonic molecules for compact optical signal processing,” Appl. Phys. Lett. 103, 201102 (2013).
[Crossref]

García, S. R.

J. Müller, F. Merget, S. Sharif Azadeh, J. Hauck, S. R. García, B. Shen, and J. Witzens, “Optical peaking enhancement in high-speed ring modulators,” Sci. Rep. 4, 6310 (2014).
[Crossref] [PubMed]

Gentry, C. M.

Haelterman, M.

Haldar, R.

Hamam, R. E.

R. E. Hamam, A. Karalis, J. Joannopoulos, and M. Soljačić, “Efficient weakly-radiative wireless energy transfer: An EIT-like approach,” Ann. Phys. 324, 1783–1795 (2009).
[Crossref]

Hauck, J.

J. Müller, F. Merget, S. Sharif Azadeh, J. Hauck, S. R. García, B. Shen, and J. Witzens, “Optical peaking enhancement in high-speed ring modulators,” Sci. Rep. 4, 6310 (2014).
[Crossref] [PubMed]

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 and W. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[Crossref]

Hsu, C. W.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Hu, X.

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested Configuration of Silicon Microring Resonator With Multiple Coupling Regimes,” IEEE Phot. Tech. Lett. 25, 580–583 (2013).
[Crossref]

Huang, C.

C. Huang, J. Fan, and L. Zhu, “Dynamic nonlinear thermal optical effects in coupled ring resonators,” AIP Adv. 2, 032131 (2012).
[Crossref]

Huang, W.

H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[Crossref]

Igarashi, Y.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Janunts, N.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[Crossref]

Jarschel, P. F.

L. A. M. Barea, F. Vallini, P. F. Jarschel, and N. C. Frateschi, “Silicon technology compatible photonic molecules for compact optical signal processing,” Appl. Phys. Lett. 103, 201102 (2013).
[Crossref]

Jia, Y.

Y. Jia, M. Qiu, H. Wu, Y. Cui, S. Fan, and Z. Ruan, “Theory of Half-Space Light Absorption Enhancement for Leaky Mode Resonant Nanowires,” Nano Lett. 15, 5513–5518 (2015).
[Crossref] [PubMed]

Jiang, X.

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested Configuration of Silicon Microring Resonator With Multiple Coupling Regimes,” IEEE Phot. Tech. Lett. 25, 580–583 (2013).
[Crossref]

Joannopoulos, J.

R. E. Hamam, A. Karalis, J. Joannopoulos, and M. Soljačić, “Efficient weakly-radiative wireless energy transfer: An EIT-like approach,” Ann. Phys. 324, 1783–1795 (2009).
[Crossref]

Joannopoulos, J. D.

A. Karalis and J. D. Joannopoulos, “Temporal coupled-mode theory model for resonant near-field thermophotovoltaics,” Appl. Phys. Lett. 107, 141108 (2015).
[Crossref]

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Kaminer, I.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Karalis, A.

A. Karalis and J. D. Joannopoulos, “Temporal coupled-mode theory model for resonant near-field thermophotovoltaics,” Appl. Phys. Lett. 107, 141108 (2015).
[Crossref]

R. E. Hamam, A. Karalis, J. Joannopoulos, and M. Soljačić, “Efficient weakly-radiative wireless energy transfer: An EIT-like approach,” Ann. Phys. 324, 1783–1795 (2009).
[Crossref]

Käsebier, T.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[Crossref]

Kivshar, Y. S.

Klein, A.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[Crossref]

Kley, E.-B.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[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]

Lederer, F.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[Crossref]

Lepage, G.

Li, F.

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested Configuration of Silicon Microring Resonator With Multiple Coupling Regimes,” IEEE Phot. Tech. Lett. 25, 580–583 (2013).
[Crossref]

Liebsch, M.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[Crossref]

Lipson, M.

M. Zhang, S. Shah, J. Cardenas, and M. Lipson, “Synchronization and Phase Noise Reduction in Micromechanical Oscillator Arrays Coupled through Light,” Phys. Rev. Lett. 115, 163902 (2015).
[Crossref] [PubMed]

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]

Lu, L.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Mancinelli, M.

Manolatou, C.

Massar, S.

Merget, F.

J. Müller, F. Merget, S. Sharif Azadeh, J. Hauck, S. R. García, B. Shen, and J. Witzens, “Optical peaking enhancement in high-speed ring modulators,” Sci. Rep. 4, 6310 (2014).
[Crossref] [PubMed]

Mesaritakis, C.

Müller, J.

J. Müller, F. Merget, S. Sharif Azadeh, J. Hauck, S. R. García, B. Shen, and J. Witzens, “Optical peaking enhancement in high-speed ring modulators,” Sci. Rep. 4, 6310 (2014).
[Crossref] [PubMed]

Pan, T.

Pantouvaki, M.

Papataxiarhis, V.

Pavesi, L.

Pertsch, T.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[Crossref]

Pick, A.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Poon, J. K. S.

Popovic, M.

Popovic, M. A.

Qiu, C.

Qiu, M.

Y. Jia, M. Qiu, H. Wu, Y. Cui, S. Fan, and Z. Ruan, “Theory of Half-Space Light Absorption Enhancement for Leaky Mode Resonant Nanowires,” Nano Lett. 15, 5513–5518 (2015).
[Crossref] [PubMed]

Ramiro-Manzano, F.

Rezende, G. F. M.

M. C. M. M. Souza, G. F. M. Rezende, L. A. M. Barea, A. A. G. von Zuben, G. S. Wiederhecker, and N. C. Frateschi, “Spectral engineering with coupled microcavities: active control of resonant mode-splitting,” Opt. Lett. 40, 3332–3335 (2015).
[Crossref] [PubMed]

M. C. M. M. Souza, L. A. M. Barea, F. Vallini, G. F. M. Rezende, G. S. Wiederhecker, and N. C. Frateschi, “Embedded coupled microrings with high-finesse and close-spaced resonances for optical signal processing,” Opt. Express 22, 20179 (2014).
[Crossref]

Ruan, Z.

Y. Jia, M. Qiu, H. Wu, Y. Cui, S. Fan, and Z. Ruan, “Theory of Half-Space Light Absorption Enhancement for Leaky Mode Resonant Nanowires,” Nano Lett. 15, 5513–5518 (2015).
[Crossref] [PubMed]

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From Electromagnetically Induced Transparency to Superscattering with a Single Structure : A Coupled-Mode Theory for Doubly Resonant Structures,” Phys. Rev. Lett. 108, 083902 (2012).
[Crossref]

Sacher, W. D.

Sandhu, S.

Scheuer, J.

Schmidt, C.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[Crossref]

Shah, S.

M. Zhang, S. Shah, J. Cardenas, and M. Lipson, “Synchronization and Phase Noise Reduction in Micromechanical Oscillator Arrays Coupled through Light,” Phys. Rev. Lett. 115, 163902 (2015).
[Crossref] [PubMed]

Sharif Azadeh, S.

J. Müller, F. Merget, S. Sharif Azadeh, J. Hauck, S. R. García, B. Shen, and J. Witzens, “Optical peaking enhancement in high-speed ring modulators,” Sci. Rep. 4, 6310 (2014).
[Crossref] [PubMed]

Shen, B.

J. Müller, F. Merget, S. Sharif Azadeh, J. Hauck, S. R. García, B. Shen, and J. Witzens, “Optical peaking enhancement in high-speed ring modulators,” Sci. Rep. 4, 6310 (2014).
[Crossref] [PubMed]

Smerieri, A.

Soljacic, M.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

R. E. Hamam, A. Karalis, J. Joannopoulos, and M. Soljačić, “Efficient weakly-radiative wireless energy transfer: An EIT-like approach,” Ann. Phys. 324, 1783–1795 (2009).
[Crossref]

Souza, M. C. M. M.

M. C. M. M. Souza, G. F. M. Rezende, L. A. M. Barea, A. A. G. von Zuben, G. S. Wiederhecker, and N. C. Frateschi, “Spectral engineering with coupled microcavities: active control of resonant mode-splitting,” Opt. Lett. 40, 3332–3335 (2015).
[Crossref] [PubMed]

M. C. M. M. Souza, L. A. M. Barea, F. Vallini, G. F. M. Rezende, G. S. Wiederhecker, and N. C. Frateschi, “Embedded coupled microrings with high-finesse and close-spaced resonances for optical signal processing,” Opt. Express 22, 20179 (2014).
[Crossref]

Su, Y.

J. Wu, P. Cao, T. Pan, Y. Yang, C. Qiu, C. Tremblay, and Y. Su, “Compact on-chip 1×2 wavelength selective switch based on silicon microring resonator with nested pairs of subrings,” Photon. Res. 3, 9–14 (2015).
[Crossref]

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested Configuration of Silicon Microring Resonator With Multiple Coupling Regimes,” IEEE Phot. Tech. Lett. 25, 580–583 (2013).
[Crossref]

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]

Sukhorukov, A. A.

Syvridis, D.

Tremblay, C.

Tünnermann, A.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[Crossref]

Vallini, F.

M. C. M. M. Souza, L. A. M. Barea, F. Vallini, G. F. M. Rezende, G. S. Wiederhecker, and N. C. Frateschi, “Embedded coupled microrings with high-finesse and close-spaced resonances for optical signal processing,” Opt. Express 22, 20179 (2014).
[Crossref]

L. A. M. Barea, F. Vallini, P. F. Jarschel, and N. C. Frateschi, “Silicon technology compatible photonic molecules for compact optical signal processing,” Appl. Phys. Lett. 103, 201102 (2013).
[Crossref]

Van Campenhout, J.

Vandoorne, K.

Varshney, S. K.

Verheyen, P.

Verslegers, L.

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From Electromagnetically Induced Transparency to Superscattering with a Single Structure : A Coupled-Mode Theory for Doubly Resonant Structures,” Phys. Rev. Lett. 108, 083902 (2012).
[Crossref]

Vinckier, Q.

von Zuben, A. A. G.

Wang, T.

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested Configuration of Silicon Microring Resonator With Multiple Coupling Regimes,” IEEE Phot. Tech. Lett. 25, 580–583 (2013).
[Crossref]

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]

Watts, M.

Wiederhecker, G. S.

M. C. M. M. Souza, G. F. M. Rezende, L. A. M. Barea, A. A. G. von Zuben, G. S. Wiederhecker, and N. C. Frateschi, “Spectral engineering with coupled microcavities: active control of resonant mode-splitting,” Opt. Lett. 40, 3332–3335 (2015).
[Crossref] [PubMed]

M. C. M. M. Souza, L. A. M. Barea, F. Vallini, G. F. M. Rezende, G. S. Wiederhecker, and N. C. Frateschi, “Embedded coupled microrings with high-finesse and close-spaced resonances for optical signal processing,” Opt. Express 22, 20179 (2014).
[Crossref]

Witzens, J.

J. Müller, F. Merget, S. Sharif Azadeh, J. Hauck, S. R. García, B. Shen, and J. Witzens, “Optical peaking enhancement in high-speed ring modulators,” Sci. Rep. 4, 6310 (2014).
[Crossref] [PubMed]

Wu, H.

Y. Jia, M. Qiu, H. Wu, Y. Cui, S. Fan, and Z. Ruan, “Theory of Half-Space Light Absorption Enhancement for Leaky Mode Resonant Nanowires,” Nano Lett. 15, 5513–5518 (2015).
[Crossref] [PubMed]

Wu, J.

J. Wu, P. Cao, T. Pan, Y. Yang, C. Qiu, C. Tremblay, and Y. Su, “Compact on-chip 1×2 wavelength selective switch based on silicon microring resonator with nested pairs of subrings,” Photon. Res. 3, 9–14 (2015).
[Crossref]

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested Configuration of Silicon Microring Resonator With Multiple Coupling Regimes,” IEEE Phot. Tech. Lett. 25, 580–583 (2013).
[Crossref]

Xu, M.

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested Configuration of Silicon Microring Resonator With Multiple Coupling Regimes,” IEEE Phot. Tech. Lett. 25, 580–583 (2013).
[Crossref]

Xu, Q.

Yang, Y.

Yariv, A.

P. Chak, J. K. S. Poon, and A. Yariv, “Optical bright and dark states in side-coupled resonator structures,” Opt. Lett. 32, 1785 (2007).
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A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
[Crossref]

Yu, H.

Yu, Z.

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From Electromagnetically Induced Transparency to Superscattering with a Single Structure : A Coupled-Mode Theory for Doubly Resonant Structures,” Phys. Rev. Lett. 108, 083902 (2012).
[Crossref]

Zhang, M.

M. Zhang, S. Shah, J. Cardenas, and M. Lipson, “Synchronization and Phase Noise Reduction in Micromechanical Oscillator Arrays Coupled through Light,” Phys. Rev. Lett. 115, 163902 (2015).
[Crossref] [PubMed]

Zhen, B.

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Zhou, L.

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested Configuration of Silicon Microring Resonator With Multiple Coupling Regimes,” IEEE Phot. Tech. Lett. 25, 580–583 (2013).
[Crossref]

Zhu, L.

C. Huang, J. Fan, and L. Zhu, “Dynamic nonlinear thermal optical effects in coupled ring resonators,” AIP Adv. 2, 032131 (2012).
[Crossref]

AIP Adv. (1)

C. Huang, J. Fan, and L. Zhu, “Dynamic nonlinear thermal optical effects in coupled ring resonators,” AIP Adv. 2, 032131 (2012).
[Crossref]

Ann. Phys. (1)

R. E. Hamam, A. Karalis, J. Joannopoulos, and M. Soljačić, “Efficient weakly-radiative wireless energy transfer: An EIT-like approach,” Ann. Phys. 324, 1783–1795 (2009).
[Crossref]

Appl. Phys. Lett. (2)

L. A. M. Barea, F. Vallini, P. F. Jarschel, and N. C. Frateschi, “Silicon technology compatible photonic molecules for compact optical signal processing,” Appl. Phys. Lett. 103, 201102 (2013).
[Crossref]

A. Karalis and J. D. Joannopoulos, “Temporal coupled-mode theory model for resonant near-field thermophotovoltaics,” Appl. Phys. Lett. 107, 141108 (2015).
[Crossref]

Electron. Lett. (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
[Crossref]

IEEE J. Quantum Electron. (1)

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]

IEEE Phot. Tech. Lett. (1)

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested Configuration of Silicon Microring Resonator With Multiple Coupling Regimes,” IEEE Phot. Tech. Lett. 25, 580–583 (2013).
[Crossref]

J. Lightwave Technol. (2)

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]

R. Haldar, S. Das, and S. K. Varshney, “Theory and Design of Off-Axis Microring Resonators for High-Density On-Chip Photonic Applications,” J. Lightwave Technol. 31, 3976–3985 (2013).
[Crossref]

J. Opt. Soc. Am. B (2)

Nano Lett. (1)

Y. Jia, M. Qiu, H. Wu, Y. Cui, S. Fan, and Z. Ruan, “Theory of Half-Space Light Absorption Enhancement for Leaky Mode Resonant Nanowires,” Nano Lett. 15, 5513–5518 (2015).
[Crossref] [PubMed]

Nature (1)

B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (5)

Optica (1)

Photon. Res. (1)

Phys. Rev. A (1)

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[Crossref]

Phys. Rev. Lett. (2)

M. Zhang, S. Shah, J. Cardenas, and M. Lipson, “Synchronization and Phase Noise Reduction in Micromechanical Oscillator Arrays Coupled through Light,” Phys. Rev. Lett. 115, 163902 (2015).
[Crossref] [PubMed]

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From Electromagnetically Induced Transparency to Superscattering with a Single Structure : A Coupled-Mode Theory for Doubly Resonant Structures,” Phys. Rev. Lett. 108, 083902 (2012).
[Crossref]

Proc. IEEE (1)

H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[Crossref]

Sci. Rep. (1)

J. Müller, F. Merget, S. Sharif Azadeh, J. Hauck, S. R. García, B. Shen, and J. Witzens, “Optical peaking enhancement in high-speed ring modulators,” Sci. Rep. 4, 6310 (2014).
[Crossref] [PubMed]

Other (1)

https://optics.synopsys.com/

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Figures (6)

Fig. 1
Fig. 1

Experimental data (a–d), TMM model (e–h) and TCMT model (i–l) of a three-ring resonator system. (a) Fabricated device and (b) transmission spectrum showing a triplet with high-Q quasi-dark state in the center when the three rings are degenerate. (c) IR-micrograph of the scattered light at each resonance. (d) Anti-crossing obtained when the outer ring is detuned using a microheater (H1 in the inset micrograph). An overall red-shift is present due to thermal crosstalk affecting the embedded rings. (e) TMM parameters: sin/sout are the input/output fileds; k1 and k2 are coupling coefficients and ϕi and Pi are the accumulated phase and attenuation in each microring, respectively. (f) TMM triplet similar to the experimental observation and (g) intracavity power spectrum with high power enhancement for the central resonance, in which case light is confined to the embedded rings. |A1|2, |A2|2 and |A3|2 represent the power circulating in the outer ring, first and second embedded rings, respectively. The blue and green curves closely overlap. (h) TMM anti-crossing showing the evolution of the supermode resonances in the absence of thermal crosstalk. (i) TCMT model (parameters described in the text). In contrast to the experimental data and TMM, no central resonance is observed in the (j) transmission spectrum and (k) intracavity power spectrum. (l) TCMT anti-crossing obtained from the transmission spectrum and from the eigenvalues of Ω (dashed-blue lines). The central mode is predicted by the eigenvalues but not excited, constituting a dark state in the TCMT model. Inset: spatial distribution of each supermode at degeneracy, representing the eigenvectors of Ω.

Fig. 2
Fig. 2

2D-FDTD simulations. (a,b) Transmission spectrum and (c,d) steady-state electric field amplitude of the supermodes of the three-ring device at degeneracy in two different configurations. (a,c) When the embedded rings coupled to the outer ring at different positions a weak field circulates in the outer ring allowing the excitation of the quasi-dark state (ii). (b,d) When the embedded rings couple to the outer ring at the same position the destructive interference in the outer ring prevents the excitation of supermode (ii), originating a dark state.

Fig. 3
Fig. 3

Comparison between modified-TCMT (m-TCMT) and TMM. (a) Transmission spectrum and (b,c) intracavity power spectrum calculated with the parameters used in Fig. 1. Insets: detail of the central peaks. (d) Average intracavity power for the quasi-dark state calculated with TMM and m-TCMT. (e) Resonance splitting for the triplet and (f) extinction ration, (g) linewidth and (h) average intracavity power for the quasi-dark state calculated for various coupling coefficients. The m-TCMT calculations agree with TMM for a wide range of coupling strengths, while in the standard TCMT curves (f–h) would vanish.

Fig. 4
Fig. 4

Comparison between m-TCMT and 2D-FDTD simulations. (a) Transmission spectrum and (b–e) transient intracavity power evolution. The FDTD results are presented in (b) for the lateral resonances and in (d) for the quasi-dark state, while the corresponding m-TCMT solutions are presented in (c) and (e). The m-TCMT model reproduces the transient evolution for the lateral modes including the fast oscillations presented in the FDTD simulation. For the quasi-dark state it describes the average power circulating in the resonator.

Fig. 5
Fig. 5

(a) Schematics of the three-ring resonator with parameters used in the TMM model. (b) TMM fitting of the triplet and adjacent outer ring resonances to extract the group index for wavelengths around 1580 nm.

Fig. 6
Fig. 6

Representation of a general resonant system with N resonant modes (ai) and M input/output ports ( s i i n / s i o u t ).

Equations (31)

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d a d t = ( j Ω Γ ) a + K T s i n
s o u t = C s i n + K a
Ω = ( ω 1 κ κ κ ω 0 0 κ 0 ω 0 ) ;
K = ( j μ 1 0 0 )
ω 1 b = ω 0 + 2 κ , b 1 = ( 1 2 1 2 1 2 ) T ω 2 b = ω 0 , b 2 = ( 0 1 2 1 2 ) T ω 3 b = ω 0 2 κ , b 3 = ( 1 2 1 2 1 2 ) T .
K b = j μ 1 2 ( 1 0 1 )
μ 1 = k 1 v g L 1
K b ( K 1 b K 2 b K 3 b ) T = ( j 1 2 ( 2 μ 1 μ 2 μ 3 ) j ( μ 2 μ 3 ) 2 j 1 2 ( 2 μ 1 + μ 2 + μ 3 ) ) T
K b = j 1 2 ( μ 1 2 μ 2 μ 1 ) .
2 μ 2 2 | b 2 ( t ) | 2 = k b 2 | B 2 ( t ) | 2 .
| B 2 ( t ) | 2 = | b 2 ( t ) | 2 v g 2 L 2
μ 2 = k b 2 v g L 2
| B 2 | 2 = k 2 2 | A 1 π | 2 .
| A 1 | 2 = | j k 1 P 1 e j ϕ 1 1 t 1 P 1 e j ϕ 1 | 2 | s i n | 2 ϕ 1 π k 1 1 , P 1 1 | A 1 π | 2 = k 1 2 4 | s i n | 2
μ 1 = k 1 v g L 1 , μ 2 = k 1 k 2 4 v g L 2 .
Γ p o r t = ( μ 1 2 2 μ 1 μ 2 2 μ 1 μ 2 2 μ 1 μ 2 2 μ 2 2 2 μ 2 2 2 μ 1 μ 2 2 μ 2 2 2 μ 3 2 2 )
| A 3 | 2 | A 2 | 2 = P 1 | χ 2 | 2
T = | t 1 P 1 e j ϕ 1 χ 2 χ 3 1 t 1 P 1 e j ϕ 1 χ 2 χ 3 | 2
A 1 = | j k 1 P 1 e j ϕ 1 χ 2 χ 3 1 t 1 P 1 e j ϕ 1 χ 2 χ 3 | 2
A 2 = | j k 1 j k 2 P 2 e j ϕ 2 1 t 2 P 2 e j ϕ 2 ( P 1 e j ϕ 1 ) 1 / 4 1 t 1 P 1 e j ϕ 1 χ 2 χ 3 | 2
A 3 = | j k 1 j k 3 P 3 e j ϕ 3 1 t 3 P 3 e j ϕ 3 ( P 1 e j ϕ 1 ) 3 / 4 χ 2 1 t 1 P 1 e j ϕ 1 χ 2 χ 3 | 2
χ i = t i P i e j ϕ i 1 t i P i e j ϕ i .
n e f f ( λ ) n g ( λ 0 ) + λ n e f f ( λ 0 ) / λ
d a d t = ( j Ω Γ ) a + K T s i n
s o u t = C s i n + K a .
| A i ( t ) | 2 = | a i ( t ) | 2 v g i L i
C K = K
K K = 2 Γ p o r t
γ i = v g i α i 2 .
μ q i = k q i v g i L i
κ i j = k i j v g i v g j L i L j .

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