Abstract

New types of dual microring resonators coupled via 3×3 couplers are proposed. By employing the transfer matrix method, a model for these four types is developed and analytical expressions for characterizing their transmissions are derived. The first two types show a coupled-resonator-induced-transparency-like transmission spectrum at the through port. The third type holds the same transmission spectrum, while the last type simultaneously realizes a first-order and a second-order filters at two drop ports. The effects of coupling coefficients on their transmission spectra are investigated in more detail. Moreover, the effects of loss are also discussed. These proposed types can be found applications in fields such as sensors and filters.

© 2007 Optical Society of America

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  1. S. T. Chu, B. E. Little, W. Pan, T. Kaneko, and Y. Kokubun, "Second-order filter response from parallel coupled glass microring resonators," IEEE Photon. Technol. Lett. 11, 1426-1428 (1999).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2007 (2)

2006 (7)

2005 (5)

Y. M. Landobasa, S. Darmawan, and M.-K. Chin, "Matrix analysis of 2-d microresonator lattice optical filters," IEEE J. Quantum Electron. 41, 1410-1418 (2005).
[CrossRef]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Phys. Rev. A 71, 043804 (2005).
[CrossRef]

S. J. Emelett, and R. A. Soref, "Synthesis of dual-microring-resonator cross-connect filters," Opt. Express 13, 4439-4456 (2005).
[CrossRef] [PubMed]

S. Darmawan, Y. M. Landobasa, and M. K. Chin, "Phase engineering for ring enhanced Mach-Zehnder interferometers," Opt. Express 13, 4580-4588 (2005).
[CrossRef] [PubMed]

2004 (5)

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters the WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

D. S. David, C. Hongrok, A. F. Kirk, A. T. Rosenberger, and W. B. Robert, "Coupled-resonator-induced transparency," Phys. Rev. A 69, 063804 (2004).
[CrossRef]

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, "A fast low-power optical memory based on coupled micro-ring lasers," Nature 432, 206-209 (2004).
[CrossRef] [PubMed]

L. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, "Tunable delay line with interacting whispering-gallery-mode resonators," Optics Letters 29, 626-628 (2004).
[CrossRef] [PubMed]

2003 (2)

C.-Y. Chao, and L. J. Guo, "Biochemical sensors based on polymer microrings with sharp asymmetrical resonance," Applied Physics Letters 83, 1527-1529 (2003).
[CrossRef]

J. Yang, Q. Zhou, F. Zhao, X. Jiang, B. Howley, M. Wang, and R. T. Chen, "Characteristics of optical bandpass filters employing series-cascaded double-ring resonators," Opt. Commun. 228, 91-98 (2003).
[CrossRef]

2000 (1)

A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
[CrossRef]

1999 (1)

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, and Y. Kokubun, "Second-order filter response from parallel coupled glass microring resonators," IEEE Photon. Technol. Lett. 11, 1426-1428 (1999).
[CrossRef]

1994 (1)

R. W. C. Vance, and J. D. Love, "Design procedures for passive planar coupled waveguide devices," IEE Proc. Optoelectron. 141, 231-241 (1994).
[CrossRef]

Absil, P. P.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters the WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

Barwicz, T.

Binsma, H.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, "A fast low-power optical memory based on coupled micro-ring lasers," Nature 432, 206-209 (2004).
[CrossRef] [PubMed]

Cai, X.

Chao, C.-Y.

C.-Y. Chao, and L. J. Guo, "Biochemical sensors based on polymer microrings with sharp asymmetrical resonance," Applied Physics Letters 83, 1527-1529 (2003).
[CrossRef]

Chen, R. T.

J. Yang, Q. Zhou, F. Zhao, X. Jiang, B. Howley, M. Wang, and R. T. Chen, "Characteristics of optical bandpass filters employing series-cascaded double-ring resonators," Opt. Commun. 228, 91-98 (2003).
[CrossRef]

Chin, M. K.

Chin, M.-K.

Y. M. Landobasa, S. Darmawan, and M.-K. Chin, "Matrix analysis of 2-d microresonator lattice optical filters," IEEE J. Quantum Electron. 41, 1410-1418 (2005).
[CrossRef]

Chu, S. T.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters the WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, and Y. Kokubun, "Second-order filter response from parallel coupled glass microring resonators," IEEE Photon. Technol. Lett. 11, 1426-1428 (1999).
[CrossRef]

Darmawan, S.

David, D. S.

D. S. David, C. Hongrok, A. F. Kirk, A. T. Rosenberger, and W. B. Robert, "Coupled-resonator-induced transparency," Phys. Rev. A 69, 063804 (2004).
[CrossRef]

De Leonardis, F.

De Vries, T.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, "A fast low-power optical memory based on coupled micro-ring lasers," Nature 432, 206-209 (2004).
[CrossRef] [PubMed]

Den Besten, J. H.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, "A fast low-power optical memory based on coupled micro-ring lasers," Nature 432, 206-209 (2004).
[CrossRef] [PubMed]

Dorren, H. J. S.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, "A fast low-power optical memory based on coupled micro-ring lasers," Nature 432, 206-209 (2004).
[CrossRef] [PubMed]

Dumeige, Y.

Emelett, S. J.

Fan, S.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, "Experimental Realization of an On-Chip All-Optical Analogue to Electromagnetically Induced Transparency," Phys. Rev. Lett. 96, 123901-123904 (2006).
[CrossRef] [PubMed]

Farca, G.

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Phys. Rev. A 71, 043804 (2005).
[CrossRef]

Feron, P.

Fujimoto, J. G.

Ghisa, L.

Gill, D.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters the WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Guo, L. J.

C.-Y. Chao, and L. J. Guo, "Biochemical sensors based on polymer microrings with sharp asymmetrical resonance," Applied Physics Letters 83, 1527-1529 (2003).
[CrossRef]

Hill, M. T.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, "A fast low-power optical memory based on coupled micro-ring lasers," Nature 432, 206-209 (2004).
[CrossRef] [PubMed]

Hongrok, C.

D. S. David, C. Hongrok, A. F. Kirk, A. T. Rosenberger, and W. B. Robert, "Coupled-resonator-induced transparency," Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Howley, B.

J. Yang, Q. Zhou, F. Zhao, X. Jiang, B. Howley, M. Wang, and R. T. Chen, "Characteristics of optical bandpass filters employing series-cascaded double-ring resonators," Opt. Commun. 228, 91-98 (2003).
[CrossRef]

Hryniewicz, J. V.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters the WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Huang, D.

Ilchenko, V. S.

L. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, "Tunable delay line with interacting whispering-gallery-mode resonators," Optics Letters 29, 626-628 (2004).
[CrossRef] [PubMed]

Ippen, E. P.

Jiang, X.

J. Yang, Q. Zhou, F. Zhao, X. Jiang, B. Howley, M. Wang, and R. T. Chen, "Characteristics of optical bandpass filters employing series-cascaded double-ring resonators," Opt. Commun. 228, 91-98 (2003).
[CrossRef]

Johnson, F. G.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters the WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Kaneko, T.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, and Y. Kokubun, "Second-order filter response from parallel coupled glass microring resonators," IEEE Photon. Technol. Lett. 11, 1426-1428 (1999).
[CrossRef]

Kartner, F. X.

Khoe, G.-D.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, "A fast low-power optical memory based on coupled micro-ring lasers," Nature 432, 206-209 (2004).
[CrossRef] [PubMed]

King, O.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters the WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Kirk, A. F.

D. S. David, C. Hongrok, A. F. Kirk, A. T. Rosenberger, and W. B. Robert, "Coupled-resonator-induced transparency," Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Kokubun, Y.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, and Y. Kokubun, "Second-order filter response from parallel coupled glass microring resonators," IEEE Photon. Technol. Lett. 11, 1426-1428 (1999).
[CrossRef]

Landobasa, Y. M.

Leijtens, X. J. M.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, "A fast low-power optical memory based on coupled micro-ring lasers," Nature 432, 206-209 (2004).
[CrossRef] [PubMed]

Lipson, M.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, "Experimental Realization of an On-Chip All-Optical Analogue to Electromagnetically Induced Transparency," Phys. Rev. Lett. 96, 123901-123904 (2006).
[CrossRef] [PubMed]

Q. Xu, J. Shakya, and M. Lipson, "Direct measurement of tunable optical delays on chip analogue to electromagnetically induced transparency," Opt. Express 14, 6463-6468 (2006).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

Little, B. E.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters the WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, and Y. Kokubun, "Second-order filter response from parallel coupled glass microring resonators," IEEE Photon. Technol. Lett. 11, 1426-1428 (1999).
[CrossRef]

Love, J. D.

R. W. C. Vance, and J. D. Love, "Design procedures for passive planar coupled waveguide devices," IEE Proc. Optoelectron. 141, 231-241 (1994).
[CrossRef]

Maleki, L.

L. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, "Tunable delay line with interacting whispering-gallery-mode resonators," Optics Letters 29, 626-628 (2004).
[CrossRef] [PubMed]

Manolatou, C.

Mashanovich, G. Z.

Matsko, A. B.

L. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, "Tunable delay line with interacting whispering-gallery-mode resonators," Optics Letters 29, 626-628 (2004).
[CrossRef] [PubMed]

Nasu, Y.

Naweed, A.

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Phys. Rev. A 71, 043804 (2005).
[CrossRef]

Oel, Y.-S.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, "A fast low-power optical memory based on coupled micro-ring lasers," Nature 432, 206-209 (2004).
[CrossRef] [PubMed]

Pan, W.

S. T. Chu, B. E. Little, W. Pan, T. Kaneko, and Y. Kokubun, "Second-order filter response from parallel coupled glass microring resonators," IEEE Photon. Technol. Lett. 11, 1426-1428 (1999).
[CrossRef]

Panepucci, R. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

Passaro, V. M. N.

Popovic, M. A.

Povinelli, M. L.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, "Experimental Realization of an On-Chip All-Optical Analogue to Electromagnetically Induced Transparency," Phys. Rev. Lett. 96, 123901-123904 (2006).
[CrossRef] [PubMed]

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Rakich, P. T.

Robert, W. B.

D. S. David, C. Hongrok, A. F. Kirk, A. T. Rosenberger, and W. B. Robert, "Coupled-resonator-induced transparency," Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Rosenberger, A. T.

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Phys. Rev. A 71, 043804 (2005).
[CrossRef]

D. S. David, C. Hongrok, A. F. Kirk, A. T. Rosenberger, and W. B. Robert, "Coupled-resonator-induced transparency," Phys. Rev. A 69, 063804 (2004).
[CrossRef]

Sandhu, S.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, "Experimental Realization of an On-Chip All-Optical Analogue to Electromagnetically Induced Transparency," Phys. Rev. Lett. 96, 123901-123904 (2006).
[CrossRef] [PubMed]

Savchenkov, A. A.

L. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, "Tunable delay line with interacting whispering-gallery-mode resonators," Optics Letters 29, 626-628 (2004).
[CrossRef] [PubMed]

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Seiferth, F.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters the WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Shakya, J.

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, "Experimental Realization of an On-Chip All-Optical Analogue to Electromagnetically Induced Transparency," Phys. Rev. Lett. 96, 123901-123904 (2006).
[CrossRef] [PubMed]

Q. Xu, J. Shakya, and M. Lipson, "Direct measurement of tunable optical delays on chip analogue to electromagnetically induced transparency," Opt. Express 14, 6463-6468 (2006).
[CrossRef] [PubMed]

Sharma, V.

Shopova, S. I.

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Phys. Rev. A 71, 043804 (2005).
[CrossRef]

Smalbrugge, B.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, "A fast low-power optical memory based on coupled micro-ring lasers," Nature 432, 206-209 (2004).
[CrossRef] [PubMed]

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M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, "A fast low-power optical memory based on coupled micro-ring lasers," Nature 432, 206-209 (2004).
[CrossRef] [PubMed]

Smith, H. I.

Socci, L.

Soref, R. A.

Suzuki, K.

Trakalo, M.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters the WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

Van, V.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters the WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
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R. W. C. Vance, and J. D. Love, "Design procedures for passive planar coupled waveguide devices," IEE Proc. Optoelectron. 141, 231-241 (1994).
[CrossRef]

Wang, M.

J. Yang, Q. Zhou, F. Zhao, X. Jiang, B. Howley, M. Wang, and R. T. Chen, "Characteristics of optical bandpass filters employing series-cascaded double-ring resonators," Opt. Commun. 228, 91-98 (2003).
[CrossRef]

Watts, M. R.

Xu, Q.

Q. Xu, J. Shakya, and M. Lipson, "Direct measurement of tunable optical delays on chip analogue to electromagnetically induced transparency," Opt. Express 14, 6463-6468 (2006).
[CrossRef] [PubMed]

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, "Experimental Realization of an On-Chip All-Optical Analogue to Electromagnetically Induced Transparency," Phys. Rev. Lett. 96, 123901-123904 (2006).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Yang, J.

J. Yang, Q. Zhou, F. Zhao, X. Jiang, B. Howley, M. Wang, and R. T. Chen, "Characteristics of optical bandpass filters employing series-cascaded double-ring resonators," Opt. Commun. 228, 91-98 (2003).
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A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
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J. Yang, Q. Zhou, F. Zhao, X. Jiang, B. Howley, M. Wang, and R. T. Chen, "Characteristics of optical bandpass filters employing series-cascaded double-ring resonators," Opt. Commun. 228, 91-98 (2003).
[CrossRef]

Zhou, Q.

J. Yang, Q. Zhou, F. Zhao, X. Jiang, B. Howley, M. Wang, and R. T. Chen, "Characteristics of optical bandpass filters employing series-cascaded double-ring resonators," Opt. Commun. 228, 91-98 (2003).
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Applied Physics Letters (1)

C.-Y. Chao, and L. J. Guo, "Biochemical sensors based on polymer microrings with sharp asymmetrical resonance," Applied Physics Letters 83, 1527-1529 (2003).
[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]

IEE Proc. Optoelectron. (1)

R. W. C. Vance, and J. D. Love, "Design procedures for passive planar coupled waveguide devices," IEE Proc. Optoelectron. 141, 231-241 (1994).
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Y. M. Landobasa, S. Darmawan, and M.-K. Chin, "Matrix analysis of 2-d microresonator lattice optical filters," IEEE J. Quantum Electron. 41, 1410-1418 (2005).
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V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, "A fast low-power optical memory based on coupled micro-ring lasers," Nature 432, 206-209 (2004).
[CrossRef] [PubMed]

Opt. Commun. (1)

J. Yang, Q. Zhou, F. Zhao, X. Jiang, B. Howley, M. Wang, and R. T. Chen, "Characteristics of optical bandpass filters employing series-cascaded double-ring resonators," Opt. Commun. 228, 91-98 (2003).
[CrossRef]

Opt. Express (8)

Opt. Lett. (2)

Optics Letters (1)

L. Maleki, A. B. Matsko, A. A. Savchenkov, and V. S. Ilchenko, "Tunable delay line with interacting whispering-gallery-mode resonators," Optics Letters 29, 626-628 (2004).
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Phys. Rev. A (2)

D. S. David, C. Hongrok, A. F. Kirk, A. T. Rosenberger, and W. B. Robert, "Coupled-resonator-induced transparency," Phys. Rev. A 69, 063804 (2004).
[CrossRef]

A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Phys. Rev. A 71, 043804 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, "Experimental Realization of an On-Chip All-Optical Analogue to Electromagnetically Induced Transparency," Phys. Rev. Lett. 96, 123901-123904 (2006).
[CrossRef] [PubMed]

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Figures (12)

Fig. 1.
Fig. 1.

(a).All-pass and (b)add-drop microring resonators (c) 2×2 coupler (d) 3×3 coupler. The dashed boxes are coupling regions correspond to couplers. Arrows represent optical transmission directions

Fig. 2.
Fig. 2.

Schematic diagram and model of 3×3 coupler based dual microring resonators (a) combination of the all-pass and the add-drop types with the input at the center port (type I). (b) combination of the all-pass and the add-drop types with the input at the upper port (type II). (c) combination of two add-drop types with the input at the center port (type III). (d) combination of two add-drop types with the input at the upper port (type IV). (e) Transfer matrix model for these types. The dashed line exists for type III and IV corresponding to the lower bus waveguide. The dashed boxes correspond to the 2×2 and 3×3 couplers. Arrows represent optical transmission directions.

Fig. 3.
Fig. 3.

Transmission spectra of (a) type I (b) type II (c) type III (d) type IV; Phase responses of (e) type I (f) type II (g) type III (h)type IV. Parameters are chosen as τ = 1 ,k 0 = 0.5 and k = 0.4

Fig. 4.
Fig. 4.

Effects of coupling coefficients on θ + (a) Contour plot of θ + /k as functions of k 0 and k . The dashed lines (i)–(iii) correspond to k 0 = 0.2, 0.5 and 0.8, respectively. The solid line (iv) corresponds to k 0=k . (b) θ + as a function of k when k 0 = 0.2, 0.5, 0.8 and k , respectively. The inset shows θ + when k 0=k and k near 0.995 .

Fig. 5.
Fig. 5.

Effects of coupling coefficients on D 1 max . (a) Contour plot of D 1 max as functions of k 0 and k . The dashed lines (i)–(iii) correspond to k 0 = 0.2, 0.5 and 0.8, respectively. The solid line (iv) corresponds to k 0=k . (b) D 1 max as a function of k when k 0 = 0.2, 0.5, 0.8 and k , respectively.

Fig. 6.
Fig. 6.

(a)-(c) Contour plots of D 1 as functions of k when k 0 equals to 0.2, 0.5 and 0.8, respectively. The dashed lines (i)–(iii) correspond to k = 0.2, 0.5 and 0.8, respectively, when k 0 =0.5 . (d) D 1 under k= 0.2, 0.5 and 0.8, respectively, when k 0 =0.5 .(e) D 1 under k = k 0 equaling to 0.2, 0.5, 0.8, 0.9, 0.95 and 0.995, respectively.

Fig. 7.
Fig. 7.

(a). Contour plot of 1 /π as functions of k 0 and k . (b) 1 as a function of k when k 0 = 0.2, 0.5, 0.8 and k , respectively. (c) Contour plot of 2/π as functions of k 0 and k . (d) 2 as a function of k when k 0 = 0.2, 0.5, 0.8 and k , respectively. The inset shows 2 when 0.7 < k < 0.9 . The dashed lines (i)-(iii) correspond to k 0 = 0.2, 0.5 and 0.8, respectively. The solid line (iv) corresponds to k 0=k .

Fig. 8.
Fig. 8.

(a) Contour plot of 1 max as functions of k 0 and k . (b) 1 max as a function of k when k 0 = 0.2, 0.5, 0.8 and k , respectively. (c) Contour plot of 2 max as functions of k 0 and k . (d) 2 max as a function of k 0 when k 0 = 0.2, 0.5, 0.8 and k , respectively. The dashed lines (i)-(iii) correspond to k 0 = 0.2, 0.5 and 0.8, respectively. The solid line (iv) corresponds to k 0=k .

Fig. 9.
Fig. 9.

(a)-(c) Contour plots of 1 as functions of k when k 0 equals to 0.2, 0.5 and 0.8, respectively. The dashed lines (i)-(iii) correspond to k = 0.2, 0.5 and 0.8, respectively. (d) 1 under k equaling to 0.2, 0.5 and 0.8, respectively, when k 0=0.5 . (e) 1 under k = k 0 equaling to 0.2, 0.5 and 0.8, respectively.

Fig. 10.
Fig. 10.

(a)-(c) Contour plots of 2 as functions of k when k 0 equals to 0.2, 0.5 and 0.8, respectively. The dashed lines (i)–(iii) correspond to k = 0.2, 0.5 and 0.8, respectively. (d) 2 under k equaling to 0.2, 0.5 and 0.8, respectively, when k 0=0.5 . (e) 2 under k = k 0 equaling to 0.2, 0.5 and 0.8, respectively.

Fig. 11.
Fig. 11.

Effects of loss for types I and II (a) D 1 when τ is adopted as 1.0(lossless), 0.9 and 0.8, respectively. (b) D 1 max and θ + as a function of τ. k=k 0= 0.5

Fig. 12.
Fig. 12.

Effects of loss for types III and IV (a) 1 when τ is adopted as 1.0(lossless), 0.9 and 0.8, respectively. (b) max and 1 as a function of τ (c) 2 when τ is adopted as 1.0(lossless), 0.9 and 0.8, respectively. (d) 2 max and 2 are as a function of τ . k = k 0 =0.5

Equations (29)

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

[ c n e n ] = [ t n ik n ik n t n ] [ d n f n ]
[ b 1 b 0 b 2 ] = [ s 1 s 2 s 3 s 2 s 4 s 2 s 3 s 2 s 1 ] [ a 1 a 0 a 2 ]
d n = p n b n
a n = p n c n
ξ 0 ξ 1 ξ 2 = b 0 e 1 e 2 a 0 = s 4 s 1 ( t 1 p 1 2 + t 2 p 2 2 ) + t 1 t 2 p 1 2 p 0 2 is 2 k 1 p 1 ( 1 t 2 p 2 2 ) is 2 k 2 p 2 ( 1 t 1 p 1 2 ) S
ξ ˜ 0 ξ ˜ 1 ξ ˜ 2 = e 1 b 0 e 2 f 1 = t 1 s 1 ( p 1 2 + t 1 t 2 p 2 2 ) + s 4 t 2 p 1 2 p 2 2 is 2 k 1 p 1 ( 1 t 2 p 2 2 ) s 3 k 2 k 2 p 1 p 2 S
T = ξ 0 2 , D 1 = ξ 1 2 , D 2 = ξ 2 2
ϕ 0 = Arg ( ξ 0 ) , ϕ 1 = Arg ( ξ 1 ) , ϕ 2 = Arg ( ξ 2 )
T ˜ = ξ ˜ 0 2 , D ˜ 1 = ξ ˜ 1 2 , D ˜ 2 = ξ ˜ 2 2
ϕ ˜ 0 = Arg ( ξ ˜ 0 ) , ϕ ˜ 1 = Arg ( ξ ˜ 1 ) , ϕ ˜ 2 = Arg ( ξ ˜ 2 )
D 1 = D ˜ 1 , ϕ 1 = ϕ ˜ 1
T + D 1 + D 2 = 1 for types I and III
T ˜ + D ˜ 1 + D ˜ 2 = 1 for types I and IV
ξ 0 = ( t 0 ( 1 + t 0 ) ( 1 + t ) p 2 2 + t p 4 ) ( 1 ( 1 + t 0 ) ( 1 + t ) p 2 2 + t 0 t p 4 )
ξ 1 = ( k 0 kp ( 1 p 2 ) 2 ) ( 1 ( 1 + t 0 ) ( 1 + t ) p 2 2 + t 0 t p 4 )
ξ ˜ 0 = ( t ( 1 + t 0 ) ( 1 + t ) p 2 2 + t 0 p 4 ) ( 1 ( 1 + t 0 ) ( 1 + t ) p 2 2 + t 0 t p 4 )
ξ ˜ 1 = ( k 0 kp ( 1 p 2 ) 2 ) ( 1 ( 1 + t 0 ) ( 1 + t ) p 2 2 + t 0 p 4 )
ξ 0 = ( t 0 tp 2 ) ( 1 t 0 tp 2 )
ξ 1 = ( k 0 kp 2 ) ( 1 t 0 tp 2 )
ξ 2 = ( k 0 kp 2 ) ( 1 t 0 tp 2 )
ξ ˜ 0 = ( t ( 1 + t 0 ) ( 1 + t 2 ) p 2 2 + t 0 t p 4 ) ( 1 ( 1 + t 0 tp 2 ) + t 0 t 2 p 4 )
ξ ˜ 1 = ( k 0 kp 2 ) ( 1 t 0 tp 2 )
ξ ˜ 2 = ( ( t 0 1 ) k 2 p 2 2 ) ( 1 ( 1 + t 0 ) tp 2 + t 0 t 2 p 4 )
θ ± = ± accros ( 1 ε 0 )
D 1 max = k 0 2 k 2 ( 2 t 0 t ( 1 t 0 ) ( 1 t ) + ( 1 + t 0 t ) ( 1 + t 0 ) ( 1 + t ) 8 t 0 t )
W ˜ 1 = 4 arcsin ( ( 1 t 0 t ) 2 ( 1 + t 0 2 t 2 ) )
W ˜ 1 = 2 arcsin ( B B 2 4 AC ( 2 A ) )
D ˜ 1 max = k 0 2 k 2 ( 2 ( 1 t 0 t ) 2 )
D ˜ 1 max = ( 1 t 0 ) 2 ( 1 + t ) 2 ( 4 ( 1 t 0 t ) 2 )

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