Abstract

Coupled microresonators exhibit great potential for nonlinear applications. In the present work, we explore the nonlinear performance of an embedded ring resonator analogous to an electromagnetically induced transparency (EIT) medium, also known as coupled resonator induced transparency (CRIT). Interestingly, an EIT-like amplitude response can have a remarkably different power enhancement factor that varies by more than one order of magnitude, which is attributed to the different phase regimes of the embedded micro-ring resonators. In addition to the non-monotonic phase profile reported in atomic EIT systems, the phase responses featuring 2π and 4π monotonic transitions are identified and analyzed. We also present an interesting phenomenon, in which the power enhancement changes greatly, even with the same transfer function (both intensity and phase responses). This reveals that wisely choosing the operating regime is critical to optimize nonlinear performance of the embedded double resonator system, without adding to design or fabrication difficulty.

© 2013 OSA

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Errata

Xiaoyan Zhou, Lin Zhang, Andrea M. Armani, Raymond G. Beausoleil, Alan E. Willner, and Wei Pang, "Power enhancement and phase regimes in embedded microring resonators in analogy with electromagnetically induced transparency: erratum," Opt. Express 21, 28414-28414 (2013)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-21-23-28414

References

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  2. I. Chremmos, O. Schwelb, and N. Uzunoglu, Photonic microresonator research and applications. (Springer, 2010).
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    [CrossRef] [PubMed]
  4. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics4(1), 37–40 (2010).
    [CrossRef]
  5. C. Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett.83(8), 1527–1529 (2003).
    [CrossRef]
  6. L. Maleki, A. A. Savchenkov, A. B. Matsko, and V. S. Ilchenko, “Tunable filters and time delays with coupled whispering gallery mode resonators,” Proc. SPIE5435, 178–186 (2004).
    [CrossRef]
  7. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett.31(4), 456–458 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  22. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett.36(4), 321–322 (2000).
    [CrossRef]
  23. L. Zhang, M. Song, T. Wu, L. Zou, R. G. Beausoleil, and A. E. Willner, “Embedded ring resonators for microphotonic applications,” Opt. Lett.33(17), 1978–1980 (2008).
    [CrossRef] [PubMed]

2012

M. S. Luchansky and R. C. Bailey, “High-Q optical sensors for chemical and biological analysis,” Anal. Chem.84(2), 793–821 (2012).
[CrossRef] [PubMed]

C. Qiu, P. Yu, T. Hu, F. Wang, X. Jiang, and J. Yang, “Asymmetric Fano resonance in eye-like microring system,” Appl. Phys. Lett.101(2), 021110 (2012).
[CrossRef]

2011

2010

J. Scheuer, A. A. Sukhorukov, and Y. S. Kivshar, “All-optical switching of dark states in nonlinear coupled microring resonators,” Opt. Lett.35(21), 3712–3714 (2010).
[CrossRef] [PubMed]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics4(1), 37–40 (2010).
[CrossRef]

2009

A. L. Washburn, L. C. Gunn, and R. C. Bailey, “Label-free quantitation of a cancer biomarker in complex media using silicon photonic microring resonators,” Anal. Chem.81(22), 9499–9506 (2009).
[CrossRef] [PubMed]

2008

2007

X. Zhang, D. Huang, and X. Zhang, “Transmission characteristics of dual microring resonators coupled via 3x3 couplers,” Opt. Express15(21), 13557–13573 (2007).
[CrossRef] [PubMed]

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys.3(6), 406–410 (2007).
[CrossRef]

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett.98(21), 213904 (2007).
[CrossRef] [PubMed]

2006

2005

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in Coherent Media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

2004

L. Maleki, A. A. Savchenkov, A. B. Matsko, and V. S. Ilchenko, “Tunable filters and time delays with coupled whispering gallery mode resonators,” Proc. SPIE5435, 178–186 (2004).
[CrossRef]

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A69(6), 063804 (2004).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett.93(8), 083904 (2004).
[CrossRef] [PubMed]

2003

Y. Chen and S. Blair, “Nonlinear phase shift of cascaded microring resonators,” J. Opt. Soc. Am. B20(10), 2125–2132 (2003).
[CrossRef]

C. Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett.83(8), 1527–1529 (2003).
[CrossRef]

2000

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett.36(4), 321–322 (2000).
[CrossRef]

1999

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(11), 1426–1428 (1999).
[CrossRef]

Bailey, R. C.

M. S. Luchansky and R. C. Bailey, “High-Q optical sensors for chemical and biological analysis,” Anal. Chem.84(2), 793–821 (2012).
[CrossRef] [PubMed]

A. L. Washburn, L. C. Gunn, and R. C. Bailey, “Label-free quantitation of a cancer biomarker in complex media using silicon photonic microring resonators,” Anal. Chem.81(22), 9499–9506 (2009).
[CrossRef] [PubMed]

Beausoleil, R. G.

Blair, S.

Boyd, R. W.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A69(6), 063804 (2004).
[CrossRef]

Chang, H.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A69(6), 063804 (2004).
[CrossRef]

Chao, C. Y.

C. Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett.83(8), 1527–1529 (2003).
[CrossRef]

Chen, Y.

Chu, S. 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(11), 1426–1428 (1999).
[CrossRef]

DeRose, G. A.

Ding, D.

Dong, P.

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys.3(6), 406–410 (2007).
[CrossRef]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in Coherent Media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

Foster, M. A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics4(1), 37–40 (2010).
[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. A69(6), 063804 (2004).
[CrossRef]

Gaeta, A. L.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics4(1), 37–40 (2010).
[CrossRef]

Gill, D. M.

Gondarenko, A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics4(1), 37–40 (2010).
[CrossRef]

Gunn, L. C.

A. L. Washburn, L. C. Gunn, and R. C. Bailey, “Label-free quantitation of a cancer biomarker in complex media using silicon photonic microring resonators,” Anal. Chem.81(22), 9499–9506 (2009).
[CrossRef] [PubMed]

Guo, L. J.

C. Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett.83(8), 1527–1529 (2003).
[CrossRef]

Hu, T.

C. Qiu, P. Yu, T. Hu, F. Wang, X. Jiang, and J. Yang, “Asymmetric Fano resonance in eye-like microring system,” Appl. Phys. Lett.101(2), 021110 (2012).
[CrossRef]

Huang, D.

Ilchenko, V. S.

L. Maleki, A. A. Savchenkov, A. B. Matsko, and V. S. Ilchenko, “Tunable filters and time delays with coupled whispering gallery mode resonators,” Proc. SPIE5435, 178–186 (2004).
[CrossRef]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in Coherent Media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

Integlia, R. A.

Jiang, W.

Jiang, X.

C. Qiu, P. Yu, T. Hu, F. Wang, X. Jiang, and J. Yang, “Asymmetric Fano resonance in eye-like microring system,” Appl. Phys. Lett.101(2), 021110 (2012).
[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(11), 1426–1428 (1999).
[CrossRef]

Kippenberg, T. J.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett.93(8), 083904 (2004).
[CrossRef] [PubMed]

Kivshar, Y. S.

Kobayashi, N.

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett.98(21), 213904 (2007).
[CrossRef] [PubMed]

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(11), 1426–1428 (1999).
[CrossRef]

Levy, J. S.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics4(1), 37–40 (2010).
[CrossRef]

Li, Y.

Lipson, M.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics4(1), 37–40 (2010).
[CrossRef]

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys.3(6), 406–410 (2007).
[CrossRef]

Little, B. E.

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(11), 1426–1428 (1999).
[CrossRef]

Luchansky, M. S.

M. S. Luchansky and R. C. Bailey, “High-Q optical sensors for chemical and biological analysis,” Anal. Chem.84(2), 793–821 (2012).
[CrossRef] [PubMed]

Maleki, L.

L. Maleki, A. A. Savchenkov, A. B. Matsko, and V. S. Ilchenko, “Tunable filters and time delays with coupled whispering gallery mode resonators,” Proc. SPIE5435, 178–186 (2004).
[CrossRef]

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in Coherent Media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

Matsko, A. B.

L. Maleki, A. A. Savchenkov, A. B. Matsko, and V. S. Ilchenko, “Tunable filters and time delays with coupled whispering gallery mode resonators,” Proc. SPIE5435, 178–186 (2004).
[CrossRef]

Pan, D. Z.

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(11), 1426–1428 (1999).
[CrossRef]

Poon, J. K. S.

Qiu, C.

C. Qiu, P. Yu, T. Hu, F. Wang, X. Jiang, and J. Yang, “Asymmetric Fano resonance in eye-like microring system,” Appl. Phys. Lett.101(2), 021110 (2012).
[CrossRef]

Rosenberger, A. T.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A69(6), 063804 (2004).
[CrossRef]

Savchenkov, A. A.

L. Maleki, A. A. Savchenkov, A. B. Matsko, and V. S. Ilchenko, “Tunable filters and time delays with coupled whispering gallery mode resonators,” Proc. SPIE5435, 178–186 (2004).
[CrossRef]

Scheuer, J.

Smith, D. D.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A69(6), 063804 (2004).
[CrossRef]

Song, M.

Spillane, S. M.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett.93(8), 083904 (2004).
[CrossRef] [PubMed]

Sukhorukov, A. A.

Tomita, M.

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett.98(21), 213904 (2007).
[CrossRef] [PubMed]

Totsuka, K.

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett.98(21), 213904 (2007).
[CrossRef] [PubMed]

Turner-Foster, A. C.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics4(1), 37–40 (2010).
[CrossRef]

Vahala, K. J.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett.93(8), 083904 (2004).
[CrossRef] [PubMed]

Wang, F.

C. Qiu, P. Yu, T. Hu, F. Wang, X. Jiang, and J. Yang, “Asymmetric Fano resonance in eye-like microring system,” Appl. Phys. Lett.101(2), 021110 (2012).
[CrossRef]

Washburn, A. L.

A. L. Washburn, L. C. Gunn, and R. C. Bailey, “Label-free quantitation of a cancer biomarker in complex media using silicon photonic microring resonators,” Anal. Chem.81(22), 9499–9506 (2009).
[CrossRef] [PubMed]

Willner, A. E.

Wu, T.

Xu, Q.

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys.3(6), 406–410 (2007).
[CrossRef]

Yang, J.

C. Qiu, P. Yu, T. Hu, F. Wang, X. Jiang, and J. Yang, “Asymmetric Fano resonance in eye-like microring system,” Appl. Phys. Lett.101(2), 021110 (2012).
[CrossRef]

Yang, J.-Y.

Yariv, A.

J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett.31(4), 456–458 (2006).
[CrossRef] [PubMed]

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett.36(4), 321–322 (2000).
[CrossRef]

Yin, L.

Yu, P.

C. Qiu, P. Yu, T. Hu, F. Wang, X. Jiang, and J. Yang, “Asymmetric Fano resonance in eye-like microring system,” Appl. Phys. Lett.101(2), 021110 (2012).
[CrossRef]

Zhang, L.

Zhang, X.

Zhu, L.

Zou, L.

Anal. Chem.

M. S. Luchansky and R. C. Bailey, “High-Q optical sensors for chemical and biological analysis,” Anal. Chem.84(2), 793–821 (2012).
[CrossRef] [PubMed]

A. L. Washburn, L. C. Gunn, and R. C. Bailey, “Label-free quantitation of a cancer biomarker in complex media using silicon photonic microring resonators,” Anal. Chem.81(22), 9499–9506 (2009).
[CrossRef] [PubMed]

Appl. Phys. Lett.

C. Y. Chao and L. J. Guo, “Biochemical sensors based on polymer microrings with sharp asymmetrical resonance,” Appl. Phys. Lett.83(8), 1527–1529 (2003).
[CrossRef]

C. Qiu, P. Yu, T. Hu, F. Wang, X. Jiang, and J. Yang, “Asymmetric Fano resonance in eye-like microring system,” Appl. Phys. Lett.101(2), 021110 (2012).
[CrossRef]

Electron. Lett.

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett.36(4), 321–322 (2000).
[CrossRef]

IEEE Photon. Technol. Lett.

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(11), 1426–1428 (1999).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Photonics

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics4(1), 37–40 (2010).
[CrossRef]

Nat. Phys.

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys.3(6), 406–410 (2007).
[CrossRef]

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. A69(6), 063804 (2004).
[CrossRef]

Phys. Rev. Lett.

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett.98(21), 213904 (2007).
[CrossRef] [PubMed]

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett.93(8), 083904 (2004).
[CrossRef] [PubMed]

Proc. SPIE

L. Maleki, A. A. Savchenkov, A. B. Matsko, and V. S. Ilchenko, “Tunable filters and time delays with coupled whispering gallery mode resonators,” Proc. SPIE5435, 178–186 (2004).
[CrossRef]

Rev. Mod. Phys.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in Coherent Media,” Rev. Mod. Phys.77(2), 633–673 (2005).
[CrossRef]

Other

A. B. Matsko, Practical applications of microresonators in optics and photonics (CRC Press, 2009).

I. Chremmos, O. Schwelb, and N. Uzunoglu, Photonic microresonator research and applications. (Springer, 2010).

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Figures (7)

Fig. 1
Fig. 1

(a) Schematic of the embedded rings. (b) and (c) Equivalent split-models of power distribution in Case I (m1 - m2 = odd number) with t2 = t4.

Fig. 2
Fig. 2

Transfer characteristics of EIT-like effect and power distributions at resonance wavelength in Case I, with the same ring-ring coupling coefficients, i.e., t2 = t4 = 0.2. (b) and (c) have the same ring-waveguide coupling coefficients: t1 = 0.3 and t3 = 0.4. (d) and (e) also have the same ring-waveguide coupling coefficients: t1 = 0.4 and t3 = 0.3.

Fig. 3
Fig. 3

Transfer characteristics of EIT-like effect and power distributions at resonance wavelength in Case I, with different ring-ring coupling coefficients, i.e., t2t4. (b), (c), and (d) have the same ring-waveguide coupling coefficients: t1 = 0.3 and t3 = 0.4, where t2 and t4 are switched in (c) and (d). (e), (f), and (g) also have the same ring-waveguide coupling coefficients: t1 = 0.4 and t3 = 0.3, where t2 and t4 are switched in (f) and (g).

Fig. 4
Fig. 4

(a) Transmission spectrums with different value of t2t4. (b) Changes of transmission and power enhancement with t2t4 at 1.55 μm.

Fig. 5
Fig. 5

Transfer characteristics of EIT-like effect and power distributions at resonance wavelength in Case II, with different ring-ring coupling coefficients, i.e., t2t4. (b), (c), and (d) have the same ring-waveguide coupling coefficients: t1 = 0.3 and t3 = 0.4, where t2 and t4 are switched in (c) and (d). (e), (f), and (g) also have the same ring-waveguide coupling coefficients: t1 = 0.4 and t3 = 0.3, where t2 and t4 are switched in (f) and (g).

Fig. 6
Fig. 6

(a) Transmission spectrums evolving from the EIT-like effect to mode splitting as t2 (t4) increases. (b) Changes of transmission and power enhancement with t2 (t4) at 1.55 μm.

Fig. 7
Fig. 7

Changes of transmission and power enhancement with loss for Case I and Case II. (a) and (c) Case I: parameters in Case IA and Case IB correspond with configurations in Figs. 2(c) and 2(e), respectively. (b) and (d) Case II: parameters in Case IIA, Case IIB, Case IIC, and Case IID correspond with configurations in Figs. 5(c), 5(d), 5(f), and 5(g), respectively.

Equations (4)

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T Through = r 1 r 1 r 2 r 4 e 2 2 + r 1 r 3 t 2 t 4 e 1 2 e 2 r 2 r 3 r 4 e 1 4 + r 3 e 1 4 e 2 2 + t 2 t 4 e 1 2 e 2 1 r 2 r 4 e 2 2 + r 1 r 3 e 1 4 e 2 2 +( r 1 + r 3 ) t 2 t 4 e 1 2 e 2 r 1 r 2 r 3 r 4 e 1 4
T Drop = r 4 t 1 t 3 e 1 2 e 2 2 r 2 t 1 t 3 e 1 2 1 r 2 r 4 e 2 2 + r 1 r 3 e 1 4 e 2 2 +( r 1 + r 3 ) t 2 t 4 e 1 2 e 2 r 1 r 2 r 3 r 4 e 1 4
P Ring2_lower P Ring1_lower = 1 t 2 2
P Ring1_lower P in = t 1 2 (1 r 1 ) 2 = t 1 2 2 t 1 2 2 1 t 1 2

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