Abstract

We present both theoretically and experimentally the existence of defect modes in side-coupled and mutually coupled microresonator arrays. The qualitative difference between the two types of defect modes is investigated. The Q factor of both defect modes for varying defect sizes is characterized, and an enhancement of 30× relative to individual loaded resonators is demonstrated. The defect modes are then compared with coupled resonator–induced transparency (CRIT), indicating that the defect modes based on side-coupled microresonator arrays are actually the extension of the CRIT resonance in two-resonator structures.

© 2012 Optical Society of America

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2011 (1)

2010 (3)

X. Tu, L. Y. Mario, and T. Mei, “Coupled Fano resonator,” Opt. Express 18, 18820–18831 (2010).
[CrossRef]

O. Schwelb and I. Chremmos, “Defect assisted coupled resonator optical waveguide: weak perturbations,” Opt. Commun. 283, 3686–3690 (2010).
[CrossRef]

I. Chremmos and O. Schwelb, “Diatomic coupled-resonator optical waveguide,” J. Opt. Soc. B 27, 1242–1251 (2010).
[CrossRef]

2009 (1)

X. Yang, M. Yu, D.-L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102, 173902(2009).
[CrossRef]

2008 (5)

2007 (3)

F. Xia, M. Rooks, L. Sekaric, and Y. Vlassov, “Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects,” Opt. Express 15, 11934–11941 (2007).
[CrossRef]

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nature 1, 65–71 (2007).
[CrossRef]

L. Y. Mario, D. C. S. Lim, and M. K. Chin, “Proposal of an ultra-narrow passband using two-coupled rings,” IEEE Photon. Technol. Lett. 19, 1688–1690 (2007).
[CrossRef]

2006 (3)

2005 (7)

2004 (4)

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]

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical delay lines,” J. Opt. Soc. B 21, 1665–1673 (2004).
[CrossRef]

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

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator sequence for linear and nonlinear optics,” J. Opt. Soc. Am. B 21, 1818–1832 (2004).
[CrossRef]

2003 (1)

Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E 68, 066616 (2003).
[CrossRef]

2002 (1)

A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002).
[CrossRef]

2001 (2)

1999 (2)

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

1997 (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
[CrossRef]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

Absil, P. P.

Assefa, S.

J. R. Ong, M. L. Cooper, G. Gupta, W. M. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Intra- and inter-band four-wave mixing in silicon coupled resonator optical waveguides,” in CLEO:2011–Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuW1.

Baets, R.

Barwicz, T.

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Bettotti, P.

Bienstman, P.

Boyd, R. W.

Chak, P.

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]

Chin, M. K.

L. Y. M. Tobing, P. Dumon, R. Baets, and M. K. Chin, “Boxlike filter response based on complementary photonic bandgap in two-dimensional microresonator arrays,” Opt. Lett. 33, 2512–2514 (2008).
[CrossRef]

L. Y. M. Tobing, P. Dumon, R. Baets, and M. K. Chin, “Demonstration of defect modes in coupled microresonator arrays fabricated in silicon-on-insulator technology,” Opt. Lett. 33, 1939–1941 (2008).
[CrossRef]

L. Y. M. Tobing, P. Dumon, R. Baets, D. C. S. Lim, and M. K. Chin, “Finesse enhancement in silicon-on-insulator two-ring resonator system,” Appl. Phys. Lett. 92, 101122 (2008).
[CrossRef]

L. Y. Mario, D. C. S. Lim, and M. K. Chin, “Proposal of an ultra-narrow passband using two-coupled rings,” IEEE Photon. Technol. Lett. 19, 1688–1690 (2007).
[CrossRef]

Y. M. Landobasa and M. K. Chin, “Defect modes in micro-ring resonator arrays,” Opt. Express 13, 7800–7815 (2005).
[CrossRef]

Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix analysis of 2-D micro-resonator lattice optical filters,” IEEE J. Quantum Electron. 41, 1410–1418 (2005).
[CrossRef]

Chin, M.-K.

L. Y. Mario, D. C. S. Lim, P. Dumon, R. Baets, and M.-K. Chin, “Experimental verification of finesse enhancement scheme in two-ring resonator system,” Proc. SPIE 6996, 69960B (2008).

Chremmos, I.

I. Chremmos and O. Schwelb, “Diatomic coupled-resonator optical waveguide,” J. Opt. Soc. B 27, 1242–1251 (2010).
[CrossRef]

O. Schwelb and I. Chremmos, “Defect assisted coupled resonator optical waveguide: weak perturbations,” Opt. Commun. 283, 3686–3690 (2010).
[CrossRef]

Cooper, M. L.

J. R. Ong, M. L. Cooper, G. Gupta, W. M. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Intra- and inter-band four-wave mixing in silicon coupled resonator optical waveguides,” in CLEO:2011–Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuW1.

Darmawan, S.

Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix analysis of 2-D micro-resonator lattice optical filters,” IEEE J. Quantum Electron. 41, 1410–1418 (2005).
[CrossRef]

Dumon, P.

L. Y. M. Tobing, P. Dumon, R. Baets, D. C. S. Lim, and M. K. Chin, “Finesse enhancement in silicon-on-insulator two-ring resonator system,” Appl. Phys. Lett. 92, 101122 (2008).
[CrossRef]

L. Y. M. Tobing, P. Dumon, R. Baets, and M. K. Chin, “Demonstration of defect modes in coupled microresonator arrays fabricated in silicon-on-insulator technology,” Opt. Lett. 33, 1939–1941 (2008).
[CrossRef]

L. Y. Mario, D. C. S. Lim, P. Dumon, R. Baets, and M.-K. Chin, “Experimental verification of finesse enhancement scheme in two-ring resonator system,” Proc. SPIE 6996, 69960B (2008).

L. Y. M. Tobing, P. Dumon, R. Baets, and M. K. Chin, “Boxlike filter response based on complementary photonic bandgap in two-dimensional microresonator arrays,” Opt. Lett. 33, 2512–2514 (2008).
[CrossRef]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

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 (2006).
[CrossRef]

M. F. Yanik and S. Fan, “Stopping and storing light coherently,” Phys. Rev. A 71, 013803 (2005).
[CrossRef]

Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E 68, 066616 (2003).
[CrossRef]

Farca, N. G.

N. 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]

Fedelli, J.-M.

Frigyes, I.

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]

Green, W. M.

J. R. Ong, M. L. Cooper, G. Gupta, W. M. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Intra- and inter-band four-wave mixing in silicon coupled resonator optical waveguides,” in CLEO:2011–Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuW1.

Guider, R.

Gupta, G.

J. R. Ong, M. L. Cooper, G. Gupta, W. M. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Intra- and inter-band four-wave mixing in silicon coupled resonator optical waveguides,” in CLEO:2011–Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuW1.

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
[CrossRef]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Heebner, J. E.

Ho, P.-T.

Hryniewicz, J. V.

Ilchenko, V. S.

Ippen, E. P.

Khan, M. H.

Khurgin, J. B.

Kwong, D.-L.

X. Yang, M. Yu, D.-L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102, 173902(2009).
[CrossRef]

Landobasa, Y. M.

Y. M. Landobasa and M. K. Chin, “Defect modes in micro-ring resonator arrays,” Opt. Express 13, 7800–7815 (2005).
[CrossRef]

Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix analysis of 2-D micro-resonator lattice optical filters,” IEEE J. Quantum Electron. 41, 1410–1418 (2005).
[CrossRef]

Lee, R. K.

Li, X.

Lim, D. C. S.

L. Y. M. Tobing, P. Dumon, R. Baets, D. C. S. Lim, and M. K. Chin, “Finesse enhancement in silicon-on-insulator two-ring resonator system,” Appl. Phys. Lett. 92, 101122 (2008).
[CrossRef]

L. Y. Mario, D. C. S. Lim, P. Dumon, R. Baets, and M.-K. Chin, “Experimental verification of finesse enhancement scheme in two-ring resonator system,” Proc. SPIE 6996, 69960B (2008).

L. Y. Mario, D. C. S. Lim, and M. K. Chin, “Proposal of an ultra-narrow passband using two-coupled rings,” IEEE Photon. Technol. Lett. 19, 1688–1690 (2007).
[CrossRef]

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 (2006).
[CrossRef]

Lu, Y.

Maes, B.

Maleki, L. B.

Mancinelli, M.

Mario, L. Y.

X. Tu, L. Y. Mario, and T. Mei, “Coupled Fano resonator,” Opt. Express 18, 18820–18831 (2010).
[CrossRef]

L. Y. Mario, D. C. S. Lim, P. Dumon, R. Baets, and M.-K. Chin, “Experimental verification of finesse enhancement scheme in two-ring resonator system,” Proc. SPIE 6996, 69960B (2008).

L. Y. Mario, D. C. S. Lim, and M. K. Chin, “Proposal of an ultra-narrow passband using two-coupled rings,” IEEE Photon. Technol. Lett. 19, 1688–1690 (2007).
[CrossRef]

Masi, M.

Matsko, A. B.

Mei, T.

Mookherjea, S.

J. R. Ong, M. L. Cooper, G. Gupta, W. M. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Intra- and inter-band four-wave mixing in silicon coupled resonator optical waveguides,” in CLEO:2011–Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuW1.

Ong, J. R.

J. R. Ong, M. L. Cooper, G. Gupta, W. M. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Intra- and inter-band four-wave mixing in silicon coupled resonator optical waveguides,” in CLEO:2011–Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuW1.

Pavesi, L.

Pereira, S.

Poon, J. K. S.

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical delay lines,” J. Opt. Soc. B 21, 1665–1673 (2004).
[CrossRef]

Popovic, M.

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 (2006).
[CrossRef]

Qi, M.

Rakich, P. T.

Rooks, M.

Rosenberger, A. T.

N. 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. 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]

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 (2006).
[CrossRef]

Savchenkov, A. A.

Scherer, A.

Scheuer, J.

J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical delay lines,” J. Opt. Soc. B 21, 1665–1673 (2004).
[CrossRef]

Schwelb, O.

O. Schwelb and I. Chremmos, “Defect assisted coupled resonator optical waveguide: weak perturbations,” Opt. Commun. 283, 3686–3690 (2010).
[CrossRef]

I. Chremmos and O. Schwelb, “Diatomic coupled-resonator optical waveguide,” J. Opt. Soc. B 27, 1242–1251 (2010).
[CrossRef]

O. Schwelb and I. Frigyes, “All-optical tunable filters built with discontinuity-assisted ring resonators,” J. Lightwave Technol. 19, 380–386 (2001).
[CrossRef]

Sekaric, L.

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 (2006).
[CrossRef]

Shen, H.

Shopova, S. I.

N. 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]

Sipe, J. E.

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]

Smith, H. I.

Tobing, L. Y. M.

Tu, X.

Van, V.

Vanacharla, M. R.

Vlasov, Y.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nature 1, 65–71 (2007).
[CrossRef]

Vlasov, Y. A.

F. Xia, L. Sekaric, and Y. A. Vlasov, “Mode conversion losses in silicon-on-insulator photonic wire based racetrack resonators,” Opt. Express 14, 3872–3886 (2006).
[CrossRef]

J. R. Ong, M. L. Cooper, G. Gupta, W. M. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Intra- and inter-band four-wave mixing in silicon coupled resonator optical waveguides,” in CLEO:2011–Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuW1.

Vlassov, Y.

Wang, P.

Wang, Z.

Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E 68, 066616 (2003).
[CrossRef]

Watts, M. R.

Wong, C. W.

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F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nature 1, 65–71 (2007).
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F. Xia, M. Rooks, L. Sekaric, and Y. Vlassov, “Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects,” Opt. Express 15, 11934–11941 (2007).
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F. Xia, L. Sekaric, and Y. A. Vlasov, “Mode conversion losses in silicon-on-insulator photonic wire based racetrack resonators,” Opt. Express 14, 3872–3886 (2006).
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J. R. Ong, M. L. Cooper, G. Gupta, W. M. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Intra- and inter-band four-wave mixing in silicon coupled resonator optical waveguides,” in CLEO:2011–Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuW1.

Xiao, S.

Xu, Q.

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 (2006).
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J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical delay lines,” J. Opt. Soc. B 21, 1665–1673 (2004).
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X. Yang, M. Yu, D.-L. Kwong, and C. W. Wong, “All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities,” Phys. Rev. Lett. 102, 173902(2009).
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M. F. Yanik and S. Fan, “Stopping and storing light coherently,” Phys. Rev. A 71, 013803 (2005).
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Appl. Phys. Lett. (1)

L. Y. M. Tobing, P. Dumon, R. Baets, D. C. S. Lim, and M. K. Chin, “Finesse enhancement in silicon-on-insulator two-ring resonator system,” Appl. Phys. Lett. 92, 101122 (2008).
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Other (3)

P. Yeh, Optical Waves in Layered Media (Wiley, 2005).

www.epixfab.eu .

J. R. Ong, M. L. Cooper, G. Gupta, W. M. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Intra- and inter-band four-wave mixing in silicon coupled resonator optical waveguides,” in CLEO:2011–Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CTuW1.

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Figures (14)

Fig. 1.
Fig. 1.

A single (point) defect embedded in a (a) SCISSOR (Type I), and (b) CROW (Type II). (c) The Fabry–Perot cavity formed by the defect and the two subarrays.

Fig. 2.
Fig. 2.

(a) Frequencies of the defect modes versus the defect length for infinite Type I periodic structure. (b) Defect transmission | T d | 2 for N = 5 , when L d = 0.52 L cav . The coupling coefficient and the mode number are r = 0.85 and m = 10 , respectively.

Fig. 3.
Fig. 3.

(a) The resonant frequency of defect modes versus normalized defect length L d / L cav . (b) Defect transmission | T d | 2 for N = 3 . The defect is a racetrack resonator whose cavity length is L d = 1.05 L cav . The coupling coefficient and the mode number is r = 0.65 and m = 10 , respectively.

Fig. 4.
Fig. 4.

Absolute amplitude of the field distribution calculated by FDTD for (a) Type I ( L d / L cav = 0.75 ) and (b) Type II ( L d / L cav = 1.1 ).

Fig. 5.
Fig. 5.

Relationship between coupler length ( L C ), coupling coefficient ( r ), and Q factor ( Q ). The cavity loss is assumed to be a 0.994 . The directional coupler and the racetrack ring resonator are shown in the insets.

Fig. 6.
Fig. 6.

Optical micrograph of fabricated Type I defect mode structures. The subarrays are designed to have (a) four and (b) six unit cells. The defect is introduced by making the interresonator length longer than the regular ones. The optical waveguide is designed to be curved to prevent length mismatch between the interresonator spacing and the cavity length.

Fig. 7.
Fig. 7.

Offset dB through transmission of Type I defect modes for different normalized defect sizes ( k d ). The subarray cell number is N = 4 . The photonic bandgap is indicated by shaded region.

Fig. 8.
Fig. 8.

Q factor versus normalized detuning relative to midgap wavelength for N = 4 (left) and N = 6 (right). The case of N = 6 shows a less sensitive Q factor with respect to normalized detuning; however, it comes at the expense of more enhanced loss.

Fig. 9.
Fig. 9.

Transmission contrast versus measured Q factor of Type I defect modes for N = 4 (left) and N = 6 (right).

Fig. 10.
Fig. 10.

Optical micrographs of the fabricated Type II defect mode structures for N = 2 and N = 3 .

Fig. 11.
Fig. 11.

Offset transmission of Type II defect mode structure for different defect sizes and subarray unit cells. The elongation lengths corresponding to each defect size are 0 μm, 4.2 μm, 8.3 μm, 12.4 μm, 16.6 μm, and 20.7 μm, respectively.

Fig. 12.
Fig. 12.

Contrast versus Q factor in Type II defect mode for different units of subarrays. The Q factors are measured in 0.005 nm wavelength resolution.

Fig. 13.
Fig. 13.

Resemblance between Type I defect mode and EIT-like resonance. The dotted lines denote the field distributions. The line thickness signifies the amplitudes.

Fig. 14.
Fig. 14.

(a) Type I and (b) Type II coupled resonator structures.

Equations (12)

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T d = T 1 T 2 exp ( i δ d / 2 ) [ R 1 R 2 exp ( i δ d ) ] m = T 1 T 2 exp ( i δ d / 2 ) 1 R 1 R 2 exp ( i δ d ) ,
2 ϕ R = δ d ,
R I = m 12 m 11 exp ( i k Λ ) , R II = p 12 p 11 exp ( i k Λ ) .
r = cos [ π L C / ( 2 L π ) + ϕ 0 ]
F = π a r 2 1 a r 2 ,
Q λ Δ λ = ( λ Δ λ FSR ) ( Δ λ FSR Δ λ ) = m F .
F int = π a 1 a .
T = r a r exp ( i δ ) 1 a r 2 exp ( i δ ) , R = ( 1 r 2 ) exp ( i δ / 2 ) 1 a r 2 exp ( i δ ) ,
[ d b ] = 1 T [ 1 R R T 2 R 2 ] [ c a ] ,
[ d b ] n = [ exp ( i δ B ) 0 0 exp ( i δ B ) ] [ c a ] n + 1 .
[ c a ] n + 1 [ m 11 m 12 m 21 m 22 ] [ c a ] n = [ 1 T exp ( i δ B ) R T exp ( i δ B ) R T exp ( i δ B ) 1 T * exp ( i δ B ) ] [ c a ] n .
[ a b ] n + 1 [ p 11 p 12 p 21 p 22 ] [ a b ] n = 1 i 1 r 2 [ exp ( i δ 2 ) r exp ( i δ 2 ) r exp ( i δ 2 ) exp ( i δ 2 ) ] [ a b ] n .

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