Abstract

Sequences of optical microresonators can be used to construct densely integrated structures that display slow group velocity, ultrahigh or low dispersion of controllable sign, enhanced self-phase modulation, and nonlinear optical switching. We consider four archetypal geometries consisting of effectively one-dimensional sequences of coupled microresonators. Two of these cases exhibit distributed feedback such as is found in a traditional multilayered structure supporting photonic bandgaps. The other two exhibit localized feedback and resonant enhancement but are free from photonic bandgaps. All of these structures offer unique properties useful for controlling the propagation of light pulses on a chip.

© 2004 Optical Society of America

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2004 (1)

2003 (3)

2002 (22)

R. W. Boyd and D. J. Gauthier, “Slow and fast light,” Prog. Opt. 33, 497–530 (2002).
[CrossRef]

K. Djordjev, S. Choi, and P. D. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 828–830 (2002).
[CrossRef]

J. E. Heebner and R. W. Boyd, “Slow and fast light in resonator-coupled waveguides,” J. Mod. Opt. 49, 2629–2636 (2002).
[CrossRef]

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, “Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR Filters,” J. Lightwave Technol. 20, 900–905 (2002).
[CrossRef]

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[CrossRef] [PubMed]

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs–AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14, 74–77 (2002).
[CrossRef]

P. Chak, J. E. Sipe, and S. Pereira, “Lorentzian model for nonlinear switching in a microresonator structure,” Opt. Commun. 213, 163–171 (2002).
[CrossRef]

K. Djordjev, S. Choi, and P. D. Dapkus, “High-Q vertically coupled InP microdisk resonators,” IEEE Photon. Technol. Lett. 14, 331–333 (2002).
[CrossRef]

J. E. Heebner, R. W. Boyd, and Q. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

J. M. Harbold, F. O. Ilday, F. W. Wise, J. S. Sanghera, V. Q. Nguyen, L. B. Shaw, and I. D. Aggarwal, “Highly nonlinear As–S–Se glasses for all-optical switching,” Opt. Lett. 27, 119–121 (2002).
[CrossRef]

S. Blair, J. E. Heebner, and R. W. Boyd, “Beyond the absorption-limited nonlinear phase shift with microring resonators,” Opt. Lett. 27, 357–359 (2002).
[CrossRef]

S. Spalter, H. Y. Wang, J. Zimmermann, G. Lenz, T. Katsufuji, S.-W. Cheong, and R. E. Slusher, “Strong self-phase modulation in planar chalcogenide glass waveguides,” Opt. Lett. 27, 363–365 (2002).
[CrossRef]

S. Pereira, J. E. Sipe, J. E. Heebner, and R. W. Boyd, “Gap solitons in a two-channel SCISSOR structure,” Opt. Lett. 27, 536–538 (2002).
[CrossRef]

M. D. Rahn, A. M. Fox, M. S. Skolnick, and T. F. Krauss, “Propagation of ultrashort nonlinear pulses through two-dimensional AlGaAs high-contrast photonic crystal waveguides,” J. Opt. Soc. Am. B 19, 716–721 (2002).
[CrossRef]

J. E. Heebner, R. W. Boyd, and Q. Park, “SCISSOR solitons and other propagation effects in microresonator modified waveguides,” J. Opt. Soc. Am. B 19, 722–731 (2002).
[CrossRef]

D. N. Christodoulides and N. K. Efremidis, “Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals,” Opt. Lett. 27, 568–570 (2002).
[CrossRef]

S. Blair, “Nonlinear sensitivity enhancement with one-dimensional photonic bandgap microcavity arrays,” Opt. Lett. 27, 613–615 (2002).
[CrossRef]

T. A. Ibrahim, V. Van, and P.-T. Ho, “All-optical time-division demultiplexing and spatial pulse routing with a GaAs/AlGaAs microring resonator,” Opt. Lett. 27, 803–805 (2002).
[CrossRef]

S. Mookherjea, D. S. Cohen, and A. Yariv, “Nonlinear dispersion in a coupled-resonator optical waveguide,” Opt. Lett. 27, 933–935 (2002).
[CrossRef]

M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002).
[CrossRef]

S. Pereira, P. Chak, and J. E. Sipe, “Gap-soliton switching in short microresonator structures,” J. Opt. Soc. Am. B 19, 2191–2202 (2002).
[CrossRef]

S. Mingaleev and Y. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B 19, 2241–2249 (2002).
[CrossRef]

2001 (3)

A. Melloni, “Synthesis of a parallel-coupled ring-resonator filter,” Opt. Lett. 26, 917–919 (2001).
[CrossRef]

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525–532 (2001).
[CrossRef]

2000 (10)

Y. Xu, R. K. Lee, and A. Yariv, “Scattering theory analysis of waveguide-resonator coupling,” Phys. Rev. E 62, 7389–7404 (2000).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12, 320–322 (2000).
[CrossRef]

G. Griffel, “Synthesis of optical filters using ring resonator arrays,” IEEE Photon. Technol. Lett. 12, 810–812 (2000).
[CrossRef]

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef] [PubMed]

B. E. Little and S. T. Chu, “Toward very large-scale integrated photonics,” Opt. Photonics News 11 (11), 24–29 (2000).
[CrossRef]

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-galley mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17, 387–400 (2000).
[CrossRef]

G. Lenz, J. Zimmermann, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spalter, R. E. Slusher, S.-W. Cheong, J. S. Sanghera, and I. D. Aggarwal, “Large Kerr effect in bulk Se-based chalcogenide glasses,” Opt. Lett. 25, 254–256 (2000).
[CrossRef]

B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).
[CrossRef]

P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Jonekis, and P.-T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554–556 (2000).
[CrossRef]

1999 (6)

J.-P. Laine, B. E. Little, and H. A. Haus, “Etch-eroded fiber coupler for whispering-gallery-mode excitation in high-Q silica microspheres,” IEEE Photon. Technol. Lett. 11, 1429–1430 (1999).
[CrossRef]

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]

C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783–785 (1999).
[CrossRef]

J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. 24, 847–849 (1999).
[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]

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[CrossRef]

1998 (5)

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM Systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

C. K. Madsen, “Efficient architectures for exactly realizing optical filters with optimum bandpass designs,” IEEE Photon. Technol. Lett. 10, 1136–1138 (1998).
[CrossRef]

C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10, 994–996 (1998).
[CrossRef]

D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23, 247–249 (1998).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express 3 (7), 4–11 (1998).
[CrossRef] [PubMed]

1997 (4)

D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho, and R. C. Tiberio, “Waveguide-coupled AlGaAs/GaAs microcavity ring and disk resonators with high finesse and 21.6-nm free-spectral range,” Opt. Lett. 22, 1244–1246 (1997).
[CrossRef] [PubMed]

J. Popp, M. H. Fields, and R. K. Chang, “Q-switching by saturable absorption in microdroplets: elastic scattering and laser emission,” Opt. Lett. 22, 1296–1298 (1997).
[CrossRef]

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]

F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
[CrossRef]

1996 (3)

1995 (3)

N. Dubreuil, J. C. Knight, D. K. Leventhal, V. Sandoghdar, J. Hare, and V. Lefevre, “Eroded monomode optical fiber for whispering-gallery mode excitation in fused-silica microspheres,” Opt. Lett. 20, 813–815 (1995).
[CrossRef] [PubMed]

J. P. Zhang, D. Y. Chu, S. L. Wu, S. T. Ho, W. G. Bi, C. W. Tu, and R. C. Tiberio, “Photonic-wire laser,” Phys. Rev. Lett. 75, 2678–2681 (1995).
[CrossRef] [PubMed]

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple ring resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447–1449 (1995).
[CrossRef]

1994 (1)

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-Hemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below the half-gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

1993 (1)

Y. Yamamoto and R. E. Slusher, “Optical processes in microcavities,” Phys. Today 46 (6), 66–74 (1993).
[CrossRef]

1992 (1)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

1991 (2)

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, “Large nonlinear phase shifts in low-loss AlGaAs waveguides near half-gap,” Appl. Phys. Lett. 59, 2558–2560 (1991).
[CrossRef]

S. Arnold, C. T. Liu, W. B. Whitten, and J. M. Ramsey, “Room-temperature microparticle-based persistent spectral hole burning memory,” Opt. Lett. 16, 420–422 (1991).
[CrossRef] [PubMed]

1989 (1)

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

1987 (2)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

V. B. Braginsky and V. S. Ilchenko, “Properties of optical dielectric microresonators,” Sov. Phys. Dokl. 32, 306–307 (1987).

1985 (1)

G. I. Stegeman and C. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, 57 (1985).
[CrossRef]

Absil, P.

Absil, P. P.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, “Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR Filters,” J. Lightwave Technol. 20, 900–905 (2002).
[CrossRef]

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs–AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14, 74–77 (2002).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12, 320–322 (2000).
[CrossRef]

B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).
[CrossRef]

Aggarwal, I. D.

Aitchison, J. S.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-Hemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below the half-gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Al-Hemyari, K.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-Hemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below the half-gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Arnold, S.

Becher, C.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef] [PubMed]

Bennett, C. V.

Bi, W. G.

J. P. Zhang, D. Y. Chu, S. L. Wu, S. T. Ho, W. G. Bi, C. W. Tu, and R. C. Tiberio, “Photonic-wire laser,” Phys. Rev. Lett. 75, 2678–2681 (1995).
[CrossRef] [PubMed]

Blair, S.

Blom, F. C.

F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
[CrossRef]

Boyd, R. W.

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

V. B. Braginsky and V. S. Ilchenko, “Properties of optical dielectric microresonators,” Sov. Phys. Dokl. 32, 306–307 (1987).

Bruce, A. J.

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[CrossRef]

Cai, M.

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-galley mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
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Calhoun, L. C.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, “Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR Filters,” J. Lightwave Technol. 20, 900–905 (2002).
[CrossRef]

Cappuzzo, M. A.

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[CrossRef]

Chak, P.

Chang, R. K.

Chen, W.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Cheong, S.-W.

Cho, P. S.

Choi, S.

K. Djordjev, S. Choi, and P. D. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 828–830 (2002).
[CrossRef]

K. Djordjev, S. Choi, and P. D. Dapkus, “High-Q vertically coupled InP microdisk resonators,” IEEE Photon. Technol. Lett. 14, 331–333 (2002).
[CrossRef]

Christodoulides, D. N.

Chu, D. Y.

J. P. Zhang, D. Y. Chu, S. L. Wu, S. T. Ho, W. G. Bi, C. W. Tu, and R. C. Tiberio, “Photonic-wire laser,” Phys. Rev. Lett. 75, 2678–2681 (1995).
[CrossRef] [PubMed]

Chu, S. T.

B. E. Little and S. T. Chu, “Toward very large-scale integrated photonics,” Opt. Photonics News 11 (11), 24–29 (2000).
[CrossRef]

B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).
[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]

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]

B. E. Little and S. T. Chu, “Estimating surface roughness loss and output coupling in microdisk resonators,” Opt. Lett. 21, 1390–1392 (1996).
[CrossRef] [PubMed]

Cohen, D. S.

Dapkus, P. D.

K. Djordjev, S. Choi, and P. D. Dapkus, “High-Q vertically coupled InP microdisk resonators,” IEEE Photon. Technol. Lett. 14, 331–333 (2002).
[CrossRef]

K. Djordjev, S. Choi, and P. D. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 828–830 (2002).
[CrossRef]

de Sterke, C. M.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Djordjev, K.

K. Djordjev, S. Choi, and P. D. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 828–830 (2002).
[CrossRef]

K. Djordjev, S. Choi, and P. D. Dapkus, “High-Q vertically coupled InP microdisk resonators,” IEEE Photon. Technol. Lett. 14, 331–333 (2002).
[CrossRef]

Driessen, A.

F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
[CrossRef]

Dubreuil, N.

Efremidis, N. K.

Eggleton, B. J.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525–532 (2001).
[CrossRef]

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM Systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Fan, S.

Fields, M. H.

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]

Fox, A. M.

Gauthier, D. J.

R. W. Boyd and D. J. Gauthier, “Slow and fast light,” Prog. Opt. 33, 497–530 (2002).
[CrossRef]

Giles, C. R.

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM Systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

Goldhar, J.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs–AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14, 74–77 (2002).
[CrossRef]

Gomez, L. T.

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[CrossRef]

Gorodetsky, M. L.

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[CrossRef] [PubMed]

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

Grant, R. S.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-Hemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below the half-gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Griffel, G.

G. Griffel, “Synthesis of optical filters using ring resonator arrays,” IEEE Photon. Technol. Lett. 12, 810–812 (2000).
[CrossRef]

Grover, R.

J. E. Heebner, N. N. Lepeshkin, A. Schweinsberg, G. W. Wicks, R. W. Boyd, R. Grover, and P.-T. Ho, “Enhanced linear and nonlinear optical phase response of AlGaAs microring resonators,” Opt. Lett. 29, 769–771 (2004).
[CrossRef] [PubMed]

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs–AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14, 74–77 (2002).
[CrossRef]

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, “Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR Filters,” J. Lightwave Technol. 20, 900–905 (2002).
[CrossRef]

Hagness, S. C.

Harbold, J. M.

Hare, J.

Haus, H. A.

J.-P. Laine, B. E. Little, and H. A. Haus, “Etch-eroded fiber coupler for whispering-gallery-mode excitation in high-Q silica microspheres,” IEEE Photon. Technol. Lett. 11, 1429–1430 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express 3 (7), 4–11 (1998).
[CrossRef] [PubMed]

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]

Heebner, J. E.

Ho, P.-T.

J. E. Heebner, N. N. Lepeshkin, A. Schweinsberg, G. W. Wicks, R. W. Boyd, R. Grover, and P.-T. Ho, “Enhanced linear and nonlinear optical phase response of AlGaAs microring resonators,” Opt. Lett. 29, 769–771 (2004).
[CrossRef] [PubMed]

T. A. Ibrahim, V. Van, and P.-T. Ho, “All-optical time-division demultiplexing and spatial pulse routing with a GaAs/AlGaAs microring resonator,” Opt. Lett. 27, 803–805 (2002).
[CrossRef]

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, “Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR Filters,” J. Lightwave Technol. 20, 900–905 (2002).
[CrossRef]

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs–AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14, 74–77 (2002).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12, 320–322 (2000).
[CrossRef]

P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Jonekis, and P.-T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554–556 (2000).
[CrossRef]

Ho, S. T.

D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A. Stair, S. T. Ho, and R. C. Tiberio, “Waveguide-coupled AlGaAs/GaAs microcavity ring and disk resonators with high finesse and 21.6-nm free-spectral range,” Opt. Lett. 22, 1244–1246 (1997).
[CrossRef] [PubMed]

J. P. Zhang, D. Y. Chu, S. L. Wu, S. T. Ho, W. G. Bi, C. W. Tu, and R. C. Tiberio, “Photonic-wire laser,” Phys. Rev. Lett. 75, 2678–2681 (1995).
[CrossRef] [PubMed]

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, “Large nonlinear phase shifts in low-loss AlGaAs waveguides near half-gap,” Appl. Phys. Lett. 59, 2558–2560 (1991).
[CrossRef]

Hobson, W. S.

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, “Large nonlinear phase shifts in low-loss AlGaAs waveguides near half-gap,” Appl. Phys. Lett. 59, 2558–2560 (1991).
[CrossRef]

Hoekstra, H. J.

F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
[CrossRef]

Hryniewicz, J. V.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, “Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR Filters,” J. Lightwave Technol. 20, 900–905 (2002).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12, 320–322 (2000).
[CrossRef]

P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Jonekis, and P.-T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554–556 (2000).
[CrossRef]

B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).
[CrossRef]

Hu, E.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef] [PubMed]

Hwang, H. Y.

Ibanescu, M.

Ibrahim, T. A.

T. A. Ibrahim, V. Van, and P.-T. Ho, “All-optical time-division demultiplexing and spatial pulse routing with a GaAs/AlGaAs microring resonator,” Opt. Lett. 27, 803–805 (2002).
[CrossRef]

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, “Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR Filters,” J. Lightwave Technol. 20, 900–905 (2002).
[CrossRef]

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs–AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14, 74–77 (2002).
[CrossRef]

Ilchenko, V. S.

D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23, 247–249 (1998).
[CrossRef]

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[CrossRef] [PubMed]

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

V. B. Braginsky and V. S. Ilchenko, “Properties of optical dielectric microresonators,” Sov. Phys. Dokl. 32, 306–307 (1987).

Ilday, F. O.

Imamoglu, A.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef] [PubMed]

Ippen, E.

Ironside, C. N.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-Hemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below the half-gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Islam, M. N.

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, “Large nonlinear phase shifts in low-loss AlGaAs waveguides near half-gap,” Appl. Phys. Lett. 59, 2558–2560 (1991).
[CrossRef]

Joannopoulos, J. D.

Johnson, F. G.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs–AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14, 74–77 (2002).
[CrossRef]

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, “Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR Filters,” J. Lightwave Technol. 20, 900–905 (2002).
[CrossRef]

Johnson, S. G.

Jonekis, L. G.

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]

Kang, J.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-Hemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below the half-gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Katsufuji, T.

Kennedy, G. T.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-Hemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below the half-gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Kimble, H. J.

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[CrossRef] [PubMed]

Kiraz, A.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef] [PubMed]

Kivshar, Y.

Knight, J. C.

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]

Kolner, B. H.

Krauss, T. F.

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Laine, J.-P.

J.-P. Laine, B. E. Little, and H. A. Haus, “Etch-eroded fiber coupler for whispering-gallery-mode excitation in high-Q silica microspheres,” IEEE Photon. Technol. Lett. 11, 1429–1430 (1999).
[CrossRef]

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]

Lee, R. K.

Lefevre, V.

Lenz, G.

S. Spalter, H. Y. Wang, J. Zimmermann, G. Lenz, T. Katsufuji, S.-W. Cheong, and R. E. Slusher, “Strong self-phase modulation in planar chalcogenide glass waveguides,” Opt. Lett. 27, 363–365 (2002).
[CrossRef]

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525–532 (2001).
[CrossRef]

G. Lenz, J. Zimmermann, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spalter, R. E. Slusher, S.-W. Cheong, J. S. Sanghera, and I. D. Aggarwal, “Large Kerr effect in bulk Se-based chalcogenide glasses,” Opt. Lett. 25, 254–256 (2000).
[CrossRef]

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[CrossRef]

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM Systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10, 994–996 (1998).
[CrossRef]

Lepeshkin, N. N.

Leventhal, D. K.

Levi, A. F. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, “Large nonlinear phase shifts in low-loss AlGaAs waveguides near half-gap,” Appl. Phys. Lett. 59, 2558–2560 (1991).
[CrossRef]

Lin, C. H.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-Hemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below the half-gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Lin, H. H.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-Hemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below the half-gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Lines, M. E.

Little, B. E.

P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Jonekis, and P.-T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554–556 (2000).
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B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12, 320–322 (2000).
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B. E. Little and S. T. Chu, “Toward very large-scale integrated photonics,” Opt. Photonics News 11 (11), 24–29 (2000).
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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]

J.-P. Laine, B. E. Little, and H. A. Haus, “Etch-eroded fiber coupler for whispering-gallery-mode excitation in high-Q silica microspheres,” IEEE Photon. Technol. Lett. 11, 1429–1430 (1999).
[CrossRef]

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]

B. E. Little and S. T. Chu, “Estimating surface roughness loss and output coupling in microdisk resonators,” Opt. Lett. 21, 1390–1392 (1996).
[CrossRef] [PubMed]

Liu, C. T.

Logan, R. A.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Mabuchi, H.

Madsen, C. K.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525–532 (2001).
[CrossRef]

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[CrossRef]

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM Systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10, 994–996 (1998).
[CrossRef]

C. K. Madsen, “Efficient architectures for exactly realizing optical filters with optimum bandpass designs,” IEEE Photon. Technol. Lett. 10, 1136–1138 (1998).
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A. Melloni, F. Morichetti, and M. Martinelli, “Optical slow wave structures,” Opt. Photon. News 14, 44–48 (2003).
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S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
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A. Melloni, F. Morichetti, and M. Martinelli, “Optical slow wave structures,” Opt. Photon. News 14, 44–48 (2003).
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A. Melloni, “Synthesis of a parallel-coupled ring-resonator filter,” Opt. Lett. 26, 917–919 (2001).
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P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
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W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
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Mookherjea, S.

Morichetti, F.

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Nguyen, V. Q.

Notomi, M.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Orta, R.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple ring resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447–1449 (1995).
[CrossRef]

Painter, O.

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-galley mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[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]

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J. E. Heebner, R. W. Boyd, and Q. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
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J. E. Heebner, R. W. Boyd, and Q. Park, “SCISSOR solitons and other propagation effects in microresonator modified waveguides,” J. Opt. Soc. Am. B 19, 722–731 (2002).
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Pearton, S. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Pereira, S.

Petroff, P. M.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef] [PubMed]

Popma, Th. J. A.

F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
[CrossRef]

Popp, J.

Rafizadeh, D.

Rahn, M. D.

Ramsey, J. M.

Ritter, K.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs–AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14, 74–77 (2002).
[CrossRef]

Sandoghdar, V.

Sanghera, J. S.

Savchenkov, A. A.

Savi, P.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple ring resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447–1449 (1995).
[CrossRef]

Scherer, A.

Schoenfeld, W. V.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef] [PubMed]

Schweinsberg, A.

Scotti, R. E.

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[CrossRef]

Seaton, C.

G. I. Stegeman and C. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, 57 (1985).
[CrossRef]

Shaw, L. B.

Shinya, A.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Sibbett, W.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-Hemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below the half-gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Sipe, J. E.

Skolnick, M. S.

Slusher, R. E.

S. Spalter, H. Y. Wang, J. Zimmermann, G. Lenz, T. Katsufuji, S.-W. Cheong, and R. E. Slusher, “Strong self-phase modulation in planar chalcogenide glass waveguides,” Opt. Lett. 27, 363–365 (2002).
[CrossRef]

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525–532 (2001).
[CrossRef]

G. Lenz, J. Zimmermann, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spalter, R. E. Slusher, S.-W. Cheong, J. S. Sanghera, and I. D. Aggarwal, “Large Kerr effect in bulk Se-based chalcogenide glasses,” Opt. Lett. 25, 254–256 (2000).
[CrossRef]

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM Systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Y. Yamamoto and R. E. Slusher, “Optical processes in microcavities,” Phys. Today 46 (6), 66–74 (1993).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, “Large nonlinear phase shifts in low-loss AlGaAs waveguides near half-gap,” Appl. Phys. Lett. 59, 2558–2560 (1991).
[CrossRef]

Soccolich, C. E.

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, “Large nonlinear phase shifts in low-loss AlGaAs waveguides near half-gap,” Appl. Phys. Lett. 59, 2558–2560 (1991).
[CrossRef]

Soljacic, M.

Spalter, S.

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[CrossRef] [PubMed]

Stair, K. A.

Stegeman, G. I.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-Hemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below the half-gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

G. I. Stegeman and C. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, 57 (1985).
[CrossRef]

Streed, E. W.

Taflove, A.

Takahashi, C.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Takahashi, J.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Tascone, R.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple ring resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447–1449 (1995).
[CrossRef]

Tiberio, R. C.

Trinchero, D.

R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple ring resonator filters for optical systems,” IEEE Photon. Technol. Lett. 7, 1447–1449 (1995).
[CrossRef]

Tu, C. W.

J. P. Zhang, D. Y. Chu, S. L. Wu, S. T. Ho, W. G. Bi, C. W. Tu, and R. C. Tiberio, “Photonic-wire laser,” Phys. Rev. Lett. 75, 2678–2681 (1995).
[CrossRef] [PubMed]

Vahala, K.

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-galley mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[CrossRef] [PubMed]

Van, V.

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs–AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14, 74–77 (2002).
[CrossRef]

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, “Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR Filters,” J. Lightwave Technol. 20, 900–905 (2002).
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T. A. Ibrahim, V. Van, and P.-T. Ho, “All-optical time-division demultiplexing and spatial pulse routing with a GaAs/AlGaAs microring resonator,” Opt. Lett. 27, 803–805 (2002).
[CrossRef]

van Dijk, D. R.

F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
[CrossRef]

Vernooy, D. W.

Villeneuve, A.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-Hemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below the half-gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Villeneuve, P. R.

Wang, H. Y.

Whitten, W. B.

Wicks, G. W.

Wilson, R. A.

P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Jonekis, and P.-T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554–556 (2000).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12, 320–322 (2000).
[CrossRef]

Wise, F. W.

Wu, S. L.

J. P. Zhang, D. Y. Chu, S. L. Wu, S. T. Ho, W. G. Bi, C. W. Tu, and R. C. Tiberio, “Photonic-wire laser,” Phys. Rev. Lett. 75, 2678–2681 (1995).
[CrossRef] [PubMed]

Xu, Y.

Yamada, K.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Yamamoto, Y.

Y. Yamamoto and R. E. Slusher, “Optical processes in microcavities,” Phys. Today 46 (6), 66–74 (1993).
[CrossRef]

Yang, C. C.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-Hemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below the half-gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Yariv, A.

Yokohama, I.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Zhang, J. P.

Zhang, L.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef] [PubMed]

Zimmermann, J.

Appl. Phys. Lett. (3)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

F. C. Blom, D. R. van Dijk, H. J. Hoekstra, A. Driessen, and Th. J. A. Popma, “Experimental study of integrated-optics microcavity resonators: toward an all-optical switching device,” Appl. Phys. Lett. 71, 747–749 (1997).
[CrossRef]

S. T. Ho, C. E. Soccolich, M. N. Islam, W. S. Hobson, A. F. J. Levi, and R. E. Slusher, “Large nonlinear phase shifts in low-loss AlGaAs waveguides near half-gap,” Appl. Phys. Lett. 59, 2558–2560 (1991).
[CrossRef]

IEEE J. Quantum Electron. (2)

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM Systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525–532 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (11)

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999).
[CrossRef]

K. Djordjev, S. Choi, and P. D. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 828–830 (2002).
[CrossRef]

C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10, 994–996 (1998).
[CrossRef]

V. Van, T. A. Ibrahim, K. Ritter, P. P. Absil, F. G. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs–AlGaAs microring resonators,” IEEE Photon. Technol. Lett. 14, 74–77 (2002).
[CrossRef]

K. Djordjev, S. Choi, and P. D. Dapkus, “High-Q vertically coupled InP microdisk resonators,” IEEE Photon. Technol. Lett. 14, 331–333 (2002).
[CrossRef]

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Other (4)

We define the finesse as the free spectral range (FSR) divided by the full width at half-depth (FWHD) of the resonance peak. Applying this definition to either the normalized group delay or the intensity buildup of an all-pass resonator, the finesse is calculated as F =FSR FWHD =2π 2 arccos 2r 1+r2 r≈1 π 1−r. In the case of the add–drop resonator, r is replaced with r1 r2.

Strictly speaking, the first configuration is constructed from two-port resonators and is a special case.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, Calif., 2001).

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, San Diego, Calif., 2003).

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Figures (12)

Fig. 1
Fig. 1

Illustrations of four archetypal ring-resonator unit cells for (a) a CROW, (b) a single-channel SCISSOR, (c) a double-channel SCISSOR, and (d) a twisted double-channel SCISSOR, along with their cavity versions. These unit cells can serve as building blocks for constructing sequences of resonators that serve as novel photonic guidance architectures.

Fig. 2
Fig. 2

Amplitude transmission (solid curves) and effective phase shift (dashed curves) for (a) an all-pass resonator, (c) through port of an add-drop resonator, and (e) drop port of an add–drop resonator. Plots (b), (d), and (f) display the coherent intensity buildup (solid curves) and group delay normalized with respect to single-pass transit time (dashed curves) for the same ports as in (a), (c), and (e), respectively. The independent variable on all plots is the normalized detuning (radian frequency multiplied by ring transit time), and all coupling coefficients are t2=0.1814.

Fig. 3
Fig. 3

Schematic depicting the fields associated with a resonator-to-waveguide coupling point. The field amplitudes across the coupling point are related by the self- and cross-coupling coefficients (r and t) as described in the text.

Fig. 4
Fig. 4

Dispersion relation, normalized group index, and GVD for (a) a low-finesse and (b) a high-finesse coupled-resonator optical waveguide. The dispersion relation is analogous to that of a multilayered structure with alternating layer indices and/or thicknesses. Bandgaps are always of the direct type and result from distributed Bragg reflection. Parameters include a refractive index of n=3.1, alternating radii of R1=2.5 µm, and R2=1.5R1. Resonances mR1=31 and 32 and mR2=46, 47, and 48 are shown. In (a), a high coupling strength, t2=0.75, results in narrow bandgaps, while in (b), a low coupling strength, t2=0.1814, results in wide bandgaps. These qualitative features are directly opposite those found in the double-channel SCISSOR.

Fig. 5
Fig. 5

Dispersion relation, normalized group index, and GVD for (a) a low-finesse and (b) a high-finesse single-channel SCISSOR. Note that the dispersion relation does not display photonic bandgaps. Nevertheless, at the resonances (λm=2πnR/m), the group index (and the intensity buildup) is maximized. Parameters include a refractive index of n=3.1 and a radius of R=2.5 µm. Resonances mR=31 and 32 at 1.571 µm and 1.522 µm are shown. In (a), a high coupling strength, t2=0.75, results in a wide bandwidth, while in (b), a low coupling strength, t2=0.1814, results in a narrow bandwidth. To avoid redundancy, and because the forward- and backward-traveling waves do not couple, only the dispersion relation for the forward-traveling wave is shown.

Fig. 6
Fig. 6

Dispersion relation, normalized group index, and GVD for (a) a low-finesse and (b) high-finesse double-channel SCISSOR. Note that unlike the dispersion relation for the single-guide SCISSOR, the double-guide variety displays photonic bandgaps. Two qualitatively different bandgaps manifest themselves. For spectral components satisfying the Bragg condition (λmB=2nL/mB), the bandgap is direct and results from distributed Bragg reflection. At the resonances of the rings (λmR=2πnR/mR), the bandgap is indirect and results from strong resonator-mediated backcoupling. Parameters were chosen such that one Bragg gap was coincident with one resonator gap within the figure: refractive index n=3.1, radii R=2.5 µm, and spacing L=1.5πR. Resonator resonances mR=31 and 32 and Bragg resonances mB=46, 47 and 48 are shown. The coincident resonator (mR=32) and Bragg (mB=48) resonance results in a wide direct gap. In (a), a high coupling strength, t2=0.75, results in wide bandgaps, while in (b), a low coupling strength, t2=0.1814, results in narrow bandgaps.

Fig. 7
Fig. 7

Dispersion relation, normalized group index, and GVD for (a) a low-finesse and (b) a high-finesse twisted double-channel SCISSOR. Bandgaps are absent in the dispersion relation, which resembles that of the single-channel SCISSOR, but with the presence of a second branch. The two branches correspond to the two decoupled forward-traveling normal modes. Near the ring resonances (λm=2πnR/m), the two branches are strongly coupled, as in the case of a directional coupler. Parameters used are the same as in Fig. 6 except that there are two resonators, each half in circumference and 100% coupled. Resonances mR=31 and 32 at 1.571 µm and 1.522 µm are shown. In (a), a high coupling strength, t2=0.75, results in wide-bandwidth channel-to-channel coupling, while in (b) a low coupling strength, t2=0.1814, results in narrow-bandwidth channel-to-channel coupling. To avoid redundancy, and because the two forward- and two backward-traveling waves do not couple, only the two forward-traveling dispersion relation branches are shown.

Fig. 8
Fig. 8

Qualitative comparison of the transmission properties of three structures possessing bandgaps: (a) a Fabry-Perot, (b) a multilayered stack, (c) an add–drop resonator, (d) a CROW, (e) a double-channel resonator, and (f) an add–drop resonator (reoriented). Note that the qualitative features of the transmission peaks and valleys are equivalent for multilayered stacks and CROWs but reversed for double-channel SCISSORs. With increasing finesse, the bandgap widths increase in both multilayered stacks and CROWs, while they decrease in double-channel SCISSORs.

Fig. 9
Fig. 9

Transmission spectra for a low-finesse (t2=0.75) finite double-channel SCISSOR and finite CROW. Parameters are the same as in Figs. 4 and 6. Here, five unit cells are used to approximate the structures. For comparison, the transmission for a single resonator is shown in each case, and shaded regions correspond to one-dimensional photonic bandgaps in the corresponding infinite structure. The labels B and R correspond to the double-channel SCISSOR’s Bragg and resonator gaps, while the labels R1 and R2 correspond to the alternating CROW resonances.

Fig. 10
Fig. 10

Transmission spectra for a high-finesse (t2=0.1814) finite double-channel SCISSOR and finite one-dimensional CROW. Parameters are the same as in Figs. 4 and 6. Here, five unit cells are used to approximate the structures. For comparison, the transmission for a single resonator is shown in each case, and shaded regions correspond to one-dimensional photonic bandgaps in the corresponding infinite structure. The labels B and R correspond to the double-channel SCISSOR’s Bragg and resonator gaps, while the labels R1 and R2 correspond to the alternating CROW resonances.

Fig. 11
Fig. 11

Inherent trade-off between bandwidth and energy required to achieve a π nonlinear phase shift per resonator in a single-channel SCISSOR structure. The diagonal lines correspond to constant resonator diameter for AlGaAs or chalcogenide-based systems near 1.55 µm. Increasing finesse is directly proportional to decreasing energy.

Fig. 12
Fig. 12

Nonlinear pulse simulation assuming a Kerr nonlinearity in a double-channel SCISSOR structure with ten unit cells. A schematic of the structure is shown in (a). A 100-ps Gaussian (FWHM) pulse is injected into the two-channel structure at the input port (lower channel). Parameters include a refractive index of 3.1, radii of R=4.1 µm, and spacing L=16 µm. The apodization profile is discussed in the text. The carrier wavelength of the pulse is at λ=1.58 µm, which is close to the 51st resonance of the resonator. A plot of the linear transmission spectrum of the structure is displayed in (b). The carrier wavelength of the input pulse is indicated by a black dot in the figure. A plot of the transmission versus peak input intensity for the Gaussian pulse is displayed in (c). Notice that the switching threshold is at approximately 15 MW/cm2 and may be obtained with picosecond pulses with energies less than 0.15 pJ when the effective areas of the guides are less than one square micron.

Tables (1)

Tables Icon

Table 1 Matrix Elements Mij, for the Four Coupled Microresonator Unit Cells Shown in Fig. 1a

Equations (16)

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E3=rE1+itE2,
E4=itE1+rE2.
Aj+1Bj+1=M11M12M21M22AjBj,
Aj+1Bj+1=exp(ikeffL)AjBj.
detM11-exp(ikeffL)M12M21M22-exp(ikeffL)=0.
exp(i2keffL)-(M11+M22)exp(ikeffL)+(M11M22
-M12M21)=0,
keff=1L arg(M11+M22)2±(M11-M22)24+M12M211/2.
keff=-i2(R1+R2) arg-1t2 cosϕ1+ϕ22+rt2 cosϕ2-ϕ12±-1t2 cosϕ1+ϕ22+rt2 cosϕ2-ϕ122-11/2,
keff=ncω+1L argr-exp(iϕ)1-r exp(iϕ)=ncω+Φ(ω)L,
keff=1L arg12 1-r1r2 exp(-iϕ)r1-r2 exp(-iϕ) exp(iθ)+12 1-r1r2 exp(iϕ)r2-r1 exp(iϕ) exp(-iθ)±12 1-r1r2 exp(-iϕ)r1-r2 exp(-iϕ) exp(iθ)-12 1-r1r2 exp(iϕ)r2-r1 exp(iϕ) exp(-iθ)2+(1-r12)(1-r22)[r2-r1 exp(iϕ)][r1-r2 exp(-iϕ)]1/2,
keff=1L argexp(iθ)+r1-r2 exp(iϕ)1-r1r2 exp(iϕ)+r2-r1 exp(iϕ)1-r1r2 exp(iϕ)2±{[r1-r2 exp(iϕ)]-[r2-r1 exp(iϕ)]}2+4(1-r12)(1-r22)exp(iϕ)4[1-r1r2 exp(iϕ)]21/2,
Δν=c/(2πRnF).
PπλAeff4F2n2R.
Eπ=λ3π ln(2)16Fnn2c.
F=FSRFWHD=2π2 arccos2r1+r2r1π1-r.

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