Abstract

In contrast to recent reports of localization-impaired transport in long slow-light waveguides, we demonstrate light transport in silicon coupled-resonator optical waveguides (CROWs) consisting of up to 235 coupled microrings without localization over frequency bands that are several hundred gigahertz wide. Furthermore, from the unique statistical signatures provided by time-domain propagation delay measurements, we demonstrate the spectrally correlated nature of light propagation in CROWs.

© 2010 OSA

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  33. H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
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  36. J. A. Sánchez-Gil, V. Freilikher, I. Yurkevich, and A. A. Maradudin, “Coexistence of Ballistic Transport, Diffusion, and Localization in Surface Disordered Waveguides,” Phys. Rev. Lett. 80(5), 948–951 (1998).
    [CrossRef]
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    [CrossRef]
  39. M. L. Cooper, G. Gupta, J. S. Park, M. A. Schneider, I. B. Divliansky, and S. Mookherjea, “Quantitative infrared imaging of silicon-on-insulator microring resonators,” Opt. Lett. 35(5), 784–786 (2010).
    [CrossRef] [PubMed]
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    [CrossRef]
  41. A. A. Chabanov and A. Z. Genack, “Statistics of dynamics of localized waves,” Phys. Rev. Lett. 87(23), 233903 (2001).
    [CrossRef] [PubMed]
  42. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer Microring Coupled-Resonator Optical Waveguides,” J. Lightwave Technol. 24(4), 1843–1849 (2006).
    [CrossRef]
  43. C. K. Madsen and J. H. Zhao, “Postfabrication optimization of an autoregressive planar waveguide lattice filter,” Appl. Opt. 36(3), 642–647 (1997).
    [CrossRef] [PubMed]
  44. T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15(11), 2107–2113 (1997).
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  47. B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(6), 7166–7172 (1999).
    [CrossRef]
  48. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
    [CrossRef] [PubMed]
  49. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12(1), 90–103 (2004).
    [CrossRef] [PubMed]

2010

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010).
[CrossRef] [PubMed]

S. Mazoyer, P. Lalanne, J. C. Rodier, J. P. Hugonin, M. Spasenović, L. Kuipers, D. M. Beggs, and T. F. Krauss, “Statistical fluctuations of transmission in slow light photonic-crystal waveguides,” Opt. Express 18(14), 14654–14663 (2010).
[CrossRef] [PubMed]

H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
[CrossRef] [PubMed]

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, D. K. Gifford, and S. Mookherjea, “Waveguide dispersion effects in silicon-on-insulator coupled-resonator optical waveguides,” Opt. Lett. 35(18), 3030–3032 (2010).
[CrossRef] [PubMed]

M. L. Cooper, G. Gupta, J. S. Park, M. A. Schneider, I. B. Divliansky, and S. Mookherjea, “Quantitative infrared imaging of silicon-on-insulator microring resonators,” Opt. Lett. 35(5), 784–786 (2010).
[CrossRef] [PubMed]

2009

X. Luo and A. W. Poon, “Many-element coupled-resonator optical waveguides using gapless-coupled microdisk resonators,” Opt. Express 17(26), 23617–23628 (2009).
[CrossRef]

C. Ferrari, F. Morichetti, and A. Melloni, “Disorder in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B 26(4), 858–866 (2009).
[CrossRef]

2008

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Y. A. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[CrossRef]

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008).
[CrossRef] [PubMed]

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
[CrossRef]

D. Dai, L. Yang, and S. He, “Ultrasmall Thermally Tunable Microring Resonator With a Submicrometer Heater on Si Nanowires,” J. Lightwave Technol. 26(6), 704–709 (2008).
[CrossRef]

S. Mookherjea, J. S. Park, S. H. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photonics 2(2), 90–93 (2008).
[CrossRef]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
[CrossRef]

2007

J. Topolancik, B. Ilic, and F. Vollmer, “Experimental observation of strong photon localization in disordered photonic crystal waveguides,” Phys. Rev. Lett. 99(25), 253901 (2007).
[CrossRef]

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

2006

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]

F. Morichetti, A. Melloni, and M. Martinelli, “Effects of polarization rotation in optical ring-resonator-based devices,” J. Lightwave Technol. 24(1), 573–585 (2006).
[CrossRef]

J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer Microring Coupled-Resonator Optical Waveguides,” J. Lightwave Technol. 24(4), 1843–1849 (2006).
[CrossRef]

2005

T. Barwicz and H. A. Haus, “Three-dimensional analysis of scattering losses due to sidewall roughness, in microphotonic waveguides,” J. Lightwave Technol. 23(9), 2719–2732 (2005).
[CrossRef]

D. K. Gifford, B. J. Soller, M. S. Wolfe, and M. E. Froggatt, “Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion,” Appl. Opt. 44(34), 7282–7286 (2005).
[CrossRef] [PubMed]

P. Markoš and C. M. Soukoulis, “Intensity distribution of scalar waves propagating in random media,” Phys. Rev. B 71(5), 054201 (2005).
[CrossRef]

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005).
[CrossRef]

J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22(5), 1062–1074 (2005).
[CrossRef]

2004

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B 21(10), 1818–1832 (2004).
[CrossRef]

A. Melloni, F. Morichetti, and M. Martinelli, “Polarization conversion in ring resonator phase shifters,” Opt. Lett. 29(23), 2785–2787 (2004).
[CrossRef] [PubMed]

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by sidewall roughness in ultrasmall SOI waveguides,” IEEE Photon. Technol. Lett. 16(7), 1661–1663 (2004).
[CrossRef]

J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12(1), 90–103 (2004).
[CrossRef] [PubMed]

2003

M. H. Eiselt, C. B. Clausen, and R. W. Tkach, “Performance characterization of components with group delay fluctuations,” IEEE Photon. Technol. Lett. 15(8), 1076–1078 (2003).
[CrossRef]

G. T. Paloczi, Y. Y. Huang, A. Yariv, and S. Mookherjea, “Polymeric Mach-Zehnder interferometer using serially coupled microring resonators,” Opt. Express 11(21), 2666–2671 (2003).
[CrossRef] [PubMed]

2002

H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
[CrossRef]

2001

A. A. Chabanov and A. Z. Genack, “Statistics of dynamics of localized waves,” Phys. Rev. Lett. 87(23), 233903 (2001).
[CrossRef] [PubMed]

1999

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(6), 7166–7172 (1999).
[CrossRef]

M. Stoytchev and A. Z. Genack, “Observations of non-Rayleigh statistics in the approach to photon localization,” Opt. Lett. 24(4), 262–264 (1999).
[CrossRef]

C. Texier and A. Comtet, “Universality of the Wigner time delay distribution for one-dimensional random potentials,” Phys. Rev. Lett. 82(21), 4220–4223 (1999).
[CrossRef]

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, “Statistics of Wave Dynamics in Random Media,” Phys. Rev. Lett. 82(4), 715–718 (1999).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
[CrossRef]

1998

J. A. Sánchez-Gil, V. Freilikher, I. Yurkevich, and A. A. Maradudin, “Coexistence of Ballistic Transport, Diffusion, and Localization in Surface Disordered Waveguides,” Phys. Rev. Lett. 80(5), 948–951 (1998).
[CrossRef]

1997

C. K. Madsen and J. H. Zhao, “Postfabrication optimization of an autoregressive planar waveguide lattice filter,” Appl. Opt. 36(3), 642–647 (1997).
[CrossRef] [PubMed]

T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15(11), 2107–2113 (1997).
[CrossRef]

1995

E. Kogan and M. Kaveh, “Random-matrix-theory approach to the intensity distributions of waves propagating in a random medium,” Phys. Rev. B Condens. Matter 52(6), R3813–R3815 (1995).
[CrossRef] [PubMed]

1994

R. Adar, C. H. Henry, M. A. Milbrodt, and R. C. Kistler, “Phase coherence of optical waveguides,” J. Lightwave Technol. 12(4), 603–606 (1994).
[CrossRef]

1967

B. Rosén, “On the central limit theorem for sums of dependent random variables,” Probab. Theory Relat. Fields 7, 48–82 (1967).

Adar, R.

R. Adar, C. H. Henry, M. A. Milbrodt, and R. C. Kistler, “Phase coherence of optical waveguides,” J. Lightwave Technol. 12(4), 603–606 (1994).
[CrossRef]

Assefa, S.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, D. K. Gifford, and S. Mookherjea, “Waveguide dispersion effects in silicon-on-insulator coupled-resonator optical waveguides,” Opt. Lett. 35(18), 3030–3032 (2010).
[CrossRef] [PubMed]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
[CrossRef]

Baets, R.

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005).
[CrossRef]

Bandaru, P. R.

S. Mookherjea, J. S. Park, S. H. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photonics 2(2), 90–93 (2008).
[CrossRef]

Barwicz, T.

T. Barwicz and H. A. Haus, “Three-dimensional analysis of scattering losses due to sidewall roughness, in microphotonic waveguides,” J. Lightwave Technol. 23(9), 2719–2732 (2005).
[CrossRef]

Beggs, D. M.

S. Mazoyer, P. Lalanne, J. C. Rodier, J. P. Hugonin, M. Spasenović, L. Kuipers, D. M. Beggs, and T. F. Krauss, “Statistical fluctuations of transmission in slow light photonic-crystal waveguides,” Opt. Express 18(14), 14654–14663 (2010).
[CrossRef] [PubMed]

Bogaerts, W.

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005).
[CrossRef]

Boyd, R. W.

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B 21(10), 1818–1832 (2004).
[CrossRef]

Cassan, E.

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by sidewall roughness in ultrasmall SOI waveguides,” IEEE Photon. Technol. Lett. 16(7), 1661–1663 (2004).
[CrossRef]

Chabanov, A. A.

A. A. Chabanov and A. Z. Genack, “Statistics of dynamics of localized waves,” Phys. Rev. Lett. 87(23), 233903 (2001).
[CrossRef] [PubMed]

Chak, P.

J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B 21(10), 1818–1832 (2004).
[CrossRef]

Chotard, H.

H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
[CrossRef]

Christodoulides, D. N.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Clausen, C. B.

M. H. Eiselt, C. B. Clausen, and R. W. Tkach, “Performance characterization of components with group delay fluctuations,” IEEE Photon. Technol. Lett. 15(8), 1076–1078 (2003).
[CrossRef]

Comtet, A.

C. Texier and A. Comtet, “Universality of the Wigner time delay distribution for one-dimensional random potentials,” Phys. Rev. Lett. 82(21), 4220–4223 (1999).
[CrossRef]

Cooper, M. L.

M. L. Cooper, G. Gupta, J. S. Park, M. A. Schneider, I. B. Divliansky, and S. Mookherjea, “Quantitative infrared imaging of silicon-on-insulator microring resonators,” Opt. Lett. 35(5), 784–786 (2010).
[CrossRef] [PubMed]

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, D. K. Gifford, and S. Mookherjea, “Waveguide dispersion effects in silicon-on-insulator coupled-resonator optical waveguides,” Opt. Lett. 35(18), 3030–3032 (2010).
[CrossRef] [PubMed]

Dai, D.

D. Dai, L. Yang, and S. He, “Ultrasmall Thermally Tunable Microring Resonator With a Submicrometer Heater on Si Nanowires,” J. Lightwave Technol. 26(6), 704–709 (2008).
[CrossRef]

DeRose, G. 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]

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C. Ferrari, F. Morichetti, and A. Melloni, “Disorder in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B 26(4), 858–866 (2009).
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A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008).
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F. Morichetti, A. Melloni, and M. Martinelli, “Effects of polarization rotation in optical ring-resonator-based devices,” J. Lightwave Technol. 24(1), 573–585 (2006).
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A. Melloni, F. Morichetti, and M. Martinelli, “Polarization conversion in ring resonator phase shifters,” Opt. Lett. 29(23), 2785–2787 (2004).
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H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
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H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
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J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12(1), 90–103 (2004).
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M. L. Cooper, G. Gupta, J. S. Park, M. A. Schneider, I. B. Divliansky, and S. Mookherjea, “Quantitative infrared imaging of silicon-on-insulator microring resonators,” Opt. Lett. 35(5), 784–786 (2010).
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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).
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J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer Microring Coupled-Resonator Optical Waveguides,” J. Lightwave Technol. 24(4), 1843–1849 (2006).
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J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12(1), 90–103 (2004).
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H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
[CrossRef] [PubMed]

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S. Mazoyer, P. Lalanne, J. C. Rodier, J. P. Hugonin, M. Spasenović, L. Kuipers, D. M. Beggs, and T. F. Krauss, “Statistical fluctuations of transmission in slow light photonic-crystal waveguides,” Opt. Express 18(14), 14654–14663 (2010).
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[CrossRef]

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P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005).
[CrossRef]

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L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010).
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A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
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J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12(1), 90–103 (2004).
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M. L. Cooper, G. Gupta, J. S. Park, M. A. Schneider, I. B. Divliansky, and S. Mookherjea, “Quantitative infrared imaging of silicon-on-insulator microring resonators,” Opt. Lett. 35(5), 784–786 (2010).
[CrossRef] [PubMed]

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, D. K. Gifford, and S. Mookherjea, “Waveguide dispersion effects in silicon-on-insulator coupled-resonator optical waveguides,” Opt. Lett. 35(18), 3030–3032 (2010).
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A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, “Statistics of Wave Dynamics in Random Media,” Phys. Rev. Lett. 82(4), 715–718 (1999).
[CrossRef]

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(6), 7166–7172 (1999).
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F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007).
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H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
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Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

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J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B 21(10), 1818–1832 (2004).
[CrossRef]

Smolka, S.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010).
[CrossRef] [PubMed]

Soller, B. J.

D. K. Gifford, B. J. Soller, M. S. Wolfe, and M. E. Froggatt, “Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion,” Appl. Opt. 44(34), 7282–7286 (2005).
[CrossRef] [PubMed]

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Soukoulis, C. M.

P. Markoš and C. M. Soukoulis, “Intensity distribution of scalar waves propagating in random media,” Phys. Rev. B 71(5), 054201 (2005).
[CrossRef]

Spasenovic, M.

S. Mazoyer, P. Lalanne, J. C. Rodier, J. P. Hugonin, M. Spasenović, L. Kuipers, D. M. Beggs, and T. F. Krauss, “Statistical fluctuations of transmission in slow light photonic-crystal waveguides,” Opt. Express 18(14), 14654–14663 (2010).
[CrossRef] [PubMed]

Stobbe, S.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010).
[CrossRef] [PubMed]

Stoytchev, M.

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, “Statistics of Wave Dynamics in Random Media,” Phys. Rev. Lett. 82(4), 715–718 (1999).
[CrossRef]

M. Stoytchev and A. Z. Genack, “Observations of non-Rayleigh statistics in the approach to photon localization,” Opt. Lett. 24(4), 262–264 (1999).
[CrossRef]

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(6), 7166–7172 (1999).
[CrossRef]

Sugita, A.

T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15(11), 2107–2113 (1997).
[CrossRef]

Suzuki, S.

T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15(11), 2107–2113 (1997).
[CrossRef]

Tanabe, T.

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
[CrossRef]

Texier, C.

C. Texier and A. Comtet, “Universality of the Wigner time delay distribution for one-dimensional random potentials,” Phys. Rev. Lett. 82(21), 4220–4223 (1999).
[CrossRef]

Thyrrestrup, H.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010).
[CrossRef] [PubMed]

Tkach, R. W.

M. H. Eiselt, C. B. Clausen, and R. W. Tkach, “Performance characterization of components with group delay fluctuations,” IEEE Photon. Technol. Lett. 15(8), 1076–1078 (2003).
[CrossRef]

Topolancik, J.

J. Topolancik, B. Ilic, and F. Vollmer, “Experimental observation of strong photon localization in disordered photonic crystal waveguides,” Phys. Rev. Lett. 99(25), 253901 (2007).
[CrossRef]

Trepanier, F.

H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
[CrossRef]

Van Thourhout, D.

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005).
[CrossRef]

van Tiggelen, B. A.

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, “Statistics of Wave Dynamics in Random Media,” Phys. Rev. Lett. 82(4), 715–718 (1999).
[CrossRef]

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, and A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(6), 7166–7172 (1999).
[CrossRef]

Varghese, L. T.

H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
[CrossRef] [PubMed]

Vivien, L.

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by sidewall roughness in ultrasmall SOI waveguides,” IEEE Photon. Technol. Lett. 16(7), 1661–1663 (2004).
[CrossRef]

Vlasov, Y.

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

Vlasov, Y. A.

Y. A. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[CrossRef]

Vollmer, F.

J. Topolancik, B. Ilic, and F. Vollmer, “Experimental observation of strong photon localization in disordered photonic crystal waveguides,” Phys. Rev. Lett. 99(25), 253901 (2007).
[CrossRef]

Wolfe, M. S.

D. K. Gifford, B. J. Soller, M. S. Wolfe, and M. E. Froggatt, “Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion,” Appl. Opt. 44(34), 7282–7286 (2005).
[CrossRef] [PubMed]

Xia, F.

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, D. K. Gifford, and S. Mookherjea, “Waveguide dispersion effects in silicon-on-insulator coupled-resonator optical waveguides,” Opt. Lett. 35(18), 3030–3032 (2010).
[CrossRef] [PubMed]

Y. A. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[CrossRef]

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

Xu, Y.

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
[CrossRef]

Xuan, Y.

H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
[CrossRef] [PubMed]

Yang, L.

D. Dai, L. Yang, and S. He, “Ultrasmall Thermally Tunable Microring Resonator With a Submicrometer Heater on Si Nanowires,” J. Lightwave Technol. 26(6), 704–709 (2008).
[CrossRef]

Yang, S. H.

S. Mookherjea, J. S. Park, S. H. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photonics 2(2), 90–93 (2008).
[CrossRef]

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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]

J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer Microring Coupled-Resonator Optical Waveguides,” J. Lightwave Technol. 24(4), 1843–1849 (2006).
[CrossRef]

J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12(1), 90–103 (2004).
[CrossRef] [PubMed]

G. T. Paloczi, Y. Y. Huang, A. Yariv, and S. Mookherjea, “Polymeric Mach-Zehnder interferometer using serially coupled microring resonators,” Opt. Express 11(21), 2666–2671 (2003).
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A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
[CrossRef]

Yurkevich, I.

J. A. Sánchez-Gil, V. Freilikher, I. Yurkevich, and A. A. Maradudin, “Coexistence of Ballistic Transport, Diffusion, and Localization in Surface Disordered Waveguides,” Phys. Rev. Lett. 80(5), 948–951 (1998).
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C. K. Madsen and J. H. Zhao, “Postfabrication optimization of an autoregressive planar waveguide lattice filter,” Appl. Opt. 36(3), 642–647 (1997).
[CrossRef] [PubMed]

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H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
[CrossRef] [PubMed]

Zhu, L.

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]

J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer Microring Coupled-Resonator Optical Waveguides,” J. Lightwave Technol. 24(4), 1843–1849 (2006).
[CrossRef]

Appl. Opt.

D. K. Gifford, B. J. Soller, M. S. Wolfe, and M. E. Froggatt, “Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion,” Appl. Opt. 44(34), 7282–7286 (2005).
[CrossRef] [PubMed]

C. K. Madsen and J. H. Zhao, “Postfabrication optimization of an autoregressive planar waveguide lattice filter,” Appl. Opt. 36(3), 642–647 (1997).
[CrossRef] [PubMed]

IEEE Photon. Technol. Lett.

M. H. Eiselt, C. B. Clausen, and R. W. Tkach, “Performance characterization of components with group delay fluctuations,” IEEE Photon. Technol. Lett. 15(8), 1076–1078 (2003).
[CrossRef]

H. Chotard, Y. Painchaud, A. Mailloux, M. Morin, F. Trepanier, and M. Guy, “Group delay ripple of cascaded Bragg grating gain flattening filters,” IEEE Photon. Technol. Lett. 14(8), 1130–1132 (2002).
[CrossRef]

F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by sidewall roughness in ultrasmall SOI waveguides,” IEEE Photon. Technol. Lett. 16(7), 1661–1663 (2004).
[CrossRef]

P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photon. Technol. Lett. 17(6), 1199–1201 (2005).
[CrossRef]

J. Lightwave Technol.

D. Dai, L. Yang, and S. He, “Ultrasmall Thermally Tunable Microring Resonator With a Submicrometer Heater on Si Nanowires,” J. Lightwave Technol. 26(6), 704–709 (2008).
[CrossRef]

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[CrossRef]

T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15(11), 2107–2113 (1997).
[CrossRef]

J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Polymer Microring Coupled-Resonator Optical Waveguides,” J. Lightwave Technol. 24(4), 1843–1849 (2006).
[CrossRef]

J. Opt. Soc. Am. B

C. Ferrari, F. Morichetti, and A. Melloni, “Disorder in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B 26(4), 858–866 (2009).
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J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22(5), 1062–1074 (2005).
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J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B 21(10), 1818–1832 (2004).
[CrossRef]

Nat. Photonics

Y. A. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics 2(4), 242–246 (2008).
[CrossRef]

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
[CrossRef]

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

S. Mookherjea, J. S. Park, S. H. Yang, and P. R. Bandaru, “Localization in silicon nanophotonic slow-light waveguides,” Nat. Photonics 2(2), 90–93 (2008).
[CrossRef]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008).
[CrossRef]

Opt. Express

S. Mazoyer, P. Lalanne, J. C. Rodier, J. P. Hugonin, M. Spasenović, L. Kuipers, D. M. Beggs, and T. F. Krauss, “Statistical fluctuations of transmission in slow light photonic-crystal waveguides,” Opt. Express 18(14), 14654–14663 (2010).
[CrossRef] [PubMed]

G. T. Paloczi, Y. Y. Huang, A. Yariv, and S. Mookherjea, “Polymeric Mach-Zehnder interferometer using serially coupled microring resonators,” Opt. Express 11(21), 2666–2671 (2003).
[CrossRef] [PubMed]

X. Luo and A. W. Poon, “Many-element coupled-resonator optical waveguides using gapless-coupled microdisk resonators,” Opt. Express 17(26), 23617–23628 (2009).
[CrossRef]

H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18(17), 18067–18076 (2010).
[CrossRef] [PubMed]

J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12(1), 90–103 (2004).
[CrossRef] [PubMed]

Opt. Lett.

M. L. Cooper, G. Gupta, J. S. Park, M. A. Schneider, I. B. Divliansky, and S. Mookherjea, “Quantitative infrared imaging of silicon-on-insulator microring resonators,” Opt. Lett. 35(5), 784–786 (2010).
[CrossRef] [PubMed]

M. Stoytchev and A. Z. Genack, “Observations of non-Rayleigh statistics in the approach to photon localization,” Opt. Lett. 24(4), 262–264 (1999).
[CrossRef]

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008).
[CrossRef] [PubMed]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
[CrossRef]

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]

M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, D. K. Gifford, and S. Mookherjea, “Waveguide dispersion effects in silicon-on-insulator coupled-resonator optical waveguides,” Opt. Lett. 35(18), 3030–3032 (2010).
[CrossRef] [PubMed]

A. Melloni, F. Morichetti, and M. Martinelli, “Polarization conversion in ring resonator phase shifters,” Opt. Lett. 29(23), 2785–2787 (2004).
[CrossRef] [PubMed]

Phys. Rev. B

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[CrossRef]

Phys. Rev. Lett.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
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[CrossRef]

C. Texier and A. Comtet, “Universality of the Wigner time delay distribution for one-dimensional random potentials,” Phys. Rev. Lett. 82(21), 4220–4223 (1999).
[CrossRef]

A. Z. Genack, P. Sebbah, M. Stoytchev, and B. A. van Tiggelen, “Statistics of Wave Dynamics in Random Media,” Phys. Rev. Lett. 82(4), 715–718 (1999).
[CrossRef]

J. Topolancik, B. Ilic, and F. Vollmer, “Experimental observation of strong photon localization in disordered photonic crystal waveguides,” Phys. Rev. Lett. 99(25), 253901 (2007).
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L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes,” Science 327(5971), 1352–1355 (2010).
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Figures (8)

Fig. 1
Fig. 1

Coupled-resonator optical waveguides (CROWs). a, Silicon microring CROWs consisting of 35, 65, 95, 135 and 235 coupled microrings were fabricated using a CMOS compatible process on 200 mm silicon-on-insulator wafers and cleaved into 4 mm long chips, as shown in this optical microscope image. b, Scanning electron micrographs of selected regions, including the input/output waveguides and microring resonators formed from single-mode silicon wire waveguides (width = 0.5 μm) in the racetrack configuration (bending radius = 6.5 μm). c, Transmission (insertion loss) spectrum, i.e., the drop port response normalized to the input port (black: 35 ring structure; gray: 235 ring structure, + 7 dB net external amplification using an L-band EDFA).

Fig. 2
Fig. 2

Transmission spectra. Transmission (insertion loss) spectra were measured with resolution 1.4 pm, for 35, 65, 95, 135, and 235 ring waveguides. Shown here are the averages of 128 measured traces. Relative to the 35-ring CROW, measurements for 65, 95, 135 and 235 ring CROWs were amplified by 7, 4, 3, and 7 dB, respectively in order to boost the power level detected at the photoreceiver.

Fig. 3
Fig. 3

Propagation (group) delay spectra. Group delay spectra were measured with resolution 1.4 pm, for 35, 65, 95, 135, and 235 ring waveguides, over the same range of wavelengths as in Fig. 2. Spectral regions of large variation in delay correspond exactly to the stopbands of the intensity spectrum shown in Fig. 2.

Fig. 4
Fig. 4

Infrared images of CROW waveguide modes. a, Transmission (insertion loss) spectrum for a single passband of a 235 ring CROW, with measurements at selected wavelengths labeled (i)-(v). b, Intensity profiles of the eigenmodes at the wavelengths (ii)-(iv) measured with an infrared camera technique [39] show that non-localized excitations (extended throughout the entire waveguide length) were observed throughout the passband, in contrast with out-of-band (i) and band-edge (v) wavelengths. No correction was made in these images for the absorptive decay of intensity with length of the propagating modes [see Fig. 5(a)].

Fig. 5
Fig. 5

Transmission (intensity) measurements. a, The mid-band average of the transmitted intensity (in dB), measured without amplification, decreased linearly with length (−0.08 dB/resonator), except for one anomalous waveguide on the measured chip (65 rings) as discussed in the text. The errorbars represent the standard deviation, i.e., ripple, in the measured intensity over the flat portion of the band. b, The probability distribution function (PDFs) of the normalized intensity transmission ( I ^ I / I ) for the all the CROWs show agreement with the Rayleigh distribution, shown by the dashed lines, indicating non-localized transport through the waveguide. In contrast, the localized regime would show considerably different (long-tailed) statistics [8,34,40].

Fig. 6
Fig. 6

Propagation (time) delay measurements. a, The measured propagation delay averaged over the middle of a transmission band <τ> (units of picoseconds) increased linearly with length (L = 35, 65, …, 235 rings). The errorbars represent the standard deviation, i.e., ripple, in the measured delay over the flat portion of the band. b, The probability distributions of normalized delay τ ^ τ/<τ> were peaked at unity (i.e., τ = <τ>). The dashed line is a Gaussian (normal distribution) fit to the data, which indicates ballistic propagation statistics and the absence of localization, previously estimated to be a severe constraint on achieving >100 resonator lengths of CROWs. In fact, the self-averaging properties of longer chains of resonators yielded better fits to normal statistics than for the shorter waveguides, where finite-size effects caused an asymmetric lineshape in the tails of the distributions.

Fig. 7
Fig. 7

Scaling of delay statistics with CROW length. a, The measured probability distribution functions (PDFs) of the normalized group delay τ ^ τ/<τ> are shown, using a logarithmic scale on the vertical axis for clarity, for the waveguides labeled (3)-(5) in Fig. 5(a). With increasing length, the distributions converged to a single-parameter Gaussian distribution, shown by the dashed black line. b, The variance of the measured delay (ps2) increased with the square of the number of resonators (N), as shown by the dashed fit, var τ = 0.0346 N2 + (12.9 ps)2 where the second term was the typically measured group delay ripple of the measurement apparatus. This scaling behavior was different from that of conventional waveguides or cascaded fiber Bragg gratings, and as discussed in the text, demonstrated that the individual resonator excitations are mutually correlated.

Fig. 8
Fig. 8

Simulation results of group delay ripple in disordered CROWs. The yellow dots are obtained from simulations of the calculated complex transmission coefficient over the passband, in the same way as done for the analysis of measured data. The dashed line shows that a quadratic function can be fit through these points. The inset shows the probability density distribution of the variance of the group delay, from Monte Carlo simulations at a particular value of N, the number of resonators.

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