Abstract

Whispering gallery modes supported by open circular dielectric cavities are embedded into a nonparametric two-dimensional frequency domain hybrid coupled mode theory framework. Regular aggregates of these cavities, including straight access channels, are investigated. The model enables convenient studies of the guided wave scattering process, the response of the circuit to guided wave excitation. Transmission resonances can be characterized directly in terms of resonance frequency and linewidth by computing supermodes of the entire composite circuits, comprising both cavities and bus waveguides. Examples of single ring and disk filters, a coupled-resonator optical waveguide, and a three-cavity photonic molecule in a reflector configuration allow the approach to be assessed.

© 2013 Optical Society of America

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2012 (2)

L. Y. M. Tobing, L. Tjahjana, S. Darmawan, and D. H. Zhang, “Numerical and experimental studies of coupling-induced phase shift in resonator and interferometric integrated optics devices,” Opt. Express 20, 5789–5801 (2012).
[CrossRef]

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[CrossRef]

2011 (2)

B. Gallinet and O. J. F. Martin, “Ab initio theory of Fano resonances in plasmonic nanostructures and metamaterials,” Phys. Rev. B 83, 235427 (2011).
[CrossRef]

S. I. Schmid, K. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011).
[CrossRef]

2010 (4)

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

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[CrossRef]

Q. Li, T. Wang, Y. Su, M. Yan, and M. Qiu, “Coupled mode theory analysis of mode splitting in coupled cavity system,” Opt. Express 18, 8367–8382 (2010).
[CrossRef]

M. Hammer, “HCMT models of optical microring-resonator circuits,” J. Opt. Soc. Am. B 27, 2237–2246 (2010).
[CrossRef]

2009 (2)

Q. Li, M. Soltani, A. H. Atabaki, S. Yegnanarayanan, and A. Adibi, “Quantitative modeling of coupling-induced resonance frequency shift in microring resonators,” Opt. Express 17, 23474–23486 (2009).
[CrossRef]

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

2008 (3)

O. Schwelb, “On the nature of resonance splitting in coupled multiring optical resonators,” Opt. Commun. 281, 1065–1071 (2008).
[CrossRef]

M. Maksimovic, M. Hammer, and E. van Groesen, “Field representation for optical defect microcavities in multilayer structures using quasi-normal modes,” Opt. Commun. 281, 1401–1411 (2008).
[CrossRef]

M. Maksimovic, M. Hammer, and E. van Groesen, “Coupled optical defect microcavities in 1d photonic crystals and quasi-normal modes,” Opt. Eng. 47, 114601 (2008).
[CrossRef]

2007 (3)

2006 (4)

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency-domain coupled mode theory,” Opt. Commun. 257, 277–297 (2006).
[CrossRef]

J. Čtyroký, I. Richter, and M. Šiňor, “Dual resonance in a waveguide-coupled ring microresonator,” Opt. Quantum Electron. 38, 781–797 (2006).
[CrossRef]

S. V. Boriskina, “Theoretical prediction of a dramatic Q-factor enhancement and degeneracy removal of whispering gallery modes in symmetrical photonic molecules,” Opt. Lett. 31, 338–340 (2006).
[CrossRef]

M. A. Popović, C. Manolatou, and M. R. Watts, “Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters,” Opt. Express 14, 1208–1222 (2006).
[CrossRef]

2005 (1)

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[CrossRef]

2004 (2)

L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
[CrossRef]

J. K. S. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16, 1331–1333 (2004).
[CrossRef]

2002 (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

2000 (2)

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000).
[CrossRef]

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

1999 (2)

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (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]

1994 (1)

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[CrossRef]

1993 (1)

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proc. J 140, 177–188 (1993).
[CrossRef]

Adibi, A.

Atabaki, A. H.

Bahlmann, N.

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
[CrossRef]

Benson, T. M.

Bertolotti, M.

M. Bertolotti, “Linear one dimensional resonant cavities,” in Microresonators as Building Blocks for VLSI Photonics, F. Michelotti, A. Driessen, and M. Bertolotti, eds., Vol. 709 of AIP Conference Proceedings (American Institute of Physics, 2004), pp. 19–47.

Boriskina, S. V.

Breda, A.

Canciamilla, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[CrossRef]

F. Morichetti, A. Melloni, A. Breda, A. Canciamilla, C. Ferrari, and M. Martinelli, “A reconfigurable architecture for continuously variable optical slow-wave delay lines,” Opt. Express 15, 17273–17281 (2007).
[CrossRef]

Chipouline, A.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[CrossRef]

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Chremmos, I.

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

O. Schwelb and I. Chremmos, “Band-limited microresonator reflectors and mirror structures,” in Photonic Microresonator Research and Applications, I. Chremmos, N. Uzunoglu, and O. Schwelb, eds., Vol. 156 of Springer Series in Optical Sciences (Springer, 2010), pp. 139–163.

Chu, S. T.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000).
[CrossRef]

Ctyroký, J.

J. Čtyroký, I. Richter, and M. Šiňor, “Dual resonance in a waveguide-coupled ring microresonator,” Opt. Quantum Electron. 38, 781–797 (2006).
[CrossRef]

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[CrossRef]

L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
[CrossRef]

Darmawan, S.

De La Rue, R. M.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[CrossRef]

Etrich, C.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[CrossRef]

Evers, J.

S. I. Schmid, K. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011).
[CrossRef]

Ferrari, C.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[CrossRef]

F. Morichetti, A. Melloni, A. Breda, A. Canciamilla, C. Ferrari, and M. Martinelli, “A reconfigurable architecture for continuously variable optical slow-wave delay lines,” Opt. Express 15, 17273–17281 (2007).
[CrossRef]

Fink, Y.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992).

Franchimon, E.

E. Franchimon, “Modelling circular optical microresonators using whispering gallery modes,” M.Sc. thesis (University of Twente, 2010).

Gallinet, B.

B. Gallinet and O. J. F. Martin, “Ab initio theory of Fano resonances in plasmonic nanostructures and metamaterials,” Phys. Rev. B 83, 235427 (2011).
[CrossRef]

Hammer, M.

M. Hammer, “HCMT models of optical microring-resonator circuits,” J. Opt. Soc. Am. B 27, 2237–2246 (2010).
[CrossRef]

M. Maksimovic, M. Hammer, and E. van Groesen, “Field representation for optical defect microcavities in multilayer structures using quasi-normal modes,” Opt. Commun. 281, 1401–1411 (2008).
[CrossRef]

M. Maksimovic, M. Hammer, and E. van Groesen, “Coupled optical defect microcavities in 1d photonic crystals and quasi-normal modes,” Opt. Eng. 47, 114601 (2008).
[CrossRef]

M. Hammer, “Hybrid analytical/numerical coupled-mode modeling of guided wave devices,” J. Lightwave Technol. 25, 2287–2298 (2007).
[CrossRef]

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency-domain coupled mode theory,” Opt. Commun. 257, 277–297 (2006).
[CrossRef]

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[CrossRef]

Hertel, P.

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
[CrossRef]

Hiremath, K. R.

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency-domain coupled mode theory,” Opt. Commun. 257, 277–297 (2006).
[CrossRef]

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[CrossRef]

Hubálek, M.

L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
[CrossRef]

Ibanescu, M.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

Janunts, N.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Johnson, S. G.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Käsebier, T.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[CrossRef]

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Kettner, B.

B. Kettner, “Detection of spurious modes in resonance mode computations—Pole condition method,” PhD Dissertation (Freie Universität zu Berlin, 2012).

Klein, A.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[CrossRef]

Kley, E.-B.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[CrossRef]

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Kokubun, Y.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000).
[CrossRef]

Lederer, F.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[CrossRef]

Lee, R. K.

Leung, P. T.

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[CrossRef]

Li, Q.

Liebsch, M.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[CrossRef]

Little, B. E.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000).
[CrossRef]

Liu, S. Y.

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[CrossRef]

Lohmeyer, M.

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
[CrossRef]

Love, J. D.

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proc. J 140, 177–188 (1993).
[CrossRef]

Maksimovic, M.

M. Maksimovic, M. Hammer, and E. van Groesen, “Field representation for optical defect microcavities in multilayer structures using quasi-normal modes,” Opt. Commun. 281, 1401–1411 (2008).
[CrossRef]

M. Maksimovic, M. Hammer, and E. van Groesen, “Coupled optical defect microcavities in 1d photonic crystals and quasi-normal modes,” Opt. Eng. 47, 114601 (2008).
[CrossRef]

M. Maksimovic, “Optical resonances in multilayer structures,” Ph.D. thesis (University of Twente, 2008).

Manolatou, C.

Martin, O. J. F.

B. Gallinet and O. J. F. Martin, “Ab initio theory of Fano resonances in plasmonic nanostructures and metamaterials,” Phys. Rev. B 83, 235427 (2011).
[CrossRef]

Martinelli, M.

Melloni, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[CrossRef]

F. Morichetti, A. Melloni, A. Breda, A. Canciamilla, C. Ferrari, and M. Martinelli, “A reconfigurable architecture for continuously variable optical slow-wave delay lines,” Opt. Express 15, 17273–17281 (2007).
[CrossRef]

Morichetti, F.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[CrossRef]

F. Morichetti, A. Melloni, A. Breda, A. Canciamilla, C. Ferrari, and M. Martinelli, “A reconfigurable architecture for continuously variable optical slow-wave delay lines,” Opt. Express 15, 17273–17281 (2007).
[CrossRef]

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

Pan, W.

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000).
[CrossRef]

Pertsch, T.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[CrossRef]

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Pishko, S. V.

Poon, J. K. S.

J. K. S. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16, 1331–1333 (2004).
[CrossRef]

Popovic, M. A.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992).

Prkna, L.

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[CrossRef]

L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
[CrossRef]

Qiu, M.

Richter, I.

J. Čtyroký, I. Richter, and M. Šiňor, “Dual resonance in a waveguide-coupled ring microresonator,” Opt. Quantum Electron. 38, 781–797 (2006).
[CrossRef]

Rowland, D. R.

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proc. J 140, 177–188 (1993).
[CrossRef]

Samarelli, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[CrossRef]

Scherer, A.

Scheuer, J.

J. K. S. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16, 1331–1333 (2004).
[CrossRef]

Schmid, S. I.

S. I. Schmid, K. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011).
[CrossRef]

Schmidt, C.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[CrossRef]

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Schwelb, O.

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

O. Schwelb, “On the nature of resonance splitting in coupled multiring optical resonators,” Opt. Commun. 281, 1065–1071 (2008).
[CrossRef]

O. Schwelb and I. Chremmos, “Band-limited microresonator reflectors and mirror structures,” in Photonic Microresonator Research and Applications, I. Chremmos, N. Uzunoglu, and O. Schwelb, eds., Vol. 156 of Springer Series in Optical Sciences (Springer, 2010), pp. 139–163.

Sewell, P. D.

Šinor, M.

J. Čtyroký, I. Richter, and M. Šiňor, “Dual resonance in a waveguide-coupled ring microresonator,” Opt. Quantum Electron. 38, 781–797 (2006).
[CrossRef]

Skorobogatiy, M. A.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Soltani, M.

Sorel, M.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[CrossRef]

Stoffer, R.

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency-domain coupled mode theory,” Opt. Commun. 257, 277–297 (2006).
[CrossRef]

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[CrossRef]

Su, Y.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992).

Tjahjana, L.

Tobing, L. Y. M.

Torregiani, M.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[CrossRef]

Tünnermann, A.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[CrossRef]

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

van Groesen, E.

M. Maksimovic, M. Hammer, and E. van Groesen, “Coupled optical defect microcavities in 1d photonic crystals and quasi-normal modes,” Opt. Eng. 47, 114601 (2008).
[CrossRef]

M. Maksimovic, M. Hammer, and E. van Groesen, “Field representation for optical defect microcavities in multilayer structures using quasi-normal modes,” Opt. Commun. 281, 1401–1411 (2008).
[CrossRef]

Vassallo, C.

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992).

Wang, T.

Watts, M. R.

Weisberg, O.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Xia, K.

S. I. Schmid, K. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011).
[CrossRef]

Xu, Y.

Yan, M.

Yariv, A.

J. K. S. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16, 1331–1333 (2004).
[CrossRef]

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

Yegnanarayanan, S.

Young, K.

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[CrossRef]

Zhang, D. H.

Zschiedrich, L.

L. Zschiedrich, “Transparent boundary conditions for Maxwells equations: numerical concepts beyond the PML method,” PhD Dissertation (Freie Universität zu Berlin, 2009).

Electron. Lett. (1)

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

IEE Proc. J (1)

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proc. J 140, 177–188 (1993).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

B. E. Little, S. T. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000).
[CrossRef]

J. K. S. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett. 16, 1331–1333 (2004).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. (1)

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12, 104008 (2010).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (5)

O. Schwelb, “On the nature of resonance splitting in coupled multiring optical resonators,” Opt. Commun. 281, 1065–1071 (2008).
[CrossRef]

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

K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency-domain coupled mode theory,” Opt. Commun. 257, 277–297 (2006).
[CrossRef]

M. Maksimovic, M. Hammer, and E. van Groesen, “Field representation for optical defect microcavities in multilayer structures using quasi-normal modes,” Opt. Commun. 281, 1401–1411 (2008).
[CrossRef]

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
[CrossRef]

Opt. Eng. (1)

M. Maksimovic, M. Hammer, and E. van Groesen, “Coupled optical defect microcavities in 1d photonic crystals and quasi-normal modes,” Opt. Eng. 47, 114601 (2008).
[CrossRef]

Opt. Express (5)

Opt. Lett. (2)

Opt. Quantum Electron. (3)

J. Čtyroký, I. Richter, and M. Šiňor, “Dual resonance in a waveguide-coupled ring microresonator,” Opt. Quantum Electron. 38, 781–797 (2006).
[CrossRef]

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Čtyroký, “Analytical approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. 37, 37–61 (2005).
[CrossRef]

L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
[CrossRef]

Phys. Rev. A (4)

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057–3067 (1994).
[CrossRef]

S. I. Schmid, K. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011).
[CrossRef]

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Käsebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tünnermann, and T. Pertsch, “Near-field mapping of optical eigenstates in coupled disk microresonators,” Phys. Rev. A 85, 033827 (2012).
[CrossRef]

Phys. Rev. B (1)

B. Gallinet and O. J. F. Martin, “Ab initio theory of Fano resonances in plasmonic nanostructures and metamaterials,” Phys. Rev. B 83, 235427 (2011).
[CrossRef]

Phys. Rev. E (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Other (17)

PhoeniX BV, P. O. Box 545, 7500 AM Enschede, The Netherlands, http://www.phoenixbv.com/http://www.phoenixbv.com/ .

JCMwave GmbH, Haarer Str. 14a, 85640 Putzbrunn/Munich, Germany, http://www.jcmwave.com .

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992).

M. Bertolotti, “Linear one dimensional resonant cavities,” in Microresonators as Building Blocks for VLSI Photonics, F. Michelotti, A. Driessen, and M. Bertolotti, eds., Vol. 709 of AIP Conference Proceedings (American Institute of Physics, 2004), pp. 19–47.

M. Maksimovic, “Optical resonances in multilayer structures,” Ph.D. thesis (University of Twente, 2008).

E. Franchimon, “Modelling circular optical microresonators using whispering gallery modes,” M.Sc. thesis (University of Twente, 2010).

F. Michelotti, A. Driessen, and M. Bertolotti, eds., Microresonators as Building Blocks for VLSI Photonics, Vol. 709 of AIP Conference Proceedings (American Institute of Physics, 2004).

I. Chremmos, N. Uzunoglu, and O. Schwelb, eds., Photonic Microresonator Research and Applications, Vol. 156 of Springer Series in Optical Sciences (Springer, 2010).

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

D. E. Amos, “A portable package for Bessel functions of a complex argument and nonnegative order,” http://www.netlib.org/amos .

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J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

B. Kettner, “Detection of spurious modes in resonance mode computations—Pole condition method,” PhD Dissertation (Freie Universität zu Berlin, 2012).

L. Zschiedrich, “Transparent boundary conditions for Maxwells equations: numerical concepts beyond the PML method,” PhD Dissertation (Freie Universität zu Berlin, 2009).

O. Schwelb and I. Chremmos, “Band-limited microresonator reflectors and mirror structures,” in Photonic Microresonator Research and Applications, I. Chremmos, N. Uzunoglu, and O. Schwelb, eds., Vol. 156 of Springer Series in Optical Sciences (Springer, 2010), pp. 139–163.

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Figures (9)

Fig. 1.
Fig. 1.

Resonance wavelengths λr and quality factors Q of WGMs supported by 2-D dielectric rings (a) and disks (b) with varying radii R in a wavelength region around 1.55 μm; parameters as given for Fig. 2. (c) Profile of the ring WGM indicated by the marker in (a), for R=7.5μm; absolute value (top) and time snapshot (bottom) of the principal electric component Ey. See Figs. 6(a) and 6(d) for the profiles of the disk WGMs marked in (b).

Fig. 2.
Fig. 2.

Filter device consisting of a cavity ring between two straight bus waveguides, a 2-D configuration described in Cartesian coordinates x, z or polar coordinates r, θ. Parameters: refractive indices ng=1.5 (guiding regions), nb=1.0 (background), cavity radius R=7.5μm, core width d=0.75μm, bus waveguides, core width w=0.6μm, and gaps g=0.3μm. All simulations are restricted to 2-D, uniform for TE polarization, where the single principal electric field component Ey is perpendicular to the xz plane of interest. We consider a spectral region around the target vacuum wavelength λ1.55μm.

Fig. 4.
Fig. 4.

Transmission properties of microring (a) and microdisk resonators (b); directly transmitted power T (top) and dropped optical power D (bottom) as a function of the excitation wavelength λ. Parameters are as given for Fig. 2. Results of different methods are compared, as explained further in the text: WGM-HCMT (continuous line, the approach of this paper), conv-CMT (dashed line, conventional CMT [34]), BM-HCMT (dash-dotted line, (a) only, bend-mode-based HCMT [4]), FEM (dash-dotted line, (b) only, FE solver, commercial [32]), and FDTD (markers, FDTD, commercial [31]). Resonances are classified by the dominant contributing WGM. The marker lines in the central part are positioned at the respective resonance wavelengths: separate WGMs (light gray, continuous line), WGMs perturbed by the bus waveguide permittivities (light gray, dashed line), and HCMT supermodes (black, lower bars indicate the linewidths).

Fig. 5.
Fig. 5.

Field pattern related to the WGM(0, 39) resonance of the ring filter in Fig. 4(a). Panels (a)–(c) correspond to the transmission problem (tr), where a guided wave excitation is present in the upper left channel. Bend-mode-based HCMT results [4] (a) serve as a reference. The WGM-HCMT simulations for (b) and (c) differ with respect to the HCMT template: Plots (b) are based on a template that includes unidirectional fields for both bus waveguides and merely the one WGM(0, 39) for the cavity. Simulations (c) also take WGMs(0, 37–41) into account. Panels (d)–(f) show the results of supermode calculations (sm), based on either a unidirectional HCMT template (d) or on a template where bidirectional variants of all fields are supplied (e) and (f).

Fig. 3.
Fig. 3.

WGM resonance wavelengths λr versus core refractive index ng, for rings with the parameters of Fig. 1(a) with R=7.5μm. The dashed lines indicate the slopes as predicted by Eq. (13), evaluated with the fields of the reference structure for ng=1.5 (markers).

Fig. 6.
Fig. 6.

Field pattern related to central resonances of the disk filter in Fig. 4(b). Panels (b) and (c) correspond to the radial fundamental mode WGM(0, 40) (a), and panels (e) and (f) show the pattern for the first-order WGM(1, 35) (d). HCMT results for the resonant transmission (b) and (e) are compared with the supermode fields of fundamental (c) and first order (f). All HCMT fields rely on a template that includes the WGMs(0, 38–42) and WGMs(1, 34–37).

Fig. 7.
Fig. 7.

Resonance wavelengths λr and Q factors of ring resonators as in Fig. 2, for varying gap g (a) and bus width w (b). The horizontal lines indicate the resonance wavelengths of the isolated cavity ring.

Fig. 8.
Fig. 8.

Coupled-resonator optical waveguide (CROW), a series of cavities as introduced in Fig. 2. (a) Spectrum of the device, relative transmission T, and power drop D as a function of the excitation wavelength λ; unidirectional HCMT simulations based on templates with one WGM per cavity (bold lines) and with five WGMs per cavity (thin dashed curves). The central lines give the positions of the resonance wavelength of one separate ring (single bold line, gray), of the resonances for the series of rings without bus waveguides (dashed line), and of the resonances of the complete system (continuous line; here also the respective linewidths are shown), in all cases computed using templates with one WGM(0, 39) per ring. (b) Supermode patterns for the system with bus waveguides (bold lines), and for the ring series only (thinner, mostly shadowed lines). Real parts (dashed lines), imaginary parts (dash-dotted lines), and absolute values (continuous lines) of the amplitudes attributed to the WGMs of neighboring cavities are connected to clarify the systematics.

Fig. 9.
Fig. 9.

(a) Three-ring photonic molecule, excited by a straight bus waveguide. (b) Relative guided transmitted (T) and reflected optical (R) power, as a function of excitation wavelength λ; bidirectional HCMT simulations with the WGMs(0, ±39) included for each ring. The vertical markers indicate the resonance WGM(0, 39) of an individual ring (single bold gray line), the HCMT supermodes computed for the three-ring molecule only, without the bus channel (dashed line), and the supermodes for the entire compound, consisting of molecule and waveguide (continuous line; here also the respective linewidths are shown). (c) Supermode profiles predicted for the three-ring molecule, time snapshots of the standing wave pattern (large panels) and absolute values (smaller insets) of the principal electric field component.

Tables (1)

Tables Icon

Table 1. Resonance Wavelengths λr or Peak Locations, Respectively, in Case of the Transmission Problems (tr), Quality Factors Q, and Spectral Linewidths Δλ Associated with Selected Resonances of Figs. 1, 46a

Equations (13)

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

×Hiωϵ0ϵE=0,×Eiωμ0H=0
(EH)(x,z)=f(z)ψf(x,z)+b(z)ψb(x,z)+jcjψjc(r,θ),
(EH)(x,z)=kak(EkHk)(x,z)
A(F,G;E,H)dxdzωB(F,G;E,H)dxdz=0for allF,G,
A(F,G;E,H)=F*·(×H)G*·(×E),B(F,G;E,H)=iϵ0ϵF*·E+iμ0G*·H.
k(AlkωBlk)ak=0,for alll
Alk=A(El,Hl;Ek,Hk)dxdz,Blk=B(El,Hl;Ek,Hk)dxdz.
[(AuuAugAguAgg)ω(BuuBugBguBgg)](ug)=0,
orKuu=KggwithKu=(AuuωBuuAguωBgu),Kg=(AugωBugAggωBgg).
KuKuu=KuKgg.
×Hiωsϵ0ϵE=0,×Eiωsμ0H=0,
Auuu=ωsBuuu.
Δω=ωoϵ0Δϵ|Eo|2dxdz(ϵoϵ0|Eo|2+μ0|Ho|2)dxdz.

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