Abstract

Circuits of dielectric integrated optical microring resonators are addressed through a two-dimensional hybrid analytical/numerical coupled mode theory (HCMT) model. Analytical modes of all straight and curved cores form templates for the optical fields of the entire circuits. Our variational technique then generates solutions for the amplitude functions in their natural Cartesian and polar coordinates, discretized by one-dimensional finite elements. Bidirectional wave propagation through all channels and pronounced reflections can be taken into account. The series of examples includes rings coupled in parallel, rows of cavities (coupled resonator optical waveguides) of varying lengths, a triangular photonic molecule, and a resonator with a slit ring to illustrate the role of intra-cavity reflections.

© 2010 Optical Society of America

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  18. 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]
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  21. K. R. Hiremath, “CIRCURS—Circular resonator simulator,” http://www.math.utwente.nl/aamp/FormMem/Hiremath/circurs/.
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    [CrossRef]
  29. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
    [CrossRef]
  30. M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35, 1451–1460 (1999).
    [CrossRef]
  31. S. V. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis,” J. Opt. Soc. Am. 23, 1565–1573 (2006).
    [CrossRef]
  32. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with whispering-gallery modes,” Opt. Lett. 31, 921–923 (2006).
    [CrossRef] [PubMed]
  33. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: Quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
    [CrossRef]
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  35. This might well relate to actual physical effects, e.g., to a pronounced surface roughness .
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    [CrossRef] [PubMed]
  37. Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41, 689–697 (2010) (available online).
    [CrossRef]
  38. M. Hammer, “Quadridirectional eigenmode expansion scheme for 2-D modeling of wave propagation in integrated optics,” Opt. Commun. 235, 285–303 (2004).
    [CrossRef]
  39. L. Prkna, J. Čtyroký, and M. Hubálek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
    [CrossRef]

2010 (1)

Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41, 689–697 (2010) (available online).
[CrossRef]

2009 (1)

M. Hammer, “Chains of coupled square dielectric optical microcavities,” Opt. Quantum Electron. 40, 821–835 (2009).
[CrossRef]

2007 (2)

2006 (4)

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with whispering-gallery modes,” Opt. Lett. 31, 921–923 (2006).
[CrossRef] [PubMed]

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]

S. V. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis,” J. Opt. Soc. Am. 23, 1565–1573 (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]

2005 (4)

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: Quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial frequency domain coupled mode theory,” Opt. Commun. 256, 46–67 (2005).
[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]

M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13, 1515–1530 (2005).
[CrossRef] [PubMed]

2004 (3)

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

M. Hammer, “Quadridirectional eigenmode expansion scheme for 2-D modeling of wave propagation in integrated optics,” Opt. Commun. 235, 285–303 (2004).
[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]

2003 (1)

2002 (1)

2001 (1)

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, “Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology,” Appl. Phys. B 73, 603–608 (2001).
[CrossRef]

2000 (1)

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

1999 (4)

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[CrossRef]

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35, 1451–1460 (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]

1998 (1)

1993 (1)

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

1982 (1)

Absil, P. P.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Arbabi, A.

Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41, 689–697 (2010) (available online).
[CrossRef]

Balistreri, M. L. M.

Benson, T. M.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with whispering-gallery modes,” Opt. Lett. 31, 921–923 (2006).
[CrossRef] [PubMed]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: Quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

Blom, F. C.

Boriskina, S. V.

S. V. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis,” J. Opt. Soc. Am. 23, 1565–1573 (2006).
[CrossRef]

Borselli, M.

Breda, A.

Bulthuis, H. F.

D. J. W. Klunder, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, B. Docter, H. J. W. M. Hoekstra, and A. Driessen, “Experimental and numerical study of SiON microresonators with air and polymer cladding,” J. Lightwave Technol. 21, 1099–1110 (2003).
[CrossRef]

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, “Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology,” Appl. Phys. B 73, 603–608 (2001).
[CrossRef]

Canciamilla, A.

Chin, M. K.

Chodorow, M.

Chu, S. T.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[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]

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial frequency domain coupled mode theory,” Opt. Commun. 256, 46–67 (2005).
[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]

Docter, B.

Driessen, A.

Fan, S.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[CrossRef]

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35, 1451–1460 (1999).
[CrossRef]

Ferrari, C.

Flannery, B. P.

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

Gill, D.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Goddard, L. L.

Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41, 689–697 (2010) (available online).
[CrossRef]

Hammer, M.

M. Hammer, “Chains of coupled square dielectric optical microcavities,” Opt. Quantum Electron. 40, 821–835 (2009).
[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]

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial frequency domain coupled mode theory,” Opt. Commun. 256, 46–67 (2005).
[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]

M. Hammer, “Quadridirectional eigenmode expansion scheme for 2-D modeling of wave propagation in integrated optics,” Opt. Commun. 235, 285–303 (2004).
[CrossRef]

M. Hammer, K. R. Hiremath, and R. Stoffer, “Analytical approaches to the description of optical microresonator devices,” in Microresonators as Building Blocks for VLSI Photonics, Vol. 709 of AIP Conference Proceedings, M.Bertolotti, A.Driessen, and F.Michelotti, eds. (American Institute of Physics, 2004), pp. 48–71.

M. Hammer, “METRIC—Mode expansion tools for 2D rectangular integrated optical circuits,” http://www.math.utwente.nl/~hammerm/Metric/.

Haus, H. A.

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35, 1451–1460 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[CrossRef]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (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]

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial frequency domain coupled mode theory,” Opt. Commun. 256, 46–67 (2005).
[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]

M. Hammer, K. R. Hiremath, and R. Stoffer, “Analytical approaches to the description of optical microresonator devices,” in Microresonators as Building Blocks for VLSI Photonics, Vol. 709 of AIP Conference Proceedings, M.Bertolotti, A.Driessen, and F.Michelotti, eds. (American Institute of Physics, 2004), pp. 48–71.

K. R. Hiremath, “CIRCURS—Circular resonator simulator,” http://www.math.utwente.nl/aamp/FormMem/Hiremath/circurs/.

K. R. Hiremath, “Coupled mode theory based modeling and analysis of circular optical microresonators,” Ph.D. dissertation (University of Twente, 2005).

Ho, S. T.

Hoekstra, H. J. W. M.

Hryniewicz, J. V.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[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]

Joannopoulos, J. D.

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35, 1451–1460 (1999).
[CrossRef]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[CrossRef]

Johnson, F. G.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Johnson, T. J.

Kang, Y. M.

Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41, 689–697 (2010) (available online).
[CrossRef]

Khan, M. J.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[CrossRef]

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35, 1451–1460 (1999).
[CrossRef]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

King, O.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Klunder, D. J. W.

Krioukov, E.

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, “Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology,” Appl. Phys. B 73, 603–608 (2001).
[CrossRef]

Kuipers, L.

Lee, R. K.

Little, B. E.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[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: Optoelectron. 140, 177–188 (1993).
[CrossRef]

Manolatou, C.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35, 1451–1460 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[CrossRef]

Martinelli, M.

Melloni, A.

Morichetti, F.

Nosich, A. I.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with whispering-gallery modes,” Opt. Lett. 31, 921–923 (2006).
[CrossRef] [PubMed]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: Quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

Okamoto, K.

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

Otto, C.

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, “Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology,” Appl. Phys. B 73, 603–608 (2001).
[CrossRef]

Painter, O.

Press, W. H.

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

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial frequency domain coupled mode theory,” Opt. Commun. 256, 46–67 (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]

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: Optoelectron. 140, 177–188 (1993).
[CrossRef]

Scherer, A.

Seiferth, F.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Sengo, G.

D. J. W. Klunder, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, B. Docter, H. J. W. M. Hoekstra, and A. Driessen, “Experimental and numerical study of SiON microresonators with air and polymer cladding,” J. Lightwave Technol. 21, 1099–1110 (2003).
[CrossRef]

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, “Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology,” Appl. Phys. B 73, 603–608 (2001).
[CrossRef]

Sewell, P.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with whispering-gallery modes,” Opt. Lett. 31, 921–923 (2006).
[CrossRef] [PubMed]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: Quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

Shaw, H. J.

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]

Smotrova, E. I.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with whispering-gallery modes,” Opt. Lett. 31, 921–923 (2006).
[CrossRef] [PubMed]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: Quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[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]

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial frequency domain coupled mode theory,” Opt. Commun. 256, 46–67 (2005).
[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]

M. Hammer, K. R. Hiremath, and R. Stoffer, “Analytical approaches to the description of optical microresonator devices,” in Microresonators as Building Blocks for VLSI Photonics, Vol. 709 of AIP Conference Proceedings, M.Bertolotti, A.Driessen, and F.Michelotti, eds. (American Institute of Physics, 2004), pp. 48–71.

Stokes, L. F.

Tan, F. S.

D. J. W. Klunder, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, B. Docter, H. J. W. M. Hoekstra, and A. Driessen, “Experimental and numerical study of SiON microresonators with air and polymer cladding,” J. Lightwave Technol. 21, 1099–1110 (2003).
[CrossRef]

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, “Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology,” Appl. Phys. B 73, 603–608 (2001).
[CrossRef]

Teukolsky, S. A.

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

Trakalo, M.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

Van, V.

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

van der Veen, T.

D. J. W. Klunder, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, B. Docter, H. J. W. M. Hoekstra, and A. Driessen, “Experimental and numerical study of SiON microresonators with air and polymer cladding,” J. Lightwave Technol. 21, 1099–1110 (2003).
[CrossRef]

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, “Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology,” Appl. Phys. B 73, 603–608 (2001).
[CrossRef]

van Hulst, N. F.

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 Press, 1992).

Villeneuve, P. R.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[CrossRef]

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35, 1451–1460 (1999).
[CrossRef]

Xu, Y.

Yariv, A.

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]

Appl. Phys. B (1)

D. J. W. Klunder, E. Krioukov, F. S. Tan, T. van der Veen, H. F. Bulthuis, G. Sengo, C. Otto, H. J. W. M. Hoekstra, and A. Driessen, “Vertically and laterally waveguide-coupled cylindrical microresonators in Si3N4 on SiO2 technology,” Appl. Phys. B 73, 603–608 (2001).
[CrossRef]

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: Optoelectron. (1)

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

IEEE J. Quantum Electron. (2)

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35, 1451–1460 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: Quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. 16, 2263–2265 (2004).
[CrossRef]

J. Lightwave Technol. (4)

J. Opt. Soc. Am. (1)

S. V. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis,” J. Opt. Soc. Am. 23, 1565–1573 (2006).
[CrossRef]

Opt. Commun. (3)

M. Hammer, “Quadridirectional eigenmode expansion scheme for 2-D modeling of wave propagation in integrated optics,” Opt. Commun. 235, 285–303 (2004).
[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]

R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial frequency domain coupled mode theory,” Opt. Commun. 256, 46–67 (2005).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Opt. Quantum Electron. (5)

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

Y. M. Kang, A. Arbabi, and L. L. Goddard, “A microring resonator with an integrated Bragg grating: a compact replacement for a sampled grating distributed Bragg reflector,” Opt. Quantum Electron. 41, 689–697 (2010) (available online).
[CrossRef]

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

M. Hammer, “Chains of coupled square dielectric optical microcavities,” Opt. Quantum Electron. 40, 821–835 (2009).
[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]

Phys. Rev. B (1)

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[CrossRef]

Other (13)

M. Hammer, K. R. Hiremath, and R. Stoffer, “Analytical approaches to the description of optical microresonator devices,” in Microresonators as Building Blocks for VLSI Photonics, Vol. 709 of AIP Conference Proceedings, M.Bertolotti, A.Driessen, and F.Michelotti, eds. (American Institute of Physics, 2004), pp. 48–71.

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

D.G.Hall and B.J.Thompson, eds., Selected Papers on Coupled-Mode Theory in Guided-Wave Optics, Vol. MS 84 of SPIE Milestone Series (SPIE Optical Engineering, 1993).

K. R. Hiremath, “Coupled mode theory based modeling and analysis of circular optical microresonators,” Ph.D. dissertation (University of Twente, 2005).

M. Hammer, “METRIC—Mode expansion tools for 2D rectangular integrated optical circuits,” http://www.math.utwente.nl/~hammerm/Metric/.

K. R. Hiremath, “CIRCURS—Circular resonator simulator,” http://www.math.utwente.nl/aamp/FormMem/Hiremath/circurs/.

The fields ψf, ψb, ψc that constitute template satisfy formally the same equations, but each with different permittivity functions that represent the upper core, the lower core, or the cavity only, always on a homogeneous background.

One might observe that this interpolation process is quite analogous to the usual analytical evaluation of parametric CMT models , where one takes into account first or second order wavelength derivatives of effective mode indices and neglects the wavelength dependence of coupling coefficients.

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

M.Bertolotti, A.Driessen, and F.Michelotti, 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, Springer Series in Optical Sciences, Vol. 156 (Springer, 2010).
[CrossRef]

This might well relate to actual physical effects, e.g., to a pronounced surface roughness .

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

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Figures (12)

Fig. 1
Fig. 1

Circular cavity between two bus channels, unidirectional HCMT model, schematically. Guided light enters from the left through the upper core; interest is in the relative transmitted and dropped guided optical powers T and D, and in their spectral properties. Ring-resonator functionality is established by the evanescent interaction of the directional guided modes ψ f , ψ b associated with the bus channels and the bend mode ψ c supported by the cavity core. Cartesian coordinates x, z and polar coordinates r, θ will be used. Parameters: cavity radius R = 7.5 μ m (outer rim), width of the cavity ring d = 0.75 μ m , waveguide width w = 0.6 μ m , gap between ring and bus waveguides g = 0.3 μ m , refractive index of all guiding regions n g = 1.5 and of the background n b = 1.0 , target vacuum wavelength λ 1.56 μ m .

Fig. 2
Fig. 2

Power spectrum of the single-ring filter as introduced in Fig. 1: relative guided transmitted T and dropped optical power D versus the vacuum wavelength λ. Bold lines: direct computation; thin lines: interpolated spectral scan according to the procedure of Subsection 2C.

Fig. 3
Fig. 3

For the single-ring filter of Fig. 2: resonance at λ = 1.5623 μ m . Top row: coupled mode amplitudes f ( z ) , b ( z ) , and c ( θ ) associated with the waves in the upper and lower bus channels, and with the cavity mode, respectively. Bottom row: time snapshot of the physical electrical field and field modulus, principal component E y of the TE polarized waves.

Fig. 4
Fig. 4

Spectral properties of CROW structures with two, three, four, and nine evanescently coupled rings, relative directly transmitted T, and dropped guided optical power D.

Fig. 5
Fig. 5

Field patterns associated with the resonances as indicated (letters) in the spectra of Fig. 4; the plots show the absolute value of the principal electric component of the TE fields.

Fig. 6
Fig. 6

Two microrings coupled in parallel. Parameters are as in Fig. 1, with a gap s = 0.25 μ m between the rings. Upon excitation with wavelength λ in the upper left port, the power is distributed to outlets T, R, A, and B.

Fig. 7
Fig. 7

Field patterns (absolute values and time snapshots of the principal electric component) observed for the parallel ring configuration of Fig. 6 at three selected wavelengths. The overall phase for the plots in the lower row has been chosen (roughly) such that the maximum amplitude becomes visible.

Fig. 8
Fig. 8

Triangular arrangement of coupled rings, accessed by a single bus waveguide. Transmission T and reflectivity R versus the wavelength λ.

Fig. 9
Fig. 9

Resonant fields (absolute values and time snapshots of the principal electric field component) supported by the triple ring configuration of Fig. 8.

Fig. 10
Fig. 10

Ring resonator with a slit. Transmission T and reflection R versus wavelength λ. Parameters are as in Fig. 1, with a slit s = 0.2 μ m . Bold continuous lines: excitation in the left port only; thin lines, mostly shadowed: symmetric (dashed) and antisymmetric (dashed-dotted) excitation in both ports.

Fig. 11
Fig. 11

Slit ring resonator. Field pattern associated with the extremal configurations indicated in Fig. 10. (s) symmetrical excitation from both the left and right ports; (l) an incoming wave from the left only, two different wavelengths close to the (s) and (a) resonances; (a) antisymmetric excitation.

Fig. 12
Fig. 12

Straight waveguide with a hole. A guided wave enters from the left and is partly reflected; rigorous reference calculation (QUEP [38]) and HCMT model. The plots show the absolute value | E y | of the principal electric field component of the TE waves.

Equations (15)

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ψ f , b ( x , z ) = ( E ̃ H ̃ ) f , b ( x ) e i β z .
ψ c ( r , θ ) = ( E ̃ H ̃ ) c ( r ) e i γ R θ ,
( E H ) ( x , z ) = f ( z ) ψ f ( x , z ) + b ( z ) ψ b ( x , z ) + c ( θ ) ψ c ( r , θ ) ,
κ = floor ( Re   γ R + 1 / 2 ) / R
f ( z ) = j = 0 N f j α j ( z ) .
( E H ) ( x , z ) = k a k ( α ( ) ψ ( , ) ) k a k ( E k H k ) ( x , z ) ,
× H i ω ϵ 0 ϵ E = 0 ,     × E i ω μ 0 H = 0 ,
Ω K ( E , H ; E , H ) d x d y d z = 0     for   all   E , H ,
K ( E , H ; E , H ) = ( E ) ( × H ) ( H ) ( × E ) i ω ϵ 0 ϵ ( E ) E i ω μ 0 ( H ) H .
k K l k a k = 0 ,
K l k = Ω K ( E l , H l ; E k , H k ) d x d y d z .
( K u u K u g K g u K g g ) ( u g ) = 0 ,
K u u = K g g     with   K u = ( K u u K g u ) ,     K g = ( K u g K g g ) .
K u K u u = K u K g g .
Δ λ λ 2 2 π 2 N r R | ρ | | τ | = λ π γ R | ρ | | τ |

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