Abstract

The resonant properties of a closed and symmetric cyclic array of N coupled microring resonators (coupled-microring resonator regular N-gon) are for the first time determined analytically by applying the transfer matrix approach and Floquet theorem for periodic propagation in cylindrically symmetric structures. By solving the corresponding eigenvalue problem with the field amplitudes in the rings as eigenvectors, it is shown that, for even or odd N, this photonic molecule possesses 1 + N/2 or 1+N resonant frequencies, respectively. The condition for resonances is found to be identical to the familiar dispersion equation of the infinite coupled-microring resonator waveguide with a discrete wave vector. This result reveals the so far latent connection between the two optical structures and is based on the fact that, for a regular polygon, the field transfer matrix over two successive rings is independent of the polygon vertex angle. The properties of the resonant modes are discussed in detail using the illustration of Brillouin band diagrams. Finally, the practical application of a channel-dropping filter based on polygons with an even number of rings is also analyzed.

© 2007 Optical Society of America

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References

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  1. T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, "Tight-binding photonic molecule modes of resonant bispheres," Phys. Rev. Lett. 82, 4623-4626 (1999).
    [CrossRef]
  2. Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, "Photonic molecule lasing," Opt. Lett. 28, 2437-2439 (2003).
    [CrossRef] [PubMed]
  3. B. Möller, U. Woggon, M. Artemyev, and R. Wannemacher, "Photonic molecules doped with semiconductor nanocrystals," Phys. Rev. B 70, 115323 (2004).
    [CrossRef]
  4. Y. P. Rakovich, J. F. Donegan, M. Gerlach, A. L. Bradley, T. M. Connolly, J. J. Boland, N. Gaponik, and A. Rogach, "Fine structure of coupled optical modes in photonic molecules," Phys. Rev. A 70, 051801(R) (2004).
    [CrossRef]
  5. Y. P. Rakovich, J. J. Boland, and J. F. Donegan, "Tunable photon lifetime in photonic molecules: a concept for delaying an optical signal," Opt. Lett. 30, 2775-2777 (2005).
    [CrossRef] [PubMed]
  6. A. L. Burin, "Bound whispering gallery modes in circular arrays of dielectric spherical particles," Phys. Rev. E 73, 066614 (2006).
    [CrossRef]
  7. A. Nakagawa, S. Ishii, and T. Baba, "Photonic molecule laser composed of GaInAsP microdisks," Appl. Phys. Lett. 86, 041112 (2005).
    [CrossRef]
  8. 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] [PubMed]
  9. S. Boriskina, "Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis," J. Opt. Soc. Am. B 23, 1565-1573 (2006).
    [CrossRef]
  10. S. Ishii, A. Nakagawa, and T. Baba, "Modal characteristics and bistability in twin microdisk photonic molecule lasers," IEEE J. Sel. Top. Quantum Electron. 12, 71-77 (2006).
    [CrossRef]
  11. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, "Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing," IEEE J. Sel. Top. Quantum Electron. 12, 78-85 (2006).
    [CrossRef]
  12. 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]
  13. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, "Ultralow lasing thresholds of π-type supermodes in cyclic photonic molecules composed of submicron disks with monopole and dipole modes," IEEE Photon. Technol. Lett. 18, 1993-1995 (2006).
    [CrossRef]
  14. S. V. Boriskina, "Coupling of whispering-gallery modes in size-mismatched microdisk photonic molecules," Opt. Lett. 32, 1557-1559 (2007).
    [CrossRef] [PubMed]
  15. M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, "Optical modes in photonic molecules," Phys. Rev. Lett. 81, 2582-2586 (1998).
    [CrossRef]
  16. G. Guttroff, M. Bayer, A. Forchel, P. A. Knipp, and T. L. Reinecke, "Isomeric photonic molecules formed from coupled microresonators," Phys. Rev. E 63, 036611 (2001).
    [CrossRef]
  17. N.-S. Zhao, H. Zhou, Q. Guo, W. Hu, X.-B. Yang, S. Lan, and X.-S. Lin, "Design of highly efficient optical diodes based on the dynamics of nonlinear photonic crystal molecules," J. Opt. Soc. Am. B 23, 2434-2440 (2006).
    [CrossRef]
  18. S. V. Boriskina, T. M. Benson, P. D. Sewell, and A. I. Nosich, "Directional emission, increased free spectral range and mode Q-factors in 2-D wavelength-scale optical microcavity structures," IEEE J. Sel. Top. Quantum Electron. 12, 1175-1182 (2006).
    [CrossRef]
  19. 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]
  20. S. Olivier, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdre, and U. Oesterle, "Miniband transmission in a photonic crystal coupled-resonator optical waveguide," Opt. Lett. 26, 1019-1021 (2001).
    [CrossRef]
  21. M. Bayindir, S. Tanriseven, and E. Ozbay, "Propagation of light through localized coupled-cavity modes in one dimensional photonic band-gap structures," Appl. Phys. A 72, 117-119 (2001).
    [CrossRef]
  22. V. N. Astratov, J. P. Franchak, and S. P. Ashili, "Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder," Appl. Phys. Lett. 85, 5508-5510 (2004).
    [CrossRef]
  23. Z. Chen, A. Taflove, and V. Backman, "Highly efficient optical coupling and transport phenomena in chains of dielectric microspheres," Opt. Lett. 31, 389-391 (2006).
    [CrossRef] [PubMed]
  24. J. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, "Matrix analysis of microring coupled-resonator optical waveguides," Opt. Express 12, 90-103 (2004).
    [CrossRef] [PubMed]
  25. 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, 456-458 (2006).
    [CrossRef] [PubMed]
  26. I. D. Chremmos and N. K. Uzunoglu, "Integral equation analysis of microring and microdisk coupled-resonator optical waveguides," IEEE Photon. Technol. Lett. 18, 1173-1175 (2006).
    [CrossRef]
  27. M. Sumetsky, "Vertically-stacked multi-ring resonator," Opt. Express 13, 6354-6375 (2005).
    [CrossRef] [PubMed]
  28. S. A. Maier, P. G. Kik, and H. A. Atwater, "Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss," Appl. Phys. Lett. 81, 1714-1716 (2002).
    [CrossRef]
  29. J. Poon, J. Scheuer, Y. Xu, and A. Yariv, "Designing coupled-resonator optical waveguide delay lines," J. Opt. Soc. Am. B 21, 1665-1673 (2004).
    [CrossRef]
  30. J. 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]
  31. J. Scheuer and A. Yariv, "Sagnac effect in coupled-resonator slow-light waveguide structures," Phys. Rev. Lett. 96, 053901 (2006).
    [CrossRef] [PubMed]
  32. 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]
  33. Y. Yang and Y. Huang, "Symmetry analysis and numerical simulation of mode characteristics for equilateral-polygonal optical microresonators," Phys. Rev. A 76, 023822 (2007).
    [CrossRef]

2007 (2)

S. V. Boriskina, "Coupling of whispering-gallery modes in size-mismatched microdisk photonic molecules," Opt. Lett. 32, 1557-1559 (2007).
[CrossRef] [PubMed]

Y. Yang and Y. Huang, "Symmetry analysis and numerical simulation of mode characteristics for equilateral-polygonal optical microresonators," Phys. Rev. A 76, 023822 (2007).
[CrossRef]

2006 (13)

J. Scheuer and A. Yariv, "Sagnac effect in coupled-resonator slow-light waveguide structures," Phys. Rev. Lett. 96, 053901 (2006).
[CrossRef] [PubMed]

Z. Chen, A. Taflove, and V. Backman, "Highly efficient optical coupling and transport phenomena in chains of dielectric microspheres," Opt. Lett. 31, 389-391 (2006).
[CrossRef] [PubMed]

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, 456-458 (2006).
[CrossRef] [PubMed]

I. D. Chremmos and N. K. Uzunoglu, "Integral equation analysis of microring and microdisk coupled-resonator optical waveguides," IEEE Photon. Technol. Lett. 18, 1173-1175 (2006).
[CrossRef]

A. L. Burin, "Bound whispering gallery modes in circular arrays of dielectric spherical particles," Phys. Rev. E 73, 066614 (2006).
[CrossRef]

N.-S. Zhao, H. Zhou, Q. Guo, W. Hu, X.-B. Yang, S. Lan, and X.-S. Lin, "Design of highly efficient optical diodes based on the dynamics of nonlinear photonic crystal molecules," J. Opt. Soc. Am. B 23, 2434-2440 (2006).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. D. Sewell, and A. I. Nosich, "Directional emission, increased free spectral range and mode Q-factors in 2-D wavelength-scale optical microcavity structures," IEEE J. Sel. Top. Quantum Electron. 12, 1175-1182 (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] [PubMed]

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

S. Ishii, A. Nakagawa, and T. Baba, "Modal characteristics and bistability in twin microdisk photonic molecule lasers," IEEE J. Sel. Top. Quantum Electron. 12, 71-77 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, "Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing," IEEE J. Sel. Top. Quantum Electron. 12, 78-85 (2006).
[CrossRef]

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, "Ultralow lasing thresholds of π-type supermodes in cyclic photonic molecules composed of submicron disks with monopole and dipole modes," IEEE Photon. Technol. Lett. 18, 1993-1995 (2006).
[CrossRef]

2005 (3)

2004 (6)

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

J. Poon, J. Scheuer, Y. Xu, and A. Yariv, "Designing coupled-resonator optical waveguide delay lines," J. Opt. Soc. Am. B 21, 1665-1673 (2004).
[CrossRef]

J. 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]

V. N. Astratov, J. P. Franchak, and S. P. Ashili, "Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder," Appl. Phys. Lett. 85, 5508-5510 (2004).
[CrossRef]

B. Möller, U. Woggon, M. Artemyev, and R. Wannemacher, "Photonic molecules doped with semiconductor nanocrystals," Phys. Rev. B 70, 115323 (2004).
[CrossRef]

Y. P. Rakovich, J. F. Donegan, M. Gerlach, A. L. Bradley, T. M. Connolly, J. J. Boland, N. Gaponik, and A. Rogach, "Fine structure of coupled optical modes in photonic molecules," Phys. Rev. A 70, 051801(R) (2004).
[CrossRef]

2003 (1)

2002 (1)

S. A. Maier, P. G. Kik, and H. A. Atwater, "Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss," Appl. Phys. Lett. 81, 1714-1716 (2002).
[CrossRef]

2001 (3)

S. Olivier, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdre, and U. Oesterle, "Miniband transmission in a photonic crystal coupled-resonator optical waveguide," Opt. Lett. 26, 1019-1021 (2001).
[CrossRef]

M. Bayindir, S. Tanriseven, and E. Ozbay, "Propagation of light through localized coupled-cavity modes in one dimensional photonic band-gap structures," Appl. Phys. A 72, 117-119 (2001).
[CrossRef]

G. Guttroff, M. Bayer, A. Forchel, P. A. Knipp, and T. L. Reinecke, "Isomeric photonic molecules formed from coupled microresonators," Phys. Rev. E 63, 036611 (2001).
[CrossRef]

1999 (3)

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]

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, "Tight-binding photonic molecule modes of resonant bispheres," Phys. Rev. Lett. 82, 4623-4626 (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]

1998 (1)

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, "Optical modes in photonic molecules," Phys. Rev. Lett. 81, 2582-2586 (1998).
[CrossRef]

Appl. Phys. A (1)

M. Bayindir, S. Tanriseven, and E. Ozbay, "Propagation of light through localized coupled-cavity modes in one dimensional photonic band-gap structures," Appl. Phys. A 72, 117-119 (2001).
[CrossRef]

Appl. Phys. Lett. (3)

V. N. Astratov, J. P. Franchak, and S. P. Ashili, "Optical coupling and transport phenomena in chains of spherical dielectric microresonators with size disorder," Appl. Phys. Lett. 85, 5508-5510 (2004).
[CrossRef]

S. A. Maier, P. G. Kik, and H. A. Atwater, "Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: Estimation of waveguide loss," Appl. Phys. Lett. 81, 1714-1716 (2002).
[CrossRef]

A. Nakagawa, S. Ishii, and T. Baba, "Photonic molecule laser composed of GaInAsP microdisks," Appl. Phys. Lett. 86, 041112 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

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. (3)

S. Ishii, A. Nakagawa, and T. Baba, "Modal characteristics and bistability in twin microdisk photonic molecule lasers," IEEE J. Sel. Top. Quantum Electron. 12, 71-77 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, "Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing," IEEE J. Sel. Top. Quantum Electron. 12, 78-85 (2006).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. D. Sewell, and A. I. Nosich, "Directional emission, increased free spectral range and mode Q-factors in 2-D wavelength-scale optical microcavity structures," IEEE J. Sel. Top. Quantum Electron. 12, 1175-1182 (2006).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, "Ultralow lasing thresholds of π-type supermodes in cyclic photonic molecules composed of submicron disks with monopole and dipole modes," IEEE Photon. Technol. Lett. 18, 1993-1995 (2006).
[CrossRef]

J. 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]

I. D. Chremmos and N. K. Uzunoglu, "Integral equation analysis of microring and microdisk coupled-resonator optical waveguides," IEEE Photon. Technol. Lett. 18, 1173-1175 (2006).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Express (2)

Opt. Lett. (9)

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, 456-458 (2006).
[CrossRef] [PubMed]

Z. Chen, A. Taflove, and V. Backman, "Highly efficient optical coupling and transport phenomena in chains of dielectric microspheres," Opt. Lett. 31, 389-391 (2006).
[CrossRef] [PubMed]

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]

S. Olivier, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdre, and U. Oesterle, "Miniband transmission in a photonic crystal coupled-resonator optical waveguide," Opt. Lett. 26, 1019-1021 (2001).
[CrossRef]

S. V. Boriskina, "Coupling of whispering-gallery modes in size-mismatched microdisk photonic molecules," Opt. Lett. 32, 1557-1559 (2007).
[CrossRef] [PubMed]

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]

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] [PubMed]

Y. P. Rakovich, J. J. Boland, and J. F. Donegan, "Tunable photon lifetime in photonic molecules: a concept for delaying an optical signal," Opt. Lett. 30, 2775-2777 (2005).
[CrossRef] [PubMed]

Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, "Photonic molecule lasing," Opt. Lett. 28, 2437-2439 (2003).
[CrossRef] [PubMed]

Phys. Rev. A (2)

Y. P. Rakovich, J. F. Donegan, M. Gerlach, A. L. Bradley, T. M. Connolly, J. J. Boland, N. Gaponik, and A. Rogach, "Fine structure of coupled optical modes in photonic molecules," Phys. Rev. A 70, 051801(R) (2004).
[CrossRef]

Y. Yang and Y. Huang, "Symmetry analysis and numerical simulation of mode characteristics for equilateral-polygonal optical microresonators," Phys. Rev. A 76, 023822 (2007).
[CrossRef]

Phys. Rev. B (1)

B. Möller, U. Woggon, M. Artemyev, and R. Wannemacher, "Photonic molecules doped with semiconductor nanocrystals," Phys. Rev. B 70, 115323 (2004).
[CrossRef]

Phys. Rev. E (2)

A. L. Burin, "Bound whispering gallery modes in circular arrays of dielectric spherical particles," Phys. Rev. E 73, 066614 (2006).
[CrossRef]

G. Guttroff, M. Bayer, A. Forchel, P. A. Knipp, and T. L. Reinecke, "Isomeric photonic molecules formed from coupled microresonators," Phys. Rev. E 63, 036611 (2001).
[CrossRef]

Phys. Rev. Lett. (3)

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, "Optical modes in photonic molecules," Phys. Rev. Lett. 81, 2582-2586 (1998).
[CrossRef]

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, "Tight-binding photonic molecule modes of resonant bispheres," Phys. Rev. Lett. 82, 4623-4626 (1999).
[CrossRef]

J. Scheuer and A. Yariv, "Sagnac effect in coupled-resonator slow-light waveguide structures," Phys. Rev. Lett. 96, 053901 (2006).
[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

Regular (a) pentagon and (b) hexagon of coupled ring resonators. (c) Propagation and coupling of waves in two adjacent rings.

Fig. 2
Fig. 2

Brillouin diagram of resonances for a regular (a) octagon ( N = 8 ) and (b) pentagon ( N = 5 ) with interring coupling coefficient κ = 0.2 . The dashed curves are the continuous dispersion lines of the infinite ring CROW.

Fig. 3
Fig. 3

Pairs ( λ m , Γ m ) of eigenvalue and field amplitude ratio for resonances with the same frequency detuning from ω 0 .

Fig. 4
Fig. 4

Field symmetry for the standing modes and (a) odd N = 5 or (b) even N = 6 . The dashed lines are planes of zero field.

Fig. 5
Fig. 5

Channel dropping filter of a regular hexagon symmetrically coupled to two waveguides.

Fig. 6
Fig. 6

Dropped power | T d | 2 for a regular hexagon ( N = 6 ) versus δ ϕ = ϕ ϕ 0 , in two cases of ring-waveguide coupling coefficient κ w = 0.25 (dashed curve) and 0.35 (solid curve). The interring coupling coefficient is κ = 0.2 .

Fig. 7
Fig. 7

Dropped power | T d | 2 for a regular octagon ( N = 8 ) versus δϕ ( κ = 0.2 , κ w = 0.25 ).

Fig. 8
Fig. 8

Dropped power | T d | 2 for a regular decagon ( N = 10 ) versus δϕ ( κ = 0.2 , κ w = 0.25 ).

Fig. 9
Fig. 9

Dropped power | T d | 2 for a regular dodecagon ( N = 12 ) versus δϕ ( κ = 0.2 , κ w = 0.25 ).

Equations (18)

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

( a 2 a 1 ) = [ e j ϕ N 0 0 e j ( ϕ ϕ N ) ] ( a 1 a 2 ) , ( b 1 a 1 ) = [ t j κ j κ t ] ( b 2 a 2 ) ,
b = C P ( ϕ N ) a .
P ( ϕ N ) = [ e j ϕ N 0 0 e j ( ϕ ϕ N ) ] , C = 1 j κ [ 1 t t 1 ] ,
T = C P ( ϕ ϕ N ) C P ( ϕ N ) = 2 e j ( π ϕ ) / 2 sin ( ϕ / 2 ) κ 2
× [ 1 t e j ϕ t e j ϕ ] I ,
T a = e j w m a ,   w m = { 4 m π / N , N even 2 m π / N , N odd ,
sin ( ϕ m / 2 ) = ± κ cos ( w m / 2 ) .
δ ω m = ω 0 q π × { sin 1 [ κ cos ( 2 m π / N ) ] , m = 0 , 1 ,   , N / 2 , N even sin 1 [ κ cos ( m π / N ) ] , m = 0 , 1 ,   , N , N odd .
a 2 a 1 Γ m = e j ϕ m / 2 t [ cos ( ϕ m / 2 ) ± κ sin ( w m / 2 ) ] ,
b = a e j ( ϕ m / N w m / 2 ) ,
a = ( 1 0 ) T b = j e j ϕ 0 / N κ 1 ( 1 t ) T ,
a = ( 0 1 ) T b = + j e j ϕ 0 / N κ 1 ( t 1 ) T .
a 1 e j ϕ N / 2 = b 2 e + j ϕ N / 2 = a 2 e j ( ϕ ϕ N ) / 2 = b 1 e j ( ϕ ϕ N ) / 2 .
f = D w a drop , a = D w a in , D w = 1 j κ w [ t w 1 1 t w ] ,
f = A d + B e , e 2 = d 1 e j ϕ N a = A b + B c , c 2 = b 1 e j ϕ N ,
    A = [ 0 e j ( ϕ ϕ N ) / 2 0 0 ] ,     B = [ 0 0 e j ( ϕ ϕ N ) / 2 0 ] .
d = S b = S D b , c = S e = S D e ,
S = { C P ( ϕ N ) T p 1 , N = 4 p T p , N = 4 p + 2 ,

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