Abstract

An analytical approach for obtaining linear and nonlinear design parameters of microresonators is presented. The eigenmode/eigenfrequency problem of planar resonators is considered in detail, with an analytical closed-form approximation derived for resonators possessing a large radius to width ratio. The analysis permits the resonant frequencies and mode profiles to be determined together with the dispersion properties. The dependence of the effective nonlinear Kerr coefficient on the mode volume is further considered, and also the waveguide coupling together with estimates of the Q-value. Examples, which are in good agreement with numerical simulations, are presented for silicon resonators. The approach can be used for designing planar microring resonators for nonlinear four-wave mixing applications, such as optical Kerr frequency comb generation.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
    [CrossRef]
  2. F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
    [CrossRef]
  3. A. Schliesser, N. Picqué, and T. W. Hänsch, “Mid-infrared frequency combs,” Nat. Photonics 6, 440–449 (2012).
    [CrossRef]
  4. R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
    [CrossRef]
  5. T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
    [CrossRef]
  6. Y. Okawachi, K. Saha, J. S. Levy, Y. H. Wen, M. Lipson, and A. L. Gaeta, “Octave-spanning frequency comb generation in a silicon nitride chip,” Opt. Lett. 36, 3398–3400 (2011).
    [CrossRef]
  7. S. Azzini, D. Grassani, M. Galli, L. C. Andreani, M. Sorel, M. J. Strain, L. G. Helt, J. E. Sipe, M. Liscidini, and D. Bajoni, “From classical four-wave mixing to parametric fluorescence in silicon microring resonators,” Opt. Lett. 37, 3807–3809 (2012).
    [CrossRef]
  8. A. Biberman, M. J. Shaw, E. Timurdogan, J. B. Wright, and M. R. Watts, “Ultralow-loss silicon ring resonators,” Opt. Lett. 37, 4236–4238 (2012).
    [CrossRef]
  9. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
    [CrossRef]
  10. M. Hossein-Zadeh and K. J. Vahala, “Free ultra-high-Q microtoroid: a tool for designing photonic devices,” Opt. Express 15, 166–175 (2007).
    [CrossRef]
  11. H. Tavernier, P. Salzenstein, K. Volyanskiy, Y. K. Chembo, and L. Larger, “Magnesium fluoride whispering gallery mode disk-resonators for microwave photonics applications,” IEEE Photon. Technol. Lett. 22, 1629–1631 (2010).
  12. M. Ferrera, D. Duchesne, L. Razzari, M. Peccianti, R. Morandotti, P. Cheben, S. Janz, D. X. Xu, B. E. Little, S. Chu, and D. J. Moss, “Low power four-wave mixing in an integrated, micro-ring resonator with Q = 1.2 million,” Opt. Express 17, 14098–14103 (2009).
    [CrossRef]
  13. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
  14. A. R. Johnson, Y. Okawachi, J. S. Levy, J. Cardenas, K. Saha, M. Lipson, and A. L. Gaeta, “Chip-based frequency combs with sub-100  GHz repetition rates,” Opt. Lett. 37, 875–877 (2012).
    [CrossRef]
  15. C. Y. Wang, T. Herr, P. Del’Haye, A. Schliesser, J. Hofer, R. Holzwarth, T. W. Hänsch, N. Picqué, and T. J. Kippenberg, “Mid-infrared optical frequency combs at 2.5  μm based on crystalline microresonators,” Nat. Commun. 4, 1345 (2013).
    [CrossRef]
  16. T. Hansson, D. Modotto, and S. Wabnitz, “Dynamics of the modulational instability in microresonator frequency combs,” Phys. Rev. A 88, 023819 (2013).
    [CrossRef]
  17. A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).
  18. M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering-gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39, (2006).
  19. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
    [CrossRef]
  20. M. K. Chin and S. T. Ho, “Design and modeling of waveguide-coupled single-mode microring resonators,” J. Lightwave Technol. 16, 1433–1446 (1998).
    [CrossRef]
  21. 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).
  22. O. Schwelb, “Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters—a tutorial overview,” J. Lightwave Technol. 22, 1380–1394 (2004).
    [CrossRef]
  23. S. J. Emelett and R. Soref, “Design and simulation of silicon microring optical routing switches,” J. Lightwave Technol. 23, 1800–1807 (2005).
    [CrossRef]
  24. A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. 36, 2845–2847 (2011).
    [CrossRef]
  25. S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato–Lefever model,” Opt. Lett. 38, 37–39 (2013).
    [CrossRef]
  26. A. Matsko, A. Savchenkov, D. Strekalov, V. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: threshold and phase diffusion,” Phys. Rev. A 71, 033804 (2005).
    [CrossRef]
  27. Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
    [CrossRef]
  28. M. Sumetsky, “Whispering-gallery-bottle microcavities: the three-dimensional etalon,” Opt. Lett. 29, 8–10 (2004).
    [CrossRef]
  29. E. A. J. Marcatili, “Bends in optical dielectric guides,” AT&T Tech. J. 48, 2103–2132 (1969).
  30. K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed., (Academic, 2006).
  31. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1998).
  32. M. Sumetsky, Y. Dulashko, and R. S. Windeler, “Super free spectral range tunable optical microbubble resonator,” Opt. Lett. 35, 1866–1868, (2010).
    [CrossRef]
  33. M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microwave Theor. Tech. 55, 1209–1218 (2007).
  34. H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
    [CrossRef]
  35. M. Kuznetsov, “Expressions for the coupling coefficient of a rectangular-waveguide directional coupler,” Opt. Lett. 8, 499–501 (1983).
    [CrossRef]
  36. F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness induced backscattering in optical silicon waveguides,” Phys. Rev. Lett. 104, 033902 (2010).
    [CrossRef]

2013 (3)

C. Y. Wang, T. Herr, P. Del’Haye, A. Schliesser, J. Hofer, R. Holzwarth, T. W. Hänsch, N. Picqué, and T. J. Kippenberg, “Mid-infrared optical frequency combs at 2.5  μm based on crystalline microresonators,” Nat. Commun. 4, 1345 (2013).
[CrossRef]

T. Hansson, D. Modotto, and S. Wabnitz, “Dynamics of the modulational instability in microresonator frequency combs,” Phys. Rev. A 88, 023819 (2013).
[CrossRef]

S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato–Lefever model,” Opt. Lett. 38, 37–39 (2013).
[CrossRef]

2012 (4)

2011 (3)

2010 (4)

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness induced backscattering in optical silicon waveguides,” Phys. Rev. Lett. 104, 033902 (2010).
[CrossRef]

M. Sumetsky, Y. Dulashko, and R. S. Windeler, “Super free spectral range tunable optical microbubble resonator,” Opt. Lett. 35, 1866–1868, (2010).
[CrossRef]

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
[CrossRef]

H. Tavernier, P. Salzenstein, K. Volyanskiy, Y. K. Chembo, and L. Larger, “Magnesium fluoride whispering gallery mode disk-resonators for microwave photonics applications,” IEEE Photon. Technol. Lett. 22, 1629–1631 (2010).

2009 (1)

2007 (3)

M. Hossein-Zadeh and K. J. Vahala, “Free ultra-high-Q microtoroid: a tool for designing photonic devices,” Opt. Express 15, 166–175 (2007).
[CrossRef]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microwave Theor. Tech. 55, 1209–1218 (2007).

2006 (2)

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering-gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39, (2006).

2005 (2)

A. Matsko, A. Savchenkov, D. Strekalov, V. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: threshold and phase diffusion,” Phys. Rev. A 71, 033804 (2005).
[CrossRef]

S. J. Emelett and R. Soref, “Design and simulation of silicon microring optical routing switches,” J. Lightwave Technol. 23, 1800–1807 (2005).
[CrossRef]

2004 (2)

2003 (1)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

2002 (1)

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
[CrossRef]

2000 (1)

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef]

1999 (1)

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).

1998 (2)

M. K. Chin and S. T. Ho, “Design and modeling of waveguide-coupled single-mode microring resonators,” J. Lightwave Technol. 16, 1433–1446 (1998).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

1987 (1)

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

1983 (1)

1969 (1)

E. A. J. Marcatili, “Bends in optical dielectric guides,” AT&T Tech. J. 48, 2103–2132 (1969).

Andreani, L. C.

Arcizet, O.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

Azzini, S.

Bajoni, D.

Biberman, A.

Canciamilla, A.

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness induced backscattering in optical silicon waveguides,” Phys. Rev. Lett. 104, 033902 (2010).
[CrossRef]

Cardenas, J.

Cheben, P.

Chembo, Y. K.

H. Tavernier, P. Salzenstein, K. Volyanskiy, Y. K. Chembo, and L. Larger, “Magnesium fluoride whispering gallery mode disk-resonators for microwave photonics applications,” IEEE Photon. Technol. Lett. 22, 1629–1631 (2010).

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
[CrossRef]

Chen, L.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Chin, M. K.

Chu, S.

Chu, S. T.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Coen, S.

Del’Haye, P.

C. Y. Wang, T. Herr, P. Del’Haye, A. Schliesser, J. Hofer, R. Holzwarth, T. W. Hänsch, N. Picqué, and T. J. Kippenberg, “Mid-infrared optical frequency combs at 2.5  μm based on crystalline microresonators,” Nat. Commun. 4, 1345 (2013).
[CrossRef]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Duchesne, D.

Dulashko, Y.

Emelett, S. J.

Erkintalo, M.

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).

Ferdous, F.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Ferrari, C.

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness induced backscattering in optical silicon waveguides,” Phys. Rev. Lett. 104, 033902 (2010).
[CrossRef]

Ferrera, M.

Fomin, A. E.

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering-gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39, (2006).

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Foresi, J. S.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).

Gaeta, A. L.

Galli, M.

Gorodetsky, M. L.

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering-gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39, (2006).

Grassani, D.

Greene, W.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).

Hänsch, T. W.

C. Y. Wang, T. Herr, P. Del’Haye, A. Schliesser, J. Hofer, R. Holzwarth, T. W. Hänsch, N. Picqué, and T. J. Kippenberg, “Mid-infrared optical frequency combs at 2.5  μm based on crystalline microresonators,” Nat. Commun. 4, 1345 (2013).
[CrossRef]

A. Schliesser, N. Picqué, and T. W. Hänsch, “Mid-infrared frequency combs,” Nat. Photonics 6, 440–449 (2012).
[CrossRef]

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
[CrossRef]

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef]

Hansson, T.

T. Hansson, D. Modotto, and S. Wabnitz, “Dynamics of the modulational instability in microresonator frequency combs,” Phys. Rev. A 88, 023819 (2013).
[CrossRef]

Haus, H. A.

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).

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

Helt, L. G.

Herr, T.

C. Y. Wang, T. Herr, P. Del’Haye, A. Schliesser, J. Hofer, R. Holzwarth, T. W. Hänsch, N. Picqué, and T. J. Kippenberg, “Mid-infrared optical frequency combs at 2.5  μm based on crystalline microresonators,” Nat. Commun. 4, 1345 (2013).
[CrossRef]

Ho, S. T.

Hofer, J.

C. Y. Wang, T. Herr, P. Del’Haye, A. Schliesser, J. Hofer, R. Holzwarth, T. W. Hänsch, N. Picqué, and T. J. Kippenberg, “Mid-infrared optical frequency combs at 2.5  μm based on crystalline microresonators,” Nat. Commun. 4, 1345 (2013).
[CrossRef]

Holzwarth, R.

C. Y. Wang, T. Herr, P. Del’Haye, A. Schliesser, J. Hofer, R. Holzwarth, T. W. Hänsch, N. Picqué, and T. J. Kippenberg, “Mid-infrared optical frequency combs at 2.5  μm based on crystalline microresonators,” Nat. Commun. 4, 1345 (2013).
[CrossRef]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
[CrossRef]

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef]

Hossein-Zadeh, M.

Huang, W. P.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

Ilchenko, V.

A. Matsko, A. Savchenkov, D. Strekalov, V. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: threshold and phase diffusion,” Phys. Rev. A 71, 033804 (2005).
[CrossRef]

Ilchenko, V. S.

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. 36, 2845–2847 (2011).
[CrossRef]

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).

Ippen, E. P.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).

Janz, S.

Joannopoulos, J. D.

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).

Johnson, A. R.

Kawakami, S.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

Khan, M. J.

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).

Kimerling, L. C.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).

Kippenberg, T. J.

C. Y. Wang, T. Herr, P. Del’Haye, A. Schliesser, J. Hofer, R. Holzwarth, T. W. Hänsch, N. Picqué, and T. J. Kippenberg, “Mid-infrared optical frequency combs at 2.5  μm based on crystalline microresonators,” Nat. Commun. 4, 1345 (2013).
[CrossRef]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

Knight, J. C.

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef]

Kuznetsov, M.

Laine, J. P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Larger, L.

H. Tavernier, P. Salzenstein, K. Volyanskiy, Y. K. Chembo, and L. Larger, “Magnesium fluoride whispering gallery mode disk-resonators for microwave photonics applications,” IEEE Photon. Technol. Lett. 22, 1629–1631 (2010).

Leaird, D. E.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Levy, J. S.

Liang, W.

Lipson, M.

Liscidini, M.

Little, B. E.

M. Ferrera, D. Duchesne, L. Razzari, M. Peccianti, R. Morandotti, P. Cheben, S. Janz, D. X. Xu, B. E. Little, S. Chu, and D. J. Moss, “Low power four-wave mixing in an integrated, micro-ring resonator with Q = 1.2 million,” Opt. Express 17, 14098–14103 (2009).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Maleki, L.

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. 36, 2845–2847 (2011).
[CrossRef]

A. Matsko, A. Savchenkov, D. Strekalov, V. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: threshold and phase diffusion,” Phys. Rev. A 71, 033804 (2005).
[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).

Marcatili, E. A. J.

E. A. J. Marcatili, “Bends in optical dielectric guides,” AT&T Tech. J. 48, 2103–2132 (1969).

Martinelli, M.

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness induced backscattering in optical silicon waveguides,” Phys. Rev. Lett. 104, 033902 (2010).
[CrossRef]

Matsko, A.

A. Matsko, A. Savchenkov, D. Strekalov, V. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: threshold and phase diffusion,” Phys. Rev. A 71, 033804 (2005).
[CrossRef]

Matsko, A. B.

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. 36, 2845–2847 (2011).
[CrossRef]

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).

Melloni, A.

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness induced backscattering in optical silicon waveguides,” Phys. Rev. Lett. 104, 033902 (2010).
[CrossRef]

Miao, H.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Modotto, D.

T. Hansson, D. Modotto, and S. Wabnitz, “Dynamics of the modulational instability in microresonator frequency combs,” Phys. Rev. A 88, 023819 (2013).
[CrossRef]

Morandotti, R.

Morichetti, F.

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness induced backscattering in optical silicon waveguides,” Phys. Rev. Lett. 104, 033902 (2010).
[CrossRef]

Moss, D. J.

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed., (Academic, 2006).

Okawachi, Y.

Oxborrow, M.

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microwave Theor. Tech. 55, 1209–1218 (2007).

Peccianti, M.

Picqué, N.

C. Y. Wang, T. Herr, P. Del’Haye, A. Schliesser, J. Hofer, R. Holzwarth, T. W. Hänsch, N. Picqué, and T. J. Kippenberg, “Mid-infrared optical frequency combs at 2.5  μm based on crystalline microresonators,” Nat. Commun. 4, 1345 (2013).
[CrossRef]

A. Schliesser, N. Picqué, and T. W. Hänsch, “Mid-infrared frequency combs,” Nat. Photonics 6, 440–449 (2012).
[CrossRef]

Randle, H. G.

Razzari, L.

Russell, P. St. J.

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef]

Saha, K.

Salzenstein, P.

H. Tavernier, P. Salzenstein, K. Volyanskiy, Y. K. Chembo, and L. Larger, “Magnesium fluoride whispering gallery mode disk-resonators for microwave photonics applications,” IEEE Photon. Technol. Lett. 22, 1629–1631 (2010).

Savchenkov, A.

A. Matsko, A. Savchenkov, D. Strekalov, V. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: threshold and phase diffusion,” Phys. Rev. A 71, 033804 (2005).
[CrossRef]

Savchenkov, A. A.

Schliesser, A.

C. Y. Wang, T. Herr, P. Del’Haye, A. Schliesser, J. Hofer, R. Holzwarth, T. W. Hänsch, N. Picqué, and T. J. Kippenberg, “Mid-infrared optical frequency combs at 2.5  μm based on crystalline microresonators,” Nat. Commun. 4, 1345 (2013).
[CrossRef]

A. Schliesser, N. Picqué, and T. W. Hänsch, “Mid-infrared frequency combs,” Nat. Photonics 6, 440–449 (2012).
[CrossRef]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Schwelb, O.

Seidel, D.

Shaw, M. J.

Sipe, J. E.

Soref, R.

Sorel, M.

Spillane, S. M.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

Srinivasan, K.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Steinmeyer, G.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).

Strain, M. J.

Strekalov, D.

A. Matsko, A. Savchenkov, D. Strekalov, V. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: threshold and phase diffusion,” Phys. Rev. A 71, 033804 (2005).
[CrossRef]

Sumetsky, M.

Sylvestre, T.

Tavernier, H.

H. Tavernier, P. Salzenstein, K. Volyanskiy, Y. K. Chembo, and L. Larger, “Magnesium fluoride whispering gallery mode disk-resonators for microwave photonics applications,” IEEE Photon. Technol. Lett. 22, 1629–1631 (2010).

Thoen, E. R.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).

Timurdogan, E.

Torregiani, M.

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness induced backscattering in optical silicon waveguides,” Phys. Rev. Lett. 104, 033902 (2010).
[CrossRef]

Udem, T.

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
[CrossRef]

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef]

Vahala, K. J.

M. Hossein-Zadeh and K. J. Vahala, “Free ultra-high-Q microtoroid: a tool for designing photonic devices,” Opt. Express 15, 166–175 (2007).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

Varghese, L. T.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

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).

Volyanskiy, K.

H. Tavernier, P. Salzenstein, K. Volyanskiy, Y. K. Chembo, and L. Larger, “Magnesium fluoride whispering gallery mode disk-resonators for microwave photonics applications,” IEEE Photon. Technol. Lett. 22, 1629–1631 (2010).

Wabnitz, S.

T. Hansson, D. Modotto, and S. Wabnitz, “Dynamics of the modulational instability in microresonator frequency combs,” Phys. Rev. A 88, 023819 (2013).
[CrossRef]

Wadsworth, W. J.

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef]

Wang, C. Y.

C. Y. Wang, T. Herr, P. Del’Haye, A. Schliesser, J. Hofer, R. Holzwarth, T. W. Hänsch, N. Picqué, and T. J. Kippenberg, “Mid-infrared optical frequency combs at 2.5  μm based on crystalline microresonators,” Nat. Commun. 4, 1345 (2013).
[CrossRef]

Wang, J.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Watts, M. R.

Weiner, A. M.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

Wen, Y. H.

Whitaker, N. A.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5, 16–23 (1987).
[CrossRef]

Wilken, T.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Windeler, R. S.

Wright, J. B.

Xu, D. X.

Yu, N.

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
[CrossRef]

AT&T Tech. J. (1)

E. A. J. Marcatili, “Bends in optical dielectric guides,” AT&T Tech. J. 48, 2103–2132 (1969).

IEEE J. Quantum Electron. (1)

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).

IEEE J. Sel. Top. Quantum Electron. (2)

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering-gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39, (2006).

IEEE Photon. Technol. Lett. (2)

H. Tavernier, P. Salzenstein, K. Volyanskiy, Y. K. Chembo, and L. Larger, “Magnesium fluoride whispering gallery mode disk-resonators for microwave photonics applications,” IEEE Photon. Technol. Lett. 22, 1629–1631 (2010).

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).

IEEE Trans. Microwave Theor. Tech. (1)

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microwave Theor. Tech. 55, 1209–1218 (2007).

J. Lightwave Technol. (5)

Nat. Commun. (1)

C. Y. Wang, T. Herr, P. Del’Haye, A. Schliesser, J. Hofer, R. Holzwarth, T. W. Hänsch, N. Picqué, and T. J. Kippenberg, “Mid-infrared optical frequency combs at 2.5  μm based on crystalline microresonators,” Nat. Commun. 4, 1345 (2013).
[CrossRef]

Nat. Photonics (2)

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011).
[CrossRef]

A. Schliesser, N. Picqué, and T. W. Hänsch, “Mid-infrared frequency combs,” Nat. Photonics 6, 440–449 (2012).
[CrossRef]

Nature (3)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416, 233–237 (2002).
[CrossRef]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lett. (9)

A. R. Johnson, Y. Okawachi, J. S. Levy, J. Cardenas, K. Saha, M. Lipson, and A. L. Gaeta, “Chip-based frequency combs with sub-100  GHz repetition rates,” Opt. Lett. 37, 875–877 (2012).
[CrossRef]

Y. Okawachi, K. Saha, J. S. Levy, Y. H. Wen, M. Lipson, and A. L. Gaeta, “Octave-spanning frequency comb generation in a silicon nitride chip,” Opt. Lett. 36, 3398–3400 (2011).
[CrossRef]

S. Azzini, D. Grassani, M. Galli, L. C. Andreani, M. Sorel, M. J. Strain, L. G. Helt, J. E. Sipe, M. Liscidini, and D. Bajoni, “From classical four-wave mixing to parametric fluorescence in silicon microring resonators,” Opt. Lett. 37, 3807–3809 (2012).
[CrossRef]

A. Biberman, M. J. Shaw, E. Timurdogan, J. B. Wright, and M. R. Watts, “Ultralow-loss silicon ring resonators,” Opt. Lett. 37, 4236–4238 (2012).
[CrossRef]

M. Sumetsky, “Whispering-gallery-bottle microcavities: the three-dimensional etalon,” Opt. Lett. 29, 8–10 (2004).
[CrossRef]

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. 36, 2845–2847 (2011).
[CrossRef]

S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato–Lefever model,” Opt. Lett. 38, 37–39 (2013).
[CrossRef]

M. Kuznetsov, “Expressions for the coupling coefficient of a rectangular-waveguide directional coupler,” Opt. Lett. 8, 499–501 (1983).
[CrossRef]

M. Sumetsky, Y. Dulashko, and R. S. Windeler, “Super free spectral range tunable optical microbubble resonator,” Opt. Lett. 35, 1866–1868, (2010).
[CrossRef]

Phys. Rev. A (3)

A. Matsko, A. Savchenkov, D. Strekalov, V. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: threshold and phase diffusion,” Phys. Rev. A 71, 033804 (2005).
[CrossRef]

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
[CrossRef]

T. Hansson, D. Modotto, and S. Wabnitz, “Dynamics of the modulational instability in microresonator frequency combs,” Phys. Rev. A 88, 023819 (2013).
[CrossRef]

Phys. Rev. Lett. (2)

F. Morichetti, A. Canciamilla, C. Ferrari, M. Torregiani, A. Melloni, and M. Martinelli, “Roughness induced backscattering in optical silicon waveguides,” Phys. Rev. Lett. 104, 033902 (2010).
[CrossRef]

R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000).
[CrossRef]

Other (2)

K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed., (Academic, 2006).

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1998).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1.

Cross section showing layer structure of the ring waveguide.

Fig. 2.
Fig. 2.

Effective refractive index as a function of wavelength for a silicon ring resonator with a=49.5μm, b=50.5μm, and h=0.5μm. Fully drawn lines are explicit analytical approximations, while dotted lines are numerical solutions of the consistency requirements; dashed–dotted lines are numerical mode solver solutions. The upper solutions (blue color) signify the E11x mode and the lower solution (red color) the E11y mode.

Fig. 3.
Fig. 3.

Differences between the analytical approximations and mode solver solution for the effective refractive index plotted in Fig. 2. The fully drawn (blue) line signifies the difference of the explicit analytical approximation of the E11x mode, while the dashed (blue) line is the difference of the numerical solution of the consistency requirement for the E11x mode. The dashed–dotted (red) line is the corresponding difference of the explicit analytical approximation of the E11y mode while the dotted (red) line is the difference of the numerical solution of the consistency requirement for the E11y mode.

Fig. 4.
Fig. 4.

Dispersion D for the E11x mode of a silicon ring resonator with a=49.5μm, b=50.5μm, and h=0.5μm. Fully drawn lines are explicit analytical approximations, while dotted lines are numerical solutions of the consistency requirements and dashed lines are numerical mode solver solutions.

Fig. 5.
Fig. 5.

Dispersion D for the E11y mode of a silicon ring resonator with the same parameters as in Fig. 4.

Fig. 6.
Fig. 6.

Comparison of cylindrical whispering-gallery-modes, explicit analytical solutions using the rectangular approximation, and numerical mode solver solutions for the normalized mode profiles of resonant modes at λ=2.5μm in the x-direction of a silicon ring resonator. Dotted lines are WGMs, fully drawn lines are explicit analytical approximations, and dashed lines are numerical mode solver solutions. The lower solution (blue color) signifies the Hy component of the E11x mode and the upper solution (red color) the Hx component of the E11y mode. The x-scale is relative to the ring radius r0.

Fig. 7.
Fig. 7.

Comparison of explicit analytical and mode solver solutions for the normalized mode profiles in the y-direction for resonant modes at λ=2.5μm of the same silicon ring resonator as in Fig. (6). The upper solution (blue color) signifies the Hy component of the E11x mode and the lower solution (red color) the Hx component of the E11y mode.

Equations (43)

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

(2n2c22t2)H=0,
[2r2+1rr+2y2+(k02n2m2r2)]Hy(r,y)=0,
1rddr(rdRdr)+(k02ne2m2r2)R=0,
d2Ydy2+k02(n2ne2)Y=0,
R0(r)=AJm(krr)+BYm(krr),arb,
R(r)=CIm(αrr),0ra,
R+(r)=DKm(αrr),rb,
Eϕ(r,y)=iωε0n2Hy(r,y)r,Hϕ(r,y)=irβHy(r,y)y,
Eϕ(r,y)=iωε0n2Hr(r,y)y,Hϕ(r,y)=iβ(rHr(r,y))r.
B=AJm(kra)η(Im(αra)/Im(αra))Jm(kra)Ym(kra)η(Im(αra)/Im(αra))Ym(kra),
C=AJm(kra)+BYm(kra)Im(αra),D=AJm(krb)+BYm(krb)Km(αrb),
[Jm(kra)ηIm(αra)Im(αra)Jm(kra)][Ym(krb)ηKm(αrb)Km(αrb)Ym(krb)][Ym(kra)ηIm(αra)Im(αra)Ym(kra)][Jm(krb)ηKm(αrb)Kβ(αrb)Jm(krb)]=0,
Y+(y)=cos(kyhψ)eα0(yh),yh,
Y0(y)=cos(kyyψ),0yh,
Y(y)=cos(ψ)eαsy,y0,
kyh=(q1)π+arctan(ξ1αsky)+arctan(ξ2α0ky),
n1(k0)=B0+B1k02+B2k04,
[d2Rdr2+1rdRdr+(k02ne2m2r2)R]r=r0kx2+ikxr0+(k02ne2m2r02)0.
d2Xdx2+(k02ne2β2)X=0,
X=Acos(kxaϕ)eαx(xa),0xa,
X0=Acos(kxxϕ),axb,
X+=Acos(kxbϕ)eαx(xb),xb.
kx(ba)=(p1)π+2arctan(ηαxkx),
neff=β/k0=2πm/(k0L).
ne2(k0)=n12(k0)+F[n1(k0)2n02+ns22,1,k0h],
neff2(k0)=ne2(k0)+F[ne2(k0)n02,n12(k0)n02,k0(ba)],
ne2(k0)=n12(k0)+F[n12(k0)n04+ns4n02+ns2,2n12(k0)n02+ns2,k0h],
neff2(k0)=ne2(k0)+F[ne2(k0)n02,1,k0(ba)],
F[a,b,c]=9b22c2(3b2)(1+c2a9(1+c2a9)2+4(3b2)c2a9b2).
βn=dnβdωnω=ωm0,
ωm=ωm0+D1(mm0)+12D2(mm0)2+16D3(mm0)3+.
γ=ωm0n2cAm0L,
Am=(|Hx,y(x,y)|2dA)2|Hx,y(x,y)|4dA.
Am49((ba)+2ηαxkx2+η2αx2)(h+ξ1αskz2+ξ12αs2+ξ2α0kz2+ξ22α02),
κ^[s(ϕ)]=ωε04I(ne2n02)E1*·E2dx,
I=14(E1*×H1+E1×H1*)·ϕ^dx,
κ=κ^[s(ϕ)]eiΔβϕdϕ,
κ^y=2kx2αxβ(kx2+αx2)(ba2+2αx)exp[αx(2s(ba))],
κ^x=(ne/n0)22kx2αxβ(kx2+(ne/n0)4αx2)(ba2+2αxne2n02kx2+αx2kx2+(ne/n0)4αx2)exp[αx(2s(ba))],
s(ϕ)=s0,|ϕ|l0s(ϕ)=s0+(|ϕ|l0)2/(2r0),|ϕ|l0
κx,y=κ^x,y(s0)(2l0+πr0αx).
Qr2π2neffr0κ2λ0.
Qi2πneffα^λ0.

Metrics