Abstract

Nonlinear interactions within compact, on-chip microring resonant cavities is a topic of increasing interest in current silicon photonics research. Frequency combs, one of the emerging nonlinear applications in microring optics, offers great potential from both scientific and practical perspectives. However, the mechanisms of comb formation appear to differ from traditional frequency combs formed by pulsed femtosecond lasers, and thus require detailed elucidation through theory and simulation. Here we propose a technique to mimic the accuracy of finite-difference time domain (FDTD) full wave nonlinear optical simulations with only a small fraction of the computational resources. Our new hybrid approach combines a single linear FDTD simulation of the key interaction parameters, then directly inserts them into a coupled-mode theory simulation. Comparison of the hybrid approach and full FDTD shows a good match both in frequency domain and in time domain. Thus, it retains the advantage of FDTD in terms of direct connection with experimental designs, while finishing much faster and sidestepping stability issues associated with direct simulation of nonlinear phenomena. The hybrid technique produces several key results explored in this paper, including: demonstrating that comb formation can occur with both anomalous and normal dispersion; suggesting a new mechanism for incoherent (Type II) frequency comb formation; and illustrating a method for creating soliton-like pulses in on-chip microresonators.

© 2014 Optical Society of America

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    [CrossRef]

2014

T. Hansson, D. Modotto, and S. Wabnitz, “On the numerical simulation of Kerr frequency combs using coupled mode equations,” Opt. Commun. 312, 134–136 (2014).
[CrossRef]

2013

2012

M. Peccianti, A. Pasquazi, Y. Park, B.E. Little, S.T. Chu, D.J. Moss, and R. Morandotti, “Demonstration of a stable ultrafast laser based on a nonlinear microcavity,” Nature Comm. 3, 765 (2012).
[CrossRef]

P. Del’Haye, S. B. Papp, and S. A. Diddams, “Hybrid electro-optically modulated microcombs,” Phys. Rev. Lett. 109, 263901 (2012).
[CrossRef]

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photon. 6, 480–487 (2012).
[CrossRef]

2011

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. Photon. 5, 770–776 (2011).
[CrossRef]

S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused-quartz-microresonator optical frequency comb,” Phys. Rev. A 84, 053833 (2011).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef] [PubMed]

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] [PubMed]

2010

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photon. 4, 37–40 (2010).
[CrossRef]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photon. 4, 41–45 (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]

F. Leo, S. Coen, P. Kockaert, S. P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon. 4, 471–476 (2010).
[CrossRef]

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

2009

2007

A. Rodriguez, M. Soljacic, J. Joannopoulos, and S. Johnson, “Chi(2) and Chi(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15, 7303–7318 (2007).
[CrossRef] [PubMed]

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]

2005

Y. Dumeige, C. Arnaud, and P. Feron, “Combining FDTD with coupled mode theories for bistability in micro-ring resonators,” Opt. Commun. 250, 376–383 (2005).
[CrossRef]

2004

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]

W. Yang, A. Josh, and X. Min, “Effects of side-coupling on the phase response of cascaded microring all-pass filters,” Opt. Commun. 232, 209–216 (2004).
[CrossRef]

T. Barwicz, M. A. Popović, P. T. Rakich, M. R. Watts, H. A. Haus, E. P. Ippen, and H. I. Smith, “Microring-resonator-based add-drop filters in SiN: fabrication and analysis,” Opt. Express 12, 1437–1442 (2004).
[CrossRef] [PubMed]

2001

V. A. Mandelshtam, “FDM: the filter diagonalization method for data processing in NMR experiments,” Prog. Nucl. Mag. Res. Sp. 38, 159–196 (2001).
[CrossRef]

1997

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]

1992

M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun. 91, 401–407 (1992).
[CrossRef]

1987

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[CrossRef] [PubMed]

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]

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communications Systems, 3rd ed. (John Wiley, 2002).
[CrossRef]

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]

Arnaud, C.

Y. Dumeige, C. Arnaud, and P. Feron, “Combining FDTD with coupled mode theories for bistability in micro-ring resonators,” Opt. Commun. 250, 376–383 (2005).
[CrossRef]

Barwicz, T.

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Chembo, Y. K.

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. Photon. 5, 770–776 (2011).
[CrossRef]

Chu, S.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photon. 4, 41–45 (2010).
[CrossRef]

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]

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]

Chu, S.T.

M. Peccianti, A. Pasquazi, Y. Park, B.E. Little, S.T. Chu, D.J. Moss, and R. Morandotti, “Demonstration of a stable ultrafast laser based on a nonlinear microcavity,” Nature Comm. 3, 765 (2012).
[CrossRef]

Coen, S.

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] [PubMed]

F. Leo, S. Coen, P. Kockaert, S. P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon. 4, 471–476 (2010).
[CrossRef]

Coutts, D. W.

Cundiff, S. T.

J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Kluwer, 2005).
[CrossRef]

Del’Haye, P.

P. Del’Haye, S. B. Papp, and S. A. Diddams, “Hybrid electro-optically modulated microcombs,” Phys. Rev. Lett. 109, 263901 (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]

Diddams, S. A.

P. Del’Haye, S. B. Papp, and S. A. Diddams, “Hybrid electro-optically modulated microcombs,” Phys. Rev. Lett. 109, 263901 (2012).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef] [PubMed]

S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused-quartz-microresonator optical frequency comb,” Phys. Rev. A 84, 053833 (2011).
[CrossRef]

Duchesne, D.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photon. 4, 41–45 (2010).
[CrossRef]

Dumeige, Y.

Y. Dumeige, C. Arnaud, and P. Feron, “Combining FDTD with coupled mode theories for bistability in micro-ring resonators,” Opt. Commun. 250, 376–383 (2005).
[CrossRef]

Emplit, P.

F. Leo, S. Coen, P. Kockaert, S. P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon. 4, 471–476 (2010).
[CrossRef]

Erkintalo, M.

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. Photon. 5, 770–776 (2011).
[CrossRef]

Feron, P.

Y. Dumeige, C. Arnaud, and P. Feron, “Combining FDTD with coupled mode theories for bistability in micro-ring resonators,” Opt. Commun. 250, 376–383 (2005).
[CrossRef]

Ferrera, M.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photon. 4, 41–45 (2010).
[CrossRef]

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]

Foster, M. A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photon. 4, 37–40 (2010).
[CrossRef]

Gaeta, A. L.

M. R. Lamont, Y. Okawachi, and A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” Opt. Lett. 38, 3478–3481 (2013).
[CrossRef] [PubMed]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photon. 4, 37–40 (2010).
[CrossRef]

Gavartin, E.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photon. 6, 480–487 (2012).
[CrossRef]

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]

Gondarenko, A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photon. 4, 37–40 (2010).
[CrossRef]

Gorodetsky, M. L.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photon. 6, 480–487 (2012).
[CrossRef]

Gorza, S. P.

F. Leo, S. Coen, P. Kockaert, S. P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon. 4, 471–476 (2010).
[CrossRef]

Granados, E.

Grudinin, I. S.

Haelterman, M.

F. Leo, S. Coen, P. Kockaert, S. P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon. 4, 471–476 (2010).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun. 91, 401–407 (1992).
[CrossRef]

Hansson, T.

T. Hansson, D. Modotto, and S. Wabnitz, “On the numerical simulation of Kerr frequency combs using coupled mode equations,” Opt. Commun. 312, 134–136 (2014).
[CrossRef]

Hartinger, K.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photon. 6, 480–487 (2012).
[CrossRef]

Haus, H. A.

Herr, T.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photon. 6, 480–487 (2012).
[CrossRef]

Holzwarth, R.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photon. 6, 480–487 (2012).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef] [PubMed]

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]

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]

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Ilchenko, V. S.

Ippen, E. P.

Jiang, W. C.

W. C. Jiang, X. Lu, J. Zhang, O. Painter, and Q. Lin, “Ultra-bright photon-pair generation on a silicon chip,” in Frontiers in Optics 2012/Laser Science XXVIII, OSA Technical Digest (online) (Optical Society of America, 2012), paper FW6C.10.
[CrossRef]

Joannopoulos, J.

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Commun. 181, 687–702 (2010).
[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, S.

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Josh, A.

W. Yang, A. Josh, and X. Min, “Effects of side-coupling on the phase response of cascaded microring all-pass filters,” Opt. Commun. 232, 209–216 (2004).
[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]

Kippenberg, T. J.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photon. 6, 480–487 (2012).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef] [PubMed]

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]

Kockaert, P.

F. Leo, S. Coen, P. Kockaert, S. P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon. 4, 471–476 (2010).
[CrossRef]

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]

Lamont, M. R.

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. Photon. 5, 770–776 (2011).
[CrossRef]

Lefever, R.

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[CrossRef] [PubMed]

Leo, F.

F. Leo, S. Coen, P. Kockaert, S. P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon. 4, 471–476 (2010).
[CrossRef]

Levy, J. S.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photon. 4, 37–40 (2010).
[CrossRef]

Liang, W.

Lin, Q.

W. C. Jiang, X. Lu, J. Zhang, O. Painter, and Q. Lin, “Ultra-bright photon-pair generation on a silicon chip,” in Frontiers in Optics 2012/Laser Science XXVIII, OSA Technical Digest (online) (Optical Society of America, 2012), paper FW6C.10.
[CrossRef]

Lipson, M.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photon. 4, 37–40 (2010).
[CrossRef]

Little, B. E.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photon. 4, 41–45 (2010).
[CrossRef]

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]

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]

Little, B.E.

M. Peccianti, A. Pasquazi, Y. Park, B.E. Little, S.T. Chu, D.J. Moss, and R. Morandotti, “Demonstration of a stable ultrafast laser based on a nonlinear microcavity,” Nature Comm. 3, 765 (2012).
[CrossRef]

Lu, X.

W. C. Jiang, X. Lu, J. Zhang, O. Painter, and Q. Lin, “Ultra-bright photon-pair generation on a silicon chip,” in Frontiers in Optics 2012/Laser Science XXVIII, OSA Technical Digest (online) (Optical Society of America, 2012), paper FW6C.10.
[CrossRef]

Lugiato, L. A.

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[CrossRef] [PubMed]

Maleki, L.

Mandelshtam, V. A.

V. A. Mandelshtam, “FDM: the filter diagonalization method for data processing in NMR experiments,” Prog. Nucl. Mag. Res. Sp. 38, 159–196 (2001).
[CrossRef]

Matsko, A. B.

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. Photon. 5, 770–776 (2011).
[CrossRef]

Min, X.

W. Yang, A. Josh, and X. Min, “Effects of side-coupling on the phase response of cascaded microring all-pass filters,” Opt. Commun. 232, 209–216 (2004).
[CrossRef]

Modotto, D.

T. Hansson, D. Modotto, and S. Wabnitz, “On the numerical simulation of Kerr frequency combs using coupled mode equations,” Opt. Commun. 312, 134–136 (2014).
[CrossRef]

Morandotti, R.

M. Peccianti, A. Pasquazi, Y. Park, B.E. Little, S.T. Chu, D.J. Moss, and R. Morandotti, “Demonstration of a stable ultrafast laser based on a nonlinear microcavity,” Nature Comm. 3, 765 (2012).
[CrossRef]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photon. 4, 41–45 (2010).
[CrossRef]

Moss, D. J.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photon. 4, 41–45 (2010).
[CrossRef]

Moss, D.J.

M. Peccianti, A. Pasquazi, Y. Park, B.E. Little, S.T. Chu, D.J. Moss, and R. Morandotti, “Demonstration of a stable ultrafast laser based on a nonlinear microcavity,” Nature Comm. 3, 765 (2012).
[CrossRef]

Okawachi, Y.

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Painter, O.

W. C. Jiang, X. Lu, J. Zhang, O. Painter, and Q. Lin, “Ultra-bright photon-pair generation on a silicon chip,” in Frontiers in Optics 2012/Laser Science XXVIII, OSA Technical Digest (online) (Optical Society of America, 2012), paper FW6C.10.
[CrossRef]

Papp, S. B.

P. Del’Haye, S. B. Papp, and S. A. Diddams, “Hybrid electro-optically modulated microcombs,” Phys. Rev. Lett. 109, 263901 (2012).
[CrossRef]

S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused-quartz-microresonator optical frequency comb,” Phys. Rev. A 84, 053833 (2011).
[CrossRef]

Park, Y.

M. Peccianti, A. Pasquazi, Y. Park, B.E. Little, S.T. Chu, D.J. Moss, and R. Morandotti, “Demonstration of a stable ultrafast laser based on a nonlinear microcavity,” Nature Comm. 3, 765 (2012).
[CrossRef]

Pasquazi, A.

M. Peccianti, A. Pasquazi, Y. Park, B.E. Little, S.T. Chu, D.J. Moss, and R. Morandotti, “Demonstration of a stable ultrafast laser based on a nonlinear microcavity,” Nature Comm. 3, 765 (2012).
[CrossRef]

Peccianti, M.

M. Peccianti, A. Pasquazi, Y. Park, B.E. Little, S.T. Chu, D.J. Moss, and R. Morandotti, “Demonstration of a stable ultrafast laser based on a nonlinear microcavity,” Nature Comm. 3, 765 (2012).
[CrossRef]

Popovic, M.

M. Popovic, Theory and design of High-Index-Contrast microphotonic circuits (MIT libraries thesis collection, Cambridge, MA, 2008).

Popovic, M. A.

Rakich, P. T.

Randle, H. G.

Razzari, L.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photon. 4, 41–45 (2010).
[CrossRef]

Riemensberger, J.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photon. 6, 480–487 (2012).
[CrossRef]

Rodriguez, A.

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Savchenkov, A. A.

Schliesser, A.

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]

Seidel, D.

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]

Smith, H. I.

Soljacic, M.

Spence, D. J.

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. Photon. 5, 770–776 (2011).
[CrossRef]

Sylvestre, T.

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]

Trillo, S.

M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun. 91, 401–407 (1992).
[CrossRef]

Turner-Foster, A. C.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photon. 4, 37–40 (2010).
[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]

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. Photon. 5, 770–776 (2011).
[CrossRef]

Wabnitz, S.

T. Hansson, D. Modotto, and S. Wabnitz, “On the numerical simulation of Kerr frequency combs using coupled mode equations,” Opt. Commun. 312, 134–136 (2014).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun. 91, 401–407 (1992).
[CrossRef]

Wang, C. Y.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photon. 6, 480–487 (2012).
[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. Photon. 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. Photon. 5, 770–776 (2011).
[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]

Yang, W.

W. Yang, A. Josh, and X. Min, “Effects of side-coupling on the phase response of cascaded microring all-pass filters,” Opt. Commun. 232, 209–216 (2004).
[CrossRef]

Yariv, A.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University, 2007).

Ye, J.

J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Kluwer, 2005).
[CrossRef]

Yeh, P.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University, 2007).

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]

I. S. Grudinin, N. Yu, and L. Maleki, “Generation of optical frequency combs with a CaF2 resonator,” Opt. Lett. 34, 878–880 (2009).
[CrossRef] [PubMed]

Zhang, J.

W. C. Jiang, X. Lu, J. Zhang, O. Painter, and Q. Lin, “Ultra-bright photon-pair generation on a silicon chip,” in Frontiers in Optics 2012/Laser Science XXVIII, OSA Technical Digest (online) (Optical Society of America, 2012), paper FW6C.10.
[CrossRef]

Comp. Phys. Commun.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

IEEE Photon. Technol. Lett.

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.

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]

Nat. Photon.

F. Leo, S. Coen, P. Kockaert, S. P. Gorza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photon. 4, 471–476 (2010).
[CrossRef]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photon. 4, 37–40 (2010).
[CrossRef]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photon. 4, 41–45 (2010).
[CrossRef]

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. Photon. 5, 770–776 (2011).
[CrossRef]

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photon. 6, 480–487 (2012).
[CrossRef]

Nature

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]

Nature Comm.

M. Peccianti, A. Pasquazi, Y. Park, B.E. Little, S.T. Chu, D.J. Moss, and R. Morandotti, “Demonstration of a stable ultrafast laser based on a nonlinear microcavity,” Nature Comm. 3, 765 (2012).
[CrossRef]

Opt. Commun.

Y. Dumeige, C. Arnaud, and P. Feron, “Combining FDTD with coupled mode theories for bistability in micro-ring resonators,” Opt. Commun. 250, 376–383 (2005).
[CrossRef]

T. Hansson, D. Modotto, and S. Wabnitz, “On the numerical simulation of Kerr frequency combs using coupled mode equations,” Opt. Commun. 312, 134–136 (2014).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun. 91, 401–407 (1992).
[CrossRef]

W. Yang, A. Josh, and X. Min, “Effects of side-coupling on the phase response of cascaded microring all-pass filters,” Opt. Commun. 232, 209–216 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Phys Rev. A

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]

Phys. Rev. A

S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused-quartz-microresonator optical frequency comb,” Phys. Rev. A 84, 053833 (2011).
[CrossRef]

Phys. Rev. Lett.

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[CrossRef] [PubMed]

P. Del’Haye, S. B. Papp, and S. A. Diddams, “Hybrid electro-optically modulated microcombs,” Phys. Rev. Lett. 109, 263901 (2012).
[CrossRef]

Prog. Nucl. Mag. Res. Sp.

V. A. Mandelshtam, “FDM: the filter diagonalization method for data processing in NMR experiments,” Prog. Nucl. Mag. Res. Sp. 38, 159–196 (2001).
[CrossRef]

Science

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef] [PubMed]

Other

W. C. Jiang, X. Lu, J. Zhang, O. Painter, and Q. Lin, “Ultra-bright photon-pair generation on a silicon chip,” in Frontiers in Optics 2012/Laser Science XXVIII, OSA Technical Digest (online) (Optical Society of America, 2012), paper FW6C.10.
[CrossRef]

J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Kluwer, 2005).
[CrossRef]

M. Popovic, Theory and design of High-Index-Contrast microphotonic circuits (MIT libraries thesis collection, Cambridge, MA, 2008).

Rosen Center for Advanced Computing: Carter User Information., http://www.rcac.purdue.edu/userinfo/resources/carter/ , last accessed on May 31, 2013.

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A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University, 2007).

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Figures (9)

Fig. 1
Fig. 1

(a) Radial distribution of the normalized amplitude function |g(r)| for the fundamental normal dispersion mode (b) Radial distribution for the order 1 anomalous dispersion mode. The boundaries of the microring core region are shown as dotted lines.

Fig. 2
Fig. 2

Distribution of the electric field of a mode with a fundamental radial profile. The component of the field going into the plane (Ez) is shown. Red designates positive values; blue negative; white zero.

Fig. 3
Fig. 3

(a) Dispersion parameter data for normal dispersion region of fundamental radial (order 0) profile modes; (b) the anomalous dispersion region of order 1 modes.

Fig. 4
Fig. 4

Comparison of FDTD and CMT results for the normal dispersion case: (a) FDTD time domain; (b) CMT time domain; (c) FDTD frequency domain; and (d) CMT frequency domain.

Fig. 5
Fig. 5

Comparison of FDTD and CMT results for the anomalous dispersion case: (a) FDTD time domain; (b) CMT time domain; (c) FDTD frequency domain; and (d) CMT frequency domain.

Fig. 6
Fig. 6

Time domain envelopes of different coupling and excitation regimes in FDTD and CMT: (a) Comparison of FDTD and CMT envelopes for different coupling mechanisms: (solid blue) FDTD results, (solid red) CMT results, (dashed black) CMT results with only self-phase modulation (SPM) terms, (dotted green) CMT results with only SPM and cross-phase modulation (XPM) terms, (solid cyan) amplitude scaled source transient; (b) Comparison of FDTD and CMT for different excitation regimes: (dotted green) uniform source, (solid red) source with σ = 3.14, (solid black) CMT results with only SPM terms and σ = 0.1.

Fig. 7
Fig. 7

(a) Dispersion parameter data for order 0 modes (dotted blue) having normal dispersion and order 1 modes (solid red) with anomalous dispersion; (b) Field profile of the traveling pulses corresponding to order 0 modes and order 1 modes. All simulations were performed in the ”cold” cavity.

Fig. 8
Fig. 8

(a) Time domain traveling pulses in the ”cold” microring cavity with an order 1 source profile. (b) Corresponding frequency domain data, showing that the broadband excitation spectrum excites both order 0 and order 1 modes.

Fig. 9
Fig. 9

Evolution of the propagating pulse in the cases of zero and X(3) = 0.004 Kerr nonlinearity: (a) FDTD calculation (b) corresponding CMT simulation. A good match is observed. In both cases, pulse spreading is dramatically suppressed with appropriate level of nonlinearity, much like in a soliton.

Tables (1)

Tables Icon

Table 1 Resonant mode parameters from FDTD and Haminv used in CMT

Equations (21)

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

H = d r [ ε E ( r ) 2 + 1 μ B ( r ) 2 ]
ε ( r ) = ε o ( r ) + ε 2 | E ( r ) | 2
H = d r [ ε o ( r ) + ε 2 | E ( r ) | 2 ] [ E ( r ) 2 + 1 μ ε B ( r ) 2 ]
E k ( r ) = C k g k ( r ) ( a k + a k )
H e l = C k 2 ε 0 ( r ) g k ( r ) 2 ( a k + a k ) 2 d r
H e l = C k 2 ε 0 ( r ) g k ( r ) 2 d r ( a k 2 + a k a k + a k a k + a k 2 )
H = 2 C k 2 ε 0 ( r ) g k ( r ) 2 d r ( a k a k + a k a k )
H = 1 2 ω k ( a k a k + a k a k )
C k = 1 2 ω k ε 0 ( r ) g k ( r ) 2 d r
E ( r ) = k 1 2 ω k ε 0 ( r ) g k ( r ) 2 d r g k ( r ) ( a k + a k )
H = k ω k ( a k a k + 1 2 ) + 2 ε 2 2 i , j , k , l ω i ω j ω k ω l M i j k l a i a j a k a l e i ( ω k + ω l ω i ω j ) t
M i j k l = g i ( r ) g j ( r ) g k ( r ) g l ( r ) d r [ ε 0 ( r ) g i ( r ) 2 d r ] 1 / 2 [ ε 0 ( r ) g j ( r ) 2 d r ] 1 / 2 [ ε 0 ( r ) g k ( r ) 2 d r ] 1 / 2 [ ε 0 ( r ) g l ( r ) 2 d r ] 1 / 2
d a i d t = 1 i [ a i , H ]
[ a i , a j ] = δ i j
[ a i , a j ] = 0
d a i d t = i ω i a i i 2 ε 2 2 ω i ω j ω k ω l M i j k l e i ( ω k + ω l ω i ω j ) t a j a k a l ,
M i i i i = g i ( r ) 4 d r ( ε 0 ( r ) g i ( r ) 2 d r ) 2
M i j i j = g i 2 ( r ) g j 2 ( r ) d r ε 0 ( r ) g i ( r ) 2 d r ε 0 ( r ) g j ( r ) 2 d r
d a i d t = i ω i a i i 2 ε 2 2 ω i ω j ω k ω l M i j k l e i ( ω k + ω l ω i ω j ) t a j a k a l + s ,
s = E s e ( t t o ) 2 / 2 τ 2 e i ω s t
ω n i = ω p + ξ i + n σ , n = 0 , ± 1 , ± 2

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