Abstract

A dynamic model for the transmission of a microring modulator based on changes in the refractive index, loss, or waveguide-ring coupling strength is derived to investigate the limitations to the intensity modulation bandwidth. Modulation bandwidths approaching the free spectral range frequency are possible if the waveguide-ring coupling strength is varied, rather than the refractive index or loss of the ring. The results illustrate that via controlled coupling, resonant modulators with high quality factors can be designed to operate at frequencies much larger than the resonator linewidth.

© 2008 Optical Society of America

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  1. P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, "Polymer micro-ring filters and modulators," J. Lightwave Technol. 20, 1968-1975 (2002).
    [CrossRef]
  2. A. Guarino, G. Poberaj, D. Rezzonico, R. Degl???Innocenti, and P. Gunter, "Electrooptically tunable microring resonators in lithium niobate," Nat. Photonics 1, 407 - 410 (2007).
    [CrossRef]
  3. Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
    [CrossRef] [PubMed]
  4. Y. Vlasov, W. M. J. Green, and F. Xia, "High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks," Nat. Photonics 2, 242-246 (2008).
    [CrossRef]
  5. T. A. Ibrahim, V. Van, and P.-T. Ho, "All-optical time-division demultiplexing and spatial pulse routing with a GaAs/AlGaAs microring resonator," Opt. Lett. 27, 803-805 (2002).
    [CrossRef]
  6. D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, "High-Q channel-dropping filters using ring resonators with integrated SOAs," IEEE Photon. Technol. Lett. 14, 1442-1444 (2002).
    [CrossRef]
  7. T. Sadagopan, S. J. Choi, S. J. Choi, K. Djordjev, and P. D. Dapkus, "Carrier-induced refractive index changes in InP-based circular microresonators for low-voltage high-speed modulation," IEEE Photon. Technol. Lett. 17, 414-416 (2005).
    [CrossRef]
  8. L. Zhang, J.-Y. Yang, M. Song, Y. Li, B. Zhang, R. G. Beausoleil, and A. E. Willner, "Microring-based modulation and demodulation of DPSK signal," Opt. Express 15, 11,564-11,569 (2007).
  9. I. L. Gheorma and R. M. Osgood, "Fundamental limitations of optical resonator based high-speed EO modulators," J. Lightwave Technol. 14, 795-797 (2002).
  10. K. Djordjev, S.-J. Choi, S.-J. Choi, and P. D. Dapkus, "Active semiconductor microdisk devices," J. Lightwave Technol. 20, 105-113 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. B. Bortnik, Y.-C. Hung, H. Tazawa, J. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104 - 110 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  16. M. Lee, H. E. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, "Broadband modulation of light by using an electro-optic polymer," Science 298, 1401-1403 (2002).
    [CrossRef] [PubMed]
  17. A. Yariv, "Critical coupling and its control in optical waveguide-ring resonator systems," IEEE Photon. Technol. Lett. 14, 483-485 (2002).
    [CrossRef]
  18. W. M. J. Green, R. K. Lee, G. A. DeRose, A. Scherer, and A. Yariv, "Hybrid InGaAsP-InP Mach-Zehnder racetrack resonator for thermooptic switching and coupling control," Opt. Express 13, 1651-1659 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  22. Y. Li, L. Zhang, M. Song, B. Zhang, J. Y. Yang, R. G. Beausoleil, A. E. Willner, and P. D. Dapkus, "Coupledring- resonator-based silicon modulator for enhanced performance," Opt. Express 16, 13342-13348 (2008).
    [CrossRef] [PubMed]

2008

Y. Vlasov, W. M. J. Green, and F. Xia, "High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks," Nat. Photonics 2, 242-246 (2008).
[CrossRef]

Y. Li, L. Zhang, M. Song, B. Zhang, J. Y. Yang, R. G. Beausoleil, A. E. Willner, and P. D. Dapkus, "Coupledring- resonator-based silicon modulator for enhanced performance," Opt. Express 16, 13342-13348 (2008).
[CrossRef] [PubMed]

2007

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl???Innocenti, and P. Gunter, "Electrooptically tunable microring resonators in lithium niobate," Nat. Photonics 1, 407 - 410 (2007).
[CrossRef]

L. Zhang, J.-Y. Yang, M. Song, Y. Li, B. Zhang, R. G. Beausoleil, and A. E. Willner, "Microring-based modulation and demodulation of DPSK signal," Opt. Express 15, 11,564-11,569 (2007).

B. Bortnik, Y.-C. Hung, H. Tazawa, J. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104 - 110 (2007).
[CrossRef]

C. Li, L. Zhou, and A. W. Poon, "Silicon microring carrier-injection-based modulators/switches with tunable extinction ratios and OR-logic switching by using waveguide cross-coupling," Opt. Express 15, 5069-5076 (2007).
[CrossRef] [PubMed]

L. Zhou and A. W. Poon, "Electrically reconfigurable silicon microring resonator-based filter with waveguidecoupled feedback," Opt. Express 15, 9194-9204 (2007).
[CrossRef] [PubMed]

W. M. J. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, "Optical modulation using anti-crossing between paired amplitude and phase resonators," Opt. Express 15, 17264-17272 (2007).
[CrossRef] [PubMed]

2005

T. Sadagopan, S. J. Choi, S. J. Choi, K. Djordjev, and P. D. Dapkus, "Carrier-induced refractive index changes in InP-based circular microresonators for low-voltage high-speed modulation," IEEE Photon. Technol. Lett. 17, 414-416 (2005).
[CrossRef]

Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

W. M. J. Green, R. K. Lee, G. A. DeRose, A. Scherer, and A. Yariv, "Hybrid InGaAsP-InP Mach-Zehnder racetrack resonator for thermooptic switching and coupling control," Opt. Express 13, 1651-1659 (2005).
[CrossRef] [PubMed]

2002

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, "High-Q channel-dropping filters using ring resonators with integrated SOAs," IEEE Photon. Technol. Lett. 14, 1442-1444 (2002).
[CrossRef]

I. L. Gheorma and R. M. Osgood, "Fundamental limitations of optical resonator based high-speed EO modulators," J. Lightwave Technol. 14, 795-797 (2002).

M. Lee, H. E. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, "Broadband modulation of light by using an electro-optic polymer," Science 298, 1401-1403 (2002).
[CrossRef] [PubMed]

A. Yariv, "Critical coupling and its control in optical waveguide-ring resonator systems," IEEE Photon. Technol. Lett. 14, 483-485 (2002).
[CrossRef]

K. Djordjev, S.-J. Choi, S.-J. Choi, and P. D. Dapkus, "Active semiconductor microdisk devices," J. Lightwave Technol. 20, 105-113 (2002).
[CrossRef]

T. A. Ibrahim, V. Van, and P.-T. Ho, "All-optical time-division demultiplexing and spatial pulse routing with a GaAs/AlGaAs microring resonator," Opt. Lett. 27, 803-805 (2002).
[CrossRef]

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, "Polymer micro-ring filters and modulators," J. Lightwave Technol. 20, 1968-1975 (2002).
[CrossRef]

2001

2000

A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
[CrossRef]

1997

D. Chen, H. R. Fetterman, A. Chen, W. H. Steier, L. R. Dalton, W. Wang, and Y. Shi, "Demonstration of 110 GHz electro-optic polymer modulators," Appl. Phys. Lett. 70, 3335-3337 (1997).
[CrossRef]

1986

Beausoleil, R. G.

Y. Li, L. Zhang, M. Song, B. Zhang, J. Y. Yang, R. G. Beausoleil, A. E. Willner, and P. D. Dapkus, "Coupledring- resonator-based silicon modulator for enhanced performance," Opt. Express 16, 13342-13348 (2008).
[CrossRef] [PubMed]

L. Zhang, J.-Y. Yang, M. Song, Y. Li, B. Zhang, R. G. Beausoleil, and A. E. Willner, "Microring-based modulation and demodulation of DPSK signal," Opt. Express 15, 11,564-11,569 (2007).

Bortnik, B.

B. Bortnik, Y.-C. Hung, H. Tazawa, J. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104 - 110 (2007).
[CrossRef]

Chen, A.

D. Chen, H. R. Fetterman, A. Chen, W. H. Steier, L. R. Dalton, W. Wang, and Y. Shi, "Demonstration of 110 GHz electro-optic polymer modulators," Appl. Phys. Lett. 70, 3335-3337 (1997).
[CrossRef]

Chen, D.

D. Chen, H. R. Fetterman, A. Chen, W. H. Steier, L. R. Dalton, W. Wang, and Y. Shi, "Demonstration of 110 GHz electro-optic polymer modulators," Appl. Phys. Lett. 70, 3335-3337 (1997).
[CrossRef]

Choi, J. M.

Choi, S. J.

T. Sadagopan, S. J. Choi, S. J. Choi, K. Djordjev, and P. D. Dapkus, "Carrier-induced refractive index changes in InP-based circular microresonators for low-voltage high-speed modulation," IEEE Photon. Technol. Lett. 17, 414-416 (2005).
[CrossRef]

T. Sadagopan, S. J. Choi, S. J. Choi, K. Djordjev, and P. D. Dapkus, "Carrier-induced refractive index changes in InP-based circular microresonators for low-voltage high-speed modulation," IEEE Photon. Technol. Lett. 17, 414-416 (2005).
[CrossRef]

Choi, S.-J.

Crosignani, B.

Dalton, L. R.

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, "Polymer micro-ring filters and modulators," J. Lightwave Technol. 20, 1968-1975 (2002).
[CrossRef]

D. Chen, H. R. Fetterman, A. Chen, W. H. Steier, L. R. Dalton, W. Wang, and Y. Shi, "Demonstration of 110 GHz electro-optic polymer modulators," Appl. Phys. Lett. 70, 3335-3337 (1997).
[CrossRef]

Dapkus, P. D.

DeRose, G. A.

Djordjev, K.

T. Sadagopan, S. J. Choi, S. J. Choi, K. Djordjev, and P. D. Dapkus, "Carrier-induced refractive index changes in InP-based circular microresonators for low-voltage high-speed modulation," IEEE Photon. Technol. Lett. 17, 414-416 (2005).
[CrossRef]

K. Djordjev, S.-J. Choi, S.-J. Choi, and P. D. Dapkus, "Active semiconductor microdisk devices," J. Lightwave Technol. 20, 105-113 (2002).
[CrossRef]

Erben, C.

M. Lee, H. E. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, "Broadband modulation of light by using an electro-optic polymer," Science 298, 1401-1403 (2002).
[CrossRef] [PubMed]

Fetterman, H. R.

B. Bortnik, Y.-C. Hung, H. Tazawa, J. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104 - 110 (2007).
[CrossRef]

D. Chen, H. R. Fetterman, A. Chen, W. H. Steier, L. R. Dalton, W. Wang, and Y. Shi, "Demonstration of 110 GHz electro-optic polymer modulators," Appl. Phys. Lett. 70, 3335-3337 (1997).
[CrossRef]

Gheorma, I. L.

I. L. Gheorma and R. M. Osgood, "Fundamental limitations of optical resonator based high-speed EO modulators," J. Lightwave Technol. 14, 795-797 (2002).

Gill, D. M.

M. Lee, H. E. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, "Broadband modulation of light by using an electro-optic polymer," Science 298, 1401-1403 (2002).
[CrossRef] [PubMed]

Gopalan, P.

M. Lee, H. E. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, "Broadband modulation of light by using an electro-optic polymer," Science 298, 1401-1403 (2002).
[CrossRef] [PubMed]

Green, W. M. J.

Guarino, A.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl???Innocenti, and P. Gunter, "Electrooptically tunable microring resonators in lithium niobate," Nat. Photonics 1, 407 - 410 (2007).
[CrossRef]

Hamacher, M.

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, "High-Q channel-dropping filters using ring resonators with integrated SOAs," IEEE Photon. Technol. Lett. 14, 1442-1444 (2002).
[CrossRef]

Heber, J. D.

M. Lee, H. E. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, "Broadband modulation of light by using an electro-optic polymer," Science 298, 1401-1403 (2002).
[CrossRef] [PubMed]

Heidrich, H.

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, "High-Q channel-dropping filters using ring resonators with integrated SOAs," IEEE Photon. Technol. Lett. 14, 1442-1444 (2002).
[CrossRef]

Ho, P.-T.

Hung, Y.-C.

B. Bortnik, Y.-C. Hung, H. Tazawa, J. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104 - 110 (2007).
[CrossRef]

Ibrahim, T. A.

Jen, A. K.-Y.

B. Bortnik, Y.-C. Hung, H. Tazawa, J. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104 - 110 (2007).
[CrossRef]

Katz, H. E.

M. Lee, H. E. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, "Broadband modulation of light by using an electro-optic polymer," Science 298, 1401-1403 (2002).
[CrossRef] [PubMed]

Lee, M.

M. Lee, H. E. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, "Broadband modulation of light by using an electro-optic polymer," Science 298, 1401-1403 (2002).
[CrossRef] [PubMed]

Lee, R. K.

Li, C.

Li, Y.

Y. Li, L. Zhang, M. Song, B. Zhang, J. Y. Yang, R. G. Beausoleil, A. E. Willner, and P. D. Dapkus, "Coupledring- resonator-based silicon modulator for enhanced performance," Opt. Express 16, 13342-13348 (2008).
[CrossRef] [PubMed]

L. Zhang, J.-Y. Yang, M. Song, Y. Li, B. Zhang, R. G. Beausoleil, and A. E. Willner, "Microring-based modulation and demodulation of DPSK signal," Opt. Express 15, 11,564-11,569 (2007).

Lipson, M.

Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Luo, J.

B. Bortnik, Y.-C. Hung, H. Tazawa, J. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104 - 110 (2007).
[CrossRef]

McGee, D. J.

M. Lee, H. E. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, "Broadband modulation of light by using an electro-optic polymer," Science 298, 1401-1403 (2002).
[CrossRef] [PubMed]

Osgood, R. M.

I. L. Gheorma and R. M. Osgood, "Fundamental limitations of optical resonator based high-speed EO modulators," J. Lightwave Technol. 14, 795-797 (2002).

Poberaj, G.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl???Innocenti, and P. Gunter, "Electrooptically tunable microring resonators in lithium niobate," Nat. Photonics 1, 407 - 410 (2007).
[CrossRef]

Poon, A. W.

Pradhan, S.

Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Rabiei, P.

Rabus, D. G.

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, "High-Q channel-dropping filters using ring resonators with integrated SOAs," IEEE Photon. Technol. Lett. 14, 1442-1444 (2002).
[CrossRef]

Rezzonico, D.

A. Guarino, G. Poberaj, D. Rezzonico, R. Degl???Innocenti, and P. Gunter, "Electrooptically tunable microring resonators in lithium niobate," Nat. Photonics 1, 407 - 410 (2007).
[CrossRef]

Rooks, M. J.

Sadagopan, T.

T. Sadagopan, S. J. Choi, S. J. Choi, K. Djordjev, and P. D. Dapkus, "Carrier-induced refractive index changes in InP-based circular microresonators for low-voltage high-speed modulation," IEEE Photon. Technol. Lett. 17, 414-416 (2005).
[CrossRef]

Scherer, A.

Schmidt, B.

Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Sekaric, L.

Shi, Y.

D. Chen, H. R. Fetterman, A. Chen, W. H. Steier, L. R. Dalton, W. Wang, and Y. Shi, "Demonstration of 110 GHz electro-optic polymer modulators," Appl. Phys. Lett. 70, 3335-3337 (1997).
[CrossRef]

Song, M.

Y. Li, L. Zhang, M. Song, B. Zhang, J. Y. Yang, R. G. Beausoleil, A. E. Willner, and P. D. Dapkus, "Coupledring- resonator-based silicon modulator for enhanced performance," Opt. Express 16, 13342-13348 (2008).
[CrossRef] [PubMed]

L. Zhang, J.-Y. Yang, M. Song, Y. Li, B. Zhang, R. G. Beausoleil, and A. E. Willner, "Microring-based modulation and demodulation of DPSK signal," Opt. Express 15, 11,564-11,569 (2007).

Steier, W. H.

B. Bortnik, Y.-C. Hung, H. Tazawa, J. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104 - 110 (2007).
[CrossRef]

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, "Polymer micro-ring filters and modulators," J. Lightwave Technol. 20, 1968-1975 (2002).
[CrossRef]

D. Chen, H. R. Fetterman, A. Chen, W. H. Steier, L. R. Dalton, W. Wang, and Y. Shi, "Demonstration of 110 GHz electro-optic polymer modulators," Appl. Phys. Lett. 70, 3335-3337 (1997).
[CrossRef]

Tazawa, H.

B. Bortnik, Y.-C. Hung, H. Tazawa, J. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104 - 110 (2007).
[CrossRef]

Troppenz, U.

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, "High-Q channel-dropping filters using ring resonators with integrated SOAs," IEEE Photon. Technol. Lett. 14, 1442-1444 (2002).
[CrossRef]

Van, V.

Vlasov, Y.

Y. Vlasov, W. M. J. Green, and F. Xia, "High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks," Nat. Photonics 2, 242-246 (2008).
[CrossRef]

Vlasov, Y. A.

Wang, W.

D. Chen, H. R. Fetterman, A. Chen, W. H. Steier, L. R. Dalton, W. Wang, and Y. Shi, "Demonstration of 110 GHz electro-optic polymer modulators," Appl. Phys. Lett. 70, 3335-3337 (1997).
[CrossRef]

Willner, A. E.

Y. Li, L. Zhang, M. Song, B. Zhang, J. Y. Yang, R. G. Beausoleil, A. E. Willner, and P. D. Dapkus, "Coupledring- resonator-based silicon modulator for enhanced performance," Opt. Express 16, 13342-13348 (2008).
[CrossRef] [PubMed]

L. Zhang, J.-Y. Yang, M. Song, Y. Li, B. Zhang, R. G. Beausoleil, and A. E. Willner, "Microring-based modulation and demodulation of DPSK signal," Opt. Express 15, 11,564-11,569 (2007).

Xia, F.

Y. Vlasov, W. M. J. Green, and F. Xia, "High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks," Nat. Photonics 2, 242-246 (2008).
[CrossRef]

Xu, Q. F.

Q. F. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Yang, J. Y.

Yang, J.-Y.

L. Zhang, J.-Y. Yang, M. Song, Y. Li, B. Zhang, R. G. Beausoleil, and A. E. Willner, "Microring-based modulation and demodulation of DPSK signal," Opt. Express 15, 11,564-11,569 (2007).

Yariv, A.

Zhang, B.

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of a ring resonator modulator.

Fig. 2.
Fig. 2.

Modulation depths of a microring resonator with sinusoidal loss modulation between 2 dB/cm and 5 dB/cm. a 0=0.9975, a =0.0011, and σ=0.9928. (a): The input is on resonance. (b): Detuned input, with the modulation resonance frequency at f m .

Fig. 3.
Fig. 3.

Modulation depths of a microring resonator with a sinusoidal index modulation. ϕ 0=0.039477 and ϕ =0.005. The input is detuned from resonance, with the modulation resonance frequency at 10 GHz.

Fig. 4.
Fig. 4.

Modulation depths of a microring resonator with a sinusoidal modulation of the coupling strength. Over-coupled: σ =0.0013 and σ 0=0.9902. Under-coupled: σ =3.5×10-4 and σ 0=0.999. The loss of the ring is 4 dB/cm, a=0.9971. (a): The input is on resonance. (b): The input is detuned from resonance, with the modulation resonance frequency at 5 GHz.

Fig. 5.
Fig. 5.

Device parameters (top) and the corresponding output intensities (bottom) versus time for single-pulse modulated microring resonators. (a), (d): Loss modulation, σ=0.9928, and the input is resonant. (b), (e): Index modulation, ϕ 0=0.039477, ϕ=0.9928, and a loss of 4 dB/cm. (c), (f): Coupling modulation, the loss is 4 dB/cm, and the input is resonant.

Equations (45)

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ϕ ( t , ω ) = ω τ + ω n t τ t η ( t ) d t ,
a ( t ) = a 0 + 1 τ t τ t γ ( t ) d t ,
n i ( t ) = n + η ( t ) ,
a i ( t ) = a + γ ( t )
D ( t ) = a ( t ) 2 π C ˜ ( Ω ) exp [ i ϕ ( t , ω ) ] d Ω ,
D ( t ) = a ( t ) exp [ i ϕ ( t , ω 0 ) ] C ( t τ ) .
B ( t ) = σ ( t ) A + i κ ( t ) a ( t ) exp [ i ϕ ( t ) ] C ( t τ ) ,
i κ ( t ) C ( t ) = σ ( t ) B ( t ) A ,
T s s B A = σ a exp ( i ϕ ) 1 a σ exp ( i ϕ ) .
T s s 2 = σ 2 + a 2 2 a σ cos ( ϕ ) 1 + a 2 σ 2 2 a σ cos ( ϕ ) .
T ( t ) B ( t ) A = σ ( t ) + κ ( t ) κ ( t τ ) a ( t ) exp [ i ϕ ( t ) ] [ σ ( t τ ) T ( t τ ) 1 ] .
T ( t ) = σ ( t ) κ ( t ) κ ( t τ ) a ( t ) exp [ i ϕ ( t ) ] +
κ ( t + τ ) κ ( t ) a ( t + τ ) σ ( t ) exp [ i ϕ ( t + τ ) ] δ [ t ( t τ ) ] T ( t ) d t .
T a ( t ) = σ a ( t ) e i ϕ + n = 1 σ n e i n ϕ [ σ a ( t n τ ) e i ϕ ] m = 0 n 1 a ( t m τ ) .
a ˜ ( Ω ) = a 0 δ ( Ω ) + a 2 [ δ ( Ω Ω m ) + δ ( Ω Ω m ) ] .
T ˜ a ( Ω ) [ 1 a 0 σ e i ( ϕ + Ω τ ) ] a 2 σ e i ( ϕ + Ω τ ) [ T ˜ a ( Ω Ω m ) e i Ω m τ + T ˜ a ( Ω + Ω m ) e i Ω m τ ]
= ( σ a 0 e i ϕ ) δ ( Ω ) a 2 e i ϕ [ δ ( Ω Ω m ) + δ ( Ω + Ω m ) ] ,
T a ˜ ( 0 ) = σ a 0 e i ϕ 1 a 0 σ e i ϕ δ ( 0 ) ,
T a ˜ ( Ω m ) = a 2 e i ϕ [ σ T ˜ a ( 0 ) δ ( 0 ) ] 1 σ a 0 e i ( ϕ + Ω m τ )
T ˜ a ( Ω m ) = a 2 e i ϕ [ σ T ˜ a ( 0 ) δ ( 0 ) ] 1 σ a 0 e i ( ϕ Ω m τ ) .
Δ = f ( t ) max f ( t ) min f ( t ) max + f ( t ) min ,
Δ = 2 T ˜ * ( Ω m ) T ˜ * ( 0 ) + T ˜ ( Ω m ) T ˜ ( 0 ) ,
Δ a = 2 a ( 1 σ 2 ) σ cos ϕ a 0 + σ a 0 e i Ω m τ ( a 0 cos ϕ σ ) ( a 0 2 + σ 2 2 a 0 σ cos ϕ ) ( 1 + a 0 2 σ 2 e i 2 Ω m τ 2 a 0 σ cos ϕ e i Ω m τ ) .
Δ a , res = 2 a ( 1 σ 2 ) σ a 0 [ 1 ( 1 a 0 σ ) 2 + a 0 σ ( Ω m τ ) 2 ] 1 2 ,
Ω a , 3 dB , res = 1 a 0 σ τ 3 a 0 σ .
T ϕ ( t ) = σ a e i ϕ ( t ) +     n = 1 σ n a n [ σ a e i ϕ ( t n τ ) ] m = 0 n 1 e i ϕ ( t m τ ) .
ϕ ( t ) = ϕ 0 ϕ cos ( Ω m t ) .
e i ϕ ( t ) = e i ϕ 0 n = i n J n ( ϕ ) e i n Ω m t .
e i ϕ ( t ) e i ϕ 0 + i ϕ e i ϕ 0 cos ( Ω m t ) .
T ˜ ϕ ( Ω ) [ 1 a σ e i ( ϕ 0 + Ω τ ) ] i ϕ 2 a σ e i ( ϕ 0 + Ω τ ) [ T ˜ ϕ ( Ω Ω m ) e i Ω τ + T ˜ ϕ ( Ω + Ω m ) e i Ω τ ]
= ( σ a e i ϕ 0 ) δ ( Ω ) i ϕ 2 a e i ϕ 0 [ δ ( Ω Ω m ) + δ ( Ω + Ω m ) ] .
T ˜ ϕ ( Ω m ) = i ϕ 2 a e i ϕ 0 [ σ T ˜ ϕ ( 0 ) δ ( 0 ) ] 1 σ a e i ( ϕ 0 + Ω m τ ) ,
T ˜ ϕ ( Ω m ) = i ϕ 2 a e i ϕ 0 [ σ T ˜ ϕ ( 0 ) δ ( 0 ) ] 1 σ a e i ( ϕ 0 Ω m τ ) .
Δ ϕ = 2 ϕ [ σ a ( 1 σ 2 ) sin ( ϕ 0 ) σ 2 + a 2 2 σ a cos ( ϕ 0 ) ] ×
[ 1 + a 4 2 a 2 cos ( Ω m τ ) ( 1 σ 2 a 2 ) 2 + 4 σ 2 a 2 [ cos ( ϕ 0 ) cos ( Ω m τ ) 2 4 σ a ( 1 σ a ) 2 cos ( ϕ 0 ) cos ( Ω m τ ) ] ] 1 2 .
T σ ( t ) = σ ( t ) κ ( t ) κ ( t τ ) a e i ϕ + κ ( t ) n = 1 a n e i n ϕ κ ( t n τ )
[ σ ( t n τ ) κ ( t n τ ) k ( t ( n + 1 ) τ ) a e i ϕ ] m = 1 n σ ( t m τ ) .
κ ( t ) = κ 0 + κ cos ( Ω m t ) ,
σ ( t ) = σ 0 + σ cos ( Ω m t ) ,
κ 0 T ˜ σ ( Ω ) [ 1 a σ 0 e i ( ϕ + Ω τ ) ] + κ 2 T ˜ σ ( Ω Ω m ) [ e i Ω m τ a e i ( ϕ + Ω τ ) ( σ 0 e i Ω m τ κ 0 2 σ 0 ) ]
+ κ 2 T ˜ σ ( Ω + Ω m ) [ e i Ω m τ a e i ( ϕ + Ω τ ) ( σ 0 e i Ω m τ κ 0 2 σ 0 ) ]
= κ 0 [ σ 0 e i Ω τ a e i ϕ ] δ ( Ω ) + κ 2 ( σ 0 e i Ω τ κ 0 2 σ 0 e i ( Ω Ω m ) τ a e i ϕ ) δ ( Ω Ω m )
+ κ 2 ( σ 0 e i Ω τ κ 0 2 σ 0 e i ( Ω + Ω m ) τ a e i ϕ ) δ ( Ω + Ω m ) .
Δ σ = 2 σ ( 1 a 2 e i Ω m τ ) [ σ 0 a cos ϕ + a σ 0 ( a σ 0 cos ϕ ) e i Ω m τ ] ( σ 0 2 + a 2 2 a σ 0 cos ϕ ) ( 1 + a 2 σ 0 2 e 2 i Ω m τ 2 a σ 0 cos ϕ e i Ω m τ ) .
Δ σ , r e s = 2 σ [ ( 1 a 2 ) 2 + a 2 ( Ω m τ ) 2 ( σ 0 a ) 2 [ ( 1 a σ 0 ) 2 + a σ 0 ( Ω m τ ) 2 ] ] 1 2 .

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