Abstract

We present a theoretical model describing the dynamics of the electromagnetic field in an optical resonator undergoing refractive index changes. We use an operator formulation of Maxwell’s equations with a standard time- dependent perturbation theory to derive the dynamic mode-amplitude equations that govern the response of a resonator to a perturbing dipole-moment density. We show that in the case of time-dependent changes in the refractive index, a coupling matrix Γkm(t) that appears in the equations accounts for all novel physical processes that can be expected to occur. In particular, the phenomenon of adiabatic wavelength conversion is governed by the diagonal elements of this matrix, and the off-diagonal elements are responsible for the transfer of energy from an excited resonator mode into its neighboring modes. Our model clearly shows that the latter process can occur only when the index changes are spatially nonuniform. We discuss the spatially uniform and nonuniform cases separately and compare the predictions of our model with experimental data available in the literature. The overall good agreement suggests that this model should be useful in the study of dynamic optical resonators. Moreover, since we do not make any assumptions about the type of dielectric cavity used, the width of input pulses, or the speed with which the refractive index is changed, this model should be applicable under most experimental situations.

© 2011 Optical Society of America

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References

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  1. M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. A 73, 051803(2006).
    [CrossRef]
  2. M. F. Yanik and S. Fan, “Dynamic photonic structures: stopping, storage, and time reversal of light,” Stud. Appl. Math. 115, 233–253 (2005).
    [CrossRef]
  3. S. F. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photon. 1, 293–296 (2007).
    [CrossRef]
  4. S. F. Preble and M. Lipson, “Conversion of a signal wavelength in a dynamically tuned resonator,” in Integrated Photonics Research and Applications/Nanophotonics, Technical Digest (CD) (Optical Society of America, 2006), paper IMC5.
  5. Z. Gaburro, M. Ghulinyan, F. Riboli, L. Pavesi, A. Recati, and I. Carusotto, “Photon energy lifter,” Opt. Express 14, 7270–7278(2006).
    [CrossRef] [PubMed]
  6. T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
    [CrossRef]
  7. Y. Xiao, D. N. Maywar, and G. P. Agrawal, “Optical pulse propagation in dynamic Fabry-Perot resonators,” J. Opt. Soc. Am. B 28, 1685–1692 (2011).
    [CrossRef]
  8. M. W. McCutcheon, A. G. Pattantyus-Abraham, G. W. Rieger, and J. F. Young, “Emission spectrum of electromagnetic energy stored in a dynamically perturbed optical microcavity,” Opt. Express 15, 11472–11480 (2007).
    [CrossRef] [PubMed]
  9. Q. Lin, T. J. Johnson, C. P. Michael, and O. Painter, “Adiabatic self-tuning in a silicon microdisk optical resonator,” Opt. Express 16, 14801–14811 (2008).
    [CrossRef] [PubMed]
  10. T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of short pulse from ultrahigh-Q nanocavities via adiabatic wavelength conversion,” in Conference on Lasers and Electro-Optics, Technical Digest (CD) (Optical Society of America, 2008), paper QPDB1.
  11. T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
    [CrossRef] [PubMed]
  12. A. W. Elshaari and S. F. Preble, “Active optical isolator using adiabatic wavelength conversion in microcavities,” in Frontiers in Optics, Technical Digest (CD) (Optical Society of America, 2009), paper FThU4.
  13. E. E. Hach, III, A. W. Elshaari, and S. F. Preble, “Fully quantum-mechanical dynamic analysis of single-photon transport in a single-mode waveguide coupled to a traveling-wave resonator,” Phys. Rev. A 82, 063839 (2010).
    [CrossRef]
  14. Y. Xiao, G. P. Agrawal, and D. N. Maywar, “Spectral and temporal changes of optical pulses propagating through time-varying linear media,” Opt. Lett. 36, 505–507 (2011).
    [CrossRef] [PubMed]
  15. P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904(2008).
    [CrossRef] [PubMed]
  16. J. W. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
    [CrossRef]
  17. N. Malkova, S. Kim, and V. Gopalan, “Jahn-Teller effect intwo-dimensional photonic crystals,” Phys. Rev. B 68, 045105(2003).
    [CrossRef]
  18. Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E 62, 7389–7404(2000).
    [CrossRef]
  19. Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J. Painter, “A proposal for highly tunable optical parametric oscillation in silicon micro-resonators,” Opt. Express 16, 10596–10610 (2008).
    [CrossRef] [PubMed]
  20. T. J. Johnson, “Silicon microdisk resonators for nonlinear optics and dynamics,” Ph.D. dissertation (California Institute of Technology, 2009).
  21. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987).
    [CrossRef]
  22. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

2011

2010

E. E. Hach, III, A. W. Elshaari, and S. F. Preble, “Fully quantum-mechanical dynamic analysis of single-photon transport in a single-mode waveguide coupled to a traveling-wave resonator,” Phys. Rev. A 82, 063839 (2010).
[CrossRef]

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

2009

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

2008

2007

2006

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. A 73, 051803(2006).
[CrossRef]

Z. Gaburro, M. Ghulinyan, F. Riboli, L. Pavesi, A. Recati, and I. Carusotto, “Photon energy lifter,” Opt. Express 14, 7270–7278(2006).
[CrossRef] [PubMed]

2005

M. F. Yanik and S. Fan, “Dynamic photonic structures: stopping, storage, and time reversal of light,” Stud. Appl. Math. 115, 233–253 (2005).
[CrossRef]

2003

N. Malkova, S. Kim, and V. Gopalan, “Jahn-Teller effect intwo-dimensional photonic crystals,” Phys. Rev. B 68, 045105(2003).
[CrossRef]

2000

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E 62, 7389–7404(2000).
[CrossRef]

1999

J. W. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[CrossRef]

1987

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987).
[CrossRef]

Agrawal, G. P.

Beggs, D. M.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

Bennett, B. R.

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987).
[CrossRef]

Carusotto, I.

Dong, P.

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904(2008).
[CrossRef] [PubMed]

Elshaari, A. W.

E. E. Hach, III, A. W. Elshaari, and S. F. Preble, “Fully quantum-mechanical dynamic analysis of single-photon transport in a single-mode waveguide coupled to a traveling-wave resonator,” Phys. Rev. A 82, 063839 (2010).
[CrossRef]

A. W. Elshaari and S. F. Preble, “Active optical isolator using adiabatic wavelength conversion in microcavities,” in Frontiers in Optics, Technical Digest (CD) (Optical Society of America, 2009), paper FThU4.

Fan, S.

M. F. Yanik and S. Fan, “Dynamic photonic structures: stopping, storage, and time reversal of light,” Stud. Appl. Math. 115, 233–253 (2005).
[CrossRef]

J. W. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[CrossRef]

Gaburro, Z.

Ghulinyan, M.

Gopalan, V.

N. Malkova, S. Kim, and V. Gopalan, “Jahn-Teller effect intwo-dimensional photonic crystals,” Phys. Rev. B 68, 045105(2003).
[CrossRef]

Hach, E. E.

E. E. Hach, III, A. W. Elshaari, and S. F. Preble, “Fully quantum-mechanical dynamic analysis of single-photon transport in a single-mode waveguide coupled to a traveling-wave resonator,” Phys. Rev. A 82, 063839 (2010).
[CrossRef]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

Ippen, E. P.

J. W. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[CrossRef]

Joannopoulos, J. D.

J. W. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[CrossRef]

Johnson, T. J.

Kampfrath, T.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

Kim, S.

N. Malkova, S. Kim, and V. Gopalan, “Jahn-Teller effect intwo-dimensional photonic crystals,” Phys. Rev. B 68, 045105(2003).
[CrossRef]

Krauss, T. F.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

Kuipers, L.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

Kuramochi, E.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of short pulse from ultrahigh-Q nanocavities via adiabatic wavelength conversion,” in Conference on Lasers and Electro-Optics, Technical Digest (CD) (Optical Society of America, 2008), paper QPDB1.

Lee, R. K.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E 62, 7389–7404(2000).
[CrossRef]

Li, Y.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E 62, 7389–7404(2000).
[CrossRef]

Lin, Q.

Lipson, M.

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904(2008).
[CrossRef] [PubMed]

S. F. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photon. 1, 293–296 (2007).
[CrossRef]

S. F. Preble and M. Lipson, “Conversion of a signal wavelength in a dynamically tuned resonator,” in Integrated Photonics Research and Applications/Nanophotonics, Technical Digest (CD) (Optical Society of America, 2006), paper IMC5.

Malkova, N.

N. Malkova, S. Kim, and V. Gopalan, “Jahn-Teller effect intwo-dimensional photonic crystals,” Phys. Rev. B 68, 045105(2003).
[CrossRef]

Manipatruni, S.

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904(2008).
[CrossRef] [PubMed]

Maywar, D. N.

McCutcheon, M. W.

Melloni, A.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

Michael, C. P.

Mitsugi, S.

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. A 73, 051803(2006).
[CrossRef]

Notomi, M.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. A 73, 051803(2006).
[CrossRef]

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of short pulse from ultrahigh-Q nanocavities via adiabatic wavelength conversion,” in Conference on Lasers and Electro-Optics, Technical Digest (CD) (Optical Society of America, 2008), paper QPDB1.

Painter, O.

Painter, O. J.

Pattantyus-Abraham, A. G.

Pavesi, L.

Perahia, R.

Preble, S. F.

E. E. Hach, III, A. W. Elshaari, and S. F. Preble, “Fully quantum-mechanical dynamic analysis of single-photon transport in a single-mode waveguide coupled to a traveling-wave resonator,” Phys. Rev. A 82, 063839 (2010).
[CrossRef]

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904(2008).
[CrossRef] [PubMed]

S. F. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photon. 1, 293–296 (2007).
[CrossRef]

A. W. Elshaari and S. F. Preble, “Active optical isolator using adiabatic wavelength conversion in microcavities,” in Frontiers in Optics, Technical Digest (CD) (Optical Society of America, 2009), paper FThU4.

S. F. Preble and M. Lipson, “Conversion of a signal wavelength in a dynamically tuned resonator,” in Integrated Photonics Research and Applications/Nanophotonics, Technical Digest (CD) (Optical Society of America, 2006), paper IMC5.

Recati, A.

Riboli, F.

Rieger, G. W.

Robinson, J. T.

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904(2008).
[CrossRef] [PubMed]

Soref, R. A.

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987).
[CrossRef]

Tanabe, T.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of short pulse from ultrahigh-Q nanocavities via adiabatic wavelength conversion,” in Conference on Lasers and Electro-Optics, Technical Digest (CD) (Optical Society of America, 2008), paper QPDB1.

Taniyama, H.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of short pulse from ultrahigh-Q nanocavities via adiabatic wavelength conversion,” in Conference on Lasers and Electro-Optics, Technical Digest (CD) (Optical Society of America, 2008), paper QPDB1.

White, T. P.

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

Winn, J. W.

J. W. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[CrossRef]

Xiao, Y.

Xu, Q.

S. F. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photon. 1, 293–296 (2007).
[CrossRef]

Xu, Y.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E 62, 7389–7404(2000).
[CrossRef]

Yanik, M. F.

M. F. Yanik and S. Fan, “Dynamic photonic structures: stopping, storage, and time reversal of light,” Stud. Appl. Math. 115, 233–253 (2005).
[CrossRef]

Yariv, A.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E 62, 7389–7404(2000).
[CrossRef]

Young, J. F.

IEEE J. Quantum Electron.

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Photon.

S. F. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photon. 1, 293–296 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. A 73, 051803(2006).
[CrossRef]

T. Kampfrath, D. M. Beggs, T. P. White, A. Melloni, T. F. Krauss, and L. Kuipers, “Ultrafast adiabatic manipulation of slow light in a photonic crystal,” Phys. Rev. A 81, 043837 (2010).
[CrossRef]

E. E. Hach, III, A. W. Elshaari, and S. F. Preble, “Fully quantum-mechanical dynamic analysis of single-photon transport in a single-mode waveguide coupled to a traveling-wave resonator,” Phys. Rev. A 82, 063839 (2010).
[CrossRef]

Phys. Rev. B

J. W. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen, “Interband transitions in photonic crystals,” Phys. Rev. B 59, 1551–1554 (1999).
[CrossRef]

N. Malkova, S. Kim, and V. Gopalan, “Jahn-Teller effect intwo-dimensional photonic crystals,” Phys. Rev. B 68, 045105(2003).
[CrossRef]

Phys. Rev. E

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E 62, 7389–7404(2000).
[CrossRef]

Phys. Rev. Lett.

P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, and M. Lipson, “Inducing photonic transitions between discrete modes in a silicon optical microcavity,” Phys. Rev. Lett. 100, 033904(2008).
[CrossRef] [PubMed]

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[CrossRef] [PubMed]

Stud. Appl. Math.

M. F. Yanik and S. Fan, “Dynamic photonic structures: stopping, storage, and time reversal of light,” Stud. Appl. Math. 115, 233–253 (2005).
[CrossRef]

Other

S. F. Preble and M. Lipson, “Conversion of a signal wavelength in a dynamically tuned resonator,” in Integrated Photonics Research and Applications/Nanophotonics, Technical Digest (CD) (Optical Society of America, 2006), paper IMC5.

A. W. Elshaari and S. F. Preble, “Active optical isolator using adiabatic wavelength conversion in microcavities,” in Frontiers in Optics, Technical Digest (CD) (Optical Society of America, 2009), paper FThU4.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of short pulse from ultrahigh-Q nanocavities via adiabatic wavelength conversion,” in Conference on Lasers and Electro-Optics, Technical Digest (CD) (Optical Society of America, 2008), paper QPDB1.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

T. J. Johnson, “Silicon microdisk resonators for nonlinear optics and dynamics,” Ph.D. dissertation (California Institute of Technology, 2009).

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Figures (5)

Fig. 1
Fig. 1

Schematic of a silicon add/drop resonator

Fig. 2
Fig. 2

Comparison with experimental results from the literature for AWC in a silicon ring resonator. Measured drop-port spectra from [3] are compared with theoretical drop-port spectra predicted by Eq. (14) under the same experimental conditions. No fitting parameters were used in the simulations; all of the necessary information was reported in [3]. The absorbed pump energies in (a) and (b) are 0.419 and 1.38 pJ respectively. All other parameters are recorded in Table 1. The experimental spectra are adapted from [3] with permission from Macmillan Publishers Ltd., copyright 2007.

Fig. 3
Fig. 3

Comparison of measured drop-port spectra from [3] with theoretical drop-port spectra predicted by Eq. (14) for three different experimental scenarios: (a) absorbed pump energy of 0.7 pJ , (b) probe pulses detuned from resonance by 0.25 nm , (c) pump pulses broadened to 26 ps duration. In (c) a pump-probe delay of 20 ps was chosen. No other fitting parameters were used in the simulations; all of the necessary information was reported in [3] and is recorded in Table 1. The experimental spectra are adapted from [3] with permission from Macmillan Publishers Ltd., copyright 2007.

Fig. 4
Fig. 4

Ring-resonator configuration for dynamic mode coupling. The arc along the ring (purple region when viewed in color) indicates the portion of the resonator over which the refractive index is changed.

Fig. 5
Fig. 5

Mode coupling in a silicon ring resonator undergoing spatially nonuniform refractive index changes. The (a) drop-port spectrum measured in [15] is compared with the (b) predicted spectrum under the same experimental conditions. Simulation param eters are recorded in Table 2. No fitting parameters were used in the calculation. Figure 5a is adapted from [15] with permission from the American Physical Society, copyright 2008.

Tables (2)

Tables Icon

Table 1 Parameter Values a Used for Figs. 2, 3

Tables Icon

Table 2 Parameter Values a Used for Fig. 5

Equations (38)

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

ϵ ( r , t ) = ϵ ( r ) + Δ ϵ ( r , t ) ,
× E = μ 0 H t , × H = ϵ 0 ϵ ( r ) E t + P ( p ) t .
i t | ψ = M ^ | ψ + | V ( t ) .
| ψ = ( ϵ 0 E μ 0 H ) ,
ψ a | ψ b = 1 4 [ ϵ 0 ϵ ( r ) E a * · E b + μ 0 H a * · H b ] d 3 r .
M ^ = ( 0 i c ϵ × i c × 0 ) .
| V ( t ) = ( i ϵ ϵ 0 P ( p ) t 0 ) .
M ^ | ω k = ω k | ω k .
| ω k = ( ϵ 0 e k μ 0 h k ) .
ω k | ω m = N m δ k m ,
N m = 1 2 ε 0 ϵ ( r ) | e m | 2 d 3 r .
| ψ ( t ) = m a m ( t ) N m | ω m ,
d a k d t = i ω k a k 1 4 N k e k * · P ( p ) t d 3 r .
d a k d t = i ω k a k 1 2 τ p h k a k + κ k A i n ( t ) 1 4 N k e k * · P ( p ) t d 3 r ,
P ( p ) ( r , t ) = ϵ 0 Δ ϵ ( r , t ) E ( r , t ) = ϵ 0 Δ ϵ ( r , t ) m a m ( t ) N m e m ( r ) .
d a k d t = i ω k a k 1 2 τ p h k a k + κ k A i n ( t ) m ( d Γ k m d t a m + Γ k m d a m d t ) ,
Γ k m ( t ) = Δ ϵ ( r , t ) e k * ( r ) · e m ( r ) d 3 r ( 4 ϵ ( r ) | e k ( r ) | 2 d 3 r ϵ ( r ) | e m ( r ) | 2 d 3 r ) 1 / 2 .
d a k d t = i ω k a k 1 2 τ p h k a k + κ k A i n ( t ) + m i Γ k m ω m a m .
[ 1 + Γ ( t ) ] d a q d t = i ω q a q ( 1 2 τ p h q + d Γ d t ) a q + κ q A i n ( t ) ,
Γ ( t ) = Δ ϵ ( r , t ) | e q | 2 d 3 r 2 ϵ ( r ) | e q | 2 d 3 r .
d a q d t = i ω q ( t ) a q γ ( t ) a q ,
ω q ( t ) = ω q 1 + Γ ( t ) , γ ( t ) = ( 1 2 τ p h q + d Γ d t ) 1 1 + Γ ( t ) .
ω q ( t ) ω q [ 1 Γ ( t ) ] .
U ( t ) = 1 4 [ ϵ 0 ϵ ( r , t ) | E | 2 + μ 0 | H | 2 ] d 3 r = [ 1 + Γ ( t ) ] | a q | 2 .
d U d t = U d Γ / d t 1 + Γ ( t ) U τ p h q ( 1 + Γ ) .
U ( t ) = U 0 1 + Γ ( t ) exp [ 1 τ p h q 0 t d t 1 + Γ ( t ) ] .
N p = U ( t ) / ω q ( t ) = U 0 / ω q ,
Δ n = σ n e N ( σ n h N ) 4 / 5 + i σ a 2 k 0 N ,
d N d t = ξ P p ( t ) V cav ω p N τ f c ,
A d r ( t ) = κ q a q ( t ) ,
e k ( r , θ , z ) = E k u ( r , θ , z ) e i k θ .
Γ k m = ( f / n 0 ) Δ n ( t ) sinc [ ( m k ) π f ] e i ( m k ) π f ,
a k ( t ) = τ r A k ( t ) e i ω 0 t .
A i n ( t ) = A i n ( t ) e i ω 0 t .
A k ( t ) = η k A k ( t τ r ) e i Δ ω τ r + T k A i n ( t ) ,
d A k d t A k ( t ) A k ( t τ r ) τ r ,
d A k d t = 1 τ r η k ( e i Δ ω τ r η k ) A k + T k τ r η k A i n e i Δ ω τ r .
d a k d t = i ω k a k 1 2 τ p h k a k + κ k A i n ,

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