Abstract

An analytical method for transient dynamics description in microresonators is used to characterize and visualize the transient effects. In the frame of this method, the pulsed complex source point concept is used to simulate an incident transient beam. The excited fields in the microcavity are described by means of a rigorous mathematical approach that is based on the analytical solution in the Laplace transform domain and accurate evaluation of residues at singular points corresponding to the excited modes. The effects of transient mode beating and ultrashort pulse splitting inside the microresonator for short pulse excitation are discussed.

© 2012 Optical Society of America

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  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]
  2. M. Fujita, A. Sakai, and T. Baba, “Ultrasmall and ultralow threshold GaInAsP-InP microdisk injection lasers: design, fabrication, lasing characteristics and spontaneous emission factor,” IEEE J. Sel. Top. Quantum Electron. 5, 673–681 (1999).
    [CrossRef]
  3. K. Djordjev, S.-J. Choi, S.-J. Choi, and R. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 823–830 (2002).
    [CrossRef]
  4. M. Lee and P. Fauchet, “Nanoscale microcavity sensor for single particle detection,” Opt. Lett. 32, 3284–3286 (2007).
    [CrossRef]
  5. M. Hill, H. Dorren, T. Vries, X. Leitjens, J. Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. Smit, “A fast low-power optical memory based upon coupled micro-ring lasers,” Nature 432, 206–209 (2004).
    [CrossRef]
  6. R. Shaw, W. Whitten, M. Barnes, and J. Ramsey, “Time-domain observation of optical pulse propagation in whispering-gallery modes of glass spheres,” Opt. Lett. 23, 1301–1303 (1998).
    [CrossRef]
  7. H. Gersen, D. Klunder, J. Korterik, A. Driessen, N. Hulst, and L. Kuipers, “Propagation of a femtosecond pulse in a microresonator visualized in time,” Opt. Lett. 29, 1291–1293 (2004).
    [CrossRef]
  8. M. Balistreri, D. Klunder, F. Blom, A. Driessen, H. Hoekstra, J. Korterik, L. Kuipers, and N. Hulst, “Visualizing the whispering-gallery modes in a cylindrical optical microcavity,” Opt. Lett. 24, 1829–1831 (1999).
    [CrossRef]
  9. A. Driessen, D. H. Geuzebroek, E. J. Klein, R. Dekker, R. Stoffer, and C. Bornholdt, “Propagation of short light pulses in microring resonators: Ballistic transport versus interference in the frequency domain,” Opt. Commun. 270, 217–224 (2007).
    [CrossRef]
  10. A. V. Boriskin, S. V. Boriskina, A. Rolland, R. Sauleau, and A. I. Nosich, “Test of the FDTD accuracy in the analysis of the scattering resonances associated with high-Q whispering-gallery modes of a circular cylinder,” J. Opt. Soc. Am. A 25, 1169–1173 (2008).
    [CrossRef]
  11. C. Baum, “On singularity expansion method for the solution of electromagnetic interaction problems,” Interact. Notes 88, 1–111 (1971).
  12. C. Baum, “Discrimination of buried targets via the singularity expansion,” Inverse Probl. 13, 557–570 (1997).
    [CrossRef]
  13. L. Marin, “Natural-mode representation of transient scattered fields,” IEEE Trans. Antennas Propag. AP-21, 809–818 (1973).
    [CrossRef]
  14. N. Sakhnenko and A. Nerukh, “Rigorous analysis of whispering gallery mode frequency conversion due to time variation of refractive index in a spherical resonator,” J. Opt. Soc. Am. A 29, 99–104 (2012).
    [CrossRef]
  15. N. K. Sakhnenko, A. G. Nerukh, T. Benson, and P. Sewell, “Frequency conversion and field pattern rotation in WGM resonator with transient inclusion,” Opt. Quantum Electron. 39, 761–771 (2007).
    [CrossRef]
  16. L. Felsen, “Complex-point source solutions of the field equations and their relation to the propagation and scattering of the Gaussian beams,” Simposia Math. 18, 40–56 (1976).
  17. E. Heyman and L. Felsen, “Complex-source pulsed beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
    [CrossRef]
  18. E. Heyman and L. Felsen, “Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. A 18, 1588–1611 (2001).
    [CrossRef]
  19. C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, V. Shuvayev, L. I. Deych, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
    [CrossRef]
  20. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
    [CrossRef]
  21. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice Hall, 1973).
  22. A. V. Boriskin and A. I. Nosich, “Whispering-gallery and Luneburg-lens effects in a beam-fed circularly layered dielectric cylinder,” IEEE Trans. Antennas Propag. 50, 1245–1249 (2002).
    [CrossRef]

2012 (1)

2009 (1)

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, V. Shuvayev, L. I. Deych, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

2008 (1)

2007 (3)

M. Lee and P. Fauchet, “Nanoscale microcavity sensor for single particle detection,” Opt. Lett. 32, 3284–3286 (2007).
[CrossRef]

N. K. Sakhnenko, A. G. Nerukh, T. Benson, and P. Sewell, “Frequency conversion and field pattern rotation in WGM resonator with transient inclusion,” Opt. Quantum Electron. 39, 761–771 (2007).
[CrossRef]

A. Driessen, D. H. Geuzebroek, E. J. Klein, R. Dekker, R. Stoffer, and C. Bornholdt, “Propagation of short light pulses in microring resonators: Ballistic transport versus interference in the frequency domain,” Opt. Commun. 270, 217–224 (2007).
[CrossRef]

2005 (1)

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

2004 (2)

M. Hill, H. Dorren, T. Vries, X. Leitjens, J. Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. Smit, “A fast low-power optical memory based upon coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

H. Gersen, D. Klunder, J. Korterik, A. Driessen, N. Hulst, and L. Kuipers, “Propagation of a femtosecond pulse in a microresonator visualized in time,” Opt. Lett. 29, 1291–1293 (2004).
[CrossRef]

2002 (2)

K. Djordjev, S.-J. Choi, S.-J. Choi, and R. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 823–830 (2002).
[CrossRef]

A. V. Boriskin and A. I. Nosich, “Whispering-gallery and Luneburg-lens effects in a beam-fed circularly layered dielectric cylinder,” IEEE Trans. Antennas Propag. 50, 1245–1249 (2002).
[CrossRef]

2001 (1)

1999 (2)

M. Balistreri, D. Klunder, F. Blom, A. Driessen, H. Hoekstra, J. Korterik, L. Kuipers, and N. Hulst, “Visualizing the whispering-gallery modes in a cylindrical optical microcavity,” Opt. Lett. 24, 1829–1831 (1999).
[CrossRef]

M. Fujita, A. Sakai, and T. Baba, “Ultrasmall and ultralow threshold GaInAsP-InP microdisk injection lasers: design, fabrication, lasing characteristics and spontaneous emission factor,” IEEE J. Sel. Top. Quantum Electron. 5, 673–681 (1999).
[CrossRef]

1998 (1)

1997 (2)

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]

C. Baum, “Discrimination of buried targets via the singularity expansion,” Inverse Probl. 13, 557–570 (1997).
[CrossRef]

1989 (1)

1976 (1)

L. Felsen, “Complex-point source solutions of the field equations and their relation to the propagation and scattering of the Gaussian beams,” Simposia Math. 18, 40–56 (1976).

1973 (1)

L. Marin, “Natural-mode representation of transient scattered fields,” IEEE Trans. Antennas Propag. AP-21, 809–818 (1973).
[CrossRef]

1971 (1)

C. Baum, “On singularity expansion method for the solution of electromagnetic interaction problems,” Interact. Notes 88, 1–111 (1971).

Baba, T.

M. Fujita, A. Sakai, and T. Baba, “Ultrasmall and ultralow threshold GaInAsP-InP microdisk injection lasers: design, fabrication, lasing characteristics and spontaneous emission factor,” IEEE J. Sel. Top. Quantum Electron. 5, 673–681 (1999).
[CrossRef]

Balistreri, M.

Barnes, M.

Baum, C.

C. Baum, “Discrimination of buried targets via the singularity expansion,” Inverse Probl. 13, 557–570 (1997).
[CrossRef]

C. Baum, “On singularity expansion method for the solution of electromagnetic interaction problems,” Interact. Notes 88, 1–111 (1971).

Benson, T.

N. K. Sakhnenko, A. G. Nerukh, T. Benson, and P. Sewell, “Frequency conversion and field pattern rotation in WGM resonator with transient inclusion,” Opt. Quantum Electron. 39, 761–771 (2007).
[CrossRef]

Benson, T. M.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

Besten, J.

M. Hill, H. Dorren, T. Vries, X. Leitjens, J. Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. Smit, “A fast low-power optical memory based upon coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Binsma, H.

M. Hill, H. Dorren, T. Vries, X. Leitjens, J. Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. Smit, “A fast low-power optical memory based upon coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Blom, F.

Boriskin, A. V.

A. V. Boriskin, S. V. Boriskina, A. Rolland, R. Sauleau, and A. I. Nosich, “Test of the FDTD accuracy in the analysis of the scattering resonances associated with high-Q whispering-gallery modes of a circular cylinder,” J. Opt. Soc. Am. A 25, 1169–1173 (2008).
[CrossRef]

A. V. Boriskin and A. I. Nosich, “Whispering-gallery and Luneburg-lens effects in a beam-fed circularly layered dielectric cylinder,” IEEE Trans. Antennas Propag. 50, 1245–1249 (2002).
[CrossRef]

Boriskina, S. V.

Bornholdt, C.

A. Driessen, D. H. Geuzebroek, E. J. Klein, R. Dekker, R. Stoffer, and C. Bornholdt, “Propagation of short light pulses in microring resonators: Ballistic transport versus interference in the frequency domain,” Opt. Commun. 270, 217–224 (2007).
[CrossRef]

Chipouline, A.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, V. Shuvayev, L. I. Deych, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Choi, S.-J.

K. Djordjev, S.-J. Choi, S.-J. Choi, and R. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 823–830 (2002).
[CrossRef]

K. Djordjev, S.-J. Choi, S.-J. Choi, and R. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 823–830 (2002).
[CrossRef]

Chu, S. T.

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]

Dapkus, R.

K. Djordjev, S.-J. Choi, S.-J. Choi, and R. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 823–830 (2002).
[CrossRef]

Dekker, R.

A. Driessen, D. H. Geuzebroek, E. J. Klein, R. Dekker, R. Stoffer, and C. Bornholdt, “Propagation of short light pulses in microring resonators: Ballistic transport versus interference in the frequency domain,” Opt. Commun. 270, 217–224 (2007).
[CrossRef]

Deych, L. I.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, V. Shuvayev, L. I. Deych, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Djordjev, K.

K. Djordjev, S.-J. Choi, S.-J. Choi, and R. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 823–830 (2002).
[CrossRef]

Dorren, H.

M. Hill, H. Dorren, T. Vries, X. Leitjens, J. Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. Smit, “A fast low-power optical memory based upon coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Driessen, A.

Fauchet, P.

Felsen, L.

E. Heyman and L. Felsen, “Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. A 18, 1588–1611 (2001).
[CrossRef]

E. Heyman and L. Felsen, “Complex-source pulsed beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
[CrossRef]

L. Felsen, “Complex-point source solutions of the field equations and their relation to the propagation and scattering of the Gaussian beams,” Simposia Math. 18, 40–56 (1976).

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice Hall, 1973).

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]

Fujita, M.

M. Fujita, A. Sakai, and T. Baba, “Ultrasmall and ultralow threshold GaInAsP-InP microdisk injection lasers: design, fabrication, lasing characteristics and spontaneous emission factor,” IEEE J. Sel. Top. Quantum Electron. 5, 673–681 (1999).
[CrossRef]

Gersen, H.

Geuzebroek, D. H.

A. Driessen, D. H. Geuzebroek, E. J. Klein, R. Dekker, R. Stoffer, and C. Bornholdt, “Propagation of short light pulses in microring resonators: Ballistic transport versus interference in the frequency domain,” Opt. Commun. 270, 217–224 (2007).
[CrossRef]

Haus, H. A.

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]

Heyman, E.

Hill, M.

M. Hill, H. Dorren, T. Vries, X. Leitjens, J. Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. Smit, “A fast low-power optical memory based upon coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Hoekstra, H.

Hulst, N.

Käsebier, T.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, V. Shuvayev, L. I. Deych, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Khoe, G.-D.

M. Hill, H. Dorren, T. Vries, X. Leitjens, J. Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. Smit, “A fast low-power optical memory based upon coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Klein, E. J.

A. Driessen, D. H. Geuzebroek, E. J. Klein, R. Dekker, R. Stoffer, and C. Bornholdt, “Propagation of short light pulses in microring resonators: Ballistic transport versus interference in the frequency domain,” Opt. Commun. 270, 217–224 (2007).
[CrossRef]

Kley, E.-B.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, V. Shuvayev, L. I. Deych, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Klunder, D.

Korterik, J.

Kuipers, L.

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]

Lee, M.

Leitjens, X.

M. Hill, H. Dorren, T. Vries, X. Leitjens, J. Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. Smit, “A fast low-power optical memory based upon coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Little, B. E.

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]

Marcuvitz, N.

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice Hall, 1973).

Marin, L.

L. Marin, “Natural-mode representation of transient scattered fields,” IEEE Trans. Antennas Propag. AP-21, 809–818 (1973).
[CrossRef]

Nerukh, A.

Nerukh, A. G.

N. K. Sakhnenko, A. G. Nerukh, T. Benson, and P. Sewell, “Frequency conversion and field pattern rotation in WGM resonator with transient inclusion,” Opt. Quantum Electron. 39, 761–771 (2007).
[CrossRef]

Nosich, A. I.

A. V. Boriskin, S. V. Boriskina, A. Rolland, R. Sauleau, and A. I. Nosich, “Test of the FDTD accuracy in the analysis of the scattering resonances associated with high-Q whispering-gallery modes of a circular cylinder,” J. Opt. Soc. Am. A 25, 1169–1173 (2008).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

A. V. Boriskin and A. I. Nosich, “Whispering-gallery and Luneburg-lens effects in a beam-fed circularly layered dielectric cylinder,” IEEE Trans. Antennas Propag. 50, 1245–1249 (2002).
[CrossRef]

Oei, Y.-S.

M. Hill, H. Dorren, T. Vries, X. Leitjens, J. Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. Smit, “A fast low-power optical memory based upon coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Pertsch, T.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, V. Shuvayev, L. I. Deych, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Ramsey, J.

Rolland, A.

Sakai, A.

M. Fujita, A. Sakai, and T. Baba, “Ultrasmall and ultralow threshold GaInAsP-InP microdisk injection lasers: design, fabrication, lasing characteristics and spontaneous emission factor,” IEEE J. Sel. Top. Quantum Electron. 5, 673–681 (1999).
[CrossRef]

Sakhnenko, N.

Sakhnenko, N. K.

N. K. Sakhnenko, A. G. Nerukh, T. Benson, and P. Sewell, “Frequency conversion and field pattern rotation in WGM resonator with transient inclusion,” Opt. Quantum Electron. 39, 761–771 (2007).
[CrossRef]

Sauleau, R.

Schmidt, C.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, V. Shuvayev, L. I. Deych, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Sewell, P.

N. K. Sakhnenko, A. G. Nerukh, T. Benson, and P. Sewell, “Frequency conversion and field pattern rotation in WGM resonator with transient inclusion,” Opt. Quantum Electron. 39, 761–771 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

Shaw, R.

Shuvayev, V.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, V. Shuvayev, L. I. Deych, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Smalbrugge, B.

M. Hill, H. Dorren, T. Vries, X. Leitjens, J. Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. Smit, “A fast low-power optical memory based upon coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Smit, M.

M. Hill, H. Dorren, T. Vries, X. Leitjens, J. Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. Smit, “A fast low-power optical memory based upon coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Smotrova, E. I.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

Stoffer, R.

A. Driessen, D. H. Geuzebroek, E. J. Klein, R. Dekker, R. Stoffer, and C. Bornholdt, “Propagation of short light pulses in microring resonators: Ballistic transport versus interference in the frequency domain,” Opt. Commun. 270, 217–224 (2007).
[CrossRef]

Tünnermann, A.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, V. Shuvayev, L. I. Deych, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Vries, T.

M. Hill, H. Dorren, T. Vries, X. Leitjens, J. Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. Smit, “A fast low-power optical memory based upon coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Whitten, W.

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

M. Fujita, A. Sakai, and T. Baba, “Ultrasmall and ultralow threshold GaInAsP-InP microdisk injection lasers: design, fabrication, lasing characteristics and spontaneous emission factor,” IEEE J. Sel. Top. Quantum Electron. 5, 673–681 (1999).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

K. Djordjev, S.-J. Choi, S.-J. Choi, and R. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 823–830 (2002).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

L. Marin, “Natural-mode representation of transient scattered fields,” IEEE Trans. Antennas Propag. AP-21, 809–818 (1973).
[CrossRef]

A. V. Boriskin and A. I. Nosich, “Whispering-gallery and Luneburg-lens effects in a beam-fed circularly layered dielectric cylinder,” IEEE Trans. Antennas Propag. 50, 1245–1249 (2002).
[CrossRef]

Interact. Notes (1)

C. Baum, “On singularity expansion method for the solution of electromagnetic interaction problems,” Interact. Notes 88, 1–111 (1971).

Inverse Probl. (1)

C. Baum, “Discrimination of buried targets via the singularity expansion,” Inverse Probl. 13, 557–570 (1997).
[CrossRef]

J. Lightwave Technol. (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]

J. Opt. Soc. Am. A (4)

Nature (1)

M. Hill, H. Dorren, T. Vries, X. Leitjens, J. Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. Smit, “A fast low-power optical memory based upon coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Opt. Commun. (1)

A. Driessen, D. H. Geuzebroek, E. J. Klein, R. Dekker, R. Stoffer, and C. Bornholdt, “Propagation of short light pulses in microring resonators: Ballistic transport versus interference in the frequency domain,” Opt. Commun. 270, 217–224 (2007).
[CrossRef]

Opt. Lett. (4)

Opt. Quantum Electron. (1)

N. K. Sakhnenko, A. G. Nerukh, T. Benson, and P. Sewell, “Frequency conversion and field pattern rotation in WGM resonator with transient inclusion,” Opt. Quantum Electron. 39, 761–771 (2007).
[CrossRef]

Phys. Rev. A (1)

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, V. Shuvayev, L. I. Deych, and T. Pertsch, “Observation of optical coupling in microdisk resonators,” Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Simposia Math. (1)

L. Felsen, “Complex-point source solutions of the field equations and their relation to the propagation and scattering of the Gaussian beams,” Simposia Math. 18, 40–56 (1976).

Other (1)

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice Hall, 1973).

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Figures (10)

Fig. 1.
Fig. 1.

Schematic diagram of the experimental layout considered in the paper: (a) tapered fiber coupled with the microresonator and (b) CSP beam placed at the point x0, y0 models the presence of a tapered fiber. The microresonator is a microdisk with effective refractive index nd and radius R, a point on the plane is specified by a radial coordinate r and an angle φ in polar coordinates or by x, y in the Cartesian coordinates.

Fig. 2.
Fig. 2.

Excitation of the cavity by harmonic CSPB, although the beam wavelength of λ=1499.8nm coincides with the WGH110,1 mode, several higher order modes (covered by the bandwidth of the excitation) and the absence of phase matching lead to a complex field pattern.

Fig. 3.
Fig. 3.

Normalized intensity versus time inside the microresonator for an exciting pulse shorter than one round-trip time (τ=0.1ps, λ=1500nm). At early times (left inset), single pulses are observed, whereas for longer times (right inset) a spreading and splitting of the pulse can be seen, indicated by two successive intensity maxima.

Fig. 4.
Fig. 4.

Ultrashort pulse traveling inside a microcavity: (a) t=2ps, (b) t=6ps, and (c) t=15ps. The exciting pulse spectrum covers several modes from which only the two highest Q modes survive resulting in the observed interference pattern (c).

Fig. 5.
Fig. 5.

Spectral density versus the wavelength for a pulse excitation of the microdisk with a pulse duration of τ=5ps (a) and τ=25ps (b). The excitation beam wavelengths marked by the arrows are the same as resonant wavelengths of the mode shown in the figure legend. The strong peak leftmost in (a) corresponds to the not directly excited WGH89,5 mode and appears for all cases. The excitation of the WGH93,4 mode is best for the chosen pulse and microresonator parameters (a). A longer pulse (b) leads to an enhancement of the mode at the center wavelength of the pulse because of a narrower pulse spectrum.

Fig. 6.
Fig. 6.

Normalized intensity versus time in two different time scales (a) and (b) for two different pulse durations of τ=2ps and τ=5ps and beam wavelength λ=1501.5nm. The round-trip time of 0.6 ps is visible in both cases as the distance between maxima in (b). For the longer pulse, an overall larger intensity can be observed. The inset in (a) shows the intensity distribution along the x axis inside the microresonator at t=30ps.

Fig. 7.
Fig. 7.

Mode-beating spectrum for an excitation pulse at λ=1500nm and different durations showing the two excited modes WGH89,5 (λ=1497.7nm) and WGH93,4 (λ=1501.5nm) of the microresonator.

Fig. 8.
Fig. 8.

Mode-beating pattern at the moment of time t=33ps for the τ=5ps pulse from Fig. 7.

Fig. 9.
Fig. 9.

Accuracy of calculation estimation: (a) spectral density versus wavelength and (b) intensity of the excited field versus time when evaluating residues for |λ0λ|<Δ (λ0 is the incident beam wavelength, λ is resonant wavelength), Δ1=100nm, Δ2=150nm, Δ3=250nm, and Δ4=500nm; pulse duration is 5 ps, the excitation beam wavelength λ0=1498.1nm, resonant with WGH98,3.

Fig. 10.
Fig. 10.

Contribution of the integral along the branch-cut into the total field. The center frequency of the pulse is the same as in Fig. 9; its time dependence is presented by Eq. (18). The vertical line represents the ratio of the integral along the branch-cut to the value of total field calculated as the residue sum and integral along the branch-cut.

Equations (19)

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(Δndc22t2)h(t,r)=0,ifrR,
(Δ1c22t2)h(t,r)=0,ifr>R,
h(t<0,r)=0,th(t<0,r)=0.
{xcs=x0+ibcosβ,ycs=y0+ibsinβ,
Δh0(t,r)1c22t2h0(t,r)=ε0tj^(t,r).
g(t,t,r,rs)=12πθ(tt|rrs|/c)(tt)2|rrs|2/c2,
h0(t,r)=12π0dt0rdr02πdφθ(tt|rr|/c)(tt)2|rr|2/c2ε0tj^(t,r).
h0(t,r)=ε00dtθ(tt|rrs|/c)(tt)2|rrs|2/c2tj(t).
H0(p,r)=ε0/(2π)K0(|rrs|p/c)pJ(p),
K0(|rrs|p/c)π2πpr/cexp(pc(rr0))exp(ipcbcos(φβ)),r.
f(t,r,rcs)=12πiiiF(p,r,rcs)exp(p(tt1))dp,
K0(p|rrcs|/c)=k=eik(φφcs)(Ik(prcs/c)Kk(pr/c)Θ(r|rcs|)+Ik(pr/c)Kk(prcs/c)Θ(|rcs|r)),
H(p,r)=k=Ak(p)Ik(ndpr/c)eik(φφcs),ifrR,
H(p,r)=k=Bk(p)Kk(pr/c)eik(φφcs),ifr>R.
H(p,r=R0)=H(p,r=R+0),rH(p,r=R0)=ndrH(p,r=R+0).
Ak=ε02πcpRndIk(ndpR/c)Kk(pR/c)ndKk(pR/c)Ik(ndpR/c)Kk(prcs/c)pJ(p),
Bk=ε02πndIk(pR/c)Ik(ndpR/c)Ik(ndpR/c)Ik(pR/c)Ik(ndpR/c)Kk(pR/c)ndKk(pR/c)Ik(ndpR/c)Kk(prcs/c)pJ(p).
j(t)=eiω0tΘ(t),
j(t)=exp(iω0t)(1cosπt2τ)(Θ(t)Θ(tτ)).

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