Abstract

A waveguide is bridged between two metallically hybrid microdisks for achieving nonlocal optical processing. The whispering gallery mode (WGM) is split into two modes, one is intrinsic WGM and the other is oscillating mode (OM) whose photons oscillate between two microdisks through the waveguide. The characteristics of OM on the waveguide’s width and length are studied. The results show the fundamental WGM oscillation only happens at certain lengths and widths of the waveguide, and strong optical interaction between two microdisks is accompanied by a high-loss waveguide. The optical processing through a waveguide between two individual resonators or photon producers may be applied to dynamic control on the photon states.

© 2013 Optical Society of America

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2013 (3)

2012 (2)

2011 (3)

2010 (4)

2008 (2)

A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, “Low threshold gain metal coated laser nanoresonators,” Opt. Lett. 33, 1261–1263 (2008).
[CrossRef]

L. I. Deych, C. Schmidt, A. Chipouline, T. Pertsch, and A. Tünnermann, “Optical coupling of fundamental whispering-gallery modes in bispheres,” Phys. Rev. A 77, 051801 (2008).
[CrossRef]

2007 (4)

2006 (6)

S. V. Boriskina, “Theoretical prediction of a dramatic Q-factor enhancement and degeneracy removal of whispering gallery modes in symmetrical photonic molecules,” Opt. Lett. 31, 338–340 (2006).
[CrossRef]

S. P. Ashili, V. N. Astratov, and E. C. H. Sykes, “The effects of inter-cavity separation on optical coupling in dielectric bispheres,” Opt. Express 14, 9460–9466 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

S. Ishii, A. Nakagawa, and T. Baba, “Modal characteristics and bistability in twin microdisk photonic molecule lasers,” IEEE J. Sel. Top. Quantum Electron. 12, 71–77 (2006).
[CrossRef]

J. W. Ryu, S. Y. Lee, C. M. Kim, and Y. J. Park, “Directional interacting whispering-gallery modes in coupled dielectric microdisks,” Phys. Rev. A 74, 013804 (2006).
[CrossRef]

A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88, 111111 (2006).
[CrossRef]

2004 (1)

2003 (1)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[CrossRef]

2001 (1)

W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223–225 (2001).
[CrossRef]

1998 (1)

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett, 81, 2582–2585 (1998).
[CrossRef]

1983 (1)

Alexander, R. W.

Ashili, S. P.

Astratov, V. N.

Baba, T.

S. Ishii, A. Nakagawa, and T. Baba, “Modal characteristics and bistability in twin microdisk photonic molecule lasers,” IEEE J. Sel. Top. Quantum Electron. 12, 71–77 (2006).
[CrossRef]

Bayer, M.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett, 81, 2582–2585 (1998).
[CrossRef]

Bell, R. J.

Bell, R. R.

Bell, S. E.

Benson, T. M.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

Benyoucef, M.

Beveratos, A.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaître, and I. Sagnes, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[CrossRef]

Boriskina, S. V.

Cai, W.

A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88, 111111 (2006).
[CrossRef]

S. Z. Deng, W. Cai, and V. N. Astratov, “Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides,” Opt. Express 12, 6468–6480 (2004).
[CrossRef]

Cai, Z. P.

K. J. Che, M. X. Lei, and Z. P. Cai, “Optical coupling and emission of metal-insulator confined circular resonators,” Opt. Express 21, 4979–4985 (2013).
[CrossRef]

K. J. Che, Q. F. Yao, Y. Z. Huang, Z. P. Cai, Y. D. Yang, and Y. Du, “Multiple-port InP/InGaAsP square-resonator microlasers,” IEEE J. Sel. Top. Quantum Electron. 17, 1656–1661 (2011).
[CrossRef]

Che, K. J.

K. J. Che, M. X. Lei, and Z. P. Cai, “Optical coupling and emission of metal-insulator confined circular resonators,” Opt. Express 21, 4979–4985 (2013).
[CrossRef]

K. J. Che, Q. F. Yao, Y. Z. Huang, Z. P. Cai, Y. D. Yang, and Y. Du, “Multiple-port InP/InGaAsP square-resonator microlasers,” IEEE J. Sel. Top. Quantum Electron. 17, 1656–1661 (2011).
[CrossRef]

Chipouline, A.

L. I. Deych, C. Schmidt, A. Chipouline, T. Pertsch, and A. Tünnermann, “Optical coupling of fundamental whispering-gallery modes in bispheres,” Phys. Rev. A 77, 051801 (2008).
[CrossRef]

Cho, J. H.

Deng, S. Z.

deWaardt, H.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. deWaardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Ntzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

Deych, L. I.

L. I. Deych, C. Schmidt, A. Chipouline, T. Pertsch, and A. Tünnermann, “Optical coupling of fundamental whispering-gallery modes in bispheres,” Phys. Rev. A 77, 051801 (2008).
[CrossRef]

Dousse, A.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaître, and I. Sagnes, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[CrossRef]

Dremin, A. A.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett, 81, 2582–2585 (1998).
[CrossRef]

Du, X.

Du, Y.

K. J. Che, Q. F. Yao, Y. Z. Huang, Z. P. Cai, Y. D. Yang, and Y. Du, “Multiple-port InP/InGaAsP square-resonator microlasers,” IEEE J. Sel. Top. Quantum Electron. 17, 1656–1661 (2011).
[CrossRef]

Eijkemans, T. J.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. deWaardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Ntzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

Fainman, Y.

Feng, L.

Forchel, A.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett, 81, 2582–2585 (1998).
[CrossRef]

Geluk, E. J.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. deWaardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Ntzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

Guo, W. H.

W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223–225 (2001).
[CrossRef]

Gutbrod, T.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett, 81, 2582–2585 (1998).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics-The Finite-Difference Time-Domain Method, 3rd ed. (Artech, 2005).

He, L. N.

L. N. He, Ş. K. Özdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photonics Rev. 7, 60–82 (2013).
[CrossRef]

Hetterich, M.

Hill, M. T.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. deWaardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Ntzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

Huang, Y. Z.

K. J. Che, Q. F. Yao, Y. Z. Huang, Z. P. Cai, Y. D. Yang, and Y. Du, “Multiple-port InP/InGaAsP square-resonator microlasers,” IEEE J. Sel. Top. Quantum Electron. 17, 1656–1661 (2011).
[CrossRef]

S. J. Wang, Y. D. Yang, and Y. Z. Huang, “Analysis of coupled microcircular resonators coupled to a bus waveguide with high output efficiency,” Opt. Lett. 35, 1953–1955 (2010).
[CrossRef]

J. J. Li, J. X. Wang, and Y. Z. Huang, “Mode coupling between first- and second-order whispering-gallery modes in coupled microdisks,” Opt. Lett. 32, 1563–1565 (2007).
[CrossRef]

W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223–225 (2001).
[CrossRef]

Ishii, S.

S. Ishii, A. Nakagawa, and T. Baba, “Modal characteristics and bistability in twin microdisk photonic molecule lasers,” IEEE J. Sel. Top. Quantum Electron. 12, 71–77 (2006).
[CrossRef]

Kalt, H.

Kanaev, A. V.

A. V. Kanaev, V. N. Astratov, and W. Cai, “Optical coupling at a distance between detuned spherical cavities,” Appl. Phys. Lett. 88, 111111 (2006).
[CrossRef]

Karl, M.

Kim, C. M.

J. W. Ryu, J. H. Cho, C. M. Kim, S. Shinohara, and S. W. Kim, “Terahertz beat frequency generation from two-mode lasing operation of coupled microdisk laser,” Opt. Lett. 37, 3210–3212 (2012).
[CrossRef]

J. W. Ryu, S. Y. Lee, C. M. Kim, and Y. J. Park, “Directional interacting whispering-gallery modes in coupled dielectric microdisks,” Phys. Rev. A 74, 013804 (2006).
[CrossRef]

Kim, S. W.

Klembt, S.

Knipp, P. A.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett, 81, 2582–2585 (1998).
[CrossRef]

Krebs, O.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaître, and I. Sagnes, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[CrossRef]

Kruse, C.

Kulakovskii, V. D.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett, 81, 2582–2585 (1998).
[CrossRef]

Kwon, S. H.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. deWaardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Ntzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

Lee, S. Y.

J. W. Ryu, S. Y. Lee, C. M. Kim, and Y. J. Park, “Directional interacting whispering-gallery modes in coupled dielectric microdisks,” Phys. Rev. A 74, 013804 (2006).
[CrossRef]

Lee, Y. H.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. deWaardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Ntzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

Lei, M. X.

Lemaître, A.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaître, and I. Sagnes, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[CrossRef]

Li, J. J.

Li, Q.

Li, S.

Li, W. J.

W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223–225 (2001).
[CrossRef]

Löffler, W.

Lomakin, V.

Long, L. L.

Mizrahi, A.

Nakagawa, A.

S. Ishii, A. Nakagawa, and T. Baba, “Modal characteristics and bistability in twin microdisk photonic molecule lasers,” IEEE J. Sel. Top. Quantum Electron. 12, 71–77 (2006).
[CrossRef]

Negro, L. D.

Nezhad, M. P.

Nosich, A. I.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

Ntzel, R.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. deWaardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Ntzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

Oei, Y. S.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. deWaardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Ntzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

Ordal, M. A.

Özdemir, S. K.

L. N. He, Ş. K. Özdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photonics Rev. 7, 60–82 (2013).
[CrossRef]

B. Peng, Ş. K. Özdemir, J. G. Zhu, and L. Yang, “Photonic molecules formed by coupled hybrid resonators,” Opt. Lett. 37, 3435–3437 (2012).
[CrossRef]

Park, Y. J.

J. W. Ryu, S. Y. Lee, C. M. Kim, and Y. J. Park, “Directional interacting whispering-gallery modes in coupled dielectric microdisks,” Phys. Rev. A 74, 013804 (2006).
[CrossRef]

Passow, T.

Peng, B.

Pertsch, T.

L. I. Deych, C. Schmidt, A. Chipouline, T. Pertsch, and A. Tünnermann, “Optical coupling of fundamental whispering-gallery modes in bispheres,” Phys. Rev. A 77, 051801 (2008).
[CrossRef]

Qiu, M.

Reinecke, T. L.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett, 81, 2582–2585 (1998).
[CrossRef]

Reithmaier, J. P.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, “Optical modes in photonic molecules,” Phys. Rev. Lett, 81, 2582–2585 (1998).
[CrossRef]

Ryu, J. W.

J. W. Ryu, J. H. Cho, C. M. Kim, S. Shinohara, and S. W. Kim, “Terahertz beat frequency generation from two-mode lasing operation of coupled microdisk laser,” Opt. Lett. 37, 3210–3212 (2012).
[CrossRef]

J. W. Ryu, S. Y. Lee, C. M. Kim, and Y. J. Park, “Directional interacting whispering-gallery modes in coupled dielectric microdisks,” Phys. Rev. A 74, 013804 (2006).
[CrossRef]

Sagnes, I.

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaître, and I. Sagnes, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
[CrossRef]

Schmidt, C.

L. I. Deych, C. Schmidt, A. Chipouline, T. Pertsch, and A. Tünnermann, “Optical coupling of fundamental whispering-gallery modes in bispheres,” Phys. Rev. A 77, 051801 (2008).
[CrossRef]

Schmidt, O. G.

Sebald, K.

Sewell, P.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

Seyfried, M.

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L. I. Deych, C. Schmidt, A. Chipouline, T. Pertsch, and A. Tünnermann, “Optical coupling of fundamental whispering-gallery modes in bispheres,” Phys. Rev. A 77, 051801 (2008).
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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. deWaardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Ntzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. deWaardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Ntzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. deWaardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Ntzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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Appl. Opt. (1)

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

S. Ishii, A. Nakagawa, and T. Baba, “Modal characteristics and bistability in twin microdisk photonic molecule lasers,” IEEE J. Sel. Top. Quantum Electron. 12, 71–77 (2006).
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L. N. He, Ş. K. Özdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photonics Rev. 7, 60–82 (2013).
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Nat. Photonics (1)

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. deWaardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Ntzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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Nature (2)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[CrossRef]

A. Dousse, J. Suffczynski, A. Beveratos, O. Krebs, A. Lemaître, and I. Sagnes, “Ultrabright source of entangled photon pairs,” Nature 466, 217–220 (2010).
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[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of a metallically hybrid microdisk.

Fig. 2.
Fig. 2.

Resonant wavelength and quality factor (Q) of TE(10,1) versus the thickness of the insulator for a metallically hybrid microdisk. The inset shows the electric field pattern as di=0.1μm.

Fig. 3.
Fig. 3.

(a) Resonant spectra of the optical system, the dashed and solid lines show the wavelength information in waveguide and microdisks, respectively. (b) Normalized electric field densities for split two states.

Fig. 4.
Fig. 4.

Propagating constant β versus the width of waveguide W at the resonant wavelength of TE(10,1), the solid, dashed and dotted lines corresponding to di=0.1, 0.15, and 0.20 μm.

Fig. 5.
Fig. 5.

(a) Q factor of the resonant mode in the whole optical system versus the length of bridged waveguide L, the solid squares, circles, and asterisks denote the results as W is 0.20, 0.25, and 0.30 μm. (b) Shift of resonant wavelength. (c) Electric field densities of resonant modes as L=2.25, 2.95, and 3.55 μm.

Fig. 6.
Fig. 6.

(a) Needed L for fundamental WGM resonance of OM versus W. (b) Q factor of OM and the recorded ratio of field density in waveguide to microdisk versus W.

Equations (3)

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

Φ(r,φ,t)=m=+Cmψm(r)ei(mφωmt),
ψm(r)={Jm(nskr)0rRA1Jm(nikr)+B1Ym(nikr)RrR+diA2Im(nmikr)+B2Km(nmikr)R+dirR+di+dmCHm1(nakr)rR+di+dm,
L2d=mπβ.

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