Abstract

We study the interaction between a planar Fabry–Perot microcavity and a single embedded gold nanoparticle. We record spatially resolved white-light transmission spectra and show that the interaction between microcavity and nanoparticle gives rise to a redshift of the cavity modes, in accordance with the Bethe–Schwinger cavity perturbation formula. The spectral tuning of cavity modes by discrete nanoparticles can be exploited for applications such as optical and micromechanical sensing.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  20. H. Becker, S. E. Burns, N. Tessler, and R. H. Friend, J. Appl. Phys. 81, 2825 (1997).
    [CrossRef]

2009 (1)

M. Humar, M. Ravnik, S. Pajk, and I. Musevic, Nat. Photonics 3, 595 (2009).
[CrossRef]

2008 (3)

G. Anetsberger, R. Riviere, A. Schlieer, O. Arcizet, and T. Kippenberg, Nat. Photonics 2, 627 (2008).
[CrossRef]

F. Vollmer and S. Arnold, Nat. Methods 5, 591 (2008).
[CrossRef] [PubMed]

D. A. Bergstein, E. Ozkumur, A. C. Wu, A. Yalcin, J. Needham, R. Irani, J. Gershoni, B. B. Goldberg, C. DeLisi, M. F. Ruane, and M. S. Ünlü, IEEE J. Sel. Top. Quantum Electron. 14, 131 (2008).
[CrossRef] [PubMed]

2007 (2)

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, Science 317, 783 (2007).
[CrossRef] [PubMed]

T. Wang, C. Chu, and C. Lin, Opt. Lett. 32, 2777 (2007).
[CrossRef] [PubMed]

2006 (1)

A. Kiraz, A. Kurt, and M. A. Dündar, Appl. Phys. Lett. 102, 093713 (2006).

2005 (1)

M. Steiner, F. Schleifenbaum, C. Stupperich, A. V. Failla, A. Hartschuh, and A. J. Meixner,ChemPhysChem 6, 2190 (2005).
[CrossRef] [PubMed]

2004 (2)

D. Armani, B. Min, A. Martin, and K. J. Vahala, Appl. Phys. Lett. 85, 5439 (2004).
[CrossRef]

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, L. V. K. S. Reitzenstein, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, Nature 432, 197 (2004).
[CrossRef] [PubMed]

2003 (2)

B. Maune, R. Lawson, C. Gunn, A. Scherer, and L. Dalton, Appl. Phys. Lett. 83, 4689 (2003).
[CrossRef]

K. J. Vahala, Nature 424, 839 (2003).
[CrossRef] [PubMed]

2001 (2)

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefevre-Seguin, New J. Phys. 3, 14.1 (2001).
[CrossRef]

Y. Hibino, T. Maruno, and K. Okamoto, NTT Tech. Rev. 13, 4 (2001).

1997 (2)

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, J. Lightwave Technol. 15, 2154 (1997).
[CrossRef]

H. Becker, S. E. Burns, N. Tessler, and R. H. Friend, J. Appl. Phys. 81, 2825 (1997).
[CrossRef]

1996 (1)

P. Heimala, P. Katila, J. Aarnio, and A. Heinämäki, J. Lightwave Technol. 14, 2260 (1996).
[CrossRef]

1995 (1)

M. S. Ünlü and S. C. Strite, J. Appl. Phys. 78, 607 (1995).
[CrossRef]

1971 (1)

W. Hauser, Introduction to the Principles of Electromagnetism (Addison-Wesley, 1971).

1943 (1)

J. Schwinger, M.I.T. Radiation Lab. Report (MIT, 1943), Vol. 43-34.

Aarnio, J.

P. Heimala, P. Katila, J. Aarnio, and A. Heinämäki, J. Lightwave Technol. 14, 2260 (1996).
[CrossRef]

Anetsberger, G.

G. Anetsberger, R. Riviere, A. Schlieer, O. Arcizet, and T. Kippenberg, Nat. Photonics 2, 627 (2008).
[CrossRef]

Arcizet, O.

G. Anetsberger, R. Riviere, A. Schlieer, O. Arcizet, and T. Kippenberg, Nat. Photonics 2, 627 (2008).
[CrossRef]

Armani, A. M.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, Science 317, 783 (2007).
[CrossRef] [PubMed]

Armani, D.

D. Armani, B. Min, A. Martin, and K. J. Vahala, Appl. Phys. Lett. 85, 5439 (2004).
[CrossRef]

Arnold, S.

F. Vollmer and S. Arnold, Nat. Methods 5, 591 (2008).
[CrossRef] [PubMed]

Becker, H.

H. Becker, S. E. Burns, N. Tessler, and R. H. Friend, J. Appl. Phys. 81, 2825 (1997).
[CrossRef]

Bergstein, D. A.

D. A. Bergstein, E. Ozkumur, A. C. Wu, A. Yalcin, J. Needham, R. Irani, J. Gershoni, B. B. Goldberg, C. DeLisi, M. F. Ruane, and M. S. Ünlü, IEEE J. Sel. Top. Quantum Electron. 14, 131 (2008).
[CrossRef] [PubMed]

Burns, S. E.

H. Becker, S. E. Burns, N. Tessler, and R. H. Friend, J. Appl. Phys. 81, 2825 (1997).
[CrossRef]

Chu, C.

Dalton, L.

B. Maune, R. Lawson, C. Gunn, A. Scherer, and L. Dalton, Appl. Phys. Lett. 83, 4689 (2003).
[CrossRef]

DeLisi, C.

D. A. Bergstein, E. Ozkumur, A. C. Wu, A. Yalcin, J. Needham, R. Irani, J. Gershoni, B. B. Goldberg, C. DeLisi, M. F. Ruane, and M. S. Ünlü, IEEE J. Sel. Top. Quantum Electron. 14, 131 (2008).
[CrossRef] [PubMed]

Dündar, M. A.

A. Kiraz, A. Kurt, and M. A. Dündar, Appl. Phys. Lett. 102, 093713 (2006).

Failla, A. V.

M. Steiner, F. Schleifenbaum, C. Stupperich, A. V. Failla, A. Hartschuh, and A. J. Meixner,ChemPhysChem 6, 2190 (2005).
[CrossRef] [PubMed]

Flagan, R. C.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, Science 317, 783 (2007).
[CrossRef] [PubMed]

Forchel, A.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, L. V. K. S. Reitzenstein, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, Nature 432, 197 (2004).
[CrossRef] [PubMed]

Fraser, S. E.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, Science 317, 783 (2007).
[CrossRef] [PubMed]

Friend, R. H.

H. Becker, S. E. Burns, N. Tessler, and R. H. Friend, J. Appl. Phys. 81, 2825 (1997).
[CrossRef]

Gershoni, J.

D. A. Bergstein, E. Ozkumur, A. C. Wu, A. Yalcin, J. Needham, R. Irani, J. Gershoni, B. B. Goldberg, C. DeLisi, M. F. Ruane, and M. S. Ünlü, IEEE J. Sel. Top. Quantum Electron. 14, 131 (2008).
[CrossRef] [PubMed]

Goldberg, B. B.

D. A. Bergstein, E. Ozkumur, A. C. Wu, A. Yalcin, J. Needham, R. Irani, J. Gershoni, B. B. Goldberg, C. DeLisi, M. F. Ruane, and M. S. Ünlü, IEEE J. Sel. Top. Quantum Electron. 14, 131 (2008).
[CrossRef] [PubMed]

Gunn, C.

B. Maune, R. Lawson, C. Gunn, A. Scherer, and L. Dalton, Appl. Phys. Lett. 83, 4689 (2003).
[CrossRef]

Hagness, S. C.

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, J. Lightwave Technol. 15, 2154 (1997).
[CrossRef]

Hare, J.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefevre-Seguin, New J. Phys. 3, 14.1 (2001).
[CrossRef]

Hartschuh, A.

M. Steiner, F. Schleifenbaum, C. Stupperich, A. V. Failla, A. Hartschuh, and A. J. Meixner,ChemPhysChem 6, 2190 (2005).
[CrossRef] [PubMed]

Hauser, W.

W. Hauser, Introduction to the Principles of Electromagnetism (Addison-Wesley, 1971).

Heimala, P.

P. Heimala, P. Katila, J. Aarnio, and A. Heinämäki, J. Lightwave Technol. 14, 2260 (1996).
[CrossRef]

Heinämäki, A.

P. Heimala, P. Katila, J. Aarnio, and A. Heinämäki, J. Lightwave Technol. 14, 2260 (1996).
[CrossRef]

Hibino, Y.

Y. Hibino, T. Maruno, and K. Okamoto, NTT Tech. Rev. 13, 4 (2001).

Ho, S. T.

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, J. Lightwave Technol. 15, 2154 (1997).
[CrossRef]

Hofmann, C.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, L. V. K. S. Reitzenstein, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, Nature 432, 197 (2004).
[CrossRef] [PubMed]

Humar, M.

M. Humar, M. Ravnik, S. Pajk, and I. Musevic, Nat. Photonics 3, 595 (2009).
[CrossRef]

Ilchenko, V. S.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefevre-Seguin, New J. Phys. 3, 14.1 (2001).
[CrossRef]

Irani, R.

D. A. Bergstein, E. Ozkumur, A. C. Wu, A. Yalcin, J. Needham, R. Irani, J. Gershoni, B. B. Goldberg, C. DeLisi, M. F. Ruane, and M. S. Ünlü, IEEE J. Sel. Top. Quantum Electron. 14, 131 (2008).
[CrossRef] [PubMed]

Katila, P.

P. Heimala, P. Katila, J. Aarnio, and A. Heinämäki, J. Lightwave Technol. 14, 2260 (1996).
[CrossRef]

Kippenberg, T.

G. Anetsberger, R. Riviere, A. Schlieer, O. Arcizet, and T. Kippenberg, Nat. Photonics 2, 627 (2008).
[CrossRef]

Kiraz, A.

A. Kiraz, A. Kurt, and M. A. Dündar, Appl. Phys. Lett. 102, 093713 (2006).

Kuhn, S.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, L. V. K. S. Reitzenstein, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, Nature 432, 197 (2004).
[CrossRef] [PubMed]

Kulakovskii, V. D.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, L. V. K. S. Reitzenstein, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, Nature 432, 197 (2004).
[CrossRef] [PubMed]

Kulkarni, R. P.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, Science 317, 783 (2007).
[CrossRef] [PubMed]

Kurt, A.

A. Kiraz, A. Kurt, and M. A. Dündar, Appl. Phys. Lett. 102, 093713 (2006).

Lawson, R.

B. Maune, R. Lawson, C. Gunn, A. Scherer, and L. Dalton, Appl. Phys. Lett. 83, 4689 (2003).
[CrossRef]

Lefevre-Seguin, V.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefevre-Seguin, New J. Phys. 3, 14.1 (2001).
[CrossRef]

Lin, C.

Löffler, A.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, L. V. K. S. Reitzenstein, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, Nature 432, 197 (2004).
[CrossRef] [PubMed]

Long, R.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefevre-Seguin, New J. Phys. 3, 14.1 (2001).
[CrossRef]

Martin, A.

D. Armani, B. Min, A. Martin, and K. J. Vahala, Appl. Phys. Lett. 85, 5439 (2004).
[CrossRef]

Maruno, T.

Y. Hibino, T. Maruno, and K. Okamoto, NTT Tech. Rev. 13, 4 (2001).

Maune, B.

B. Maune, R. Lawson, C. Gunn, A. Scherer, and L. Dalton, Appl. Phys. Lett. 83, 4689 (2003).
[CrossRef]

Meixner, A. J.

M. Steiner, F. Schleifenbaum, C. Stupperich, A. V. Failla, A. Hartschuh, and A. J. Meixner,ChemPhysChem 6, 2190 (2005).
[CrossRef] [PubMed]

Min, B.

D. Armani, B. Min, A. Martin, and K. J. Vahala, Appl. Phys. Lett. 85, 5439 (2004).
[CrossRef]

Musevic, I.

M. Humar, M. Ravnik, S. Pajk, and I. Musevic, Nat. Photonics 3, 595 (2009).
[CrossRef]

Needham, J.

D. A. Bergstein, E. Ozkumur, A. C. Wu, A. Yalcin, J. Needham, R. Irani, J. Gershoni, B. B. Goldberg, C. DeLisi, M. F. Ruane, and M. S. Ünlü, IEEE J. Sel. Top. Quantum Electron. 14, 131 (2008).
[CrossRef] [PubMed]

Okamoto, K.

Y. Hibino, T. Maruno, and K. Okamoto, NTT Tech. Rev. 13, 4 (2001).

Ozkumur, E.

D. A. Bergstein, E. Ozkumur, A. C. Wu, A. Yalcin, J. Needham, R. Irani, J. Gershoni, B. B. Goldberg, C. DeLisi, M. F. Ruane, and M. S. Ünlü, IEEE J. Sel. Top. Quantum Electron. 14, 131 (2008).
[CrossRef] [PubMed]

Pajk, S.

M. Humar, M. Ravnik, S. Pajk, and I. Musevic, Nat. Photonics 3, 595 (2009).
[CrossRef]

Rafizadeh, D.

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, J. Lightwave Technol. 15, 2154 (1997).
[CrossRef]

Ravnik, M.

M. Humar, M. Ravnik, S. Pajk, and I. Musevic, Nat. Photonics 3, 595 (2009).
[CrossRef]

Reinecke, T. L.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, L. V. K. S. Reitzenstein, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, Nature 432, 197 (2004).
[CrossRef] [PubMed]

Reithmaier, J. P.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, L. V. K. S. Reitzenstein, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, Nature 432, 197 (2004).
[CrossRef] [PubMed]

Reitzenstein, L. V. K. S.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, L. V. K. S. Reitzenstein, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, Nature 432, 197 (2004).
[CrossRef] [PubMed]

Riviere, R.

G. Anetsberger, R. Riviere, A. Schlieer, O. Arcizet, and T. Kippenberg, Nat. Photonics 2, 627 (2008).
[CrossRef]

Ruane, M. F.

D. A. Bergstein, E. Ozkumur, A. C. Wu, A. Yalcin, J. Needham, R. Irani, J. Gershoni, B. B. Goldberg, C. DeLisi, M. F. Ruane, and M. S. Ünlü, IEEE J. Sel. Top. Quantum Electron. 14, 131 (2008).
[CrossRef] [PubMed]

Scherer, A.

B. Maune, R. Lawson, C. Gunn, A. Scherer, and L. Dalton, Appl. Phys. Lett. 83, 4689 (2003).
[CrossRef]

Schleifenbaum, F.

M. Steiner, F. Schleifenbaum, C. Stupperich, A. V. Failla, A. Hartschuh, and A. J. Meixner,ChemPhysChem 6, 2190 (2005).
[CrossRef] [PubMed]

Schlieer, A.

G. Anetsberger, R. Riviere, A. Schlieer, O. Arcizet, and T. Kippenberg, Nat. Photonics 2, 627 (2008).
[CrossRef]

Schwinger, J.

J. Schwinger, M.I.T. Radiation Lab. Report (MIT, 1943), Vol. 43-34.

Sek, G.

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, L. V. K. S. Reitzenstein, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, Nature 432, 197 (2004).
[CrossRef] [PubMed]

Steiner, M.

M. Steiner, F. Schleifenbaum, C. Stupperich, A. V. Failla, A. Hartschuh, and A. J. Meixner,ChemPhysChem 6, 2190 (2005).
[CrossRef] [PubMed]

Strite, S. C.

M. S. Ünlü and S. C. Strite, J. Appl. Phys. 78, 607 (1995).
[CrossRef]

Stupperich, C.

M. Steiner, F. Schleifenbaum, C. Stupperich, A. V. Failla, A. Hartschuh, and A. J. Meixner,ChemPhysChem 6, 2190 (2005).
[CrossRef] [PubMed]

Taflove, A.

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, J. Lightwave Technol. 15, 2154 (1997).
[CrossRef]

Tessler, N.

H. Becker, S. E. Burns, N. Tessler, and R. H. Friend, J. Appl. Phys. 81, 2825 (1997).
[CrossRef]

Ünlü, M. S.

D. A. Bergstein, E. Ozkumur, A. C. Wu, A. Yalcin, J. Needham, R. Irani, J. Gershoni, B. B. Goldberg, C. DeLisi, M. F. Ruane, and M. S. Ünlü, IEEE J. Sel. Top. Quantum Electron. 14, 131 (2008).
[CrossRef] [PubMed]

M. S. Ünlü and S. C. Strite, J. Appl. Phys. 78, 607 (1995).
[CrossRef]

Vahala, K. J.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, Science 317, 783 (2007).
[CrossRef] [PubMed]

D. Armani, B. Min, A. Martin, and K. J. Vahala, Appl. Phys. Lett. 85, 5439 (2004).
[CrossRef]

K. J. Vahala, Nature 424, 839 (2003).
[CrossRef] [PubMed]

Vollmer, F.

F. Vollmer and S. Arnold, Nat. Methods 5, 591 (2008).
[CrossRef] [PubMed]

von Klitzing, W.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefevre-Seguin, New J. Phys. 3, 14.1 (2001).
[CrossRef]

Wang, T.

Wu, A. C.

D. A. Bergstein, E. Ozkumur, A. C. Wu, A. Yalcin, J. Needham, R. Irani, J. Gershoni, B. B. Goldberg, C. DeLisi, M. F. Ruane, and M. S. Ünlü, IEEE J. Sel. Top. Quantum Electron. 14, 131 (2008).
[CrossRef] [PubMed]

Yalcin, A.

D. A. Bergstein, E. Ozkumur, A. C. Wu, A. Yalcin, J. Needham, R. Irani, J. Gershoni, B. B. Goldberg, C. DeLisi, M. F. Ruane, and M. S. Ünlü, IEEE J. Sel. Top. Quantum Electron. 14, 131 (2008).
[CrossRef] [PubMed]

Appl. Phys. Lett. (3)

D. Armani, B. Min, A. Martin, and K. J. Vahala, Appl. Phys. Lett. 85, 5439 (2004).
[CrossRef]

A. Kiraz, A. Kurt, and M. A. Dündar, Appl. Phys. Lett. 102, 093713 (2006).

B. Maune, R. Lawson, C. Gunn, A. Scherer, and L. Dalton, Appl. Phys. Lett. 83, 4689 (2003).
[CrossRef]

ChemPhysChem (1)

M. Steiner, F. Schleifenbaum, C. Stupperich, A. V. Failla, A. Hartschuh, and A. J. Meixner,ChemPhysChem 6, 2190 (2005).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

D. A. Bergstein, E. Ozkumur, A. C. Wu, A. Yalcin, J. Needham, R. Irani, J. Gershoni, B. B. Goldberg, C. DeLisi, M. F. Ruane, and M. S. Ünlü, IEEE J. Sel. Top. Quantum Electron. 14, 131 (2008).
[CrossRef] [PubMed]

J. Appl. Phys. (2)

M. S. Ünlü and S. C. Strite, J. Appl. Phys. 78, 607 (1995).
[CrossRef]

H. Becker, S. E. Burns, N. Tessler, and R. H. Friend, J. Appl. Phys. 81, 2825 (1997).
[CrossRef]

J. Lightwave Technol. (2)

P. Heimala, P. Katila, J. Aarnio, and A. Heinämäki, J. Lightwave Technol. 14, 2260 (1996).
[CrossRef]

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, J. Lightwave Technol. 15, 2154 (1997).
[CrossRef]

Nat. Methods (1)

F. Vollmer and S. Arnold, Nat. Methods 5, 591 (2008).
[CrossRef] [PubMed]

Nat. Photonics (2)

G. Anetsberger, R. Riviere, A. Schlieer, O. Arcizet, and T. Kippenberg, Nat. Photonics 2, 627 (2008).
[CrossRef]

M. Humar, M. Ravnik, S. Pajk, and I. Musevic, Nat. Photonics 3, 595 (2009).
[CrossRef]

Nature (2)

K. J. Vahala, Nature 424, 839 (2003).
[CrossRef] [PubMed]

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, L. V. K. S. Reitzenstein, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, Nature 432, 197 (2004).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Experimental configuration. White light from a fiber-optic illuminator (a) is coupled into a multimode fiber, collimated and then focused by an air objective (b) onto the microcavity (c). The microcavity is raster-scanned across the focus using a x y piezo stage (d). The transmitted light from the microcavity is collected by a high-NA oil objective (e) and directed through a confocal imaging pinhole (f) to a spectrometer (g). The inset shows the structural details of the microcavity.

Fig. 2
Fig. 2

(a) Spectrum of the white-light source employed in the experiments. (b) Cavity transmission spectrum in the absence [black (blue online)] and presence [gray (red online)] of a single gold nanoparticle. The solid lines are Lorentzian fits to the experimental data (black dots).

Fig. 3
Fig. 3

Spatial maps of the center frequencies of m = 2 and m = 3 cavity modes. (a) m = 3 mode, (b) m = 2 mode. Scalebar: 2 μ m .

Equations (3)

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ω ω m ω m = V [ E o * ϵ o Δ ϵ ( r ) E + H o * μ o Δ μ ( r ) H ] d V V [ ϵ o ϵ ( r ) E o * E + μ o μ ( r ) H o * H ] d V .
λ λ m λ = Δ ϵ Δ V V ,
n pol L cos θ = λ 2 [ m i Δ ϕ i ( d i , θ , λ ) 2 π ] ,

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