Abstract

We show that time-independent scattering coefficients calculated from the standard extrapolation of Mie theory to the gain regime have physical meaning up to the laser threshold. The theoretical width of a resonance decreases linearly with increasing gain and becomes zero at the laser threshold. We performed experiments on dielectric microspheres with gain, trapped with optical tweezers. The width of the mode was measured to narrow as a function of the gain up to the lasing threshold, confirming both the validity of the extrapolation of Mie theory to the gain regime below threshold and our interpretation of its point of divergence as the laser threshold.

© 2006 Optical Society of America

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  1. M.Kerker, ed., Selected Papers on Light Scattering, Parts 1 and 2 (SPIE, 1988).
  2. H.-B. Lin, J. D. Eversole, and A. J. Campillo, J. Opt. Soc. Am. B 9, 43 (1992).
    [CrossRef]
  3. M. Kuwata-Gonokami, K. Takeda, H. Yasuda, and K. Ema, Jpn. J. Appl. Phys. 31, L99 (1992).
    [CrossRef]
  4. V. Sandoghdar, F. Treussart, J. Hare, V. Lefévre-Seguin, J.-M. Raimond, and S. Haroche, Phys. Rev. A 54, R1777 (1996).
    [CrossRef] [PubMed]
  5. A. J. Campillo, J. D. Eversole, and H.-B. Lin, Phys. Rev. Lett. 67, 437 (1991).
    [CrossRef] [PubMed]
  6. M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, Opt. Lett. 25, 1430 (2000).
    [CrossRef]
  7. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  8. H. Chew, J. Chem. Phys. 87, 1355 (1987).
    [CrossRef]
  9. H. Chew, Phys. Rev. A 38, 3410 (1988).
    [CrossRef] [PubMed]
  10. J. Ripoll, C. M. Soukoulis, and E. N. Economou, J. Opt. Soc. Am. A 21, 141 (2004).
    [CrossRef]
  11. V. V. Datsyuk, J. Opt. Soc. Am. B 19, 142 (2002).
    [CrossRef]
  12. B. A. Hunter, M. A. Box, and B. Maier, J. Opt. Soc. Am. A 5, 1281 (1988).
    [CrossRef]
  13. M. Kerker, Appl. Opt. 18, 1180 (1979).
    [CrossRef] [PubMed]
  14. N. G. Alexoupoulos and N. K. Uzunoglu, Appl. Opt. 17, 235 (1978).
    [CrossRef]
  15. M. Kerker, Appl. Opt. 17, 3337 (1978).
    [CrossRef] [PubMed]
  16. K. J. Vahala, Nature 424, 839 (2003).
    [CrossRef] [PubMed]
  17. J. Heinrichs, Phys. Rev. B 56, 8674 (1997).
    [CrossRef]
  18. C. W. J. Beenakker, J. C. J. Paaschens, and P. W. Brouwer, Phys. Rev. Lett. 76, 1368 (1996).
    [CrossRef] [PubMed]
  19. H. M. Lai, C. C. Lam, P. T. Leung, and K. Young, J. Opt. Soc. Am. B 8, 1962 (1991).
    [CrossRef]
  20. H. M. Nussenzveig, J. Math. Phys. 10, 82 (1969).
    [CrossRef]
  21. A. Bott and W. Zdunkowski, J. Opt. Soc. Am. A 4, 1361 (1987).
    [CrossRef]
  22. X. Ma, J. Q. Lu, R. Scott Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, Phys. Med. Biol. 48, 4165 (2003).
    [CrossRef]
  23. A. E. Siegman, Lasers (University Science, 1986).

2004

J. Ripoll, C. M. Soukoulis, and E. N. Economou, J. Opt. Soc. Am. A 21, 141 (2004).
[CrossRef]

2003

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

X. Ma, J. Q. Lu, R. Scott Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

2002

2000

1997

J. Heinrichs, Phys. Rev. B 56, 8674 (1997).
[CrossRef]

1996

C. W. J. Beenakker, J. C. J. Paaschens, and P. W. Brouwer, Phys. Rev. Lett. 76, 1368 (1996).
[CrossRef] [PubMed]

V. Sandoghdar, F. Treussart, J. Hare, V. Lefévre-Seguin, J.-M. Raimond, and S. Haroche, Phys. Rev. A 54, R1777 (1996).
[CrossRef] [PubMed]

1992

H.-B. Lin, J. D. Eversole, and A. J. Campillo, J. Opt. Soc. Am. B 9, 43 (1992).
[CrossRef]

M. Kuwata-Gonokami, K. Takeda, H. Yasuda, and K. Ema, Jpn. J. Appl. Phys. 31, L99 (1992).
[CrossRef]

1991

A. J. Campillo, J. D. Eversole, and H.-B. Lin, Phys. Rev. Lett. 67, 437 (1991).
[CrossRef] [PubMed]

H. M. Lai, C. C. Lam, P. T. Leung, and K. Young, J. Opt. Soc. Am. B 8, 1962 (1991).
[CrossRef]

1988

1987

1979

1978

1969

H. M. Nussenzveig, J. Math. Phys. 10, 82 (1969).
[CrossRef]

Alexoupoulos, N. G.

Beenakker, C. W. J.

C. W. J. Beenakker, J. C. J. Paaschens, and P. W. Brouwer, Phys. Rev. Lett. 76, 1368 (1996).
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Bott, A.

Box, M. A.

Brouwer, P. W.

C. W. J. Beenakker, J. C. J. Paaschens, and P. W. Brouwer, Phys. Rev. Lett. 76, 1368 (1996).
[CrossRef] [PubMed]

Cai, M.

Campillo, A. J.

H.-B. Lin, J. D. Eversole, and A. J. Campillo, J. Opt. Soc. Am. B 9, 43 (1992).
[CrossRef]

A. J. Campillo, J. D. Eversole, and H.-B. Lin, Phys. Rev. Lett. 67, 437 (1991).
[CrossRef] [PubMed]

Chew, H.

H. Chew, Phys. Rev. A 38, 3410 (1988).
[CrossRef] [PubMed]

H. Chew, J. Chem. Phys. 87, 1355 (1987).
[CrossRef]

Datsyuk, V. V.

Economou, E. N.

J. Ripoll, C. M. Soukoulis, and E. N. Economou, J. Opt. Soc. Am. A 21, 141 (2004).
[CrossRef]

Ema, K.

M. Kuwata-Gonokami, K. Takeda, H. Yasuda, and K. Ema, Jpn. J. Appl. Phys. 31, L99 (1992).
[CrossRef]

Eversole, J. D.

H.-B. Lin, J. D. Eversole, and A. J. Campillo, J. Opt. Soc. Am. B 9, 43 (1992).
[CrossRef]

A. J. Campillo, J. D. Eversole, and H.-B. Lin, Phys. Rev. Lett. 67, 437 (1991).
[CrossRef] [PubMed]

Hare, J.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefévre-Seguin, J.-M. Raimond, and S. Haroche, Phys. Rev. A 54, R1777 (1996).
[CrossRef] [PubMed]

Haroche, S.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefévre-Seguin, J.-M. Raimond, and S. Haroche, Phys. Rev. A 54, R1777 (1996).
[CrossRef] [PubMed]

Heinrichs, J.

J. Heinrichs, Phys. Rev. B 56, 8674 (1997).
[CrossRef]

Hu, X.-H.

X. Ma, J. Q. Lu, R. Scott Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Hunter, B. A.

Jacobs, K. M.

X. Ma, J. Q. Lu, R. Scott Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

Kerker, M.

Kuwata-Gonokami, M.

M. Kuwata-Gonokami, K. Takeda, H. Yasuda, and K. Ema, Jpn. J. Appl. Phys. 31, L99 (1992).
[CrossRef]

Lai, H. M.

Lam, C. C.

Lefévre-Seguin, V.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefévre-Seguin, J.-M. Raimond, and S. Haroche, Phys. Rev. A 54, R1777 (1996).
[CrossRef] [PubMed]

Leung, P. T.

Lin, H.-B.

H.-B. Lin, J. D. Eversole, and A. J. Campillo, J. Opt. Soc. Am. B 9, 43 (1992).
[CrossRef]

A. J. Campillo, J. D. Eversole, and H.-B. Lin, Phys. Rev. Lett. 67, 437 (1991).
[CrossRef] [PubMed]

Lu, J. Q.

X. Ma, J. Q. Lu, R. Scott Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

Ma, X.

X. Ma, J. Q. Lu, R. Scott Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

Maier, B.

Nussenzveig, H. M.

H. M. Nussenzveig, J. Math. Phys. 10, 82 (1969).
[CrossRef]

Paaschens, J. C. J.

C. W. J. Beenakker, J. C. J. Paaschens, and P. W. Brouwer, Phys. Rev. Lett. 76, 1368 (1996).
[CrossRef] [PubMed]

Painter, O.

Raimond, J.-M.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefévre-Seguin, J.-M. Raimond, and S. Haroche, Phys. Rev. A 54, R1777 (1996).
[CrossRef] [PubMed]

Ripoll, J.

J. Ripoll, C. M. Soukoulis, and E. N. Economou, J. Opt. Soc. Am. A 21, 141 (2004).
[CrossRef]

Sandoghdar, V.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefévre-Seguin, J.-M. Raimond, and S. Haroche, Phys. Rev. A 54, R1777 (1996).
[CrossRef] [PubMed]

Scott Brock, R.

X. Ma, J. Q. Lu, R. Scott Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

Sercel, P. C.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Soukoulis, C. M.

J. Ripoll, C. M. Soukoulis, and E. N. Economou, J. Opt. Soc. Am. A 21, 141 (2004).
[CrossRef]

Takeda, K.

M. Kuwata-Gonokami, K. Takeda, H. Yasuda, and K. Ema, Jpn. J. Appl. Phys. 31, L99 (1992).
[CrossRef]

Treussart, F.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefévre-Seguin, J.-M. Raimond, and S. Haroche, Phys. Rev. A 54, R1777 (1996).
[CrossRef] [PubMed]

Uzunoglu, N. K.

Vahala, K. J.

Yang, P.

X. Ma, J. Q. Lu, R. Scott Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

Yasuda, H.

M. Kuwata-Gonokami, K. Takeda, H. Yasuda, and K. Ema, Jpn. J. Appl. Phys. 31, L99 (1992).
[CrossRef]

Young, K.

Zdunkowski, W.

Appl. Opt.

J. Chem. Phys.

H. Chew, J. Chem. Phys. 87, 1355 (1987).
[CrossRef]

J. Math. Phys.

H. M. Nussenzveig, J. Math. Phys. 10, 82 (1969).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys.

M. Kuwata-Gonokami, K. Takeda, H. Yasuda, and K. Ema, Jpn. J. Appl. Phys. 31, L99 (1992).
[CrossRef]

Nature

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

Opt. Lett.

Phys. Med. Biol.

X. Ma, J. Q. Lu, R. Scott Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, Phys. Med. Biol. 48, 4165 (2003).
[CrossRef]

Phys. Rev. A

H. Chew, Phys. Rev. A 38, 3410 (1988).
[CrossRef] [PubMed]

V. Sandoghdar, F. Treussart, J. Hare, V. Lefévre-Seguin, J.-M. Raimond, and S. Haroche, Phys. Rev. A 54, R1777 (1996).
[CrossRef] [PubMed]

Phys. Rev. B

J. Heinrichs, Phys. Rev. B 56, 8674 (1997).
[CrossRef]

Phys. Rev. Lett.

C. W. J. Beenakker, J. C. J. Paaschens, and P. W. Brouwer, Phys. Rev. Lett. 76, 1368 (1996).
[CrossRef] [PubMed]

A. J. Campillo, J. D. Eversole, and H.-B. Lin, Phys. Rev. Lett. 67, 437 (1991).
[CrossRef] [PubMed]

Other

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

M.Kerker, ed., Selected Papers on Light Scattering, Parts 1 and 2 (SPIE, 1988).

A. E. Siegman, Lasers (University Science, 1986).

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Figures (3)

Fig. 1
Fig. 1

(a) Experimental emission spectrum of a microsphere excited with a low pumping power of 1 nJ pulse (normalized to the peak intensity of the TE 95 mode). The dashed curve shows a theoretical fit of Mie theory. (b) Experimental spectrum of the same sphere measured at 100 nJ pulse , above threshold.

Fig. 2
Fig. 2

Measured integrated intensity in the TE 95 mode versus luminance of a sphere with gain (the lines are a guide to the eye). The laser threshold is clearly visible.

Fig. 3
Fig. 3

Width of the TE 95 mode versus the gain index. The line represents the calculated width from Mie theory. The squares are data points from fits to the measured spectra. The error bars for the width are taken from the fits; the error bars for the gain index are taken from cw experiments. The gray area indicates the nonphysical area. Above the laser threshold the measured data points are not presented, as the gain saturates in the experiment.

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