Abstract

Evanescent coupling of light from glass channel waveguides into the whispering gallery modes of glass microspheres of radius 15 μm and 100 μm is studied experimentally at wavelengths near 1550 nm. Fitting the positions, widths and heights of resonances in the experimental spectra to the characteristic equation for microsphere modes and to universal coupled microresonator theory, we establish sphere radius and index, identify mode numbers, and determine losses. The results provide detailed information for the design of optical devices incorporating microsphere resonators in planar lightwave circuits.

© 2008 Optical Society of America

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References

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  1. J. P. Laine, B. E. Little, D. R. Lim, H. C. Tapalian, L. C. Kimerling and H. A. Haus, "Microsphere resonator mode characterization by pedestal anti-resonant reflecting waveguide coupler," IEEE Photon. Technol. Lett. 12, 1004-1006 (2000).
    [CrossRef]
  2. B. E. Little, J. P. Laine, D. R. Lim, H. A. Haus, L. C. Kimerling and S. T. Chu, "Pedestal antiresonant reflecting waveguides for robust coupling to microsphere resonators and for microphotonic circuits," Opt. Lett. 25, 73-75 (2000).
    [CrossRef]
  3. J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, "Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper," Opt. Lett. 22, 1129-1131 (1997).
    [CrossRef] [PubMed]
  4. J. P. Laine, C. Tapalian, B. Little, and H. Haus, "Acceleration sensor based on high-Q optical microsphere resonator and pedestal antiresonant reflecting waveguide coupler," Sens. Actuators A93, 1-7 (2001).
    [CrossRef]
  5. B. E. Little, J. P. Laine, and H. A. Haus, "Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators," J. Lightwave Technol. 17, 704-715 (1999).
    [CrossRef]
  6. A. W. Snyder and A. Ankiewicz, "Optical fiber couplers- optimum solution for unequal cores," J. Lightwave Technol. 6, 463-474 (1988).
    [CrossRef]
  7. A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
    [CrossRef]
  8. M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, "Fiber-coupled microsphere laser," Opt. Lett. 25, 1430-1432 (2000).
    [CrossRef]
  9. V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A 54, R1777-R1780 (1996).
    [CrossRef]
  10. G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006).
    [CrossRef]
  11. http://www.Schott.com.
  12. G. R. Elliott, D.W. Hewak, G. S. Murugan, and J. S. Wilkinson, "Chalcogenide glass microspheres; their production, characterization and potential," Opt. Express 15, 17542-17553 (2007).
    [CrossRef] [PubMed]
  13. A. E. Siegman, Lasers (University Science Books, 1986).

2007 (1)

G. R. Elliott, D.W. Hewak, G. S. Murugan, and J. S. Wilkinson, "Chalcogenide glass microspheres; their production, characterization and potential," Opt. Express 15, 17542-17553 (2007).
[CrossRef] [PubMed]

2006 (1)

G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006).
[CrossRef]

2001 (1)

J. P. Laine, C. Tapalian, B. Little, and H. Haus, "Acceleration sensor based on high-Q optical microsphere resonator and pedestal antiresonant reflecting waveguide coupler," Sens. Actuators A93, 1-7 (2001).
[CrossRef]

2000 (4)

J. P. Laine, B. E. Little, D. R. Lim, H. C. Tapalian, L. C. Kimerling and H. A. Haus, "Microsphere resonator mode characterization by pedestal anti-resonant reflecting waveguide coupler," IEEE Photon. Technol. Lett. 12, 1004-1006 (2000).
[CrossRef]

B. E. Little, J. P. Laine, D. R. Lim, H. A. Haus, L. C. Kimerling and S. T. Chu, "Pedestal antiresonant reflecting waveguides for robust coupling to microsphere resonators and for microphotonic circuits," Opt. Lett. 25, 73-75 (2000).
[CrossRef]

A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
[CrossRef]

M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, "Fiber-coupled microsphere laser," Opt. Lett. 25, 1430-1432 (2000).
[CrossRef]

1999 (1)

B. E. Little, J. P. Laine, and H. A. Haus, "Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators," J. Lightwave Technol. 17, 704-715 (1999).
[CrossRef]

1997 (1)

J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, "Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper," Opt. Lett. 22, 1129-1131 (1997).
[CrossRef] [PubMed]

1996 (1)

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A 54, R1777-R1780 (1996).
[CrossRef]

1988 (1)

A. W. Snyder and A. Ankiewicz, "Optical fiber couplers- optimum solution for unequal cores," J. Lightwave Technol. 6, 463-474 (1988).
[CrossRef]

Ankiewicz, A.

A. W. Snyder and A. Ankiewicz, "Optical fiber couplers- optimum solution for unequal cores," J. Lightwave Technol. 6, 463-474 (1988).
[CrossRef]

Berneschi, S.

G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006).
[CrossRef]

Birks, T. A.

J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, "Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper," Opt. Lett. 22, 1129-1131 (1997).
[CrossRef] [PubMed]

Brenci, M.

G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006).
[CrossRef]

Cai, M.

M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, "Fiber-coupled microsphere laser," Opt. Lett. 25, 1430-1432 (2000).
[CrossRef]

Cheung, G.

J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, "Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper," Opt. Lett. 22, 1129-1131 (1997).
[CrossRef] [PubMed]

Chiasera, A.

G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006).
[CrossRef]

Chu, S. T.

B. E. Little, J. P. Laine, D. R. Lim, H. A. Haus, L. C. Kimerling and S. T. Chu, "Pedestal antiresonant reflecting waveguides for robust coupling to microsphere resonators and for microphotonic circuits," Opt. Lett. 25, 73-75 (2000).
[CrossRef]

Dumeige, Y.

G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006).
[CrossRef]

Elliott, G. R.

G. R. Elliott, D.W. Hewak, G. S. Murugan, and J. S. Wilkinson, "Chalcogenide glass microspheres; their production, characterization and potential," Opt. Express 15, 17542-17553 (2007).
[CrossRef] [PubMed]

Feron, P.

G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006).
[CrossRef]

Ferrari, M.

G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006).
[CrossRef]

Ghisa, L.

G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006).
[CrossRef]

Hare, J.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A 54, R1777-R1780 (1996).
[CrossRef]

Haus, H.

J. P. Laine, C. Tapalian, B. Little, and H. Haus, "Acceleration sensor based on high-Q optical microsphere resonator and pedestal antiresonant reflecting waveguide coupler," Sens. Actuators A93, 1-7 (2001).
[CrossRef]

Haus, H. A.

B. E. Little, J. P. Laine, D. R. Lim, H. A. Haus, L. C. Kimerling and S. T. Chu, "Pedestal antiresonant reflecting waveguides for robust coupling to microsphere resonators and for microphotonic circuits," Opt. Lett. 25, 73-75 (2000).
[CrossRef]

J. P. Laine, B. E. Little, D. R. Lim, H. C. Tapalian, L. C. Kimerling and H. A. Haus, "Microsphere resonator mode characterization by pedestal anti-resonant reflecting waveguide coupler," IEEE Photon. Technol. Lett. 12, 1004-1006 (2000).
[CrossRef]

B. E. Little, J. P. Laine, and H. A. Haus, "Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators," J. Lightwave Technol. 17, 704-715 (1999).
[CrossRef]

Hewak, D.W.

G. R. Elliott, D.W. Hewak, G. S. Murugan, and J. S. Wilkinson, "Chalcogenide glass microspheres; their production, characterization and potential," Opt. Express 15, 17542-17553 (2007).
[CrossRef] [PubMed]

Jacques, F.

J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, "Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper," Opt. Lett. 22, 1129-1131 (1997).
[CrossRef] [PubMed]

Kimerling, L. C.

J. P. Laine, B. E. Little, D. R. Lim, H. C. Tapalian, L. C. Kimerling and H. A. Haus, "Microsphere resonator mode characterization by pedestal anti-resonant reflecting waveguide coupler," IEEE Photon. Technol. Lett. 12, 1004-1006 (2000).
[CrossRef]

B. E. Little, J. P. Laine, D. R. Lim, H. A. Haus, L. C. Kimerling and S. T. Chu, "Pedestal antiresonant reflecting waveguides for robust coupling to microsphere resonators and for microphotonic circuits," Opt. Lett. 25, 73-75 (2000).
[CrossRef]

Knight, J. C.

J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, "Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper," Opt. Lett. 22, 1129-1131 (1997).
[CrossRef] [PubMed]

Laine, J. P.

J. P. Laine, C. Tapalian, B. Little, and H. Haus, "Acceleration sensor based on high-Q optical microsphere resonator and pedestal antiresonant reflecting waveguide coupler," Sens. Actuators A93, 1-7 (2001).
[CrossRef]

B. E. Little, J. P. Laine, D. R. Lim, H. A. Haus, L. C. Kimerling and S. T. Chu, "Pedestal antiresonant reflecting waveguides for robust coupling to microsphere resonators and for microphotonic circuits," Opt. Lett. 25, 73-75 (2000).
[CrossRef]

J. P. Laine, B. E. Little, D. R. Lim, H. C. Tapalian, L. C. Kimerling and H. A. Haus, "Microsphere resonator mode characterization by pedestal anti-resonant reflecting waveguide coupler," IEEE Photon. Technol. Lett. 12, 1004-1006 (2000).
[CrossRef]

B. E. Little, J. P. Laine, and H. A. Haus, "Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators," J. Lightwave Technol. 17, 704-715 (1999).
[CrossRef]

Lef, V.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A 54, R1777-R1780 (1996).
[CrossRef]

Lim, D. R.

J. P. Laine, B. E. Little, D. R. Lim, H. C. Tapalian, L. C. Kimerling and H. A. Haus, "Microsphere resonator mode characterization by pedestal anti-resonant reflecting waveguide coupler," IEEE Photon. Technol. Lett. 12, 1004-1006 (2000).
[CrossRef]

B. E. Little, J. P. Laine, D. R. Lim, H. A. Haus, L. C. Kimerling and S. T. Chu, "Pedestal antiresonant reflecting waveguides for robust coupling to microsphere resonators and for microphotonic circuits," Opt. Lett. 25, 73-75 (2000).
[CrossRef]

Little, B.

J. P. Laine, C. Tapalian, B. Little, and H. Haus, "Acceleration sensor based on high-Q optical microsphere resonator and pedestal antiresonant reflecting waveguide coupler," Sens. Actuators A93, 1-7 (2001).
[CrossRef]

Little, B. E.

B. E. Little, J. P. Laine, D. R. Lim, H. A. Haus, L. C. Kimerling and S. T. Chu, "Pedestal antiresonant reflecting waveguides for robust coupling to microsphere resonators and for microphotonic circuits," Opt. Lett. 25, 73-75 (2000).
[CrossRef]

J. P. Laine, B. E. Little, D. R. Lim, H. C. Tapalian, L. C. Kimerling and H. A. Haus, "Microsphere resonator mode characterization by pedestal anti-resonant reflecting waveguide coupler," IEEE Photon. Technol. Lett. 12, 1004-1006 (2000).
[CrossRef]

B. E. Little, J. P. Laine, and H. A. Haus, "Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators," J. Lightwave Technol. 17, 704-715 (1999).
[CrossRef]

Murugan, G. S.

G. R. Elliott, D.W. Hewak, G. S. Murugan, and J. S. Wilkinson, "Chalcogenide glass microspheres; their production, characterization and potential," Opt. Express 15, 17542-17553 (2007).
[CrossRef] [PubMed]

Nunzi Conti, G.

G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006).
[CrossRef]

Painter, O.

M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, "Fiber-coupled microsphere laser," Opt. Lett. 25, 1430-1432 (2000).
[CrossRef]

Pelli, S.

G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006).
[CrossRef]

Righini, G. C.

G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006).
[CrossRef]

Sandoghdar, V.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A 54, R1777-R1780 (1996).
[CrossRef]

Sebastiani, S.

G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006).
[CrossRef]

Sercel, P. C.

M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, "Fiber-coupled microsphere laser," Opt. Lett. 25, 1430-1432 (2000).
[CrossRef]

Snyder, A. W.

A. W. Snyder and A. Ankiewicz, "Optical fiber couplers- optimum solution for unequal cores," J. Lightwave Technol. 6, 463-474 (1988).
[CrossRef]

Tapalian, C.

J. P. Laine, C. Tapalian, B. Little, and H. Haus, "Acceleration sensor based on high-Q optical microsphere resonator and pedestal antiresonant reflecting waveguide coupler," Sens. Actuators A93, 1-7 (2001).
[CrossRef]

Tapalian, H. C.

J. P. Laine, B. E. Little, D. R. Lim, H. C. Tapalian, L. C. Kimerling and H. A. Haus, "Microsphere resonator mode characterization by pedestal anti-resonant reflecting waveguide coupler," IEEE Photon. Technol. Lett. 12, 1004-1006 (2000).
[CrossRef]

Treussart, F.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A 54, R1777-R1780 (1996).
[CrossRef]

Vahala, K. J.

M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, "Fiber-coupled microsphere laser," Opt. Lett. 25, 1430-1432 (2000).
[CrossRef]

Wilkinson, J. S.

G. R. Elliott, D.W. Hewak, G. S. Murugan, and J. S. Wilkinson, "Chalcogenide glass microspheres; their production, characterization and potential," Opt. Express 15, 17542-17553 (2007).
[CrossRef] [PubMed]

Yariv, A.

A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
[CrossRef]

Electron. Lett. (1)

A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. P. Laine, B. E. Little, D. R. Lim, H. C. Tapalian, L. C. Kimerling and H. A. Haus, "Microsphere resonator mode characterization by pedestal anti-resonant reflecting waveguide coupler," IEEE Photon. Technol. Lett. 12, 1004-1006 (2000).
[CrossRef]

J. Lightwave Technol. (2)

B. E. Little, J. P. Laine, and H. A. Haus, "Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators," J. Lightwave Technol. 17, 704-715 (1999).
[CrossRef]

A. W. Snyder and A. Ankiewicz, "Optical fiber couplers- optimum solution for unequal cores," J. Lightwave Technol. 6, 463-474 (1988).
[CrossRef]

J. Non-Cryst. Solids (1)

G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006).
[CrossRef]

Opt. Express (1)

G. R. Elliott, D.W. Hewak, G. S. Murugan, and J. S. Wilkinson, "Chalcogenide glass microspheres; their production, characterization and potential," Opt. Express 15, 17542-17553 (2007).
[CrossRef] [PubMed]

Opt. Lett. (3)

M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, "Fiber-coupled microsphere laser," Opt. Lett. 25, 1430-1432 (2000).
[CrossRef]

B. E. Little, J. P. Laine, D. R. Lim, H. A. Haus, L. C. Kimerling and S. T. Chu, "Pedestal antiresonant reflecting waveguides for robust coupling to microsphere resonators and for microphotonic circuits," Opt. Lett. 25, 73-75 (2000).
[CrossRef]

J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, "Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper," Opt. Lett. 22, 1129-1131 (1997).
[CrossRef] [PubMed]

Phys. Rev. A (1)

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A 54, R1777-R1780 (1996).
[CrossRef]

Sens. Actuators (1)

J. P. Laine, C. Tapalian, B. Little, and H. Haus, "Acceleration sensor based on high-Q optical microsphere resonator and pedestal antiresonant reflecting waveguide coupler," Sens. Actuators A93, 1-7 (2001).
[CrossRef]

Other (2)

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

http://www.Schott.com.

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Figures (9)

Fig. 1.
Fig. 1.

Coupling configuration

Fig. 2.
Fig. 2.

WGM radial field distributions

Fig. 3.
Fig. 3.

WGM and waveguide fields

Fig. 4.
Fig. 4.

Coupling factor vs separation

Fig. 5.
Fig. 5.

Circulating power vs wavelength

Fig. 6.
Fig. 6.

Circulating power vs coupling factor

Fig. 7.
Fig. 7.

Measurement apparatus

Fig. 8.
Fig. 8.

Measured scattered power for 15 μm BK7 microspheres

Fig. 9.
Fig. 9.

Resonance spectrum for the 100 μm microsphere

Tables (2)

Tables Icon

Table 1. Experimental and theoretical resonant wavelengths and assigned mode numbers l,n for the 15 μm sphere

Tables Icon

Table 2. Experimental and theoretical resonant wavelengths and assigned mode numbers l,n for the 100 μm sphere

Equations (10)

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

( η s α s + l R o ) jl ( kn s R o ) = kn s j l + 1 ( kn s R o ) ,
η s = { 1 TE mode n s 2 n o 2 TM mode ,
α s = [ β l 2 k 2 n o 2 ] 1 2 ,
β l = [ l ( l + 1 ) ] 1 2 R o ,
k = 2 π λ .
κ 2 = K o 2 2 π R o γ wg exp ( R o Δ β 2 γ wg ) ,
K o = π n s λ sphere ( n s 2 n o 2 ) E s E w ( D teflon ) dxdy ,
φ = 4 π 2 n eff R o λ
E r E i 2 = a 2 a 1 2 = α 2 κ 2 1 2 α [ 1 κ 2 ] 1 2 cos ( φ + φ t ) + α 2 ( 1 κ 2 ) ,
FSR = λ 2 2 π n eff R o .

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