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 A 93, 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. Lefévre-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. SiegmanLasers (University Science Books, (1986).

2007 (1)

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 A 93, 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. Lefévre-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]

Conti, G.Nunzi

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]

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.

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. Lefévre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev A  54, R1777–R1780 (1996).
[CrossRef]

Haroche, S.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefévre-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 A 93, 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.

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.

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]

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 A 93, 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]

Lefévre-Seguin, V.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefévre-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 A 93, 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.

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]

Raimond, J. M.

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

Siegman, A. E.

A. E. SiegmanLasers (University Science Books, (1986).

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 A 93, 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. Lefévre-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.

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)

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 (1)

V. Sandoghdar, F. Treussart, J. Hare, V. Lefévre-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 A 93, 1–7 (2001).
[CrossRef]

Other (2)

A. E. SiegmanLasers (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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