Abstract

A SNAP (Surface Nanoscale Axial Photonics) device consists of an optical fiber with introduced nanoscale effective radius variation, which is coupled to transverse input/output waveguides. The input waveguides excite whispering gallery modes circulating near the fiber surface and slowly propagating along the fiber axis. In this paper, the theory of SNAP devices is developed and applied to the analysis of transmission amplitudes of simplest SNAP models exhibiting a variety of asymmetric Fano resonances and also to the experimental characterization of a SNAP bottle microresonator and to a chain of 10 coupled microresonators. Excellent agreement between the theory and the experiment is demonstrated.

© 2012 OSA

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References

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  1. M. Sumetsky, “Localization of light in an optical fiber with nanoscale radius variation,” in CLEO/Europe and EQEC 2011 Conference Digest, postdeadline paper PDA_8.
  2. M. Sumetsky and J. M. Fini, “Surface nanoscale axial photonics,” Opt. Express 19(27), 26470–26485 (2011).
    [CrossRef] [PubMed]
  3. M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, E. M. Monberg, and T. F. Taunay, “Surface nanoscale axial photonics: robust fabrication of high-quality-factor microresonators,” Opt. Lett. 36(24), 4824–4826 (2011).
    [CrossRef] [PubMed]
  4. M. Sumetsky, K. Abedin, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, E. Monberg, and E. M. Monberg, “Coupled high Q-factor surface nanoscale axial photonics (SNAP) microresonators,” Opt. Lett. 37(6), 990–992 (2012).
    [CrossRef] [PubMed]
  5. M. Wilson, “Optical fiber microcavities reach angstrom-scale precision,” Phys. Today 65(2), 14–16 (2012).
    [CrossRef]
  6. M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, X. Liu, E. M. Monberg, and T. F. Taunay, “Photo-induced SNAP: fabrication, trimming, and tuning of microresonator chains,” Opt. Express 20(10), 10684–10691 (2012).
    [CrossRef] [PubMed]
  7. F. Luan, E. Magi, T. Gong, I. Kabakova, and B. J. Eggleton, “Photoinduced whispering gallery mode microcavity resonator in a chalcogenide microfiber,” Opt. Lett. 36(24), 4761–4763 (2011).
    [CrossRef] [PubMed]
  8. H. G. Limberger, P.-Y. Fonjallaz, R. P. Salathé, and F. Cochet, “Compaction‐ and photoelastic‐induced index changes in fiber Bragg gratings,” Appl. Phys. Lett. 68(22), 3069–3071 (1996).
    [CrossRef]
  9. A. D. Yablon, M. F. Yan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84(1), 19–21 (2004).
    [CrossRef]
  10. A. D. Stone and A. Szafer, “What is measured when you measure a resistance?-The Landauer formula revisited,” IBM J. Res. Develop. 32(3), 384–413 (1988).
    [CrossRef]
  11. H. U. Baranger, R. A. Jalabert, and A. D. Stone, “Quantum-chaotic scattering effects in semiconductor microstructures,” Chaos 3(4), 665–682 (1993).
    [CrossRef] [PubMed]
  12. S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, UK, 1997).
  13. F.-M. Dittes, “The decay of quantum systems with a small number of open channels,” Phys. Rep. 339(4), 215–316 (2000).
    [CrossRef]
  14. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. Khan, C. Manolatou, and H. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59(24), 15882–15892 (1999).
    [CrossRef]
  15. Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(5), 7389–7404 (2000).
    [CrossRef] [PubMed]
  16. M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express 11(4), 381–391 (2003).
    [CrossRef] [PubMed]
  17. S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics (American Mathematical Society, Providence, 2004).
  18. Yu. N. Demkov and V. N. Ostrovskii, Zero-Range Potentials and their Applications in Atomic Physics (Plenum, New York, 1988).
  19. M. Sumetsky, “Whispering-gallery-bottle microcavities: the three-dimensional etalon,” Opt. Lett. 29(1), 8–10 (2004).
    [CrossRef] [PubMed]
  20. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
    [CrossRef] [PubMed]
  21. A. E. Miroshnichenko, S. Flach, and Yu. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
    [CrossRef]
  22. T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. 12(2), 182–183 (2000).
    [CrossRef]
  23. M. Sumetsky and Y. Dulashko, “Radius variation of optical fibers with angstrom accuracy,” Opt. Lett. 35(23), 4006–4008 (2010).
    [CrossRef] [PubMed]
  24. J. Ziman, Elements of Advanced Quantum Theory (Cambridge University Press, London, 1969).

2012

2011

2010

A. E. Miroshnichenko, S. Flach, and Yu. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[CrossRef]

M. Sumetsky and Y. Dulashko, “Radius variation of optical fibers with angstrom accuracy,” Opt. Lett. 35(23), 4006–4008 (2010).
[CrossRef] [PubMed]

2004

M. Sumetsky, “Whispering-gallery-bottle microcavities: the three-dimensional etalon,” Opt. Lett. 29(1), 8–10 (2004).
[CrossRef] [PubMed]

A. D. Yablon, M. F. Yan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84(1), 19–21 (2004).
[CrossRef]

2003

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[CrossRef] [PubMed]

M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express 11(4), 381–391 (2003).
[CrossRef] [PubMed]

2000

T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. 12(2), 182–183 (2000).
[CrossRef]

F.-M. Dittes, “The decay of quantum systems with a small number of open channels,” Phys. Rep. 339(4), 215–316 (2000).
[CrossRef]

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(5), 7389–7404 (2000).
[CrossRef] [PubMed]

1999

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. Khan, C. Manolatou, and H. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59(24), 15882–15892 (1999).
[CrossRef]

1996

H. G. Limberger, P.-Y. Fonjallaz, R. P. Salathé, and F. Cochet, “Compaction‐ and photoelastic‐induced index changes in fiber Bragg gratings,” Appl. Phys. Lett. 68(22), 3069–3071 (1996).
[CrossRef]

1993

H. U. Baranger, R. A. Jalabert, and A. D. Stone, “Quantum-chaotic scattering effects in semiconductor microstructures,” Chaos 3(4), 665–682 (1993).
[CrossRef] [PubMed]

1988

A. D. Stone and A. Szafer, “What is measured when you measure a resistance?-The Landauer formula revisited,” IBM J. Res. Develop. 32(3), 384–413 (1988).
[CrossRef]

Abedin, K.

Baranger, H. U.

H. U. Baranger, R. A. Jalabert, and A. D. Stone, “Quantum-chaotic scattering effects in semiconductor microstructures,” Chaos 3(4), 665–682 (1993).
[CrossRef] [PubMed]

Birks, T. A.

T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. 12(2), 182–183 (2000).
[CrossRef]

Cochet, F.

H. G. Limberger, P.-Y. Fonjallaz, R. P. Salathé, and F. Cochet, “Compaction‐ and photoelastic‐induced index changes in fiber Bragg gratings,” Appl. Phys. Lett. 68(22), 3069–3071 (1996).
[CrossRef]

DiGiovanni, D. J.

DiMarcello, F. V.

A. D. Yablon, M. F. Yan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84(1), 19–21 (2004).
[CrossRef]

Dimmick, T. E.

T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. 12(2), 182–183 (2000).
[CrossRef]

Dittes, F.-M.

F.-M. Dittes, “The decay of quantum systems with a small number of open channels,” Phys. Rep. 339(4), 215–316 (2000).
[CrossRef]

Dulashko, Y.

Eggleton, B.

Eggleton, B. J.

Fan, S.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. Khan, C. Manolatou, and H. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59(24), 15882–15892 (1999).
[CrossRef]

Fini, J. M.

Flach, S.

A. E. Miroshnichenko, S. Flach, and Yu. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[CrossRef]

Fleming, J. W.

A. D. Yablon, M. F. Yan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84(1), 19–21 (2004).
[CrossRef]

Fonjallaz, P.-Y.

H. G. Limberger, P.-Y. Fonjallaz, R. P. Salathé, and F. Cochet, “Compaction‐ and photoelastic‐induced index changes in fiber Bragg gratings,” Appl. Phys. Lett. 68(22), 3069–3071 (1996).
[CrossRef]

Gong, T.

Haus, H.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. Khan, C. Manolatou, and H. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59(24), 15882–15892 (1999).
[CrossRef]

Jalabert, R. A.

H. U. Baranger, R. A. Jalabert, and A. D. Stone, “Quantum-chaotic scattering effects in semiconductor microstructures,” Chaos 3(4), 665–682 (1993).
[CrossRef] [PubMed]

Jasapara, J.

A. D. Yablon, M. F. Yan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84(1), 19–21 (2004).
[CrossRef]

Joannopoulos, J. D.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. Khan, C. Manolatou, and H. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59(24), 15882–15892 (1999).
[CrossRef]

Kabakova, I.

Khan, M.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. Khan, C. Manolatou, and H. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59(24), 15882–15892 (1999).
[CrossRef]

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[CrossRef] [PubMed]

Kivshar, Yu. S.

A. E. Miroshnichenko, S. Flach, and Yu. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[CrossRef]

Knight, J. C.

T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. 12(2), 182–183 (2000).
[CrossRef]

Lee, R. K.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(5), 7389–7404 (2000).
[CrossRef] [PubMed]

Li, Y.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(5), 7389–7404 (2000).
[CrossRef] [PubMed]

Limberger, H. G.

H. G. Limberger, P.-Y. Fonjallaz, R. P. Salathé, and F. Cochet, “Compaction‐ and photoelastic‐induced index changes in fiber Bragg gratings,” Appl. Phys. Lett. 68(22), 3069–3071 (1996).
[CrossRef]

Lines, M. E.

A. D. Yablon, M. F. Yan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84(1), 19–21 (2004).
[CrossRef]

Liu, X.

Luan, F.

Magi, E.

Manolatou, C.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. Khan, C. Manolatou, and H. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59(24), 15882–15892 (1999).
[CrossRef]

Miroshnichenko, A. E.

A. E. Miroshnichenko, S. Flach, and Yu. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[CrossRef]

Monberg, E.

Monberg, E. M.

Painter, O. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[CrossRef] [PubMed]

Reed, W. A.

A. D. Yablon, M. F. Yan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84(1), 19–21 (2004).
[CrossRef]

Salathé, R. P.

H. G. Limberger, P.-Y. Fonjallaz, R. P. Salathé, and F. Cochet, “Compaction‐ and photoelastic‐induced index changes in fiber Bragg gratings,” Appl. Phys. Lett. 68(22), 3069–3071 (1996).
[CrossRef]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[CrossRef] [PubMed]

Stone, A. D.

H. U. Baranger, R. A. Jalabert, and A. D. Stone, “Quantum-chaotic scattering effects in semiconductor microstructures,” Chaos 3(4), 665–682 (1993).
[CrossRef] [PubMed]

A. D. Stone and A. Szafer, “What is measured when you measure a resistance?-The Landauer formula revisited,” IBM J. Res. Develop. 32(3), 384–413 (1988).
[CrossRef]

Sumetsky, M.

Szafer, A.

A. D. Stone and A. Szafer, “What is measured when you measure a resistance?-The Landauer formula revisited,” IBM J. Res. Develop. 32(3), 384–413 (1988).
[CrossRef]

Taunay, T. F.

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[CrossRef] [PubMed]

Villeneuve, P. R.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. Khan, C. Manolatou, and H. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59(24), 15882–15892 (1999).
[CrossRef]

Wilson, M.

M. Wilson, “Optical fiber microcavities reach angstrom-scale precision,” Phys. Today 65(2), 14–16 (2012).
[CrossRef]

Wisk, P.

A. D. Yablon, M. F. Yan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84(1), 19–21 (2004).
[CrossRef]

Xu, Y.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(5), 7389–7404 (2000).
[CrossRef] [PubMed]

Yablon, A. D.

A. D. Yablon, M. F. Yan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84(1), 19–21 (2004).
[CrossRef]

Yan, M. F.

A. D. Yablon, M. F. Yan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84(1), 19–21 (2004).
[CrossRef]

Yariv, A.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(5), 7389–7404 (2000).
[CrossRef] [PubMed]

Appl. Phys. Lett.

H. G. Limberger, P.-Y. Fonjallaz, R. P. Salathé, and F. Cochet, “Compaction‐ and photoelastic‐induced index changes in fiber Bragg gratings,” Appl. Phys. Lett. 68(22), 3069–3071 (1996).
[CrossRef]

A. D. Yablon, M. F. Yan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84(1), 19–21 (2004).
[CrossRef]

Chaos

H. U. Baranger, R. A. Jalabert, and A. D. Stone, “Quantum-chaotic scattering effects in semiconductor microstructures,” Chaos 3(4), 665–682 (1993).
[CrossRef] [PubMed]

IBM J. Res. Develop.

A. D. Stone and A. Szafer, “What is measured when you measure a resistance?-The Landauer formula revisited,” IBM J. Res. Develop. 32(3), 384–413 (1988).
[CrossRef]

IEEE Photon. Technol. Lett.

T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. 12(2), 182–183 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

F.-M. Dittes, “The decay of quantum systems with a small number of open channels,” Phys. Rep. 339(4), 215–316 (2000).
[CrossRef]

Phys. Rev. B

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. Khan, C. Manolatou, and H. Haus, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59(24), 15882–15892 (1999).
[CrossRef]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(5), 7389–7404 (2000).
[CrossRef] [PubMed]

Phys. Rev. Lett.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91(4), 043902 (2003).
[CrossRef] [PubMed]

Phys. Today

M. Wilson, “Optical fiber microcavities reach angstrom-scale precision,” Phys. Today 65(2), 14–16 (2012).
[CrossRef]

Rev. Mod. Phys.

A. E. Miroshnichenko, S. Flach, and Yu. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[CrossRef]

Other

S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics (American Mathematical Society, Providence, 2004).

Yu. N. Demkov and V. N. Ostrovskii, Zero-Range Potentials and their Applications in Atomic Physics (Plenum, New York, 1988).

S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, UK, 1997).

J. Ziman, Elements of Advanced Quantum Theory (Cambridge University Press, London, 1969).

M. Sumetsky, “Localization of light in an optical fiber with nanoscale radius variation,” in CLEO/Europe and EQEC 2011 Conference Digest, postdeadline paper PDA_8.

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Figures (7)

Fig. 1
Fig. 1

Illustration of a SNAP device. Red: WGM microresonators. WG1 and WG2 are the waveguide 1 and waveguide 2 coupled to the SF. Inset: magnified nanometer-scale fiber radius variation.

Fig. 2
Fig. 2

Surface plots of the transmission amplitude as a function of distance along the fiber and wavelength deviation for λ = 1.5 μm, γres = 0.1 pm, nf0 = 1.5, and different coupling parameters shown on the plots.

Fig. 3
Fig. 3

Surface plot of the transmission amplitude through a uniform SF as a function of wavelength deviation and coupling parameters. (a), (b) – lossless transmission (b),(c) – lossy transmission. The SF parameters are λ = 1.5 μm, γres = 0.1 pm, and nf0 = 1.5. Other parameters are given in this Fig. 3.

Fig. 4
Fig. 4

Surface plots of the transmission amplitude S11(λ) through a uniform SF coupled to two waveguides, WG1 and WG2, as a function of wavelength deviation and the coupling parameter |C1|2 to WG1. Coupling to WG2 is assumed to be large compared to coupling to WG1. The waveguides are spaced by 150 µm. (a) – D1 = i|C1|2/2 (b) – D1 = (1 + i/2)|C1|2. The SF parameters are λ = 1.5 μm, γres = 0.1 pm, and nf0 = 1.5.

Fig. 5
Fig. 5

(a) – The surface plot of the experimentally measured resonant transmission amplitude through the microfiber scanned along the SNAP bottle resonator with 2 µm steps. (b) – Surface plot of the theoretically calculated transmission amplitude fitting the experimental data of Fig. 5(a).

Fig. 6
Fig. 6

(a) – The experimental transmission amplitudes of the bottle microresonator from Fig. 5(a) measured at the axial microfiber positions 50 μm, 94 μm, and 120 μm. (b) – The bare Green’s function calculated with the radius dependence defined by Eq. (27) fitting narrow experimental resonances in Fig. 6(a). (c) – Fitting the experimental transmission amplitude at axial position 120 µm.

Fig. 7
Fig. 7

(a) – The surface plot of the experimentally measured resonant transmission amplitude through the microfiber scanned along the chain of 10 SNAP microresonators with 2 µm steps. (b) – Surface plot of Fig. 7(a) magnified in the region of the fundamental transmission band. (c) – Surface plot of the theoretically calculated transmission amplitude fitting the experimental data of Fig. 7(a). (d) – Surface plot of Fig. 7(c) magnified in the region of the fundamental transmission band. (e) – Surface plot of Fig. 7(d) perturbed by the effective radius variation shown at the bottom of Fig. 7(e).

Equations (49)

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

U m,p,q (r)= Ψ m,p,q (z) Ξ m,p (ρ)exp(imφ)
d 2 Ψ d z 2 + β 2 (λ,z)Ψ=0, β 2 (λ,z)=E(λ)V(z).
E(λ)=2 k 2 ( λ res ) λ λ res i γ res λ res ,V(z)=2 k 2 ( λ res ) Δ r eff (z) r 0 , Δ r eff (z) r 0 = Δr(z) r 0 + Δ n f (z) n f0 , k( λ res )= 2π n f0 λ res .
d 2 Ψ(z) d z 2 +[ β 2 (λ,z)+ n=1 N D n δ(z z n ) ]Ψ(z)=0.
S 11 (λ)= S 11 (0) i| C 1 | 2 G ¯ (λ, z 1 , z 1 ),
S 1n (λ)= S 1n (0) i C 1 C n * G ¯ (λ, z 1 , z n ),
G ¯ (λ, z 1 , z 2 )= G(λ, z 1 , z 2 ) 1+ D 1 G(λ, z 1 , z 1 ) .
S 11 (λ)= S 11 (0) i| C 1 | 2 G(λ, z 1 , z 1 ) 1+ D 1 G(λ, z 1 , z 1 ) .
S 11 (λ)= 1+( Re D 1 i 2 | C 1 | 2 )G(λ, z 1 , z 1 ) 1+( Re D 1 + i 2 | C 1 | 2 )G(λ, z 1 , z 1 ) .
| S 11 (0) |<1,
Im( D 1 )>| C 1 | 2 1Re( S 11 (0) ) 1| S 11 (0) | 2 .
G(λ, z 1 , z 2 )= Ψ n ( z 1 ) Ψ n ( z 2 ) E(λ) E n +i Γ 0 ,
S 11 (λ)= S 11 (0) i Λ n (E(λ) E n Δ n )+i( Γ 0 + Σ n ) , Λ n =| C 1 | 2 Ψ n 2 ( z 1 ), Δ n =Re( D 1 ) Ψ n 2 ( z 1 ), Σ n =Im( D 1 ) Ψ n 2 ( z 1 ).
S 11 (λ)= (E(λ) E n Δ n )+i( Γ 0 1 2 Λ n ) (E(λ) E n Δ n )+i( Γ 0 + 1 2 Λ n ) ,
S 11 (λ)= (λ λ res )i( γ res γ C exp[ (z/ z w ) 2 ]) (λ λ res )i( γ res + γ C exp[ (z/ z w ) 2 ]) ,
G(λ, z 1 , z 1 )= n | Ψ n ( z 1 ) | 2 E(λ) E n +i Γ n
G(λ, z 1 , z 2 )= exp[i (E(λ) E 0 +i Γ 0 ) 1/2 | z 2 z 1 |] 2i (E(λ) E 0 +i Γ 0 ) 1/2 .
G ¯ (λ, z 1 , z 1 )= G(λ, z 1 , z 1 )+ D 2 ( G(λ, z 1 , z 1 )G(λ, z 2 , z 2 ) G 2 (λ, z 1 , z 2 ) ) ( 1+ D 1 G(λ, z 1 , z 1 ) )( 1+ D 2 G(λ, z 2 , z 2 ) ) D 1 D 2 G 2 (λ, z 1 , z 2 ) ,
G ¯ (λ, z 1 , z 2 )= G(λ, z 1 , z 2 ) ( 1+ D 1 G(λ, z 1 , z 1 ) )( 1+ D 2 G(λ, z 2 , z 2 ) ) D 1 D 2 G 2 (λ, z 1 , z 2 ) ,
Im D n = 1 2 | C n | 2 .
S 11 (λ)= E(λ) E n + Δ n +i( Γ 0 + Σ n (2) Σ n (1) ) E(λ) E n + Δ n +i( Γ 0 + Σ n (2) + Σ n (1) ) ,
S 12 (λ)= 2i Ξ n E(λ) E n + Δ n +i( Γ 0 + Σ n (1) + Σ n (2) ) ,
Σ n (j) = 1 2 | C j | 2 Ψ n 2 ( z j ), Ξ n = C 1 C 2 * Ψ n ( z 1 ) Ψ n ( z 2 ), Δ n =Re( D 1 ) Ψ n 2 ( z 1 )+Re( D 2 ) Ψ n 2 ( z 2 ).
S 11 (λ)= G(λ, z 2 , z 2 )(Re D 1 i 2 | C 1 | 2 )( G(λ, z 1 , z 1 )G(λ, z 2 , z 2 ) G 2 (λ, z 1 , z 2 ) ) G(λ, z 2 , z 2 )(Re D 1 + i 2 | C 1 | 2 )( G(λ, z 1 , z 1 )G(λ, z 2 , z 2 ) G 2 (λ, z 1 , z 2 ) ) .
Δ r eff (z)=Δ r 0 [ exp( (z z 0 ) 2 ζ 2 )+ ε 1+[ (z z 0 ) 2 / ζ 2 ] ]
Δ r 0 =14.1 nm,ζ=10.9 μm, z 0 =120 μm,ε=0.22,
S 11 (0) =0.8790.084i,| C 1 | 2 =0.026 μ m 1 , D 1 =0.022+0.02i μ m 1 .
Δ r eff (z)=Δ r 0 { exp( z 2 / ζ 2 ), z<0, a 2 2 π 2 ζ 2 ( cos( 2πz/a )1 )+1, 0<z<Na, exp{ [z(N1)a] 2 / ζ 2 }, z>(N1)a,
Δ r 0 =6.9 nm,ζ=19.4 μm,a=50μm,N=10.
S 11 (0) =0.830.15i,| C 1 | 2 =0.028 μ m 1 , D 1 =0.022+0.026i μ m 1 .
U m,p,q (ren) (r)= Ψ m,p,q (ren) (z) Ξ m,p (ρ)exp(imφ)
G(λ, r 1 , r 2 )= m,p,q U m,p,q (ren) ( r 1 ) U m,p,q (ren)* ( r 2 ) E(λ) E m,p,q
G(λ, r 1 , r 2 )= m,p G m,p (ren) (λ, z 1 , z 2 ) Ξ m,p ( ρ 1 ) Ξ m,p ( ρ 2 )exp[im( φ 1 φ 2 )] ,
S=1iT=1i(V+VGV)
S 11 (λ)=1i d r 1 | χ 1 ( r 1 ) | 2 V( r 1 ) i d r 1 d r 2 χ 1 ( r 1 )V( r 1 )G(λ, r 1 , r 2 )V( r 2 ) χ 1 * ( r 2 )
S 11 (λ)= S 11 (0) i| C 1 | 2 G m 0 , p 0 (ren) (λ, z 1 , z 1 ), C 1 = dr χ 1 (r)V(r) Ξ m 0 , p 0 (ρ)exp(i m 0 φ) .
S 1n (λ)= S 1n (0) i C 1 C n * G m 0 , p 0 (ren) (λ, z 1 , z n ), C n = dr χ n (r)V(r) Ξ m 0 , p 0 (ρ)exp(i m 0 φ) ,
Ψ 1z (λ,z) Ψ 2 (λ,z) Ψ 1 (λ,z) Ψ 2z (λ,z)=1,
G(λ, z 1 , z 2 )={ Ψ 1 (λ, z 1 ) Ψ 2 (λ, z 2 ) z 1 > z 2 Ψ 1 (λ, z 2 ) Ψ 2 (λ, z 1 ) z 1 < z 2 .
G ¯ (λ,z, z n )| z z n G ¯ (λ,z, z n )| z z n =0,n=1,2,...,N G ¯ z (λ,z, z 1 )| z z 1 G ¯ z (λ,z, z 1 )| z z 1 =1+i D 1 G ¯ (λ,z, z 1 ) G ¯ z (λ,z, z 1 )| z z n G ¯ z (λ,z, z 1 )| z z n =i D n G ¯ (λ, z 2 , z 1 ) ,n=2,3,...N
G ¯ (λ,z, z 1 )={ [ A 1 Ψ 1 (λ, z 1 )+ A 2 Ψ 2 (λ, z 1 )] Ψ 1 (λ,z) Ψ 2 (λ, z 2 ), [ A 1 Ψ 1 (λ,z)+ A 2 Ψ 2 (λ,z)] Ψ 1 (λ, z 1 ) Ψ 2 (λ, z 2 ), [ A 1 Ψ 1 (λ, z 2 )+ A 2 Ψ 2 (λ, z 2 )] Ψ 1 (λ, z 1 ) Ψ 2 (λ,z), z< z 1 , z 1 <z< z 2 , z> z 2 .
Λ 0 +2Re( Λ 1 G)+ Λ 0 1 (| Λ 1 | 2 | C 1 | 4 )|G | 2 >0
Λ 0 =1| S 11 (0) | 2 , Λ 1 =i| C 1 | 2 ( S 11 (0) ) * i Λ 0 D 1 .
Λ 0 +2Re Λ 1 ReG+ Λ 0 1 (| Λ 1 | 2 | C 1 | 4 ) (ReG) 2 >0,
Λ 0 >0,
|Im Λ 1 |>| C 1 | 2 .
Ξ 0 = Λ 0 ( 1 Re( Λ 1 ) | Λ 1 | 2 | C 1 | 4 ).
Ξ 0 2Im Λ 1 ImG+ Λ 0 1 (| Λ 1 | 2 | C 1 | 4 ) (ImG) 2 >0
Im Λ 1 >| C 1 | 2 ,

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