Abstract

Propagation of whispering gallery modes in a microfiber coil having an n-fold helical symmetry is considered. The n-fold helically symmetric deformations of a coiled microfiber is similar to a long period grating introduced by periodic microfiber bending with the azimuthal angle period 2π/n. The considered modes are localized near a geodesic situated at the peripheral part of the microfiber surface. Using the perturbation theory, a simple condition for the stability of this geodesic is found. Violation of this condition happens in a small region localized near a point determined by the phase matching condition for the introduced long period grating. In the classical limit, the unstable region corresponds to the parametric resonance and stochastization of whispering gallery rays. Generally, this region corresponds to resonance intermode coupling, which may result either in the periodic transmission of radiation power between two modes or in the aperiodic wave parametric resonance, which causes simultaneous coupling and power transfer between numerous whispering gallery modes. The obtained results are important for engineering of miniature optical fiber coils and analysis of their propagation loss.

© 2010 OSA

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References

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  1. M. Berry, “Interpreting the anholonomy of coiled light,” Nature 326(6110), 277–278 (1987).
    [CrossRef]
  2. R. Y. Chiao and Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57(8), 933–936 (1986).
    [CrossRef] [PubMed]
  3. G. B. Malykin and V. I. Pozdnyakova, “Geometric phases in single mode fiber light guides and fiber ring interferometers,” Phys-Usp. 47(3), 289–308 (2004).
    [CrossRef]
  4. M. Sumetsky, “Optical fiber microcoil resonators,” Opt. Express 12(10), 2303–2316 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=OPEX-12-10-2303 .
    [CrossRef] [PubMed]
  5. D. J. Hill, “The evolution and exploitation of the fibre-optic hydrophone,” Proc. SPIE 6619, 661907 (2007).
    [CrossRef]
  6. C. N. Alexeyev, B. A. Lapin, and M. A. Yavorsky, “Optical vortices and topological phase in strongly anisotropic coiled few-mode optical fibers,” J. Opt. Soc. Am. B 24(10), 2666–2675 (2007).
    [CrossRef]
  7. V. Atanasov and R. Dandoloff, “Curvature-induced quantum behaviour on a helical nanotube,” Phys. Lett. A 372(40), 6141–6144 (2008).
    [CrossRef]
  8. M. Sumetsky, “Basic elements for microfiber photonics: micro/nanofibers and microfiber coil resonators,” J. Lightwave Technol. 26(1), 21–27 (2008).
    [CrossRef]
  9. N. G. Broderick, “Optical snakes and ladders: dispersion and nonlinearity in microcoil resonators,” Opt. Express 16(20), 16247–16254 (2008), http://www.opticsinfobase.org/abstract.cfm?&uri=oe-16-20-16247 .
    [CrossRef] [PubMed]
  10. J. Scheuer, “Fiber microcoil optical gyroscope,” Opt. Lett. 34(11), 1630–1632 (2009).
    [CrossRef] [PubMed]
  11. M. Sumetsky, “Optical microfiber loop and coil resonators,” in Practical applications of microresonators in optics and photonics, Ed., A. Matsko (CRC Press, 2009) p. 355–384.
  12. G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. P. Sessions, E. Koukharenko, X. Feng, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1(1), 107–161 (2009).
    [CrossRef]
  13. L. Yao, T. A. Birks, and J. C. Knight, “Low bend loss in tightly-bent fibers through adiabatic bend transitions,” Opt. Express 17(4), 2962–2967 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-4-2962 .
    [CrossRef] [PubMed]
  14. M. Sumetsky, “Optical microfiber coil delay line,” Opt. Express 17(9), 7196–7205 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-9-7196 .
    [CrossRef] [PubMed]
  15. M. Sumetsky, Y. Dulashko, and S. Ghalmi, “Fabrication of miniature optical fiber and microfiber coils,” Opt. Lasers Eng. 48 , 272-275 (2010)
    [CrossRef]
  16. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
    [CrossRef]
  17. S. Ramachandran, “Dispersion-tailored few-mode fibers: a versatile platform for in-fiber photonic devices,” J. Lightwave Technol. 23(11), 3426–3443 (2005).
    [CrossRef]
  18. H. Xuan, W. Jin, and M. Zhang, “CO2 laser induced long period gratings in optical microfibers,” Opt. Express 17(24), 21882–21890 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-24-21882 .
    [CrossRef] [PubMed]
  19. V. M. Babich, and V. S. Buldyrev, Short-Wavelength Diffraction Theory. Asymptotic Methods. (Springer, Berlin, 1991).
  20. M. Berry and K. Mount, “Semiclassical approximation in wave mechanics,” Rep. Prog. Phys. 35(1), 315–397 (1972).
    [CrossRef]
  21. The Gaussian curvature of a surface is the inversed product of its principal radii of curvature [22].
  22. See, e.g., M. P. Do Carmo, Differential geometry of curves and surfaces (Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1976).
  23. L. D. Landau, and E. M. Lifshitz, Mechanics (Pergamon Press, 1976).
  24. A. M. Perelomov, and Y. B. Zeldovich, Quantum mechanics: selected topics (World Scientific Publishing Co, Singapore, 1998).
  25. Quantum chaos, http://en.wikipedia.org/wiki/Quantum_chaos .

2010

M. Sumetsky, Y. Dulashko, and S. Ghalmi, “Fabrication of miniature optical fiber and microfiber coils,” Opt. Lasers Eng. 48 , 272-275 (2010)
[CrossRef]

2009

H. Xuan, W. Jin, and M. Zhang, “CO2 laser induced long period gratings in optical microfibers,” Opt. Express 17(24), 21882–21890 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-24-21882 .
[CrossRef] [PubMed]

J. Scheuer, “Fiber microcoil optical gyroscope,” Opt. Lett. 34(11), 1630–1632 (2009).
[CrossRef] [PubMed]

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. P. Sessions, E. Koukharenko, X. Feng, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1(1), 107–161 (2009).
[CrossRef]

L. Yao, T. A. Birks, and J. C. Knight, “Low bend loss in tightly-bent fibers through adiabatic bend transitions,” Opt. Express 17(4), 2962–2967 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-4-2962 .
[CrossRef] [PubMed]

M. Sumetsky, “Optical microfiber coil delay line,” Opt. Express 17(9), 7196–7205 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-9-7196 .
[CrossRef] [PubMed]

2008

V. Atanasov and R. Dandoloff, “Curvature-induced quantum behaviour on a helical nanotube,” Phys. Lett. A 372(40), 6141–6144 (2008).
[CrossRef]

M. Sumetsky, “Basic elements for microfiber photonics: micro/nanofibers and microfiber coil resonators,” J. Lightwave Technol. 26(1), 21–27 (2008).
[CrossRef]

N. G. Broderick, “Optical snakes and ladders: dispersion and nonlinearity in microcoil resonators,” Opt. Express 16(20), 16247–16254 (2008), http://www.opticsinfobase.org/abstract.cfm?&uri=oe-16-20-16247 .
[CrossRef] [PubMed]

2007

D. J. Hill, “The evolution and exploitation of the fibre-optic hydrophone,” Proc. SPIE 6619, 661907 (2007).
[CrossRef]

C. N. Alexeyev, B. A. Lapin, and M. A. Yavorsky, “Optical vortices and topological phase in strongly anisotropic coiled few-mode optical fibers,” J. Opt. Soc. Am. B 24(10), 2666–2675 (2007).
[CrossRef]

2005

S. Ramachandran, “Dispersion-tailored few-mode fibers: a versatile platform for in-fiber photonic devices,” J. Lightwave Technol. 23(11), 3426–3443 (2005).
[CrossRef]

2004

G. B. Malykin and V. I. Pozdnyakova, “Geometric phases in single mode fiber light guides and fiber ring interferometers,” Phys-Usp. 47(3), 289–308 (2004).
[CrossRef]

M. Sumetsky, “Optical fiber microcoil resonators,” Opt. Express 12(10), 2303–2316 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=OPEX-12-10-2303 .
[CrossRef] [PubMed]

1997

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

1987

M. Berry, “Interpreting the anholonomy of coiled light,” Nature 326(6110), 277–278 (1987).
[CrossRef]

1986

R. Y. Chiao and Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57(8), 933–936 (1986).
[CrossRef] [PubMed]

1972

M. Berry and K. Mount, “Semiclassical approximation in wave mechanics,” Rep. Prog. Phys. 35(1), 315–397 (1972).
[CrossRef]

Alexeyev, C. N.

C. N. Alexeyev, B. A. Lapin, and M. A. Yavorsky, “Optical vortices and topological phase in strongly anisotropic coiled few-mode optical fibers,” J. Opt. Soc. Am. B 24(10), 2666–2675 (2007).
[CrossRef]

Atanasov, V.

V. Atanasov and R. Dandoloff, “Curvature-induced quantum behaviour on a helical nanotube,” Phys. Lett. A 372(40), 6141–6144 (2008).
[CrossRef]

Berry, M.

M. Berry, “Interpreting the anholonomy of coiled light,” Nature 326(6110), 277–278 (1987).
[CrossRef]

M. Berry and K. Mount, “Semiclassical approximation in wave mechanics,” Rep. Prog. Phys. 35(1), 315–397 (1972).
[CrossRef]

Birks, T. A.

L. Yao, T. A. Birks, and J. C. Knight, “Low bend loss in tightly-bent fibers through adiabatic bend transitions,” Opt. Express 17(4), 2962–2967 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-4-2962 .
[CrossRef] [PubMed]

Brambilla, G.

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. P. Sessions, E. Koukharenko, X. Feng, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1(1), 107–161 (2009).
[CrossRef]

Broderick, N. G.

N. G. Broderick, “Optical snakes and ladders: dispersion and nonlinearity in microcoil resonators,” Opt. Express 16(20), 16247–16254 (2008), http://www.opticsinfobase.org/abstract.cfm?&uri=oe-16-20-16247 .
[CrossRef] [PubMed]

Chiao, R. Y.

R. Y. Chiao and Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57(8), 933–936 (1986).
[CrossRef] [PubMed]

Dandoloff, R.

V. Atanasov and R. Dandoloff, “Curvature-induced quantum behaviour on a helical nanotube,” Phys. Lett. A 372(40), 6141–6144 (2008).
[CrossRef]

Dulashko, Y.

M. Sumetsky, Y. Dulashko, and S. Ghalmi, “Fabrication of miniature optical fiber and microfiber coils,” Opt. Lasers Eng. 48 , 272-275 (2010)
[CrossRef]

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

Feng, X.

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. P. Sessions, E. Koukharenko, X. Feng, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1(1), 107–161 (2009).
[CrossRef]

Ghalmi, S.

M. Sumetsky, Y. Dulashko, and S. Ghalmi, “Fabrication of miniature optical fiber and microfiber coils,” Opt. Lasers Eng. 48 , 272-275 (2010)
[CrossRef]

Hill, D. J.

D. J. Hill, “The evolution and exploitation of the fibre-optic hydrophone,” Proc. SPIE 6619, 661907 (2007).
[CrossRef]

Horak, P.

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. P. Sessions, E. Koukharenko, X. Feng, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1(1), 107–161 (2009).
[CrossRef]

Jin, W.

H. Xuan, W. Jin, and M. Zhang, “CO2 laser induced long period gratings in optical microfibers,” Opt. Express 17(24), 21882–21890 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-24-21882 .
[CrossRef] [PubMed]

Jung, Y.

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. P. Sessions, E. Koukharenko, X. Feng, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1(1), 107–161 (2009).
[CrossRef]

Knight, J. C.

L. Yao, T. A. Birks, and J. C. Knight, “Low bend loss in tightly-bent fibers through adiabatic bend transitions,” Opt. Express 17(4), 2962–2967 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-4-2962 .
[CrossRef] [PubMed]

Koizumi, F.

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. P. Sessions, E. Koukharenko, X. Feng, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1(1), 107–161 (2009).
[CrossRef]

Koukharenko, E.

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. P. Sessions, E. Koukharenko, X. Feng, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1(1), 107–161 (2009).
[CrossRef]

Lapin, B. A.

C. N. Alexeyev, B. A. Lapin, and M. A. Yavorsky, “Optical vortices and topological phase in strongly anisotropic coiled few-mode optical fibers,” J. Opt. Soc. Am. B 24(10), 2666–2675 (2007).
[CrossRef]

Malykin, G. B.

G. B. Malykin and V. I. Pozdnyakova, “Geometric phases in single mode fiber light guides and fiber ring interferometers,” Phys-Usp. 47(3), 289–308 (2004).
[CrossRef]

Mount, K.

M. Berry and K. Mount, “Semiclassical approximation in wave mechanics,” Rep. Prog. Phys. 35(1), 315–397 (1972).
[CrossRef]

Murugan, G. S.

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. P. Sessions, E. Koukharenko, X. Feng, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1(1), 107–161 (2009).
[CrossRef]

Pozdnyakova, V. I.

G. B. Malykin and V. I. Pozdnyakova, “Geometric phases in single mode fiber light guides and fiber ring interferometers,” Phys-Usp. 47(3), 289–308 (2004).
[CrossRef]

Ramachandran, S.

S. Ramachandran, “Dispersion-tailored few-mode fibers: a versatile platform for in-fiber photonic devices,” J. Lightwave Technol. 23(11), 3426–3443 (2005).
[CrossRef]

Richardson, D. J.

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. P. Sessions, E. Koukharenko, X. Feng, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1(1), 107–161 (2009).
[CrossRef]

Scheuer, J.

J. Scheuer, “Fiber microcoil optical gyroscope,” Opt. Lett. 34(11), 1630–1632 (2009).
[CrossRef] [PubMed]

Sessions, N. P.

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. P. Sessions, E. Koukharenko, X. Feng, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1(1), 107–161 (2009).
[CrossRef]

Sumetsky, M.

M. Sumetsky, Y. Dulashko, and S. Ghalmi, “Fabrication of miniature optical fiber and microfiber coils,” Opt. Lasers Eng. 48 , 272-275 (2010)
[CrossRef]

M. Sumetsky, “Optical microfiber coil delay line,” Opt. Express 17(9), 7196–7205 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-9-7196 .
[CrossRef] [PubMed]

M. Sumetsky, “Basic elements for microfiber photonics: micro/nanofibers and microfiber coil resonators,” J. Lightwave Technol. 26(1), 21–27 (2008).
[CrossRef]

M. Sumetsky, “Optical fiber microcoil resonators,” Opt. Express 12(10), 2303–2316 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=OPEX-12-10-2303 .
[CrossRef] [PubMed]

Wilkinson, J. S.

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. P. Sessions, E. Koukharenko, X. Feng, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1(1), 107–161 (2009).
[CrossRef]

Wu, Y. S.

R. Y. Chiao and Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57(8), 933–936 (1986).
[CrossRef] [PubMed]

Xu, F.

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. P. Sessions, E. Koukharenko, X. Feng, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1(1), 107–161 (2009).
[CrossRef]

Xuan, H.

H. Xuan, W. Jin, and M. Zhang, “CO2 laser induced long period gratings in optical microfibers,” Opt. Express 17(24), 21882–21890 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-24-21882 .
[CrossRef] [PubMed]

Yao, L.

L. Yao, T. A. Birks, and J. C. Knight, “Low bend loss in tightly-bent fibers through adiabatic bend transitions,” Opt. Express 17(4), 2962–2967 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-4-2962 .
[CrossRef] [PubMed]

Yavorsky, M. A.

C. N. Alexeyev, B. A. Lapin, and M. A. Yavorsky, “Optical vortices and topological phase in strongly anisotropic coiled few-mode optical fibers,” J. Opt. Soc. Am. B 24(10), 2666–2675 (2007).
[CrossRef]

Zhang, M.

H. Xuan, W. Jin, and M. Zhang, “CO2 laser induced long period gratings in optical microfibers,” Opt. Express 17(24), 21882–21890 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-24-21882 .
[CrossRef] [PubMed]

Adv. Opt. Photon.

G. Brambilla, F. Xu, P. Horak, Y. Jung, F. Koizumi, N. P. Sessions, E. Koukharenko, X. Feng, G. S. Murugan, J. S. Wilkinson, and D. J. Richardson, “Optical fiber nanowires and microwires: fabrication and applications,” Adv. Opt. Photon. 1(1), 107–161 (2009).
[CrossRef]

J. Lightwave Technol.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

S. Ramachandran, “Dispersion-tailored few-mode fibers: a versatile platform for in-fiber photonic devices,” J. Lightwave Technol. 23(11), 3426–3443 (2005).
[CrossRef]

M. Sumetsky, “Basic elements for microfiber photonics: micro/nanofibers and microfiber coil resonators,” J. Lightwave Technol. 26(1), 21–27 (2008).
[CrossRef]

J. Opt. Soc. Am. B

C. N. Alexeyev, B. A. Lapin, and M. A. Yavorsky, “Optical vortices and topological phase in strongly anisotropic coiled few-mode optical fibers,” J. Opt. Soc. Am. B 24(10), 2666–2675 (2007).
[CrossRef]

Nature

M. Berry, “Interpreting the anholonomy of coiled light,” Nature 326(6110), 277–278 (1987).
[CrossRef]

Opt. Express

M. Sumetsky, “Optical fiber microcoil resonators,” Opt. Express 12(10), 2303–2316 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=OPEX-12-10-2303 .
[CrossRef] [PubMed]

N. G. Broderick, “Optical snakes and ladders: dispersion and nonlinearity in microcoil resonators,” Opt. Express 16(20), 16247–16254 (2008), http://www.opticsinfobase.org/abstract.cfm?&uri=oe-16-20-16247 .
[CrossRef] [PubMed]

H. Xuan, W. Jin, and M. Zhang, “CO2 laser induced long period gratings in optical microfibers,” Opt. Express 17(24), 21882–21890 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-24-21882 .
[CrossRef] [PubMed]

L. Yao, T. A. Birks, and J. C. Knight, “Low bend loss in tightly-bent fibers through adiabatic bend transitions,” Opt. Express 17(4), 2962–2967 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-4-2962 .
[CrossRef] [PubMed]

M. Sumetsky, “Optical microfiber coil delay line,” Opt. Express 17(9), 7196–7205 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-9-7196 .
[CrossRef] [PubMed]

Opt. Lasers Eng. 48

M. Sumetsky, Y. Dulashko, and S. Ghalmi, “Fabrication of miniature optical fiber and microfiber coils,” Opt. Lasers Eng. 48 , 272-275 (2010)
[CrossRef]

Opt. Lett.

J. Scheuer, “Fiber microcoil optical gyroscope,” Opt. Lett. 34(11), 1630–1632 (2009).
[CrossRef] [PubMed]

Phys-Usp.

G. B. Malykin and V. I. Pozdnyakova, “Geometric phases in single mode fiber light guides and fiber ring interferometers,” Phys-Usp. 47(3), 289–308 (2004).
[CrossRef]

Phys. Lett. A

V. Atanasov and R. Dandoloff, “Curvature-induced quantum behaviour on a helical nanotube,” Phys. Lett. A 372(40), 6141–6144 (2008).
[CrossRef]

Phys. Rev. Lett.

R. Y. Chiao and Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57(8), 933–936 (1986).
[CrossRef] [PubMed]

Proc. SPIE

D. J. Hill, “The evolution and exploitation of the fibre-optic hydrophone,” Proc. SPIE 6619, 661907 (2007).
[CrossRef]

Rep. Prog. Phys.

M. Berry and K. Mount, “Semiclassical approximation in wave mechanics,” Rep. Prog. Phys. 35(1), 315–397 (1972).
[CrossRef]

Other

The Gaussian curvature of a surface is the inversed product of its principal radii of curvature [22].

See, e.g., M. P. Do Carmo, Differential geometry of curves and surfaces (Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1976).

L. D. Landau, and E. M. Lifshitz, Mechanics (Pergamon Press, 1976).

A. M. Perelomov, and Y. B. Zeldovich, Quantum mechanics: selected topics (World Scientific Publishing Co, Singapore, 1998).

Quantum chaos, http://en.wikipedia.org/wiki/Quantum_chaos .

M. Sumetsky, “Optical microfiber loop and coil resonators,” in Practical applications of microresonators in optics and photonics, Ed., A. Matsko (CRC Press, 2009) p. 355–384.

V. M. Babich, and V. S. Buldyrev, Short-Wavelength Diffraction Theory. Asymptotic Methods. (Springer, Berlin, 1991).

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Figures (5)

Fig. 1
Fig. 1

(a) – An MFC with an infinite helical symmetry; (b) – An MFC with a 5-fold helical symmetry; (c) – MFC parameters and local coordinate system along the geodesic s.

Fig. 2
Fig. 2

An MFC with r 0 = 10 μm and R 0 = 230 μm. The inset shows the geodesic s and the launched ray spaced from the geodesic by 0.07 r 0.

Fig. 3
Fig. 3

(a)-(g) – Fragments of the Poincaré surfaces of sections (left hand side figures) and the relative deviation of the reflection points (right hand side figures) versus the number of turns for the launched ray shown in Fig. 2. (h) and (i) – The sine of incidence angle at the reflection point and the relative deviation of the same point, respectively, after completion of 200 turns as a function of the microfiber radius r o.

Fig. 4
Fig. 4

Transmission power P of the fundamental WGM: (a) – as a function of r 0 for λ = 1.5 μm (inset – magnified region between r 0 = 10 μm and r 0 = 14 μm); (b) – as a function of radiation wavelength λ for r 0 = 11.53 μm and r 0 = 12.1 μm.

Fig. 5
Fig. 5

Transmission power of the fundamental WGM as a function of propagation distance (upper plots) and corresponding distribution of the radiation along the microfiber (lower plots). Inset surface plots are the cross-sectional radiation distribution at the positions indicated by arrows.

Equations (13)

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

E m l ( s , p , q ) ρ 1 / 6 ( s ) σ 1 / 2 ( s ) exp [ i s β m l ( s ) d s + β 0 2 σ ( s ) ( i d σ d s 1 σ ( s ) ) q 2 ] ×                  H m ( β 0 1 / 2 q σ ( s ) ) A i [ ( 2 β 0 2 ρ ( s ) ) 1 / 3 p ς l ] ,          β 0 = 2 π n f λ .
β m l ( s ) = β 0 + ς l ( β 0 2 ρ 2 ( s ) ) 1 / 3 + ( m + 1 2 ) 1 σ 2 ( s )
σ ( s ) = i , j = 1 2 a i j θ i ( s ) θ j ( s ) ,
d 2 θ i d s 2 + K ( s ) θ i = 0.
det a i j ( θ 1 d θ 2 d s θ 2 d θ 1 d s ) 2 = 1.
E m l ( s , p , q ) exp [ i β m l s 1 2 ( β 0 2 R 0 r 0 ) 1 2 q 2 ] H m [ ( β 0 2 R 0 r 0 ) 1 4 q ] A i [ ( 2 β 0 2 R 0 ) 1 3 p ς l ] , β m l = β 2 1 3 ς l β 0 1 3 R 0 2 3 ( m + 1 2 ) ( R 0 r 0 ) 1 2 , R 0 = R 0 + 2 r 0 .
x = [ R ( φ ) + r ( φ ) ( 1 + cos ( θ ) ) ] cos ( φ ) y = [ R ( φ ) + r ( φ ) ( 1 + cos ( θ ) ) ] sin ( φ ) z = r ( φ ) sin ( θ ) + tan ( γ ) 0 φ [ R ( φ ) + r ( φ ) ] d φ
R ( φ ) = R 0 + Δ R ( φ ) , r ( φ ) = r 0 + Δ r ( φ ) , Δ R ( φ ) < < R 0 , Δ r ( φ ) < < r 0 .
x = R 0 cos ( φ ) , R 0 = R 0 + 2 r 0 y = R 0 sin ( φ ) z = ( R 0 r 0 ) tan ( γ ) φ
E m 0 | d E m 0 d s = n ( n 2 + 2 ) Δ R 0 cos ( n s / R 0 ) 8 R 0 2 { ( m + 1 ± 1 ) ( m ± 1 ) , m = m ± 2 , 0 , m m ± 2 ,
{ d a 00 d s = κ exp [ i s ( 2 ( R 0 r 0 ) 1 / 2 n R 0 ) ] a 20 d a 20 d s = κ exp [ i s ( 2 ( R 0 r 0 ) 1 / 2 n R 0 ) ] a 00 , κ = n ( n 2 + 2 ) Δ R 0 2 7 / 2 R 0 2 .
S p = π κ = 2 7 / 2 π R 0 2 n ( n 2 + 2 ) Δ R 0 .
R l o c ( φ ) = R 0 + n = 0 Δ R l o c ( n ) cos ( i n φ + ξ n ) ,

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