Abstract

In a tapered optical fiber there exist localized light structures that, in analogy to the magnetic bottles used in plasma fusion, can be called whispering-gallery bottles (WGBs). These essentially three-dimensional structures are formed by the spiral rays that experience total internal reflection at the fiber surface and that also bounce along the fiber axis in response to reflection from the regions of tapering. It is shown that the Wentzel–Kramers–Brillouin quantization rules for the strongly prolate WGBs can be inversed exactly, thus determining the cavity shape from its spectrum. The approximation considered allows one to find the shape of the etalon bottle, which, similar to the one-dimensional Fabry–Perot etalon, contains an unlimited number of equally spaced wave-number eigenvalues. The problem of determining such a non-one-dimensional cavity is not trivial, because such a cavity does not exist among the uniformly filled cavities such as rectangular boxes, cylinders, and spheroids that allow separation of variables. The etalon cavity corresponds to the fiber radius variation ρz=ρ0cosΔkz, where Δk is the wave-number spacing. The latter result is in excellent agreement with ray-dynamics numerical modeling.

© 2004 Optical Society of America

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References

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2003

V. I. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, J. Opt. Soc. Am. A 20, 157 (2003).
[CrossRef]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef]

2001

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, New J. Phys. 3, 14.1 (2001).
[CrossRef]

G. Kakaranzas, T. E. Dimmick, T. A. Birks, R. Le Roux, and P. St. J. Russell, Opt. Lett. 26, 1137 (2001).
[CrossRef]

2000

1997

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981).

Babich, V. M.

V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory (Springer-Verlag, Berlin, 1991).
[CrossRef]

Birks, T. A.

Buldyrev, V. S.

V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory (Springer-Verlag, Berlin, 1991).
[CrossRef]

Cheung, G.

Chu, S. T.

Dimmick, T. E.

Gorenflo, R.

R. Gorenflo and S. Vessella, Abel Integral Equations, Vol. 1461 of Lecture Notes in Mathematics Series, A. Dold, B. Eckmann, and F. Takens, eds. (Springer-Verlag, Berlin, 1991).

Hare, J.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, New J. Phys. 3, 14.1 (2001).
[CrossRef]

Haus, H. A.

Ilchenko, V. I.

Ilchenko, V. S.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, New J. Phys. 3, 14.1 (2001).
[CrossRef]

Jacques, F.

Kakaranzas, G.

Katsenelenbaum, B. Z.

B. Z. Katsenelenbaum, L. Mercader del Rio, M. Pereyaslavets, M. Sorolla Ayza, and M. Thumm, Theory of Nonuniform Waveguides: The Cross-Section Method (Institution of Electrical Engineers, London, 1998).
[CrossRef]

Kimerling, L. C.

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef]

Knight, J. C.

Laine, J.-P.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1958).

Le Roux, R.

Lefèvre-Seguin, V.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, New J. Phys. 3, 14.1 (2001).
[CrossRef]

Lichtenberg, A. J.

A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983).
[CrossRef]

Lieberman, M. A.

A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983).
[CrossRef]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1958).

Lim, D. R.

Little, B. E.

Long, R.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, New J. Phys. 3, 14.1 (2001).
[CrossRef]

Maleki, L.

Matsko, A. B.

Mercader del Rio, L.

B. Z. Katsenelenbaum, L. Mercader del Rio, M. Pereyaslavets, M. Sorolla Ayza, and M. Thumm, Theory of Nonuniform Waveguides: The Cross-Section Method (Institution of Electrical Engineers, London, 1998).
[CrossRef]

Nöckel, J. U.

J. U. Nöckel and A. D. Stone, Nature 385, 45 (1997).
[CrossRef]

Painter, O. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef]

Pereyaslavets, M.

B. Z. Katsenelenbaum, L. Mercader del Rio, M. Pereyaslavets, M. Sorolla Ayza, and M. Thumm, Theory of Nonuniform Waveguides: The Cross-Section Method (Institution of Electrical Engineers, London, 1998).
[CrossRef]

Russell, P. St. J.

Savchenkov, A. A.

Sorolla Ayza, M.

B. Z. Katsenelenbaum, L. Mercader del Rio, M. Pereyaslavets, M. Sorolla Ayza, and M. Thumm, Theory of Nonuniform Waveguides: The Cross-Section Method (Institution of Electrical Engineers, London, 1998).
[CrossRef]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef]

Stone, A. D.

J. U. Nöckel and A. D. Stone, Nature 385, 45 (1997).
[CrossRef]

Sturrock, P. A.

P. A. Sturrock, Plasma Physics (Cambridge University, Cambridge, UK, 1994).
[CrossRef]

Thumm, M.

B. Z. Katsenelenbaum, L. Mercader del Rio, M. Pereyaslavets, M. Sorolla Ayza, and M. Thumm, Theory of Nonuniform Waveguides: The Cross-Section Method (Institution of Electrical Engineers, London, 1998).
[CrossRef]

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef]

Vessella, S.

R. Gorenflo and S. Vessella, Abel Integral Equations, Vol. 1461 of Lecture Notes in Mathematics Series, A. Dold, B. Eckmann, and F. Takens, eds. (Springer-Verlag, Berlin, 1991).

von Klitzing, W.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, New J. Phys. 3, 14.1 (2001).
[CrossRef]

J. Opt. Soc. Am. A

Nature

J. U. Nöckel and A. D. Stone, Nature 385, 45 (1997).
[CrossRef]

New J. Phys.

W. von Klitzing, R. Long, V. S. Ilchenko, J. Hare, and V. Lefèvre-Seguin, New J. Phys. 3, 14.1 (2001).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef]

Other

V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory (Springer-Verlag, Berlin, 1991).
[CrossRef]

B. Z. Katsenelenbaum, L. Mercader del Rio, M. Pereyaslavets, M. Sorolla Ayza, and M. Thumm, Theory of Nonuniform Waveguides: The Cross-Section Method (Institution of Electrical Engineers, London, 1998).
[CrossRef]

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981).

R. Gorenflo and S. Vessella, Abel Integral Equations, Vol. 1461 of Lecture Notes in Mathematics Series, A. Dold, B. Eckmann, and F. Takens, eds. (Springer-Verlag, Berlin, 1991).

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1958).

A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983).
[CrossRef]

P. A. Sturrock, Plasma Physics (Cambridge University, Cambridge, UK, 1994).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Illustration of the WGB: (a) projection of the WGB on plane ρ,φ, (b) projection of the WGB on the plane z,ρ.

Fig. 2
Fig. 2

Poincaré surfaces of sections for the nonprolate (Δkρ0=1) and prolate (Δkρ0=0.1) cosine-shaped microcavities defined by Eq. (6) for different angular momenta M.

Fig. 3
Fig. 3

Dependencies of the period of oscillations along the z axis on the angular momentum, M, and the angle between the initial direction of trajectory and the z axis, θ, for the cosine-shaped cavity: (a) nonprolate cavity (Δkρ0=1), (b) prolate cavity (Δkρ0=0.1).

Equations (7)

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

z1z2kmpq2-μmp2ρw2z1/2dz=πq+12,
zU=120UdEΔkmpkmpE-1U-E-1/2×E+μmp2ρ02-1/2,
ρw,mpz=μmpUz+μmp2ρ02-1/2.
ρw,mpz=ρ01+Δkz2-1/2,  Δk=ΔEρ02μmp
kmpq=μmp2ρ02+q+12ΔE1/2.
ρwz=ρ0cosΔkz.
kmpq=μmpρ0+q+12Δk.

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