Abstract

The classical rays propagating along a conical surface are bounded on the narrower side of the cone and unbounded on its wider side. In contrast, it is shown here that a dielectric cone with a small half-angle γ can perform as a high Q-factor optical microresonator which completely confines light. The theory of the discovered localized conical states is confirmed by the experimental demonstration, providing a unique approach for accurate local characterization of optical fibers (which usually have γ105 or less) and a new paradigm in the field of high Q-factor resonators.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. J. Vahala, Nature 424, 839 (2003).
    [CrossRef] [PubMed]
  2. N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).
  3. A.B.Matsko, ed., Practical Applications of Microresonators in Optics and Photonics (Taylor and Francis, 2009).
    [CrossRef]
  4. M. L. Irons, Geodesics on a cone, http://www.rdrop.com/~half/Creations/Puzzles/cone.geodesics/index.html.
  5. J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, Opt. Lett. 22, 1129 (1997).
    [CrossRef] [PubMed]
  6. M. Sumetsky, Opt. Lett. 35, 2385 (2010).
    [CrossRef] [PubMed]
  7. M. Gutzwiller, J. Math. Phys. 12, 343 (1971).
    [CrossRef]
  8. P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay, Chaos: Classical and Quantum, http://ChaosBook.org (Niels Bohr Institute, 2010).
  9. For a small cone half-angle γ, the radial dependence of the propagating beam (i.e., dependence on the coordinate normal to the unfolded surface) remains unchanged and is omitted for brevity.
  10. T. A. Birks, J. C. Knight, and T. E. Dimmick, IEEE Photonics Technol. Lett. 12, 182 (2000).
    [CrossRef]
  11. M. Sumetsky and Y. Dulashko, Opt. Lett. 35, 4006 (2010).
    [CrossRef] [PubMed]
  12. The resonance shape in Fig.  is reversed compared to those in Fig.  owing to the negative sign in the relation (λ−λq)/λq=−Δβ¯δβ0/βq.

2010

2003

K. J. Vahala, Nature 424, 839 (2003).
[CrossRef] [PubMed]

2000

T. A. Birks, J. C. Knight, and T. E. Dimmick, IEEE Photonics Technol. Lett. 12, 182 (2000).
[CrossRef]

1997

1971

M. Gutzwiller, J. Math. Phys. 12, 343 (1971).
[CrossRef]

Artuso, R.

P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay, Chaos: Classical and Quantum, http://ChaosBook.org (Niels Bohr Institute, 2010).

Birks, T. A.

T. A. Birks, J. C. Knight, and T. E. Dimmick, IEEE Photonics Technol. Lett. 12, 182 (2000).
[CrossRef]

J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, Opt. Lett. 22, 1129 (1997).
[CrossRef] [PubMed]

Cheung, G.

Cvitanovic, P.

P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay, Chaos: Classical and Quantum, http://ChaosBook.org (Niels Bohr Institute, 2010).

Dimmick, T. E.

T. A. Birks, J. C. Knight, and T. E. Dimmick, IEEE Photonics Technol. Lett. 12, 182 (2000).
[CrossRef]

Dulashko, Y.

Gutzwiller, M.

M. Gutzwiller, J. Math. Phys. 12, 343 (1971).
[CrossRef]

Hodgson, N.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

Irons, M. L.

M. L. Irons, Geodesics on a cone, http://www.rdrop.com/~half/Creations/Puzzles/cone.geodesics/index.html.

Jacques, F.

Knight, J. C.

T. A. Birks, J. C. Knight, and T. E. Dimmick, IEEE Photonics Technol. Lett. 12, 182 (2000).
[CrossRef]

J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, Opt. Lett. 22, 1129 (1997).
[CrossRef] [PubMed]

Mainieri, R.

P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay, Chaos: Classical and Quantum, http://ChaosBook.org (Niels Bohr Institute, 2010).

Sumetsky, M.

Tanner, G.

P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay, Chaos: Classical and Quantum, http://ChaosBook.org (Niels Bohr Institute, 2010).

Vahala, K. J.

K. J. Vahala, Nature 424, 839 (2003).
[CrossRef] [PubMed]

Vattay, G.

P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay, Chaos: Classical and Quantum, http://ChaosBook.org (Niels Bohr Institute, 2010).

Weber, H.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

IEEE Photonics Technol. Lett.

T. A. Birks, J. C. Knight, and T. E. Dimmick, IEEE Photonics Technol. Lett. 12, 182 (2000).
[CrossRef]

J. Math. Phys.

M. Gutzwiller, J. Math. Phys. 12, 343 (1971).
[CrossRef]

Nature

K. J. Vahala, Nature 424, 839 (2003).
[CrossRef] [PubMed]

Opt. Lett.

Other

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

A.B.Matsko, ed., Practical Applications of Microresonators in Optics and Photonics (Taylor and Francis, 2009).
[CrossRef]

M. L. Irons, Geodesics on a cone, http://www.rdrop.com/~half/Creations/Puzzles/cone.geodesics/index.html.

P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay, Chaos: Classical and Quantum, http://ChaosBook.org (Niels Bohr Institute, 2010).

For a small cone half-angle γ, the radial dependence of the propagating beam (i.e., dependence on the coordinate normal to the unfolded surface) remains unchanged and is omitted for brevity.

The resonance shape in Fig.  is reversed compared to those in Fig.  owing to the negative sign in the relation (λ−λq)/λq=−Δβ¯δβ0/βq.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

(a) Geodesic propagating along the conical surface; (b) localized WGM.

Fig. 2
Fig. 2

Unfolded conical surface with a geodesic S m ( φ , z 1 , z 2 ) connecting points ( 0 , z 1 ) and ( φ , z 2 ) after completing m turns. The inset shows a cone segment with the same geodesic.

Fig. 3
Fig. 3

Inset, illustration of a dielectric cone coupled to a microfiber. (a) Surface plot of | I ( z ¯ , Δ β ¯ ) | . (b) Distribution of the conical WGMs for different values of Δ β ¯ along the lines 1–4 in Fig. 3a. (c) Plots of the real and imaginary parts of I ( 0 , Δ β ¯ ) .

Fig. 4
Fig. 4

Inset, radius variation of a 50 mm optical fiber segment obtained with the microfiber probe method [10, 11]. (a), (b) Resonant transmission spectra of the optical fiber at positions a and b, respectively, shown in the inset. (c) Plot (a), magnified near a single resonance. (d) Plot (b), magnified near a single resonance.

Equations (8)

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

S m ( φ , z 1 , z 2 ) 2 π r m π 3 3 γ 2 r m 3 π m γ ( z 2 + z 1 ) + ( z 2 z 1 ) 2 4 π r m + φ r .
Ψ ( φ , z , β + i α ) m | 2 S m ( φ , z 2 , z 1 ) z 1 z 2 | z 2 = z z 1 = 0 1 / 2 exp [ i ( β + i α ) S m ( φ , 0 , z ) ] = m ( 2 π m r ) 1 / 2 exp [ i ( β + i α ) S m ( φ , 0 , z ) ] .
γ π 3 / 2 ( β q r ) 1 / 2 ,
Δ β ¯ = ( β β q ) / δ β , α ¯ = α / δ β , z ¯ = z / δ z , δ β = r 2 / 3 β 1 / 3 γ 2 / 3 , δ z = r 1 / 3 β 2 / 3 γ 1 / 3 ,
Ψ ( φ , z , β + i α ) exp ( i q φ ) I ( z ¯ , Δ β ¯ + i α ¯ ) ,
I ( z ¯ , Δ β ¯ ) = 0 d m ¯ m ¯ 1 / 2 exp [ i ( 2 Δ β ¯ m ¯ 1 3 m ¯ 3 z ¯ m ¯ + z ¯ 2 4 m ¯ ) ] ,
γ α 3 / 2 β q 1 / 2 r ,
Δ β ¯ n 1 2 ( 9 π 4 + 3 π n ) 2 / 3 , n = 0 , 1 , 2 , ,

Metrics