Abstract

As opposed to the modes in an optical spherical/spheroidal microresonator, the whispering gallery modes in a long cylindrical microresonator are delocalized. Consequently, a circulating light beam that is evanescently coupled into the cylinder and experiences total internal reflection eventually radiates out along the cylinder axis. However, the self-interference of such a beam can produce a resonant mode that is strongly localized along the axial direction. Specifically, the mode characteristic width is (αβ)1/2, where α and β are the attenuation and propagation constants of the cylinder material. The Q-factor of this mode can be almost as large as the Q-factor of modes in a spheroidal microresonator with the same α divided by 2.542.

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2010 (2)

2009 (3)

2008 (1)

2007 (1)

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, Proc. SPIE 6452, 64520M (2007).
[CrossRef]

2006 (2)

A. B. Matsko and V. S. Ilchenko, IEEE J. Sel. Top. Quantum Electron. 12, 3 (2006).
[CrossRef]

I. M. White, H. Oveys, and X. Fan, Opt. Lett. 31, 1319 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (2)

2003 (2)

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

D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, Nature 421, 925 (2003).
[CrossRef] [PubMed]

2001 (1)

2000 (2)

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, Phys. Rev. E 62, 7389 (2000).
[CrossRef]

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

1998 (1)

1997 (2)

J. A. Lock, J. Opt. Soc. Am. A 14, 653 (1997).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
[CrossRef]

1987 (1)

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, J. Lightwave Technol. 5, 16 (1987).
[CrossRef]

Andrés, M. V.

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, Nature 421, 925 (2003).
[CrossRef] [PubMed]

Birks, T. A.

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

Chang, R. K.

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
[CrossRef]

Díez, A.

Dimmick, T. E.

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

Dulashko, Y.

M. Sumetsky, Y. Dulashko, and R. S. Windeler, Opt. Lett. 35, 898 (2010).
[CrossRef] [PubMed]

M. Sumetsky and Y. Dulashko, in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, 2006), paper OTuL6.

Fan, X.

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, Proc. SPIE 6452, 64520M (2007).
[CrossRef]

I. M. White, H. Oveys, and X. Fan, Opt. Lett. 31, 1319 (2006).
[CrossRef] [PubMed]

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
[CrossRef]

Gimeno, B.

Gorodetsky, M. L.

Han, Y.

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
[CrossRef]

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, J. Lightwave Technol. 5, 16 (1987).
[CrossRef]

Huang, W. P.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, J. Lightwave Technol. 5, 16 (1987).
[CrossRef]

Ilchenko, V. S.

A. B. Matsko and V. S. Ilchenko, IEEE J. Sel. Top. Quantum Electron. 12, 3 (2006).
[CrossRef]

V. S. Ilchenko, M. L. Gorodetsky, X. S. Yao, and L. Maleki, Opt. Lett. 26, 256 (2001).
[CrossRef]

Kawakami, S.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, J. Lightwave Technol. 5, 16 (1987).
[CrossRef]

Kippenberg, T. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, Nature 421, 925 (2003).
[CrossRef] [PubMed]

Knight, J. C.

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

Laine, J.-P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
[CrossRef]

Lee, R. K.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, Phys. Rev. E 62, 7389 (2000).
[CrossRef]

Li, Y.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, Phys. Rev. E 62, 7389 (2000).
[CrossRef]

Little, B. E.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
[CrossRef]

Lock, J. A.

Maleki, L.

Matsko, A. B.

A. B. Matsko and V. S. Ilchenko, IEEE J. Sel. Top. Quantum Electron. 12, 3 (2006).
[CrossRef]

Murugan, G. S.

O’Shea, D.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, Phys. Rev. Lett. 103, 053901 (2009).
[CrossRef] [PubMed]

Oveys, H.

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, Proc. SPIE 6452, 64520M (2007).
[CrossRef]

I. M. White, H. Oveys, and X. Fan, Opt. Lett. 31, 1319 (2006).
[CrossRef] [PubMed]

Pöllinger, M.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, Phys. Rev. Lett. 103, 053901 (2009).
[CrossRef] [PubMed]

Poon, A. W.

Rauschenbeutel, A.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, Phys. Rev. Lett. 103, 053901 (2009).
[CrossRef] [PubMed]

Schwefel, H. G. L.

Schwelb, O.

Spillane, S. M.

D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, Nature 421, 925 (2003).
[CrossRef] [PubMed]

Stone, A. D.

Sumetsky, M.

M. Sumetsky, Opt. Express 18, 2413 (2010).
[CrossRef] [PubMed]

M. Sumetsky, Y. Dulashko, and R. S. Windeler, Opt. Lett. 35, 898 (2010).
[CrossRef] [PubMed]

M. Sumetsky, Opt. Lett. 29, 8 (2004).
[CrossRef] [PubMed]

M. Sumetsky and Y. Dulashko, in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, 2006), paper OTuL6.

Suter, J. D.

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, Proc. SPIE 6452, 64520M (2007).
[CrossRef]

Tureci, H. E.

Vahala, K. J.

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

D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, Nature 421, 925 (2003).
[CrossRef] [PubMed]

Warken, F.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, Phys. Rev. Lett. 103, 053901 (2009).
[CrossRef] [PubMed]

Whitaker, N. A.

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, J. Lightwave Technol. 5, 16 (1987).
[CrossRef]

White, I. M.

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, Proc. SPIE 6452, 64520M (2007).
[CrossRef]

I. M. White, H. Oveys, and X. Fan, Opt. Lett. 31, 1319 (2006).
[CrossRef] [PubMed]

Wilkinson, J. S.

Windeler, R. S.

Xu, Y.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, Phys. Rev. E 62, 7389 (2000).
[CrossRef]

Yao, X. S.

Yariv, A.

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, Phys. Rev. E 62, 7389 (2000).
[CrossRef]

Zamora, V.

Zervas, M. N.

Zhang, H.

Zhu, H.

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, Proc. SPIE 6452, 64520M (2007).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. B. Matsko and V. S. Ilchenko, IEEE J. Sel. Top. Quantum Electron. 12, 3 (2006).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

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

J. Lightwave Technol. (3)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
[CrossRef]

H. A. Haus, W. P. Huang, S. Kawakami, and N. A. Whitaker, J. Lightwave Technol. 5, 16 (1987).
[CrossRef]

O. Schwelb, J. Lightwave Technol. 22, 1380 (2004).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (2)

Nature (2)

D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, Nature 421, 925 (2003).
[CrossRef] [PubMed]

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

Opt. Express (2)

Opt. Lett. (6)

Phys. Rev. E (1)

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, Phys. Rev. E 62, 7389 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, Phys. Rev. Lett. 103, 053901 (2009).
[CrossRef] [PubMed]

Proc. SPIE (1)

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, Proc. SPIE 6452, 64520M (2007).
[CrossRef]

Other (2)

M. Sumetsky and Y. Dulashko, in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, 2006), paper OTuL6.

The difference between a plain mirror resonator and a cylindrical resonator considered here is that the amplitude of a Gaussian beam propagating in three dimensions between two plain mirrors decreases with reflection number N as 1/N. In a cylinder, a Gaussian beam is not expanding in the radial direction and its amplitude decreases much slower as 1/N. For this reason, in a cylinder, the sum of the interfering components has a much stronger divergence.

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Figures (3)

Fig. 1
Fig. 1

(a) Spheroidal microresonator coupled to a microfiber. (b) Cylindrical microresonator coupled to a microfiber.

Fig. 2
Fig. 2

(a) Introduced coordinate system ( s , z , ρ ) . (b) Graphical illustration of terms in Eq. (4).

Fig. 3
Fig. 3

(a) Comparison of a conventional Lorentzian transmission resonance determined by Eq. (1) and a resonance for a cylindrical microresonator determined by Eq. (6). (b) Distribution of the resonant mode amplitude as a function of dimensionless axial coordinate ( α β 0 ) 1 / 2 z and dimensionless deviation from the resonance Δ β / ( 2 α ) .

Equations (9)

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P sph ( Δ β ) = 1 4 γ α Δ β 2 + α 2 ,
Δ β sph ( FWHM ) = 2 α .
G ml ( z , ρ , s ) = q ( s ) m / 2 H m ( β 0 1 / 2 z / q 1 / 2 ( s ) ) G 0 l ( s , z , ρ ) , G 0 l ( z , ρ , s ) = q ( s ) 1 / 2 exp [ i ( β 0 + i α ς l ) s ] exp ( β 0 z 2 / q ( s ) ) Ξ l ( ρ ) , q ( s ) = η + 2 i s .
t ( Δ β ) = t 00 ( 0 ) + t 01 ( 1 ) exp [ ( i Δ β α ) s 0 ] t 10 ( 1 ) + t 01 ( 1 ) exp [ ( i Δ β α ) s 0 ] t 11 ( 1 ) exp [ ( i Δ β α ) s 0 ] t 10 ( 2 ) + ... + t 01 ( 1 ) exp [ ( N + 1 ) ( i Δ β α ) s 0 ] t 10 ( N + 1 ) t 11 ( 1 ) t 11 ( 2 ) ... t 11 ( N ) + ... ,
P cyl ( Δ β ) = | t ( Δ β ) | 2 | Δ β s 0 | , | α s 0 | 1 1 | t 01 ( 1 ) | 2 2 | t 01 ( 1 ) | 2 Re { 0 d N N 1 / 2 exp [ ( i Δ β α ) s 0 N ] } .
P cyl ( Δ β ) 1 | t 01 ( 1 ) | 2 [ ( Δ β 2 + α 2 ) 1 / 2 + α R ( Δ β 2 + α 2 ) ] 1 / 2 .
Δ β cyl ( FWHM ) = 2 ( 3 + 2 · 3 1 / 2 ) 1 / 2 α = 5.084 α .
U l q ( z , ρ , φ ) Ξ l ( ρ ) e i q φ 0 d N N 1 / 2 exp [ i ( Δ β + i α ) s 0 N + i β 0 z 2 2 N s 0 ] = Ξ l ( ρ ) e i q φ ( i π ( Δ β + i α ) s 0 ) 1 / 2 exp { | z | [ ( Δ β + i α ) β 0 ] 1 / 2 } ,
Δ z ( FWHM ) = 2 ln ( 2 ) ( α β 0 ) 1 / 2 .

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