Abstract

We measure experimentally the spatial intensity profiles and resonant frequencies of the transverse modes of nearly hemispherical microcavities with cavity length and mirror curvatures below 10μm. These resonators possess axially symmetric Gauss–Laguerre-like modes, but do not display the frequency degeneracies typical of large-scale optical cavities. It is possible to interpret these results using a paraxial model of cavity propagation that includes nonparabolic optical elements.

© 2007 Optical Society of America

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2006

2005

G. V. Prakash, M. Kaczmarek, A. Dyadyusha, J. J. Baumberg, and G. D'Alessandro, Opt. Express 13, 2201 (2005).
[CrossRef] [PubMed]

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, Appl. Phys. Lett. 87, 211106 (2005).
[CrossRef]

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, J. Appl. Phys. 97, 023101 (2005).
[CrossRef]

2004

2003

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

2000

J. U. Nockel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, Phys. Rev. E 62, 8677 (2000).
[CrossRef]

T. Ackemann, S. Barland, J. R. Tredicce, M. Cara, S. Balle, R. Jager, M. Grabherr, M. Miller, and K. J. Ebeling, Opt. Lett. 25, 814 (2000).
[CrossRef]

1999

H. M. Gibbs, G. Khitrova, F. Jahnke, M. Kira, and S. W. Koch, Rev. Mod. Phys. 71, 1591 (1999).
[CrossRef]

1997

R. P. Wang and M.-M. Dumitrescu, Opt. Lett. 81, 3391 (1997).

1996

H. Saito, K. Nishi, I. Ogura, S. Sugou, and Y. Sugimoto, Appl. Phys. Lett. 69, 3140 (1996).
[CrossRef]

1990

1987

F. De Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, Phys. Rev. Lett. 59, 2955 (1987).
[CrossRef] [PubMed]

1986

A. Siegman, Lasers (University Science Books, 1986).

1964

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, 1964).

1961

A. G. Fox and T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Abdelsalam, M. E.

Abram, I.

J. U. Nockel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, Phys. Rev. E 62, 8677 (2000).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, 1964).

Ackemann, T.

Adams, R.

J. U. Nockel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, Phys. Rev. E 62, 8677 (2000).
[CrossRef]

Balle, S.

Barland, S.

Bartlett, P. N.

Baumberg, J. J.

Baumberg, Jeremy J.

Benyoucef, M.

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, J. Appl. Phys. 97, 023101 (2005).
[CrossRef]

Besombes, L.

Bhongale, S.

Bourdon, G.

J. U. Nockel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, Phys. Rev. E 62, 8677 (2000).
[CrossRef]

Cara, M.

Chatterjee, S.

Cui, G.

Curtis, E. A.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, Appl. Phys. Lett. 87, 211106 (2005).
[CrossRef]

D'Alessandro, G.

De Martini, F.

F. De Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, Phys. Rev. Lett. 59, 2955 (1987).
[CrossRef] [PubMed]

Dumitrescu, M.-M.

R. P. Wang and M.-M. Dumitrescu, Opt. Lett. 81, 3391 (1997).

Dyadyusha, A.

Ebeling, K. J.

Eriksson, S.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, Appl. Phys. Lett. 87, 211106 (2005).
[CrossRef]

Forchel, A.

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, J. Appl. Phys. 97, 023101 (2005).
[CrossRef]

Foster, D. H.

D. H. Foster and J. U. Nockel, Opt. Commun. 234, 351 (2004).
[CrossRef]

Fox, A. G.

A. G. Fox and T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Gibbs, H. M.

Grabherr, M.

Hannigan, J. M.

Hinds, E. A.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, Appl. Phys. Lett. 87, 211106 (2005).
[CrossRef]

Holland, M.

Innocenti, G.

F. De Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, Phys. Rev. Lett. 59, 2955 (1987).
[CrossRef] [PubMed]

Jacobovitz, G. R.

F. De Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, Phys. Rev. Lett. 59, 2955 (1987).
[CrossRef] [PubMed]

Jager, R.

Jahnke, F.

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, J. Appl. Phys. 97, 023101 (2005).
[CrossRef]

H. M. Gibbs, G. Khitrova, F. Jahnke, M. Kira, and S. W. Koch, Rev. Mod. Phys. 71, 1591 (1999).
[CrossRef]

Kaczmarek, M.

Kelf, T.

Khitrova, G.

Kira, M.

H. M. Gibbs, G. Khitrova, F. Jahnke, M. Kira, and S. W. Koch, Rev. Mod. Phys. 71, 1591 (1999).
[CrossRef]

Koch, S. W.

H. M. Gibbs, G. Khitrova, F. Jahnke, M. Kira, and S. W. Koch, Rev. Mod. Phys. 71, 1591 (1999).
[CrossRef]

Kraft, M.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, Appl. Phys. Lett. 87, 211106 (2005).
[CrossRef]

Kukharenka, E.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, Appl. Phys. Lett. 87, 211106 (2005).
[CrossRef]

Le Ru, E.

J. U. Nockel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, Phys. Rev. E 62, 8677 (2000).
[CrossRef]

Li, T.

A. G. Fox and T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Loeckenhoff, R.

Lugiato, L. A.

Mataloni, P.

F. De Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, Phys. Rev. Lett. 59, 2955 (1987).
[CrossRef] [PubMed]

Matinaga, F. M.

Michler, P.

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, J. Appl. Phys. 97, 023101 (2005).
[CrossRef]

Miller, M.

Moison, J.-M.

J. U. Nockel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, Phys. Rev. E 62, 8677 (2000).
[CrossRef]

Moktadir, Z.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, Appl. Phys. Lett. 87, 211106 (2005).
[CrossRef]

Mosor, S.

Narducci, L. M.

Nishi, K.

H. Saito, K. Nishi, I. Ogura, S. Sugou, and Y. Sugimoto, Appl. Phys. Lett. 69, 3140 (1996).
[CrossRef]

Nockel, J. U.

D. H. Foster and J. U. Nockel, Opt. Commun. 234, 351 (2004).
[CrossRef]

J. U. Nockel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, Phys. Rev. E 62, 8677 (2000).
[CrossRef]

Ogura, I.

H. Saito, K. Nishi, I. Ogura, S. Sugou, and Y. Sugimoto, Appl. Phys. Lett. 69, 3140 (1996).
[CrossRef]

Oppo, G. L.

Pernigo, M. A.

Prakash, G. V.

Raymer, M. G.

Robert, I.

J. U. Nockel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, Phys. Rev. E 62, 8677 (2000).
[CrossRef]

Saito, H.

H. Saito, K. Nishi, I. Ogura, S. Sugou, and Y. Sugimoto, Appl. Phys. Lett. 69, 3140 (1996).
[CrossRef]

Siegman, A.

A. Siegman, Lasers (University Science Books, 1986).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, 1964).

Sugimoto, Y.

H. Saito, K. Nishi, I. Ogura, S. Sugou, and Y. Sugimoto, Appl. Phys. Lett. 69, 3140 (1996).
[CrossRef]

Sugou, S.

H. Saito, K. Nishi, I. Ogura, S. Sugou, and Y. Sugimoto, Appl. Phys. Lett. 69, 3140 (1996).
[CrossRef]

Tredicce, J. R.

Trupke, M.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, Appl. Phys. Lett. 87, 211106 (2005).
[CrossRef]

Ulrich, S. M.

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, J. Appl. Phys. 97, 023101 (2005).
[CrossRef]

Vahala, K. J.

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

Wang, R. P.

R. P. Wang and M.-M. Dumitrescu, Opt. Lett. 81, 3391 (1997).

Wiersig, J.

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, J. Appl. Phys. 97, 023101 (2005).
[CrossRef]

Appl. Phys. Lett.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, Appl. Phys. Lett. 87, 211106 (2005).
[CrossRef]

H. Saito, K. Nishi, I. Ogura, S. Sugou, and Y. Sugimoto, Appl. Phys. Lett. 69, 3140 (1996).
[CrossRef]

Bell Syst. Tech. J.

A. G. Fox and T. Li, Bell Syst. Tech. J. 40, 453 (1961).

J. Appl. Phys.

M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, J. Appl. Phys. 97, 023101 (2005).
[CrossRef]

J. Opt. Soc. Am. B

Nature

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

Opt. Commun.

D. H. Foster and J. U. Nockel, Opt. Commun. 234, 351 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. E

J. U. Nockel, G. Bourdon, E. Le Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, Phys. Rev. E 62, 8677 (2000).
[CrossRef]

Phys. Rev. Lett.

F. De Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, Phys. Rev. Lett. 59, 2955 (1987).
[CrossRef] [PubMed]

Rev. Mod. Phys.

H. M. Gibbs, G. Khitrova, F. Jahnke, M. Kira, and S. W. Koch, Rev. Mod. Phys. 71, 1591 (1999).
[CrossRef]

Other

A. Siegman, Lasers (University Science Books, 1986).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (Dover, 1964).

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Figures (3)

Fig. 1
Fig. 1

(a) Geometry of microcavity (R, radius of curvature; L, cavity length; a, radius of flat bottom; d, dish thickness) and experimental setup, with (b) typical transmission spectrum spanning two longitudinal mode families.

Fig. 2
Fig. 2

Radial intensity profiles of the transmitted light as a function of energy, showing families of p modes for a cavity L = 6.5 μ m , R = 10 μ m . Inset: spatial intensity distributions at energies shown by arrows labeled a–d.

Fig. 3
Fig. 3

Experimental (red circles) and theoretical: dish model (blue crosses), aperture model (black triangles). (a) Transverse micromode splitting, Δ ω ̃ q T ; (b) degeneracy lifting, δ ϵ q ; (c) finesse. L = 9.7 μ m , R = 10 μ m , p = 0 , and experimental flat radius a = 2.5 μ m , with model a as indicated.

Equations (2)

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

G p q ( r , ϕ , z ) = ( 2 r ) q 2 π w 0 L p q ( 2 r 2 w 0 2 ) e r 2 w 0 2 + i q ϕ ,
ω ̃ n p q = n θ Au + ( 2 p + q + 1 ) Δ ω ̃ T ,

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