Abstract

We demonstrate numerically calculated electromagnetic eigenmodes of a 3D dome cavity resonator that owe their shape and character entirely to the Goos–Hänchen effect. The V-shaped modes, which have purely TE or TM polarization, are well described by a 2D billiard map with the Goos–Hänchen shift included. A phase space plot of this augmented billiard map reveals a saddle-node bifurcation; the stable periodic orbit that is created in the bifurcation corresponds to the numerically calculated eigenmode, dictating the angle of its “V.” A transition from a fundamental Gaussian to a TM V mode has been observed as the cavity is lengthened to become nearly hemispherical.

© 2007 Optical Society of America

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  1. F. Goos and H. Hänchen, Ann. Phys. 1, 333 (1947).
    [CrossRef]
  2. N. H. Tran, L. Dutriaux, P. Balcou, A. Le Floch, and F. Bretenaker, Opt. Lett. 20, 1233 (1995).
    [CrossRef] [PubMed]
  3. F. Bretenaker, A. Le Floch, and L. Dutriaux, Phys. Rev. Lett. 68, 931 (1992).
    [CrossRef] [PubMed]
  4. L. Dutriaux, A. Le Floch, and F. Bretenaker, J. Opt. Soc. Am. B 12, 2283 (1992).
    [CrossRef]
  5. D. Q. Chowdhurry, D. H. Leach, and R. K. Chang, J. Opt. Soc. Am. A 11, 1110 (1994).
    [CrossRef]
  6. D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, IEEE J. Quantum Electron. 29, 1950 (1993).
    [CrossRef]
  7. D. H. Foster and J. U. Nöckel, Opt. Commun. 234, 351 (2004).
    [CrossRef]
  8. K. V. Artmann, Ann. Phys. 2, 87 (1948).
    [CrossRef]
  9. W. Nasalski, J. Opt. Soc. Am. A 13, 172 (1996).
    [CrossRef]
  10. D. H. Foster and J. U. Nöckel, Opt. Lett. 29, 2788 (2004).
    [CrossRef] [PubMed]
  11. D. H. Foster, 'Fabry-Perot and whispering gallery modes in realistic resonator models,' Ph.D. thesis, (University of Oregon, 2006), http://hdl.handle.net/1794/3778.
  12. R. Blümel, T. M. Antonsen, B. Georgeot, E. Ott, and R. E. Prange, Phys. Rev. Lett. 76, 2476 (1996).
    [CrossRef] [PubMed]

2004

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

D. H. Foster and J. U. Nöckel, Opt. Lett. 29, 2788 (2004).
[CrossRef] [PubMed]

1996

W. Nasalski, J. Opt. Soc. Am. A 13, 172 (1996).
[CrossRef]

R. Blümel, T. M. Antonsen, B. Georgeot, E. Ott, and R. E. Prange, Phys. Rev. Lett. 76, 2476 (1996).
[CrossRef] [PubMed]

1995

1994

1993

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, IEEE J. Quantum Electron. 29, 1950 (1993).
[CrossRef]

1992

F. Bretenaker, A. Le Floch, and L. Dutriaux, Phys. Rev. Lett. 68, 931 (1992).
[CrossRef] [PubMed]

L. Dutriaux, A. Le Floch, and F. Bretenaker, J. Opt. Soc. Am. B 12, 2283 (1992).
[CrossRef]

1948

K. V. Artmann, Ann. Phys. 2, 87 (1948).
[CrossRef]

1947

F. Goos and H. Hänchen, Ann. Phys. 1, 333 (1947).
[CrossRef]

Antonsen, T. M.

R. Blümel, T. M. Antonsen, B. Georgeot, E. Ott, and R. E. Prange, Phys. Rev. Lett. 76, 2476 (1996).
[CrossRef] [PubMed]

Artmann, K. V.

K. V. Artmann, Ann. Phys. 2, 87 (1948).
[CrossRef]

Babic, D. I.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, IEEE J. Quantum Electron. 29, 1950 (1993).
[CrossRef]

Balcou, P.

Blümel, R.

R. Blümel, T. M. Antonsen, B. Georgeot, E. Ott, and R. E. Prange, Phys. Rev. Lett. 76, 2476 (1996).
[CrossRef] [PubMed]

Bowers, J. E.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, IEEE J. Quantum Electron. 29, 1950 (1993).
[CrossRef]

Bretenaker, F.

N. H. Tran, L. Dutriaux, P. Balcou, A. Le Floch, and F. Bretenaker, Opt. Lett. 20, 1233 (1995).
[CrossRef] [PubMed]

L. Dutriaux, A. Le Floch, and F. Bretenaker, J. Opt. Soc. Am. B 12, 2283 (1992).
[CrossRef]

F. Bretenaker, A. Le Floch, and L. Dutriaux, Phys. Rev. Lett. 68, 931 (1992).
[CrossRef] [PubMed]

Chang, R. K.

Chowdhurry, D. Q.

Chung, Y.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, IEEE J. Quantum Electron. 29, 1950 (1993).
[CrossRef]

Dagli, N.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, IEEE J. Quantum Electron. 29, 1950 (1993).
[CrossRef]

Dutriaux, L.

N. H. Tran, L. Dutriaux, P. Balcou, A. Le Floch, and F. Bretenaker, Opt. Lett. 20, 1233 (1995).
[CrossRef] [PubMed]

L. Dutriaux, A. Le Floch, and F. Bretenaker, J. Opt. Soc. Am. B 12, 2283 (1992).
[CrossRef]

F. Bretenaker, A. Le Floch, and L. Dutriaux, Phys. Rev. Lett. 68, 931 (1992).
[CrossRef] [PubMed]

Foster, D. H.

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

D. H. Foster and J. U. Nöckel, Opt. Lett. 29, 2788 (2004).
[CrossRef] [PubMed]

D. H. Foster, 'Fabry-Perot and whispering gallery modes in realistic resonator models,' Ph.D. thesis, (University of Oregon, 2006), http://hdl.handle.net/1794/3778.

Georgeot, B.

R. Blümel, T. M. Antonsen, B. Georgeot, E. Ott, and R. E. Prange, Phys. Rev. Lett. 76, 2476 (1996).
[CrossRef] [PubMed]

Goos, F.

F. Goos and H. Hänchen, Ann. Phys. 1, 333 (1947).
[CrossRef]

Hänchen, H.

F. Goos and H. Hänchen, Ann. Phys. 1, 333 (1947).
[CrossRef]

Le Floch, A.

N. H. Tran, L. Dutriaux, P. Balcou, A. Le Floch, and F. Bretenaker, Opt. Lett. 20, 1233 (1995).
[CrossRef] [PubMed]

L. Dutriaux, A. Le Floch, and F. Bretenaker, J. Opt. Soc. Am. B 12, 2283 (1992).
[CrossRef]

F. Bretenaker, A. Le Floch, and L. Dutriaux, Phys. Rev. Lett. 68, 931 (1992).
[CrossRef] [PubMed]

Leach, D. H.

Nasalski, W.

Nöckel, J. U.

D. H. Foster and J. U. Nöckel, Opt. Lett. 29, 2788 (2004).
[CrossRef] [PubMed]

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

Ott, E.

R. Blümel, T. M. Antonsen, B. Georgeot, E. Ott, and R. E. Prange, Phys. Rev. Lett. 76, 2476 (1996).
[CrossRef] [PubMed]

Prange, R. E.

R. Blümel, T. M. Antonsen, B. Georgeot, E. Ott, and R. E. Prange, Phys. Rev. Lett. 76, 2476 (1996).
[CrossRef] [PubMed]

Tran, N. H.

Ann. Phys.

F. Goos and H. Hänchen, Ann. Phys. 1, 333 (1947).
[CrossRef]

K. V. Artmann, Ann. Phys. 2, 87 (1948).
[CrossRef]

IEEE J. Quantum Electron.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, IEEE J. Quantum Electron. 29, 1950 (1993).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

L. Dutriaux, A. Le Floch, and F. Bretenaker, J. Opt. Soc. Am. B 12, 2283 (1992).
[CrossRef]

Opt. Commun.

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

Opt. Lett.

Phys. Rev. Lett.

F. Bretenaker, A. Le Floch, and L. Dutriaux, Phys. Rev. Lett. 68, 931 (1992).
[CrossRef] [PubMed]

R. Blümel, T. M. Antonsen, B. Georgeot, E. Ott, and R. E. Prange, Phys. Rev. Lett. 76, 2476 (1996).
[CrossRef] [PubMed]

Other

D. H. Foster, 'Fabry-Perot and whispering gallery modes in realistic resonator models,' Ph.D. thesis, (University of Oregon, 2006), http://hdl.handle.net/1794/3778.

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Figures (4)

Fig. 1
Fig. 1

(a) Diagram of the simple ray model for the V mode, in a dome cavity, defining the polar angle θ and the mirror labels M 1 , M 2 . (b) Gray-scale plot of a VTE 0 1 mode at mirror offset z 1 = 0.4 μ m . (c) Angular spectrum of this mode.

Fig. 2
Fig. 2

(a) Comparison of θ k with χ as a function of z 1 for a VTE 0 1 mode. The wavenumber k used in the χ s calculation is taken from the actual mode. (b) Left-hand side (LHS) and right-hand side (RHS) of Eq. (2) (using Δ s ) with respect to χ (deg), with fixed λ. The solutions corresponding to stable (s) and unstable (u) orbits alternate.

Fig. 3
Fig. 3

Billiard map with GH shift for TE polarization at z 1 = 0.55 , showing creation of the stable V-ray island and separatrix.

Fig. 4
Fig. 4

Angular spectra showing the creation of the VTM 0 1 mode from the fundamental Gaussian as z 1 decreases. Dashed curves, ψ p 2 ; solid curves, ψ s 2 .

Equations (4)

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Δ s p ( θ k ; k ) = 1 k cos θ k d ϕ s p ( θ k ; k ) d θ k ,
Δ s p ( χ ; k ) = 2 z 1 tan χ ,
T : ( p θ ( n ) , θ ( n ) ) ( p θ ( n + 1 ) , θ ( n + 1 ) ) ,
QTE ( M ) N m = LG ( N m 1 ) 2 m 1 σ + ± LG ( N m + 1 ) 2 m + 1 σ

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