Abstract

We demonstrate that the lifetimes and emission patterns of the optical modes in generic (asymmetric) microresonators are strongly affected by the phenomenon of chaos-assisted tunneling and develop a theory of the effect.

© 2005 Optical Society of America

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  1. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
    [CrossRef] [PubMed]
  2. S.-X. Qian, J. Snow, H.-M. Tzeng, and R. K. Chang, Science 231, 486 (1986).
    [CrossRef] [PubMed]
  3. J. U. Nöckel and A. D. Stone, Nature 385, 45 (1997).
    [CrossRef]
  4. V. Podolskiy, E. Narimanov, W. Fang, and H. Cao, Proc. Natl. Acad. Sci. USA 101, 10498 (2004).
    [CrossRef]
  5. C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, Opt. Lett. 27, 824 (2002).
    [CrossRef]
  6. N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).
    [CrossRef]
  7. M. J. Davis and E. J. Heller, J. Chem. Phys. 75, 246 (1981).
  8. M. Hentschel and J. U. Nöckel, “The sequential-reflection model in deformed dielectric cavities,” preprint http://xxx.lanl.gov/abs/physics/0203064 .
  9. O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223(2), 43 (1993).
    [CrossRef]
  10. S. Tomsovic and D. Ullmo, Phys. Rev. E 50, 145 (1994).
    [CrossRef]
  11. F. Leyvraz and D. Ullmo, J. Phys. A 29, 2529 (1996).
    [CrossRef]
  12. H. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E. Narimanov, Opt. Express 10, 752 (2002), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  13. V. A. Podolskiy and E. E. Narimanov, Phys. Rev. Lett. 91, 263601 (2003).
    [CrossRef]
  14. M. Berry, J. Phys. A 10, 2083 (1977).
    [CrossRef]
  15. A. Mouchet, C. Miniatura, R. Kaiser, B. Grémaud, and D. Delande, Phys. Rev. E 64, 016221 (2001).
    [CrossRef]
  16. J. Zakrzewski, D. Delande, and A. Buchleitner, Phys. Rev. E 57, 1458 (1998).
    [CrossRef]
  17. M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  18. E. Doron and U. Smilansky, Phys. Rev. Lett. 68, 1255 (1992).
    [CrossRef] [PubMed]
  19. O. A. Starykh, P. R. J. Jacquod, E. E. Narimanov, and A. D. Stone, Phys. Rev. E 62, 2078 (2000).
    [CrossRef]
  20. H. E. Tureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” arXiv.org e-Print archive, , August4, 2003, http://arxiv.org/abs/physics/0308016 .
  21. E. J. Heller, P. W. O’Connor, and J. Gehlen, Phys. Scr. 40, 354 (1989).
    [CrossRef]

2004 (1)

V. Podolskiy, E. Narimanov, W. Fang, and H. Cao, Proc. Natl. Acad. Sci. USA 101, 10498 (2004).
[CrossRef]

2003 (1)

V. A. Podolskiy and E. E. Narimanov, Phys. Rev. Lett. 91, 263601 (2003).
[CrossRef]

2002 (3)

2001 (1)

A. Mouchet, C. Miniatura, R. Kaiser, B. Grémaud, and D. Delande, Phys. Rev. E 64, 016221 (2001).
[CrossRef]

2000 (1)

O. A. Starykh, P. R. J. Jacquod, E. E. Narimanov, and A. D. Stone, Phys. Rev. E 62, 2078 (2000).
[CrossRef]

1998 (2)

J. Zakrzewski, D. Delande, and A. Buchleitner, Phys. Rev. E 57, 1458 (1998).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef] [PubMed]

1997 (1)

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

1996 (1)

F. Leyvraz and D. Ullmo, J. Phys. A 29, 2529 (1996).
[CrossRef]

1994 (1)

S. Tomsovic and D. Ullmo, Phys. Rev. E 50, 145 (1994).
[CrossRef]

1993 (1)

O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223(2), 43 (1993).
[CrossRef]

1992 (1)

E. Doron and U. Smilansky, Phys. Rev. Lett. 68, 1255 (1992).
[CrossRef] [PubMed]

1989 (1)

E. J. Heller, P. W. O’Connor, and J. Gehlen, Phys. Scr. 40, 354 (1989).
[CrossRef]

1986 (1)

S.-X. Qian, J. Snow, H.-M. Tzeng, and R. K. Chang, Science 231, 486 (1986).
[CrossRef] [PubMed]

1981 (1)

M. J. Davis and E. J. Heller, J. Chem. Phys. 75, 246 (1981).

1977 (1)

M. Berry, J. Phys. A 10, 2083 (1977).
[CrossRef]

Baillargeon, J. N.

Berry, M.

M. Berry, J. Phys. A 10, 2083 (1977).
[CrossRef]

Bohigas, O.

O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223(2), 43 (1993).
[CrossRef]

Buchleitner, A.

J. Zakrzewski, D. Delande, and A. Buchleitner, Phys. Rev. E 57, 1458 (1998).
[CrossRef]

Cao, H.

V. Podolskiy, E. Narimanov, W. Fang, and H. Cao, Proc. Natl. Acad. Sci. USA 101, 10498 (2004).
[CrossRef]

Capasso, F.

C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, Opt. Lett. 27, 824 (2002).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef] [PubMed]

Chang, R. K.

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).
[CrossRef]

S.-X. Qian, J. Snow, H.-M. Tzeng, and R. K. Chang, Science 231, 486 (1986).
[CrossRef] [PubMed]

Cho, A. Y.

C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, Opt. Lett. 27, 824 (2002).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef] [PubMed]

Davis, M. J.

M. J. Davis and E. J. Heller, J. Chem. Phys. 75, 246 (1981).

Delande, D.

A. Mouchet, C. Miniatura, R. Kaiser, B. Grémaud, and D. Delande, Phys. Rev. E 64, 016221 (2001).
[CrossRef]

J. Zakrzewski, D. Delande, and A. Buchleitner, Phys. Rev. E 57, 1458 (1998).
[CrossRef]

Doron, E.

E. Doron and U. Smilansky, Phys. Rev. Lett. 68, 1255 (1992).
[CrossRef] [PubMed]

Faist, J.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef] [PubMed]

Fang, W.

V. Podolskiy, E. Narimanov, W. Fang, and H. Cao, Proc. Natl. Acad. Sci. USA 101, 10498 (2004).
[CrossRef]

Gehlen, J.

E. J. Heller, P. W. O’Connor, and J. Gehlen, Phys. Scr. 40, 354 (1989).
[CrossRef]

Gmachl, C.

C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, Opt. Lett. 27, 824 (2002).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef] [PubMed]

Grémaud, B.

A. Mouchet, C. Miniatura, R. Kaiser, B. Grémaud, and D. Delande, Phys. Rev. E 64, 016221 (2001).
[CrossRef]

Heller, E. J.

E. J. Heller, P. W. O’Connor, and J. Gehlen, Phys. Scr. 40, 354 (1989).
[CrossRef]

M. J. Davis and E. J. Heller, J. Chem. Phys. 75, 246 (1981).

Hentschel, M.

M. Hentschel and J. U. Nöckel, “The sequential-reflection model in deformed dielectric cavities,” preprint http://xxx.lanl.gov/abs/physics/0203064 .

Jacquod, P.

H. E. Tureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” arXiv.org e-Print archive, , August4, 2003, http://arxiv.org/abs/physics/0308016 .

Jacquod, P. R. J.

O. A. Starykh, P. R. J. Jacquod, E. E. Narimanov, and A. D. Stone, Phys. Rev. E 62, 2078 (2000).
[CrossRef]

Kaiser, R.

A. Mouchet, C. Miniatura, R. Kaiser, B. Grémaud, and D. Delande, Phys. Rev. E 64, 016221 (2001).
[CrossRef]

Leyvraz, F.

F. Leyvraz and D. Ullmo, J. Phys. A 29, 2529 (1996).
[CrossRef]

Miniatura, C.

A. Mouchet, C. Miniatura, R. Kaiser, B. Grémaud, and D. Delande, Phys. Rev. E 64, 016221 (2001).
[CrossRef]

Mouchet, A.

A. Mouchet, C. Miniatura, R. Kaiser, B. Grémaud, and D. Delande, Phys. Rev. E 64, 016221 (2001).
[CrossRef]

Narimanov, E.

V. Podolskiy, E. Narimanov, W. Fang, and H. Cao, Proc. Natl. Acad. Sci. USA 101, 10498 (2004).
[CrossRef]

Narimanov, E. E.

V. A. Podolskiy and E. E. Narimanov, Phys. Rev. Lett. 91, 263601 (2003).
[CrossRef]

H. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E. Narimanov, Opt. Express 10, 752 (2002), http://www.opticsexpress.org .
[CrossRef] [PubMed]

C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, Opt. Lett. 27, 824 (2002).
[CrossRef]

O. A. Starykh, P. R. J. Jacquod, E. E. Narimanov, and A. D. Stone, Phys. Rev. E 62, 2078 (2000).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef] [PubMed]

Nöckel, J. U.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef] [PubMed]

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

M. Hentschel and J. U. Nöckel, “The sequential-reflection model in deformed dielectric cavities,” preprint http://xxx.lanl.gov/abs/physics/0203064 .

O’Connor, P. W.

E. J. Heller, P. W. O’Connor, and J. Gehlen, Phys. Scr. 40, 354 (1989).
[CrossRef]

Podolskiy, V.

V. Podolskiy, E. Narimanov, W. Fang, and H. Cao, Proc. Natl. Acad. Sci. USA 101, 10498 (2004).
[CrossRef]

Podolskiy, V. A.

V. A. Podolskiy and E. E. Narimanov, Phys. Rev. Lett. 91, 263601 (2003).
[CrossRef]

Qian, S.-X.

S.-X. Qian, J. Snow, H.-M. Tzeng, and R. K. Chang, Science 231, 486 (1986).
[CrossRef] [PubMed]

Rex, N. B.

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).
[CrossRef]

Schwefel, H. G. L.

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).
[CrossRef]

H. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E. Narimanov, Opt. Express 10, 752 (2002), http://www.opticsexpress.org .
[CrossRef] [PubMed]

H. E. Tureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” arXiv.org e-Print archive, , August4, 2003, http://arxiv.org/abs/physics/0308016 .

Sivco, D. L.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef] [PubMed]

Smilansky, U.

E. Doron and U. Smilansky, Phys. Rev. Lett. 68, 1255 (1992).
[CrossRef] [PubMed]

Snow, J.

S.-X. Qian, J. Snow, H.-M. Tzeng, and R. K. Chang, Science 231, 486 (1986).
[CrossRef] [PubMed]

Starykh, O. A.

O. A. Starykh, P. R. J. Jacquod, E. E. Narimanov, and A. D. Stone, Phys. Rev. E 62, 2078 (2000).
[CrossRef]

Stone, A. D.

H. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E. Narimanov, Opt. Express 10, 752 (2002), http://www.opticsexpress.org .
[CrossRef] [PubMed]

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).
[CrossRef]

O. A. Starykh, P. R. J. Jacquod, E. E. Narimanov, and A. D. Stone, Phys. Rev. E 62, 2078 (2000).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef] [PubMed]

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

H. E. Tureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” arXiv.org e-Print archive, , August4, 2003, http://arxiv.org/abs/physics/0308016 .

Tomsovic, S.

S. Tomsovic and D. Ullmo, Phys. Rev. E 50, 145 (1994).
[CrossRef]

O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223(2), 43 (1993).
[CrossRef]

Tureci, H. E.

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).
[CrossRef]

H. E. Tureci, H. G. L. Schwefel, A. D. Stone, and E. E. Narimanov, Opt. Express 10, 752 (2002), http://www.opticsexpress.org .
[CrossRef] [PubMed]

H. E. Tureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” arXiv.org e-Print archive, , August4, 2003, http://arxiv.org/abs/physics/0308016 .

Tzeng, H.-M.

S.-X. Qian, J. Snow, H.-M. Tzeng, and R. K. Chang, Science 231, 486 (1986).
[CrossRef] [PubMed]

Ullmo, D.

F. Leyvraz and D. Ullmo, J. Phys. A 29, 2529 (1996).
[CrossRef]

S. Tomsovic and D. Ullmo, Phys. Rev. E 50, 145 (1994).
[CrossRef]

O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223(2), 43 (1993).
[CrossRef]

Zakrzewski, J.

J. Zakrzewski, D. Delande, and A. Buchleitner, Phys. Rev. E 57, 1458 (1998).
[CrossRef]

J. Chem. Phys. (1)

M. J. Davis and E. J. Heller, J. Chem. Phys. 75, 246 (1981).

J. Phys. A (2)

F. Leyvraz and D. Ullmo, J. Phys. A 29, 2529 (1996).
[CrossRef]

M. Berry, J. Phys. A 10, 2083 (1977).
[CrossRef]

Nature (1)

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

Opt. Express (1)

Opt. Lett. (1)

Phys. Rep. (1)

O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223(2), 43 (1993).
[CrossRef]

Phys. Rev. E (4)

S. Tomsovic and D. Ullmo, Phys. Rev. E 50, 145 (1994).
[CrossRef]

A. Mouchet, C. Miniatura, R. Kaiser, B. Grémaud, and D. Delande, Phys. Rev. E 64, 016221 (2001).
[CrossRef]

J. Zakrzewski, D. Delande, and A. Buchleitner, Phys. Rev. E 57, 1458 (1998).
[CrossRef]

O. A. Starykh, P. R. J. Jacquod, E. E. Narimanov, and A. D. Stone, Phys. Rev. E 62, 2078 (2000).
[CrossRef]

Phys. Rev. Lett. (3)

E. Doron and U. Smilansky, Phys. Rev. Lett. 68, 1255 (1992).
[CrossRef] [PubMed]

V. A. Podolskiy and E. E. Narimanov, Phys. Rev. Lett. 91, 263601 (2003).
[CrossRef]

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, Phys. Rev. Lett. 88, 094102 (2002).
[CrossRef]

Phys. Scr. (1)

E. J. Heller, P. W. O’Connor, and J. Gehlen, Phys. Scr. 40, 354 (1989).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

V. Podolskiy, E. Narimanov, W. Fang, and H. Cao, Proc. Natl. Acad. Sci. USA 101, 10498 (2004).
[CrossRef]

Science (2)

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef] [PubMed]

S.-X. Qian, J. Snow, H.-M. Tzeng, and R. K. Chang, Science 231, 486 (1986).
[CrossRef] [PubMed]

Other (3)

M. Hentschel and J. U. Nöckel, “The sequential-reflection model in deformed dielectric cavities,” preprint http://xxx.lanl.gov/abs/physics/0203064 .

M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

H. E. Tureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” arXiv.org e-Print archive, , August4, 2003, http://arxiv.org/abs/physics/0308016 .

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Figures (3)

Fig. 1
Fig. 1

Ray dynamics inside (a) circular and (b) asymmetric microcavities visualized through a Poincaré SOS. The wave-optical modes closely follow classical ray trajectories both in phase space [(c), (d) Husimi projection shown] and in real space [insets of (c) and (d)]. Because of conservation of angular momentum, all trajectories in the circular microcavity have a quasi-periodic whispering-gallery structure [left inset of (a)]. The trajectories above the line of total internal reflection (green) are classically trapped inside the cavity and can leave only via dynamic tunneling (red arrow). Deformation of the cavity (b) preserves the stability of some whispering-gallery orbits (red), whereas it introduces unstable (chaotic) orbits (blue); note that the chaotic trajectory can leave the cavity even if its initial angle of incidence (black square) is much higher than that of the total internal reflection. Although the regular trajectories are still classically trapped inside the cavity, the dynamic barrier that separates them from the escape is now related only to the size of the corresponding island of stability in the phase space and not to the actual distance to the critical line [once the ray leaves the island of stability and enters the chaotic region it refractively escapes the cavity, as shown by the blue trajectory in (b)]. Thus the tunneling distance is much shorter [small red arrow pointing to the square in (b)] and the tunneling probability is much larger than in the case of the circular resonator.

Fig. 2
Fig. 2

Mean width of the resonance of the diamond-shaped mode as a function of nkR, calculated by S-matrix numerical simulations (filled circles), direct evanescent escape theory8 (solid blue line), and as presented in Eq. (3) (red dashed curve). Inset, excellent agreement between Eq. (3) and the results of numerical simulations can be seen; n=2.

Fig. 3
Fig. 3

Effect of CAT on the emission directionality is shown on the example of the double-triangle mode in the dielectric cavity with quadrupole deformation =0.07; nkR=110; n=2.6 (left). Since the island of the stability of this trajectory is above the critical line (green), the mode cannot leave the resonator by refraction (right). However, in contrast with the expected tangential evanescent escape (inset and red lines in the main plot), the emission of the mode has a maximum of 90°—clear evidence of a CAT-dominated regime (the escape-pattern peak at 90° is the characteristic feature of the emission from the chaotic mode).3

Equations (4)

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

Hˆ=RER|ψRψR|+CEC|ψCψC|+RC{VRC|ψRψC|+c.c.},
V2VRC2=c0nkR2ΓAnkRπ,2AnkRπΓAnkRπ+1,0,
γCAT=c0ΓAnkRπ,2AnkRπΓAnkRπ+1,0,
rϕ=R1+ cos2ϕ.

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