Abstract

A multimode optical fiber with a D-shaped cross section is an experimental paradigm of a wave system with chaotic ray dynamics. We show that seldom but usable modes, called scar modes, localized along some particular direction of the geometric trajectories, can be selectively excited. We report numerical simulations that demonstrate the importance of the so-called self-focal point in the scar mode selection process. We use a localized illumination in a passive fiber, or a localized gain in a ytterbium-doped fiber, located in the vicinity of this special point to control scar mode selection.

© 2009 Optical Society of America

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References

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  1. H.-J. Stöckmann, Quantum Chaos: an Introduction (Cambridge U. Press1999).
  2. K. Schaadt, T. Guhr, C. Ellegaard, and M. Oxborrow, “Experiments on elastomechanical wave functions in chaotic plates and their statistical features,” Phys. Rev. E 68, 036205 (2003).
    [CrossRef]
  3. H.-J. Stöckmann and J. Stein, ““Quantum chaos” in billiard studied by microwave absorption,” Phys. Rev. Lett. 64, 2215-2218 (1990).
    [CrossRef] [PubMed]
  4. S. Sridhar, “Experimental observation of scarred eigenfunctions of chaotic microwave cavities,” Phys. Rev. Lett. 67, 785-788 (1991).
    [CrossRef] [PubMed]
  5. V. Doya, O. Legrand, F. Mortessagne, and Ch. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E 65, 056223 (2002).
    [CrossRef]
  6. V. Doya, O. Legrand, and F. Mortessagne, “Optimized absorption in a chaotic double-clad fiber amplifier,” Opt. Lett. 26, 872-874 (2001).
    [CrossRef]
  7. T. Harayama, T. Fukushima, P. Davis, P. O. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, “Lasing on scar modes in fully chaotic microcavities,” Phys. Rev. E 67, 015207 (2003).
    [CrossRef]
  8. W. Fang and H. Cao, “Wave interference effect on polymer microstadium laser,” Appl. Phys. Lett. 91, 041108 (2007).
    [CrossRef]
  9. W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
    [CrossRef]
  10. M. V. Berry, “Regular and irregular semiclassical wavefunctions,” J. Phys. A 10, 2083-2092 (1977).
    [CrossRef]
  11. E. J. Heller, “Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits,” Phys. Rev. Lett. 53, 1515-1518 (1984).
    [CrossRef]
  12. M. V. Berry, “Quantum scars of classical closed orbits in phase space,” Proc. R. Soc. London Ser. A 423, 219-231(1989).
    [CrossRef]
  13. L. Kaplan, “Scars in quantum chaotic wavefunctions,” Nonlinearity 12, R1-R40 (1999).
    [CrossRef]
  14. B. Li, “Numerical study of scars in a chaotic billiard,” Phys. Rev. E 55, 5376-5379 (1997).
    [CrossRef]
  15. E. B. Bogomolny, “Smoothed wave functions of chaotic quantum systems,” Physica D: Nonlinear Phenomena 31, 169-189 (1988).
    [CrossRef]
  16. M. D. Feit and J. A. Fleck, Jr., “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19, 1154-1164 (1980).
    [CrossRef] [PubMed]
  17. C. Michel, V. Doya, O. Legrand, and F. Mortessagne, “Selective amplification of scars in a chaotic optical fiber,” Phys. Rev. Lett. 99, 224101 (2007).
    [CrossRef]
  18. M. D. Feit and J. A. Fleck, Jr., “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990-3998(1978).
    [CrossRef] [PubMed]
  19. P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002).
    [CrossRef]
  20. V. Doya, O. Legrand, and F. Mortessagne, “Optimized absorption in a chaotic double-clad fiber amplifier,” Opt. Lett. 26, 872-874 (2001).
    [CrossRef]
  21. P. Leproux, V. Doya, P. Roy, D. Pagnoux, F. Mortessagne, and O. Legrand, “Experimental study of pump power absorption along rare-earth-doped double clad optical fibres,” Opt. Commun. 218, 249-254 (2003).
    [CrossRef]
  22. This step has been achieved in collaboration with the Optical Development Workshop at the Institute of Plasma Physics, Prague, Czech Republic.

2007 (2)

W. Fang and H. Cao, “Wave interference effect on polymer microstadium laser,” Appl. Phys. Lett. 91, 041108 (2007).
[CrossRef]

C. Michel, V. Doya, O. Legrand, and F. Mortessagne, “Selective amplification of scars in a chaotic optical fiber,” Phys. Rev. Lett. 99, 224101 (2007).
[CrossRef]

2005 (1)

W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
[CrossRef]

2003 (3)

T. Harayama, T. Fukushima, P. Davis, P. O. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, “Lasing on scar modes in fully chaotic microcavities,” Phys. Rev. E 67, 015207 (2003).
[CrossRef]

K. Schaadt, T. Guhr, C. Ellegaard, and M. Oxborrow, “Experiments on elastomechanical wave functions in chaotic plates and their statistical features,” Phys. Rev. E 68, 036205 (2003).
[CrossRef]

P. Leproux, V. Doya, P. Roy, D. Pagnoux, F. Mortessagne, and O. Legrand, “Experimental study of pump power absorption along rare-earth-doped double clad optical fibres,” Opt. Commun. 218, 249-254 (2003).
[CrossRef]

2002 (2)

P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002).
[CrossRef]

V. Doya, O. Legrand, F. Mortessagne, and Ch. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E 65, 056223 (2002).
[CrossRef]

2001 (2)

1999 (1)

L. Kaplan, “Scars in quantum chaotic wavefunctions,” Nonlinearity 12, R1-R40 (1999).
[CrossRef]

1997 (1)

B. Li, “Numerical study of scars in a chaotic billiard,” Phys. Rev. E 55, 5376-5379 (1997).
[CrossRef]

1991 (1)

S. Sridhar, “Experimental observation of scarred eigenfunctions of chaotic microwave cavities,” Phys. Rev. Lett. 67, 785-788 (1991).
[CrossRef] [PubMed]

1990 (1)

H.-J. Stöckmann and J. Stein, ““Quantum chaos” in billiard studied by microwave absorption,” Phys. Rev. Lett. 64, 2215-2218 (1990).
[CrossRef] [PubMed]

1989 (1)

M. V. Berry, “Quantum scars of classical closed orbits in phase space,” Proc. R. Soc. London Ser. A 423, 219-231(1989).
[CrossRef]

1988 (1)

E. B. Bogomolny, “Smoothed wave functions of chaotic quantum systems,” Physica D: Nonlinear Phenomena 31, 169-189 (1988).
[CrossRef]

1984 (1)

E. J. Heller, “Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits,” Phys. Rev. Lett. 53, 1515-1518 (1984).
[CrossRef]

1980 (1)

1978 (1)

1977 (1)

M. V. Berry, “Regular and irregular semiclassical wavefunctions,” J. Phys. A 10, 2083-2092 (1977).
[CrossRef]

Aida, T.

T. Harayama, T. Fukushima, P. Davis, P. O. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, “Lasing on scar modes in fully chaotic microcavities,” Phys. Rev. E 67, 015207 (2003).
[CrossRef]

Berry, M. V.

M. V. Berry, “Quantum scars of classical closed orbits in phase space,” Proc. R. Soc. London Ser. A 423, 219-231(1989).
[CrossRef]

M. V. Berry, “Regular and irregular semiclassical wavefunctions,” J. Phys. A 10, 2083-2092 (1977).
[CrossRef]

Bogomolny, E. B.

E. B. Bogomolny, “Smoothed wave functions of chaotic quantum systems,” Physica D: Nonlinear Phenomena 31, 169-189 (1988).
[CrossRef]

Cao, H.

W. Fang and H. Cao, “Wave interference effect on polymer microstadium laser,” Appl. Phys. Lett. 91, 041108 (2007).
[CrossRef]

W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
[CrossRef]

Davis, P.

T. Harayama, T. Fukushima, P. Davis, P. O. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, “Lasing on scar modes in fully chaotic microcavities,” Phys. Rev. E 67, 015207 (2003).
[CrossRef]

Doya, V.

C. Michel, V. Doya, O. Legrand, and F. Mortessagne, “Selective amplification of scars in a chaotic optical fiber,” Phys. Rev. Lett. 99, 224101 (2007).
[CrossRef]

P. Leproux, V. Doya, P. Roy, D. Pagnoux, F. Mortessagne, and O. Legrand, “Experimental study of pump power absorption along rare-earth-doped double clad optical fibres,” Opt. Commun. 218, 249-254 (2003).
[CrossRef]

V. Doya, O. Legrand, F. Mortessagne, and Ch. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E 65, 056223 (2002).
[CrossRef]

V. Doya, O. Legrand, and F. Mortessagne, “Optimized absorption in a chaotic double-clad fiber amplifier,” Opt. Lett. 26, 872-874 (2001).
[CrossRef]

V. Doya, O. Legrand, and F. Mortessagne, “Optimized absorption in a chaotic double-clad fiber amplifier,” Opt. Lett. 26, 872-874 (2001).
[CrossRef]

Ellegaard, C.

K. Schaadt, T. Guhr, C. Ellegaard, and M. Oxborrow, “Experiments on elastomechanical wave functions in chaotic plates and their statistical features,” Phys. Rev. E 68, 036205 (2003).
[CrossRef]

Fang, W.

W. Fang and H. Cao, “Wave interference effect on polymer microstadium laser,” Appl. Phys. Lett. 91, 041108 (2007).
[CrossRef]

W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Fukushima, T.

T. Harayama, T. Fukushima, P. Davis, P. O. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, “Lasing on scar modes in fully chaotic microcavities,” Phys. Rev. E 67, 015207 (2003).
[CrossRef]

Guhr, T.

K. Schaadt, T. Guhr, C. Ellegaard, and M. Oxborrow, “Experiments on elastomechanical wave functions in chaotic plates and their statistical features,” Phys. Rev. E 68, 036205 (2003).
[CrossRef]

Harayama, T.

T. Harayama, T. Fukushima, P. Davis, P. O. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, “Lasing on scar modes in fully chaotic microcavities,” Phys. Rev. E 67, 015207 (2003).
[CrossRef]

Heller, E. J.

E. J. Heller, “Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits,” Phys. Rev. Lett. 53, 1515-1518 (1984).
[CrossRef]

Kaplan, L.

L. Kaplan, “Scars in quantum chaotic wavefunctions,” Nonlinearity 12, R1-R40 (1999).
[CrossRef]

Legrand, O.

C. Michel, V. Doya, O. Legrand, and F. Mortessagne, “Selective amplification of scars in a chaotic optical fiber,” Phys. Rev. Lett. 99, 224101 (2007).
[CrossRef]

P. Leproux, V. Doya, P. Roy, D. Pagnoux, F. Mortessagne, and O. Legrand, “Experimental study of pump power absorption along rare-earth-doped double clad optical fibres,” Opt. Commun. 218, 249-254 (2003).
[CrossRef]

V. Doya, O. Legrand, F. Mortessagne, and Ch. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E 65, 056223 (2002).
[CrossRef]

V. Doya, O. Legrand, and F. Mortessagne, “Optimized absorption in a chaotic double-clad fiber amplifier,” Opt. Lett. 26, 872-874 (2001).
[CrossRef]

V. Doya, O. Legrand, and F. Mortessagne, “Optimized absorption in a chaotic double-clad fiber amplifier,” Opt. Lett. 26, 872-874 (2001).
[CrossRef]

Leproux, P.

P. Leproux, V. Doya, P. Roy, D. Pagnoux, F. Mortessagne, and O. Legrand, “Experimental study of pump power absorption along rare-earth-doped double clad optical fibres,” Opt. Commun. 218, 249-254 (2003).
[CrossRef]

Li, B.

B. Li, “Numerical study of scars in a chaotic billiard,” Phys. Rev. E 55, 5376-5379 (1997).
[CrossRef]

Michel, C.

C. Michel, V. Doya, O. Legrand, and F. Mortessagne, “Selective amplification of scars in a chaotic optical fiber,” Phys. Rev. Lett. 99, 224101 (2007).
[CrossRef]

Miniatura, Ch.

V. Doya, O. Legrand, F. Mortessagne, and Ch. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E 65, 056223 (2002).
[CrossRef]

Miyasaka, T.

T. Harayama, T. Fukushima, P. Davis, P. O. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, “Lasing on scar modes in fully chaotic microcavities,” Phys. Rev. E 67, 015207 (2003).
[CrossRef]

Mortessagne, F.

C. Michel, V. Doya, O. Legrand, and F. Mortessagne, “Selective amplification of scars in a chaotic optical fiber,” Phys. Rev. Lett. 99, 224101 (2007).
[CrossRef]

P. Leproux, V. Doya, P. Roy, D. Pagnoux, F. Mortessagne, and O. Legrand, “Experimental study of pump power absorption along rare-earth-doped double clad optical fibres,” Opt. Commun. 218, 249-254 (2003).
[CrossRef]

V. Doya, O. Legrand, F. Mortessagne, and Ch. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E 65, 056223 (2002).
[CrossRef]

V. Doya, O. Legrand, and F. Mortessagne, “Optimized absorption in a chaotic double-clad fiber amplifier,” Opt. Lett. 26, 872-874 (2001).
[CrossRef]

V. Doya, O. Legrand, and F. Mortessagne, “Optimized absorption in a chaotic double-clad fiber amplifier,” Opt. Lett. 26, 872-874 (2001).
[CrossRef]

Nishimura, T.

T. Harayama, T. Fukushima, P. Davis, P. O. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, “Lasing on scar modes in fully chaotic microcavities,” Phys. Rev. E 67, 015207 (2003).
[CrossRef]

Oxborrow, M.

K. Schaadt, T. Guhr, C. Ellegaard, and M. Oxborrow, “Experiments on elastomechanical wave functions in chaotic plates and their statistical features,” Phys. Rev. E 68, 036205 (2003).
[CrossRef]

Pagnoux, D.

P. Leproux, V. Doya, P. Roy, D. Pagnoux, F. Mortessagne, and O. Legrand, “Experimental study of pump power absorption along rare-earth-doped double clad optical fibres,” Opt. Commun. 218, 249-254 (2003).
[CrossRef]

Roy, P.

P. Leproux, V. Doya, P. Roy, D. Pagnoux, F. Mortessagne, and O. Legrand, “Experimental study of pump power absorption along rare-earth-doped double clad optical fibres,” Opt. Commun. 218, 249-254 (2003).
[CrossRef]

Schaadt, K.

K. Schaadt, T. Guhr, C. Ellegaard, and M. Oxborrow, “Experiments on elastomechanical wave functions in chaotic plates and their statistical features,” Phys. Rev. E 68, 036205 (2003).
[CrossRef]

Sebbah, P.

P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002).
[CrossRef]

Sridhar, S.

S. Sridhar, “Experimental observation of scarred eigenfunctions of chaotic microwave cavities,” Phys. Rev. Lett. 67, 785-788 (1991).
[CrossRef] [PubMed]

Stein, J.

H.-J. Stöckmann and J. Stein, ““Quantum chaos” in billiard studied by microwave absorption,” Phys. Rev. Lett. 64, 2215-2218 (1990).
[CrossRef] [PubMed]

Stöckmann, H.-J.

H.-J. Stöckmann and J. Stein, ““Quantum chaos” in billiard studied by microwave absorption,” Phys. Rev. Lett. 64, 2215-2218 (1990).
[CrossRef] [PubMed]

H.-J. Stöckmann, Quantum Chaos: an Introduction (Cambridge U. Press1999).

Vaccaro, P. O.

T. Harayama, T. Fukushima, P. Davis, P. O. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, “Lasing on scar modes in fully chaotic microcavities,” Phys. Rev. E 67, 015207 (2003).
[CrossRef]

Vanneste, C.

P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002).
[CrossRef]

Yamilov, A.

W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

W. Fang and H. Cao, “Wave interference effect on polymer microstadium laser,” Appl. Phys. Lett. 91, 041108 (2007).
[CrossRef]

J. Phys. A (1)

M. V. Berry, “Regular and irregular semiclassical wavefunctions,” J. Phys. A 10, 2083-2092 (1977).
[CrossRef]

Nonlinearity (1)

L. Kaplan, “Scars in quantum chaotic wavefunctions,” Nonlinearity 12, R1-R40 (1999).
[CrossRef]

Opt. Commun. (1)

P. Leproux, V. Doya, P. Roy, D. Pagnoux, F. Mortessagne, and O. Legrand, “Experimental study of pump power absorption along rare-earth-doped double clad optical fibres,” Opt. Commun. 218, 249-254 (2003).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
[CrossRef]

Phys. Rev. B (1)

P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002).
[CrossRef]

Phys. Rev. E (4)

B. Li, “Numerical study of scars in a chaotic billiard,” Phys. Rev. E 55, 5376-5379 (1997).
[CrossRef]

T. Harayama, T. Fukushima, P. Davis, P. O. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, “Lasing on scar modes in fully chaotic microcavities,” Phys. Rev. E 67, 015207 (2003).
[CrossRef]

K. Schaadt, T. Guhr, C. Ellegaard, and M. Oxborrow, “Experiments on elastomechanical wave functions in chaotic plates and their statistical features,” Phys. Rev. E 68, 036205 (2003).
[CrossRef]

V. Doya, O. Legrand, F. Mortessagne, and Ch. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E 65, 056223 (2002).
[CrossRef]

Phys. Rev. Lett. (4)

H.-J. Stöckmann and J. Stein, ““Quantum chaos” in billiard studied by microwave absorption,” Phys. Rev. Lett. 64, 2215-2218 (1990).
[CrossRef] [PubMed]

S. Sridhar, “Experimental observation of scarred eigenfunctions of chaotic microwave cavities,” Phys. Rev. Lett. 67, 785-788 (1991).
[CrossRef] [PubMed]

E. J. Heller, “Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits,” Phys. Rev. Lett. 53, 1515-1518 (1984).
[CrossRef]

C. Michel, V. Doya, O. Legrand, and F. Mortessagne, “Selective amplification of scars in a chaotic optical fiber,” Phys. Rev. Lett. 99, 224101 (2007).
[CrossRef]

Physica D: Nonlinear Phenomena (1)

E. B. Bogomolny, “Smoothed wave functions of chaotic quantum systems,” Physica D: Nonlinear Phenomena 31, 169-189 (1988).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

M. V. Berry, “Quantum scars of classical closed orbits in phase space,” Proc. R. Soc. London Ser. A 423, 219-231(1989).
[CrossRef]

Other (2)

H.-J. Stöckmann, Quantum Chaos: an Introduction (Cambridge U. Press1999).

This step has been achieved in collaboration with the Optical Development Workshop at the Institute of Plasma Physics, Prague, Czech Republic.

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Figures (8)

Fig. 1
Fig. 1

(a) Transverse cross section of D-shaped multimode fiber with (b) a typical ray trajectory in the associated D-shaped billiard and (c)  generic speckle mode of a D-shaped chaotic optical fiber.

Fig. 2
Fig. 2

Scar modes along the two-bounce PO. Integer p labels the order of the scar mode.

Fig. 3
Fig. 3

(a) Transverse wavenumber spectrum for Gaussian beam excitation and (b) spatial overlap between the Gaussian beam and the modes. The number on each peak labels the order of the scar mode. The first peak corresponds to the fundamental mode.

Fig. 4
Fig. 4

(a) Radial integrated far field at the fiber input and (b) the maximum amplification length. The inset shows the whole far-field pattern. We note z max as the length of maximum amplification.

Fig. 5
Fig. 5

Overlap of the field of each scar mode with the doped area (squares) in comparison with the amplification coefficient (crosses).

Fig. 6
Fig. 6

Exponential evolution of the power associated with the scar of order 4.

Fig. 7
Fig. 7

Absorption spectrum measured in the fiber.

Fig. 8
Fig. 8

Experimental measurement of the gain in the ytterbium doped D-shaped fiber with an off-center core.

Tables (1)

Tables Icon

Table 1 Theoretical Value of the Transverse Wavenumber for Each Scar Mode of Order p a

Equations (3)

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

κ p L Δ φ π 2 = 2 π p ,
C ( κ , z ) = z ( L z / 2 ) z + ( L z / 2 ) d z d r ψ * ( r , 0 ) ψ ( r , z ) e i β ( κ ) z ,
P p ( z ) = P p ( 0 ) exp ( α p z ) .

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