Abstract

Measurements of the optical spectra of semiconductor injection lasers with deformed cylinder resonators show strong indications of the classical Kolmogorov–Arnold–Moser transition from integrability to chaos for devices with small deformation. At larger deformation, evidence for laser action on scar modes is obtained. The diode lasers operate with TE polarization, resulting in laser action on (partially) chaotic whispering-gallery modes for all deformations.

© 2002 Optical Society of America

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References

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  1. J. U. Noeckel and A. D. Stone, Nature 385, 45 (1997).
    [CrossRef]
  2. A. Mekis, J. U. Noeckel, G. Chen, A. D. Stone, and R. K. Chang, Phys. Rev. Lett. 75, 2682 (1995).
    [CrossRef] [PubMed]
  3. M. Berggren, A. Dodabalapur, Z. Bao, and R. E. Slusher, Adv. Mater. 9, 968 (1997).
    [CrossRef]
  4. S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
    [CrossRef]
  5. T. Fukushima, J. Lightw. Technol. 18, 2208 (2000).
    [CrossRef]
  6. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1989).
  7. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Noeckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
    [CrossRef] [PubMed]
  8. S. Chang, N. B. Rex, R. K. Chang, G. Chong, and L. J. Guido, Appl. Phys. Lett. 75, 166 (1999).
    [CrossRef]
  9. By use of conventional, manual wet-chemical etching as a fabrication method for the cylinders, an appropriate amount of shape asymmetry is automatically introduced, as was confirmed by plain microscopy.
  10. A. J. Lichtenberg and M. A. Liberman, Regular and Chaotic Dynamics (Springer-Verlag, Berlin, 1991).
  11. P. LeBoeuf and M. Saraceno, J. Phys. A 23, 1745 (1990).
    [CrossRef]
  12. Note that for the mode doublets related to each other by a discrete symmetry, there is also an intrinsic splitting similar to double-well energy splitting in quantum mechanics. Such splittings, however, are very small and below the experimental resolution.
  13. R. J. Heller, in Chaos and Quantum Physics, M. J. Giannoni, A. Voros, and J. Zinn-Justin, eds. (Elsevier, Amsterdam, 1991), pp. 547–663.
  14. Note that once it is established the average small spacing is not very sensitive to the further increase of the global deformation, as the degree of asymmetry does not vary significantly with e.
  15. N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” http://xxx.lanl.gov/abs/physics/0105089.

2000

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

T. Fukushima, J. Lightw. Technol. 18, 2208 (2000).
[CrossRef]

1999

S. Chang, N. B. Rex, R. K. Chang, G. Chong, and L. J. Guido, Appl. Phys. Lett. 75, 166 (1999).
[CrossRef]

1998

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

1997

M. Berggren, A. Dodabalapur, Z. Bao, and R. E. Slusher, Adv. Mater. 9, 968 (1997).
[CrossRef]

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

1995

A. Mekis, J. U. Noeckel, G. Chen, A. D. Stone, and R. K. Chang, Phys. Rev. Lett. 75, 2682 (1995).
[CrossRef] [PubMed]

1991

A. J. Lichtenberg and M. A. Liberman, Regular and Chaotic Dynamics (Springer-Verlag, Berlin, 1991).

R. J. Heller, in Chaos and Quantum Physics, M. J. Giannoni, A. Voros, and J. Zinn-Justin, eds. (Elsevier, Amsterdam, 1991), pp. 547–663.

1990

P. LeBoeuf and M. Saraceno, J. Phys. A 23, 1745 (1990).
[CrossRef]

1989

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1989).

Arnold, V. I.

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1989).

Bao, Z.

M. Berggren, A. Dodabalapur, Z. Bao, and R. E. Slusher, Adv. Mater. 9, 968 (1997).
[CrossRef]

Berggren, M.

M. Berggren, A. Dodabalapur, Z. Bao, and R. E. Slusher, Adv. Mater. 9, 968 (1997).
[CrossRef]

Capasso, F.

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

Chang, R. K.

S. Chang, N. B. Rex, R. K. Chang, G. Chong, and L. J. Guido, Appl. Phys. Lett. 75, 166 (1999).
[CrossRef]

A. Mekis, J. U. Noeckel, G. Chen, A. D. Stone, and R. K. Chang, Phys. Rev. Lett. 75, 2682 (1995).
[CrossRef] [PubMed]

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” http://xxx.lanl.gov/abs/physics/0105089.

Chang, S.

S. Chang, N. B. Rex, R. K. Chang, G. Chong, and L. J. Guido, Appl. Phys. Lett. 75, 166 (1999).
[CrossRef]

Chen, G.

A. Mekis, J. U. Noeckel, G. Chen, A. D. Stone, and R. K. Chang, Phys. Rev. Lett. 75, 2682 (1995).
[CrossRef] [PubMed]

Cho, A. Y.

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

Chong, G.

S. Chang, N. B. Rex, R. K. Chang, G. Chong, and L. J. Guido, Appl. Phys. Lett. 75, 166 (1999).
[CrossRef]

Dodabalapur, A.

M. Berggren, A. Dodabalapur, Z. Bao, and R. E. Slusher, Adv. Mater. 9, 968 (1997).
[CrossRef]

Faist, J.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

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

Fukushima, T.

T. Fukushima, J. Lightw. Technol. 18, 2208 (2000).
[CrossRef]

Gianordoli, S.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

Gmachl, C.

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

Gornik, E.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

Guido, L. J.

S. Chang, N. B. Rex, R. K. Chang, G. Chong, and L. J. Guido, Appl. Phys. Lett. 75, 166 (1999).
[CrossRef]

Heller, R. J.

R. J. Heller, in Chaos and Quantum Physics, M. J. Giannoni, A. Voros, and J. Zinn-Justin, eds. (Elsevier, Amsterdam, 1991), pp. 547–663.

Hvozdara, L.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

LeBoeuf, P.

P. LeBoeuf and M. Saraceno, J. Phys. A 23, 1745 (1990).
[CrossRef]

Liberman, M. A.

A. J. Lichtenberg and M. A. Liberman, Regular and Chaotic Dynamics (Springer-Verlag, Berlin, 1991).

Lichtenberg, A. J.

A. J. Lichtenberg and M. A. Liberman, Regular and Chaotic Dynamics (Springer-Verlag, Berlin, 1991).

Mekis, A.

A. Mekis, J. U. Noeckel, G. Chen, A. D. Stone, and R. K. Chang, Phys. Rev. Lett. 75, 2682 (1995).
[CrossRef] [PubMed]

Narimanov, E. E.

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

Noeckel, J. U.

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

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

A. Mekis, J. U. Noeckel, G. Chen, A. D. Stone, and R. K. Chang, Phys. Rev. Lett. 75, 2682 (1995).
[CrossRef] [PubMed]

Rex, N. B.

S. Chang, N. B. Rex, R. K. Chang, G. Chong, and L. J. Guido, Appl. Phys. Lett. 75, 166 (1999).
[CrossRef]

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” http://xxx.lanl.gov/abs/physics/0105089.

Saraceno, M.

P. LeBoeuf and M. Saraceno, J. Phys. A 23, 1745 (1990).
[CrossRef]

Schrenk, W.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

Schwefel, H. G. L.

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” http://xxx.lanl.gov/abs/physics/0105089.

Sivco, D. L.

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

Slusher, R. E.

M. Berggren, A. Dodabalapur, Z. Bao, and R. E. Slusher, Adv. Mater. 9, 968 (1997).
[CrossRef]

Stone, A. D.

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

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

A. Mekis, J. U. Noeckel, G. Chen, A. D. Stone, and R. K. Chang, Phys. Rev. Lett. 75, 2682 (1995).
[CrossRef] [PubMed]

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” http://xxx.lanl.gov/abs/physics/0105089.

Strasser, G.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

Tureci, H. E.

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” http://xxx.lanl.gov/abs/physics/0105089.

Adv. Mater.

M. Berggren, A. Dodabalapur, Z. Bao, and R. E. Slusher, Adv. Mater. 9, 968 (1997).
[CrossRef]

Appl. Phys. Lett.

S. Chang, N. B. Rex, R. K. Chang, G. Chong, and L. J. Guido, Appl. Phys. Lett. 75, 166 (1999).
[CrossRef]

IEEE J. Quantum Electron.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, IEEE J. Quantum Electron. 36, 458 (2000).
[CrossRef]

J. Lightw. Technol.

T. Fukushima, J. Lightw. Technol. 18, 2208 (2000).
[CrossRef]

J. Phys. A

P. LeBoeuf and M. Saraceno, J. Phys. A 23, 1745 (1990).
[CrossRef]

Nature

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

Phys. Rev. Lett.

A. Mekis, J. U. Noeckel, G. Chen, A. D. Stone, and R. K. Chang, Phys. Rev. Lett. 75, 2682 (1995).
[CrossRef] [PubMed]

Science

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

Other

Note that for the mode doublets related to each other by a discrete symmetry, there is also an intrinsic splitting similar to double-well energy splitting in quantum mechanics. Such splittings, however, are very small and below the experimental resolution.

R. J. Heller, in Chaos and Quantum Physics, M. J. Giannoni, A. Voros, and J. Zinn-Justin, eds. (Elsevier, Amsterdam, 1991), pp. 547–663.

Note that once it is established the average small spacing is not very sensitive to the further increase of the global deformation, as the degree of asymmetry does not vary significantly with e.

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” http://xxx.lanl.gov/abs/physics/0105089.

By use of conventional, manual wet-chemical etching as a fabrication method for the cylinders, an appropriate amount of shape asymmetry is automatically introduced, as was confirmed by plain microscopy.

A. J. Lichtenberg and M. A. Liberman, Regular and Chaotic Dynamics (Springer-Verlag, Berlin, 1991).

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1989).

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic device structure of a deep-etched quadrupolar cylinder laser. (b) Shape of a quadrupole with =0.16. The schematic represents cylindrical coordinates r,φ and angle of incidence χ at the boundary. The thick solid and the dashed gray lines indicate the two triangular orbits.

Fig. 2
Fig. 2

(a) Spectra of a circular and four quadrupolar deformed single-quantum-well cylinder lasers obtained at 200 K heat-sink temperature and with pulsed peak currents of 120 mA, i.e., approximately twice the threshold current. The center wavelength is 950 nm. (b) Magnitude of the mode splitting as a function of deformation for one set of lasers. , values of individual mode doublets measured close to laser threshold; ×, data obtained at approximately twice the threshold current.

Fig. 3
Fig. 3

Husimi phase space projections of the wave equation solutions for quadrupolar deformed lasers with (a) =0.03, (b) =0.06, and (c) =0.12. The sine of angle of incidence χ and the polar angle of bounce point φ are oriented as in Fig. 1(b). (a) For small deformation 0.05, the modes are constrained to unbroken KAM tori. (b) As the system is driven to chaos, the KAM tori break and the mode is now localized at a stable periodic orbit. Finally, at high deformations, the originally stable periodic orbits bifurcate and become unstable. (c) The mode is still localized near the orbit, a phenomenon known as wave-function scarring. These scarred states are also present at higher deformations, i.e., >0.12.

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