Abstract

We suggest semianalytic estimates for the Q-switching instability boundary of the continuous-wave (cw) mode-locking regime domain for a ring-cavity semiconductor laser. We use a differential delay laser model that allows us to assume large gain and loss in the cavity, which is a typical situation for this class of lasers. The Q-switching instability boundary is obtained as a Neimark–Sacker bifurcation curve of a map describing the transformation of pulse parameters after a round trip in the cavity. We study the dependence of this boundary on laser parameters and show that our theoretical results are in qualitative agreement with the experimental data obtained with a passively mode-locked monolithic semiconductor laser.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. A. Haus, "Parameter ranges for cw passive mode locking," IEEE J. Quantum Electron. 12, 169-176 (1976).
    [CrossRef]
  2. J. Palaski and K. Lau, "Parameter ranges for ultrahigh frequency mode locking of semiconductor lasers," Appl. Phys. Lett. 59, 7-9 (1991).
    [CrossRef]
  3. F. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid-state laser dynamics by semiconductor devices," Opt. Eng. 34, 2024-2036 (1995).
  4. J. L. A. Dubbeldam, J. A. Leegwater, and D. Lenstra, "Theory of mode-locked semi-conductor lasers with finite relaxation times," Appl. Phys. Lett. 70, 1938-1940 (1997).
    [CrossRef]
  5. C. Hönniger, R. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, "Q-switching stability limits of continuous-wave passive mode locking," J. Opt. Soc. Am. B 16, 46-56 (1999).
  6. R. Paschotta and U. Keller, "Passive mode-locking with slow saturable absorbers," Appl. Phys. B 73, 653-662 (2001).
    [CrossRef]
  7. T. Kolokolnikov, Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5; T. Erneux, Optique Nonlinéaire Théorique, Université Libre de Bruxelles, Campus Plaine CP 231, B-1050 Bruxelles, Belgium; N. Joly, Department of Physics, Centre for Photonics and Photonic Materials, University of Bath, Bath BA2 7AY, United Kingdom; and S. Bielawski, Laboratoire de Physique des Lasers, Atomes et Molécules, UMR CNRS 8523, Centre d'Études et de Recherches Lasers et Applications, FR CNRS 2416, Université des Sciences et Technologies de Lille, F-59655 Villeneuve D'Ascq Cedex, France, are preparing a manuscript titled, "The Q-switching instability in passively mode-locked lasers."
  8. E. Avrutin, J. Marsh, and E. Portnoi, "Monolithic and multi-GigaHertz mode-locked semiconductor lasers: constructions, experiments, models, and applications," IEE Proc.: Optoelectron. 147, 251-278 (2000).
    [CrossRef]
  9. A. Vladimirov, D. Turaev, and G. Kozyreff, "Delay differential equations for mode-locked semiconductor lasers," Opt. Lett. 29, 1221-1223 (2004).
    [CrossRef] [PubMed]
  10. G. H. C. New, "Pulse evolution in mode-locked quasi-continuous lasers," IEEE J. Quantum Electron. 10, 115-124 (1974).
    [CrossRef]
  11. P. Mandel and T. Erneux, "Stationary, harmonic, and pulsed operations of an optically bistable laser with saturable absorber: I," Phys. Rev. A 30, 1893-1901 (1984).
    [CrossRef]
  12. T. Erneux, "Q-switching bifurcation in a laser with a saturable absorber," J. Opt. Soc. Am. B 5, 1063-l069 (1988).
    [CrossRef]
  13. M. Yamada, "A theoretical analysis of self-sustained pulsation phenomena in narrow-stripe semiconductor lasers," IEEE J. Quantum Electron. QE-29, 1330-1336 (1993).
    [CrossRef]
  14. H. Haus, "Theory of mode locking with a slow saturable absorber," IEEE J. Quantum Electron. 11, 736-746 (1975).
    [CrossRef]
  15. A. Vladimirov and D. Turaev, "Model for passive mode-locking in semiconductor lasers," Phys. Rev. A 72, 033808 (2005).
    [CrossRef]
  16. A. Vladimirov and D. Turaev, "A new model for a mode-locked semiconductor laser," Radiophys. Quantum Electron. 47, 857-865 (2004).
    [CrossRef]
  17. B. Hüttl, R. Kaiser, W. Rehbein, H. Stolpe, C. Kindel, S. Fidorra, A. Steffan, A. Umbach, and H. Heidrich, "Low noise monolithic 40GHz mode-locked DBR lasers based on GaInAsP/InP," presented at the 17th Indium Phosphide and Related Materials Conference, Glasgow, UK, 8-12 May 2005.

2005 (1)

A. Vladimirov and D. Turaev, "Model for passive mode-locking in semiconductor lasers," Phys. Rev. A 72, 033808 (2005).
[CrossRef]

2004 (2)

A. Vladimirov and D. Turaev, "A new model for a mode-locked semiconductor laser," Radiophys. Quantum Electron. 47, 857-865 (2004).
[CrossRef]

A. Vladimirov, D. Turaev, and G. Kozyreff, "Delay differential equations for mode-locked semiconductor lasers," Opt. Lett. 29, 1221-1223 (2004).
[CrossRef] [PubMed]

2001 (1)

R. Paschotta and U. Keller, "Passive mode-locking with slow saturable absorbers," Appl. Phys. B 73, 653-662 (2001).
[CrossRef]

2000 (1)

E. Avrutin, J. Marsh, and E. Portnoi, "Monolithic and multi-GigaHertz mode-locked semiconductor lasers: constructions, experiments, models, and applications," IEE Proc.: Optoelectron. 147, 251-278 (2000).
[CrossRef]

1999 (1)

1997 (1)

J. L. A. Dubbeldam, J. A. Leegwater, and D. Lenstra, "Theory of mode-locked semi-conductor lasers with finite relaxation times," Appl. Phys. Lett. 70, 1938-1940 (1997).
[CrossRef]

1995 (1)

F. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid-state laser dynamics by semiconductor devices," Opt. Eng. 34, 2024-2036 (1995).

1993 (1)

M. Yamada, "A theoretical analysis of self-sustained pulsation phenomena in narrow-stripe semiconductor lasers," IEEE J. Quantum Electron. QE-29, 1330-1336 (1993).
[CrossRef]

1991 (1)

J. Palaski and K. Lau, "Parameter ranges for ultrahigh frequency mode locking of semiconductor lasers," Appl. Phys. Lett. 59, 7-9 (1991).
[CrossRef]

1988 (1)

1984 (1)

P. Mandel and T. Erneux, "Stationary, harmonic, and pulsed operations of an optically bistable laser with saturable absorber: I," Phys. Rev. A 30, 1893-1901 (1984).
[CrossRef]

1976 (1)

H. A. Haus, "Parameter ranges for cw passive mode locking," IEEE J. Quantum Electron. 12, 169-176 (1976).
[CrossRef]

1975 (1)

H. Haus, "Theory of mode locking with a slow saturable absorber," IEEE J. Quantum Electron. 11, 736-746 (1975).
[CrossRef]

1974 (1)

G. H. C. New, "Pulse evolution in mode-locked quasi-continuous lasers," IEEE J. Quantum Electron. 10, 115-124 (1974).
[CrossRef]

Avrutin, E.

E. Avrutin, J. Marsh, and E. Portnoi, "Monolithic and multi-GigaHertz mode-locked semiconductor lasers: constructions, experiments, models, and applications," IEE Proc.: Optoelectron. 147, 251-278 (2000).
[CrossRef]

Bielawski, S.

T. Kolokolnikov, Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5; T. Erneux, Optique Nonlinéaire Théorique, Université Libre de Bruxelles, Campus Plaine CP 231, B-1050 Bruxelles, Belgium; N. Joly, Department of Physics, Centre for Photonics and Photonic Materials, University of Bath, Bath BA2 7AY, United Kingdom; and S. Bielawski, Laboratoire de Physique des Lasers, Atomes et Molécules, UMR CNRS 8523, Centre d'Études et de Recherches Lasers et Applications, FR CNRS 2416, Université des Sciences et Technologies de Lille, F-59655 Villeneuve D'Ascq Cedex, France, are preparing a manuscript titled, "The Q-switching instability in passively mode-locked lasers."

Brovelli, L.

F. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid-state laser dynamics by semiconductor devices," Opt. Eng. 34, 2024-2036 (1995).

Calasso, I.

F. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid-state laser dynamics by semiconductor devices," Opt. Eng. 34, 2024-2036 (1995).

Dubbeldam, J. L. A.

J. L. A. Dubbeldam, J. A. Leegwater, and D. Lenstra, "Theory of mode-locked semi-conductor lasers with finite relaxation times," Appl. Phys. Lett. 70, 1938-1940 (1997).
[CrossRef]

Erneux, T.

T. Erneux, "Q-switching bifurcation in a laser with a saturable absorber," J. Opt. Soc. Am. B 5, 1063-l069 (1988).
[CrossRef]

P. Mandel and T. Erneux, "Stationary, harmonic, and pulsed operations of an optically bistable laser with saturable absorber: I," Phys. Rev. A 30, 1893-1901 (1984).
[CrossRef]

T. Kolokolnikov, Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5; T. Erneux, Optique Nonlinéaire Théorique, Université Libre de Bruxelles, Campus Plaine CP 231, B-1050 Bruxelles, Belgium; N. Joly, Department of Physics, Centre for Photonics and Photonic Materials, University of Bath, Bath BA2 7AY, United Kingdom; and S. Bielawski, Laboratoire de Physique des Lasers, Atomes et Molécules, UMR CNRS 8523, Centre d'Études et de Recherches Lasers et Applications, FR CNRS 2416, Université des Sciences et Technologies de Lille, F-59655 Villeneuve D'Ascq Cedex, France, are preparing a manuscript titled, "The Q-switching instability in passively mode-locked lasers."

Fidorra, S.

B. Hüttl, R. Kaiser, W. Rehbein, H. Stolpe, C. Kindel, S. Fidorra, A. Steffan, A. Umbach, and H. Heidrich, "Low noise monolithic 40GHz mode-locked DBR lasers based on GaInAsP/InP," presented at the 17th Indium Phosphide and Related Materials Conference, Glasgow, UK, 8-12 May 2005.

Haus, H.

H. Haus, "Theory of mode locking with a slow saturable absorber," IEEE J. Quantum Electron. 11, 736-746 (1975).
[CrossRef]

Haus, H. A.

H. A. Haus, "Parameter ranges for cw passive mode locking," IEEE J. Quantum Electron. 12, 169-176 (1976).
[CrossRef]

Heidrich, H.

B. Hüttl, R. Kaiser, W. Rehbein, H. Stolpe, C. Kindel, S. Fidorra, A. Steffan, A. Umbach, and H. Heidrich, "Low noise monolithic 40GHz mode-locked DBR lasers based on GaInAsP/InP," presented at the 17th Indium Phosphide and Related Materials Conference, Glasgow, UK, 8-12 May 2005.

Hönniger, C.

Hüttl, B.

B. Hüttl, R. Kaiser, W. Rehbein, H. Stolpe, C. Kindel, S. Fidorra, A. Steffan, A. Umbach, and H. Heidrich, "Low noise monolithic 40GHz mode-locked DBR lasers based on GaInAsP/InP," presented at the 17th Indium Phosphide and Related Materials Conference, Glasgow, UK, 8-12 May 2005.

Joly, N.

T. Kolokolnikov, Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5; T. Erneux, Optique Nonlinéaire Théorique, Université Libre de Bruxelles, Campus Plaine CP 231, B-1050 Bruxelles, Belgium; N. Joly, Department of Physics, Centre for Photonics and Photonic Materials, University of Bath, Bath BA2 7AY, United Kingdom; and S. Bielawski, Laboratoire de Physique des Lasers, Atomes et Molécules, UMR CNRS 8523, Centre d'Études et de Recherches Lasers et Applications, FR CNRS 2416, Université des Sciences et Technologies de Lille, F-59655 Villeneuve D'Ascq Cedex, France, are preparing a manuscript titled, "The Q-switching instability in passively mode-locked lasers."

Kaiser, R.

B. Hüttl, R. Kaiser, W. Rehbein, H. Stolpe, C. Kindel, S. Fidorra, A. Steffan, A. Umbach, and H. Heidrich, "Low noise monolithic 40GHz mode-locked DBR lasers based on GaInAsP/InP," presented at the 17th Indium Phosphide and Related Materials Conference, Glasgow, UK, 8-12 May 2005.

Kamp, M.

F. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid-state laser dynamics by semiconductor devices," Opt. Eng. 34, 2024-2036 (1995).

Kärtner, F.

F. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid-state laser dynamics by semiconductor devices," Opt. Eng. 34, 2024-2036 (1995).

Keller, U.

R. Paschotta and U. Keller, "Passive mode-locking with slow saturable absorbers," Appl. Phys. B 73, 653-662 (2001).
[CrossRef]

C. Hönniger, R. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, "Q-switching stability limits of continuous-wave passive mode locking," J. Opt. Soc. Am. B 16, 46-56 (1999).

F. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid-state laser dynamics by semiconductor devices," Opt. Eng. 34, 2024-2036 (1995).

Kindel, C.

B. Hüttl, R. Kaiser, W. Rehbein, H. Stolpe, C. Kindel, S. Fidorra, A. Steffan, A. Umbach, and H. Heidrich, "Low noise monolithic 40GHz mode-locked DBR lasers based on GaInAsP/InP," presented at the 17th Indium Phosphide and Related Materials Conference, Glasgow, UK, 8-12 May 2005.

Kolokolnikov, T.

T. Kolokolnikov, Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5; T. Erneux, Optique Nonlinéaire Théorique, Université Libre de Bruxelles, Campus Plaine CP 231, B-1050 Bruxelles, Belgium; N. Joly, Department of Physics, Centre for Photonics and Photonic Materials, University of Bath, Bath BA2 7AY, United Kingdom; and S. Bielawski, Laboratoire de Physique des Lasers, Atomes et Molécules, UMR CNRS 8523, Centre d'Études et de Recherches Lasers et Applications, FR CNRS 2416, Université des Sciences et Technologies de Lille, F-59655 Villeneuve D'Ascq Cedex, France, are preparing a manuscript titled, "The Q-switching instability in passively mode-locked lasers."

Kopf, D.

F. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid-state laser dynamics by semiconductor devices," Opt. Eng. 34, 2024-2036 (1995).

Kozyreff, G.

Lau, K.

J. Palaski and K. Lau, "Parameter ranges for ultrahigh frequency mode locking of semiconductor lasers," Appl. Phys. Lett. 59, 7-9 (1991).
[CrossRef]

Leegwater, J. A.

J. L. A. Dubbeldam, J. A. Leegwater, and D. Lenstra, "Theory of mode-locked semi-conductor lasers with finite relaxation times," Appl. Phys. Lett. 70, 1938-1940 (1997).
[CrossRef]

Lenstra, D.

J. L. A. Dubbeldam, J. A. Leegwater, and D. Lenstra, "Theory of mode-locked semi-conductor lasers with finite relaxation times," Appl. Phys. Lett. 70, 1938-1940 (1997).
[CrossRef]

Mandel, P.

P. Mandel and T. Erneux, "Stationary, harmonic, and pulsed operations of an optically bistable laser with saturable absorber: I," Phys. Rev. A 30, 1893-1901 (1984).
[CrossRef]

Marsh, J.

E. Avrutin, J. Marsh, and E. Portnoi, "Monolithic and multi-GigaHertz mode-locked semiconductor lasers: constructions, experiments, models, and applications," IEE Proc.: Optoelectron. 147, 251-278 (2000).
[CrossRef]

Morier-Genoud, F.

Moser, M.

New, G. H. C.

G. H. C. New, "Pulse evolution in mode-locked quasi-continuous lasers," IEEE J. Quantum Electron. 10, 115-124 (1974).
[CrossRef]

Palaski, J.

J. Palaski and K. Lau, "Parameter ranges for ultrahigh frequency mode locking of semiconductor lasers," Appl. Phys. Lett. 59, 7-9 (1991).
[CrossRef]

Paschotta, R.

Portnoi, E.

E. Avrutin, J. Marsh, and E. Portnoi, "Monolithic and multi-GigaHertz mode-locked semiconductor lasers: constructions, experiments, models, and applications," IEE Proc.: Optoelectron. 147, 251-278 (2000).
[CrossRef]

Rehbein, W.

B. Hüttl, R. Kaiser, W. Rehbein, H. Stolpe, C. Kindel, S. Fidorra, A. Steffan, A. Umbach, and H. Heidrich, "Low noise monolithic 40GHz mode-locked DBR lasers based on GaInAsP/InP," presented at the 17th Indium Phosphide and Related Materials Conference, Glasgow, UK, 8-12 May 2005.

Steffan, A.

B. Hüttl, R. Kaiser, W. Rehbein, H. Stolpe, C. Kindel, S. Fidorra, A. Steffan, A. Umbach, and H. Heidrich, "Low noise monolithic 40GHz mode-locked DBR lasers based on GaInAsP/InP," presented at the 17th Indium Phosphide and Related Materials Conference, Glasgow, UK, 8-12 May 2005.

Stolpe, H.

B. Hüttl, R. Kaiser, W. Rehbein, H. Stolpe, C. Kindel, S. Fidorra, A. Steffan, A. Umbach, and H. Heidrich, "Low noise monolithic 40GHz mode-locked DBR lasers based on GaInAsP/InP," presented at the 17th Indium Phosphide and Related Materials Conference, Glasgow, UK, 8-12 May 2005.

Turaev, D.

A. Vladimirov and D. Turaev, "Model for passive mode-locking in semiconductor lasers," Phys. Rev. A 72, 033808 (2005).
[CrossRef]

A. Vladimirov and D. Turaev, "A new model for a mode-locked semiconductor laser," Radiophys. Quantum Electron. 47, 857-865 (2004).
[CrossRef]

A. Vladimirov, D. Turaev, and G. Kozyreff, "Delay differential equations for mode-locked semiconductor lasers," Opt. Lett. 29, 1221-1223 (2004).
[CrossRef] [PubMed]

Umbach, A.

B. Hüttl, R. Kaiser, W. Rehbein, H. Stolpe, C. Kindel, S. Fidorra, A. Steffan, A. Umbach, and H. Heidrich, "Low noise monolithic 40GHz mode-locked DBR lasers based on GaInAsP/InP," presented at the 17th Indium Phosphide and Related Materials Conference, Glasgow, UK, 8-12 May 2005.

Vladimirov, A.

A. Vladimirov and D. Turaev, "Model for passive mode-locking in semiconductor lasers," Phys. Rev. A 72, 033808 (2005).
[CrossRef]

A. Vladimirov and D. Turaev, "A new model for a mode-locked semiconductor laser," Radiophys. Quantum Electron. 47, 857-865 (2004).
[CrossRef]

A. Vladimirov, D. Turaev, and G. Kozyreff, "Delay differential equations for mode-locked semiconductor lasers," Opt. Lett. 29, 1221-1223 (2004).
[CrossRef] [PubMed]

Yamada, M.

M. Yamada, "A theoretical analysis of self-sustained pulsation phenomena in narrow-stripe semiconductor lasers," IEEE J. Quantum Electron. QE-29, 1330-1336 (1993).
[CrossRef]

Appl. Phys. B (1)

R. Paschotta and U. Keller, "Passive mode-locking with slow saturable absorbers," Appl. Phys. B 73, 653-662 (2001).
[CrossRef]

Appl. Phys. Lett. (2)

J. Palaski and K. Lau, "Parameter ranges for ultrahigh frequency mode locking of semiconductor lasers," Appl. Phys. Lett. 59, 7-9 (1991).
[CrossRef]

J. L. A. Dubbeldam, J. A. Leegwater, and D. Lenstra, "Theory of mode-locked semi-conductor lasers with finite relaxation times," Appl. Phys. Lett. 70, 1938-1940 (1997).
[CrossRef]

IEE Proc.: Optoelectron. (1)

E. Avrutin, J. Marsh, and E. Portnoi, "Monolithic and multi-GigaHertz mode-locked semiconductor lasers: constructions, experiments, models, and applications," IEE Proc.: Optoelectron. 147, 251-278 (2000).
[CrossRef]

IEEE J. Quantum Electron. (4)

M. Yamada, "A theoretical analysis of self-sustained pulsation phenomena in narrow-stripe semiconductor lasers," IEEE J. Quantum Electron. QE-29, 1330-1336 (1993).
[CrossRef]

H. Haus, "Theory of mode locking with a slow saturable absorber," IEEE J. Quantum Electron. 11, 736-746 (1975).
[CrossRef]

H. A. Haus, "Parameter ranges for cw passive mode locking," IEEE J. Quantum Electron. 12, 169-176 (1976).
[CrossRef]

G. H. C. New, "Pulse evolution in mode-locked quasi-continuous lasers," IEEE J. Quantum Electron. 10, 115-124 (1974).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Eng. (1)

F. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid-state laser dynamics by semiconductor devices," Opt. Eng. 34, 2024-2036 (1995).

Opt. Lett. (1)

Phys. Rev. A (2)

P. Mandel and T. Erneux, "Stationary, harmonic, and pulsed operations of an optically bistable laser with saturable absorber: I," Phys. Rev. A 30, 1893-1901 (1984).
[CrossRef]

A. Vladimirov and D. Turaev, "Model for passive mode-locking in semiconductor lasers," Phys. Rev. A 72, 033808 (2005).
[CrossRef]

Radiophys. Quantum Electron. (1)

A. Vladimirov and D. Turaev, "A new model for a mode-locked semiconductor laser," Radiophys. Quantum Electron. 47, 857-865 (2004).
[CrossRef]

Other (2)

B. Hüttl, R. Kaiser, W. Rehbein, H. Stolpe, C. Kindel, S. Fidorra, A. Steffan, A. Umbach, and H. Heidrich, "Low noise monolithic 40GHz mode-locked DBR lasers based on GaInAsP/InP," presented at the 17th Indium Phosphide and Related Materials Conference, Glasgow, UK, 8-12 May 2005.

T. Kolokolnikov, Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5; T. Erneux, Optique Nonlinéaire Théorique, Université Libre de Bruxelles, Campus Plaine CP 231, B-1050 Bruxelles, Belgium; N. Joly, Department of Physics, Centre for Photonics and Photonic Materials, University of Bath, Bath BA2 7AY, United Kingdom; and S. Bielawski, Laboratoire de Physique des Lasers, Atomes et Molécules, UMR CNRS 8523, Centre d'Études et de Recherches Lasers et Applications, FR CNRS 2416, Université des Sciences et Technologies de Lille, F-59655 Villeneuve D'Ascq Cedex, France, are preparing a manuscript titled, "The Q-switching instability in passively mode-locked lasers."

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Q-switching instability curves (QS) and background instability boundaries (L and T) of a ML pulse. L, leading-edge; T, trailing-edge instability boundary. Solid curves are obtained using Eqs. (11, 12, 15). Dashed curves are obtained from Eqs. (11, 12, 19, 20). Dots show Q-switching and background instability boundaries calculated by direct numerical integration of the laser equations (1, 2, 3). The straight line th indicates the linear lasing threshold. Parameters are as in Table 1.

Fig. 2
Fig. 2

Background stability 1 and ML Q-switching instability 2 domains. (a) s κ = 5 and (b) s κ = 1.3 . Solid and dashed curves present the Q-switching and background instability boundaries of the ML solution calculated for s = 35 and s = 15 , respectively. Thin curves in (a) indicate the Q-switching instability of the cw regime. This latter instability does not exist for s κ = 1.3 . Other parameters are the same as in Fig. 1.

Fig. 3
Fig. 3

(a) Normalized difference between the ML pulse repetition period T * and the cavity round-trip time T. (b) Normalized width τ * of a ML pulse. Curves L (T) correspond to the leading (trailing) edge instability boundaries shown in Fig. 1. Solid and dotted curves correspond to s κ = 5 and s κ = 1.3 , respectively. Other parameters are the same as in Fig. 1.

Fig. 4
Fig. 4

Experimentally measured ML domain 1. Different pulse widths are shown by different depth of the gray color. In the white area 2 below the ML domains and above the threshold line indicated Th the Q-switched ML regimes have been observed. (a) Laser with normal losses. (b) Laser with extra losses introduced by applying additional current to the passive section.

Fig. 5
Fig. 5

The same as Fig. 4 but for two lasers with different number of quantum wells in both gain and absorber sections. (a) A laser with three quantum wells; (b) a laser with six quantum wells. The ML area is much larger for the three quantum-well laser than for the six quantum-well one.

Tables (1)

Tables Icon

Table 1 Parameter Set for Eqs. (1, 2, 3)

Equations (26)

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

γ 1 A ̇ + A = κ e [ ( 1 i α g ) G ( t T ) 2 ( 1 i α q ) Q ( t T ) 2 + i ϕ ] A ( t T ) ,
G ̇ = g 0 γ g G e Q ( e G 1 ) A 2 ,
Q ̇ = q 0 γ q Q s ( 1 e Q ) A 2 ,
g 0 γ g G e Q ( e G 1 ) A 0 2 = 0 , q 0 γ q Q s ( 1 e Q ) A 0 2 = 0 ,
κ e G Q 1 Ω 2 = 0 ,
Ω + tan [ γ T Ω + ( α g G α q Q ) 2 ϕ ] = 0 ,
T P ̇ = P + e G Q + ln κ P ,
G ̇ = g 0 γ g G e Q ( e G 1 ) P ,
Q ̇ = q 0 γ q Q s ( 1 e Q ) P .
G ( p ) = ln [ 1 1 e G n ( 1 + e s p Q n e Q n ) 1 s ] ,
Q ( p ) = ln [ 1 + e s p ( e Q n 1 ) ] ,
G ( t ) = G ( P n ) e γ g t + g 0 γ g ( 1 e γ g t ) ,
Q ( t ) = Q ( P n ) e γ q t + q 0 γ q ( 1 e γ q t ) ,
G n + 1 = e γ g T ln [ 1 1 e G n ( 1 + e s P n Q n e Q n ) 1 s ] + ( 1 e γ g T ) g 0 γ g ,
Q n + 1 = e γ q T ln [ 1 + e s P n ( e Q n 1 ) ] + ( 1 e γ q T ) q 0 γ q .
γ 1 A ̇ n + 1 ( t γ 1 δ n ) + A n + 1 ( t γ 1 δ n ) = κ e [ 1 i α g 2 G ( t ) 1 i α q 2 Q ( t ) ] A n ( t ) .
γ 2 0 τ n + 1 A ̇ n + 1 2 d t + P n + 1 = κ 0 P n e G ( p ) Q ( p ) d p ,
P n + 1 = κ ln [ 1 e G n + e G n ( 1 + e s P n Q n e Q n ) 1 s ] .
( κ 1 e q 0 γ q ) tanh γ q T 2 > s ( 1 e q 0 γ q ) tanh γ g T 2 ,
G * Q * + ln κ = 0 , G ̃ * Q ̃ * + ln κ = 0 .
A n ( t ) = P n γ 2 τ n sech ( γ t τ n ) ,
P n + 1 3 τ n + 1 2 + P n + 1 = κ ln [ 1 e G n + e G n ( 1 + e s P n Q n e Q n ) 1 s ] .
τ n + 1 P n + 1 = κ τ n P n [ 1 π 0 P n Φ ( p , Q n , G n ) p ( P n p ) d p ] 2 ,
Φ ( p , Q n , G n ) = [ 1 + e s p ( e Q n 1 ) ] 1 2 [ 1 1 e G n ( 1 + e s p Q n e Q n ) 1 s ] 1 2 .
γ 1 A ̇ ( t γ 1 δ * ) + A ( t γ 1 δ * ) = κ e [ G ( t ) Q ( t ) ] 2 A ( t ) .
δ * = 1 + τ * κ π 0 P * Φ ( p , Q * , G * ) p ( P * p ) arctanh ( 2 p P * 1 ) d p ,

Metrics