Abstract

We report the first theoretical investigation of passive mode-locking in photonic crystal mode-locked lasers. Related work has investigated coupled-resonator-optical-waveguide structures in the regime of active mode-locking [Opt. Express 13, 4539–4553 (2005)]. An extensive numerical investigation of the influence of key parameters of the active sections and the photonic crystal cavity on the laser performance is presented. The results show the possibility of generating stable and high quality pulses in a large parameter region. For optimized dispersion properties of the photonic crystal waveguide cavity, the pulses have sub picosecond widths and are nearly transform limited.

© 2010 Optical Society of America

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    [CrossRef]
  2. J. Mulet, and J. Mørk, “Analysis of timing jitter in external-cavity mode-locked semiconductor lasers,” IEEE J. Quantum Electron. 42, 249–256 (2006).
    [CrossRef]
  3. K. Yvind, D. Larsson, L. J. Christiansen, C. Angelo, L. K. Oxenlowe, J. Mørk, D. Birkedal, J. Hvam, and J. Hanberg, “Low-jitter and high-power 40-GHz all-active mode-locked lasers,” IEEE Photon. Technol. Lett. 16, 975–977 (2004).
    [CrossRef]
  4. M. Soljacic, and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
    [CrossRef] [PubMed]
  5. E. A. Avrutin, J. H. Marsh, and E. L. Portnoi, “Monolithic and multi-gigahertz mode-locked semiconductor lasers: constructions, experiments, models and applications,” IEE Proc., Optoelectron. 147, 251–278 (2000).
    [CrossRef]
  6. M. G. Thompson, A. R. Rae, M. Xia, R. V. Penty, and I. H. White, “InGaAs Quantum-Dot Mode-Locked Laser Diodes,” IEEE J. Sel. Top. Quantum Electron. 15, 661–672 (2009).
    [CrossRef]
  7. R. Hao, E. Cassan, H. Kurt, X. L. Roux, D. Marris-Morini, L. Vivien, H. Wu, Z. Zhou, and X. Zhang, “Novel slow light waveguide with controllable delay-bandwidth product and ultra-low dispersion,” Opt. Express 18(6), 5942–5950 (2010).
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    [CrossRef] [PubMed]
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    [CrossRef]
  10. S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. M. Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
    [CrossRef]
  11. A. G. Vladimirov, A. S. Pimenov, and D. Rachinskii, “Numerical Study of Dynamical Regimes in a Monolithic Passively Mode-Locked Semiconductor Laser,” IEEE J. Quantum Electron. 45, 462–468 (2009).
    [CrossRef]
  12. R. G. M. P. Koumans, and R. van Roijen, “Theory for passive mode-locking in semiconductor laser structures including the effects of self-phase modulation, dispersion, and pulse collisions,” IEEE J. Quantum Electron. 32, 478–492 (1996).
    [CrossRef]
  13. J. Mulet, M. Kroh, and J. Mørk, “Pulse properties of external-cavity mode-locked semiconductor lasers,” Opt. Express 14, 1119–1124 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  19. S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
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  22. N. Cheng, and J. C. Cartledge, “Measurement-based model for MQW electroabsorption modulators,” J. Lightwave Technol. 23, 4265–4269 (2005).
    [CrossRef]

2010 (1)

2009 (3)

M. G. Thompson, A. R. Rae, M. Xia, R. V. Penty, and I. H. White, “InGaAs Quantum-Dot Mode-Locked Laser Diodes,” IEEE J. Sel. Top. Quantum Electron. 15, 661–672 (2009).
[CrossRef]

A. G. Vladimirov, A. S. Pimenov, and D. Rachinskii, “Numerical Study of Dynamical Regimes in a Monolithic Passively Mode-Locked Semiconductor Laser,” IEEE J. Quantum Electron. 45, 462–468 (2009).
[CrossRef]

S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

2008 (1)

2007 (1)

2006 (4)

M. J. R. Heck, E. A. J. M. Bente, Y. Barbarin, D. Lenstra, and M. K. Smit, “Simulation and design of integrated femtosecond passively mode-locked semiconductor ring lasers including integrated passive pulse shaping components,” IEEE J. Sel. Top. Quantum Electron. 12, 265–276 (2006).
[CrossRef]

L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14, 9444–9450 (2006).
[CrossRef] [PubMed]

J. Mulet, and J. Mørk, “Analysis of timing jitter in external-cavity mode-locked semiconductor lasers,” IEEE J. Quantum Electron. 42, 249–256 (2006).
[CrossRef]

J. Mulet, M. Kroh, and J. Mørk, “Pulse properties of external-cavity mode-locked semiconductor lasers,” Opt. Express 14, 1119–1124 (2006).
[CrossRef] [PubMed]

2005 (2)

N. Cheng, and J. C. Cartledge, “Measurement-based model for MQW electroabsorption modulators,” J. Lightwave Technol. 23, 4265–4269 (2005).
[CrossRef]

A. G. Vladimirov, and D. Turaev, “Model for passive mode locking in semiconductor lasers,” Phys. Rev. A 72(3), 033808 (2005).
[CrossRef]

2004 (2)

K. Yvind, D. Larsson, L. J. Christiansen, C. Angelo, L. K. Oxenlowe, J. Mørk, D. Birkedal, J. Hvam, and J. Hanberg, “Low-jitter and high-power 40-GHz all-active mode-locked lasers,” IEEE Photon. Technol. Lett. 16, 975–977 (2004).
[CrossRef]

M. Soljacic, and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
[CrossRef] [PubMed]

2000 (1)

E. A. Avrutin, J. H. Marsh, and E. L. Portnoi, “Monolithic and multi-gigahertz mode-locked semiconductor lasers: constructions, experiments, models and applications,” IEE Proc., Optoelectron. 147, 251–278 (2000).
[CrossRef]

1996 (2)

J. A. Leegwater, “Theory of mode-locked semiconductor lasers,” IEEE J. Quantum Electron. 32, 1782–1790 (1996).
[CrossRef]

R. G. M. P. Koumans, and R. van Roijen, “Theory for passive mode-locking in semiconductor laser structures including the effects of self-phase modulation, dispersion, and pulse collisions,” IEEE J. Quantum Electron. 32, 478–492 (1996).
[CrossRef]

1995 (1)

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. M. Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

1993 (1)

H. A. Haus, and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

Angelo, C.

K. Yvind, D. Larsson, L. J. Christiansen, C. Angelo, L. K. Oxenlowe, J. Mørk, D. Birkedal, J. Hvam, and J. Hanberg, “Low-jitter and high-power 40-GHz all-active mode-locked lasers,” IEEE Photon. Technol. Lett. 16, 975–977 (2004).
[CrossRef]

Avrutin, E. A.

E. A. Avrutin, J. H. Marsh, and E. L. Portnoi, “Monolithic and multi-gigahertz mode-locked semiconductor lasers: constructions, experiments, models and applications,” IEE Proc., Optoelectron. 147, 251–278 (2000).
[CrossRef]

Barbarin, Y.

M. J. R. Heck, E. A. J. M. Bente, Y. Barbarin, D. Lenstra, and M. K. Smit, “Simulation and design of integrated femtosecond passively mode-locked semiconductor ring lasers including integrated passive pulse shaping components,” IEEE J. Sel. Top. Quantum Electron. 12, 265–276 (2006).
[CrossRef]

Beggs, D. M.

S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

Bente, E. A. J. M.

M. J. R. Heck, E. A. J. M. Bente, Y. Barbarin, D. Lenstra, and M. K. Smit, “Simulation and design of integrated femtosecond passively mode-locked semiconductor ring lasers including integrated passive pulse shaping components,” IEEE J. Sel. Top. Quantum Electron. 12, 265–276 (2006).
[CrossRef]

Birkedal, D.

K. Yvind, D. Larsson, L. J. Christiansen, C. Angelo, L. K. Oxenlowe, J. Mørk, D. Birkedal, J. Hvam, and J. Hanberg, “Low-jitter and high-power 40-GHz all-active mode-locked lasers,” IEEE Photon. Technol. Lett. 16, 975–977 (2004).
[CrossRef]

Bischoff, S.

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. M. Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

Borel, P. I.

Brorson, S. D.

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. M. Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

Cartledge, J. C.

Cassan, E.

Cheng, N.

Christiansen, L. J.

K. Yvind, D. Larsson, L. J. Christiansen, C. Angelo, L. K. Oxenlowe, J. Mørk, D. Birkedal, J. Hvam, and J. Hanberg, “Low-jitter and high-power 40-GHz all-active mode-locked lasers,” IEEE Photon. Technol. Lett. 16, 975–977 (2004).
[CrossRef]

Engelen, R. J. P.

Fage-Pedersen, J.

Franck, T.

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. M. Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

Frandsen, L. H.

Gomez-Iglesias, A.

Hanberg, J.

K. Yvind, D. Larsson, L. J. Christiansen, C. Angelo, L. K. Oxenlowe, J. Mørk, D. Birkedal, J. Hvam, and J. Hanberg, “Low-jitter and high-power 40-GHz all-active mode-locked lasers,” IEEE Photon. Technol. Lett. 16, 975–977 (2004).
[CrossRef]

Hao, R.

Haus, H. A.

H. A. Haus, and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

Heck, M. J. R.

M. J. R. Heck, E. A. J. M. Bente, Y. Barbarin, D. Lenstra, and M. K. Smit, “Simulation and design of integrated femtosecond passively mode-locked semiconductor ring lasers including integrated passive pulse shaping components,” IEEE J. Sel. Top. Quantum Electron. 12, 265–276 (2006).
[CrossRef]

Hvam, J.

K. Yvind, D. Larsson, L. J. Christiansen, C. Angelo, L. K. Oxenlowe, J. Mørk, D. Birkedal, J. Hvam, and J. Hanberg, “Low-jitter and high-power 40-GHz all-active mode-locked lasers,” IEEE Photon. Technol. Lett. 16, 975–977 (2004).
[CrossRef]

Joannopoulos, J. D.

M. Soljacic, and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
[CrossRef] [PubMed]

Koumans, R. G. M. P.

R. G. M. P. Koumans, and R. van Roijen, “Theory for passive mode-locking in semiconductor laser structures including the effects of self-phase modulation, dispersion, and pulse collisions,” IEEE J. Quantum Electron. 32, 478–492 (1996).
[CrossRef]

Krauss, T. F.

Kroh, M.

Kuipers, L.

Kurt, H.

Larsen, A. M.

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. M. Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

Larsson, D.

K. Yvind, D. Larsson, L. J. Christiansen, C. Angelo, L. K. Oxenlowe, J. Mørk, D. Birkedal, J. Hvam, and J. Hanberg, “Low-jitter and high-power 40-GHz all-active mode-locked lasers,” IEEE Photon. Technol. Lett. 16, 975–977 (2004).
[CrossRef]

Lavrinenko, A. V.

Leegwater, J. A.

J. A. Leegwater, “Theory of mode-locked semiconductor lasers,” IEEE J. Quantum Electron. 32, 1782–1790 (1996).
[CrossRef]

Lenstra, D.

M. J. R. Heck, E. A. J. M. Bente, Y. Barbarin, D. Lenstra, and M. K. Smit, “Simulation and design of integrated femtosecond passively mode-locked semiconductor ring lasers including integrated passive pulse shaping components,” IEEE J. Sel. Top. Quantum Electron. 12, 265–276 (2006).
[CrossRef]

Li, J.

Marris-Morini, D.

Marsh, J. H.

E. A. Avrutin, J. H. Marsh, and E. L. Portnoi, “Monolithic and multi-gigahertz mode-locked semiconductor lasers: constructions, experiments, models and applications,” IEE Proc., Optoelectron. 147, 251–278 (2000).
[CrossRef]

Mecozzi, A.

H. A. Haus, and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

Michaeli, A.

Mørk, J.

J. Mulet, M. Kroh, and J. Mørk, “Pulse properties of external-cavity mode-locked semiconductor lasers,” Opt. Express 14, 1119–1124 (2006).
[CrossRef] [PubMed]

J. Mulet, and J. Mørk, “Analysis of timing jitter in external-cavity mode-locked semiconductor lasers,” IEEE J. Quantum Electron. 42, 249–256 (2006).
[CrossRef]

K. Yvind, D. Larsson, L. J. Christiansen, C. Angelo, L. K. Oxenlowe, J. Mørk, D. Birkedal, J. Hvam, and J. Hanberg, “Low-jitter and high-power 40-GHz all-active mode-locked lasers,” IEEE Photon. Technol. Lett. 16, 975–977 (2004).
[CrossRef]

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. M. Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

Mulet, J.

J. Mulet, M. Kroh, and J. Mørk, “Pulse properties of external-cavity mode-locked semiconductor lasers,” Opt. Express 14, 1119–1124 (2006).
[CrossRef] [PubMed]

J. Mulet, and J. Mørk, “Analysis of timing jitter in external-cavity mode-locked semiconductor lasers,” IEEE J. Quantum Electron. 42, 249–256 (2006).
[CrossRef]

Nielsen, J. M.

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. M. Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

O’Faolain, L.

S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16(9), 6227–6232 (2008).
[CrossRef] [PubMed]

Oxenlowe, L. K.

K. Yvind, D. Larsson, L. J. Christiansen, C. Angelo, L. K. Oxenlowe, J. Mørk, D. Birkedal, J. Hvam, and J. Hanberg, “Low-jitter and high-power 40-GHz all-active mode-locked lasers,” IEEE Photon. Technol. Lett. 16, 975–977 (2004).
[CrossRef]

Penty, R. V.

M. G. Thompson, A. R. Rae, M. Xia, R. V. Penty, and I. H. White, “InGaAs Quantum-Dot Mode-Locked Laser Diodes,” IEEE J. Sel. Top. Quantum Electron. 15, 661–672 (2009).
[CrossRef]

Pimenov, A. S.

A. G. Vladimirov, A. S. Pimenov, and D. Rachinskii, “Numerical Study of Dynamical Regimes in a Monolithic Passively Mode-Locked Semiconductor Laser,” IEEE J. Quantum Electron. 45, 462–468 (2009).
[CrossRef]

Portnoi, E. L.

E. A. Avrutin, J. H. Marsh, and E. L. Portnoi, “Monolithic and multi-gigahertz mode-locked semiconductor lasers: constructions, experiments, models and applications,” IEE Proc., Optoelectron. 147, 251–278 (2000).
[CrossRef]

Rachinskii, D.

A. G. Vladimirov, A. S. Pimenov, and D. Rachinskii, “Numerical Study of Dynamical Regimes in a Monolithic Passively Mode-Locked Semiconductor Laser,” IEEE J. Quantum Electron. 45, 462–468 (2009).
[CrossRef]

Rae, A. R.

M. G. Thompson, A. R. Rae, M. Xia, R. V. Penty, and I. H. White, “InGaAs Quantum-Dot Mode-Locked Laser Diodes,” IEEE J. Sel. Top. Quantum Electron. 15, 661–672 (2009).
[CrossRef]

Roux, X. L.

Salib, M.

Schulz, S.

S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

Settle, M. D.

Smit, M. K.

M. J. R. Heck, E. A. J. M. Bente, Y. Barbarin, D. Lenstra, and M. K. Smit, “Simulation and design of integrated femtosecond passively mode-locked semiconductor ring lasers including integrated passive pulse shaping components,” IEEE J. Sel. Top. Quantum Electron. 12, 265–276 (2006).
[CrossRef]

Soljacic, M.

M. Soljacic, and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
[CrossRef] [PubMed]

Sørensen, M. P.

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. M. Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

Thompson, M. G.

M. G. Thompson, A. R. Rae, M. Xia, R. V. Penty, and I. H. White, “InGaAs Quantum-Dot Mode-Locked Laser Diodes,” IEEE J. Sel. Top. Quantum Electron. 15, 661–672 (2009).
[CrossRef]

Turaev, D.

A. G. Vladimirov, and D. Turaev, “Model for passive mode locking in semiconductor lasers,” Phys. Rev. A 72(3), 033808 (2005).
[CrossRef]

van Roijen, R.

R. G. M. P. Koumans, and R. van Roijen, “Theory for passive mode-locking in semiconductor laser structures including the effects of self-phase modulation, dispersion, and pulse collisions,” IEEE J. Quantum Electron. 32, 478–492 (1996).
[CrossRef]

Vivien, L.

Vladimirov, A. G.

A. G. Vladimirov, A. S. Pimenov, and D. Rachinskii, “Numerical Study of Dynamical Regimes in a Monolithic Passively Mode-Locked Semiconductor Laser,” IEEE J. Quantum Electron. 45, 462–468 (2009).
[CrossRef]

A. G. Vladimirov, and D. Turaev, “Model for passive mode locking in semiconductor lasers,” Phys. Rev. A 72(3), 033808 (2005).
[CrossRef]

White, I. H.

M. G. Thompson, A. R. Rae, M. Xia, R. V. Penty, and I. H. White, “InGaAs Quantum-Dot Mode-Locked Laser Diodes,” IEEE J. Sel. Top. Quantum Electron. 15, 661–672 (2009).
[CrossRef]

White, T. P.

S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80, 195305 (2009).
[CrossRef]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16(9), 6227–6232 (2008).
[CrossRef] [PubMed]

Wu, H.

Xia, M.

M. G. Thompson, A. R. Rae, M. Xia, R. V. Penty, and I. H. White, “InGaAs Quantum-Dot Mode-Locked Laser Diodes,” IEEE J. Sel. Top. Quantum Electron. 15, 661–672 (2009).
[CrossRef]

Yvind, K.

K. Yvind, D. Larsson, L. J. Christiansen, C. Angelo, L. K. Oxenlowe, J. Mørk, D. Birkedal, J. Hvam, and J. Hanberg, “Low-jitter and high-power 40-GHz all-active mode-locked lasers,” IEEE Photon. Technol. Lett. 16, 975–977 (2004).
[CrossRef]

Zhang, X.

Zhou, Z.

Appl. Phys. Lett. (1)

S. Bischoff, M. P. Sørensen, J. Mørk, S. D. Brorson, T. Franck, J. M. Nielsen, and A. M. Larsen, “Pulse-shaping mechanism in colliding-pulse mode-locked laser diodes,” Appl. Phys. Lett. 67, 3877–3879 (1995).
[CrossRef]

IEE Proc., Optoelectron. (1)

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Figures (10)

Fig. 1.
Fig. 1.

Illustration of the PC ML laser design. The gain and absorber sections are placed in a line defect cavity, which is created by removing holes of the perfectly periodic PC structure. By altering the shape and position of holes close to the cavity, it is possible to engineer the dispersion properties of the cavity.

Fig. 2.
Fig. 2.

Illustration of the ring cavity model. The pulse circulates in the cavity, passing through each element.

Fig. 3.
Fig. 3.

Left: Phase diagram of possible solution types as a function of the unsaturated gain, g 0, and loss, q 0. The green domain represents ML solutions, red is intensity modulated pulses, blue is irregular solutions with no periodicity, yellow is higher harmonic solutions and orange is CW solutions. Pulse properties of solutions restricted to the black line are shown in Fig. 4. Right: Plots of the different solution types depending on the value of (q 0,g 0). I: (q 0,g 0) = (5,1.5), II: (q 0,g 0) = (5,2.5), III: (q 0,g 0) = (5,4.2), IV: (q 0,g 0) = (5,5).

Fig. 4.
Fig. 4.

Variation of the pulse width (black) and duty cycle (red) for (q 0,g 0) restricted to the line g 0 = 0.65q 0−0.2 in the left plot of Fig. 3.

Fig. 5.
Fig. 5.

(a) A plot of the Taylor expansion (solid black) of the dispersion relation with parameter values from Table 1. The group velocity is constant along the dashed black line. The frequencies on the abscissa are fω 0/2π, were ω 0 is the fast oscillating part of the total field, ( z , t ) = E ( z , t ) e i k 0 z i ω 0 t . (b) Variation of the pulse width in the time (black) and frequency domain (red) with the filter bandwidth, ∆f PCW.

Fig. 6.
Fig. 6.

(a) Variation of the pulse width (black) and the repetition rate, fr , (red) with the length of the PCW, 2L PCW. The dashed blue line shows fr from Eq. (9). (b) Variation of the pulse width (black) and length reduction factor, ρR , (red) as the length of the PCW and the slowdown factor are varied simultaneously, while keeping their product S × 2L PCW = 1200 µm constant.

Fig. 7.
Fig. 7.

Variation of the pulse width (black) and repetition rate (red) as a function of the slowdown factor. The dashed blue line shows fr from Eq. (9).

Fig. 8.
Fig. 8.

Pulse shape for different values of the second and third order expansion coefficients in Eq. (3). The values of ( k ω 0 ( 3 ) , k ω 0 ( 2 ) ) for the different curves are shown in Fig. 9 by the location of the corresponding roman numerals. The dashed red line shows a Gaussian fit to pulse I.

Fig. 9.
Fig. 9.

(a) The dependence of the ratio between the trailing and main pulse peak powers on the second and third order expansion coefficients in Eq. (3). (b) Pulse width as a function of the second and third order expansion coefficients in Eq. (3).

Fig. 10.
Fig. 10.

(a) A phase diagram of solution types found by varying the linewidth enhancement factors. The color code is the same as in Fig. 3. (b) Variation of the pulse width (black) and time-bandwidth product, ∆t FWHMf FWHM, (red) as a function of the linewidth enhancement factors. The values of [ k ω 0 ( 1 ) ] 2 k 0 + k ω 0 ( 2 ) are −4×106 ps2/km (dashed lines) and 2×106 ps2/km (solid lines).

Tables (1)

Tables Icon

Table 1. Parameter values used unless otherwise stated in the figure captions.

Equations (14)

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z E ( z , t ) + 1 ν g b t E ( z , t ) = g r Γ r 2 ( 1 i α r ) [ N r ( z , t ) N r t r ] E ( z , t ) .
F ˜ PCW ( ω ω 0 ) = e i ϕ ( ω ω 0 ) f H ( ω ω 0 ) , F ˜ PCW ( ω ) = + F PCW ( t ) e i ω t
ϕ ( ω ω 0 ) = 2 L PCW 2 k 0 ( [ p = 0 3 1 p ! k ω 0 ( p ) ( ω ω 0 ) p ] 2 k 0 2 ) , k ω 0 ( p ) = d p k ( ω ) d ω P ω = ω 0
f H ( ω ω 0 ) = p = 1 3 a p exp ( b p 2 [ ( ω ω 0 ) 2 c p 2 ] ) .
+ f H ( u ) cos [ ϕ ( u ) ] π ( ω u ) d u = f H ( ω ) sin [ ϕ ( ω ) ] .
d τ G ( τ ) = g 0 Γ G ( τ ) e Q ( τ ) ( e G ( τ ) 1 ) A ( τ ) 2
d τ Q ( τ ) = q 0 Q ( τ ) s ( 1 e Q ( τ ) ) A ( τ ) 2
A ( τ ) = τ F PCW ( τ u ) R ( u T 0 ) A ( u T 0 ) du
R ( τ ) = κ exp [ ( 1 i α g ) G ( τ ) 2 ( 1 i α q ) Q ( τ ) 2 ] .
ζ = γ q ν g b z , τ = γ q ( t z ν g b ) , Γ = γ g γ q , s = g q Γ q g g Γ g
A ( ζ , τ ) = E ( ζ , τ ) ν g b γ q g g Γ g , A ( τ ) = A ( ζ 0 , τ )
G ( τ ) = z g z g + L g g g Γ g [ N g N g t r ] d z Q ( τ ) = z q z q + L q g q Γ q [ N q N q tr ] dz
g 0 = z g z g + L g ( J g γ g N g tr ) g g Γ g γ q dz q 0 = z q z q + L q N q tr g q Γ q dz .
f r = 1 T = [ 1 ν g b ( S × 2 L PCW + 2 L tot ) ] 1 .

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