Abstract

Optical Frequency Comb (OFC) generated by semiconductor lasers are currently widely used in the extremely timely field of high capacity optical interconnects and high precision spectroscopy. In the last decade, several experimental evidences of spontaneous OFC generation have been reported in single section Quantum Dot (QD) lasers. Here we provide a physical understanding of these self-organization phenomena by simulating the multi-mode dynamics of a single section Fabry-Perot (FP) QD laser using a Time-Domain Traveling-Wave (TDTW) model that properly accounts for coherent radiation-matter interaction in the semiconductor active medium and includes the carrier grating generated by the optical standing wave pattern in the laser cavity. We show that the latter is the fundamental physical effect at the origin of the multi-mode spectrum appearing just above threshold. A self-mode-locking regime associated with the emission of OFC is achieved for higher bias currents and ascribed to nonlinear phase sensitive effects as Four Wave Mixing (FWM). Our results explain in detail the behaviour observed experimentally by different research groups and in different QD and Quantum Dash (QDash) devices.

© 2017 Optical Society of America

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2017 (1)

S. M. Link, D. J. H. C. Maas, D. Waldburger, and U. Keller, “Dual-comb spectroscopy of water vapor with a free-running semiconductor disk laser,” Science 356, 1164–1168 (2017).
[Crossref] [PubMed]

2016 (4)

N. Vukovic, J. Radovanovic, V. Milanovic, and D. L. Boiko, “Analytical expression for Risken-Nummedal-Graham-Haken instability threshold in quantum cascade lasers,” Opt. Express 24, 26911–26929 (2016).
[Crossref] [PubMed]

P. Tzenov, D. Burghoff, Q. Hu, and C. Jirauschek, “Time domain modeling of terahertz quantum cascade lasers for frequency comb generation,” Opt. Express 24, 23232–23247 (2016).
[Crossref] [PubMed]

V. Panapakkam, A. P. Anthur, V. Vujicic, R. Zhou, Q. Gaimard, K. Merghem, G. Aubin, F. Lelarge, E. A. Viktorov, L. P. Barry, and A. Ramdane, “Amplitude and phase noise of frequency combs generated by single-section InAs/InP quantum-dash-based passively and actively mode-locked lasers,” IEEE J. Quantum Electron. 52, 1–7 (2016).
[Crossref]

J. Faist, G. Villares, G. Scalari, M. Rösch, C. Bonzon, A. Hugi, and M. Beck, “Quantum cascade laser frequency combs,” Nanophotonics 5, 272–291 (2016).
[Crossref]

2015 (4)

G. Villares and J. Faist, “Quantum cascade laser combs: effects of modulation and dispersion,” Opt. Express 23, 1651–1669 (2015).
[Crossref] [PubMed]

M. Gioannini, P. Bardella, and I. Montrosset, “Time-domain traveling-wave analysis of the multimode dynamics of quantum dot Fabry-Perot lasers,” IEEE J. Sel. Topics Quantum Electron. 21, 698–708 (2015).
[Crossref]

C.-H. Chen, M. A. Seyedi, M. Fiorentino, D. Livshits, A. Gubenko, S. Mikhrin, V. Mikhrin, and R. G. Beausoleil, “A comb laser-driven dwdm silicon photonic transmitter based on microring modulators,” Opt. Express 23, 21541–21548 (2015).
[Crossref] [PubMed]

J. Müller, J. Hauck, B. Shen, S. Romero-García, E. Islamova, S. Azadeh, S. Joshi, N. Chimot, A. Moscoso-Mártir, F. F. Merget, F. Lelarge, and J. Witzens, “Silicon photonics WDM transmitter with single section semiconductor mode-locked laser,” Adv. Opt. Techn. 4, 119–145 (2015).

2014 (4)

A. C. O. Karni, G. Eisenstein, V. Sichkovskyi, V. Ivanov, and J. P. Reithmaier, “Coherent control in a semiconductor optical amplifier operating at room temperature,” Nat. Commun. 52952 (2014).
[PubMed]

D. Lenstra and M. Yousefi, “Rate-equation model for multi-mode semiconductor lasers with spatial hole burning,” Opt. Express 22, 8143–8149 (2014).
[Crossref] [PubMed]

A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Directional reversals and multimode dynamics in semiconductor ring lasers,” Phys. Rev. A 89, 023818 (2014).
[Crossref]

K. Merghem, C. Calò, R. Rosales, X. Lafosse, G. Aubin, A. Martinez, F. Lelarge, and A. Ramdane, “Stability of optical frequency comb generated with InAs/InP quantum-dash-based passive mode-locked lasers,” IEEE J. Quant. Electron. 50, 275–280 (2014).
[Crossref]

2013 (3)

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7, 977–981 (2013).
[Crossref]

R. Maldonado-Basilio, J. Parra-Cetina, S. Latkowski, N. Calabretta, and P. Landais, “Experimental investigation of the optical injection locking dynamics in single-section quantum-dash Fabry-Pérot laser diode for packet-based clock recovery applications,” J. Lightw. Tech. 31, 860–865 (2013).
[Crossref]

M. Kolarczik, N. Owschimikow, J. Korn, B. Lingnau, Y. Kaptan, D. Bimberg, E. Scholl, K. Lüdge, and U. Woggon, “Quantum coherence induces pulse shape modification in a semiconductor optical amplifier at room temperature,” Nat. Commun. 42953 (2013).
[Crossref] [PubMed]

2012 (3)

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
[Crossref] [PubMed]

R. Rosales, S. G. Murdoch, R. Watts, K. Merghem, A. Martinez, F. Lelarge, A. Accard, L. P. Barry, and A. Ramdane, “High performance mode locking characteristics of single section quantum dash lasers,” Opt. Express 20, 8649–8657 (2012).
[Crossref] [PubMed]

R. Rosales, K. Merghem, C. Calò, G. Bouwmans, I. Krestnikov, A. Martinez, and A. Ramdane, “Optical pulse generation in single section InAs/GaAs quantum dot edge emitting lasers under continuous wave operation,” Appl. Phys. Lett. 101, 221113 (2012).
[Crossref]

2011 (3)

R. Rosales, K. Merghem, A. Martinez, A. Akrout, J. P. Tourrenc, A. Accard, F. Lelarge, and A. Ramdane, “InAs/InP quantum-dot passively mode-locked lasers for 1.55-μm applications,” IEEE J. Sel. Top. Quantum Electron. 17, 1292–1301 (2011).
[Crossref]

M. Rossetti, P. Bardella, and I. Montrosset, “Time-domain travelling-wave model for quantum dot passively mode-locked lasers,” IEEE J. Quantum Electron. 47, 139–150 (2011).
[Crossref]

A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Longitudinal mode multistability in ring and Fabry-Perot lasers: the effect of spatial hole burning,” Opt. Express 19, 3284–3289 (2011).
[Crossref]

2010 (3)

J. Javaloyes and S. Balle, “Mode-locking in semiconductor Fabry-Perot lasers,” IEEE J. Quantum Electron. 46, 1023–1030 (2010).
[Crossref]

S. Latkowski, J. Parra-Cetina, R. Maldonado-Basilio, P. Landais, G. Ducournau, A. Beck, E. Peytavit, T. Akalin, and J.-F. Lampin, “Analysis of a narrowband terahertz signal generated by a unitravelling carrier photodiode coupled with a dual-mode semiconductor Fabry-Pérot laser,” Appl. Phys. Lett. 96, 241106 (2010).
[Crossref]

R. Maldonado-Basilio, J. Parra-Cetina, S. Latkowski, and P. Landais, “Timing-jitter, optical, and mode-beating linewidths analysis on subpicosecond optical pulses generated by a quantum-dash passively mode-locked semiconductor laser,” Opt. Lett. 35, 1184–1186 (2010).
[Crossref] [PubMed]

2009 (4)

2008 (4)

Z. Lu, J. Liu, S. Raymond, P. Poole, P. Barrios, and D. Poitras, “312-fs pulse generation from a passive C-band InAs/InP quantum dot mode-locked laser,” Opt. Express 16, 10835–10840 (2008).
[Crossref] [PubMed]

J. Liu, Z. Lu, S. Raymond, P. J. Poole, P. J. Barrios, and D. Poitras, “Dual-wavelength 92.5 GHz self-mode-locked InP-based quantum dot laser,” Opt. Lett. 33, 1702–1704 (2008).
[Crossref] [PubMed]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 1–18 (2008).
[Crossref]

S. Latkowski, F. Surre, and P. Landais, “Terahertz wave generation from a dc-biased multimode laser,” Appl. Phys. Lett. 92, 081109 (2008).
[Crossref]

2007 (2)

J. Renaudier, G. H. Duan, P. Landais, and P. Gallion, “Phase correlation and linewidth reduction of 40 GHz self-pulsation in distributed bragg reflector semiconductor lasers,” IEEE J.Quantum Electron. 43, 147–156 (2007).
[Crossref]

F. Lelarge, B. Dagens, J. Renaudier, R. Brenot, A. Accard, F. v. Dijk, D. Make, O. L. Gouezigou, J. G. Provost, F. Poingt, J. Landreau, O. Drisse, E. Derouin, B. Rousseau, F. Pommereau, and G. H. Duan, “Recent advances on InAs/InP quantum dash based semiconductor lasers and optical amplifiers operating at 1.55 μm,” IEEE Journal of Selected Topics in Quantum Electronics 13, 111–124 (2007).
[Crossref]

2006 (3)

P. J. Delfyett, S. Gee, M.-T. Choi, H. Izadpanah, W. Lee, S. Ozharar, F. Quinlan, and T. Yilmaz, “Optical frequency combs from semiconductor lasers and applications in ultrawideband signal processing and communications,” IEEE J. Lightwave Technol. 24, 2701–2719 (2006).
[Crossref]

T. W. Hänsch, “Nobel lecture: Passion for precision,” Rev. Mod. Phys. 78, 1297–1309 (2006).
[Crossref]

C. Gosset, K. Merghem, A. Martinez, G. Moreau, G. Patriarche, G. Aubin, A. Ramdane, J. Landreau, and F. Lelarge, “Subpicosecond pulse generation at 134 GHz using a quantum-dash-based Fabry-Perot laser emitting at 1.56μm,” Appl. Phys. Lett. 88, 241105 (2006).
[Crossref]

2004 (1)

L. Furfaro, F. Pedaci, M. Giudici, X. Hachair, J. Tredicce, and S. Balle, “Mode-switching in semiconductor lasers,” IEEE J. Quantum Electron. 40, 1365–1376 (2004).
[Crossref]

2001 (1)

K. Sato, “100 GHz optical pulse generation using Fabry-Perot laser under continuous wave operation,” Electron. Lett. 37, 763–764 (2001).
[Crossref]

1996 (1)

M. Homar, J. V. Moloney, and M. S. Miguel, “Travelling wave model of a multimode Fabry-Perot laser in free running and external cavity configurations,” IEEE J. of Quant. Electron. 32, 553–566 (1996).
[Crossref]

1993 (1)

A. Mecozzi, A. D’Ottavi, and R. Hui, “Nearly degenerate four-wave mixing in distributed feedback semiconductor lasers operating above threshold,” IEEE J. Quantum Electron. 29, 1477–1487 (1993).
[Crossref]

1969 (1)

S. Stenholm and W. E. Lamb, “Semiclassical theory of a high-intensity laser,” Phys. Rev. 181, 618–635 (1969).
[Crossref]

1968 (1)

H. Risken and K. Nummedal, “Self pulsating in lasers,” J. App. Phys. 39, 4662–4672 (1968).
[Crossref]

Accard, A.

R. Rosales, S. G. Murdoch, R. Watts, K. Merghem, A. Martinez, F. Lelarge, A. Accard, L. P. Barry, and A. Ramdane, “High performance mode locking characteristics of single section quantum dash lasers,” Opt. Express 20, 8649–8657 (2012).
[Crossref] [PubMed]

R. Rosales, K. Merghem, A. Martinez, A. Akrout, J. P. Tourrenc, A. Accard, F. Lelarge, and A. Ramdane, “InAs/InP quantum-dot passively mode-locked lasers for 1.55-μm applications,” IEEE J. Sel. Top. Quantum Electron. 17, 1292–1301 (2011).
[Crossref]

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R. Maldonado-Basilio, J. Parra-Cetina, S. Latkowski, and P. Landais, “Timing-jitter, optical, and mode-beating linewidths analysis on subpicosecond optical pulses generated by a quantum-dash passively mode-locked semiconductor laser,” Opt. Lett. 35, 1184–1186 (2010).
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S. Latkowski, R. Maldonado-Basilio, and P. Landais, “Sub-picosecond pulse generation by 40-GHz passively mode-locked quantum-dash 1-mm-long Fabry-Pérot laser diode,” Opt. Express 17, 19166–19172 (2009).
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S. Latkowski, F. Surre, and P. Landais, “Terahertz wave generation from a dc-biased multimode laser,” Appl. Phys. Lett. 92, 081109 (2008).
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K. Merghem, C. Calò, R. Rosales, X. Lafosse, G. Aubin, A. Martinez, F. Lelarge, and A. Ramdane, “Stability of optical frequency comb generated with InAs/InP quantum-dash-based passive mode-locked lasers,” IEEE J. Quant. Electron. 50, 275–280 (2014).
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R. Rosales, S. G. Murdoch, R. Watts, K. Merghem, A. Martinez, F. Lelarge, A. Accard, L. P. Barry, and A. Ramdane, “High performance mode locking characteristics of single section quantum dash lasers,” Opt. Express 20, 8649–8657 (2012).
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K. Merghem, A. Akrout, A. Martinez, G. Aubin, A. Ramdane, F. Lelarge, and G.-H. Duan, “Pulse generation at 346 GHz using a passively mode locked quantum-dash-based laser at 1.55μm,” Appl. Phys. Lett. 94, 021107 (2009).
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[Crossref]

C. Gosset, K. Merghem, A. Martinez, G. Moreau, G. Patriarche, G. Aubin, A. Ramdane, J. Landreau, and F. Lelarge, “Subpicosecond pulse generation at 134 GHz using a quantum-dash-based Fabry-Perot laser emitting at 1.56μm,” Appl. Phys. Lett. 88, 241105 (2006).
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Leuther, A.

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Lingnau, B.

M. Kolarczik, N. Owschimikow, J. Korn, B. Lingnau, Y. Kaptan, D. Bimberg, E. Scholl, K. Lüdge, and U. Woggon, “Quantum coherence induces pulse shape modification in a semiconductor optical amplifier at room temperature,” Nat. Commun. 42953 (2013).
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Link, S. M.

S. M. Link, D. J. H. C. Maas, D. Waldburger, and U. Keller, “Dual-comb spectroscopy of water vapor with a free-running semiconductor disk laser,” Science 356, 1164–1168 (2017).
[Crossref] [PubMed]

Liu, H. C.

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
[Crossref] [PubMed]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 1–18 (2008).
[Crossref]

Liu, J.

Livshits, D.

Lopez-Diaz, D.

S. Koenig, D. Lopez-Diaz, J. Antes, F. Boes, R. Henneberger, A. Leuther, A. Tessmann, R. Schmogrow, D. Hillerkuss, R. Palmer, T. Zwick, C. Koos, W. Freude, O. Ambacher, J. Leuthold, and I. Kallfass, “Wireless sub-THz communication system with high data rate,” Nat. Photonics 7, 977–981 (2013).
[Crossref]

Lu, Z.

Lüdge, K.

M. Kolarczik, N. Owschimikow, J. Korn, B. Lingnau, Y. Kaptan, D. Bimberg, E. Scholl, K. Lüdge, and U. Woggon, “Quantum coherence induces pulse shape modification in a semiconductor optical amplifier at room temperature,” Nat. Commun. 42953 (2013).
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Maas, D. J. H. C.

S. M. Link, D. J. H. C. Maas, D. Waldburger, and U. Keller, “Dual-comb spectroscopy of water vapor with a free-running semiconductor disk laser,” Science 356, 1164–1168 (2017).
[Crossref] [PubMed]

Maier, T.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: From coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 1–18 (2008).
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Make, D.

F. Lelarge, B. Dagens, J. Renaudier, R. Brenot, A. Accard, F. v. Dijk, D. Make, O. L. Gouezigou, J. G. Provost, F. Poingt, J. Landreau, O. Drisse, E. Derouin, B. Rousseau, F. Pommereau, and G. H. Duan, “Recent advances on InAs/InP quantum dash based semiconductor lasers and optical amplifiers operating at 1.55 μm,” IEEE Journal of Selected Topics in Quantum Electronics 13, 111–124 (2007).
[Crossref]

Maldonado-Basilio, R.

R. Maldonado-Basilio, J. Parra-Cetina, S. Latkowski, N. Calabretta, and P. Landais, “Experimental investigation of the optical injection locking dynamics in single-section quantum-dash Fabry-Pérot laser diode for packet-based clock recovery applications,” J. Lightw. Tech. 31, 860–865 (2013).
[Crossref]

S. Latkowski, J. Parra-Cetina, R. Maldonado-Basilio, P. Landais, G. Ducournau, A. Beck, E. Peytavit, T. Akalin, and J.-F. Lampin, “Analysis of a narrowband terahertz signal generated by a unitravelling carrier photodiode coupled with a dual-mode semiconductor Fabry-Pérot laser,” Appl. Phys. Lett. 96, 241106 (2010).
[Crossref]

R. Maldonado-Basilio, J. Parra-Cetina, S. Latkowski, and P. Landais, “Timing-jitter, optical, and mode-beating linewidths analysis on subpicosecond optical pulses generated by a quantum-dash passively mode-locked semiconductor laser,” Opt. Lett. 35, 1184–1186 (2010).
[Crossref] [PubMed]

S. Latkowski, R. Maldonado-Basilio, and P. Landais, “Sub-picosecond pulse generation by 40-GHz passively mode-locked quantum-dash 1-mm-long Fabry-Pérot laser diode,” Opt. Express 17, 19166–19172 (2009).
[Crossref]

Marconi, M.

M. Marconi, J. Javaloyes, P. Hamel, F. Raineri, G. Beaudoin, I. Sagnes, A. Levenson, and A. M. Yacomotti, “Quench dynamics in strongly coupled laser cavities,” ArXiv e-prints (2017).

Martinez, A.

K. Merghem, C. Calò, R. Rosales, X. Lafosse, G. Aubin, A. Martinez, F. Lelarge, and A. Ramdane, “Stability of optical frequency comb generated with InAs/InP quantum-dash-based passive mode-locked lasers,” IEEE J. Quant. Electron. 50, 275–280 (2014).
[Crossref]

R. Rosales, K. Merghem, C. Calò, G. Bouwmans, I. Krestnikov, A. Martinez, and A. Ramdane, “Optical pulse generation in single section InAs/GaAs quantum dot edge emitting lasers under continuous wave operation,” Appl. Phys. Lett. 101, 221113 (2012).
[Crossref]

R. Rosales, S. G. Murdoch, R. Watts, K. Merghem, A. Martinez, F. Lelarge, A. Accard, L. P. Barry, and A. Ramdane, “High performance mode locking characteristics of single section quantum dash lasers,” Opt. Express 20, 8649–8657 (2012).
[Crossref] [PubMed]

R. Rosales, K. Merghem, A. Martinez, A. Akrout, J. P. Tourrenc, A. Accard, F. Lelarge, and A. Ramdane, “InAs/InP quantum-dot passively mode-locked lasers for 1.55-μm applications,” IEEE J. Sel. Top. Quantum Electron. 17, 1292–1301 (2011).
[Crossref]

K. Merghem, A. Akrout, A. Martinez, G. Aubin, A. Ramdane, F. Lelarge, and G.-H. Duan, “Pulse generation at 346 GHz using a passively mode locked quantum-dash-based laser at 1.55μm,” Appl. Phys. Lett. 94, 021107 (2009).
[Crossref]

C. Gosset, K. Merghem, A. Martinez, G. Moreau, G. Patriarche, G. Aubin, A. Ramdane, J. Landreau, and F. Lelarge, “Subpicosecond pulse generation at 134 GHz using a quantum-dash-based Fabry-Perot laser emitting at 1.56μm,” Appl. Phys. Lett. 88, 241105 (2006).
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Figures (9)

Fig. 1
Fig. 1 Schematic of the simulated single section QD laser (a) and schematic of the electron dynamics as considered in our model (b).
Fig. 2
Fig. 2 Temporal evolution of the total optical power (P(t)) (a) and of the modal optical power (Pq(t)) (b,c) obtained by numerical integration of Eqs. (6a)(6h). The simulation shows the effect of SHB, switched on at t = 0, in the destabilization of the single mode lasing emission (q = 0) and the excitation of sides modes (q ≠ 0).
Fig. 3
Fig. 3 Curve of the average optical power vs. bias current for the simulated QD laser. The insets show representative optical spectra normalized to their maximum value for ΔI = 40 mA and ΔI =200 mA.
Fig. 4
Fig. 4 Unlocked multi-mode dynamics for a pump current 40 mA. Temporal evolution of the modal power (Pq(t)) (a) and of phase difference between adjacent modes (ΔΦq(t)) (b) for selected modes. The average values (μPq and μΔΦq) and standard deviations (σδPq and σΔΦq) are reported next to the right vertical axis. In panel (c) these statistical moments are reported for all the modes within −10 dB bandwidth in the optical spectrum. The temporal average of the frequency separations between couples of adjacent modes (Δμfq) (left vertical axis) and its difference with respect to the overall average over the selected modes (MΔμf) (right vertical axis) are represented in panel (d).
Fig. 5
Fig. 5 Self-generated OFC regime for a pump current of 200 mA. Temporal evolution of the modal power (Pq(t)) (a) and of phase difference between adjacent modes (ΔΦq(t)) (b) for selected modes. The average values (μPq and μΔΦq) and standard deviations (σδPq and σΔΦq) are reported next to the right vertical axis. In panel (c) these statistical moments are reported for all the modes within −10 dB bandwidth in the optical spectrum. The temporal average of the frequency separations between couples of adjacent modes (Δμfq) (left vertical axis) and its difference with respect to the overall average over the selected modes (MΔμf) (right vertical axis) are represented in panel (d).
Fig. 6
Fig. 6 RIN spectra associated to the total power (violet/cyan lines) and the modal power of the central mode in the optical spectrum (q = 0) (red lines) for ΔI =40 mA (a) and Δ = 200 mA (b). First beat-note (c) and total power spectrum (d) for ΔI = 40 mA (violet line) and ΔI = 200 mA (cyan line).
Fig. 7
Fig. 7 Average intensity noise (MσδP) (a) and average differential phase noise (MσΔΦ) (b) for increasing values of the bias current. For the same values we plot the integrated RIN (MiRIN) (c) and the average frequency separation between couples of adjacent mode minus its mean over the selected modes (ΔμfqMΔμf) (d).
Fig. 8
Fig. 8 Beat note spectrum (normalized for each current, with respect to its maximum) vs. bias current above threshold. The green dashed line traces the 0.9 contour level
Fig. 9
Fig. 9 Zoom of the optical spectra (normalized to their maximum) around the mode q = 0 for ΔI = 40 mA (violet line), ΔI = 60 mA (green line), ΔI = 80 mA (black line) and ΔI = 200 mA (cyan line). The main lasing line and the side bands generated by FWM are evident in the unlocked regimes (ΔI < 100 mA) while it disappears when the modes are equally distant, i.e in the locked regime (ΔI > 200 mA). The inset in (d) shows the quadratic dependence of the lasing lines width on to their frequency.

Tables (1)

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Table 1 Main Model Parameters

Equations (29)

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E ^ ( x , y , z , t ) = ϕ ( x , y ) { [ E ( z , t ) + exp ( j k 0 z + j ω 0 t ) + E ( z , t ) exp ( + j k 0 z + j ω 0 t ) ] + c . c . }
1 v g E ± t ± E ± z = α w g 2 E ± j ω 0 2 c η 0 Γ x y < P exp ( ± j k 0 z ) > + S sp ±
E + ( 0 , t ) = r R E ( 0 , t ) , E ( L , t ) = r F E + ( L , t )
P ( z , t ) = N D h QD m = GS , ES μ m d m * p m ( z , t )
p m ( z , t ) t = [ j ( ω m ω 0 ) Γ ] p m + j d m ( 2 ρ m 1 ) [ E + exp ( j k 0 z ) + E exp ( j k 0 z ) ]
E ± E ± c η 0 w h QD η L 2 Γ x y , P ± P ± c η 0 w h QD η L 2 Γ x y
ρ GS ( z , t ) t = ρ GS τ nr GS ρ GS 2 τ sp GS ρ GS ( 1 ρ ES ) τ e GS + μ ES μ GS ρ ES ( 1 ρ GS ) τ C GS R ¯ GS st N D μ GS η L
ρ ES ( z , t ) t = ρ E S τ nr ES + D WL μ ES N d ρ WL ( 1 ρ ES ) τ C ES ρ ES ( 1 ρ WL ) τ e ES + μ GS μ ES ρ GS ( 1 ρ ES ) τ e GS ρ ES ( 1 ρ GS ) τ C GS
ρ WL ( z , t ) t = Λ ρ WL τ nr WL + [ ρ WL τ C ES ( 1 ρ ES ) + μ ES N D D WL ρ E S τ e ES ( 1 ρ WL ) ]
R ¯ GS st = j β GS 2 × [ ( E + * ρ GS E ρ GS * ) exp ( j k 0 z ) + ( E * ρ GS E + ρ GS * ) exp ( j k 0 z ) ]
β GS = 2 Γ g 0 , GS w ω GS d GS , g 0 , GS = ω 0 N D Γ x y μ GS d GS 2 2 c η 0 Γ h QD .
p GS ( z , t ) = p GS + ( z , t ) exp ( j k 0 z ) + p GS ( z , t ) exp ( j k 0 z ) = exp ( j k 0 z ) n = 0 + p n , GS + ( z , t ) exp ( 2 n j k 0 z ) + exp ( j k 0 z ) n = 0 + p n , GS ( z , t ) exp ( 2 n j k 0 z ) ρ m ( z , t ) = ρ 0 , m ( z , t ) + n = 1 + [ ρ n , m ( z , t ) exp ( 2 n j k 0 z ) + c . c . ]
1 v g E ± t ± E ± z = α w g 2 E ± j ω 0 2 c η 0 Γ x y μ GS d GS N D h QD p 0 , GS ± + S sp ±
p 0 , GS ± ( z , t ) t = [ j ( ω GS ω 0 ) Γ ] p 0 , GS ± + j d GS [ ( 2 ρ 0 , GS 1 ) E ± + 2 ρ GS ± E ]
ρ 0 , GS ( z , t ) t = ρ 0 , GS τ nr GS ρ 0 , GS 2 τ sp GS ρ 0 , GS ( 1 ρ 0 , E S ) τ e GS + μ ES μ GS ρ 0 , E S ( 1 ρ 0 , GS ) τ C GS β GS μ GS N D η L Im ( E + * ρ 0 , GS + + E * ρ 0 , GS )
ρ GS + ( z , t ) t = ρ GS + τ nr GS ρ GS + ( 1 ρ 0 , E S ) τ e GS + ρ 0 , GS ρ ES + τ e GS μ ES μ GS ρ 0 , E S ρ GS + τ C GS + μ ES μ GS ρ ES + ( 1 ρ 0 , GS ) τ C GS ρ 0 , GS ρ GS + τ sp GS + j β GS 2 μ GS N D η L ( E * p 0 , GS + E + p 0 , GS * )
ρ 0 , E S ( z , t ) t = ρ 0 , E S τ nr ES + D WL μ ES N d ρ 0 , W L ( 1 ρ 0 , E S ) τ C ES ρ 0 , E S ( 1 ρ 0 , W L ) τ e ES + μ GS μ ES ρ 0 , GS ( 1 ρ 0 , E S ) τ e GS ρ 0 , E S ( 1 ρ 0 , GS ) τ C GS
ρ ES + ( z , t ) t = ρ ES + τ nr ES + μ GS μ ES ρ GS + ( 1 ρ 0 , E S ) τ e GS μ GS μ ES ρ 0 , GS ρ ES + τ e GS + ρ 0 , E S ρ GS + τ C GS ρ ES + ( 1 ρ 0 , GS ) τ C GS D WL μ ES N D [ ρ 0 , W L ρ ES + τ C ES ρ WL + ( 1 ρ 0 , E S ) τ C ES ] + ρ 0 , E S ρ WL + τ e ES ρ ES + ( 1 ρ 0 , W L ) τ e ES
ρ 0 , W L ( z , t ) t = Λ ρ 0 , W L τ nr WL ρ 0 , W L τ C ES ( 1 ρ 0 , E S ) + μ GS N D D WL ρ 0 , E S τ e ES ( 1 ρ 0 , W L )
ρ WL + ( z , t ) t = ρ WL + τ nr WL + ρ 0 , W L τ C ES ρ ES + ρ WL + τ C ES ( 1 ρ 0 , E S ) μ ES N D D WL ( ρ 0 , E S ρ WL + τ e ES ρ ES + ( 1 ρ 0 , W L ) τ e ES )
ρ m ( z , t ) ρ 0 , m ( z , t ) + ρ m + ( z , t ) exp ( 2 k 0 z ) + ρ m ( z , t ) exp ( 2 k 0 z ) ,
RIN ( f ) = | { P ( t ) < P ( t ) > } | 2 / < P ( t ) > 2 , R I N q ( f ) = | { P q ( t ) μ P q } | 2 / μ P q 2
iRIN = B RIN ( f ) d f B , i RIN q = B RIN q ( f ) d f B ;
p k , j ± = j d GS Γ [ ( 2 ρ 0 GS k , j 1 ) I k , j + 2 ρ GS k , j ± I k , j ]
I ± = e Γ Δ t I k , j 1 ± + Γ Δ t 2 e Γ Δ t E k , j 1 ± + Γ Δ t 2 E k , j ±
E k , j ± = E k 1 , j 1 ± + Δ z { α w g 2 E k , j ± + E k 1 , j 1 ± 2 j g 0 , GS Γ d GS p k , j ± + p k 1 , j 1 ± 2 }
E k , j ± = E k 1 , j p ± + Δ z 2 { α w g 2 [ E k , j ± + E k ± 1 , j 1 ± ] + G 0 ; k , j [ e Γ Δ t I k , j 1 + + Γ Δ t 2 e Γ Δ t E k , j 1 ± + Γ Δ t 2 E k , j ± ] + G 0 ; k 1 , j 1 I k ± 1 , j 1 ± + G k , j ± [ e Γ Δ t I k , j 1 + Γ Δ t 2 e Γ Δ t E k , j 1 + Γ Δ t 2 E k , j ] + G k 1 , j 1 ± I k 1 , j 1 }
C ( k 1 , k ) , ( j , j 1 ) ± = Δ z z { α w g 2 E k ± 1 , j 1 ± + G 0 ; k , j [ e Γ Δ t I k , j 1 ± + Γ Δ t 2 e Γ Δ t E k , j 1 ± ] + G k , j ± [ e Γ Δ t I k , j 1 + Γ Δ t 2 e Γ Δ t E k , j 1 ] + G 0 ; k 1 , j 1 I k 1 , j 1 ± + G k 1 , j 1 ± I k 1 , j 1 }
[ 1 A 0 ; k , j A k , j + A k , j + * 1 A 0 ; k , j ] [ E k , j + E k , j ] = [ C ( k 1 , k ) , ( j , j 1 ) + + E k 1 , j 1 + C ( k + 1 , k ) , ( j , j 1 ) + E k + 1 , j 1 ]

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