Abstract

We demonstrate class I excitability in optically injected microdisk lasers, and propose a possible optical spiking neuron design. The neuron has a clear threshold and an integrating behavior, leading to an output rate-input rate dependency that is comparable to the characteristic of sigmoidal artificial neurons. We also show that the optical phase of the input pulses has influence on the neuron response, and can be used to create inhibitory, as well as excitatory perturbations.

© 2013 OSA

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2013 (4)

M. A. Nahmias, B. J. Shastri, A. N. Tait, S. Member, and P. R. P. Fellow, “A leaky integrate-and-fire laser neuron for ultrafast cognitive computing,” IEEE J. Sel. Top. Quantum Electron.16, 1–12 (2013).

M. Turconi, B. Garbin, M. Feyereisen, M. Giudici, and S. Barland, “Control of excitable pulses in an injection-locked semiconductor laser,” Phys. Rev. E88, 022923 (2013).
[CrossRef]

P. Mechet, S. Verstuyft, T. D. Vries, T. Spuesens, P. Regreny, D. V. Thourhout, G. Roelkens, and G. Morthier, “Unidirectional III–V microdisk lasers heterogeneously integrated on SOI,” Opt. Express21, 1988–1990 (2013).

M. R. Watts, J. Sun, C. DeRose, D. C. Trotter, R. W. Young, and G. N. Nielson, “Adiabatic thermo-optic Mach-Zehnder switch,” Opt. Lett.38, 733–5 (2013).
[CrossRef] [PubMed]

2012 (3)

2011 (3)

S. Stankovicić, R. Jones, J. Heck, M. Sysak, D. Van Thourhout, and G. Roelkens, “Die-to-die adhesive bonding procedure for evanescently-coupled photonic devices,” Electrochem. Solid-State Lett.14, H326 (2011).
[CrossRef]

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 1–8 (2011).
[CrossRef]

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw.22, 1469–1481 (2011).
[CrossRef] [PubMed]

2010 (4)

W. Coomans, S. Beri, G. V. D. Sande, L. Gelens, and J. Danckaert, “Optical injection in semiconductor ring lasers,” Phys. Rev. A81, 033802 (2010).
[CrossRef]

L. Liu, R. Kumar, K. Huybrechts, G. Roelkens, E.-j. Geluk, T. Spuesens, T. D. Vries, P. Regreny, D.V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4, 1–6 (2010).
[CrossRef]

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A374, 739–743 (2010).
[CrossRef]

L. Gelens, S. Beri, G. Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. Journ. D58, 197–207 (2010).
[CrossRef]

2009 (2)

L. Gelens, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Appl. Phys.1–9 (2009).

S. Ghosh-Dastidar and H. Adeli, “Spiking neural networks,” Int. J. Neural Syst.19, 295–308 (2009).
[CrossRef] [PubMed]

2008 (2)

B. Schrauwen, M. D’Haene, D. Verstraeten, and J. V. Campenhout, “Compact hardware liquid state machines on FPGA for real-time speech recognition,” Neural Netw.21, 511–23 (2008).
[CrossRef] [PubMed]

J. Van Campenhout, P. Romeo, D. Van Thourhout, C. Seassal, P. Regreny, L. Di Cioccio, J.-M. Fedeli, and R. Baets, “Design and optimization of electrically injected InP-based microdisk lasers integrated on and coupled to a SOI waveguide circuit,” J. Lightw. Technol.26, 52–63 (2008).
[CrossRef]

2007 (1)

D. Goulding, S. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

2005 (1)

S. Wieczorek, B. Krauskopf, T. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rept.416, 1–128 (2005).
[CrossRef]

2004 (1)

H. Jaeger, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science304, 78–80 (2004).
[CrossRef] [PubMed]

2003 (1)

M. Sorel, G. Giuliani, a. Scire, R. Miglierina, S. Donati, and P. Laybourn, “Operating regimes of gaas-algaas semiconductor ring lasers: experiment and model,” IEEE J. Quant. Electron.39, 1187–1195 (2003).
[CrossRef]

2002 (3)

M. Sorel, P. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett.27, 1992–1994 (2002).
[CrossRef]

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: a new framework for neural computation based on perturbations,” Neural Comput.14, 2531–2560 (2002).
[CrossRef] [PubMed]

S. Wieczorek, B. Krauskopf, and D. Lenstra, “Multipulse excitability in a semiconductor laser with optical injection,” Phys. Rev. Lett.88, 063901 (2002).
[CrossRef] [PubMed]

1997 (1)

W. Maass, “Networks of spiking neurons: the third generation of neural network models,” Neural Netw.10, 1659–1671 (1997).
[CrossRef]

Adeli, H.

S. Ghosh-Dastidar and H. Adeli, “Spiking neural networks,” Int. J. Neural Syst.19, 295–308 (2009).
[CrossRef] [PubMed]

Akopyan, F.

P. Merolla, J. Arthur, F. Akopyan, N. Imam, R. Manohar, and D. S. Modha, “A digital neurosynaptic core using embedded crossbar memory with 45pj per spike in 45nm,” in “Custom Integrated Circuits Conference (CICC), 2011 IEEE,” (IEEE, San Jose, 2011), 1–4.
[CrossRef]

Arthur, J.

P. Merolla, J. Arthur, F. Akopyan, N. Imam, R. Manohar, and D. S. Modha, “A digital neurosynaptic core using embedded crossbar memory with 45pj per spike in 45nm,” in “Custom Integrated Circuits Conference (CICC), 2011 IEEE,” (IEEE, San Jose, 2011), 1–4.
[CrossRef]

Baets, R.

L. Liu, R. Kumar, K. Huybrechts, G. Roelkens, E.-j. Geluk, T. Spuesens, T. D. Vries, P. Regreny, D.V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4, 1–6 (2010).
[CrossRef]

J. Van Campenhout, P. Romeo, D. Van Thourhout, C. Seassal, P. Regreny, L. Di Cioccio, J.-M. Fedeli, and R. Baets, “Design and optimization of electrically injected InP-based microdisk lasers integrated on and coupled to a SOI waveguide circuit,” J. Lightw. Technol.26, 52–63 (2008).
[CrossRef]

Y. De Koninck, K. Huybrechts, G. Van der Sande, J. Danckaert, R. Baets, and G. Morthier, “Nonlinear dynamics of asymmetrically coupled microdisk lasers,” in “LEOS Annual Meeting Conference Proceedings, 2009. LEOS’09. IEEE,” (IEEE, 2009), 503–504.
[CrossRef]

Balle, S.

Barland, S.

M. Turconi, B. Garbin, M. Feyereisen, M. Giudici, and S. Barland, “Control of excitable pulses in an injection-locked semiconductor laser,” Phys. Rev. E88, 022923 (2013).
[CrossRef]

Beri, S.

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 1–8 (2011).
[CrossRef]

L. Gelens, S. Beri, G. Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. Journ. D58, 197–207 (2010).
[CrossRef]

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A374, 739–743 (2010).
[CrossRef]

W. Coomans, S. Beri, G. V. D. Sande, L. Gelens, and J. Danckaert, “Optical injection in semiconductor ring lasers,” Phys. Rev. A81, 033802 (2010).
[CrossRef]

L. Gelens, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Appl. Phys.1–9 (2009).

Bienstman, P.

Caluwaerts, K.

Campenhout, J. V.

B. Schrauwen, M. D’Haene, D. Verstraeten, and J. V. Campenhout, “Compact hardware liquid state machines on FPGA for real-time speech recognition,” Neural Netw.21, 511–23 (2008).
[CrossRef] [PubMed]

Coomans, W.

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 1–8 (2011).
[CrossRef]

W. Coomans, S. Beri, G. V. D. Sande, L. Gelens, and J. Danckaert, “Optical injection in semiconductor ring lasers,” Phys. Rev. A81, 033802 (2010).
[CrossRef]

D’Haene, M.

B. Schrauwen, M. D’Haene, D. Verstraeten, and J. V. Campenhout, “Compact hardware liquid state machines on FPGA for real-time speech recognition,” Neural Netw.21, 511–23 (2008).
[CrossRef] [PubMed]

Dambre, J.

Danckaert, J.

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 1–8 (2011).
[CrossRef]

L. Gelens, S. Beri, G. Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. Journ. D58, 197–207 (2010).
[CrossRef]

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A374, 739–743 (2010).
[CrossRef]

W. Coomans, S. Beri, G. V. D. Sande, L. Gelens, and J. Danckaert, “Optical injection in semiconductor ring lasers,” Phys. Rev. A81, 033802 (2010).
[CrossRef]

L. Gelens, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Appl. Phys.1–9 (2009).

Y. De Koninck, K. Huybrechts, G. Van der Sande, J. Danckaert, R. Baets, and G. Morthier, “Nonlinear dynamics of asymmetrically coupled microdisk lasers,” in “LEOS Annual Meeting Conference Proceedings, 2009. LEOS’09. IEEE,” (IEEE, 2009), 503–504.
[CrossRef]

De Koninck, Y.

Y. De Koninck, K. Huybrechts, G. Van der Sande, J. Danckaert, R. Baets, and G. Morthier, “Nonlinear dynamics of asymmetrically coupled microdisk lasers,” in “LEOS Annual Meeting Conference Proceedings, 2009. LEOS’09. IEEE,” (IEEE, 2009), 503–504.
[CrossRef]

DeRose, C.

Di Cioccio, L.

J. Van Campenhout, P. Romeo, D. Van Thourhout, C. Seassal, P. Regreny, L. Di Cioccio, J.-M. Fedeli, and R. Baets, “Design and optimization of electrically injected InP-based microdisk lasers integrated on and coupled to a SOI waveguide circuit,” J. Lightw. Technol.26, 52–63 (2008).
[CrossRef]

Donati, S.

M. Sorel, G. Giuliani, a. Scire, R. Miglierina, S. Donati, and P. Laybourn, “Operating regimes of gaas-algaas semiconductor ring lasers: experiment and model,” IEEE J. Quant. Electron.39, 1187–1195 (2003).
[CrossRef]

M. Sorel, P. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett.27, 1992–1994 (2002).
[CrossRef]

Fedeli, J.-M.

J. Van Campenhout, P. Romeo, D. Van Thourhout, C. Seassal, P. Regreny, L. Di Cioccio, J.-M. Fedeli, and R. Baets, “Design and optimization of electrically injected InP-based microdisk lasers integrated on and coupled to a SOI waveguide circuit,” J. Lightw. Technol.26, 52–63 (2008).
[CrossRef]

Fellow, P. R. P.

M. A. Nahmias, B. J. Shastri, A. N. Tait, S. Member, and P. R. P. Fellow, “A leaky integrate-and-fire laser neuron for ultrafast cognitive computing,” IEEE J. Sel. Top. Quantum Electron.16, 1–12 (2013).

Feyereisen, M.

M. Turconi, B. Garbin, M. Feyereisen, M. Giudici, and S. Barland, “Control of excitable pulses in an injection-locked semiconductor laser,” Phys. Rev. E88, 022923 (2013).
[CrossRef]

Fiers, M.

Garbin, B.

M. Turconi, B. Garbin, M. Feyereisen, M. Giudici, and S. Barland, “Control of excitable pulses in an injection-locked semiconductor laser,” Phys. Rev. E88, 022923 (2013).
[CrossRef]

Gelens, L.

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 1–8 (2011).
[CrossRef]

L. Gelens, S. Beri, G. Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. Journ. D58, 197–207 (2010).
[CrossRef]

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A374, 739–743 (2010).
[CrossRef]

W. Coomans, S. Beri, G. V. D. Sande, L. Gelens, and J. Danckaert, “Optical injection in semiconductor ring lasers,” Phys. Rev. A81, 033802 (2010).
[CrossRef]

L. Gelens, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Appl. Phys.1–9 (2009).

Geluk, E.-j.

L. Liu, R. Kumar, K. Huybrechts, G. Roelkens, E.-j. Geluk, T. Spuesens, T. D. Vries, P. Regreny, D.V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4, 1–6 (2010).
[CrossRef]

Ghosh-Dastidar, S.

S. Ghosh-Dastidar and H. Adeli, “Spiking neural networks,” Int. J. Neural Syst.19, 295–308 (2009).
[CrossRef] [PubMed]

Giudici, M.

M. Turconi, B. Garbin, M. Feyereisen, M. Giudici, and S. Barland, “Control of excitable pulses in an injection-locked semiconductor laser,” Phys. Rev. E88, 022923 (2013).
[CrossRef]

Giuliani, G.

M. Sorel, G. Giuliani, a. Scire, R. Miglierina, S. Donati, and P. Laybourn, “Operating regimes of gaas-algaas semiconductor ring lasers: experiment and model,” IEEE J. Quant. Electron.39, 1187–1195 (2003).
[CrossRef]

M. Sorel, P. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett.27, 1992–1994 (2002).
[CrossRef]

Goulding, D.

D. Goulding, S. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Greene, G.

D. Goulding, S. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Hartnett, M.

D. Goulding, S. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Heck, J.

S. Stankovicić, R. Jones, J. Heck, M. Sysak, D. Van Thourhout, and G. Roelkens, “Die-to-die adhesive bonding procedure for evanescently-coupled photonic devices,” Electrochem. Solid-State Lett.14, H326 (2011).
[CrossRef]

Hegarty, S.

D. Goulding, S. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Huybrechts, K.

L. Liu, R. Kumar, K. Huybrechts, G. Roelkens, E.-j. Geluk, T. Spuesens, T. D. Vries, P. Regreny, D.V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4, 1–6 (2010).
[CrossRef]

Y. De Koninck, K. Huybrechts, G. Van der Sande, J. Danckaert, R. Baets, and G. Morthier, “Nonlinear dynamics of asymmetrically coupled microdisk lasers,” in “LEOS Annual Meeting Conference Proceedings, 2009. LEOS’09. IEEE,” (IEEE, 2009), 503–504.
[CrossRef]

Huyet, G.

D. Goulding, S. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Imam, N.

P. Merolla, J. Arthur, F. Akopyan, N. Imam, R. Manohar, and D. S. Modha, “A digital neurosynaptic core using embedded crossbar memory with 45pj per spike in 45nm,” in “Custom Integrated Circuits Conference (CICC), 2011 IEEE,” (IEEE, San Jose, 2011), 1–4.
[CrossRef]

Izhikevich, E. M.

E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (Computational Neuroscience) (The MIT Press, 2006), 1st ed.

Jaeger, H.

H. Jaeger, “Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication,” Science304, 78–80 (2004).
[CrossRef] [PubMed]

Jones, R.

S. Stankovicić, R. Jones, J. Heck, M. Sysak, D. Van Thourhout, and G. Roelkens, “Die-to-die adhesive bonding procedure for evanescently-coupled photonic devices,” Electrochem. Solid-State Lett.14, H326 (2011).
[CrossRef]

Keyvaninia, S.

Krauskopf, B.

S. Wieczorek, B. Krauskopf, T. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rept.416, 1–128 (2005).
[CrossRef]

S. Wieczorek, B. Krauskopf, and D. Lenstra, “Multipulse excitability in a semiconductor laser with optical injection,” Phys. Rev. Lett.88, 063901 (2002).
[CrossRef] [PubMed]

Kumar, R.

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express20, 20292–20308 (2012).
[CrossRef] [PubMed]

L. Liu, R. Kumar, K. Huybrechts, G. Roelkens, E.-j. Geluk, T. Spuesens, T. D. Vries, P. Regreny, D.V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4, 1–6 (2010).
[CrossRef]

Laybourn, P.

M. Sorel, G. Giuliani, a. Scire, R. Miglierina, S. Donati, and P. Laybourn, “Operating regimes of gaas-algaas semiconductor ring lasers: experiment and model,” IEEE J. Quant. Electron.39, 1187–1195 (2003).
[CrossRef]

M. Sorel, P. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett.27, 1992–1994 (2002).
[CrossRef]

Lenstra, D.

S. Wieczorek, B. Krauskopf, T. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rept.416, 1–128 (2005).
[CrossRef]

S. Wieczorek, B. Krauskopf, and D. Lenstra, “Multipulse excitability in a semiconductor laser with optical injection,” Phys. Rev. Lett.88, 063901 (2002).
[CrossRef] [PubMed]

Liu, L.

L. Liu, R. Kumar, K. Huybrechts, G. Roelkens, E.-j. Geluk, T. Spuesens, T. D. Vries, P. Regreny, D.V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4, 1–6 (2010).
[CrossRef]

Maass, W.

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: a new framework for neural computation based on perturbations,” Neural Comput.14, 2531–2560 (2002).
[CrossRef] [PubMed]

W. Maass, “Networks of spiking neurons: the third generation of neural network models,” Neural Netw.10, 1659–1671 (1997).
[CrossRef]

Manohar, R.

P. Merolla, J. Arthur, F. Akopyan, N. Imam, R. Manohar, and D. S. Modha, “A digital neurosynaptic core using embedded crossbar memory with 45pj per spike in 45nm,” in “Custom Integrated Circuits Conference (CICC), 2011 IEEE,” (IEEE, San Jose, 2011), 1–4.
[CrossRef]

Markram, H.

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: a new framework for neural computation based on perturbations,” Neural Comput.14, 2531–2560 (2002).
[CrossRef] [PubMed]

Mashall, L.

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A374, 739–743 (2010).
[CrossRef]

McInerney, J.

D. Goulding, S. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Mechet, P.

P. Mechet, S. Verstuyft, T. D. Vries, T. Spuesens, P. Regreny, D. V. Thourhout, G. Roelkens, and G. Morthier, “Unidirectional III–V microdisk lasers heterogeneously integrated on SOI,” Opt. Express21, 1988–1990 (2013).

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express20, 20292–20308 (2012).
[CrossRef] [PubMed]

Melnik, S.

D. Goulding, S. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Member, S.

M. A. Nahmias, B. J. Shastri, A. N. Tait, S. Member, and P. R. P. Fellow, “A leaky integrate-and-fire laser neuron for ultrafast cognitive computing,” IEEE J. Sel. Top. Quantum Electron.16, 1–12 (2013).

Merolla, P.

P. Merolla, J. Arthur, F. Akopyan, N. Imam, R. Manohar, and D. S. Modha, “A digital neurosynaptic core using embedded crossbar memory with 45pj per spike in 45nm,” in “Custom Integrated Circuits Conference (CICC), 2011 IEEE,” (IEEE, San Jose, 2011), 1–4.
[CrossRef]

Mezosi, G.

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A374, 739–743 (2010).
[CrossRef]

Miglierina, R.

M. Sorel, G. Giuliani, a. Scire, R. Miglierina, S. Donati, and P. Laybourn, “Operating regimes of gaas-algaas semiconductor ring lasers: experiment and model,” IEEE J. Quant. Electron.39, 1187–1195 (2003).
[CrossRef]

M. Sorel, P. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett.27, 1992–1994 (2002).
[CrossRef]

Modha, D. S.

P. Merolla, J. Arthur, F. Akopyan, N. Imam, R. Manohar, and D. S. Modha, “A digital neurosynaptic core using embedded crossbar memory with 45pj per spike in 45nm,” in “Custom Integrated Circuits Conference (CICC), 2011 IEEE,” (IEEE, San Jose, 2011), 1–4.
[CrossRef]

Morthier, G.

P. Mechet, S. Verstuyft, T. D. Vries, T. Spuesens, P. Regreny, D. V. Thourhout, G. Roelkens, and G. Morthier, “Unidirectional III–V microdisk lasers heterogeneously integrated on SOI,” Opt. Express21, 1988–1990 (2013).

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express20, 20292–20308 (2012).
[CrossRef] [PubMed]

L. Liu, R. Kumar, K. Huybrechts, G. Roelkens, E.-j. Geluk, T. Spuesens, T. D. Vries, P. Regreny, D.V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4, 1–6 (2010).
[CrossRef]

Y. De Koninck, K. Huybrechts, G. Van der Sande, J. Danckaert, R. Baets, and G. Morthier, “Nonlinear dynamics of asymmetrically coupled microdisk lasers,” in “LEOS Annual Meeting Conference Proceedings, 2009. LEOS’09. IEEE,” (IEEE, 2009), 503–504.
[CrossRef]

Muneeb, M.

Nahmias, M. A.

M. A. Nahmias, B. J. Shastri, A. N. Tait, S. Member, and P. R. P. Fellow, “A leaky integrate-and-fire laser neuron for ultrafast cognitive computing,” IEEE J. Sel. Top. Quantum Electron.16, 1–12 (2013).

M. A. Nahmias, A. N. Tait, B. J. Shastri, and P. R. Prucnal, “An evanescent hybrid silicon laser neuron,” in “Proc. IEEE Photonics Conference (IPC),” (IEEE, Seattle, 2013), 93–94.

Natschläger, T.

W. Maass, T. Natschläger, and H. Markram, “Real-time computing without stable states: a new framework for neural computation based on perturbations,” Neural Comput.14, 2531–2560 (2002).
[CrossRef] [PubMed]

Nielson, G. N.

Prucnal, P. R.

M. A. Nahmias, A. N. Tait, B. J. Shastri, and P. R. Prucnal, “An evanescent hybrid silicon laser neuron,” in “Proc. IEEE Photonics Conference (IPC),” (IEEE, Seattle, 2013), 93–94.

Rachinskii, D.

D. Goulding, S. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Rasskazov, O.

D. Goulding, S. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Regreny, P.

P. Mechet, S. Verstuyft, T. D. Vries, T. Spuesens, P. Regreny, D. V. Thourhout, G. Roelkens, and G. Morthier, “Unidirectional III–V microdisk lasers heterogeneously integrated on SOI,” Opt. Express21, 1988–1990 (2013).

L. Liu, R. Kumar, K. Huybrechts, G. Roelkens, E.-j. Geluk, T. Spuesens, T. D. Vries, P. Regreny, D.V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4, 1–6 (2010).
[CrossRef]

J. Van Campenhout, P. Romeo, D. Van Thourhout, C. Seassal, P. Regreny, L. Di Cioccio, J.-M. Fedeli, and R. Baets, “Design and optimization of electrically injected InP-based microdisk lasers integrated on and coupled to a SOI waveguide circuit,” J. Lightw. Technol.26, 52–63 (2008).
[CrossRef]

Roelkens, G.

P. Mechet, S. Verstuyft, T. D. Vries, T. Spuesens, P. Regreny, D. V. Thourhout, G. Roelkens, and G. Morthier, “Unidirectional III–V microdisk lasers heterogeneously integrated on SOI,” Opt. Express21, 1988–1990 (2013).

S. Keyvaninia, M. Muneeb, S. Stanković, P. J. Van Veldhoven, D. Van Thourhout, and G. Roelkens, “Ultra-thin DVS-BCB adhesive bonding of III–V wafers, dies and multiple dies to a patterned silicon-on-insulator substrate,” Opt. Mater. Express3, 35 (2012).
[CrossRef]

S. Stankovicić, R. Jones, J. Heck, M. Sysak, D. Van Thourhout, and G. Roelkens, “Die-to-die adhesive bonding procedure for evanescently-coupled photonic devices,” Electrochem. Solid-State Lett.14, H326 (2011).
[CrossRef]

L. Liu, R. Kumar, K. Huybrechts, G. Roelkens, E.-j. Geluk, T. Spuesens, T. D. Vries, P. Regreny, D.V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4, 1–6 (2010).
[CrossRef]

Romeo, P.

J. Van Campenhout, P. Romeo, D. Van Thourhout, C. Seassal, P. Regreny, L. Di Cioccio, J.-M. Fedeli, and R. Baets, “Design and optimization of electrically injected InP-based microdisk lasers integrated on and coupled to a SOI waveguide circuit,” J. Lightw. Technol.26, 52–63 (2008).
[CrossRef]

Sande, G.

L. Gelens, S. Beri, G. Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. Journ. D58, 197–207 (2010).
[CrossRef]

Sande, G. V. D.

W. Coomans, S. Beri, G. V. D. Sande, L. Gelens, and J. Danckaert, “Optical injection in semiconductor ring lasers,” Phys. Rev. A81, 033802 (2010).
[CrossRef]

Schrauwen, B.

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express20, 20292–20308 (2012).
[CrossRef] [PubMed]

M. Fiers, T. Van Vaerenbergh, K. Caluwaerts, D. Vande Ginste, B. Schrauwen, J. Dambre, and P. Bienstman, “Time-domain and frequency-domain modeling of nonlinear optical components at the circuit-level using a node-based approach,” J. Opt. Soc. Am. B29, 896–900 (2012).
[CrossRef]

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw.22, 1469–1481 (2011).
[CrossRef] [PubMed]

B. Schrauwen, M. D’Haene, D. Verstraeten, and J. V. Campenhout, “Compact hardware liquid state machines on FPGA for real-time speech recognition,” Neural Netw.21, 511–23 (2008).
[CrossRef] [PubMed]

Scire, a.

M. Sorel, G. Giuliani, a. Scire, R. Miglierina, S. Donati, and P. Laybourn, “Operating regimes of gaas-algaas semiconductor ring lasers: experiment and model,” IEEE J. Quant. Electron.39, 1187–1195 (2003).
[CrossRef]

Scirè, A.

Seassal, C.

J. Van Campenhout, P. Romeo, D. Van Thourhout, C. Seassal, P. Regreny, L. Di Cioccio, J.-M. Fedeli, and R. Baets, “Design and optimization of electrically injected InP-based microdisk lasers integrated on and coupled to a SOI waveguide circuit,” J. Lightw. Technol.26, 52–63 (2008).
[CrossRef]

Shastri, B. J.

M. A. Nahmias, B. J. Shastri, A. N. Tait, S. Member, and P. R. P. Fellow, “A leaky integrate-and-fire laser neuron for ultrafast cognitive computing,” IEEE J. Sel. Top. Quantum Electron.16, 1–12 (2013).

M. A. Nahmias, A. N. Tait, B. J. Shastri, and P. R. Prucnal, “An evanescent hybrid silicon laser neuron,” in “Proc. IEEE Photonics Conference (IPC),” (IEEE, Seattle, 2013), 93–94.

Simpson, T.

S. Wieczorek, B. Krauskopf, T. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rept.416, 1–128 (2005).
[CrossRef]

Sorel, M.

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A374, 739–743 (2010).
[CrossRef]

M. Sorel, G. Giuliani, a. Scire, R. Miglierina, S. Donati, and P. Laybourn, “Operating regimes of gaas-algaas semiconductor ring lasers: experiment and model,” IEEE J. Quant. Electron.39, 1187–1195 (2003).
[CrossRef]

M. Sorel, P. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett.27, 1992–1994 (2002).
[CrossRef]

Spuesens, T.

P. Mechet, S. Verstuyft, T. D. Vries, T. Spuesens, P. Regreny, D. V. Thourhout, G. Roelkens, and G. Morthier, “Unidirectional III–V microdisk lasers heterogeneously integrated on SOI,” Opt. Express21, 1988–1990 (2013).

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express20, 20292–20308 (2012).
[CrossRef] [PubMed]

L. Liu, R. Kumar, K. Huybrechts, G. Roelkens, E.-j. Geluk, T. Spuesens, T. D. Vries, P. Regreny, D.V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4, 1–6 (2010).
[CrossRef]

Stankovic, S.

Stankovicic, S.

S. Stankovicić, R. Jones, J. Heck, M. Sysak, D. Van Thourhout, and G. Roelkens, “Die-to-die adhesive bonding procedure for evanescently-coupled photonic devices,” Electrochem. Solid-State Lett.14, H326 (2011).
[CrossRef]

Sun, J.

Sysak, M.

S. Stankovicić, R. Jones, J. Heck, M. Sysak, D. Van Thourhout, and G. Roelkens, “Die-to-die adhesive bonding procedure for evanescently-coupled photonic devices,” Electrochem. Solid-State Lett.14, H326 (2011).
[CrossRef]

Tait, A. N.

M. A. Nahmias, B. J. Shastri, A. N. Tait, S. Member, and P. R. P. Fellow, “A leaky integrate-and-fire laser neuron for ultrafast cognitive computing,” IEEE J. Sel. Top. Quantum Electron.16, 1–12 (2013).

M. A. Nahmias, A. N. Tait, B. J. Shastri, and P. R. Prucnal, “An evanescent hybrid silicon laser neuron,” in “Proc. IEEE Photonics Conference (IPC),” (IEEE, Seattle, 2013), 93–94.

Thourhout, D. V.

P. Mechet, S. Verstuyft, T. D. Vries, T. Spuesens, P. Regreny, D. V. Thourhout, G. Roelkens, and G. Morthier, “Unidirectional III–V microdisk lasers heterogeneously integrated on SOI,” Opt. Express21, 1988–1990 (2013).

Thourhout, D.V.

L. Liu, R. Kumar, K. Huybrechts, G. Roelkens, E.-j. Geluk, T. Spuesens, T. D. Vries, P. Regreny, D.V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4, 1–6 (2010).
[CrossRef]

Trotter, D. C.

Turconi, M.

M. Turconi, B. Garbin, M. Feyereisen, M. Giudici, and S. Barland, “Control of excitable pulses in an injection-locked semiconductor laser,” Phys. Rev. E88, 022923 (2013).
[CrossRef]

Van Campenhout, J.

J. Van Campenhout, P. Romeo, D. Van Thourhout, C. Seassal, P. Regreny, L. Di Cioccio, J.-M. Fedeli, and R. Baets, “Design and optimization of electrically injected InP-based microdisk lasers integrated on and coupled to a SOI waveguide circuit,” J. Lightw. Technol.26, 52–63 (2008).
[CrossRef]

J. Van Campenhout, “Thin-film microlasers for the integration of electronic and photonic integrated circuits,” Ph.D. thesis, UGent (2007–2008).

Van der Sande, G.

W. Coomans, L. Gelens, S. Beri, J. Danckaert, and G. Van der Sande, “Solitary and coupled semiconductor ring lasers as optical spiking neurons,” Phys. Rev. E84, 1–8 (2011).
[CrossRef]

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A374, 739–743 (2010).
[CrossRef]

Y. De Koninck, K. Huybrechts, G. Van der Sande, J. Danckaert, R. Baets, and G. Morthier, “Nonlinear dynamics of asymmetrically coupled microdisk lasers,” in “LEOS Annual Meeting Conference Proceedings, 2009. LEOS’09. IEEE,” (IEEE, 2009), 503–504.
[CrossRef]

Van Thourhout, D.

S. Keyvaninia, M. Muneeb, S. Stanković, P. J. Van Veldhoven, D. Van Thourhout, and G. Roelkens, “Ultra-thin DVS-BCB adhesive bonding of III–V wafers, dies and multiple dies to a patterned silicon-on-insulator substrate,” Opt. Mater. Express3, 35 (2012).
[CrossRef]

S. Stankovicić, R. Jones, J. Heck, M. Sysak, D. Van Thourhout, and G. Roelkens, “Die-to-die adhesive bonding procedure for evanescently-coupled photonic devices,” Electrochem. Solid-State Lett.14, H326 (2011).
[CrossRef]

J. Van Campenhout, P. Romeo, D. Van Thourhout, C. Seassal, P. Regreny, L. Di Cioccio, J.-M. Fedeli, and R. Baets, “Design and optimization of electrically injected InP-based microdisk lasers integrated on and coupled to a SOI waveguide circuit,” J. Lightw. Technol.26, 52–63 (2008).
[CrossRef]

Van Vaerenbergh, T.

Van Veldhoven, P. J.

Vande Ginste, D.

Vandoorne, K.

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw.22, 1469–1481 (2011).
[CrossRef] [PubMed]

Verschaffelt, G.

L. Gelens, S. Beri, G. Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. Journ. D58, 197–207 (2010).
[CrossRef]

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A374, 739–743 (2010).
[CrossRef]

Verstraeten, D.

K. Vandoorne, J. Dambre, D. Verstraeten, B. Schrauwen, and P. Bienstman, “Parallel reservoir computing using optical amplifiers,” IEEE Trans. Neural Netw.22, 1469–1481 (2011).
[CrossRef] [PubMed]

B. Schrauwen, M. D’Haene, D. Verstraeten, and J. V. Campenhout, “Compact hardware liquid state machines on FPGA for real-time speech recognition,” Neural Netw.21, 511–23 (2008).
[CrossRef] [PubMed]

Verstuyft, S.

P. Mechet, S. Verstuyft, T. D. Vries, T. Spuesens, P. Regreny, D. V. Thourhout, G. Roelkens, and G. Morthier, “Unidirectional III–V microdisk lasers heterogeneously integrated on SOI,” Opt. Express21, 1988–1990 (2013).

Vries, T. D.

P. Mechet, S. Verstuyft, T. D. Vries, T. Spuesens, P. Regreny, D. V. Thourhout, G. Roelkens, and G. Morthier, “Unidirectional III–V microdisk lasers heterogeneously integrated on SOI,” Opt. Express21, 1988–1990 (2013).

L. Liu, R. Kumar, K. Huybrechts, G. Roelkens, E.-j. Geluk, T. Spuesens, T. D. Vries, P. Regreny, D.V. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4, 1–6 (2010).
[CrossRef]

Watts, M. R.

Wieczorek, S.

S. Wieczorek, B. Krauskopf, T. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Phys. Rept.416, 1–128 (2005).
[CrossRef]

S. Wieczorek, B. Krauskopf, and D. Lenstra, “Multipulse excitability in a semiconductor laser with optical injection,” Phys. Rev. Lett.88, 063901 (2002).
[CrossRef] [PubMed]

Young, R. W.

Appl. Phys. (1)

L. Gelens, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Appl. Phys.1–9 (2009).

Electrochem. Solid-State Lett. (1)

S. Stankovicić, R. Jones, J. Heck, M. Sysak, D. Van Thourhout, and G. Roelkens, “Die-to-die adhesive bonding procedure for evanescently-coupled photonic devices,” Electrochem. Solid-State Lett.14, H326 (2011).
[CrossRef]

Eur. Phys. Journ. D (1)

L. Gelens, S. Beri, G. Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. Journ. D58, 197–207 (2010).
[CrossRef]

IEEE J. Quant. Electron. (1)

M. Sorel, G. Giuliani, a. Scire, R. Miglierina, S. Donati, and P. Laybourn, “Operating regimes of gaas-algaas semiconductor ring lasers: experiment and model,” IEEE J. Quant. Electron.39, 1187–1195 (2003).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

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Figures (6)

Fig. 1
Fig. 1

Bifurcation diagram of the microdisk laser. For both output powers Pout, i = |Eout, i|2, the extrema are plotted. If the output is constant, maximum and minimum power are equal and the markers overlap. If the output power oscillates, two markers are plotted per current, per mode. Inset: geometry of the microdisk laser.

Fig. 2
Fig. 2

Left: Bifurcation lines and locking regions for I = 2.3 mA, red lines represent Hopfbifurcations, blue lines are limit points. Time traces: Crossing the SN1 bifurcation at Δω = −15 ns−1, | E in | = 2.77 μ W. At t = 10 ns the locking amplitude is raised from 2.65 to 2.76 μ W, at t = 20 ns, the bifurcation is crossed by again increasing |Ein| to 2.78 μ W.

Fig. 3
Fig. 3

The ‘neuron’ topology; using a constant locking signal (CW), the microdisk gets locked just above the SN1-bifurcation. Pulses at the other input of the splitter, perturb the microdisk, possibly causing excitation. The energy peak in the E mode that comes with this excitation, can be seen as the output pulse.

Fig. 4
Fig. 4

a)–c) Response to pulses of fixed length (0.2 ns), for different pulse powers, out of phase with the locking signal. a: Time traces. b: Output peak power as a function of input peak power. c: Pulse latency as a function of input peak power. d)–f) Response to pulses of fixed peak power and length (1.4 μW, 0.24 ns), but varying phase. d: Time traces. e: Output peak power as a function of phase. f: Pulse latency as a function of input pulse phase.

Fig. 5
Fig. 5

a) Input rate-output rate characteristic of the neuron, when the input pulse train consists of subthreshold pulses (|Ein|2 = 1 μW peak power, 1 ns length). b)–c) time-traces, for input rates 1 GHz (a), and 4 GHz (b). The arrows show how they are linked to points in the first graph.

Fig. 6
Fig. 6

Output pulse rate as response to the superposition of two input pulse streams, one stream is constant (6 GHz, 1 μW peak power and 0.1 ns duration) and purely excitatory (relative phase with respect to the CW is π). The other pulse stream has a varying rate r2, the horizontal axis represents its phase. Peak power and duration are the same as for the first stream. The error bars give the 80% certainty interval for the inverse interpulse delays.

Tables (1)

Tables Icon

Table 1 Model parameters used in this paper are based on the values and definitions proposed in [27] and [17]. Comparable values can be found in [19].

Equations (6)

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d E + d t = 1 2 ( 1 j α ) ( G + 1 τ p ) E + + j Δ ω E + + C E j κ τ ω 0 E in , 1
d E _ d t = 1 2 ( 1 j α ) ( G 1 τ p ) E + j Δ ω E + C E + j κ τ ω 0 E in , 2
d N d t = η I q N τ c G + | E + | 2 G | E 2 |
G ± = Γ g N ( N N 0 ) 1 + Γ ε N L ( | E ± | 2 + 2 | E | 2 )
E out , 1 = E in , 2 j κ ω 0 τ E
E out , 2 = E in , 1 j κ ω 0 τ E +

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