Abstract

With the development of new applications using semiconductor ring lasers (SRLs) subject to optical feedback, the stability properties of their outputs becomes a crucial issue. We propose a systematic bifurcation analysis in order to properly identify the best parameter ranges for either steady or self-pulsating periodic regimes. Unlike conventional semiconductor lasers, we show that SRLs exhibit both types of outputs for large and well defined ranges of the feedback strength. We determine the stability domains in terms of the pump parameter and the feedback phase. We find that the feedback phase is a key parameter to achieve a stable steady output. We demonstrate that the self-pulsating regime results from a particular Hopf bifurcation mechanism referred to as bifurcation bridges. These bridges connect two distinct external cavity modes and are fully stable, a scenario that was not possible for diode lasers under the same conditions.

© 2017 Optical Society of America

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References

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    [Crossref]
  3. J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70(10), 1192–1194 (1997).
    [Crossref]
  4. M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27(22), 1992–1994 (2002).
    [Crossref]
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    [Crossref]
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    [Crossref]
  23. M. Khoder, G. Friart, J. Danckaert, T. Erneux, G. Van der Sande, and G. Verschaffelt, “Dynamics of semiconductor microring lasers subject to on-chip filtered optical feedback,” Proc. SPIE 9892, 98921J (2016).
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]

2016 (2)

M. Khoder, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Effect of external optical feedback on tunable micro-ring lasers using on-chip filtered feedback,” IEEE Photon. Technol. Lett. 28(9), 959–962 (2016).
[Crossref]

M. Khoder, G. Friart, J. Danckaert, T. Erneux, G. Van der Sande, and G. Verschaffelt, “Dynamics of semiconductor microring lasers subject to on-chip filtered optical feedback,” Proc. SPIE 9892, 98921J (2016).
[Crossref]

2014 (1)

M. Khoder, R. M. Nguimdo, X. Leijtens, J. Bolk, J. Danckaert, and G. Verschaffelt, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” IEEE Photon. Technol. Lett. 26(5), 520–523 (2014).
[Crossref]

2013 (3)

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Digitally tunable dual wavelength emission from semiconductor ring lasers with filtered optical feedback,” Laser Phys. Lett. 10(7), 075804 (2013).
[Crossref]

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Controlled multiwavelength emission using semiconductor ring lasers with on-chip filtered optical feedback,” Opt. Lett. 38(14), 2608–2610 (2013).
[Crossref] [PubMed]

G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013).
[Crossref]

2012 (1)

2011 (2)

2009 (1)

J. Javaloyes and S. Balle, “Emission directionality of semiconductor ring lasers: A traveling-wave description,” IEEE J. Quantum Electron. 45(5), 431–438 (2009).
[Crossref]

2008 (1)

G. Van der Sande, L. Gelens, P. Tassin, A. Scirè, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behaviour of a semiconductor ring laser,” J. Phys. B 41(9), 095402(2008).
[Crossref]

2004 (2)

O. Ushakov, S. Bauer, O. Brox, H.-J. Wünsche, and F. Henneberger, “Self-organization in semiconductor lasers with ultrashort optical feedback,” Phys. Rev. Lett. 92, 043902 (2004).
[Crossref] [PubMed]

S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radziunas, J. Sieber, H.-J. Wünsche, and F. Henneberger, “Nonlinear dynamics of semiconductor lasers with active optical feedback,” Phys. Rev. E 69, 016206 (2004).
[Crossref]

2003 (2)

O. Brox, S. Bauer, M. Radziunas, M. Wolfrum, J. Sieber, J. Kreissl, B. Sartorius, and H. J. Wunsche, “High-frequency pulsations in dfb lasers with amplified feedback,” IEEE J. Quantum Electron. 39(11), 1381–1387 (2003).
[Crossref]

M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

2002 (3)

M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27(22), 1992–1994 (2002).
[Crossref]

M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Mégret, and M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A 65, 041801 (2002).
[Crossref]

T. Erneux, A. Gavrielides, and M. Sciamanna, “Stable microwave oscillations due to external-cavity-mode beating in laser diodes subject to optical feedback,” Phys. Rev. A 66, 033809 (2002).
[Crossref]

2001 (1)

D. Pieroux, T. Erneux, B. Haegeman, K. Engelborghs, and D. Roose, “Bridges of periodic solutions and tori in semiconductor lasers subject to delay,” Phys. Rev. Lett. 87, 193901 (2001).
[Crossref] [PubMed]

2000 (1)

T. Erneux, F. Rogister, A. Gavrielides, and V. Kovanis, “Bifurcation to mixed external cavity mode solutions for semiconductor lasers subject to optical feedback,” Opt. Commun. 183, 467–477 (2000).
[Crossref]

1997 (1)

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70(10), 1192–1194 (1997).
[Crossref]

1992 (1)

J. P. Hohimer, G. A. Vawter, D. C. Craft, and G. R. Hadley, “Improving the performance of semiconductor ring lasers by controlled reflection feedback,” Appl. Phys. Lett. 61(9), 1013–1015 (1992).
[Crossref]

1990 (2)

T. Krauss, P. J. R. Laybourn, and J. Roberts, “Cw operation of semiconductor ring lasers,” Electron. Lett. 26(25), 2095–2097 (1990).
[Crossref]

R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42, 4315–4324 (1990).
[Crossref] [PubMed]

1986 (1)

W. R. Christian and L. Mandel, “Frequency-dependence of a ring laser with backscattering,” Phys. Rev. A 34, 3932–3939 (1986).
[Crossref]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[Crossref]

Arai, K.

Ballantyne, J. M.

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70(10), 1192–1194 (1997).
[Crossref]

Balle, S.

J. Javaloyes and S. Balle, “Emission directionality of semiconductor ring lasers: A traveling-wave description,” IEEE J. Quantum Electron. 45(5), 431–438 (2009).
[Crossref]

M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27(22), 1992–1994 (2002).
[Crossref]

Bauer, S.

O. Ushakov, S. Bauer, O. Brox, H.-J. Wünsche, and F. Henneberger, “Self-organization in semiconductor lasers with ultrashort optical feedback,” Phys. Rev. Lett. 92, 043902 (2004).
[Crossref] [PubMed]

S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radziunas, J. Sieber, H.-J. Wünsche, and F. Henneberger, “Nonlinear dynamics of semiconductor lasers with active optical feedback,” Phys. Rev. E 69, 016206 (2004).
[Crossref]

O. Brox, S. Bauer, M. Radziunas, M. Wolfrum, J. Sieber, J. Kreissl, B. Sartorius, and H. J. Wunsche, “High-frequency pulsations in dfb lasers with amplified feedback,” IEEE J. Quantum Electron. 39(11), 1381–1387 (2003).
[Crossref]

Blondel, M.

M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Mégret, and M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A 65, 041801 (2002).
[Crossref]

Bolk, J.

M. Khoder, R. M. Nguimdo, X. Leijtens, J. Bolk, J. Danckaert, and G. Verschaffelt, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” IEEE Photon. Technol. Lett. 26(5), 520–523 (2014).
[Crossref]

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Digitally tunable dual wavelength emission from semiconductor ring lasers with filtered optical feedback,” Laser Phys. Lett. 10(7), 075804 (2013).
[Crossref]

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Controlled multiwavelength emission using semiconductor ring lasers with on-chip filtered optical feedback,” Opt. Lett. 38(14), 2608–2610 (2013).
[Crossref] [PubMed]

Brox, O.

S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radziunas, J. Sieber, H.-J. Wünsche, and F. Henneberger, “Nonlinear dynamics of semiconductor lasers with active optical feedback,” Phys. Rev. E 69, 016206 (2004).
[Crossref]

O. Ushakov, S. Bauer, O. Brox, H.-J. Wünsche, and F. Henneberger, “Self-organization in semiconductor lasers with ultrashort optical feedback,” Phys. Rev. Lett. 92, 043902 (2004).
[Crossref] [PubMed]

O. Brox, S. Bauer, M. Radziunas, M. Wolfrum, J. Sieber, J. Kreissl, B. Sartorius, and H. J. Wunsche, “High-frequency pulsations in dfb lasers with amplified feedback,” IEEE J. Quantum Electron. 39(11), 1381–1387 (2003).
[Crossref]

Christian, W. R.

W. R. Christian and L. Mandel, “Frequency-dependence of a ring laser with backscattering,” Phys. Rev. A 34, 3932–3939 (1986).
[Crossref]

Craft, D. C.

J. P. Hohimer, G. A. Vawter, D. C. Craft, and G. R. Hadley, “Improving the performance of semiconductor ring lasers by controlled reflection feedback,” Appl. Phys. Lett. 61(9), 1013–1015 (1992).
[Crossref]

Danckaert, J.

M. Khoder, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Effect of external optical feedback on tunable micro-ring lasers using on-chip filtered feedback,” IEEE Photon. Technol. Lett. 28(9), 959–962 (2016).
[Crossref]

M. Khoder, G. Friart, J. Danckaert, T. Erneux, G. Van der Sande, and G. Verschaffelt, “Dynamics of semiconductor microring lasers subject to on-chip filtered optical feedback,” Proc. SPIE 9892, 98921J (2016).
[Crossref]

M. Khoder, R. M. Nguimdo, X. Leijtens, J. Bolk, J. Danckaert, and G. Verschaffelt, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” IEEE Photon. Technol. Lett. 26(5), 520–523 (2014).
[Crossref]

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Digitally tunable dual wavelength emission from semiconductor ring lasers with filtered optical feedback,” Laser Phys. Lett. 10(7), 075804 (2013).
[Crossref]

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Controlled multiwavelength emission using semiconductor ring lasers with on-chip filtered optical feedback,” Opt. Lett. 38(14), 2608–2610 (2013).
[Crossref] [PubMed]

L. Mashal, G. Van der Sande, L. Gelens, J. Danckaert, and G. Verschaffelt, “Square-wave oscillations in semiconductor ring lasers with delayed optical feedback,” Opt. Express 20, 22503–22516 (2012).
[Crossref] [PubMed]

G. Van der Sande, L. Gelens, P. Tassin, A. Scirè, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behaviour of a semiconductor ring laser,” J. Phys. B 41(9), 095402(2008).
[Crossref]

Davis, P.

Deparis, O.

M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Mégret, and M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A 65, 041801 (2002).
[Crossref]

Donati, S.

M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27(22), 1992–1994 (2002).
[Crossref]

Eliel, E. R.

R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42, 4315–4324 (1990).
[Crossref] [PubMed]

Engelborghs, K.

D. Pieroux, T. Erneux, B. Haegeman, K. Engelborghs, and D. Roose, “Bridges of periodic solutions and tori in semiconductor lasers subject to delay,” Phys. Rev. Lett. 87, 193901 (2001).
[Crossref] [PubMed]

Erneux, T.

M. Khoder, G. Friart, J. Danckaert, T. Erneux, G. Van der Sande, and G. Verschaffelt, “Dynamics of semiconductor microring lasers subject to on-chip filtered optical feedback,” Proc. SPIE 9892, 98921J (2016).
[Crossref]

T. Erneux, A. Gavrielides, and M. Sciamanna, “Stable microwave oscillations due to external-cavity-mode beating in laser diodes subject to optical feedback,” Phys. Rev. A 66, 033809 (2002).
[Crossref]

M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Mégret, and M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A 65, 041801 (2002).
[Crossref]

D. Pieroux, T. Erneux, B. Haegeman, K. Engelborghs, and D. Roose, “Bridges of periodic solutions and tori in semiconductor lasers subject to delay,” Phys. Rev. Lett. 87, 193901 (2001).
[Crossref] [PubMed]

T. Erneux, F. Rogister, A. Gavrielides, and V. Kovanis, “Bifurcation to mixed external cavity mode solutions for semiconductor lasers subject to optical feedback,” Opt. Commun. 183, 467–477 (2000).
[Crossref]

T. Erneux and P. Glorieux, Laser Dynamics (Cambridge University Press, 2010).
[Crossref]

Friart, G.

M. Khoder, G. Friart, J. Danckaert, T. Erneux, G. Van der Sande, and G. Verschaffelt, “Dynamics of semiconductor microring lasers subject to on-chip filtered optical feedback,” Proc. SPIE 9892, 98921J (2016).
[Crossref]

Gavrielides, A.

T. Erneux, A. Gavrielides, and M. Sciamanna, “Stable microwave oscillations due to external-cavity-mode beating in laser diodes subject to optical feedback,” Phys. Rev. A 66, 033809 (2002).
[Crossref]

T. Erneux, F. Rogister, A. Gavrielides, and V. Kovanis, “Bifurcation to mixed external cavity mode solutions for semiconductor lasers subject to optical feedback,” Opt. Commun. 183, 467–477 (2000).
[Crossref]

Gelens, L.

L. Mashal, G. Van der Sande, L. Gelens, J. Danckaert, and G. Verschaffelt, “Square-wave oscillations in semiconductor ring lasers with delayed optical feedback,” Opt. Express 20, 22503–22516 (2012).
[Crossref] [PubMed]

G. Van der Sande, L. Gelens, P. Tassin, A. Scirè, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behaviour of a semiconductor ring laser,” J. Phys. B 41(9), 095402(2008).
[Crossref]

Giuliani, G.

M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27(22), 1992–1994 (2002).
[Crossref]

Glorieux, P.

T. Erneux and P. Glorieux, Laser Dynamics (Cambridge University Press, 2010).
[Crossref]

Hadley, G. R.

J. P. Hohimer, G. A. Vawter, D. C. Craft, and G. R. Hadley, “Improving the performance of semiconductor ring lasers by controlled reflection feedback,” Appl. Phys. Lett. 61(9), 1013–1015 (1992).
[Crossref]

Haegeman, B.

D. Pieroux, T. Erneux, B. Haegeman, K. Engelborghs, and D. Roose, “Bridges of periodic solutions and tori in semiconductor lasers subject to delay,” Phys. Rev. Lett. 87, 193901 (2001).
[Crossref] [PubMed]

Harayama, T.

Henneberger, F.

S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radziunas, J. Sieber, H.-J. Wünsche, and F. Henneberger, “Nonlinear dynamics of semiconductor lasers with active optical feedback,” Phys. Rev. E 69, 016206 (2004).
[Crossref]

O. Ushakov, S. Bauer, O. Brox, H.-J. Wünsche, and F. Henneberger, “Self-organization in semiconductor lasers with ultrashort optical feedback,” Phys. Rev. Lett. 92, 043902 (2004).
[Crossref] [PubMed]

Hohimer, J. P.

J. P. Hohimer, G. A. Vawter, D. C. Craft, and G. R. Hadley, “Improving the performance of semiconductor ring lasers by controlled reflection feedback,” Appl. Phys. Lett. 61(9), 1013–1015 (1992).
[Crossref]

Javaloyes, J.

J. Javaloyes and S. Balle, “Emission directionality of semiconductor ring lasers: A traveling-wave description,” IEEE J. Quantum Electron. 45(5), 431–438 (2009).
[Crossref]

Khoder, M.

M. Khoder, G. Friart, J. Danckaert, T. Erneux, G. Van der Sande, and G. Verschaffelt, “Dynamics of semiconductor microring lasers subject to on-chip filtered optical feedback,” Proc. SPIE 9892, 98921J (2016).
[Crossref]

M. Khoder, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Effect of external optical feedback on tunable micro-ring lasers using on-chip filtered feedback,” IEEE Photon. Technol. Lett. 28(9), 959–962 (2016).
[Crossref]

M. Khoder, R. M. Nguimdo, X. Leijtens, J. Bolk, J. Danckaert, and G. Verschaffelt, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” IEEE Photon. Technol. Lett. 26(5), 520–523 (2014).
[Crossref]

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Digitally tunable dual wavelength emission from semiconductor ring lasers with filtered optical feedback,” Laser Phys. Lett. 10(7), 075804 (2013).
[Crossref]

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Controlled multiwavelength emission using semiconductor ring lasers with on-chip filtered optical feedback,” Opt. Lett. 38(14), 2608–2610 (2013).
[Crossref] [PubMed]

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[Crossref]

Kovanis, V.

T. Erneux, F. Rogister, A. Gavrielides, and V. Kovanis, “Bifurcation to mixed external cavity mode solutions for semiconductor lasers subject to optical feedback,” Opt. Commun. 183, 467–477 (2000).
[Crossref]

Krauss, T.

T. Krauss, P. J. R. Laybourn, and J. Roberts, “Cw operation of semiconductor ring lasers,” Electron. Lett. 26(25), 2095–2097 (1990).
[Crossref]

Kreissl, J.

S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radziunas, J. Sieber, H.-J. Wünsche, and F. Henneberger, “Nonlinear dynamics of semiconductor lasers with active optical feedback,” Phys. Rev. E 69, 016206 (2004).
[Crossref]

O. Brox, S. Bauer, M. Radziunas, M. Wolfrum, J. Sieber, J. Kreissl, B. Sartorius, and H. J. Wunsche, “High-frequency pulsations in dfb lasers with amplified feedback,” IEEE J. Quantum Electron. 39(11), 1381–1387 (2003).
[Crossref]

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[Crossref]

Lau, S. T.

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70(10), 1192–1194 (1997).
[Crossref]

Laybourn, P. J. R.

M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27(22), 1992–1994 (2002).
[Crossref]

T. Krauss, P. J. R. Laybourn, and J. Roberts, “Cw operation of semiconductor ring lasers,” Electron. Lett. 26(25), 2095–2097 (1990).
[Crossref]

Leary, M. H.

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70(10), 1192–1194 (1997).
[Crossref]

Leijtens, X.

M. Khoder, R. M. Nguimdo, X. Leijtens, J. Bolk, J. Danckaert, and G. Verschaffelt, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” IEEE Photon. Technol. Lett. 26(5), 520–523 (2014).
[Crossref]

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Digitally tunable dual wavelength emission from semiconductor ring lasers with filtered optical feedback,” Laser Phys. Lett. 10(7), 075804 (2013).
[Crossref]

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Controlled multiwavelength emission using semiconductor ring lasers with on-chip filtered optical feedback,” Opt. Lett. 38(14), 2608–2610 (2013).
[Crossref] [PubMed]

Leijtens, X. J. M.

X. J. M. Leijtens, “JePPIX: The platform for indium phosphide-based photonics,” IET Optoelectron. 5, 202–206 (2011).
[Crossref]

Liang, J. J.

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70(10), 1192–1194 (1997).
[Crossref]

Mandel, L.

W. R. Christian and L. Mandel, “Frequency-dependence of a ring laser with backscattering,” Phys. Rev. A 34, 3932–3939 (1986).
[Crossref]

Mashal, L.

Mechet, P.

G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013).
[Crossref]

Mégret, P.

M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Mégret, and M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A 65, 041801 (2002).
[Crossref]

Miglierina, R.

M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27(22), 1992–1994 (2002).
[Crossref]

Morthier, G.

G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013).
[Crossref]

Neelen, R. C.

R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42, 4315–4324 (1990).
[Crossref] [PubMed]

Nguimdo, R. M.

M. Khoder, R. M. Nguimdo, X. Leijtens, J. Bolk, J. Danckaert, and G. Verschaffelt, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” IEEE Photon. Technol. Lett. 26(5), 520–523 (2014).
[Crossref]

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Digitally tunable dual wavelength emission from semiconductor ring lasers with filtered optical feedback,” Laser Phys. Lett. 10(7), 075804 (2013).
[Crossref]

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Controlled multiwavelength emission using semiconductor ring lasers with on-chip filtered optical feedback,” Opt. Lett. 38(14), 2608–2610 (2013).
[Crossref] [PubMed]

Pieroux, D.

D. Pieroux, T. Erneux, B. Haegeman, K. Engelborghs, and D. Roose, “Bridges of periodic solutions and tori in semiconductor lasers subject to delay,” Phys. Rev. Lett. 87, 193901 (2001).
[Crossref] [PubMed]

Radziunas, M.

S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radziunas, J. Sieber, H.-J. Wünsche, and F. Henneberger, “Nonlinear dynamics of semiconductor lasers with active optical feedback,” Phys. Rev. E 69, 016206 (2004).
[Crossref]

O. Brox, S. Bauer, M. Radziunas, M. Wolfrum, J. Sieber, J. Kreissl, B. Sartorius, and H. J. Wunsche, “High-frequency pulsations in dfb lasers with amplified feedback,” IEEE J. Quantum Electron. 39(11), 1381–1387 (2003).
[Crossref]

Roberts, J.

T. Krauss, P. J. R. Laybourn, and J. Roberts, “Cw operation of semiconductor ring lasers,” Electron. Lett. 26(25), 2095–2097 (1990).
[Crossref]

Rogister, F.

M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Mégret, and M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A 65, 041801 (2002).
[Crossref]

T. Erneux, F. Rogister, A. Gavrielides, and V. Kovanis, “Bifurcation to mixed external cavity mode solutions for semiconductor lasers subject to optical feedback,” Opt. Commun. 183, 467–477 (2000).
[Crossref]

Roose, D.

D. Pieroux, T. Erneux, B. Haegeman, K. Engelborghs, and D. Roose, “Bridges of periodic solutions and tori in semiconductor lasers subject to delay,” Phys. Rev. Lett. 87, 193901 (2001).
[Crossref] [PubMed]

Sartorius, B.

S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radziunas, J. Sieber, H.-J. Wünsche, and F. Henneberger, “Nonlinear dynamics of semiconductor lasers with active optical feedback,” Phys. Rev. E 69, 016206 (2004).
[Crossref]

O. Brox, S. Bauer, M. Radziunas, M. Wolfrum, J. Sieber, J. Kreissl, B. Sartorius, and H. J. Wunsche, “High-frequency pulsations in dfb lasers with amplified feedback,” IEEE J. Quantum Electron. 39(11), 1381–1387 (2003).
[Crossref]

Sciamanna, M.

T. Erneux, A. Gavrielides, and M. Sciamanna, “Stable microwave oscillations due to external-cavity-mode beating in laser diodes subject to optical feedback,” Phys. Rev. A 66, 033809 (2002).
[Crossref]

M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Mégret, and M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A 65, 041801 (2002).
[Crossref]

Scirè, A.

G. Van der Sande, L. Gelens, P. Tassin, A. Scirè, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behaviour of a semiconductor ring laser,” J. Phys. B 41(9), 095402(2008).
[Crossref]

M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27(22), 1992–1994 (2002).
[Crossref]

Sieber, J.

S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radziunas, J. Sieber, H.-J. Wünsche, and F. Henneberger, “Nonlinear dynamics of semiconductor lasers with active optical feedback,” Phys. Rev. E 69, 016206 (2004).
[Crossref]

O. Brox, S. Bauer, M. Radziunas, M. Wolfrum, J. Sieber, J. Kreissl, B. Sartorius, and H. J. Wunsche, “High-frequency pulsations in dfb lasers with amplified feedback,” IEEE J. Quantum Electron. 39(11), 1381–1387 (2003).
[Crossref]

Sorel, M.

M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27(22), 1992–1994 (2002).
[Crossref]

Spreeuw, R. J. C.

R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42, 4315–4324 (1990).
[Crossref] [PubMed]

Sunada, S.

Tassin, P.

G. Van der Sande, L. Gelens, P. Tassin, A. Scirè, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behaviour of a semiconductor ring laser,” J. Phys. B 41(9), 095402(2008).
[Crossref]

Tsuzuki, K.

Uchida, A.

Ushakov, O.

O. Ushakov, S. Bauer, O. Brox, H.-J. Wünsche, and F. Henneberger, “Self-organization in semiconductor lasers with ultrashort optical feedback,” Phys. Rev. Lett. 92, 043902 (2004).
[Crossref] [PubMed]

Van der Sande, G.

M. Khoder, G. Friart, J. Danckaert, T. Erneux, G. Van der Sande, and G. Verschaffelt, “Dynamics of semiconductor microring lasers subject to on-chip filtered optical feedback,” Proc. SPIE 9892, 98921J (2016).
[Crossref]

M. Khoder, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Effect of external optical feedback on tunable micro-ring lasers using on-chip filtered feedback,” IEEE Photon. Technol. Lett. 28(9), 959–962 (2016).
[Crossref]

L. Mashal, G. Van der Sande, L. Gelens, J. Danckaert, and G. Verschaffelt, “Square-wave oscillations in semiconductor ring lasers with delayed optical feedback,” Opt. Express 20, 22503–22516 (2012).
[Crossref] [PubMed]

G. Van der Sande, L. Gelens, P. Tassin, A. Scirè, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behaviour of a semiconductor ring laser,” J. Phys. B 41(9), 095402(2008).
[Crossref]

van Druten, N. J.

R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42, 4315–4324 (1990).
[Crossref] [PubMed]

Vawter, G. A.

J. P. Hohimer, G. A. Vawter, D. C. Craft, and G. R. Hadley, “Improving the performance of semiconductor ring lasers by controlled reflection feedback,” Appl. Phys. Lett. 61(9), 1013–1015 (1992).
[Crossref]

Verschaffelt, G.

M. Khoder, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Effect of external optical feedback on tunable micro-ring lasers using on-chip filtered feedback,” IEEE Photon. Technol. Lett. 28(9), 959–962 (2016).
[Crossref]

M. Khoder, G. Friart, J. Danckaert, T. Erneux, G. Van der Sande, and G. Verschaffelt, “Dynamics of semiconductor microring lasers subject to on-chip filtered optical feedback,” Proc. SPIE 9892, 98921J (2016).
[Crossref]

M. Khoder, R. M. Nguimdo, X. Leijtens, J. Bolk, J. Danckaert, and G. Verschaffelt, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” IEEE Photon. Technol. Lett. 26(5), 520–523 (2014).
[Crossref]

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Digitally tunable dual wavelength emission from semiconductor ring lasers with filtered optical feedback,” Laser Phys. Lett. 10(7), 075804 (2013).
[Crossref]

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Controlled multiwavelength emission using semiconductor ring lasers with on-chip filtered optical feedback,” Opt. Lett. 38(14), 2608–2610 (2013).
[Crossref] [PubMed]

L. Mashal, G. Van der Sande, L. Gelens, J. Danckaert, and G. Verschaffelt, “Square-wave oscillations in semiconductor ring lasers with delayed optical feedback,” Opt. Express 20, 22503–22516 (2012).
[Crossref] [PubMed]

Woerdman, J. P.

R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42, 4315–4324 (1990).
[Crossref] [PubMed]

Wolfrum, M.

O. Brox, S. Bauer, M. Radziunas, M. Wolfrum, J. Sieber, J. Kreissl, B. Sartorius, and H. J. Wunsche, “High-frequency pulsations in dfb lasers with amplified feedback,” IEEE J. Quantum Electron. 39(11), 1381–1387 (2003).
[Crossref]

Wunsche, H. J.

O. Brox, S. Bauer, M. Radziunas, M. Wolfrum, J. Sieber, J. Kreissl, B. Sartorius, and H. J. Wunsche, “High-frequency pulsations in dfb lasers with amplified feedback,” IEEE J. Quantum Electron. 39(11), 1381–1387 (2003).
[Crossref]

Wünsche, H.-J.

S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radziunas, J. Sieber, H.-J. Wünsche, and F. Henneberger, “Nonlinear dynamics of semiconductor lasers with active optical feedback,” Phys. Rev. E 69, 016206 (2004).
[Crossref]

O. Ushakov, S. Bauer, O. Brox, H.-J. Wünsche, and F. Henneberger, “Self-organization in semiconductor lasers with ultrashort optical feedback,” Phys. Rev. Lett. 92, 043902 (2004).
[Crossref] [PubMed]

Xue, P.-P.

P.-P. Xue, J.-Z. Zhang, and L.-Z. Yang, “Synchronization and optimization of semiconductor ring lasers with self-feedback,” AIP Advances5, (2015).
[Crossref]

Yang, L.-Z.

P.-P. Xue, J.-Z. Zhang, and L.-Z. Yang, “Synchronization and optimization of semiconductor ring lasers with self-feedback,” AIP Advances5, (2015).
[Crossref]

Yoshimura, K.

Zhang, J.-Z.

P.-P. Xue, J.-Z. Zhang, and L.-Z. Yang, “Synchronization and optimization of semiconductor ring lasers with self-feedback,” AIP Advances5, (2015).
[Crossref]

Appl. Phys. Lett. (2)

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70(10), 1192–1194 (1997).
[Crossref]

J. P. Hohimer, G. A. Vawter, D. C. Craft, and G. R. Hadley, “Improving the performance of semiconductor ring lasers by controlled reflection feedback,” Appl. Phys. Lett. 61(9), 1013–1015 (1992).
[Crossref]

Electron. Lett. (1)

T. Krauss, P. J. R. Laybourn, and J. Roberts, “Cw operation of semiconductor ring lasers,” Electron. Lett. 26(25), 2095–2097 (1990).
[Crossref]

IEEE J. Quantum Electron. (5)

M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

J. Javaloyes and S. Balle, “Emission directionality of semiconductor ring lasers: A traveling-wave description,” IEEE J. Quantum Electron. 45(5), 431–438 (2009).
[Crossref]

G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013).
[Crossref]

O. Brox, S. Bauer, M. Radziunas, M. Wolfrum, J. Sieber, J. Kreissl, B. Sartorius, and H. J. Wunsche, “High-frequency pulsations in dfb lasers with amplified feedback,” IEEE J. Quantum Electron. 39(11), 1381–1387 (2003).
[Crossref]

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[Crossref]

IEEE Photon. Technol. Lett. (2)

M. Khoder, R. M. Nguimdo, X. Leijtens, J. Bolk, J. Danckaert, and G. Verschaffelt, “Wavelength switching speed in semiconductor ring lasers with on-chip filtered optical feedback,” IEEE Photon. Technol. Lett. 26(5), 520–523 (2014).
[Crossref]

M. Khoder, G. Van der Sande, J. Danckaert, and G. Verschaffelt, “Effect of external optical feedback on tunable micro-ring lasers using on-chip filtered feedback,” IEEE Photon. Technol. Lett. 28(9), 959–962 (2016).
[Crossref]

IET Optoelectron. (1)

X. J. M. Leijtens, “JePPIX: The platform for indium phosphide-based photonics,” IET Optoelectron. 5, 202–206 (2011).
[Crossref]

J. Phys. B (1)

G. Van der Sande, L. Gelens, P. Tassin, A. Scirè, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behaviour of a semiconductor ring laser,” J. Phys. B 41(9), 095402(2008).
[Crossref]

Laser Phys. Lett. (1)

M. Khoder, G. Verschaffelt, R. M. Nguimdo, X. Leijtens, J. Bolk, and J. Danckaert, “Digitally tunable dual wavelength emission from semiconductor ring lasers with filtered optical feedback,” Laser Phys. Lett. 10(7), 075804 (2013).
[Crossref]

Opt. Commun. (1)

T. Erneux, F. Rogister, A. Gavrielides, and V. Kovanis, “Bifurcation to mixed external cavity mode solutions for semiconductor lasers subject to optical feedback,” Opt. Commun. 183, 467–477 (2000).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (4)

M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Mégret, and M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A 65, 041801 (2002).
[Crossref]

R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42, 4315–4324 (1990).
[Crossref] [PubMed]

W. R. Christian and L. Mandel, “Frequency-dependence of a ring laser with backscattering,” Phys. Rev. A 34, 3932–3939 (1986).
[Crossref]

T. Erneux, A. Gavrielides, and M. Sciamanna, “Stable microwave oscillations due to external-cavity-mode beating in laser diodes subject to optical feedback,” Phys. Rev. A 66, 033809 (2002).
[Crossref]

Phys. Rev. E (1)

S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radziunas, J. Sieber, H.-J. Wünsche, and F. Henneberger, “Nonlinear dynamics of semiconductor lasers with active optical feedback,” Phys. Rev. E 69, 016206 (2004).
[Crossref]

Phys. Rev. Lett. (2)

D. Pieroux, T. Erneux, B. Haegeman, K. Engelborghs, and D. Roose, “Bridges of periodic solutions and tori in semiconductor lasers subject to delay,” Phys. Rev. Lett. 87, 193901 (2001).
[Crossref] [PubMed]

O. Ushakov, S. Bauer, O. Brox, H.-J. Wünsche, and F. Henneberger, “Self-organization in semiconductor lasers with ultrashort optical feedback,” Phys. Rev. Lett. 92, 043902 (2004).
[Crossref] [PubMed]

Proc. SPIE (1)

M. Khoder, G. Friart, J. Danckaert, T. Erneux, G. Van der Sande, and G. Verschaffelt, “Dynamics of semiconductor microring lasers subject to on-chip filtered optical feedback,” Proc. SPIE 9892, 98921J (2016).
[Crossref]

Other (2)

T. Erneux and P. Glorieux, Laser Dynamics (Cambridge University Press, 2010).
[Crossref]

P.-P. Xue, J.-Z. Zhang, and L.-Z. Yang, “Synchronization and optimization of semiconductor ring lasers with self-feedback,” AIP Advances5, (2015).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Schematic representation of the device. The abreviations used are AWG: Arrayed Waveguide Grating, LF: Lensed Fiber, Det: fast detector, OSA: Optical Spectrum Analyzer, I ring : bias current of the SRL, and I gate : pump current to one of the SOA gate.
Fig. 2
Fig. 2 (a) Time trace of the SRL output subject to optical feedback. The CW and CCW modes are in mauve and green respectively. They oscillate in anti-phase. The CCW mode is offseted in the vertical direction for clarity. (b), (c), and (d) Electrical spectra of the laser output with respect to the experimental setup’s noise background for three increasing feedback strengths.
Fig. 3
Fig. 3 Top: bifurcation diagram of the extrema of |Ecw|2 obtained by numerical integration of Eqs. (1)(3). Parameters values are μ = 1.2, and θ = 3.712. Bottom: Power of the CW mode as a function of η for the In-phase (mauve) and Out-phase (green) ECMs given by Eqs (14) and (15), respectively. The arrows indicate the crossings between the modes, corresponding to stable bifurcation bridges in the laser output. Along these bridges the CW and CCW modes oscillate in antiphase.
Fig. 4
Fig. 4 Power of the CW mode (full line) and the CCW mode (dashed line) along the first bridge of Fig. 3, for η = 0.002. The two directional mode oscillate in anti-phase with a period T = 140.
Fig. 5
Fig. 5 Hopf bifurcations lines (black) and crossings between In-phase and Out-phase ECMs (red) in the feedback strength versus feedback phase diagram. Full red lines indicate mode-mode crossings while the dashed red line indicates crossings between a mode and an anti-mode. Full and dashed black lines indicate stable and unstable bifurcations, respectively. Anti-phase oscillations appear in the grey region (bifurcation bridge region). The arrow indicates the point where the Hopf lines cross each other and the red line. Fixed parameters are the same as in Fig. 2.
Fig. 6
Fig. 6 Bifurcation diagrams of the extrema of |Ecw|2 obtained by numerical integration of Eqs. (1)(3) for different values of the feedback phase: (a) θ = 4.3, (b) θ = 3.6, and (c) θ = 3.5. Fixed parameters are the same as in Fig. 3.
Fig. 7
Fig. 7 Hopf bifurcations lines (black) in the feedback strength versus feedback phase diagram. Full and dashed lines indicate stable and unstable bifurcations, respectively. The pump parameter value has been increased to μ = 1.3. Anti-phase oscillations appear in the grey region (bifurcation bridge region) which now surrounds η = 0.
Fig. 8
Fig. 8 Bifurcation diagrams of the extrema of |Ecw|2 obtained by numerical integration of Eqs. (1)(3) for different values of the pump: (a) μ = 1.2, and (b) μ = 1.3. The phase is θ = 0.

Tables (1)

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Table 1 Values of the model parameters used in the simulations

Equations (19)

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d E cw d t = ( 1 + i α ) [ N G cw 1 ] E cw ( k d + i k c ) E ccw + η E cw ( t τ ) e i θ ,
d E ccw d t = ( 1 + i α ) [ N G ccw 1 ] E ccw ( k d + i k c ) E cw + η E ccw ( t τ ) e i θ ,
d N d t = γ [ μ N N G cw | E cw | 2 N G ccw | E ccw | 2 ] .
G cw = ( 1 s | E cw | 2 c | E ccw | 2 ) ,
G ccw = ( 1 s | E ccw | 2 c | E cw | 2 ) ,
E cw = I cw e i ( ω t + ϕ cw ) ,
E ccw = I ccw e i ( ω t + ϕ ccw ) ,
N = A ,
0 = [ A ( 1 ( s + c ) I 2 ) 1 ] k d cos ( ϕ ) k c sin ( ϕ ) + η cos ( θ ω τ ) ,
ω = α [ A ( 1 ( s + c ) I 2 ) 1 ] k c cos ( ϕ ) + k d sin ( ϕ ) + η sin ( θ ω τ ) ,
0 = [ A ( 1 ( s + c ) I 2 ) 1 ] k d cos ( ϕ ) + k c sin ( ϕ ) + η cos ( θ ω τ ) ,
ω = α [ A ( 1 ( s + c ) I 2 ) 1 ] k c cos ( ϕ ) k d sin ( ϕ ) + η sin ( θ ω τ ) ,
0 = μ A 2 A ( 1 ( s + c ) I 2 ) I 2 .
( a ) ϕ = 0 , ω = α k d k c + η [ sin ( θ ω τ ) α cos ( θ ω τ ) ] .
b ( ϕ ) = π , ω = α k d + k c + η [ sin ( θ ω τ ) α cos ( θ ω τ ) ] .
E cw I 1 e i ( ω 1 t + ϕ cw 1 ) + I 2 e i ( ω 2 t + ϕ cw 2 ) ,
E ccw I 1 e i ( ω 1 t + ϕ ccw 1 ) + I 2 e i ( ω 2 t + ϕ ccw 2 ) ,
| ω 1 ω 2 | 2 | k c α k d | ,
ω H = 2 k c 2 k d 2 2 α k d k c .

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