Abstract

We theoretically investigate a system of two mutually delay-coupled semiconductor lasers, in a face to face configuration for integration in a photonic integrated circuit. This system is described by single-mode rate equations, which are a system of delay differential equations with one fixed delay. Several bifurcation scenarios involving multistabilities are presented, followed by a comprehensive frequency analysis of the symmetric and symmetry-broken, one-color and two-color states.

© 2018 Optical Society of America

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References

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  1. H. G. Schuster, Nonlinear Laser Dynamics: from Quantum Dots to Cryptography (Wiley, 2012).
  2. T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86, 795–798 (2001).
    [Crossref]
  3. M. C. Soriano, J. Garca-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013).
    [Crossref]
  4. M. C. Soriano, V. Flunkert, and I. Fischer, “Relation between delayed feedback and delay-coupled systems and its application to chaotic lasers,” Chaos: Interdiscip. J. Nonlinear Sci. 23, 043133 (2013).
    [Crossref]
  5. S. Yanchuk, K. R. Schneider, and L. Recke, “Dynamics of two mutually coupled semiconductor lasers: Instantaneous coupling limit,” Phys. Rev. E 69, 056221 (2004).
    [Crossref]
  6. J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B: Quantum Semiclassical Opt. 6, 97–105 (2004).
    [Crossref]
  7. H. Erzgräber, D. Lenstra, B. Krauskopf, and I. Fischer, “Dynamical properties of mutually delayed coupled semiconductor lasers,” Proc. SPIE 5452352–362 (2004).
    [Crossref]
  8. H. Erzgräber, B. Krauskopf, and D. Lenstra, “Mode structure of delay-coupled semiconductor lasers: influence of the pump current,” J. Opt. B: Quantum Semiclassical Opt. 7, 361–371 (2005).
    [Crossref]
  9. H. Erzgräber, B. Krauskopf, and D. Lenstra, “Compound laser modes of mutually delay-coupled lasers,” SIAM J. Appl. Dyn. Syst. 5, 30–65 (2006).
    [Crossref]
  10. H. Erzgräber, E. Wille, B. Krauskopf, and I. Fischer, “Amplitude-phase dynamics near the locking region of two delay-coupled semiconductor lasers,” Nonlinearity 22, 585–600 (2009).
    [Crossref]
  11. E. Clerkin, S. O’Brien, and A. Amann, “Multistabilities and symmetry-broken one-color and two-color states in closely coupled single-mode lasers,” Phys. Rev. E 89, 032919 (2014).
    [Crossref]
  12. K. Engelborghs, T. Luzyanina, and G. Samaey, “DDE-BIFTOOL: a MATLAB package for bifurcation analysis of delay differential equations,” (2000).
  13. F. Rogister and J. Garcia-Ojalvo, “Symmetry breaking and high-frequency periodic oscillations in mutually coupled laser diodes,” Opt. Lett. 28, 1176–1178 (2003).
    [Crossref]
  14. M. Seifikar, A. Amann, and F. H. Peters, “Emergence of stable two-colour states in mutually delay-coupled lasers,” EPJ Web Conf. 139, 00010 (2017).
    [Crossref]

2017 (1)

M. Seifikar, A. Amann, and F. H. Peters, “Emergence of stable two-colour states in mutually delay-coupled lasers,” EPJ Web Conf. 139, 00010 (2017).
[Crossref]

2014 (1)

E. Clerkin, S. O’Brien, and A. Amann, “Multistabilities and symmetry-broken one-color and two-color states in closely coupled single-mode lasers,” Phys. Rev. E 89, 032919 (2014).
[Crossref]

2013 (2)

M. C. Soriano, J. Garca-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013).
[Crossref]

M. C. Soriano, V. Flunkert, and I. Fischer, “Relation between delayed feedback and delay-coupled systems and its application to chaotic lasers,” Chaos: Interdiscip. J. Nonlinear Sci. 23, 043133 (2013).
[Crossref]

2009 (1)

H. Erzgräber, E. Wille, B. Krauskopf, and I. Fischer, “Amplitude-phase dynamics near the locking region of two delay-coupled semiconductor lasers,” Nonlinearity 22, 585–600 (2009).
[Crossref]

2006 (1)

H. Erzgräber, B. Krauskopf, and D. Lenstra, “Compound laser modes of mutually delay-coupled lasers,” SIAM J. Appl. Dyn. Syst. 5, 30–65 (2006).
[Crossref]

2005 (1)

H. Erzgräber, B. Krauskopf, and D. Lenstra, “Mode structure of delay-coupled semiconductor lasers: influence of the pump current,” J. Opt. B: Quantum Semiclassical Opt. 7, 361–371 (2005).
[Crossref]

2004 (3)

S. Yanchuk, K. R. Schneider, and L. Recke, “Dynamics of two mutually coupled semiconductor lasers: Instantaneous coupling limit,” Phys. Rev. E 69, 056221 (2004).
[Crossref]

J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B: Quantum Semiclassical Opt. 6, 97–105 (2004).
[Crossref]

H. Erzgräber, D. Lenstra, B. Krauskopf, and I. Fischer, “Dynamical properties of mutually delayed coupled semiconductor lasers,” Proc. SPIE 5452352–362 (2004).
[Crossref]

2003 (1)

2001 (1)

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86, 795–798 (2001).
[Crossref]

Amann, A.

M. Seifikar, A. Amann, and F. H. Peters, “Emergence of stable two-colour states in mutually delay-coupled lasers,” EPJ Web Conf. 139, 00010 (2017).
[Crossref]

E. Clerkin, S. O’Brien, and A. Amann, “Multistabilities and symmetry-broken one-color and two-color states in closely coupled single-mode lasers,” Phys. Rev. E 89, 032919 (2014).
[Crossref]

Clerkin, E.

E. Clerkin, S. O’Brien, and A. Amann, “Multistabilities and symmetry-broken one-color and two-color states in closely coupled single-mode lasers,” Phys. Rev. E 89, 032919 (2014).
[Crossref]

Elsässer, W.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86, 795–798 (2001).
[Crossref]

Engelborghs, K.

K. Engelborghs, T. Luzyanina, and G. Samaey, “DDE-BIFTOOL: a MATLAB package for bifurcation analysis of delay differential equations,” (2000).

Erzgräber, H.

H. Erzgräber, E. Wille, B. Krauskopf, and I. Fischer, “Amplitude-phase dynamics near the locking region of two delay-coupled semiconductor lasers,” Nonlinearity 22, 585–600 (2009).
[Crossref]

H. Erzgräber, B. Krauskopf, and D. Lenstra, “Compound laser modes of mutually delay-coupled lasers,” SIAM J. Appl. Dyn. Syst. 5, 30–65 (2006).
[Crossref]

H. Erzgräber, B. Krauskopf, and D. Lenstra, “Mode structure of delay-coupled semiconductor lasers: influence of the pump current,” J. Opt. B: Quantum Semiclassical Opt. 7, 361–371 (2005).
[Crossref]

H. Erzgräber, D. Lenstra, B. Krauskopf, and I. Fischer, “Dynamical properties of mutually delayed coupled semiconductor lasers,” Proc. SPIE 5452352–362 (2004).
[Crossref]

Fischer, I.

M. C. Soriano, J. Garca-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013).
[Crossref]

M. C. Soriano, V. Flunkert, and I. Fischer, “Relation between delayed feedback and delay-coupled systems and its application to chaotic lasers,” Chaos: Interdiscip. J. Nonlinear Sci. 23, 043133 (2013).
[Crossref]

H. Erzgräber, E. Wille, B. Krauskopf, and I. Fischer, “Amplitude-phase dynamics near the locking region of two delay-coupled semiconductor lasers,” Nonlinearity 22, 585–600 (2009).
[Crossref]

H. Erzgräber, D. Lenstra, B. Krauskopf, and I. Fischer, “Dynamical properties of mutually delayed coupled semiconductor lasers,” Proc. SPIE 5452352–362 (2004).
[Crossref]

J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B: Quantum Semiclassical Opt. 6, 97–105 (2004).
[Crossref]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86, 795–798 (2001).
[Crossref]

Flunkert, V.

M. C. Soriano, V. Flunkert, and I. Fischer, “Relation between delayed feedback and delay-coupled systems and its application to chaotic lasers,” Chaos: Interdiscip. J. Nonlinear Sci. 23, 043133 (2013).
[Crossref]

Garca-Ojalvo, J.

M. C. Soriano, J. Garca-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013).
[Crossref]

Garcia-Ojalvo, J.

Heil, T.

J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B: Quantum Semiclassical Opt. 6, 97–105 (2004).
[Crossref]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86, 795–798 (2001).
[Crossref]

Krauskopf, B.

H. Erzgräber, E. Wille, B. Krauskopf, and I. Fischer, “Amplitude-phase dynamics near the locking region of two delay-coupled semiconductor lasers,” Nonlinearity 22, 585–600 (2009).
[Crossref]

H. Erzgräber, B. Krauskopf, and D. Lenstra, “Compound laser modes of mutually delay-coupled lasers,” SIAM J. Appl. Dyn. Syst. 5, 30–65 (2006).
[Crossref]

H. Erzgräber, B. Krauskopf, and D. Lenstra, “Mode structure of delay-coupled semiconductor lasers: influence of the pump current,” J. Opt. B: Quantum Semiclassical Opt. 7, 361–371 (2005).
[Crossref]

H. Erzgräber, D. Lenstra, B. Krauskopf, and I. Fischer, “Dynamical properties of mutually delayed coupled semiconductor lasers,” Proc. SPIE 5452352–362 (2004).
[Crossref]

Lenstra, D.

H. Erzgräber, B. Krauskopf, and D. Lenstra, “Compound laser modes of mutually delay-coupled lasers,” SIAM J. Appl. Dyn. Syst. 5, 30–65 (2006).
[Crossref]

H. Erzgräber, B. Krauskopf, and D. Lenstra, “Mode structure of delay-coupled semiconductor lasers: influence of the pump current,” J. Opt. B: Quantum Semiclassical Opt. 7, 361–371 (2005).
[Crossref]

H. Erzgräber, D. Lenstra, B. Krauskopf, and I. Fischer, “Dynamical properties of mutually delayed coupled semiconductor lasers,” Proc. SPIE 5452352–362 (2004).
[Crossref]

Luzyanina, T.

K. Engelborghs, T. Luzyanina, and G. Samaey, “DDE-BIFTOOL: a MATLAB package for bifurcation analysis of delay differential equations,” (2000).

Mirasso, C.

J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B: Quantum Semiclassical Opt. 6, 97–105 (2004).
[Crossref]

Mirasso, C. R.

M. C. Soriano, J. Garca-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013).
[Crossref]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86, 795–798 (2001).
[Crossref]

Mulet, J.

J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B: Quantum Semiclassical Opt. 6, 97–105 (2004).
[Crossref]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86, 795–798 (2001).
[Crossref]

O’Brien, S.

E. Clerkin, S. O’Brien, and A. Amann, “Multistabilities and symmetry-broken one-color and two-color states in closely coupled single-mode lasers,” Phys. Rev. E 89, 032919 (2014).
[Crossref]

Peters, F. H.

M. Seifikar, A. Amann, and F. H. Peters, “Emergence of stable two-colour states in mutually delay-coupled lasers,” EPJ Web Conf. 139, 00010 (2017).
[Crossref]

Recke, L.

S. Yanchuk, K. R. Schneider, and L. Recke, “Dynamics of two mutually coupled semiconductor lasers: Instantaneous coupling limit,” Phys. Rev. E 69, 056221 (2004).
[Crossref]

Rogister, F.

Samaey, G.

K. Engelborghs, T. Luzyanina, and G. Samaey, “DDE-BIFTOOL: a MATLAB package for bifurcation analysis of delay differential equations,” (2000).

Schneider, K. R.

S. Yanchuk, K. R. Schneider, and L. Recke, “Dynamics of two mutually coupled semiconductor lasers: Instantaneous coupling limit,” Phys. Rev. E 69, 056221 (2004).
[Crossref]

Schuster, H. G.

H. G. Schuster, Nonlinear Laser Dynamics: from Quantum Dots to Cryptography (Wiley, 2012).

Seifikar, M.

M. Seifikar, A. Amann, and F. H. Peters, “Emergence of stable two-colour states in mutually delay-coupled lasers,” EPJ Web Conf. 139, 00010 (2017).
[Crossref]

Soriano, M. C.

M. C. Soriano, J. Garca-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013).
[Crossref]

M. C. Soriano, V. Flunkert, and I. Fischer, “Relation between delayed feedback and delay-coupled systems and its application to chaotic lasers,” Chaos: Interdiscip. J. Nonlinear Sci. 23, 043133 (2013).
[Crossref]

Wille, E.

H. Erzgräber, E. Wille, B. Krauskopf, and I. Fischer, “Amplitude-phase dynamics near the locking region of two delay-coupled semiconductor lasers,” Nonlinearity 22, 585–600 (2009).
[Crossref]

Yanchuk, S.

S. Yanchuk, K. R. Schneider, and L. Recke, “Dynamics of two mutually coupled semiconductor lasers: Instantaneous coupling limit,” Phys. Rev. E 69, 056221 (2004).
[Crossref]

Chaos: Interdiscip. J. Nonlinear Sci. (1)

M. C. Soriano, V. Flunkert, and I. Fischer, “Relation between delayed feedback and delay-coupled systems and its application to chaotic lasers,” Chaos: Interdiscip. J. Nonlinear Sci. 23, 043133 (2013).
[Crossref]

EPJ Web Conf. (1)

M. Seifikar, A. Amann, and F. H. Peters, “Emergence of stable two-colour states in mutually delay-coupled lasers,” EPJ Web Conf. 139, 00010 (2017).
[Crossref]

J. Opt. B: Quantum Semiclassical Opt. (2)

J. Mulet, C. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B: Quantum Semiclassical Opt. 6, 97–105 (2004).
[Crossref]

H. Erzgräber, B. Krauskopf, and D. Lenstra, “Mode structure of delay-coupled semiconductor lasers: influence of the pump current,” J. Opt. B: Quantum Semiclassical Opt. 7, 361–371 (2005).
[Crossref]

Nonlinearity (1)

H. Erzgräber, E. Wille, B. Krauskopf, and I. Fischer, “Amplitude-phase dynamics near the locking region of two delay-coupled semiconductor lasers,” Nonlinearity 22, 585–600 (2009).
[Crossref]

Opt. Lett. (1)

Phys. Rev. E (2)

E. Clerkin, S. O’Brien, and A. Amann, “Multistabilities and symmetry-broken one-color and two-color states in closely coupled single-mode lasers,” Phys. Rev. E 89, 032919 (2014).
[Crossref]

S. Yanchuk, K. R. Schneider, and L. Recke, “Dynamics of two mutually coupled semiconductor lasers: Instantaneous coupling limit,” Phys. Rev. E 69, 056221 (2004).
[Crossref]

Phys. Rev. Lett. (1)

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86, 795–798 (2001).
[Crossref]

Proc. SPIE (1)

H. Erzgräber, D. Lenstra, B. Krauskopf, and I. Fischer, “Dynamical properties of mutually delayed coupled semiconductor lasers,” Proc. SPIE 5452352–362 (2004).
[Crossref]

Rev. Mod. Phys. (1)

M. C. Soriano, J. Garca-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85, 421–470 (2013).
[Crossref]

SIAM J. Appl. Dyn. Syst. (1)

H. Erzgräber, B. Krauskopf, and D. Lenstra, “Compound laser modes of mutually delay-coupled lasers,” SIAM J. Appl. Dyn. Syst. 5, 30–65 (2006).
[Crossref]

Other (2)

K. Engelborghs, T. Luzyanina, and G. Samaey, “DDE-BIFTOOL: a MATLAB package for bifurcation analysis of delay differential equations,” (2000).

H. G. Schuster, Nonlinear Laser Dynamics: from Quantum Dots to Cryptography (Wiley, 2012).

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Figures (15)

Fig. 1.
Fig. 1. Schematic of two semiconductor lasers, with cavity lengths L, and mirror reflectivities R1 and R2, which are separated by a variable optical attenuator (VOA) section.
Fig. 2.
Fig. 2. Time traces of (a) field intensities and (b) inversions; (c),(d) optical frequency spectra relative to the central frequency. Parameters used are τ=0.2, κ=0.2, and Cp=0.1π. Initial conditions: R(E1(t=0))=0.4, I(E1(t=0))=0, R(E2(t=0))=0.4, I(E2(t=0))=0, N1(t=0)=0.05, and N2(t=0)=0.05.
Fig. 3.
Fig. 3. (a) Time trace; (b) inversions; and (c),(d) frequency spectra of lasers 1 and 2, for τ=0.2, κ=0.4, and Cp=0.46π.
Fig. 4.
Fig. 4. (a) Time trace; (b) inversions; and (c),(d) frequency spectra of lasers 1 and 2, for τ=0.2, κ=0.2, and Cp=0.3π.
Fig. 5.
Fig. 5. (a) Time trace; (b) inversions; and (c),(d) frequency spectra of lasers 1 and 2, for τ=0.2, κ=0.2, and Cp=0.50433π.
Fig. 6.
Fig. 6. Hopf (dashed lines) and pitchfork (solid lines) bifurcation lines in the (Cp,κ) plane for various values of τ.
Fig. 7.
Fig. 7. Bifurcations and stability regions in phase Cp and coupling κ plane, for τ=0.2.
Fig. 8.
Fig. 8. Comparison between the frequencies of (a),(b) one-color symmetric and (c),(d) two-color symmetry-broken states, for τ=0.2, κ=0.2, and Cp=0.25π. (a) and (c) show frequencies of laser 1, and (b) and (d) show frequencies of laser 2. Initial values of DDEs are the same as Fig. 2, for (c) and (d). For (a) and (b), the initial value of N2 is different and is N2=0.05.
Fig. 9.
Fig. 9. Comparison between the frequencies of (a),(b) symmetric and (c),(d) symmetry-broken one-color states, for τ=0.2, κ=0.32, and Cp=0.36π. (a) and (c) show frequencies of laser 1, and (b) and (d) show frequencies of laser 2. The initial values for symmetry-broken states are the same as Fig. 2, while for symmetric states it is N2=0.05.
Fig. 10.
Fig. 10. Bifurcations and stability regions in phase Cp and coupling κ plane, for τ=1.
Fig. 11.
Fig. 11. Frequencies of laser 1 (upper figure) and laser 2 (lower figure) for τ=0.2 and κ=0.1. The dashed white curves display the frequencies calculated analytically.
Fig. 12.
Fig. 12. Frequencies of laser 1 versus phase, Cp, for τ=0.2 and κ=0.2 (upper figure), and κ=0.3 (lower figure).
Fig. 13.
Fig. 13. Frequencies of laser 1 versus phase, Cp, calculated analytically for τ=0.2 and coupling κ=0.10.5.
Fig. 14.
Fig. 14. Frequencies of laser 1 versus phase, Cp, for τ=1 and κ=0.1 (upper figure), 0.2 (middle figure), and κ=0.3 (lower figure).
Fig. 15.
Fig. 15. Frequency versus coupling κ for τ=1, and different values of phase Cp=0.2π, 0.4π, 0.5π, and 0.6π.

Equations (9)

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

dE1(t)dt=(1+iα)N1(t)E1(t)+κeiCpE2(tτ),
dE2(t)dt=(1+iα)N2(t)E2(t)+κeiCpE1(tτ),
TdN1(t)dt=PN1(t)(1+2N1(t))|E1(t)|2,
TdN2(t)dt=PN2(t)(1+2N2(t))|E2(t)|2.
E1(t)=A1exp(iωst),E2(t)=A2exp(iωst+iσ),N1(t)=N1s,N2(t)=N2s.
ωs=±κ1+α2sin(Cp+ωsτ+arctan(α)),
E1(t)=A1exp(iωAt)+B1exp(iωBt),E2(t)=A2exp(iωAt+iσA)+B2exp(iωBt+iσB),N1(t)=N1s,N2(t)=N2s,
N1/2s(ωA/B)=I[α^*QA/B]2ωA/B±(I[α^*QA/B]2ωA/B)2R[QA/B],
QA/B=(κeiCpeiωA/Bτ)2+ωA/B2α^.

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