Abstract

The response of an optically injected quantum-dot semiconductor laser (SL) is studied both experimentally and theoretically. In particular, the nature of the locking boundaries is investigated, revealing features more commonly associated with Class A lasers rather than conventional Class B SLs. Experimentally, two features stand out; the first is an absence of instabilities resulting from relaxation oscillations, and the second is the observation of a region of bistability between two locked solutions. Using rate equations appropriate for quantum-dot lasers, we analytically determine the stability diagram in terms of the injection rate and frequency detuning. Of particular interest are the Hopf and saddle-node locking boundaries that explain how the experimentally observed phenomena appear.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]

2009

2008

K. Lüdge, M. J. P. Bormann, E. Malic, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, Phys. Rev. B 78, 035316 (2008).
[CrossRef]

2007

T. Erneux, E. A. Viktorov, and P. Mandel, Phys. Rev. A 76, 023819 (2007).
[CrossRef]

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, Phys. Rev. Lett. 98, 153903 (2007).
[CrossRef] [PubMed]

S. P. Hegarty, D. Goulding, B. Kelleher, G. Huyet, M. T. Todaro, A. Salhi, A. Passaseo, and M. De Vittorio, Opt. Lett. 32, 3245 (2007).
[CrossRef] [PubMed]

2006

2005

S. Wieczorek, B. Krauskopf, T. Simpson, and D. Lenstra, Phys. Rep. 416, 1 (2005).
[CrossRef]

2004

2003

T. B. Simpson, Opt. Commun. 215, 135 (2003).
[CrossRef]

2002

C. Mayol, R. Toral, C. R. Mirasso, and M. A. Natiello, Phys. Rev. A 66, 013808 (2002).
[CrossRef]

1999

D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (Wiley, 1999).

1995

A. Hohl, H. J. C. van der Linden, R. Roy, G. Goldsztein, F. Broner, and S. H. Strogatz, Phys. Rev. Lett. 74, 2220 (1995).
[CrossRef] [PubMed]

Alouini, M.

G. Baili, M. Alouini, T. Malherbe, D. Dolfi, I. Sagnes, and F. Bretanker, Europhys. Lett. 87, 44005 (2009).
[CrossRef]

G. Baili, M. Alouini, C. Moronvalle, D. Dolfi, and F. Bretenaker, Opt. Lett. 31, 62 (2006).
[CrossRef] [PubMed]

Baili, G.

G. Baili, M. Alouini, T. Malherbe, D. Dolfi, I. Sagnes, and F. Bretanker, Europhys. Lett. 87, 44005 (2009).
[CrossRef]

G. Baili, M. Alouini, C. Moronvalle, D. Dolfi, and F. Bretenaker, Opt. Lett. 31, 62 (2006).
[CrossRef] [PubMed]

Bimberg, D.

K. Lüdge, M. J. P. Bormann, E. Malic, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, Phys. Rev. B 78, 035316 (2008).
[CrossRef]

D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (Wiley, 1999).

Bormann, M. J. P.

K. Lüdge, M. J. P. Bormann, E. Malic, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, Phys. Rev. B 78, 035316 (2008).
[CrossRef]

Bretanker, F.

G. Baili, M. Alouini, T. Malherbe, D. Dolfi, I. Sagnes, and F. Bretanker, Europhys. Lett. 87, 44005 (2009).
[CrossRef]

Bretenaker, F.

Broner, F.

A. Hohl, H. J. C. van der Linden, R. Roy, G. Goldsztein, F. Broner, and S. H. Strogatz, Phys. Rev. Lett. 74, 2220 (1995).
[CrossRef] [PubMed]

Cong, Ding-Yi

De Vittorio, M.

Dolfi, D.

G. Baili, M. Alouini, T. Malherbe, D. Dolfi, I. Sagnes, and F. Bretanker, Europhys. Lett. 87, 44005 (2009).
[CrossRef]

G. Baili, M. Alouini, C. Moronvalle, D. Dolfi, and F. Bretenaker, Opt. Lett. 31, 62 (2006).
[CrossRef] [PubMed]

Erneux, T.

T. Erneux, E. A. Viktorov, and P. Mandel, Phys. Rev. A 76, 023819 (2007).
[CrossRef]

Fischer, M.

Gerschütz, F.

Goldsztein, G.

A. Hohl, H. J. C. van der Linden, R. Roy, G. Goldsztein, F. Broner, and S. H. Strogatz, Phys. Rev. Lett. 74, 2220 (1995).
[CrossRef] [PubMed]

Goulding, D.

Greene, G.

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, Phys. Rev. Lett. 98, 153903 (2007).
[CrossRef] [PubMed]

Grundmann, M.

D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (Wiley, 1999).

Hartnett, M.

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, Phys. Rev. Lett. 98, 153903 (2007).
[CrossRef] [PubMed]

Hegarty, S. P.

Hohl, A.

A. Hohl, H. J. C. van der Linden, R. Roy, G. Goldsztein, F. Broner, and S. H. Strogatz, Phys. Rev. Lett. 74, 2220 (1995).
[CrossRef] [PubMed]

Hövel, P.

K. Lüdge, M. J. P. Bormann, E. Malic, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, Phys. Rev. B 78, 035316 (2008).
[CrossRef]

Huyet, G.

Kelleher, B.

Knorr, A.

K. Lüdge, M. J. P. Bormann, E. Malic, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, Phys. Rev. B 78, 035316 (2008).
[CrossRef]

Koeth, J.

Krauskopf, B.

S. Wieczorek, B. Krauskopf, T. Simpson, and D. Lenstra, Phys. Rep. 416, 1 (2005).
[CrossRef]

Kuntz, M.

K. Lüdge, M. J. P. Bormann, E. Malic, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, Phys. Rev. B 78, 035316 (2008).
[CrossRef]

Ledentsov, N. N.

D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (Wiley, 1999).

Lemaître, A.

Lenstra, D.

S. Wieczorek, B. Krauskopf, T. Simpson, and D. Lenstra, Phys. Rep. 416, 1 (2005).
[CrossRef]

Lüdge, K.

K. Lüdge, M. J. P. Bormann, E. Malic, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, Phys. Rev. B 78, 035316 (2008).
[CrossRef]

Malherbe, T.

G. Baili, M. Alouini, T. Malherbe, D. Dolfi, I. Sagnes, and F. Bretanker, Europhys. Lett. 87, 44005 (2009).
[CrossRef]

Malic, E.

K. Lüdge, M. J. P. Bormann, E. Malic, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, Phys. Rev. B 78, 035316 (2008).
[CrossRef]

Mandel, P.

T. Erneux, E. A. Viktorov, and P. Mandel, Phys. Rev. A 76, 023819 (2007).
[CrossRef]

Martinez, A.

Mayol, C.

C. Mayol, R. Toral, C. R. Mirasso, and M. A. Natiello, Phys. Rev. A 66, 013808 (2002).
[CrossRef]

McInerney, J. G.

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, Phys. Rev. Lett. 98, 153903 (2007).
[CrossRef] [PubMed]

Melnik, S.

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, Phys. Rev. Lett. 98, 153903 (2007).
[CrossRef] [PubMed]

Mirasso, C. R.

C. Mayol, R. Toral, C. R. Mirasso, and M. A. Natiello, Phys. Rev. A 66, 013808 (2002).
[CrossRef]

Moronvalle, C.

Natiello, M. A.

C. Mayol, R. Toral, C. R. Mirasso, and M. A. Natiello, Phys. Rev. A 66, 013808 (2002).
[CrossRef]

O'Brien, D.

Passaseo, A.

Rachinskii, D.

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, Phys. Rev. Lett. 98, 153903 (2007).
[CrossRef] [PubMed]

Ramdane, A.

Rasskazov, O.

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, Phys. Rev. Lett. 98, 153903 (2007).
[CrossRef] [PubMed]

Roy, R.

A. Hohl, H. J. C. van der Linden, R. Roy, G. Goldsztein, F. Broner, and S. H. Strogatz, Phys. Rev. Lett. 74, 2220 (1995).
[CrossRef] [PubMed]

Sagnes, I.

G. Baili, M. Alouini, T. Malherbe, D. Dolfi, I. Sagnes, and F. Bretanker, Europhys. Lett. 87, 44005 (2009).
[CrossRef]

Salhi, A.

Schöll, E.

K. Lüdge, M. J. P. Bormann, E. Malic, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, Phys. Rev. B 78, 035316 (2008).
[CrossRef]

Simpson, T.

S. Wieczorek, B. Krauskopf, T. Simpson, and D. Lenstra, Phys. Rep. 416, 1 (2005).
[CrossRef]

Simpson, T. B.

T. B. Simpson, Opt. Commun. 215, 135 (2003).
[CrossRef]

Strogatz, S. H.

A. Hohl, H. J. C. van der Linden, R. Roy, G. Goldsztein, F. Broner, and S. H. Strogatz, Phys. Rev. Lett. 74, 2220 (1995).
[CrossRef] [PubMed]

Todaro, M. T.

Toral, R.

C. Mayol, R. Toral, C. R. Mirasso, and M. A. Natiello, Phys. Rev. A 66, 013808 (2002).
[CrossRef]

Uskov, A. V.

van der Linden, H. J. C.

A. Hohl, H. J. C. van der Linden, R. Roy, G. Goldsztein, F. Broner, and S. H. Strogatz, Phys. Rev. Lett. 74, 2220 (1995).
[CrossRef] [PubMed]

Viktorov, E. A.

T. Erneux, E. A. Viktorov, and P. Mandel, Phys. Rev. A 76, 023819 (2007).
[CrossRef]

Wieczorek, S.

S. Wieczorek, B. Krauskopf, T. Simpson, and D. Lenstra, Phys. Rep. 416, 1 (2005).
[CrossRef]

Europhys. Lett.

G. Baili, M. Alouini, T. Malherbe, D. Dolfi, I. Sagnes, and F. Bretanker, Europhys. Lett. 87, 44005 (2009).
[CrossRef]

Opt. Commun.

T. B. Simpson, Opt. Commun. 215, 135 (2003).
[CrossRef]

Opt. Lett.

Phys. Rep.

S. Wieczorek, B. Krauskopf, T. Simpson, and D. Lenstra, Phys. Rep. 416, 1 (2005).
[CrossRef]

Phys. Rev. A

C. Mayol, R. Toral, C. R. Mirasso, and M. A. Natiello, Phys. Rev. A 66, 013808 (2002).
[CrossRef]

T. Erneux, E. A. Viktorov, and P. Mandel, Phys. Rev. A 76, 023819 (2007).
[CrossRef]

Phys. Rev. B

K. Lüdge, M. J. P. Bormann, E. Malic, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, Phys. Rev. B 78, 035316 (2008).
[CrossRef]

Phys. Rev. Lett.

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, Phys. Rev. Lett. 98, 153903 (2007).
[CrossRef] [PubMed]

A. Hohl, H. J. C. van der Linden, R. Roy, G. Goldsztein, F. Broner, and S. H. Strogatz, Phys. Rev. Lett. 74, 2220 (1995).
[CrossRef] [PubMed]

Other

D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (Wiley, 1999).

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Figures (3)

Fig. 1
Fig. 1

Experimental stability diagram. The solid (black) lines are saddle-node bifurcations, while the dashed (red) lines are Hopf bifurcations. The injection strength is defined as the power of the light injected into the cavity divided by the power in the cavity when free running. One inset shows switching between two locked states with different intensities; the other inset shows a zoom of the region of low injection strength where the locking is via a saddle-node bifurcation for both signs of the detuning.

Fig. 2
Fig. 2

Analytic stability diagram. SN and H denote the saddle-node and Hopf bifurcation points, respectively. The shaded region corresponds to the domain of steady-state bistability. The values of the parameters are g = 1.01 , B = 10 2 , η = 2 × 10 3 , α = 1.2 , and J = 1.2 J th = 4.8 ( J th = 4 ) . The dots are fold-Hopf points where Hopf and SN bifurcation lines merge. Inset, stability diagram for an injected Class A laser (Eq. (1) in [2] with Γ κ 1 = 1.2 , α = 1.2 , β = 1 , Γ σ S κ 1 , Δ Δ Ω κ 1 , and t κ t ).

Fig. 3
Fig. 3

Bifurcation diagrams of the stable steady-state and time-periodic solutions. The extrema of R are shown as functions of the detuning Δ. The complete S-shaped branch of steady states is shown by a broken curve. The values of the parameters are the same as in the previous figure. (a) Regions of coexistence between two locked states and coexistence between a locked state and an unlocked limit cycle both exist for Γ = 0.0012 and (b) coexistence between two locked solutions only for Γ = 0.002 . The figures were obtained by scanning the detuning back and forth.

Equations (7)

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

E = 1 2 ( 1 + i α ) [ 1 + g ( 2 ρ 1 ) ] E + Γ exp ( i Δ t ) ,
ρ = η [ B n ( 1 ρ ) ρ ( 2 ρ 1 ) | E | 2 ] ,
n = η [ J n 2 B n ( 1 ρ ) ] .
E = 1 2 ( 1 + i α ) [ 1 + 2 u ( 1 + ϵ ) ] E + γ exp ( i δ s ) ,
u = ϵ 2 η [ B ϵ n u 1 ϵ u ( 1 + 2 u ϵ ) | E | 2 ] ,
n = ϵ 1 η [ J n + 2 B ϵ n u ] ,
λ 3 + a 1 λ 2 + a 2 λ + a 3 = 0

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