Sensitivity of Quantum Dot Lasers to Delayed Optical Feedback Christian Otto, Kathy L¨udge, Roland Aust, Niels Majer and Eckehard Sch¨oll Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin Introduction
Schematic setup of the considered device j
Quantum dot (QD) lasers display low feedback sensitivity in comparison to quantum well (QW) lasers. This advantageous dynamical behavior is ascribed to their smaller phase-amplitude coupling that is expressed in small linewidth enhancement factors α [HUY04]. Therefore QD lasers are excellent candidates for directly modulated lasers needed for future telecom applications. We investigate the complex dynamics of a semiconductor QD laser subjected to optical feedback from a distant mirror. In dependence of the feedback strength we obtain complex bifurcation scenarios for the intensity of the emitted laser light.
mirror
• The light is amplified in the active region Vact that consists of 15 layers with QDs. • After one roundtrip in the external cavity the light is partially reflected back into the active region with a phase-shift Θ and a delay-time τec .
e iΘE(t − τec )
Vact Vcav 0
−L
l
z
Microscopic QD model combined with Lang-Kobayashi like field equation
Equation of motion for electrical field amplitude • Total field E (t) = E(t)e −i2πνth t is normalized: Nph = |E |2
o n ˙ = 1 − iα 1 ΓWA ne (t) + nh (t) − N QD − 2κ E(t) E(t) | {z } 2 inversion
+
K iΘ e E(t − τec ) + FE (t) τin
External Cavity Modes (ECMs)
• Equation for frequency deviations:
=
p
0 4
0.74 0.72 0.90 0.84
0 305
315
0
0
4 8 12 υ [GHz]
20 40 60 nph [104 cm-2]
2.16 2.22 2.19
0
20 40 60 nph [104 cm-2]
3.5 3.2 3
0.1
(iii)
α
0.08
(ii)
2
0.07 0.06
1.5
(iii)
0.9 1
0.05 0.04
0.002
0.004
0.006
0.008
0.01
Γ
(ii) Γ = 0.00225 2 a⋅1/(1+α )+b
Kc
0.12 0.08
Parameter: a = 0.12 b = 0.037
]
(d)
1 0.9
Kc
(iii)
2 α
3 3.2
4
5
α = 3.2 α = 0.9
0 0.00225
0.005 Γ
0.0075
0.01
FeWL (we )−EeQD kT EhQD −FhWL (wh ) kT
Se/hin/out [ps-1]
QD • Ee/h : QD carrier energy WL Fe/h : quasi-Fermi levels
(QD laser)
K = 0.2295
0
315
0.8 0.88
2
0 310 Time [ns]
0
4 υ [GHz]
8
2.24
2.67
0.42 0.48 0.46
0.84
0.44
0.8
0.42 0
5 10 15 20 4 -2 nph [10 cm ]
(a)
2.64 2.22
0.44
2.20 2.24 2.22
5 10 15 20 4 -2 nph [10 cm ]
2.58 2.67
(b)
2.64 2.61 2.58
2.20 0
2.61
0
5 10 15 20 4 -2 nph [10 cm ]
0
5 10 15 20 4 -2 nph [10 cm ]
2.50 0
20 40 60 nph [104 cm-2]
0
20 40 60 nph [104 cm-2]
• In the multistability region the timeseries shows regular intensity pulsation (pulse packages) corresponding to a motion on a delay-induced limit cycle in phasespace. 16
Dependence of Kc on α-factor • Analytical expression for lowest bound of Kc at which first ECM loses stability [LEV95]: 1 Kc ∼ 1+α 2 (red line in (ii)) • Kc decreases with increasing α-factor. ⇒ QW lasers are more sensitive to optical feedback than QD lasers.
• Parabolic shape of Kc as function of Γ is due to two competing processes [LEV95]:
2. Kc increases with differential gain ΓWA and thus with Γ-factor.
Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin
T
Scaling of regular pulse packages
data a⋅ln|K-Kbif|
12
1. Kc decreases with increasing carrier in out lifetimes τe,h = (Se,h (we , wh ) + Se,h (we , wh ))−1 .
0.1
0 20
0 305
2.60
Dependence of Kc on Γ-factor
0.04
Shin = Shout exp
0.46
0.84
2.48
Kc
0.09 2.5
• Sein = Seout exp
0.48 0.88 2
-2
(c)
2.50
• Laser loses stability in a Hopf bifurcation at the critical feedback strength Kc . Kc is plotted as a function of the α- and the Γ-factor.
0.11
100
Detailed balance condition
Rind = WA(ne + nh − N QD )nph W Rsp = QD ne nh N R˜sp = B S we wh
K = 0.123
20
First laser instability at Kc 0.00225
80
• Cascade of period doubling bifurcations starting at K = 0.12 leads to chaos. • Newly created ECM-pair provokes multistability at the end of the bifurcation cascade.
2.55
2.16
0.39
0.72
0 310 Time [ns]
0.45 0.42
0.78
30
0.4
Radiative rates
Bifurcation diagram for small α = 0.9
2.52 wh / Nsum
K = 0.79
60
ne / NQD
nph(υ) a.u.
0
2.54 2.17
0.42 nh / NQD
] -2 4
nph (t) [ 10 cm
0.76
60
we/h [1011 cm-2]
• Γ = Vact/Vcav = Γg N QD /N sum is a measure for overlap of electronically active area Vact and optically active area Vcav: QD lasers exhibit smaller Γ-factors than QW lasers
(QW laser)
0.44
0.78 4
we / Nsum
K = 0.66
30
40
• α is a measure for the phase-amplitude coupling: Large α → QW laser, small α → QD laser
• Local extrema of the photon density nph are plotted over the feedback strength K . • Multistability is caused by the creation of ECM-pairs with negative frequency deviation ∆νs . • In the first bifurcation cascade the system becomes chaotic by period doubling. • System takes a quasiperiodic route to chaos in the third bifurcation cascade. 60
20
Optical confinement factor Γ
• ECMs in dependance of K are created pairwise in saddle-node bifurcations [ROT07].
Bifurcation diagram for large α = 3.2
x10
0
Linewidth enhancement factor α
⊤ Nph,s e −i2π∆νs t , ne,s , nh,s , we,s , wh,s K √ ∆νs = ν − νth = − 1 + α2 sin 2π∆νs τec + Θ + arctan(α) 2πτin ⊤
• Rotating wave ansatz: E (t), ne (t), nh (t), we (t), wh(t)
5
x10
sum
10
wh / N
8
0.3 0.2 0.1 0
we / N
4 6 Time [ns]
Sein Seout Shin Shout
0.5 0.4
sum
2
Microscopic scattering rates
QD
0 0
QD
0
ne / N
5
nph(υ) a.u.
1 0.5
4
10
n˙ ph = −2κnph + ΓRind(ne , nh , nph ) + βRsp(ne , nh ) q K +2 nph (t − τec )nph cos(φ − φ(t − τec ) + Θ) τin o n α ˙ ΓWA ne + nh − N QD − 2κ φ= 2 s K nph (t − τec ) − sin(φ − φ(t − τec ) + Θ) τin nph ne n˙ e = − + Sein (we , wh )N QD − Rind(ne , nh , nph ) − Rsp(ne , nh ) τ (we , wh ) nh n˙ h = − + Shin (we , wh )N QD − Rind(ne , nh , nph ) − Rsp(ne , nh ) τh (we , wh ) j(t) ne N sum w˙ e = + − Sein (we , wh )N sum − R˜sp(we , wh ) e0 τe (we , wh ) N QD j(t) N sum nh w˙ h = − Shin (we , wh )N sum − R˜sp(we , wh ) + e0 τh (we , wh ) N QD
nph (t) [ 10 cm
j / jth
15
• Response of photon density nph to current pulse with feedback (K = 0.98) and without feedback (K = 0).
nph [ 104 cm -2 ]
current pulse K=0 K = 0.98
2 1.5
Band structure
Rate equations for QD laser [LUE09] 20
2.5
nh / N
Turn-on dynamics
8 4
slope: a = -1.23
0 -7 10
10
-6
-5
-4
10 10 ln|K - Kbif|
10
-3
10
-2
• System undergoes a homoclinic bifurcation of limit cycles at K = 0.2301. ⇒ Inter pulse interval time T scales logarithmic with the distance from the bifurcation point Kbif : T ∝ a ln|K − Kbif | .
Conclusion • Combining a Lang-Kobayashi like field equation with microscopically based carrier rate equations, we can explain the reduced feedback sensitivity found in QD devices by the relatively small number of ECMs in comparison to QW devices. • The small number of ECMs originates from a weaker phase amplitude coupling in QD laser that is modeled by smaller α-factors.
References [HUY04] G. Huyet at al.: Quantum dot semiconductor lasers with optical feedback, phys. stat. sol. (b), 201, 345 (2004) [LUE09] K. L¨udge and E. Sch¨oll: Quantum dot lasers – desynchronized nonlinear dynamics of electrons and holes, IEEE J. Quantum Electron. (2009), in print. [ROT07] V. Rottsch¨afer and B. Krauskopf: The ECM-backbone of the Lang-Kobayashi equations: A geometric picture, Int. J. Bif. Chaos 17, 1575 (2007). [LEV95] A. M. Levine, G. H. M. van Tartwijk, D. Lenstra, and T. Erneux: Diode lasers with optical feedback: Stability of the maximum gain mode, Phys. Rev. A 52, R3436 (1995).
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