Effect of negative gain suppression on the stability of laser diodes
S. Bennett,
a)
C. M. Snowden, and S. Iezekiel
Department of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, England
Received 14 November 1994; accepted for publication 15 February 1995
The stability of homogeneously pumped single-section laser diodes is analyzed theoretically and it
is found that under certain bias conditions negative gain suppression factor laser diodes NGSFLDs
self-pulsate. An analytical condition for self-pulsation in homogeneously pumped single-section
laser diodes is derived. The dynamics of self-pulsating NGSFLDs under direct modulation are
analyzed and a quasiperiodic route to chaos is found. © 1995 American Institute of Physics.
Laser diodes with wavelength constraints such as verti-
cal cavity surface emitting lasers VCSELs
1
and distributed
feedback DFB laser diodes are currently of considerable
interest because of their inherent single-mode operation and
in the case of VCSELs their ability to be integrated in more
than one dimension. Recently, it has been shown that for
laser diodes with a lasing wavelength at the longer wave-
length side of the laser gain profile a negative gain suppres-
sion factor is possible.
2
This could occur in laser diodes with
wavelength constraints such as DFB laser diodes and VC-
SELs and indeed has been observed in VCSELs by Wang
et al.
2
The dynamics of laser diodes are known to be strongly
affected by the gain suppression factor in the rate
equations.
3–5
In this letter, we analyze the dynamics of laser
diodes exhibiting a negative gain suppression factor. A sta-
bility analysis is performed to investigate the possibility of
self-pulsation. The dynamics of the negative gain suppres-
sion factor laser diode NGSFLD under direct modulation
are then analyzed.
The dynamics of a semiconductor laser can be described
by the following single-mode rate equations
dN
dt
=
I
-
N
n
- N -N
t
1 - S S , 1
dS
dt
= N -N
t
1 - S S -
S
p
+
N
n
, 2
where S and N are the photon and carrier densities,
n
and
p
are carrier and photon lifetimes, is the product of active
region volume and electronic charge, is the mode confine-
ment factor, is the fraction of spontaneous emission enter-
ing the lasing mode, N
t
is the carrier density required for
transparency, is the optical gain coefficient, I is the current
entering the active region, and is the gain suppression fac-
tor where the scaling factor 1-Swhich approximates
1/1+S for S1 models the gain saturation due to lateral
carrier diffusion.
The equilibrium solution N =N
0
, S =S
0
can be
found from Eqs. 1 and 2 by setting the left-hand side of
the equations to zero. There are three equilibrium solutions
only one of which is physical. The stability of the equilib-
rium solution is now studied by linearizing Eqs. 1 and 2.
The linearization F =( N , S )
T
of the equilibrium solution
satisfies dF / dt =JF where J, the Jacobian of Eqs. 1 and
2 about ( N
0
, S
0
) is given by
- 1 - S
0
S
0
-
1
n
- N
0
-N
t
1 - S
0
2
1 - S
0
S
0
+
n
N
0
-N
t
1 - S
0
2
-
1
p
.
The nature of the eigenvalues
i
( i =1,2) of the Jacobian
determines the stability of the equilibrium solution. If
Re
i
0 for all
i
, then any perturbation grows with time
and the solution is unstable. If Re
i
0 for all
i
, then
all sufficiently small perturbations tend toward 0 as t→ and
the solution is stable. If there exists i and j such that Re
i
0 and Re
j
0, then the solution is said to be
nonstable.
6
The laser studied here has the following parameters:
=1.5810
-35
Am
3
s, =1.010
-5
,
=8.6910
-13
s
-1
m
3
, =-2.210
-23
m
3
,
n
=1.49 ns,
p
=3.89 ps, N
t
=9.2610
23
m
-3
,
=0.986, and the threshold current, I
th
=13 mA. The pa-
rameters are those of a typical laser diode with the exception
of . This value was set to the value observed by Wang et al.
2
Figure 1 shows the real and imaginary parts of the two eigen-
values. It is observed that the laser undergoes a Hopf
bifurcation.
7
If Re
i
0 then self-pulsation exists. At the
onset of self-pulsation (
i
=0), the angular frequency of
self-pulsation is equal to | Im
i
| . For the device consid-
a
Electronic mail: eensb@leeds.ac.uk
FIG. 1. The a real and b imaginary parts of the two eigenvalues E1 and
E2 of the Jacobian for three different values of the gain suppression factor;
1
=7.4410
-24
,
2
=0, and
3
=-2.210
-23
. Re0 implies
self-pulsation.
1874 Appl. Phys. Lett. 66 (15), 10 April 1995 0003-6951/95/66(15)/1874/3/$6.00 © 1995 American Institute of Physics