Download Gain Compression and Above-Threshold Linewidth

GRILLOT et al.: GAIN COMPRESSION AND ABOVE-THRESHOLD LINEWIDTH ENHANCEMENT FACTOR IN 1.3- m InAs–GaAs QUANTUM-DOT LASERS
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change of the GS index caused by the ES gain, and
is related to the GS index change caused by the GS gain variation.
keeps increasing
When the laser operates above threshold,
according to (4), as previously shown for the case of QW devices. Let us note that the differential gain at GS, , can also
be simply expressed as a function of the gain compression coand
. Indeed, it has been shown in
efficient as well as
[25] that the gain saturation in a QD media can be described by
the following equation:
(7)
where is the carrier density and
is the transparency carrier
density. When the laser operates above threshold, the differential gain for the GS lasing is defined as follows:
Fig. 5. Stability map based on the normalized linewidth-enhancement factor
= in the (P=P ; g =g ) plane.
(8)
where
is the uncompressed material gain
increasing with the output power. Equation (8) leads to
(9)
is the differential gain at threshold. Then, using (4),
where
(6), and (9), the linewidth-enhancement factor can be written as
(10)
where
. The first term in (10) denotes the gain
compression effect at the GS (similar to QWs) while the second
is the contribution from the carrier filling in the ES that is related
to the gain saturation in the GS. For the case of strong gain
saturation, (10) can be reduced to
(11)
In Fig. 5, the normalized linewidth enhancement factor
is calculated through (11) and reported in the (
) plane with
and
. This picture acts as a stability
map that simply shows that a larger maximum gain is absolutely
ratio. For instance, let us
required for a lower and stable
: at low output
consider the situation for which
, the normalized
-factor remains conpowers, i.e.,
since gain compression is negligible. On
stant
the other hand, as soon as the output power approaches
and
is increased. Gain compression
goes beyond, the ratio
-factor,
effects lead to an enhancement of the normalized
level of injection for
which can go up to 10 for
which the ES occurs.
, Fig. 5 shows that
On the other hand, assuming
the effects of gain compression are significantly attenuated since
remains almost constant over a wider range of
the ratio
output power. The level at which gain compression starts being
Fig. 6. Calculated GS -factor versus the bias current (black dots). Superimposed red stars correspond to experimental data from [23].
critical is now shifted to
instead of
. Let
us also stress that, at a certain level of injection, the normalized
GS -factor can even become negative. This effect has already
been experimentally reported in [23] and they occur when the
GS gain collapses, e.g., when ES lasing wavelength occurs.
-factor (black dots) of the QD
In Fig. 6, the calculated GS
laser under study is depicted as a function of the bias current.
Red stars superimposed correspond to data measurements from
[23], which have been obtained via the AM/FM technique. This
method consists of an interferometric method: the output optical
signal from the laser operated under small-signal direct modulation is filtered in a 0.2-nm resolution monochromator and
sent in a tunable Mach–Zehnder interferometer. From separate
measurements on opposite slopes of the interferometer transfer
function, phase and amplitude deviations are extracted against
the modulating frequency, in the 50-MHz–20-GHz range [28].
The LEF is given by the phase to amplitude responses ratio at
the highest frequencies, in the limits of the device modulation
bandwidth
Thus, a qualitative agreement between simulations and mea-factor increases
surements is obtained. As expected, the GS
with the injected current due to the filling of the excited states as
well as carrier filling of the nonlasing states (higher lying energy