7
Above-Threshold Analysis of
Various DFB Laser Structures
Using the TMM
7.1 INTRODUCTION
The above-threshold lasing characteristics of DFB semiconductor laser diodes were
presented in the previous chapter using a modified transfer matrix. Instead of using an
averaged carrier concentration, the inclusion of the actual carrier distribution allows
phenomena such as the spatial hole burning effect and non-linear gain to be included. In the
analysis, a parabolic gain model and high-order carrier recombination were assumed. Lasing
mode characteristics such as the longitudinal distribution of carrier density, photon density,
refractive index and the internal field intensity were shown for various laser structures. In
this chapter, results obtained from the lasing mode characteristics will be used to determine
the mode stability and noise characteristics of DFB LDs.
For a coherent optical communication system, it is essential that the LD used oscillates at
a stable single mode and that a narrow spectral linewidth is achieved. Using the information
obtained for the lasing mode characteristics, a method derived from the above-threshold
transfer matrix model will be introduced in section 7.2 which allows the gain margin to be
evaluated. By introducing an imaginary wavelength into the transfer matrix equation, chara-
cteristics of other non-lasing side modes can be evaluated and hence the single-mode stability
can be obtained. Numerical results obtained using this method will be presented in section 7.3.
In section 7.4, an alternative method which allows the theoretical prediction of the above-
threshold spontaneous emission will be presented. Based on the Green’s function method,
one can use transfer matrices to help determine the single-mode stability of a DFB laser
structure by inspecting the spectral components of oscillating modes.
The TMM also allows the noise characteristics of the DFB LD to be evaluated. In
section 7.5, it will be shown that various contributions to the spectral linewidth can be
determined using the information obtained from the above-threshold transfer matrices. In
the analysis, the effective linewidth enhancement factor [1] has been used instead of the
material-based linewidth enhancement [2]. Using a more realistic effective linewidth
enhancement factor, impacts caused by structural changes can be investigated in a

discussed in the previous chapter.
Single-mode stability implies the suppression of non-lasing side modes. There are two
possible ways to demonstrate single-mode stability in DFB LDs. The first approach involves
the evaluation of the normalised gain margin ÁL between the lasing mode and the probable
non-lasing side modes. The single-mode stability is said to be threatened if the gain margin,
Á, drops below 5 cm
À1
for a 500 mm length laser cavity. An alternative method to check
the stability of the device involves the measurement of the spectral characteristics. With the
help of an optical spectrum analyser, the measured intensity difference between the lasing
mode and the side modes will give single-mode stability. The second approach is often used
to measure the single-mode stability of DFB LDs. In this section, we will concentrate on the
first approach which leads to the evaluation of the above-threshold gain margin.
From the numerical method discussed in the previous chapter, oscillation characteristics
of the lasing mode were obtained at a fixed biasing current. By dividing the DFB laser into a
large number of smaller sections, longitudinal distributions like the carrier and photon
densities were determined. Since the laser cavity is now dominated by the lasing mode, the
characteristics of other non-lasing side modes should be derived from the lasing mode. In
order to evaluate the characteristics of other non-lasing side modes, the dominant lasing
mode has to be suppressed in a mathematical way. In the analysis, an imaginary wavelength

i
is introduced [6]. As a result, the complex wavelength 
c
of an unknown side mode
becomes

c
¼  þ j
i

Þ at the left
facet are found which serve as the input electric field to the transfer matrix chain.
4. By passing the electric field through the transfer matrix chain, the output electric field at
the right laser facet is determined. The discrepancy with the right facet boundary
condition is then evaluated and stored.
SINGLE-MODE STABILITY IN DFB LD
173
5. Steps (2) to (4) are then repeated with other pairs of (; 
i
) obtained from the 5 Â 5
mathematical grid. By comparing the discrepancy obtained from each of these (; 
i
)
pairs, the one showing the minimum discrepancy is then selected. Depending on the
position of the selected (; 
i
) on the mathematical grid, a new mathematical grid is
formed ready for the next iteration.
6. Procedures shown above are then repeated until the discrepancy with the right facet
boundary condition falls below 10
À14
. The final  obtained becomes the non-lasing side
mode and distributions of the amplitude gain ðzÞ and the detuning coefficient ðzÞ are
stored.
7. The average values of
"

SM
and
"

ð j ¼ 1toNÞ are the amplitude gain value and the detuning value
obtained from each transfer matrix, and N is the total number of transfer matrices.
8. The whole numerical procedure can be repeated for other non-lasing side modes. All
"

SM
obtained are then sorted in increasing order. The one showing the smallest value
becomes the most probable side mode. Characteristics of a new dominant lasing mode
must be loaded every time a new injection current is used.
From the result obtained, the gain margin between the lasing mode and the most probable
non-lasing side mode can be evaluated as
Á ¼
"

L
À
"

SM
ð7:4Þ
To maintain a stable single-mode oscillation, one must ensure that ÁL > 0:25 for a 500 mm
length laser cavity.
7.3 NUMERICAL RESULTS ON THE GAIN MARGIN
OF DFB LDs
In this section, numerical results obtained for various DFB LDs, including the QWS, 3PS
and the DCC þ QWS laser structures, will be presented. Figure 7.2 shows the characteristics
of the lasing mode (0) and side modes (Æ1) in the
"
L;
"

With multiple phase shifts introduced along the corrugation, the characteristics of the 3PS
structure are shown in Fig. 7.5. In the analysis, the 3PS DFB is assumed to be anti-reflection
coated. Phase shifts 
2
¼ 
3
¼ 
4
¼ =3 and PSP ¼ 0:5 are assumed for the 500 mm
long cavity. Compared with the QWS structure, the 3PS structure shows a smaller shift in
mode characteristics. This may be clearer when the variation of both amplitude gain and the
detuning coefficient are shown as a function of normalised injection current. From Fig. 7.6
where the amplitude gain change is shown, the injection current alters the oscillating mode
in a different way. It can be observed that the gain margin between the lasing mode and the
Figure 7.2 Lasing characteristics of the QWS DFB laser structure in the (
"
L, "L) plane showing two
values of normalised injection current.
NUMERICAL RESULTS ON THE GAIN MARGIN OF DFB LDS
175
Figure 7.3 Average amplitude gain
"
L of the QWS DFB laser structure versus the normalised
injection current. Results for both the lasing mode and non-lasing side modes (Æ1) are shown.
Figure 7.4 Average detuning coefficient
"
L of the QWS DFB laser structure versus the normalised
injection current. Results for both the lasing mode and non-lasing side modes (Æ1) are shown.
Figure 7.5 Lasing characteristics of the 3PS DFB laser structure in the ð
"

Figure 7.11 shows the normalised gain margin (ÁL) between the lasing mode and the
most probable side mode. Results obtained from QWS, 3PS and DCC þ QWS structures are
shown. The gain margin of both the QWS and the DCC þ QWS structures reduces when
the biasing current increases. From the above-threshold analysis, these structures are
Figure 7.7 Average detuning coefficient
"
L of the 3PS DFB laser structure versus the normalised
injection current. Results for both the lasing mode and non-lasing side modes (Æ1) are shown.
178
ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES
Figure 7.8 Lasing characteristics of the DCC þ QWS DFB laser structure in the ð
"
L;
"
LÞ plane
showing two values of normalised injection current.
Figure 7.9 Average amplitude gain
"
L of the DCC þ QWS DFB laser structure versus the nor-
malised injection current. Results for both the lasing mode and non-lasing side modes (Æ1) are shown.