3
Structural Impacts on the Solutions
of Coupled Wave Equations:
An Overview
3.1 INTRODUCTION
The introduction of semiconductor lasers has boosted the development of coherent optical
communication systems. With the built-in wavelength selection mechanism, distributed
feedback semiconductor laser diodes with a higher gain margin are superior to the Fabry–
Perot laser in that a single longitudinal mode of lasing can be achieved.
In this chapter, results obtained from the threshold analysis of conventional and single-
phase-shifted DFB lasers will be investigated. In particular, structural impacts on the
threshold characteristic will be discussed in a systematic way. The next two sections of this
chapter present solutions of the coupled wave equations in DFB laser diode structures. In
section 3.4 the concepts of mode discrimination and gain margin are discussed. The
threshold analysis of a conventional DFB laser diode is studied in section 3.5, whilst the
impact of corrugation phase at the DFB laser diode facets is discussed in section 3.6. By
introducing a phase shift along the corrugations of DFB LDs, the degenerate oscillating
characteristic of the conventional DFB LD can be removed. In section 3.7, structural
impacts due to the phase shift and the corresponding phase shift position (PSP) will be
considered.
As mentioned earlier in Chapter 2, the introduction of the coupling coefficient  into the
coupled wave equations plays a vital role since it measures the strength of feedback provided
by the corrugation. In section 3.7, the effect of the selection of corrugation shape on the
magnitude of  will be presented. With a =2 phase shift fabricated at the centre of the DFB
cavity, the quarterly-wavelength-shifted (QWS) DFB LD oscillates at the Bragg wavelength.
However, the deterioration of gain margin limits its use as the current injection increases.
This phenomenon induced by the spatial hole burning effect, which is the major drawback of
the QWS laser structure, will be examined at the end of this chapter. The limited
application of the eigenvalue equation in solving the coupled wave equations will also be
considered.
Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz

0
z
ð3:2Þ
where the coefficients RðzÞ and SðzÞ are given as [1]
RðzÞ¼R
1
e
ðgzÞ
þ R
2
e
ðÀgzÞ
ð3:3aÞ
and
SðzÞ¼S
1
e
ðgzÞ
þ S
2
e
ðÀgzÞ
ð3:3bÞ
In the above equations, R
1
, R
2
, S
1
and S

 R
2
¼ je
Àj
S
2
ð3:4bÞ
 S
1
¼ je
j
R
1
ð3:4cÞ
^
 S
2
¼ je
j
R
2
ð3:4dÞ
where
^
 ¼ 
s
À jd À g ð3:5aÞ
 ¼ 
s
À jd þ g ð3:5bÞ

Similarly, by equating eqns (3.4a) and (3.4c), one obtains
g
2
¼ð
s
À j dÞ
2
þ 
2
ð3:8Þ
It is important that the dispersion equation shown above is independent of the residue
corrugation phase .
With a finite laser cavity length L extending from z ¼ z
1
to z ¼ z
2
(where both z
1
and z
2
are assumed to be greater than zero), the boundary conditions at the terminating facets
become
Rðz
1
Þ e
Àj b
0
z
1
¼

where
^
r
1
and
^
r
2
are amplitude reflection coefficients at the laser facets z
1
and z
2
, respectively.
According to eqns (3.3) and (3.4), the above equations could be expanded in such a way that
R
2
¼
ð1 À  r
1
Þ e
2g z
1
r
1
= À 1
Á R
1
ð3:10aÞ
R
2

1
e
j
¼
^
r
1
e
j 
1
ð3:11aÞ
r
2
¼
^
r
2
e
À2jb
0
z
2
e
Àj
¼
^
r
2
e
j 

Then the above equation can be solved for  and 1= whilst employing the relation
g ¼
Àj
2
 À
1


ð3:13Þ
SOLUTIONS OF THE COUPLED WAVE EQUATIONS
81
derived from eqns (3.5a) and (3.5b). After some lengthy manipulation [2], one ends up with
an eigenvalue equation
gL ¼
Àj sinhðLÞ
D
Á r
1
þ r
2
ðÞ1 À r
1
r
2
ðÞcoshðgLÞÆ 1 þ r
1
r
2
ðÞÁ
1

2
ðg LÞð3:15bÞ
r
1
¼
^
r
1
e
2jb
0
z
1
e
j
¼
^
r
1
e
j
1
ð3:15cÞ
r
2
¼
^
r
2
e

1
þ r
2
Þ
2
ð1 À r
1
r
2
Þ gL sinhðgLÞ coshðgLÞ¼0 ð3:16Þ
In the above equation, there are four parameters which govern the threshold characteristics
of DFB laser structures. These are the coupling coefficient , the laser cavity length L and
the complex facet reflectivities r
1
and r
2
. Due to the complex nature of the above equation,
numerical methods like the Newton–Raphson iteration technique can be used, provided that
the Cauchy–Riemann condition on complex analytical functions is satisfied.
Before starting the Newton–Raphson iteration, an initial value of ð; Þ
ini
is chosen from a
selected range of ð; Þ values. Usually, the first selected guess will not be a solution of the
threshold equation and hence the iteration continues. At the end of the first iteration, a new
pair of ð
0
;
0
Þ will be generated and checked to see if it satisfies the threshold equation. The
iteration will continue until the newly generated ð

@W
@z
¼
@U
@z
þ j
@V
@z
¼
@U
@x
þ j
@V
@x
ð3:19Þ
The second equality sign can be obtained using the chain rule. Applying the Taylor series,
the functions U(z) and V(z) can be approximated about the exact solution ðx
req
, y
req
Þ such
that
Uðx
req
; y
req
Þ¼Ux; yðÞþ
@U
@x
ðx

req
¼ x þ
Vðx; yÞ
@U
@y
À Uðx; yÞ
@V
@y
Det
ð3:22Þ
y
req
¼ y þ
Uðx; yÞ
@V
@x
À Vðx; yÞ
@U
@x
Det
ð3:23Þ
where
Det ¼
@U
@x

2
þ
@V
@y

x
req
¼ x À
Vðx; yÞ
@V
@x
þ Uðx; yÞ
@U
@x
Det
ð3:27Þ
Here, only the first-order derivative terms @U=@x and @V=@x are used. These can be
determined from the complex function of eqn (3.19).
Given an initial guess of ðx; yÞ, the numerical iteration process then starts. A new guess is
generated by following eqns (3.23), (3.26) and (3.27). Unless the new guess is sufficiently
close to the exact solution (within 10
À9
, let’s say), the new guess solution formed will
become the initial guess of the next iteration. The iteration process continues until
approximate solutions of ðx
req
, y
req
Þ appear.
The advantages of this method are its speed and flexibility. In addition, the derivative term
@W=@z is found analytically first, before any numerical iteration is started. Using this
method, one can avoid any errors associated with other numerical methods such as
numerical differentiation.
3.4 CONCEPTS OF MODE DISCRIMINATION
AND GAIN MARGIN

STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS
(return to zero, RZ, or non-return to zero, NRZ), transmission rate, the biasing condition of
the laser sources, the length and characteristics of the single-mode fibre (SMF) used. A
simulation based on a 20 km dispersive SMF [6] indicated that a Á of 5 cm
À1
is required
for a 2.4 Gb s
À1
data transmission in order that a bit error rate, BER < 10
À9
can be achieved.
A detailed analysis of the requirement of Á under different system configurations is clearly
beyond the scope of the present analysis. On the other hand, from the above data one can get
some idea of the typical values of gain margin required in a coherent optical communication
system.
The value of the gain margin, however, is difficult to measure directly from an experiment.
An alternative method is to measure the spontaneous emission spectrum. For a stable SLM
source, a minimum side mode suppression ratio (SMSR) of 25 dB [7] between the power of
the lasing mode and the most probable side mode is necessary.
3.5 THRESHOLD ANALYSIS OF A CONVENTIONAL DFB LASER
For a conventional DFB laser having zero facet reflection, the threshold equation (3.16)
becomes
j g L ¼ÆL sinhðLÞð3:28Þ
Using the Newton–Raphson iteration approach, the eigenvalue equation can be solved as a
fixed coupling coefficient. Results obtained for the above equation are shown in Fig. 3.2. All
Figure 3.1 A simplified – plot showing the mode spectrum and the oscillating mode of a DFB LD.
Different symbols are used to show longitudinal modes obtained from various  values.
THRESHOLD ANALYSIS OF A CONVENTIONAL DFB LASER
85
parameters used have been normalised with respect to the overall cavity length L. Discrete