INTRODUCTION
Nowadays, optical communication systems play a key role in computer networks,
optical receivers using conventional submicron CMOS technology, image transfer
and people communication (Temmar and Belghoraf, 2006;
Asadpour and Golnabi, 2008; Touati
et al., 2007). In order to increase operational speed in optical
communication systems, alloptical devices should be used. Lasers are one of
the most applicable devices that act as sources for optical communications,
robot and camera network, wheat germination and optical elements (Chu
et al., 2011; Agrawal, 2002; AbuElsaoud
et al., 2008; Liang et al., 2010).
In some applications, for example when fiber loss became vital or in a system
that coherent detection techniques are used, operation of laser as a single
frequency source with high power is necessary. There are some ways to provide
single mode operation. The way that is commonly used, is periodic structures.
Distributed Feedback (DFB) laser (Shahshahani and Ahmadi,
2008; Bazhdanzadeh et al., 2008; Morgado
et al., 2010) and Distributed Bragg Reflector (DBR) laser (Signoret
et al., 2004) are two types of applicable structures. In DBR lasers,
active region and spatial grating are longitudinally separated. But in DFB laser
these two sections are combined. Fabrication of DFB lasers is simpler than DBR
lasers because of longitudinally uniform structure. Instead, DFB analysis is
more complicated as a result of dependency between gain and phase condition.
Many DFB structures are fabricated with uniform grating in laser section to
achieve maximum optical output power. Sometimes laser characteristics including
Side Mode Suppression Ratio (SMSR), lasing wavelength and optical output power
are unstable because of phase uncertainty in grating at the facets. As a result,
although the device is fabricated on the same wafer, transform operation of
conventional DFB lasers with uniform grating changes (Bazhdanzadeh
et al., 2008).
To obtain a single mode source power, in some structures, grating of DFB laser
has a λ/4 phases shift in the middle of device. In uniform grating DFB
diode lasers, there is a limit to increase coupling coefficient (κL); because
in high coupling coefficient, Longitudinally Spatial Hole Burning (LSHB) causes
spectral instability. But, λ/4 phase shifted DFB lasers operate in single
mode even though with relatively large κL. This is another advantage of
DFB laser with λ/4 phase shift. Also, when putting in combined structures
with modulator, DFB lasers large κL have high immunity against optical
feedback caused by facets reflections (Jia et al.,
2007).
To overcome these problems, inserting spatial grating chirp in distributed
feedback laser structure is another way to improve its output parameters such
as power, SMSR and single mode output power (Hillmer et
al., 2004). So in this study, first we have simulated the conventional
DFB laser and also have compared their results with the other studies. Then,
by using the mentioned results, the effects of spatial chirped grating on the
dynamic large signal response is investigated.
DFB LASER STRUCTURE
In order to study the fundamental characteristics of a DFB laser, a DFBlaser has been considered to be uniform (conventional) and nonuniform (linear chirp) grating operating at 1550 nm. The device structure and its parameters, used in simulation, are shown in Fig. 1 and Table 1, respectively. The model structure is assumed to be a travelingwave type DFB laser with no facets reflections merely supporting a single transverse mode.
Furthermore, the current is considered to be uniformly injected into the entire area of the device, while the local carrier densities vary with position in the active region. Thus, the carrier density in each section can be derived from the corresponding rate equations.

Fig. 1: 
Schematic model of a DFB laser structure and operational
configuration 
Table 1: 
Device parameters used in simulation 

GENERAL FORMALISM
The timedomain traveling wave model based on the coupled wave equations is well established for simulating DFB structures. In the spatial domain, such devices exhibit nonuniform carrier and photon distributions along the propagation direction. Therefore, the governing equations have to be discretized along the active layer in a DFB laser, to treat these variations. Obviously, these equations and the discretization scheme can also be adopted to describe the processes in DFB laser. DFB structures are normally designed for singlemode operation. Therefore, it is sufficient that the governing equations are solved only in a narrow spectral range near the lasing wavelength, where the phase information is preserved. The electric field in this type of waveguide is expressed as:
where, β_{0}=π/Λ is the propagation constant at the
Bragg wavelength, Λ is the grating period, ω_{r} is the reference
frequency and φ(x,y) is the transverse field profile. F(z,t) and R(z,t)
represent the complex electrical field envelopes of forward and backward traveling
waves, respectively. Substitution of Eq. 1 into Maxwell’s
equations yields the following timedependent coupled wave equations governing
the lasing mode field propagating in the DFB laser active layer (Jabbari
et al., 2009):
and:
where, v_{g} is the group velocity, α_{s} accounts for
internal loss which is assumed to be negligible, κ denotes the grating
coupling coefficient and s_{r} and s_{f} represent the spontaneous
emission noise and will be simulated by a Gaussiandistributed random number
generator satisfying the condition (Park et al.,
2003):
where, L is the length of the active section and γ is the coupling coefficient
of the spontaneous emission rate R_{sp} into the waveguide. The expression
for R_{sp} will be given later in our optical gain model. The phase
detuning factor from the Bragg wavelength is given as (Jabbari
et al., 2009):
where, β (λ_{s}) = 2pn_{eff}/λ_{s} is the signal wave propagation constant and λ_{s} is the signal wavelength, a_{m} denotes the linewidth enhancement factor and g_{m }(z,t) is the material gain which depends on the carrier density, N(z, t) and wavelength.
In the nonuniform DFB laser the variations in the period of the grating are
small. So, the coupledmode Eq. 2a and 2b
remain unaltered except for the detuning parameter d which now depends on z.
For a linear spatial chirp, detuning factor is given by (Jabbari
et al., 2009):
where, δ_{0} is the constant phase detuning which should be smaller
than that for which the Bragg resonances occur, as for the uniform DFB laser
and C is the chirp parameter. In order to model the asymmetric gain profile,
the gain spectrum is assumed to be cubic and the material gain is approximately
by Jabbari et al. (2009):
where, a_{1}, a_{2} and a_{3} are gain constant and
N_{0} is the carrier density at transparency for the peak wavelength
λ_{N}, which is assumed to shift linearly with the carrier density
, i.e., λ_{N}=λ_{0}a_{4}(NN_{0}),
with λ_{0} and a_{4}, being the peak wavelength at transparency
and gain constant. Neglecting the facets reflectivity coefficients (i.e., R_{1
}= R_{2 }= 0), the boundary conditions reduce to:
where, E_{in} is the input optical field. In an active layer whose thickness (d) and width (W) are both much larger than the carrier diffusion length, the rate equation for the carrier density becomes:
where, J = I/(WL) is the current density, L is the active region length, σ
is the mode cross section and τ_{c} is the carrier life time:
where, A_{nrad} is the nonradiative recombination rate due to ShockleyReadHall
(SRH) process, B_{rad} is the total radiative recombination rate and
C_{Aug} is the total Auger recombination rates.
SIMULATION
Finite difference time domain is one of the powerful method to analyze all
optical active and passive devices and photonic crystal such as optical filter,
optical coupler and waveguide bending, (Dekkiche and Naoum,
2007, 2008; Djavid et al.,
2008). In this study, the performance of DFB laser with uniform and nonuniform
gratings, using FDTD method have been simulated. As a starting point, by solving
the carrier density rate equation in each section, (z,t) have been obtained.
This has allowed us to calculate the material gain coefficient, g_{m},
in the corresponding section. Then, it is assumed that the duration of the input
pulse is much longer than the roundtrip time in the cavity. Such an assumption
has enabled us to ignore the spatial derivative term in eq. 2.
Then, to solve for the forward and backward waves by FDTD method, with an appropriate
accuracy, the length, L, have been divided into M = 30 equal sections. It is
also assumed that the distribution of carrier and photon densities in each section
to be uniform and have applied Lax average technique to the time dependent coupled
wave equations (Carroll et al., 1998). Thus:
where, T and Z are the time and length steps, respectively, while the coefficients A, B, D, E, C_{1} and C_{2 }are:
where, s = L/M is the length of each section and:

Fig. 2: 
Optical output power versus time from the emission facet in
the DFB laser. (insert: mode beating). Injection current is the above threshold
condition (42.5 mA) 
In order that to compare the results, the parameters such as coupling coefficient, length, spatial chirp coefficient and injection current are chosen κL = 2.5, 300 μm, 1.5x10^{3} and 90 mA, respectively. The other required structural parameters are given in Table 1.
By adjusting the injection current above threshold condition, the optical output powers from the both facets begin to lasing. As is shown in the Fig. 2, optical output power versus time from the emission facet of the conventional DFB laser is plotted. Lasing starts after passing needed time to satisfy the threshold condition. But output power does not have specific amount because of modes beating. To show the effects of modes beating in the optical output power responses, part of output power is given in the right up corner of the Fig. 2, in which existence of different frequencies is obvious. In other words, laser does not operate as a single frequency source. The output power spectrum is shown in Fig. 3. There are side modes in addition to the main one. Side mode’s amplitudes are comparable to the main mode ones and they cannot be ignored. Hence, laser moves among side mode’s peaks continuously.
The effects of spatial hole burning in the DFB lasers make local variations in the carrier density and hence variations in real refractive index and gain, give rise to change in magnitude and phase of the feedback from each section of the grating. Hence, the longitudinal mode intensity distribution and also alters the gain suppression of side modes relative to the lasing mode are changed. The lasing mode then exhibits a nonlinear light/current characteristic which is accompanied by a frequency shift or chirp.
As is shown in Fig. 4, spatial hole burning changes the magnitude of the forward, backward and total internal power along the length of waveguide. At the middle of the cavity, spatial hole burning causes reduction in the carrier density and hence, carrier density at the middle of cavity is decreased relative to the facets. Therefore, internal optical power increase at the middle of the DFB laser and decrease at the facets as is shown in Fig. 4.
However, to have a single mode operation laser and to increase the optical
output power, the effect of hole burning should be minimized. One way to achieve
single mode output power, is inserting λ/4 phase shift in the middle of
the device or spatial chirp into the grating structure.

Fig. 3: 
Optical output power versus wavelength of the conventional
DFB laser without phase shift in the middle of device. Injection current
is adjusted about 42.5 mA above the threshold condition 

Fig. 4: 
Output power versus length of waveguide of conventional DFB
laser 
So, the DFB laser with λ/4 phase shift and spatial chirp grating is simulated.
To compare the results, the other structural parameters are kept the same as
previous structure. To show the effects of λ/4 phase shift and spatial
chirp grating in the DFB laser on the responses, optical output power versus
time is plotted in Fig. 5. The structure has a specific output
power in its stable situation. The reason is that, in one hand the reflection
of optical wave experience higher feedback toward the rear facet and lower feedback
toward emission facet and on the other hand the λ/4 phase shift in the
middle of the DFB laser reduce the optical power in the middle of device and
hence the laser acts as a single mode source and there isn’t any side mode
in the spectrum as is shown in the Fig. 6.

Fig. 5: 
Large signal output power versus time in the spatial chirped
grating DFB laser with λ/4 phase shift in the middle of device. Injection
current is the same as conventional DFB laser 

Fig. 6: 
Optical output power versus wavelength of the spatial chirped
grating DFB laser with λ/4 phase shift in the middle of device 
Furthermore side mode suppression ratio (SMSR) is more than 60 dB due to the
single mode output power.
Finally, to show the effects of the spatial chirped grating and phase shift
on the DFB laser the carrier density and forward, backward and total internal
power have been plotted versus DFB laser length along the waveguide in the Fig.
7a and b, respectively. Due to the reduction of spatial
hole burning, carrier density is maximum at the center and minimum at the facets
as is shown in Fig. 7a.

Fig. 7: 
(a)Carrier density and (b) forward, backward and total internal
power along the spatial chirped grating DFB laser with λ/4 phase shift 
Output power is increased by reduction in carrier density at the facets. Hence,
optical output power is increased (about 60 mW) relation to the uniform structure
(42.5 mW).
CONCLUSION
The results of simulation show that in uniform DFB laser with high κL, DFB laser is not in single mode operation because optical output laser oscillate between both modes near the Bragg wavelength. Hence, by adding λ/λ 4 phase shift and spatial chirp into the grating structure, DFB laser acts as a single frequency source and output power and SMSR will be more than 50 mW and 60 dB, respectively.