INTRODUCTION
Laser communication in free space offers an attractive alternative for transferring
highbandwidth data when optical fiber cable is either impractical or not viable.
Here, wireless optical connectivity can be used as the last mile to connect
fiber backbone to end users, such as from building to building, due to the cost
and timeconsumption on top of the impossibility and impracticality in laying
down optic fibers (Agrawal, 2001; Arnon,
2003). Other advantages of adopting the optical wireless communication systems,
also termed as Free Space Optics (FSO) or lasercom (laser communications), includes
(Potasek and Kim, 2001):
• 
No licensing or tariffs fees required for its utilization
(Potasek, 1999) 
• 
Small, lightweight and compact 
• 
Ease of installation and deployment (digging up of road is unnecessary) 
• 
It offers very high data rates due to its large bandwidth 
• 
High security fears (the extremely directional, narrow beam optical link
makes eavesdropping and jamming nearly impossible) 
• 
It operates at low power consumption 
• 
There are no rf radiation hazards (eyesafe power levels are maintained) 
However, random fluctuations in the atmosphere’s refractive index can
severely degrade the wave front of a signalcarrying laser beam, causing the
receiver to suffer from intensity fading. This results in increased system Bit
Error Rates (BERs) particularly, along horizontal propagation paths (Ricklin
and Davidson, 2003; Bouchet et al., 2006).
Research related to pulse propagation in both fiber optic and FSO showed the
propagating pulse is affected by both linear and nonlinear elements. The linear
effects include the Group Velocity Dispersion (GVD) or Second Order Dispersion
(SOD) and Third Order Dispersion (TOD), while the nonlinear effects comprise
of Self Phase Modulation (SPM). Both, the linear and nonlinear effects are responsible
for pulse broadening as well as distortion (Agrawal, 2002).
Based on the severity of these effects, data reliability can be compromised
and may lead to the increase in BER (Garlington et al.,
2005). In fiber optic, the extent of these effects can be estimated and
anticipated through numerous literatures and research. Unfortunately for FSO,
the extent of these effects cannot be estimated easily due to the random nature
of the atmosphere. Thus, it is important to have an accurate prediction model
to estimate pulse behavior in the atmosphere.
In this study, the simulation on FSO is carried out without the nonlinear effects. The nonlinear Schrödinger equation is briefly discussed while two types of pulse are used in the second order dispersion simulation.
NONLINEAR SCHRÖDINGER EQUATION MODEL
The Nonlinear Schrödinger Equation (NLSE) is used to mathematically explain
varying pulse envelope propagating in a medium with linear and nonlinear elements.
Thus, NLSE is suitable for describing pulse propagation in free space. Numerical
solution for NLSE can be obtained by applying Split Step Fourier (SSF) or Beam
Propagation Method (BPM) (Bogomolov and Yunakosky, 2006;
Wang et al., 2006).
Equation 1 represents the generalized form of NLSE for complex
envelope A(z, t). The β_{2} and β_{3} are the quadratic
and cubic dispersion coefficient respectively, α is the attenuation factor
and γ is the nonlinear coefficient (Agrawal, 2002).
Under the assumption that the electric field of the light in free space has a slowly varying envelope A(z,t) and that the free space medium has an instantaneous nonlinear response, Eq. 1 can be rewritten as:
where, Z is propagation length in free space, is
the complex electric field envelope A(z,t) normalized to the absolute amplitude
of the field ,
P_{0} is the peak power, t is time normalized to a convenient time scale
T_{0} measured in a reference frame moving with the group velocity of
the pulse [τ = (tz/v_{g})/T_{0}].
The NLSE is composed of linear and nonlinear terms and can be written in operator form as:
where, and
are
linear (consist of SOD, TOD and attenuation) and nonlinear (SPM) operators respectively.
Thus, taking the nonlinear part into calculation and then the dispersion term,
the solution below is obtained:
where, F is the Fourier transform and ω is the Fourier frequency.
TYPE OF PULSE
Two types of pulse were used in the simulation. They are the chirped Gaussian
pulse and the chirped hyperbolic secant pulse (Agrawal, 2002)
as shown in Eq. 6 and 7 respectively as
initial pulses:
where, t is time period, T_{0} is the halfwidth at 1/e intensity point and C is the frequency chirp. All of the simulations were carried out using the parameter values of, T_{0} = 2 ps, P_{0} = 1W and C = 0 (unchirped).
SECOND ORDER DISPERSION SIMULATION RESULTS
Second Order Dispersion (SOD) is a linear effect and the primary source of
pulse broadening. From Eq. 2, SOD is governed by β_{2},
known as the Group Velocity Dispersion (GVD). GVD represents dispersion of group
velocity that determines the broadening characteristic of the pulse. The frequency
dependence of the group velocity leads to pulse broadening simply because different
component of the pulse disperse during propagation and do not arrive simultaneously
(Agrawal, 2002). Pulse broadening occurs due to frequency
chirps generated by the GVD induced phase shift. GVD changes the phase of each
spectral component of the pulse by an amount that depends on the frequency and
the propagated distance (Bandelow et al., 2003).
The generated frequency chirps changes the velocity of each spectral components
causing them to travel in different velocity. Spectral components at the leading
edge travel faster compare to the trailing edges. This causes a delay on the
pulse arrival. Pulse broadening is dependent on the delay and linearly correlated
with distance. The pulse broadening does not rely on the sign of β_{2}.
To observe the effect of SOD alone, β_{3} and γ in Eq.
1 are set to zero while GVD, β_{2 }= 21 ps^{2} km^{1}
(Potasek, 1999; Alexeev et al.,
2004).
The broadening experienced by both pulses can be observed in Fig.
1, where both pulses show a significant amount of broadening. Pulse broadening
is linearly correlated with the propagated distance. As the pulse propagates,
constant phase shift cause a constant increase in chirp.

Fig. 1:  Pulse
propagation at z = 0, 0.5, 1, 1.5 km with β_{2 }= 21 ps^{2}
km^{1} for (a) unchipred Gaussian pulse and (b) unchipred hyperbolic
secant pulse 
The increase in chirp affects the velocity and the arrival of the pulse spectral
components. The change of velocity consequentially increases delay and cause
further broadening. The magnitude of delay increases with the distance. These
effects can be observed in Fig. 1a and b.
From Fig. 2a and b, the waterfall plot
for both pulses showed similar characteristics in pulse broadening. It is obvious
SOD induced broadening increase linearly with propagating distance. Nevertheless,
both pulses have displayed different broadening rates. Hyperbolic secant pulse
reveals a lower broadening rate compare to Gaussian. This can be observed as
Gaussian pulse exhibits wider broadening and lower pulse amplitude as it propagates,
in comparison to hyperbolic secant pulse. This implies that both pulses have
different effect to GVD. There is one important attribute; hyperbolic secant
pulse shows a faint distortion at both edges of its pulse. Distortion can be
seen between distances 0.3 and 0.5 km but disappears as the pulse propagates;
as can be observed in Fig. 2b.
Broadening rate for both Gaussian and hyperbolic secant pulse can be observed
in Fig. 3. Hyperbolic secant pulse shows lower broadening
rate at about 34.5% compare to Gaussian pulse. The difference in broadening
rate can be traced back to the difference in the pulse shape. The pulse shape
is defined by the pulse equation and both pulses manifest differently over the
same parameters as can be seen in the Gaussian pulse which is presented by Eq.
4 and hyperbolic secant pulse as in Eq. 5.

Fig. 2:  Pulse
propagation at z = 1 km and with β_{2 } = 21 ps^{2}
km^{1} for (a) unchirped Gaussian pulse and (b) unchirped hyperbolic
secant pulse 

Fig. 3:  Broadening
factor for unchirped Gaussian and unchirped Hyperbolic Secant pulse over
the distance, z = 1 km, with β_{2 }= 21 ps^{2} km^{1} 
These differences create variations and rare anomalies.
CONCLUSION
In this study, the dispersion effects were simulated and shown individually
in 1D and 2D graphical representation. Simulations were done in order to observe
pulse behavior and response to linear parameters. The SOD effects on the pulse
propagation in free space was observed and the pulse behavior was discussed.
Simulation result may serve as a prediction model that can be used to estimate
or predict to an extent the actual pulse behavior in free space.
ACKNOWLEDGMENT
This study is supported by the Internal Grant of Universiti Teknikal Malaysia Melaka under grant PJP/2009/FKEKK (20D) S614.