Implementation and comparison of classical and the doubly periodic dispersion mapping for the 40gb/s single channel long haul optical transmission system for on-off keying(OOK) modulation, the system has been analyzed using in-line and post-compensation scheme setting pre-compensation to zero with maintaining system’s net residual dispersion to zero in the nonlinear regime.

1. Investigating the single and doubly periodic
mapping in fully Dispersion managed system
CHIKKPETI Sachidanand
Optical Communication
Abstract—Implementation and comparison of classical and the
doubly periodic dispersion mapping for the 40gb/s single channel
long haul optical transmission system for on-off keying(OOK)
modulation, the system has been analyzed using in-line and
post-compensation scheme setting pre-compenstion to zero with
maintaining system’s net residual dispersion to zero in the
nonlinear regime. Both the schemes are compared in two terms,
Firstly various RDPS values are taken changing the RDPP value
for each value of Nspan for doubly periodic scheme, neglect
RDPP in single periodic scheme and another term by various
spans are taken with changing power values setting RDPS
constant and RDPP to zero in doubly periodic, the behavior of
both the scheme are analyzed by Sensitivity penalty (SP) changes
such that the minimum value of the SP with respect to back to
back transmission at Bit Error Rate (BER) = 10-9 is moderate
around (0.9 ÷ 3) dB.
Index Terms—Dispersion Managed, Residual dispersion per
span (RDPS),Residual dispersion per path (RDPP),Dispersion
managed link (DM), Dispersion compensating fiber (DCF), Single
mode fiber (SMF), Sensitivity penalty (SP), On-Off Keying
(OOK),Group velocity dispersion (GVD),Post-compensation.
I. INTRODUCTION
The influence to signal transmission quality in optical fiber
communication systems is mainly caused by fiber loss, dis-
persion and nonlinear effect. However, with the development
and usage of the Erbium Doped Fiber Amplifier (EDFA)
fiber loss has no longer become the main factor to limit
fiber transmission length. As a result of that, dispersion and
nonlinear effect become two hot spots for researchers, The
dispersion due to the difference between the propagation
velocities of different spectral components is called the Group
Velocity Dispersion (GVD).There are two main types of
dispersion: the Intermodal Dispersion (between the various
modes of propagation) and intramodal dispersion (under the
same mode of propagation).Intramodal dispersion/chromatic
dispersion is a frequency dependent group delay because each
frequency travels at different speed hence destroying the shape
of the signal and induces Inter symbol interference (ISI) in
SMF. Discarding higher order dispersion effects, the parameter
which describes the fiber dispersion is given by
Dλ =
−2πC
λ2
β2 (1)
Where D (ps/nm/km) is chromatic dispersion coefficient and
β2(ps/nm2) is second order dispersion. In the SMF the chro-
matic dispersion is further divided into three regions. In
normal dispersion region, one has Dλ < 0 = β2 > 0,where
group velocity decreases with the frequency, In the anomalous
dispersion region, one has Dλ > 0 = β2 < 0, the group
velocity increases with the frequency
The basic propagation equation governing pulse evolution
inside a SMF resulting time domain equation is given by
∂A
∂z
=
α
2
A − β1
∂A
∂t
+ j
β2
2
∂2
A
∂t2
+
β3
6
∂3
A
∂t3
− j ⋎ |A|2
A (2)
where the terms in the equation gives the fiber losses, group
velocity, second and third order dispersion which are respon-
sible for pulse broadening (GVD) in optical fiber, the last term
provide the non-linearity of the SMF(spectra broadening). As
discussed above β2 is negative for the fiber at 1.55µm region,
dispersion compensation fiber(DCFs) with large positive value
of β2 developed for sole purpose for dispersion compensation,
the use of DCFs provides optical technique to overcome the
detrimental effects of chromatic dispersion in fiber, provided
the average signal power is low enough, the periodic ar-
rangement of the transmission fiber and DCFs refers to the
dispersion mapping. There are mainly three possible dispersion
mapping 1)Singly periodic, 2)Doubly periodic. 3)Pre-channel
dispersion mapping, Further these dispersion mapping are
incorporated with the different scheme a)Pre-compensation(at
the transmitter end), b)post-compensation(at the receiver end),
c)In-line compensation(periodically along the link), the pre-
compensation is set zero in our model. The wave propagation
along fiber dependent on z(distance) is given by
A(z, t) =
1
2π
∞
Z
−∞
Ã(0, ω)exp(
i
2
β2ω2
z +
i
6
β3ω3
z − iωt)dω,
(3)
where Ã(0, ω) is Fourier transform of Ã(0, t), one can
think of fiber as an optical filter with the transfer function
Hf (z, ω) = exp( i
2 β2ω2
z + i
6 β3ω3
z). All Dispersion man-
aged(DM) scheme implement optical dispersion-compensating
”filter” who’s transfer function is chosen to cancel the phase
factor associated to fiber H∗
f (z, ω). Assume the optical bit
stream propagating through two fiber segment of length
L1, L2, L2 is DCFs. Each fiber’s transfer function is given
by
A(z, t) =
1
2π
∞
Z
−∞
Ã(0, ω)Hf1(L1, ω)Hf2(L2, ω)exp(−iωt)dω,
(4)
Where L = L1 + L2 is the total length, If β2j are the
GVD of the two fiber j = (1, 2) the condition for perfect
compensation is given byβ21L1 + β22L2 = 0,in dispersion

2. parameter D21L1 + D22L2 = 0, when we add RDPS to his
equation it becomes D21L1 + D22L2 + Din = 0. The above
scheme is applied only for a single span now extending this
scheme to multiple span the total accumulated dispersion upto
distance z becomes da(z) =
∞
R
0
D(z
′
)dz
′
. Any dispersion map
for which da(z) = 0 at the end of link z would recover the
original pulse, no matter how much distorted along the way.
However the non-linear effects are always present, impact due
to power launched in the fiber link.
we introduce a normalized amplitude U as
A(z, τ) =
p
P0exp(−αz/2)U(z, τ) (5)
P0is average peak power of the incident pulse, using (2)
U(z, τ) found to satisfy
i
∂U
∂z
=
sign(β2)
2LD
∂2
U
∂τ2
−
exp(−αz)
2LNL
|U|2
U, (6)
where signβ2 = ±1 depending on GVD sign, length scale
over which dispersion and nonlinear effects become important
for pulse evolution respected lengths are given by LD =
(T2
0 /|β2|), LNL = (⋎P0)−1
The non-linearity of the fiber
is self-induced phase modulation phenomenon responsible
for self-phase modulation the nonlinear phase shift ϕNL is
ϕNL(L, T) = |U(0, t)|2
(Leff )/(LNL), Leff = 1 − eαz
/α
is effective length upto which power is assumed to be con-
stant. The new frequency are generated continuously as signal
propagates induce spectral broadening and increase the signal
bandwidth. The SPM depends on input pulse and instantaneous
power level within pulse, all nonlinear effects are weak depend
on long interaction length to build up, the SPM can be
improved by adjusting the net residual dispersion(NRD) of
the system. SPM induced chirp is positive in nature, so when
there is normal GVD then pulse broadens and its in anomalous
GVD then pulse comperes to form a soliton.
A Singly periodic dispersion map is characterized by a con-
stant RDPS with combination of pre and post-compensation
fiber can be incorporated to facilitate single RDPS, the RDPS
can be chosen such that the positive dispersion walk-off of the
RDPS balances the nonlinear shaping due to SPM along the
fiber. In Doubly periodic maps provide the extra degree of free-
dom that can be exploited to achieve improved performance
over the singly periodic at the cost of greater complexity,
this map uses constant RDPS through some number of nodes
followed by new compensation value (RDPP) and the pattern
repeats.
II. STIMULATION MODEL
Simulation is carried out using Optilux which is an open
source collection of tools that offer techniques to design,
simulate and examine optical communication systems. The
model performs with Single-Step Fourier Method(SSFM)
with(Nsymb= 64) are the number of transmission symbols in
the system, for binary this is number of bits, In addition (NT
= 32) provides number of discrete points in symbol, which
TABLE I
PARAMETERS USED IN SMF AND DCFS
parameters SMF DCF1 DCF1
alphadB (attenuation) 0.2 [dB/km] 0.6 [dB/km] 0.6 [dB/km]
aeff (effective area) 80[um2] 20[um2] 20[um2]
wavelength [nm] @ disper-
sion
1550[nm] 1550[nm] 1550[nm]
dispersion @ wavelength 17 [ps/nm/km] -100
[ps/nm/km]
-150
[ps/nm/km]
Slope (slope @ wavelength) 0
[ps/nm2/km]
0
[ps/nm2/km]
0
[ps/nm2/km]
Non-linear index n2 2.7e − 20 2.7e − 20 2.7e − 20
takes into account all linear and nonlinear effects, except po-
larization mode dispersion. For variation of the instantaneous
input power along the fiber cumulative non linear phase is
given in the range of(0.3-0.9*π)[rad] these values are varied
for comparison 0.3*pi is used in the model. The modulation
formats from the external modulator are assumed to be return-
to-zero(RZ) OOK using an additional mach-zehnder driven by
frequency equal to the symbol rate, with double bandwidth
at receiver compared to (NRZ). The symbol rate and the
pattern for the modulation where 40Gbit/s pseudo-random
bit sequence and De Burjin sequence with extension ratio
of 13[dB], 0.5 duty cycle, roll off of 0.2, chirp free pulse
propagating within a single channel system consisting of the
periodic arrangement of the single mode fiber (SMF) and
in-line dispersion compensation fiber1 (DCF1/2) which has
attenuation and operates in non linear regime incorporated
along with EDFA amplifiers to amplify the optical signal
along the link of various spans, the DM system performance
is limited by the optical signal to noise degradation induced
by Amplified spontaneous emission (ASE), the accumulation
of ASE with the dispersion and nonlinear effects affects the
optical pulse and induce not only energy fluctuation but also
shifting each pulse in random fashion from its original location
of bit slot. For implementation of the DP map a compensation
fiber2 (DCF2) with the extra negative dispersion fiber is
implemented after each 2 spans in this model, The parameters
of SMF and DCFs used in the stimulation is given in the
table 1. The length DCF keeps on changing as we provide the
variation in Din(RDPS) according to the given equation.
LDCF s = Dins − Dtx(Ltx)/DDCF s (7)
The nonlinear effects of the fibers are obtained by the given
equation.γ = 2πn2
λ0Aeff
DCFs have a larger non-linearity param-
eter because of the smaller core size, pulse are also compressed
to have much peak power, at the end of fiber the signal is
weak enough as even the non linear effects are less important
at that instant and placing the amplifier after each DCFs. The
optimization of the system performance using the different
map has been the subject of study. Perfect compensation of the
GVD in each map period is not the best solution in presence
of the nonlinear effects, In general the local GVD should
be maintained relatively high, while minimizing the average
dispersion of the channel,due to intrachannel pulse broadening
in the single channel. Implementation of DP map with com-

3. paring to the SP map, DP have the extra degree of freedom as
Din2(RDPP) which has the high negative dispersion then SP
map, the compensation length various according to the equa-
tion, even the gain of the amplifier is varied in accordance with
value of the RDPP. So the amplifier assign after each DCFs
(separate for Din1 & Din2) is different. As the cumulative
dispersion from the transmitter to receiver in DM system is
zero Dtot(ps/nm) = Dpre + NRD + Dpost = 0, considering
pre-compensation set to zero and post-compensation fiber
at the receiver end with λ and slope same as DCFs fiber
is design to compensate total Net residual dispersion(NRD)
Dpost = −NRD (Sum of accumulated dispersion by RDPS
and RDPP in DP map), NRD is calculated by the given
formula below separately for the RDPS(Din1) & RDPP(Din2)
spans assigned 2 and 1 respectively in DP map, for SP map
just the Din1.
NRD1/2 = Dtx∗Ltx+DDCF s∗LDCF s+(Nspan−1)Din1/2
(8)
The GVD and the SPM is maintained as the length of the
link various accordingly LD << L >> LNL. At the receiver
(Rx) after post-compensation fiber consist of the optical filter
of type ”gauss”, PIN diode, and pulse shaping with electrical
filterof type ”Bessel5” which are 1.8dB and 0.65dB bandwidth
normalized to the symbol rate. The performance analyze of the
singly and doubly periodic mapping system is done by using
the optilux function to evaluate the ECP under condition target
bit error rate of 10−9
with karhunen loeve method. The Eye
Closure Penalty(ECP)is used to assess the system performance
of both the mapping at its receiving signals, according to
the equation. ECP[dB] = 10log10
EOb2b
EOlink
. Where EOlink
is the Eye Opening of the receiver optical pulse and EOb2b
of the Eye opening of the input pulse respectively. The Eye
opening is defined as difference between the minimum power
of the 1’s and maximum power of the 0’s optical pulse, So
the good optical transmission ECP value should be near or
equal to 0. The aim of this project was to analyze impact of
GVD and SPM induced Eye closure penalty by comparing
the two dispersion mapping when we placed DCFs after the
transmission fiber and this study has been performed with the
following conditions:single channel system with various RDPS
and inner RDPP behavior at random values setting constant
span, on-off keying (OOK) modulation, The graphs are plotted
by varying Number of spans, signal power and the non linear
phase of the signal. The (ECP) at the bit-error rate (BER) of
10−9
is evaluated .The transmission quality is assessed by the
received (ECP), which is given in above equation. So the ECP
values for a good transmission should be as close as possible
to 0, when the value goes below 0 then the transmission is
better then the back to back case.
III. STIMULATED RESULTS AND DISCUSSION
The results obtained from the computer simulations per-
formed using the split-step Fourier method for the system
plotted in Figures. During the process only GVD and SPM
was consider for the better analyzing the nonlinear effects.
First investigating the comparison with taking RDPS(Din1 =
[0 : 20 : 100]) for SP map and varying the values of
RDPP(Din2) = [0 : 20 : 80] for DP map, where keeping
spans value to 12 with the peak power to create unique field.
The ECP is calculated in accordance with the Din1 with y
and x axies respectively. As the signal propagates in the fiber
the pulse get broaden in accordance with the GVD due to
this the power goes on depleting in SMF when it enters to
the DCF fiber with low power pulse the non-linearity effects
are lowered at the entry but being very low effective area of
the DCFs the non-linearity enhances at the starting depending
on its effective length, At point ′
0′
RDPS being zero the
RDPP for every 3rd span adds up to provide the high ECP
Dpost = −
P
RDPP as the RDPP value increases. At the
end of the DCFs fiber nonlinear effect lowers and the negative
dispersion cancel out positive dispersion leaving some RDPS
as the values assigned, for the higher RDPS pulse power is
lower so further propagation in nonlinear regime non-linearity
is less effected, where further addition with low RDPP to
lower the dispersion, pulse compresses so the non-linearity
in the higher span enhances accordingly. As you can clearly
observe in the figure. In SP map b∗ GVD increases as the
RDPS increase upto 60 which has the worst bpost, beyond
that value SPM comes to an act to chirp out the negative
with the positive one for pulse compression, but as that of
in the DP map the lower value of RDPP may degrade the
system by lower residual dispersion even average accumulated
dispersion may be lowered where bpost may not be optimal.
As optical pulses spread considerably outside their assigned bit
slot in 40gb/s symbol rate systems, they overlap considerably
and interact with each other through the nonlinear term in
the NLS equation, where the intra channel phase effects only
the phase of the each pulse due to which SPM effect is
observed lately, even intra four wave mixing may be present, It
enhancing pulse power by keeping low RDPP. As the residual
dispersion is maintained large locally for RDPP(40,60,80), the
pulse observe non-linearity equally to the SP map at Din1 60
after that DP map has lower chirping compensation then the
SP as Din1 grows. So in the single channel the pulse stretching
is larger in DP map, leading to large interaction between the
neighboring pulse so system is degraded by using DP map.
In the figure2 the system performance is observed by various
number of spans Nspan = [3 : 3 : 30] for both the maps with
RDPS set to 60 compared by varying RDPP in DP map as
used in the previous observation. As we see in figure for lower
number of span both maps are almost same, but at 9th span at
lower RDPP(0,20,40) have low penalty then the SP map, As
the number of span increases at 12th span SP and other higher
RDPP(60,80) in DP attain peak penalty and starts observing
the added up nonlinear effect for pulse compression to get low
penalty values, at RDPP 60 DP overlaps SP and at 80 the DP
perform better then SP. For lower value of RDPP the peak is
attained at 18th span leaving RDPP 40 and non-linear effects
are added up accordingly and effects are observed strongly as
the RDPP is increased and system performs well for higher
RDPP as the number of span increases and system degrades

4. Fig. 1. Eye Closure penalty vs RDPS, Singly periodic map in blue colour
compared with different value of RDPP in the DP map having red, magenta,
cyan, green, black, with value RDPP of 0, 20, 40, 60, 80 respectively
for lower value of RDPP when compared with the SP map.
In the figure3 both the maps are compared by power
variation from 1mW to 10mW by setting the RDPS for both
maps to 60 and making RDPP to 0 in DP map, the number of
span Nspan[3, 6, 12, 24] are varied to observe the behavior of
the both maps. At first Nspan = 3 there is no variation between
both the maps, As the spans increases the variation in nonlinear
effect also increases due to ϕNL = Nspan|A(0)|2
Leff /LNL,
So in order to increase the span number the power provided
should be low. Where at Nspan 6 DP is performing well by
very small margin, which can be clearly observed in next
higher span at 12 DP have low penalty then SP, after 7mW
the signal is totally distorted, In 24 spans the signal is not
observed beyond 5 in SP where as the DP has higher penalty
and does no go beyond 4mW but even at the higher power
value at 9mW it attains the negative penalty value. By this we
can say that in the single channel transmission DP performance
is degraded compared to the SP mapping.
IV. CONCLUSION
The analysis on the impact of Eye Closure Penalty by
changing the dispersion mapping with various value of RDPS,
RDPP Number of span, and Power the fiber transmission was
validated. The RDPS can be chosen such that the positive dis-
persion walk-off of the RDPS balances the nonlinear shaping
due to self phase modulation along the transmission path with
the fixed RDPS in both the maps for each case of study with
different value of RDPP in DP mapping which provide an
extra degree of freedom that can be exploited to achieve the
improved performance, but due to large interaction between
the neighboring pulse at 40gb/s symbol rate DP map degrades
the performance when compared to SP map in the single
channel transmission.
Fig. 2. Eye Closure penalty vs Nspan, Singly periodic map in blue color
compared with different value of RDPP in the DP map having red, magenta,
cyan, green, black, with value RDPP of 0, 20, 40, 60, 80 respectively
Fig. 3. Eye Closure penalty vs Power variation, Singly periodic map in blue
colour compared with the value of RDPP = 0 in the DP map having red
REFERENCES
[1] Lightwave Technology: Components and Devices, Volume 1 Govind P.
Agrawal - 2004
[2] ”Optical Communication” course by P.Serena ’University of Parma’
[3] “Optical Solitons in Fibers, Akira Hasegawa, Masayuki Matsumoto -
2012 page no 167
[4] Nonlinear Fiber Optics Govind Agrawal - 2013, page no 109-112.
[5] P.Serena,M.Bertolini, A.vannucci,“ Optilux Toolbox Doc,” March, 2009.
[6] Optical Fiber Telecommunications VB: Systems and Networks Ivan
Kaminow, Tingye Li, Alan E. Willner - 2010, page no-317