# Influence of Even-Order Dispersion on

Authors: Aleksandra Panajotovic; Daniela Milovic; Anjan Biswas; Essaid Zerrad
Journal: Research Letters in Optics
 (2008)

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## Abstract

The transmission speed of optical network strongly depends on the impact of higher order dispersion. In presence of coherent crosstalk, which cannot be otherwise controlled by optical filtering, the impact of higher order dispersions becomes more pronounced. In this paper, the general expressions, that describe pulse deformation due to second- and fourth-order dispersions in a single-mode fiber, are given. The responses for such even-order dispersions, in presence of coherent crosstalk, are characterized by waveforms with long trailing edges. The transmission quality of optical pulses, due to both individual and combined influence of second- and fourth-order dispersions,  is studied in this paper. Finally, the pulse shape and eye diagrams are obtained.


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## Body

## 1. Introduction

Intramodal or chromatic
dispersion is a major limiting factor in ultrahigh-speed
optical transmission and may occur in all types of optical fibers. It originates
from the finite spectral width of the optical source [1–5] and causes
broadening of each transmitted mode (LP01 in single-mode fiber), and
thus induces an effect that is known as intersymbol interference (ISI). At
high transmission speed, higher order
dispersion becomes a severe limiting factor and must thus be taken into
consideration.The strong harmful effect of second-order
dispersion is well known in literature and linear commercial systems use dispersion shifted fibers with minimal
second-order dispersion in the 1.55-micrometer region [1].
Investigation of new methods and new technologies that could reduce and
minimize degrading influence of the second-order dispersion was of interest in the
past decade. Different techniques
may extend the transmission distance of a dispersion-limited system by a factor
of two or so but are unsuitable for long-haul systems for which group velocity
dispersion (GVD) has to be compensated along the transmission line in a
periodic fashion. Some of the techniques are dispersion-shifted fiber
(DSF), dispersion-flattened fiber (DFF), dispersion-compensating fiber (DCF),
dispersion-managed optical fiber, temporal optical conjugation (TOC), and so
forth. Chromatic dispersion in
optical fibers can add additional noise through phenomena such as intersymbol
interference and mode-partition noise.In the optical transmission system with
bit rate higher than hundreds of Mb/s, the influence of fourth-order dispersion
on pulse propagation must not be neglected [1, 2]. In some cases, it is
necessary to investigate the combined effect of second- and fourth-order dispersions
on transmission quality of signal. In this paper, we investigate the super-sech
optical soliton transmission under the influence of such even-order dispersions
assuming that odd-order dispersion is fully compensated. Soliton transmission
systems can potentially provide exceptionally high information transmission
capacity over long distances. Unfortunately for ultralong distance systems,
however, the maximum information bit rate depends strongly on the dispersion
and intersymbol interference.Another limiting factor in optical
communication systems is optical crosstalk. It represents leakage from other
optical signal to useful optical signal. The effects of optical crosstalk were
first studied by Moura et al. in 1997 [5]. Two types of crosstalk can occur,
which depends on whether the crosstalk channel is a different wavelength to the
signal (incoherent or out-of-band) or at the same wavelength (coherent or
in-band crosstalk). Incoherent crosstalk can in most cases be kept under
control with optical filtering at the receiver. Coherent crosstalk [3] is far more
detrimental than incoherent crosstalk because it cannot be controlled in this
way and thus it influences optical pulse propagation along the optical fiber
under chromatic dispersion (second- or fourth-order).This paper clarifies optical
pulse deformation caused by even-order dispersion. First, the intrinsic fiber impulsive
response for the higher
dispersion orders is expressed in general terms. Next, the impulsive response of
the linear fiber to an ultrashort pulse corrupted by coherent crosstalk is
given. Finally, numerically obtained results show the impact of coherent crosstalk
on system transmission quality by eye diagrams.


## 2. Theory

Ultrashort pulses from lasers often have a
temporal shape that can be analytically modeled by general form of super-sech
function given by(1)s(t)=P0⁢sechm⁡(tT0)⋅ejω0t, where ω0 is optical carrier
frequency, P0
is optical
pulse peak power, and
T0
is pulse width, and
m  represents the parameter that determines the super-sech pulse shape. This is
mathematical model that is found to be most appropriate to describe laser-induced
ultrashort optical pulse shape.From the original point of view, coherent crosstalk
is at the same frequency as useful signal but is often time and phase shifted
from the useful optical signal. Taking this fact into consideration, the model
of coherent crosstalk can be described as(2)si(t)=Pi⁢sechm⁡(tT0−b')⋅ej(ω0t+ϕ),
where Pi is crosstalk signal peak power and
b(b=b′T0)
is crosstalk time shift. The value of time
shift (propagation delay) depends on the nature of coherent crosstalk (inband
crosstalk or multipath reflection) [3, 4]. The phase shift φ varies in random way due to temperature
variation and wavelength variation in the range of [0, π].Coherent crosstalk influence on the
initial optical pulse is equal to a linear superposition of useful optical
pulse and crosstalk. The linear superposition is due to the fact that coherent crosstalk
is at the same frequency as useful optical signal. Hence, the resulting pulsesr(t) that will propagate through linear fiber can be expressed
completely by its envelope and phase as(3)|sr(t)|=Q,ψ(t)=arctg[Pi⁢sechm(t/T0−b')sin⁡ϕP0⁢sechm(t/T0)+Pi⁢sechm(t/T0−b')cos⁡ϕ],   where Q denotes (P0⁢sechm(t/T0))2+2P0⁢sechm(t/T0)⋅Pi⁢sechm(t/T0−b')cos⁡ϕ+(Pi⁢sechm(t/T0−b'))2.For an arbitrary input optical pulse, a general expression
of fiber responser(t,L)
is given in the literature [1] for an arbitrary input pulse as(4)r(t,L)=12πe−αLej(ω0t−β0L)×∫−∞∞F(ω)ej(ωt−ωβ1L−(1/2!)β2Lω2−(1/3!)β3Lω3−⋯)dω, where βi(i=1,2,…)  represents ith
order dispersion, α is attenuation constant, and L is propagation distance. F(ω) is Fourier transform of input pulse
and for even values of exponent m is
obtained in terms of Gauss’ hypergeometric function as (5)F(ω)=2mT0(F21(m,(m−iT0ω)/2,1+(m−iT0ω)/2,−1)m−iT0ω+F21(m,(m+iT0ω)/2,1+(m+iT0ω)/2,−1)m+iT0ω), where the Gauss’ hypergeometric function is defined as (6)F21(a,b;c;z)=∑k=0∞(a)k(b)k(c)kzkk!.The Fourier transform of
input optical pulse for odd m values is(7)F(ω)=1π2m/2−2e−(1/4)π((k−1)i+2ω)×(B−1(14(k−1−2iω),12(3−k))+eπωB−1(14(k−1+2iω),12(3−k))), where(8)Bz(a,b)=∫0zta−1(1−t)b−1dt is the incomplete beta function. Both hypergeometric
functions and incomplete beta functions can be numerically evaluated with any arbitrary
precision.Depending on the crosstalk phase shift value, coherent
crosstalk can be constructive or destructive. Let us investigate the worst case
crosstalk influence, that is, the destructive coherent crosstalk assuming that
it appears at the beginning of the optical fiber (e.g., double reflection or in—band
crosstalk resulting of WDM components used for routing and switching along the
optical network [3]). The received pulse shape under above mentioned crosstalk
influence and under the influence of second- or fourth-order
dispersion is(9)r2,4(t,L)=P0T02e−αLej(ω0t−β0L+θ(t))I12(t)+I22(t),θ(τ)=arctgI2(τ)I1(τ), with(10)I1(t)=∫−∞∞sechm(πT0ω2)×[(1−PiP0cos⁡(b'T0ω))cos⁡(ωt−b2ω2−b4ω4)−PiP0sin⁡(b'T0ω)sin⁡(ωt−b2ω2−b4ω4)]dω,I2(t)=∫−∞∞sechm(πT0ω2)×[(1−PiP0cos⁡(b'T0ω))sin⁡(ωt−b2ω2−b4ω4)+PiP0sin⁡(b'T0ω)cos⁡(ωt−b2ω2−b4ω4)]dω,       where(11)bn=(βnLn!),n=2,4.If we consider only the
influence of second- and fourth-order dispersion, in (9)–(11) we set b2=b4=0.


## 3. The Influence of Even-Order Dispersion on Super-Sech Pulse Propagation

For a binary data sequence with 0 and 1
transmitted along the optical fiber, the digital bit rate (B) must be less than
the reciprocal of the broadened pulse duration. This is the main condition for
nonoverlapping of light pulses down an optical fiber link. In the following
figures (Figures1, 2,
and 3) we consider such cases for the super-sech optical
pulse and crosstalk given by (1) and (2), respectively, with m=2. Depending on the value of crosstalk
time shift, coherent crosstalk can be left (b<0) or right (b>0) shifted in
regard to center of data in binary sequences. In all the figures in this paper,
fiber length is expressed in terms of dispersion length which is for nth-order dispersion that is given by(12)LD=(T0)n|βn|.
Signal-to-interference
ratio (SIR) is defined as follows:(13)SIR=10log⁡|r2,4(t,L)|2P0.Figure 1
Pulse shape at the end of the optical fiber(L=3LD) under
second-order dispersion for SIR=10⁢dB and different time shifts b>0.Figure 2
Pulse shape at the end of the optical fiber(L=3LD) under
fourth-order dispersion for SIR=10⁢dB and different time shifts b>0.Figure 3
Pulse shape at the end of the optical fiber(L=3LD) under
second- and fourth-order dispersion for SIR=10⁢dB and different time shifts b>0.Second-order dispersion induces asymmetrical
broadening of pulse and there is no need to investigate influence of bothb>0 and b<0.
Figure 1 shows the strong influence of crosstalk even for b>T0. Great time shift of crosstalk induces the
asymmetrical pulse deformation and occurrence of trailing edges. Long trailing
edges may give rise to intersymbol interference (ISI) and thus limits the
transmission length. Because of both, the noisy nature of the input to
clock-recovery circuit and noise produced by optical amplifiers, timing jitter
can also be produced. Then, as previously mentioned, asymmetrical pulse
deformation may be dangerous. The worst case in detection process occurs for b=0. This situation is very often observed
in switching systems. The eye diagram for the worst case is depicted in Figure
4.Figure 4
Eye diagram for individual influence of second-order dispersion in the
presence of worst case crosstalk.There are methods that can compensate for
degrading influence of second- and third-order dispersion. In such cases, fourth-order
dispersion will have the great influence on pulse shape. Fourth-order
dispersion distorts the pulse shape symmetrically around zero and as a result
it produces long trailing edges just like for the second-order dispersion.
Because of symmetrical deformation of pulse induced by fourth-order dispersion,
the greatest error in the detection process will appear for zero crosstalk
timing shift. Great absolute values of timing shift increase ISI, if B is high enough
to induce considerable overlapping of pulses. Figure5 compares eye diagram for
both in absence of crosstalk and in the presence of the most destructive crosstalk.
It can be seen that eye opening is 10 times decreased in presence of crosstalk
which unquestionably leads to bit error rate degradation.Eye diagram for individual influence of fourth-order dispersion(L=3LD):
(a) in absence of crosstalk, (b) in the presence of worst case crosstalk.
(a)(b)Figures3 and 6 show joint effect of second-
and fourth-order dispersions. Super-sech pulse at the receiver
(Figure 3) is
broadened by second- and fourth-order dispersion interplay and it has a long
trailing edge as a result of crosstalk. Eye diagram for this case is shown in
Figure 6.Figure 6
Eye diagram for joint influence of second- and fourth-order dispersion (L=3LD) in the presence of worst case crosstalk.


## 4. Conclusion

This paper studied the quality of
transmission of super-sech solitons, in a single-mode fiber, in presence of
second- and fourth-order dispersions. Coherent crosstalk is considered as it is
one of the most detrimental effects on the soliton transmission quality and
performance enhancement of optical systems. The analytical expressions of the
pulse shape, in presence of coherent crosstalk, in optical fibers were derived.
Finally, numerical simulations and eye diagrams are obtained.



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*Source: 613986-2008-05-07.xml*