APPLICATION OF DIGITAL CELLULAR RADIO FOR MOBILE LOCATION ESTIMATION
IIUM Engineering Journal, Vol. 10, No. 2, 2009 Islam and Majumder
87
INFLUENCE OF POLARIZATION MODE DISPERSION ON
THE EFFECT OF CROSS-PHASE MODULATION IN
INTENSITY MODULATION-DIRECT DETECTION WDM
TRANSMISSION SYSTEM
M. S. ISLAM1* AND S. P. MAJUMDER2
1Institute of Information and Communication Technology
2Department of Electrical and Electronic Engineering
Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh.
*
1. INTRODUCTION
E-mail: mdsaifulislam@iict.buet.ac.bd
ABSTRACT: Cross-phase modulation (XPM) changes the state-of-polarization (SOP) of the
channels through nonlinear polarization rotation and induces nonlinear time dependent phase shift
for polarization components that leads to amplitude modulation of the propagating waves in a
wavelength division multiplexing (WDM) system. Due to the presence of birefringence, the angle
between the SOP changes randomly and as a result polarization mode dispersion (PMD) causes
XPM modulation amplitude fluctuation random in the perturbed channel. In this paper we
analytically determine the probability density function of the random angle between the SOP of
pump and probe, and evaluate the impact of polarization mode dispersion on XPM in terms of bit
error rate, channel spacing etc for a two channel intensity modulation-direct detection WDM
system at 10 Gb/s. It is found that the XPM induced crosstalk is polarization independent for
channel spacing greater than 3 nm or PMD coefficient larger than 2 ps/√km. We also investigate
the dependence of SOP variance on PMD coefficient and channel spacing.
KEYWORDS: Polarization Mode Dispersion, Cross-Phase Modulation, State of Polarization,
Birefringence, Bit Error Rate.
Cross-phase modulation (XPM) is a nonlinear phenomena occurring in optical fibers
when two or more optical fields are transmitted through a fiber simultaneously. It has an
important impact on the performance of high-speed wavelength division multiplexing
(WDM) in optical fiber communication systems [1]. In the last few years, many research
works have been carried out on the interaction among fiber nonlinearity, polarization
variation and polarization mode dispersion (PMD) [2-8]. Physically, PMD has its origin
in the birefringence that is present in any optical fiber. Just like signal distortion due to
chromatic dispersion and nonlinearity accumulate along the length of the communication
link, so does the polarization and PMD. Polarization fluctuations and PMD become
increasingly important as the per-channel data rates have increased and now arguably the
most important of the polarization effects. Lin and Agrawal [4], developed a vector theory
of XPM in optical fiber and used to investigate the effect of PMD on XPM crosstalk in a
WDM system in terms of amplitude of the probe fluctuation induced by a co-propagating
pump channel and found that PMD reduces the difference in the average crosstalk level
IIUM Engineering Journal, Vol. 10, No. 2, 2009 Islam and Majumder
88
between cases of copolarized and orthogonally polarized channels. Zhang et al. [5],
experimentally reported that self-phase modulation (SPM) can suppress PMD penalty and
XPM-induced polarization scattering can reduce the PMD impairment in single channel
and dense WDM transmission system respectively. Here, we focus for simplicity on the
pump-probe configuration and neglect pulse broadening induced by group velocity
dispersion (GVD), but includes the group velocity mismatch between the pump and probe
owing to their different wavelength. We use Jones-matrix formalism for pump- or probe
field evolution and Stokes-vector formalism for SOP of pump and probe change on the
Poincare sphere.
PMD-induced fluctuation is the relative orientation between the pump and probe SOP
which causes the XPM modulation amplitude to be random. In this paper we analytically
determined the probability density function (pdf) of the random angle ( )zθ between the
pump and probe SOP fluctuation that produce intensity dependent pulse distortion on a
propagating signal. Using the pdf we analyzed the impact of PMD on XPM in terms of
average bit error rate (BER) for an intensity modulation-direct detection (IM-DD) WDM
system at a bit rate of 10 Gb/s. We also find the variance of the SOP between pump and
probe with respect to fiber link and channel spacing. To our knowledge the influence of
PMD on XPM in terms of BER is yet to be reported.
2. SYSTEM MODEL
The block diagram of a pump-probe configuration WDM system with EDFA in
cascade used for theoretical analysis is shown in Fig. 1. The pump and probe are
multiplexed by WDM MUX and the composite signal is transmitted through a standard
single mode fiber (SMF). To describe the effect of PMD on XPM, we assume that the
pump, 1A act as channel 1, which modulates the transmitted ( 1f ) data from the laser
diode at wavelength, 1λ and the probe, 2A act as channel 2, which is a low-power
continuous wave (cw) at wavelength, 2λ . The in-line EDFA’s are used in cascade to
compensate the fiber losses. Finally, the composite signal is demultiplexed at WDM
DEMUX and from the modulated carrier; 2λ original signal 1f is recovered through IM-
DD method.
Fig. 1: Block diagram of a 2-channel WDM system with EDFA in cascade.
IIUM Engineering Journal, Vol. 10, No. 2, 2009 Islam and Majumder
89
3. THEORETICAL ANALYSIS
3.1 Modulation of the Probe Field
The probe is assumed to be weak enough that the SPM, XPM and intrachannel PMD
induced by it can be neglected. Although these effects broadens pulse in each channel,
they barely affect the interchannel XPM interaction because the channel spacing typically
is much larger than channel bandwidth and the evolution of SOP of the channels is mainly
governed by the birefringence. The XPM induces a time dependent phase shift as well as
nonlinear polarization rotation on the probe channel that cause the XPM induced
modulation amplitude to be random. The cw probe field is modulated by the pump field by
the combination of XPM and PMD.
To study the temporal modulation of cw probe field as a consequence of pump field, let
us linearize the probe field 2A by assuming unperturbed, 20A and perturbed, 21A
probe fields respectively. With these simplifications, we can write the following equations
for pump and probe field evolutions as [4],
11
111 AA
AAA
01
1
2
2
21
1 22
Pi
i
z
ε−
α
−
τ∂
∂β
−
τ∂
∂
δβ−=
∂
∂
(1)
2020
20 Ab.σA
A
Ω−
α
−=
∂
∂
22
2 i
z
(2)
[ ])(3
22 00
2
22
2
22 zcosθSP
ii
z
+
ε
+α−
τ∂
∂β
−=
∂
∂
2120
21202120 AA
AAAA
(3)
where, 11 AA=0P and 2020 AAS =0 are the pump and unperturbed probe power;
0
ˆ PP 11 AσA= and 0ˆ SAσA 2020=S are the unit vector representing the SOP of
the pump and unperturbed probe on the Poincare sphere; )( 12 ω−ω=Ω is the channel
spacing, 11 98 γ=ε and 22 98 γ=ε are the effective nonlinear parameters, )(ˆˆ)( zSPz .=θ
is the random angle between the pump and probe SOP, )( 12111 β−β=δβ is the group
velocity mismatch between the two channels and )( 12 zt β−=τ is the reduced temporal
variable.
From (1) we see that the pump polarization P̂ remain fixed in the rotating frame.
However, PMD changes the relative orientation between the pump and probe stokes
vectors at a rate dictated by the magnitude of channel spacing and relative birefringence
( bΩ ). The total optical power of the probe field is given by,
[ ] 002120202022 δSSAAAAAA +≡++≈ .. cc (4)
where 0δS is the modulation amplitude, which is a measure of the XPM induced
crosstalk. PMD randomly changes the angle between P̂ and (z)Ŝ along the fiber and thus
probe field modulation amplitude becomes a random quantity. Solving equation (3)
IIUM Engineering Journal, Vol. 10, No. 2, 2009 Islam and Majumder
90
analytically and introducing the normalized modulation amplitude as,
)(),(
~
)(
~
00 LSLSX ωδ=ωδ ; we obtain,
[ ] dzdzzzθcoszz
L
z
L
X
βω+ωε−=ωδ ∫∫ 11222
0
2 )(2
1
sin)(3),(
~
)()(
~
0P (5)
where a tilde denotes the Fourier transform and ),(
~
ωz0P is the Fourier spectrum of the
pump power at a distance z inside the fiber and 22β is the GVD coefficient of the pump.
3.2 Pdf of )(zθ , SOP and Crosstalk Variance
Since the pump polarization P̂ remains constant and the random variation of (z)Ŝ (unit
vector of SOP of the probe) will be responsible for the randomness of )(zθ . Thus to find
the pdf of the angle )(zθ between pump and probe, we assume that it is driven by the
white noise process and can be written as,
)(ˆ zS
dz
d
=
θ
(6)
where, 0)(ˆ =zS (7)
)()(ˆ)(ˆ 12
2
21 zzzSzS −δη= (8)
where 31 22Ω==η pd DL ; 12 ω−ω=Ω is the channel spacing and pD is the PMD
parameter. This model captures all the essentials of the more realistic case in which both
the angle )(zθ and birefringence vary randomly. From equation (8) a diffusion equation
for the probability distribution of )(zθ or simply θ can be obtained [9],
2
2
2
2
1
θ∂
∂
η=
∂
∂ p
z
p
(9)
where ),( zp θ is π2 periodic in θ and )(),0( 0θ−θδ=zp . 0θ is the value of θ at
0=z , 0θ is the relative angle between the pump and probe SOPs at the input end of the
fiber. By solving the equation (9) we get the pdf )(zθ and expressed as,
)(cos)
2
1
exp(
1
2
1
),( 0
22
1
θ−θη−
π
+
π
=θ ∑
∞
=
mzmzp
m
(10)
The SOP of the pump and probe depends on the power and bit pattern of the pump. The
SOP fluctuates with time because of its doubly in nature. In the absence of residual
birefringence, randomness comes only from the bit pattern. The SOP of pump and probe
variance can be obtained by taking the inverse Fourier transform of equation (8) along the
fiber link length L,
IIUM Engineering Journal, Vol. 10, No. 2, 2009 Islam and Majumder
91
{ }
−δηωω
π
=
=σ
∫ ∫∫∫
∫∫
∞
∞
∞
∞
−
L L
LL
sop
zzdzdzdd
zSzSFdzdz
0 0
21
2
21
_
2
_
12
21
1
0
2
0
1
2
)(
4
1
)(ˆ)(ˆ
(11)
To find the ),(
~
ωz0P , we assume that the effects of dispersion and nonlinearities do not
significantly change the pulse shape of the pump channel along the fiber and analytically
solving(1), we then obtain,
[ ]
δβω+α−ω=ω ∫
z
dzzizz
0
1111100 )()(exp),0(
~
),(
~
PP (12)
Substituting this expression in (5), we obtain the following analytic result for the XPM-
induced crosstalk, [ ] dzzFzcosθ
L
X )()(3),0(
~
)(
~
0
∫ +ω=ωδ 0P (13)
where the function )(zF takes into account loss and dispersion variation along the
fiber link and is given by,
[ ] dzdzzdzzizzzF
L
z
z
βω
δβω+α−ε−= ∫∫ 11222
0
111112 )(2
1
sin)()(exp)()( (14)
The crosstalk level changes with time depending on the bit pattern in the pump channel.
Thus, )(τδ X fluctuates with time and the crosstalk variance, also called XPM-induced
interference can given by,
[ ]
ω−ω
ωδωδωω
π
=
ττδ≡σ
∫∫
∫
∞
∞−
∞
∞−
−
2
)(
)(
~
)(
~
4
1
)(
1
21
2
*
1212
2
2
22
T
incsdd
d
T
XX
T
T
Xm
(15)
where, T is the time interval of measurement. Usually the measurement time T is small
compared with the fluctuation time of PMD and birefringence fluctuations remain frozen
during measurement.
3.3 Bit Error Rate (BER) Expression
The output noise contributed by the photodetector quantum shot noise, the receiver
thermal noise, the interferometric noise due to input intensity fluctuation. The different
spectral densities of the noises are:
IIUM Engineering Journal, Vol. 10, No. 2, 2009 Islam and Majumder
92
BWiePRfS sdpd 22)( == (16)
L
thth R
BWKT
ifS
4
)( 2 == (17)
)]()([5.0)( 222 fxfSPRfS xsdpdi δ
−
−= (18)
where )( fS x and
−
x represent the PSD and the mean value of x(t) respectively and δ(f)
is a delta function in frequency. Denoting the total noise power spectral density by,
)()()()( fSfSfSfS pdithpdn ++= (19)
The total noise power (noise variance) at the receiver output can be obtained as,
dffHfS Rnn
22 )()(∫
∞
∞−
=σ (20)
where, )( fH R is the transfer function of a Gaussian low-pass filter in the receiver of
3-dB bandwidth equal to 0.75 B. The signal to crosstalk noise ratio (SCNR) can be written
as,
22
2
0 )(
nm
d SRSCNR
σ+σ
= (21)
where 2mσ crosstalk variance due to PMD, dR is the responsivity and 0S is the probe
power and 0SRI ds = . Thus,
2
0
2
2
0 )(
)(
nm
sIangleforSCNR
σ+θσ
=θ (22)
For a given value of 0θ , the conditional BER can be expressed as
θ
=θ
2
)(
5.0)( 00
angleforSCNR
erfcBER (23)
Because of environmental changes (i.e., temperature, pressure, vibration, stress, twisting
etc.), PMD fluctuates randomly. Generally, the impact of PMD fluctuations as well as the
variation of angle, )(zθ between the pump and probe SOP typically occurs on a time scale
of milliseconds. Thus the average BER is given by,
θθθ= ∫
π
π−
dzpBERBER ),()( 0 (24)
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93
4. RESULTS AND DISCUSSION
Following the analytical approach, we investigate the influence of PMD on XPM of the
pump-probe configuration of IM-DD WDM system at a bit rate of 10 Gb/s. The pdf of the
random polarization angle between pump and probe is evaluated at various length of
single mode fiber and shown in Fig. 2. From the figure, it is found that for particular
channel spacing the pdf is a delta function at 0=z and becomes flat as the link length
increases. It is observed that for any lightwave system of fiber length much larger than the
diffusion length ( =Eh 280 m - 1.5 km), pdf become Gaussian like distribution.
Fig. 2: Plots of pdf of )(zθ at different length of fiber link.
Figure 3 shows the variance of pump and probe polarization SOP versus fiber link
length for different PMD coefficient. It is noticed that as the PMD coefficient increases,
the variance of SOP also increases linearly. Thus the fibers which have higher values of
PMD (usually older fiber) will subject to larger amount of amplitude modulation
fluctuation.
Fig. 3: Plots of variance of SOP against fiber link length for different PMD parameters.
IIUM Engineering Journal, Vol. 10, No. 2, 2009 Islam and Majumder
94
The average BER versus pump power is plotted in Fig. 4. It is observed that interaction
of PMD and XPM increases as the channel spacing decreases for a specific pump power
and system suffers a large amount of BER. The significant dependence of XPM effect on
BER as a function of channel spacing comes from the fact that the PMD diffusion length
is inversely proportional to the square of the channel spacing. However, at increased pump
power the BER increases and when pump power is about 16 dBm the BER curves makes a
floor at 10-1 irrespective of channel spacing.
Fig. 4: Plots of average BER against pump power for different channel spacing.
The plots of average BER versus probe power is shown in Fig.5. It is seen that at high
pump and probe power the BER performance of the fiber link deteriorates sharply. This
happens due to the strong interaction of PMD and XPM on each other channel. For
example, when the pump power is -15 dBm, we can achieve a BER of 10-9 for a 6.25 dBm
probe power. Keeping the probe power same, it is found that at 25 dBm pump power the
BER drops to a value of 10-1 only. This happens due to the higher modulation amplitude
fluctuation as a result of high pump power.
Fig. 5: Plots of average BER vs probe power for different pump power.
IIUM Engineering Journal, Vol. 10, No. 2, 2009 Islam and Majumder
95
Figure 6 depicts the variance of SOP as a function of channel spacing for different
length. For smaller channel spacing and short length of fiber link the variance is very
negligible. As the link length increases and so does the channel spacing, the variances
increases exponentially for larger values of channel spacing.
Fig. 6: Plots of variance of SOP against channel spacing ( λ∆ ) for different link length.
5. CONCLUSION
An analytical approach is used to investigate the effect of PMD and XPM interaction in
a two channel pump-probe configuration WDM system using SMF. We determine the
probability distribution function of the random SOP angle between the pump and probe in
an IM-DD WDM system, which causes the XPM induced modulation amplitude to be
random. The strength of the XPM effect is strongly influenced by the evolution of light-
polarization of the WDM carriers. Our results show that the XPM induced crosstalk
becomes polarization independent when channel spacing is large ( i.e., λ∆ > 3 nm) or
when the fiber has a relatively large value of PMD coefficient (i.e., kmpsD p /2> ) .
Thus at relatively high pump the combination of nonlinearity and PMD can limit the
performance of a dense WDM fiber link significantly.
ACKNOWLEDGEMENT
This research work has been carried out as a part of Ph.D Thesis work in the
Department of Electrical and Electrical Engineering (EEE) of Bangladesh University of
Engineering and Technology (BUET), Dhaka, Bangladesh.
IIUM Engineering Journal, Vol. 10, No. 2, 2009 Islam and Majumder
96
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2.2 Two MEMS Bridge Parallel with the Resonator
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2.4 MEMS Bridges Parallel and Covering All the Resonators
ACKNOWLEDGEMENT
REFERENCES
KEYWORDS : Quantum Cryptography, Quantum Key Distribution, Decoy State Protocol, Optical
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2. EXPERIMENTAL SETUP
INFLUENCE OF POLARIZATION MODE DISPERSION ON THE EFFECT OF CROSS-PHASE MODULATION IN INTENSITY MODULATION-DIRECT DETECTION wdm TRANSMISSION SYSTEM
INTRODUCTION
SYSTEM MODEL
THEORETICAL ANALYSIS
3.1 Modulation of the Probe Field
RESULTS AND DISCUSSION
CONCLUSION
ACKNOWLEDGEMENT
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