Dispersion
1. Introduction
2. Modal
dispersion
3. Material
dispersion
3.1
Introductions.
3.2
Material dispersion
derivations:
3.3
Material dispersion
examples:
4. Waveguide
Dispersion.
5. Dispersion
Shifted Fiber – DSF
5.1 Design of DSF
6. Dispersion compensation
7. Polarization
Mode Dispersion
7.1 What is PMD
7.2 Causes of PMD
7.3 Birefringence
7.4 Mode coupling
7.5 Statistical Nature of PMD
7.6 PMD compensation
techniques
7.6.1 Electrical PMD
compensation
7.6.2 Optical PMD
compensation
REFERENCES
1. Introduction
Dispersion defined as the signal broadening or spreading while it is propagates inside the fiber.
There are many different types of dispersion in optical fibers. These types are:
- Intermodal, or modal, dispersion occurs only in multimode fibers.
- Intramodal, or chromatic, dispersion occurs in all types of fibers. It is cases by two type of dispersion:
i. Material dispersion.
ii. Waveguide dispersion.
- Polarization mode dispersion PMD.
Each type of dispersion mechanism leads to pulse spreading. As a pulse spreads, energy is overlapped. This condition is shown in figure 1. The spreading of the optical pulse as it travels along the fiber will effect the whole system where:
- Dispersion will case the signal to spread out, lose it is shape and become difficult to detect by receivers at the end of a fiber span, thus dispersion will causes bit error rates (BER) to increase to unacceptable levels.
- The spreading of the optical pulse as it travels along the fiber limits the information capacity of the fiber.
- Dispersion results in intersymbol interference (ISI) and hence power penalty.
- Dispersion induces coherent cross-talk between channels in multiplexed transmission systems.
- Dispersion causes pulse spreading and distortion and thus can lead to system penalties.
Figure 1. - Pulse overlap.
2. Modal dispersion
Occurs because each mode travels a different distance over the same time span, as shown in figure 2. The modes of a light pulse that enter the fiber at one time exit the fiber a different times. This condition causes the light pulse to spread. As the length of the fiber increases, modal dispersion increases.
Modal dispersion is the dominant source of dispersion in multimode fibers. Modal dispersion does not exist in single mode fibers. Single mode fibers propagate only the fundamental mode. Therefore, single mode fibers exhibit the lowest amount of total dispersion. Single mode fibers also exhibit the highest possible bandwidth.
The solution for this type of dispersion is to use the single mode fiber optic; also the graded index fiber will reduce this effect.
Where the graded-index fiber is an optical fiber with a core having a refractive index that decreases with increasing radial distance from the fiber axis. The most common refractive index profile for a graded-index fiber is very nearly parabolic. The parabolic profile results in continual refocusing of the rays in the core; because the inverse relation between velocity and refractive index the velocity of the modes that travel more distance in the fiber will be faster in graded index rather step.
Figure (3) below shows how graded index enhance the signal by removing dispersion depending on variation of signal speed.
Figure (3)
3. Material dispersion:
3.1 introductions:
Every laser source has a range of optical wavelengths; figure 3 show examples for LD and LED laser sources.
Figure (3)
Also Figure 4 illustrates the refractive index as it changes with wavelength for silica material.
Figure (4)
Since a pulse of light from the laser usually contains several wavelengths, and these wavelengths have different refractive indexes we expect different speeds for it is component that will case spread out in time after traveling some distance in the fiber
Note the relation of the velocity and it dependency on refractive index
Thus multiple wavelengths mean multiple nonlinear refractive indexes, which
mean different speeds for this pulse component which case the pulse spreading.
The refractive index of fiber decreases as wavelength increases see figure 2, so longer wavelengths travel faster. The net result is that the received pulse is wider than the transmitted one, or more precisely, is a superposition of the variously delayed pulses at the different wavelengths.
A further complication is that lasers, when they are being turned on, have a tendency to shift slightly in wavelength, effectively adding some Frequency Modulation (FM) to the signal. This effect, called “chirp,” causes the laser to have an even wider optical line width.
3.2 material dispersion
derivations:
To understand the effect of
dispersion, consider the group delay. This is the time delay per unit
length of energy propagating through a transmission system. We can assume each
spectral component travels independently and undergoes its own time delay, 
.
Let L be the transmission distance and 
the group velocity. Then:
Where
And
Let define
Thus
Note that
From equation 2 and 3
We define the material dispersion D as:
From equation 1, 4, and 5
The figure (5) below shows the first and second derivative of refractive index with respect to wavelength for pure silica material.
Figure
(5)
The resulted material dispersion is
shown in the figure below (this result can be obtained by multiplying the
values found in figure 5(c) by 
).
Figure (6)
Material dispersion is quoted in ps/(nm*km). That means that the total Dispersion of a fiber link depends on the distance (km) as well as the Laser Bandwidth (nm) of the Transmitter Laser.
To get minimum bit error rate in practical systems, the total delay caused by chromatic dispersion should not be bigger than 20% of the Bit period
3.3 Material dispersion
examples:
Example:
STM-16 (2,5Gbit/s): Bit period is 400ps ŕ max. Delay caused by CD is 80ps
STM-64 (10Gbit/s): Bit period is 100ps ŕ max. Delay caused by CD is 20ps
Example – max. Length Calculation
Parameters:
Laser Bandwidth 0,1nm
CD of Fiber is 20ps/(nm*km)
Bit rate: STM-16
For STM-16: 80ps
Maximum Delay = (CD of Fiber * Laser Bandwidth * maximum Length of fiber).
ŕ Maximum Fiber Length = Maximum Delay/(CD of Fiber * Laser Bandwidth) = 40km
Thus after 40km a Regenerator is needed. If CD of the fiber is only 16.5ps/(nm*km) the maximum distance is 48km. Therefore it is important to know the CD value of the installed fiber in order to install the regenerator at the right place.
Furthermore if Laser Bandwidth is only 0,015nm, CD of the fiber is 20ps/(nm*km) the maximum distance is around 364km. Again it is important to know the CD of the fiber in order to select a proper Transmitter. Because a better Transmitter (smaller Laser bandwidth) could save some expensive regenerators. Or on the other hand a better transmitter might be needed in order to cover even bigger distances. Any regenerators. In such a case countermeasures needs to taken in order to reduce CD. This is done via so-called Dispersion Compensation.
These parameters can be controlled in such way to control the material dispersion:
- Using of fiber has minimum dispersion, this value depend on fiber material.
- Choosing source that has lower bandwidth.
- Using regenerator.
- Using dispersion compensation.
The figure below shows the chromatic dispersion where it is the total sum of material and waveguide dispersion
Figure (7)
4. Waveguide Dispersion
Even if there were no material dispersion, there would still be some pulse spreading due to ‘waveguide dispersion’, which occurs because β for a given mode varies with wavelength.
The effective refractive index of the fiber lies between the refractive indices of the core and the cladding. The effective refractive index varies with the wavelength:
At short wavelengths, the effects of diffraction are small and the light in the fiber is confined well within the core of the fiber. The effective refractive index is very close to the refractive index of the core.
At medium wavelengths, as the wavelength increases, the effects of diffraction become more important and the light spreads slightly into the cladding of the fiber. The effective refractive index decreases towards the refractive index of the cladding.
But at long wavelengths, the effects of diffraction dominate and the light in the fiber spreads well into the cladding of the fiber. The effective refractive index is very close to the refractive index of the cladding.
The figure below shows the light distribution inside the fiber (in the core and cladding) for different wavelengths.
figure (8)
How does wavelength affect dispersion?
Consider first order mode at different wavelengths.
figure (9)
β( λ1 ) > β( λ2 ) for the first order mode of same waveguide.
figure (10)
Thus long λŕmore light in the cladding and thus different dispersion curve. We can note that shift will increase as the difference in refractive index increase; also as core diameter decrease the shift will increase. Waveguide dispersion shifts delay minimum to longer wavelengths.
To derive waveguide dispersion equation we start for the group velocity equation:
But a guided mode propagates as if it had a k of β for a guided mode
The relation between v and b given in figure below for different wavelengths
Figure (11)
The figure below represents the first and second derivative for parameter V with respect to b, where it is needed for general waveguide relation.
Figure (12)
Expressed in ρs/km.nm.
Example
This Numerical example shows how to calculate waveguide dispersion for a fiber:
Let n2 = 1.5
D = 0.0015
l = 1.5m = 1.5 * 10^-4 cm = 1500nm
c = 3*10^8 m/s
ŕ V= 1.72
For figure 5
The image below show the waveguide dispersion for different core diameter, note that as core diameter increase the dispersion decrease.
Figure (13)
Wave guide dispersion depend on many parameter:
- Dn, refractive index difference between core and cladding.
- Core diameter, as core diameter decreases larger dispersion will occurs.
- Source line spectrum, as it is increase waveguide dispersion increase.
- Fiber distance, as distance increases the waveguide dispersion increase.
- Fabrication of the fiber (core and cladding shape)
5. Dispersion Shifted Fiber – DSF
Because of the chromatic dispersion is the sum of material and waveguide effects, thus in order to shift the dispersion zero point to 1.5µm, the waveguide component of dispersion can be changed.
Waveguide dispersion arises from the way that the light propagating in the fiber is distributed.
At short l’s, much of the light is confined in the high index core of the fiber than Longer l’s. The speed of light in the fiber is thus a function of l.
Changing the design of the core/cladding interface can changes the magnitude of this effect.
Figure (14)
5.1 Design of DSF
Design illustrated above moves the zero dispersion point to 1.55µm – ideal for telecommunications. This is called zero dispersion-shifted fiber (ZDSF).
Often require the dispersion shifted so that the zero point is at a longer wavelength than 1.55mm to eliminate four-wave mixing. This is called non-zero dispersion shifted fiber (NZDSF).
Dispersion Flattened Fibers
If we could tailor the WG dispersion to exactly (or almost) cancel the material dispersion, then dispersion would “cease” to be a problem.
In the figure below you will find many techniques used to design DSF or DFF
Figure (15)
Figure (16)
6. Dispersion compensation
Much existing fiber is designed for
zero-dispersion around 1.3µm, Actually want to operate around 1.5µm due to low losses and
available amplifiers. So we need to compensate for dispersion effects. So we
design what is called dispersion compensating fiber (DCF).
To do that design fiber with a large negative waveguide dispersion. Place the required length to balance the positive dispersion from the conventional fiber afterwards. thus net dispersion is zero.
Problem is that DCF has higher attenuation and is susceptible to bend losses.
7. Polarization
Mode Dispersion
The rapid increasing in the demand for bandwidth is
driving most telecommunication operators toward the deployment of large capacity
transmission systems. The data rates per channel in commercial systems have
increased to10 Gbit/s. These systems are deployed and
systems with channel rates of 40 Gbit/s are being
announced for deployment. However, high-speed optical transmission systems have
to face limitations imposed by physical properties of the transmission fiber.
There are four principal impairments in optical fiber transmission: chromatic
dispersion, nonlinearity, polarization effects and amplified spontaneous
emission noise which is defined as A background
noise mechanism common to all types of. It contributes to the noise figure of
the EDFA which causes loss of signal-to-noise ratio (SNR).
Polarization implies a definite direction and phase
relationship between electric fields of a propagating wave. Polarization
effects have historically played a minor role in the development of light wave
systems because commercial optical receivers are insensitive to polarization as
they detect optical power rather than the optical field. But because of the
recent developments like the optical amplifier, which has dramatically
increased the optical path lengths achievable, and new transmitter and receiver
technologies, which have pushed the capacity of optical fiber to its limit,
small effects such as PMD.
7.1 What is PMD?
PMD is actually a form of material dispersion. SMF
supports one mode, which consists of two orthogonal polarization modes.
Ideally, the core of an optical fiber has an index of refraction that is
uniform over the entire cross section; however, mechanical stresses and
external environmental effects can cause slight changes in the core of the
fiber, which causes a change in index of refraction in one dimension. This can
cause one of the orthogonal polarization modes to travel faster than the other,
hence causing dispersion of the optical pulse.
7.2 Causes of PMD
PMD is caused by asymmetries and stress distribution in
the core of the fiber, which locally leads to birefringence, i.e. a
polarization-dependent refractive index. Globally, the birefringence is
combined with random polarization mode coupling. To understand how PMD arises
in optical fiber, it is best to start by considering a short section with
uniform birefringence within a long fiber span. Any SMF can then be modeled as
a concatenation of many such arbitrarily oriented birefringent
elements.
7.3 Birefringence
Optical birefringence is the origin of PMD. In an ideal
circularly symmetric fiber, there are two orthogonally polarized 
modes, which have the same group delay. They
may also experience non uniform stress such that the cylindrical symmetry of
the fiber is broken. The asymmetry
breaks the degeneracy of the orthogonally polarized of the modes-a difference in the phase and group velocities of
the two modes. Optical fiber
birefringence is caused by both intrinsic and extrinsic perturbations. The
permanent intrinsic perturbations in the fiber are set up from imperfections in
the manufacturing process. When fiber is
spooled, cabled or embedded in the ground, birefringence can be induced from a
number of extrinsic perturbations, including lateral stress, bending or
twisting. These perturbations are
changing as the fiber’s external environment changes. The birefringence can be considered uniform
in a short section of fiber. The
difference between the propagation constants of the fast and slow modes can be
defined as:
(1)
Where w is
the angular optical frequency, c is the speed of light, and 
is the effective differential refractive index between the
slow and fast modes.
7.4 Mode coupling
There are random variations in the axes of the
birefringence along the fiber length, as illustrated in Figure 4 causing
polarization mode coupling, wherein the fast and slow polarization modes from
one segment each decompose into both the fast and slow modes of the next
segment. Polarization-mode coupling
results from localized stress during spooling, cabling, deployment, from
splices and components, from variations in the fiber drawing process and from
intentional fiber “spinning” during drawing, which induces mode coupling at
“meter” length. Long fibers are
frequently modeled as a concatenation of birefringent
sections model.
Figure 17: A concatenation of birefringent sections model.
As mentioned earlier, modeling of birefringence with the
length of fiber gets complicated because of mode coupling. To understand the
concept of mode coupling, consider a light pulse that is plane polarized in the
fast-axis injected into the fiber. As the pulse propagates across the fiber,
some of the energy will couple into the orthogonal slow-axis polarization
state, this in turn will also couple back into the original state until
eventually, for a sufficiently long distance, both states are equally
populated, as illustrated in figure18.
Figure 18: Decorrelation
of polarization in long fibers (Poole and Nagel, 1997).
The length of the fiber at which the ensemble average
power in one orthogonal polarization mode is within 1/ 
of the power in the
starting mode is called the coupling length or correlation length
. It
is a statistical parameter that varies with wavelength, position along the
length of the fiber and temperature. Typical values of coupling length range
from tens of meters to almost a kilometer.
Figure 19: Spatial evolution of
polarization caused by uniform birefringence.
Beat length
, is
defined as the propagation distance for which a 2p phase difference accumulates
between the two modes or, equivalently, the polarization rotates through a full
cycle and is given by = l/Dn, where l is optical wavelength and Dn is the differential
refractive index. A typical value of beat length is ~10 m for standard
telecommunications-type fibers.
Due to the absence of mode coupling in the short fibers,
differential group delay (DGD) (Dt) accumulates linearly with fiber length. Dt can be found from the
frequency derivative of the difference in propagation constants as:
(2)
The effect of PMD in the time-domain in a short fiber is
illustrated in Figure 20, where a pulse launched with equal power on the two birefringent axes results in two pulses at the output
separated by the DGD.
Figure 20: Time-domain
effect of PMD in a short fiber
fiber PMD is often specified using
a PMD coefficient having units of
. Fibers manufactured today have mean PMD coefficients
smaller than 0.1 
whereas “legacy” fibers installed in the 1980s may exhibit
PMD coefficients greater than 0.8 
. In the frequency domain, as the wavelength is varied, the
output state of polarization (SOP) of a long fiber will trace out an irregular
path on the Poincar'e sphere. Any portion of this
path, over a small wavelength interval, can be represented as an arc of a
circle, the center of which when projected normal to the plane of the circle to
the surfaces of the sphere, locates two diametrically opposed, orthogonal
states of polarization known as principal states of polarization (PSPs).
Figure 21: Model for a long fiber as a
concatenation of birefringent sections with
Birefringence axes and magnitudes that change randomly along the
fiber.
As discussed in the earlier sections, PMD in a fiber
varies randomly with wavelength and also with environmental conditions. This is
because of the randomness of mode coupling and core deformation due to external
stress in the fiber. The PMD vector can be decomposed into three orthogonal
vectors along the axes of the Poincar’e sphere, each
of which is an independent random variable with zero mean and can be described
statistically by the Gaussian distribution. The magnitude of the PMD-vector is
the DGD given by the square root of the sum of the squares of the orthogonal
components. The impact of PMD on telecommunication systems can be predicted
from the distribution of DGD (Δτ).
It has been shown that in the random mode-coupled or long fiber regime, Δτ follows a Maxwellian
distribution, given by :
(3)
And shown in Figure 22. This means that the
distribution of values of Δτ measured over
a wide wavelength range or over time at a fixed wavelength, if a changing
environment acts upon the path, will be Maxwellian As
a result of this variability, the PMD of a path is expressed statistically as
mean DGD
(4)
For 0 < Dt< +Ą where Dt is DGD and 
is the variance.
The probability density function of 
has been
derived
(35)
Where 
is the mean DGD
Figure 22: Normalized Maxwellian
distribution, with magnified tail portion
7.6 PMD
compensation techniques
PMD is the primary barrier to achieve single-channel data rates at 10 Gbit/s and beyond in installed terrestrial fiber systems. By inducing polarization dependent propagation, PMD generates multiple images of light pulse carrying the information and leads to ISI. PMD is a particular difficult phenomenon to assess because of its statistical nature with characteristics varying in time and frequency. The stochastic nature of PMD and its higher-orders are such that, reducing the impact of PMD does not necessarily imply the complete cancellation of the effect, but the reduction of the outage probability due to PMD as Figure 23 shows.
Figure 23: Pulse widths evolution as a function of
propagation Distance
Hence, this process is called PMD mitigation. Several PMD compensation techniques have been proposed in the past few years that can be separated into two main categories.
-Electrical PMD compensation
-Optical PMD compensation
These following two sections present a brief discussion
of these techniques
7.6.1
Electrical PMD compensation
Electrical compensation of PMD
involves equalizing the electrical signal after the photodiode. This
equalization can be implemented in many ways. One way is to use a linear
transversal filter (TF) that is realized using a tapped delay line model. The
TF divides the signal, delays the copies by constant delay stages and
superimposes the differentially delayed signals at the output port. The tap
weights are adjusted to maximize the receive signal quality. Another way of
equalization is to use non-linear decision feedback equalizer (DFE). The basic principle of DFE is
that once a decision is made on a particular bit as a one or zero, the ISI that
this bit induces on future bits can be subtracted out before deciding on future
bits. This method requires fast signal processing for coupling the decided bit
back in time One other way of equalization is based on phase diversity detection
where two photo diodes detect the signals on the PSPs
of the fiber and infer a DGD value and a controlled delay is generated on one
arm before electrical recombination.
Electrical compensation schemes, in general, are robust
and will improve the signal against all kinds of transmission impairments. On
the other hand, they do not perform as good as optical PMD compensators and
also they require high-speed electronics for better performance.
7.6.2 Optical
PMD compensation
The goal of optical PMD compensation is to reduce the
total PMD impairment caused by the transmission fiber and the compensator. The
block diagram of a general optical PMD compensation scheme is shown in Figure
24. It has an adaptive counter element, a feedback signal and a control algorithm.
Figure 24: General Scheme for optical PMD compensation
The adaptive counter element is the core of any PMD
compensator. It must be able to counteract PMD impairments and be tunable. High
birefringent elements like PM fibers, LiNbO3 delays,
and Bragg gratings etc. separated by polarization controllers are used as
adaptive counter elements. Different schemes vary in the number of birefringent elements used, their tunability
and the technology used for polarization control. The feedback signal is required
to provide the PMD information to the controlling algorithm of the compensator.
Important characteristics of a feedback signal are (i)
its sensitivity to PMD, (ii) BER and (iii) its response time. The common
feedback signals used in various schemes are the degree of polarization (DOP),
RF spectral width, total RF power and eye opening. Of all these, DOP feedback
signal has the advantages of being bit rate independent and having quick
response time. Finally, the controlling algorithm controls the adaptive counter
element based on the feedback signal. It is generally implemented as some kind
of gradient search method. The parameters of the algorithm are chosen carefully
to avoid the sub-optimum operating points, which can have disastrous impact on
the transmission quality.
The present of unique way of classifying the optical
compensators based on the number of degrees of freedom (DOF) in the
compensator, which is the number of elements or parameters in the compensating
element that are controlled by the control algorithm. The DOF is a good
estimate of the complexity and hence the cost of the compensator. Figure 16
shows six different PMD compensation methods and their corresponding DOF. The
PSP method shown in Figure16a is a pre-transmission compensation technique in
which a PC is used to align the SOP(State of
polarization) with one
of the input PSPs(Principal states of
polarization) of the
fiber link. It is a first order compensator with two DOF. A first-order
post-compensator with a PC and a fixed time delay (2 DOF) is shown in Figure
16b this is sometimes referred to as a half-order compensator. In Figure 25c, a
post-compensator with a PC and a variable delay (3 DOF) is demonstrated. Figure 25d shows a
double-stage compensator with two
PCs and two fixed delays
(4 DOF). Another double-stage
compensator with two PCs and one fixed and one variable delay (5 DOF) is
illustrated in Figure 25e. The double stage compensators can compensate for
higher order PMD. Figure 25f shows a PC and a polarizer in combination (2 DOF),
where the average power through the polarizer could act as an error signal.
comp (2 DOF), (c) 3 DOF post comp, (d) 4 DOF post comp, (e) 5 DOF post comp,
(f) polarizer method (2 DOF).
By means of numerical simulations, concludes that a
single-stage compensator with a variable delay (3 DOF), shown in Figure 25c
performs better than the first-order compensators of Figure 25a and b. This is
because it has a well-defined optimum and it can compensate for higher-order
PMD to some extent. The double-stage compensators (4 DOF, 5 DOF), shown in
Figure 25d and e, have several sub-optima. They are better than first-order
compensators when operating at the global optimum, but they could be trapped in
a local optimum unless a good way to reach the global optimum is reached.
Figure 25: Optical compensation schemes. (a) PSP method (2 DOF),
(b) 1st order
Post comp (2 DOF), (c) 3 DOF post comp, (d) 4 DOF post
comp,(e) 5 DOF post
comp, (f) polarizer method (2
DOF).
As a consequence, these schemes perform worse than first-order compensator at low PMD (relative to bit period), but still better at high PMD. The polarizer method (2 DOF) shown in Figure 25f has only one optimum and it performs better than first-order compensators at large PMD because it does not add any DGD. At low PMD, the performance is worse than first-order compensators but an improvement is achieved compared to the uncompensated case .Of all the compensators discussed in the section, the single-stage compensator with a variable delay line is optimum with respect to the performance and complexity. To conclude, use of any of the compensation techniques discussed in this section does not necessarily guarantee complete cancellation of PMD, but can reduce the probability of outage due to PMD. Using PMD compensators is an expensive proposition, especially in WDM systems. This is because PMD compensators are band-limited and so each compensator can work at the most on a few consecutive channels and therefore many compensators have to be used to compensate for PMD on all the channels of a WDM system.
REFERENCES
Hasan
Farahneh.
D.C, 2004, “Measuring
Polarization Mode Dispersion In Optical Fibers Using Four-wave Mixing Nonlinear
Effect”,
Agarwal. D.C,
(1993), “Fiber optic communication, 2nd edition,” Wheeler Publishing.
Born . M and Wolf. E,
(1980) “Principles of optics,” Pergammon
Press,
Geoff. Snell, (1996), “An Introduction
to Fiber Optics and Broadcasting,” SMPTE Journal, January.