Statistical Model for Polarization Mode Dispersion in Single Mode Fibers

: As the bit rate of fiber optic transmission systems is increased to more than Gbps 10 , the system will suffer from an important random phenomena, which is called polarization mode dispersion. This phenomenon contributes effectively to: increasing pulse width, power decreasing, time jittering, and shape distortion. The time jittering means that the pulse center will shift to left or right. So that, time jittering leads to interference between neighboring pulses. On the other hand, increasing bit period will prevent the possibility of sending high rates. In this paper, an accurate mathematical analysis to increase the rates of transmission, which contain all physical random variables that contribute to determine the transmission rates, is presented. Thereafter, new mathematical expressions for: pulse power, peak power, time jittering, pulse width, and power penalty are derived. On the basis of these formulas, one can choose a certain operating values to reduce or prevent the effects of polarization mode dispersion.


Introduction:
Polarization mode dispersion (PMD) arises in single mode fiber and fiber optic components due to a small difference in refractive index (birefringence) for a particular pair of orthogonal polarization states [1,2].This index difference results in a difference in the propagation time called differential group delay (DGD) for waves traveling in these two polarization modes [3].The propagation of a pulse through a long fiber can be very complicated since the birefringence varies randomly along the fiber.However, there are two special orthogonal polarization states, called principal states of polarization (PSP's), at the fiber input for which the output pulse is undistorted to first order, in spite of random changes in fiber birefringence [4,5].An optical pulse polarized along a PSP does not split into two parts and maintains its shape.In practice, the launched pulses are rarely polarized along one of PSP's, each pulse then splits into two parts that are delayed with respect to each other by a random amount [6,7].
This kind of PMD is commonly known as first-order PMD.Under firstorder PMD, a pulse at the input of a fiber can be decomposed into two pulses with orthogonal states of polarization (SOP).Both pulses will arrive at the output of the fiber undistorted and polarized along different SOP's, the output SOP's being orthogonal [8,9].Both the PSP's and the DGD are assumed to be frequency independent when only firstorder PMD is being considered [10].Second-order PMD effects account for the frequency dependence of the DGD and the PSP's.The frequency dependence of the DGD introduces an effective chromatic dispersion of *Thi-Qar University, Science College 1255 opposite sign on the signals polarized along the output PSP's [11].PMD induced pulse broadening can move bits outside of their allocated time slots, resulting in errors and system failure in an unpredictable manner [1,5].
When the signal channel bit rates reached beyond 10 Gb/s, PMD became interesting to a larger technical community.PMD is now regarded as a major limitation in optical transmission systems in general, and an ultimate limitation for ultra-high speed signal channel systems based on standard single mode fibers [7,12].PMD arises in optical fibers when the cylindrical symmetry is broken due to noncircular symmetric stress.The loss of such symmetry destroys the degeneracy of the two eigen-polarization modes in fiber, which will cause different group velocity dispersion parameters for these modes.In standard single mode fibers, PMD is random, i.e. it varies from fiber to fiber.Moreover, in the same fiber PMD will vary randomly with respect to wavelength and ambient temperature [6].
Disorder in single mode fibers arises in many different ways and has a negative effect.For example, amplifier noise [9], and random fiber birefringence (PMD) [8] that lead to random shifts in the pulse position (timing jitter), pulse broadening, and so to cause inter-symbol interference (ISI) impairment of a single digital transmission channel .The ISI impairment is caused by the DGD,   , between the two pulses propagating in the fiber when the input polarization of the signal does not match one of the PSP's of the fiber PMD impairments due to inter-channel effects that occur in polarization-multiplexed transmission systems.However, all PMD effects eventually cause destruction of bit patterns and lead to an increase of bit error rate, the most important parameter describing performance in fiber communications systems [8,10].The description of data stream degradation requires the use of statistical methods and opens a new field that may be called statistical physics of fiber-optic communication.
Our objective in this paper, is to model the PMD (at first order in w ) density distribution and determined the probability of DGD that exceeds a particular value.Initially, the treatment will be quite general, involving standard equations for impulse response and density distribution of PMD.This general treatment will make it possible to estimate the density distribution of impulse response and power penalty.We will then consider the effect of all random variables on the output pulse.Due to the difficulty of the mathematical analysis, assuming the pulses are Gaussian form.The parameters that maximize density distributions of PMD, impulse response, and power penalty are determined.We proposed that the time jittering and pulse broadening effects may be reduced to minimum values by controlling on the angle between the PMD and input SOP vectors.

Theory
The effects of PMD are usually treated by means of the threedimensional PMD vector that is defined as pmd p

 
, where p is a unit vector pointing in the direction of slow PSP, and pmd  is the DGD between the fast and slow components which is defined as [4] The , so that no changes in output polarization can be observed close to these states at first order in w .
To the first order, the impulse response of an optical fiber with PMD is defined as [7] where   are the splitting ratios.The is the Pauli spin vector, which is their components are defined as [8] It is important to note that the angle between p ˆ and s ˆ in Stokes vector is  , while the angle between


. That is; if two vectors are perpendicular in Jones space then the corresponding two vectors in Stokes space are antiparallel.
Consider that the PSP's occur with a uniform distribution over the Poincare sphere, and that ŝ is aligned with the north pole of the sphere as shown in Fig. (1).The probability density of PSP's being in the range d  about the angle  relative to ŝ is proportional to the differential area , | p   , and | p   are the input SOP and the two PSP's vectors.If the PSP's are defined as , where t is the matrix transpose, then we can write where Comparing Eqs.( 4) and ( 5), we can extract In turn, the splitting ratios can be calculated by using Eq.( 6) and the fact that To this end, the relationship between the splitting ratios and elevation angle was calculated.Also, showing how the angle will be determined the values of   that we have been calculated for this point.

Statistics of DGD
Throughout this subsection, the PMD statistics have been carefully analyzed that causes the variation in pulse properties.A proper measure of pulse width for pulses of arbitrary shapes is the root-mean square (rms) width of the pulse defined as  [6] where c is the correlation length that is defined as the length over which two polarization components remain correlated, , a reasonable estimate of pulse broadening was obtained by taking the limit c L  in Eq.( 9).The result is given by [7] where p D is known as the PMD parameter that takes the values (0.01 10) / ps km  .The variable pmd  has been determined to obey a Maxwellian distribution of the form [6] The mean of pmd  is done simply as follows Using Eq.( 12), the Maxwellian distribution will take the form A cursory inspection of Eq.( 13) reveals that the ) (

Statistics of Impulse Response
The rms width of the impulse response, eff  , can be readily calculated by substituting Eq.( 2

Power Penalty
In the first-order picture, PMD splits the input signal entering the fiber into two orthogonally polarized components that are delayed by pmd  relative to each other during transmission.The impairment caused by this effect can be expressed as a power penalty of the form [12] where the penalty expressed in dB is assumed to be small, T is the bit interval, and K is a dimensionless parameter takes the value from 10 to 70 depend on pulse shape, modulation format, and specific receiver characteristics.It is straightforward to note that the penalty will minimize by decreasing DGD, increasing bit period, and making the elevation angle around 0   or    .The mean penalty parameter can be determined using Eqs.( 22

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Using the PSP's as an orthogonal basis set, any input or output polarization can be expressed as the vector sum of two components, each aligned with a PSP.Within the realm of the first-order PMD, the output electric field from a fiber with PMD has the form [9] where .Therefore, according to Eq.( 29), the shifted pulses will reshape as where 0 t is the initial pulse width.Substituting Eqs.( 8) and ( 30) into (29), using the output power definition, using the orthogonality properties of Jones vectors, and simplified the result, we obtain the following expression The width of the output pulse 1 t can be determined using Eq.( 31) as follows The time jittering of the pulse can be found by determining the maximum value of ) (t P out .This maximum value will happen at The peak power, as a function of DGD and an angle  , at the pulse center can be determined by substituting Eq.(33) into Eq.(31) as follows At this point, we drive a formulas for the output power form, final width, time jittering (shifting), and peak power as functions of the random physical variables  and pmd  .
Eqs.(31-34) are considered as the main achievement of this work.

Special cases
Now, using Eqs.(31) to (34), very important special cases may be illustrated 1.For 0 ,   pmd random   (SOP enters with random angle, no PMD) Note that, this case does not represented any physical fact but it introduced for illustration only.

Results and Discussion:
The parameters used in the simulation are as follows: .By rising  , the pulse width and distortion will increase, while the power and shifting will decrease.


, which is very small and does not effect system properties.
We can not fail to mention that the increasing of PMD parameter and fiber length will lead to increase the values above and thus adversely affecting the functioning of the system specifications.Finally, the angle  can control an important factors that are affecting all physical properties of the system, where the best values are   , 0  . Table (1) summarizes all cases of Figs.( 3) and (4).will make the system to achieve the desired objectives.
fiber and its associated stresses, where the splitting ratios can range from zero to one.Note that, the function () pmd ht is normalized in the range (  to  ).
the difference in group velocities along the two PSP's.For distances 1 L km  Eq.(14) provides a method for calculating the maximum likelihood value of pmd  if one knows pmd  .

(
    to yield The result may be simplified by substituting Eqs.(8) into (16) to yield Using Eqs.(4) and (17), one can transform the density distribution for  to the density for eff  as follows It is important to note that the probability density is a function of eff  and pmd  .As a consequence of this dependence, Eq.(18) can not be integrated to determine eff  because presence of the other variable pmd  .So, in the next, we seek about () .(13)and (18) as follows Return to Eq.(17), it may be written as 2 sin level, Eq.(18) is the same as Eq.(20) but the latter is a function of eff  only, which can be integrated to obtain eff  .However, both equations are normalized properly.The mean value of eff  can be determined as So, Eq.(20) may be written as


) and (25) as follows This allow us to easily transform the Rayleigh density, i.e.Eq.(22), of eff  to the density for  and obtain the following distribution Using this distribution, the probability of  exceeding a particular value can be are depended on the parameters p D and L .For illustration, the density distributions of pmd  , eff  , and  are computed in Fig.(2).

Fig.( 2 )
Fig.(2): density distributions ) ( pmd p  represent distinct bits.Fig.(3) represents the pulses shape, where the solid line is the original pulse, while the discrete lines are the resulted pulses with different values of pmd  ranging from 0 to ps 8 , where the closest to 0  t is the pulse that has least value of pmd  .Fig.(4) represents pulse width, time jittering (shifting), peak power, and penalty as functions of pmd  and  .At the angle 0   , one note that the pulse is faced only by a displacement to right at 2

Fig.( 3 )
Fig.(3): pulse shape with different value of pmd  and  , the solid line represents the original pulse while the discrete lines represent the resulted pulses for different values of pmd  , where the lower value of

Table ( 1): pulse properties with different values of
 and pmd  .