## Polarization mode dispersion

In ideal single-mode fibers, propagation constants of the two polarization Eigenmodes

are degenerate. In real telecommunications fibers, perturbations act on the fiber in a

way that it induces a birefringence, which splits this degeneracy. Consequently, when

a pulse is launched in a fiber, it gives rise to a differential group delay between the two

polarization Eigenmodes. The stochastic behavior of these perturbations is an issue

because it yields to the phenomenon of random mode coupling which makes

impossible the basic definition of the differential group delay.

## Principal states of polarization

The Principal States of Polarization (PSP) model [Poole and Wagner, 1986[14]] is

based on the observation that at any given optical frequency, there exists a set of two

mutually orthogonal input states of polarization for which the corresponding output

states of polarization are independent of frequency to first order. The Differential

Group Delay (DGD) resulting from Polarization Mode Dispersion (PMD) is then

defined between the two output principal PSPs.

The birefringence in telecommunication single-mode fibers varies randomly along the

fiber length, an artifact of variation in the drawing and cabling process. Furthermore,

owing to the temperature dependence of many of the perturbations that act on the

fiber, the transmission properties typically vary with ambient temperature. In practice,

fluctuations in temperature strongly affect PMD time evolution. To evaluate properties

of long fiber spans, one adopts a statistical approach. In this case of long span fibers,

the polarization Eigenstates can only be defined locally and the birefringence vector

has to be considered as stochastic.

## Dispersion vector definition

In the time domain, the Polarization Mode Dispersion (PMD) induces a time shift

between the two Principal States of Polarization (PSP).

In the frequency domain, the output state of polarization undergoes a rotation on the

Poincaré sphere about an axis connecting the two PSPs.

The rate and direction of rotation is given by the dispersion vector Ω(ω, *z*) given by:

where *P _{b}* – represents the negative output principal state.

The strength of the dispersion vector Ω(ω, *z*) is equal to the differential delay time Δτ

between the two output principal states, where its combined Stokes vector

corresponds to the Stokes vector of the negative output principal state. The direction

of the Dispersion Vector Ω defines an axis whose two intercepts with the surface of the Pointcaré sphere correspond to the two principal states of polarization at the fiber

output.

## Poincaré sphere

The Poincaré sphere is a graphical tool that allows convenient description of polarized

signals and polarization transformations during propagation. A point within a unit

sphere can uniquely represent any state of polarization, where circular states of

polarization are located at the poles. The coordinates of a point within or on the

Poincaré sphere are the normalized Stokes parameters.

## Ensemble simulation

Based on the Principal States of Polarization (PSP) model, one can model the whole

fiber as a sum of N trunks where each trunk represents a uniform and constant

birefringent device. A recursive formula describing the evolution of the dispersion

vector for N and N-1 concatenated trunks is the basis for the calculus of the first and

second order of the Differential Group Delay (DGD) induced by the Polarization Mode

Dispersion (PMD).

In the ensemble simulation model, the PMD calculation is repeated in a number of

runs. First, a set of concatenated fiber trunks is generated randomly. Then the

polarization mode dispersion is calculated. In the second run, a second set of trunks

is generated, followed by the PMD calculations. The process is repeated as many

times as it is required. Statistics of the PMD are calculated using the same number of

runs as the ensemble representation.

## Spectral simulation

In the frequency domain, the Polarization Mode Dispersion (PMD) causes the state

of polarization at the output of a fiber to vary with frequency for a fixed input

polarization, which occurs in a cyclic fashion. When displayed on the Poincaré

sphere, the polarization at the output moves on a circle on the surface of the sphere

as the optical frequency is varied.

In the spectral simulation, a set of concatenated fiber trunks is generated randomly.

The calculations of polarization mode dispersion are performed over a range of

wavelengths.

## First order dispersion definition

The Differential Group Delay (DGD) that is calculated based on a stochastic fiber

model (first order PMD) quantifies the first order Polarization Mode Dispersion.

This latter represents the discrete model for Polarization Mode Dispersion (PMD) and

is given by the following recursive formula [Bruyère, 1994[3]]

where

In the above, the symbol W with index N represents the birefringence of the fiber. It

consists of a background linear birefringence Δβ on top of which is added a

perturbation birefringence δβ. On the Poincaré sphere, the Δβ vector is

(Δβ, 0, 0) and the δβ vector is (δβ1, δβ2, δβ3), respectively. The above

recursive formula represents the discrete version of the Poole’s dynamical equation

[Poole et al., 1991[16]].

## Mean value of the first order PMD

For a long fiber span, the average Differential Group Delay (DGD) has a square root

dependence on fiber length z as follows:

## Root Mean Square (RMS) of the first order PMD

The root mean square differential delay time has also a square root of length

dependence:

The probability Density Function (PDF) of the differential delay time is Maxwellian:

## Second order dispersion definition

The second order Polarization Mode Dispersion (second order PMD) is defined by the

following expression [Foschini and Poole, 1991[6]]:

where Ωω(ω, *z*) the first frequency derivative of the dispersion vector Ω:

The dispersion vector is defined in the section: Dispersion vector definition.

## Effective Nonlinear Refractive Index

Usually one of the design goals when constructing a fiber is to minimize its

nonlinearities. The effective nonlinear coefficients of optical fibers depend on the

nonlinear indices of the bulk materials building the fiber and on its waveguiding

properties: shape of modes, degree of confinement, etc. As a result it can vary within

broad limits. OptiFiber calculates the eff. nonlinear coefficient as [Marcuse, 1991[8]]:

where *n _{2}(x, y)* is the user-defined spatially dependent nonlinear refractive index of

the various fiber layers and

*F(x, y)*is the normalized mode field pattern.

## Interference (speckle) patterns (linear superposition/composition of modal fields)

With OptiFiber the user can calculate the superposition/interference pattern of an

arbitrary group of modes. Each of the modes has an arbitrary user defined guided

power and initial phase (a complex weighting coefficient assigned). This feature

targets applications such as coupling efficiency calculations, switching phenomena in

a few mode fiber sensors, etc.

## Coupling efficiencies (linear decomposition of an input beam)

OptiFiber calculates the per-mode coupling efficiencies (the linear decomposition) of

a user defined input beam into an arbitrarily selected group of supported modes.

The decomposition coefficient is calculated using the following well-known

expression:

where *F _{i}(x, y)* is the input field,

*Fm. n(x, y)*is the modal field.

## Scanning

One of the most powerful features of OptiFiber software is the design optimization by

means of parameter scanning.“Scanning” is the name of the parametrical calculations

of some fiber characteristic versus certain technological properties or quantities of the

fiber design: for example the core/cladding refractive index difference, the spatial

width or some region, etc. It predicts automatically how any given fiber could be

optimized versus a design goal, for example small, but non-zero dispersion and

maximal mode area.

## Compatibility with the NR-9200 Optical Fiber Analyzer from EXFO Inc.

Version 1.5 and later versions of OptiFiber are designed and developed to be fully

compatible with NR-9200 in the sense that it directly imports the ASCII files with the

experimental refractive index profile data, recorded by the low level drivers of the

device. It is as a software tool that supplements and extends the fiber characterization

capabilities of the NR-9200.

When the user chooses a NR-9200 Data File with a valid format the experimental data

is processed and the following parameters are extracted and imported in OptiFiber:

- Fiber Data: Fiber name, Fiber type, Cladding index
- Measurement Info: Operator, Date, Measurement wavelength, Comments
- Scanning details: Scan type, Data type, Number of points, Scan step, Data format
- The actual data

The input files may contain just one scan (X) or two scans (X and Y).

After that, the experimental data is processed and a single, axially symmetric profile

is formed from the available raw data.

The details of this process are controlled by the “Refractive index profile

preconditioning dialog” described in the “Dialog Boxes Reference”. It offers two

features:

- The ability to define the step (inversely proportional to the resolution) of the

numerical approximation of the refractive index. This is the minimal refractive index difference between two adjacent experimental points that are to be

considered belonging to different reference index regions in OptiFiber. It is

entered as a percentage of the maximal core-cladding difference. Lower values

mean finer approximation/more layers/longer computations and vice-versa. The

value of the step should be the best trade-off between accuracy and numerical

efficiency. - It offers the user the (optional) choice to reduce the ripples, imperfections and

spurious mechanical or optical noises of the data by means of low-pass Fourier

filtering and/or averaging. The bandwidth of the filter is given in percentage of the

highest frequency of the profile curve as a function of the point number. The

averaging substitutes every data point with the symmetrical mean value of the

respective number of adjacent points. When this is not possible (for example the

first point lacks left-standing neighboring points) the number of averaged points

is reduced accordingly. The recommended values of the filtering and averaging

parameters are 10 –15% and 2-3 respectively, however the overall quality of

smoothing depends also on the value of refractive index step (as described

above), so the user might prefer to fine-tune these three values for optimal results

in his particular case.

When the experimental data is imported, the user can proceed with defining other

parameters of the fiber, altering the profile data manually (for example by merging

some marginal layers of the cladding) and with running the various models currently

supported by OptiFiber.

The process of importing NR-9200 data is described in detail in “Tutorial

Lessons/Lesson 2”. See also the description of the “Menu/Import Profile” command

and of the “Refractive index profile preconditioning” dialog box.