IEC TR 61282-3:2002

Fibre optic communication system design guides - Part 3: Calculation of polarization mode dispersion

This technical report provides guidelines for the calculation of polarisation mode dispersion (PMD) in fibre optic systems. It accomodates the statistical variation of PMD and differential group delay (DGD) of optical fibre cables and optical components in an optical link. Example calculations are given, including concatenating individual optical fibre cables. The calculations describe first order PMD only.

General Information

• Status: Published
• Publication Date: 13-Aug-2002
• ICS: 33.180.01 - Fibre optic systems in general
• Technical Committee: SC 86C - Fibre optic systems, sensing and active devices
• Drafting Committee: WG 1 - TC 86/SC 86C/WG 1
• Current Stage: DELPUB - Deleted Publication
• Start Date: 25-Oct-2006
• Completion Date: 26-Oct-2025

Relations

• Revised: IEC TR 61282-3:2006 - Fibre optic communication system design guides - Part 3: Calculation of link polarization mode dispersion
• Effective Date: 05-Sep-2023

Standards Content (Sample)

RAPPORT CEI
TECHNIQUE IEC
TR 61282-3
TECHNICAL
REPORT
First edition
2002-08
Fibre optic communication system
design guides –
Part 3:
Calculation of polarization mode dispersion
Guides de conception des systèmes
de communication à fibres optiques –
Partie 3:
Calcul de la dispersion en mode de polarisation

Numéro de référence
Reference number
CEI/IEC/TR 61282-3:2002
Publication numbering
As from 1 January 1997 all IEC publications are issued with a designation in the

60000 series. For example, IEC 34-1 is now referred to as IEC 60034-1.

Consolidated editions
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edition numbers 1.0, 1.1 and 1.2 refer, respectively, to the base publication, the

base publication incorporating amendment 1 and the base publication incorporating

amendments 1 and 2.
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RAPPORT CEI
TECHNIQUE IEC
TR 61282-3
TECHNICAL
REPORT
First edition
2002-08
Fibre optic communication system
design guides –
Part 3:
Calculation of polarization mode dispersion
Guides de conception des systèmes
de communication à fibres optiques –
Partie 3:
Calcul de la dispersion en mode de polarisation

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– 2 – TR 61282-3  IEC:2002(E)

CONTENTS
FOREWORD . 3

INTRODUCTION .5

1 Scope and object . 6

2 Basic design models for total system PMD performance . 6

2.1 Notation. 6

2.2 Calculation of system PMD . 7

2.2.1 System maximum PMD . 8

2.2.2 Calculation of system maximum DGD. 8
3 Calculation of cabled fibre PMD.10
3.1 Method 1: Calculating PMD , the PMD link design value.11
Q
3.1.1 Determining the probability distribution of the link PMD coefficients .11
3.1.2 Determining the value of PMD .13
Q
3.2 Method 2: Calculating the probability of exceeding DGDmax.15
3.2.1 Combining link and Maxwell variations.16
3.2.2 Convolution: Theory of method 2 .17
3.3 Equivalence of methods .18
3.3.1 Equivalence of the default specifications .19
3.3.2 Discussion regarding the basis of the default specifications
for method 2 .20
3.3.3 Calculation of the parameters of figure 4 .20
4 Calculation of optical component PMD.20
4.1 Calculation for random components .21
4.2 Calculation for deterministic components .21
4.2.1 Worse case calculation .21
4.2.2 Calculation for embedded deterministic components.22
5 Summary of acronyms and symbols .22
Annex A (informative) PMD concatenation fundamentals.24
A.1 Definitions .24
A.2 Concatenation – General .25
A.3 Application to random elements .25

A.4 Application to deterministic elements .26
Annex B (informative) Combining Maxwell extrema from two populations .28
B.1 Maxwell distribution definitions.28
B.2 Convolution definition .29
B.3 Convolution of optical fibre cable and random components .29
B.4 Evaluation of the double convolution .30
Annex C (informative) Worked example.32
Annex D (informative) Relationship of probability to system performance .33
Annex E (informative) Concatenation experiment.34
Bibliography .36

TR 61282-3  IEC:2002(E) – 3 –

INTERNATIONAL ELECTROTECHNICAL COMMISSION

____________
FIBRE OPTIC COMMUNICATION SYSTEM DESIGN GUIDES –

Part 3: Calculation of polarization mode dispersion

FOREWORD
1) The IEC (International Electrotechnical Commission) is a worldwide organization for standardization comprising
all national electrotechnical committees (IEC National Committees). The object of the IEC is to promote
international co-operation on all questions concerning standardization in the electrical and electronic fields. To
this end and in addition to other activities, the IEC publishes International Standards. Their preparation is
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participate in this preparatory work. International, governmental and non-governmental organizations liaising
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for Standardization (ISO) in accordance with conditions determined by agreement between the two
organizations.
2) The formal decisions or agreements of the IEC on technical matters express, as nearly as possible, an
international consensus of opinion on the relevant subjects since each technical committee has representation
from all interested National Committees.
3) The documents produced have the form of recommendations for international use and are published in the form
of standards, technical specifications, technical reports or guides and they are accepted by the National
Committees in that sense.
4) In order to promote international unification, IEC National Committees undertake to apply IEC International
Standards transparently to the maximum extent possible in their national and regional standards. Any
divergence between the IEC Standard and the corresponding national or regional standard shall be clearly
indicated in the latter.
5) The IEC provides no marking procedure to indicate its approval and cannot be rendered responsible for any
equipment declared to be in conformity with one of its standards.
6) Attention is drawn to the possibility that some of the elements of this technical report may be the subject of
patent rights. The IEC shall not be held responsible for identifying any or all such patent rights.
The main task of IEC technical committees is to prepare International Standards. However, a
technical committee may propose the publication of a technical report when it has collected
data of a different kind from that which is normally published as an International Standard, for
example “state of the art”.
Technical reports do not necessarily have to be reviewed until the data they provide are
considered to be no longer valid or useful by the maintenance team.
IEC 61282-3, which is a technical report, has been prepared by subcommittee 86C: Fibre optic
systems and active devices, of IEC technical committee 86: Fibre optics.

The text of this technical report is based on the following documents:
Enquiry draft Report on voting
86C/296/DTR 86C/346/RVC
Full information on the voting for the approval of this technical report can be found in the report
on voting indicated in the above table.
Annexes A, B, C, D and E are for information only.
This publication has been drafted in accordance with the ISO/IEC Directives, Part 3.
This document, which is purely informative, is not to be regarded as an International Standard.

– 4 – TR 61282-3  IEC:2002(E)

The committee has decided that the contents of this publication will remain unchanged until

2006 At this date, the publication will be

• reconfirmed;
• withdrawn;
• replaced by a revised edition, or

• amended.
A bilingual version of this publication may be issued at a later date.

TR 61282-3  IEC:2002(E) – 5 –

INTRODUCTION
Polarization mode dispersion (PMD) is usually described in terms of a differential group delay

(DGD), which is the time difference between the principal states of polarization of an optical

signal at a particular wavelength and time. PMD in cabled fibres and optical components

causes an optical pulse to spread in the time domain, which may impair the performance of

a fibre optic telecommunication system, as defined in IEC 61281-1.

– 6 – TR 61282-3  IEC:2002(E)

FIBRE OPTIC COMMUNICATION SYSTEM DESIGN GUIDES –

Part 3: Calculation of polarization mode dispersion

1 Scope
The purpose of this technical report is to provide guidelines for the calculation of polarization

mode dispersion (PMD) in fibre optic systems to accommodate the statistical variation of PMD
and differential group delay (DGD) in optical fibre cables and components.
This guideline describes methods for calculating PMD due to optical fibre cables and optical
components in an optical link. Example calculations are given to illustrate the methods for
calculating total optical link PMD from typical cable and optical component data. The
calculations include the statistics of concatenating individual optical fibre cables drawn from
a specified distribution. The calculations assume that all components have PMD equal to the
maximum specified value.
NOTE The statistical specification of the distribution of the PMD of optical fibre cables is a current work item to
amend IEC 60794-3, in SC86A/WG3 [2] . The agreements following the last ballot (86A/501/CD) are aligned with
the methods given in this technical report.
The calculations described cover first order PMD only. This study of PMD continues to evolve,
therefore the material in this technical report may be modified in the future. The following
subject areas are currently beyond the scope of this technical report, but remain under study:
– calculation of second and higher order PMD;
– accommodation of components with polarization dependent loss (PDL) – if it is assumed
that PDL is negligible in optical fibre cables;
– system impairments (power penalty) due to PMD;
– interaction with chromatic dispersion and other nonlinear effects.
Measurement of PMD is beyond the scope of this technical report. Guidelines on the
measurement of PMD of optical fibre and cable are given in IEC 61941. The measurement of
optical amplifier PMD will be documented in IEC 61290-11-1 . The measurement of component
PMD will be documented in IEC 61300-3-32 .
2 Basic design models for total system PMD performance

2.1 Notation
For cabled fibre and components with randomly varying DGD, the PMD frequency domain
measurement is based on averaging the individual DGD values for a range of wavelengths.
The probability density function of DGD values is known to be Maxwell for fibre, and is
assumed to be Maxwell for random components. The single parameter for the Maxwell
distribution scales with the PMD value.
___________
Figures in brackets refer to the bibliography.
To be published
To be published
TR 61282-3  IEC:2002(E) – 7 –

For long fibre and cable (typically longer than 500 m to 1000 m), the PMD value is divided by

the square root of the length to obtain the PMD coefficient. For components, the PMD value is

reported without normalization. The following terms and meanings will be used to distinguish

the various expressions:
– DGD value The differential group delay at a time and wavelength (ps)

– PMD value The wavelength average of DGD values (ps)

– PMD coefficient The length normalized PMD (ps/sqrt(km))

– DGD coefficient The length normalized DGD (ps/sqrt(km))

NOTE The term “DGD coefficient” is used only in some of the calculations. The physical square root length
dependence of the PMD value does not apply to DGD.

Deterministic components are those for which the DGD may vary with wavelength, but not
appreciably with time. The variation in wavelength may be complex, depending on the number
and characteristics of the sub-components within. For these types of components, either the
maximum DGD is reported or the wavelength average is reported as the PMD value.
2.2 Calculation of system PMD
PMD values of randomly varying elements can be added in quadrature. Annex A shows the
basis of this, as well as one basis for concluding that the Maxwell distribution is appropriate to
describe the distribution of DGD values. Annex A describes the concatenation in terms of the
addition of rotated polarization dispersion vectors (pdv) which are, for randomly varying
components, assumed to be random in magnitude and direction over both time and
wavelength.
For deterministic components, the evolution of the pdv with wavelength may be quite complex,
but for each wavelength, there is a value that does not vary appreciably with time. Analysis of
the relationships in annex A shows that if all deterministic components are at the end of the
system and all their pdvs are aligned, the total contribution to the link DGD at a particular
wavelength is equal to the sum of the individual DGD values of each deterministic component.
The worst case contribution across all wavelengths is therefore the sum of maximum DGD
values.
For randomly varying components such as fibre, the statistics of DGD variation imply that there
is little wavelength dependence of the PMD value. This leads to an equivalence between PMD
measurement methods such as Jones Matrix Eigenanalysis (JME) and interferometric methods
(IT) where the wavelength ranges of the two are different. For deterministic elements, there
can be distinct dependence of both the DGD and PMD on the wavelength range. Therefore for
these elements, the wavelength range must be specified. When doing calculations which
combine both randomly varying and deterministic elements, the combined values are only
representative of the wavelength overlap.

The relationships of annex A also show an analysis for a more realistic assumption: the
deterministic components are embedded within the system and randomly aligned. For this
assumption, the DGD values are time randomized across the wavelengths by the downstream
fibre. Furthermore, the random alignment of these components with respect to the other
elements leads to the following conclusions for embedded deterministic components.
– The quadrature addition of PMD values can be used to calculate the contribution to system
PMD.
– The Maxwell distribution can conservatively be used to describe the variation in DGD
across time and wavelength.
– 8 – TR 61282-3  IEC:2002(E)

The following two subclauses provide equations to calculate: a) the maximum PMD value for

the system, b) the maximum DGD value for the system. In both cases, the maximum is defined

in terms of a probability level that takes into account the statistics of the concatenation of

individual cables drawn from a specified distribution of optical fibre cable. For maximum DGD,

these statistics are combined with the Maxwell statistics of DGD variation. Clause 3 provides

methods of calculating the relevant statistics for the contribution of optical fibre cable, which
are used in combination with the component values below.

2.2.1 System maximum PMD
The total maximum PMD value of a fibre optic system including optical fibre cable and other

optical components is given by one of the following, depending on the placement of

deterministic components:
1/ 2
 
2 2
PMD =L PMD + PMD  + PMD (1a)
tot link Q Ci Dj
∑ ∑
 
 i  j
1/ 2
 
2 2 2
 
PMD = L PMD + PMD + PMD + PMD (1b)
tot link Q∑∑Ci Dj Dlast
 
ij
 
where
PMD is the total system PMD value (ps);
tot
PMD is the link design value of the concatenated optical fibre cable (ps/√km);
Q
L is the link length (km);
link
th
PMD is the PMD value of the i randomly varying optical component (ps);
Ci
th
PMD is the PMD value of the j deterministic optical component;
Dj
PMD is the PMD value of the last non-embedded deterministic component.
Dlast
The link design value, PMD , (see 3.1) defines a maximum in terms of the probability, Q, for
Q
links with at least M individual cable sections.
NOTE The PMD parameter is not related to the Q factor used in bit error ratio calculations.
Q
The validity of these equations has been demonstrated empirically for systems composed of
concatenated optical fibre cables [2]. Equation (1a) is relevant assuming that all deterministic
components are at the end of the system. Equation (1b) is relevant assuming that most
deterministic components are embedded.

2.2.2 Calculation of system maximum DGD
The total maximum DGD value of a fibre optic system including optical fibre cable and other
optical components is given by one of the following, depending on the placement of
deterministic components:
1/ 2
 
2 2 2
DGDmax =DGDmax + S PMD  + DGDmax (2a)
tot F ∑ Ci ∑ Dj
 
 i  j
1/ 2
 
 
2 2 2 2 
 
DGD max = DGD max + S PMD + PMD + DGD max (2b)
tot F∑∑Ci Dj Dlast
 
 
 
ij
 
 
TR 61282-3  IEC:2002(E) – 9 –

where
DGDmax is the maximum system DGD (ps);
tot
DGDmax is the maximum concatenated optical fibre cable DGD (ps) (see below);
F
S is the Maxwell adjustment factor (see below);
th
PMD is the PMD value of the i random component (ps);
Ci
th
DGDmax is the maximum DGD of the j deterministic component (ps);
Dj
th
PMD is the PMD value of the j embedded deterministic component (ps);

Dj
DGDmax is the maximum DGD of the last non-embedded deterministic component (ps).
Dlast
The maximum DGD for optical fibre cable (see 3.2) is defined by a probability, P and
F,
reference length. It is computed from the convolution of the distribution of the concatenated
link PMD distribution and the Maxwell distribution of DGD values.
For components, the S parameter relates to the probability, P , that a random component DGD
C
value exceeds S ⋅⋅⋅⋅PMD , assuming the Maxwell distribution. The following table shows the
C
relationship of S to probability when the PMD value is defined as the wavelength average.
Table 1 – Probability based on wavelength average
S Probability
3,0 4,2E-05
3,1 2,0E-05
3,2 9,2E-06
3,3 4,1E-06
3,4 1,8E-06
3,5 7,7E-07
3,6 3,2E-07
3,7 1,3E-07
3,775 6,5E-08
3,8 5,1E-08
3,9 2,0E-08
4,0 7,4E-09
4,1 2,7E-09
4,2 9,6E-10
4,3 3,3E-10
4,4 1,1E-10
4,5 3,7E-11
Annex B shows that the probability that a system DGD value, DGD , exceeds DGDmax is
tot tot
bounded by the sum of the two probabilities as:
P()DGD > DGD max ≤ P + P (3)
tot tot F C
NOTE The notation P( ) indicates a probability statement relative to the inequality within the parenthesis.
The above equations are applicable to all links with length less than the reference length. An
adjustment for longer lengths is included in 3.2. Equation (2a) is relevant for the assumption
that all deterministic components are aligned and at the end of the system. Equation (2b) is
relevant for the assumption that almost all deterministic components are randomly aligned and
embedded in the system. The multiplication of the deterministic PMD values with the

– 10 – TR 61282-3  IEC:2002(E)

S parameter treats these elements as though their DGD values are distributed as Maxwell –

a conservative assumption that allows the quadrature addition. Because the Maxwell

approximation for deterministic elements is conservative, if equation (2a) yields a DGDmax
tot
value less than equation (2b), then equation (2a) value should be used (see annex E and [10]).

NOTE 1 The assumption of quadrature addition of DGD values of cabled fibre and randomly varying optical

components is subject to experimental verification.

NOTE 2 While it is possible to combine the statistical distributions of random components with cabled fibre, it

would require access to information that may not be generally available to any single vendor or customer.

NOTE 3 The DGD specified for deterministic components is assumed to be the maximum across the relevant
wavelength range and environmental conditions

Equation (3) illustrates that the total probability of exceeding some overall maximum can be

bounded by an addition that does not depend on the relative magnitude of DGDmax and
F
S ⋅⋅⋅⋅PMD . Given an overall probability target, one approach is to allocate half the overall allowed
C
probability to fibre and half to components. Annex C provides a worked example for both
equations (2a) and (2b).
3 Calculation of cabled fibre PMD
PMD is a stochastic attribute that varies in magnitude randomly over time and wavelength. The
variation in the DGD value is described by a Maxwell probability density function that can be
characterized by a single parameter, the PMD value (see equation (15) in 3.2.1). This
parameter may be the average of the DGD values measured across a wavelength band, or it
may be the rms value of these DGD values, depending on the definition chosen. For mode
coupled fibre, the PMD coefficient is the PMD value divided by the square root of length.
In accordance with ballot 86A/501/CD, the PMD of cabled fibre should be
specified/characterized on a statistical basis, not on an individual fibre basis. Two methods for
this specification are proposed: method 1 can be used to obtain PMD , used in 2.2.1, and
Q
method 2 can be used to obtain DGDmax and P , used in 2.2.2. The method and specification
F F
values chosen shall be agreed upon between the buyer and the cable manufacturer.
Paragraph 3.3 shows how specification values for each method can be selected so the two
methods are nearly equivalent.
Method 1 relies on the fact that the mean PMD coefficient of an optical link is the root mean
square (quadrature average) of the mean PMD coefficients of the cabled fibres comprising the
link. Method 2 assumes the same relationship.
th
Let x and L be the PMD coefficient (ps/√km) and length, respectively, of a fibre in the i cable
i i
in a concatenated link of N cables. The PMD coefficient, x (ps/√km), of this link is:
N
1/ 2
N
 
 L x 
1/ 2
i i
∑
N
 
 
i=1 2
x = = L x  (4)
N   i i
∑
N
L
 
Link
 
i=1
 
L
i
∑
 
i=1
 
If one assumes that all cable section lengths are less than some common value, L , and
Cab
simultaneously reducing the number of assumed cable sections to M = L /L then, for a link
Link Cab,
comprised of equal-length cables, L = L equation (4) becomes
i cable,
TR 61282-3  IEC:2002(E) – 11 –

1/ 2 1/ 2
M M
   
L 1
Cab 2 2
x ≤ x = x  = x  (5)
N M i i
∑ ∑
L M
   
Link
i=1 i=1
   
The variation in the concatenated link PMD coefficient, x , will be less than the variation in the
M
individual cable sections, x , because of the averaging of the concatenated fibres.
i
Method 1 should be used with equation (1) of 2.2.1. In method 1, the manufacturer supplies a

maximum PMD link design value, PMD , that serves as a statistical upper bound for the PMD
Q
coefficient of the concatenated fibres comprising an optical cable link. For this case, the upper

bound for the PMD value of the concatenation of optical fibre cables, PMD , in equation (1)
FTot
becomes:
PMD = PMD L (6)
FTot Q Link
Unless otherwise specified in the detail specification, the PMD link design value shall be less
than 0,5 ps/√km, and the probability that a PMD coefficient of a link comprised of at least
–4
20 cables will exceed the link design value shall be less than 10 . The link design value shall
be computed using a method agreed upon between the buyer and cable manufacturer (see 3.1
for examples).
Because method 1 provides a statistical upper bound on the PMD of concatenated links,
approved PMD measurement methods can be used on installed cable links to determine
whether their PMD complies with the statistical upper bound stated by the manufacturer.
Furthermore, the upper bound can be used to compute the effect of the link PMD on
the performance of any type of transmission system and is a more realistic indication of the
maximum PMD likely to be encountered in a concatenated link than the value that would be
obtained using a worst-case PMD value.
Method 2 should be used with equations (2) and (3) of 2.2.2. Method 2 combines the PMD
density function of the concatenated links with the Maxwell probability density function of
DGD values to compute an estimate of the probability that the DGD of a concatenated link at
a given wavelength exceeds a specified value for a defined reference link.
The specification is that the probability that the DGD over the link exceeds a given value,
DGDmax , shall be less than some maximum, P . One useful reference system consists of
F F
a concatenated link of 400 km comprised of forty 10 km cable sections. For such a link, the
buyer and cable manufacturer may agree on specifying values such as DGDmax = 25 ps for
F
–8
P ≤ 6,5 ⋅⋅⋅⋅ 10 . The particular statistical methodology for their calculation shall be agreed
F
between the buyer and cable manufacturer (see 3.2).

NOTE Subclause 3.3 shows conditions under which the specifications of the two methods are nearly equivalent.
3.1 Method 1: Calculating PMD , the PMD link design value
Q
3.1.1 Determining the probability distribution of the link PMD coefficients
Equation (5) shows that the PMD coefficient, x , of a particular concatenated link can be
M
derived from the PMD coefficients of the individual cable sections, x , comprising that link. The
i
probability distribution of the link PMD coefficients depends on the distribution of the cable
PMD coefficients and the number of cable sections comprising the link.

– 12 – TR 61282-3  IEC:2002(E)

The following paragraphs describe three methods that can be used to estimate the distribution

of the link PMD coefficients. One method is numerical [1] and two are analytic [4]. Of the two

analytic methods, the first assumes a specific analytic function for the distribution of the cable

PMD coefficients, while the second method makes no such assumptions but invokes an

extension of the central limit theorem.

3.1.1.1 Monte Carlo numeric method [1]

The Monte Carlo method can be used to determine the probability density, f , of the
link
concatenated link PMD coefficients without making any assumption about its functional form.
This method simulates the process of building links by sampling the measured cable population

repeatedly. PMD coefficients are measured on a sufficiently large number of cabled fibres so

as to characterize the underlying distribution. This data is then used to compute the PMD
coefficient for a single fibre-path in a concatenated link.
Computation of the link PMD coefficient is made by randomly selecting M values from the
measured cabled PMD coefficients, and adding them on an rms basis (in quadrature)
according to equation (5). The computed link PMD coefficient is placed in a table or a
histogram of values derived from other random samplings. The process is repeated until a
sufficient number of link PMD values has been computed to produce a high density
(0,001 ps/√km) histogram of the concatenated link PMD coefficient distribution. If used directly,
without any additional characterization, the number of resamples should be at least 100 000.
Because of the central limit theorem, the histogram of link PMD coefficients will tend to
converge to distributions that can be described with a minimum of two parameters. Hence, the
histogram can be fit to a parametric distribution that enables extrapolation to probability levels
that are smaller than what would be implied by the sample size. The two parameters will
invariably represent two aspects of the distributions: the central value and the variability about
the central value. A choice of probability distributions can be made on the basis of the shape of
the histogram. Typical distributions could include lognormal (the log of the link PMD
coefficients is Gaussian) or one that is derived from the Gamma distribution.
3.1.1.2 Gamma distribution analytic method [4]
The Gamma family of distributions can often be used to represent the distributions of both the
measured cable PMD coefficients and the link PMD coefficients. If one assumes that the
square of the measured cable PMD coefficients, x , is distributed as a Gamma random variable,
i
the probability density of the cabled PMD coefficients is given by
α 2α −1
2β x
f ()x;α, β = exp()− βx (7)
cable
Γ()α
where x is a possible value of the cable PMD coefficient, Γ( ) is the Gamma function, and the
two parameters α and β control the shape of the density. Standard fitting techniques, such as
the method of maximum likelihood, can be used to fit equation (7) to measured cable PMD
data to find values for α and β.
The probability density of the link PMD coefficients, x , of M concatenated equal cable lengths
M
has the same form as equation (7), but with α and β replaced by Mα and Mβ:

TR 61282-3  IEC:2002(E) – 13 –

Mα 2Mα −1
2()Mβ x
f()x;α, β, M = exp()− Mβx (8)
Link
Γ()Mα
Consequently, the α and β parameters found by fitting equation (7) to the measured cable

PMD coefficients can be used in equation (8) to describe the probability density of the link PMD

coefficients.
3.1.1.3 Model-independent analytic method [4]

A more general alternative to the one described in 3.1.1.2 can be used that does not make any
assumptions regarding the form of the density function that describes the measured PMD

coefficients of the cabled fibre.
After measuring the PMD coefficients, x , on N cabled fibres, compute the mean, variance and
i
third moment of their squares
N N N
2 3
1 1 1
2 2 2
μ = x μ = ()x − μ  μ = ()x − μ (9)
2 i 1 3 i 1
∑ i ∑ ∑
N N −1 N −1
i =1 i =1
i=1
Let x be a random variable representing the link PMD coefficient of a fibre-path formed from
M
the concatenation of M equal-length cables, and let u be a possible value of x . Invoking the
M
extended central limit theorem [5], it can be shown that the distribution of the link PMD
coefficients is approximated by:
t
f ()u; M = Φ[]z(u) + φ[]z()u[]1− z()u (10)
Link
1/ 2
M
where
z
 
1 z
 
φ()z = exp −  Φ()z = φ(y)dy (10a)
∫
 
2π
 
−∞
and
1/ 2
 M  μ
2 3
 
z()u = ()u − μ t = (10b)
 
3 / 2
μ
 2 6μ
Differentiating equation (10) with respect to u provides an approximation to the link PMD
coefficient probability density function.
3.1.2 Determining the value of PMD
Q
The density functions found for the link PMD coefficients using one of the three methods
described in 3.1.1 are now be used to compute the PMD link design value. For a concatenated
link comprised of M cables, the PMD link design value, PMD , is defined as the value that the
Q
link PMD coefficient, x , exceeds with probability Q:
M
P(x > PMD ) = Q (11a)
M Q
It follows, that for N > M, the probability that x exceeds PMD is less than Q:
N Q
P(x > PMD ) < Q (11b)
N Q
– 14 – TR 61282-3  IEC:2002(E)

For discussion purposes, an assumption is made that N ≥ 20 (the link contains at least
–4
20 cables) and that Q = 10 (the probability that the link PMD exceeds the PMD design value
is less than 0,0001). However, the actual values for M and Q shall be agreed upon between the

buyer and seller. The following subclauses discuss how PMD can be found using the cable
Q
PMD density functions obtained in 3.1.1.

3.1.2.1 Determining the PMD link design value from

the Monte Carlo density of 3.1.1.1

–4
To obtain probability levels of Q = 10 using a pure numeric approach requires Monte Carlo

simulations of at least 10 samples. Once this is complete, PMD can be interpolated from the
Q
associated cumulative probability density function.
Alternatively, the histogram of the link PMD coefficients can be fit with a parametric distribution
to enable extrapolation to lower probability levels than the measurement resampling would
otherwise allow. A choice of probability distributions can be made on the basis of the shape of
the histogram. Typical distributions could include lognormal (the log of the link PMD
coefficients is Gaussian) or one that is derived from the Gamma distribution. After the function
th
is fit, the value for PMD at the Q quantile can be computed.
Q
3.1.2.2 Determining the PMD link design value from the Gamma density of 3.1.1.2
–4
An excellent approximation for the link PMD coefficient, x , for M cables at the 10 quantile is
Q
given by:
2,004 + 0,975 Mα
PMD = (12)
Q
Mβ
where the α and β parameters were those found in 3.1.1.2.
For the 288 randomly selected scaled cabled fibres reported in [6], α = 0,979 and β = 48,6, and
PMD = 0,20 ps/√km.
Q
3.1.2.3 Determining the PMD link design value using
the model-independent method of 3.1.1.3
The moments computed in 3.1.1.3 can be used to compute the link PMD coefficient, x . For a
Q
th
link comprised of M cables, x at the Q quantile can be approximated by [5]:
Q
1/ 2
1/ 2
 
μ μ
 
2 3 2
 
PMD = μ + z   + ()z −1 (13)
Q 1 Q Q
 
M 6Mμ
 
 
 
th
where z is the Q quantile of the standard normal distribution. For N > M = 20 cables and
Q
–4
Q=10 , z = 3,72, the PMD design value becomes:
Q
1/ 2
 μ 
PMD = μ + 0,832 μ + 0,107 (14)
Q  1 2 
μ
 2
TR 61282-3  IEC:2002(E) – 15 –

For the 288 randomly selected scaled cabled fibres reported in [6],

−2 −4 −5
μ = 2,2 ⋅ 10   μ = 7,43 ⋅ 10  μ = 8,26 ⋅ 10
1 2 3
and equation (14) produces PMD = 0,23 ps/√km.
Q
3.2 Method 2: Calculating the probability of exceeding DGD
max
PMD induced impairment of an optical signal occurs when the DGD at the signal’s wavelength

is too high. Since DGD varies randomly with time and wavelength, some means of imposing an
upper limit, defined in terms of a low probability value, is necessary for system design. This
upper limit is usually associated with a receiver sensitivity penalty. The probability can be
associated with a potential PMD-induced impairment time (min/year/circuit). See annex D.
One means of calculating an upper limit on DGD is to multiply the upper limit on the PMD value
by a Maxwell adjustment factor, i.e. 3 (see table 1 of 2.2.2). This could also be done with the
upper limit represented by PMD . When this is done, one is in effect, assuming that the bulk of
Q
the distribution is very close to the upper limit. In reality, the bulk of the distribution is usually
well away from the upper limit. Method 2 is intended to provide metrics and methods to take
this into account. Because method 2 takes into account the statistics of the individual optical
fibre cables and their concatenation, as well as the combined statistics of DGD variation, the
values calculated for system design (equation (2b)) are substantially reduced from the “worst-
case” values – both in the value of maximum DGD and the probability of exceeding it.
The following figure shows several distributions of concatenated link PMD coefficient. Each
–4
distribution just passes the default criteria of M = 20, Q = 10 , and PMD ≤ 0,5 ps/√km.
Q
0,07
0,06
0,05
0,04
0,03
0,02
0,01
0 0,1 0,2 0,3 0,4 0,5 0,6
Concatenated link PMD coefficient
IEC  1896/02
NOTE The above distributions are representative of the Gamma type distribution defined in 3.1.1.2.
Figure 1 – Various passing distributions
The leftmost distributions should provide better DGD performance than the rightmost
distribution. Method 2 assigns value to producing a distribution that is more to the left.
3.3 provides a means to link method 1 and method 2 so that, for most practical situations,
passing the default method 1 criterion will imply passing a default method 2 criterion.
Relative frequecy
– 16 – TR 61282-3  IEC:2002(E)

3.2.1 Combining link and Maxwell variations

DGD coefficient (ps/√km) values, X , vary randomly with time and wavelength according to the
M
Maxwell probability density function:

3 / 2
2  
 
X  X 
4 4
M M
 
 
f()X ; x = 2 exp −   (15)
Max M M
 
 
Γ()3 / 2  π x 
πx
 M 
 M 
 
where x is the PMD coefficient of a concatenated link comprised of M cables as given by
M
equations (4) or (5). The distribution of DGD values (over the length) is obtained by multiplying

the DGD coefficient values with the square root of the link length.
To combine the variations in the concatenated link PMD coefficient with the Maxwell variation
into a value to be used in a system design, a reference link is defined. Performance on the
reference link can then be generalized to other links. The reference link is defined with two
parameters, the overall link length, L , and the cable section length, L , which is assumed
REF Cab
to be constant for all cable sections.
Let f (xi) be the discretized probability density function (histogram) of the values of the
Link
concatenated link PMD coefficient values defined by the analysis of the distribution of
the measured PMD coefficient values and equation (4). Any of the methods of 3.1.1 for
determining the probability density function of the concatenated link can be used.
Let X be some DGD coefficient value (ps/√km) that is to be used in system design. The
max
probability, P , of exceeding X is:
F max
 
4 X 
M
 
 
 
π x
X
   M 
max 3 / 2−1
 
y
 
P = f ()x 1− f (Y; x)dY = f ()x 1− exp()− y dy (16)
F ∑ Link i Max i ∑ Link i
∫ ∫
  Γ()3 / 2
 
i i
0 0
 
 
 
 
NOTE The rightmost integral is just the standard gamma function.
The maximum DGD over the link derived from optical fibre cable, DGDmax , is the product of
F
the square root of the reference link length and X :
Max
DGD max = X L (17)
F Max REF
For method 1, the probability is pre-set and the associated PMD value is calculated and
Q
required to be less than a specified value. For method 2, DGDmax is pre-set and the
F
probability value, P is calculated and required to be less than a specified value.
F
For link lengths less than the reference length, the DGD and probability relationship will be
conservative as long as either the installed lengths are less than L or the cable lengths
Cab
measured to obtain the distribution are less than L The reduction in averaging because of
Cab.
the reduced number of cable lengths is offset by the decrease in overall length.

TR 61282-3  IEC:2002(E) – 17 –

For link lengths greater than the reference length, the maximum DGD to optical fibre cable

should be adjusted as:
L
Link
DGDadj = DGD max (18)
F F
L
REF
3.2.2 Convolution: Theory of method 2

The calculation principle is derived from extending the worst case approach. With this

approach, the link PMD distribution is assumed to be a dirac function and the DGD distribution
is represented as a Maxwell distribution. The probability that the Maxwell distribution exceeds
DGDmax yields P . These distributions are represented in figure 2.
F F
0,07
0,06
Link PMD
Link DGD
0,05
0,04
0,03
0,02
0,01
0 1020 3040
DGD/PMD link value
IEC  1897/02
Figure 2 – Worst case approach assumption
NOTE Though not shown, the dirac function illustrated in figure 2 extends to a relative frequency value of 1,0.
Suppose the link PMD distribution could be represented by two dirac functions, each with a
magnitude of 0,5. This would represent a situation where half the links were at one value and
the other half at another value. The DGD probability density function of the combined
distribution would be the weighted total of the two individual Maxwell distributions. Figure 3
illustrates this case.
Relative frequency
– 18 – TR 61282-3  IEC:2002(E)

0,07
0,06
Link PMD A
0,05
Link DGD A
Link PMD B
0,04
Link DGD B
0,03
Link DGD tot
0,02
0,01
010 20 30 40
DGD/PMD value (ps)
IEC  1898/02
Figure 3 – Convolution of two diracs
NOTE Though not sh
...