IEC TR 61282-9:2016

Fibre optic communication system design guides - Part 9: Guidance on polarization mode dispersion measurements and theory

IEC TR 61282-9:2016(E) which is a Technical Report, describes effects and theory of polarization mode dispersion (PMD) and provides guidance on PMD measurements. This second edition cancels and replaces the first edition published in 2006. This second edition includes the following significant technical changes with respect to the previous edition:
- much of the theory has been condensed - focusing only on content that is needed to explain the test method;
- symbols have been removed, but abbreviations are retained;
- the material in the Clause 5 has been significantly reduced in an effort to avoid repeating what is already in the actual International Standards. Instead, the focus is on explaining the International Standards;
- measurement methods that are not found in International Standards have been removed;
- there are significant corrections to the modulation phase shift method, particularly in regard to the Mueller set technique;
- there are significant corrections to the polarization phase shift method;
- the proof of the GINTY interferometric method is presented. This proof also extends to the Fixed Analyser Cosine transfer technique;
- another Fixed Analyser method is suggested. This is based on the proof of the GINTY method and is called "spectral differentiation method";
- Clause 6 has been renamed "Limitations" and refocused on the limitations of the test methods. This Technical Report is not intended to be an engineering manual;
- the annexes have been removed;
- the bibliography has been much reduced in size;
- the introduction has been expanded to include some information on system impairments. Keywords: polarization mode dispersion (PMD)

General Information

• Status: Published
• Publication Date: 23-Mar-2016
• 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: PPUB - Publication issued
• Start Date: 24-Mar-2016
• Completion Date: 31-May-2016

Relations

• Revises: IEC TR 61282-9:2006 - Fibre optic communication system design guides - Part 9: Guidance on polarization mode dispersion measurements and theory
• Effective Date: 05-Sep-2023

Standards Content (Sample)

IEC TR 61282-9 ®
Edition 2.0 2016-03
TECHNICAL
REPORT
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inside
Fibre optic communication system design guides –
Part 9: Guidance on polarization mode dispersion measurements and theory
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IEC TR 61282-9 ®
Edition 2.0 2016-03
TECHNICAL
REPORT
colour
inside
Fibre optic communication system design guides –

Part 9: Guidance on polarization mode dispersion measurements and theory

INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
ICS 33.180.01 ISBN 978-2-8322-3236-1

– 2 – IEC TR 61282-9:2016 © IEC 2016
CONTENTS
FOREWORD . 4
INTRODUCTION . 6
1 Scope . 7
2 Normative references. 7
3 Terms, definitions, and abbreviations . 7
3.1 Terms and definitions . 7
3.2 Abbreviations . 8
4 Theoretical framework . 8
4.1 Limitations and outline . 8
4.2 Optical field and state of polarization . 8
4.3 SOP measurements, Stokes vectors, and Poincaré sphere rotations . 11
4.4 First order polarization mode dispersion . 14
4.5 Birefringence vector, concatenations, and mode coupling . 16
4.6 The statistics of PMD and second order PMD . 17
4.7 Managing time . 20
5 Measurement methods . 20
5.1 General . 20
5.2 Stokes parameter evaluation . 22
5.2.1 Equipment setup and procedure . 22
5.2.2 Jones matrix eigenanalysis . 23
5.2.3 Poincaré sphere analysis . 24
5.2.4 One ended measurements based on SPE [3] . 26
5.3 Phase shift based measurement methods . 27
5.3.1 General . 27
5.3.2 Modulation phase shift – Full search . 28
5.3.3 Modulation phase shift method – Mueller set analysis [4] . 29
5.3.4 Polarization phase shift measurement method[5] . 31
5.4 Interferometric measurement methods . 33
5.4.1 General . 33
5.4.2 Generalized interferometric method [6] . 35
5.4.3 Traditional interferometric measurement method . 40
5.5 Fixed analyser . 41
5.5.1 General . 41
5.5.2 Extrema counting . 42
5.5.3 Fourier transform . 43
5.5.4 Cosine Fourier transform . 44
5.5.5 Spectral differentiation . 45
5.6 Wavelength scanning OTDR and SOP analysis (WSOSA) method [7] . 46
5.6.1 General . 46
5.6.2 Continuous model . 48
5.6.3 Large difference model . 49
5.6.4 Scrambling factor derivation . 50
6 Limitations . 53
6.1 General . 53
6.2 Amplified spontaneous emission and degree of polarization . 53
6.3 Polarization dependent loss (or gain) . 53

6.4 Coherence effects and multiple path interference . 54
6.5 Test lead fibres . 54
6.6 Aerial cables testing . 55
Bibliography . 56

Figure 1 – Two electric field vector polarizations of the HE mode in a SMF . 10
Figure 2 – A rotation on the Poincaré sphere . 13
Figure 3 – Strong mode coupling – Frequency evolution of the SOP . 16
Figure 4 – Random DGD variation vs. wavelength . 18
Figure 5 – Histogram of DGD values from Figure 4 . 18
Figure 6 – SPE equipment diagram . 22
Figure 7 – Relationship of orthogonal output SOPs to the PDV . 24
Figure 8 – Stokes vector rotation with frequency change . 25
Figure 9 – Setup for modulation phase shift . 27
Figure 10 – Setup for polarization phase shift . 28
Figure 11 – Output SOP relation to the PSP . 30
Figure 12 – Interferometric measurement setup . 33
Figure 13 – Interferogram relationships . 35
Figure 14 – Mean square envelopes . 38
Figure 15 – Fixed analyser setup . 41
Figure 16 – Fixed analyser ratio . 42
Figure 17 – Power spectrum . 44
Figure 18 – Fourier transform . 44
Figure 19 – WSOSA setup . 46
Figure 20 – Frequency grid . 47

Table 1 – Map of test methods and International Standards . 22
Table 2 – Mueller SOPs . 29

– 4 – IEC TR 61282-9:2016 © IEC 2016
INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
FIBRE OPTIC COMMUNICATION SYSTEM DESIGN GUIDES –

Part 9: Guidance on polarization mode dispersion
measurements and theory
FOREWORD
1) The International Electrotechnical Commission (IEC) is a worldwide organization for standardization comprising
all national electrotechnical committees (IEC National Committees). The object of IEC is to promote
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8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is
indispensable for the correct application of this publication.
9) Attention is drawn to the possibility that some of the elements of this IEC Publication may be the subject of
patent rights. 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".
IEC TR 61282-9, which is a Technical Report, has been prepared by subcommittee 86C: Fibre
optic systems and active devices, of IEC technical committee 86: Fibre optics.
This second edition cancels and replaces the first edition published in 2006.
This second edition includes the following significant technical changes with respect to the
previous edition:
a) much of the theory has been condensed – focusing only on content that is needed to
explain the test method;
b) symbols have been removed, but abbreviations are retained;

c) the material in the Clause 5 has been significantly reduced in an effort to avoid repeating
what is already in the actual International Standards. Instead, the focus is on explaining
the International Standards;
d) measurement methods that are not found in International Standards have been removed;
e) there are significant corrections to the modulation phase shift method, particularly in
regard to the Mueller set technique;
f) there are significant corrections to the polarization phase shift method;
g) the proof of the GINTY interferometric method is presented. This proof also extends to the
Fixed Analyser Cosine transfer technique;
h) another Fixed Analyser method is suggested. This is based on the proof of the GINTY
method and is called "spectral differentiation method";
i) Clause 6 has been renamed "Limitations" and refocused on the limitations of the test
methods. This Technical Report is not intended to be an engineering manual;
j) the annexes have been removed;
k) the bibliography has been much reduced in size;
l) the introduction has been expanded to include some information on system impairments.
The text of this Technical Report is based on the following documents:
Enquiry draft Report on voting
86C/1342/DTR 86C/1366/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.
This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.
A list of all parts in the IEC 61282 series, published under the general title Fibre optic
communication system design guides, can be found on the IEC website.
The committee has decided that the contents of this publication will remain unchanged until
the stability date indicated on the IEC website under "http://webstore.iec.ch" in the data
related to the specific publication. 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.

IMPORTANT – The 'colour inside' logo on the cover page of this publication indicates
that it contains colours which are considered to be useful for the correct
understanding of its contents. Users should therefore print this document using a
colour printer.
– 6 – IEC TR 61282-9:2016 © IEC 2016
INTRODUCTION
This Technical Report is complementary to the International Standards describing PMD
procedures (IEC 60793-1-48, IEC 61280-4-4, IEC 61290-11-1, IEC 61290-11-2 and
IEC 61300-3-32) and other design guides on PMD (IEC 61282-3 and IEC 61292-5), as well as
ITU-T Recommendation G.650.2.
The system power penalty associated with PMD varies depending on transmission format and
bit rate. It also varies with optical frequency and state of polarization (SOP) of the light
source. At the output of a link, the signal can shift from a maximum delay to a minimum delay
as a result of using different SOPs at the source. The difference in these delays is called the
differential group delay (DGD), which is associated with two extremes of input SOP. At these
extremes, a signal in the form of a single pulse appears shifted up or down by half the DGD,
about a midpoint, at the output. At intermediate SOPs, the single pulse appears as a weighted
total of two pulses at the output, one shifted up by half the DGD and one shifted down by half
the DGD. This weighted total of two shifted pulses is what causes signal distortion.
The system power penalty is partly defined in terms of a maximum allowed bit error rate and a
minimum received power. In the absence of distortion, there is a minimum received power
that will produce the maximum allowed bit error rate. In the presence of distortion, the
received power should be increased to produce the maximum bit error rate. The magnitude of
the required increase of received power is the power penalty of the distortion.
The term PMD is used to describe two distinctly different ideas.
One idea is associated with the signal distortion induced by transmission media for which the
output SOP varies with optical frequency. This is the fundamental source of signal distortion.
The other idea is that of a number (value) associated with the measurement of a single-mode
fibre transmission link or element of that link. There are several measurement methods with
different strengths and capabilities. They are all based on quantifying the magnitude of
possible variation in output SOP with optical frequency. The objective of this Technical Report
is to explain the commonality of the different methods.
The DGD at the source’s optical frequency is what controls the maximum penalty across all
possible SOPs. However, in most links, the DGD varies randomly across optical frequency
and time. The PMD value associated with measurements, and which is specified, is a
statistical metric that describes the DGD distribution. There are two main metrics, linear
average and root-mean square (RMS), that exist in the literature and in the measurement
methods. For most situations, one metric can be calculated from the other using a conversion
formula. The reason for the dual metrics is an accident of history. If history could be
corrected, the RMS definition would be the most suitable.
For the non-return to zero transmission format, DGD equal to 0,3 of the bit period yields
approximately 1 dB maximum penalty. Because DGD varies randomly, a rule of thumb
emerged in the system standardization groups: keep PMD less than 0,1 of the bit period for
less than 1 dB penalty. This assumes that DGD larger than three times the PMD, and that the
source output SOP produces the worst case distortion, is not very likely. For 10 Gbit/s non-
return to zero, this rule yields a design rule: keep the link PMD less than 10 ps.
ITU-T G.sup.39 [1] has more information on the relationship of PMD and system penalties.

___________
Numbers in square brackets refer to the Bibliography.

FIBRE OPTIC COMMUNICATION SYSTEM DESIGN GUIDES –

Part 9: Guidance on polarization mode dispersion
measurements and theory
1 Scope
This part of IEC 61282, which is a Technical Report, describes effects and theory of
polarization mode dispersion (PMD) and provides guidance on PMD measurements.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and
are indispensable for its application. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any
amendments) applies.
IEC 60793-1-48, Optical fibres – Part 1-48: Measurement methods and test procedures –
Polarization mode dispersion
IEC 61280-4-4, Fibre optic communication subsystem test procedures – Part 4-4: Cable
plants and links – Polarization mode dispersion measurement for installed links
IEC 61290-11-1, Optical amplifier – Test methods – Part 11-1: Polarization mode dispersion
parameter – Jones matrix eigenanalysis (JME)
IEC 61290-11-2, Optical amplifier – Test methods – Part 11-1: Polarization mode dispersion
parameter – Poincaré sphere analysis method
IEC 61300-3-32, Fibre optic interconnecting devices and passive components – Basic tests
and measurement procedures – Part 3-32: Examinations and measurements – Polarization
mode dispersion measurement for passive optical components
3 Terms, definitions, and abbreviations
3.1 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
3.1.1
PMD phenomenon
polarization mode dispersion phenomenon
signal of fibre-optic transmission signal induced by variation in the signal output state of
polarization with optical frequency
Note 1 to entry: PMD can limit the bit rate-length product of digital systems.
3.1.2
PMD value
polarization mode dispersion value
magnitude of polarization mode dispersion phenomenon associated with a single-mode fibre,
optical component and sub-system, or installed link

– 8 – IEC TR 61282-9:2016 © IEC 2016
Note 1 to entry: The polarization mode dispersion value is usually expressed in ps.
3.2 Abbreviations
CFT cosine Fourier transform
DGD differential group delay
DOP degree of polarization
EC extrema counting
FA fixed analyser
FT Fourier transform
GINTY general interferometric method
JME Jones matrix eigenanalysis
MPS modulation phase shift
OTDR optical time domain reflectometer
PDL polarization dependent loss
PDV polarization dispersion vector
PMD polarization mode dispersion
PPS polarization phase shift
PSA Poincaré sphere analysis
PSP principal state of polarization
SMF single-mode fibre
SOP state of polarization
SPE Stokes parameter evaluation
TINTY traditional interferometric method
WSOSA wavelength scanning OTDR and SOP analysis method
4 Theoretical framework
4.1 Limitations and outline
The theory presented in Clause 4 does not include the effects of polarization dependent loss
or gain, or nonlinear effects. See 6.3 for information on polarization dependent loss.
The outline for Clause 4 is
• optical field and state of polarization;
• measurement of SOP, Stokes vector, and rotation;
• first order polarization mode dispersion;
• birefringence vector, concatenations, and mode coupling;
• the statistics of PMD and second order PMD.
4.2 Optical field and state of polarization
This subclause is intended to show the linkage between the propagation of an optical field in
a single-mode fibre (SMF) and the transmission signal state of polarization (SOP). This
information is fundamental to the PMD phenomena because variation in output SOP with
optical frequency is the distortion inducing mechanism.
The solution of the wave equation has degenerated eigenvalues. This means that even the
fundamental solution is degenerated. A SMF supports a pair of polarization modes for a
monochromatic light source. In particular, the lowest order mode, namely the fundamental

mode HE (LP ) can be defined to have its transverse electric field predominately along the
11 01
x-direction; the orthogonal polarization is an independent mode, as shown in Figure 1.
In a lossless SMF, the electric field vector of a monochromatic electromagnetic wave
propagating along the z-direction can be described by a linear superposition of these two
modes in the x-y transverse plane as shown in Equation (1) and in Figure 1.
{ ( )}
E = [ j exp(iβ z)]+ [j exp β z ] exp(− iϖt)
x x y y
= [j exp(− ∆βz / 2) + j exp(i∆βz / 2)]exp[− i(ϖt − βz)]
x y
(1)
where
j and j are complex coefficients describing the amplitude and phase of the x/y initial
x y
SOPs;
E (x,y) and
x
E (x,y) are the spatial variation (in the x-y transverse plane) of the E vector of the
y
PM along the x/y-direction (see Figure 1);
β and β are the propagation constants (also called effective index or wavenumber)
x y
of the PM along the x/y-directions with the index of refraction n /n . Using
x y
i = x or y, β = k⋅n . The index of refraction has a dependence on frequency
i i
ϖ, frequency ν, or wavelength λ;
∆β is the difference of β and β ;
x y
β
is the average of βx and βy;
k is the propagation constant with the wavelength λ in vacuum
(= 2πν/c = 2π λ = ω c);
/ /
–1
ν is the frequency in s or Hz;
ϖ is the angular frequency in rad s (the bar indicates absolute frequency
/
rather than deviation from some particular value);
c is the speed of light in vacuum (299792458 m/s);
∆’ is the birefringence coefficient (s/m), = ∆n/c
z is the distance (m) in the DUT along the optical axis (axis of propagation);
z = L at the output of the DUT with length L.

– 10 – IEC TR 61282-9:2016 © IEC 2016
Axis of propagation, z
Fibre
core
E (x,y);β
x x
E (x,y);β
y y
Axis of
polarization, x
Axis of polarization, y
Cladding
IEC
Figure 1 – Two electric field vector polarizations of the HE mode in a SMF
The complex pair, [j exp(− i∆βz / 2), j exp(i∆βz / 2)], describes the SOP defined in the x-y plane
x y
of the wave propagating along the z-direction. This pair can be considered as a vector, and is
often called a Jones vector.
In the case where the transmission media is an ideal SMF with perfect circular symmetry,
β = β ,
y x
• the two polarization modes are degenerate (when two solutions have the same
eigenvalue, they are said to be degenerate);
• any wave with a defined input SOP will propagate unchanged along the z-direction
throughout the output of the SMF.
However, in a practical SMF, the circular symmetry is broken by imperfections produced by
the fabrication process, cabling, field installation/use or the installation environment:
• β ≠ β , implying a phase difference, an index-of-refraction difference ∆n, and a phase-
y x
velocity difference between the two PMs;
• the degeneracy of the two polarization modes is lifted;
• the SOP of an input wave will change along the z-direction throughout the output of the
SMF.
The difference between β and β , namely ∆β, is called the phase birefringence or simply the
y x
−1
birefringence and has units of inverse length (m ). Birefringence may also be referred to as
the index difference, ∆n, or as the birefringence coefficient, the difference divided by length.
Birefringence coefficient values typically vary between 0,25 fs/m and 2,5 fs/m in commonly
available SMFs.
As the SOPs travel through the fibre, they will return to the initial state at positions that are
increments of 2π/∆β. At these positions, the two field components will beat. The difference
between these positions is called the beat length.
Birefringence can be induced by a number of factors such as core non-circularity or
asymmetric stresses that can be induced by bends, twist, and compression. These factors
change over the length of the fibre and can change over time due to changes in configuration,
or temperature. These factors will also vary with optical frequency. Equation (1) is also
defined with an arbitrary coordinate system that will not generally correspond to the laboratory
or field test equipment coordinate system. The mapping of the input SOP to the output SOP

over a particular length and optical frequency, L and ϖ , is represented with the Jones matrix,
 
T, and the input and output Jones vectors, j and j , as:
IN 0
 
j = Tj (2)
0 IN
where
↑
T = V S V (3)
T T T
exp(− ix / 2) 0 
T
S =
(4)
T  
0 exp(ix / 2)
 T 
cosθ exp(− iµ / 2) − sinθ exp(− iµ / 2)
T T T T
V = = S(iµ / 2)R(θ ) (5)
T   T T
sinθ exp(iµ / 2) cosθ exp(iµ / 2)
 T T T T 
NOTE 1 The subscript T is used to distinguish the matrix parameters from similar parameters used later.
The main operation of Equation (3), corresponding to Equation (1), is found in the matrix, S .
T
↑ ↑
Pre and post-multiplying by V and V is a change of coordinates. The notation V indicates
T T
the transpose conjugate for matrices and vectors.
The diagonal expressions in Equation (4) are intended to show the connection to Equation (1).
Equation (1), however, is only applicable locally, while Equation (3) is used to indicate change
that is
over the entire transmission media. There is another expression that uses γ = x / 2
T T
found in ITU-T Recommendation G.650.2. On the Poincaré sphere, the action of Equation (4)
is a rotation of γ from an input SOP to an output SOP. When equations are found in both
T
documents, the γ notation will be used.
T
Unit Jones vectors, which are one way to represent SOPs, can be specified with a (θ,µ) pair
as:

cosθ exp(− iµ / 2)
j = (6)
 
sinθ exp(iµ / 2)
 
↑
The Jones matrix, T, is unitary in that T T = I , the identity matrix. The columns of V are the
T
eigenvectors, and the diagonal elements of S are the eigenvalues. When the input Jones
T
vector is equal to either of the eigenvectors, the output SOP is the same as the input SOP
because the SOP is not affected by a multiplication by a constant. These states are
sometimes called the eigenstates.
NOTE 2 All the parameters of T can change with optical frequency, as well as with changes caused by fibre
movement over time or temperature change.
4.3 SOP measurements, Stokes vectors, and Poincaré sphere rotations
The SOP is usually measured with a polarimeter, which yields a Stokes vector. The
measurement is actually done with a series of power measurement differences through
various states of a polarizer/analyser.
An ideal polarizer may be defined as:

– 12 – IEC TR 61282-9:2016 © IEC 2016
 
cos θ sin2θ exp(− iµ )
P P P
 
1 0
↑
Pol(θ ,µ ) = V V =
(7)
 
P P Pol   Pol
0 0
 
 
sin2θ exp(iµ ) sin θ
P P P
 
 
where V is of the same form as Equation (5), but with different parameters.
Pol
For a given Jones vector, the power through the polarizer is:
 
↑ ↑
P(θ ,µ ) = j Pol (θ ,µ )Pol(θ ,µ )j (8)
P P P P P P
The measured Stokes vector is given as:
 P(0,0)+ P(π / 2,0) 
 

P(0,0)− P(π / 2,0)
 
S = indexed with j = 0,1,2,3 (9)
 
P(π / 4,0)− P(− π / 4,0)
 
P(π / 4,π / 2)− P(π / 4,−π / 2)
 

The normalized Stokes vector is designated with a small s and has three elements indexed:
1, 2, and 3. The normalized Stokes vector elements are the measured Stokes vector element
values with the same index, divided by S . The normalized Stokes vector has a length of one.
SOPs where s is zero are linear states. When s is nonzero but with absolute value less than
3 3
one, the polarization is elliptical. When s is ±1, the polarization is circular.
NOTE 1 In the rest of this Technical Report, the normalized Stokes vector will be referred to simply as the Stokes
vector.
The degree of polarization (DOP) is normally expressed as a per cent and is equal to
100 ⋅ S / P , where P is the power without the polarizer.
A unit Jones vector can be represented either as an x/y pair or as Equation (6). The
relationship of the normalized Stokes vector to the Jones vector is given as:
* *
 
xx − yy  cos2θ 
  
 
* *
s = xy + yx = sin2θ cos µ (10)
 
 
 * * 
 
i(xy − yx ) sin2θ sin µ
 
 
 
There is an ambiguity in trying to calculate the Jones vector from the Stokes vector. One must
assume something like 0<θ<π. This is due to the fact that the Stokes vector is not affected by
*
multiplying the Jones vector by any unit complex number (a number, c, for which cc = 1),
including ±1. This can be called a one π ambiguity. This property is one reason to think of the
Stokes vector as the primary definition of the SOP: the SOP is not changed when either of the
eigenstates are used as inputs, but the output Jones vector is multiplied by exp(±iξ /2).
T
Unit three term vectors can be represented on a sphere. In the case of Stokes vectors, the
sphere is called the Poincaré sphere.
Examination of the rightmost expression of Equation (10) and the different parts of Equations
(3), (4) and (5) shows that the action of T is consistent with the following right-hand-rule
rotations applied to the input Stokes vector that corresponds to the input Jones vector:

a) anti-rotation of µ about the (1,0,0) vector, which describes a linear SOP;
T
b) anti-rotation of 2θ about the (0,0,1) vector, which describes a circular SOP;
T
c) rotation of ξ (= 2γ ) about the (1,0,0) vector;
T T
d) rotation of 2θ about the (0,0,1) vector;
T
e) rotation of µ about the (1,0,0) vector.
T
These steps can be combined into a single rotation, from input Stokes vector to output Stokes
vector, as:
 
s = R s (11)
0 T IN
where
 
T
R = yy (1− cosx )+ I cosx + [y ×]sinx (12)
T T T T



T
where y is the transpose of y , I is the identity matrix, and [y ×] is the cross product operator,
 0 − y y 
3 2
 
y 0 − y .
3 1
 
 
− y y 0
2 1
 

This is a rotation of ξ about the rotation vector, y . The rotation vector is found by converting
T
the first column of V to a Stokes vector. Figure 2 illustrates a rotation on the Poincaré
T
sphere. This is a 2π rotation about the (1,0,0) axis.
s
(d)
(c),(g)
(a),(e)
s s
1 2
(b),(f)
IEC
Figure 2 – A rotation on the Poincaré sphere
The T matrix can now be written in a simplified form:
cosγ − iy sinγ − (y + iy )sinγ 
T 1 T 3 2 T
T =
 
(y − iy )sinγ cosγ + iy sinγ
 3 2 T T 1 T 
– 14 – IEC TR 61282-9:2016 © IEC 2016
y y − iy
 
1 2 3
= cosγ I − i sinγ (13)
T T  
y + iy − y
2 3 1
 
NOTE 2 The form of this equation re-emerges in Equation (20), in which it is pointed out that the rightmost matrix
is the weighted sum of Pauli matrices.
4.4 First order polarization mode dispersion
First order PMD is induced by the variance in the output Jones vector with optical frequency.
This is the same as the variance of the output Stokes vector with frequency, which is one way
to measure it, but the signal distortion must be understood in terms of the Jones calculus. To
distinguish the variance of SOP with optical frequency from the input-to-output Jones matrix
of the prior clauses, the frequency transfer matrix is designated with J(ω) as
  
exp(− i∆τω / 2) 0 
↑
j (ω) = J (ω)j = V V j (14)
OUT 0 J   J 0
0 exp(i∆τω / 2)
 
where
V is the same form as Equation (5);
J
ω is deviation from the particular frequency, ϖ ;

j is the output SOP at that frequency;

j (ω) is the output Jones vector at the frequency, ϖ +ω;
OUT
∆τ is the differential group delay (DGD).
The exponent of the ratio of eigenvalues of J yields ∆τ∆ω for a finite frequency increment.
The action of J(ω) is a rotation on the Poincaré sphere from the output Stokes vector at ϖ to
the Stokes vector at ϖ + ω. The rotation angle is ∆τω, and the rotation vector is formed by
transforming the first column of V into a Stokes vector. This rotation vector is called the
J

principal state of polarization (PSP) and is later designated as a vector, .
p
The distortion can be understood by considering the Fourier transform of the signal field, H(ω)
and its inverse transform, h(t). Time shifting of the inverse transform is given as transform pair
as:
exp(it ω)H (ω) ⇔ h(t − t ) (15)
0 0
2π
The output Jones vector in the time domain is therefore given as:
 
h(t − ∆τ / 2) 0 
↑
j (t) = V V j (16)
OUT J   J 0
2π 0 h(t + ∆τ / 2)
 
The output power is the conjugate transpose of the output Jones vector times itself. When V
J

is the identity matrix and the elements of j are both equal to 2 , the output pulse is the
sum of squares of two field pulses separated by ∆τ. This represents a worst case for distortion
within the first order formulation.

When is equal to either column of V , there is no distortion, but the signal is shifted in the
j
0 J
time domain by ±∆τ/2. At intermediate polarization states, the output pulse is the total of two
time shifted pulses of different magnitudes. In the worst case, the magnitudes are equal.

In the Stokes formulation, the action of Equation (14) is a rotation, R , with rotation angle,
J

∆τω, about rotation vector, . The rotation vector is found by converting the first column of V
p
J

to a Stokes vector. This rotation vector is also called the fast PSP, because when s is equal
to this vector, there is, to the first order, no change in the output Stokes vector with
frequency. Since it is aligned with –i∆τω/2, it is the early arriving, or fast, part of the signal.
For completeness, R is of the form of Equation (12), but with different parameters, and the
J
relationship of the output Stokes vector as a function of frequency is:
 
s (ω) = R (ω)s
(17)
OUT J 0
Two important objects in the PMD literature are the polarization dispersion vector (PDV),

designated as Ω and the frequency transfer function differential operator, a matrix
designated as D. These are defined as the following, considering that all parameters of both
RT and T are functions of frequency:


   
ds (ω) dR (ω)
T
OUT T
(18)
= R (0) s = [Ω ×]s = [(∆τp)×]s
0 0 T 0 0 0
dω dω

When the output Stokes vector at ϖ , designated as s , is aligned with the PSP or PDV, there
0 0
is no change in the output Stokes vector with respect to frequency, to the first order.

   
dj (ω) dT (ω) dJ (ω) − i∆τ / 2 0 
−1 ↑
Out
= Dj = T (0)j = j = V V j
0 0 0 0 0 0 J   J 0
dω dω dω 0 i∆τ / 2
 

p p − ip
∆τ  
1 2 3
= −i j (19)
  0
p + ip − p
 2 3 1 
The matrix expression in the last term is the weighted sum of the three Pauli operators:
1 0  0 1 0 − i
σ = σ = σ = (20)
1 2 3
     
0 −1 1 0 i 0
     
The eigenvalues of D are ±i∆τ/2.
Expansion of the expressions containing the parameters of T shows the following:
• the equations are consistent;
• a differential equation that links the parameters and derivative of T to the PDV emerges.
 

 
 
dγ dy dy
T
 
Ω = 2 y + sin2γ + 2sin γ y × (21)
T T
 
dϖ dϖ dϖ
 
Given the PDV as a function of optical frequency and suitable boundary conditions, the
parameters of T, hence T can be solved using numerical techniques. This was important in
simulating the interferometric measurement method.

– 16 – IEC TR 61282-9:2016 © IEC 2016
First order PMD means that the exponent of the diagonal matrix in Equation (14) is linear in
frequency. In reality, there are limitations in the first order projections because the PSP and
DGD actually vary with frequency in most (long) transmission media.
4.5 Birefringence vector, concatenations, and mode coupling
4.4 indicated that the important parameters, the PSP and DGD, also vary with optical
frequency. 4.5 will give some background as to why this is so.
The birefringence vector is defined by replacing the frequency (ω) derivatives with length (z)
derivatives in Equations (18) and (21). Also, replace the symbol for the PDV with the symbol
for the birefringence vector. The evolution of the output Stokes vector through a length of
transmission media can be calculated from the modified Equation (18) and a set of
assumptions about the birefringence vector. These assumptions can include factors such as
twist, which induces circular birefringence, bend, lateral load, and spin.
A simple birefringent element can be defined from Equations (3) through (5) with the
following:
Set ξ = ϖz∆’ and µ = 0 (22)
T T
where ∆’ is the birefringence coefficient (s/m).
A concatenation of a number of randomly rotated simple birefringent elements will induce
mode coupling that varies with optical frequency. As a result, depending on the number and
length of such elements, the output SOPs no longer follow simple rotations with frequency,
and the PDV becomes random. Figure 3 shows an example.
IEC
Figure 3 – Strong mode coupling – Frequency evolution of the SOP
Negligible mode coupling would display a less complicated evolution, such as what is shown
in Figure 2. However, there is no firm boundary between the different regimes.
The concatenation of randomly rotated simple elements is not a complete model for optical
fibre. The PMD behaviour of optical fibre and cable is in the realm of strong random mode
coupling. Most optical fibre is manufactured using a spinning technique to deliberately
introduce strong mode coupling.

Consequences of mode coupling include:
• the connect
...