IEC TR 61282-9:2006

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

This technical report applies to all commercially available fibre optic products sensitive to polarization mode dispersion (PMD). This report is complementary to the International Standards describing the 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).

General Information

• Status: Published
• Publication Date: 10-Jul-2006
• 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: 24-Mar-2016
• Completion Date: 26-Oct-2025

Relations

• Revised: IEC TR 61282-9:2016 - Fibre optic communication system design guides - Part 9: Guidance on polarization mode dispersion measurements and theory
• Effective Date: 05-Sep-2023

Standards Content (Sample)

TECHNICAL IEC
REPORT TR 61282-9
First edition
2006-07
Fibre optic communication system
design guides –
Part 9:
Guidance on polarization mode
dispersion measurements and theory
Reference number
IEC/TR 61282-9:2006(E)
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TECHNICAL IEC
REPORT TR 61282-9
First edition
2006-07
Fibre optic communication system
design guides –
Part 9:
Guidance on polarization mode
dispersion measurements and theory

 IEC 2006  Copyright - all rights reserved
No part of this publication may be reproduced or utilized in any form or by any means, electronic or
mechanical, including photocopying and microfilm, without permission in writing from the publisher.
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International Electrotechnical Commission
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For price, see current catalogue

– 2 – TR 61282-9  IEC:2006(E)
CONTENTS
FOREWORD.5
1 Scope.7
2 Normative references .7
3 Acronyms and abbreviations.8
4 General information.9
4.1 Polarization modes.9
4.2 Birefringence.11
4.3 Beat length.12
4.4 Polarization transfer function.12
4.5 Stokes parameters and the Poincaré sphere .13
4.6 Principal states of polarization.15
4.7 Differential group delay .15
4.8 Polarization mode dispersion.15
4.9 Polarization dispersion and birefringence vectors .16
4.10 Polarization mode coupling.18
4.11 Second-order polarization mode dispersion .23
5 Mathematical formulations of the polarization mode dispersion test methods.26
5.1 Stokes parameter evaluation .26
5.2 Modulation phase shift .39
5.3 Polarization phase shift .42
5.4 Fixed analyser.45
5.5 Interferometric method .59
5.6 Poincaré sphere arc method.84
5.7 Poole formula method .87
5.8 Single-end test methods.88
6 Measurement issues.95
6.1 Degree of polarization and amplified spontaneous emission.95
6.2 Suppression of amplified spontaneous emission using optical or electrical filtering .97
6.3 The use of a broadband source .98
6.4 The Nyquist theorem and optical measurements .99
6.5 Continuously swept tuneable laser source and sampling theory.106
6.6 Tuneable laser source and noise.107
6.7 The selection of the states of polarization.107
6.8 Coherence interference effects and multiple path interference.107
6.9 Fibre pigtails .108
6.10 Power measurement resolution and linearity .108
6.11 Calibration of test instruments .108

Annex A (informative) Summary of various PMD test methods found in IEC standards.109
Annex B (Informative) Summary of key definitions .111
Annex C (Informative) Calculation of polarization mode dispersion value .114
Annex D (informative) Generalised Parseval theorem .117
Annex E (informative) Open issues .119

Bibliography.125

TR 61282-9  IEC:2006(E) – 3 –
Figure 1 – Two electric field vector polarizations of the HE mode in an optical fibre
along the a) x-direction and b) y-direction .10
Figure 2 – Cartesian and elliptical representation of a state of polarization .12
Figure 3 – Poincaré sphere representation of states of polarization .14
Figure 4 – Effect of polarization mode dispersion on transmission of an information-bit
pulse in a device.16
Figure 5 – Polarization dispersion vector and principal states of polarization .17
Figure 6 – No or negligible mode coupling .18
Figure 7 – Random mode coupling.19
Figure 8 – Statistics of differential group delay and related Maxwell distribution [15].22
Figure 9 – Polarization mode dispersion and differential group delay in negligible
mode coupling .23
Figure 10 – Effects of first-order polarization mode dispersion (PMD ) and second-
order polarization mode dispersion (PMD ) on the output state of polarization on the
Poincaré sphere.24
Figure 11 – Rectangular system of co-ordinates defined by the response Stokes
vectors, and direction angles of the polarization dispersion vector .29
Figure 12 – Arc of a circle described by the output state of polarization in the
ω, ω+Δω] .30
increment [
Figure 13 – Functional diagram of Stokes parameter evaluation .36
Figure 14 – a) Differential group delay (DGD) as a function of the optical frequency (f)
obtained through Poincaré sphere analysis (PSA) and Jones matrix analysis (JME),
and b) Difference of DGDs.38
Figure 15 – Trajectories of the principal states of polarization on the Poincaré sphere

from a) Jones matrix eigenanalysis (JME) and b) Poincaré sphere analysis (PSA).39
Figure 16 – Mueller states on Poincaré sphere .40
Figure 17 – Example of random mode coupling result with fixed analyser using Fourier
transform technique [15] .49
Figure 18 – Polarization mode dispersion by Fourier analysis .54
Figure 19 – Mean cross-correlation and autocorrelation functions.58
Figure 20 – Generic set-up for the measurement of polarization mode dispersion using
the interferometric test method .59
Figure 21 – Schematic diagram for GINTY analysis using input/output state-of-
polarization scrambling .65
Figure 22 – Comparison between single-scan and scrambling uncertainties .69
a) With a polarization maintaining fibre and one I/O SOP).70
b) With I/O-SOP scrambling (L/h << 1, DGD = 0,732 ps, σ = 50 fs, DGD/σ ~ 14,7) .70
A A
Figure 23 – Example of negligible-mode-coupling result using a) TINTY analysis and
b) GINTY analysis.70
Figure 24 – Example of random-mode-coupling result using TINTY analysis.71
Figure 25 – Example of random-mode-coupling result using GINTY analysis with I/O-
SOP scrambling.75
Figure 26 – Equivalence between a) Stokes parameter evaluation method PSA
analysis and b) GINTY analysis .76
Figure 27 – Example of mixed-mode-coupling result using GINTY analysis.79
Figure 28 – Comparison between polarization mode dispersion results from TINTY
and GINTY analyses .81
Figure 29 – Relationship between beat length and state of polarization .85

– 4 – TR 61282-9  IEC:2006(E)
Figure 30 – Relationship between Stokes parameter and state of polarization on the
Poincaré sphere.85
Figure 31 – Relationship between fixed analyser method with circular analyser (―)
and Poincaré sphere arc method (---) .86
Figure 32 – Relationship of state of polarization (SOP) analysis to normalised Stokes
s parameter.87
a) For wide SOP-to-birefringence axis angle .93
b) For a small SOP-to-birefringence axis angle .93
Figure 33 – Backscattered state of polarization (SOP) for short and long pulses versus
distance.93
Figure 34 – Degree of polarization (DOP) vs. distance for a concatenation of three
500-m fibres, with a centre fibre that exhibits a high h value .94
Figure 35 – Power spectrum of a typical optical fibre amplifier output showing the
amplified signal, the amplified spontaneous emission (ASE), and the optical signal-to-
noise ratio (OSNR) .95
Figure 36 – Power ratio of amplified signal to total amplified spontaneous emission
(ASE) versus the optical signal-to-noise ratio (OSNR) in 0,1-nm resolution bandwidth
(RBW).96
Figure 37 – Time varying signal .99
Figure 38 – Frequency spectrum of Figure 37 signal (its Fourier transform) a) in linear;
b) in log scales .100
Figure 39 – Filtered signal with Nyquist-based defined bandwidth.101
Under-sampled signal (factor 2) .101
Figure 40 – Unfiltered signal with under-sampled (factor 2) defined bandwidth a .101
Figure 41 – Figure 37 signal with noise.102
Figure 42 – Frequency spectrum of Figure 41 signal.102
Figure 43 – Filtered noisy signal respecting the Nyquist theorem .103
Figure 44 – Un-filtered noisy signal spectrum not respecting the Nyquist theorem .103
Figure 45 – Signal frequency spectrum (of a filter as an example) .104
Figure 46 – Impulse response from Figure 45 signal .104
a) Signal frequency spectrum reconstructed following the Nyquist theorem.105
b) Under-sampled signal frequency spectrum .105
c) Under-sampled noisy signal spectrum.105
d) Reconstructed frequency spectrum of signal shown in (c) .105
Figure 47 – Reconstructed signal based on Figure 45 a) following the Nyquist
theorem; b) not following the Nyquist theorem; c) including noise; d) based on c) .105

Table 1 – Example of Mueller set.41
Table 2 – Cosine Fourier transform calculations .57
Table 3 – Values of g for various types or degrees of coupling.80
S
Table 4 – Values of 45 ° power, Stokes parameters and SOP corresponding to Figure 29 .85
Table A.1 – Applicability matrix for various polarization-mode-dispersion test methods.109
Table D.1 – Definition of the Fourier transform pairs .117

TR 61282-9  IEC:2006(E) – 5 –
INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
FIBRE OPTIC COMMUNICATION SYSTEM DESIGN GUIDES –

Part 9: Guidance on polarization mode dispersion
measurements and theory
FOREWORD
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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 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.
The text of this technical report is based on the following documents:
Enquiry draft Report on voting
86C/696/DTR 86C/703/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.

– 6 – TR 61282-9  IEC:2006(E)
This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.
A list of all parts of 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 maintenance result date indicated on the IEC web site 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.

TR 61282-9  IEC:2006(E) – 7 –
FIBRE OPTIC COMMUNICATION SYSTEM DESIGN GUIDES –

Part 9: Guidance on polarization mode dispersion
measurements and theory
1 Scope
This technical report applies to all commercially available fibre optic products sensitive to
polarization mode dispersion (PMD).
This report presents general information about PMD, the mathematical formulation related to
the application of the generally accepted methods to test PMD, and some considerations
related to the sampling theory regarding the use of different light sources and detection
systems.
This report is complementary to the International Standards describing the 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).
2 Normative references
The following referenced documents are indispensable for the application of this document.
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 fibre amplifier test methods – Part 11-1: Polarization mode
dispersion – Jones matrix eigenanalysis method (JME)
IEC 61290-11-2: Optical amplifiers – Test methods – Part 11-2: Polarization mode dispersion
parameter – Poincaré sphere analysis method
IEC 61300-3-2: Fibre optic interconnecting devices and passive components – Basic test and
measurement procedures – Part 3-2: Examinations and measurements – Polarization
dependence of attenuation in a single-mode fibre optic device
IEC 61300-3-32: Fibre optic interconnecting devices and passive components – Basic test
and measurement procedures – Part 3-32: Examinations and measurements – Polarization
mode dispersion for passive optical components
IEC/TR 61282-3: Fibre optic communication system design guides – Part 3: Calculation of
polarization mode dispersion
IEC/TR 61292-5: Optical amplifiers – Part 5: Polarization mode dispersion parameter –
General information
—————————
To be published
– 8 – TR 61282-9  IEC:2006(E)
3 Acronyms and abbreviations
ASE  amplified spontaneous emission
AWG  array waveguide grating
BBS broadband source
BER  bit error rate or bit error ratio
Cl h  circ ul ar left handed
Crh circular right handed
CSO  composite second-order beat noise
DAS  differential attenuation slope
DGD  differential group delay
DOP  degree of polarization
DUT  device under test
DVV  polarization dispersion vector velocity
DWDM dense wavelength division multiplexing
EC  extrema counting
EDFA erbium doped fibre amplifier
FA  fixed analyser
FAFT fixed analyser Fourier transform
FAEC  fixed analyser extrema counting
FCFT fast cosine Fourier transform
ffs  for further study
FT  Fourier transform
FWHM  full width at half the maximum
GINTY general analysis for the interferometric method
HMD  harmonic distortion
IMD  intermodulation distortion
INTY  interferometry
I/O  input/output
ISI  inter-symbol interference
JME  Jones matrix eigenanalysis
L  linear
LH  linear horizontal
LV  linear vertical
MMA  Mueller matrix analysis
MPS  modulation phase shift
OA  optical amplifier
OFA optical fibre amplifier
OSA  optical spectrum analyser
OTDR optical time domain reflectometer
P-  parallel polarization (to the plane of incidence)
PDCD  polarization dependent chromatic dispersion
PDD polarization dependent delay
PDG  polarization dependent gain

TR 61282-9  IEC:2006(E) – 9 –
PDL  polarization dependent loss
PDV  polarization dispersion vector
PM  polarization mode
PMD  polarization mode dispersion
PMD first-order polarization mode dispersion
PMD second-order polarization mode dispersion
PMF  polarization maintaining fibre
POTDR polarization optical time domain reflectometer
POWA  planar optical waveguide amplifier
PPS polarization phase shift
PS  Poincaré sphere
PSA  Poincaré sphere analysis
PSP  principal states of polarization
RBW  resolution bandwidth
RMS  root mean square
RTM reference test method
S  source
SMSR  side mode suppression ratio
SOP  state of polarization
SPE  Stokes parameter evaluation
SRM standard reference material
SSE  source spontaneous emission
TINTY traditional analysis for the interferometric method
TLS  tuneable laser source
TSSE  total source spontaneous emission
WDM  wavelength division multiplexing
4 General information
The following text provides general information concerning the PMD theory and phenomenon.
In that context, the word “device” or “device under test” (DUT) is used throughout the text in
the sense of an optical path with an input and an output interface, such as an optical fibre, an
optical fibre cable, an optical component, an optical amplifier, etc. The device may be
connectorised.
4.1 Polarization modes
The solution of the wave equation has degenerated eigenvalues. This means that even the
fundamental solution is degenerated. A single-mode fibre can therefore support several
modes, the polarization modes (PM), and by analogy can be considered as a multi-
(polarization) mode fibre. In particular, the lowest order mode, namely the fundamental HE
(LP ) mode, can be chosen to have its transverse electric field predominately along the
x-direction; the orthogonal polarization is an independent mode, as shown in Figure 1.

– 10 – TR 61282-9  IEC:2006(E)
IEC  1246/06
Figure 1 – Two electric field vector polarizations of the HE mode
in an optical fibre along the a) x-direction and b) y-direction
In a lossless optical fibre, the electric field vector of a monochromatic electromagnetic wave
propagating along the z-direction can be described by a linear superposition of these two PM
in the x-y transverse plane as shown in Equation (1) and in Figure 1 [1,2] .
−iωt
{}[][]
E = A (z) ⋅ E (x,y) + A (z))⋅ E (x,y) ⋅ e (1)
x x y y
where
iβ z
x
A (z) = E e , is the complex coefficient describing the amplitude E and the phase β of
x x x x
the PM along the x-direction propagating in the z-direction;
iβ z
y
A (z) = E e , is the complex coefficient describing the amplitude E and the phase β of
y
y y y
the PM along the y-direction propagating in the z-direction;
E (x,y) is the spatial variation (in the x-y transverse plane) of the electric field vector of the
x
PM along the x-direction (see Figure 1(a);
E (x,y) is the spatial variation (in the x-y transverse plane) of the electric field vector of the
y
PM along the y-direction (see Figure 1(b);
β = kn , is the propagation constant (also called effective index or wavenumber) of the
x x
PM along the x-direction with the index of refraction n ; β , through n has a
x x x
dependence on optical angular frequency ω, or optical frequency ν, or wavelength λ;
β = kn , is the propagation constant (also called effective index or wavenumber) of the
y y
PM along the y-direction with the index of refraction n ; β , through n has a
y y y
dependence on angular optical frequency ω, or optical frequency ν, or wavelength λ;
k = 2πν = 2π/λ = ω/c, is the propagation constant with the wavelength λ in vacuum;
–1
ν is the optical frequency in s or Hz;
ω is the angular optical frequency in rad/s;
c is the speed of light in vacuum;
z is the distance in the device along the optical axis; z = L at the output of the device
with length L.
The complex ratio A (z)/A (z) describes the state of polarization (SOP) defined in the x-y plane
x y
of the wave propagating along the z-direction.
—————————
Figures in square brackets refer to the Bibliography.

TR 61282-9  IEC:2006(E) – 11 –
In an ideal optical fibre with perfect circular symmetry,
• β = β ;
y x
• the two PM are degenerate;
• consequently, any wave with a defined input SOP will propagate unchanged along the z-
direction throughout the output of the fibre.
However, in a practical optical fibre, imperfections produced by the fabrication process,
cabling, field installation/use or the environment break the circular symmetry,
• β ≠ β , implying a phase difference, and index of refraction difference Δn and a phase
y x
velocity difference Δv between the two PM;
• the degeneracy of the two PM is lifted;
• consequently, the SOP of an input wave will change along the z-direction throughout the
output of the fibre.
The difference between β and β , namely Δβ, is called the phase birefringence or simply the
y x
birefringence and has units of inverse length. Birefringence may also be referred to as the
–7 –5
index difference, Δn. Index differences typically vary between 10 and 10 in commonly
available single-mode fibres [2].
4.2 Birefringence
Birefringence is produced by an anisotropic distribution of the index of refraction in the
propagating region of an optical device medium. As such, any device is susceptible to
birefringence. For instance, birefringence is produced by asymmetry in the optical fibre core,
meaning when the circular symmetry of the fibre core is broken [1,3].
It is well known that the asymmetry provides an index of refraction that is smaller along an
axis, and as such that smaller index provides a faster phase velocity along that axis compared
to the other one. That axis is consequently called the fast axis as opposed to the other one,
which is called the slow axis, corresponding to a larger index and a slower phase velocity.
The slower wave is also said to be retarded compared to the other one.
The asymmetry can result from geometrical deformation of the medium or material anisotropy
through various elasto-optic, magneto-optic or electro-optic changes of the index of refraction.
Geometrical deformation and asymmetric lateral stress via elasto-optic index changes may be
produced during fabrication, for instance, and will typically produce linear birefringence.
Bending, kinks and electro-optic Kerr effect will also typically produce linear birefringence.
Twist and magneto-optic effect via Faraday effect will produce circular birefringence.
Several of these mechanisms may coexist and may be present in various numbers, strengths
and distributions; and may vary with time and over environmental conditions. This makes the
SOP along the propagating medium and at the output unpredictable and unstable and,
consequently, PMD compensation difficult to achieve.
Birefringence can, however, be imposed, such as is the case for polarization maintaining fibre
(PMF). This type of fibre has a strong birefringence maintained along the fibre length which
can be created, for example, by introducing stress in one plane. This will keep the two PM
non-degenerate. Any SOP launched aligned with the axis of one of these two PM will be
maintained throughout the fibre. As opposed to PMF, commonly available single-mode fibres
used for fibre optic transmissions in the field are weakly birefringent fibres.

– 12 – TR 61282-9  IEC:2006(E)
4.3 Beat length
Birefringence makes the two PM slip in phase relative to one another as they propagate at
different phase velocities. When the phase difference between the two PM is equal to an
integer number of 2π, the two PM beat with one another, and at that periodic point along the
z-direction, the input SOP is reproduced. The length corresponding to that periodicity is called
the beat length L .
b
2π
L = in units of length (2)
b
Δβ
Beat lengths typically vary between 2 m and15 m in commonly available single-mode fibres
[2].
Beat length should not be confused with coupling length, which will be explained later.
4.4 Polarization transfer function
Polarized light and its related SOP can be represented using Jones calculus [4] as a complex
vector illustrating the x- and y- components of the electric field of the SOP at a point in space
(see Figure 2).
IEC  1247/06
Figure 2 – Cartesian and elliptical representation of a state of polarization
NOTE In the presence of anisotropy, the y axis is the slow axis in Figure 2, while the x axis is the fast axis, and
the wave propagating in the y-z plane is retarded. The resulting wave from the vector combination of the y-z plane
wave and the x-z plane wave is shown in the x-y plane in the lower right corner of Figure 2
The Jones vector has the form (not considering any unit vector):
iδ
 x
A E e
 
x
x  
 
E = = (3)
 iδ 
 
y
A
y  
  E e
y
 
Equation (3) can easily be related to Equation (1) in case of anisotropy when δ becomes β. As
polarized light traverses any DUT such as an optical fibre, its SOP undergoes a
transformation. This transformation is described in Jones calculus by the complex 2x2 Jones
matrix, T, so that the output Jones vector E is related to the input Jones vector E through
out in
E = T E . The Jones matrix is determined by measuring the output Jones vectors in
out in
response to any three unique input vectors. The calculation is simplest when the stimuli are
o o
linear horizontal (LH) 0 (E = 0; δ = 0), linear vertical (LV) 90 (E = 0, δ = 0) and linear (L)
y x x y
o
+45 (E = E ; δ = δ = 0) SOPs. If the responses to the three stimuli are:
y x y x
TR 61282-9  IEC:2006(E) – 13 –
 X  X  X 
1 2 3
   
, , (4)
   
Y Y Y
 1 2  3 
the Jones matrix is
K K K 
1 4 2
T = C * (5)
 
K 1
 4 
K − K
X X X
3 2
1 2
K = ; K = ; K = ; K = (6)
1 2 3 4
Y Y Y K − K
1 2 3 1 3
where C is a complex constant.
The magnitude of C can be calculated from intensities measured with the DUT removed from
the optical path. Most measurements do not require the calculation of C.
4.5 Stokes parameters and the Poincaré sphere
While Jones calculus is very useful in explaining SOPs, it is problematic in practice to
measure the electric field of a light wave. An alternative expression for the SOPs is the
Stokes vector, as shown in Equation (7):
S 
 
S
 
S =
(7)
 
S
 
S
 3
where
2 2
S = E + E is the total power including the polarized and unpolarized parts;
x y
NOTE The degree of polarization (DOP) is equal to the ratio of the polarized part of the power to S .
2 2
S = E − E is the power difference between the LH and LV SOPs;
x y
*
o o
S = 2Re(E E ) is the power difference between the L +45 and the L –45 SOPs;
x y
*
S = 2Im(E E ) is the power difference between the circular right-handed (Crh) and circular
x y
left-handed (Clh) SOPs.
The magnitude of the Stokes vector is the square root of the sum of the squares of S , S and
1 2
S .
The Poincaré sphere uses the normalised S parameters s (s = S /S , s = S /S , s = S /S ) as
1 1 0 2 2 0 3 3 0
a three-real component unit vector co-ordinate system to represent any possible SOP [5,6].
o o o o
The LH 0 SOP is represented as (1,0,0), LV 90 as (–1,0,0), L +45 as (0,1,0), L –45 as
(0,-1,0), Crh as (0,0,1), and Clh as (0,0,-1).

– 14 – TR 61282-9  IEC:2006(E)
2 2
2 2
 
 
E + E
S  E + E
x y
x y
 
 
  2 2
iδx
2 2
 
S  E − E 
E ∗ e  − 
x y E E
x 1
  x y
 
= ⇒ = =
E (8)
 
iδy  
∗
   
S
E ∗ e Δ
2Re(E E ) 2E E cos δ
y 2
 x y  x y
   
 
 
∗
S  
2E E sin Δδ
 3 2Im(E E )
x y
x y  
 
2 2 2 2 2 2 2 2
   
(E − E ) (E + E ) (E − E ) (E + E )
s  S S 
x y x y x y x y
1 1 0
   
   
∗ 2 2 2 2
s = S S =2Re(E E ) (E + E ) =2E E cos Δδ (E + E ) (9)
2 2 0 x y x y x y x y
   
   
∗ 2 2 2 2
   
s S S
2Im(E E ) (E + E ) 2E E sin Δδ (E + E )
3 3 0
   
 x y x y   x y x y 
   
In polar coordinates,
−iµ/ 2
 
()cosθ e
iφ
E = e (10)
 
iµ/ 2
()sinθ e
 
 
s  cos2θ 
   
s =()sin2θ(cosµ) (11)
   
   
s ()sin2θ(sinµ)
   
where Δδ = δ – δ , is the difference in phase between the x (LH) and y (LV) SOPs.
x y
The use of the Stokes vector parameters is best illustrated via the use of the Poincaré sphere
(see Figure 3). Stokes vectors can be converted to Jones vectors using the respective E-field
formulae.
IEC  1248/06
Figure 3 – Poincaré sphere representation of states of polarization
NOTE The LH axis usually faces the observer/receiver

TR 61282-9  IEC:2006(E) – 15 –
2 2 2
For a fully polarized field (DOP = 100 %), s + s + s is unitary, and a SOP is represented
1 2 3
by a dot on the surface of the sphere (the tip of the SOP vector). The observer looks at the LH
SOP. Any SOP laying on the equator of the sphere is linear. The north pole of the sphere is
Crh SOP, and the south pole is Clh SOP. Any other SOP anywhere else on the sphere is
elliptical.
4.6 Principal states of polarization
In a device that exhibits PMD, there are two SOPs called principal states of polarization
(PSPs) [7,8] having the property that the SOP measured at the output of the device is
invariant with optical frequency (or wavelength) to first order (over a small optical frequency
or wavelength increment) for any output SOP aligned with these PSPs. Of the two PSPs, the
one called the fast PSP gives the fastest group velocity and consequently the shortest
propagation delay; the other, called the slow PSP, gives the slowest group velocity and
consequently the longest propagation delay. These two PSPs will be orthogonal in the
absence of polarization dependent loss (PDL) or polarization dependent gain (PDG) and non-
linear effects. In a device with polarization-mode coupling such as a single-mode optical fibre
used to transport fibre optic telecommunications, the PSPs are, in general, different from the
eigenstates of the device; in a non-mode-coupled device such as a PMF, the PSPs are
identical to the eigenstates of the device. The eigenstates are the eigenvectors of the Jones
matrix, i.e. the two SOPs which propagate through the device without modification.
NOTE 1 The two PSPs are an intrinsic function of the material birefringence and induced external and internal
stresses acting on the device.
NOTE 2 An output signal whose SOP is aligned with one of the two PSPs will have been undistorted by PMD, at
least to first order.
4.7 Differential group delay
The difference between the two PSP arrival times is the differential group delay (DGD). The
resultant pulse broadening or splitting is the source of signal distortion.
st
The definition provided above is in the context of 1 order PMD without PDL. In the case of
PDL, there is an open issue regarding the definition (see also E.2.1).
NOTE Depending on the device, DGD varies with wavelength, and for a randomly mode coupled device, the
pattern of variation changes randomly with device deployment and small temperature changes (~0,1 °C). The
average DGD across a wavelength range remains moderately stable, depending on the device construction and the
wavelength range that is sampled.
4.8 Polarization mode dispersion
PMD is the temporal distortion of an optical signal induced by the interaction of the source
polarization and spectral characteristics with the transport of the signal through a device that
includes two orthogonal SOPs (i.e. PSPs) with differing group velocities. The term "PMD" is
used both in this general sense of a “phenomenon” of pulse width broadening, as well as
specific “numeric values” of broadening (in time) that characterise a device.
4.8.1 Phenomenon
When an optical signal pulse passes through a device, the pulse may change. PMD refers to
the change in shape, particularly in the root-mean-square (RMS) width of the pulse, due to the
average propagation delay difference between the two PSPs, i.e. DGD, and/or waveform
distortion for each PSP (see Figure 4). PMD together with PDL and PDG, when applicable,
may introduce waveform distortion leading to an unacceptable increase in bit error ratio
(BER).
– 16 – TR 61282-9  IEC:2006(E)

IEC  1249/06
a) One birefringent section/two pulses/with distortion

IEC  1250/06
b) Multi-birefringent sections/one pulse/no distortion (only broadening)
Figure 4 – Effect of polarization mode dispersion on transmission
of an information-bit pulse in a device
4.8.2 Numerical value
The PMD value (in units of time) is specified and measured depending on the type of device
and its contribution to the overall signal link distortion. In some cases, either the average or
RMS DGD over an optical frequency (or wavelength) range is used to define the PMD value.
In other cases, the maximum DGD occurring in the optical frequency or wavelength range is
used.
4.9 Polarization dispersion and birefringence vectors
ˆ
From Poole [6,9,10], the polarization dispersion vector (PDV), Ω , and the birefringence
ˆ
ˆ
vector, Δβ are a compact way of representing PMD. The variation of the SOP, s, as a
function of the optical angular frequency, ω and z, inside the device obeys the equations of
motion described in (12) and (13) respectively (see Figure 5).

TR 61282-9  IEC:2006(E) – 17 –
ˆ
ds()ω
ˆ
ˆ
() ( )
= Ω ω x s ω (12)
dω
ˆ
ds()ω
ˆ
β()ω x sˆ()ω (13)
= Δ
dz
where
ˆ
s is the SOP vector with three components, s , s and s defined in
...