Iec Tr 61282-9-2016.Pdf

IEC TR 61 282 9 Edition 2 0 201 6 03 TECHNICAL REPORT Fibre optic communication system design guides – Part 9 Guidance on polarization mode dispersion measurements and theory IE C T R 6 1 2 8 2 9 2 0[.]

IEC TR 61 282- 9

Edit io 2.0 2 16-0

Fibre opt ic communication sy st em design guides –

Part 9: Guidance on polariz t ion mode dispersion measurement s and t heory

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IEC TR 61 282- 9

Edit io 2.0 2 16-0

Fibre opt ic communicat ion syst em design guides –

CONTENTS

FOREWORD 4

INTRODUCTION 6

1 Sco e

7 2 Normative ref eren es 7

3 Terms, def i ition , an a breviation 7

3.1 Terms an def i ition 7

3.2 Ab reviation 8

4 The retical framework 8

4.1 Limitation an outl ne 8

4.2 Optical f ield an state of p larization 8

4.3 SOP me s rements, Stokes vectors, an Poin aré sphere rotation 1

4.4 First order p larization mode disp rsion 14 4.5 Birefrin en e vector, con atenation , an mode coupl n 16 4.6 The statistic of PMD an secon order PMD 17 4.7 Managin time 20 5 Me s rement method 2

5.1 General 2

5.2 Stokes p rameter evaluation 2

5.2.1 Eq ipment setup an proced re 2

5.2.2 Jones matrix eigenanaly is 2

5.2.3 Poin aré sphere analy is 2

5.2.4 One en ed me s rements b sed on SPE [3] 2

5.3 Phase s if t b sed me s rement method 2

5.3.1 General 2

5.3.2 Mod lation phase s if t – Ful se rc 2

5.3.3 Mod lation phase s if t method – Muel er set analy is [4] 2

5.3.4 Polarization phase s if t me s rement method[5] 31

5.4 Interferometric me s rement method 3

5.4.1 General 3

5.4.2 General zed interf erometric method [6] 3

5.4.3 Traditional interferometric me s rement method 4

5.5 Fixed analy er 41 5.5.1 General 41

5.5.2 Extrema cou tin 42 5.5.3 F urier tran form 4

5.5.4 Cosine Fourier tran f orm 4

5.5.5 Sp ctral dif f erentiation 4

5.6 Wavelen th s an in OTDR an SOP analy is (WSOSA) method [7] 4

5.6.1 General 4

5.6.2 Contin ou model 4

5.6.3 L rge dif feren e model 4

5.6.4 Scrambl n factor derivation 5

6 Limitation 5

6.1 General 5

6.2 Ampl f ied sp ntane u emis ion an degre of p larization 5

6.3 Polarization de en ent los (or gain) 5

6.4 Coheren e ef f ects an multiple p th interferen e 5

6.5 Test le d f ibres 54 6.6 Aerial ca les testin 55 Bibl ogra h 5

Fig re 1 – Two electric f ield vector p larization of the HE 1 mode in a SMF 10 Fig re 2 – A rotation on the Poin aré sphere 13 Fig re 3 – Stron mode coupl n – Freq en y evolution of the SOP 16 Fig re 4 – Ran om DGD variation v wavelen th 18 Fig re 5 – Histogram of DGD values f rom Fig re 4 18 Fig re 6 – SPE eq ipment diagram 2

Fig re 7 – Relation hip of orthogonal output SOPs to the PDV 24 Fig re 8 – Stokes vector rotation with f req en y c an e 2

Fig re 9 – Setup for mod lation phase s if t 2

Fig re 10 – Setup for p larization phase s if t 2

Fig re 1 – Output SOP relation to the PSP 3

Fig re 12 – Interferometric me s rement setup 3

Fig re 13 – Interferogram relation hips 3

Fig re 14 – Me n s uare en elo es 3

Fig re 15 – Fixed analy er setup 41

Fig re 16 – Fixed analy er ratio 42 Fig re 17 – Power sp ctrum 44 Fig re 18 – Fourier tran form 4

Fig re 19 – WSOSA setup 46 Fig re 2 – Freq en y grid 47 Ta le 1 – Ma of test method an International Stan ard 22 Ta le 2 – Muel er SOPs 2

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tec nical commite may pro ose the publ cation of a tec nical re ort when it has col ected

data of a different kin f rom that whic is normal y publ s ed as an International Stan ard, f or

example "state of the art"

IEC TR 612 2-9, whic is a Tec nical Re ort, has b en pre ared by s bcommit e 8 C: Fibre

o tic s stems an active devices, of IEC tec nical commite 8 : Fibre o tic

This secon edition can els an re laces the f irst edition publs ed in 2 0

This secon edition in lu es the fol owin sig ifi ant tec nical c an es with resp ct to the

previou edition:

a) mu h of the the ry has b en con en ed – foc sin only on content that is ne ded to

explain the test method;

b) s mb ls have b en removed, but a breviation are retained;

c) the material in the Clau e 5 has b en sig if i antly red ced in an ef f ort to avoid re e tin

what is alre d in the actual International Stan ard In te d, the foc s is on explainin

the International Stan ard ;

d) me s rement method that are not fou d in International Stan ard have b en removed;

e) there are sig if i ant cor ection to the mod lation phase s if t method, p rtic larly in

regard to the Muel er set tec niq e;

f ) there are sig if i ant cor ection to the p larization phase s ift method;

g) the pro f of the GINTY interferometric method is presented This pro f also exten s to the

Fixed Analy er Cosine tran fer tec niq e;

h) another Fixed Analy er method is s g ested This is b sed on the pro f of the GINTY

method an is caled "sp ctral diff erentiation method";

i) Clau e 6 has b en renamed "Limitation " an refoc sed on the l mitation of the test

method This Tec nical Re ort is not inten ed to b an en ine rin man al;

j) the an exes have b en removed;

k) the bibl ogra h has b en mu h red ced in size;

l) the introd ction has b en exp n ed to in lu e some information on s stem imp irments

The text of this Tec nical Re ort is b sed on the f ol owin doc ments:

En uiry draft Re ort o v tin

Ful information on the votin for the a proval of this Tec nical Re ort can b f ou d in the

re ort on votin in icated in the a ove ta le

This publcation has b en draf ted in ac ordan e with the ISO/IEC Directives, Part 2

A l st of al p rts in the IEC 612 2 series, publ s ed u der the general title Fibre o tic

c mmu ic tio s stem desig g id es, can b f ou d on the IEC we site

The commit e has decided that the contents of this publcation wi remain u c an ed u ti

the sta i ty date in icated on the IEC we site un er "htp:/we store.iec.c " in the data

related to the sp cifi publ cation At this date, the publ cation wi b

• recon rmed,

• with rawn,

• re laced by a revised edition, or

A bi n ual version of this publ cation may b is ued at a later date

IMPORTANT – The 'colour in ide' logo on the cov r pa e of this publ c tion indic te

that it contains colours whic are consid re to be u ef ul f or the cor e t

understa ding of its conte ts Us rs s ould theref ore print this doc me t using a

colour printer

This Tec nical Re ort is complementary to the International Stan ard des ribin PMD

proced res (IEC 6 7 3-1-4 , IEC 612 0-4-4, IEC 612 0-1 -1, IEC 612 0-1 -2 an

IEC 613 0-3-3 ) an other desig g ides on PMD (IEC 612 2-3 an IEC 612 2-5), as wel as

ITU-T Recommen ation G.6 0.2

The s stem p wer p nalty as ociated with PMD varies de en in on tran mis ion f ormat an

bit rate It also varies with o tical f req en y an state of p larization (SOP) of the l g t

source At the output of a l n , the sig al can s if t f rom a maximum delay to a minimum delay

as a res lt of u in dif ferent SOPs at the source The dif f eren e in these delay is caled the

dif f erential group delay (DGD), whic is as ociated with two extremes of input SOP At these

extremes, a sig al in the f orm of a sin le pulse a p ars s if ted up or down by half the DGD,

a out a midp int, at the output At intermediate SOPs, the sin le pulse a p ars as a weig ted

total of two pulses at the output, one s if ted up by half the DGD an one s if ted down by half

the DGD This weig ted total of two s if ted pulses is what cau es sig al distortion

The s stem p wer p nalty is p rtly def i ed in terms of a maximum al owed bit er or rate an a

minimum received p wer In the a sen e of distortion, there is a minimum received p wer

that wi prod ce the maximum al owed bit er or rate In the presen e of distortion, the

received p wer s ould b in re sed to prod ce the maximum bit er or rate The mag itu e of

the req ired in re se of received p wer is the p wer p nalty of the distortion

The term PMD is u ed to des rib two distin tly dif ferent ide s

One ide is as ociated with the sig al distortion in u ed by tran mis ion media f or whic the

output SOP varies with o tical freq en y This is the fun amental source of sig al distortion

The other ide is that of a n mb r (value) as ociated with the me s rement of a sin le-mode

f ibre tran mis ion l n or element of that l n There are several me s rement method with

dif ferent stren th an ca a i ties They are al b sed on q antif yin the mag itu e of

p s ible variation in output SOP with o tical freq en y The o jective of this Tec nical Re ort

is to explain the commonal ty of the dif f erent method

The DGD at the source’s o tical f req en y is what controls the maximum p nalty acros al

p s ible SOPs However, in most l n s, the DGD varies ran omly acros o tical freq en y

an time The PMD value as ociated with me s rements, an whic is sp cified, is a

statistical metric that des rib s the DGD distribution There are two main metric , l ne r

average an ro t-me n s uare (RMS), that exist in the l terature an in the me s rement

method For most situation , one metric can b calc lated f rom the other u in a con ersion

f ormula The re son f or the d al metric is an ac ident of history If history could b

cor ected, the RMS def i ition would b the most s ita le

For the non- eturn to zero tran mis ion format, DGD eq al to 0,3 of the bit p riod yield

a proximately 1 dB maximum p nalty Becau e DGD varies ran omly, a rule of thumb

emerged in the s stem stan ardization groups: ke p PMD les than 0,1 of the bit p riod for

les than 1 dB p nalty This as umes that DGD larger than thre times the PMD, an that the

source output SOP prod ces the worst case distortion, is not very l kely For 10 Gbit s non

-return to zero, this rule yield a desig rule: ke p the l n PMD les than 10 ps

FIBRE OPTIC COMMUNICATION SYSTEM DESIGN GUIDES –

Part 9: Guidance on polarization mode dispersion

measurements and theory

This p rt of IEC 612 2, whic is a Tec nical Re ort, des rib s ef fects an the ry of

p larization mode disp rsion (PMD) an provides g idan e on PMD me s rements

The f olowin doc ments, in whole or in p rt, are normatively ref eren ed in this doc ment an

are in isp n a le f or its a pl cation For dated ref eren es, only the edition cited a pl es For

u dated ref eren es, the latest edition of the referen ed doc ment (in lu in an

amen ments) a pl es

IEC 6 7 3-1-4 , Optic l fibre – P art 1-4 : Mea ureme t methods a d te t pro edure –

P olariz tio mod e d is ersio

IEC 612 0-4-4, Fibre o tic c mmu ic tio s b y tem test pro edure – P art 4-4: Ca le

pla ts a d li k – P olariz tio mode d is ersio me s reme t for instaled ln s

IEC 612 0-1 -1, Optic l amplifier – Te t meth ds – P art 1 -1: P olariz tio mode dis ersio

p rameter – J on s matrix eig n n ly is (J M E)

IEC 612 0-1 -2, Optic l amp fier – Te t meth ds – P art 1 -1: P olariz tio mode d is ersio

p rameter – P oin aré s h re a aly is meth d

IEC 613 0-3-3 , Fibre o tic interc n e tin devic s a d p s iv c mp n nts – Ba ic te ts

a d me s reme t pro edure – P art 3-3 : Ex aminatio s a d me s reme ts – P olariz tio

mod e d is ersio me s reme t for p s iv o tic l c mp n nts

3 Terms, de nitions, and a breviations

3.1 Terms a d def initions

For the purp ses of this doc ment, the fol owin terms an def i ition a ply

3.1.1

PMD phe ome on

polarization mode dispersion phe ome on

sig al of f ibre-o tic tran mis ion sig al in u ed by variation in the sig al output state of

p larization with o tical freq en y

Note 1 to e try: PMD c n lmit th bit rate-le gth pro u t of digital s stems

3.1.2

PMD v lue

polarization mode dispersion v lue

mag itu e of p larization mode disp rsion phenomenon as ociated with a sin le-mode fibre,

o tical comp nent an s b-s stem, or in tal ed l n

3.2 Ab reviations

CFT cosine Fourier tran f orm

DGD dif f erential group delay

MPS mod lation phase s if t

OTDR o tical time domain ref lectometer

PDL p larization de en ent los

PDV p larization disp rsion vector

PMD p larization mode disp rsion

PPS p larization phase s if t

PSA Poin aré sphere analy is

PSP prin ip l state of p larization

SMF sin le-mode f ibre

SOP state of p larization

SPE Stokes p rameter evaluation

TINTY traditional interf erometric method

WSOSA wavelen th s an in OTDR an SOP analy is method

4 Theoretical f ramework

4.1 Limitations a d outl ne

The the ry presented in Clau e 4 do s not in lu e the ef fects of p larization de en ent los

or gain, or nonl ne r ef fects Se 6.3 f or inf ormation on p larization dep n ent los

The outl ne for Clau e 4 is

• o tical f ield an state of p larization;

• me s rement of SOP, Stokes vector, an rotation;

• f irst order p larization mode disp rsion;

• birefrin en e vector, con atenation , an mode coupl n ;

• the statistic of PMD an secon order PMD

4.2 Optic l f ield a d state of p larization

This s bclau e is inten ed to s ow the l n age b twe n the pro agation of an o tical f ield in

a sin le-mode f ibre (SMF) an the tran mis ion sig al state of p larization (SOP) This

information is fun amental to the PMD phenomena b cau e variation in output SOP with

o tical f req en y is the distortion in u in mec anism

The solution of the wave eq ation has degenerated eigen alues This me n that even the

f un amental solution is degenerated A SMF s p orts a p ir of p larization modes f or a

monoc romatic l g t source In p rtic lar, the lowest order mode, namely the fun amental

mode HE

1(LP

01can b def i ed to have its tran verse electric f ield predominately alon the

x-direction; the orthogonal p larization is an in e en ent mode, as s own in Fig re 1

In a los les SMF, the electric f ield vector of a monoc romatic electromag etic wave

pro agatin alon the z-direction can b des rib d by a l ne r s p rp sition of these two

modes in the x-y tran verse plane as s own in Eq ation (1) an in Fig re 1

ϖβ

βϖβ

∆β

(x,y) are the sp tial variation (in the x-y tran verse plane) of the E vector of the

PM alon the x/y-direction (se Fig re 1);

β

x

y

are the pro agation con tants (also cal ed ef fective in ex or wavenumb r)

of the PM alon the x/y-direction with the in ex of refraction n

x/n

y Usin

s ( he b r in icates a solute f req en y

rather than deviation from some p rtic lar value);

∆’ is the birefrin en e co f f icient (s/m), = ∆n/

z is the distan e (m) in the DUT alon the o tical axis (axis of pro agation);

z = L at the output of the DUT with len th L

Figure 1 – Two ele tric f ield v ctor polarizations of th HE

1

mode in a SMF

The complex p ir, [j ex p( i z/2), j ex p(i z/2)]

yx

β

∆β

∆

of the wave pro agatin alon the z-direction This p ir can b con idered as a vector, an is

of ten caled a Jones vector

In the case where the tran mis ion media is an ide l SMF with p rfect circ lar s mmetry,

β

y

= β

x,

• the two p larization modes are degenerate (when two solution have the same

eigen alue, they are said to b degenerate);

• an wave with a def i ed input SOP wi pro agate u c an ed alon the z-direction

throu hout the output of the SMF

However, in a practical SMF, the circ lar s mmetry is broken by imp rf ection prod ced by

the fabrication proces , ca l n , field in tal ation/u e or the in tal ation en ironment:

y

≠ β

x, implyin a phase dif f eren e, an in ex-of -ref raction dif feren e ∆n, an a phase-

velocity dif feren e b twe n the two PMs;

• the degenerac of the two p larization modes is l f ted;

• the SOP of an input wave wi c an e alon the z-direction throu hout the output of the

birefrin en e an has u its of in erse len th (m

−1

Birefrin en e may also b refer ed to as

the in ex diff eren e, ∆n, or as the biref rin en e co f ficient, the dif feren e divided by len th

Birefrin en e co f f icient values typical y vary b twe n 0,2 f s/m an 2,5 f s/m in commonly

avai a le SMFs

As the SOPs travel throu h the f ibre, they wi return to the initial state at position that are

in rements of 2 /∆β At these p sition , the two f ield comp nents wi b at The dif feren e

b twe n these p sition is caled the b at len th

Birefrin en e can b in u ed by a n mb r of f actors s c as core non-circ larity or

as mmetric stres es that can b in u ed by b n s, twist, an compres ion These f actors

c an e over the len th of the f ibre an can c an e over time d e to c an es in con g ration,

or temp rature These f actors wi also vary with o tical f req en y Eq ation (1) is also

defi ed with an arbitrary co rdinate s stem that wi not general y cor esp n to the la oratory

or f ield test eq ipment co rdinate s stem The ma pin of the input SOP to the output SOP

x

E

y(x,y);β

over a p rtic lar len th an o tical freq en y, L an ϖ

0, is re resented with the Jones matrix,

T, an the input an output Jones vectors,

INj



an

0j



, as:

INjTj

ex p0

02

/

ex p

TT

T

ii

S

xx

TT

TT

TT

TT

T

RiS

ii

ii

µθ

µθ

µθ

µθ

2/

2/

ex p

c os2

/

ex psin

2/

ex psin2/

NOT 1 Th s b cript T is u e to distin uis th matrix p rameters fom simiar p rameters u e later

The main o eration of Eq ation (3), cor esp n in to Eq ation (1), is f ou d in the matrix, S

Pre an p st-multiplyin by V

Tan

↑

TV

is a c an e of co rdinates The notation

↑

V in icates

the tran p se conju ate f or matrices an vectors

The diagonal expres ion in Eq ation (4) are inten ed to s ow the con ection to Eq ation (1)

Eq ation (1), however, is only a pl ca le localy, whie Eq ation (3) is u ed to in icate c an e

over the entire tran mis ion media There is another expres ion that u es /2

TTx

ex p

sin

2/

ex p

c os

µθ

µθ

ii

eigen ectors, an the diagonal elements of S are the eigen alues When the input Jones

vector is eq al to either of the eigen ectors, the output SOP is the same as the input SOP

b cau e the SOP is not af f ected by a multipl cation by a con tant These states are

sometimes cal ed the eigen tates

NOT 2 Al th p rameters of T c n c a g with o tic l fe u n y, a wel a with c a g s c u e b fibre

mo eme t o er time or temp rature c a g

4.3 SOP me s reme ts, Stok s v ctors, a d Poinc ré sphere rotations

The SOP is u ual y me s red with a p larimeter, whic yield a Stokes vector The

me s rement is actualy done with a series of p wer me s rement dif feren es throu h

variou states of a p larizer/analy er

An ide l p larizer may b def i ed as:

PP

P

PolPol

PP

i

i

VV

Pol

θµ

θ

µθ

θ

µθ

22

sin

ex p2sin

21

ex p2sin

21

c os

00

01

,

(7)

where V

Pol

is of the same f orm as Eq ation (5), but with dif ferent p rameters

For a given Jones vector, the p wer throu h the p larizer is:

P

PPPPP

P





µθµθµ

=

2/,4/2/,4/

0,4/0

,4/

0,2/0,0

0,2/0,0

πππ

π

ππ

ππ

PP

PP

PP

PP

an has thre elements in exed:

1, 2, an 3 The normal zed Stokes vector elements are the me s red Stokes vector element

values with the same in ex, divided by S

0 The normalzed Stokes vector has a len th of one

SOPs where s

3

is zero are l ne r states When s

3

is non ero but with a solute value les than

one, the p larization is el ptical When s

3

is ±1, the p larization is circ lar

NOT 1 In th re t of this Te h ic l Re ort, th n rmalz d Sto e v ctor wi b refere to simply a th Sto e

⋅ , where P is the p wer without the p larizer

A u it Jones vector can b re resented either as an x/y p ir or as Eq ation (6) The

relation hip of the normal zed Stokes vector to the Jones vector is given as:

−

=

µθ

µθθ

sin2sin

c os2sin

There is an ambig ity in tryin to calc late the Jones vector f rom the Stokes vector One mu t

as ume somethin lke 0 θ π This is d e to the f act that the Stokes vector is not aff ected by

multiplyin the Jones vector by an u it complex n mb r (a n mb r, c, f or whic c

*

= 1),

in lu in ±1 This can b cal ed a one

π ambig ity This pro erty is one re son to thin of the

Stokes vector as the primary def i ition of the SOP: the SOP is not c an ed when either of the

eigen tates are u ed as inputs, but the output Jones vector is multipl ed by exp(±iξ

T/2)

Unit thre term vectors can b re resented on a sphere In the case of Stokes vectors, the

sphere is cal ed the Poin aré sphere

Examination of the rig tmost expres ion of Eq ation (10) an the dif ferent p rts of Eq ation

(3), (4) an (5) s ows that the action of T is con istent with the fol owin rig t-han - ule

rotation a pl ed to the input Stokes vector that cor esp n s to the input Jones vector:

T

yI

12

13

23

yy

yy

yy

Figure 2 – A rotation on the Poinc ré sphere

The T matrix can now b writen in a simplf ied f orm:

T

TT

T

i y

i yy

i yy

i y

T

γγ

γ

γγ

γ

sin

c ossin

sin

sin

c os

12

3

231

−

=

13

2

321

sin

c os

y

i yy

i yyy

iI

TT

γγ

(13)

NOT 2 Th form of this e u tio re-emerg s in Eq atio (2 ), in whic it is p inte o t th t th rig tmo t matrix

is th weig te s m of Pa l matric s

4.4 First ord r polarization mode dispersion

First order PMD is in u ed by the varian e in the output Jones vector with o tical f req en y

This is the same as the varian e of the output Stokes vector with f req en y, whic is one way

to me s re it, but the sig al distortion mu t b u dersto d in terms of the Jones calc lu To

distin uis the varian e of SOP with o tical freq en y from the input-to-output Jones matrix

of the prior clau es, the f req en y tran f er matrix is desig ated with J(ω) as

00

2/

ex p0

02

/

ex p

jV

ii

VjJj

JJ

(14)

where

V

J

is the same f orm as Eq ation (5);

ω is deviation from the p rtic lar f req en y, ϖ

∆τ is the dif f erential group delay (DGD)

The exp nent of the ratio of eigen alues of J yield ∆τ∆ω f or a f i ite f req en y in rement

The action of J(ω) is a rotation on the Poin aré sphere f rom the output Stokes vector at ϖ

0

to

the Stokes vector at ϖ

0+ ω The rotation an le is ∆τ ω, an the rotation vector is formed by

tran formin the f irst column of V

J

into a Stokes vector This rotation vector is cal ed the

prin ip l state of p larization (PSP) an is later desig ated as a vector, p



The distortion can b u dersto d by con iderin the Fourier tran form of the sig al f ield, H(ω)

an its in erse tran f orm, h(t) Time s if tin of the in erse tran form is given as tran f orm p ir

as:

00

21

πωω

02

/

21

jV

th

th

Vt

j

JJ



are b th eq al to 2 , the output pulse is the

s m of s uares of two f ield pulses se arated by ∆τ This re resents a worst case f or distortion

within the first order f ormulation

When

0j



is eq al to either column of V

J, there is no distortion, but the sig al is s if ted in the

time domain by ±∆τ /2 At intermediate p larization states, the output pulse is the total of two

time s if ted pulses of diff erent mag itu es In the worst case, the mag itu es are eq al

In the Stokes f ormulation, the action of Eq ation (14) is a rotation, R

J, with rotation an le,



is eq al

to this vector, there is, to the f irst order, no c an e in the output Stokes vector with

f req en y Sin e it is al g ed with –i∆τ ω/2, it is the e rly ar ivin , or fast, p rt of the sig al

For completenes , R

J

is of the f orm of Eq ation (12), but with diff erent p rameters, an the

relation hip of the output Stokes vector as a fun tion of freq en y is:

0sRs

an the f req en y tran fer f un tion dif f erential o erator, a matrix

desig ated as D These are defi ed as the f ol owin , con iderin that al p rameters of b th

RT an T are f un tion of freq en y:

00

00

0

R

dd

d

sd

T

TT

ωω

(18)

When the output Stokes vector at ϖ

0, desig ated as

0s

00

1

00

0

2/0

02/

ii

Vj

d

dJ

jT

ddT

jD

d

jd

JJ

ωω

ωω

0

13

2

321

2

j

p

i pp

i ppp

−

−

=τ

01

1σ

10

2σ

3

ii

σ

(2 )

The eigenvalues of D are ±i∆τ /2

Exp n ion of the expres ion containin the p rameters of T s ows the f olowin :

• the eq ation are con istent;

• a dif f erential eq ation that l n s the p rameters an derivative of T to the PDV emerges

+

=

ϖγ

ϖγ

ϖγ

Ω

d

yd

y

d

yd

y

dd

TT

sin

Given the PDV as a fun tion of o tical freq en y an s ita le b u dary con ition , the

p rameters of T, hen e T can b solved u in n merical tec niq es This was imp rtant in

simulatin the interferometric me s rement method

First order PMD me n that the exp nent of the diagonal matrix in Eq ation (14) is lne r in

f req en y In re l ty, there are l mitation in the f irst order projection b cau e the PSP an

DGD actual y vary with freq en y in most (lon ) tran mis ion media

4.5 Birefringe c v ctor, conc te ations, a d mode coupl n

4.4 in icated that the imp rtant p rameters, the PSP an DGD, also vary with o tical

f req en y 4.5 wi give some b c grou d as to wh this is so

The biref rin en e vector is def i ed by re lacin the f req en y (ω) derivatives with len th (z)

derivatives in Eq ation (18) an (21) Also, re lace the s mb l for the PDV with the s mb l

f or the birefrin en e vector The evolution of the output Stokes vector throu h a len th of

tran mis ion media can b calc lated f rom the modif ied Eq ation (18) an a set of

as umption a out the birefrin en e vector These as umption can in lu e f actors s c as

twist, whic in u es circ lar birefrin en e, b n , lateral lo d, an spin

A simple biref rin ent element can b defi ed f rom Eq ation (3) throu h (5) with the

where ∆’ is the birefrin en e co f f icient (s/m)

A con atenation of a n mb r of ran omly rotated simple birefrin ent elements wi in u e

mode coupl n that varies with o tical f req en y As a res lt, de en in on the n mb r an

len th of s c elements, the output SOPs no lon er fol ow simple rotation with freq en y,

an the PDV b comes ran om Fig re 3 s ows an example

Figure 3 – Strong mode coupl ng – Fre ue c e olution of the SOP

Neglgible mode coupl n would display a les compl cated evolution, s c as what is s own

in Fig re 2 However, there is no f irm b u dary b twe n the dif ferent regimes

The con atenation of ran omly rotated simple elements is not a complete model for o tical

f ibre The PMD b haviour of o tical f ibre an ca le is in the re lm of stron ran om mode

coupl n Most o tical f ibre is man factured u in a spin in tec niq e to delb rately

introd ce stron mode coupl n

IEC

Con eq en es of mode coupl n in lu e:

• the con ection b twe n the biref rin en e vector an the PDV is lost The birefrin en e

vector is a local phenomenon, whi e the PDV is a res lt of tran mis ion over the whole of

the media;

• DGD varies ran omly with freq en y;

• the PSP varies ran omly with freq en y;

• the f irst order PMD model is an a proximation that is only a pl ca le for a nar ow

f req en y interval;

• de en in on the me s rement typ , f req en y domain or time domain, the sampl n

den ity an /or extent mu t take into ac ou t the ran om nature of the phenomenon;

• the PMD value that is re orted mu t b derived f rom a statistical calc lation;

• PMD in re ses with the s uare ro t of len th, le din to the defi ition of the PMD

co f ficient as the PMD me s red value divided by s uare ro t of the tran mis ion len th;

• there is a phenomenon cal ed secon order PMD, whic is a me s re of how the DGD an

PSP vary with f req en y;

• third- an hig er order PMD may have an ef f ect on the distortion at very hig bit rates or

u der p rtic lar tran mis ion f ormats but have not b en stu ied

4.6 The statistic of PMD a d s cond order PMD

This s bclau e as umes the tran mis ion media is in the ran om mode coupl n regime

A f irst con eq en e of ran om mode coupl n is that the thre elements of the PDV b come

in e en ently distributed Gau sian ran om varia les if the f req en y p ints are far enou h

a art The average of these distribution is zero, an there is a common stan ard deviation,

σ The DGD,

∆τ , is the s uare ro t of the s m of s uares of the thre elements This me n

that the distribution of DGD values is ide l y Maxwel an, with pro a i ty den ity f un tion

32

2

ex p2

• l ne r average eq als a simple average of DGD values acros freq en ies

Ex e tion to these statistic may b fou d for some comp nents for whic the negl gible

mode coupl n regime a pl es In these cases, the maximum DGD can b re orted A PMD

can sti b re orted, but the Maxwel a proximation do s not hold

The RMS statistic is the most "natural" for PMD (se Eq ation (2 ) , but the lne r average

was stan ardized for f ibre an ca led f ibre me s rements b f ore PMD was more completely

u dersto d This is of p rtic lar imp rtan e f or the interferometric me s rement method ,

where the value re orted is eq ivalent to the RMS statistic

The RMS value is expres ed with the exp cted value o erator as

21

2

∆ The l ne r average

is expres ed as ∆τ De en in on whic statistic is re orted, the σ p rameter f ou d in

Eq ation (2 ) can b re laced with either the expres ion in Eq ation (2 ) f or RMS or with

Eq ation (2 ) f or l ne r average

31

τ

∆

8π

τ

From these eq ation , it can b ded ced that the RMS value is a proximately

83π

Fig re 5 – Histogram of DGD v lue f rom Figure 4

For the freq en y domain me s rements, there is a s an of freq en ies f rom ϖ

1

to ϖ

2

with an

in rement of ∆ω The extent an sampl n den ity ne ded de en on the exp cted PMD of

the tran mis ion media b in me s red

NOT For time d main me s reme ts, th time in reme t is in ers ly pro ortio al to th fe u n y e te t a d

th time e te t is in ers ly pro ortio al to th fe u n y in reme t

Guidan e on the f req en y extent can b derived from the f i ed analy er me s rement

method (se 5.5) This method relates the RMS PMD to the n mb r of extrema, N

e, cou ted

2/1

2

83

ϖϖ

ππ

(2 )

As the n mb r of extrema in re ses, the re rod cibi ty of the PMD me s rements improves

One way to in re se the n mb r of extrema is to me s re acros a wider freq en y interval

Eq ation (2 ), f rom Gisin [2] relates the freq en y extent to the u certainty (one stan ard

deviation) of the me s rement comp red to the res lt one mig t o tain with a very large

(a pro c in in nity) freq en y extent

2

2/1

22/1

2

9,0

1

ϖϖτ

∆

τ

∆τ

∆

(2 )

If the freq en y extent is 16 THz, cor esp n in to 12 nm arou d 1 5 0 nm, an the PMD is

1 ps, the me s rement u certainty is a out 9 % However, if the wavelen th ran e were

in re sed from 1 310 nm to 1 6 5 nm, the u certainty would b red ced to arou d ± %

One re son f or the u certainty situation is secon order PMD, whic is expres ed as the

f req en y derivative of the PDV, i.e

ωΩ



It is related to the PMD ac ordin to the folowin

22

τ

∆Ω

2

ωω

ω

τ

∆τ

31

ΩΩ

Ωτ

2

2

28

ΩΩ

τ

∆

ωω

(3 )

Secon order PMD RMS (s uare ro t of Eq ation (3 ) in re ses as the s uare of PMD, an

the res lt is dominated by the p rt as ociated with the PSP (Eq ation (3 ) On the other

han , f or very low PMD, a me s rement at a sin le freq en y mig t not b mu h dif f erent

f rom me s rements acros a ran e that is practical

Another outcome of the u certainty is that o tical fibre ca le is sp cif ied statistical y u in a

p rameter cal ed the l n desig value or PMDQ IEC TR 612 2-3 [9] explain this statistic

more completely, but it is an up er con den e l mit, at 9 ,9 % pro a i ty, on the

con atenated PMD co f ficient of 2 ca les The statistic takes b th average an varian e of

the PMD distribution into ac ou t The varian e in lu es me s rement u certainty, so the

statistical sp cif i ation is a primary way of managin me s rement u certainty

4.7 Ma a ing time

The output SOP can b modif ied by f actors other than o tical f req en y These f actors can

c an e with time Factors in lu e:

• Temp rature

Even for buried ca l n , smal temp rature c an es over lon times can prod ce c an es

in the DGD that are a p rently ran om when the times b twe n meas rements are

This is imp rtant for active s stem monitorin

Chan es in these f actors over the time of me s rement can res lt in er ors, de en in on the

me s rement method Ex e t f or u control ed c an es in the input SOP, the main ef f ects of

these sources of variation are c an es in mode coupl n

For freq en y domain me s rements, at le st a p rt of the output SOP s al b me s red at

either a p ir of f req en ies, otherwise a me n of evaluatin the f req en y derivative s al b

provided in order to o tain an estimate of the DGD The me s rement time for s c a p ir

s ould b smal in comp rison to the factor variation rate, e.g 1 ms for vibration noise in the

ran e of 5 Hz to 6 Hz

When s c a smal interval is not practical, there are alternatives:

• wait unti the disruption are not c an in , e.g the win sto s, or the train is gone;

• u e a time domain, interferometric me s rement

The slow variation with smal temp rature c an es prod ces a dif ferent ran om DGD c rve

v o tical f req en y at one time v another This gives rise to the term "in tantane u

DGD"

De en in on the typ of ph sical p rturb tion , the statistic of the DGD acros time at a

given freq en y are simi ar to the statistic of the DGD acros f req en ies at a sin le time

This is an asp ct that can b exploited in s stem monitorin

Verif i ation of the sta i ty of the output Stokes vector as a f un tion of time is a pru ent thin

to do, p rtic larly for me s rement of in tal ed l n s There is, however, no f ormal

req irement to do this c ec or a stan ardized criterion f or "sta i ty"

There are several international publ cation that des rib some common me s rement

method Eac s c publ cation is normative to the prod cts to whic the publ cation a pl es

The prod ct categories an normative test publ cation are:

• o tical f ibre an ca le: IEC 6 7 3-1-4 ;

• o tical ampl fiers: IEC 612 0-1 -1 an IEC 612 0-1 -2 (IEC TR 612 2-5 [10] f or

information);

• p s ive o tical comp nents: IEC 613 0-3-3 ;

• in tal ed l n s: IEC 612 0-4-4

These normative test proced res diff er in resp ct to detai s that are esp cial y critical to the

prod ct For example, the ampl f ier doc ments p y p rtic lar at ention to ampl fied

sp ntane u emis ion The p s ive comp nent doc ment provides some g idan e on

p larization de en ent los for devices with smal levels of PMD an whic are in the

negl gible mode coupl n regime

Another asp ct relevant to o tical comp nents is that some comp nents s c a wavelen th

division multiplexin devices have multiple o tical p th that req ire sp cial proced res

Provision for devices with multiple o tical p th are not in the s o e of this Tec nical

Re ort

Eac normative test publcation has one or more me s rement method The same

me s rement method a p ar in multiple normative publ cation The intent of this Tec nical

Re ort is to explain how the dif ferent me s rement method relate to the u derlyin the ry It

is not the intent of this Tec nical Re ort to re e t the tec nical req irements f ou d in the

in ivid al normative test proced res lsted a ove In some cases, however, some of the

req irements mu t b re e ted in order to provide b c grou d neces ary f or the explanation

The method in lu e

• Stokes p rameter evaluation (SPE):

– Jones matrix eigenanaly is (JME);

– Poin aré sphere analy is (PSA)

• Phase b sed me s rements:

– mod lation phase s if t (MPS);

– p larization phase s if t (PPS)

• Interferometric method :

– traditional interf erometric method (TINTY);

– general interf erometric method (GINTY)

• Fixed analy er (FA):

– extrema cou tin (EC);

– Fourier tran f orm (FT);

– cosine Fourier tran f orm (CFT)

• Wavelen th s an in OTDR an SOP analy is method (WSOSA)

The main l st items u e common eq ipment an proced res, whi e the in ented l st items

dif fer mainly in the analy is There are some eq ipment modif i ation to the TINTY, f or

example, that are ne ded to complete the GINTY me s rement, but most of the eq ipment

ne d are q ite simiar

SPE, FA, an phase b sed me s rements in olve a s an of o tical f req en y from a lower

freq en y to an up er f req en y at a freq en y in rement F r SPE an phase b sed

me s rements, the DGD is re orted f or e c in rement, an the PMD value is b sed on DGD

statistic , either l ne r average or RMS

Se 4.6 for information on freq en y ran e an in rement The dif ferent publ cation have

req irements f or the freq en y in rement b sed on a req irement that the output Stokes

rotation from one freq en y to the next s ould not ex e d 18 ° (on the Poin aré sphere), but

con iderin secon order PMD, this l mit is to large f or ran omly coupled tran mis ion

media In te d, a rule b sed on a p rent contin ity of the res lt s ould b u ed In an case,

the larger the PMD b in me s red, the smal er the in rement s al b

For FA an interf erometric method , the PMD as a whole is re orted For interferometric

method , the PMD is re orted as RMS DGD value For FA, PMD is re orted as l ne r average

DGD Eq ation (2 ) an (2 ) can b u ed to con ert either way

Ta le 1 s ows whic test method are defi ed in whic International Stan ard An X denotes

a normative method; a Y denotes an informative method

Table 1 – Map of te t methods a d International Sta dards

NOT Whie WSOSA h s a s p rate cla s , it is c te oriz d a a Sto e Parameter Ev lu tio meth d

b c u e of th simiarity to th oth rs in this c te ory It is a fe u n y d main meth d th t is b s d o

diff ere c s b twe n Sto e c ara teristic of a ja e t fe u n y p irs

5.2 Stok s parameter e aluation

• bro db n source an tu e ble f ilter that could b u ed f or either PSA or JME;

• tu e ble laser that could also b u ed f or either PSA or JME

Either the f ilter or tu e ble laser is s an ed acros the f req en y ran e, ϖ1 to ϖ2, in

in rements of ∆ω At e c freq en y, the p larization control er is swe t throu h thre input

l ne r SOPs: θ = 0, π/4, an π/2, with µ = 0 (se Eq ation (6) an (10) The normal zed

output Stokes vectors as ociated with these input states are me s red an cal ed H

, resp ctively

The rotation of the output SOP from one freq en y to another is ded ced u in either the

Jones formulation of Eq ation (16), or the eq ivalent rotation on the Poin aré sphere directly

u in calc lation in 5.2.2 an 5.2.3 These calc lation method are exactly eq ivalent for

DGD

The DGD is re orted f or e c adjacent p ir of f req en ies, an f i al y, the PMD is calc lated

u in either the RMS or l ne r average of the DGD values

The output Stokes vectors are con erted to complex Jones vectors:

Eq ation (6) an (10) for e c f req en y u in an as umption that θ π

The T matrices are calc lated f or e c freq en y Given the f ol owin , one mig t thin that

the elements of the T matrix could b taken directly from the elements of the horizontal an

vertical outputs:

01

Tvh





(3 )

Due to the one π ambig ity in the con ersion of Stokes vectors to Jones vectors, Eq ation

(3 ) is not q ite a val d way of estimatin T In te d, the f ol owin ratios, whic are not

af f ected by the ambig ity, are calc lated:

yx

hh

k =

1

yx

vv

3

TT

TT

qq

k

yx

++

=

=

110

31

23

4

TT

kk

kk

−

−

For the last two eq ation in Eq ation (3 ), the terms on the lef t are n mb rs an the terms

on the f ar rig t are relation hips The T matrix estimate is now given as the f ol owin :

yxyx

vvvh

vvvh

k

kkk

T

//

//

1

~

4

241

(3 )

It is stated that the actual T matrix is eq al to this estimate times an arbitrary complex

con tant

The first matrix expres ion of Eq ation (3 ) provides a n merical estimate of T The secon

matrix expres ion s ows that the my teriou con tant is ±v

y This expres ion res lts f rom

exp n in the expres ion f or k

4 The one π ambig ity is now isolated to the sig of the matrix,

whic can b arbitrary Given a seq en es of f req en ies, the sig s could b c osen for

contin ity

NOT Mo t impleme tatio s of th JME d n t c ry o t th multiplc tio b ±v Th y d n t n e to

The freq en y ma pin matrix, J, of Eq ation (14) is now given as

00

00

01

ϖω

∆ϖ

ϖω

∆ϖω

vv

jJj

yy







−+

In p ction of Eq ation (14) s ows that the a solute value of the arg ment of the ratio of

eigen alues of J is eq al to ∆ω∆ Sin e multipl cation by an arbitrary con tant do s not affect

the ratio of eigen alues, the ratio of the eigen alues of the expres ion u in the T estimates

in ide Eq ation (3 ) also yield the value of this ratio

5.2.3 Poinc ré sphere a aly is

This analy is tec niq e ded ces the rotation an le, ∆ω∆ directly

Sin e the input SOPs, 0 an π/2 (sphere co rdinates) are orthogonal, an the action of the

input to output T matrix is that of a rotation, the as ociated output SOPs s ould b orthogonal

on the sphere A third orthogonal normal zed Stokes vector re resentin what would b output

f rom circ lar input SOP is calc lated with the cros prod ct o erator The me s red output

SOPs f or b th freq en ies, ϖ

0

0+∆ω, are adju ted to en ure orthogonal ty with the

f ol owin

Hh





QH

QH

Vq

ccvqh







,,

is explained u in Fig res 7 an 8 Fig re 7 s ows a p s ible

relation hip of the PSP to the vectors o tained from Eq ation (3 )

Fig re 7 – Relations ip of orthogon l output SOPs to th PDV

Fig re 8 i u trates the rotation f rom one freq en y to the next f or an arbitrary output Stokes

vector

IEC h

ˆ

qˆ

qhc

ˆˆ

ˆ

×

=

Figure 8 – Stok s v ctor rotation with fre u nc c a ge

ϖω

∆ϖ

=

From ge metrical con ideration :

αsin

=

⊥s



αθθ

2sin2

2sin

=

⊥ss

∆

∆

∆

222

sin

2

sin4

hc

222

22

sinsin

sin

2sin

=++

cq

h

αα

αθ

22

22

c os

c os

c os3

=

(4 )

The el mination of the s m of cosine terms is d e to the cosine rule of thre orthogonal

vectors with resp ct to a common f ourth vector The s m of the cosine s uared terms is eq al

=

=

−

2/1

222

1

8sin

∆θ

IEC

(ω)

⊥sˆ

(ω+ ∆ω)

⊥sˆ

sˆ

∆

ωsˆ

(ω+∆ω)s

ˆ

−

−

2/1

222

12

/1

222

1

8sin

8sin

NOT It is e siy d mo strate th t th P A is als e uiv le t to th Mu ler matrix meth d

5.2.4 One e de me s reme ts ba e on SPE [3]

IEC 612 0-4-4 in lu es an informative me s rement method b sed on me s rements that

can b done from one en of the tran mis ion media It is b sed on analy is of the output

SOPs that are ref lected f rom the far en an re-tran mited b c toward the source where the

ref lected SOPs are me s red with a p larimeter

A directional coupler is in erted b twe n the p larization control er an the tran mis ion

media The tran mis ion media is at ac ed to the me s rement eq ipment with a low los

an led con ector The ref lected lg t go s throu h the directional coupler an into the

p larimeter

The f ar en of the tran mis ion media s al ref lect s f f icient lg t so a f lat en c t that is

p rp n ic lar to the f ibre en mig t ne d to b pre ared

The ref lected l g t is the s m of the Rayleig s aterin l g t an the l g t ref lected f rom the

f ar en The f ol owin l mitation mu t b evaluated:

• The f ar en ref lected p wer dominates This places a practical l mit on the len th

me s red

• The Rayleig s at erin wi normal y b de olarized d e to stron mode coupl n Degre

of p larization of 9 % or more is pref er ed, but degre of p larization as low as 2 % can

b me s red − with red ced ac urac

The b sis of this tec niq e is to write out the matrix, R

B, that ma s input Stokes to output

ref lected Stokes as Eq ation (4 )

MR

MRR

T

B

Where R is the forward tran mis ion matrix an Mis a diagonal matrix with elements (1,1,-1)

Like f orward tran mis ion, the DGD ef f ect is ded ced from the derivative expres ion:

00

0

ss

R

dd

dds

BT

BB

×

=

ωω

21

ΩΩ

Sin e the action of the rotation in Eq ation (4 ) have no ef fect on the len th of the res ltin

vector, we can se that, in the context of stron mode coupl n , the exp cted value of the

len th of this vector is related to the PMD RMS, hen e lne r average PMD, as Eq ation (4 )

22

3

24

τ

∆πτ

∆τ

BB

τ

∆

πτ

(4 )

Combining Eq ation (4 ) an (4 ) yield Eq ation (4 )

Bτ

∆

πτ

∆2

M mo ulator (o tio al)

PC p lariz tio c ntroler

A

Ref

TL tu e ble la er

M mo ulator (o tio al)

PC p lariz tio c ntroler

Pol p lariz r

ORx o tic l re eiv r

G HF o ci ator tu e ble in ra g 0,01 to 10 GHz

A ampltu e / p a e me s reme t a aly er

Fig re 10 – Setup f or p larization ph s s if t

For these method , there is a s an of o tical f req en y yieldin a DGD v freq en y plot For

e c o tical f req en y, there are a n mb r of input SOPs that are lau c ed For e c lau c ,

the phase s if t of a mod lated o tical sig al is me s red F r the MPS me s rements, e c

o tical f req en y res lts in a DGD value For the PPS me s rement, e c p ir of adjacent

o tical f req en ies prod ces a DGD value

The main dif f eren e in the eq ipment is the p larization b am splt er u ed in the PPS

method

The in ivid al test method stan ard have req irements on the mod lation f req en y that

de en on the maximum DGD to b me s red an the minimum resolution

5.3.2 Modulation ph s s if t – Ful s arc

The input SOP is varied over the ful Poin aré sphere as p irs of orthogonal states The

phase diff eren e for e c orthogonal p ir is calc lated The switc in b twe n orthogonal

p irs may b mod lated so phase dif f eren e me s rements wi not b af f ected by thermal

drif t

The se rc is for the input p ir that prod ces the maximum phase dif feren e

When the two orthogonal p irs are al g ed with the f ast an slow PSPs, the phase dif f eren e

is maximal A f i e degre dif f eren e b twe n the true PSPs an the input SOPs wi yield a

DGD er or of a proximately 0,4 % This forms a l mit on the se rc den ity

The DGD (ps) is o tained f rom the maximum phase dif f eren e, ∆

ma(rad), an the

mod lation freq en y, f(GHz) as

fπ