Modal dynamics in multimode fibers

0.51 MB·

by  Moti Fridman

Modal dynamics in multimode fibers
Moti Fridman, Haim Suchowski, Micha Nixon, Asher A. Friesem, and Nir Davidson∗
Weizmann Institute of Science, Dept. of Physics of Complex Systems, Rehovot 76100, Israel
∗Corresponding author: [email protected]
CompiledJanuary28,2011
Thedynamicsofmodesandtheirstatesofpolarizationsinmultimodefibersasafunctionoftime,space,and
wavelengthareexperimentallyandtheoreticallyinvestigated.Theresultsrevealthatthestatesofpolarizations
are displaced in Poincare sphere representation when varying the angular orientations of the polarization at
the incident light. Such displacements, which complicates the interpretation of the results, are overcome by
resortingtomodifiedPoincarespheres representation.Withsuchmodificationitshouldbepossibleto predict
1 the output modes and their state of polarization when the input mode and state of polarization are known.
1 (cid:13)c 2011 OpticalSocietyofAmerica
0 OCIS codes: 140.3290,140.3510.
2

n
a 1. Introduction sentation can be simplified to obtain a more convenient
J andintuitiveunderstanding.Finally,basedontheresults
7 Multimodefibershavehighcouplingefficiencies,canop- from the model, we show how to obtain such simplified
2 erate over a wide range of wavelengths, and have rela-
representation experimentally.
tively low susceptibility to degrading nonlinear effects.

]
s As a result they are widely used in mid and short com- 2. Experimental configuration and results
c municationranges,highpowerlasersandamplifiers,and
i The experimental configuration for measuring the out-
t in transporting high optical powers from one location to
p put polarization from a multimode fiber is presented in
another [1,2]. Yet, while in single mode fibers the po-
o
Fig. 1. A linearly polarized light from a tunable fiber
. larization dispersion is low and can be significantly re-
s laser propagates through a half wave plate and a quar-
duced either by using polarization maintaining fibers or
c
terwaveplateinordertoobtainlightofanydesiredSOP
i by operating at the principal state of polarization [3–9],
s and wavelength at the input. The light is then launched
y in multimode fibers the mode dispersion is high result-
into a multimode fiber, and the SOP at the output as a
h ing in significant reduction of performance [10,11]. Con-
p sequently, it is important to continuously analyze and functionofspace,timeandwavelengthismeasuredusing
[ a real-time space-variant polarization measurement sys-
monitor the dynamics of the modes propagating in mul-
tem [16]. In our experiments the multimode fiber length
2 timode fibers.
v When analyzing the dynamics of modes in multimode was 10m, core diameter 20µm and numerical aperture
7 0.07. It mainly supported the TE and TM modes
fibers, it is common to separate the dynamics of the 01 01
0 which in the weakly guided approximation can be ex-
transverse modes and the dynamics of the states of po-
2
pressed as orthogonal sets of linearly polarized modes.
larization [12–15]. Unfortunately, since there is coupling
1
Theelectricfieldintensitydistributionsandpolarization
. between transverse modes with different states of po-
2 directions for these modes are presented in Fig. 2. As
larization, such a separation is not natural and results
1
shown, there are four distinct modes of LP , each with
0 in coupled equations that are difficult to analyze. Here 11
differentintensitydistributionordifferentorientationsof
1 we present a new approach where the complicated dy-
: namics of the modes in multimode fibers are separated the polarization direction. The white arrows at the cen-
v
ter of the individual intensity distributions denote the
i and represented by two uncoupled equations of motion
X direction of the polarizations in each mode.
that lead to simple geometric plots on modified and un-
r coupled Poincare spheres. Such plots fully describe the
a
modal dynamics in multimode fibers, and can be con- HWP QWP Multimode fiber
veniently used for monitoring, analyzing and predicting Tunable fiber Real time
laser polarization
them. measurement
We begin by describing our experimental configura-
tion and present experimental results of the state of po- Fig. 1. Experimental configuration for measuring the
larization (SOP) at the output from multimode fiber as output polarization from a multimode fiber as a func-
a function of wavelength for different angular orienta- tion of the input wavelength for different input states of
tions of light with linear polarization at the input. The polarization.
results reveal that the dynamics of the modes and their
SOP at the output from multimode fibers are relatively Westartedourexperimentsbyverifyingthatthecou-
complicated.Thenwedevelopedananalyticalmodelfor plingintothefiberdoesnotchangethemodeortheSOP
calculating these dynamics, and show how their repre- of the input. This was done by determining that the in-
1
E E
1 2
u °
2 60
°
50
30° 40°
°
u u 20
3 3 10°
E3 E4 0° S
3
S
u 2
2
S
1
(a) (b)
Fig.3.PoincaresphererepresentationoftheSOPatthe
Fig. 2. Intensity distributions together with the orien- output as a function of the wavelength for different an-
tations of the polarization direction for the orthogonal gularorientationsofthelinearpolarizationattheinput.
sets of linearly polarized modes LP11. The white arrows (a) Experimental results; (b) calculated results. The in-
inthecenteroftheindividualintensitydistributionsde- putwavelengthrangedfrom1062nmto1066nmforeach
note the polarization directions. The black arrows and angular orientation.
the terms u and u denote the coupling between the
2 3
different modes. 1
3
S
n
o
toeuntspiutyt adriesttrhibeustaiomneoafstthheosmeoadtethaendintphuet SafOtePr partotphae- zation 0.5 30°
gatingthroughashort7cmfiber.Then,welaunchedthe ari
ol
samelinearlypolarizedlightwithauniformintensitydis- ut p 90°
tributionintothe10mlongfiber,andmeasuredtheSOP p 0
ut
at the output while varying the input wavelength from e o
h
1062nm to 1066nm. We repeated these measurements, of t
each time after rotating the linearly polarized light at n −0.5
o
the input by 10◦. cti
e
TheresultsarepresentedinFig.3asaPoincaresphere oj
representationoftheSOPattheoutputfordifferentan- Pr
−1
gularorientationsofthelinearpolarizationattheinput. 1062 1063 1064 1065
Input wavelength [nm]
As evident, the circle of smallest size is obtain when the
linearly polarized input beam is oriented at 30◦, indi-
Fig. 4. Projection of the output polarization on the S
cating where the SOP of the output is nearly invariant 1
axes as a function of the input wavelength for two ori-
totheinputwavelengths.Namely,atanangularorienta-
tionofabout30◦ wehaveaprincipalmode(PM).Asthe entationsoftheinputpolarization;asterisks(blue)-30◦
deviation from 30◦ increases, the circles become larger orientation, close to the PM; circles (red) - 90◦ orienta-
tion, far from the PM.
indicating that more spectral dispersion occurs. Unlike
increasing concentric circles around the principal state
ofpolarizationlocationinsinglemodefiber,hereweob- modes composed from a superposition of the four fields
tained increasing circles whose centers continuously de- E = [E ,E ,E ,E ]T shown in Fig. 2. The re-
in,out 1 2 3 4 in,out
viate from the PM location as the difference between lation between them is
the input and the PM increases. We also present the
projectionoftheoutputpolarizationontheS1 axesasa Eout =eiβUEin, (1)
function of the input wavelength for two input polariza-
where β is real and U is a unitary matrix. To explic-
tioninFig.4.Asevident,whentheinputpolarizationis
itly define U, we assume that birefringence and twists
close to the PM the output polarization is less sensitive
in the fiber are the main sources for coupling between
to wavelength variations.
the modes. Birefringence in the fiber leads to coupling
of modes with the same intensity distributions but with
3. Model and calculated results
orthogonal SOP. Accordingly, the coupling between the
We developed a model for calculating the dynamics fieldsE1 andE2 canbedescribedbya2by2matrix,as
of the SOP at the output of multimode fibers, in or-
(cid:20) (cid:21)
u (ω) u (ω)
der to gain a better physical intuition about the evo- U(2)(ω)= 1 2 , (2)
−u∗(ω) u∗(ω)
lution of modes in a multimode fiber and to under- 2 1
stand what causes the deviations from the PM loca- whereu denotechangesinthephaseofeachmodewhile
1
tion. We define E and E as the input and output propagatinginthefiberandu thecouplingstrengthbe-
in out 2
2
tweenthemodes.Thismatrixisthesameasthatforcou- where
pSdliiimsntrigliaborurlyttih,oosnigsnocnaeanlEdp3aoaslsanurdmizEaint4igoantlsshoainthabsviinreegtflrheienmgseoandmceeefiiibnsteeersnsss[ei5nt]y-. ΩL(ω)=(cid:34) −|u˜τcb∗3o|ss(i|nτb(||ωτb)|ω) |u˜τbc3|ossi(n|τ(b|τ|ωb|)ω) (cid:35)
tiallyconstantacrossarelativelysmallfibercore[17,18],
and
we can describe the coupling between E and E with
3 4
the same unitary matrix U(2)(ω). Twists in the fiber ro-
Ω (ω)=
R
taantdebtheetwmeeondeEs2anadndleEad4swtoithcocuopulipnlginbgesttwreeenngtEh1daenndotEed3 (cid:34) cos(|τa|ω)+ |uτ˜a1|sin(|τa|ω) −|τu˜a∗2| sin(|τa|ω) (cid:35)
as u3. We also assume that coupling between E1 and |uτ˜a2|sin(|τa|ω) cos(|τa|ω)− |uτ˜a1|sin(|τa|ω)
E and between E and E are negligible. Taking bire-
4 2 3
with τ ≡(b ,b ,a ) and τ ≡(0,a ,b ).
fringence and twists into account, the overall coupling a 1 2 2 b 3 3
Equation (9) provides an analytic solution of the out-
matrix between all four fields is then U(ω) = exp(H)
put mode distribution and its SOP as a function of the
where H is skew-Hermitian matrix as
wavelength when the input SOP is known. It involves a
 
u (ω) u (ω) u (ω) 0
1 2 3 multiplication of two matrices Ω and Ω . The matrix
H(ω)= −−uu∗3∗2((ωω)) u∗10(ω) u10(ω) uu32((ωω)) . (3) pΩlRingisbceotmwepeonseedacfhropmaiur1ofamndodue2ssRwoiitthrtehpereLssaemntesitnhteencsoiuty-
0 −u∗(ω) −u∗(ω) u∗(ω)
3 2 1 distributionsbutwithdifferentSOPwhereasthematrix
Following the derivation of Poole and Wagner [5] and ΩL is composed only from u3 so it represents the cou-
extending it to our multimode fibers [14,15], leads to plingbetweeneachpairofmodeswithdifferentintensity
four principal modes (PMs) where each is insensitive to distributionsanddifferentSOP.UsingEq.(9),wecalcu-
variations in the input wavelength up to the first order. latedtheevolutionoftheoutputSOPasafunctionofthe
ThefourPMs,denotedas(cid:15) wheren=1,2,3and4,are input wavelength for different angular orientations of a
n
the solutions of linearlypolarizedlightattheinput.Theresultsarepre-
sentedinFig.3,withthecouplingparametersu ,u and
1 2
[U(cid:48)−iknU](cid:15)n =[H−iknI]U(cid:15)n =0, (4) u3, as fitting parameters. As evident, there is a qualita-
tiveagreementbetweentheexperimentalandcalculated
where the prime denotes differentiation with respect to
results, indicating that our assumptions in developing
frequency, and the four k are
n
the model are justified. We attribute the difference be-
(cid:113)
(cid:112) tweenthe calculated results and the measured results to
k =±i |u |2+|u |2+|u |2±2|u | |u |2+|u |2.
n 1 2 3 3 1 2
smalldeviationsinthepolarizationmeasurementandto
(5)
inaccurate coupling parameters.
These four PMs will propagate in the multimode fiber
with minimal dispersion.
4. Modified Poincare spheres representation
To analyze the dynamics of modes other than PMs
as a function of the input wavelength, we differentiated In order to simplify the Poincare sphere representation
Eq.(1)withrespecttofrequency,andusedthedefinition of the SOP dynamics at the output, we resorted to
of U to obtain modified Stokes parameters S ≡ (S(1),S(2),S(3)) and
a a a a
dE S ≡(S(1),S(2),S(3)) where
out =HE . (6) b b b b
dω out
S(1) =|E |2−|E |2+|E |2−|E |2
Using Pauli matrices, σ , σ and σ , we can decompose a 1 2 3 4
x y z S(2) =2|E ||E |cos(ϕ )+2|E ||E |cos(ϕ ) (10)
H to a 1 2 12 3 4 34
S(3) =2|E ||E |sin(ϕ )+2|E ||E |sin(ϕ )
a 1 2 12 3 4 34
H=(ib σ +ia σ +ib σ )⊗I+I⊗(ib σ +ia σ ),
1 z 2 y 2 x 3 x 3 y
(7) and
where ai denote the real part of ui and bi the imaginary S(1) =|E |2+|E |2−|E |2−|E |2
part of u . Equation (7) defines rotation in four dimen- b 1 2 3 4
i S(2) =2|E ||E |cos(ϕ )+2|E ||E |cos(ϕ ) (11)
sions, one part for left rotation and the other for right b 1 3 13 2 4 24
rotation [20,21]. Now, in order to find analytic solution Sb(3) =2|E1||E3|sin(ϕ13)+2|E2||E4|sin(ϕ24)
to Eq. (6) we redefine the vectors E and E to a 2
in out where ϕ denote the phase difference between E and
by 2 matrices, ij i
E . Now, Eq. (6) can be separated into two uncoupled
j
(cid:20) (cid:21)
E E torque equations, as
E˜ = 1 2 , (8)
in,out E E
3 4 in,out S˙ =τ ×S
a a a (12)
and substituting Eqs. (7) and (8) into Eq. (4) leads to S˙ =τ ×S .
b b b
the solution of E˜ as a function of frequency as
out
Note, the torque vectors depend only on the parameters
E˜ (ω)=Ω (ω)E˜ Ω (ω), (9) of the fiber and not on the input SOP.
out L in R
3
Using the two sets of modified Stokes parameters we
recalculated the evolution of the output SOP as a func-
tion of the input wavelength for different angular ori-
entations of a linearly polarized light at the input. The
results are presented in Fig. 6. As evident, the use of
the modified Stokes parameters and the separation of
Eq. (6) into two uncoupled equations, resulted in two
modifiedPoincaresphereswheretheSOPsineachsphere
formconcentric,ratherthandisplaced,circlesaroundthe (a) (b)
PM when varying the input wavelength. The modified
Poincare spheres representations involve natural modes Fig. 6. The average polarization when varying the in-
ofafiber.Specifically,S representsthemodesTE and put wavelength for different input polarizations on (a)
a 01
TM , and S represents the modes HE and HE . the original Poincare sphere and on (b) the modified
01 b 21 −21
Theseresultsindicatethatwhenevaluatingthepropaga- Poincare sphere.
tionoflightinmultimodefiberitispreferableandmore
convenienttoresorttoouruncoupledmodifiedPoincare
spheres representation rather than the usual Poincare 5. Concluding remarks
spheres representation [12,13].
To conclude, we presented experimental and calculated
results about the dynamics of modes and their SOP in
multimode fibers. We show how the representation of
thesedynamicscanbesimplifiedsoastoobtainadirect
measure of the SOP at the output when knowing the
inputSOP.Theresultsofthismeasurecanthenbeused
S3 S3 for determining the SOP for an arbitrary input SOP.
aS2 bS2
a b
S1 S1 Acknowledgements
a (a) (b) b
This research was supported by the Israeli Ministry of
Fig. 5. Calculated modified Poincare sphere representa- Science and Technology and by the USA-Israel Bina-
tion of the SOP at the output as a function of the wave- tional Science Foundation.
length for different angular orientations of linearly po-
larized light at the input. (a) Using the modified Stokes
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