Cross talk analysis in multicore optical fibers by supermode theory

Abstract

We discuss the theoretical aspects of core-to-core power transfer in multicore fibers relying on supermode theory. Based on a dual core fiber model, we investigate the consequences of this approach, such as the influence of initial excitation conditions on cross talk. Supermode interpretation of power coupling proves to be intuitive and thus may lead to new concepts of multicore fiber-based devices. As a conclusion, we propose a definition of a uniform cross talk parameter that describes multicore fiber design.

Full text

Cross talk analysis in multicore optical fibers by
supermode theory
LUKASZ SZOSTKIEWICZ,2,*MAREK NAPIERALA,1,3 ANNA ZIOLOWICZ,1,3 ANNA PYTEL,1TADEUSZ TENDERENDA,1
AND TOMASZ NASILOWSKI1
1InPhoTech sp. z o. o., 17 Slominskiego St 31, 00-195 Warsaw, Poland
2Polish Centre for Photonics and Fibre Optics, Rogoznica 312, 36-060 Glogow Malopolski, Poland
3Faculty of Physics, Warsaw University of Technology, 75 Koszykowa St, 00-662 Warsaw, Poland
*Corresponding author: lszostkiewicz@pcfs.org.pl
Received 12 April 2016; revised 7 July 2016; accepted 20 July 2016; posted 20 July 2016 (Doc. ID 262939); published 4 August 2016
We discuss the theoretical aspects of core-to-core power
transfer in multicore fibers relying on supermode theory.
Based on a dual core fiber model, we investigate the con-
sequences of this approach, such as the influence of initial
excitation conditions on cross talk. Supermode interpreta-
tion of power coupling proves to be intuitive and thus may
lead to new concepts of multicore fiber-based devices. As a
conclusion, we propose a definition of a uniform cross talk
parameter that describes multicore fiber design. © 2016
Optical Society of America
OCIS codes: (060.0060) Fiber optics and optical communications;
(060.2310) Fiber optics; (060.2400) Fiber properties; (060.2280)
Fiber design and fabrication.
http://dx.doi.org/10.1364/OL.41.003759
Multicore optical fibers (MCFs) are considered to be the can-
didate of choice for the next generation of telecommunication
networks based on space division multiplexing [1], and in the
next few years MCF-based networks will undoubtedly be
implemented. They are also proposed for sensing applications
[2], in which they enable building, e.g., shape sensors [3]or
multiparameter sensors [4]. To allow independent propagation
of signals in the cores, they are isolated by different means, such
as their distancing, differentiating [5], insulating by trench
[6,7] or airholes [8–10] or a combination of these techniques
[11]. The parameter that describes the isolation of the cores in a
quantitative manner is core-to-core cross talk (XT). Although
XT is an intuitive parameter—it defines the relative amount of
power that is transferred from the excited core to another
core—there are many discrepancies in the literature in terms
of XT definition, measurement methodology, and even the unit
in which it is expressed [12–14]. As a consequence, there is no
reliable way of designing MCFs in terms of the XT parameter,
which is confirmed by disagreement between the XT values
calculated and measured for fabricated fibers [15].
This article focuses on a model of XT, which is free from
simplifications and is suitable for calculating XT in complex
MCFs, regardless of the core number and the way of their
isolation. The model is based on the basic fact that any light
field can be represented as a sum of orthogonal states. In a vac-
uum, any light field can be represented as a sum of plane waves
[16]. The same methodology stays in force when one considers
light distribution in a complex medium. The only difference is
that there are different solutions of the wave equation, especially
in waveguides, in which the role of a plane wave is carried out by
modes. As the space is no longer infinite, the number of
solutions (modes) that can be supported by the given structure
is limited. All modes are orthogonal and their superposition rep-
resents an actual field distribution in a fiber. This also applies to
MCFs, in which we are talking about supermodes [17]insteadof
modes. Supermodes have different propagation constants just
like modes in standard multimode fibers, and therefore their
superposition produces a different power distribution along
the fiber. In theory, this is the only source of XT in MCFs.
In actual fibers, there is also a second origin of XT, namely,
the power coupling between supermodes, which results from im-
perfections in the fiber, and is thus random in nature. Since the
modes’coupling is strongly dependent on the fiber fabrication
process, its influence on XT cannot be quantified at the stage of
fiber design. In addition, XT induced by the coupling of superm-
odes plays an important role mainly when analyzing propagation
in MCFs with strongly isolated cores over very large distances.
Since at the stage of fiber design only XT induced by the
interference of supermodes can be taken into account, in
the further considerations we consider a theoretical case, in
which the power coupling between the modes is not allowed.
In such a case, the transfer of power between cores is periodic
and can be characterized by two parameters, namely, the maxi-
mum value of cross talk (XTmax) and the beat length (Lb).
XTmax is expressed in decibels for any pair of cores, and this
parameter is independent of the fiber length.
The methodology for calculating XTmax in an arbitrary
structure is presented below. First, one has to solve the follow-
ing wave equation directly:
ΔEx;y−k2
0n2λ;x;yEx;y0;(1)
where Ex;ystands for the electric field amplitude at a certain
point of the fiber, k0is the wave vector, and nλ;x;yis the
Letter Vol. 41, No. 16 / August 15 2016 / Optics Letters 3759
0146-9592/16/163759-04 Journal © 2016 Optical Society of America

refractive index at a certain point and at a given wavelength λ.
Equation (1) is solved for the eigenvalue-iβ. As a result, one can
obtain profiles of supermodes in MCFs. Each single mode core
guarantees the existence of two orthogonally polarized superm-
odes, which are locally similar to the fundamental modes of a
single-core structure. Supermodes can take many various
forms (Fig. 1).
The electric field amplitude at any point can be expressed as
a superposition of all supermodes propagating in the MCF:
Ex;y;zX
n
j1
AjEjx;ye−iβjziϕj;(2)
where Ex;y;zis the electric field amplitude at a certain point
of the fiber, Ejx;yis the supermode field distribution func-
tion, which is directly derived from Eq. (1); βjis the propaga-
tion constant of the jth supermode; ϕjis the initial phase of jth
supermode; Ajrepresents the relative jth supermode amplitude
and is expressed as the overlap integral between the initially
injected wavefront (indexed with p) and jth supermode:
Aj
Re"RR Ep×H
jdSRR Ej×H
pdS
RR Ep×H
pdS#1
ReRR Ej×H
jdS
:
(3)
The level of each supermode excitation may be different and
is strongly dependent on the initial field distribution. By exam-
ining the Pcore-nzfunction
Pcore-nzZZ dScore-nX
k
j1
AjEjx;ye−iβjziϕj2
;(4)
which indicates the amount of power present in a specific core
of the MCF (with an area of Score ), it is possible to obtain
detailed characteristics of the power transfer between the cores
over the fiber length.
Thus, the final part of determining XTmax is to find
the maximum power resulting from the superposition of the
supermodes in the core, which is not directly excited.
Therefore, XTmax [dB] is defined as the relation between the
maximum power level in the investigated core (indexed
with n) and the maximum power level in the initially excited
core (indexed with m) according to the equation
XTmax 10 log maxPnz
maxPmz :(5)
Beat length (Lb) is then defined as the propagation length over
which the initial phase difference between supermodes is recon-
structed. In other words, power distribution in a fiber cross
section is always periodic, and the beat length is understood
as being this period.
Considering the strict model of core-to-core power transfer
presented, it is worth noting that analyzing XT at one point of
the MCF is not viable. Even if a multicore fiber is treated asbeing
an extremely weakly coupled coupler and XTmax is low, the phe-
nomenon is never a linear function of fiber length. To measure
XTmax, one ought to find the point of maximum power transfer.
For multicore fiber components, which have strictly defined op-
tical length, it is necessary to distinguish between XTmax of the
used fiber and the observed XT of the fiber component.
Equation (5) and beat length characterizes a two-dimensional
MCF structure and is thus independent of fiber length.
Both XTmax and Lbare highly dependent on structure
parameters and wavelength, but the result of the supermode
superposition also depends strongly on the chosen pair of cores,
so for MCFs with more than two cores, it is important to note
between which cores the XTmax is being calculated or mea-
sured. Furthermore, XTmax is not the only value that depends
on the chosen pair of cores—the same situation is observed in
the terms of Lb, which also varies for different pairs of cores.
This results in the different lengths over which the initial phase
difference of supermodes is reproduced.
The presented model is suitable for any number of cores,
but for a detailed discussion, we will analyze the behavior of
supermodes in dual-core fiber. The model being investigated
consists of two cores separated by an area of decreased refractive
index—naand diameter—da(Fig. 2). To clarify the influence
of splice imperfections to XT, we also introduce an initial field
offset directed to the center of the fiber (Fig. 2.). Generally, any
effects resulting from changing the refractive index naare in-
deed equivalent to distancing the cores or changing the diam-
eter of the area with decreased refractive index (da). All
calculations were performed for the wavelength of 1550 nm
by the finite element method with at least 4·105finite ele-
ments to ensure convergence. The initial field distribution is
considered as a fundamental mode of a single-core fiber with
a core radius of 4.1 μm and core doping of 3.5 mol. % GeO2.
A dual-core structure supports two orthogonal supermodes,
namely, symmetric and antisymmetric (Fig. 3). An effect of the
interference of these supermodes is presented in Fig. 4. By def-
inition, two orthogonal supermodes possess different phase
velocities, which results in periodic switching of power between
cores along the fiber. Therefore, although there is no power
coupling between supermodes, their superposition results in
core-to-core cross talk. In the presented case (Figs. 3and 4),
strong power transfer between cores arises from a large spatial
overlap of excited supermodes. Symmetric refractive index
distribution supports the symmetric and antisymmetric
supermodes and will always lead to a complete power transfer.
According to the definition of XTmax (5), such a fiber has
Fig. 1. Normalized electric field amplitude distribution of exem-
plary supermodes (2 of 14), which are present in seven-core fiber with
airhole isolation [9].
Fig. 2. Dual-core fiber model with dcore 8.2 μm,da6μm,
2Λ20 μm, and variable na.
3760 Vol. 41, No. 16 / August 15 2016 / Optics Letters Letter

XTmax equal to 0 dB, which means that the power transfer is
complete over Lb∕2—the entire power that is introduced to
one core is transferred to the other one.
So far we have discussed the symmetric case, in which sym-
metric refractive index distribution forces the mode profile to
be symmetric or antisymmetric as the following relation implies
for all supermodes in the fiber:
E2
xE2
−xif nxn−x:(6)
Thus, only propagation constants and consequently Lbare
changed while increasing isolation of the cores. When we decide
to break the symmetry in the fiber refractive index distribution, it
will lead us to a nonidentical supermode power distribution. In
Fig. 5, the case is presented in which the symmetry is broken
by a slight change of one core diameter (dcore1 8.18 μm)with
respect to the other (dcore2 8.2 μm). This in turn causes that
quasi-symmetric and quasi-antisymmetric supermodes begin to
be clearly distinguishable, not only in terms of intensity but also
powerdistribution.Those nonsymmetric modeprofilescan be fur-
ther differentiated by decreasing na[Figs. 5(a),5(b),and5(c)].
When refractive index distribution supports quasi-
symmetric or quasi-antisymmetric supermodes, core-to-core
power switching is not complete (Fig. 6). In such case, accord-
ing to the definition, XTmax is no longer equal to 0 dB but
obtains lower values.
As symmetry breaking introduces spatial nonuniformity
of the power distribution of quasi-symmetric and quasi-
antisymmetric supermodes, their overlap integral with the ini-
tial field introduced to the fiber is no longer equal. This leads to
different XTmax values for various initial excitation conditions,
e.g., splicing with the offset (see Fig. 7) or with different fibers.
Figure 8presents XTmax for different isolation (understood
as refractive index Δof the isolating part) of cores as a function
of excitation offset. One can find that increasing the difference
in the spatial power overlap of supermode power profiles (e.g.,
by decreasing na) causes XTmax to be more sensitive to the
change in the excitation field.
Another important aspect, which is worth mentioning when
discussing XT in MCFs, is the influence of fiber bending on
XT. Unless we consider a diabatic transformation of the fiber,
the bending changes the intensity distribution of each super-
mode without power transfer between supermodes. It also
changes the difference between the effective refractive indices
of supermodes and thus also the beat length of the power trans-
fer between cores. One may consider this as a bend-induced
cross talk (due to the change of the beat length, the power
distribution at the fiber output is also changed) but, in fact,
-1
-0.5
0
0.5
1
-1.00
-0.50
0.00
0.50
1.00
-30 -15 0 15 30
Δrefractive index [%]
normalized electric field
amplitude [a.u.]
horizontal axis [µm]
symmetric supermode
antisymmetric supermode
refractive index
Fig. 3. Symmetric and antisymmetric supermodes’profiles with the
marked refractive index profile supporting them. The nain this case is
equal to the refractive index of the fiber cladding, and the cores have
exactly the same refractive indices.
Fig. 4. Power transferring for the structure presented in Fig. 3.
(a) Pictorial power distribution in particular cores over fiber length.
(b) Normalized power in particular cores over fiber length expressed
in beat lengths.
Fig. 5. Quasi-symmetric and quasi-antisymmetric modes’profiles
and refractive index profiles. The difference in dcore between the cores
is at the level of 0.25%. The relative difference between naand the
refractive index of the cladding is equal to 0%, −0.07%, and
−0.97% for graphs (a), (b), and (c), respectively.
Letter Vol. 41, No. 16 / August 15 2016 / Optics Letters 3761

as long as we couple light into the straight part of the fiber and
capture light from the straight part of the fiber, which is almost
always the case, there is no change in the maximum level of
XT, which can be observed at the fiber output.
In conclusion, there are two sources of core-to-core power
transfer in MCFs. One source is fiber imperfections, which re-
sult in the coupling of supermodes between each other and is
sometimes discussed from the statistical point of view [18]. The
second origin is a result of the superposition of supermodes
having different propagation constants, which is the key
consideration while designing MCFs.
In this Letter we have discussed the description of core-
to-core power transfer with the use of two parameters
XTmax and Lb. In addition, we have demonstrated a way of
calculating these parameters. The method is based on the analy-
sis of the supermode interference and opens up new possibilities
for designing and analyzing different MCF structures with
complex refractive index distribution. The model presented
does not require high computational power as it relies on
two-dimensional calculations. At the same time, it is free from
simplifications, which means we can obtain exact results.
Moreover, perturbations such as fiber bending or splice offset,
which play a crucial role in fiber performance, could also be
easily investigated. Since the method relies on fully vectoral
solutions it can also take into account polarization effects
and is not limited to low-refractive-index contrast structures.
Funding. Narodowe Centrum Nauki (NCN) (2013/09/D/
ST7/03961); Ministerstwo Nauki i Szkolnictwa Wyższego
(MNiSzW) (DI2013 019343); Polska Agencja Rozwoju
Przedsiębiorczości (PARP) (POIG.01.04.00-06-017/11-00).
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Fig. 6. Power transferring for the structure presented in Fig. 5(c).
(a) Pictorial power distribution in particular cores over fiber length.
(b) Normalized power in particular cores over fiber length expressed
in beat lengths.
Fig. 7. Power transferring for the structure presented in Fig. 5(c) for
two different excitation conditions (offsets).
Fig. 8. Initial field introduction offset increases the XTmax value.
3762 Vol. 41, No. 16 / August 15 2016 / Optics Letters Letter