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# [10GMMF] Polarization effects for 10GBASE-LRM

```David,

following the long and interesting conference call held this afternoon
on Task 4, I would like to clarify the background theory I shortly
sketched during the discussion. I appreciated your comments and I
understand you disagree on almost any polarization-dependent effect in
MMF transmission, at least assuming circular symmetric refractive index
without any birefringence. Nevertheless, since I am fully convinced of
its relevance, at least starting from our experimental evidence, please
find below some more detailed reasoning.

Let us review first few basic assumptions of optical fiber modal theory:

*       The fiber has circular symmetry, a circular cross section and
straight line geometry. No bending effect nor birefringence, neither
core ovality. I am considering the "classical" MMF model of every
textbook assuming the refractive index is a scalar quantity depending on
the radial coordinate only.
*       The exciting electric field is linearly polarized. Let us choose
the Cartesian reference system to represent the vector field components
Ex, Ey and Ez and let us choose the cylindrical (r,phi,z) coordinate
system for representing the position. Let us define the x-axis along the
polarization axis. Modes are the eigensolution of the scalar wave
equation (Helmoltz equation)
*       The modal field can be separated into the product R(r)F(phi) of
the radial component R(r) by the harmonic term F(phi) (sine and cosine).
The latter dependence is only due to the circular symmetry. The radial
dependence of the refractive index infer only on the radial component of
the field.
*       The whole bound modes set constitutes a complete orthonormal
basis for representing any bound energy propagation. Each LP(l,m) mode
presents a degeneracy of order 2 according to the two allowable "sine"
and "cosine" solutions. I would prefer to avoid confusion in identifying
them as two orthogonal polarizations. For the assumed polarization
(x-axis oriented), the axial symmetry produces an intrinsic degeneration
of factor 2 according to the exchanging role of the Ex and Ey respect
their azimuthal dependence (sine or cosine dependence). This
degeneration is a consequence of the circular symmetry only.
*       Assuming the weakly guiding approximation WGA still valid, a
second degeneration holds, namely originating the mode group concept.
Different mode solutions belongs to the "same" propagation constant
(Beta) and they propagate with the "same" group velocity  (Quotation
marks refers to the WGA assumption). Since the radial component of the
two intrinsically degenerate mode solutions (sine and cosine) is the
same, the intensity of each LP(l,m) must have circular symmetry, no
matter how large or small mode numbers could be.
*       Since mode are orthogonal, results in addition that each mode
group must have still circular symmetry. As already stated, each mode
group propagates at a fixed and characteristic velocity. Two different
mode groups will propagate with different velocities.

Up to this point I guess you agree on those basic assumptions of the
modal analysis. In order to compute the amount of modal field excitation
due to a Gaussian beam incident on the launching cross-section we have
to introduce the overlapping integral. Those integrals represent proper
field coupling coefficients by virtue of the abovementioned mode
orthogonality.

What happens now if we compute the overlapping integral with a small and
exocentric (some offset) Gaussian spot? The basic conclusion is that the
axial symmetry will be definitely broken. The overlapping integrals deal
with the mode fields, not intensities. The coupling coefficient will
depend on the cylindrical coordinates on the fiber cross section,
including both the radial and the angular one. For each given mode
LP(l,m) the overlapping integral leads to two different values for each
of the two intrinsic degenerate solutions (sine and cosine). The
"weight" of the sine and cosine terms will be no more the same due to
the broken circular symmetry and when the intensity of LP(l,m) is
computed it will be no more a constant. It will be dependent on both
cylindrical coordinates, r and phi. In other words, the amount of field
coupled depends on the relative orientation of the offset and the light
polarization.

We nned only one more brick to close the wall: since the fiber is
assumed highly dispersive compared to the bit-rate, with a not optimized
refractive index profile (multiple alpha, kinks,...), the different
power contributions to the launched pulse will travel at different
speeds reaching the final section at different time instants and leading

For a given refractive index profile and fiber length, the amount of
pulse broadening is therefore a function of the coupling coefficient
distribution (overlapping integrals) and definitely of the relative
angle between the offset position and the light orientation.

Of course, fixing the offset position and rotating the light
polarization leads to the same conclusion and we are facing with the
polarization induced pulse broadening under offset launching condition.
This does not deal at all with modal noise. No connector was never been
involved into all the abovementioned discussion.

Please, letr me know you feedback. Hopefully, formal theory could follow
in few weeks...

Best regards

Stefano

Dr. Ing. Stefano Bottacchi
Senior Technical Consultant
Concept Engineering
Infineon Technologies Fiber Optics GmbH
Wernerwerkdamm 16, 13623 Berlin
Phone  +49 (0)30 85400 1930
Mobile: +49 (0)160 8 81 20 94

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