Re: [10GMMF] Polarization effects for 10GBASE-LRM
I agree with Stefano Bottachi that this was an interesting conference call.
I would like to confirm & note a couple of points.
[1] It was my impression from the call that both Infineon and Agilent have seen an effect on eye closure experimentally as they change the polarization of the offset launch. Infineon has provided some *.pdfs with pictures of the eye diagrams which are helpful.
[2] David Cunningham suggested the reference Saijonmaa et al., Applied Optics Vol. 19 No.14 (15Jul1980)p.2442, which I also recommend. In addition I would refer to the well-known reference by Snyder&Love, Optical Waveguide Theory, Chapman&Hall 1983, where section 20 deals with illuminations, tilts & offsets. On page 426 Snyder & Love explicitly state "By symmetry, the power excited in each mode of a circular fiber is independent of the polarization of the beam fields, and therefore does not depend on our choice of an x-polarized electric field in Eq. (20-7a)." So it is worthwile to go over this carefully.
JSA
-----Original Message-----
From: Bottacchi.external@INFINEON.COM [mailto:Bottacchi.external@INFINEON.COM]
Sent: Tuesday, October 12, 2004 3:46 AM
To: STDS-802-3-10GMMF@listserv.ieee.org
Subject: [10GMMF] Polarization effects for 10GBASE-LRM
Importance: High
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 to pulse broadening.
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