Re: [10GMMF] Polarization effects for 10GBASE-LRM
David,
please find my comments in blue.
> -----Original Message-----
> From: CUNNINGHAM,DAVID (A-England,ex1)
> [mailto:david_cunningham@agilent.com]
> Sent: Dienstag, 12. Oktober 2004 22:45
> To: Bottacchi Stefano (IFFO MOD CE external);
> david_cunningham@agilent.com
> Cc: STDS-802-3-10GMMF@listserv.ieee.org
> Subject: RE: Polarization effects for 10GBASE-LRM
>
> Stefano,
>
> RE: 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.
>
> Ans: This is not quite correct. My concern is about how the task
> group proposes to model the following:
>
> 1) The excitation of the modes of multimode fibre of circular cross
> section with a monomode, monochromatic source.
[Stefano Bottacchi] Please specify, since it not clear to me
the reason for your concern.
>
> 2) Birefringence of multimode fibre of circular cross-section.
[Stefano Bottacchi] I agree with you, birefringence of normal
circular cross section fiber is not responsible for the polarization
effects we experienced. To clean any doubt, Joerg disassembled the spool
to lay down the fiber in order to cancel out any potentially stressed
condition. The measured polarization effects were exactly the same of
the wounded fiber.
> I am not satisfied with the sketchy models that have been briefly
> outlined so far. Since no detailed mathematics has been presented I
> cannot check if this is simply a miss-understanding.
>
> But from what I have understood so far I am compelled to say I do not
> agree with what has been outlined.
>
> RE: 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)
>
> ANS: The scalar wave equation contains no information about
> polarization. Polarization effects must be added by some other means
> for example the polarization of the modal fields must be set by
> inspection or other knowledge - it is not an output of the scalar wave
> equation. Also, as far as propagation constants are concerned if the
> scalar wave equation is solved then small corrections could be applied
> to account for polarization - but at a fundamental level scalar wave
> solvers do not include polarization effects.
[Stefano Bottacchi] The solution of the scalar wave equation
represents the complete set of bound modes allowable by that waveguide
geometry. The choice for x and y axis is arbitrary assuming the
refractive index is a scalar with no birefringence components. If the
exciting electric field is linearly polarized along some direction in
the cross section, we can conveniently choose that direction as one
coordinate axis (x-axis). Of course, we could also choose x and y axis
independently from any polarization assumption. In that case we will
deal with two orthogonal field components acting along respective
coordinate axis. This gauge just makes mathematics more complicated but
does not change the physical solution. Now, rotating the linear
polarized field respect a fixed observer makes the whole mode set
rotating in the same way respect the fixed reference system. Let us
assume the fixed spatial orientation is the direction defined by the
offset coordinate respect the center of the fiber. This is the picture
we have in mind to justify how polarization takes place in multimode
fiber when a fixed offset direction is involved. If the refractive index
is a scalar and the fiber exhibits no asymmetries nor stress, there is
no reason for any polarization effect, even negligible small. Even in
single mode fiber, PMD arise just because the fundamental mode
degeneration breaks into two very slightly different modes due to
birefringence. This is not the case we are considering. In our basic
model the multimode fiber is ideal, circular with a true scalar
refractive index. Polarization effect arise just during the coupling
between an offset fiber and a given linear polarization.
> If the vector wave equation is solved then there are small (very, very
> small) differences in the propagation constants even for modes within
> the same group due to the optical polarization of the modes.
>
> I would recommend reading Snyder&Love, Optical Waveguide Theory,
> Chapman & Hall 1983 sections:
> 11-15, 13-6,13-7,13-11 for more information.
[Stefano Bottacchi] I have that book (very nice and complete)
and I red it as my first reference on fiber optic. I will reconsider the
sections you mentioned anywise.
> Therefore, I accept polarization effects can exist even in circular
> fibres. But, this is unlikely to be the cause of the larger
> birefringence we observe in the lab.
>
> RE: 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 fibre
> 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.
>
> ANS: I do not agree. Suggest you read the following papers:
>
> 1) Saijonmaa et al., Applied Optics Vol. 19 No.14
> (15Jul1980)p.2442-2452.
>
> 2) Grau et al., " Mode Excitation in Parabolic Index Fibres by
> Gaussian Beams" AEU, Band 34, 1980, Heft 6 pages 259-265. see Appendix
> 2: Connection between the power coupling coefficients and
> polarization.
>
> In words these references show that the input optical beam
> polarization can be decomposed into two equivalent orthogonal
> components (reference 2 is much more explicit here):
>
> A first one parallel to the line from the centre of the exciting laser
> spot to the optical centre of the fibre.
> A second parallel a line through the centre of the exciting laser spot
> parallel to the tangent to the core/cladding interface of the fibre.
>
> Furthermore, it is shown that the normalized coupling strength to each
> mode of the fibre is the same for each polarization component of the
> beam.
>
> The two directions, which are defined by the geometry of the
> excitation, define the relevant optical axis for the underlying basis
> modes of the fibre.
>
> Changing the orientation of the polarization of the input beam changes
> the relative power in each of the underlying orthogonal components of
> the beam. This leads to an asymmetry in the power coupled to the
> basis modes of the fibre. That is the same modes are always excited
> in each basis set but the relative excitation of each basis set is
> dependent on the angle of polarization.
>
> Hence the mode power distribution (MPD) is constant - the same modes
> are always excited.
>
> There is no rotation of modes at any point in this process - only a
> change in orientation of the polarization of the input beam.
>
> I don't expect you to believe my words but if I am to accept another
> model I need to know where these
[Stefano Bottacchi] The basic difference is that now we have to
consider THE POLARIZATION RESPECT TO THE OFFSET DIRECTION. It is this
angle which is responsible for breaking circular symmetry originating
polarization dependent coupling coefficients.
>
>
> RE: 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.
>
> ANS: Perturbations which break symmetry are very likely to induce
> polarization splitting of mode groups that have large amounts of power
> in the region of the perturbation.
[Stefano Bottacchi] Why do you introduce perturbations? Offset
and polarization state are not perturbations.
> If the vector wave equation is solved, with the asymmetric
> perturbation included, this splitting will arise as part of the model.
> If the scalar wave equation is solved correction terms need to be
> calculated.
>
> BUT IN OUR MODELS THE UNDERLYING REFRACTIVE INDEX PROFILES ARE
> RADIALLY SYMMETRIC. THEREFORE, EVEN IF THE VECTOR WAVE EQUATION IS
> SOLVED THERE ARE NO SIGNIFICANT POLARIZATION EFFECTS.
>
> I agree that real fibres are exhibiting significant birefringence.
>
> I DISAGREE THAT THESE EFFECTS CAN BE MODELLED WITH THE CURRENT
> RADIALLY SYMMETRIC REFRACTIVE INDEX PROFILES.
[Stefano Bottacchi] David, I guess there is some
misunderstanding. I am not assuming any perturbation here. I just
repeated above the fiber model is pretty ideal, basic like in standard
textbooks. In order to simplify mathematics we could reach same
qualitative polarization sensitivity even working with the simpler step
index fiber.
>
>
> RE: For a given refractive index profile and fibre 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.
>
> ANS: I disagree, you are almost right. The following is what standard
> theory would say:
>
> For a multimode fibre excited with an offset single mode beam
>
> The input beam can be decomposed into two equivalent orthogonal
> components:
> A first one parallel to the line from the centre of the exciting laser
> spot to the optical centre of the fibre.
> A second parallel a line through the centre of the exciting laser spot
> parallel to the tangent to the core of the fibre.
[Stefano Bottacchi] OK, it is an assumption.
> The two directions define the effective optical axis of the underlying
> basis modes of the fibre.
[Stefano Bottacchi] What are more explicitly those "effective
optical axis"? The exciting electric field is oscillating along its
polarization axis. Of course I can decompose it along the two direction
you mentioned above but why should I do this if the fiber is circular
symmetric?
> The normalized coupling strength to each equivalently polarized mode
> of the fibre is the same for each of the two equivalent orthogonal
> components of the beam.
[Stefano Bottacchi] OK, the fiber is isotropic.
> Changing the orientation of the polarization of the input beam changes
> the relative power in each of the underlying orthogonal components of
> the beam.
[Stefano Bottacchi] OK
> This leads to an asymmetry in the power coupled to the basis modes of
> the fibre.
[Stefano Bottacchi] OK, the source power distribution between
the two axis is changing accordingly.
> That is the same mode groups are always excited with the same mode
> power distribution in each basis set but the excitation of one basis
> set relative to the other is dependent on the angle of polarization.
[Stefano Bottacchi] This is not clear to me. Do you assert each
mode group receive the SAME power independently from the polarization?
What do you mean with "one basis set relative to the other..." Which
would be the "other" basis set?
> Therefore, the total normalized MPD remains constant but the relative
> power in each basis mode set varies with the orientation of the
> polarization.
>
> Therefore, since the MPD remains constant, in models that have radial
> symmetric refractive index profiles, since the propagation constants
> for the two underlying modal polarizations are the same (to very, very
> high precision) there is no polarization splitting.
>
> But, if non symmetric refractive index perturbations are included in
> the model and either the vector wave equation is solved or
> polarization corrections are used then birefringence will result.
> Modes with light in the area of the perturbation will be "split" the
> most.
>
> RE: 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.
>
> ANS: I assume you mean artificially rotating the underlying basis
> modes in order to change the coupling coefficients and hence compute a
> different MPD. If this is what you mean then this is exactly what I
> disagree with. I do not agree it is a valid model or even a valid
> approximation. The MPD does not change.
>
> Once again, I don't expect you to believe my words but if I am to
> accept another model I need to know where these papers are wrong.
>
> Regards,
> David
[Stefano Bottacchi]
In conclusion I identify two basic points in our discussion:
* Is the modal field solution rotating according to the input
linear polarization orientation? Paper from Saijonmaa et al., Applied
Optics Vol. 19, No.14 (15Jul1980)p.2442-2452, refers to x-polarized
light and it does not specify any polarization changing effect.
* Once the overlapping integral have been computed, the intensity
for each mode group is independent from the polarization orientation
respect to the offset axis?
Best regards
Stefano
> -----Original Message-----
> From: Bottacchi Stefano (IFFO MOD CE external)
> [mailto:Bottacchi.external@infineon.com]
> Sent: 12 October 2004 08:46
> To: david_cunningham@agilent.com
> Cc: STDS-802-3-10GMMF@listserv.ieee.org
> Subject: 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
>