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At the 10/25 Task 4 Meeting David asked for comments on his notes.
A. I've attached a *.pdf rather than a long email. I am curious about plotting figure 3b in a different way to see if the two Hermite Gauss modes HMG(2,0)+HMG(0,2) show up as having a constant total power. This is curiously related to discussions we have had in Task4 on coherent and incoherent modes:
HMG(2,0)^2 + HMG(0,2)^2 does not stay constant with polarization;
(HMG(2,0)+HMG(0,2))^2 does stay constant, because it is proportional {I think} to the nu=0 radial mode for the group in the Laguerre-Gauss formulation.
B. I worry about some of the conclusions because I think the degenerate modes all have the same propagation constant & mode delays if the profile is axisymmetric and one uses the scalar wave equation -- so there can be no effect upon the impulse response by changing the polarization angle, since it would seem to merely shift power between the degenerate modes.
I continue to be very curious about the experimental results.
John A.
> -----Original Message-----
> From: CUNNINGHAM,DAVID (A-England,ex1)
> [mailto:david_cunningham@agilent.com]
> Sent: Donnerstag, 21. Oktober 2004 20:55
> To: Bottacchi Stefano (IFFO MOD CE external); ysun@optiumcorp.com
> Cc: STDS-802-3-10GMMF@listserv.ieee.org
> Subject: RE: Polarization effects for 10GBASE-LRM
>
> Dear Stefano and Yu,
>
> I attach a PDF file which documents an analysis of SMF offset launch
> as a function of polarisation direction.
>
> Stefano: I now agree with you.
> The mathematical analysis I attach shows the effects that you describe
> and it is perfectly consistent within the references I quoted. I now
> agree that it is likely to be the launch in combination with
> incomplete mode mixing between the modes within groups that is likely
> to be causing the change in impulse response (IPR) with polarisation
> rotation. The variation in IPR will be largest when the index
> perturbations (which are cylindrically symmetric) cause delay
> splitting within groups and significant light power is launched into
> the modes with such splitting. I also agree that the scalar wave
> equation is sufficient to explain these effects.
>
> Yu: On your call on Monday I said I would provide example calculations
> for coupling into the fibre. Please accept this as my example - the
> extension to connectors is straightforward. It is school mid-term
> break here in the UK so I'm taking tomorrow off work. So this is as
> much as I can get done on this this week. I thought it best to send
> you what I had done so far.
>
> Obviously, having worked through the analysis I now agree with your
> general method for analyzing connectors too.
>
> Stefano and Yu: Please confirm that my analysis is indeed in agreement
> with the concepts you have been advocating to the group.
>
> Sorry for the stress I may have caused you during this debate, David
>
>
> << File: LaunchMPDdgc1.pdf >>
>
>
>
>
> -----Original Message-----
> From: Bottacchi Stefano (IFFO MOD CE external)
> [mailto:Bottacchi.external@infineon.com]
> Sent: 18 October 2004 11:35
> To: david_cunningham@agilent.com
> Cc: STDS-802-3-10GMMF@listserv.ieee.org
> Subject: RE: Polarization effects for 10GBASE-LRM
> Importance: High
>
> 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
>
Polarization_effects_JSA_1026.pdf