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*To*: STDS-802-3-10GMMF@xxxxxxxxxxxxxxxxx*Subject*: Re: [10GMMF] Polarization effects for 10GBASE-LRM*From*: Yu Sun <yusun@xxxxxxxx>*Date*: Wed, 27 Oct 2004 17:16:13 -0400*Importance*: Normal*In-Reply-To*: <6BD8B3E713606246B10066BF3C312FFC945332@blnse201.eu.infineon.com>*Reply-To*: "IEEE P802.3aq 10GBASE-LRM"<stds-802-3-10gmmf@xxxxxxxx>*Sender*: owner-stds-802-3-10gmmf@xxxxxxxx

Dear Stefano, Thank you very much for your comments. It is indeed our pleasure to work with experts like you in the group. Please let me go though your comments one by one. 1. Yes, the modal field, as well as the input fields is normalized so that the total energy in one mode is a unit. 2. It is known that a perfect multimode fiber transmits its guided modes without energy conversion to the other possible guided modes or continuous spectrum. One can group the modes with the same propagation constant as one modal group. As you commented, "all modes belonging to the same mode group behave coherently since they have no relative phase shift during strictly axial propagation." This is a very important point when we consider the mode coupling at the connectors. I hope we will have chance for a detailed discussion in next Monday meeting(Nov. 1st). 3. The electrical field in the fiber can always be expressed in terms of the fiber guided modes polarized in x and y directions (please see (1.3)). The input field, which has an arbitrary polarization state, can be written as the linear combination of the fields in x and y direction respectively. The corresponding coefficients are complex numbers, with amplitude and phase. When the overlap integral is calculated, the inner product of the input field vector and modal field vector is taken. To simplify the discussion, (1.6) gives an example. It shows that the phase term in the input beam coefficient will introduce a rotation of the modal field. This is consistent with David's work. 4. I apologize about the confusion. The mode coupling coefficient serves as the amplitude coefficient of the modal field. The power is calculated by integration of the intensity. Since the transverse modal fields form an orthogonal basis, the integration of the square of the modal field gives the square of the mode coupling coefficient. If the fiber is perfect symmetric, the variation of the polarization rotation of the input light changes the power partition in the degenerate modes within one mode. Since these degenerate modes have the same propagation constant, it would not affect the pulse shape at the end of the fiber. However, the fiber may not be perfect. The asymmetric fiber core or the perturbation of the index may induce the mode selective loss (mode mixing is very similar as one mode sees loss and other modes see gain), polarization dependent loss, as well as PMD (I do not think PMD plays a big role at this moment, since the modal delay in general is much larger than PMD). These effects combined with the energy transfer between degenerate modes, will cause the power variation of different mode group. In our simulation, we assume that the mode selective loss is lumped at the connector or receiver. For instance, if we introduce excess loss at sita = 0, the cos modes will experience larger loss than the sin modes. Of course, this approach may be over-simplified the problem, although, we observed similar behavior in the simulation as experiments. I hope this clarify some confusions. I sincerely appreciate your further comments. Best regards, Yu -----Original Message----- From: owner-stds-802-3-10gmmf@IEEE.ORG [mailto:owner-stds-802-3-10gmmf@IEEE.ORG] On Behalf Of Bottacchi.external@INFINEON.COM Sent: Tuesday, October 26, 2004 3:49 AM To: STDS-802-3-10GMMF@listserv.ieee.org Subject: Re: [10GMMF] Polarization effects for 10GBASE-LRM Importance: High Dear Yu, I really appreciated your excellent contribution on polarization effect in multimode fiber link. Mathematics and assumptions looks quite clear and "conventional making the contest well readable. Let me go through your work more carefully, with some comments about. 1 - In order to associate to the square value of the coupling coefficient the meaning of power amount coupled to the corresponding mode it is required that both the selected mode and the input field were properly normalized. This can be implicitly assumed through (1.1) and (1.5). 2 - According to the weakly guiding approximation all mode are linearly polarized in the fiber cross-section and fiber modes can be grouped into degenerate subsets (mode groups) characterized by the same eigenvalues or propagation constant. All modes belonging to the same mode group behave therefore coherently since they have no relative phase shift during strictly axial propagation. 3 - Equation (2.5) reports the coupling coefficient between the input electric field (2.1) and the LP(l,m) mode, according to the modal field (2.2). Cylindrical coordinates ro and phi define the position of the center of the input Gaussian beam respect the reference position of the modal field which corresponds with the polarization orientation. As I stated earlier to David, this is still an assumption and it should be proved or at least well justified. Let us assume it is true for the moment: the modal field is oriented as the linearly polarized input beam. 4 - The coupling coefficient (2.5) is a complex function of the angle phi and its square modulus gives the relative power amount coupled to LP(l,m) mode. Is it correct? This means the intensity transferred to the LP(l,m) mode from the normalized input beam is given by the square modulus of the integral (2.5) and not, this is the important point, by the integral of the square modulus, as you report indeed. I guess this is not correct. Although (2.5) is correct, the calculation of the modulus (power) of the coupling coefficient cannot go through the integration of the square value of the real and imaginary parts, but instead it must go through the sum of the square value of the integral of the real and imaginary parts. I apologize if the used terminology induce some confusion but the meaning is quite correct. 5 - As a consequence of the above sketched calculation, I do not see why the total intensity should remain constant changing the angle phi. My position is that we should expect that the power transferred to the mode group depends on the angle phi, making DMD responsible for the polarization induced pulse distortion. Sincerely I do not see how single mode could experience different group delay within the same mode group, in order to justify quantitatively the large polarization effect we measured during experiment. For the moment I prefer to concentrate upon polarization and offset launch, neglecting any additional connector effect. We can work clearly without any connector and any fiber asymmetries, canceling all PDL contribution. I hope my comments are clear enough to be replied. How do you justify quantitatively the polarization induced pulse distortion in low bandwidth MMF link without invoking polarization selective mode group power transfer? Best regards Stefano > -----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 >

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