Mode coupling at connectors in mode-division multiplexed transmission over few-mode fiber Jordi Vuong,1 Petros Ramantanis,2,* Yann Frignac,1 Massimiliano Salsi,2 Philippe Genevaux,2 Djalal F. Bendimerad1 and Gabriel Charlet2 1
Institut Mines-Telecom, Telecom SudParis, CNRS SAMOVAR UMR5157, 9 rue Charles Fourier, 91011 Evry, France 2 Alcatel-Lucent Bell Labs, Route de Villejust, 91620 Nozay, France * [email protected]
Abstract: In mode-division multiplexed (MDM) transmission systems, mode coupling is responsible for inter-modal crosstalk. We consider the transmission of modulated signals over a few-mode fiber (FMF) having low mode coupling and large differential mode group delay in the presence of a non-ideal fiber connection responsible for extra mode coupling. In this context, we first analytically derive the coupling matrix of the multimode connector and we numerically study the dependence of the matrix coefficients as a function of the butt-joint connection characteristics. The numerical results are then validated through an experiment with a fivemode setup. Finally, through numerical simulations, we assess the impact of the connector on the signal quality investigating different receiver digital signal processing (DSP) schemes. ©2015 Optical Society of America OCIS codes: (060.2340) Fiber optics components; (060.4370) Nonlinear optics, fibers; (060.2330) Fiber optics communications; (060.2360) Fiber optics links and subsystems.
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1. Introduction In order to avoid the foreseen capacity crunch in optical long-haul transmission systems based on single-mode fibers [1], it has been recently proposed to use a combination of Spatial Division Multiplexing (SDM) over few-mode or multicore fibers and Multiple Input Multiple Output (MIMO) DSP at the receiver side [2–4]. So far, MDM transmission exploiting, respectively, 5 and 6 Linearly Polarized (LP) modes [5] and reaching over 708 km [6] has been reported. In general, coherent MIMO-SDM experiments embody two opposite strategies which differ from one another by their management of mode coupling. In [7], spatial modes propagate over a graded-index fiber, characterized by limited Differential Mode Group Delay (DMGD) and strong mode coupling. The corresponding strategy consists of tolerating mode coupling and enabling equalization through joint MIMO processing of all interfering modes. In this scenario, due to the potentially high system memory, reducing DMGD (for example by new fiber design) or applying DMGD management [8] become crucial to maintain a low Digital Signal Processing (DSP) complexity. On the other hand, in [5], spatial modes propagate over a Step-Index (SI) fiber designed to limit mode coupling below a certain threshold [9]. This strategy allows for modes to be optically separated and detected independently by performing MIMO over only degenerate modes. While, for both the above strategies all sources of modal crosstalk are important and must therefore be identified and analyzed, for the second scenario using stepindex fiber, the analysis of modal crosstalk is critical. The most prominent sources of crosstalk are the transmission medium itself and the connections. In this paper we focus on the crosstalk coming from the connectors. The concatenation of connectors (splices) in the context of gradient-index multimode fiber systems and their impact on system performance is discussed in [10]. In this paper, by means of numerical simulations and experiments, we provide insights into the energy coupled between modes at the junction of two step-index FMFs in the presence of a transverse misalignment, as similarly done in the context of single mode fibers [11]. Such transverse
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misalignment is most likely to occur in connectors of any type and also in a free-space optics setup such as a spatial multiplexer (MUX) or demultiplexer (DEMUX) 2. Problem statement and theory formalism 2.1 Theory In the following, we model the connection as a “butt joint” between two identical step-index (SI) fibers, as shown in Fig. 1(a). Both fibers have the same core radius rcore and the same core and cladding refractive indices nco and ncl (Fig. 1(b)). The input fiber is laterally misaligned with respect to the output fiber by a distance d, while no axial gaps or angular tilts are considered. Such a connector can be fully described by the parameters couple {V, D}, where 2π rcore oo d V= nco2 − ncl2 is the normalized frequency of both fibers and D = = 1 2 is the λ rcore rcore normalized distance between the core centers noted as O1 and O2 respectively. In addition we define the coordinate system O1 xy of Fig. 1(c), referred to in the following as coordinate system of the connector, as a coordinate system with its origin coinciding with the core center of the incoming fiber O1 and its x axis coinciding with the direction of the misalignment. As for this modeling of the connector, the incoming polarization is conserved in the second fiber with no possible coupling between orthogonal polarizations of any mode, in the following we may focus on a scalar electric field only, while an orthogonal polarization would be practically treated in the same way. We note that in the general case, the coordinate system and the modal bases of both the incoming and the outgoing fibers may be independently chosen, without any loss of generality. Our particular choice of the coordinate system x-axis coinciding with the direction of the lateral misalignment is a natural choice that facilitates calculations, as it detailed later on. Likewise, even if the modal bases of the incoming and outgoing fiber could have been chosen to be different, choosing them to practically coincide when d → 0 (as hinted in Figs. 1(a) and (c)) may also lead to simpler expressions.
Fig. 1. a) Geometry and b) Refractive index profile in the cross section of a laterally misaligned butt-joint between identical SI-FMF. c) Geometry after rotation around the Z axis so that the lateral misalignment coincides with the X axis. In Figs. (a) and (c) the appearing LP11 mode illustrates our choice for the modal basis of the incoming and outgoing fiber (referred to as reference modal basis).
Both fibers have a relatively low refractive index difference (Δn = nco-ncib>c>0.With the help of the subspace matrices introduced in Eq. (19), we first concentrate on the diagonal elements Γlp −lp of the matrix Γ (ϕ ) . We first note that Γ 01− 01 does not depend on ϕ and therefore, the origin azimuthal offsets have no influence on the energy coupled to this subspace, while this remark could be generalized for all modes LP0p. Furthermore, within the limits of our simplification, for the two degenerate modes of the subspace LP21, the origin azimuthal offsets have no influence either. Similarly as before, this remark can be generalized for all modes LPlp, with l>1. Finally, the 2 × 2 matrix of the LP11 subspace is equivalent to the Jones matrix of a PDL element [20] after a coordinate transformation. This highlights the fact that a simple laterally misaligned FMF connector is enough to generate MDL, even between degenerate modes. This is also true for all the subspaces LPlp, with l = 1. Moving to the non-diagonal matrices, we first note that the elements Γ 01− 21 and Γ 21− 01 are zero. This means that no energy is practically exchanged between the subspaces LP01 and LP21. Nevertheless, the matrices Γ 01−11 = −Γ 01−11T are not zero, indicating the nature of the energy exchange between LP01 and LP11 that depends on the origin azimuthal offset of LP11. Finally, concerning the energy exchange between the subspaces LP11 and LP21 we see that the sub-matrix Γ 21−11 = −Γ11− 21 corresponds to the rotation matrix, depending on the angle (ϕ11 − 2ϕ21 ) . In the following section we discuss the
impact of the abovementioned coefficients on the global system performance. 3.2 Lab measurements The experimental setup described in Fig. 7 has been built to assess the crosstalk generated by the aforementioned non-ideal connector for various couples of distinct spatial modes, i.e. experimentally validate the coupling coefficients of the connector matrix. For this measurement a single-mode, single-channel (1550 nm) 112 Gb/s PDM-QPSK signal is generated and divided in 5 SMFs with different lengths to induce a de-correlation of some hundreds of symbols between the data streams, as it is also considered in [5]. Then, the signal in the first fiber (fundamental mode, LP01) is directly coupled to the FMF fiber, while for the other fibers a mode conversion is applied to generate the 4 other spatial mode profiles, i.e. LP11a, LP11b, LP21a, and LP21b. Mode conversion basically relies on SMF-to-FMF mode conversions using 4f correlators within which phase masks are being placed. These masks are illustrated in Fig. 7(b) with the red filled circles indicating where incoming beams hit the phase mask to get different mode conversions. All modes are then coupled to the FMF using the free-space setup of Fig. 7(a). The FMF connector is emulated by a butt joint of two 5-m long FMFs (Fig. 7(c)) with a variable lateral misalignment. Mode DEMUX is achieved by using a symmetric setup to the MUX. Crosstalk is measured by injecting only in one of the 5 modes and measuring the received power to all 5 modes using a set of power-meters.
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Received 24 Sep 2014; revised 17 Nov 2014; accepted 28 Nov 2014; published 21 Jan 2015 26 Jan 2015 | Vol. 23, No. 2 | DOI:10.1364/OE.23.001438 | OPTICS EXPRESS 1449
Fig. 7. Experimental setup for the crosstalk ratios measurements.
In our experiment, the FMF link is limited to two butt-jointed 5-meter long FMF strands with a view to exclusively emulate a single connector. Indeed, such short fiber lengths prevent important propagative crosstalk from building up under the effect of both intrinsic and extrinsic fiber perturbations. Moreover, both strands come from the same FMF spool to guarantee the identity of their optical and geometrical properties. The FMF has a step-index (SI) refractive index profile with rcore = 7,5 µm and Δn = 0.0102. More specifically, its design is the one suggested in [5,9] which aims at strongly guiding 3 spatial LP modes. It should be stated that in the weakly guiding approximation, LP21 and LP02 share the same cutoff value. Thus, our SI-FMF can guide 4 LP modes. However, in our MDM transmission system implementation, for reasons of experimental simplicity, LP02 is not used. Finally, the capabilities of a splice machine were used to control the lateral alignment of the upstream and downstream FMFs in the butt-joint.
Fig. 8. Experimental versus numerical estimation of the matrix crosstalk coefficients.
Commenting on the experimental issues affecting the measurements, we first note that in practice, a - fully or partially - depolarized signal has proven to significantly enhance power measurement stability owing to an averaging over polarization states which restrains the impact of polarization-dependent components combined with polarization changes. Secondly, the splicing machine gave us the possibility of the variation of the lateral misalignment with a minimum possible step of approximately 0.26 µm, therefore inducing a possible imprecision of about ± 0.015 in terms of normalized distance D. Thirdly, the MUX/DEMUX free-space setup introduces an additional mode-dependent loss that was a posteriori subtracted from the direct measurements. Finally, since the MUX/DEMUX free-space setup also introduces a low inter-modal crosstalk (typical values may be found in Table 1 of [5]), very low values of connector-induced crosstalk (about −25 dB) cannot be measured. This is critical for connector crosstalk measurements with a lateral misalignment lower than about 10% of the core radius. In Fig. 8 we plot the crosstalk coefficient as a function of the lateral misalignment for both the numerical evaluation and the experiment for the modes LP01, LP11a, LP11b, LP21a and LP21b. We observe that, for every inspected power crosstalk ratio, numerical and experimental
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curves are in good agreement. This measurement validates the form of the connector matrix presented in the section 3.1. 4. Back-to-back numerical investigation of the connector impact
While the connector matrix expression was established in the previous section, its influence on system performance is not straightforward. While a comprehensive study of the connector system impact is beyond the scope of the present paper, in this section we provide some insights based on a simplified numerical simulation setup. In Fig. 9, we show our numerical simulation setup. Each of the five modes is carrying a single-channel Root Raised Cosine (RRC) QPSK signal with a roll-off factor of 0.5, data based on different pseudorandom sequences of 4096 symbols, modulated at 32Gbaud, with the same power Ps. All modes have been assumed to have the same initial phase. This assumption can be justified from the intuitions appearing in Fig. 3(b) of the reference [21] from which we can infer that the influence of a random phase difference between different modes should be limited in the case where a MIMO with a sufficient number of taps is applied. All modes are multiplexed in the same FMF, followed by the multimode connector under study, considering for this example a normalized lateral misalignment D = 0.2. In order to exclusively focus on the impact of a single connector, the FMF of the simulation setup are considered very short (i.e. a few meters long) and DMGD has been neglected. Then, a multimode black-box amplifier is used to load a noise power Pn for each mode bringing each mode up to an OSNR0.1nm = 10 dB. Finally, the modes are de-multiplexed, filtered by a RRC matched filter and treated simultaneously by a coherent receiver. For the sake of simplicity we consider ideal multiplexer, de-multiplexer, amplifier and coherent receiver.
Fig. 9. Simplified system for the performance analysis of the connector.
For the signal processing at the receiver side we consider the two configurations presented above, i.e. the global MIMO receiver scheme and the partial MIMO receiver scheme. We note here that, if one considers a realistic transmission over FMF, a low number of taps may be enough to separate modes with low DMGD, a significantly higher number of taps (that may quickly become unreasonable) is required for modes with high DMGD, depending on the transmission length. Nevertheless, in this back-to-back investigation of the connector impact, both partial and global MIMO reception can be performed with a reasonable number of taps. The source separation is achieved by a Constant Modulus Algorithm (CMA). Its taps are initialized with the channel matrix estimated by the Equivariant Adaptive Separation via Independence (EASI) algorithm [22] in order to avoid the singularity problems of standard CMA. Finally, the BER is converted to an equivalent Q2 factor.
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Received 24 Sep 2014; revised 17 Nov 2014; accepted 28 Nov 2014; published 21 Jan 2015 26 Jan 2015 | Vol. 23, No. 2 | DOI:10.1364/OE.23.001438 | OPTICS EXPRESS 1451
Fig. 10. Q2 for the five modes after a misaligned butt-joint connector as a function of the origin azimuthal offset φ11, for the partial MIMO receiver scheme.
In Fig. 10 we show the Q2 factor after a partial MIMO equalizer scheme for each mode (considering separately the degenerate modes), the average quality of each LP mode and the average quality of all five modes as a function of the azimuthal offset φ11, fixing φ21 to 0 rad. First we observe that LP01 yields the highest performance, LP11 yields an average performance (over the two degenerate modes) of about 2.1 dB lower than LP01 and LP21 exhibits an intermediate performance, about 1 dB higher than LP11. This result may be better understood in light of the Eq. (22), reminding that a>b>c. Indeed, the high performance of LP01 stems from both its high transmission coefficient through the connector and also the lowest overall coupled energy from the other modes. Equation (22) also indicates that LP11 presents an average transmission coefficient higher than the one of LP21. Nevertheless, in Fig. 10 we observe that LP11 performs worse than LP21 which can be understood from the fact that LP11 is also impacted by a higher overall coupled energy from the other modes, compared to LP21 which only interferes with LP11. This further suggests that, in this receiver scheme, if we decide to transmit information transmission over three modes only, it should be preferable to use LP01 and LP21 rather than LP01 and LP11. Commenting now on the quality evolution of each degenerate mode separately, we note that all modes present a periodic oscillation with respect to φ11, with the LP11 modes, however, presenting a peak-to-peak amplitude variation of about 3.5 dB, while a variation of less than 0.5 dB is observed for the other modes. This can be justified in view of Eq. (22) by considering the combined effect of the transmission and coupling coefficients for each mode. Indeed, it can be verified that for LP01, LP21a and LP21b both the transmission coefficient and the total coupled power are independent of both φ11 and φ21. It can also be verified that the total coupled energy for the modes LP11a and LP11b is also independent on φ21, both depending however on φ11. We suggest that the residual oscillations of LP01, LP21a and LP21b as a function of φ11 may be qualitatively understood using the analysis of [23], stating that the degradation brought by QPSK interfering signals is aggravated for an increasing number of interferes while the total interfering signal power remains constant. Nevertheless, we underline the fact that the average quality of all five modes is independent of φ11 and approximately about 6.5 dB. It can be easily verified that qualitatively similar results can be drawn for other values of φ21. Finally, we note that while the performance of each degenerate mode LP11 strongly depends on φ11, the average performance of LP11a and LP11b is practically independent of φ11, as it is also the case for all the other modes.
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Received 24 Sep 2014; revised 17 Nov 2014; accepted 28 Nov 2014; published 21 Jan 2015 26 Jan 2015 | Vol. 23, No. 2 | DOI:10.1364/OE.23.001438 | OPTICS EXPRESS 1452
Fig. 11. Global MIMO performance vs. φ11.
In Fig. 11 we show the quality for a global MIMO receiver scheme as a function of φ11 and φ21 = 0 rad. First we note that in this case, the quality of all modes in terms of Q2 factor is limited within a range of about 1 dB (between 9.2 and 10.2 dB). As before, LP01 performs better than the other modes and a periodic quality inversion is observed for the modes LP11a and LP11b. Nevertheless, this time LP11 performs better than LP21 (9.8 against 9.2 dB), since the interference is well compensated by the global MIMO, with the dominant influence coming this time from the power transmission coefficient (higher for LP11 compared to LP21), together with the residual MDL between degenerate modes. Finally, the average quality of all five modes is about 9.6, i.e. about 3 dB higher than the average quality of the partial MIMO receiver scheme. 5. Conclusion
In recent assessments of slightly multimode optical fiber systems, system designers have highlighted the necessity to perform further investigations of the mode coupling that may impair the overall MDM transmission quality. In that sense, we have proposed here a theoretical, numerical and experimental analysis of the mode coupling experienced by optical signals passing through a misaligned butt-joint connection between two identical SI-FMFs. We have arrived at a simple analytical model based on a connector transfer matrix for which coefficients are derived from mode overlap integrals and proposed simplifications based on symmetry, orders of magnitude of the various possible mode coupling and numerical estimation of matrix coefficients. Numerical coefficients estimated for LP01, LP11a&b and LP21a&b modes have also been validated through FMF experiments. First observations on estimated coefficients yield the information that more than 0.2 dB of mode losses may be reached by such a connector when the fiber misalignment exceeds 18% of the core radius. Furthermore, the transmission coefficients of all modes are assessed, indicating that the modes suffering from higher losses are, as expected, LP21 modes, then LP11 and finally LP01, quantifying the loss evolution for an increasing offset between fiber core centers. From these coefficients, one can easily derive the MDL introduced by the connector. On the other hand, we also quantify the coupling coefficients evolution as a function of D. For this aspect, LP11 is shown to potentially get the maximum coupling power from the other modes at the connector. In addition, we quantify the evolution of both MDL and mode coupling as a function of the relative angle of the connection with respect to the axis of misalignment ( ϕ ). On that point, LP11 modes appear to be highly dependent on this angle when D increases. This also highlights the fact that a statistical MDL between two degenerate modes (like LP11a and LP11b) may be critical for the MDM transmission quality. Finally, we have also estimated the quality impairments caused by such connectors using a numerical estimation of BERs of various RRC-QPSK signals launched in the five modes. When considering a receiver with a partial MIMO DSP, LP11 exhibits the highest #223753 - $15.00 USD © 2015 OSA
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impairments through the connection also varying with the highest amplitude as a function of φ11. When considering a global MIMO at the receiver side, the connector impairment is significantly mitigated while the resulting discrepancy between the mode transmission qualities stem for the sole MDL of the connection. Consequently, in that latter context, LP21 shows the poorest performance. Further investigations need to be performed, with cascaded connectors and propagation with DMGD between connections to yield a better statistical estimation of these mode coupling impairments on the overall system quality. Appendix 1: Primarily-coupled modes in a SI fiber
For a given mode of the incoming fiber, we are interested in approximately determining the modes of the outcoming fiber towards which, the light of the first mode is primarily coupled. More precisely, we investigate the possibility of the analysis of [19], developed in the case of graded-index fibers, being qualitatively applicable for the first modes of a step-index fiber. To do this we verify if we can approximate the LP modes with those of an “equivalent” Infinite-Parabolic-Index (IPI), using the Gaussian approximation studied in [11]. Indeed, in [24], the waist of the fundamental Gaussian beam is empirically found to approximate the actual fundamental LP01 mode field of a Step-Index fiber as a function of the fiber normalized frequency V, with the validity domain of this approximation being restricted to low-order modes. The approximate transverse modal fields of the LP modes fields of a matched IPI fiber [12] reads in polar coordinates: l
Flp( s ) (r , θ ) ∝ exp ( −V ρ 2 / 2 ) ⋅ (V ρ 2 ) 2 ⋅ Llp −1 (V ρ 2 ) f l ( s ) (θ )
(23)
where l stands for the LP mode azimuthal index, p its radial index, V the fiber normalized frequency, ρ ( = r/rcore) the normalized radial polar coordinate and Lαβ is the generalized Laguerre polynomial of order α and degree β. Replacing in the previous equation V ρ 2 with 2
ρ2 w0
transforms these LP mode fields into
w0 , where w0 rcore is the beam waist. The correspondence between a set of LPlp modes of a single IPI fiber and a set (with the same waist) of LGmn beams is given by:
a set of orthonormal Laguerre-Gaussian (LG) beams of normalized waist W0 =
l=m p = n +1 V=
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2 W02
Received 24 Sep 2014; revised 17 Nov 2014; accepted 28 Nov 2014; published 21 Jan 2015 26 Jan 2015 | Vol. 23, No. 2 | DOI:10.1364/OE.23.001438 | OPTICS EXPRESS 1454
Fig. 12. Transmission coefficient or matching parameter T as a function of V for the modes LP11, LP21, LP02 and LP31.
As a justification of the generalized Gaussian approximation within our theoretical analysis, we must indicate the matching quality and the tolerance with respect to our specific application. For a hypothetical connector between a step-index and a ISI fiber as in [24], we plot in Fig. 12 the power transmission coefficient (noted as matching parameter T) between the transverse fields of the best-fitting LGl,p-1 beam and the actual LPlp SI-fiber mode for the first ten LP modes and for a normalized frequency V spanning from 2.5 to 8. With the Gaussian approximation already shown to be accurate for LP01 [24], we note a matching over 99.5% for the modes of interest, i.e. LP11 and LP21 and V=5.1. Nevertheless, it could be verified that T drops significantly when p increases, as it can be seen for the example of LP02 in Fig. 12. In conclusion, the analysis of [19] may be used to qualitatively estimate the relative weight of the mode coupling coefficients in few mode SI fibers. Acknowledgement
The authors would like to thank the French government for supporting a part of this work through the ANR agency under the project STRADE (ANR-09-VERS-010).
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Institut Mines-Telecom, Telecom SudParis, CNRS SAMOVAR UMR5157, 9 rue Charles Fourier, 91011 Evry, France 2 Alcatel-Lucent Bell Labs, Route de Villejust, 91620 Nozay, France * [email protected]
Abstract: In mode-division multiplexed (MDM) transmission systems, mode coupling is responsible for inter-modal crosstalk. We consider the transmission of modulated signals over a few-mode fiber (FMF) having low mode coupling and large differential mode group delay in the presence of a non-ideal fiber connection responsible for extra mode coupling. In this context, we first analytically derive the coupling matrix of the multimode connector and we numerically study the dependence of the matrix coefficients as a function of the butt-joint connection characteristics. The numerical results are then validated through an experiment with a fivemode setup. Finally, through numerical simulations, we assess the impact of the connector on the signal quality investigating different receiver digital signal processing (DSP) schemes. ©2015 Optical Society of America OCIS codes: (060.2340) Fiber optics components; (060.4370) Nonlinear optics, fibers; (060.2330) Fiber optics communications; (060.2360) Fiber optics links and subsystems.
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9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost, “Few-mode fiber for uncoupled modedivision multiplexing transmissions,” in 37th European Conference on Optical Communication (Optical Society of America, 2011), paper Tu5LeCervin7. S. Warm and K. Petermann, “Splice loss requirements in multi-mode fiber mode-division-multiplex transmission links,” Opt. Express 21(1), 519–532 (2013). J. Cook, W. L. Mammel, and R. J. Grow, “Effect of misalignments on coupling efficiency of single-mode optical,” Bell Syst. Tech. J. 52(8), 1439–1448 (1973). A. Snyder and J. Love, Optical waveguide theory. (Springer, 1983). B. Saleh and M. Teich, Fundamentals of Photonics, (John Wiley & Sons Inc, 2007). A. D. Yablon, Optical fiber fusion splicing, (Springer, 2005). C. Simos, P. Leproux, P. D. Bin, and P. Facq, “Influence of mode orientations on power transfer at misaligned fibre connections,” J. Opt. A, Pure Appl. Opt. 4(1), 8–15 (2002). H. Kogelnik and P. J. Winzer, “Modal birefringence in weakly guiding fibers,” J. Lightwave Technol. 30(14), 2240–2245 (2012). P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19(17), 16680–16696 (2011). T. Duthel, C. Fludger, J. Geyer, and C. Schulien, “Impact of polarisation dependent loss on coherent POLMUXNRZ-DQPSK,” in Optical Fiber Communication Conference (Optical Society of America, 2008), paper OThU5. D. Pagnoux, J. Blondy, P. Di Bin, P. Faugeras, N. Katcharov, and P. Facq, “Modal effects in optical fibre connections: theory and experimentation,” Ann. Telecommun. 49, 619–628 (1994). J. Damask, Polarization Optics In Telecommunications, (Springer, 2005). J. Vuong, P. Ramantanis, A. Seck, D. Bendimerad, and Y. Frignac, “Understanding discrete linear mode coupling in few-mode fiber transmission systems”, in 37th European Conference and Exhibition on Optical Communication (ECOC 2011), paper Tu5B2. J.-F. Cardoso and B. H. Laheld, “Equivariant adaptive source separation,” IEEE Trans. Signal Process. 44(12), 3017–3030 (1996). M. Chiani and A. Giorgetti, “Statistical analysis of asynchronous QPSK cochannel interference,” in Proceedings of IEEE Global Telecommunications Conference (IEEE, 2002), pp. 1855–1859. D. Marcuse, Light transmission optics, (Van Nostrand Reinhold New York, 1982).
1. Introduction In order to avoid the foreseen capacity crunch in optical long-haul transmission systems based on single-mode fibers [1], it has been recently proposed to use a combination of Spatial Division Multiplexing (SDM) over few-mode or multicore fibers and Multiple Input Multiple Output (MIMO) DSP at the receiver side [2–4]. So far, MDM transmission exploiting, respectively, 5 and 6 Linearly Polarized (LP) modes [5] and reaching over 708 km [6] has been reported. In general, coherent MIMO-SDM experiments embody two opposite strategies which differ from one another by their management of mode coupling. In [7], spatial modes propagate over a graded-index fiber, characterized by limited Differential Mode Group Delay (DMGD) and strong mode coupling. The corresponding strategy consists of tolerating mode coupling and enabling equalization through joint MIMO processing of all interfering modes. In this scenario, due to the potentially high system memory, reducing DMGD (for example by new fiber design) or applying DMGD management [8] become crucial to maintain a low Digital Signal Processing (DSP) complexity. On the other hand, in [5], spatial modes propagate over a Step-Index (SI) fiber designed to limit mode coupling below a certain threshold [9]. This strategy allows for modes to be optically separated and detected independently by performing MIMO over only degenerate modes. While, for both the above strategies all sources of modal crosstalk are important and must therefore be identified and analyzed, for the second scenario using stepindex fiber, the analysis of modal crosstalk is critical. The most prominent sources of crosstalk are the transmission medium itself and the connections. In this paper we focus on the crosstalk coming from the connectors. The concatenation of connectors (splices) in the context of gradient-index multimode fiber systems and their impact on system performance is discussed in [10]. In this paper, by means of numerical simulations and experiments, we provide insights into the energy coupled between modes at the junction of two step-index FMFs in the presence of a transverse misalignment, as similarly done in the context of single mode fibers [11]. Such transverse
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Received 24 Sep 2014; revised 17 Nov 2014; accepted 28 Nov 2014; published 21 Jan 2015 26 Jan 2015 | Vol. 23, No. 2 | DOI:10.1364/OE.23.001438 | OPTICS EXPRESS 1439
misalignment is most likely to occur in connectors of any type and also in a free-space optics setup such as a spatial multiplexer (MUX) or demultiplexer (DEMUX) 2. Problem statement and theory formalism 2.1 Theory In the following, we model the connection as a “butt joint” between two identical step-index (SI) fibers, as shown in Fig. 1(a). Both fibers have the same core radius rcore and the same core and cladding refractive indices nco and ncl (Fig. 1(b)). The input fiber is laterally misaligned with respect to the output fiber by a distance d, while no axial gaps or angular tilts are considered. Such a connector can be fully described by the parameters couple {V, D}, where 2π rcore oo d V= nco2 − ncl2 is the normalized frequency of both fibers and D = = 1 2 is the λ rcore rcore normalized distance between the core centers noted as O1 and O2 respectively. In addition we define the coordinate system O1 xy of Fig. 1(c), referred to in the following as coordinate system of the connector, as a coordinate system with its origin coinciding with the core center of the incoming fiber O1 and its x axis coinciding with the direction of the misalignment. As for this modeling of the connector, the incoming polarization is conserved in the second fiber with no possible coupling between orthogonal polarizations of any mode, in the following we may focus on a scalar electric field only, while an orthogonal polarization would be practically treated in the same way. We note that in the general case, the coordinate system and the modal bases of both the incoming and the outgoing fibers may be independently chosen, without any loss of generality. Our particular choice of the coordinate system x-axis coinciding with the direction of the lateral misalignment is a natural choice that facilitates calculations, as it detailed later on. Likewise, even if the modal bases of the incoming and outgoing fiber could have been chosen to be different, choosing them to practically coincide when d → 0 (as hinted in Figs. 1(a) and (c)) may also lead to simpler expressions.
Fig. 1. a) Geometry and b) Refractive index profile in the cross section of a laterally misaligned butt-joint between identical SI-FMF. c) Geometry after rotation around the Z axis so that the lateral misalignment coincides with the X axis. In Figs. (a) and (c) the appearing LP11 mode illustrates our choice for the modal basis of the incoming and outgoing fiber (referred to as reference modal basis).
Both fibers have a relatively low refractive index difference (Δn = nco-ncib>c>0.With the help of the subspace matrices introduced in Eq. (19), we first concentrate on the diagonal elements Γlp −lp of the matrix Γ (ϕ ) . We first note that Γ 01− 01 does not depend on ϕ and therefore, the origin azimuthal offsets have no influence on the energy coupled to this subspace, while this remark could be generalized for all modes LP0p. Furthermore, within the limits of our simplification, for the two degenerate modes of the subspace LP21, the origin azimuthal offsets have no influence either. Similarly as before, this remark can be generalized for all modes LPlp, with l>1. Finally, the 2 × 2 matrix of the LP11 subspace is equivalent to the Jones matrix of a PDL element [20] after a coordinate transformation. This highlights the fact that a simple laterally misaligned FMF connector is enough to generate MDL, even between degenerate modes. This is also true for all the subspaces LPlp, with l = 1. Moving to the non-diagonal matrices, we first note that the elements Γ 01− 21 and Γ 21− 01 are zero. This means that no energy is practically exchanged between the subspaces LP01 and LP21. Nevertheless, the matrices Γ 01−11 = −Γ 01−11T are not zero, indicating the nature of the energy exchange between LP01 and LP11 that depends on the origin azimuthal offset of LP11. Finally, concerning the energy exchange between the subspaces LP11 and LP21 we see that the sub-matrix Γ 21−11 = −Γ11− 21 corresponds to the rotation matrix, depending on the angle (ϕ11 − 2ϕ21 ) . In the following section we discuss the
impact of the abovementioned coefficients on the global system performance. 3.2 Lab measurements The experimental setup described in Fig. 7 has been built to assess the crosstalk generated by the aforementioned non-ideal connector for various couples of distinct spatial modes, i.e. experimentally validate the coupling coefficients of the connector matrix. For this measurement a single-mode, single-channel (1550 nm) 112 Gb/s PDM-QPSK signal is generated and divided in 5 SMFs with different lengths to induce a de-correlation of some hundreds of symbols between the data streams, as it is also considered in [5]. Then, the signal in the first fiber (fundamental mode, LP01) is directly coupled to the FMF fiber, while for the other fibers a mode conversion is applied to generate the 4 other spatial mode profiles, i.e. LP11a, LP11b, LP21a, and LP21b. Mode conversion basically relies on SMF-to-FMF mode conversions using 4f correlators within which phase masks are being placed. These masks are illustrated in Fig. 7(b) with the red filled circles indicating where incoming beams hit the phase mask to get different mode conversions. All modes are then coupled to the FMF using the free-space setup of Fig. 7(a). The FMF connector is emulated by a butt joint of two 5-m long FMFs (Fig. 7(c)) with a variable lateral misalignment. Mode DEMUX is achieved by using a symmetric setup to the MUX. Crosstalk is measured by injecting only in one of the 5 modes and measuring the received power to all 5 modes using a set of power-meters.
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Received 24 Sep 2014; revised 17 Nov 2014; accepted 28 Nov 2014; published 21 Jan 2015 26 Jan 2015 | Vol. 23, No. 2 | DOI:10.1364/OE.23.001438 | OPTICS EXPRESS 1449
Fig. 7. Experimental setup for the crosstalk ratios measurements.
In our experiment, the FMF link is limited to two butt-jointed 5-meter long FMF strands with a view to exclusively emulate a single connector. Indeed, such short fiber lengths prevent important propagative crosstalk from building up under the effect of both intrinsic and extrinsic fiber perturbations. Moreover, both strands come from the same FMF spool to guarantee the identity of their optical and geometrical properties. The FMF has a step-index (SI) refractive index profile with rcore = 7,5 µm and Δn = 0.0102. More specifically, its design is the one suggested in [5,9] which aims at strongly guiding 3 spatial LP modes. It should be stated that in the weakly guiding approximation, LP21 and LP02 share the same cutoff value. Thus, our SI-FMF can guide 4 LP modes. However, in our MDM transmission system implementation, for reasons of experimental simplicity, LP02 is not used. Finally, the capabilities of a splice machine were used to control the lateral alignment of the upstream and downstream FMFs in the butt-joint.
Fig. 8. Experimental versus numerical estimation of the matrix crosstalk coefficients.
Commenting on the experimental issues affecting the measurements, we first note that in practice, a - fully or partially - depolarized signal has proven to significantly enhance power measurement stability owing to an averaging over polarization states which restrains the impact of polarization-dependent components combined with polarization changes. Secondly, the splicing machine gave us the possibility of the variation of the lateral misalignment with a minimum possible step of approximately 0.26 µm, therefore inducing a possible imprecision of about ± 0.015 in terms of normalized distance D. Thirdly, the MUX/DEMUX free-space setup introduces an additional mode-dependent loss that was a posteriori subtracted from the direct measurements. Finally, since the MUX/DEMUX free-space setup also introduces a low inter-modal crosstalk (typical values may be found in Table 1 of [5]), very low values of connector-induced crosstalk (about −25 dB) cannot be measured. This is critical for connector crosstalk measurements with a lateral misalignment lower than about 10% of the core radius. In Fig. 8 we plot the crosstalk coefficient as a function of the lateral misalignment for both the numerical evaluation and the experiment for the modes LP01, LP11a, LP11b, LP21a and LP21b. We observe that, for every inspected power crosstalk ratio, numerical and experimental
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Received 24 Sep 2014; revised 17 Nov 2014; accepted 28 Nov 2014; published 21 Jan 2015 26 Jan 2015 | Vol. 23, No. 2 | DOI:10.1364/OE.23.001438 | OPTICS EXPRESS 1450
curves are in good agreement. This measurement validates the form of the connector matrix presented in the section 3.1. 4. Back-to-back numerical investigation of the connector impact
While the connector matrix expression was established in the previous section, its influence on system performance is not straightforward. While a comprehensive study of the connector system impact is beyond the scope of the present paper, in this section we provide some insights based on a simplified numerical simulation setup. In Fig. 9, we show our numerical simulation setup. Each of the five modes is carrying a single-channel Root Raised Cosine (RRC) QPSK signal with a roll-off factor of 0.5, data based on different pseudorandom sequences of 4096 symbols, modulated at 32Gbaud, with the same power Ps. All modes have been assumed to have the same initial phase. This assumption can be justified from the intuitions appearing in Fig. 3(b) of the reference [21] from which we can infer that the influence of a random phase difference between different modes should be limited in the case where a MIMO with a sufficient number of taps is applied. All modes are multiplexed in the same FMF, followed by the multimode connector under study, considering for this example a normalized lateral misalignment D = 0.2. In order to exclusively focus on the impact of a single connector, the FMF of the simulation setup are considered very short (i.e. a few meters long) and DMGD has been neglected. Then, a multimode black-box amplifier is used to load a noise power Pn for each mode bringing each mode up to an OSNR0.1nm = 10 dB. Finally, the modes are de-multiplexed, filtered by a RRC matched filter and treated simultaneously by a coherent receiver. For the sake of simplicity we consider ideal multiplexer, de-multiplexer, amplifier and coherent receiver.
Fig. 9. Simplified system for the performance analysis of the connector.
For the signal processing at the receiver side we consider the two configurations presented above, i.e. the global MIMO receiver scheme and the partial MIMO receiver scheme. We note here that, if one considers a realistic transmission over FMF, a low number of taps may be enough to separate modes with low DMGD, a significantly higher number of taps (that may quickly become unreasonable) is required for modes with high DMGD, depending on the transmission length. Nevertheless, in this back-to-back investigation of the connector impact, both partial and global MIMO reception can be performed with a reasonable number of taps. The source separation is achieved by a Constant Modulus Algorithm (CMA). Its taps are initialized with the channel matrix estimated by the Equivariant Adaptive Separation via Independence (EASI) algorithm [22] in order to avoid the singularity problems of standard CMA. Finally, the BER is converted to an equivalent Q2 factor.
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Received 24 Sep 2014; revised 17 Nov 2014; accepted 28 Nov 2014; published 21 Jan 2015 26 Jan 2015 | Vol. 23, No. 2 | DOI:10.1364/OE.23.001438 | OPTICS EXPRESS 1451
Fig. 10. Q2 for the five modes after a misaligned butt-joint connector as a function of the origin azimuthal offset φ11, for the partial MIMO receiver scheme.
In Fig. 10 we show the Q2 factor after a partial MIMO equalizer scheme for each mode (considering separately the degenerate modes), the average quality of each LP mode and the average quality of all five modes as a function of the azimuthal offset φ11, fixing φ21 to 0 rad. First we observe that LP01 yields the highest performance, LP11 yields an average performance (over the two degenerate modes) of about 2.1 dB lower than LP01 and LP21 exhibits an intermediate performance, about 1 dB higher than LP11. This result may be better understood in light of the Eq. (22), reminding that a>b>c. Indeed, the high performance of LP01 stems from both its high transmission coefficient through the connector and also the lowest overall coupled energy from the other modes. Equation (22) also indicates that LP11 presents an average transmission coefficient higher than the one of LP21. Nevertheless, in Fig. 10 we observe that LP11 performs worse than LP21 which can be understood from the fact that LP11 is also impacted by a higher overall coupled energy from the other modes, compared to LP21 which only interferes with LP11. This further suggests that, in this receiver scheme, if we decide to transmit information transmission over three modes only, it should be preferable to use LP01 and LP21 rather than LP01 and LP11. Commenting now on the quality evolution of each degenerate mode separately, we note that all modes present a periodic oscillation with respect to φ11, with the LP11 modes, however, presenting a peak-to-peak amplitude variation of about 3.5 dB, while a variation of less than 0.5 dB is observed for the other modes. This can be justified in view of Eq. (22) by considering the combined effect of the transmission and coupling coefficients for each mode. Indeed, it can be verified that for LP01, LP21a and LP21b both the transmission coefficient and the total coupled power are independent of both φ11 and φ21. It can also be verified that the total coupled energy for the modes LP11a and LP11b is also independent on φ21, both depending however on φ11. We suggest that the residual oscillations of LP01, LP21a and LP21b as a function of φ11 may be qualitatively understood using the analysis of [23], stating that the degradation brought by QPSK interfering signals is aggravated for an increasing number of interferes while the total interfering signal power remains constant. Nevertheless, we underline the fact that the average quality of all five modes is independent of φ11 and approximately about 6.5 dB. It can be easily verified that qualitatively similar results can be drawn for other values of φ21. Finally, we note that while the performance of each degenerate mode LP11 strongly depends on φ11, the average performance of LP11a and LP11b is practically independent of φ11, as it is also the case for all the other modes.
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Received 24 Sep 2014; revised 17 Nov 2014; accepted 28 Nov 2014; published 21 Jan 2015 26 Jan 2015 | Vol. 23, No. 2 | DOI:10.1364/OE.23.001438 | OPTICS EXPRESS 1452
Fig. 11. Global MIMO performance vs. φ11.
In Fig. 11 we show the quality for a global MIMO receiver scheme as a function of φ11 and φ21 = 0 rad. First we note that in this case, the quality of all modes in terms of Q2 factor is limited within a range of about 1 dB (between 9.2 and 10.2 dB). As before, LP01 performs better than the other modes and a periodic quality inversion is observed for the modes LP11a and LP11b. Nevertheless, this time LP11 performs better than LP21 (9.8 against 9.2 dB), since the interference is well compensated by the global MIMO, with the dominant influence coming this time from the power transmission coefficient (higher for LP11 compared to LP21), together with the residual MDL between degenerate modes. Finally, the average quality of all five modes is about 9.6, i.e. about 3 dB higher than the average quality of the partial MIMO receiver scheme. 5. Conclusion
In recent assessments of slightly multimode optical fiber systems, system designers have highlighted the necessity to perform further investigations of the mode coupling that may impair the overall MDM transmission quality. In that sense, we have proposed here a theoretical, numerical and experimental analysis of the mode coupling experienced by optical signals passing through a misaligned butt-joint connection between two identical SI-FMFs. We have arrived at a simple analytical model based on a connector transfer matrix for which coefficients are derived from mode overlap integrals and proposed simplifications based on symmetry, orders of magnitude of the various possible mode coupling and numerical estimation of matrix coefficients. Numerical coefficients estimated for LP01, LP11a&b and LP21a&b modes have also been validated through FMF experiments. First observations on estimated coefficients yield the information that more than 0.2 dB of mode losses may be reached by such a connector when the fiber misalignment exceeds 18% of the core radius. Furthermore, the transmission coefficients of all modes are assessed, indicating that the modes suffering from higher losses are, as expected, LP21 modes, then LP11 and finally LP01, quantifying the loss evolution for an increasing offset between fiber core centers. From these coefficients, one can easily derive the MDL introduced by the connector. On the other hand, we also quantify the coupling coefficients evolution as a function of D. For this aspect, LP11 is shown to potentially get the maximum coupling power from the other modes at the connector. In addition, we quantify the evolution of both MDL and mode coupling as a function of the relative angle of the connection with respect to the axis of misalignment ( ϕ ). On that point, LP11 modes appear to be highly dependent on this angle when D increases. This also highlights the fact that a statistical MDL between two degenerate modes (like LP11a and LP11b) may be critical for the MDM transmission quality. Finally, we have also estimated the quality impairments caused by such connectors using a numerical estimation of BERs of various RRC-QPSK signals launched in the five modes. When considering a receiver with a partial MIMO DSP, LP11 exhibits the highest #223753 - $15.00 USD © 2015 OSA
Received 24 Sep 2014; revised 17 Nov 2014; accepted 28 Nov 2014; published 21 Jan 2015 26 Jan 2015 | Vol. 23, No. 2 | DOI:10.1364/OE.23.001438 | OPTICS EXPRESS 1453
impairments through the connection also varying with the highest amplitude as a function of φ11. When considering a global MIMO at the receiver side, the connector impairment is significantly mitigated while the resulting discrepancy between the mode transmission qualities stem for the sole MDL of the connection. Consequently, in that latter context, LP21 shows the poorest performance. Further investigations need to be performed, with cascaded connectors and propagation with DMGD between connections to yield a better statistical estimation of these mode coupling impairments on the overall system quality. Appendix 1: Primarily-coupled modes in a SI fiber
For a given mode of the incoming fiber, we are interested in approximately determining the modes of the outcoming fiber towards which, the light of the first mode is primarily coupled. More precisely, we investigate the possibility of the analysis of [19], developed in the case of graded-index fibers, being qualitatively applicable for the first modes of a step-index fiber. To do this we verify if we can approximate the LP modes with those of an “equivalent” Infinite-Parabolic-Index (IPI), using the Gaussian approximation studied in [11]. Indeed, in [24], the waist of the fundamental Gaussian beam is empirically found to approximate the actual fundamental LP01 mode field of a Step-Index fiber as a function of the fiber normalized frequency V, with the validity domain of this approximation being restricted to low-order modes. The approximate transverse modal fields of the LP modes fields of a matched IPI fiber [12] reads in polar coordinates: l
Flp( s ) (r , θ ) ∝ exp ( −V ρ 2 / 2 ) ⋅ (V ρ 2 ) 2 ⋅ Llp −1 (V ρ 2 ) f l ( s ) (θ )
(23)
where l stands for the LP mode azimuthal index, p its radial index, V the fiber normalized frequency, ρ ( = r/rcore) the normalized radial polar coordinate and Lαβ is the generalized Laguerre polynomial of order α and degree β. Replacing in the previous equation V ρ 2 with 2
ρ2 w0
transforms these LP mode fields into
w0 , where w0 rcore is the beam waist. The correspondence between a set of LPlp modes of a single IPI fiber and a set (with the same waist) of LGmn beams is given by:
a set of orthonormal Laguerre-Gaussian (LG) beams of normalized waist W0 =
l=m p = n +1 V=
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2 W02
Received 24 Sep 2014; revised 17 Nov 2014; accepted 28 Nov 2014; published 21 Jan 2015 26 Jan 2015 | Vol. 23, No. 2 | DOI:10.1364/OE.23.001438 | OPTICS EXPRESS 1454
Fig. 12. Transmission coefficient or matching parameter T as a function of V for the modes LP11, LP21, LP02 and LP31.
As a justification of the generalized Gaussian approximation within our theoretical analysis, we must indicate the matching quality and the tolerance with respect to our specific application. For a hypothetical connector between a step-index and a ISI fiber as in [24], we plot in Fig. 12 the power transmission coefficient (noted as matching parameter T) between the transverse fields of the best-fitting LGl,p-1 beam and the actual LPlp SI-fiber mode for the first ten LP modes and for a normalized frequency V spanning from 2.5 to 8. With the Gaussian approximation already shown to be accurate for LP01 [24], we note a matching over 99.5% for the modes of interest, i.e. LP11 and LP21 and V=5.1. Nevertheless, it could be verified that T drops significantly when p increases, as it can be seen for the example of LP02 in Fig. 12. In conclusion, the analysis of [19] may be used to qualitatively estimate the relative weight of the mode coupling coefficients in few mode SI fibers. Acknowledgement
The authors would like to thank the French government for supporting a part of this work through the ANR agency under the project STRADE (ANR-09-VERS-010).
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Received 24 Sep 2014; revised 17 Nov 2014; accepted 28 Nov 2014; published 21 Jan 2015 26 Jan 2015 | Vol. 23, No. 2 | DOI:10.1364/OE.23.001438 | OPTICS EXPRESS 1455
