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 step-index 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 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 r_core and the same core and cladding refractive indices n_co and n_cl (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 $V=\frac{2\pi r_{core}}{\lambda }\sqrt{n_{co}^2-n_{cl}^2}$ is the normalized frequency of both fibers and $D=\frac{d}{r_{core}}=\frac{o_1{o}_2}{r_{core}}$ is the normalized distance between the core centers noted as O₁ and O₂ respectively. In addition we define the coordinate system $O_{1}xy$ 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 $d\to 0$ (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).

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Both fibers have a relatively low refractive index difference (Δn = n_co-n_ci<<1) and for simplicity we use a description of the guided optical field based on LP modes [12]. Considering our coordinate system $O_{1}xy$, for a given angular frequency ω, the transverse distribution $F_{lp}^{(s)}$ of an LP mode guided by the fiber is determined by its azimuthal index l = 0,1,2…, its radial index p = 1,2,…, and a parity tag (s) that may be “a”, ”b” or omitted, indicating its symmetry with respect to the angular polar coordinate θ. More precisely, with “a” we denote symmetric (even) modes, with “b” anti-symmetric (odd) modes, while the tag is omitted for circular-symmetric modes. Modes with a circular symmetry are also even and therefore, in the following we refer to modes with opposite parities when one of them is either circular-symmetric or even and the other is odd; otherwise we refer to modes with the same parity. The normalized mode transverse distribution of a given mode LP_lp(s) is given by the relations [13, 14]:

Suppose now that our fiber supports a limited number of LP modes LP₀₁, LP₁₁, LP₂₁,…, denoted N_g, appearing in an ascending order of their cut-off normalized frequency values x_lp. In this case, the corresponding functions $F_{lp}^{(s)}(r,\theta )$ form a modal basis, i.e.

The field guided in this fiber may be decomposed onto this modal basis and, mathematically, it may be written as:

From Eq. (4) it is evident that the orthogonality between modes is conserved, even if is replaced in the above expressions by θ-φ, where φ is an arbitrary positive angle (Fig. 1(c)), referred to in the following as origin azimuthal offset [15]. In other words, if $\left(l,p\right)\ne \left(l^{\prime },p^{\prime }\right)$ and for any $\phi ,{\phi }^{\prime },s,s^{\prime }$:

Therefore, a given modal basis depends on the actual origin azimuthal offsets for each LP mode. In this work, we base our analysis on the assumption that the incoming LP modes preserve their shape at the connector interface (therefore we neglect all effects of intramodal dispersion [16]), but we consider that, due to the lateral offset (Fig. 1(a)), each LP mode has an origin azimuthal offset at the connector interface, grouped for convenience in the vector ${\overrightarrow{\phi }}_{ref}={\left[{\phi }_{01},{\phi }_{11},{\phi }_{21},\mathrm{...}\right]}^T$. We refer to this modal basis as reference modal basis. We also associate to the connector the origin azimuthal offset vector ${\overrightarrow{\phi }}_{con}={\left[0,0,0,\mathrm{...}\right]}^T$with its corresponding modal basis referred to as connector modal basis.

Using Eq. (10), the matrix expression of Eq. (9) may be considerably simplified. First, the coefficients describing the coupling between modes with opposite parities are 0. This property can be illustrated by considering the mode symmetries with respect to the axis of the misalignment, as shown in Fig. 2. In Fig. 2(a), considering the overlaps of the upper and lower lobes of LP_11b (outgoing mode) with the mode LP_11a (incoming mode), we observe that they yield the same overlap integral with the opposite sign and the sum of the two is zero, i.e. $c_{11a-11b}=0$. Similarly, it can be shown that the overlap integral of the incoming mode LP_11a and the outgoing LP_11a (Fig. 2(b)) is not zero, while the pair LP_11b and LP_11b (Fig. 2(c)) also yields a non-zero coefficient that is not necessarily equal to the one of Fig. 2(b). It is evident that according to the above, the diagonal elements of C_con are non-zero. Finally, one more simplification of C_con could be achieved by pointing out the following relation between non-diagonal elements: $c_{lp(s)-l^{\prime }{p}^{\prime }(s^{\prime })}={(-1)}^{l+l^{\prime }}{c}_{l^{\prime }{p}^{\prime }\left(s^{\prime }\right)-lp(s)}$, (illustrated in Figs. 2(d), (e), (f) and (g)). More precisely, in Fig. 2(d) and (e) we illustrate the case where the two coefficients have the same absolute value and an opposite sign, showing both the coupling between an incoming LP_11a with the outgoing LP₀₁ that has an azimuth index with a different parity (Fig. 2(d)) and the inverse coupling between an incoming LP₀₁ with an outgoing LP_11a (e). On the other hand, the case where both coefficients have the same absolute value and the same sign is illustrated in Fig. 2(f) and (g). In view of the above properties, C_con may be simplified to:

 

Fig. 2 Schematic view of overlap integrals between incoming (solid contours) and outgoing modes (dashed contours), leading to a simplification of the coupling matrix coefficients. (a) example of even and odd modes that are not coupled (b)-(c): example of a pair of even modes and a pair of odd modes with a different positive coupling coefficient, (d)-(e) example of incoming and outgoing modes with overlap integrals changing signs when switching places for incoming and outgoing modes (f)-(g) example of incoming and outgoing modes with equal overlap integrals when switching places for incoming and outgoing modes.

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We note that the above matrix is non-unitary, a direct consequence of the inherent Mode Dependent Loss (MDL) of such a misaligned connector. This unavoidably implies performance penalties due to loss of mode orthogonality and SNR variation between modes [17], as it is also true for the qualitatively similar case of Polarization Dependent Loss (PDL) [18].

In an attempt to further simplify the matrix of Eq. (11) we adapt the rules presented in [19] for graded-index fibers, in our context of step-index fibers (see the appendix I). In Fig. 3 we illustrate its principal results concerning the coupling towards the outgoing mode LP_lp coming from the incoming modes, or in other words, we provide more information on the relative weight between the different coupling coefficients of a given row of the matrix in Eq. (11). In Fig. 3(a), the filled round marker with coordinates (l,p) indicate our considered outgoing mode LP_lp, the empty round markers indicate the modes that are going to be mainly coupled with LP_lp (with corresponding coefficients referred to as first-order coupling coefficients), the empty triangle markers indicate the modes that are less efficiently coupled to LP_lp with coefficients referred to as second-order coupling coefficients, while finally when no marker is present the modes are practically not coupled to LP_lp. Besides, depending on the normalized index V, some modes may not be guided by the fiber (illustrated by dashed-line markers in Fig. 3(a)), whereas some other modes are certainly guided (illustrated by solid line markers). Following the analysis of the appendix I, it can be shown that the second-order coupling coefficients are approximately equal to the square of the first-order coupling coefficients, indeed being weaker (given that $c_{lps}<1,\forall l,p,s$). Practically, there is a difference of at least one order of magnitude (or even several orders of magnitude for a small offset d) between first and second order coupling coefficients and therefore, as a first approximation, we may neglect the second order coupling coefficients. For example, considering the first row of the matrix of Eq. (11), the mode LP₀₁ will be principally coupled only with LP₁₁ modes, while it will be very weakly coupled to LP₂₁ or LP₀₂ modes, given of course that the mentioned modes are guided by the fiber. In other words, $c_{21a-01}\approx 0$, $c_{02-01}\approx 0$, while all the other higher order coupling coefficients are practically zero, as well. However, if we focus on the outgoing mode LP_11a (i.e. the second row of the matrix), we note that this mode is principally coupled with the mode LP₀₁, while it may also be principally coupled with LP₂₁ and LP₀₂ (Fig. 3(b)).

 

Fig. 3 (a) Primary (round markers) and secondary (triangle markers) ross-talk sources coupled upon the outgoing mode LP_lp (filled round marker). Modes that are definitely guided are indicated by solid contours, while modes that are potentially not supported are indicated by dashed contours. (b) Application of (a) for the outgoing mode LP₁₁. The numbers indicate the order of appearance for an increasing normalized frequency V.

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Now, we come back to our initial assumption that the LP modes LP₀₁, LP₁₁, LP₂₁ etc. appear at the interface with an origin azimuthal offset vector ${\overrightarrow{\phi }}_{ref}={\left[{\phi }_{01},{\phi }_{11},{\phi }_{21},\mathrm{...}\right]}^T$. For each LP_lp mode we define the corresponding subspace decomposition matrix

In other words, for circular-symmetric modes $(l=0)$, R_lp is equal to the scalar 1 and φ_0p can be fixed at an arbitrary value, while, when $(l\ne 0)$ R_lp is a $2\times 2$ decomposition matrix of the reference modal basis onto the connector modal basis. Using the subspace decomposition matrices we may also define the global decomposition matrix

Then, it can be shown (see appendix II) that the connector transmission matrix $\Gamma (\overrightarrow{\phi })$ in our reference modal basis is given by the expression

Considering a cascade of n connectors with different azimuth vectors ${\overrightarrow{\phi }}_1,{\overrightarrow{\phi }}_2,\mathrm{...},{\overrightarrow{\phi }}_n$ and normalized distances $D_1,D_2,\mathrm{...},D_n$, the overall matrix ${\Gamma }_{tot}$ may be found by multiplying the matrices of each individual connector, i.e. ${\Gamma }_{tot}={\Gamma }_n(\overrightarrow{{\phi }_n})\cdots {\Gamma }_2(\overrightarrow{{\phi }_2})\cdot {\Gamma }_1(\overrightarrow{{\phi }_1})$ . In the following we just focus on the impact of only one connector, while the analysis of a cascade of n connectors may be the object a future investigation.

Finally, we introduce the crosstalk induced by an incoming mode LP_lp(s) with an origin azimuthal offset φ_lp towards an outgoing subspace LP_l’p’. Using the Eq. (14), the power transfer coefficient is given by

2.2 Receiver considerations

The analysis of the impact that a lateral offset at any coupling interface may have on the performance of a transmission system depends on the reception scheme. As discussed in the introduction, the two basic strategies consist in applying coherent reception either with an independent block of MIMO for each of the N_g LP modes (including possible degenerate modes) referred to as partial MIMO receiver scheme, or alternatively, a unique $N_m\times N_m$ MIMO for all exploited modes, referred to as global MIMO receiver scheme. In the global MIMO receiver scheme, the MIMO block is able to cope with the crosstalk between the spatial tributaries, if we use a sufficient number of taps with respect to the maximum accumulated DMGD. In the global MIMO receiver scheme, the system performance will principally depend on the fraction of energy of each incoming mode that will be coupled to the guided outgoing modes, referred to as global transmittance and expressed (using Eq. (17)) as

In the partial MIMO receiver scheme, MIMO processing is able to mitigate the crosstalk between the two degenerate modes, whereas it cannot mitigate the crosstalk coming from other LP modes (which is thus seen as a source of noise).

In Fig. 4 we represent the coupling matrix of the connector for a fiber supporting N_g = 3 LP modes, i.e. LP₀₁, LP₁₁ and LP₂₁. Regrouping the terms ${\gamma }_{lp(s)-l^{\prime }{p}^{\prime }(s^{\prime })}$ as shown in Fig. 4, we can rewrite Eq. (14) with the help of the sub-matrices ${\Gamma }_{lp-l^{\prime }{p}^{\prime }}$ that describe the coupling of the subspace LP_lp towards the subspace LP_l’p’:

 

Fig. 4 Connector coupling or transmission matrix.

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In Eq. (19), the dimensionality of the matrix ${\Gamma }_{lp-l^{\prime }{p}^{\prime }}$ is $M\times N$, where M is the number of (possibly degenerate) LP_l’p’ modes and N is the number of (possibly degenerate) LP_lp modes.

In the partial MIMO receiver scheme we consider 3 MIMO blocks, one for LP₀₁, one for LP₁₁ and one for LP₂₁. Each MIMO block will compensate for the crosstalk within its subspace. Therefore, for each LP mode, the performance will be determined by the crosstalk coming from other LP modes and by the transmittance of the LP mode itself, referred to as subspace transmittance and expressed as

In the following sections we present numerical and experimental results, based on the coefficients of the matrices C_con and $\Gamma \left({\overrightarrow{\phi }}_{ref}\right)$ for variable parameters V and D.

3. Evaluation of connector transmission and coupling coefficients

3.1 Numerical evaluation

In this section we consider the example of a FMF with V = 5.1 supporting 4 LP modes, i.e. LP₀₁, LP₁₁, LP₂₁ and LP₀₂. This could correspond to a specific SI fiber design of r_core = 7.5μm and Δn = 10⁻² [9]. In the following we analyze the coefficients $c_{lp(s)-l^{\prime }{p}^{\prime }(s^{\prime })}$ of the matrix C_con (Eq. (11)), numerically calculating the integral of Eq. (10). We underline the fact that in the numerical analysis of this section, the matrix is evaluated in the abovementioned connector modal basis, i.e. with ${\overrightarrow{\phi }}_{ref}=\overrightarrow{0}$. Next, we focus on just the first 3 LP modes of the abovementioned fiber.

In Fig. 5, we plot the power self-coupling coefficients (also referred to as transmission coefficients) ${\left[c_{lp(s)-lp(s)}\right]}^2$ as a function of the normalized distance between the core centers . Furthermore, since the insertion loss introduced by a single-mode fibers’ connector would preferably not exceed 0.2 dB, we also indicate it as reference with a solid line at the level of 0.2 dB. Commenting on Fig. 5, we first note that for all considered values of D, ${\left[c_{01-01}\right]}^2\approx {\left[c_{11b-11b}\right]}^2=a^2$ and ${\left[c_{11a-11a}\right]}^2\approx {\left[c_{21a-21a}\right]}^2\approx {\left[c_{21b-21b}\right]}^2=b^2$. We also note that the pair of numbers ${\left[c_{01-01}\right]}^2,{\left[c_{11b-11b}\right]}^2$ reaches the insertion loss reference of single-mode fibers for $D\approx 0.18$, while for the same D, all the other coefficients present a transmission coefficient of about -1dB. It is also interesting to note that there is an important difference between the coefficients of the subspace LP₁₁, i.e. ${\left[c_{11b-11b}\right]}^2$ and ${\left[c_{11a-11a}\right]}^2$ that significantly increases for increasing D, while, on the other hand, the coefficients of the subspace LP₂₁ have the same order of magnitude. It has been shown [19] that such a differential attenuation between degenerate LP_lp modes, is significant only for modes with an azimuthal index l = 1. In other words, the connector acts as an intra-subspace MDL element. Moreover, it could be also shown that the average over the two coefficients of the same subspace globally decreases as the mode order increases.

 

Fig. 5 Power self-coupling (or transmission) coefficients ${\left[c_{lp(s)-lp(s)}\right]}^2$.

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In Fig. 6, we plot the distinct power cross-coupling coefficients ${\left[c_{lp(s)-l^{\prime }{p}^{\prime }(s^{\prime })}\right]}^2$ for $(l,p,s)=(l^{\prime },p^{\prime },s^{\prime })$, being reduced to 4 by taking into account the fact that we have zero and symmetric (or antisymmetric) elements in the matrix C_con. We can distinguish two sets of coefficients. On one hand ${\left[c_{11a-01}\right]}^2\approx {\left[c_{21a-11a}\right]}^2\approx {\left[c_{21b-11b}\right]}^2=c^2$ with these coefficients exhibiting approximately the same evolution as a function of D for our considered range of values and being equal to about -12dB for D = 0.18. On the other hand, the coefficient ${\left[c_{21a-01}\right]}^2$ is about 20 dB lower than the other coefficients and therefore, in agreement with the analysis of Fig. 3, it could be safely neglected when compared to the other terms, i.e. ${\left[c_{21a-01}\right]}^2\approx 0$.

 

Fig. 6 Power cross-coupling coefficients ${\left[c_{lp(s)-l^{\prime }{p}^{\prime }(s^{\prime })}\right]}^2$ as a function of the normalized offset D.

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Including the above simplifications (valid for low values of D), the matrix C_con (see Eq. (11)) reads:

As mentioned in section 2, the matrix C_con is the coupling coefficient matrix in the special case when the origin azimuthal offsets of all LP modes are zero, i.e. $\overrightarrow{\phi }=\overrightarrow{0}$. In practice, for a non-zero $\overrightarrow{\phi }$, $\Gamma \left(\overrightarrow{\phi }\right)$ has to be considered instead of C_con. Including the abovementioned simplifications for low values of D and using the Eq. (14), $\Gamma \left(\overrightarrow{\phi }\right)$ reads:

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.

 

Fig. 7 Experimental setup for the crosstalk ratios measurements.

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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.

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 r_core = 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.

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, LP_11a, LP_11b, LP_21a and LP_21b. We observe that, for every inspected power crosstalk ratio, numerical and experimental curves are in good agreement. This measurement validates the form of the connector matrix presented in the section 3.1.

 

Fig. 8 Experimental versus numerical estimation of the matrix crosstalk coefficients.

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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 P_s. 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 P_n for each mode bringing each mode up to an OSNR_0.1nm = 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.

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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.

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, LP_21a and LP_21b 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 LP_11a and LP_11b is also independent on φ₂₁, both depending however on φ₁₁. We suggest that the residual oscillations of LP₀₁, LP_21a and LP_21b 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 φ_11, the average performance of LP_11a and LP_11b is practically independent of φ_11, as it is also the case for all the other modes.

 

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.

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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 LP_11a and LP_11b. 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.

 

Fig. 11 Global MIMO performance vs. φ₁₁.

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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, LP_11a&b and LP_21a&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 ($\overrightarrow{\phi }$). 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 LP_11a and LP_11b) 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 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:

(23)$$F_{lp}^{(s)}(r,\theta )\propto \exp \left(-V{\rho }^2/2\right)\cdot {\left(V{\rho }^2\right)}^{\frac{l}{2}}\cdot L_{p-1}^l(V{\rho }^2)f_l^{(s)}(\theta )$$

where l stands for the LP mode azimuthal index, p its radial index, V the fiber normalized frequency, ρ ( = r/r_core) the normalized radial polar coordinate and $L_{\beta }^{\alpha }$ is the generalized Laguerre polynomial of order α and degree β.

Replacing in the previous equation $V{\rho }^2$ with $2\frac{{\rho }^2}{w_0}$ transforms these LP mode fields into a set of orthonormal Laguerre-Gaussian (LG) beams of normalized waist $W_0=\frac{w_0}{r_{core}}$, where w₀ is the beam waist. The correspondence between a set of LP_lp modes of a single IPI fiber and a set (with the same waist) of LG_mn beams is given by:

$$|\begin{array}{l}l=m\\ p=n+1\\ V=\frac{2}{W_0^2}\end{array}$$

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 LG_l,p-1 beam and the actual LP_lp 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.

 

Fig. 12 Transmission coefficient or matching parameter T as a function of V for the modes LP₁₁, LP₂₁, LP₀₂ and LP₃₁.

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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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