1. Introduction

Mode-division multiplexing (MDM) with few-mode circular-core fibers is being actively explored as a promising technology to increase the transmission capacity of a single strand of optical fiber [1–3]. The development of the MDM technology relies very much on the availability of key mode-controlling devices, such as mode (de)multiplexers, which serve to spatially combine or separate different fiber modes, and mode converters, which serve to convert between different fiber modes. Mode (de)multiplexers, in particular, are needed not only for (de)multiplexing modes or mode groups at the input and output ends of a few-mode fiber [1,4], but also for the compensation of mode-dependent losses and delays along the few-mode fiber link [5].

Among the reported mode multiplexers, those based on buried channel waveguides can provide good mode selectivity with compact size and flexibility in the choice of waveguide materials [6–11]. Mode converters can also be realized effectively with buried channel waveguides [12,13]. Such mode multiplexers and converters are usually formed with rectangular cores of lifted mode degeneracy to facilitate the separation of spatial modes. However, because of the geometry mismatch, butt-coupling a rectangular-core waveguide to a circular-core fiber inevitably leads to modal crosstalk, especially when the number of spatial modes is large. Although buried waveguides with nearly circular refractive-index profiles can be formed by a direct-writing technique with a femtosecond laser [11], the laser-writing process is not as precise as the conventional waveguide fabrication process based on photolithography and the laser-induced refractive-index contrast is generally restricted to less than 6 × 10⁻³ [14]. A practical way to minimize the geometry mismatch of a rectangular-core waveguide and a circular-core fiber for direct butt-coupling is to taper the rectangular core into a square core [9,15]. However, there has been no detailed analysis of the parameters of the square core required for optimizing the coupling and the minimum modal crosstalk that can be achieved. In the first part of our study, we provide such an analysis for step-index and parabolic-index circular-core fibers that support 6 spatial modes (or 12 modes if considering the polarization). We find that, to achieve low modal crosstalk (< −20 dB) for all the six spatial modes, the waveguide and the fiber should have close mode volumes and the shape of the waveguide core must be very close to a square (with an aspect ratio between 0.993 and 1.007).

To achieve stable and uncoupled transmission of individual spatial modes, elliptical-core few-mode fibers have been proposed recently. Such a fiber may have a moderate ellipticity [16–18] or a high ellipticity [19], or consist of an elliptical ring [20]. The fiber with a moderate-ellipticity core, in particular, has been demonstrated experimentally with low-cost direct detection for intra-datacenter networks [16] and the transmission capacity could be further multiplied by using a multicore elliptical-core fiber [21]. As an elliptical core and a rectangular core possess identical two-fold rotational symmetry and hence similar mode patterns, it should be possible to achieve low modal crosstalk in butt-coupling by matching the parameters of the fiber and waveguide cores. In the second part of our study, we analyze butt-coupling between a buried rectangular-core waveguide and a step-index or a parabolic-index elliptical-core fiber with the objective of identifying the conditions to achieve low modal crosstalk for the first 6 spatial modes. We find that the fiber-waveguide geometry mismatch problem can indeed be alleviated by using an elliptical-core fiber. The use of a parabolic-index elliptical-core fiber, compared with a step-index elliptical-core fiber, can further relax the tolerances on the waveguide parameters required for the achievement of low modal crosstalk in butt-coupling.

The results in our study are useful for the design of waveguide-based mode-selective devices, such as mode (de)multiplexers, mode converters, and mode switches, for MDM using either circular-core or elliptical-core few-mode fibers.

2. Coupling between a circular-core fiber and a rectangular-core waveguide

2.1 Spatial modes in a circular-core fiber and a rectangular-core waveguide

We consider the coupling of the first 6 spatial modes between a circular-core fiber and a rectangular-core waveguide. The fiber can have a step-index profile or a parabolic-index profile, while the waveguide has a step-index profile. The refractive indices of the core (or at the center of the core, in the case of a parabolic-index profile) and the cladding of the fiber are fixed at 1.454 and 1.444, respectively, as in a typical few-mode fiber design [4]. To achieve a high coupling efficiency between the fiber and the waveguide, the core-cladding refractive-index difference of the waveguide should be comparable to that of the fiber and the average refractive indices of the fiber and the waveguide should not be too different. With these considerations, we fix the refractive indices of the core and the cladding of the waveguide at 1.570 and 1.560, respectively, which are typical values for the polymer materials used in the recently demonstrated mode multiplexers [6,7] and mode converters [12,13]. Polymer waveguides are compatible with fibers in terms of both the core size and the core-cladding index difference, which makes possible direct fiber-waveguide butt-coupling and thus facilitates device packaging. High-index-difference waveguide platforms, such as silicon photonics, on the other hand, do not allow direct fiber-waveguide butt-coupling. With the refractive indices of the fiber and the waveguide fixed, our task is to analyze the effects of the geometry mismatch between the fiber and the waveguide on the performance of butt-coupling, with emphasis on the modal crosstalk incurred. In our study, we calculate the effective indices and the electric-field distributions of the modes of the fibers and the waveguides with a full-vector finite-difference mode solver (Lumerical MODE Solutions).

As an example, we calculate the modes of a step-index circular-core fiber, which has a core diameter of 13.1 µm (core area 135.0 µm²) and supports only the LP₀₁, LP_11a, LP_11b, LP_21a, LP_21b, and LP₀₂ modes at the wavelength 1550 nm. The major electric-field components of these 6 spatial modes are shown in Fig. 1(a). We should note that our modal analysis actually produces 12 true vector modes. The LP modes, which are almost linearly polarized modes, are constructed by properly combining the nearly degenerate vector modes [22]. As a result, each of the 6 LP spatial modes has a two-fold degeneracy for their two orthogonal polarizations. As the two polarized modes of the same order have practically the same butt-coupling efficiency, there is no need to differentiate them in our analysis. The LP spatial modes, with or without polarization multiplexing, are used for fiber-based MDM.

 

Fig. 1 Major electric-field components of the first 6 spatial modes in (a) a step-index circular-core fiber with a core diameter of 13.1 µm and three rectangular-core waveguides with (b) h/w = 0.95, (c) 1.00, and (d) 1.05, respectively, with h = 11.2 µm calculated at the wavelength 1550 nm, where h and w are the height and the width of the rectangular core, respectively. The red and blue colors in the mode patterns represent positive and negative electric fields, respectively, and the green arrows indicate the evolution of modes by tapering the rectangular cores into the square core.

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The rectangular-core waveguides considered in our study also support only 6 spatial modes, each of which consists of two almost degenerate orthogonal polarizations. The height and the width of the core are denoted as h and w, respectively. For a rectangular core with an aspect ratio h/w not too close to 1 (i.e., the shape is not too close to a square), the spatial modes are designated as the E_mn modes, where m and n are the numbers of the peaks in the electric-field distributions along the horizontal and vertical axes of the core, respectively. For a square core, i.e., h/w = 1, however, the modes are designated with the LP mode notations, where the LP₀₁, LP_21a, LP_21b, and LP₀₂ modes are true vector modes and the LP_11a and LP_11b modes are constructed by properly combining the nearly degenerate vector modes (which are similar to the TE₀₁, TM₀₁, HE_21e, and HE_21o vector modes in a circular-core fiber). For example, the first 6 spatial modes of three rectangular-core waveguides with h/w = 0.95, 1.00, and 1.05 at h = 11.2 µm (core area 125 µm² for the core with h/w = 1) at the wavelength 1550 nm, are shown in Figs. 1(b)–1(d), respectively. As indicated by the arrows in Figs. 1(b)–1(d), the E_mn modes of a rectangular-core waveguide evolve into the LP modes of a square-core waveguide by tapering the rectangular core into the square core [9]. For h/w < 1, the E₃₁ and E₁₃ modes evolve into the LP₀₂ and LP_21a modes of the square core, respectively, while for h/w > 1, the E₃₁ and E₁₃ modes evolve into the LP_21a and LP₀₂ mode, respectively. The correspondence of the modes in the evolution can be understood by considering the symmetry properties of the mode-field distributions and the orders of the effective indices.

Assuming perfect center-to-center alignment, the butt-coupling efficiency from a waveguide to a fiber, denoted as η, can be calculated from the overlap integral between the two modes of concern (Lumerical MODE Solutions):

As shown in Fig. 1, the E₃₁ and E₁₃ modes of the two rectangular-core waveguides (h/w = 0.95 and 1.05) do not match the LP modes of the circular-core fiber. It is impossible to achieve low modal crosstalk for all the 6 spatial modes by butt-coupling any of the two rectangular-core waveguides directly to the circular-core fiber. The geometry mismatch is minimized by using a square-core waveguide. The idea of tapering a rectangular core into a square core to facilitate butt-coupling to a circular-core fiber has in fact been demonstrated experimentally [9]. Our task here is to find the best square core for mode-selective coupling to a circular-core fiber and study the tolerances on the waveguide core required.

The symmetry properties of the modes have large effects on the modal crosstalk. Here we divide the 6 spatial modes of a circular-core fiber and a square-core waveguide into 4 classes, according to their symmetries in the spatial distributions of the major electric-field components with respect to the horizontal and vertical symmetry axes of the core: (I) the LP₀₁, LP_21a, and LP₀₂ modes, whose fields are symmetric with respect to both axes, (II) the LP_11a mode, whose field is symmetric with respect to the horizontal axis and anti-symmetric with respect to the vertical axis, (III) the LP_11b mode, whose field is anti-symmetric with respect to the horizontal axis and symmetric with respect to the vertical axis, and (IV) the LP_21b mode, whose field is anti-symmetric with respect to both axes. According to our mode classification, modes from different classes have zero overlap integral. As a result, the LP_11a, LP_11b, and LP_21b modes of a square core do not cause any modal crosstalk in a circular core, since these modes are singletons in their own classes. For the same reason, the LP₀₁, LP_21a, and LP₀₂ modes of a square core do not cause crosstalk to the LP_11a, LP_11b, and LP_21b modes of a circular core. A subtle property of the LP₀₁, LP_21a, and LP₀₂ modes of a square core in class (I) is that these modes possess only rectangular symmetry (i.e., two-fold rotational symmetry) rather than square symmetry (i.e., four-fold rotational symmetry) [23]. On the other hand, the LP₀₁ and LP₀₂ modes of a circular-core fiber possess circular symmetry and the LP_21a mode possesses square symmetry. Therefore, we only need to consider the crosstalk among the modes in Class (I), namely the following 6 crosstalk characteristics: (i) LP₀₁(w) to LP_21a(f), (ii) LP₀₁(w) to LP₀₂(f), (iii) LP_21a(w) to LP₀₁(f), (iv) LP_21a(w) to LP₀₂(f), (v) LP₀₂(w) to LP₀₁(f), and (vi) LP₀₂(w) to LP_21a(f), where “w” and “f” in the brackets specify the modes of the square-core waveguide and the circular-core fiber, respectively.

The above discussions are general and apply to both step-index and graded-index fibers.

2.2 Butt-coupling between a step-index circular-core fiber and a square-core waveguide

In this section, we consider three specific step-index circular-core fibers with the following core diameters: 12.4, 13.1, and 13.8 µm, which correspond to the core areas 120.0, 135.0, and 150.0 µm², respectively. We calculate the 6 crosstalk characteristics for the three fibers with the area of the square core as the variable. We choose the range of square-core area so that the core supports only the first 6 spatial modes. The results are shown in Fig. 2. The LP₀₁(w)-to-LP_21a(f) crosstalk is always smaller than –60 dB and, therefore, ignored. Among the remaining 5 crosstalks shown in Fig. 2, the LP_21a(w)-to-LP₀₁(f) crosstalk is always the smallest, regardless of the area of the square core. The LP_21a(w)-to-LP₀₂(f) and the LP₀₂(w)-to-LP_21a(f) crosstalk are almost the same and both increase with the area of the square core, which show little dependence on the fiber used. The LP₀₁(w)-to-LP₀₂(f) and the LP₀₂(w)-to-LP₀₁(f) crosstalk are in general more significant, but there exists a window of square-core area where these crosstalks are exceedingly small. As shown in Fig. 2, to minimize the largest crosstalk among the 5 crosstalks, we should choose a square core with an area at which the LP₀₂(w)-to-LP_21a(f) crosstalk curve intersects the LP₀₁(w)-to-LP₀₂(f) crosstalk curve (on the smaller area side). For the circular-core areas 120.0, 135.0, and 150.0 µm², respectively, the best crosstalk performances are −33.2, −29.4, and −26.7 dB when the square-core areas are 119.4, 131.0, and 141.7 µm², respectively, which are indicated by the three black arrows in Fig. 2.

 

Fig. 2 Crosstalk characteristics for butt-coupling between a step-index square-core waveguide and a step-index circular-core fiber that has a core area of (a) 120.0 µm², (b) 135.0 µm², and (c) 150.0 µm², where the black arrows indicate the conditions for the best crosstalk performance and the red arrows indicate the equal-mode-volume conditions.

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The operation condition of a step-index fiber or a waveguide is characterized by the normalized frequency V = (2π/λ)a(n_co² − n_cl²)^1/2, where λ is the free-space wavelength, a is the core radius, and n_co and n_cl are the refractive indices of the core and the cladding, respectively. The mode volume, which governs the number of modes supported by a step-index fiber, is given by V²/2 [24]. For the square-core waveguides to have the same mode volumes as the three circular-core fibers, respectively, the areas of the square cores are 111.1, 125.0, and 138.9 µm², respectively, and the corresponding crosstalk performances are −26.5, −25.7, and −25.3 dB, respectively, which are indicated by the three red arrows in Fig. 2. These values are not much larger than the best values. Therefore, as a rule of thumb for practical applications, for the achievement of low-crosstalk butt-coupling, the circular-core fiber and the square-core waveguide should have close mode volumes. The low-crosstalk coupling also leads to small coupling losses. As the crosstalks among the spatial modes are small, the mode-dependent loss induced by butt-coupling is simply given by the difference between the largest and the smallest coupling loss for the 6 spatial modes. For example, for the fiber with a 135.0-µm² core, over the low-crosstalk coupling ranges shown in Fig. 2(b), the average coupling loss over the 6 spatial modes is below 0.16 dB and the mode-dependent loss is below 0.30 dB. Similar results are obtained for the other two fibers.

In practice, it is difficult to fabricate a perfect square core for the waveguide. Figure 3 shows the characteristics of butt-coupling a rectangular-core waveguide with a core shape that differs only slightly from a square to a step-index circular-core fiber. The results in Fig. 3 apply to both h/w ≤ 1 and h/w ≥ 1 (i.e., w/h ≤ 1). The core areas of the waveguide and the fiber are fixed at 125.0 and 135.0 µm², respectively, so that they have the same mode volume. As shown in Fig. 3, the crosstalk performance is limited by the LP_21a(w)-to-LP₀₂(f) and the LP₀₂(w)-to-LP_21a(f) crosstalk, as discussed earlier. For these crosstalks to be smaller than –20 dB, the aspect ratio of the waveguide core must be within the range from 0.993 to 1.007. The tolerance on the aspect ratio of the waveguide core is about ± 0.7%. As the aspect ratio changes from h/w = 1.000 (square) to 0.993, the coupling losses for the LP₀₁, LP_11a, LP_11b, LP_21a, LP_21b, and LP₀₂ modes increase slightly from 0.02, 0.06, 0.06, 0.07, 0.03, and 0.16 dB, to 0.02, 0.10, 0.09, 0.12, 0.03, and 0.21 dB, respectively, i.e., the mode-dependent loss increases slightly from 0.14 to 0.19 dB. If the crosstalk performance is relaxed to –10 dB, the aspect ratio of the waveguide core required must be within the range from h/w = 0.98 to 1.02, which correspond to a ± 2% tolerance on the aspect ratio of waveguide core. For the aspect ratio h/w = 0.98, the coupling losses for the LP₀₁, LP_11a, LP_11b, LP_21a, LP_21b, and LP₀₂ modes are 0.02, 0.06, 0.08, 0.55, 0.03, and 0.63 dB respectively. The mode-dependent loss is 0.61 dB. As shown by the results in Fig. 3, for the achievement of low crosstalk, the tolerances on the waveguide parameters are tight.

 

Fig. 3 Coupling characteristics for a rectangular-core waveguide butt-coupled to a step-index circular-core fiber, where the core areas of the waveguide and the fiber are 125.0 µm² and 135.0 µm², respectively.

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2.3 Butt-coupling between a parabolic-index circular-core fiber and a square-core waveguide

As few-mode graded-index fibers are also commonly used in MDM [25], in this section, we analyze butt-coupling between a square-core waveguide and a parabolic-index circular-core fiber. The mode volume of a parabolic-index fiber is V²/4 [24]. For a parabolic-index fiber to have the same mode volume as a step-index fiber, the area of the parabolic-index core must be twice of that of the step-index core.

Here we consider three specific parabolic-index fibers that support only 6 spatial modes. The core areas of the three fibers are 240.0, 270.0, and 300.0 µm², respectively. The mode volumes of these three parabolic-index fibers are the same as those of the step-index fibers studied in the previous section. Figure 4 shows the variation of the crosstalk characteristics with the area of the square core. The results are similar to those shown in Fig. 2. For the circular-core areas 240.0, 270.0, and 300.0 µm², respectively, the best crosstalk performances are –34.9, –32.2, and –30.1 dB, which occur at the square-core areas 113.5, 120.3, and 126.6 µm², respectively (indicated by the three black arrows in Fig. 4). The equal-mode-volume square-core areas for the three fibers are 111.1, 125.0, and 138.9 µm², respectively, at which the largest crosstalks are –32.1, –31.0, and –27.2 dB, respectively (indicated by the three red arrows in Fig. 4), which are quite close to the minimum values. The rule of thumb, namely, the use of an equal-mode-volume square-core waveguide for butt-coupling to a circular-core fiber, also applies to a parabolic-index fiber. In fact, by using an equal-mode-volume square core for butt-coupling, the crosstalk performance of a parabolic-index fiber is better than that of a step-index fiber that has the same mode volume as the parabolic-index fiber. The low-crosstalk butt-coupling also leads to small coupling losses. For example, for the fiber with a 270.0-µm² core, over the low-crosstalk coupling ranges shown in Fig. 4(b), the average coupling loss over the 6 spatial modes is below 0.27 dB and the mode-dependent loss is below 0.50 dB. Similar results are obtained for the other two fibers.

 

Fig. 4 Crosstalk characteristics for butt-coupling between a square-core waveguide and a parabolic-index circular-core fiber that has a core area of (a) 240.0 µm², (b) 270.0 µm², and (c) 300.0 µm², where the black arrows indicate the conditions for the best crosstalk performance and the red arrows indicate the equal-mode-volume conditions.

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Figure 5 shows the characteristics of butt-coupling a rectangular-core waveguide to a parabolic-index circular-core fiber, where the core areas of the waveguide and fiber are fixed at 125.0 µm² and 270.0 µm², respectively, to ensure equal mode volume. As in the case of a step-index fiber, the crosstalk performance is limited by the LP_21a(w)-to-LP₀₂(f) and the LP₀₂(w)-to-LP_21a(f) crosstalk. For these crosstalks to be smaller than –20 dB, the aspect ratio of the waveguide core must be larger than h/w = 0.993 and smaller than 1.007. The corresponding tolerance on the aspect ratio of the waveguide core is about ± 0.7%. As the aspect ratio changes from h/w = 1.0 (square) to 0.993, the coupling losses for the LP₀₁, LP_11a, LP_11b, LP_21a, LP_21b, and LP₀₂ modes increase from 0.03, 0.16, 0.16, 0.45, 0.39, and 0.40 dB, to 0.03, 0.20, 0.19, 0.50, 0.39, and 0.45 dB, respectively, i.e., the mode-dependent loss increases slightly from 0.42 to 0.47 dB. If the crosstalk performance is relaxed to –10 dB, the aspect ratio of the waveguide core must be within the range from h/w = 0.979 to 1.021, which corresponds to a ± 2.1% tolerance on the aspect ratio of waveguide core. For the aspect ratio h/w = 0.979, the coupling losses for the LP₀₁, LP_11a, LP_11b, LP_21a, LP_21b, and LP₀₂ modes are 0.03, 0.16, 0.18, 0.95, 0.39, and 0.90 dB, respectively. The mode-dependent loss is 0.92 dB. The results are similar to those obtained for step-index fibers. Although the use of a parabolic-index profile in a circular-core fiber cannot relax the tolerances on the aspect ratio of the square core required for low-crosstalk butt-coupling, it can achieve a minimum crosstalk a few decibels smaller than can be achieved with a step-index profile.

 

Fig. 5 Coupling characteristics for a rectangular-core waveguide butt-coupled to a parabolic-index circular-core fiber, where the core areas of the waveguide and the fiber are 125.0 µm² and 270.0 µm², respectively.

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3. Coupling between an elliptical-core fiber and a rectangular-core waveguide

In this section, we analyze butt-coupling between an elliptical-core fiber and a rectangular-core waveguide, both of which support only 6 spatial modes. The elliptical core can have a step-index profile or a parabolic-index profile. The ranges of the core aspect ratio and area for supporting only 6 spatial modes are shown in Fig. 6(a) for a step-index elliptical-core fiber and a rectangular-core waveguide, and in Fig. 6(b) for a parabolic-index elliptical-core fiber, respectively, where the curves show the boundary of the allowed ranges of the aspect ratio and area of the core. When the aspect ratios of the cores are not too close to 1, the spatial modes of the fiber and the waveguide can be designated as the E₁₁, E₂₁, E₁₂, E₂₂, E₃₁, and E₁₃ modes. The transition of the rectangular-core E_mn modes into the corresponding LP modes with the aspect ratio of the core approaching unity has been shown in Fig. 1. Each spatial mode corresponds to two almost degenerate orthogonally polarized modes with major electric-field components along the horizontal and vertical symmetry axes of the core, respectively. The coupling behaviors of the two polarized modes are practically the same, so there is no need to differentiate them in the calculation of the crosstalk characteristics.

 

Fig. 6 Ranges of the aspect ratio h/w and the area of the core allowed to support only 6 spatial modes for (a) a step-index elliptical-core fiber and a rectangular-core waveguide and (b) a parabolic-index elliptical-core fiber, where the points A, B, C, D, and E in (a) and A', B', C', D', and E' in (b) mark the specific elliptical-core fibers analyzed in our study.

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3.1 Butt-coupling between a step-index elliptical-core fiber and a rectangular-core waveguide

We analyze the butt-coupling characteristics of five step-index elliptical-core fibers: Fiber A, B, C, D, and E, which cover a good range of core parameters and are marked in Fig. 6(a) as Point A, B, C, D, and E, respectively. Fiber A, B, and C have the same core aspect ratio (0.85) and different core areas (120, 135, and 160 µm²), while Fiber B, D, and E have the same core area (135 µm²) and different core aspect ratios (0.85, 0.775, and 0.70). As the results for these fibers are qualitatively similar, we only show the detailed results for Fiber B, and then summarize the important findings from the study of the five fibers. We should note that the results for the fibers with h/w < 1 can be applied to fibers with h/w > 1 by simply rotating the fiber and the waveguide of concern by 90° about their centers.

The core of Fiber B has an aspect ratio of 0.85 and an area of 135.0 µm². The coupling and crosstalk characteristics for the entire ranges of the aspect ratio and the area of the core of the 6-mode rectangular-core waveguide butted coupled to this fiber are shown in Fig. 7 for the rectangular-core h/w ≤ 1 and in Fig. 8 for h/w ≥ 1, where the colors indicate the coupling efficiencies and “E_mn(w)-to-E_pq(f)” denotes the coupling from the waveguide E_mn mode to the fiber E_pq mode. The colored areas in these figures help us visualize the ranges of the parameters of the rectangular core required for achieving high-efficiency and low-crosstalk butt-coupling.

 

Fig. 7 Coupling efficiencies for butt-coupling between a step-index elliptical-core fiber and a rectangular-core waveguide that has h/w ≤ 1: (a) E₁₁(w)-to-E₁₁(f), (b) E₁₁(w)-to-E₃₁(f), (c) E₁₁(w)-to-E₁₃(f), (d) E₃₁(w)-to-E₁₁(f), (e) E₃₁(w)-to-E₃₁(f), (f) E₃₁(w)-to-E₁₃(f), (g) E₁₃(w)-to-E₁₁(f), (h) E₁₃(w)-to-E₃₁(f) and (i) E₁₃(w)-to-E₁₃(f), where the aspect ratio and the area of the elliptical core are fixed at 0.85 and 135.0 µm² (Fiber B), respectively.

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Fig. 8 Coupling efficiencies for butt-coupling between a step-index elliptical-core fiber and a rectangular-core waveguide that has h/w ≥ 1: (a) E₁₁(w)-to-E₁₁(f), (b) E₁₁(w)-to-E₃₁(f), (c) E₁₁(w)-to-E₁₃(f), (d) E₃₁(w)-to-E₁₁(f), (e) E₃₁(w)-to-E₃₁(f), (f) E₃₁(w)-to-E₁₃(f), (g) E₁₃(w)-to-E₁₁(f), (h) E₁₃(w)-to-E₃₁(f) and (i) E₁₃(w)-to-E₁₃(f), where the aspect ratio and the area of the elliptical core are fixed at 0.85 and 135.0 µm² (Fiber B), respectively.

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We first discuss the case of using a rectangular core with h/w ≤ 1. The waveguide E₁₁, E₃₁ and E₁₃ modes always couple to the corresponding fiber modes with low loss, as shown in Figs. 8(a), 8(e), and 8(i), respectively (mainly red color). The waveguide E₁₁ mode causes small crosstalk to the fiber E₃₁ and E₁₃ modes, as shown in Figs. 7(b) and 7(c), respectively (mainly deep-blue color). The waveguide E₃₁ mode also causes small crosstalk to the fiber E₁₁ mode, as shown in Fig. 7(d) (mainly deep-blue color), but its crosstalk to the fiber E₁₃ mode is always larger than about –10 dB, as shown in Fig. 7(f) (yellow to red color). Similarly, the waveguide E₁₃ mode cause small crosstalk to the fiber E₁₁ mode, as shown in Fig. 7(g) (mainly deep-blue color), but its crosstalk to the fiber E₃₁ mode is always larger than –16 dB, as shown in Fig. 7(h) (mainly green to red color). The results are insensitive to the aspect ratio and the area of the waveguide core. As shown in Fig. 7, it is impossible to achieve a modal crosstalk smaller than about –10 dB by butt-coupling a 6-mode rectangular-core waveguide with h/w ≤ 1 to Fiber B.

We then discuss the case of using a rectangular core with h/w ≥ 1. The waveguide E₁₁, E₃₁ and E₁₃ modes always couple to the corresponding fiber modes with low loss, as shown in Figs. 8(a), 8(e), and 8(i), respectively. The largest crosstalk is determined by the coupling from the waveguide E₃₁ mode to the fiber E₁₃ mode, as shown in Fig. 8(f), or from the waveguide E₁₃ mode to the fiber E₃₁ mode, as shown in Fig. 8(h), which is smaller than about –10 dB for practically any combination of the aspect ratio (not too close to 1) and the area of the rectangular core. In fact, the largest crosstalk can be kept smaller than –20 dB, when the aspect ratio of the waveguide core is around 1.05, which is insensitive to the core area, as shown by the narrow deep-blue bands shown in Fig. 8(f) and 8(h).

According to the crosstalk characteristics shown in Fig. 7 and Fig. 8, it appears that a rectangular core with h/w > 1 in general matches the step-index elliptical core better than a rectangular core with h/w < 1. This finding is counter-intuitive. As the aspect ratio of the elliptical core is 0.85, one would expect that it should be better matched by a rectangular core with h/w < 1 instead of one with h/w > 1. In reality, the opposite is true. Figure 9 compares the field patterns of the 6 spatial modes of a step-index elliptical core with h/w = 0.85, a rectangular core with h/w = 0.85, and a rectangular core with h/w = 1.05 (the optimal aspect ratio). The elliptical core has an area of 135.0 µm² and the two rectangular cores have the same area of 125.0 µm². As shown in Fig. 9, the E₃₁ (E₁₃) mode of the rectangular core with h/w = 1.05 matches the E₃₁ (E₁₃) mode of the elliptical core better than the corresponding mode of the rectangular core with h/w = 0.85. The concept of equal mode volume is not useful in the optimization of mode-selective butt-coupling to an elliptical-core fiber, as the crosstalk performance depends much more strongly on the aspect ratio of the rectangular core.

 

Fig. 9 Spatial modes of (a) a step-index elliptical core with h/w = 0.85, (b) a rectangular core with h/w = 0.85, and (c) a rectangular core with h/w = 1.05, where the elliptical core has an area of 135.0 µm² and the two rectangular cores have the same area of 125.0 µm².

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Figure 10(a) shows the approximate ranges of the aspect ratio and the area of the rectangular core required for achieving a modal crosstalk smaller than –20 dB for butt-coupling to Fiber A, B, C, D, and E, respectively. For all these fibers, the aspect ratio of the rectangular core required should be around 1.05. The results for Fiber A, B, and C show that the allowed range of the area of the rectangular core decreases with an increase in the area of the elliptical core. On the other hand, the results for Fiber B, D, and E show that the allowed range of the parameters for the rectangular core decreases with the aspect ratio of the elliptical core. The allowed range for Fiber E is exceedingly small, which is located around the aspect ratio 1.025 and the area 148.8 µm² of the rectangular core. If we relax the crosstalk requirement to smaller than –10 dB, the allowed ranges of the aspect ratio and the area of the rectangular core become much wider, as shown in Fig. 10(b). In this case, the aspect ratio of the rectangular core required can be smaller or larger than 1 over almost the full range of the core area allowed for supporting the 6 spatial modes. Fiber C, which has the largest core area of all, is the exception, for which the crosstalk cannot be smaller than –10 dB by using a rectangular core with an aspect ratio smaller than 1, so no result for this fiber is shown in Fig. 10(b). Our results are useful not only for the choice of the parameters of the rectangular core to match a given elliptical-core fiber, but also for the choice of an elliptical-core fiber to relax the tolerances on the parameters of the rectangular core for butt-coupling. According to our results, step-index elliptical-core fibers with a moderate ellipticity and a core area not close to the upper limit allowed for supporting 6 spatial modes, such as Fiber A and B, are preferred.

 

Fig. 10 Approximate ranges of the aspect ratio and the area of the rectangular core required for achieving a modal crosstalk smaller than (a) –20 dB and (b) −10 dB for butt-coupling to the step-index elliptical-core fibers A, B, C, D, and E.

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As in the case of coupling between a circular-core fiber and a square-core waveguide, low-crosstalk butt-coupling between an elliptical-core fiber and a rectangular-core waveguide also leads to small coupling losses. For Fiber A in the low-crosstalk (–20 dB) range shown in Fig. 10(a), the average coupling loss varies between 0.18 and 0.44 dB and the mode-dependent loss varies between 0.40 and 1.8 dB. For Fiber B in the low-crosstalk range, the average coupling loss varies between 0.11 and 0.30 dB and the mode-dependent loss varies between 0.16 and 0.76 dB. Similar results are obtained for the other fibers. The average coupling loss is always below 1 dB in the low-crosstalk range. Except near the boundary of the low-crosstalk range, the mode-dependent loss is also below 1 dB.

3.2 Butt-coupling between a parabolic-index elliptical-core fiber and a rectangular-core waveguide

We analyze the butt-coupling characteristics of five parabolic-index elliptical-core fibers: Fiber A', B', C', D', and E', which are marked in Fig. 6(b) as Point A', B', C', D', and E', respectively. Fiber A', B', and C' have the same core aspect ratio (0.85) and different core areas (240, 270, and 320 µm²), while Fiber B', D', and E' have the same core area (270 µm²) and different core aspect ratios (0.85, 0.775, and 0.70). We first show the detailed results for Fiber B', and then summarize the findings for the five fibers.

The coupling and crosstalk characteristics for the entire ranges of the aspect ratio and the area of the core of the 6-mode rectangular-core waveguide butt-coupled to Fiber B' are shown in Fig. 11 for the rectangular-core h/w ≤ 1 and in Fig. 12 for h/w ≥ 1. In both cases, the waveguide E₁₁, E₃₁, and E₁₃ modes couple to the corresponding fiber modes with low loss. In the case h/w ≤ 1, as shown in Fig. 11, the largest crosstalk due to the coupling from the waveguide E₃₁ (E₁₃) mode to the fiber E₁₃ (E₃₁) mode can be much smaller than –20 dB with an aspect ratio small than about 0.9 over a wide range of the core area. In the case h/w ≥ 1, as shown in Fig. 12, the largest crosstalk due to the coupling from the waveguide E₃₁ (E₁₃) mode to the fiber E₁₃ (E₃₁) mode can be smaller than −18 dB with an aspect ratio of 1.2 – 1.3 over a wide range of the core area. According to these results, there exist many possible combinations of core area and aspect ratio for the rectangular-core waveguide to achieve high-efficiency low-crosstalk butt-coupling to a parabolic-index fiber, which is a distinct advantage of using a parabolic-index profile.

 

Fig. 11 Coupling efficiencies for butt-coupling between a parabolic-index elliptical-core fiber and a rectangular-core waveguide that has h/w ≤ 1: (a) E₁₁(w)-to-E₁₁(f), (b) E₁₁(w)-to-E₃₁(f), (c) E₁₁(w)-to-E₁₃(f), (d) E₃₁(w)-to-E₁₁(f), (e) E₃₁(w)-to-E₃₁(f), (f) E₃₁(w)-to-E₁₃(f), (g) E₁₃(w)-to-E₁₁(f), (h) E₁₃(w)-to-E₃₁(f) and (i) E₁₃(w)-to-E₁₃(f), where the aspect ratio and the area of the elliptical core are fixed at 0.85 and 270.0 µm² (Fiber B'), respectively.

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Fig. 12 Coupling efficiencies for butt-coupling between a parabolic-index elliptical-core fiber and a rectangular-core waveguide that has h/w ≥ 1: (a) E₁₁(w)-to-E₁₁(f), (b) E₁₁(w)-to-E₃₁(f), (c) E₁₁(w)-to-E₁₃(f), (d) E₃₁(w)-to-E₁₁(f), (e) E₃₁(w)-to-E₃₁(f), (f) E₃₁(w)-to-E₁₃(f), (g) E₁₃(w)-to-E₁₁(f), (h) E₁₃(w)-to-E₃₁(f) and (i) E₁₃(w)-to-E₁₃(f), where the aspect ratio and the area of the elliptical core are fixed at 0.85 and 270.0 µm² (Fiber B'), respectively.

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Figure 13(a) shows the approximate ranges of the aspect ratio and the area of the rectangular core required for achieving a modal crosstalk smaller than –20 dB for butt-coupling to Fiber A', B', C', D', and E', respectively. All the five fibers match well with rectangular cores over a wide range of aspect ratios smaller than 1 and almost the entire range of allowable core areas. Fiber C', which has the largest core area, can even match well with a range of rectangular cores with aspect ratios larger than 1. Figure 13(b) shows the results for a modal crosstalk smaller than –10 dB. Almost all 6-mode rectangular-core waveguides, except for those with aspect ratios too close to unity (the square core), can couple well to the five fibers. A comparison of the results in Fig. 10 and Fig. 13 clearly shows that the use of a parabolic-index profile in an elliptical-core fiber can greatly relax the tolerances on the waveguide parameters for the achievement of low modal-crosstalk performance. For all the five fibers in the low-crosstalk (–20 dB) ranges, the average coupling losses are below 1 dB and, except near the boundaries of the low-crosstalk ranges, the mode-dependent losses are also below 1 dB. Again, low-crosstalk coupling leads to low-loss performance.

 

Fig. 13 Approximate ranges of the aspect ratio and the area of the rectangular core required for achieving a modal crosstalk smaller than (a) –20 dB and (b) −10 dB for butt-coupling to the parabolic-index elliptical-core fibers A', B', C', D', and E'.

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4. Conclusion

With extensive numerical examples, we have analyzed in detail butt-coupling between few-mode fibers and buried rectangular-core channel waveguides with an emphasis on the identification of the conditions for achieving low modal crosstalk performance for the first 6 spatial modes. For low-crosstalk (< –20 dB) coupling to a step-index or a parabolic-index circular-core fiber, the waveguide required should have a core with a shape sufficiently close to a square (within ± 0.7% variation in its aspect ratio) and the fiber and the waveguide should have close mode volumes. The best crosstalk performance of the parabolic-index fiber is better than that of the step-index fiber by a few decibels. The use of an elliptical-core fiber instead of a circular-core fiber can provide a better geometry match to a rectangular-core waveguide and hence lead to smaller crosstalk and more flexible waveguide designs. When a step-index elliptical-core fiber is used, the one with a moderate ellipticity and a core area not close to the upper limit allowed for supporting 6 spatial modes is preferred. A counter-intuitive finding is that a step-index elliptical core can be best matched by a rectangular core with a very different aspect ratio (e.g., an aspect ratio of 0.85 for the elliptical core versus an aspect ratio of 1.05 for the rectangular core). Another important finding is that the use of a parabolic-index elliptical-core fiber can greatly relax the tolerances in the design of rectangular-core waveguides to achieve low-crosstalk (< –20 dB) performance. In general, when the fiber and the waveguide core are matched for low-crosstalk performance, the butt-coupling losses for all the 6 spatial modes as well as the mode-dependent loss are small (typically well below 1 dB). Our results are useful for the design of mode-selective waveguide devices, such as mode (de)multiplexers, mode converters, and mode switches, for fiber-based MDM applications.

In the present study, we assume perfect alignment between the fiber and the waveguide core to obtain benchmark results. Any lateral or angular misalignment can destroy the symmetry of the butt-coupling geometry and may introduce significant crosstalks among the modes that belong to different symmetry groups. Longitudinal misalignment does not change the symmetry of the butt-coupling geometry and should be much less detrimental to the coupling performance. Quantifying the effects of misalignment on the fiber-waveguide coupling performance for MDM applications is an important subject for future investigation.