1. Introduction

Mode-division multiplexing (MDM) systems using few-mode fibers (FMFs) have been regarded as a sustainable long-term solution to increase the optical network capacity in a cost-effective/energy-efficient manner [1–3]. For these systems, although the use of high-order modes of FMFs effectively enables high-bandwidth data transmission, various multiple-input-multiple-output (MIMO) digital signal processing (DSP) equalization approaches are generally needed to mitigate the crosstalk induced by modal coupling [4]. From the perspectives of both hardware cost and power consumption, the reduction in the complexity of MIMO DSP has been and always will be an essential requirement [5,6]. For existing MDM systems over conventional step-/graded-index FMFs, the required complexity of MIMO DSP increases rapidly with the number of propagation modes (information channels) [1,7]. However, when mode coupling is sufficiently minimized through specific FMF design strategies, the use of MIMO DSP can be largely simplified [1,8,9] or even avoided [10–15] over a certain transmission distance, thereby further improving the viability and applicability of MDM systems.

Recently, radially single-mode and azimuthally multimode (RSMAM) ring-core fibers (RCFs) have been theoretically [16–19] and experimentally [10,14,20–25] demonstrated to possess great potential for realizing linearly polarized (LP) MDM systems that feature a low-complexity MIMO DSP and an efficient use of weakly coupled propagating modes. This can mainly be attributed to the following four advantages of RSMAM RCFs. Firstly, in a RSMAM RCF, the difference in propagation constant between adjacent azimuthal modes significantly increases with azimuthal number, which in theory results in relatively weak (strong) modal coupling between high (low) order azimuthal modes [19,23]. The MIMO DSP used to mitigate modal coupling may be avoided for the signals carried on the high-order azimuthal modes experiencing weak modal coupling [10,14,26]. The DSP complexity can therefore be reduced by only using MIMO signal processing to recover signals carried on the lower-order azimuthal modes experiencing strong modal coupling [16,27,28], and/or the degenerated modes within a high-order modal group [21,24]. Secondly, for the MDM transmission over a RSMAM RCF, the application of a silicon photonic integration circuit (with a circular grating coupler) for modal conversion can effectively avoid the employment of bulky free-space modal conversion based on phase masks [18,19,29]. Thirdly, the RCF amplifier is, in theory, capable of providing approximately identical gain for all the propagating signal modes, due to the fact that similar overlap factors can be achieved between the erbium doped core and all the signal spatial modes [19,23,30,31]. Fourthly, for a MDM system, large mode conversion between cylindrically symmetric LP_0n (n ≥ 1) modes at discontinuous points can be completely avoided through employing a RSMAM RCF [17], because such modes in conventional step-/graded-index FMFs have a field distribution featuring a peak in the center of the core.

Currently, driven by a higher spatial information density, a smaller physical footprint of the FMF, and an efficient pumping for amplification (strong modal overlap) in MDM systems, the question of scalability (the largest possible number of usable propagating LP modes) for FMFs has been raised [1,32,33]. In this respect, RSMAM RCFs supporting 2-LP-mode [20], 3-LP-mode [13,17], 4-LP-mode [10,14,22–24], 5-LP-mode [34], 6-LP-mode [27], and 7-LP-mode [16,18,21] for MDM applications have been reported. However, in the design trade-offs that are at stake [1,35] when optimizing varieties of FMFs that support a large number of LP modes, one needs to comprehensively consider various modally dependent physical quantities (e.g., effective index difference (Δn_eff), effective area (A_eff), differential mode group delay (DMGD), macro-/micro-bending loss, and etc.) [32,33,36]. Specifically speaking, 1) although typical RSMAM RCFs have the advantage of having a relatively much larger A_eff (about 300 μm² [17,37]) for the LP₀₁ mode (compared with the conventional step-index single-mode fiber and graded-index multimode/few-mode fibers), the normalized spatial density (NSD, defined as a measure of space efficiency [33]) has not been considered in the design process, and 125 μm glass diameter FMFs with oversized (≥ 160 μm²) A_eff are known to be highly sensitive to micro-bending [38,39]; 2) even though the RSMAM RCFs with a small ring radius and a large ring thickness are usually necessary for a high average Δn_eff to produce weak modal coupling [18], the trade-off between Δn_eff and A_eff at a fixed normalized frequency V has not been fully discussed; 3) for the 4-LP-mode RSMAM RCF proposed in [22] and [23], although the large Δn_eff between the guiding ring-core modes and the cladding modes can enable low macro-bending loss sensitivity, the increased relative refractive index contrast Δ (> 1%) between the core and the cladding is undesirable from the perspectives of both Rayleigh scattering loss and optical fiber manufacturing [17], and the LP modal robustness (defined by macro-bending losses of LP modes < 10 dB/turn at a bend radius R of 10 mm [1,33]) in RSMAM RCFs supporting higher numbers of LP modes still needs to be further analyzed.

In this paper, we detail the design trade-offs and the optimization procedures of RSMAM RCFs, based on the design criteria of weakly coupled FMFs given in [1,35,38] and rigorous numerical simulations. One aim is to explain the impact of the increase of the number of weakly coupled LP modes supported by RSMAM RCFs, and further point out the key limiting factors of this increase. On the basis of such an explanation, as well as a more comprehensive consideration of various modally resolved physical quantities, another aim is to improve the weakly coupled LP modal propagation characteristics of the RSMAM RCFs proposed in [17] and [23].

2. Theoretical approach

Figure 1 displays the refractive index profiles (n(r)) of step-/graded-index RCFs with a ring thickness d and a ring radius r_a. n₀ represents the peak refractive index in the ring-core. The refractive index for the graded-index RCFs can be given as

 

Fig. 1 Refractive index profiles of step-index (α = ∞) and graded-index (α = 8, 4, & 2) RCFs.

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In this paper, bent RSMAM RCFs (R = ∞ for the case of the straight RSMSM RCFs) simulation was performed using the full-vectorial finite element method, together with the conformal mapping [33]. During this process, the circularly bent RSMAM RCF is represented by an equivalent, straight fiber, with a modified refractive index distribution, n_eq (x, y)

Based on the equivalent, straight fiber represented by Eq. (2), we can determine the LP modes of RSMAM RCF structures and simulate their propagating characteristics, under a variety of R and n(r). In such finite element method-based simulations, the computational window of 130 μm × 130 μm is set within the transverse (x-y) plane. The outer boundary region is realized by assuming a perfectly matched layer, with a width of 10 μm and a refractive index (n₁) matched to the cladding region, for implementing losses into the numerical calculations. The performance of perfectly matched layer is upgraded through establishing a scattering boundary condition on the very edge of the simulation domain for suppressing spurious reflections. By doing so, the LP modal power losses, 2α (in dB/m), can be calculated from the imaginary part of the effective mode indices, Im(n_eff),

In this paper, we optimize/design the propagation properties of high-order azimuthal modes in RSMAM RCFs, based on the weakly coupled design criteria given in [1], [35], and [38] (i.e., the minimum Δn_eff (Min|Δn_eff|) between any two non-degenerate LP modes > 1.0 × 10⁻³ to limit inter-modal crosstalk, LP modal macro-bending losses < 10 dB/turn at R = 10 mm to ensure good LP modal robustness, the minimum LP modal A_eff (Min|A_eff|) > 100 μm² to limit intra-modal nonlinearity, and the maximum LP modal A_eff (Max|A_eff|) ≤ 160 μm² to avoid micro-bending loss issues for a glass diameter D = 125 μm). It should be noted that, we have not considered the design criterion of differential modal attenuation, because this limitation can be overcome by using power splitters with different splitting ratios [40] and few-mode RCF amplifiers [30,31] in real-life weakly coupled MDM systems. Moreover, as an un-cabled coated fiber diameter of about 250 μm has been specified in industry standards, LP modes with A_eff > 160 μm² might be sensitive to micro-bending [38,39]. However, current solutions for reducing this sensitivity (e.g., the optimization of glass and dual-coated diameters, the use of a softer inner-primary coating) can be readily applied for RCFs.

We will also give a glimpse on the NSD of LP modes in RSMAM RCFs that having weak inter-modal coupling. Under the premise of D = 125 μm, the generalized form of NSD for weakly coupled modes depends on the modal multiplicity factor of the weakly coupled LP modes (including the twofold spatial degeneracy), T, and the A_eff of the t^th weakly coupled spatial mode, A_{eff t} [33],

3. Design trade-offs of RSMAM RCFs

In theory, the maximum value of m increases with increasing r_a and d, while the maximum value of n only increases with d [16]. On one hand, the core of a RCF can be considered as a slab waveguide wrapped around a line, and the condition to be radially single-mode is V ≤ π (for a graded-index RCF, V = $kd\sqrt{\left(n_0^2-n_1^2\right)\left[\alpha /\left(\alpha \text{+}2\right)\right]}$; for a step-index RCF (α = ∞), V = $kd\sqrt{\left(n_0^2-n_1^2\right)}$. k = 2π/λ is the wave number, and λ is the wavelength) [18,29]. On the other hand, a larger value of V or d is generally needed for producing a larger Δn_eff between guiding ring-core modes and radiation modes within the cladding layer [18]. Accordingly, in this paper, we have chosen V = π to guarantee robustness and good separation (i.e., high, different values of normalized propagation constant b = $\left[{\left(\beta /k\right)}^2-n_1^2\right]/\left(n_0^2-n_1^2\right)$) for the guiding LP_m1 modes while cutting off the LP_mn modes with n ≥ 2, thereby obtaining a RSMAM RCF. By doing this, Fig. 2 displays the dependence of the calculated maximum values of d (d_max) on Δ (note that similar to [17], Δ more than 1% is undesirable in this paper from the viewpoints of optical fiber manufacturing) for different values of α. At a constant Δ, the d_max of a graded-index RSMAM RCF, which is higher than that of a step-index RSMAM RCF, increases with decreasing α. It should be noted that, when Δ and the number of LP modes are both fixed, the d_max corresponding to a larger α is preferred to produce a higher effective index n_eff of all guiding LP_m1 modes [18], and thereby to improve the macro-bending losses of these modes.

 

Fig. 2 For RSMAM RCFs, dependence of d_max on Δ for α = ∞, 8, 4, and 2 (n₁ = 1.444). Note that because the values of d_max are obtained based on the radially single-mode condition, i.e., V ≤ π, the calculated d_max is independent of r_a. That is to say, the results in Fig. 2 is valid for arbitrary values of r_a.

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3.1. Influences of α and r_a in RSMAM RCFs with a fixed number of LP modes

For RSMAM RCFs with a fixed number of LP modes, given that 1) the value of α does not significantly impact the LP modal n_eff distribution and DMGD [18], 2) the determination of d_max depends entirely on the values of Δ and α (as shown in Fig. 2), in this section we focus on the impacts of α and r_a on LP modal A_eff property. In order to directly give an extended investigation into the 7-LP-mode RSMAM RCFs described in [18], here we use the refractive index profiles of these fibers (r_a = 10 μm, n₀ = 1.46, n₁ = 1.4453, Δ = 1%, d_max = 3.8, 4.3, 4.8, and 5.6 μm for α values of ∞, 8, 4, and 2) and explore their feasibilities and limiting factors. The modal A_eff for all 13 spatial modes are shown in Fig. 3, and the spatial modes with the identical values of β and DMGD form a pair of degenerated modes. It can be seen that the variation of α does not significantly affect the modal A_eff. For higher-order LP modes with large Δn_eff and weak mode coupling, the curves of modal A_eff for the small-α-value RSMAM RCF cases shift up slightly compared to that for the large-α-value RSMAM RCF and the step-index RSMAM RCF cases. In addition, it is worth noting that, the values of modal A_eff for all 13 spatial modes are obviously larger than 160 μm², implying micro-bending loss issues of the above-mentioned 7-LP-mode RSMAM RCFs under the premise of D = 125 μm [38].

 

Fig. 3 Modal A_eff of all 13 spatial modes in the step-/graded-index 7-LP-mode RSMAM RCFs (r_a = 10 μm, n₀ = 1.46, n₁ = 1.4453, Δ = 1%, d_max = 3.8, 4.3, 4.8, and 5.6 μm for α values of ∞, 8, 4, and 2, respectively).

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By noting that the increase of r_a can lead to 1) a decreased average Δn_eff for a fixed number of LP modes, or 2) an increased number of guiding LP modes in RSMAM RCFs [18], in the following we investigate the impacts of r_a on LP modal n_eff, Δn_eff between two neighboring LP modes, macro-bending sensitivity, and A_eff in RSMAM RCFs. Through using the refractive index profiles of the 7-LP-mode RSMAM RCFs given in [18], the dependences of n_eff for LP_m1 modes (m = 0 ~6) on r_a are shown in Figs. 4(a)–4(c). In the white region located on the left (right) of the colored curves, the RSMAM RCFs possess less (more) than 13 spatial modes. As shown, for fixed values of the number of spatial modes (13) and α, the n_eff of higher-order LP_m1 modes with m ≥ 3 increases when r_a increases, and the variation of r_a has a limited impact on the n_eff of lower-order LP_m1 modes with m ≤ 2. Figures 4(a)–4(c) indicate that, a larger-r_a-value and a larger-α-value can produce a higher n_eff for all 7 guiding LP_m1 modes, benefiting for the improvement of macro-bending loss tolerance. However, as indicated by [18], this will be compromised by the decreasing average Δn_eff.

 

Fig. 4 For the step-/graded-index 7-LP-mode RSMAM RCFs (n₀ = 1.46, n₁ = 1.4453, Δ = 1%, d_max = 3.8, 4.8, and 5.6 μm for α values of (a) ∞, (b) 4, and (c) 2, respectively), dependence of LP modal n_eff on r_a.

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We mainly focus on the macro-bending loss of the highest-order propagating LP modes in a RSMAM RCF, because these modes have the lowest n_eff/n₁ ratio and thereby have the highest macro-bending sensitivity [33]. In our simulations for the 7-LP-mode RSMAM RCFs described in [18], although we have adopted the largest possible values of r_a (10.3 μm) and α (∞) (in order to achieve the highest possible n_eff of all modes under the premise of Δ = 1%), the modal fields of the twofold degenerate LP₆₁ modes become evanescent when R = 10 mm, indicating an unacceptable modal robustness of the LP₆₁ modes. Figures 3 and 4 reveal that, the feasibility of the 7-LP-mode RSMAM RCFs described in [18] is still limited by 1) the oversized values of LP modal A_eff (> 160 μm²) and 2) an undesired robustness of the highest-order LP₆₁ modes (macro-bending loss >> 10 dB/turn at R = 10 mm). One possible solution to these limitations is to decrease r_a, thereby improving both the n_eff/n₁ ratio of the highest-order guiding LP modes and the Min|A_eff| characteristic. However, this will lead to a significantly reduced number of guiding LP modes (from 7 to 3 or 4), as explained in the following two subsections.

3.2. Influences of α and r_a in RSMAM RCFs

Under a given value of Δ = 1%, Figs. 5(a)–5(c) show the effects of r_a (> d_max/2) on the Δn_eff between any two neighboring LP modes (i.e., LP_m1 and LP_(m+1)1 modes, m ≥ 0), in RSMAM RCFs with α values of ∞, 4, and 2, respectively. In these subfigures, at a fixed value of r_a, the highest existing ordinate value represents the Δn_eff between the highest-order guiding LP mode and the second highest-order guiding LP mode. Hence, the number of guiding LP modes versus r_a can also be reflected in Figs. 5(a)–5(c). As shown, the variation of r_a has a significant impact on both the number of guiding LP modes and Δn_eff. To be specific, the Δn_eff between any two neighboring LP modes decreases with r_a, and the number of guiding LP modes increases with r_a. This implies that, when r_a constantly increases, an increasing extent of mode coupling between the guiding LP modes with different orders can be obtained. In this process, increasing numbers of lower-order LP modes (e.g., the LP₀₁, LP₁₁, LP₂₁, and LP₃₁ modes for r_a ≥ 10.5 μm) have become unsuitable for weakly coupled MDM systems, because for these LP modes, the Δn_eff between two neighboring LP modes becomes smaller than 1.0 × 10⁻³. It can also be seen from Figs. 5(a)–5(c) that, the variation of α has a limited impact on the Δn_eff between any two neighboring LP modes, though when r_a constantly increases, the number of guiding LP modes for α = ∞ has grown slightly faster than that for both α = 4 and α = 2.

 

Fig. 5 For the step-/graded-index RSMAM RCFs (n₀ = 1.46, n₁ = 1.4453, Δ = 1%, d_max = 3.8, 4.8, and 5.6 μm for α values of (a) ∞, (b) 4, and (c) 2, respectively), dependences of the Δn_eff between two neighboring LP modes (LP_m1 and LP_(m+1)1 modes, m ≥ 0) and the number of guiding LP modes on r_a (> d_max/2).

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Under the assumption of Δ = 1%, Figs. 6(a)–6(c) display the dependence of LP modal A_eff on r_a, for α = ∞, 4, and 2, respectively. As shown, the variation of α has a very limited impact on LP modal A_eff. When r_a ≥ 7.5 μm, the use of LP₅₁, LP₆₁ and LP₇₁ modes in weakly coupled MDM systems will be limited by their oversized A_eff (> 160 μm²). It is interesting to see that, when r_a constantly increases (≥ 5.0 μm), all the guiding LP_m1 modes with m ≥ 1 (except for the LP₀₁ mode) gradually exhibit a similar value of A_eff. At relatively larger values of r_a, the LP₀₁ mode has the largest value of A_eff. Figures 6(a)–6(c) also note that at a fixed value of r_a, the LP modal A_eff with a larger-α-value is slightly smaller than that with a smaller-α-value.

 

Fig. 6 For the step-/graded-index RSMAM RCFs (n₀ = 1.46, n₁ = 1.4453, Δ = 1%, d_max = 3.8, 4.8, and 5.6 μm for α values of (a) ∞, (b) 4, and (c) 2, respectively), dependences of LP modal A_eff on r_a.

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Under the assumption of Δ = 1%, in Figs. 7(a)–7(c), we further examine the impacts of r_a and α on the macro-bending losses of the highest-order guiding LP modes at R = 10 mm. Under a fixed value of α, these subfigures also show the number of guiding LP modes for the straight RSMAM RCF, plotted as a function of r_a. Note that when R = 10 mm, the macro-bending loss of the highest-order LP₂₁ (resp. LP₃₁) modes in 3-LP-mode (resp. step-index 4-LP-mode) RSMAM RCF is displayed only over a small range of r_a. This is because for other values of r_a, the corresponding highest-order LP_m1 modes (m = 2, 3, 4, and 5) have become evanescent when R = 10 mm and therefore disappeared from the simulation window. In addition, as shown in Figs. 7(a)–7(c), the LP_21e and LP_21o (resp. LP_31e and LP_31o) modes, which are spatially degenerate in an ideal straight 3-LP-mode (resp. step-index 4-LP-mode) RSMAM RCF, behave differently in terms of macro-bending loss at R = 10 mm. Such a difference is due to the fact that bend tends to break the rotational symmetry of the even and odd modes. That is, two of the intensity lobes of the LP_21e (LP_31e) mode are aligned in the direction of the bend (i.e., x-direction), causing this mode to experience higher macro-bending losses compared to the LP_21o (LP_31o) mode, whose intensity lobes are not aligned in x-direction. One can also see from Figs. 7(a)–7(c) that, as R = 10 mm, the increase of α can bring about a larger macro-bending loss difference between the LP_21e and LP_21o mode. This is because when increasing α, the LP_21e mode exhibits more power at the side opposing the bend center, yielding a larger degree of the bend-induced asymmetry between the LP_21e and LP_21o modes.

 

Fig. 7 For the step-/graded-index RSMAM RCFs (n₀ = 1.46, n₁ = 1.4453, Δ = 1%, d_max = 3.8, 4.8, and 5.6 μm for α values of (a) ∞, (b) 4, and (c) 2, respectively), dependence of the macro-bending loss of the highest-order propagating LP modes (at R = 10 mm) on r_a. Under a fixed value of α, the influence of changing r_a on the number of guiding LP modes for the straight RSMAM RCF is also shown.

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Figures 7(a)–7(c) suggest that, 1) for the 3-LP-mode (resp. step-index 4-LP-mode) RSMAM RCF, the increase of r_a is benefit for decreasing the macro-bending losses of the highest-order LP₂₁ (resp. LP₃₁) modes; 2) for the 5-LP-mode (resp. 6-LP-mode) RSMAM RCF, the highest-order LP₄₁ (resp. LP₅₁) modes will become evanescent as R = 10 mm, regardless of the value of r_a. The design criterion of macro-bending losses for the highest-order guiding LP modes < 10 dB/turn at R = 10 mm can only be satisfied within limited value ranges of r_a. Specifically speaking, 1) the macro-bending losses of the highest-order guiding LP₂₁ modes in 3-LP-mode RSMAM RCFs with α = ∞, 4, and 2, are smaller than 10 dB/turn when 3.85 μm ≤ r_a ≤ 4.25 μm, 4.00 μm ≤ r_a ≤ 4.30 μm, and 4.30 μm ≤ r_a ≤ 4.50 μm, respectively; 2) the macro-bending losses of the highest-order guiding LP₃₁ modes in the step-index (i.e., α = ∞) 4-LP-mode RSMAM RCF, are smaller than 10 dB/turn when 5.72 μm ≤ r_a ≤ 5.80 μm. For the 3-LP-mode RSMAM RCFs in Figs. 7(a)–7(c), the desired value range of r_a in which the macro-bending losses of the highest-order guiding LP₂₁ modes (R = 10 mm) < 10 dB/turn, increases with α. From this perspective, a 3-LP-mode RSMAM RCF with a step-index profile (i.e., α = ∞) is preferred, for obtaining the highest macro-bending tolerance to a small deviation in the value of r_a (as described in subsection 4. 1).

3.3. Limiting factors for increasing the number of LP modes in RSMAM RCFs

Ideally speaking, a high number of guiding LP modes with weak inter-modal coupling is desirable for high capacity/spectral efficiency MDM systems with low-complexity MIMO DSP [1,8,32]. Note that in this paper we do not consider the step-index 2-LP-mode RSMAM RCF because it has a larger intensity overlap and a reduced Δn_eff between the LP₀₁ and LP₁₁ modes, in comparison with the 2-LP-mode step-index FMF [20]. In the following we intend to comprehensively consider the Δn_eff, A_eff, and macro-bending sensitivity characteristics of the guiding LP modes, which have already been shown in Figs. 5, 6, and 7. By satisfying the weakly coupled design criteria previously given in section 2, the largest possible number of usable propagating modes (4 spatial modes, which composed of twofold degenerate LP₂₁ modes and twofold degenerate LP₃₁ modes) in weakly coupled MDM systems can be obtained when 5.72 μm ≤ r_a ≤ 5.80 μm (under the case of step-index 4-LP-mode RSMAM RCF with Δ = 1%).

It can also be observed from Figs. 5, 6, and 7 that, under the premise of Δ = 1%, the use of all 5 spatial modes in 3-LP-mode (i.e., LP₀₁, LP₁₁, and LP₂₁ modes) RSMAM RCFs for weakly coupled MDM systems is mainly limited by the undersized A_eff (< 80 μm²) of both the LP₁₁ and LP₂₁ modes. In addition, the applicability of n-LP-mode (n ≥ 5) RSMAM RCF for weakly coupled MDM systems is mainly limited by the undesired macro-bending sensitivity and oversized A_eff (> 160 μm²) of the LP₄₁, LP₅₁, and LP₆₁ modes.

4. Improved designs of RSMAM RCFs

Recently, the authors of [17] and [23] have proposed 3-and 4-LP-mode RCF designs, respectively, which can enable weak mode coupling between LP modes with different orders. However, by using the weakly coupled design criteria depicted in section 2 and the finite element method-based numerical calculations, we have found that, 1) for the 3-LP-mode RCFs (supporting weakly coupled LP₀₁, LP₁₁, and LP₂₁ modes, T = 5) proposed in [17], the pertaining values of V still tend to be larger than π, and the modal fields of the LP₂₁ modes become evanescent when R = 10 mm; 2) for the 4-LP-mode RSMAM RCF (supporting higher-order weakly coupled LP₂₁ and LP₃₁ modes, T = 4) proposed in [23], the relevant Δ is still larger than 1%. Therefore, next we proceed to provide improved designs for the two types of RSMAM RCFs mentioned above. In the following design processes, the cladding layer is assumed to be silica (its refractive index n₁ = 1.444 at λ = 1550 nm), consisting with [17] and [23]. Note that we also take into account the influence of variation of Δ on LP mode propagation characteristics. It is necessary to reemphasize that, for RSMAM RCFs with a step-index profile (i.e., α = ∞), the value of d_max is entirely dependent on Δ (as reflected in Fig. 2).

4.1. Improved design for the 3-LP-mode RCFs proposed in [M. Kasahara, J. Lightw. Technol. 32, 1337 (2014)]

To investigate the impact of r_a and Δ on the number of guiding LP modes, colored regions of the required r_a and Δ for obtaining 3-LP-mode (5 spatial modes composed of LP₀₁, LP_11e, LP_11o, LP_21e, and LP_21o modes) step-index RSMAM RCFs are plotted in Fig. 8. The color scales in Figs. 8(a)–8(c) indicate Min|Δn_eff| (depends entirely on the Δn_eff between LP₀₁ and LP₁₁ modes), Min|A_eff|, and Max|A_eff|, respectively. In the white regions below (above) the colored regions, the step-index RSMAM RCFs possess less (more) than 5 spatial modes. In the colored region, 1) the value of Min|Δn_eff| remains larger than 1.0 × 10⁻³, indicating that the design of step-index weakly coupled 3-LP-mode RSMAM RCFs will not be limited by Min|Δn_eff|; 2) the value of Min|Δn_eff| decreases as r_a increases, and no significant influence of the Δ variation on Min|Δn_eff| was found. Based on the results shown in Figs. 8(a)–8(c), we then define an acceptable region in which the design requirements of Min|Δn_eff| > 1.0 × 10⁻³, Min|A_eff| > 100 μm², and Max|A_eff| ≤ 160 μm² can be simultaneously satisfied.

 

Fig. 8 Dependences of (a) Min|Δn_eff| (depends entirely on the Δn_eff between LP₀₁ and LP₁₁ modes), (b) Min|A_eff|, and (c) Max|A_eff| on the required Δ and r_a for obtaining 3-LP-mode (5 spatial modes composed of LP₀₁, LP_11e, LP_11o, LP_21e, and LP_21o modes) in step-index (α = ∞) RSMAM RCFs.

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The above-mentioned acceptable region is thrice plotted as the colored regions in Figs. 9(a)–9(c). As indicated in these three subfigures, we then examine the macro-bending sensitivity of the highest-order LP modes (i.e., LP_21e and LP_21o modes) and the NSD for 3-LP-mode RSMAM RCFs, within the acceptable region. It can be seen that, the macro-bending sensitivity will no longer be a limiting factor within the acceptable region, since the macro-bending losses of both LP_21e and LP_21o modes remain smaller than 10 dB/turn (R = 10 mm) within the colored region. Therefore, as shown in Fig. 9(c), we here choose the highest achievable NSD (8.05) within the acceptable region.

 

Fig. 9 Dependences of (a) macro-bending loss of LP_21e mode at R = 10 mm, (b) macro-bending loss of LP_21o mode at R = 10 mm, and (c) NSD on Δ and r_a for obtaining weakly coupled LP₀₁, LP₁₁, and LP₂₁ modes in step-index (α = ∞) 3-LP-mode RSMAM RCFs.

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The optimized index profile of step-index weakly coupled 3-LP-mode RSMAM RCF consists of r_a = 4.8 μm, Δ = 0.65%, d = 4.68 μm. The main LP modal propagation characteristics of this optimized fiber are given in Table 1. As shown, for the weakly coupled LP₀₁, LP₁₁, and LP₂₁ modes (T = 5), Min|Δn_eff| = 1.3 × 10⁻³, macro-bending losses of the highest-order LP_21e and LP_21o modes < 10 dB/turn when R = 10 mm, Min|A_eff| = 113 μm², Max|A_eff| = 160 μm², NSD = 8.05.

Table 1. Main LP Modal Propagation Characteristics of The Optimized Step-Index Weakly Coupled 3-LP-Mode RSMAM RCF (λ = 1550 nm)^a

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4.2. Improved design for the 4-LP-mode RSMAM RCF proposed in [Y. Jung, J. Lightw. Technol. 35, 1363 (2017)]

Similar to the analysis procedure depicted in Fig. 8, colored regions of the required r_a and Δ for obtaining 4-LP-mode (7 spatial modes composed of LP₀₁, LP_11e, LP_11o, LP_21e, LP_21o, LP_31e, and LP_31o modes) step-index RSMAM RCFs are plotted in Fig. 10. The color scales in Figs. 10(a)–10(c) indicate Min|Δn_eff| (depends entirely on the Δn_eff between LP₀₁ and LP₁₁ modes), Min|A_eff| (depends entirely on the A_eff of LP₁₁ modes), and Max|A_eff| (depends entirely on the A_eff of LP₀₁ mode), respectively. In the white regions below (above) the colored regions, the step-index RSMAM RCFs have less (more) than 7 spatial modes. Our simulation results suggest that, in the colored region of Fig. 10, as the value of Δn_eff between LP₀₁ and LP₁₁ modes > 1.0 × 10⁻³, the use of all 4-LP-modes as weakly coupled spatial information channels will be limited by the oversized A_eff of LP₀₁ mode (> 160 μm²) and/or the undersized A_eff of LP₁₁ modes (< 100 μm²). This phenomenon indicates that the lower-order LP₀₁ and LP₁₁ modes in step-index 4-LP-mode RSMAM RCF are unsuitable for weakly coupled MDM systems.

 

Fig. 10 Dependences of (a) Min|Δn_eff| (depends entirely on the Δn_eff between LP₀₁ and LP₁₁ modes), (b) Min|A_eff|, and (c) Max|A_eff| on the required Δ and r_a for obtaining 4-LP-mode (7 spatial modes composed of LP₀₁, LP_11e, LP_11o, LP_21e, LP_21o, LP_31e, and LP_31o modes) in step-index (α = ∞) RSMAM RCFs.

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Our simulation results also show that, in the colored region of Fig. 10, the value of Δn_eff between LP₁₁ and LP₂₁ modes is always larger than 1.3 × 10⁻³, regardless of the value of r_a or Δ. This implies that, from the view point of weakly coupled MDM systems, one only needs to consider the Min|A_eff| and Max|A_eff| characteristics of LP₂₁ and LP₃₁ modes (in our simulation, the A_eff of LP₂₁ modes is always smaller than that of LP₃₁ modes).

Colored region of the required r_a and Δ for obtaining weakly coupled LP₂₁ and LP₃₁ modes (T = 4) are thrice plotted in Figs. 11(a)–11(c). In these three subfigures, we comprehensively consider the macro-bending sensitivities and the NSD of the weakly coupled LP₂₁ and LP₃₁ modes. Under the premise of macro-bending losses of LP₃₁ modes < 10 dB/turn at R = 10 mm, we here choose the highest achievable NSD (7.55) within the colored region.

 

Fig. 11 Dependences of (a) macro-bending loss of LP_31e mode at R = 10 mm, (b) macro-bending loss of LP_31o mode at R = 10 mm, and (c) NSD on Δ and r_a for obtaining weakly coupled LP₂₁ and LP₃₁ modes in step-index (α = ∞) 4-LP-mode RSMAM RCFs.

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The optimized index profile of step-index 4-LP-mode RSMAM RCF consists of r_a = 6.6 μm, Δ = 0.775%, and d = 4.277 μm. The main LP modal propagation characteristics of this optimized fiber are given in Table 2. As shown, for the weakly coupled LP₂₁ and LP₃₁ modes (T = 4), Min|Δn_eff| = 2.4 × 10⁻³, macro-bending losses of the highest-order LP_31e and LP_31o modes < 10 dB/turn when R = 10 mm, Min|A_eff| = the A_eff of LP₂₁ modes = 147 μm², Max|A_eff| = the A_eff of LP₃₁ modes = 155 μm², NSD = 7.55. It is worth noting that, when R is fixed at 10 mm, the calculated macro-bending loss values of the two LP₃₁ modes for the optimized 4-LP-mode RSMAM RCF are slightly higher than the measured macro-bending loss value of the LP₃₁ mode for the 4-LP-mode RSMAM RCF proposed in [23]. This is mainly because the authors of [23] have set a relatively larger Δ (> 1%) to enable low macro-bending loss sensitivities.

Table 2. Main LP Modal Propagation Characteristics of the Optimized Step-Index 4-LP-Mode RSMAM RCF (λ = 1550 nm)^a

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5. Conclusion

Based on using finite element method simulations, we have investigated the design trade-offs of RSMAM RCFs through comprehensively considering their main LP modal propagation characteristics including Min|Δn_eff|, Min|A_eff|, Max|A_eff|, macro-bending sensitivity, and NSD. From the perspective of weakly coupled FMFs adapted to weakly coupled MDM systems [1,35], we have found that the applicability of n-LP-mode (n ≥ 5) RSMAM RCF for weakly coupled MDM systems is mainly limited by the undesired macro-bending sensitivity and oversized A_eff (> 160 μm²) of the LP₄₁, LP₅₁, and LP₆₁ modes. Such a claim is based on the following two important preconditions for RSMAM RCFs: 1) Δ ≤ 1% was set for facilitating the fiber manufacturing; 2) Max|A_eff| ≤ 160 μm² was set for avoiding micro-bending loss issues for a glass diameter D = 125 μm.

In addition, we have presented improved designs for the step-index 3- and 4-LP-mode RSMAM RCFs given in [17] and [23]. For the optimized step-index weakly coupled 3-LP-mode RSMAM RCF (T = 5), Min|Δn_eff| = 1.3 × 10⁻³, Min|A_eff| = 113 μm², Max|A_eff| = 160 μm², NSD = 8.05. For the optimized step-index 4-LP-mode RSMAM RCF (T = 4), Min|Δn_eff| = 2.4 × 10⁻³, Min|A_eff| = 147 μm², Max|A_eff| = 155 μm², NSD = 7.55.