Rapid data traffic increase along a conventional single mode fiber (SMF) communication network quickly exhausts the current fiber optic transmission capacity and various methods are being intensively investigated to accommodate the data increase [1-3]. Among these methods, mode division multiplexing (MDM) based on few-mode fibers (FMFs) has recently been reported by major telecom research laboratories and fiber manufacturers [4-6]. In contrast to multi-core fibers for space division multiplexing [7], the MDM based on FMF can be a practical solution to increase the transmission capacity, because of low connection loss between FMF and SMFs, mass production capability for FMFs, and relatively easier solutions for optical amplification along FMFs.

MDM is utilizing individual orthogonal modes in FMF as separate carriers and there have been two contrasting methods experimentally demonstrated. In the first method [8, 9], the mode coupling among the modes is compensated by electronically using multiple input multiple output (MIMO) processing at the receiver. The other method [10, 11] minimizes the mode coupling along FMF in order to optically separate each mode at the receiver.

Therefore, FMF structure should be optimized depending on its usage, and especially the coupling or cross talk between the propagating modes would be very contrasting issue in FMF design. In addition to the general requirements for such as low attenuation, large effective area, and a low bending loss, FMFs should have either a low DGD, or a low coupling between the adjacent modes [4, 5]. Low DGD is required in the first MDM method to reduce the signal processing burden in MIMO. On the while, low mode coupling between the modes is a mandatory requirement in the second MDM method and this might also reduce MIMO processing burden in the first method.

In the meantime, overall chromatic dispersion should be optimized to enable dense wavelength division multiplexing (DWDM) even in FMF. In order to meet the requirements in both the low inter-symbol interference penalty and the low four-wave mixing impairments in DWDM, non-zero dispersion shifted fiber (NZDSF) has been developed and being widely used in long-haul applications, where the dispersion value within the transmission band was maintained at a non-zero low value [12]. Despite of its importance, non-zero dispersion value has not been fully addressed in prior FMF research.

In this study, we focus on FMF design for the uncoupled MDM, such that a new waveguide was designed to provide a minimum coupling along with a large effective index difference between two adjacent LP₀₁ and LP₁₁ modes. In order to achieve this goal we need to consider two key waveguide properties: the difference between the guided modes and their spatial mode overlap. It has been reported that the effective index difference (Δ_neff) between the adjacent propagation modes can play an important role in suppressing the crosstalk when Δ_neff is larger than ~10^-4 [13]. In the coupled mode theory [14], the overlap integral of two adjacent modes is known to be directly proportional to the mode coupling strength, and it should be minimized by optimal waveguide design. In order to use spatial filtering MDM at the receiver side, the overlap integral between LP₀₁ and LP₁₁ should be minimized for securing proper level of bit error rate [15, 16]. Despite the high importance in FMF design, detailed parametric analyses to reduce the overlap integral between the two modes have been very scarce in prior reports. Efforts to expand the operating spectral range of MDM to cover both C and L bands have been limited as well. We also optimized the chromatic dispersion of the two modes such that both of them satisfy the NZDSF requirements.

In this paper, a new TMF structure is proposed which is composed of the three-layered core; the central core, inner cladding layer, and the outer ring core as shown in Fig. 1 (a). This waveguide structure provides an efficient reduction of the modal overlap by confining the fundamental LP₀₁ mode into the central core while separating it from the LP₁₁ mode along the outer ring core as schematically shown in Fig. 1(b). Numerical modal analyses were performed by using full-vectorial finite element method (FEM) with the perfect matched layer (PML) boundary condition [17]. Optical properties of guided modes were optimized to satisfy ITU-T requirements in C+L band, for the first time to the best knowledge of the authors [18]. Relatively simple step index profile in the proposed waveguide could be easily mass-producible using the state of art fiber manufacturing technology and it can find practical applications in high capacity DMD.

Cross section of the proposed two-mode fiber (TMF) is shown in Fig. 1(a). The inner cladding and outer cladding material is pure silica, whose optical dispersion was calculated using a Sellmeier equation for vitreous silica glass [19]. In reference to silica, the central core of radius a has an index difference Δn₁, whose optical property was calculated by using a Sellmeier equation for GeO₂ doped silica glass [20]. The outer ring core has the inner radius b, the outer radius c, and the index difference Δn₂. Δn₂ was also assumed to be raised by GeO₂ doping in silica. The inner cladding layer between the two cores acted as an optical barrier separating the two modes to reduce the modal overlap. The structural parameters of our TMF are summarized in Fig. 1(b).

By using vectorial FEM with the PML condition, the modal analyses were carried out for our TMF. The magnetic field propagating along the z-direction in the fiber can be expressed as:

where β is the propagating constant and ω is the angular frequency. Eq. (2) shows an eigenvalue equation for magnetic field in the steady-state with a refractive index distribution, n.

where k₀ is given by ω/c and c is the speed of light. The Sellmeier equation was used to evaluate wavelength dependent refractive index for silica cladding and GeO₂-doped silica core regions to account for correct material dispersion. By using FEM, Eq. (2) is solved with triangular meshes to find β. Subsequently the effective mode index n_eff, of the guided mode was obtained by taking the real part of β/k₀, while confinement loss was obtained from its imaginary part [21]. In the analysis, boundary conditions for the inner elements were set to satisfy the continuity conditions and the outer boundary was set to be continuous with the PML [21].

Figure 2 briefly describes the role of triple-layered core in reducing the overlap integral and subsequently the mode coupling between LP₀₁ and LP₁₁ modes, illustration of normalized mode power overlap of the LP₀₁ and LP₁₁ modes in single core TMF (a) and our designed TMF (b). Inset shows the transverse electrical fields of LP₀₁, and LP₁₁. Normalized modal intensity profiles at the wavelength of λ = 1550 nm are compared for (a) the single core step index fiber with the core radius of 6 μm and Δn = 0. 36%. In Fig. 2(b), our TMF has the structural parameters: a = 4.1 μm, b = 6 μm, c = 8.2 μm, Δn₁ = 0.36%, and Δn₂ = 0.31%. It is clearly shown that the spatial overlap between LP₀₁ and LP₁₁ modes is significantly reduced almost by a factor of two in our proposed TMF, which confirms the unique role of three-layered core to spatially separate the two modes LP₀₁ and LP₁₁.

In order to confirm that our proposed waveguide structure allows two-mode guidance we numerically calculated the effective indices of LP₀₁, LP₁₁ and LP₂₁ modes and investigated their cut-off behavior. In the calculations, we fixed parameters for the central core; a = 4.1 μm and Δn₁ = 0.36%, which will allow a low connection loss with commercial SMF [22]. Figures 3 and 4 show effective mode index of each mode, LP₀₁, LP₁₁, and LP₂₁, as a function of waveguide parameters b, c, and Δn₂. We choose the wavelength 1530 and 1625 nm, the shortest wavelength of the C-band and the longest wavelength of the L-band [23]. If the two-mode condition is satisfied at these two wavelengths, it would be valid in the entire wavelength in C+L band. As shown in Figs. 3(a) and 4(a), the effective indices of the LP₀₁, and LP₁₁ modes were found to be higher than the silica cladding index for the given ranges of b, c, and Δn₂, which ensured that the two modes are guided along the proposed TMF. Figs. 3(b) and 4(b) show the effective index of the LP₂₁ mode and it should be lower than the silica cladding index to satisfy the two-mode guidance condition [20]. Based upon the modal guidance analyses in Figs. 3 and 4, we obtained the reference structural parameters a = 4.1 μm, b = 6 μm, c = 8.2 μm, Δn₁ = 0.36%, and Δn₂ = 0.31% satisfying the two-mode condition and they are summarized in Table 1.

It is wellknown that a large effective index difference Δn_eff between the adjacent propagation modes can suppress the mode coupling between propagated modes [26]. One of previous works [13] has shown that Δn_eff larger than ~10^-4 can efficiently suppress the crosstalk between the modes. Figure 5(a) shows the effective indices of the LP₀₁, and LP₁₁ modes. In the optimized TMF design, we obtained a large effective index difference between LP₀₁ and LP₁₁ with Δn_eff = ~2 × 10^-3 in the entire C+L band, which is an order of magnitude larger than the required value. In addition, Fig. 5(b) shows that effective index of the LP₂₁ mode in the given parameters of Table 1, which confirmed that this mode was not guided in our TMF.

The effective mode area A_eff of the LP₀₁ and LP₁₁ modes was calculated using the following equation:

Figure 6 shows the A_eff of the LP₀₁ and LP₁₁ in our designed fiber. A_eff of the LP₀₁ mode monotonically increased from 112 μm² to 126 μm² in the C+L band. A_eff of the LP₁₁ increased from 383 μm² to 431 μm². It is noted that A_eff of the LP₀₁ mode in our TMF was comparable to those of previous reports [27, 28], while A_eff of the LP₁₁ mode in our TMF was larger by a factor of two. Large effective area in our TMF ensures reduction of nonlinear effects and high optical power capacity in MDM applications.

The overlap integral between the LP₀₁ and LP₁₁ modes can be expressed as:

where I₀₁, and I₁₁ are the intensities of the LP₀₁ and LP₁₁ modes, respectively. In this study we used the normalized mode intensities to investigate the mode coupling along TMF.

In comparison to the single core step index fiber which has 53.9% overlap integral with the core radius of 6 μm and Δn = 0.36%. The overlap integral of the proposed fiber is estimated about 27.6% at λ = 1550 nm, which corresponds to a half the single core step index fiber as shown in Fig. 2. We further investigated the change of the overlap integral in the C+L band and the results are summarized in Fig. 7. The overlap integral monotonically increased from 27.45% to 27.94%. This low overlap integral can efficiently suppress the mode coupling in our TMF providing less difficulty in separating guided modes at the receiver side [16].

Chromatic dispersion of the LP₀₁ and LP₁₁ modes in our TMF was calculated as a function of wavelength using the real part of n_eff and Eq. (5).

The results are summarized in Fig. 8. Both the LP₀₁ and LP₁₁ modes showed negative dispersion owing to the triple layered core structure. Their values were between 3.5 and 5.5 ps/km/nm over the entire C+L band. These chromatics dispersion values are suitable for NZDSF for dense wavelength division multiplexing (DWDM) fiber optical networks and satisfy the requirement of ITU-T NZDSF recommendations [29] for the chromatic dispersion range. Utilizing these properties, it is further applicable not only for MDM but also for WDM.

Low DGD is important for relaxing the complexity of MIMO processing [24]. We defined the DGD between LP₀₁ and LP₁₁ as follows:

where n_eff,01 and n_eff,11 is the effective index of the LP₀₁, and LP₁₁ mode, respectively. Here c is the speed of light. In Fig. 8(b), DGD of our TMF is plotted over the C+L band and the value was about ~4 ps/m, which is very comparable to prior step-index few mode fibers [25].

In order to further confirm that the proposed TMF guides the two modes, we calculated the confinement loss α using full vector FEM analyses with PML condition in dB/km from the imaginary part of the propagation constant, β [21].

In order to properly guide a mode along a fiber, the confinement loss should be lower than 10^-4 dB/km in the spectral range of interest. Figure 9(a) shows the confinement loss of the LP₀₁, and the LP₁₁ mode and it was lower than 10^-4 dB/km for both modes. We could confirm that our proposed TMF would provide sufficient guidance for both the LP₀₁ and the LP₁₁ modes. In contrast, the confinement loss of the LP₂₁ was as large as few dB/km as shown in Fig. 9(b), which confirms that our TMF indeed guides only two modes, LP₀₁ and LP₁₁.

We have designed a new TMF for uncoupled MDM applications. Our proposed TMF waveguide was based on a three-layered core composed of a central core, an inner cladding, and an outer ring core. We successfully reduced the modal overlap by confining the LP₀₁ mode to the central core and the LP₁₁ mode to the outer ring core. Using a full vectorial finite element method, we optimized the TMF structural parameters so that it can guide the LP₀₁ and the LP₁₁ modes with a large effective index difference Δn_eff = ~2 × 10^-3 in the entire C+L band spectral range, which is an order of magnitude larger than the normally required value. The proposed TMF provided a low modal overlap between the LP₀₁ and the LP₁₁ mode of less than 27.9%, which could be a strong indicator to suppress the modal coupling. Additionally, the proposed TMF showed a low DGD (4.4~4.6 ps/m), a large A_eff of the LP₀₁ (112~126 μm²) and LP₁₁ (383~431 μm²), and an appropriate chromatic dispersion (3.5~5.5 ps/km/nm) in Table 2, which confirmed the proposed waveguide structure is viable, and practical MDM applications.