1. Introduction

The exponential growth of Internet data traffic puts ever-increasing pressure on modern optical telecommunication networks whose backbone mainly consists of singlemode fibers [1]. Spatial division multiplexing (SDM) is a potentially groundbreaking approach to ward off the looming capacity crunch [2]. This would be achieved by scaling the number of data paths available inside a single fiber strand through spatially independent channels [3]. One embodiment of SDM − known as mode-division multiplexing (MDM) − involves the parallel excitation and propagation of several optical modes within a few-mode fiber, which also has the potential of maximizing the overall energy efficiency of SDM links [4].

Pertaining to MDM, one possible method is to use the spatial LP modes of a few-mode fiber (FMF). However, since the scalar LP modes are actually composed of several spatial and polarization degenerate modes that randomly couple during propagation, their demultiplexing at the fiber output usually requires multiple-input and multiple-output (MIMO) signal processing whose complexity scales quadratically with the number of modes. An alternate scheme for MDM is to excite fiber modes carrying quantized states of orbital angular momentum (OAM) [5]. The OAM state of an optical mode stems from its helical phase front − also known as an optical vortex − whose formulation inside a FMF relies on utilizing distinct vector modes (i.e. the true eigenmodes) of the cylindrical fiber [6]. This attractive approach would enable one to forgo MIMO provided that modal degeneracy is lifted through appropriate fiber design.

Similarly to polarization-maintaining fibers, lifting modal degeneracy of the vector modes − and therefore mitigating modal coupling − depends on enhancing the separation, Δn_eff, between the effective indices of adjacent modes. A so-called “vortex fiber” characterized by a high-index ring was thus shown to separate the effective indices of the constitutive vector modes of the LP₁₁ group by Δn_eff ≈1.8 × 10⁻⁴ [7] which enabled OAM-based MDM terabit-scale data transmission [8]. A hollow-core high-index-ring fiber achieving Δn_eff ≈1.0 × 10⁻⁴ was shown to support 12 OAM modes [9], and a similar fiber that increased this number to over 16 OAM modes was recently demonstrated [10]. We also note a number of numerical studies that point to the potential of annular core fibers [11,12] and ring-index photonic crystal fibers [13] for the delivery and nonlinear optical interactions of OAM modes, respectively. Given the current scarcity of optical fibers that have demonstrated long-distance propagation of OAM modes, it is essential to investigate practical fiber designs optimized for OAM-based MDM.

In this work, we demonstrate that a novel type of all-glass few-mode optical fiber (first reported in [14]) is capable of large effective index separations between the vector modes (Δn_eff > 2.1 × 10⁻⁴) as predicted by theory and confirmed in experiments. The proposed FMF is characterized by an inverse-parabolic graded-refractive-index profile. We first present the inverse-parabolic graded-index fiber (IPGIF) and theoretically investigate how this fiber can achieve wide modal separations. A discussion of the fabrication procedure and the characterization methods used to confirm the large effective index splittings, is then presented. We finally demonstrate the propagation of OAM carrying modes in the IPGIF over a kilometer length, thus supporting the appeal of this design for MDM with OAM states.

2. Description of the IPGIF

In order to achieve wide modal index separations (Δn_eff >1 × 10⁻⁴) it was demonstrated that fiber designs tailored towards large refractive index gradients coupled with high modal field gradients are imperative [7]. Prior studies have shown that this may be accomplished in fiber profiles presenting a high-refractive-index contrast and sharp index structures that, in combination, promote a high concentration of fields around these features [7–12]. A fiber design that fills both criteria, but has yet to be investigated in the context of OAM waveguiding, is that of the IPGIF previously described in [15]. A generic description of a graded-index fiber presenting a parabolic profile and infinite cladding is given by:

 

Fig. 1 IPGIF refractive index profile with parameters: a = 3 μm, N = −4, Δn_max = 0.05 and n₂ = 1.4440.

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Considering a cladding with a radius of 62.5 μm and a refractive index corresponding to that of undoped silica (n₂ = 1.4440 at λ = 1550 nm), we performed numerical simulations based on the finite-element-method (FEM) with a commercial software (COMSOL). The IPGIF considered in Fig. 1 was found to support up to three mode groups (LP₀₁, LP₁₁ and LP₂₁). The field profiles of the constitutive vector modes are shown in Fig. 2.

 

Fig. 2 Transverse electric field, $\overrightarrow{e}$, amplitude (grayscale) and direction (arrows) of the guided vector modes at λ = 1550 nm in the IPGIF with parameters a = 3 μm, N = −4, Δn_max = 0.05 and n₂ = 1.4440. The graph shows the fields of the fundamental HE₁₁ mode (LP₀₁), the TE₀₁, HE₂₁, and TM₀₁ modes (LP₁₁ group), and the EH₁₁ and HE₃₁ modes (LP₂₁ group) calculated with FEM.

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3. Analysis of the effective index separation between vector modes

In this section we first develop an analytical understanding of the main physical factors that contribute towards large intermodal separations in optical fibers, and examine how these can be engineered in IPGIFs. We then provide numerical examples that validate our theoretical analysis.

3.1 Theoretical analysis of the modal separations in the IPGIF

Using the simulated mode field profiles [Fig. 2], we investigate analytically the effective index separations between the vector modes of the IPGIF. Let $\overrightarrow{e}=[e_r\widehat{r},e_{\varphi }\widehat{\varphi }]$ be the transverse field of a given vector mode, we consider a first-order perturbative analysis [16] and introduce a correction factor $\delta {\beta }^2$ for the effective index of a vector mode, $n_{eff}$, given its scalar degenerate counterpart, ${\tilde{n}}_{eff}^{}$, as related by $n_{eff}={\tilde{n}}_{eff}^{}\sqrt{1+\delta {\beta }^2/k_0^2{\tilde{n}}_{eff}^2}$ and write

3.2 Numerical analysis of the modal separations in the IPGIF

The effective index separation between vector modes was calculated using the FEM mode solver within the parameter space defined by (−4.75 ≤ N ≤ 0) and (0.02 ≤ Δn_max ≤ 0.07). For simplification purposes, the optimization solely focuses on maximizing the intermodal index separation (Δn_eff) between the modes of the LP₁₁ group (TE₀₁, HE₂₁, and TM₀₁). However, it is expected that higher-order modes sharing the same radial mode number, m = 1 (e.g. EH₁₁ and HE₃₁), will similarly achieve large intermodal separations.

Figure 3(a) plots the minimum effective index separation inside the LP₁₁ mode group as a function of the profile curvature and maximum index contrast. The results show that Δn_eff indeed scales with $\left|-N\right|$ and Δn_max, thus indicating that the sharpest profiles with the highest refractive-index-contrasts are those that enable the largest effective index separation of vector modes. Moreover, the calculations in Fig. 3(a) suggest that very large effective index separations, Δn_eff >4.0 × 10⁻⁴, may theoretically be achieved with an IPGIF profile.

 

Fig. 3 (a) Minimum effective index separation inside the LP₁₁ mode group as a function of IPGIF parameters N and Δn_max calculated by FEM. (b) Isolines along selected contrast values: Δn_max = [0.02, 0.05, 0.07]. Other simulation parameters were kept at a = 3 μm, n₂ = 1.4440 and λ = 1550 nm.

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Closer inspection of the results [Fig. 3(b)] reveals that for a given index contrast value, Δn_max, the relationship between Δn_eff and -N takes a sigmoid behavior that grows towards a saturation value, Δn_eff,max. One could argue that an optimum profile curvature is close to N = −3 since more than 95% of the Δn_eff,max value is reached at this point.

Additional FEM calculations in Fig. 4(a) confirm that for a given curvature value, N, the Δn_eff increases quasi-linearly with the relative permittivity contrast,$\Delta$, as predicted by Eq. (5). Similarly, Fig. 4(b) clearly shows that for a given value of $\Delta$, the scaling law $\Delta n_{eff}^{}\propto \left|N\right|$ applies in the case of inverse-parabolic profiles. Finally, Fig. 4(c) shows that for given values of N = −4 and $\Delta =[\text{0}\text{.010,}\text{0}\text{.015,}\text{0}\text{.020}]$, the modal separation increases towards smaller core radii, a, down to an optimal core radius, that occurs just before modal cutoff. Hence the validity of the $\Delta n_{eff}^{}\propto a_{}^{-1}$ scaling is limited to well-guided modes, as opposed to leaky modes. Figure 4(d) illustrates the same $\Delta n_{eff}^{}\propto a_{}^{-1}$ dependence for N = −4 and some practical values of the maximum refractive index contrast: $\Delta n_{\max }=[\text{0}\text{.03,}\text{0}\text{.05,}\text{0}\text{.07}]$. In particular, Fig. 4(d) indicates that the optimal core radius (found at the peak: a = a_max) for N = −4 and Δn_max = 0.05 is indeed a_max = 3 μm, as chosen in our target IPGIF design shown in Fig. 1. Inspection of our simulation data in Fig. 4(d) indicate that the minimum effective index separations occur between the TM₀₁ and HE₂₁ modes when a < a_max, and between the TE₀₁ and HE₂₁ modes when a > a_max. Crucially, it also means that all the vector modes {TE₀₁, HE₂₁ and TM₀₁} of the LP₁₁ group are evenly spaced apart at the optimal core radius a = a_max.

 

Fig. 4 Minimum effective index separation for the LP₁₁ group in an IPGIF (a = 3 μm, n₂ = 1.4440 and λ = 1550 nm) as a function of (a) the relative permittivity contrast and for various profile curvatures, and (b) as a function of the profile curvature and for various relative permittivity contrasts. Modal separation in an IPGIF (N = −4, n₂ = 1.4440 and λ = 1550 nm) as a function of the core radius size and for different values of the (c) relative permittivity contrast, and (d) maximum refractive index contrast.

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4. Calculation of bend induced losses in IPGIFs

The effect of fiber bends on the attenuation properties of IPGIFs was simulated using a standard conformal mapping approach [17] which has been shown to be fairly accurate in the case of singlemode fibers [18], FMFs [19] and even for strongly multimode fibers [20]. In this method, a coordinate transformation is performed on the original unperturbed fiber refractive index profile, n_fiber, to obtain the equivalent index profile, n_eq, generated by a fiber bend, R_bend, applied along the transverse x-axis direction:

 

Fig. 5 Confinement losses of the HE₁₁ mode induced by a R_bend = 1 cm fiber bend as a function of curvature (N) and for different refractive index contrasts (Δn_max). Other parameters are: a = 3 μm, n₂ = 1.4440 and λ = 1550 nm.

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The simulation results in Fig. 5 indicate two trends. Firstly, the bend-induced losses grow almost exponentially with negative curvature (-N). Secondly, the bend-induced losses rapidly decrease as the core-cladding index contrast Δn_max is raised. In particular for the high-index contrast profiles of practical interest (Δn_max > 0.02 and −4 ≤ N ≤ −1), modal confinement within the core is so strong that bend-induced confinement losses become negligible, in agreement with prior analysis [20]. Nevertheless, we note that actual fiber bend losses are expected to be higher due to light scattering at the core-cladding boundary, especially in the present case of high-index contrast fibers [21]. The above simulations are thus useful for directing the design of IPGIFs in the assessment of the baseline performance in terms of resilience to fiber bends.

5. Fabrication and characterization of the IPGIF

Based on the FEM calculations of Fig. 3 and Fig. 4(d) and the discussion therein, we selected the design parameters, N = −4, Δn_max = 0.05 and a = 3 μm, corresponding to the refractive index profile presented in Fig. 1. These IPGIF parameters theoretically enable large intermodal index separations of $\Delta n_{eff}^{LP11}=2.37\times {10}^{-4}$ and $\Delta n_{eff}^{LP21}=1.55\times {10}^{-4}$, while presenting a maximum refractive index contrast value (Δn_max = 0.05) that is still readily attainable in germania-doped silica glass preforms manufactured by the MCVD process.

The fiber preform was fabricated via the MCVD process by incorporating the GeO₂ dopants in gaseous state inside a fused silica tube − so as to coat the interior walls − which was then collapsed to produce the macroscopic all-glass preform. The refractive index profile was first measured on the fiber preform. The small spatial diffusion of the germania dopants (that occur during fiber drawing at 2000 °C) was then simulated through Fick's second law of diffusion and the Sellmeier equation of GeO₂-doped silica glass [22], before the ensuing macroscopic preform profile was finally scaled down to the fiber dimensions [Fig. 6]. The resulting IPGIF's refractive index profile [Fig. 6] reaches a maximum value of 1.4871 with a corresponding maximum core-cladding index contrast of Δn_max = 0.0431. The refractive index profile was also measured directly on the fabricated fiber using a refracted near-field analyzer (Exfo NR-9200HR). Although these on-fiber measurements were in agreement with the downscaled preform profile of Fig. 6, they were not retained for subsequent modal simulations because of their limited spatial resolution, 0.1 μm, which precluded from accurately resolving the sharp refractive index features of the fabricated IPGIF.

 

Fig. 6 Fabricated IPGIF refractive index profile (solid black) and E-field intensity (dashed red) of the TE₀₁ guided mode at λ = 1550 nm.

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The vector modes of the fabricated IPGIF were solved for by importing the index profile of Fig. 6 into the FEM software. The corresponding effective indices (n_eff) and group velocity dispersions (GVD) of all guided modes over the C-band are plotted in Fig. 7(a)-7(b). The dispersion curves in Fig. 7(b) were calculated with the formula $GVD=-2\pi c{\beta }_2/{\lambda }^2$ by performing a 4th-order polynomial fit on the modal propagation constants of Fig. 7(a) as expressed in a Taylor series: $\beta =n_{eff}\omega /c={\beta }_0+{\beta }_1(\omega -{\omega }_0)+\frac{1}{2}{\beta }_2{(\omega -{\omega }_0)}^2+\mathrm{...},$ where the center frequency, ${\omega }_0$, was taken at λ₀ = 1550 nm.

 

Fig. 7 (a) Effective indices of the guided vector modes (neglecting polarization degenerate modes) of the fabricated IPGIF. (b) Corresponding group velocity dispersion in ps/(km-nm).

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By inspecting the data from Fig. 7(a) we determine that the minimum effective index separation is Δn_eff ≈2.1 × 10⁻⁴ inside the LP₁₁ group (TE₀₁, HE₂₁, TM₀₁) and Δn_eff ≈1.6 × 10⁻⁴ for the LP₂₁ group (EH₁₁, HE₃₁) throughout the C-band. GVD values [Fig. 7(b)] for the fundamental HE₁₁ and the LP₁₁ mode group vary between 4 and 12 ps/(nm-km), while those of the LP₂₁ mode group lies in the strongly normal dispersion regime (<-100 ps/(nm-km)) and exhibits a pronounced negative slope. The latter behavior may be explained by the relatively close proximity of the LP₂₁ mode group to the cutoff (i.e. the index of undoped silica cladding) as shown in Fig. 7(a), hence resulting in greater group velocity dispersion.

To experimentally characterize these modal separations in the fabricated IPGIF, a fiber Bragg grating (FBG) was inscribed in a deuterium loaded fiber sample by scanning a uniform phase mask of period Λ = 1071 nm with a 244 nm UV laser beam. The reflection spectrum, shown in Fig. 8, was acquired through optical frequency domain reflectometry (OFDR) using a commercial instrument (OVA, Luna Technology). Light from the tunable laser source with a singlemode fiber output was butt coupled into the IPGIF with a small offset between the fiber cores in order to excite all vector modes of the IPGIF simultaneously.

 

Fig. 8 Reflectogram acquired by OFDR after vector modes interaction with the FBG inscribed in the IPGIF. The numbered mode groups (#1, #2 and #3) correspond respectively to the LP₀₁, LP₁₁ and LP₂₁ mode groups.

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The broader peaks identified as #1, #2 and #3 in Fig. 8, which correspond to the Bragg reflection of each mode group to itself [23], are centered at wavelengths λ_p = 1575.31, 1571.34, and 1554.01 nm respectively. The narrow peaks (λ_k-p) stem from the cross-coupling of a given mode “p” with another counter-propagating mode “k”. Values of the effective indices for each vector mode can be retrieved either through n_eff,p = λ_p/Λ (where λ_p is the wavelength at the self-reflection peak) or alternatively via n_eff,k = (2λ_k-p - λ_p)/Λ where λ_k-p is the cross-coupling peak of mode “k” with a lower-order mode “p” having a self-reflection peak at λ_p.

By applying the latter formalism on the reflectogram of Fig. 8, we find intermodal index separations of Δn_eff ≈3.9 × 10⁻⁴ inside the LP₁₁ mode group (#2), and Δn_eff ≈1.9 × 10⁻⁴ in the LP₂₁ mode group (#3). The latter extracted Δn_eff values are somewhat larger than what was predicted by FEM calculations [Fig. 7(a)]. These discrepancies can be attributed to the limited precision in the reconstructed refractive index profile of the fiber (based on the initial preform), and the inevitable presence of small longitudinal variations in the refractive index profile of the drawn fiber. Moreover, the UV side-writing process of the FBG may raise the local refractive index which can slightly perturb the ideal modal properties.

Subsequent tests that were conducted have shown that by reducing the UV exposure dose during FBG fabrication (i.e. writing a weaker FBG) it is possible to reduce this FBG-induced perturbation of the modal properties to negligible levels [24]. In the case of this weaker FBG, measurements in [24] indicate effective index separations of 2.9 × 10⁻⁴ inside the LP₁₁ mode group and 1.57 × 10⁻⁴ for the LP₂₁ mode group. Discrepancies (<0.8 × 10⁻⁴) between the effective index splittings measured with the weaker FBG and those calculated by FEM [in Fig. 7(a)] are within experimental errors $\propto {10}^{-4}$.

6. Transmission of OAM modes

Based on numerical simulations [Fig. 7], the fabricated IPGIF is expected to support up to 6 ${\text{OAM}}_{\pm \ell ,m}^{\pm }$states through coherent combinations of the even and odd hybrid modes: ${\text{OAM}}_{\pm 1,1}^{\pm }={\text{HE}}_{21}^e\mp i{\text{HE}}_{21}^o$, ${\text{OAM}}_{\mp 2,1}^{\pm }={\text{EH}}_{11}^e\mp i{\text{EH}}_{11}^o$ and ${\text{OAM}}_{\pm 2,1}^{\pm }={\text{HE}}_{31}^e\mp i{\text{HE}}_{31}^o$, where the $\pm$ superscript indicates the circular polarization state of the OAM mode, the $\pm \ell$ subscript its topological-charge and m denotes the radial mode number (i.e. the number of rings in the mode profile). Although strictly speaking the fundamental ${\text{OAM}}_{0,1}^{\pm }={\text{HE}}_{11}^{\pm }$ mode does not carry OAM, it still constitutes an important waveguide mode that can be harnessed for MDM.

The various OAM modes were successfully excited in the IPGIF using a spatial light modulator (SLM) and free space optics depicted in Fig. 9. The different linearly polarized OAM beams were first generated through the SLM. A quarter wave plate is then used to convert the linearly polarized OAM beams into circularly polarized beams, which were subsequently coupled in the fiber via a lens. A similar combination of a quarter wave plate and lens were employed at the fiber output in order to convert back the circularly polarized transmitted OAM beams into linearly polarized beams for their ensuing detection. In order to mitigate modal perturbations created by environmental fluctuations, the fiber spool was put inside an isolating styrofoam box.

 

Fig. 9 Schematic of the experimental setup used to image the OAM carrying modes at the fiber output. Legend: M1, M2, M3: mirrors, SLM: spatial light modulator, QWP: quarter wave plate, SMF: singlemode fiber, BE: Beam expander, PBS: Polarizing Beam Splitter, PC: Polarization controller.

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After 1-meter propagation, the intensity distributions [Fig. 10 (top row)] and modal interference patterns with a Gaussian reference beam [Fig. 10 (bottom row)] of the OAM_-1,1, OAM_-2,1 and OAM_+2,1 modes − excited with corresponding input OAM beams of left(-) circular polarization − were captured at the fiber output by a CCD camera. We recall that for this specific circular polarization (-sign), the detected OAM_-2,1 and OAM_+2,1 modes originate from the superposition of the even and odd π/2-shifted copies of the HE₃₁ and EH₁₁ eigenmodes, respectively.

 

Fig. 10 Output fiber modal intensity distributions (in top row) and corresponding interference patterns (bottom row) of the OAM(−1,1), OAM(−2,1) and OAM( + 2,1) modes excited by (-sign) circularly-polarized light, after 1 m propagation in the IPGIF. On the right-hand side: OAM( + 1,1) mode excited by ( + sign) circularly-polarized light after 1.1 km propagation in the IPGIF. We note that the corollary set of modes OAM( + 1,1), OAM( + 2,1) and OAM(−2,1) obtained with ( + sign) circular polarization were also recovered after 1 m distance and have similar field distributions as their counterparts shown above except that their helical phase rotates in the opposite direction.

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We were also able to excite and recover the OAM_+1,1 mode (excited by right( + ) circular polarized light) after a propagation distance of 1.1 km in the IPGIF, as shown on the right-hand side of Fig. 10. Due to the proximity of the LP₂₁ mode group to the cutoff, the OAM _{± 2,1} modes experienced higher losses that prevented us from reliably detecting their presence after >100 m distance. The propagation losses averaged over all mode groups, were measured by the cutback method and estimated at 8.6 dB/km. Finer measurements utilizing selective excitation of the fundamental HE₁₁ mode indicate that this mode sustains lower losses, on the order of 6.5 dB/km. We are currently investigating the origin of these losses and whether they could be reduced by an optimization of the fiber fabrication process.

7. Conclusion

We proposed and demonstrated a novel type of few-mode fiber, the inverse-parabolic graded-index fiber (IPGIF) that enables very large effective index separations between its supported vector modes. In particular, we numerically studied and experimentally demonstrated an IPGIF design enabling over Δn_eff >2.1 × 10⁻⁴ separation between the vector modes of the LP₁₁ mode group {TE₀₁, HE₂₁, TM₀₁}. Subsequently, we experimentally showed the transmission of the OAM _{± 1,1} mode over more than 1 km. The flexibility of this design and its remarkable ability to lift modal degeneracy and separate the vector modes into distinct communication channels, make the IPGIF a promising fiber towards achieving a practical long-distance OAM based mode-division multiplexing system.