## Abstract

A novel type of few-mode fiber, characterized by an inverse-parabolic graded-index profile, is proposed for the robust transmission of cylindrical vector modes as well as modes carrying quantized orbital angular momentum (OAM). Large effective index separations between vector modes (>2.1 × 10^{−4}) are numerically calculated and experimentally confirmed in this fiber over the whole C-band, enabling transmission of OAM(+/−1,1) modes for distances up to 1.1 km. Simple design rules are provided for the optimization of the fiber parameters.

© 2014 Optical Society of America

## 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, Δ*n _{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 LP

_{11}group by Δ

*n*≈1.8 × 10

_{eff}^{−4}[7] which enabled OAM-based MDM terabit-scale data transmission [8]. A hollow-core high-index-ring fiber achieving Δ

*n*≈1.0 × 10

_{eff}^{−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 (Δ*n _{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 (Δ*n _{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:

*n*

_{1}and

*n*

_{2}are respectively the refractive indices at the core center (

*r*= 0) and in the cladding (

*r*>

*a*). Also, $\Delta =({n}_{1}^{2}-{n}_{2}^{2})/2{n}_{1}^{2}$ is the relative permittivity contrast and $N=({n}_{1}^{}-{n}_{a}^{})/({n}_{1}^{}-{n}_{2}^{})$ is the curvature parameter, where

*n*denotes the refractive index value exactly at the core-cladding interface (

_{a}*r*=

*a*). For the inverse-parabolic profiles of interest, with

*N*<0, the maximum refractive index contrast occurs at the core-cladding interface and is given by $\Delta {n}_{\mathrm{max}}={n}_{a}^{}-{n}_{2}^{}=(1-N)({n}_{1}^{}-{n}_{2}^{})$. The special case

*N*= 0 corresponds to the conventional step-index profile. Figure 1 shows the refractive index profile of the IPGIF specifically studied in this work with

*N*= −4, Δ

*n*

_{max}= 0.05 and

*a =*3 μm.

Considering a cladding with a radius of 62.5 μm and a refractive index corresponding to that of undoped silica (*n*_{2} = 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 (LP_{01}, LP_{11} and LP_{21}). The field profiles of the constitutive vector modes are shown in Fig. 2.

## 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, ${\tilde{n}}_{eff}^{}\text{\hspace{0.05em}}$, as related by ${n}_{eff}={\tilde{n}}_{eff}^{}\text{\hspace{0.05em}}\sqrt{1+\delta {\beta}^{2}/{k}_{0}^{2}{\tilde{n}}_{eff}^{2}\text{\hspace{0.05em}}}\text{\hspace{0.17em}}$ and write

_{11}). As such, Eq. (2) indicates that substantial modal index separations can occur when large refractive index gradients $\overrightarrow{\nabla}(\mathrm{ln}{n}^{2})$ are colocated with high transverse field amplitudes $\overrightarrow{e}$ and large field variations $\overrightarrow{\nabla}\cdot \overrightarrow{e}$. For circular symmetric fibers, the terms in Eq. (2) involving the gradient operators simplify to:

_{11}mode group, Eq. (3) predicts, as expected, that the effective index of the TE

_{01}mode undergoes no vector correction since it has no radial field component (

*e*= 0) as well as a uniform azimuthally polarized field (

_{r}*e*is constant). On the other hand, the TM

_{ϕ}_{01}mode potentially accrues the largest correction term of the LP

_{11}group because of its purely radially varying field (

*e*=

_{r}*f*(

*r*) and

*e*= 0), as evidenced later in Fig. 7(a) by the fact that its effective index lies the furthest away from the uncorrected

_{ϕ}*n*value of the TE

_{eff}_{01}mode. Further insight can be gained by substituting the expression of the IPGIF's refractive index core profile [Eq. (1)] in Eq. (4), evaluated at

*r*=

*a*, as to find:

*n*

_{max}. On the other hand, the magnitude of the modal separations is predicted to vary inversely with the core radius, $\Delta {n}_{eff}^{}\propto {a}_{}^{-1}$. The underlying reason is that a larger core leads to a stronger power confinement in the center core which lowers the modal field overlap with the sharp high-index features found at the core-cladding boundaries.

#### 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 ≤ Δ*n*_{max} ≤ 0.07). For simplification purposes, the optimization solely focuses on maximizing the intermodal index separation (Δ*n _{eff}*) between the modes of the LP

_{11}group (TE

_{01}, HE

_{21}, and TM

_{01}). However, it is expected that higher-order modes sharing the same radial mode number,

*m*= 1 (e.g. EH

_{11}and HE

_{31}), will similarly achieve large intermodal separations.

Figure 3(a) plots the minimum effective index separation inside the LP_{11} mode group as a function of the profile curvature and maximum index contrast. The results show that Δ*n _{eff}* indeed scales with $\left|-N\right|$ and Δ

*n*

_{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, Δ

*n*>4.0 × 10

_{eff}^{−4}, may theoretically be achieved with an IPGIF profile.

Closer inspection of the results [Fig. 3(b)] reveals that for a given index contrast value, Δ*n*_{max}, the relationship between Δ*n _{eff}* and -

*N*takes a sigmoid behavior that grows towards a saturation value, Δ

*n*

_{eff,}_{max}. One could argue that an optimum profile curvature is close to

*N*= −3 since more than 95% of the Δ

*n*

_{eff,}_{max}value is reached at this point.

Additional FEM calculations in Fig. 4(a) confirm that for a given curvature value, *N*, the Δ*n _{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: $\Delta {n}_{\mathrm{max}}=[\text{0}\text{.03,}\text{0}\text{.05,}\text{0}\text{.07}]$. In particular, Fig. 4(d) indicates that the optimal core radius (found at the peak:

*a*=

*a*

_{max}) for

*N*= −4 and Δ

*n*

_{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 TM

_{01}and HE

_{21}modes when

*a*<

*a*

_{max}, and between the TE

_{01}and HE

_{21}modes when

*a*>

*a*

_{max}. Crucially, it also means that all the vector modes {TE

_{01}, HE

_{21}and TM

_{01}} of the LP

_{11}group are evenly spaced apart at the optimal core radius

*a*=

*a*

_{max}.

## 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*, generated by a fiber bend,

_{eq}*R*, applied along the transverse

_{bend}*x*-axis direction:

*confinement losses*of the fundamental HE

_{11}mode were numerically computed for a tight bend of

*R*= 1 cm radius. The results are shown in Fig. 5 as a function of the profile curvature and for different refractive index contrasts. We also note that bend-induced confinement losses of the higher-order modes within the LP

_{bend}_{11}group are approximately an order of magnitude larger compared to the fundamental HE

_{11}mode.

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 Δ*n*_{max} is raised. In particular for the high-index contrast profiles of practical interest (Δ*n*_{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, Δ*n*_{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 (Δ*n*_{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 GeO_{2} 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 GeO_{2}-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 Δ*n*_{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.

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: $\beta ={n}_{eff}\omega /c={\beta}_{0}+{\beta}_{1}(\omega -{\omega}_{0})+{\scriptscriptstyle \frac{1}{2}}{\beta}_{2}{(\omega -{\omega}_{0})}^{2}+\mathrm{...},$ where the center frequency, ${\omega}_{0}$, was taken at λ

_{0}= 1550 nm.

By inspecting the data from Fig. 7(a) we determine that the minimum effective index separation is Δ*n _{eff}* ≈2.1 × 10

^{−4}inside the LP

_{11}group (TE

_{01}, HE

_{21}, TM

_{01}) and Δ

*n*≈1.6 × 10

_{eff}^{−4}for the LP

_{21}group (EH

_{11}, HE

_{31}) throughout the C-band. GVD values [Fig. 7(b)] for the fundamental HE

_{11}and the LP

_{11}mode group vary between 4 and 12 ps/(nm-km), while those of the LP

_{21}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 LP

_{21}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.

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}_{-}

*) stem from the cross-coupling of a given mode “*

_{p}*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*/Λ (where

_{p}*λ*is the wavelength at the self-reflection peak) or alternatively via

_{p}*n*= (2

_{eff,k}*λ*

_{k}_{-}

*-*

_{p}*λ*)/Λ where

_{p}*λ*

_{k}_{-}

*is the cross-coupling peak of mode “*

_{p}*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 Δ*n _{eff}* ≈3.9 × 10

^{−4}inside the LP

_{11}mode group (#2), and Δ

*n*≈1.9 × 10

_{eff}^{−4}in the LP

_{21}mode group (#3). The latter extracted Δ

*n*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.

_{eff}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 LP_{11} mode group and 1.57 × 10^{−4} for the LP_{21} 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.

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 HE_{31} and EH_{11} eigenmodes, respectively.

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 LP_{21} 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 HE_{11} 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 Δ*n _{eff}* >2.1 × 10

^{−4}separation between the vector modes of the LP

_{11}mode group {TE

_{01}, HE

_{21}, TM

_{01}}. 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.

## Acknowledgments

The authors thank PhD student Charles Brunet from the COPL (Université Laval) for helping in the theoretical understanding and description of OAM modes generation/propagation in optical fibers. The authors thank PhD student Cang Jin, also from the COPL, for his help in the measurement of fiber propagation losses. The authors are grateful to Pierre-André Bélanger from the department of physics at Université Laval for insightful discussions in the application of the inverse-parabolic graded-index profile, and Adrian Lorenz from the Leibniz Institute of Photonic Technology for valuable suggestions regarding fiber bend modeling. This work was supported by the Canada Research Chair in Advanced photonics technologies for emerging communication strategies (APTECS), by the Canada Excellence Research Chair in Enabling Photonic innovations for information and communications (CERCP), and by the Natural sciences and engineering research council of Canada (NSERC). B. Ung acknowledges the Fonds de recherche du Québec - Nature et technologies for a postdoctoral fellowship.

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