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, Δneff, 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 Δneff ≈1.8 × 10−4 [7] which enabled OAM-based MDM terabit-scale data transmission [8]. A hollow-core high-index-ring fiber achieving Δneff ≈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 (Δneff > 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 (Δneff >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 [712]. 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(r)={n112NΔ(r2/a2),0ran2,r>a
where n1 and n2 are respectively the refractive indices at the core center (r = 0) and in the cladding (r>a). Also, Δ=(n12n22)/2n12 is the relative permittivity contrast and N=(n1na)/(n1n2) is the curvature parameter, where na denotes the refractive index value exactly at the core-cladding interface (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 Δnmax=nan2=(1N)(n1n2). 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, Δnmax = 0.05 and a = 3 μm.

 

Fig. 1 IPGIF refractive index profile with parameters: a = 3 μm, N = −4, Δnmax = 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, 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, Δnmax = 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 e=[err^,eϕϕ^] be the transverse field of a given vector mode, we consider a first-order perturbative analysis [16] and introduce a correction factor δβ2 for the effective index of a vector mode, neff, given its scalar degenerate counterpart, n˜eff, as related by neff=n˜eff1+δβ2/k02n˜eff2 and write

δβ2tot(e)(e(lnn2))dAtot|e|2dA
We stress that Eq. (2) only provides a reasonable estimate of the correction factor, δβ2, since the longitudinal component of the field, ezz^, is neglected here. The magnitude of δβ2 is directly linked to the intermodal index separations within a given mode group (e.g. LP11). As such, Eq. (2) indicates that substantial modal index separations can occur when large refractive index gradients (lnn2) are colocated with high transverse field amplitudes e and large field variations e. For circular symmetric fibers, the terms in Eq. (2) involving the gradient operators simplify to:
e=1r[(rer)r+(eϕ)ϕ]
(lnn2)=r^n2(r)dn2(r)dr
Pertaining to the LP11 mode group, Eq. (3) predicts, as expected, that the effective index of the TE01 mode undergoes no vector correction since it has no radial field component (er = 0) as well as a uniform azimuthally polarized field (eϕ is constant). On the other hand, the TM01 mode potentially accrues the largest correction term of the LP11 group because of its purely radially varying field (er = f(r) and eϕ = 0), as evidenced later in Fig. 7(a) by the fact that its effective index lies the furthest away from the uncorrected neff value of the TE01 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:
|(lnnIPGIF2)|Δ|N|a
Equation (5) thus predicts that the magnitude of the effective index separations inside an IPGIF directly scales with the relative permittivity contrast, ΔneffΔ, and with the profile curvature, Δneff|N|. It should be noted that the increase in either of these parameters is correlated with an enhancement of the maximum index contrast, Δnmax. On the other hand, the magnitude of the modal separations is predicted to vary inversely with the core radius, Δneffa1. 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 ≤ Δnmax ≤ 0.07). For simplification purposes, the optimization solely focuses on maximizing the intermodal index separation (Δneff) 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 Δneff indeed scales with |N| and Δnmax, 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, Δneff >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 Δnmax calculated by FEM. (b) Isolines along selected contrast values: Δnmax = [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, Δnmax, the relationship between Δneff and -N takes a sigmoid behavior that grows towards a saturation value, Δneff,max. One could argue that an optimum profile curvature is close to N = −3 since more than 95% of the Δneff,max value is reached at this point.

Additional FEM calculations in Fig. 4(a) confirm that for a given curvature value, N, the Δneff increases quasi-linearly with the relative permittivity contrast,Δ, as predicted by Eq. (5). Similarly, Fig. 4(b) clearly shows that for a given value of Δ, the scaling law Δneff|N| applies in the case of inverse-parabolic profiles. Finally, Fig. 4(c) shows that for given values of N = −4 and Δ=[0.010,0.015,0.020], 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 Δneffa1 scaling is limited to well-guided modes, as opposed to leaky modes. Figure 4(d) illustrates the same Δneffa1 dependence for N = −4 and some practical values of the maximum refractive index contrast: Δnmax=[0.03,0.05,0.07]. In particular, Fig. 4(d) indicates that the optimal core radius (found at the peak: a = amax) for N = −4 and Δnmax = 0.05 is indeed amax = 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 < amax, and between the TE01 and HE21 modes when a > amax. 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 = amax.

 

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, nfiber, to obtain the equivalent index profile, neq, generated by a fiber bend, Rbend, applied along the transverse x-axis direction:

neq(x,y)=nfiber(x,y)(1+x1.40Rbend)
where the 1.40 factor was added in the denominator to account for the additional change in refractive index due to photoelastic effects created by the local strain in bent fused silica fibers, as derived in [19]. By implementing Eq. (6) in the FEM, bend-induced confinement losses of the fundamental HE11 mode were numerically computed for a tight bend of Rbend = 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 LP11 group are approximately an order of magnitude larger compared to the fundamental HE11 mode.

 

Fig. 5 Confinement losses of the HE11 mode induced by a Rbend = 1 cm fiber bend as a function of curvature (N) and for different refractive index contrasts (Δnmax). 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 Δnmax is raised. In particular for the high-index contrast profiles of practical interest (Δnmax > 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, Δnmax = 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 ΔneffLP11=2.37×104 and ΔneffLP21=1.55×104, while presenting a maximum refractive index contrast value (Δnmax = 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 Δnmax = 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 (neff) 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=2πcβ2/λ2 by performing a 4th-order polynomial fit on the modal propagation constants of Fig. 7(a) as expressed in a Taylor series: β=neffω/c=β0+β1(ωω0)+12β2(ωω0)2+..., where the center frequency, ω0, 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 Δneff ≈2.1 × 10−4 inside the LP11 group (TE01, HE21, TM01) and Δneff ≈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 λp = 1575.31, 1571.34, and 1554.01 nm respectively. The narrow peaks (λk-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 neff,p = λp/Λ (where λp is the wavelength at the self-reflection peak) or alternatively via neff,k = (2λk-p - λp)/Λ where λk-p is the cross-coupling peak of mode “k” with a lower-order mode “p” having a self-reflection peak at λp.

By applying the latter formalism on the reflectogram of Fig. 8, we find intermodal index separations of Δneff ≈3.9 × 10−4 inside the LP11 mode group (#2), and Δneff ≈1.9 × 10−4 in the LP21 mode group (#3). The latter extracted Δneff 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 104.

6. Transmission of OAM modes

Based on numerical simulations [Fig. 7], the fabricated IPGIF is expected to support up to 6 OAM±,m±states through coherent combinations of the even and odd hybrid modes: OAM±1,1±=HE21eiHE21o, OAM2,1±=EH11eiEH11o and OAM±2,1±=HE31eiHE31o, where the ± superscript indicates the circular polarization state of the OAM mode, the ± 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 OAM0,1±=HE11± 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 Δneff >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.

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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23. T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997). [CrossRef]  

24. L. Wang, P. Vaity, B. Ung, Y. Messaddeq, L. A. Rusch, and S. LaRochelle, “Characterization of OAM fibers using fiber Bragg gratings,” Opt. Express 22(13), 15653–15661 (2014). [CrossRef]   [PubMed]  

References

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  1. R.-J. Essiambre and R. W. Tkach, “Capacity Trends and Limits of Optical Communication Networks,” Proc. IEEE 100(5), 1035–1055 (2012).
    [CrossRef]
  2. D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
    [CrossRef]
  3. P. M. Krummrich, “Spatial multiplexing for high capacity transport,” Opt. Fiber Technol. 17(5), 480–489 (2011).
    [CrossRef]
  4. P. J. Winzer, “Energy-Efficient Optical Transport Capacity Scaling Through Spatial Multiplexing,” IEEE Photon. Technol. Lett. 23(13), 851–853 (2011).
    [CrossRef]
  5. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
  6. P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006).
    [CrossRef] [PubMed]
  7. S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34(16), 2525–2527 (2009).
    [CrossRef] [PubMed]
  8. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers,” Science 340(6140), 1545–1548 (2013).
    [CrossRef] [PubMed]
  9. P. Gregg, P. Kristensen, S. E. Golowich, J. Ø. Olsen, P. Steinvurzel, and S. Ramachandran, “Stable Transmission of 12 OAM States in Air-Core Fiber,” in Proc. of CLEO: 2012, CTu2K (2013).
    [CrossRef]
  10. C. Brunet, B. Ung, Y. Messaddeq, S. LaRochelle, E. Bernier, and L. A. Rusch, “Design of an Optical Fiber Supporting 16 OAM Modes,” in Proc. of OFC: 2014, Th2A.24 (2014).
    [CrossRef]
  11. Y. Yue, Y. Yan, N. Ahmed, J.-Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. 4(2), 535–543 (2012).
    [CrossRef]
  12. S. Li and J. Wang, “A Compact Trench-Assisted Multi-Orbital-Angular-Momentum Multi-Ring Fiber for Ultrahigh-Density Space-Division Multiplexing (19 Rings × 22 Modes),” Sci. Rep. 4, 3853 (2014).
    [CrossRef] [PubMed]
  13. Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. 37(11), 1889–1891 (2012).
    [CrossRef] [PubMed]
  14. B. Ung, P. Vaity, L. Wang, Y. Messaddeq, L. A. Rusch, and S. LaRochelle, “Inverse-parabolic graded-index profile for transmission of cylindrical vector modes in optical fibers,” in Proc. of OFC: 2014, Tu3K.4 (2014).
    [CrossRef]
  15. R. L. Lachance and P.-A. Bélanger, “Modes in Divergent Parabolic Graded-Index Optical Fibers,” J. Lightwave Technol. 9(11), 1425–1430 (1991).
    [CrossRef]
  16. J. Bures, Guided Optics: Optical Fibers and All-Fiber Components (Wiley-VCH, 2009), Chap. 5.
  17. M. Heiblum and J. H. Harris, “Analysis of Curved Optical Waveguides by Conformal Transformation,” IEEE J. Quantum Electron. 11(2), 75–83 (1975).
    [CrossRef]
  18. D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21(23), 4208–4213 (1982).
    [CrossRef] [PubMed]
  19. C. Schulze, A. Lorenz, D. Flamm, A. Hartung, S. Schröter, H. Bartelt, and M. Duparré, “Mode resolved bend loss in few-mode optical fibers,” Opt. Express 21(3), 3170–3181 (2013).
    [CrossRef] [PubMed]
  20. J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express 14(1), 69–81 (2006).
    [CrossRef] [PubMed]
  21. M. E. Lines, W. A. Reed, D. J. DiGiovanni, and J. R. Hamblin, “Explanation of anomalous loss in high delta singlemode fibres,” Electron. Lett. 35(12), 1009–1010 (1999).
    [CrossRef]
  22. J. W. Fleming, “Dispersion in GeO2-SiO2 glasses,” Appl. Opt. 23(24), 4486–4493 (1984).
    [CrossRef] [PubMed]
  23. T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
    [CrossRef]
  24. L. Wang, P. Vaity, B. Ung, Y. Messaddeq, L. A. Rusch, and S. LaRochelle, “Characterization of OAM fibers using fiber Bragg gratings,” Opt. Express 22(13), 15653–15661 (2014).
    [CrossRef] [PubMed]

2014 (2)

S. Li and J. Wang, “A Compact Trench-Assisted Multi-Orbital-Angular-Momentum Multi-Ring Fiber for Ultrahigh-Density Space-Division Multiplexing (19 Rings × 22 Modes),” Sci. Rep. 4, 3853 (2014).
[CrossRef] [PubMed]

L. Wang, P. Vaity, B. Ung, Y. Messaddeq, L. A. Rusch, and S. LaRochelle, “Characterization of OAM fibers using fiber Bragg gratings,” Opt. Express 22(13), 15653–15661 (2014).
[CrossRef] [PubMed]

2013 (3)

C. Schulze, A. Lorenz, D. Flamm, A. Hartung, S. Schröter, H. Bartelt, and M. Duparré, “Mode resolved bend loss in few-mode optical fibers,” Opt. Express 21(3), 3170–3181 (2013).
[CrossRef] [PubMed]

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[CrossRef]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

2012 (3)

Y. Yue, Y. Yan, N. Ahmed, J.-Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. 4(2), 535–543 (2012).
[CrossRef]

R.-J. Essiambre and R. W. Tkach, “Capacity Trends and Limits of Optical Communication Networks,” Proc. IEEE 100(5), 1035–1055 (2012).
[CrossRef]

Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. 37(11), 1889–1891 (2012).
[CrossRef] [PubMed]

2011 (3)

P. M. Krummrich, “Spatial multiplexing for high capacity transport,” Opt. Fiber Technol. 17(5), 480–489 (2011).
[CrossRef]

P. J. Winzer, “Energy-Efficient Optical Transport Capacity Scaling Through Spatial Multiplexing,” IEEE Photon. Technol. Lett. 23(13), 851–853 (2011).
[CrossRef]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).

2009 (1)

2006 (2)

P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006).
[CrossRef] [PubMed]

J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express 14(1), 69–81 (2006).
[CrossRef] [PubMed]

1999 (1)

M. E. Lines, W. A. Reed, D. J. DiGiovanni, and J. R. Hamblin, “Explanation of anomalous loss in high delta singlemode fibres,” Electron. Lett. 35(12), 1009–1010 (1999).
[CrossRef]

1997 (1)

T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

1991 (1)

R. L. Lachance and P.-A. Bélanger, “Modes in Divergent Parabolic Graded-Index Optical Fibers,” J. Lightwave Technol. 9(11), 1425–1430 (1991).
[CrossRef]

1984 (1)

1982 (1)

1975 (1)

M. Heiblum and J. H. Harris, “Analysis of Curved Optical Waveguides by Conformal Transformation,” IEEE J. Quantum Electron. 11(2), 75–83 (1975).
[CrossRef]

Ahmed, N.

Y. Yue, Y. Yan, N. Ahmed, J.-Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. 4(2), 535–543 (2012).
[CrossRef]

Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. 37(11), 1889–1891 (2012).
[CrossRef] [PubMed]

Alhassen, F.

P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006).
[CrossRef] [PubMed]

Bartelt, H.

Bélanger, P.-A.

R. L. Lachance and P.-A. Bélanger, “Modes in Divergent Parabolic Graded-Index Optical Fibers,” J. Lightwave Technol. 9(11), 1425–1430 (1991).
[CrossRef]

Birnbaum, K. M.

Y. Yue, Y. Yan, N. Ahmed, J.-Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. 4(2), 535–543 (2012).
[CrossRef]

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

Dashti, P. Z.

P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006).
[CrossRef] [PubMed]

DiGiovanni, D. J.

M. E. Lines, W. A. Reed, D. J. DiGiovanni, and J. R. Hamblin, “Explanation of anomalous loss in high delta singlemode fibres,” Electron. Lett. 35(12), 1009–1010 (1999).
[CrossRef]

Dolinar, S.

Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. 37(11), 1889–1891 (2012).
[CrossRef] [PubMed]

Y. Yue, Y. Yan, N. Ahmed, J.-Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. 4(2), 535–543 (2012).
[CrossRef]

Duparré, M.

Erdogan, T.

T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[CrossRef]

Erkmen, B. I.

Y. Yue, Y. Yan, N. Ahmed, J.-Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. 4(2), 535–543 (2012).
[CrossRef]

Essiambre, R.-J.

R.-J. Essiambre and R. W. Tkach, “Capacity Trends and Limits of Optical Communication Networks,” Proc. IEEE 100(5), 1035–1055 (2012).
[CrossRef]

Fini, J. M.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[CrossRef]

J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express 14(1), 69–81 (2006).
[CrossRef] [PubMed]

Flamm, D.

Fleming, J. W.

Hamblin, J. R.

M. E. Lines, W. A. Reed, D. J. DiGiovanni, and J. R. Hamblin, “Explanation of anomalous loss in high delta singlemode fibres,” Electron. Lett. 35(12), 1009–1010 (1999).
[CrossRef]

Harris, J. H.

M. Heiblum and J. H. Harris, “Analysis of Curved Optical Waveguides by Conformal Transformation,” IEEE J. Quantum Electron. 11(2), 75–83 (1975).
[CrossRef]

Hartung, A.

Heiblum, M.

M. Heiblum and J. H. Harris, “Analysis of Curved Optical Waveguides by Conformal Transformation,” IEEE J. Quantum Electron. 11(2), 75–83 (1975).
[CrossRef]

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

Y. Yue, Y. Yan, N. Ahmed, J.-Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. 4(2), 535–543 (2012).
[CrossRef]

Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. 37(11), 1889–1891 (2012).
[CrossRef] [PubMed]

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34(16), 2525–2527 (2009).
[CrossRef] [PubMed]

Krummrich, P. M.

P. M. Krummrich, “Spatial multiplexing for high capacity transport,” Opt. Fiber Technol. 17(5), 480–489 (2011).
[CrossRef]

Lachance, R. L.

R. L. Lachance and P.-A. Bélanger, “Modes in Divergent Parabolic Graded-Index Optical Fibers,” J. Lightwave Technol. 9(11), 1425–1430 (1991).
[CrossRef]

LaRochelle, S.

Lee, H. P.

P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006).
[CrossRef] [PubMed]

Li, S.

S. Li and J. Wang, “A Compact Trench-Assisted Multi-Orbital-Angular-Momentum Multi-Ring Fiber for Ultrahigh-Density Space-Division Multiplexing (19 Rings × 22 Modes),” Sci. Rep. 4, 3853 (2014).
[CrossRef] [PubMed]

Lines, M. E.

M. E. Lines, W. A. Reed, D. J. DiGiovanni, and J. R. Hamblin, “Explanation of anomalous loss in high delta singlemode fibres,” Electron. Lett. 35(12), 1009–1010 (1999).
[CrossRef]

Lorenz, A.

Marcuse, D.

Messaddeq, Y.

Nelson, L. E.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[CrossRef]

Padgett, M. J.

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34(16), 2525–2527 (2009).
[CrossRef] [PubMed]

Reed, W. A.

M. E. Lines, W. A. Reed, D. J. DiGiovanni, and J. R. Hamblin, “Explanation of anomalous loss in high delta singlemode fibres,” Electron. Lett. 35(12), 1009–1010 (1999).
[CrossRef]

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

Y. Yue, Y. Yan, N. Ahmed, J.-Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. 4(2), 535–543 (2012).
[CrossRef]

Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. 37(11), 1889–1891 (2012).
[CrossRef] [PubMed]

Richardson, D. J.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[CrossRef]

Rusch, L. A.

Schröter, S.

Schulze, C.

Tkach, R. W.

R.-J. Essiambre and R. W. Tkach, “Capacity Trends and Limits of Optical Communication Networks,” Proc. IEEE 100(5), 1035–1055 (2012).
[CrossRef]

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. 37(11), 1889–1891 (2012).
[CrossRef] [PubMed]

Y. Yue, Y. Yan, N. Ahmed, J.-Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. 4(2), 535–543 (2012).
[CrossRef]

Ung, B.

Vaity, P.

Wang, J.

S. Li and J. Wang, “A Compact Trench-Assisted Multi-Orbital-Angular-Momentum Multi-Ring Fiber for Ultrahigh-Density Space-Division Multiplexing (19 Rings × 22 Modes),” Sci. Rep. 4, 3853 (2014).
[CrossRef] [PubMed]

Wang, L.

Willner, A. E.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

Y. Yue, Y. Yan, N. Ahmed, J.-Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. 4(2), 535–543 (2012).
[CrossRef]

Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. 37(11), 1889–1891 (2012).
[CrossRef] [PubMed]

Winzer, P. J.

P. J. Winzer, “Energy-Efficient Optical Transport Capacity Scaling Through Spatial Multiplexing,” IEEE Photon. Technol. Lett. 23(13), 851–853 (2011).
[CrossRef]

Yan, M. F.

Yan, Y.

Y. Yue, Y. Yan, N. Ahmed, J.-Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. 4(2), 535–543 (2012).
[CrossRef]

Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. 37(11), 1889–1891 (2012).
[CrossRef] [PubMed]

Yang, J.-Y.

Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. 37(11), 1889–1891 (2012).
[CrossRef] [PubMed]

Y. Yue, Y. Yan, N. Ahmed, J.-Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. 4(2), 535–543 (2012).
[CrossRef]

Yao, A. M.

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-Scale Orbital Angular Momentum Mode Division Multiplexing in Fibers,” Science 340(6140), 1545–1548 (2013).
[CrossRef] [PubMed]

Y. Yue, Y. Yan, N. Ahmed, J.-Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. 4(2), 535–543 (2012).
[CrossRef]

Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. 37(11), 1889–1891 (2012).
[CrossRef] [PubMed]

Zhang, L.

Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J.-Y. Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Octave-spanning supercontinuum generation of vortices in an As2S3 ring photonic crystal fiber,” Opt. Lett. 37(11), 1889–1891 (2012).
[CrossRef] [PubMed]

Y. Yue, Y. Yan, N. Ahmed, J.-Y. Yang, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. 4(2), 535–543 (2012).
[CrossRef]

Adv. Opt. Photon. (1)

Appl. Opt. (2)

Electron. Lett. (1)

M. E. Lines, W. A. Reed, D. J. DiGiovanni, and J. R. Hamblin, “Explanation of anomalous loss in high delta singlemode fibres,” Electron. Lett. 35(12), 1009–1010 (1999).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Transverse electric field, 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, Δnmax = 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.

Fig. 3
Fig. 3

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

Fig. 4
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.

Fig. 5
Fig. 5

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

Fig. 6
Fig. 6

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

Fig. 7
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).

Fig. 8
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.

Fig. 9
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.

Fig. 10
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.

Equations (6)

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n(r)={ n 1 12NΔ( r 2 / a 2 ) , 0ra n 2 , r>a
δ β 2 tot ( e )( e (ln n 2 ) )dA tot | e | 2 dA
e = 1 r [ (r e r ) r + ( e ϕ ) ϕ ]
(ln n 2 )= r ^ n 2 (r) d n 2 (r) dr
| (ln n IPGIF 2 ) | Δ| N | a
n eq (x,y)= n fiber (x,y)( 1+ x 1.40 R bend )

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