On the scalability of ring fiber designs for OAM multiplexing

S. Ramachandran; P. Gregg; P. Kristensen; S. E. Golowich

doi:10.1364/OE.23.003721

1. OAM in fiber communications

Increasing the number of bits of information that can be encoded per photon is of paramount importance to addressing the upcoming internet capacity bottleneck. Having exhausted wavelength and polarization as “axes” in which to encode information, the search for additional photonic degrees of freedom has led to the exploration of spatial modes as candidates [1]. Since spatial modes typically mix unpredictably during fiber propagation, the preferred approach to transmitting information using them borrows from concepts ubiquitous in wireless communications, involving coherently disentangling mixing in the fiber via digital signal processing [2,3]. An alternative would be a fiber in which this mode mixing is suppressed or avoided in the first place.

One promising option is the use of the orbital angular momentum (OAM) [4] of light that yields multiple orthogonal light paths. These states have been exploited extensively in free space for enhancing the information capacity of classical links [5] as well as to achieve hyper-dimensionality in quantum entangled links [6]. Light’s OAM is a conserved quantity along the propagation axis in cylindrically isotropic media, but OAM eigenmodes are easily destroyed by free-space turbulence [7] or fiber-bends [8], because anisotropic perturbations impart angular momentum [9], coupling near-degenerate OAM states. Akin to signal processing techniques used to disentangle mode mixing amongst modes in conventional fibers, pattern recognition techniques to detect the phase singularities of OAM as a means of distinguishing different states have been theoretically postulated [10] and experimentally demonstrated [11] to enable km-length information transfer. Pure OAM transfer in free-space has, however, been limited to 100-m indoor lengths for the first order OAM state [12] and meter-long laboratory environments, when an ensemble of OAM states have been used.

Fibers, being cylindrically symmetric media, may support modes carrying OAM, but in analogy to turbulence in free-space, bends and twists in fibers couple the modes because of inherent near-degeneracies [8]. Several efforts towards managing these perturbations, interferometrically tailoring the input beam or forcing modal selectivity, have succeeded in demonstrating fiber generation or transmission of the first order OAM mode (especially the subset that has cylindrical polarization, of interest in fiber laser applications) over up to meter lengths [9,13–21]. Attempts at designs for sculpting the modal density of states to lift the inherent near-degeneracies which fundamentally limit OAM stability include helicoidally microstructured fibers [22], and the class of so-called ring fibers.

The ring fiber design space has, in particular, provided a general framework to realize an ensemble of OAM modes that may be stable over km length-scales, and thus have been used in applications such as classical communications links. Figure 1 summarizes recent data transmission results with one such ring fiber that supported the two polarizations of the fundamental Gaussian-shaped mode, and two distinct modes with OAM of + 1·ħ and −1·ħ per photon, respectively [23]. The details of single wavelength 50 GBaud QPSK signals, sent through each of the four spatial modes of the fiber individually as well as simultaneously, are shown in Fig. 1(b). After 1 km of transmission, the maximum power penalty with respect to back-to-back measurements, for achieving “error-free” operation assuming FEC corrections when each channel was individually used, was 2.2 dB, and this penalty rose to 4.7 dB when all channels were simultaneously populated with decorrelated signals. The key distinction from several other multimode data transmission experiments was that no digital signal processing was used at the receiver, confirming the ability of the ring fiber to maintain OAM stability over these length scales. The ability of obtaining error-free transmission in a WDM scheme over these modes was also confirmed (Figs. 1(c) and 1(d)), although the wavelength range over which this was tested was limited to 10 nm, due to inadequate cross-talk performance outside this optimal spectral window. The cross-talk between the modes that gave rise to various power penalties or restricted the spectral windows of operation had two origins, namely imperfect mode multiplexing at the input and output, and in-fiber mode coupling. Interferometric signal measurements suggested that the former was −15 dB while the latter was lower than −40 dB. This is encouraging, since fiber mode coupling may be the more fundamental issue for such systems, whereas it is feasible that (de)multiplexing crosstalk could dramatically improve as mode conversion technologies mature. Indeed, while the aforementioned experiments utilized bulk spatial light modulators and a series of free-space beam splitters for the purposes of a proof-of-concept demonstration, it is apparent that, given the strong interest in this field, a multitude of mode conversion technologies have been concurrently developed. While a review of the variety of these technologies is out of the scope of this brief review, some noteworthy devices include mode-sorters that can theoretically losslessly (de)multiplex an ensemble of OAM states [24,25], chip-based technologies [26–28], and q-plates that can offer complex linear combinations of spin and orbital angular momentum states [29], which may prove especially compatible with OAM states in fiber (we will discuss the reasons for this in sections 3 and 4).

Fig. 1 (a) Experimental schematic for realization of data transmission with 4 fiber modes including 2 OAM modes [23]; (b) Bit error rates (BERs) for single wavelength 50Gbaud QPSK for individual modes and combinations of modes (c) power, before (top) and after (bottom) data transmission, as a function of wavelength for a 10-channel WDM experiment over a 10-nm spectral window, and (d) crosstalk showing the optimal 10-nm spectral window.

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The development of chip-based technologies along with compatible OAM fibers may also address the packing density problem with interconnects that is anticipated in future data-center networks. The real estate occupied by a fiber with N (OAM) modes, when connected with an appropriate chip-based (de)multiplexer could potentially be N times lesser than that needed for N single mode fibers (cartoon illustrating this concept shown in Fig. 2). Hence, future data center networks may benefit from networking based on fibers with multiple, cross-talk-free OAM modes for reasons other than bandwidth or capacity scaling.

Fig. 2 Schematic diagram representing a proposal for on-chip space-reduction via use of multi-channel OAM-supporting fibers. Instead of requiring N output (single-mode) fibers for N chip outputs, on-chip state multiplexing could reduce the waveguide footprint factor proportional to the number of cross-talk free OAM modes a fiber could support.

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All such exciting prospects and applications critically depend on the scalability of OAM modes in fiber, and the rest of this report discusses the prospects of the “ring” fiber design to scale in the number of stable OAM states. Section 2 discusses recent results, while section 3 points to a potentially fundamental bottleneck in the scaling laws for this design space. As of this writing, it is not clear if this bottleneck represents a fundamentally insurmountable challenge or just requires the development of appropriate (de)multiplexing technologies. We conclude with section 4 that discusses these issues and reviews other fiber design methodologies that may offer alternative routes for scaling OAM in fiber.

2. Ring fiber design: concept, designs and recent demonstrations

The full vectorial description of the transverse electric field $\vec{e_{t; L}}$ of the first radial order eigenmodes with OAM L in typical circularly symmetric fibers that satisfy the paraxial approximation may be given by Eq. (1). F_|L_|(r) (r, radial coordinate) describes the field amplitude distribution, which may change as a function of mode order (although, for many designs in the class of ring fibers, they are almost identical). Note that the phase and polarization distributions of these fiber fields are identical to those of free space Laguerre-Gaussian (LG) beams, and by analogy, carry both spin angular momentum (SAM), represented by left or right circular polarization ( ${\hat{σ}}^{\pm} = (\hat{x} \pm i \hat{y}) / \sqrt{2}$ ), as well as OAM, given by its helical phase exp( ± iLϕ) (ϕ, azimuthal coordinate). Moreover, for all modes other than TE/TM, these quantities are separable, and hence in analogy with paraxial LG beams, the total angular momentum, J, of paraxial modes in most low index contrast fibers may be separated into spin (S) and orbital (L) angular momentum components. However, unlike free space LG beams, the propagation constants (β) for the different eigenstates are different, sans the polarization degenerate modes represented by ${\hat{σ}}^{\pm}$ .

{\vec{e_{t; L}}} = F_{| L |} (r) {\begin{array}{l} {\hat{σ}}^{\pm} & \cdot e^{i β_{0} \cdot z} & L  = 0; fundamental mode \\ \frac{1}{2} [{\hat{σ}}^{-} \cdot e^{i φ} - {\hat{σ}}^{+} \cdot e^{- i φ}] & \cdot e^{i β_{1, T E} \cdot z} & | L | = 1;TE mode {(L}_{t o t a l} = 0) \\ {\hat{σ}}^{\pm} \cdot e^{\pm i φ} & \cdot e^{i β_{1, A} \cdot z} & | L | = 1;SO aligned OAM \\ \frac{- i}{2} [{\hat{σ}}^{-} \cdot e^{i φ} + {\hat{σ}}^{+} \cdot e^{- i φ}] & \cdot e^{i β_{1, T M} \cdot z} & | L | = 1;TM mode {(L}_{t o t a l} = 0) \\ {\hat{σ}}^{\pm} \cdot e^{\mp i L φ} & \cdot e^{i β_{L, A A} \cdot z} & | L | > 1;SO anti-aligned OAM \\ {\hat{σ}}^{\pm} \cdot e^{\pm i L φ} & \cdot e^{i β_{L, A} \cdot z} & | L | > 1;SO aligned OAM \end{array}}

The subscripts TE and TM stand for transverse electric and transverse magnetic fields, respectively, and A and AA stand for spin-orbit (SO) aligned (sign of SAM and OAM being the same) and spin-orbit anti-aligned states (sign of SAM and OAM opposite), respectively. The TE/TM modes are here classified as |L| = 1 due to possessing the same radial field dependence as the |L| = 1 SO aligned OAM modes, although the total (ensemble average) OAM of either mode is 0. In a typical fiber with bends and other shape deformations, the modes couple because the perturbation P_pert breaks the orthogonality between modes via

〈 \vec{e_{t; L} \cdot} \bar{\bar{P_{p e r t}}} \cdot \vec{e_{t; L^{'}}} 〉 \neq 0

. A bend perturbation typically induces birefringence (i.e. it has a matrix element converting

{\hat{σ}}^{+}

to

[a \cdot {\hat{σ}}^{+} + b \cdot {\hat{σ}}^{-}]

{a,b} ϵ complex) and imparts OAM (i.e. it has a matrix element of the form

e^{\mp i Δ L φ}

that spans all ΔL, though ΔL = 1 is often the strongest component). The amount of mode coupling is itself proportional to the aforementioned matrix element as well as a phase matching term

e^{- {[\frac{1}{2} L_{c} Δ β]}^{2}}

, where L_c is the correlation length of the perturbation and Δβ is the difference in β between the two modes, proportional to Δn_eff, the difference in effective index, n_eff, of the two modes [30].

In conventional fibers, OAM states are unstable because β's within each mode group of the same |L| are nearly degenerate. The ring fiber design breaks this near-degeneracy, hence minimizing mode coupling and yielding stable OAM modes. The design philosophy of the ring fiber has been discussed in detail elsewhere [8,31–33] and hence is only briefly summarized here: an annular index profile enhances the field of the mode at index steps, and hence TE-like fields, parallel to the index steps, accumulate different phase shifts compared to normally incident, TM-like fields. Since these phase shifts are directly related to β of the mode, high field amplitudes at large index steps lead to large Δβ, reducing mode coupling. Figure 3(a) shows a recent all-glass realization of such a fiber, and Fig. 3(b) shows the corresponding mode separation achieved (in conventional fibers, these three curves would be coincident). Figure 3(c) shows an experimentally recorded spiral pattern obtained from interfering an OAM beam that had travelled 1.1 km through this fiber with an expanded Gaussian reference beam. This fiber was used for the data transmission experiments [23] described earlier (see Fig. 1). Since Δn_eff increases with index contrast for this class of fiber designs, one attractive option may be to realize very high contrasts using an air core, used in the past for atom guidance [34] or dispersion compensation [35]. Figure 3(d) shows our realization of a suitably designed air core fiber. As is evident from the results shown in Fig. 3(e) and 3(f), many more OAM states (12 within the set of |L| = 5,6,7) propagated stably even in the presence of 3-cm-radius fiber bends [36]. In addition, 8 of these states remained more than 18 dB pure after km-length propagation (the highest order |L| = 7 states were too lossy to probe mode stability over km lengths). An important design precaution was the avoidance of accidental degeneracies, which would have caused mode coupling with a parasitic mode, typically a 2nd order radial mode (of lower L).

Fig. 3 (a) Refractive index profile and facet image of the vortex fiber used for transmission experiments in Fig. 1. (b) Effective index plot of the first order |L| = 1 states in this fiber; note that in a conventional fiber these curves would not be distinguishable. (c) Interference of fiber-output |L| = 1 OAM states with a reference beam after 1.1km of fiber propagation. (d) Refractive index profile and facet image of the air-core fiber discussed in [36]. (e) effective index splitting and (f) spiral interferences for the |L| = 5,6, and 7 OAM modes guided in the air core fiber. In (a) and (d) the refractive index steps which lead to OAM state stability are highlighted in blue.

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The relative initial success of this design class has attracted much recent interest, with several reports on, for instance, more complex designs that tailor group velocity dispersion for nonlinear optical vortex manipulation [37], designs with multicore versions [38], designs that integrate multiplexing schemes [39], designs using graded index profiles [40], and experimental tests on designs that aim to dramatically increase the number of OAM modes [41]. A common aim of designs for scaling the number of OAM modes is realizing modal Δn_eff of 10⁻⁴ or greater, which, from a phenomenological standpoint, appears to have previously sufficed for km length propagation. Achieving this while obtaining several OAM modes and also avoiding the 2nd radial order modes we mentioned in the context of the air-core fiber of [36], described in Fig. 3(d)–3(f), requires not only high index contrasts but also thin index rings. The thin rings not only cut off 2^nd order radial modes, they increase the field amplitude at index boundaries, which suggests, from the perturbative reasoning mentioned earlier, that Δn_eff increases sufficiently to support OAM modes. While this is generally true, the perturbative approach to describing the guided fields may no longer apply, and this may lead to some subtleties complicating the simple mode representations described in Eq. (1).

3. Spin-orbit coupling in ring fibers

We consider the canonical air core fiber design shown in Fig. 4(a). Figure 4(b) shows the variation of Δn_eff vs mode order for a range of inner ring radii but constant ring thickness (although ring thickness is indeed a critical parameter for achieving a large Δn_eff, it is kept constant here for illustrative purposes). It becomes immediately apparent that a large number of (low |L|) OAM states with Δn_eff > 10⁻⁴ are obtained for large inner radii (or, equivalently, for rings with thicknesses small compared to their inner radii). In addition, a thin band of higher |L| states with moderate Δn_eff also appear stable for smaller air core dimensions. This latter design space is similar to the class of fibers in which 12 modes were demonstrated to be stable (section 2, Fig. 3(d)–3(f)). But it is the large inner radii region that yields the largest number of OAM modes with high Δn_eff and this is similar to the design space recently explored by several reports on designs and attempts at stably propagating of a large ensemble of OAM states. However, OAM modes in this design space differ considerably from modes we have considered thus far, and have no direct counterpart in paraxial optics [42]. With high field amplitudes at index gradients, the perturbative approach to analyzing the mode fields is no longer valid, and the full vectorial solution for |L|>2 OAM states is given by:

\vec{e_{t; L}} = {\begin{matrix} e^{\pm i (L - 1) φ} \cdot [e_{r; L, A A} (r) \hat{r} \mp i e_{φ; L, A A} (r) \hat{ϕ}] & \cdot e^{i β_{L, A A} \cdot z} \\ e^{\pm i (L + 1) φ} \cdot [e_{r; L, A} (r) \hat{r} \mp i e_{φ; L, A} (r) \hat{ϕ}] & \cdot e^{i β_{L, A} \cdot z} \end{matrix}}

where _{${\hat{r}, \hat{ϕ}}$} are the radial and azimuthal unit vectors, e_r and e_ϕ are the field amplitudes in the corresponding directions, and all other quantities are as defined earlier. Note the distinction between Eq. (1) describing weakly guided fields and Eq. (2): instead of uniform (circular) polarization, the fields are elliptically polarized and spatially non-uniform. Inspection of the form of these fields reveals that the ellipticity depends on the ratio |e_ϕ|/|e_r|, with the solutions reducing to the uniform-polarization OAM fields of Eq. (1) as this field ratio approaches unity. Figure 4(c) shows that this effect, of spin and orbital angular momentum not being separable, is most severe for lower |L| modes. Figure 4(d) shows field distributions for an |L| = 2 spin-orbit aligned as well as anti-aligned mode – note that the SO aligned modes are almost azimuthally polarized, while the anti-aligned modes are almost radially polarized. This complicates the problem of input-coupling with most free-space devices such as spatial light modulators and mode sorters. It is conceivable that devices such as q-plates, capable of controllably converting input spin into output OAM of a predetermined topological charge, may be the ideal devices for coupling into such states. Equation (2) may also be rewritten as:

{\vec{e_{t; L}}} = {\begin{matrix} e_{r; L, A A} (r) \cdot e^{\pm i (L - 1) φ} \cdot \frac{1}{2} [{\hat{σ}}^{+} e^{- i φ} (1 \mp | \frac{e_{φ; L, A A} (r)}{e_{r; L, A A} (r)} |) + {\hat{σ}}^{-} e^{i φ} (1 \pm | \frac{e_{φ; L, A A} (r)}{e_{r; L, A A} (r)} |)] & \cdot e^{i β_{L, A A} \cdot z} \\ e_{r; L, A} (r) \cdot e^{\pm i (L + 1) φ} \cdot \frac{1}{2} [{\hat{σ}}^{+} e^{- i φ} (1 \pm | \frac{e_{φ; L, A} (r)}{e_{r; L, A} (r)} |) + {\hat{σ}}^{-} e^{i φ} (1 \mp | \frac{e_{φ; L, A} (r)}{e_{r; L, A} (r)} |)] & \cdot e^{i β_{L, A} \cdot z} \end{matrix}}

That is, the SO coupled states are substantially the old L states of paraxial free-space, but have an extra component with |L| ± 2. The fraction of this other OAM component increases with the ratio |e_ϕ/e_r| (see Fig. 4(c) and 4(d)). This means that, in addition to q-plates, interferometric combinations of two OAM beams may also enable coupling into such states.

Fig. 4 (a) Canonical air core fiber design; (b) effective index splitting for a fixed ring thickness and refractive index step for different value of the inner radius of the ring, r₁; (c) ratio between the maximum value of the radial component of the electric field to the maximum value of the azimuthal component versus OAM order. In the paraxial case (Eq. (1) the ratio should be 1; (d) 1D and 2D plots of the electric fields of the L = 2 SO aligned and SO anti-aligned fields of the waveguide simulated in (b). Unequal radial and azimuthal field components yield a spatially varying elliptical polarization state.

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The reason that predominantly low-order OAM states are affected by this spin-orbit coupling is that they more strongly feel the effect of the air-core. Like free-space LG beams, OAM states in fiber have large fields further away from the origin as |L| increases. This, in addition to the fact that the air core acts as a repulsive barrier, implies that they are unlikely to have large fields at the inner refractive index step; thus, their stability is ensured by a large Δn₁ and an appropriate ring width. The low-|L| states, however, are more likely to experience the much larger step, Δn_a-g, and consequently tend to be more TE/TM-like in electric field.

Finally, we note, from Eq. (2), that the SO coupled states can mode mix via a perturbation element P_pert ~exp( ± i2ϕ), which may occur from twists or elliptical deformations, perhaps during fiber fabrication. Thus, in addition to requiring a precise mode launching capability, ring fibers supporting a large number of OAM modes may be unstable in spite of the large Δn_eff splitting between modes. Further experiments with such fibers should clarify the role the aforementioned factors play in the practical realization of OAM scaling with these fibers.

4. Summary and impact on scalability

The class of ring fiber designs has shown much promise by enabling the first km-length stable propagation in an ensemble of OAM modes [36], as well as by demonstrating data transmission without the need for digital signal processing [23]. The use of OAM to encode information is exciting because OAM forms a countably infinite basis set, which promises scalability in the number of states in which information may be encoded. To realize this in fiber, designs that go significantly past the number of modes demonstrated to date are needed. As of this writing, to the best of our knowledge, km-length stable transmission has been achieved in 8 modes [36], though in sub-meter lengths, as many as 36 states have been generated [41]. In addition, the ring fiber design concept has been exploited to simulate or demonstrate a slew of fibers and devices that would be relevant to the general problem of constructing OAM links with fiber [20,22,37–40]. However, significant scalability in the number of OAM modes over km-lengths remains elusive. We speculate that one reason could be that outlined in section 3: in order to realize highly scalable designs, the ring thickness has to be small, and this leads to spin-orbit coupling effects that make the input/output coupling problem harder, and it may also lead to inherent mode coupling because the modes are no longer single-OAM carrying beams. The input/output coupling problem almost certainly has viable solutions, especially in light of the development of devices such as q-plates that can offer the complex phase and polarization field conversions. Once this problem is addressed, it should be feasible to test the issue of whether SO coupling leads to fundamental propagation instabilities in scalable ring fiber designs. As alternatives to the class of ring fibers, it has been postulated that inducing chiral structure in fibers can also help OAM stability, because chiral structures impart angular momentum [43], and this can, for example, be exploited to lift even the OAM degeneracies within spin-orbit aligned or anti-aligned modes [44].

The task of understanding the relative merits of different design classes of fibers supporting OAM would benefit from some standardization, through quantification, of the stability metric. Several reports on OAM generation or transmission rely on the rather qualitative observation of spiral images when the OAM mode is interfered with an expanded Gaussian. This is, however, a poor metric. Even a mode that is 40~45% impure could lead to visually “clean” spirals if the impurities are predominantly nearest-neighbors in L; the interference pattern then has the desired number of spiral arms but appears as if the reference beam is slightly tilted. There are multiple quantitative characterization tools that can clarify this issue, especially since the purity or stability metric in a fiber is with respect to the relative power in unwanted modes, and modes have distinct, well defined n_eff, group delays, and field distribution patterns. For instance, (1) fiber Bragg gratings have been used to probe the neff of different OAM modes in fiber [45] (although this characterization technique would not be able to measure length-dependent mode coupling); (2) spectral interferometry [46–49] and time-domain impulse response measurements [36,50] can very accurately characterize the group delays and amplitudes of each mode, and measure mode purities ranging from 10 to 40 dB; (3) spatial interferometry can help further separate the contributions of even degenerate modes [51]. Any subset of these or other [52,53] techniques would help compare the performance of different OAM fiber designs in a quantitative fashion.

In summary, success in the development of fibers that could support an ensemble of OAM modes would be of great interest to a variety of communications applications. While stable OAM propagation length scales demonstrated to date are of the order of a km, this limitation primarily arose from lengths of fiber available in a research environment, and not from any inherent rise in mode instabilities observed for longer lengths. Thus, we speculate that OAM carrying fibers of much longer lengths may be feasible. This may not suffice for applications in long-haul links, where the length scales are several orders of magnitude larger. On the other hand, data networking applications, with typical lengths of the order of km, may benefit from OAM fibers for capacity enhancement, solving the packing density problem, as well the problem of connecting a large number of nodes. The length scales over which OAM fibers have been demonstrated may also suggest applications in optical amplification, which requires ~10-m doped fiber lengths, where a single doped OAM ring fiber could provide gain for multiple channels without introducing cross-talk. Finally, we note that the development of OAM fiber communications links would also depend on the development of compatible (de)multiplexing technologies, and recent developments suggest that a multitude of devices may be suitable for this.

Acknowledgments

The authors wish to acknowledge N. Bozinovic, P. Steinvurzel, M.W. Grogan, Y. Yue, Y. Ren, M. Tur, H. Huang, and A.E. Willner for insightful discussions and the collaborative work that was partially reviewed here. In addition, the authors would like to acknowledge J.Ø. Olsen for help with fiber fabrication, and M. V. Pedersen for help with the numerical waveguide simulation tool. This work was funded, in part, by the DARPA InPho program under ARO grant Nos. W911NF-12-1-0323 & W911NF-13-1-0103, and Air Force contract No. FA8721-05-C-0002, as well as NSF grant No. ECCS-1310493. P.G. additionally acknowledges support from a NSF Graduate Research Fellowship under Grant No. DGE-1247312. Opinions, interpretations, conclusions, and recommendations are those of the authors and not necessarily endorsed by the United States government.

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On the scalability of ring fiber designs for OAM multiplexing

Abstract

1. OAM in fiber communications

2. Ring fiber design: concept, designs and recent demonstrations

3. Spin-orbit coupling in ring fibers

4. Summary and impact on scalability

Acknowledgments

References and links

Cited By

Figures (4)

Equations (3)

Optics Express