1. Introduction

Orbital Angular Momentum (OAM) states have recently received tremendous attention for their use as orthogonal information carriers in both the classical and quantum domains due to the potentially large dimensionality of available state space [1,2] without having to resort to digital signal processing techniques for signal recovery. OAM states carrying data at telecom rates have been transmitted over 1.1km of fiber [3], and over free-space links of approximately 100m in length [4]. Free-space OAM transmission has been demonstrated over as long as 3.1km, although at Hz rates [5]. Degenerate OAM state polarization mode dispersion (PMD) has been measured after 1.09km of propagation in fiber [6]. Up to 8 distinct higher-order OAM states have been transmitted over kilometer-length fibers and the largest number of OAM states propagated through short (~meter-length) fibers with reasonable measured purity is currently 12 [7]. Fibers supporting an even larger number of OAM states have been proposed [8] but transmission with quantifiable purity metrics through any reasonable length of fiber has not been achieved, to date [9].

While much attention has been focused on increasing the ensemble of stable OAM modes in fiber, propagation lengths have thus far been limited to ~1km. To advance our understanding of fiber propagation in this eigenbasis, and also considering that longer lengths are required for practical applications (metro network links are ≥ 10km [10]; future data centers may need ~2km fiber links [11]), OAM state propagation over longer lengths must be studied. Here we demonstrate OAM state propagation over 13.4km of air core fiber, enabled by an enhanced effective index separation ($\Delta n_{eff}=1.7\times {10}^{-4}$) between the distinct OAM modes, and a record-low loss of 0.8dB/km. Given the availability of only ~1-2km segments of fiber, we performed recirculating fiber loop experiments [12] to probe longer length transmission, the length tested being limited only by extraneous loop connection and component losses. We find that the modes remain 15dB pure after 2km of propagation, greater than 10dB pure after 5.5km, and 7.2dB pure the after the maximum length (13.4km) we were able to propagate them (before the cascaded loss from extraneous loop components became too high). We find that the intermodal crosstalk between adjacent OAM modes of the same |L| follows a tanh(hz) dependence, where h is the mode coupling rate [13] and z is fiber length. For the |L| = 7 modes, $h=1.7\times {10}^{-2}k{m}^{-1}$, comparable to elliptical core polarization maintaining (PM) fibers [14,15], confirming that birefringent perturbations, which are typically less abundant than bend/shape perturbations, are the primary cause for mode coupling between OAM modes in fibers.

2. OAM states in air core fibers

OAM states in fibers are specified by their orbital angular momentum per photon, L⋅ħ, where L may be a positive or negative integer, and their polarization (spin), either right or left circular [16]. Thus, there are four OAM states for a particular |L| (|L|>1). The states either have OAM and circular polarization of the same handedness, and are called spin-orbit aligned, or of the opposite handedness, and are called spin-orbit anti-aligned.

For these experiments we use an air core fiber designed to support several high-|L| OAM states, the details of which are presented in [7]. An end facet image and an experimentally measured refractive index profile are shown in Fig. 1(a). The air core and large refractive index step of the guiding ring region, induce a large effective index splitting between the spin-orbit aligned and spin-orbit anti-aligned modes of |L| = 5,6, and 7, and enable stable propogation. Higher |L| modes are not guided at 1550nm; lower |L| modes feature small Δn_eff and readily couple. The transmission loss for these three mode families is 1.0dB/km or lower as measured by cutback (Fig. 1(b)), with the highest order modes, the |L| = 7 family, having a loss of 0.8dB/km, the lowest value reported to date for an OAM mode in fiber. This low loss was achieved through refinements in the manufacturing process, including a reduced water concentration in the fiber. The separation in effective index between the spin-orbit aligned and spin-orbit anti-aligned modes at 1550nm for each of these states is given in Fig. 1(c), with the |L| = 7 modes possessing thelargest splitting of $\Delta n_{eff}=1.7\times {10}^{-4}$. However, in these fibers, the |L| = 6 modes are “accidentally” degenerate with the |L| = 2, m = 2 modes (i.e. the undesired second radial order modes) near 1500nm. This wavelength of degeneracy is sensitive to the inner diameter of the guiding region, and can fluctuate by 50 or 100nm over the course of the fiber draw. For this reason, we are unable to propagate |L| = 6 modes longer than 1.2km in our fiber samples [7]. Group indices and dispersions of the |L| = 5,6, and 7 modes are illustrated in Fig. 1(d) and (e).

 

Fig. 1 (a) Optical microscope image and measured refractive index profile of the air core fiber. (b) Transmission loss for each OAM mode order measured at 1550nm via fiber cutback. (c) n_eff versus wavelength for higher order OAM modes. Grey curve corresponds to parasitic EH_1,2 and HE_3,2 modes which become “accidentally” degenerate with OAM |L| = 6 around 1500nm, leading to severe mode coupling. Insets are experimentally measured images after ~2m of fiber propagation at 1550nm, except for the left-most image at 1500nm. (d) and (e) Group index and dispersion of OAM modes in the air core fiber. For accurate time of flight measurements, pulse spread due to dispersion must be significantly smaller than possible pulse spread due to differential time of flight between adjacent OAM modes.

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3. Time of flight measurements and coupled mode theory

Intermodal crosstalk is measured using time of flight measurements [19]. The spin-orbit aligned and spin-orbit anti-aligned modes of the same |L| have different group indices, and consequently pulses launched into the two modes simultaneously will have different arrival times, determined entirely by differential group delay within the fiber. Figure 2(a) illustrates this for the |L| = 7 modes. If only one of the two mode types is launched and in-fiber mode coupling occurs, it will result in a distribution of power between these two arrival times (red region in Fig. 2(b)). Integration over this distribution approximates the total power in the parasitic mode set, while integration over the feature which arrives at the arrival time of the “pure mode” approximates the power which has not been parasitically coupled (green region in Fig. 2(b)). The ratio of these two integrals measures the crosstalk.

 

Fig. 2 Time of flight measurements after 1.2km of fiber propagation. (a) impulse response on linear scale when |L| = 7 spin-orbit aligned and spin-orbit anti-aligned are launched separately. (b) Impulse response of |L| = 7 spin-orbit aligned on log scale. Oscillations occuring after peak are electrical noise. Shoulder with average local power −31dB created by intermodal coupling between spin-orbit aligned and spin-orbit anti-aligned modes over fiber length. Integration of the shoulder (red region) divided by integration of the main pulse (green) yields fraction of power depleted from desired state and manifest as crosstalk, −17dB here. (c) and (d) Impulse response for |L| = 5 and |L| = 6 after 1.2km.

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This methodology inherently makes three assumptions: strong discrete-coupling instances within the fiber are unlikely, pulse broadening due to dispersion is significantly smaller than broadening due to mode coupling, and that multiple coupling instances – light coupling from spin-orbit aligned to anti-aligned and back, are negligible. We justify the first assumption by experimental observation; unless intentionally induced, such discrete coupling instances are unlikely in these fibers. The second assumption is validated by the group index and dispersion values of the modes of interest; however, it prevents the investigation of lower-|L| modes for which the two kinds of temporal broadening are comparable. Error due to the third assumption can be calculated from theory. Using the temporal impulse response for in-fiber mode coupling in a two-mode fiber and comparing with the theoretical crosstalk in such fibers [20], we find that the shoulder-integration method is accurate to within 0.5dB unless the crosstalk is −5dB or stronger, at which point multiple-coupling instances are non-negligible and multipath interference (MPI) will be significant [21].

Time of flight measurements are realized with a pulsed laser (Pritel, pulse width ~5ps), fast detector (New Focus 1444-50 NIR Photodetector, ~20ps response time or New Focus 1611 InGaAs Photoreceiver, 1GHz frequency response) and a sampling oscilloscope (Agilient Infiniium DCA with 86109A detector module). Since the electronic receiver is not fast enough to resolve the launched pulses, ringing from the electronic impulse response is evident following each detected pulse. This implies that measurements over arrival times before the dominant pulse are more reliable, since this temporal region negligibly suffers from the impulse-response distortion. Sample time of flight measurements on logarithmic scale for the OAM modes in the air core fiber after 1.2km of propagation are shown in Figs. 2(b)-2(d), with integrated crosstalk values given in red. We find that the crosstalk strictly decreases with increasing Δn_eff, as expected.

4. Design and implementation of loop experiment

To simulate long-length OAM state propagation, we design a recirculating fiber loop, shown in Fig. 3(a). Light from a pulsed laser source, described in detail shortly, is directed to a spatial light modulator (SLM) and quarter wave plate (QWP) for free-space mode shaping, and introduced into the loop via a 3-dB beam splitter. It then couples into the fiber under test (facet A), and propagates through the fiber, leaving facet D. The output of D passes again through the same 3-dB splitter, where half of the light pumps further round trips and half leaves the loop for detection.

 

Fig. 3 (a) Picosecond pulses (70GHz bandwidth) at 1558nm from a passively mode-locked fiber laser source with variable repetition rate are shaped into free space OAM states by a spatial light modulator (SLM). They are then passed through a quarter-wave plate and coupled into the recirculating loop via a 3dB beam splitter. Fiber-to-fiber coupling loss is 1.2dB for both |L| = 5 and |L| = 7. Output passes through an acousto-optic modulator (AOM) and is sent to a fast detector and oscilloscope. “x” indicates fiber splice, loss 0.2dB or better and mode purity ~15dB. Inset illustrates quality of splice. (b) Schematic diagram of laser source block. A pulsed laser of fixed repetition rate is passed through an electro-optic modulator (EOM) driver by a digital delay generator (DDG) to selectively pass pulses, thus controlling the repetition rate. Rates used vary from 20MHz to ~100kHz. A 1% tap is taken as a trigger signal for the wide bandwidth oscilloscope, while 99% is fed to the system in (a). For some measurements, an erbium doped fiber amplifier (EDFA) is used to partially compensate system losses.

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The loop is configured by first ensuring good free space alignment (appx. 25dB of isolation between modes of the same |L| and 20dB between different |L|) into fiber segment (A-B) from the SLM. Using a second SLM and input coupling system (not shown), fiber segment C-D is excited from facet C with similar mode purity. The output from facet D is then used to align fiber-to-fiber (D to A) coupling, for which loss is 1.2dB for both |L| = 5 and |L| = 7 modes. The loop is then closed by splicing facets B and C, using time of flight for a mode launched into facet A and exiting facet D as feedback for splice quality. Splices were performed using a custom splice program on a conventional splicer (Ericsson FSU 995). Splice losses of 0.2dB or better and splice-crosstalk of −15dB are achieved by splicing at low current, thus preventing the collapse of the air core.

Conventional loop experiments for single or few mode fibers [12,22] use acousto-optic modulators (AOMs) or choppers to gate data streams into and out of the loop. AOMs for OAM modes in air core fiber are not available, and our loop lengths are too short for choppers to be used, thus necessitating the use of (inherently lossy) 3-dB beam splitters for input/output coupling (see Fig. 3(a)). We use a free-space AOM after the loop to temporally filter the pump and loop output, which co-propagate after the beam splitter. The insertion loss from the AOM combined with the input coupling loss into the 50μm diameter GRIN multimode fiber used for pickup is ~4dB for both |L| = 5 and |L| = 7. No difference in insertion loss between spin-orbit aligned and spin-orbit anti-aligned states is measurable for either L.

The pulsed laser source is illustrated in Fig. 3(b). We pass the output of a fixed repetition rate (20MHz) laser of wavelength 1558nm and pulse duration ~5ps through an electro-optic modulator (EOM). The EOM is biased at the null point and driven by a digital delay generator (DDG, Stanford Research Systems DG645) whose output frequency is varied to change the laser’s repetition rate. This is done to avoid loop output and loop pump pulses coinciding in time, and prevents multiple features of interest, for example, the first and second round trips of a mode from the loop, from overlapping temporally. Since the EOM’s extinction ratio is only 25dB, comparable to the instantaneous power of distributed coupling features of interest even without loop loss, care is exercised to adjust loop lengths by a few cm to avoid parasitic (undesired) pulses in the region of interest on our sampling scope. For some measurements an erbium doped fiber amplifier (EDFA) is used to partially offset input coupling and component losses. In all instances of out experiments, we do not observe any apparent increase in the spectral bandwidth of the output pulses, confirming that self-phase modulation effects are negligible. When the EDFA is used, we place a bandpass filter before the AOM after the loop to filter out undesired amplified spontaneous emission from the EDFA.

5. Experimental results

Loop measurements are performed on a multitude of fiber lengths, by splicing together air core fibers of lengths 0.6km, 1.2km (four samples), and 1.9km. The 0.6km sample is obtained by halving a 1.2km sample; thus, the largest possible loop is 6.7km. For measurements of crosstalk after the first round trip (up to 6.7km), integration over the distributed mode coupling shoulder as described in Section 3 is sufficient. For second round trips at longer lengths, the mode coupling shoulder is often inseparable from electronic ringing after parasitic laser peaks, which are separated by only 50ns. In this case, measurements are taken with the loop open and with the loop blocked, and the root mean squared (RMS) strength of the mode coupling shoulder is determined from these two measurements. Distributed coupling is approximated as this RMS value multiplied by the shoulder width.

For lengths 3.1km or less, the ~20ps response time (New Focus 1444-50) detector is used due to its superior temporal resolution. For lengths 3.1km and longer, the 1GHz (New Focus 1611) photoreceiver is used due to its higher sensitivity. The two detectors agree on measured crosstalk at 3.1km to within 0.5dB. One source of this (small) discrepancy may arise from the fact that the detector with a slower response inadequately samples the entire temporal regime over which distributed mode coupling occurs. This is less problematic at longer lengths, although for intermediate lengths pulse broadening due to detector bandwidth can affect measurements, and care must be exercised in measuring output responses in this regime.

Measured crosstalks versus fiber length for the |L| = 5 and |L| = 7 modes are shown in Fig. 4(a). Inset images are fiber output at various lengths, where the increasing “beadiness,” heuristically defined as a departure from a smooth, azimuthally uniform intensity profile, indicates worsening crosstalk. Crosstalk increases approximately linearly with length, at a rate of 1.5% per kilometer for the |L| = 7 modes, and at a rate of 15% per km for the |L| = 5 modes.

 

Fig. 4 (a) Measured crosstalk for |L| = 5 and |L| = 7 from loop experiment. Dashed lines are theoretical crosstalk, for given h values determined by minimization of mean-squared error from data. Inset images become progressively beadier at longer lengths, indicating stronger crosstalk. (b) Measured crosstalk for |L| = 6 from cutback from 1km to 500m. Insets are experimental images of dominant circular polarization component at fiber output.

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The |L| = 7 modes are better than −10dB pure up to 5.5km of propagation, and is better than −15dB pure after 2km of transmission. We fit this data (via minimization of mean-square error) to a coupled power theory model [20] that predicts a tanh(hz) dependence for crosstalk. We find that $h=1.7\times {10}^{-2}k{m}^{-1}$for the |L| = 7 modes and $h=1.4\times {10}^{-1}k{m}^{-1}$for the |L| = 5 modes. We measure crosstalk at several wavelengths across the telecom C-band for one of the lengths (5.5km), observing fluctuations of about 0.5dB, confirming that this crosstalk is wavelength agnostic. We do not observe distributed coupling between adjacent high-|L| OAM mode orders above the noise floor of our detector (approximately −35dB, which would yield integrated crosstalk of −19dB over 1.2km) for any loop length. However, based on the extremely large ($5\times {10}^{-3}$) n_eff separation between mode orders, we expect this crosstalk should be orders of magnitude lower than in-group crosstalk [23].

The longest length we were able to measure was 13.4km, the second round trip of the 6.7km fiber loop. We are limited to this by losses external to the loop, which combine to be more than 15dB, whereas mode coupling features of interest are typically at least 20dB weaker in peak power than pulses which show no mode coupling. These losses are only partially correctable with an EDFA before self-phase modulation in the fiber leading up to the loop becomes significant. Moreover, there are abundant parasitic pulses leaving the loop due to the low extinction ratio of the EOM used to select pulses. Each parasitic pulse will pass through the loop and be detected, but it will also enter the loop and create its own round trips. The availability of an AOM configured for fiber OAM modes would have enabled testing propagation over longer lengths, but the excellent fit of the theoretical model to the data indicates that crosstalk in OAM fibers can be accurately predicted to follow the tanh(hz) dependence predicted by coupled power theory.

Although we cannot transmit |L| = 6 over longer lengths due to accidental degeneracies with undesired higher radial order modes, we have a fiber sample not used in the loop experiment, for which |L| = 6 is supported over 1km. We cut back this sample in 50m steps and measure distributed mode coupling versus length (Fig. 4(b)). We find that the mode coupling rate for these modes again fits the coupled power theory model we successfully used for |L| = 5 and |L| = 7, with an h value of $~4.1\times {10}^{-2}k{m}^{-1}$.

6. Discussion

The measured mode coupling rates and effective index separations for |L| = 5,6, and 7 are given in Table 1. Mode coupling strictly decreases with increasing Δn_eff, with more than an order of magnitude difference between the values of h for |L| = 5 and |L| = 7, despite Δn_eff changing by only a factor of two. This underscores the importance of n_eff separation for mode stability, and the value of the general class of ring fiber designs. These results are an experimental validation of the design rule proposed in [17] based on the properties of PM fibers, which states that OAM modes separated from their neighbors of the same |L| by $\Delta n_{eff}\ge 1\times {10}^{-4}$should be stable.

Table 1. Coupling rates and Δn_eff.

View Table

Further, this work provides a system design rule for high-|L| states. Assuming that bend-induced birefringence effects, which are needed to couple between spin-orbit aligned and spin-orbit anti-aligned modes, scale as the radius of the fiber [24], we expect similar behavior to occur for high-|L| OAM states in fibers with guiding regions of comparable size. Thus using the parameters in Table 1, crosstalk can be predicted once Δn_eff is known. This may be extendable to other types of fibers in which birefringence is needed to couple between modes [25]. It may not, however, be applicable in fibers for which an OAM transition of ΔL = 2|L| can also cause spin-orbit aligned to spin-orbit anti-aligned coupling. Perturbations capable of causing such transitions are more likely for low |L| [7], for which mode coupling rates might be higher for comparable Δn_eff.

Recent measurements have studied similar mode coupling between adjacent LP mode families [23]. Despite comparable spooling tensions (~0.45N), the OAM states studied here are at least an order of magnitude more stable when compared at the same Δn_eff. We attribute this to the larger probability of perturbations such as slight bends containing the OAM ΔL = 1 component necessary to couple LP₀₁ and LP₁₁, as opposed to birefringent perturbations which strongly couple spins, which are necessary to couple the OAM states in the air-core fiber. Indeed, we find that our measured coupling rates compare well with those of modes in PM elliptical core fibers, which reinforces conclusions from our earlier demonstration that illustrated that stable OAM modes behave like PM modes even in strictly circular fibers [7].

During the course of our experiments, we respooled the fiber under test several times. We find that the |L| = 5 modes are very sensitive to spooling conditions, while the |L| = 7 modes do not change appreciably when spooled at higher tension. This agrees with the findings in [23], which suggest that above some value of Δn_eff, mode coupling will be spooling-invariant, suggesting crosstalk will be reproducible across different spooling or cabling conditions.

In the process of performing these experiments, we demonstrate that OAM-carrying fibers can be spliced with low (0.2dB or better) loss and low (−15dB or better) crosstalk, a functionality critical for scaling of OAM fiber-based data transmission. In principle, there is no reason why these numbers cannot be lower with more precise equipment, although the splice is very sensitive, as offsetting the fibers can lead to significant crosstalk.

The transmission loss measured for these fiber is, to the best of our knowledge, the lowest measured transmission loss for pure OAM modes in fiber. The so-called Vortex Fiber showed OAM transmission losses of 1.6dB/km [3], and an inverse-parabolic graded index fiber showed transmission loss of 8.6dB/km for |L| = 1 [26], while previous air core fibers of similar dimension [7] yielded transmission losses of about 2dB/km. We expect that the measured loss of 0.8dB/km could be lowered with further fabrication improvements; the fact that |L| = 5 shows higher loss than |L| = 7 may suggest that some of the loss is due to scattering off of the edge of the air core, which could be reduced. Although these fibers will intrinsically have higher loss than single mode fiber due to the large refractive index of the guiding region [27], we believe that in principle they could have losses comparable to those in dispersion compensating fiber (that have similar index contrasts), which are as low as 0.3dB/km [28].

7. Conclusion

Fiber supporting OAM modes currently enable the only demonstrated multimode fiber schematic that could provide capacity scaling via adding spatial modes (as opposed to mode groups) without resorting to digital signal processing or adaptive optics. This has been enabled by ring fiber designs that support stable OAM modes. Here we show that this design, especially one based on an air core, can achieve propagation distances of up to at least 13.4 km. Crosstalk values remain below 10 dB for 5.5-km propagation and below 15 dB for 2-km lengths, relevant for large scale data centers [11]. We measure loss values (0.8 dB/km) that we believe to be a record for OAM fibers, and show that these fibers may be spliced with low loss (0.2 dB) and crosstalk (15 dB). Crucially, we show that the mode stability (and hence crosstalk) is very well described a simple coupled power theory model, based on which we have obtained specific rules for effective index separation between fiber OAM modes to achieve desired crosstalk values (of use to developing future fiber designs). Interestingly, our analysis shows that crosstalk between OAM modes scales (with Δn_eff) as in PM fibers, where the perturbation required to induce mode coupling involves birefringence. In contrast, conventional LP mode orders are coupled via bend perturbations, and as a result, experience significantly more crosstalk compared to OAM modes at similar effective index separations. The design rules regarding the required Δn_eff for crosstalk minimization is thus radically different for intra-mode order coupling versus inter-mode order coupling. We expect that the elucidation of these design rules, combined with improvements in the manufacturing process, would enable future designs with longer length, stable propagation of OAM modes in fiber.