Simple all-fiber three-mode multiplexers were made by adiabatically merging three dissimilar single-mode cores into one multimode core. This was achieved by collapsing air holes in a photonic crystal fiber and (in a separate device) by fusing and tapering separate telecom fibers in a fluorine-doped silica capillary. In each case the LP01 mode and both LP11 modes were individually excited from three separate input cores, with losses below 0.3 and 0.7 dB respectively and mode purities exceeding 10 dB. Scaling to more modes is challenging, but would be assisted by using single-mode fibers with a smaller ratio of cladding to core diameter.
©2014 Optical Society of America
Recent interest in space-division multiplexing has highlighted the need for spatial multiplexers that couple light from several separate channels into the modes of one few-mode or multimode fiber (MMF), and vice versa. For example, information capacity is tripled using the three lowest-order spatial modes: the gaussian-like LP01 mode and the two 90°-rotated versions of the two-lobed LP11 mode [1–3]. Using optical MIMO signal processing it is sufficient to excite orthogonal combinations of modes within a given mode group (eg, LP11) [3–5]. However, MIMO requires coherent detection; for simpler direct detection schemes the MMF modes must be excited individually by a mode multiplexer, and then coupled to a suitably-asymmetric few-mode MMF that resists coupling between the modes . To minimize loss (and indeed cost) it is desirable to implement such mode multiplexers in all-fiber form.
We report three-mode all-fiber multiplexers that operate by the adiabatic propagation of light along a taper transition, in which three dissimilar input cores gradually merge into a common multimode output core. We used two different methods to make them. In our first method we post-processed a photonic crystal fiber (PCF) with three cores. Light in each input core was converted to just one of the three modes at the output, with less than 0.3 dB loss. However, the PCF's input ports were less than 20 µm apart, making some kind of fan-out necessary to address them separately. In our second method we therefore also made mode multiplexers by tapering three separate (but identical) single-mode telecommunication fibers (SMFs) in a low-index glass jacket. The three SMF inputs can be separately connected to other SMFs without any fan-out, exciting three separate modes in the output MMF port with insertion losses of 0.6-0.7 dB. The adiabaticity gave both types of multiplexer operating bandwidths exceeding 100 nm.
2. Principle of operation
The adiabatic principle behind both of our multiplexer designs was first described for integrated optics  and also underlies the operation of tapered fiber null couplers and mode convertors [9–11]. Three input cores (ideally identical and single-mode to start with) are made dissimilar somehow over a length of several centimeters. This gives the guided modes of the cores different propagation constants β - they all look like fundamental modes, but only one is the true fundamental mode of the composite three-core system. A gradual taper transition is then formed in which the three separate cores merge into one multimode core, Fig. 1. The multimode core is asymmetric to break the degeneracy of the two differently-oriented LP11 modes , ensuring that its three lowest-order modes (including LP01) also have different β. If the transition is perfectly adiabatic then, by definition, light in the single-mode core of 1st/2nd/3rd greatest β must unambiguously evolve into just the mode of 1st/2nd/3rd greatest β in the multimode core. This remains the case across all wavelengths for which the transition is adiabatic, in contrast to mode convertors (such as long-period gratings) that rely on a resonant interaction. Adiabatic mode multiplexers should therefore function over a broad band . Since linear light propagation is reciprocal, behavior should be similar in the reverse direction, ie as a mode demultiplexer. (Although we call our devices “three mode” multiplexers, it is understood that in fact six modes are involved when the polarization of the light is taken into account. However, our discussion is simpler if we include just the spatial-mode degree of freedom, as we made no attempt to exploit the polarization degree of freedom for multiplexing.)
These principles are illustrated by the PCF version of the device, shown in the top row of Fig. 2, where (a) to (d) refer to different locations along the taper transition. The three identical well-separated single-mode cores of a multicore PCF (a) are made dissimilar by allowing 2, 1 or 0 adjacent holes to collapse (b). The collapse of the holes enlarges the cores by different amounts and so makes them dissimilar. The air holes between the three cores are then gradually shrunk over a distance of a few centimeters (c) until they too have collapsed completely to form one enlarged few-moded core (d). This large core is not quite hexagonal: one “corner” hole remains open to reduce the symmetry enough to break the degeneracy of the LP11 modes. This structure could if necessary be tapered down in size to match a realistic few-mode fiber.
Propagation of light through this transition was simulated by the scalar beam propagation method (BPM) . The model transitions between (a) and (b) and between (b) and (d) were both 1 cm long. The dimensions of the model fiber matched those of the experimental fibers described later, and the wavelength was 1550 nm. For light in each of the three input cores, calculated field patterns along the device are shown in Fig. 3 and Media 1. As expected for an adiabatic transition that preserves mode order, light in the largest of the cores at (b) emerges in the LP01 mode at (d), whereas light in the other two cores emerges in LP11 modes with orthogonally-oriented lobes that respect the reflection symmetry of the core. The calculated losses between each core and the corresponding pure mode at the output were less than 0.15 dB in all three cases. This confirmed that the experimental 4-cm transition should be gradual enough to be adiabatic.
3. Experimental PCF multiplexers
We made mode multiplexers by post-processing each of two PCFs . Air holes were collapsed (by the action of surface tension) along a length of each fiber that was heated by the small flame of a fiber tapering rig. However, selected holes were kept open by pressurizing them enough to oppose surface tension. This differential pressurization was achieved by gluing up the ends of the holes we wanted to close, but not the ones we wanted to keep open. When the fiber end was connected to a supply of pressurized nitrogen gas, only the unblocked holes became pressurized. Hole closure took place gradually along the fiber by profiling the exposure of the fiber to the flame, with the aim of producing adiabatic transitions.
When we began our experiments  we did not have a suitable three-core PCF, so we took a single-core PCF (fiber A) with a hole pitch Λ = 5 µm and a hole diameter to pitch ratio d/Λ = 0.4, Fig. 2(a) middle row. We used the tapering rig to uniformly collapse 5 holes in a 12 cm section of fiber to locally create three dissimilar cores, Fig. 2(b), the biggest of which incorporated the original fiber core. Referring to these cores in decreasing order of size by the numbers 1, 2 and 3, the original core therefore becomes core 1. This device omitted the transition from (a) to (b), but the cores in (a) would have been well separated (17 µm apart).
The fiber was processed again to make 12 more holes collapse with a linear profile along 4 cm of the fiber. Finally the fiber was cleaved at both ends of the processed section, giving a simple compact device with 5 cm of uniform three-core fiber (b) at one end, 3 cm of uniform fiber with a single large core (d) at the other end, and a 4 cm transition in between. We could measure the device's loss for core 1 because it incorporated the original core of fiber A, permitting a reliable cutback measurement when the 12 cm device was cleaved at the input. The measured loss was less than 0.1 dB at the wavelength of 1550 nm, for a structure that included a rather short 3 mm transition between the original core and the enlarged core at (b).
Subsequently we made a new PCF (fiber B) with three cores but otherwise the same structure as fiber A, Fig. 2(a) bottom row. We collapsed 3 holes to make the cores locally dissimilar, Fig. 2(b). At this point the structure matched that of the fiber A device at the same stage of fabrication, and the same procedure was then followed from this point onwards to produce the device shown on the bottom row of Fig. 2. Cutback measurements could be made for all three input cores at 1550 nm, giving losses of 0.05, 0.16 and 0.27 dB respectively.
Light patterns at the output of each device were imaged in the near- and far- fields using an IR camera. The input was white light from a supercontinuum source that was passed through a 1550 nm filter with a 10 nm passband. The light was focused onto each of the three cores in turn. Light in the cladding was suppressed by applying colloidal graphite to the outside of the fiber. Translation of the beam from one core to another caused the output to go dark in between, confirming that we were able to excite the cores individually and that cladding light was (as far as we could see) being stripped.
The output near-field intensity patterns for the fiber B device are presented in Fig. 4, showing that the LP01 and LP11 modes (the latter oriented orthogonally to each other) could be individually excited from each input core. The far-field images, also shown, are valuable because they are sensitive to hidden phase discrepancies in the near fields and are also less contaminated by cladding-mode light. As expected, the outputs were in LP01 and LP11 modes for light in core 1 and cores 2/3 respectively.
Intensity profiles, Fig. 4(c), indicate that the output LP11 modes were fairly pure, the intensity minimum between the LP11 lobes being about 5% of the peak or less. However, for perfect LP11 modes the minimum should be zero. The ratio between minimum and peak intensity was used to estimate the LP11 mode purity, assuming that a non-zero minimum is due to contamination by the LP01 mode, that the spectrum of the light is broad enough to average phase, and that the modes of a multimode step-index core are good approximations for the calculation. For the centre and right intensity profiles in Fig. 4 the estimated mode purities were 12.7 and 14.6 dB respectively. The exercise was repeated for the far-field profiles, with the calculation based on the Fourier transforms of the mode fields, giving estimated mode purities of 18.0 and 15.7 dB . We expect values from far-field profiles to be more favorable (but also more representative of true performance) because cladding modes and other stray light in the near field gets the chance to diffract away. Behavior for 1650 nm light was much the same as for 1550 nm light, indicating the bandwidth benefits of a non-resonant adiabatic device.
The device made from fiber A behaved very similarly to the fiber B device . To investigate core symmetry we made an otherwise-identical device from fiber A in which the output core had complete hexagonal symmetry: like Fig. 2(d) but without the symmetry-breaking hole. The measured output near-field patterns, Fig. 5, included two annular modes rather than the more familiar two-lobed LP11 modes of Fig. 4. In a symmetric core the LP11 modes hybridize to form the TE01, HE21 and TM01 vector modes, which have ring-shaped intensity patterns. They are nearly degenerate and so are susceptible to mode coupling, hence the need for asymmetric cores for direct mode multiplexing [1,2]. In our experiments the annular modes had polarization properties indicative of the hybrid mode set .
4. Experimental SMF multiplexers
The PCF devices described in Section 3 were simple, low-loss and broad band. However, they required special PCFs and also had single-mode ports less than 20 µm apart, making a fan-out necessary to address them separately . We therefore investigated multiplexers operating by the same physical principle but made from three lengths of a standard telecom fiber (Corning SMF-28), giving identical SMF inputs that could be separately connected to other SMFs without any fan-out.
To make the three initially-identical cores dissimilar in the device transition, two of the fibers were pre-tapered along 10 cm lengths, locally reducing their diameters to 105 µm and 90 µm respectively. The third fiber was not pre-tapered and so remained 125 µm in diameter. (Such pre-tapering has been used to make wavelength-flattened  and null  fused couplers with identical ports; in effect our mode multiplexer was half of a 3 × 3 null coupler.) As with the PCF cores, we can use the numbers 1, 2 and 3 to refer to the fibers with diameters of 125, 105 and 90 µm respectively after the pre-tapering step.
The three fibers were threaded into 4-5 cm of a close-fitting capillary made of fluorine-doped silica (ID/OD of 260/330 µm), so that the pre-tapered sections were inside. The capillary had a lower refractive index than the undoped SMF cladding, with a relative numerical aperture of 0.22. It also had a lower melting temperature, so that its collapse did not deform the fibers . Such capillary jacketing was previously used to make multiport couplers  and photonic lanterns [16,18], the latter also being cleaved to yield MMF ports.
The fibers and capillary were heated and stretched together in our fiber tapering rig, the three fibers fusing to each other and to the capillary with no remaining air gaps, Fig. 6. The waist was cleaved to yield a MMF-like port, with a core of undoped silica (from the fused fibers) and a lower-index cladding of F-doped silica (from the capillary). The irregular three-lobed shape of the MMF core would be an imperfect match to common MMFs, but did ensure that its LP11 modes were non-degenerate as required. The largest residual SMF core in the MMF end had been tapered down by ~12 × and so played no role in guidance in the MMF core.
Light from a supercontinuum source was coupled via a 1310 nm bandpass filter ((Δλ = 12.5 nm) into each SMF in turn. The taper transitions were coated with colloidal graphite to strip any cladding modes. Near- and far-field images at the MMF-like output were recorded using the infrared camera, Fig. 7.
Light in the un-pretapered SMF excited a clear LP01 mode at the MMF output, and light in each other SMF excited a clear LP11 mode, the two LP11 modes being oriented at 90° to each other as expected. In all three cases the insertion loss was 0.6-0.7 dB. For the centre and right intensity profiles in Fig. 7 the mode purities estimated in the same way as for the PCF device were 8.2 and 11.1 dB respectively from the near fields and 17.3 and 19.1 dB respectively from the far fields. The discrepancy between the estimates from the near- and far-fields is large, and may be due to a greater amount of cladding-mode light in this case. BPM simulation of the modes, Fig. 7 bottom row, shows that the unequal lobe intensities in one case can simply result from the core's irregular shape. The calculated mode effective indices confirmed that the LP11 modes were non-degenerate.
The experiment was repeated with different bandpass filters, Fig. 8. Clear mode patterns were obtained for wavelengths of 1250, 1310 and 1400 nm. However, the quality of the LP11 patterns degraded at longer wavelengths, becoming somewhat annular. We believe this is because an irregular core behaves more like a circular core at longer wavelengths. This makes the LP11 modes more degenerate so that, as we saw in the PCF case, they evolve into the ring-shaped TE01, HE21 and TM01 vector modes in the limit. The effect was stronger in devices tapered to smaller diameters.
5. Scaling to more modes
Having demonstrated two types of N = 3 multiplexer exploiting the same principle of adiabatic propagation, we can consider whether we can make multiplexers for more modes. (Here N is the number of input cores and output modes.) For example, N = 6 multiplexers could be made from a PCF with six cores made dissimilar by post-processing, or from six SMFs made dissimilar by pre-tapering. However, as N increases it becomes harder to make an adiabatic transition. Below we use plausibility arguments to show that the required length of the taper increases roughly as N2, but that fiber design can help mitigate the consequences.19].
Evaluation of this length for a mode multiplexer requires a numerical analysis of the modes along the structure, as well as specific design choices among several degrees of freedom. This is beyond the scope of this paper, but we can at least qualitatively consider the form of Eq. (1a), which has two key factors. Firstly there is the reciprocal of the propagation constant difference δβ = β1 − β2 between the two local modes. At the wide/input end of the transition, the sizes of the largest and smallest cores are bounded by the need for the largest to be single-mode (normalized frequency V not too big) and the smallest to be a good waveguide (V not too small). Increasing N means we must fit more modes into the same range of β's . Hence δβ (and the maximum dρ/dz, Eq. (1a)) must scale inversely with N for a pair of modes with adjacent β values at the input. Similarly, at the narrow/output end of the transition, the area of the MMF core is proportional to the number of cores or fibers that are fused together to form it. Since mode β spacing is inversely proportional to area, δβ (and the maximum dρ/dz) must scale inversely with N for a pair of modes with adjacent β values at the output too.
Secondly, the integral in Eq. (1a) depends on how rapidly a mode changes size with ρ along the transition. If the mode has further to spread because the MMF core comprises material from more cores or fibers, the integral will increase and require a smaller dρ/dz as N increases. As a universal proxy for this expansion rate, for a centered core we calculated the rate of change with V of the LP01 Petermann I mode field diameter (MFD)  normalized to ρ. To perform the calculation, we used the analytical “bottom of band” expressions for photonic bandgap fibers in , which also represent a cylindrical finite-cladding fiber with zero field at the cladding outer boundary. The maximum MFD expansion rate is plotted in Fig. 9(a) as a function of the ratio of cladding and core areas, which is proportional to N. The expansion rate increases roughly linearly with N. The insets to Fig. 9(a) show how the field has further to spread when the MMF cladding is bigger, which (as can be seen in Media 2) means the peak expansion rate is much greater for bigger claddings.
The two effects combine to require a transition whose length increases roughly quadratically with N. This scaling with N soon makes higher-order multiplexers impractically long for fixed values of other parameters.
However, there is a lot of scope to optimize those other parameters. As Fig. 9(a) shows, there is a strong dependence on cladding/core ratio. We therefore expect adiabaticity to be easier to achieve in SMF devices if the original fibers have larger cores relative to their claddings. For a single tapered fiber, the adiabaticity criterion in the form of Eq. (1b) can be integrated without further approximation to yield the minimum possible transition length L as a function of the ratio α of cladding diameter to core diameter. This L(α) is plotted in Fig. 9(b) for λ = 1550 nm and a fiber of NA 0.11 and core diameter 9 µm. (The resulting L values seem short, but ideal taper profiles like those shown in the inset are unlikely to be achieved in practice; a real profile no steeper than the ideal profile at its most gradual will be much longer.) The standard cladding diameter of 125 µm corresponds to α = 13.9, for which L = 1.56 mm. In contrast, a fiber with α = 4 requires a transition over 20 × shorter! This big improvement in adiabaticity would allow N to be more than quadrupled (assuming the ~N2 scaling discussed above), permitting 12-mode multiplexers with better performance than our 3-mode multiplexers.
Such a fiber for making mode multiplexers with higher N and superior mode purity for λ = 1550 nm could have core and cladding diameters of 20 and 80 µm and a numerical aperture of 0.11. The fiber would be multimode, but a single-mode version with a ~40 µm outer diameter would be difficult to handle. However, a device's input fiber could be tapered down to half its diameter to locally cut-off the higher-order modes and yield a core that is matched to conventional SMFs, via a carefully fused unequal-diameter splice or a simple connector join. A fiber with an outer boundary so close to the core would have a prohibitive leakage loss of light from core to coating over kilometer lengths, but losses over the meter lengths needed for discrete components would be negligible.
We have demonstrated two types of low-loss three-mode multiplexers. Both operate by the same physical principle of adiabatic propagation through a taper transition between three dissimilar single-mode cores at one end and a common merged multimode core at the other. In the first case the cores exist in a single PCF and merge as the result of the collapse of the air holes separating them. In the second case the cores are in separate conventional single-mode fibers and merge as the result of fusion and tapering within a low-index silica jacket. Different techniques can be applied in both cases (the collapse of different numbers of adjacent holes and differential pre-tapering respectively) to have identical input ports but make them dissimilar inside the device.
The fabrication process is simple in both cases, and we showed experimentally that input light in each core excited individual LP01 and LP11 modes at the output. Asymmetry in the output cores was necessary to excite two-lobed rather than annular LP11 modes. The PCF devices were lower in insertion loss (0.3 dB versus 0.7 dB), but the SMF device was directly compatible with standard single-mode fibers without the need for a fan-out.
Scaling to a larger number of modes N is challenging because the taper length for adiabatic propagation rises rapidly with N. However, it should be possible to reduce mode coupling by using a fiber designed to match standard SMF but with a smaller ratio of cladding to core diameters, making it easier to form an adiabatic transition. Reducing this ratio from ~14 (for conventional telecom fibers) to 4 could enable a dramatic 95% reduction in the necessary transition length, providing leeway to increase the mode number while also improving mode purity and loss in a component of realistic length. Indeed the performance of other tapered fiber devices, such as fused couplers [9,15,17], photonic lanterns  and spatial multiplexers intended for MIMO  (which need only excite orthogonal combinations of modes rather than individual pure modes) could be similarly improved and/or scaled to larger N using such fibers.
The authors thank Y Chen for improving the hole inflation technique and WJ Wadsworth for providing fiber A. SY is supported by EPSRC through a PhD studentship. The research leading to these results has received funding from the European Union Seventh Framework Programme under grant agreement 312430 and from the UK STFC under grant ST/K00235X/1. We thank the Leibniz-Institut für Astrophysik Potsdam for providing the F-doped glass.
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