Characterizing the variation of propagation constants in multicore fibre

We demonstrate a numerical technique that can evaluate the core-to-core variations in propagation constant in multicore fibre. Using a Markov Chain Monte Carlo process, we replicate the interference patterns of light that has coupled between the cores during propagation. We describe the algorithm and verify its operation by successfully reconstructing target propagation constants in a fictional fibre. Then we carry out a reconstruction of the propagation constants in a real fibre containing 37 single-mode cores. We find that the range of fractional propagation constant variation across the cores is approximately $\pm2 \times 10^{-5}$.

Multicore fibre (MCF) is finding applications in a number of different areas of science and technology. Spatial-division multiplexing (SDM) with MCF provides a possible route to bypass the impending capacity crunch in optical telecommunication networks [1], with single-fibre capacity now exceeding 1 Pb/s using multicore architecture [2,3]. MCF allows a range of new capabilities in nonlinear endoscopy through propagating pulses in separate cores to enable the coherent synthesis of a high-intensity pulse at the sample [4][5][6], or by excitation of functionalised distal core tips. MCF has the potential to revolutionise astronomical instrumentation in the emerging field of astrophotonics [7]. It can be used to reformat light from a telescope's focal plane and guide it to a sensor, to filter out unwanted emission lines originating in the Earth's atmosphere without the need for large monochromators [8], and to interface multimode inputs with devices that require single-mode operation [9]. In quantum optics, coupled-core MCFs provide a platform for extending the dimensionality of quantum walks of single and entangled photons beyond the capabilities of planar [10] or direct-write waveguide technology [11,12]. MCF can also be used in optical switching and modelocking applications [13].
However, the application of MCF will always be limited by the precision that can be achieved in its fabrication. For example, if MCF is to be used as a filter for narrowband spectral features, Bragg gratings must be written into the cores; non-uniformity in the propagation constant between cores results in a spread in filter wavelength and a corresponding decrease in extinction across the device [9,14]. To carry out a photonic quantum walk in MCF the cores must be strongly coupled; any variation in propagation constant from core-to-core will accumulate phase mismatch across the MCF as the probability amplitude propagates. If this phase mismatch is too great, quantum correlations will no longer be observed at the output of the fibre. Although in the case of SDM data transmission, coupling between the cores is usually undesirable [15,16], accurate characterization of the structural uniformity of the fibre would yield important information about its transmission characteristics.
Obtaining information about the inevitable variations in fibre structure that arise during fabrication is not trivial [17]. Imaging the structure does not yield the required precision or accuracy. Over long fibre lengths where the cores are weakly coupled, information about the coupling strength may be inferred from optical time-domain reflectometry [18] though this provides no information about variations in propagation constant, and in some applications a sufficiently long length of MCF may not be available.
In this manuscript we present a method of determining the differences in propagation constant in MCF. We achieve this by using a short length of MCF in which the cores are strongly coupled. We measure output intensity distributions for various input fields, and using a Monte Carlo technique to reconstruct numerically the differences in propagation constant between the cores.

I. PROPAGATION IN MULTICORE FIBRE
The linear propagation of light at frequency ω in the z-direction in a MCF can be described by the coupled amplitude equation: The transfer matrix M M M can be written g 21 β 2 g 23 · · · g 2m g 31 g 32 β 3 g 3m . . . . . . . . .
where β i is the propagation constant in the i th core and g ij = g ji is the coupling strength between the i th and j th cores. In general the propagation constants and couplings are strong functions of frequency, however we will be considering only narrow wavelength ranges at any one time and therefore the frequency dependence of M M M has been omitted for notational convenience. The propagation constant in the i th core is defined as β i = n eff i (ω)ω/c, where n eff i (ω) is the effective refractive index in the i th core. The coupling strength g ij is defined as the overlap of the unperturbed mode profiles of the i th and j th cores,Ψ i (ω, x, y) and Ψ j (ω, x, y) respectively, with the perturbation introduced by the refractive index contrast of the j th core: Here the unperturbed mode profiles obey the normalization condition dx dyΨ 2 i (ω, x, y) = 1. We model MCF as a nominally regular array of circular cores with step-index refractive index profiles, in which the mode profiles are Bessel functions with exponentially-decaying wings. Due to the exponential decay of the mode profiles, the dynamics of these structures are dominated by nearest-neighbour couplings. Non-nearest-neighbour couplings can be ignored to an excellent approximation and we will adopt this convention henceforth. In a perfect MCF, all the fibre cores would be identical, with radius a, centre-to-centre separation d and index contrast with respect to the surrounding cladding material of ∆n = n core − n cladding . In this ideal situation the propagation constants for each core and the nearest-neighbour coupling strengths would be identical. Light coupling between two cores within the MCF would be perfectly phasematched, and, if light was input to just one core, it would spread symmetrically and eventually re-assemble in phase in the conjugate core after reflecting from the MCF boundaries.

II. IMPACT OF STRUCTURAL VARIATIONS
Any MCF that we might fabricate will not be ideal; the structure will inevitably contain some level of variation. In general this could be either in the longitudinal or transverse direction, corresponding respectively to the i th core varying along its length (β i = β i (z)) or the propagation constants and coupling strengths varying from core to core (β i = β j and g ij = g jk ; note however that for physically realistic systems we always require that the coupling is reciprocal so that g ij = g ji and so on) [19]. Variations in propagation constant and coupling strength both have the potential to influence the amplitude of light in each core at the output. Local increases in coupling strength create "preferred" routes along which the light spreads, whereas differences in propagation constant imperfect phasematching between cores resulting in incomplete transfer of light from one core to the next. Although it may seem at first glance that the latter effect is somewhat secondary, in fact -as we will see -it typically dominates the output intensity distributions.
We can calculate the relative impact of these effects on the flow of light between the cores by considering the analogous case of a two-mode coupler (equivalent to the simplest MCF containing only two cores). Following the analysis in [20], the coupling strength between a pair of identical step-index cores, reduces to where U , V , and W are the usual core, waveguide, and cladding parameters also defined in [20], K n are modified Bessel functions of the second kind, and the other symbols have the meanings previously defined in the text. The beat length between the two cores is defined is to a good approximation exponential when the cores are well-separated (a d) but the dependence on a becomes less straightforward as the cores are brought into closer proximity due to the dependence on a of many of the parameters in Eq 4.
The relative impact of variations in β and g can be found by examining the response of the two-mode coupler to changes core radii. β and g 2-core are plotted for a range of core radii in Figure 1. For parameters similar to those of the fabricated fibre presented later in this work (a 0 = 0.48 µm, d = 8.0 µm, ∆n = 0.02), we see that a change in core radius of 1% yields a fractional change in propagation constant ∆β/β ≈ 5 × 10 −5 and a fractional change in coupling strength of ∆g/g ≈ 0.08. We then solve Equation 1 for three situations: two identical cores of nominal radius a 0 with coupling g 0 and propagation constants β 0 ; two identical cores of radius a 0 + ∆a with identical propagation constants β 0 + ∆β but reduced coupling strength g 0 − ∆g; and two slightly different cores of radii a 0 and a 0 + ∆a with different propagation constants β 0 and β 0 + ∆β but coupling g 0 . For each system we input light to one core only and calculate the fraction that couples into the other after a fixed propagation length. The results of this are shown in Figure 1. It can be seen that in the parameter range of interest, the change in output intensity is dominated by the differences in propagation constant rather than those in the coupling strength.
This provides a normalised figure of merit that expresses per input state how much power on average ends up in the "wrong" core at the output. We then implement a Markov Chain Monte Carlo (MCMC) routine with acceptance criteria inspired by the Metropolis algorithm to reconstruct the differences in the propagation constants [22]. We randomly perturb the c 1 1 is a constant and n st is an integer that increments on every iteration for which the solution is stationary. We choose a temperature function that decreases exponentially with iteration number: where c 2 and c 3 are constants, and N is the total number of iterations. The MCMC routine is repeated for N iterations during which the magnitude of the random perturbations in

V. FABRICATION AND RECONSTRUCTION OF 37-CORE MCF
We fabricated a 3-ring MCF consisting of 37 cores arranged in a regular triangular array, as shown in Figure 4. Each core began as a graded-index Germanium-doped preform that was drawn into a rod, jacketed and re-drawn. The MCF preform was constructed using 37 of these twice-drawn rods stacked with an additional layer of pure silica rods forming a dummy fourth ring to limit the deformation of the third ring of cores during the fibre draw.
The MCF preform was drawn to a cane, jacketed, and finally drawn to fibre, resulting in cores with diameters of approximately 1.1 µm and separation of 8 µm. These cores can be modelled by step-index cores of radius 0.48 µm and index contrast 0.02 with an equivalent two-mode beat length of approximately 35 mm at a wavelength of 650 nm.
The measurement apparatus is shown in Figure 4 (a). A 6.9 mm length of the MCF was cleaved, taking particular care to obtain cleaves that were flat and perpendicular to the fibre axis. Using a femtosecond amplified fibre laser we generated optical supercontinuum in a photonic crystal fibre (PCF) [23] and butt-coupled it directly to the MCF. The output from the MCF passed through a 10 nm bandpass filter to select the wavelength range of interest and the output face of the MCF was imaged onto a CCD camera. By scanning the PCF across the input face of the MCF we verified that the supercontinuum was coupled into only one core at a time; due to the strong frequency-dependence of the coupling strength, the short-wavelength light reflected from the interference filter allowed us to monitor which core the light was coupled into even when the wavelength reaching the camera had spread across much of the structure. We moved the PCF between all the cores at the input and hence recorded 37 output intensity patterns {P P P ranges in the visible and near infra-red.
We then ran our reconstruction algorithm on the data measured at a wavelength of 650 nm. The results of 100 runs are displayed in Figure 4. Of these 100 runs, 95 converged and the resulting reconstructed variations in propagation constant for the 37 cores are dis-played in panel (d) along with their associated standard deviations. The residual difference F between the reconstructed and measured intensities for those runs that converged is larger than that for the simulated reconstruction in the previous section for four reasons: noise in the measured data; uncertainty in the uniform value of g used in the reconstruction (though we note that the reconstructed {β (rec) i } are robust to small errors in g); variations in g between cores in the MCF; and the effects of averaging the variation in g over the 10 nm filter bandwidth. Nevertheless the residual F for the runs that converged corresponds to a difference between the reconstructed and measured intensities that is approximately a quarter of its value were the propagation constants assumed to be uniform between all the cores, and the output intensities found using {β  Figure 5; it can be seen that, although the differences in propagation constant contain a randomly-varying element, there is also a clear pattern to the variations: cores towards the edge of the fibre tend to have a larger propagation constant than those near the centre.
This suggests a systematic variation in the structure as the MCF and if this pattern were accounted for only by differences in core size, we see from Figure 1 that it would correspond to the outer cores being slightly larger, by a factor of approximately 1.01, than those at the centre. It is also interesting to note that the two cores for which the uncertainty in reconstructed propagation constant is largest are both at the corners of the structure; corner cores have only three nearest neighbours and hence have the smallest interaction with the remainder of the structure.

VI. CONCLUSION
We have implemented a robust method of determining the variations in propagation constant in MCF that requires only straightforward measurements of intensity with a simple camera. We outlined an algorithm that allows the variations in propagation constant to be reconstructed if a constant coupling strength is assumed, and demonstrated that the algorithm successfully reconstructs target values of propagation constant. Finally, we applied our technique to a 37-core MCF fabricated in-house and found that the propagation constants vary over a fractional range of ±2 × 10 −5 . Our method relies upon the MCF cores being strongly coupled. However, if the cores are similar, there is no fundamental reason why the same technique could not be applied to find the variations in MCF designed to have low coupling strength either by testing it at a longer wavelength or by drawing a section of the fibre preform to a smaller diameter for the purposes of testing. Therefore we anticipate that this method will be of widespread use in characterizing MCF for all application areas.

VII. ACKNOWLEDGEMENTS
We gratefully acknowledge support from the UK EPSRC grant EP/K022407/1, the UK STFC grant ST/K00235X/1, and the EU 7 th Framework Programme under grant agreement 312430.