Multicore fiber (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-fiber 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 [4–6], 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 fiber. 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 fiber would yield important information about its transmission characteristics.

Obtaining information about the inevitable variations in fiber structure that arise during fabrication is not trivial [17]. Imaging the structure does not yield the required precision or accuracy. Over long fiber 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.

1. Propagation in multicore fiber

The linear propagation of light at frequency ω in the z-direction in a MCF can be described by the coupled amplitude equation:

For a given wavelength, M describes everything that we require to describe light propagation through the MCF. Given sufficiently accurate knowledge of the MCF structure we could straightforwardly calculate the propagation constants and mode coupling parameters for the fiber. This would allow us to solve the propagation Eq. 1, either through finding the matrix exponential of M or by a stepwise numerical method, and hence determine the output amplitudes for any number of fiber cores. However, direct measurement of the structure does not provide sufficient accuracy to determine the values of propagation constant and coupling strength.

In a perfect MCF, all the fiber cores would be identical, with radius a, centre-to-centre separation d and index contrast with respect to the surrounding cladding material of Δn = 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.

2. 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

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 Fig. 1. For parameters similar to those of the fabricated fiber presented later in this work (a₀ = 0.48 μm, d = 8.0 μm, Δn = 0.02), we see that a change in core radius of 1% yields a fractional change in propagation constant Δβ/β ≈ 5×10⁻⁵ and a fractional change in coupling strength of Δg/g ≈ 0.08. We then solve Eq. 1 for three situations: two identical cores of nominal radius a₀ with coupling g₀ and propagation constants β₀; two identical cores of radius a₀ + Δa with identical propagation constants β₀ + Δβ but reduced coupling strength g₀ − Δg; and two slightly different cores of radii a₀ and a₀ + Δa with different propagation constants β₀ and β₀ + Δβ but coupling g₀. 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 Fig. 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.

 

Fig. 1 Dependence of the propagation constant of an individual core (a, b) and the coupling strength between two identical cores (c, d) on core radius. (b, d) show the fitted dependencies over the typical range of variation in a seen in our MCF. (e) Evolution of intensity for propagation in two coupled cores: identical cores with radii a₀, propagation constants β₀ and coupling g₀ in black; identical cores with radii a₀ + Δa, identical propagation constants β₀ + Δβ, and modified coupling strength g₀ − Δg in red; two different cores with different propagation constants β₀ and β₀ + Δβ but coupling g₀ in blue. Parameters were a₀ = 0.48 μm, a₀ + Δa = 0.4824 μm, d = 8.0 μm, Δn = 0.02.

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3. Reconstruction algorithm

We would like to find the variations in propagation constant and coupling strength in MCF that result from small structural fluctuations introduced during the fabrication process. To do so we make the following assumptions: each core can be individually addressed at the input end of the MCF; the intensity of light exiting each core of the MCF can be measured; deviations of the output intensity pattern from the ideal are dominated by differences in propagation constant rather than coupling strength (i.e. all nearest-neighbour coupling strengths are assumed equal); non-nearest-neighbour couplings are zero; the MCF is sufficiently short that the fiber length is less than the beat length and the fiber properties can be considered constant in the longitudinal direction (i.e. transverse core-to-core variations dominate); and each core is single-moded over the wavelength range of interest. Hence to describe propagation in a MCF with m cores we need to find a single nearest-neighbour coupling strength g and m values of β_i.

By addressing each core individually we construct a set of m normalised input intensity patterns $\left\{{\mathit{P}}_{\text{in}}^{(k)}\right\}$, 1 ≤ k ≤ m with related input amplitudes $\left\{{\mathit{A}}_{\text{in}}^{(k)}\right\}$, each of which has m elements containing a single nonzero entry equal to unity. These states propagate through a length L of MCF according to Eq. 1 and form a corresponding set of m output states $\left\{{\mathit{A}}_{\text{out}}^{(k)}\right\}$, for which we can measure the power exiting each core to yield m output intensity patterns $\left\{{\mathit{P}}_{\text{out}}^{(k)}\right\}$ with elements $P_{\text{out},i}^{(k)}={\left|A_{\text{out},i}^{(k)}\right|}^2$. We need to find the M that relates $\left\{{\mathit{P}}_{\text{in}}^{(k)}\right\}$ to $\left\{{\mathit{P}}_{\text{out}}^{(k)}\right\}$.

Due to the phase insensitivity of our measurement, the correspondence between any individual pair of input intensity ${\mathit{P}}_{\text{in}}^{(l)}$ and output intensity ${\mathit{P}}_{\text{out}}^{(l)}$ is insufficient to define M uniquely. Fortunately, the additional information gained from measuring the output intensities related to all m input states provides sufficient constraints to determine all m values of β_i contained in M, however it results in a set of simultaneous equations that cannot be solved analytically. Furthermore the problem of minimising the difference between the input and output intensities is not generally convex in each individual β_i; local minima often do not correspond to the global minimum. Hence to reconstruct M we have implemented an iterative Monte Carlo method known as simulated annealing that is typically applied to multi-parameter optimization problems in condensed matter physics [21].

We know $\left\{{\mathit{P}}_{\text{in}}^{(k)}\right\}$ and $\left\{{\mathit{P}}_{\text{out}}^{(k)}\right\}$ for a MCF with unknown transfer matrix M. We can approximate the mean nearest-neighbour coupling strength g̃ and mean propagation constant β₀ of the MCF cores from our imperfect knowledge of the average MCF structure. We construct a transfer matrix M̃ that contains g̃ and a set of randomly-selected {β̃_i} that vary around β₀. The input states are propagated subject to M̃ using Eq. 1 to yield a set of output intensities $\left\{{\tilde{\mathit{P}}}_{\text{out}}^{(k)}\right\}$. The difference between the calculated output intensities and the measured output intensities is then found using

After the algorithm has finished, we are left with a set of reconstructed $\left\{{\beta }_i^{(\text{rec})}\right\}$. Note that, although the output intensity patterns give us information about the relative phase of light that has propagated through each core, there is no sensitivity to global phase. Therefore, the reconstruction cannot give any indication of the absolute magnitude of the mean value of $\left\{{\beta }_i^{(\text{rec})}\right\}$, only the differences between the individual ${\beta }_i^{(\text{rec})}$. Hence we rescale the $\left\{{\beta }_i^{(\text{rec})}\right\}$ to have a mean value equal to β₀; these then represent our best estimate of the propagation constants in the m cores. We also obtain a figure of merit for the fit, the residual value of ℱ^(rec), and the reconstruction can be repeated to find the value of the coupling constant, g^(rec), that yields the smallest residual value.

4. Simulated reconstructions

To test the performance of our algorithm we reconstructed fictional MCFs. We defined a set of target propagation constants {β′_i} containing both random and systematic variations over a range Δβ and found the associated output intensities $\left\{{\mathit{P}}_{\text{out}}^{\prime (k)}\right\}$ assuming a uniform coupling strength g′ in Eq. 1. A random variation of approximately 1% was added to the individual elements of $\left\{{\mathit{P}}_{\text{out}}^{\prime (k)}\right\}$ to simulate the effects of measurement noise. We used these output intensities in the algorithm outlined above to reconstruct a set of $\left\{{\beta }_i^{\prime (\text{rec})}\right\}$ and compared them with the known values of {β′_i}. For each set of propagation constants we ran the reconstruction algorithm 100 times to build up statistics on the quality of the reconstruction.

The results for one particular run of the reconstruction are shown in Fig. 2. The reconstruction was performed on a MCF containing 37 cores in a triangular array (a “3-ring” MCF). The total run time for the reconstruction was 60 minutes using Matlab on a standard laptop computer. It can be seen that the residual ℱ gradually decreased as the space of {β̃′_i} was sampled, eventually reaching a minimum value. Occasional increases in ℱ are observed when the algorithm accepts a bad move. The reconstructed values of propagation constant $\left\{{\beta }_i^{\prime (\text{rec})}\right\}$ are displayed beneath the target set {β′_i}.

 

Fig. 2 Typical performance of simulated reconstruction algorithm on one run. (a) Fit residual ℱ and (b) intermediate fractional values of Δβ_i as functions of iteration number. (c) Fractional values of target propagation constants {β′_i} that we aimed to reconstruct. (d) Reconstructed fractional values of propagation constants $\left\{{\beta }_i^{\prime (\text{rec})}\right\}$.

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The statistical results over 100 reconstructions of the same set of {β′_i} are displayed in Fig. 3. This set of reconstructions was carried out in parallel on a quad-core desktop PC, resulting in a run time of approximately 45 hours. We envisage that this time could be significantly reduced; beyond simple parallelisation of independent reconstructions, no particular effort was expended in optimising the code for speed. 91 of the reconstructions satisfied our convergence condition, defined as having a residual ℱ within one standard deviation of the mean ℱ̄. The final set of reconstructed propagation constants $\left\{{\overline{\beta }}_i^{\prime (\text{rec})}\right\}$ plotted in Fig. 3 are the mean for each core of the $\left\{{\beta }_i^{\prime (\text{rec})}\right\}$ values taken from those reconstructions that converged. In the vast majority of cases the target values of {β′_i} are within the standard deviation of the reconstructed values $\left\{{\overline{\beta }}_i^{\prime (\text{rec})}\right\}$, confirming the validity of the reconstruction algorithm.

 

Fig. 3 Statistics of simulated reconstruction over 100 runs. (a) Raw reconstructed differences in β_i for each of 100 runs. (b) Residual value of ℱ, with convergence cut-off illustrated by black line. (c) Reconstructed differences in β_i for each of the 91 runs that satisfied convergence condition. (d) Mean reconstructed differences in β_i (red) with error bars equal to the standard deviation of the values in (c), with target β_i plotted in black.

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5. 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 Fig. 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 fiber draw. The MCF preform was drawn to a cane, jacketed, and finally drawn to fiber, 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.

 

Fig. 4 (Top) Optical micrograph of the cleaved end face of the 37-core MCF illuminated from below, and schematic of the setup used to characterise the MCF. (Bottom) Results of reconstruction of 37-core MCF over 100 runs. (a) Raw reconstructed differences in β_i for each of 100 runs. (b) Residual value of ℱ, with convergence cut-off illustrated by black line. (c) Reconstructed differences in β_i for each of the 95 runs that satisfied convergence condition. (d) Mean reconstructed differences in β_i (red) with error bars equal to the standard deviation of the values in (c). (e) Variations in β_i plotted against position in the MCF structure; size of points represents certainty in reconstructed value.

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The measurement apparatus is shown in Fig. 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 fiber axis. Using a femtosecond amplified fiber laser we generated optical supercontinuum in a photonic crystal fiber (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 $\left\{{\mathit{P}}_{\text{out}}^{(k)}\right\}$ for the 37 possible input conditions $\left\{{\mathit{P}}_{\text{in}}^{(k)}\right\}$. With the PCF tip at the mid-point between three of the MCF cores, we recorded a background frame to subtract from the data. This was repeated for various wavelength 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 Fig. 4. Of these 100 runs, 95 converged and the resulting reconstructed variations in propagation constant for the 37 cores are displayed 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 $\left\{{\beta }_i^{(\text{rec})}\right\}$ 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 $\left\{{\overline{\beta }}_i^{(\text{rec})}\right\}$ have less than 6% of their light in the “wrong” cores.

The variations in $\left\{{\overline{\beta }}_i^{(\text{rec})}\right\}$ are plotted arranged in their position in the MCF structure in Fig. 4(e); 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 fiber 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 Fig. 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.

6. 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 fiber preform to a smaller diameter for the purposes of testing. Although the method is suitable only for short lengths of fiber, longitudinal changes in the differences between cores in a sacrificial MCF could be addressed by measuring short sections taken from different points along its length. Therefore we anticipate that this method will be of widespread use in characterizing MCF for many application areas.