1. Introduction: dense arrays of small coupled waveguides

The trend of light confinement in increasingly smaller and denser sets of optical waveguides is based on various needs of nano-optics, such as better connection to nano-objects, more efficient nonlinear effects, or denser device integration, in reference to all-optical information processing (routing, computing, image processing…). On the whole, the search for the optimally-small individual waveguide in fiber technology has succeeded. Single-mode fibers (SMF) involving only low index gradients already provided waists (radius at 1/e²) of only a few wavelengths. Microstructured fibers (MOF) using claddings with high air-filling fractions lead to waists going down to the wavelength [1,2,3]. Finally tapering silica rods [4], MOFs [5,6] or SMFs [0] has brought sizes down to the diffraction limit ~λ/2n_silica, i.e. to the minimal waist and to the maximal nonlinear efficiency [7,8]. Beyond that optimum, performance decreases.

Coupling many of these waveguides shows some fascinating prospects in discrete optical data processing, especially through changing the coupling pattern by nonlinear effects.

However increasing the density involves new challenges, with in the first place the entanglement of individual modes. Distant waveguide structures can be considered as weakly coupled waveguides, i.e. they can be described approximately on the basis of the modes of the individual waveguide. The power exchange between them takes place on long distances and is controllable, and input and output in a single channel - and hence single-channel processing - are meaningful. Indeed in the “large” waveguide domain - i.e. far from the λ/2n limit - coupling properties of one-dimension arrays of waveguides are well documented [9,10], and a few promising studies of two-dimension (2D) arrays have appeared, involving multicore fiber patterns [11,12,13,14,15] and photo-induced refractive-index modifications in silica [16] or photorefractive materials [17,18,19,20]. On the opposite, when structures come too close to each other, they behave as a whole system described by “super” modes involving the whole pattern as in photonic crystals. In this case classical data processing on individual waveguides becomes unpractical. To our knowledge, no clear attempt to tackle this limit, or even define it, has been made.

The most promising candidate for the dense array of small - diffraction-limited - waveguides in fiber technology is the honeycomb array of so-called “waveguide channels” [21], “trefoil channels” [22], “apexes” [23] or “secondary cores” [24] formed between the air holes of MOFs. This structure is indeed considered either as a collection of weakly-coupled waveguides whose individual modes are used for transmission [22] or frequency conversion [21,24] applications, or as a global medium with a complex structure of allowed and forbidden bands of Bloch modes that control losses when the medium is used as a cladding [23]. In this paper, we first evaluate from calculations to what limit the density of such 2D optimally-small waveguides can be pushed in fiber technology at λ=1.55µm, while keeping an efficient coupled-waveguide operation. We find that at least one order of magnitude can be gained over present large-waveguide results, and we demonstrate that actual arrays of trefoil channels in a MOF actually do reach this density limit. We show that in this structure problems expected but not quantitatively evaluated up to now - waveguide losses, channel inhomogeneity - do not to hinder the desired operation.

2. The maximal density of waveguides in the weak-coupling regime

The transition between the weak-coupling and strong-coupling regimes may be defined as follows. In the weak-coupling regime, tight binding or coupled-mode theory (CMT) [25] measures the coupling strength by the ratio of the inter-guide coupling constant C to the propagation constant β₀ of the mode of the individual waveguide, and C/β₀ is assumed to be much less than unity. This ratio will be used in the following to evaluate the strength of the coupling.

For evaluating the dependence of C/β₀ on the density of waveguides, we consider two representative existing designs and one ideal hypothetical one. For the silica/silica system used in the literature for designing “large” waveguides, we select the hexagonal multicore fiber structure. For the silica/air system where “small” waveguides are expected due to the much higher index gradient, we select the array of the trefoil channels, with the hypothetical hexagonal lattice of silica rods in air as a reference in the spirit of the calculation of the optimal size. We assume that the core radius is fixed, at the ultimate radius in the silica/air system (0.52 λ/n=0.6µm [7,8]) for rods or trefoil channels, and at the single-mode fiber value (5µm) for multicore fibers, and we vary the waveguide density. We estimate C and β₀ as their values in a simplified object involving only two waveguides having the same geometry as the infinite array; they are derived from the average and difference of propagation constants of the first even and odd supermodes of the two-waveguide system, obtained by the finite-element method (FEM). From the study of specific cases we find that this overestimates C/β₀ by less than a factor of 2 with respect to its value for a waveguide array. Results are shown in Fig. 1 together with the few data from the literature.

 

Fig. 1. Coupling strength vs. channel density for arrays of multicore fibers (green line), trefoil channels of MOFs (red line), and silica rods (blue line); vertical bars correspond to the two polarizations. Dots indicate reported values for multicore fibers [11,12,14] (green), photorefractive patterns [16] (purple) and trefoil channels [this work] (red).

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This approximate analysis confirms that silica/air systems (red and blue lines) can provide much higher densities than silica/silica systems (green line and green dots), by more than one decade. They also show that arrays of trefoil channels are a good approximation of the unfeasible - at present - arrays of rods. This holds up to ~0.25µm-² where modes are no longer clearly obtained for the trefoil channels, while they are up to the physical contact for the rods. Nevertheless, it must be kept in mind that when C/β₀ approaches unity, its derivation fails together with the very signification of C. As a conservative figure, we shall then state that the best realistic design in fiber technology involves arrays of trefoil channels in the 0.1–0.2 µm^-2 range i.e. a potential pixel density of ~20 Mpixels/cm², with a coupling constant in the 5mm^-1 range i.e. millimeter-long devices. This respects fairly well the weak-coupling rule, with C/β₀<0.001.

In spite of these promising specifications, backed up by preliminary observations of single-channel transmission [22], correct operation may be precluded by two problems which appear with decreasing waveguide size i.e. with the increased impact of technological imperfections, (i) the decrease of channel regularity and homogeneity which can break the uniform coupling pattern of the channels, and (ii) the increase of roughness-induced propagation losses which can lower the throughput to ineffective values. Such requirements (homogeneity, low loss), are also needed for application of the CMT and hence for simple analysis of the presumably complex behavior of waveguide arrays. No data is available at present on these issues and realization of operating waveguide arrays remains a challenge. We have therefore checked on a test MOF whether optimally-small and optimally-dense trefoil channels could indeed operate as an efficient array of coupled waveguides.

3. Design and modeling of the test channel array

A trefoil channel pattern is defined by the type of hole lattice, the diameter of the holes d and the period of the lattice Λ. We select fibers involving well-mastered fabrication processes since, as we shall see, geometry control is of the essence. This implies a hexagonal lattice of holes and hence a honeycomb lattice of channels with a limited number of holes, here 27 in 3 successive rings giving a 3-fold pattern of 37 channels (Fig. 2). The channel array will then have a finite character. A d/Λ ratio close to 0.85 is chosen to preserve the hole circularity [27]. In approximate reference to the rod calculation [7,8], we design optimal-size channels at λ=1.55µm by stating that the inner radius r of the channel is ~λ/2n_silica. This completes the design of the test fiber for which Λ=4.36µm, d=3.71µm, r=0.66µm, and r/(λ/n)=0.62. Core size is slightly larger than the optimal value: the mode of individual channels given by FEM corresponds to a Gaussian waist of ~0.75 λ/n, while the optimum for cylindrical rods is 0.65 λ/n. The channel density 0.12 µm^-2 is also near to the optimal value.

 

Fig. 2. Schematics of the test MOF fiber. Channels are marked by red triangles.

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An extensive analysis of modes in this kind of fiber has already been performed [23], displaying the whole photonic band structure of the channel array. We focus in this study on the narrow highest band of “apex” modes, for which power lies in the trefoil channels; lower modes, which are separated from them by a large forbidden band, are considered as loss modes for our purpose. The drastic CMT simplification, required for a simple description of the power exchanges between channels, states that these modes can be approximated by combinations of the fundamental modes of individual channels, with a slow phase perturbation upon propagation described by C. In order to validate this point and get a good evaluation of C, we use a refined version of the two-channel calculation. For the test patterns, 37 CMT modes are expected, with propagation coefficients β_m=β₀-k_mC, where k_m are the eigenvalues of the pattern’s adjacency matrix [28]. C and β₀ are obtained by the fit between the {β_m} set of values given by the FEM calculation of modes and the {k_m} set given by the CMT. C=3.5 mm^-1 gives a good fit between the six highest levels. The identification of CMT and FEM mode shapes is also clear up to the fifth mode. This overall agreement, together with the fulfillment of the first CMT prerequisite (C/β~6.5×10^-4), justifies the use of CMT. Note that the coupling length L_c required for transferring power from a waveguide to its neighbor is ~1/2C, i.e. here 0.14mm.

The requirements of low loss and identical channels involve the following approximate specifications. We assume that the CMT model can tolerate a 1dB loss over the ~20L_c path over which useful effects are obtained - such a loss also corresponds to a correct insertion loss - so the maximum tolerable loss rate is ~3.5dB/cm. The issue of channel homogeneity is more intricate. The efficiency of power transfer from a waveguide to its neighbors changes when local variations (Δβ, ΔC) of the propagation or coupling coefficients are too large. From the structure of the equations and analytical solutions [29], efficient coupling requires Δβ/2C and ΔC/C to be less than unity. By evaluating variations due to simple distortions (hole diameter or position) in one- (for Δβ) or two-waveguide (for ΔC) systems, we find that distortions should not be larger than ~2% at λ=1.55µm; this requirement is much more stringent in the visible range, e.g. by a factor of 4 around 0.7µm.

4. The real test channel fiber: transmission measurements

The real test fiber is fabricated from pure silica by a double-stage stack-and-draw process. Optical or scanning electron microscopy images of cleaved sections (b,c) reveal a hole pattern very close to the ideal design.

 

Fig. 3. Cleaved sections of the test fiber: (left) optical microscopy, (right) scanning electron microscopy.

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Image analysis shows that deviations from the design pattern lie in the range of the accuracy of the microscopy (~2–3%), with the exception of the outer holes which are smaller than designed; this is somewhat expected due to their specific environment. Such measurements are of course blind to possible variations along the axis. In view of the results of the previous section, efficient inter-channel coupling can be expected, except perhaps for the outermost channels, at the nominal λ=1.55µm but possibly not at λ=0.675µm which is used for alignment and comparison purposes.

This is tested by injection into a section of channel fiber by a microlensed fiber. The injection waist has been checked by near-field scanning microscopy to be less than the inter-channel distance. Taking the power injected in the channel under the tip as unity we estimate that the power injected into its nearest neighbor is less than 25% and 3% into the secondnearest neighbor, so injection conditions are near to point injection. Injection was performed using a nanometric precision XYZ positioning stage, and with this setup we were able to observe long term stable injection over at least one hour without any adjustment. The fiber output is imaged on an IR camera through a microscope objective. Most experiments have been performed on fiber lengths >1cm, i.e. propagation paths >70L_c long enough to reveal the collective nature of the channel array by the spreading out of the power among all channels. This is indeed observed at λ=1.55µm. Various maps can be obtained depending on injection conditions and fiber length but propagation always involves many channels (see an example in Fig. 4(a)). On the other hand, at λ=0.675µm light propagates mostly in the injection channel (Fig. 4(b)); this is used for selecting this channel, prior to switching the wavelength to λ=1.55µm. A fit to CMT predictions is not possible at present since the output map can vary along the propagation path over sub-millimeter distances and we cannot measure fiber lengths with this accuracy; temperature variations could possibly help scanning these maps. We can only state that some output images are fairly similar to CMT maps or modes calculated by FEM on the ideal structure. On a more quantitative basis, the extension E of the power in the images can be defined as its mean square deviation from its average position, normalized to the inter-channel spacing. If the channels are efficiently coupled, E values for the FEM modes lie between 2 and 3, and CMT predicts that after point injection E varies between 1.5 and 3 upon propagation. If they are not, output and input maps are identical, and we expect E< 0.8. Experimental values at λ=1.55µm (1.5<E<3) agree with the assumption of efficient coupling, whereas at λ=0.675µm E (1.2<E<1.5) lies below the expected range, though above the no-coupling limit. This reduction of E also shows in complementary measurements of transmission for a fixed output channel and a varying input channel. It is due to a weakening of the power transfer efficiency caused by geometrical defects, as expected from the prediction of a much greater sensitivity than at the nominal λ=1.55µm. Another indication of to the deformation of the structure is given by polarization measurements. At λ=1.55µm a partial polarization of the global output along a particular axis is observed, regardless of the input polarization, whereas isotropy is expected for a perfect structure. This deviation can be attributed to a deformation of the actual channel structure favoring a particular direction, and FEM calculations show that a 1% ellipticity of the whole structure explains the anisotropy obtained. Efficient coupling is hence obtained at λ=1.55µm in spite of technological defects, but probably with a narrow margin.

 

Fig. 4. Intensity maps of the output of a 1.3cm-long channel fiber upon light injection in the center channel. In map (b) the hole structure is revealed by an additional white-light illumination of the output face.

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Finally the propagation losses have been measured by the cutback technique (Fig. 5 (left)). For each fiber length, transmission is optimized with respect to small variations of the input parameters. The propagation loss found (~1dB/cm) lies below the loss limit evaluated in the previous section. The coupling losses on injection and detection (~28dB) are high but not optimized. Comparison to other small-size waveguides is difficult since results are scarce and rather inhomogeneous (Fig. 5 (right)). On the whole, losses increase with decreasing waveguide radius probably due to the increasing influence of interface roughness [30]. Trefoil channels behave like rods, with ~1dB/cm, similar to the loss in etched semiconductor waveguides. Tapered fibers display lower losses, by about a decade, which suggest a possible progress.

 

Fig. 5. (left) overall transmission of setup versus length of channel fiber. (right) loss for various silica waveguides (rods [4], tapered MOFs [5,6], tapered SMFs [0], and trefoil channels [22, this work]) versus waveguide radius in λ/n units. Lines are guides to the eye. The variation of the waist for silica rods [8] is recalled by the black curve.

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5. Summary and perspectives

In search for a realization of the calculated array of optimally small and optimally dense coupled waveguides, we have designed the pattern of a microstructured fiber to form a regular array of trefoil channels. In spite of the high level of geometrical specifications required, this fiber drawn using present-day technology qualifies for our purpose. Most of the power is carried by the channels, the efficiency of the coupling is not hindered by technological defects, and losses are low enough to allow operation of inter-channel coupling applications. Therefore, channels act as efficient waveguides, and propagation patterns can be analyzed within the framework of “array of weakly-coupled waveguides” and hence calculated by the coupled-mode theory.

The perspectives of using such arrays in 2D parallel all-optical data-processing devices (computing, image processing…) can be assessed as follows. As a device, the channel array is compact, easy to manipulate, and produced in large quantities by inexpensive present-day technology. The problem of practical connections to both upstream and downstream devices could be solved using developing techniques for interfacing multi-core microstructure and standard optical fibers. Specifically, practical and stable techniques could be based on either the splice-free ferrule interface method described by Leon-Saval et al. [31], spliced or physical-contact connections to tapered multicore fibers, or complementary approaches such as the growing of photopolymer microtips on the end faces of single mode fibers [32]. Although the initial fabrication of such devices would require careful design, once integrated to a multi-core waveguide array, stable and robust coupling would be expected.

The propagation loss is acceptably low, and the bandwidth is large. The pixel density compares with what can be obtained in microelectronics and all-optical processing offers very fast responses. Now defining the possible device functions calls for further advances. The relationship between output and input, over a limited number of coupling lengths, requires a deeper analysis. More important, strategies for the optical control of this relationship are needed. Simple calculations, confirmed by preliminary experiments, prove that influencing propagation patterns by optical means requires powers beyond practical values, at least with basic nonlinear effects acting on pure silica. Nevertheless, proving that such an object is within the possibility of present-day technology opens up the way to further optimizations or migration to other systems.