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The optical property map calculated for the SF57 glass and shown in Figure 3.14 easily allows one to predict that a hexagonally stacked fibre (denoted as HF1 on the map) with Λ = 1.36µm andd/Λ = 0.45 would satisfy the dispersion requirements of zero dispersion and dispersion slope at 1550 nm, with γ as large as 468 W−1km−1. This fibre is used as a reference in order to check the convergence of the inverse design method. In this case, the free-parameters to optimise are Λ andd/Λ, whileDtandDslopet are both set to

zero. The inverse simulation was found to converge to the same structural parameters of HF1 in between 20 and 70 optimisation steps , depending on the chosen starting point. Note, as a comparison, that in order to obtain a map such as the one in Figure 3.14, more than 100 calculations are generally required. An example of the evolution of Obj

when two different initial conditions are chosen is shown in Figure 4.2(a), showing that the algorithm tends to converge with an exponential behaviour. In Figure 4.2(c) the results of similar optimisations are also reported, relating to the case of a hexagonally stacked fibre in which the interstitials between the stacked capillaries would not collapse during fabrication. This situation is likely to happen, for example, if vacuum is not applied during the fibre pull, and the resulting fibre structure, shown in Figure 4.2(b), is modelled by further assuming that the external circular shape with a radius of∼Λ/2 is retained by each capillary. The optimum structures in this case, whenDslopet = 0 and D_t is set to 1.5 and 5 ps/nm/km, require slightly larger holes and a smaller pitch than the structure without interstitials.

(a) (b) (c)

Figure 4.2: (a) Typical evolution of the parameterObj (b) Structure with interstitials

with (c) two optimisation results forDt = 1.5 andDt= 5 ps/nm/km.

While these examples confirmed a correct implementation of the algorithm, they also showed that values of the pitch smaller than 2µm would be required in order to take advantage of the favourable dispersion properties of the fibre. Such a small scale can only be reached, in ordinary hexagonally stacked HFs, with a two-step drawing process, in which the cane, resulting from the first draw is subsequently inserted into a larger jacketing tube and then drawn again. This process is fairly standard now for small- core silica fibres, but it still presents technical issues in the case of soft glasses, where

the repeated heating steps required are more likely to induce crystallisation, and the accurate control of the final dimensions is more difficult.

Other fabrication procedures for soft glass HFs which may overcome these issues have been recently suggested by other research groups, and consist of either stacking rods of two different glasses and then etching one glass away [52] or of using a die-cast process whereby the alloy steel die is removed by using an acid solution [53]. Alternatively, a fabrication approach which has been referred to as the Structured Element Stacking Technique (SEST) and combines the best features of both extrusion and stacking, has been recently implemented at the ORC [182]. The SEST technique, whose fabrication scheme is shown in Figure 4.3, requires the extrusion of 3 separate types of elements: several 7-hole cladding elements, one 6-hole central element, and one jacketing tube. The structured preforms obtained from extrusion are drawn into canes with an outer diameter of∼600µm and stacked into the extruded tube. At this point, the assembled preform can be drawn to the required size, allowing micron-sized pitches to be achieved, as required in order to control the overall fibre dispersion.

Figure 4.3: SEST technique, combining extrusion and stacking to fabricate soft glass

fibres with complex profile and small dimensions.

The inverse design procedure already explained for the case of a hexagonally stacked fibre, has also been applied to the design of an optimum SEST fibre. The separation between the holes in each extruded structure, Λ, and the relative holes size, d/Λ, are considered as the only two free-parameters, as the distance between the most external holes and the boundary of each hexagonal element is assumed to be fixed to Λ/2. The target dispersion slope at 1550 nm (Dslopet) was set to 0 ps/nm2/km, while Dt was

set to 2 ps/nm/km, in order to allow for a relatively broad region with small and anomalous dispersion. The best fibre, whose structure is shown in Figure 4.4(a) and whose dispersion and confinement loss are shown in Figure 4.4(b), was obtained for Λ =

1.36µm and d/Λ = 0.46. Note that the control of the total dispersion characteristics is achieved at the expense of a reduced nonlinearity: γ is 526 W−1_km−1_{, nearly 30% of the}

maximum achievable nonlinearity in this glass. The small scale of the structure and the limited number of holes produce a rather high confinement loss of ∼20 dB/m at 1550 nm. This can be reduced to∼ 1 dB/km, well below the bulk loss of SF57 by adding a second ring of stacked elements. Note that, from a theoretical perspective, this fibre is expected to support higher order modes. However, the high confinement loss of these higher order modes (nearly 5 orders of magnitude worse than the fundamental mode) and the large difference in effective index between them and the fundamental mode, leave us confident to predict that this fibre design is likely to be effectively single mode, in practice.

(a) SEST structure with 1 (top) and 2 (bottom) rings of stacked 7-holes elements

1.4 1.5 1.6 1.7 −10 −5 0 5 10 Wavelength [µm] D [ps/nm/km] 1.4 1.5 1.6 1.7 10−4 10−2 100 102 Wavelength [µm] Conf. Loss [dB/m] Λ = 1.36 µm; d/Λ = 0.46 D t 1 ring 2 rings

(b) Optimum structure: dispersion and CL

Figure 4.4: Optimum SEST structure.

Instead of adding a second ring of stacked elements, a different fibre design can be envisaged, requiring a more complex extruded element in the centre of the stack, but allowing for a better confinement loss to be achieved with only one ring of stacked elements. Its underlying concept is shown in Figure 4.5 where only one quarter of the structure is shown for symmetry reasons: an extruded element with 2 rings of air holes and whose Λ and d/Λ are designed to obtain the target dispersion characteristics is positioned in the centre of the fibre. This allows multiple 6-hole elements with much larger hole size to be stacked all around the central element, effectively improving the modal confinement. A geometrical consideration suggests that the pitch of the internal

(Λ_int) and external (Λ) elements are simply related by: Λint=

√

3 + 1

2√3 + 1Λ≈0.68Λ, (4.2) hence the number of free-parameters for this structure can be reduced to 3: Λ, the diameter of the internal holes (d1) and the diameter of the external holes (d2).

Figure 4.5: Improved SEST design: the final structure is shown on the right, while the

additional circles in the left figure have been drawn to help visualising the geometrical relationship between Λ and Λint.

The Downhill Simplex method has been applied also to this improved SEST fibre. The optimum structure, shown in Figure 4.6(b), was found for Λ = 2.29µm,d1/Λint= 0.46 and d2/Λ = 0.64. Its dispersion is shown by the red curve in Figure 4.6(a) together with the dispersion curves of all the other structures evaluated by the algorithm during the inverse search. Although the nonlinear parameter of this fibre is slightly smaller

1.45 1.5 1.55 1.6 1.65 −8 −6 −4 −2 0 2 4 6 8 Wavelength [µm] D [ps/nm/km] (a) (b)

Figure 4.6: (a) Dispersion curves of all the fibres evaluated during the inverse opti-

misation and, in red, the optimum solution; (b) best structure and 2 dB contour plot of its fundamental mode.

than for the ordinary SEST variant (γ = 464 W−1_km−1_{), the confinement loss is rather}

small, even with only one ring of stacked elements (CL = 0.026 dB/m).