Effective mode area

In this section we investigate how the structural parameters Λ and d/Λ influence the ef- fective area of the fundamental mode of holey fibres at 1064 nm. In the first of these studies presented here, the range of fibre parameters evaluated is chosen with the aim of

encompassing single-mode structures with 80µm2_<

∼AFMeff <∼400µm2, that are also practical to fabricate. (The fabrication of holey fibres is discussed in Section 1.2.4.)

The minimum value ofd/Λ is chosen to be 0.2 due to the fact that the modal properties of large-mode-area holey fibres, such as AFM_eff, become increasingly sensitive to the fibre structure towards small values of d/Λ and this reduces tolerances during fabrication [31]. Moreover, we find that values ofd/Λ<0.2 result in impractically large bending losses (see Section 5.1.) The maximum value ofd/Λ is determined by considering the upper limit on single-mode operation. Although there is much debate over how to determine the number of modes present in a holey fibre, with the cut-off for single-mode operation quoted for values of d/Λ ranging from 0.4 to 0.45 [142, 17, 143, 144, 145, 146, 147], all reports agree that holey fibres are multi-mode for values ofd/Λ>0.5 at 1064 nm. As a result, structures with 0.2< d/Λ<0.5 are studied here. The range of values for Λ is then determined only by the range of effective areas required. In this section I choose 7µm<Λ<14µm to take a first look at the range of effective mode sizes this creates.

Fig. 5.1 shows a plot of effective mode area for the fundamental mode (AFM_eff) at 1064 nm as a function of Λ, for 7µm < Λ < 14µm for 4 values of d/Λ: 0.2, 0.3, 0.4 and 0.5 at 1064 nm. For the range of structures considered here, AFM_eff ranges from 56 to 410 µm2_. This plot shows that the mode size can be enlarged by increasing Λ, which acts to increase the core size, or by using smaller holes, which acts to lower the numerical aperture. Note that different fibre structures can result in the same values of AFM_eff. For example, an effective mode area of 155µm2 _{can be achieved with Λ = 8.0µm,d/Λ = 0.2 and with Λ = 12.0µm,}

d/Λ = 0.5, as shown in Fig. 5.1. The intensity profile of the fundamental mode for these two fibres is shown in Fig. 5.2 (a) and (b) respectively. These modal profiles show clearly that while these two fibres share the same value of effective mode area, the mode shapes are significantly different, reflecting the geometry of each fibre. In Fig. 5.2 (a), the mode extends past the first ring of air holes and as a result is filamented in shape. In contrast, the mode in Fig. 5.2 (b) is well confined to the core and presents a neater, more Gaussian-like mode shape. These differences arise from the different value of the effective cladding index (nFSM) in each fibres. For the fibre in Fig. 5.2 (a) nFSM = 1.448697, and for the fibre in Fig. 5.2 (b) n_FSM = 1.448135 (see Fig. 6.1 (b) in Chapter 6), which corresponds to a numerical aperture (NA) of 0.052 and 0.066 respectively. (the refractive index of silica is taken from the Sellmeier equation to be 1.449631, as described in Section C). In the fibre in Fig. 5.2 (a), the low NA allows the light to penetrate further into the cladding region

Figure 5.1: Points show the predicted effective mode area of the fundamental mode (AFMeff) as a function of

the hole-to-hole spacing, 7µm<Λ<14µm, for a range of relative hole sizes: 0.2< d/Λ<0.5 at 1064 nm. The solid line for each value ofd/Λ is a fitted curve of the formy=ax3_+bx2_{+cdrawn to guide the eye.} than in the fibre in Fig. 5.2 (b), which has a higher NA. As a result, the mode areas of the fibres are equal, despite the fact that the region of silica inside the inner ring of holes (i.e. the core) is smaller for the fibre in Fig. 5.2 (a) than in Fig. 5.2 (b).

In order to better visualise the relationship between the fibre structure and the mode size of the fundamental mode, I have used the data in Fig. 5.1 to create a contour plot of AFM_eff as a function of Λ andd/Λ, which is shown in Fig. 5.3. (Note that the accuracy in the fit used to construct this contour plot is ≈2−3%). This plot shows how the mode area increases with increasing values of Λ and decreasing values ofd/Λ. It can also be seen that the mode area increases more rapidly with decreasing d/Λ towards smaller values of d/Λ.

(a) (b)

Figure 5.2: Modal intensity profile at 1064 nm for a holey fibre with (a) Λ = 8.0µm,d/Λ = 0.2 and (b) Λ = 12.0µm,d/Λ = 0.5. Contours are separated by 2 dB.

This results from the fact that the modal field becomes less affected by each individual hole as d/Λ decreases, allowing the modal field to penetrate further into the cladding. By tracing the path of any given contour line on this plot it can also be clearly seen that a wide range of structures can be used to create fibres with the same mode area. However, as demonstrated by the above examples (shown in Fig. 5.2 (a) and (b)), each different structure that results in the same mode area will correspond to a fibre with a different mode shape and a different NA. (The effect of the fibre structure on mode shape, bend loss and modedness is studied in detail for AFM_eff = 190 µm2 _{in Section 5.3). The differences in} mode shape and NA may influence both the number of modes guided by the fibre and the associated bending losses. The next step in this study must therefore be to determine the bend loss and the modedness of each of the structures considered here in order to determine the best way of creating single-mode fibres with the lowest bending losses for a given mode area.