Bend loss

In this section, the critical bend radius of the fundamental mode (RFM_c ) is calculated for each of the fibres in Fig. 5.1 using the numerical techniques described in Chapter 3. The results of these calculations are plotted in Fig. 5.4. As described in Section 4.5, RFM_c is evaluated for each fibre considered by calculating the bend loss for several values of bend

250 300 350 200 200 150 150 100 100

Lambda [microns]

d/Lambda

Figure 5.3: Contour plot generated from the data in (a) showing effective mode area inµm2_{as a function}

of Λ andd/Λ.

radius above and below RFM_c . The value of RFM_c is then extracted by fitting a curve through these points, as demonstrated in Section 4.5.3. The number of calculations performed for any given fibre thus depends on the number of points required to ensure that a fitted curve is representative, which is usually somewhere between 4-6. However, towards small values of Λ and large values of d/Λ, the bend loss becomes a sharp function of bend radius and a greater number of points are required to accurately determine RFM_c (This is discussed further in Section 5.3). Note that in Fig. 5.4, the critical bend radii of the smallest mode area fibres with d/Λ = 0.5 are not shown. These calculations have been omitted for two reasons: (1) the calculation of RFM_c is time consuming for these fibres as described above and (2) the critical bend radii of these fibres are small, each less than 2 cm, and so the

bend loss is not a concern from a practical standpoint.

Figure 5.4: Points show predicted values of the critical bend radius of the fundamental mode (RFMc ) as a

function of the hole-to-hole spacing, Λ, for a range of holes 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_{+b, drawn to guide the eye. The dotted line}

marks the position of RFMc = 15 cm.

Fig. 5.4 shows RFM_c for a selection of the fibres considered here at 1064 nm, which range in AFM_eff from 56 to 410 µm2_{. For the entire range of fibres, R}FM

c ranges from below 2 cm to as high as 45 cm. However, only those fibres with practical levels of bend loss (RFM_c < 15 cm) are indicated on Fig. 5.4. This plot shows that the bend loss improves (i.e. RFM_c decreases) with decreasing Λ and increasingd/Λ. These facts demonstrate that the relationship between bend loss and the fibre structure in holey fibres is similar to that of conventional fibres, in which the bend loss can be improved by decreasing the core size

250 300 350 200 200 150 150 100 100

Lambda [microns]

d/Lambda

Figure 5.5: Thick black lines show contours of the critical bend radius of the fundamental mode (RFMc )

in cm (generated from the data in Fig. 5.4) as a function of Λ and d/Λ. The dashed red and black line indicates the RFMc = 15 cm contour line. The graded colour contours correspond to A

FM

eff inµm2, repeated

from Fig. 5.3.

and/or increasing the numerical aperture. This is further discussed in Section 5.7, in which the bending losses of holey and conventional fibres at 1064 nm are compared in detail.

As before, a contour plot of RFM_c as a function of Λ and d/Λ is constructed in order to better visualise the relationship between the fibre structure and the bend loss (note that the accuracy of the fit that is used to construct this contour plot is≈2−3%). Contour lines of RFM_c in cm are shown by the thick black lines in Fig. 5.5, which are superimposed onto the contour plot of AFM_eff, repeated from Fig. 5.3. As in Fig. 5.4, this contour plot demonstrates that the bending losses increase towards larger values of Λ and decrease for larger values of d/Λ. By plotting the AFM_eff and RFM_c together in this way it becomes easy to visualise

how the AFM_eff and RFM_c both change with respect to the structural parameters. For example, Fig. 5.5 demonstrates that for a given value of d/Λ, both AFM_eff and RFM_c increase with Λ. Similarly, for a fixed value of Λ we see that both AFM_eff and RFM_c decrease as the relative hole size is increased. Recall that in Section 5.1 the upper limit for tolerable bend loss was defined as a critical bend radius of 15 cm (shown on Fig. 5.3 by the dashed red and black contour line). By using this definition, the range of fibres that possess tolerable levels of bend loss are located above the RFM_c =15 cm contour line in Fig. 5.5. The largest mode area fibre in this range has AFM_eff ≈320µm2. However, it can be seen that for any given value of AFM_eff, the bending losses improve as the relative hole size increases. This demonstrates that the optimum way of creating large-mode-area holey fibres at 1064 nm, with low values of bend loss, is to use large values of Λ and d/Λ. (This is intuitive, since it is equivalent to using a large core and a high NA in a conventional fibre.) If this relationship holds true for Λ>14 µm, one can imagine following the contour lines of AFM_eff >320 µm2_{, across and} up to larger values of Λ and d/Λ, into a region of parameter space where RFM_c < 15 cm. However, as discussed in Chapter 1, the number of modes supported by a holey fibre increase with increasingd/Λ. It is therefore essential to know the position of the boundary between single-mode and multi-mode guidance on Fig. 5.5 before additional structures are considered. This is explored in the following section.