Accuracy

In order to validate the method, a number of convergence tests were run for the main free-parameters of the algorithm: the size of the finite element mesh and the PML parameters. As was mentioned in the previous Section, it is generally a widespread practice to fix α= 2 andb= 1 in the PMLs (see for example Ref. [152]). Therefore the parameters whose convergence will be analysed in the following are the maximum value of the imaginary part of si,amax, and the size of the finite element mesh.

The general practice to validate a numerical method would be to test its results against a known analytical result. However as has been previously observed, no analytical results are available for MOFs; furthermore comparisons with the analytical solution of, for example, a step index fibre would be inadequate for testing the imaginary part

of the propagation constant. It has become a common practice therefore, to validate a numerical code by comparison with the results obtained with the method which is recognised as the most accurate for MOF modelling: the multipole method. The most employed ‘benchmark’ for such comparisons is typically the fibre with a single ring of 6 equally spaced holes studied at λ = 1.45µm by White et al. [9] and presenting Λ = 6.75 µm and d= 5 µm. The silica refractive index is assumed to ben= 1.45.

Figure 2.2 compares the effective index calculated by the MM (n_eff= 1.445395345 +

j3.150×10−8_{) with the results obtained with the FEM for increasing values ofa} max. A

π/2 sector with BCs for the p= 4 mode, and a mesh of 28328 triangles (56907 DOFs), denser in the core, have been used for the FEM. The real part of n_eff is almost unaffected

0 5 10 15 20 1.4453940 1.4453945 1.4453950 1.4453955 amax Re(neff) 5 10 15 20 1 1.5 2 2.5 3 3.5x 10 −8 amax Im(neff) FEM MM FEM MM

Figure 2.2: Effective index dependence on amax for the fundamental mode of the

fibre studied in [9], calculated with the FEM (blue); the MM result is shown in red for comparison.

by the change ofα, while a good convergence of=m(n_eff) is observed forα larger than 5. A value of α= 10 will be therefore employed for all future simulations.

The n_eff convergence as a function of mesh density is shown in Figure 2.3 for the same fibre and with α = 10. Both <e(n_eff) and =m(n_eff) are affected by an increase in the number of points, but for a number of triangles greater than ∼40000 they both seem to have converged to a stable value. For this converged value the difference in the real part of neff between the two methods lies in the 6th significant digit (∼1×10−6), while

the error in the imaginary part is around 1×10−10_{, corresponding to a variation of less}

than 1% in the calculated confinement loss.

Finally, for an additional confirmation of the good agreement between the two methods, a comparison between the first ten modes of the fibre (see Figure 2.4), calculated with the FEM and with the MM is shown in Table 2.4. Note that, while the MM is able to correctly predict the modal degeneracy, due to the use of local cylindrical coordinates centred about each hole, in the FEM a small false birefringence is observed for the modes in class p = 3,4 and p = 5,6. This ‘numerical birefringence’ is intrinsic in the FEM procedure, where the generated meshes do not reflect the symmetries in the structure;

20000 40000 60000 1.445380 1.445385 1.445390 1.445395 1.445400

Mesh size (Number of triangles)

Re(neff) 20000 40000 60000 2.9 2.95 3 3.05 3.1 3.15 3.2 x 10−8

Mesh size (Number of triangles)

Im(neff)

FEM MM

FEM MM

Figure 2.3: Effective index dependence on the mesh size for the fundamental mode

of the fibre studied in [9], calculated with the FEM (blue); the MM result is shown in red for comparison.

FEM Class MM Class

<e(neff) =m(neff) p <e(neff) =m(neff) p

1.44539443 3.172×10−8 ₄ 1.445395345 3.150×10−8 _3,4 1.44539422 3.181×10−8 ₃ 1.43857915 5.352×10−7 _{2 1.438585801 4.986×10}−7 ₂ 1.43844140 9.666×10−7 ₆ 1.438445842 9.929×10−7 5,6 1.43844094 9.659×10−7 5 1.43836265 1.400×10−6 _{1 1.438366726 1.374×10}−6 ₁ 1.43038897 2.135×10−5 8 1.430175 2.220×10−5 8 1.42994844 1.577×10−5 ₄ 1.4299694 1.577×10−5 _3,4 1.42994834 1.578×10−5 ₃ 1.42925107 8.796×10−6 7 1.429255296 9.337×10−6 7

Table 2.4: neff of the first 10 modes of the fibre in Ref. [9]: comparison between FEM

and MM.

however for the mesh considered here (28328 elements) it is already as small as 1 to 5×10−7, and it can be further reduced by increasing the mesh size [144].

An agreement in excess of 1×10−5 _{is generally observed for <e(n}

eff) for all solutions

other than for the seventh eigenvalue, for which it is considerably larger. By analysing the results of a similar comparison between SMT and MM [143] though, a similar dis- crepancy for the same eigenvalue emerges. In this case the result of the SMT (<e(n_eff)= 1.430408204) is much closer to the FEM result; this seems to suggest a possible unin- tentional error in the determination of the eigenvalue for the MM method. The order of magnitude of =m(n_eff) agrees in all cases, and the errors in its significant digits are in the order of a few percent. The level of accuracy found for our implementation of the FEM is therefore in line with that of other numerical methods. For all the studies reported in this thesis it is well above the accuracy that can be obtained in practice when modelling fabricated fibres.

(a) p= 4 (b) p= 3

(c)p= 2 (d)p= 6

(e)p= 5 (f) p= 1

(g) p= 8 (h)p= 4

(i) p= 3 (j)p= 7

Figure 2.4: Poynting vector and vector plots of the transverse electric field of the first