Simulations using realistic cross-sections

Although the idealised fibre structure provides a simple way for estimating the principal performances of a fibre, it cannot determine those properties, such as birefringence, that result from structural asymmetries. Neither can it predict the exact location of surface modes inside the PBG, which are extremely sensitive to very fine details of the ring surrounding the core in which they are localised, as will be discussed in more detail in the next chapter. Therefore it can sometimes be necessary to simulate the performances of a structure as closely matched as is possible to the SEM of the real fibre’s cross- section. The process, already employed in several other projects reported in this thesis, generally involves multiple image-processing stages to eliminate noise, charging effects and to obtain a two-level bitmap, which is then approximated with splines and finally meshed.

Initial attempts at simulating the full structure of the fibre always predicted a PBG at much longer wavelengths than actually measured. A careful analysis of the various simulated fibres permitted us to attribute this mismatch to the limited image size. The 19 periods (Λ) along the diameter of a typical fibre are represented by∼700 pixels along the vertical dimension of the image; this limits the resolution of each individual strut to between 1 and 2 pixels, which is clearly insufficient to obtain accurate results and often led us to underestimate the correct value of d/Λ.

It was however verified that by increasing the magnification by 2-3 times as in Fig- ure 6.11(a) it is possible to determine the hole shapes and positions with sufficient ac- curacy. Although this approach implies that information about the fibre’s confinement loss is lost, since only part of the fibre is represented, the principal modal properties were found not to be significantly affected by neglecting the outer rings of holes. For example the PBG can still be accurately estimated by studying the percentage of power in the core (Pcore) of the air-guided modes. In Figure 6.11(b) we plot the calculated

Pcore of the structure represented by the yellow contours in Figure 6.11(a). The PBG

extends from nearly 1.3 to 1.6 µm. A number of SMs, which are different for the two polarisations and, as will be discussed in the next Chapter, are identifiable as dips in the

+1 pixel

Transmitted Power (dB) Measured

70 80 90 100

Power in the core (%) Simulated, Pol.1

1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 60 70 80 90 100 Wavelength [µm]

Power in the core (%) Simulated, Pol.2

(b)

−90 −80 −70 −60

Transmitted Power (dB) Measured

70 80 90 100

Power in the core (%)

Simulated, Pol.1 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 60 70 80 90 100 Wavelength [µm]

Power in the core (%)

Simulated, Pol.2

(c)

Figure 6.11: Modelling a PBGF from its SEM. (a) High magnification SEM of the

fibre, zoomed in around the core and, in yellow, the structure used in the simulations. (b) and (c) Measured spectrum of 0.68 m of the fibre and simulated percentage of power in the core for the two linearly polarised modes, corresponding to the yellow structure (b) and to a structure where the red boundary in (a) has been enlarged by 1 pixel (c).

plot, appear at the edges of the PBG, while only the fundamental and the higher order modes (not shown) are guided within its central region. This is in reasonable agreement with the measured spectrum of a short section of the fibre, shown at the top of the figure. However, due to the nature and localisation of the SMs, their exact position in the PBG is extremely dependant upon the thickness of the interface between the core and the photonic crystal. This dependence is demonstrated in Figure 6.11(c) where we plot the results of a simulation for a similar fibre, in which the thickness of the core boundary

has been expanded by just 1 pixel all around (which is within the resolution error of our SEM measurements). As a result, the SMs around 1.55 µm shift to longer wavelengths and disappear into the continuum of cladding modes, while other SMs emerge from the short wavelength edge and move towards the centre of the PBG.

This example (and similar work on other fabricated fibres) leads us to conclude that by using an SEM image with a strut resolution of at least 3-5 pixels it is possible to accurately predict the width and position of the PBG without having to determine the structural parameters via comparison with experimental data. However, it was found that an even higher image resolution would be required in order to accurately estimate also the position of SMs within the PBG. In order to confirm this, a state-of-the-art scanning electron microscope was employed to obtain images of the previous fibre. This allowed SEMs with a pixel resolution of 3072×2304 to be acquired. One example of the structural contours obtained from such high definition images, to be subsequently used in FEM simulations, is shown by the red lines in Figure 6.12.

(a) (b)

(c) (d) (e)

Figure 6.12: PBGF modelling from a high definition SEM: (a) SEM of the full struc-

ture and (b) overimposed in red the structure to be simulated. (c), (d) and (e) show details of the fibre’s cross section and, in red, the contours of the simulated structure

The high image resolution, its good contrast and the absence of charging effects, achieved as a result of the meticulous work and repeated trials by Dr. M. N. Petrovich, results in the unequivocal representation of each strut in the cladding (∼170 nm wide) by on average 7 pixels, while the thinner core boundary is constituted by∼4 pixels. The sim-

(a) 10−4 100 104 Confinement Loss [dB/m] Pol.1 Pol.2 70 80 90 100 Wavelength [µm]

Power in the core (%)

Pol.1 Pol.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 −2000 −1000 0 1000 2000 Wavelength [µm] D [ps/nm/km] Pol.1 Pol.2 (b)

Figure 6.13: PBGF modelling from a high definition SEM: (a) Poynting vector of

the simulated fundamental air-guided mode; (b) simulated optical parameters for both polarisations.

ulation results for this structure are shown in Figure 6.13. A mesh of 105000 triangular elements, denser in the core area was used, and a scan over 100 wavelengths was con- ducted, requiring nearly 4 hours of computation time on a dual-core PC with 2.5 GHz processors and 8 GB of memory. For both polarisations of the FM, anticrossing with SMs are only present near the PBG edges, in accord with the transmission measure- ments. The simulated confinement loss of∼10−4 _{dB/m is orders of magnitude smaller}

than the measured loss of the fibre, indicating that scattering loss is the predominant loss mechanism. The percentage of power in the core, dispersion and birefringence could also be obtained from the same simulation.

Despite these encouraging results in the simulation of the wavelength position of SMs, the resulting PBG was smaller and shifted to longer wavelengths than the measured one, suggesting that the actual hole size had been underestimated (see Figure 6.6 and related discussion). This was attributed to the presence of a gold coating layer, applied to reduce charging effects during the SEM acquisition, and whose thickness, ranging

between 5 and 20 nm and generally difficult to estimate accurately, can influence the simulation results. Additional simulations on samples with thicker coating were then performed, and as expected the resulting PBG was even narrower and centred at longer wavelengths.