5.3 Efficient generation of visible light
5.3.1 HF design for efficient supercontinuum generation in the visible
It is known that low and anomalous dispersion is key to an efficient SC broadening, as it enables the phase matching of the nonlinear processes broadening the spectrum to take place over a long fibre length. Optimising the fibre dispersion relaxes the need for high intensity pulses, allowing longer pulses to be used [217]. Efficient SC generation in the visible would therefore require one, or preferably two (see Section 3.3), ZDWs close to the pump, which in the following will be assumed to be at 532 nm (such as the one obtained by frequency doubling an Yb-doped fibre laser). Silica’s material dispersion at 532 nm is however strongly normal (∼-440 ps/nm/km) and a large amount of waveguide dispersion of opposite sign is necessary to obtain the required net-zero dispersion. One may wonder if any practical HF design exists, that is able to generate the desired dispersion characteristics. A literature search amongst fabricated fibres suggests that the shortest ZDW achieved so far in a HF is, to the best of my knowledge, 565 nm, obtained using a rather extreme cobweb design with a core diameter of ∼1
µm [64]. This section investigates whether (and how) this limit can be pushed to even shorter wavelengths.
It is well known from the literature that a fibre (or equivalently, a circular silica rod in air), tapered down to very small, wavelength-size diameters, can provide the required two ZDWs around 532 nm [212]. However, tapers are generally both short and fragile, and it would be desirable to obtain the same dispersion characteristics from a more structurally robust HF design. The study of the SC-HF in Section 5.2, on the other hand, seems to suggest that the shape of the core is fundamentally important to the maximum amount of waveguide dispersion obtainable: with a triangular and strongly concave core shape, it is not possible to achieve ZDWs shorter than∼600 nm. All these considerations prompted us to conduct a detailed study on how the minimum ZDW achievable in a silica waveguide depends on the shape of the waveguide itself.
Please note that this study could have been presented, in a normalised and more gen- eral way, by ignoring the material dispersion and then simply analysing the maximum waveguide contribution to the overall dispersion. However I have chosen the present approach, related to a specific application of silica fibres, because it links well with the study illustrated at the end of the chapter, presenting an alternative solution to the same design problem.
The influence of the shape and dimension of a silica rod in air on the dispersion of waveguides with small cross-section is simulated for regular polygons with a circular, hexagonal, square and triangular shape. The core shape of two realistic HFs are also studied: a hexagonal core with rounded angles and six additional thin struts, closely representing the core of a fabricated cobweb fibre [such as the one in Figure 5.8(b)]; and the core of the SC-HF analysed in the previous section and obtained from an SEM image. Figure 5.5 shows the results. Note that, in the case of the fabricated fibres,
L (a) L (b) L (c) L (d) D_L = 0.95 L (e) d (f)
Figure 5.5: Dispersion curves for small-scale silica rods whose shape is: (a) circular,
(b) hexagonal, (c) square (d) triangular and (e) hexagonal with struts. (f) shows the results for the core of the SC-HF analysed in Section 5.2.
neglecting the cladding is a reasonable approximation only in the wavelength region around the pump wavelength, where λ/Λ ' 1 and the field is highly localised in the core. For much longer wavelengths the field expands out of the core and the full fibre cross section should be analysed for accurate results.
Figure 5.5 shows that circular, hexagonal and square waveguides allow anomalous dis- persion at the pump wavelength and at the same two ZDWs at shorter and longer wavelengths to be achieved, provided the appropriate waveguide dimension is chosen. It was previously shown in Section 3.3 how this dispersion profile is desirable in order to achieve a flat and relatively pump-independent supercontinuum. Conversely, the trian- gular shape is unable to achieve two ZDWs equally spaced around the pump, and only allows one ZDW close to 532 nm to be obtained. The plots in Figure 5.5 indicate an influence of the core shape on the achievable waveguide dispersion, and they suggest that, for application to the visible SC generation targeted in this study, the waveguide shape must be carefully chosen. This is further demonstrated by the analysis of the two realistic cores. The hexagonal core of a cobweb fibre with 6 very thin struts, only wide 5% of the core dimension, can still provide anomalous dispersion at the pump wavelength, as shown in Figure 5.6(e), even though the two ZDWs are shifted to longer wavelengths if compared to the perfect hexagonal case. In contrast, as already observed, the waveguide dispersion of a triangular SC-HF is insufficient to compensate for the material dispersion at 532 nm [Figure 5.6(f)].
All of these results are summarised in Figure 5.6, which shows the anomalous dispersion region achievable as a function of the area of the central rod and for the different core shapes analysed. 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 Rod Area [µm2] Wavelength [ µ m] circular hexagonal square hex with struts triangular
1 1.1 1.2 1.3 1.4 1.5 Rod Area [µm2]
triang. with struts
Figure 5.6: Anomalous dispersion region achievable as a function of the core size and
One important conclusion is that the circular shape allows, amongst all the shapes considered, the shortest theoretical ZDW to be achieved. The minimum wavelength for which an overall zero group velocity dispersion can be achieved in a silica rod is∼0.46
µm, and it is achieved with circular waveguides of diameters around 0.55 µm. The remaining regular polygons provide an amount of waveguide dispersion which decreases with the number of symmetry axes. Furthermore, for a given symmetry (e.g. hexagonal or triangular), the maximum waveguide dispersion is decreased by the introduction of silica struts at the corners of the polygon, which increase the total concavity of the cross-section. All of these rules can be explained by considering that, for an optical mode concentrated in the core, the additional corners or struts extending out of it act like perturbations that reduce the cladding effective index. As a result the relative strength of the waveguide dispersion is also reduced.
To investigate further the influence of the struts on the achievable waveguide dispersion, a hexagonal core with 6 struts is simulated, and the strut thicknesst is varied from 5% to 10% and 15% the core dimensionL, as shown in Figure 5.7.
t
t = 0.05 L t = 0.10 L t = 0.15 LL
(a) 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.4 0.5 0.6 0.7 0.8 0.9 1 Core size L [µm] Wavelength [ µ m] strut thickness = 0.05 Λ strut thickness = 0.10 Λ strut thickness = 0.15 Λ (b)Figure 5.7: Influence of the strut thickness on the overall dispersion: hexagonal core
under analysis; (b) anomalous dispersion regions achievable.
The simulation results indicate that in order to achieve anomalous dispersion at 532 nm, care must be paid during fabrication in order to maintain the strut thickness to below 0.1 times L. Such thin struts may be difficult to achieve if the submicron core is obtained through tapering a larger core MOF, since some unavoidable hole collapse under the surface tension would tend to increase the strut thickness [212]. Although Leon-Savalet al. demonstrated SC generation in the visible, using 90 mm long MOFs tapered down to a core size of 500 and 700 nm [212], from the results in Figure 5.7 and from the SEM of their tapered cross section, one can conclude that their fibre’s dispersion was probably far from optimum.
An improved control over the strut thickness can be, in principle, achieved during the fibre drawing, provided active pressurisation is used in order to control the holes expan- sion. Therefore, based on the previous results, a cobweb fibre with a structure similar to that of fibres already fabricated at the ORC [see for example Figure 5.8(a)] and with a dispersion profile allowing two ZDWs around the 532 nm pump has been designed. Its structure, fundamental mode and optical properties are shown in Figure 5.8(b) and Fig- ure 5.8(c). (a) (b) 0.45 0.5 0.55 0.6 0.65 0.7 −200 −150 −100 −50 0 50 Wavelength [µm] D [ps/nm/km] 0.45 0.5 0.55 0.6 0.65 0.7 0.2 0.25 0.3 0.35 Wavelength [µm] Aeff [ µ m 2] (c)
Figure 5.8: (a) Example of a fabricated cobweb fibre; (b) designed cobweb fibre for
SC generation, with a core diameter of 0.55 µm (2 dB plot of the fundamental mode atλ= 532 nm, in red); (c) dispersion and effective area of the designed fibre.
Key to the achievement of the required dispersion properties are both the core diameter, which must be around 0.55µm, and the air-filling fraction, which needs to be very large, corresponding tod/Λ ∼0.95. At the short wavelengths targeted in this application the mode is extremely confined, and 3 rings of holes are sufficient to reduce confinement loss to less than 10−5 dB/m. The fibre is also strictly single mode in the whole wavelength range of interest.
5.3.2 Efficient white light generation in secondary cores of a photonic