We report the first analytical description of the drawing of fibres with holes that does not require knowledge of the viscosity (or temperature) of the glass in the furnace. The model yields expressions for the size of a hole that is isolated from other holes and small compared to the outer diameter of the fibre, but includes the effects of surface tension, pressurisation and arbitrary viscosity profiles. The effect of viscosity is represented by the fibre draw tension which, unlike viscosity, can readily be measured in practice by the fibre fabricator. The model matches experiments without recourse to any adjustable fitting parameters.
© 2013 OSA
Photonic crystal fibres (PCFs) contain air holes that run along the length of the fibre. The holes give the guided light properties that can be difficult or impossible to obtain in fibres of more conventional design. The fibres are drawn in a furnace from preforms that also contain holes. In contrast to features in ordinary solid fibres, these holes can profoundly change in size (and indeed in shape) relative to the fibre as a whole, under the influence of surface tension, glass viscosity and pressure differences. To produce PCFs of given designs it is therefore important to understand how such parameters affect the final holes.
Several published treatments of this problem range from detailed numerical studies of complex multi-hole structures to simplified analytical models whose strict validity is limited but which give insight [1–6]. For example, Fitt et al. reported an expression for the radius of a small hole in a capillary , assuming a slender draw zone and negligible inertia and gravity but including the effects of glass viscosity, surface tension and a pressure difference between the hole and the atmosphere.
However, these studies suffer from an important drawback from the experimenter's point of view: the results are expressed in terms of the viscosity of the glass in the furnace. Unfortunately the viscosity along the glass body is unknown to the fibre fabricator, not even as a single parameter along a uniform hot-zone length, let alone in its full form as a distribution μ(x) along the preform/fibre axis x. Thus authors typically make assumptions about the temperature T of the glass and deduce the viscosity from the material’s reported μ(T), or alternatively fit μ as a free parameter to their experimental data. The most thorough experiments included careful measurements of the temperature distribution T(x) along a preform in the furnace and extrapolated to the process temperature of silica glass, and use of the best available published information for μ(T) in silica .
We show that knowledge of viscosity can be completely eliminated from the analysis, at least in the case of a hole that is isolated from other holes and small compared to the outer diameter of the fibre. Instead we express the hole size in terms of the draw tension F, which is not only knowable but is routinely logged during fibre fabrication. Enough assumptions and approximations are made to render the problem analytical, while still representing the effects of surface tension, pressurisation and arbitrary viscosity profiles. The result is equivalent to Fitt et al's small-hole result, but for the first time allows direct comparisons with experiments based entirely on known experimental parameters. Indeed, the only variables relating the final and initial hole sizes are the draw stress, the draw-down ratio and the hole pressure.
We verify that our results are a good approximation for isolated holes drawn in real fibres by measuring the final diameters of three dissimilar holes. These measurements confirm our key finding, that draw stress in the fibre governs hole size irrespective of how the stress is changed. Furthermore the measured hole sizes match the predictions of our model with no introduction of any adjustable fitting parameters.
Previous reports took a general fundamental starting point such as the Navier-Stokes equation of fluid mechanics, and carefully applied various simplifications to deduce the answers of interest [2–6]. Instead, to avoid the full complexity of these equations, we start with a small number of much-simpler expressions describing individual effects such as hole collapse under surface tension and fibre drawing under an elongating force. We adopt approximations that were found allowable in the previous studies: we neglect inertia, gravity and temperature differences transverse to the draw direction x; we assume a slender neck-down region; and we assume incompressible glass with a known temperature-independent surface tension. We also restrict our study to the drawing of a fibre from a preform containing a single hole that is small compared with the outer diameter – the simplest system capable of representing the effects of viscosity distribution, surface tension and applied pressure. This approach gives a derivation that can be more easily followed by the non-mathematician while still leading to a non-trivial result.
The hole will change size when the preform is drawn. Firstly, all transverse dimensions decrease as the fibre is drawn down in size. Secondly, the hole is subject to transverse forces due to surface tension, viscosity and applied pressure differences. In the first case we consider drawdown in the absence of transverse forces (i.e., as if the hole is a solid glass feature in the preform), and in the second case we consider heating in the absence of drawdown. In a small time interval, the small diameter changes due to both can be added to form a differential equation that can be solved.
A solid preform of radius r1 is fed into a furnace at the feed rate uf and a fibre of radius r2 is drawn out at the draw speed ud, all in the direction of positive x, Fig. 1 . The furnace gives the glass a viscosity distribution μ(x), defined for all x by having infinite values outside the furnace. We assume the drawing process has reached a steady state, with a constant draw tension, and that volume of glass is conserved. The invariance of glass flux through transverse planes at arbitrary x relates the longitudinal glass speed u(x) to the local radius r(x) at x
The force F required to induce a velocity gradient du/dx along an extending viscous thread is 
By considering the elongation of a cylindrical slice of length l at x we can equate du/dx to the strain rate (1/l)dl/dt, differentiate the volume of the cylinder to relate it to the rate of change of fibre radius dr/dt, and finally use Eq. (1) to obtain the rate of change of fibre radius with x
For the case of a preform containing a hole we will use this result in two ways. Firstly, Eq. (5) describes the drawdown of the incompressible glassy portions of the fibreEq. (2) is generalised so that r values in Eqs. (2) and (3) are defined byEqs. (1) and (4).
Secondly, Eq. (5) also describes the variation of hole radius R in the absence of deforming forces, under the action of drawdown alone:
2.2 Transverse pressures
Surface tension γ at the hole boundary exherts an inward pressure pst according to the Young-Laplace equation
If the hole is changing size, viscosity μ causes an effective pressure pv that opposes the change in size. This effective pressure (defined positive if acting inwards) is8] expressed in terms of rate of change of R. If the hole is expanding (dR/dt > 0) viscosity acts like an inward pressure (pv > 0) whereas if the hole is shrinking (dR/dt < 0) it acts like an outward pressure (pv < 0). Again we can neglect viscosity at the outer boundary of the preform (more precisely, the lack of viscous forces due to the glass that isn't there between the outer radius and infinity).
Finally there may be an ordinary hydrostatic pressure difference applied between the pressure pH of the gas in the hole and the atmospheric pressure pa outside, with a net outward pressure of
Again neglecting inertia because all accelerations are small, these pressures must be balanced. The resulting equation gives the rate of change of hole size with x using Eq. (1), as we did with Eq. (5)
This equation can in principle be solved given the viscosity profile μ(x), with the drawdown profile ρ(x) obtained by solving Eq. (6), see Section 2.4. However, if instead ρ is chosen to be the independent variable with the aid of Eq. (6) we obtain
Equation (14) is a first-order linear differential equation for hole radius R(ρ) as a function of local drawdown ratio, which varies from ρ = 1 at the preform (x → −∞) to ρ = ρ 0 = r2/r1 at the final fibre (x → ∞). It can be solved by the integrating factor method to giveEqs. (15) and (17) is the draw stress in the preform.
The integral in Eq. (15) does not have elementary solutions in general. However, it can be expressed in terms of the error function erf or the imaginary error function erfi, depending on the sign of EEq. (18) can be dropped. erf is well-known from the theory of the gaussian statistical distribution. erfi is less well-known, though more important here because it describes the case where the hole has positive pressure. Both are available as built-in functions in computer algebra systems such as Mathematica , making the evaluation of Eq. (15) straightforward and quick.
Setting ρ = ρ 0 in Eq. (15) give the final hole size in terms of parameters that are readily known. In particular, it is not necessary to know anything about (or make assumptions about) the viscosity profile μ(x) along the glass body. This is because viscosity determines the dynamics of hole-size changes and fibre drawing in the same way. The relationship between the hole sizes in the preform and the fibre Eq. (15) depends only on the three easily-measurable parameters of draw stress, draw-down ratio and hole pressure, given that surface tension can (despite some contrary evidence ) be taken to be a known material constant.
Insight is gained by subtracting copies of Eq. (15) for two holes of different initial radii R0,1 and R0,2 to find their size difference in the final fibre (or indeed at intermediate points) without the integral termEq. (17), since 1 − ρ2 ≈1 at the fibre. Although pressurisation is widely used to stabilise the absolute hole size when drawing PCFs despite the collapsing effects of surface tension, according to Eq. (19) it causes holes of different sizes to diverge exponentially in size according to the ratio of the pressure difference to the draw tension. The extent to which the sizes diverge reduces with increasing draw stress (i.e., for lower temperatures, shorter hot zones or faster processes).
2.4 Comparison with previous results
Our Eq. (15) for hole size R in terms of local drawdown ratio ρ completes the solution if the profile of hole size along the preform neck within the furnace is not required. Usually only the hole pattern in the final fibre (ρ → ρ0, x → ∞) is important - or perhaps the hole patterns at intermediate points but not the values of x where they occur, for example to check whether the hole will collapse completely or explode part way along.
However, we may wish to locate the hole patterns at particular values of x, or compare our results with those reported elsewhere. It is then necessary to find how the drawdown ratio ρ depends on x by solving Eq. (6), which requires knowledge of the viscosity profile μ(x) along the glass bodyEq. (15) to give R(x).
The hot zone of the furnace is often assumed to produce a uniform viscosity μ along hot-zone length L, with infinite viscosity elsewhere . If the hot zone lies between x = 0 and x = L, Eq. (21) becomesEq. (16) to the viscosity. Equation (22) becomesEq. (15) to be expressed as a function of x. Making both substitutions, Eq. (15) matches Fitt et al's small-hole result (the un-numbered equation between Eqs. (43) and (44) in ), confirming that our less-fundamental starting point is consistent with the reasoned approximations used in that paper.
2.5 Normalised expression
Equation (15) is simple enough, and fast enough to evaluate using Eq. (18), to be readily used for quick parametric studies of hole size behaviour when drawing PCFs despite the approximation of a single isolated hole that was used to derive it. The number of degrees of freedom is reduced by introducing further normalised parameters. The hole radius R is normalised by the radius the hole would have had in the absence of surface tension and pressurisation - if the hole had instead been a solid glass element such as a doped core in the preform:Eq. (15) becomes
3. Example calculations
Our expressions reproduce well-known trends in behaviour; for example, that holes shrink due to surface tension, which can be opposed by applying an over-pressure in the holes. However, previously-reported behaviour with furnace temperature, process speeds, hot-zone length etc is subsumed within the effect of draw stress in our model. An example of the effect of draw stress S is given in Fig. 2 for certain final drawdown ratio ρ0 = 0.01 and pressure P = 0.6. (In these plots, a normalised hole radius of a = 1 means that the hole has reduced in size in direct proportion to the solid-fibre drawdown ratio ρ0.) As expected, decreasing stress allows more time for surface tension to destabilise the hole. As stress decreases the final hole radius decreases and the hole eventually closes altogether. However, for very small stresses the hole stays open and indeed quickly enlarges. This is because the chosen value of P exceeds 1/2, at which value p0 in Eq. (27) exceeds the surface tension pressure Eq. (9) in the preform. The hole therefore initially expands as the preform enters the furnace. If stress is low (a slow hot process), drawdown never overtakes this expansion as it does at higher stresses.
The effect of the exaggeration parameter E in Eq. (19) is demonstrated in Fig. 3 . For a given drawdown ratio ρ0, a given target hole size a0 can be produced by a range of draw tensions and hole pressures (as can be seen in Eq. (29) for S(E) with the required a0 as a constant). For example, Fig. 3(a) is the locus of P and S values that give a0 = 0.8 when ρ0 = 0.01 and E is varied. For high stresses a negative pressure is needed, because surface tension is not enough on its own to shrink the hole that far in the time available. Nevertheless, any of the plotted (P, S) combinations will yield the desired a0. However, different stresses have very different effects on a co-existing hole of slightly different size in the preform, according to whether E is small or large. In Figs. 3(b)-3(d) the final sizes of a reference hole are all the same. However, drawing under high stress and E = 0.2 exaggerates an 5% initial difference in hole radii in the preform to an 8% final difference in the fibre. In contrast, drawing under low stress and E = 3 exaggerates the same initial difference to a 125% final difference in the fibre.
Photonic crystal fibres are usually made by a two-step drawing process. First the preform is drawn to a stiff cane, and then the cane is drawn down to the fibre. This allows additional materials (perhaps an outer jacket) to be included, but it also helps to control the hole pattern in the fibre. This is illustrated in Fig. 4(a) for a two-step draw where the first step reduces a 12.5 mm diameter preform to a 1.25 mm cane, and the second reduces the cane to a 125 µm fibre. Figure 4(b) is the equivalent plot for a one-step draw straight from preform to fibre. In each step the applied draw stress is equivalent to a draw tension of F = 100 grams-force in the fibre, and the applied pressure is that required to preserve the normalised hole radius for a final hole diameter of 0.7 µm, with γ = 0.3 Jm−2 for silica. The two-step process converts a 10% smaller hole in the preform into a 10.6% smaller hole in the fibre - the ratio of hole sizes is barely changed. In contrast, a draw to the same target hole size in one step causes the 10% smaller hole to close completely.
The two-step scheme works because it allows the fabricator to apply a greater stress at the wide end of the glass body (fed preform or cane) for a given stress at the narrow end (drawn cane or fibre), the latter being limited by the tensile strength of the glass. In our example, the glass from ρ = 1 to ρ = 0.1 is drawn at a 100 × higher stress in the two-step process than in the one-step process, greatly reducing the scope for hole distortions that are induced by surface tension and pressurisation. In principle even greater advantage is gained by drawing in three or more steps, though with diminishing returns, and the increased handling of the glass causes problems of its own. The limiting case of increasing the number of steps is a configuration like that used for fibre tapering, where the glass body can be drawn at the breaking stress for all ρ if the temperature (hence viscosity) and strain rate Eq. (4) are constant. The optimum way to minimise hole distortions is therefore fibre tapering , though only for a very limited length of final narrowed fibre.
4. Experimental tests
We drew fibres from a silica cane containing three isolated holes of differing diameters 150, 203 and 263 µm and separated by 2.6 mm from each other, Fig. 5 . A preform was assembled from this cane, with a surrounding jacket tube to increase the outer diameter. The preform was made specifically to test our model, and was not designed to make fibres that guide light. In each draw, nitrogen gas with a given pressure po above atmospheric pressure was applied to all three holes, subjecting them to identical un-normalised draw parameters except for the different initial hole radii R0. The fibres were drawn at various draw speeds and furnace temperatures, resulting in various measured draw tensions, but in all cases the ratio of preform feed rate and fibre draw speed was kept fixed at a value that yielded a fibre diameter of 125 µm when the holes were negligible in size. This ensures that the final drawdown ratio ρ0 of Eq. (3) is fixed for all the experiments. The final hole radii were measured using a high-power optical microscope, for which we conservatively estimate the diameter uncertainty to be ± 1 µm.
The diameters of the three holes were recorded for a given applied pressure as a function of draw tension F, where F was changed in two independent ways. Firstly, the furnace temperature was adjusted with the process speeds fixed, which changes the viscosity of the glass in the furnace. Secondly, the process speed was adjusted (by changing the preform feed and fibre draw speeds by a common factor) with the temperature fixed, which keeps the viscosity constant. The measured diameter variations 2R(F), plotted as points in Fig. 6(a) , coincide for the two different ways of changing F, confirming our result that the draw stress in the fibre has the same effect regardless of how it is changed. We therefore expect that changing the draw tension by instead changing the hot-zone length or the temperature profile within the furnace would also have no effect other than via F, though we were not able to test this because only one furnace element was available to us.
Figure 6(a) also shows that the predictions of the model, plotted as continuous lines, are quantitatively matched by the experiments. We emphasise that this was achieved without the introduction of any adjustable fitting parameters.
Equation (19) was tested by varying pressure p0 for common values of the other parameters and plotting the differences in hole size, Fig. 6(b). The experiments matched Eq. (19) very closely at all pressures, and confirmed that positive pressure exponentially exaggerates hole size differences.
5. Discussion and conclusions
We have shown how the sizes of isolated holes in optical fibres can be modelled without any need to know the viscosity distribution (or associated parameters such as temperature and hot-zone length) along the hot glass body. Our model accommodates the effects of surface tension and pressurisation of the holes, and predicts that the outcome depends only on the pressure p0 in the holes, the drawdown ratio ρ0 (fixed by the ratio of the preform feed rate and the fibre draw speed) and the draw stress s in the fibre, with surface tension γ as the sole material constant. Other variables, such as feed rate, draw speed, preform diameter, fibre diameter, hot-zone length, furnace temperature and the profiles of temperature and viscosity, are relevant only in so far as they determine these three. It is thus possible for the first time to simulate the outcome of at least some PCF drawing experiments (those where the holes are well-separated ) using only quantities that can actually be measured, since unlike viscosity the draw tension in the fibre is well-defined and routinely measured and logged by fibre fabricators.
Our result Eq. (15) expressed as hole radius R(ρ), where the drawdown ratio ρ0 in the final fibre is determined by the process speeds Eq. (3), is straightforward to calculate without numerical methods using computer algebra systems with erf and erfi as built-in functions. There are many ways to draw a fibre with a hole of a given size starting with a given preform, by suitably varying both hole pressure and draw tension to give a particular R(ρ0). However, our even-simpler result Eq. (19) highlights how positive pressure, despite its widespread use to stabilise holes while drawing fibres, causes small differences in hole size to become exaggerated according to the parameter E of Eq. (17).
To maintain a uniform pattern of holes in a PCF drawn from an imperfect preform with holes of slightly different sizes, it is therefore advantageous to select the combination of pressure and draw tension that minimises E while producing the desired R, usually by drawing under higher tension. On the other hand, by suitable choice of pressure and draw tension it is often possible to produce holes of desired different sizes from a preform where the holes do not have their sizes in the desired ratio. There may even be cases where, to minimise such variations, it would be advantageous to fashion a preform with enlarged cores and draw with a carefully-chosen negative pressure to controllably shrink the holes and “de-exaggerate” their size differences according to Eq. (17). However, the only advantages that could be gained by adjusting draw tension in a particular way (eg via the furnace temperature, hot-zone length or process speeds) come from factors outside of our model, since within the model these parameters all have the same effect. The model also explains why the common practice of drawing a PCF from a preform stack in two steps, rather than directly in one draw, helps to control the hole pattern in the final fibre.
The reader will appreciate that there are many such conclusions that can be drawn. It is our intention that our results will enable readers to experiment with these various dependences for themselves.
Our model assumes a single isolated hole. Some types of PCF (for example, the endlessly single-mode fibres) contain holes that are well-separated relative to their sizes, and the model should prove quantitatively useful in such cases . Although it cannot be expected to accurately represent PCFs with closely-spaced holes, a simple and accessible analytical model is arguably a better guide to intuition than more elaborate numerical simulations even in such cases. In any case, we would expect that simulations of more complex and realistic structures could be similarly simplified by the elimination of viscosity from the results in favour of draw tension.
Y.C. acknowledges the support of the China Scholarship Council.
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