## Predicting hole sizes after fibre drawing without knowing the viscosity |

Optical Materials Express, Vol. 3, Issue 3, pp. 346-356 (2013)

http://dx.doi.org/10.1364/OME.3.000346

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### Abstract

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

## 1. Introduction

1. A. L. Yarin, P. Gospodinov, and V. I. Roussinov, “Stability loss and sensitivity in hollow fiber drawing,” Phys. Fluids **6**(4), 1454–1463 (1994). [CrossRef]

6. C. J. Voyce, A. D. Fitt, and T. M. Monro, “Mathematical modeling as an accurate predictive tool in capillary and microstructured fiber manufacture: the effects of preform rotation,” J. Lightwave Technol. **26**(7), 791–798 (2008). [CrossRef]

2. A. D. Fitt, K. Furusawa, T. M. Monro, C. P. Please, and D. J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. **43**(2/4), 201–227 (2002). [CrossRef]

*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.

## 2. Analysis

### 2.1 Drawdown

*r*

_{1}is fed into a furnace at the feed rate

*u*and a fibre of radius

_{f}*r*is drawn out at the draw speed

_{2}*u*, all in the direction of positive

_{d}*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*where the local drawdown ratio

*ρ*(

*x*) isAs

*x*→ ∞ this gives the final drawdown ratio

*ρ*

_{0}in terms of the process speeds

*F*required to induce a velocity gradient

*du*/

*dx*along an extending viscous thread is [7

7. F. T. Trouton, “On the coefficient of viscous traction and its relation to that of viscosity,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character **77**(519), 426–440 (1906). [CrossRef]

*F*is uniform along the glass body and equal to the draw tension in the fibre.

*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*

*ρ*(

*x*) of Eq. (2) is generalised so that

*r*values in Eqs. (2) and (3) are defined bywhere

*A*(

*x*) is the cross-sectional area of glass at position

*x*. In other words,

*r*is the radius the fibre would have if all holes had collapsed and the fibre had circularised. This interpretation is justified because it is really the glass cross-section

*A*that enters directly into both Eqs. (1) and (4).

*R*in the absence of deforming forces, under the action of drawdown alone:

### 2.2 Transverse pressures

*γ*at the hole boundary exherts an inward pressure

*p*according to the Young-Laplace equationSince the hole is small compared to the size of the preform, the 1/

_{st}*R*dependence means that surface tension at the outer boundary of the preform can be neglected.

*μ*causes an effective pressure

*p*that opposes the change in size. This effective pressure (defined positive if acting inwards) iswith the strain rate in [8] expressed in terms of rate of change of

_{v}*R*. If the hole is expanding (

*dR*/

*dt*> 0) viscosity acts like an inward pressure (

*p*> 0) whereas if the hole is shrinking (

_{v}*dR*/

*dt*< 0) it acts like an outward pressure (

*p*< 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).

_{v}*p*of the gas in the hole and the atmospheric pressure

_{H}*p*outside, with a net outward pressure of

_{a}### 2.3 Solution

*R*(

*x*)

*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 obtainA crucial consequence of the change of variable, along with the inclusion of the draw tension

*F*in the analysis at the outset, is that the viscosity profile

*μ*(

*x*) has now been completely eliminated from the problem! This is at the cost of losing information about length scales along

*x*, but does not prevent us finding the hole size in the final fibre because the appropriate final condition

*ρ*=

*ρ*

_{0}is well defined.

*R*(

*ρ*) as a function of local drawdown ratio, which varies from

*ρ*= 1 at the preform (

*x*→ −∞) to

*ρ*=

*ρ*

_{0}=

*r*

_{2}/

*r*

_{1}at the final fibre (

*x*→ ∞). It can be solved by the integrating factor method to givewhere

*R*

_{0}is the initial hole radius (in the preform),

*s*is the mean draw stress in the glass in the final fibreand the normalised parameter

*E*combines the excess pressure in the hole and the draw stressIt is worth noting that the combination

*sρ*

_{0}

^{2}that appears in both Eqs. (15) and (17) is the draw stress in the

*preform*.

*erf*or the imaginary error function

*erfi*, depending on the sign of

*E*For small drawdown ratios

*ρ*(fibres much thinner than their preforms) the expression further simplifies because the terms containing

*ρ*in Eq. (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 [9], making the evaluation of Eq. (15) straightforward and quick.

*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 [10

10. P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express **13**(1), 236–244 (2005). [CrossRef] [PubMed]

*R*

_{0,1}and

*R*

_{0,2}to find their size difference in the final fibre (or indeed at intermediate points) without the integral termThis shows that the hole size difference scales with the drawdown ratio

*ρ*but multiplied by an exponential factor whose size depends mainly on what can be termed the “exaggeration parameter”

*E*of Eq. (17), since 1 −

*ρ*

^{2}≈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

*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*→

*ρ*

_{0},

*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.

*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 bodyFrom its ratio with the expression for

*x*→ ∞we getwhich can be substituted into Eq. (15) to give

*R*(

*x*).

*μ*along hot-zone length

*L*, with infinite viscosity elsewhere [2

2. A. D. Fitt, K. Furusawa, T. M. Monro, C. P. Please, and D. J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. **43**(2/4), 201–227 (2002). [CrossRef]

*x*= 0 and

*x*=

*L*, Eq. (21) becomeswhich relates

*s*in Eq. (16) to the viscosity. Equation (22) becomeswhich allows Eq. (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 [2

2. A. D. Fitt, K. Furusawa, T. M. Monro, C. P. Please, and D. J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. **43**(2/4), 201–227 (2002). [CrossRef]

### 2.5 Normalised expression

*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:We introduce a normalised stressalongside the already-normalised

*E*and

*ρ*. A normalised pressurecan also be defined, though it is not independent of

*E*and

*S*. In these units, Eq. (15) becomesand the stress required to achieve a given final hole size

*a*

_{0}is

## 3. Example calculations

*S*is given in Fig. 2 for certain final drawdown ratio

*ρ*

_{0}= 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

*ρ*

_{0}.) 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

*p*

_{0}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.

*E*in Eq. (19) is demonstrated in Fig. 3 . For a given drawdown ratio

*ρ*

_{0}, a given target hole size

*a*

_{0}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

*a*

_{0}as a constant). For example, Fig. 3(a) is the locus of

*P*and

*S*values that give

*a*

_{0}= 0.8 when

*ρ*

_{0}= 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

*a*

_{0}. 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.

*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.

*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 [11

11. S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, P. St. J. Russell, and M. W. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express **12**(13), 2864–2869 (2004). [CrossRef] [PubMed]

## 4. Experimental tests

*p*above atmospheric pressure was applied to all three holes, subjecting them to identical un-normalised draw parameters except for the different initial hole radii

_{o}*R*

_{0}. 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

*ρ*

_{0}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.

*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 2

*R*(

*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.

*p*

_{0}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

*p*

_{0}in the holes, the drawdown ratio

*ρ*

_{0}(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 [12

12. W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, “Demonstration of ultra-flattened dispersion in photonic crystal fibers,” Opt. Express **10**(14), 609–613 (2002). [CrossRef] [PubMed]

*R*(

*ρ*), where the drawdown ratio

*ρ*

_{0}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*(

*ρ*

_{0}). 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).

*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.

12. W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, “Demonstration of ultra-flattened dispersion in photonic crystal fibers,” Opt. Express **10**(14), 609–613 (2002). [CrossRef] [PubMed]

## Acknowledgment

## References and links

1. | A. L. Yarin, P. Gospodinov, and V. I. Roussinov, “Stability loss and sensitivity in hollow fiber drawing,” Phys. Fluids |

2. | A. D. Fitt, K. Furusawa, T. M. Monro, C. P. Please, and D. J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. |

3. | S. C. Xue, R. I. Tanner, G. W. Barton, R. Lwin, M. C. J. Large, and L. Poladian, “Fabrication of microstructured optical fibers - Part I: problem formulation and numerical modeling of transient draw process,” J. Lightwave Technol. |

4. | S. C. Xue, M. C. J. Large, G. W. Barton, R. I. Tanner, L. Poladian, and R. Lwin, “Role of material properties and drawing conditions in the fabrication of microstructured optical fibers,” J. Lightwave Technol. |

5. | R. M. Wynne, “A fabrication process for microstructured optical fibers,” J. Lightwave Technol. |

6. | C. J. Voyce, A. D. Fitt, and T. M. Monro, “Mathematical modeling as an accurate predictive tool in capillary and microstructured fiber manufacture: the effects of preform rotation,” J. Lightwave Technol. |

7. | F. T. Trouton, “On the coefficient of viscous traction and its relation to that of viscosity,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character |

8. | R. T. Knapp, J. W. Daily, and F. G. Hammitt, |

9. | S. Wolfram, |

10. | P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express |

11. | S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, P. St. J. Russell, and M. W. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express |

12. | W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, “Demonstration of ultra-flattened dispersion in photonic crystal fibers,” Opt. Express |

**OCIS Codes**

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

(060.2290) Fiber optics and optical communications : Fiber materials

(060.4005) Fiber optics and optical communications : Microstructured fibers

(060.5295) Fiber optics and optical communications : Photonic crystal fibers

**ToC Category:**

Materials for Fiber Optics

**History**

Original Manuscript: December 21, 2012

Revised Manuscript: January 31, 2013

Manuscript Accepted: January 31, 2013

Published: February 4, 2013

**Citation**

Y. Chen and T. A. Birks, "Predicting hole sizes after fibre drawing without knowing the viscosity," Opt. Mater. Express **3**, 346-356 (2013)

http://www.opticsinfobase.org/ome/abstract.cfm?URI=ome-3-3-346

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### References

- A. L. Yarin, P. Gospodinov, and V. I. Roussinov, “Stability loss and sensitivity in hollow fiber drawing,” Phys. Fluids6(4), 1454–1463 (1994). [CrossRef]
- A. D. Fitt, K. Furusawa, T. M. Monro, C. P. Please, and D. J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math.43(2/4), 201–227 (2002). [CrossRef]
- S. C. Xue, R. I. Tanner, G. W. Barton, R. Lwin, M. C. J. Large, and L. Poladian, “Fabrication of microstructured optical fibers - Part I: problem formulation and numerical modeling of transient draw process,” J. Lightwave Technol.23(7), 2245–2254 (2005). [CrossRef]
- S. C. Xue, M. C. J. Large, G. W. Barton, R. I. Tanner, L. Poladian, and R. Lwin, “Role of material properties and drawing conditions in the fabrication of microstructured optical fibers,” J. Lightwave Technol.24(2), 853–860 (2006). [CrossRef]
- R. M. Wynne, “A fabrication process for microstructured optical fibers,” J. Lightwave Technol.24(11), 4304–4313 (2006). [CrossRef]
- C. J. Voyce, A. D. Fitt, and T. M. Monro, “Mathematical modeling as an accurate predictive tool in capillary and microstructured fiber manufacture: the effects of preform rotation,” J. Lightwave Technol.26(7), 791–798 (2008). [CrossRef]
- F. T. Trouton, “On the coefficient of viscous traction and its relation to that of viscosity,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character77(519), 426–440 (1906). [CrossRef]
- R. T. Knapp, J. W. Daily, and F. G. Hammitt, Cavitation (McGraw-Hill, 1970), p. 108.
- S. Wolfram, The Mathematica Book, 5th ed. (Wolfram Media, 2003).
- P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express13(1), 236–244 (2005). [CrossRef] [PubMed]
- S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, P. St. J. Russell, and M. W. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express12(13), 2864–2869 (2004). [CrossRef] [PubMed]
- W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, “Demonstration of ultra-flattened dispersion in photonic crystal fibers,” Opt. Express10(14), 609–613 (2002). [CrossRef] [PubMed]

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