## Drawing of the hollow all-polymer Bragg fibers

Optics Express, Vol. 14, Issue 13, pp. 5838-5852 (2006)

http://dx.doi.org/10.1364/OE.14.005838

Acrobat PDF (872 KB)

### Abstract

Drawing of the hollow all-polymer Bragg fibers based on PMMA/PS and PVDF/PC materials combinations are demonstrated. Hole collapse during drawing effects the uniformity of a photonic crystal reflector in the resultant fiber. We first investigate how the hole collapse effects fiber transmission properties. We then present modelling of fluid dynamics of hollow multilayer polymer fiber drawing. Particularly, hole collapse during drawing and layer thickness non-uniformity are investigated as a function of draw temperature, draw ratio, feeding speed, core pressurization and mismatch of material properties in a multilayer. Both the newtonian and generalized newtonian cases are considered assuming slender geometries.

© 2006 Optical Society of America

## 1. Introduction

19. M. Skorobogatiy, “Efficient anti-guiding of TE and TM polarizations in low index core waveguides without the need of omnidirectional reflector,” Opt. Lett. **30**, 2991 (2005). [CrossRef] [PubMed]

20. Y. Gao, N. Guo, B. Gauvreau, M. Rajabian, O. Skorobogata, E. Pone, O. Zabeida, L. Martinu, C. Dubois, and M. Skorobogatiy, “Consecutive Solvent Evaporation and Co-Rolling Techniques for Polymer Multilayer Hollow Fiber Preform Fabrication,” to appear in september issue of the J. Materials Research, 2006.

15. B. Temelkuran, S.D. Hart, G. Benoit, J.D. Joannopoulos, and Y. Fink “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature **420**, 650 (2002). [CrossRef] [PubMed]

21. M.R. Matovich and J.R.A. Pearson, “Spinning a molten threadline - Steady-state isothermal viscous flows,” Ind. Eng. Chem. Fundam. **8**, 512–520 (1969). [CrossRef]

22. Y.T. Shah and J.R.A. Pearson, “On the stability of nonisothermal fiber spinning,” Ind. Eng. Chem. Fundam. **11**, 145–149 (1972). [CrossRef]

25. B.D. Freeman, M.M. Denn, R. Keunings, G.E. Molau, and J. Ramos, “Profile development in drawn hollow tubes,” J. Polym. Eng. **6**, 171–186 (1986). [CrossRef]

26. 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**, 201–227 (2002). [CrossRef]

27. K. Lyytikainen, J. Canning, J. Digweed, and J. Zagari, “Geometry control of air-silica structured optical fibres using pressurisation”, IMOC Proceedings, International Microwave and Optoelectronics Conference (Cat. No.03TH8678) **2**, 1001–1005 (2003). [CrossRef]

29. 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**, 2245–2254 (2005). [CrossRef]

30. S.C. Xue, R.I. Tanner, G.W. Barton, R. Lwin, M.C.J. Large, and L. Poladian, “Fabrication of microstructured optical fibers Part II: Numerical modeling of steady-state draw process,” J. Lightwave Technol. **23**, 2255–2266 (2005). [CrossRef]

31. 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**, 853–860 (2006). [CrossRef]

29. 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**, 2245–2254 (2005). [CrossRef]

30. S.C. Xue, R.I. Tanner, G.W. Barton, R. Lwin, M.C.J. Large, and L. Poladian, “Fabrication of microstructured optical fibers Part II: Numerical modeling of steady-state draw process,” J. Lightwave Technol. **23**, 2255–2266 (2005). [CrossRef]

31. 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**, 853–860 (2006). [CrossRef]

17. Alexander Argyros, Martijn A. van Eijkelenborg, Maryanne C. J. Large, and Ian M. Bassett, “Hollow-core microstructured polymer optical fiber,” Opt. Lett. **31**, 172–174 (2006). [CrossRef] [PubMed]

## 2. Effect of a hole collapse on the transmission properties of hollow Bragg fibers

*TE*

_{01}core modes for the high index-contrast air filled fibers drawn with different values of

*C*

_{r}, while featuring the same outside diameter

*C*

_{r}=1 corresponds to a target hollow core fiber

*n*

_{c}=1 with a strictly periodic 15 layer quarter-wave reflector having material refractive indices

*n*

_{h}=2.0,

*n*

_{l}=1.5 and layer thicknesses

*µm*(for more details on design of such fibers see [32

32. Steven G. Johnson, Mihai Ibanescu, M. Skorobogatiy, Ori Weisberg, Torkel D. Engeness, Marin Soljačić, Steven A. Jacobs, J. D. Joannopoulos, and Yoel Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express **9**, 748 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748 [CrossRef] [PubMed]

*µm*,

*µm*. By design, such a fiber has a large band gap centered at

*C*

_{r}<1 (assuming the same value of a drawdown ratio

*D*) two major changes in the the fiber geometry happen. First, while the outside fiber radius is fixed

*C*

_{r}. Second, from Eq. (1) it can be shown that thicknesses of the reflector layers become non-uniform, increasing towards the fiber core, while, on average, layer thicknesses increase as

*d*

_{h,l}

^{−}~

*C*

_{r}. These geometrical changes can significantly modify fiber transmission spectra.

_{c}of a photonic bandgap is proportional to the average reflector layer thickness, then, in the presence of a hole collapse, center of a bandgap is expected to shift to the longer wavelengths

*λ*

_{c}~

*C*

_{r}(Fig. 2(a)). We find, however, that in the presence of a hole collapse the ratio of a band gap to a mid-gap (relative bandgap) stays almost uneffected. Another prominent effect of a hole collapse is on the core mode radiation losses. From ([32

32. Steven G. Johnson, Mihai Ibanescu, M. Skorobogatiy, Ori Weisberg, Torkel D. Engeness, Marin Soljačić, Steven A. Jacobs, J. D. Joannopoulos, and Yoel Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express **9**, 748 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748 [CrossRef] [PubMed]

*λ*

_{c})

^{p-1}/(

^{p}, where exponent

*p*equals 3 for the

*TE*

_{0n}modes, while for the HE, EH, and TM modes, exponent

*p*is in the range [1,3] and depends strongly upon the fiber core size. Thus, in the presence of a hole collapse, due to reduction of the core radius, and due to shift in the center of a bandgap, we expect core mode radiation loss to increase as

*Loss*~

^{(2p-1)}, which for

*TE*

_{01}mode gives

*Loss*~

*TE*

_{01}mode, actual scaling exponent varies from -5 when

*C*

_{r}≃1 to almost -7 when

*C*

_{r}≃0.5 signifying additional degradation of modal confinement due to nonuniformity in the reflector layer thicknesses.

*TE*

_{01}and

*HE*

_{11}core modes of the low index-contrast water filled fibers. In this example, a target water filled core

*n*

_{c}=1.332 fiber features a strictly periodic 16 layer quarter-wave reflector with refractive indices

*n*

_{h}=1.6,

*n*

_{l}=1.4 and layer thicknesses

*nm*,

*nm*, where band gap center frequency is

*µm*; inside and outside radii are chosen to be

*µm*,

*µm*. Behavior of modal losses as a function of a core collapse parameter in this case is similar to that of a high index-contrast system presented in Fig. 2(a). However, in this particular case due to accidental degeneracy of the studied modes with higher order ones, for some values of

*C*

_{r}modal losses can exhibit a double dip profile instead of a single dip profile corresponding to a classical band gap.

## 3. Basic equations

*r*and

*z*are the radial and axial coordinates,

*v*

_{r}and

*v*

_{z}are the

*r*and

*z*components of the velocity vector

*v*,

*ρ*is a constant density,

*p*is a pressure, τ

_{ij}is an extra-stress and

*g*is a gravitational acceleration. The components of a total stress tensor σ̿ are

_{ij}depends upon the polymer model, and is discussed in details later. For these equations, we need to specify the boundary conditions. At the interfaces between different layers the kinematic conditions are

*R*

_{j}=

*R*

_{j}(

*z*) denote the interfaces between layers and the index

*j*=1,2…

*N*is used to number them starting from the inner one. The primes denote the derivative with respect to

*z*. Since the first and the

*N*-th interfaces are external interfaces, we will distinguish them by denoting

*R*

_{i}≡

*R*

_{1}and

*R*

_{o}≡

*R*

_{N}for the inner and outer boundaries respectively.

*γ*denotes the surface tension coefficient, and

*P*

_{i}is the hole overpressure (constant ambient pressure has no effect on the flow). Outward-pointing normal at the inner boundary

**n**

_{i}is defined as:

**n**

_{o},

**t**

_{o}and

*κ*

_{o}satisfy the same equations as -

**n**

_{i}, -

**t**

_{i}and

*κ*

_{i}respectively with

*R*

_{i}replaced by

*R*

_{o}.

*V*

_{d}) and the feeding (

*V*

_{f}) velocities. Furthermore, as an initial condition, the values

*R*

_{j}(0) are known.

## 4. Thin filament equations

*ε*. In the case when

*ε*≪1 a so called

*thin filament*approximation can be used. There are two different approaches for simplifying the equations in this case. In the first approach [26

26. 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**, 201–227 (2002). [CrossRef]

33. L.J. Cummings and P.D. Howell, “On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity,” J. Fluid mech. **389**, 361–389 (1999). [CrossRef]

^{2}and only the dominant terms are retained in the equations. In the second approach [25

25. B.D. Freeman, M.M. Denn, R. Keunings, G.E. Molau, and J. Ramos, “Profile development in drawn hollow tubes,” J. Polym. Eng. **6**, 171–186 (1986). [CrossRef]

*z*.

*φ̄*(

*z*) of a variable

*φ*(

*z*) is defined as

*v*

_{z}=

*v̄*

_{z}is made explicitly. We note first that for a slow varying thin filament;

*r*-component of the momentum equation (3) by 2

*πr*

^{2}, integrating from

*R*

_{i}to

*R*

_{o}, considering

_{rr}given by Eq. (12), we obtain

*z*-component of the momentum equation (3) by 2

*π r*, integrating from

*R*

_{i}to

*R*

_{o}, using the boundary values of τ

_{rz}=σ

_{rz}given by Eq. (12) as well as the Eq. (13) and neglecting terms of relative order

*πQ*=

*π*(

*v*

_{z}is the constant volumetric flow rate. This is the axial force balance equation.

## 5. Newtonian flow

*η*is the viscosity which in our case depends on

*r*and

*z*. In this paper we consider a uniform temperature in any given cross-section, thus

*T*is only a function of

*z*. We also assume an axial variation of the temperature dependent viscosity for each constituent material. From the continuity equation one finds:

*A*=

*A*(

*z*) is a function to be determined later. The extra-stress tensor takes the form:

*A*is obtained by directly integrating the

*r*-component of the momentum equation (3) from

*R*

_{i}to

*R*

_{o}and using the boundary values of σ

_{rr}given by Eq. (12)

## 6. Numerical solution for the Newtonian case

*v*

_{z}(

*z*),

*A*(

*z*) and

*R*

_{j}(

*z*). This system of equations can be solved with an iterative method. Starting from an arbitrary initial distribution of

*v*

_{z}(

*z*), say a linear distribution between the feeding and drawing speed, the initial value equations (21) can be integrated in order to obtain

*R*

_{j}(

*z*) with the value of

*A*given by Eq. (20). This functions are then used to solve the boundary value problem Eq. (19) to obtain a new function

*v*

_{z}(

*z*) passing so at the next iteration. For the examples given later in this paper we have tested this procedure and it converges very fast (less than 200 iterations).

*C*is a constant. This equation, which is now uncoupled from the other two, can be easily integrated to give

*L*is a furnace length. Once the axial velocity is known, the initial value equations (21) can be easily integrated to obtain the profile of a drawn structure.

*C*

_{r}. Hole collapse typically results in a faster reduction of a smaller core radius compared to the larger outer radius. Thus, starting with identical thicknesses of the same material layers in a preform, in a drawn fiber the inner layers will become thicker than the outer ones. We will characterize the thickness non-uniformity by the ratio

## 6.1. Effects of draw ratio, temperature and viscosity mismatch

*T*(°C) is the temperature,

*η*

_{0}(Pa) and α(°C) are constant coefficients and

*T*

_{0}is a reference temperature. For PMMA their values are given in Ref. [35

35. H.M. Reeve, A.M. Mescher, and A.F. Emery, “Investigation of steady-state drawing force and heat transfer in polymer optical fiber manufacturing,” J. Heat Transfer **126**, 236–243 (2004). [CrossRef]

_{0}=1.506×10

^{5}Pa, α=2935°C and

*T*

_{0}=170°C.

*µ*m. The value of surface tension coefficient is considered constant for exterior interfaces

*γ*=0.032N/m [36

36. S. Wu, “Surface and interfacial tensions of polymer melts. II. Poly(methylmethacrylate), poly(nbutylmethacrylate), and polystyrene,” J. Phys. Chem. **74**, 632–638 (1970). [CrossRef]

^{3}[35

35. H.M. Reeve, A.M. Mescher, and A.F. Emery, “Investigation of steady-state drawing force and heat transfer in polymer optical fiber manufacturing,” J. Heat Transfer **126**, 236–243 (2004). [CrossRef]

*L*=30cm, a constant preform feeding velocity

*V*

_{f}=2.5

*µ*m/s and a zero pressurization

*P*

_{i}=0.

*D*

_{r}=

*V*

_{d}/

*V*

_{f}(not to be confused with a drawdown ratio defined as

*D*=

*C*

_{r}as a function of a draw ratio

*D*

_{r}=

*V*

_{d}/

*V*

_{f}for different values of the maximal temperature. Dashed lines represent the parameter curves resulting in a constant outside diameter

*D*

_{o}≡2

*R*

_{o}=125

*µm*and

*D*

_{o}=250

*µm*after the draw. For comparison, in dotted curves we present the hole collapse if no other polymer is present in the preform (drawing of a simple PMMA tube of the same inner and outer radii as a multilayer preform).

*η*

_{0}while keeping the other parameters unchanged, which corresponds to the case of using the same polymer, but with a different molecular mass. It should be mentioned here that the polymer’s molecular mass determines whether the fiber is drawable in the first place. Second, we investigate the effects of changing

*T*

_{0}which corresponds to the case of varying the polymer material. In Fig. 6 we consider drawing of preforms of various compositions at a fixed maximum furnace temperature of 190 °C and a draw rate

*D*

_{r}=30000. Multilayer preform geometry is the same as described above; mismatch in the polymer viscosities is described in terms of the ratios of the material parameters

*η*

_{0}/

*η*

_{0},

_{PMMA}and (

*T*

_{0}-

*T*

_{0},

_{PMMA})/

*T*

_{0},

_{PMMA}. From Fig. 6 we see that the hole collapse depends significatively on the viscosity of a second material and can be prevented by choosing a polymer with an appropriate viscosity.

## 6.2. Effects of the pressurization and preform feeding velocity

*D*

_{r}=5000, while the other draw parameters remain unchanged. In Fig. 7 hole collapse and layer thickness non-uniformity are presented as functions of the hole overpressure

*P*

_{i}and feeding speed

*V*

_{f}. As expected, the hole collapse is very sensitive to pressurization, and in principle, can be reduced by increasing the pressure. Time evolving transient draw simulations, not discussed in this paper, also show that above a certain critical value for an overpressure, which in our case is less than 7Pa, even if the fiber does not blow up immediately, the drawing process never reaches its steady state. A more subtle way of controlling the hole collapse is by changing the preform feeding speed, although, for a given draw ratio, this could be limited by the maximal draw velocity.

## 7. Generalized Newtonian model

*η*as a non-Newtonian viscosity or the generalized viscosity. Polymers in general are also viscoelastic, however we will neglect elasticity effects in our analysis which may be considered as a first step toward a complete modelling of polymeric flow.

37. T. Sedlacek, M. Zatloukal, P. Filip, A. Boldizar, and P. Saha “On the effect of pressure on the shear and elongational viscosities of polymer melts,” Polymer eng. sci. , **44**, 1328–1337 (2004). [CrossRef]

*η*

_{0},

*K*

_{1},

*a*, and

*n*are constant coefficients,

*II*

_{D}=

*tr*(

*D̿*·

*D̿*) is its second invariant and the term

*f*gives the temperature dependence of the viscosity in the Arrhenius form

*A*is negligible compared to

*v*

_{z}(

*z*) can be evaluated separately by using repeatedly Eq. (23) with a viscosity update according to Eq. (25), until the convergence is attained. Once

*v*

_{z}(

*z*) is known, Eq. (21) can be easily integrated.

*µ*m for all the layers. We consider

*γ*=0.032N/m,

*ρ*=1195kg/m

^{3},

*L*=30cm,

*V*

_{f}=2.5

*µ*m/s and

*P*

_{i}=0. Parameters for the non- Newtonian viscosity Eq. (25) are

*K*

_{1}=0.0861

*n*=0.1401 and

*a*=0.7347 for PMMA, and

*K*

_{1}=0.3891

*n*=0.2194 and

*a*=0.6097 for PS [37

37. T. Sedlacek, M. Zatloukal, P. Filip, A. Boldizar, and P. Saha “On the effect of pressure on the shear and elongational viscosities of polymer melts,” Polymer eng. sci. , **44**, 1328–1337 (2004). [CrossRef]

*α*=2935,

*T*

_{0}=170°C, and the value of the maximum temperature in the furnace being

*T*=180 °C.

*D*

_{r}. This can be clearly seen in Fig. 8(b) where the viscosity distribution is plotted as a function of

*z*for different draw ratios. For comparison, we have also included the Newtonian viscosity at the same temperature.

## 8. Conclusions

## Acknowledgment

## References and links

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17. | Alexander Argyros, Martijn A. van Eijkelenborg, Maryanne C. J. Large, and Ian M. Bassett, “Hollow-core microstructured polymer optical fiber,” Opt. Lett. |

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27. | K. Lyytikainen, J. Canning, J. Digweed, and J. Zagari, “Geometry control of air-silica structured optical fibres using pressurisation”, IMOC Proceedings, International Microwave and Optoelectronics Conference (Cat. No.03TH8678) |

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29. | 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. |

30. | S.C. Xue, R.I. Tanner, G.W. Barton, R. Lwin, M.C.J. Large, and L. Poladian, “Fabrication of microstructured optical fibers Part II: Numerical modeling of steady-state draw process,” J. Lightwave Technol. |

31. | 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. |

32. | Steven G. Johnson, Mihai Ibanescu, M. Skorobogatiy, Ori Weisberg, Torkel D. Engeness, Marin Soljačić, Steven A. Jacobs, J. D. Joannopoulos, and Yoel Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express |

33. | L.J. Cummings and P.D. Howell, “On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity,” J. Fluid mech. |

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35. | H.M. Reeve, A.M. Mescher, and A.F. Emery, “Investigation of steady-state drawing force and heat transfer in polymer optical fiber manufacturing,” J. Heat Transfer |

36. | S. Wu, “Surface and interfacial tensions of polymer melts. II. Poly(methylmethacrylate), poly(nbutylmethacrylate), and polystyrene,” J. Phys. Chem. |

37. | T. Sedlacek, M. Zatloukal, P. Filip, A. Boldizar, and P. Saha “On the effect of pressure on the shear and elongational viscosities of polymer melts,” Polymer eng. sci. , |

**OCIS Codes**

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

(060.2290) Fiber optics and optical communications : Fiber materials

(160.5470) Materials : Polymers

(220.4000) Optical design and fabrication : Microstructure fabrication

(230.1480) Optical devices : Bragg reflectors

(230.4170) Optical devices : Multilayers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 10, 2006

Revised Manuscript: June 7, 2006

Manuscript Accepted: June 9, 2006

Published: June 26, 2006

**Citation**

Elio Pone, Charles Dubois, Ning Gu, Yan Gao, Alexandre Dupuis, Francis Boismenu, Suzanne Lacroix, and Maksim Skorobogatiy, "Drawing of the hollow all-polymer Bragg fibers," Opt. Express **14**, 5838-5852 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-13-5838

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### References

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