OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 13 — Jun. 26, 2006
  • pp: 5838–5852
« Show journal navigation

Drawing of the hollow all-polymer Bragg fibers

Elio Pone, Charles Dubois, Ning Guo, Yan Gao, Alexandre Dupuis, Francis Boismenu, Suzanne Lacroix, and Maksim Skorobogatiy  »View Author Affiliations


Optics Express, Vol. 14, Issue 13, pp. 5838-5852 (2006)
http://dx.doi.org/10.1364/OE.14.005838


View Full Text Article

Acrobat PDF (872 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

In our research group we study fabrication of all-polymer hollow multilayer Bragg fibers. We find that polymer material combination is very well suited for this purpose. The key issue for Bragg fiber fabrication is a choice of a pair of thermo-mechanically compatible polymer materials with sufficiently different refractive indexes that can be co-drawn into a fiber. Because of the commercial abundance of various moderately priced polymer materials the process of material selection is relatively short time and low cost. Moreover, polymer thermo-mechanical properties can be fine-tuned by simple variation of their molecular mass. Many polymers are available in various physical forms such as rods, tubes, films, granules and powders which enables multiple design decisions for preform fabrication. Many polymers are swelled or solvable in organic solvent which enables incorporation of active materials in their matrix such as dopants, laser dyes, nanoparticles, etc. Polymer materials are generally bio-friendly and some of them are biodegradable which is of potential interest for fiber based bio-medical technologies [18

18. A. Dupuis, Y. Gao, N. Guo, E. Pone, N. Godbout, S. Lacroix, C. Dubois, and M. Skorobogatiy “Biodegradable, Double-Core, Porous Optical Fiber,” CLEO - Conference on Lasers and Electro-Optics, postdeadline paper Cat. No. CPDB5, (2006).

]. Finally, drawing of large core Bragg fibers does not typically require core pressurization which simplifies fabrication infrastructure. Although refractive index contrast between layers in an all-polymer Bragg fiber is relatively small (at most 1.3/1.7), as demonstrated in [19

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]

] liquid core all-polymer Bragg fibers can be designed to guide very well both TE and TM like modes, while gas filled all-polymer Bragg fibers can guide effectively a TE polarized mode. Moreover, in liquid filled core fibers high index-contrast omnidirectional designs typically hurt fiber performance making lower index-contrast systems superior in their transmission properties.

Recently, our research group has succeeded in developing two methodologies for fabrication of multilayered all-polymer hollow preforms. One approach uses consecutive deposition of layers of two different polymers by solvent evaporation on the inside of a rotating polymer cladding tube [20

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.

]. Orthogonal solvents were found, and solvent evaporation process was developed for both PMMA(Polymethyl methylacrylate)/PS(Polystyrene) and PVDF(Polyvinylidene fluoride)/PC(Polycarbonate) material combinations. In the left of Fig. 1(a), a 30cm long all-polymer preform with 10 consecutive PMMA/PS layers deposited on the inside of a PMMA cladding tube is presented; on the right, preform cross section is shown. In the left of Fig. 1(b), again a 30cm long all-polymer preform with 15 consecutive PVDF/PC layers deposited on the inside of a PC cladding tube is presented; while on the right, preform cross section is shown. Alternative preform fabrication method uses a co-rolling of two dissimilar polymer films similarly to [15

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]

], where both commercial and home-made films were used. In the left of Fig. 1(c), an end part of an all-polymer preform with 19 consecutive PVDF/PC layers is presented; in the middle of the figure a cross section of a drawn fiber is shown with a drawdown ration of 1:20; finally in the right of Fig. 1(c) another drawn fiber with 32 layers PMMA/PS layers is demonstrated with a similar drawdown ratio.

Fig. 1. a) Left - 30cm long all-polymer preform with 10 consecutive PMMA/PS layers deposited on the inside of a PMMA cladding tube. Right - PMMA/PS preform cross section. b) Left - 30cm long all-polymer preform with 15 consecutive PVDF/PC layers deposited on the inside of a PC cladding tube. Right - PVDF/PC preform cross section c) Left - end part of a rolled 19 layer PVDF/PC preform. Middle - cross section of a drawn PVDF/PC fiber with a 1:20 drawdown ratio. Right - cross section of a drawn PMMA/PS fiber with a 1:20 drawdown ratio.

After preform is fabricated, hollow MOFs are manufactured by preform heating and drawing. Geometry of the final fiber can be significantly influenced by controlling various parameters in the drawing process such as temperature distribution in a furnace, fiber drawing and preform feed velocities, as well as pressurization of the hollow core. When preform is made of polymer materials non-Newtonian nature of polymer viscosity can be of importance. Moreover, if several materials are used in a single preform, drawing process can be influenced greatly by the mismatch in the viscosities of the constitutive materials. For the problem of hollow Bragg fiber drawing, geometries of all the resultant fibers can be simply parametrized by only two parameters - a drawdown ratio, and a hole collapse parameter. We further find that fiber transmission properties depend strongly on a hole collapse parameter which can only be controlled indirectly during drawing process. The purpose of this paper is to characterize hole collapse and multilayer non-uniformity during the draw of a polymer multilayer fiber as a function of standard control variables. We investigate importance of mismatch in the viscosities of two different materials during co-drawing and compare predictions of Newtonian and generalized Newtonian flow models.

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

Fig. 2. a) Radiation loss of the bandgap guided TE 01 core modes for the high index-contrast (2.0/1.5) air filled fibers with different hole collapse ratios C r , while the same outside radii Roft . Hole collapse leads to the shift of a bandgap center into the longer wavelength, as well as to a considerable increase in the modal radiation losses. b) Radiation losses of the bandgap guided TE 01 and HE 11 core modes for the low index-contrast (1.6/1.4) water filled fibers with different hole collapse ratios.

(rf)2=(rpD)2(1Cr2)(RipD)2(Rop)2(rp)2(Rop)2(Rip)2.
(1)

To understand the effect of a hole collapse on the transmission properties of the resultant fibers, in Fig. 2(a) we first present a set of theoretical curves showing radiation losses of the 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 Rof . In this example, 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 dht=0.25λctnh2nc2=144nm, dlt=0.25λctnl2nc2=224nm, where λct =1µm (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]

]). Target fiber inside and outside radii are chosen to be Rift =5µm, Roft =12.72µm. By design, such a fiber has a large band gap centered at λct .

In the presence of a hole collapse 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 Roft , the fiber core radius is reduced Rif =Rift 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 dh,l~dh,lt/C r . These geometrical changes can significantly modify fiber transmission spectra.

From the analysis above it follows that hole collapse mainly leads to the linear shift in the bandgap frequency and a super-linear increase in the radiation losses of the core guided modes. In what follows we quantify hole collapse and layer non-uniformity as a function of the standard control parameters during drawing.

3. Basic equations

Schematic of a hollow multilayer preform profile during drawing is shown in Fig. 3. As the density of polymers varies little in the range of temperatures and pressures considered here we consider an incompressible axisymmetric steady flow. The equations for conservation of mass and momentum in cylindrical coordinates are as follows:

1r(rvr)r+vzz=0
(2)
ρ(vrvrr+vzvrz)=pr+1r(rτrr)rτθθr+τrzz
ρ(vrvzr+vzvzz)=pz+1r(rτrz)rτzzz+ρg
(3)

where 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

σij=pδij+τij
(4)

The definition of τ 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

vr=Rjvzatr=Rj
(5)

where 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 iR 1 and R oR N for the inner and outer boundaries respectively.

Fig. 3. Schematic of a hollow multilayer preform during drawing. Different colors correspond to different materials in a multilayer.

Hollow core can be pressurized in order to control its collapse under the action of a surface tension. In this case, at the inner interface the dynamic boundary conditions are

σ̿·ni=(γκiPi)ni
σ̿·ti=0
(6)

where γ denotes the surface tension coefficient, and

κi=1Ri(1+Ri2)12Ri"(1+Ri2)32
(7)

is the curvature, while 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:

niT=(nr,nθ,nz)=(11+Ri2,0,Ri1+Ri2),
(8)

while

tiT=(nz,0,nr)
(9)

is the unit tangent vector. In a similar way, the dynamic boundary conditions at the outer boundary are

σ̿·no=γκono
σ̿·to=0
(10)

where 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 .

At the interfaces between the internal layers, we will consider a continuous stress and velocity. In the axial direction the boundary conditions are the known values of the drawing (V d ) and the feeding (V f ) velocities. Furthermore, as an initial condition, the values R j (0) are known.

4. Thin filament equations

One of the basic dimensionless parameters in the problem is the ratio between the preform radius and the length of the neck down region defined as 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

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]

], the variables are expanded as power series in ε2 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]

], which we also follow in this paper, the equations are averaged over the cross-section at each value of z.

The average φ̄(z) of a variable φ(z) is defined as

φ¯(z)=1π(Ro2Ri2)RiRo2πrφ(r,z)dr
(11)

For the axial velocity, the assumption v z = z is made explicitly. We note first that for a slow varying thin filament; Rj ≪1, and by neglecting terms of the order Rj2 the boundary conditions Eqs. (6) and (10) take the following form:

σrr=γRiPi
σrz=(γRi+Pi)Riatr=Ri
σzz=(γRiPi)Ri2
σrr=γRo
σrz=γRoRoatr=Ro
σzz=γRoRo2
(12)

Multiplying the r-component of the momentum equation (3) by 2πr 2, integrating from R i to R o , considering ddzRiRo2πr2τrzdr0 (see Ref. [34

34. R.B. Bird, R.C. Amstrong, and O. Hassager. Dynamics of polymeric liquids, vol. 1, Fluid mechanics, (JohnWiley & Sons, New York, 1987).

] p.382), neglecting the inertial term because of the small value of the radial velocity and using the boundary values of σ rr given by Eq. (12), we obtain

p¯=τ¯rr+τ¯θθ2+γ(Ro+Ri)Ri2PiRo2Ri2
(13)

Multiplying the z-component of the momentum equation (3) by 2π r, integrating from R i to R o , using the boundary values of τ rzrz given by Eq. (12) as well as the Eq. (13) and neglecting terms of relative order Rj2 , we obtain

ρ¯Qvz=[Qvz(τ¯zzτ¯rr+τ¯θθ2)+γ(Ro+Ri)]+ρ¯gQvz
(14)

where πQ=π(Ro2 -Ri2 )v z is the constant volumetric flow rate. This is the axial force balance equation.

5. Newtonian flow

The constitutive equation for the Newtonian fluid is

τ̿=η(r,z)(v+vT)
(15)

where 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:

vr=rvz2+Ar
(16)

where A=A(z) is a function to be determined later. The extra-stress tensor takes the form:

τ̿=(η(vz+2Ar2)000η(vz2Ar2)0002ηvz)
(17)

where the component τrz=ηvrz has been neglected. From Eq. (17) the following relation holds for the averaged components of an extra-stress tensor

τ¯zzτ¯rr+τ¯θθ2=3η¯vz
(18)

and the axial force balance Eq. (14) can be written in the form

ρ¯Qvz=[3η¯Qvzvz+γ(Ro+Ri)]+ρ¯gQvz
(19)

The expression for 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)

A=Piγ(1Ri+1Ro)4RiRoη(r)r3dr
(20)

Combining the kinematic boundary conditions Eq. (5) with relation Eq. (16), we obtain

(Rj2vz)=2A
(21)

6. Numerical solution for the Newtonian case

In most cases of practical importance, inertial, gravitational and capillary terms in Eq. (19) can be neglected and it takes a simple form

η¯vzvz=C
(22)

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

vz(z)=exp(lnVf+0zdzη¯(z)0Ldzη¯(z)lnVdVf)
(23)

where 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.

One of the key aspects of hollow preform drawing is the partial or even complete collapse of a compressible core as a result of the surface tension forces acting at the free boundaries. As introduced earlier, we characterize the hole collapse by the parameter 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 hohi between the thickness of the outer layer and the thickness of the inner one, assuming they were equal in a preform.

Fig. 4. Temperature distribution in the furnace.

In the following we investigate how the hole collapse and layer thickness non-uniformity are influenced by various control parameters. As an example, we consider drawing of a multilayer hollow Bragg fiber preform where cladding tube and one of the two materials of a multilayer is PMMA, while the other material is a different polymer. Materials in consecutive polymer layers are alternated to create a periodic multilayer structure.

6.1. Effects of draw ratio, temperature and viscosity mismatch

In our calculations we assume a uniaxial temperature distribution with a maximum at the furnace center (Fig. 4). In the following when we vary the maximum value of the temperature, we simply rescale the whole profile. We assume that Newtonian viscosity of PMMA obeys an Arrhenius type dependency:

η(T)=η0exp[α(1T1T0)]
(24)

where 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]

]; η0=1.506×105Pa, α=2935°C and T 0=170°C.

In our first calculation viscosity of the other polymer is assumed to be simply two times higher than that of PMMA. Although we recognize that to model correctly the flow of a particular polymer we need to use its proper temperature dependent viscosity, in our first simulation we rather want to highlight the major effect of adding another material into a preform. Particularly, we want to investigate how the hole collapse is affected by the viscosity of the second material despite of its small volume fraction in the preform.

We consider drawing of a preform with external and internal diameter 31.75 and 25.4mm respectively. PMMA tube is coated on the inside with 25 alternated layers of PMMA and another polymer with a viscosity two times higher, each one of them having a thickness of 50µm. 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]

] and both densities are considered to be 1195kg/m 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]

]. We also assume a furnace length L=30cm, a constant preform feeding velocity V f =2.5µm/s and a zero pressurization P i =0.

First, we consider the effects of varying the drawing ratio defined as D r =V d /V f (not to be confused with a drawdown ratio defined as D=Rop /Rof ) and the maximum temperature in a furnace. In Fig. 5(a), solid lines represent the parameter of a hole collapse 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 ≡2R o =125µm and D o=250µm 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).

Fig. 5. (a) Hole collapse parameter Cr as a function of the draw ratio D r for different temperatures. Solid lines correspond to multilayer preform. Dotted lines correspond to a simple tube with the same thickness as the multilayer preform. Dashed lines represent the curves of a constant outside diameter. (b) Ratio h o /h i between the inner and outer layer thicknesses as a function of the draw ratio for different temperatures.

We see that to prevent hole collapse higher draw ratios and lower temperatures have to be used. Both cases demand higher draw force which might lead to fiber breakage. We also observe that the collapse of the tube with the same thickness as the multilayer preform is more pronounced because of a lower average viscosity. Effect of the temperature on a hole collapse is easy to understand as viscosity decreases rapidly with reduction of the temperature, thus hindering the hole collapse. Effect of the draw ratio is more subtle. Starting with preforms of the same diameter, draw ratio increase leads to the reduction of a resultant fiber cross-section. As a consequence, the forces of surface tension become more pronounced, thus favoring the hole collapse. This is overcompensated by the fact that increasing the draw ratio leads to the higher axial velocities, thus the time a cross-section spends in a melted zone diminishes which works against the hole collapse.

In Fig. 5(b), thickness non-uniformity parameter is presented as a function of the draw ratio for different values of the maximum temperature. The curves are similar to those for the hole collapse in Fig. 5(a). From mass conservation, the ratio between the cross-section areas of different layers remains constant from which it follows that the thickness non-uniformity parameter is proportional to the hole collapse.

We now describe in more details the effect of mismatch in the polymer viscosities when two different materials are used in the same preform. As seen in Fig. 5(a) the hole collapse is considerably less pronounced for the multilayer structure despite the fact that the higher viscosity material occupies only a very small fraction of the total volume and its viscosity is only two times higher than that of PMMA. In what follows we assume that multilayer preform is made of PMMA and another polymer. For the viscosity of a second material a similar Arrhenius law as for PMMA is assumed.

First, we investigate the effects of changing 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.

Fig. 6. Effect of mismatch in the viscosities of materials in a multilayer on hole collapse (solid lines), and layer non-uniformity (dotted lines). Maximum furnace temperature is T=190°C, draw ratio D r =30000. (a) Effects of η 0. (b) Effects of (T 0-T 0, PMMA )/T 0, PMMA .

6.2. Effects of the pressurization and preform feeding velocity

Other parameters that influence hole collapse are the hole overpressure and the feeding speed. By increasing the hole pressure we expect to reduce the hole collapse. Also, for a given draw ratio, by increasing the preform feeding speed we expect reduction of the hole collapse as fiber cross-section would spend less time in the melted zone.

Fig. 7. The hole collapse and thickness non-uniformity as a function of the hole overpressure and feeding speed. Maximal furnace temperature is T=190°C and draw ratio D r =5000. (a) Effects of hole overpressure P i . (b) Effects of feeding speed V f .

7. Generalized Newtonian model

For the non-Newtonian viscosity we will use the Carreau-Yasuda model [34

34. R.B. Bird, R.C. Amstrong, and O. Hassager. Dynamics of polymeric liquids, vol. 1, Fluid mechanics, (JohnWiley & Sons, New York, 1987).

] which was also considered in [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]

] and the values of different coefficients for some important polymers are given in that paper. According to this model the viscosity is given in the form

η(T,IID)=η0f[1+(K1f2IID)a]1na
(25)

where η 0, K 1, a, and n are constant coefficients, D̿=12(v+vT) is the rate of the deformation tensor, II D =tr(D̿·D̿) is its second invariant and the term f gives the temperature dependence of the viscosity in the Arrhenius form f=eα(1T1T0). Considering the Eqs. (17) and (15) and the fact that A is negligible compared to vz , one obtains IID=32vz2.

The numerical procedure is the same as in the Newtonian case. The system of coupled partial differential equations (19–21) can be solved again by the same iterative procedure. At each step of the iteration we must solve the boundary value problem Eq. (19) which, in this case, becomes more complex because the viscosity depends on the rate of deformation. Thus, this equation must be solved itself by an iterative procedure (so in overall there are two nested iterations). However, since the inertial and gravitational terms are usually negligible, the axial velocity 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.

As an example, we consider drawing of the same preform as for the Newtonian case, which is a PMMA tube with OD and ID diameters 31.75 and 25.4mm respectively, coated on the inside with 25 alternated layers of PMMA. Viscosity of PS is assumed to be two times higher than that of a PMMA. Layer thickness is assumed to be 50µm for all the layers. We consider γ=0.032N/m, ρ=1195kg/m3, L=30cm, V f =2.5µm/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]

]. Finally we assume α=2935, T 0=170°C, and the value of the maximum temperature in the furnace being T=180 °C.

Numerical results are presented in Fig.8(a) where hole collapse parameter is plotted as a function of the draw rate and maximum furnace temperature for Newtonian and non-Newtonian models. We observe that hole collapse is more pronounced in the non-Newtonian case because, as seen from Eq. (25), the non-Newtonian viscosity is lower than the Newtonian one, making it easier for the surface tension to reduce the hole. As expected, non-Newtonian nature of polymer viscosity becomes more apparent with increase in 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.

Fig. 8. a) Comparison between Newtonian and generalized Newtonian model. Solid lines correspond to generalized Newtonian model and dotted lines to Newtonian model. b) Viscosity distribution in the furnace for different draw ratios at T=180°C. Solid lines correspond to generalized Newtonian model and dotted line to Newtonian model.

8. Conclusions

Comparison between the experimental results and theoretical modelling is currently underway. In general, qualitative agreement between theory and experiment of the dependence of a core collapse on the draw parameters is readily observable. As the results of our numerical analysis correspond to the equilibrium drawing conditions, to make a quantitative comparison with experiments we have to ensure that our experimental drawing has reached equilibrium and is not dominated by transients.

Acknowledgment

Funding for this work comes partially from the contribution of Canada Research Chairs, NSERC, FQRNT, Canada Institute for Photonic Innovations, projects BP5 and FP3.

References and links

1.

T. Katsuyama and H. Matsumura. Infrared Optical Fibers, (Adam Hilger, Bristol, England, 1989).

2.

M. Saito and K. Kikuchi, “Infrared optical fiber sensors,” Opt. Rev. 4, 527–538 (1997). [CrossRef]

3.

J. Sanghera and I. Aggarwal. Infrared Fiber Optics, (CRC, Boca Raton, USA, 1998).

4.

S. Martellucci, A.N. Chester, and A.G. Mignani Optical Sensors and Microsystems : New Concepts, Materials, Technologies, 1st ed. (Springer, New York, USA, 2000).

5.

J.A. Harrington, “A review of IR transmitting, hollow waveguides,” Fib. Integr. Opt. 19, 211 (2000). [CrossRef]

6.

Y.W. Shi, K. Ito, Y. Matsuura, and M. Miyagi, “Multiwavelength laser light transmission of hollow optical fiber from the visible to the mid-infrared,” Opt. Lett. 30, 2867–2869 (2005). [CrossRef] [PubMed]

7.

P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

8.

J. Canning, E. Buckley, and K. Lyytikainen “Propagation in air by field superposition of scattered light within a Fresnel fiber,” Opt. Lett. 28, 230–232 (2003). [CrossRef] [PubMed]

9.

J. Canning, E. Buckley, K. Lyttikainen, and T. Ryan, “Wavelength dependent leakage in a Fresnel-based air-silica structured optical fibre,” Optics Communications 207, 35 (2002). [CrossRef]

10.

C.M. Smith, N. Venkataraman, M.T. Gallagher, D. Muller, J.A. West, N.F. Borrelli, D.C. Allan, and K.W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003). [CrossRef] [PubMed]

11.

H. Han, H. Park, M. Cho, and J. Kim “Terahertz pulse propagation in plastic photonic crystal fibers,” CLEO/Pac - Conference on Lasers and Electro-Optics, postdeadline paper Cat. No. 01TH8557, 22 (2001).

12.

J. Choi, K.Y. Kim, and U.C. Paek, POF - Proceedings of Plastic Optical Fibres, 355 (2001).

13.

M.A. van Eijkelenborg, A. Argyros, G. Barton, I.M. Bassett, M. Fellew, G. Henry, N.A. Issa, M.C.J. Large, S. Manos, W. Padden, L. Poladian, and J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication and characterisation,” Opt. Fiber Techn. 9, 199–209 (2003). [CrossRef]

14.

T. Katagiri, Y. Matsuura, and M. Miyagi, “Photonic bandgap fiber with a silica core and multilayer dielectric cladding,” Opt. Lett. 29, 557–559 (2004). [CrossRef] [PubMed]

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]

16.

G. Vienne, Y. Xu, C. Jakobsen, H.J. Deyerl, J. Jensen, T. Sorensen, T. Hansen, Y. Huang, M. Terrel, R. Lee, N. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports,” Opt. Express 12, 3500–3508 (2004). [CrossRef] [PubMed]

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]

18.

A. Dupuis, Y. Gao, N. Guo, E. Pone, N. Godbout, S. Lacroix, C. Dubois, and M. Skorobogatiy “Biodegradable, Double-Core, Porous Optical Fiber,” CLEO - Conference on Lasers and Electro-Optics, postdeadline paper Cat. No. CPDB5, (2006).

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.

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]

23.

J.A. Burgman “Liquid glass jets in the forming of continuous fibers,” Glass Technol. 11, 110–116 (1970).

24.

F.T. Geyling, “Basic fluid dynamic consideration in the drawing of optical fibers,” Bell Sys. Tech. J. 55, 1011–1056 (1976).

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]

28.

K. Lyytikainen. Control of complex structural geometry in optical fibre drawing, (Phd Thesis, University of Sydney, 2005).

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]

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]

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]

34.

R.B. Bird, R.C. Amstrong, and O. Hassager. Dynamics of polymeric liquids, vol. 1, Fluid mechanics, (JohnWiley & Sons, New York, 1987).

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]

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]

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]

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. T. Katsuyama and H. Matsumura. Infrared Optical Fibers, (Adam Hilger, Bristol, England, 1989).
  2. M. Saito and K. Kikuchi, "Infrared optical fiber sensors," Opt. Rev. 4, 527-538 (1997). [CrossRef]
  3. J. Sanghera and I. Aggarwal. Infrared Fiber Optics, (CRC, Boca Raton, USA, 1998).
  4. S. Martellucci (editor), A.N. Chester (editor), and A.G. Mignani (editor). Optical Sensors and Microsystems : New Concepts, Materials, Technologies, 1st ed. (Springer, New York, USA, 2000).
  5. J.A. Harrington, "A review of IR transmitting, hollow waveguides," Fib. Integr. Opt. 19, 211 (2000). [CrossRef]
  6. Y.W. Shi, K. Ito, Y. Matsuura, M. Miyagi, "Multiwavelength laser light transmission of hollow optical fiber from the visible to the mid-infrared," Opt. Lett. 30, 2867-2869 (2005). [CrossRef] [PubMed]
  7. P. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003). [CrossRef] [PubMed]
  8. J. Canning, E. Buckley, and K. Lyytikainen "Propagation in air by field superposition of scattered light within a Fresnel fiber," Opt. Lett. 28, 230-232 (2003). [CrossRef] [PubMed]
  9. J. Canning, E. Buckley, K. Lyttikainen, and T. Ryan, "Wavelength dependent leakage in a Fresnel-based air-silica structured optical fibre," Optics Communications 207, 35 (2002). [CrossRef]
  10. C.M. Smith, N. Venkataraman, M.T. Gallagher, D. Muller, J.A. West, N.F. Borrelli, D.C. Allan, K.W. Koch, "Low-loss hollow-core silica/air photonic bandgap fibre," Nature 424, 657-659 (2003). [CrossRef] [PubMed]
  11. H. Han, H. Park, M. Cho and J. Kim "Terahertz pulse propagation in plastic photonic crystal fibers," CLEO/Pac - Conference on Lasers and Electro-Optics, postdeadline paper Cat. No. 01TH8557, 22 (2001).
  12. J. Choi, K.Y. Kim, U.C. Paek, POF - Proceedings of Plastic Optical Fibres, 355 (2001).
  13. M.A. van Eijkelenborg, A. Argyros, G. Barton, I.M. Bassett, M. Fellew, G. Henry, N.A. Issa, M.C.J. Large, S. Manos, W. Padden, L. Poladian, J. Zagari, "Recent progress in microstructured polymer optical fibre fabrication and characterisation," Opt. Fiber Techn. 9, 199-209 (2003). [CrossRef]
  14. T. Katagiri, Y. Matsuura, M. Miyagi, "Photonic bandgap fiber with a silica core and multilayer dielectric cladding," Opt. Lett. 29, 557-559 (2004). [CrossRef] [PubMed]
  15. B. Temelkuran, S.D. Hart, G. Benoit, J.D. Joannopoulos, Y. Fink "Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission," Nature 420, 650 (2002). [CrossRef] [PubMed]
  16. G. Vienne, Y. Xu, C. Jakobsen, H.J. Deyerl, J. Jensen, T. Sorensen, T. Hansen, Y. Huang, M. Terrel, R. Lee, N. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, "Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports," Opt. Express 12, 3500-3508 (2004). [CrossRef] [PubMed]
  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]
  18. A. Dupuis, Y. Gao, N. Guo, E. Pone, N. Godbout, S. Lacroix, C. Dubois and M. Skorobogatiy "Biodegradable, Double-Core, Porous Optical Fiber," CLEO - Conference on Lasers and Electro-Optics, postdeadline paper Cat. No. CPDB5, (2006).
  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, 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.
  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]
  23. J.A. Burgman "Liquid glass jets in the forming of continuous fibers," Glass Technol. 11, 110-116 (1970).
  24. F.T. Geyling, "Basic fluid dynamic consideration in the drawing of optical fibers," Bell Sys. Tech. J. 55, 1011-1056 (1976).
  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, 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]
  28. K. Lyytikainen. Control of complex structural geometry in optical fibre drawing, (Phd Thesis, University of Sydney, 2005).
  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]
  32. StevenG. 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]
  33. L.J. Cummings, 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]
  34. R.B. Bird, R.C. Amstrong, and O. Hassager. Dynamics of polymeric liquids, vol. 1, Fluid mechanics, (JohnWiley & Sons, New York, 1987).
  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]
  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]
  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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited