We investigate the refractive index profile of the cross-section of fused type fiber-optic couplers by solving the convective diffusion equation. We assume the refractive index to be a linear function of the dopant concentration. The viscous sintering of the optical fibers is considered as the motion of an incompressible Newtonian fluid which is driven by the surface tension acting at the free boundary. The internal velocity field is obtained using conformal mapping methods. We present numerical solutions of the resulting equations and compare them with experimental observations.
©2004 Optical Society of America
Fused-tapered fiber couplers are widely used in all-fiber devices such as multiplexers, demultiplexers, filters, etc. They are fabricated using a combination of fusion and tapering. Two or more optical fibers have their coatings removed and are then brought into contact along their length. The adhering fibers are then heated until a certain amount of coalescence takes place due to the surface tension. Then the fibers are drawn along their length under tension into a taper until the required functionality of the coupler is achieved. To accurately analyse optical properties of fused-tapered fiber couplers, a model for the refractive index profile must be determined.
A simple model for the refractive index profile is given by Lacroix et al.  where the cross-section interface is represented by two intersecting circles and by taking into account only the material conservation. The circular cross-section of the cores is assumed to remain unchanged, apart from the re-scaling of the core radius to account for mass conservation under tapering and no diffusion of core dopants is considered implying an unchanged core refractive index. An improvement of this model is made by Farget et al.  where fusion and tapering are considered as coupled phenomena and a simple unidirectional rheological model is used to describe the tapering of the coupler, but its cross-section is still represented as in .
The coupling between fusion and tapering is also investigated by Cummings and Howell , where the problem is tackled with the equations of fluid mechanics. They show that the evolution of the cross-section during the tapering of the coupler is the same as in the simple fusion process except for the fact that the cross-section area changes with time. However their study is restricted to the determination of external interface evolution.
The purpose of this paper is to provide a realistic model of the refractive index profile evolution during the fusion process, a phenomenon known as viscous sintering. Understanding this evolution is important for coupler designer because this evolution combined with the results of  makes possible to determine the fiber core distribution along the tapering and waist region of the fused coupler.
We assume the refractive index to be a linear function of the dopant concentration. Thus to determine the evolution of the cross-section of the coupler, one has to solve the diffusion-convection equation for different dopants within optical fiber cores. We use a complex-variable method to determine the internal velocity field in order to solve this equation. This method is introduced by Hopper  and then reformulated and extended by Richardson [5–7]. However this method is developed to give the external interface evolution, so some manipulations are needed to derive a equation for the internal velocity.
In the following section we first consider the diffusion-convection equation and we explain the Hopper-Richardson method. Then in Section 3 we give the results of our calculations and compare them with cross-section photos of real fused couplers. Mathematical details of the method are given in the appendices.
2. Viscous sintering
The dopant diffusion in a liquid (softened glass in our case) is governed by the equation
where c, t,u and D denote the dopant concentration, the time, the velocity vector and the diffusion coefficient respectively. We must determine the velocity field u in order to solve this equation, and we use the complex-variable method introduced by Hopper and Richardson [4–6]. To do so, we assume that softened glass may be considered as a viscous incompressible Newtonian fluid, which is surrounded entirely by a free surface. The motion is driven solely by the action of a constant surface tension acting at that free surface. The magnitudes of inertial and gravitational forces are considered negligible compared with viscous or capillary forces. The governing equations (momentum and continuity) are in this case
where p and µ denote the pressure and the viscosity respectively. The stress boundary condition is that the surface traction is in the outward normal direction and of magnitude γκ, where γ is the coefficient of surface tension and κ is the mean curvature of the external interface.
In this paper we assume the viscosity to be a function of the temperature alone. Because of the reduced transversal dimensions of the optical fibers, thermal conduction maintains a constant temperature in the cross-section. Therefore we consider a constant average viscosity.
The mathematical analysis of the Hopper-Richardson method involves the complex variable theory. The evolution of the cross-section is considered in the complex η-plane and is described in terms of a time-dependent conformal mapping η=Ω(ζ, t) from a fixed reference domain, the unit disc |ζ|≤1, in the complex ζ-plane. This conformal mapping is shown schematically in Fig.1. The zero time is taken as the initial contact of the two fibers. The analytical expression of the mapping is given by Eq. (10) in Appendix A.
The components ux and uy, along two perpendicular axes x and y, of the velocity vector u, are considered as the real and the imaginary part of a complex velocity u=ux+iuy in η-plane. We derive in Appendix A the Eq. (17) which gives the velocity in the form u=u(ζ, t). So to evaluate the velocity at a point η(x,y), at a particular time, one has to evaluate first ζ=Ω-1(x,y) and then insert this value in Eq. (17). Although the velocity in that equation is given in an analytical form, there are several Cauchy integrals or their derivatives which must be evaluated numerically. Finally, to solve Eq. (1) we apply an explicit finite difference scheme.
3. Results and comparisons
Consider first how the cross-section of a coupler composed by two different multi-layer fibers will evolve in the case of no dopant diffusion (D=0). To determine the evolution of these layers instead of solving Eq. (1) we follow the trajectories of some markers which are distributed along the layers boundaries. In Fig. 2, we present the initial configuration and our numerical solution for the structure of the coupler cross-section at a fusion degree f=0.9 (the definition of the fusion degree is given by Eq. (14) in Appendix A).
Consider now the fusion of two standard SMF28 Corning fibers where the dopant is germanium. The cladding and core diameters are 125 µm and 9 µm respectively. We assume an initial uniform distribution of the dopant in the core with c=3.4%. In Appendix B we estimate the viscosity to be µ=1.15×105Pa s at our heating temperature and for the coefficient of surface tension.
We present in Figs. [3–5] the results of our computations, the photographs and the comparisons for the cross-section of the coupler at three different values of fusion degree. In order to match the cores width, we have assumed for the diffusion coefficient. The seemingly exact agreement between calculated core positions and shapes, and the experiment confirms the accuracy of our model.
The temperature is the only fabrication parameter one can control during the fusion process, so it is natural to investigate it’s effect on the cross-section profile.
The Eqs. (2) can be non-dimensionalized by setting
where an asterisk denotes the real variable and R a typical dimension of the cross-section. The governing equations become
We see that neither µ nor γ enter in these equations or the boundary conditions. Suppose we consider only the material convection without the dopant diffusion. In this case the real values of the viscosity and the coefficient of surface tension are needed only to determine the time it takes to arrive at a given fusion degree. The structure of the cross-section, once arrived at that fusion degree is the same for any value of these parameters.
If the diffusion is considered, the dimensionless diffusion coefficient is given by
where D*is the real diffusion coefficient. So the cross-section evolution is determined by the value of . As we discuss in Appendix B, γ varies little with the temperature, but the diffusion coefficient and the viscosity vary exponentially. However, the viscosity and the diffusion coefficient vary in opposite directions keeping D*µ almost constant.
We fused the SMF28 fibers several times to the same fusion degree at different temperatures. The only difference we could observe was the time of fusion.
Our heating source is a propane micro torch. When the distance of the torch from the fibers is 4mm where we estimate the temperature to be about 1400 °C, it takes 30s to arrive at a fusion degree f=0.355. At the distances 5mm, 6mm and 7mm it takes 48.5s, 99s and 314s respectively to arrive at that fusion degree. At the distances greater than 9mm where we estimate the temperature to be lower than 900 °C the fibers almost do not fuse at all. When the temperature increases, the viscosity decreases and the time of fusion decreases too. The relation between the viscosity and the time of fusion is given in Appendix B.
A realistic model for the refractive index profile of fused-fiber couplers cross-section has been described. The material convection and the dopant diffusion are taken into account. In order to exploit the complex-variable methods the dependence of viscosity on the chemical composition is ignored and we have considered an average viscosity. A good agreement is observed between the numerical results and the photographs of real fused couplers. The temperature effects only the time of fusion. We have presented here only the case of couplers with only two fibers but this model is valid for couplers with any number of fibers and we have already performed such computations. If in the initial configuration of the cross-section we have a double-connected region, that is, there is a hole in the initial configuration (e.g. the coupler 3×3 in triangle), the equations of Appendix A must be modified as shown by Richardson .
5. Appendix A. The internal velocity
In the following, primes denote complex derivatives with respect to the independent complex variable, and an overbar, the complex conjugate. The Eqs. (2) are non-dimensionalized as shown in Section 3. Hopper  shows that the complex velocity in dimensionless form can be expressed in terms of two analytic functions ϕ(ζ) and ψ(ζ) in the form
An expression for ψ will be given later in Eq. (15). The function ϕ(ζ) is shown to be
where the function F (ζ, t) is defined by the Cauchy integral
The contour integral is along the unit circle, |σ|=1. Thus, σ indicates points on the unit circle of ζ-plane, while ζ is used for general points on this plane.
According to Richardson , in a general situation with N-touching fibers the form of mapping is given by: where the 2N parameters, βj and λj, for j=1,2, ⋯,N, with |λj|<1, are subject to the 2N conditions
where rj are the initial radii of fibers. In the case of two fibers we have real parameters which verify the relations: β̄1=β 1>0, 1=0, β̄2=β 2>0, 0≤ 2=λ2≡λ<1 and finally the mapping has the form
In the above expression, λ, β 1 and β 2 are evaluated from cross-section width and initial radii of the fibers as we show in the following.
The image of the boundary of the reference domain gives the boundary of the cross-section. So, if we replace ζ=eiθ in Eq. (10) we find these parametric equations for the cross-section cladding-air interface
The fiber’s centers are on the x-axis. Thus, the cross-section width is given by
We know that, at the initial (t=0) and final (t=∞) instant, we have W(t 0)=2r 1+2r 2 and . So, the fusion degree is defined as
To obtain ψ (ζ) in |ζ|<1 we exploit the analytical continuation of ψ. After developing (15) the only non analytical term is σ̄. So we replace it by 1/σ (at the boundary we have σ=eiθ), and then ψ (ζ) is simply obtained by replacing σ by ζ. So, the analytical continuation of in |ζ|<1 is
Finally, substituting φ (ζ), ψ (ζ) and Ω(σ) in Eq. (6), we obtain for the complex velocity
The time derivatives can be calculated using the second equation in (9). Though straightforward the computations in (17) are complicated and tiresome.
6. Appendix B. The fiber viscosity
We present here a simple method to evaluate the fiber viscosity at a given temperature. The second equation in (9) in the case of two coalescing fibers has the form
As explained by Hopper  and Richardson [5, 6], if the initial contact of the fibers is at t=0 (λ=1), the solution is valid for t>0 (λ<1). Since F(λ) is positive, we see from Eq. (18) that λ decreases when t increases. We consider λ=0.9999 as the start value. Integrating Eq. (18) and switching to real variables we obtain for the real total time
From this equation we can evaluate the viscosity if we know t, γ and λ.
Unlike the viscosity, the coefficient of surface tension γ is a slowly varying function of the temperature. For silica, according to Ref.  we have
where the temperature T is in °C. So an approximate value of the temperature is sufficient for a reasonable estimation of γ. For the couplers presented in Section 3 we estimate the heating temperature to be about 1400 °C, so we may consider .
For the cross-section shown in Fig. 3 (fiber diameter 125µm) it takes 30s to arrive at the fusion degree 0.355 at our heating temperature, which according to Eqs. (11),(13) and (14) corresponds to λ=0.8686. Substituting these values in Eq. (19) and evaluating numerically the integrals we find for the viscosity: µ=1.15×105 Pa s.
References and links
2. C. Farget, J.P. Meunier, and P.E. Bonneau, “An Efficient Taper Shape Model for Fused Optical Fiber Components,” Int. Conf. Fiber Opt. Photon. Photonics-96” 1141–1146 (1996).
3. 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]
4. R. W. Hopper, “Plane Stokes flow driven by capillarity on a free surface,” J. Fluid Mech. 213, 349–375 (1990). [CrossRef]
5. S. Richardson, “Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension,” Eur. J. Appl. Math. 3, 193–207 (1992). [CrossRef]
6. S. Richardson, “Two-dimensional Stokes flows with time-dependent free boundaries driven by surface tension,” Eur. J. Appl. Math. 8, 311–329 (1997).
7. S. Richardson, “Plane Stokes flows with time-dependent free boundaries in which the fluid occupies a doubly-connected region,” Eur. J. Appl. Math. 11, 249–269 (2000). [CrossRef]
8. Glass: Science and Technology (Academic Press, 1986)