We investigate the refractive index profile evolution during the tapering of fused couplers. We assume the refractive index to be a linear function of the dopant concentration. The evolution of the refractive index profile is obtained by solving the diffusion-convection equation. The couplers are cleaved at different positions along the coupler axis, and the real refractive index profiles, measured using Refracted-Near-Field technique, are compared with the numerical solutions.
©2004 Optical Society of America
Fused fiber couplers have a wide range of uses in the optical networking industry. They are used as power splitters, wavelength-division multiplexers, polarisation beam splitters, mode splitters, filters etc. The power exchange along a coupler can be described either in terms of the coupling of the fundamental modes of the individual guides  or in terms of the beating of the lowest-order symmetric (even) and antisymmetric (odd) modes of the composite waveguide . The knowledge of propagation constants is necessary to calculate the coupling coefficient and a model for the refractive index profile must be determined.
The weakly fused tapered coupler, when properly designed, can function as a polarization beam splitter [3–5]. The form birefringence which corresponds to the difference in the coupling coefficients between orthogonal polarization states causes the polarization splitting in the coupler. The form birefringence is small, and a accurate model of the refractive index profile is crucial in the calculation of such small quantities.
Fused couplers are fabricated by fusing and tapering two fibers together. In a recent paper , we present a model for the fusion of the fibers, a process known as viscous sintering. The fusion 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 purpose of this paper is to provide a realistic model for the tapering of the coupler. This model is important for coupler designer in order to determine the fiber core distribution along the tapering and waist region of the fused coupler.
A simple model for the tapering of the coupler is given by Lacroix et al.  where only the mass conservation is taken into account. The tapering of the coupler is also investigated by Farget et al.  where a simple unidirectional rheological model is used. Cummings and Howell  investigate tapering in a more rigorous way by using 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 two-dimensional fusion process, except for the fact that the cross-section convects with the fluid in the axial direction and its area changes over time. However their study is restricted to the determination of the governing equations for the external interface evolution.
The inclusion of the dopant diffusion in the model is important [10–13]. So, the evolution of refractive profile is investigated by solving the diffusion-convection equation for different dopants within optical fiber cores. During the tapering of the coupler the fusion continues. Thus, we also investigate how the cross-sections along the coupler axis evolve during the tapering of the coupler.
In the following section we present the equations governing the tapering of the coupler and we explain how to determine the parameters that enter in these equations. Then in Section 3 we give the numerical solutions of these equations for an arbitrary fabrication recipe and compare them with the refractive index profile of real fused couplers. Finally, in Section 4, the effect of the control parameters is investigated.
2. Governing equations
Fused-tapered couplers 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. We use a travelling flame as heat source and the ends of the coupler are drawn with a constant velocity.
We assume that softened glass may be considered as a viscous incompressible Newtonian liquid. The magnitudes of inertial and gravitational forces in comparison to viscous and capillary forces are negligible and we may consider the tapering of the coupler as a Stokes flow.
In this paper we assume the viscosity to be a function of the temperature alone, ignoring its dependence on chemical composition. We consider a varying axial temperature, but its value across the cross-section will be considered uniform. Because of the reduced transversal dimensions of the optical fibers, thermal conduction maintains a constant temperature across the cross-section. This dependence is also considered constant in time, that is, decoupled from the flow equations. Such an assumption is only valid when radiative heat loss dominates convection and conduction.
Cummings and Howell  show that the three dimensional Stokes equation can be simplified if the slenderness of the geometry is taken into account. They derive these two important equations:
The axial stress balance equation is
where z is the axial coordinate and µ, S, v, γ and Γ, which are all z-dependent, denote the viscosity, the cross-section area, the axial velocity, the surface tension coefficient and the circumference of the cross-section respectively.
The equation representing global conservation of mass is given by
where t denotes the time.
These equations are obtained considering a general case (Navier-Stokes equations) and a uniform viscosity. Thus, we have simplified them because we consider a Stokes flow, but we have inserted a varying axial viscosity.
In order to obtain the time evolution of S(z), v(z) and Γ(z), we need an additional relation between these variables. This relation can be obtained between S(z) and Γ(z), if the evolution of the cross-section shape is known. Another conclusion of Ref.  is that the coupler cross-section evolves according to a classical two-dimensional fusion problem, provided that the cross-section is convected with the fluid in the axial direction and its area is re-scaled according to Eq. (2). The solution of the two-dimensional fusion problem is given in Ref. , completing so the system of equations.
To proceed with a numerical solution, we separate the process in small time steps. From the initial configuration of the coupler S(z) and Γ(z), and the boundary conditions for v (the constant pulling speed), we numerically solve Eq. (1) to find v(z). Then we solve Eq. (2) to find S(z) at the next time step. Γ(z) at the next time step is evaluated by evolving the cross-sections along the coupler axis according to the model of Ref. , taking into account the cross-section z-coordinate (the values of µ(z), γ(z)) and its area.
A concern is how to determine the values of µ(z) and γ(z) which are temperature dependent. In order to solve the convective diffusion equation for the cross-sections along the coupler axis (we neglect the inter-section diffusion) the value of diffusion coefficient D(z) is also needed. Normally, one has to measure the temperature distribution T(z) along the coupler axis and then use the relations µ(T), γ(T) and D(T). The main drawback using a travelling flame as the heat source is that the temperature cannot be monitored during the coupler fabrication. Furthermore, µ(T), γ(T) and D(T) are not well known. Thus, we propose the following procedure.
In order to determine µ(z)/γ, two fibers are heated in the same way as the coupler during the tapering, but no drawing is applied. From the distribution of the degree of fusion, or equivalently the distribution of the fused structure width, we can easily calculate µ(z)/γ as explained in Appendix B of Ref. . The degree of fusion is defined as f=[W(t 0)-W]/[W(t 0)-W(t ∞)], where W denotes the major cross-sectional axis dimension, W(t 0) its value when the fibers first enter in contact and W(t ∞) its value when the fibers are completely fused into a cylinder.
The remaining D(z) is determined by heating a single fiber in the same way. Then, the fiber is cleaved at some different positions along its axis and a one-dimensional measurement of the refractive index profile is taken. From this measurement and the time of heating, the value of D is obtained at these positions. Then the value of D(z) is approximated by a smooth curve passing at the measured values.
3. Solutions and comparisons
We now apply the method described in the previous section to the following arbitrary recipe of tapering:
Two SMF28 Corning fibers are fused to a degree of fusion f=0.358. Then the tapering begins by heating a zone of 3 mm with a flame travelling at 2 mm/s. The ends of the coupler are drawn with a pulling velocity 50 µm/s. The tapering process ends after 30.5 s. The distance between the torch and the fibers is 3 mm.
In order to determine the evolution of the cross-sections during the tapering, we need to know their initial profile, that is, the profile after the fusion to f=0.358. This profile is obtained using the model presented in Ref. . In Fig. 1 we give the real refractive index profile measured using Refracted-Near-Field (RNF) technique , the results of our computations and their superposition. As in Ref.  a good agreement is observed. The difference between the core and cladding refractive index of a normal SMF28 fiber is considered as corresponding to a 3.4% value of the dopant concentration.
We now need the values of µ(z)/γ and D(z). After heating two fibers as we do during the tapering, but without drawing their ends, we obtain the structure shown in Fig. 2. From the width distribution of this structure we obtain a distribution for the degree of fusion shown in Fig. 3(a) which in its turn gives a distribution for given in Fig. 3(b).
We heat a single fiber in the same way, and we cleave it at three different positions. Then we measure the radial (one-dimensional) index profile using the RNF technique. From this profile and the time of heating, we can easily evaluate the value of diffusion coefficient. A smooth curve passing at the measured values is considered as D(z). We have considered a Gaussian distribution for log[D(z)], as shown in Fig. 4 for the half of the coupler.
Knowing the initial configuration of the coupler and the values of µ(z)/γ and D(z), we proceed now with the numerical solution as explained in the previous section. The obtained results are given in Fig. 5, where we present the normalized width and degree of fusion distribution, for the half of the coupler, after the tapering.
The normalized width is defined as the ratio between the major cross-sectional axis dimension and its value for the untapered coupler. The major cross-sectional axis profile is what we see in photographs like that presented in Fig. 2. This profile reflects the changes in both cross-sectional area and degree of fusion. The fact that normalized width and the degree of fusion, both dimesionless quantities, have values between 0 and 1 enables us to plot them in the same graph.
We have also inserted in Fig. 5 the measured normalized width and degree of fusion at some positions on the coupler axis, where we have cleaved the coupler. The coordinate z of these cross-sections is obtained by matching the measured degree of fusion with the calculated one. A good agreement between the measured and the calculated cross-section width can be observed.
We remark that the differences between measured and calculated cross-section width, although small, are more evident near the center of the coupler, where the temperature is higher. We attribute these differences to the loss of some surface material, as a consequence of the interaction between the flame and the fibers, at very high temperatures. We observe a very good agreement between the measured and calculated cores distance for all the measured cross-sections. For two cases, the worst and best match in Fig. 5 for the cross-section width, we present in Figs. 6 and 7, a detailed comparison between the results of our computations and the measured refractive index profile. A good agreement between the measured and calculated refractive index profile is observed for all the cross-sections. The measured refractive index contours are non-smooth in the center of the core because few measured points are there.
We have presented here a case where the coupler is heated in a rather exaggerated way compared to the fabrication of real fused couplers. This allows us to test the model in an extreme case, and also to highlight the fact that the fusion continues much more rapidly in the center of the coupler. Two factors are responsible for the acceleration of the fusion in the center of the coupler. First, the viscosity is much lower there, and second, the dimensions of the cross-section get smaller as the tapering goes on. In Eq. (3) of Ref.  it is shown that the real time of fusion is linearly proportional to both of them.
For other tapering recipes where lower temperatures are used, we expect negligible loss of surface material and a relatively better agreement between the model and the experiment.
4. Control parameters
We now investigate the effect of the control parameters, which are the temperature distribution and the pulling velocity.
The effect of temperature distribution is investigated by observing how the results of the previous section will change if two other temperature distributions are considered. First, we consider the same shape of the temperature distribution, but at lower values. And second, we consider the same maximum value for the temperature but a flat distribution in the center with rather abrupt changes of the slope at the ends of the coupler. These two situations are presented in Fig. 8.
In Fig. 9 we present the results of our calculations for the two situations. It is seen that, when lower temperatures are considered, lower degree of fusion values are obtained, but the width profile doesn’t change too much. Thus, if a small variation for the degree of fusion is needed, the temperature must be as low as possible. Also, as seen in Fig. 9(b), the width and the degree of fusion profile follow the temperature distribution. These examples give an idea how the temperature distribution affects the coupler shape.
We consider now the effect of the pulling speed by observing how the results of the previous section will change if the ends of the coupler are drawn with twice the velocity. The tapering time is taken the half of the previous one in order to have the same coupler length. The results are presented in Fig. 10. It is seen that, when higher pulling speed are applied, lower values of the degree of fusion are obtained because the degree of fusion is proportional to the elapsed time. The coupler width profile is not affected too much by the pulling speed.
A realistic model for the evolution of refractive index profile of fused-fiber couplers during the tapering process has been described. The material convection and the dopant diffusion are taken into account. Two preliminary measurements are necessary to determine the parameters that enter in the governing equations. A good agreement is observed between the numerical results and the refractive index profiles of real fused couplers measured with RNF technique. The model presented here for the tapering together with the model for the fusion  give a complete description of refractive index profile of the fused-tapered couplers.
We neglected in this paper the viscosity dependence on the chemical composition. This resulted in good predictions for the couplers composed by single-mode fibers where the core diameter is small. For the couplers composed by large core fibers, the non-uniformity of the viscosity across the cross-section may slightly affect the accuracy of the computations.
Also, if more than one dopant is present in the fibers core the determination of the diffusion coefficients distribution becomes more complicated. In this case, perhaps it is better to determine an approximate temperature distribution and to use the values of D(T) found in the literature.
References and links
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