We present a model which allows us to accurately simulate the fabrication process of complex-shaped tapered fibers. The range of possible profiles is only limited by the properties of the heat source used to shape the fiber. The model takes into account the motion of the heat source relative to the fiber as well as its temperature distribution. Our measurements and corresponding finite element method (FEM) simulations have shown a strong dependency of the temperature distribution along the fiber axis on the actual diameter of the fiber. The inclusion of this relation in the model proved to be crucial for the accuracy of the results. Our model has been verified experimentally by fabricating tapered fibers with a sinusoidally modulated waist. A comparison to the profile predicted by our model reveals an excellent agreement.
© 2010 Optical Society of America
Tapered optical fibers have gained a lot of interest in the last years due to their unique physical and optical properties not found in standard fibers. This opens up possibilities for many new applications. As it is well known, the reduction of the fiber diameter leads to an enhancement of the light intensity in the taper region. Hence the efficiency of nonlinear processes taking place during the propagation through the fiber raises dramatically, by up to four orders of magnitude. The fact that the waveguide dispersion of optical fibers depends strongly on the diameter  yields the possibility to design and fabricate fibers whose dispersion exactly matches the requirements of a certain application by tapering down to the corresponding diameter . In particular, this allows us to shift the zero dispersion wavelength so that the overall dispersion becomes anomalous at visible or near-infrared wavelengths of commonly used laser pump sources. The combination of one of these efficient nonlinear effects, namely self-phase modulation, and the propagation in the anomalous dispersion regime enables the formation of optical solitons , whose fission  may trigger the development of an octave-spanning supercontinuum , given a carefully chosen set of experimental parameters . An additional possibility to influence the optical properties of the fiber is given by using a suitable Photonic Crystal Fiber (PCF)  instead of a standard step-index fiber as the fiber to be tapered down [8, 9].
If the diameter of the fiber is decreased down into the range of the laser wavelength coupled in, we should pay closer attention on the evanescent fields outside the fiber. These fields can be used to trap and detect single atoms or molecules with high sensitivity . If a significant fraction of the light intensity is contained in these evanescent fields, the influence of the surrounding medium on the optical properties of the fiber can be taken as a tool to fine-tune the overall dispersion . Using the self-modulated taper drawing technique [12, 13], even tapered fibers with sub-wavelength diameters in the range of tens of nanometers have been demonstrated.
A common setup, easy to implement, for fabricating fibers consists of a motor which pulls apart the fixed ends of the decoated fiber while a heating device is travelling along the fiber axis. The heating device moves according to a certain program and softens the fiber at its current position. The resulting shape of the tapered fiber is dependent on its initial geometry, the temperature distribution of the heating device along the fiber axis, and particularly the motion of the heating device with respect to the fiber during the pulling process. Throughout the process, we keep the parameters of the heating device constant, except its position. This assumption is valid for a flame with constant, stable gas flow used at the flame-brushing technique , as well as for a ceramic or resistive element kept at a certain temperature by a given heating current [15, 16], or also for a high power laser with fixed optical parameters . In the latter case, however, additional attention is needed since the conversion from the energy of the laser light into thermal energy by absorption depends on the current shape of the fiber [18, 19]. A ceramic tube around the fiber that absorbs the laser power and gives a constant heat to the fiber can aid this process .
To tailor a tapered fiber with a given profile along the fiber axis one needs to find an algorithm for the movement of the heating device with respect to the fiber so that the desired shape is fabricated. Birks and Li  have presented a simple model using the assumption of an ideal rectangular distribution of temperature with a variable width, which is capable of delivering the program of motion of the heating device producing tapered fibers which consist of a homogeneous waist connecting two identical decreasing taper transitions. Although this kind of tapered fibers is widely used successfully in the fields of the applications summarized above, there are other possible applications demanding a different shape. For example, a sinusoidally modulated waist has been proposed for tailoring the spectral properties of a supercontinuum generated by the fiber . Another advantage of this shape is the avoidance of multiple splicings during the fabrication of dispersion-alternating fibers for ultrashort laser pulses , which introduce notable losses. Another concept, inserting a barrel-like bulge serving as a bottle resonator, is predicted to excite whispering gallery modes [24, 25], which enhance the sensitivity of the fiber when probing for single atoms or molecules .
Within this work, we present an easy-to-use model describing the drawing process taking into account the motion of the heating device as well as its temperature distribution. We show that without complicated fluid mechanics we are able to deduce programs of motion to fabricate new types of tapered fibers consisting for example of sinusoidally modulated or conically shaped waists.
2. The model
To keep the setup as simple as possible, we assume an asymmetric tapering rig consisting of a motor which pulls the fixed fiber with a constant speed v0 to one direction and a second motor which moves the heating device along the fiber axis, determining the shape of the resulting tapered fiber. The commonly used setup where two motors pull the fiber symmetrically can easily be simulated by adding a constant v0/2 to the speed of the motor moving the heating device. This transformation moves the coordinates from the lab frame of reference to the fiber frame of reference.
Assuming cylindrical symmetry, we choose to describe the shape of the fiber by its profile of the radius r(z) along the fiber axis (z-axis). The progress during the tapering is given by the distance x the pulling motor has moved since the beginning. This x-coordinate can also be regarded as time coordinate given by the transformation t = x/v0. Thus, r(z,x) denotes the shape of the fiber throughout the complete process. Let x 0 be the final elongation of the fiber. The considered interval [0,z 0] of the z-axis has to be set large enough that r(z = 0,x) = r(z = z 0,x) = r 0 is valid for all x, where r 0 denotes the constant initial radius of the fiber. Since the volume of the total fiber is conserved, we can state that by elongating the fiber by an amount of dx, we always remove the same volume from the interval [0,z 0], namely dV = −π r 0 2 dx. Thus, the change of the volume within this interval is given by
To take the effect of the heating device into account we introduce a function Θ(z) which is assumed to include the temperature distribution of the heating device as well as the fiber’s properties, e.g., its viscosity. Hence, Θ(z) can be regarded as a measure of the fiber’s deformability caused by the heating device. First, we have assumed it to be independent of the current state of the fiber, but as described in the next section, it turned out to be dependent on the current fiber diameter. The specific shape of this function has to be calibrated for each setup as described in the next section. We choose to set Θ(z) = 0 outside the heating zone and normalize it without loss of generality according to
During the pulling of the fiber the parts of the fiber within the heating zone are stretched, the parts towards the fixed end stay unchanged, whereas the part at the side of the pulling motor is just shifted. This behavior can be described by the equation
Here, A serves as a normalization introduced to ensure volume conservation; f(z) determines the type of change of the fiber shape: For f(z) = 0 the fiber stays unchanged, whereas a constant f(z) > 0 shifts the fiber shape to larger z-values. For a non-constant f(z) the fiber is stretched. The larger the slope of f(z) the larger is the amount of the stretching. These facts invite to connect f(z) and Θ(z) by
These settings deliver the desired behavior during the pulling as described above and summarized in the table Tab. 1.
If we now Taylor-expand the lefthand side of Eq. 3, we obtain
Inserting this result into Eq. 1, we find that for A = 1/2 the requirement of volume conservation is fulfilled with the given assumptions (f(z = 0) = 0, f(z = z 0) = 1 and r(z = z 0,x) = r 0).
In a numerical simulation with a finite elongation step size ∆x, the evolution of the fiber profile can now be calculated in a linear approximation step by step using Eq. 5, starting from a homogeneous fiber with a constant radius r 0:
Here, the derivative ∂r/∂z has been replaced by the difference quotient, which can be calculated easily after performing a discretization of the z coordinate with step size ∆z. In our simulations, we have applied for the calculation of the derivative more advanced numerical methods like nonlinear interpolation which are provided by the used software. When the final elongation x 0 is reached, the procedure is stopped. The possibility to change Θ(z) from step to step enables us to simulate a travelling heating device by setting Θ(z) ≡ Θ(z + p(x)), where p(x) denotes the corresponding position of the center of the heating device’s profile with respect to the fiber.
Fig. 1 depicts an example using the described simulation method. In the bottom diagram, the motion of the heating device p(x) is shown. In the upper part of the figure, three profiles evolve during the drawing process, resulting in a tapered fiber which includes a homogeneous waist with a diameter of 3 μm. ∆x and ∆z are chosen in the order of tens of micrometers, depending on the complexity of the profile. This leads to simulation times smaller than one minute using a standard office computer.
To test our algorithm, we now assume the special case of a rectangular distribution for Θ(z,x) with a width of L(x), yielding a height of 1/L(x). With the heating device standing in the reference frame of the fiber, this kind of thermal profile leads to a waist whose radius is independent of z within the heating zone. With Eq. 5 we obtain
which is exactly the equation derived analytically in , proved to work very well in many experiments.
More generally, for an arbitrary distribution Θ(z,x) and again the case of a standing heating device in the reference frame of the fiber, we obtain for the position of the maximum of Θ denoted by ΘM
Assuming a x-independent ΘM, this equation can be simplified to
3. Remarks on the distribution Θ(z)
The following parametrization of the measure of deformability Θ(z) refers to the heating source we use in our tapering rig, in particular a propane-butane-oxygen flame. Other heating devices might require a different shape of the distribution. As an empirical approximation for Θ(z), we set
Here, N is a normalization constant given by the condition of Eq. 2, ±Θ0 denote the first roots of the distribution, whereas ν stands for the half width of half maximum (HWHM) relative to Θ0. This function is designed to offer full control over roots and width. Furthermore the first and second derivatives equal 0 at the first roots ±Θ0. These properties and setting the function to 0 beyond ±Θ0 help to reduce numerical artifacts during the simulation (see Fig. 2 a)).
Using such a kind of distribution, we have to find the appropriate parameters Θ0 and ν associated with a given heating device by drawing tapers with different final elongation x 0 while the heating device is at rest in the reference frame of the fiber, and we then have to compare the measured minimal diameter  with the minimal diameter simulated by our model for different values of Θ0 and ν. Using this procedure to calibrate Θ(z) for the propane-butane-oxygen flame used in our tapering rig, no common set of these values could be found for all chosen minimal diameters. However, in the experiment we have observed a clear tendency that a smaller final fiber diameter yields a smaller width of the distribution. This made us carry out a simulation of the temperature distribution inside the fiber for a given diameter d. By means of the finite element method (FEM), we investigate the stationary heat equation
where T denotes the temperature and k the thermal conductivity of the fiber material. The boundary conditions are given on the one hand by the axial symmetry at r = 0 and on the other hand by allowing heat flux at the fiber-air interface described by the equation
where n is the normal vector, σ the Stefan-Boltzmann constant, ε the emissivity of the fiber material, and T 0 the ambient temperature, which is set to 300 K . Over a length of 1 mm along the fiber-air interface, the temperature is fixed to a value of Tfix=2150 K, simulating the flame touching the fiber. For the far ends of the fiber, each at a distance of 50 mm, we assume thermal insulation (See Fig. 2 b)). Since the temperature varies within a large interval, the material properties k and ε have to be treated as temperature-dependent . These empirically found dependencies for fused silica as the fiber material are taken from reference :
Furthermore, we assume thermal equilibrium, which can be maintained when pulling the fiber very slowly, i.e. v0 < 2 mm/s.
The results of this FEM simulation are shown in Fig. 3 and confirm the experimental findings of smaller widths of the temperature distribution for smaller final fiber diameters. Inside the heating zone, the temperature is almost equal to Tfix across the fiber. Here, this balance of temperature is governed by heat conduction within the softened part of the fiber. We assume that convection can be neglected for our process, which is supported by the fact that the radial core and cladding refractive index profile stays the same in the tapered region. Outside the heating zone, the temperature of the solid fiber parts decays mainly due to reemission to the surrounding medium. The full width of half maximum (FWHM) of the temperature decay along the fiber axis at r = 0 in dependence of the diameter is fitted very well by a square root function. This behavior is reproduced for different values of Tfix. The effective width of the heating device then is determined by the width of the temperature distribution for temperatures exceeding the melting temperature of the fiber.
Taking this result into account, we allow Θ0, and hence Θ(z) to be dependent on the fiber radius at the current position of the heating device p, setting
Repeating the calibration described above with these new settings, we now find a set of parameters, resulting in an excellent agreement of the simulated and experimental fiber shape. For our flame, the parameters are ν = 0.51, B 0=0.32 mm, and B 1=0.11 mm, when r is given in micrometers.
The result of a diameter-dependent width of the temperature distribution can be proven experimentally by tapering fibers with different initial diameters. In this experiment, the heating device is fixed in the reference frame of the fiber, and the final elongation x 0 is set to 3 mm for all tests. As starting points we have produced fibers containing homogenous waists with lengths of several centimeters and different diameters. These waists serve as initial fibers for the test drawing. This initial diameters as well as the final ones have been measured by using a microscope or the refraction method described in Ref. . According to Eq. 9, the ratio between final (d) and initial diameter (d 0) can be expressed by
where L 0 = 1/ΘM denotes the effective width of the heating zone. For a diameter-independent width of the temperature distribution, one would expect a constant L 0, and hence a constant ratio of d and d 0. Fig. 4 shows the experimental results. Each color/symbol stands for a certain distance between the fiber and the flame of our tapering rig. These distances have been varied to supply different effective widths L 0 of the temperature distribution. For all of these distances the ratio d/d 0 becomes significantly smaller for smaller final diameters d. According to Eq. 16, this corresponds to smaller L 0 for smaller final diameters, which exactly emphasizes the findings of our simulation and calibration process.
The number of possible shapes of the fiber is restricted by the properties of the heating source, namely the distribution Θ(z). For example, the narrowest symmetric profile evolves by pulling with a standing heating device in the reference frame of the fiber. The resulting shape along the fiber has a FWHM z̅ of
where x 0 is the final elongation of the fiber and
denotes the average of all Θ0 appearing during the pulling process.
4. Experimental results and conclusion
Starting from a tapered fiber with a homogeneous waist produced by motion of the heating device as suggested in  and shown at the bottom diagram of Fig. 1, we now can use the calibrated model to iteratively approximate the fiber shape given by the model to the desired one. This can be implemented by varying the positions of the heating device p(x) for given ∆x-steps during the simulation and using an optimization procedure to find the motion which yields the best fitting shape. To warrant the convergence of this iterative method we use the fact that a slower burner motion at a certain position results in a thinner fiber diameter at this position.
As an example, we present a fiber with a sinusoidally modulated waist. The desired parameters for the sine function are an amplitude of the diameter of 0.25 μm and a period of 3.05 mm. As mean value for the diameter we set 2.75 μm, whereas the initial waist diameter is 3.3 μm. The resulting simulated shape is shown in Fig. 5 a), which shows that the desired parameters are reproduced very well.
A second possible shape is a conically formed waist as demonstrated in Fig. 5 b). Here, the set parameters are 2.5 μm for the minimal diameter and 3.0 μm for the maximal diameter of the waist. The simulation of a profile composed of three continuously connected quadratic functions (Fig. 5 c) is another representative for a shape with a bulged waist center. The settings for this profile are a waist length of 120 mm, a minimal diameter of 1 μm and a maximal diameter of 2.5 μm. In all figures, the average movement of the heating device is towards smaller z-values, whereas the fiber is pulled towards larger z-values. Therefore, the starting position of the heating device seems to be located at the middle of the waist region, but, if the elongation of the fiber is taken into account, it is actually set to the end of the waist region at the beginning of the pulling process.
As criterion for stopping the iterative procedure of the simulation process we don’t allow the radius difference between simulated and desired profile to be larger than 10 nm along the whole waist. This accepted error is well below the accuracy of the methods we use to measure the fiber diameter, and the hereby induced changes of the optical properties of the fiber are negligible.
To verify our model, we have produced a sinusoidally modulated fiber according to Fig. 5 a). The resulting profile is shown as black dots in Fig. 6 a), measured by a diffraction method . The inset demonstrates a magnification of the region from z=105 to 125 mm. Fig. 6 b) shows a captured image of the diffraction pattern generated by a laser line focus hitting the fiber perpendicularly . A fitted sine function (red) exhibits only small deviations from the desired values. Note that even the small steps in the taper transitions, located at the turning points of the oscillating movement of the heating device, are predicted by the model very precisely.
In conclusion, we have presented a model which is capable of simulating the drawing process of a fiber with a high accuracy taking into account the motion of the heating device. A method for the implementation and calibration of the properties of the heating source has been demonstrated and supported by numerical simulations of the heat transport inside the fiber using FEM. Applying the model, we are able to successfully deduce the motion of the heating source necessary to create a desired, complex shape of the tapered fiber as shown by the given examples of fibers with sinusoidally modulated or conically shaped waists.
The authors would like to thank the Landesgraduiertenförderung of Baden-Württemberg for support of this work. We also would like to thank Dr. Wolfgang Alt for helpful advice.
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