This work reports a modified flame-brush technique to fabricate fiber tapers with arbitrary waist profiles. The flame-brush approach is used to produce small step reductions in the fiber diameter, or step-tapers, with a constant speed flame brush sweep, while the fiber is uniformly stretched. Arbitrary waist profiles in tapers are fabricated by approximating the taper diameter function to any monotonic function of the fiber length while combining a superposition of step-tapers. This method to produce the arbitrary profiles is described and a set of tapers with dissimilar transition regions are fabricated for its validation.
© 2012 OSA
The fabrication of fiber optic tapers is related to a mature technology development, namely, the flame-brush technique, originally proposed to fabricate fiber couplers by stretching a pair of fibers while heating a section of their length with a flame brush. By properly configuring the sweeping length, speed and size of the flame-brush, arbitrary tapers can be fabricated, or, more specifically, arbitrary taper transition regions and waist profiles. In this case, a taper can be characterized by two transition regions usually represented by a decreasing function of the radius with the length of the fiber. The transition function reaches a uniform region called taper waist that has a constant diameter. The resulting device could also be characterized as a biconical taper , if both transitions are equal and connected to a uniform waist with a diameter having a different value from non-processed fiber. With such techniques, the fabrication of waists with diameters reaching a minimum of 1μm is a commonplace. When requiring the production of longer and thinner uniform nanotapers, specific techniques are required to, among other things, stabilize the heating process and isolate the fiber from environmental disturbances during the process [3, 4, 5]. Another aspect of the nanotaper fabrication techniques is the possibility of using a draw-tower that would allow the fabrication of short or long tapers with dissimilar and abrupt transitions[6, 7].
The literature reports a large number of applications of such devices, ranging from the production of near-field optical probes  to applications of micro- and nanotapers in fiber optic sensing . The control of the dynamics of soliton propagation and supercontinuum generation  would be a direct application, since in this case the transition of the taper must be tailored to produce the required group-velocity and nonlinearity [11, 12]. In this work, a technique to fabricate microtapers with dissimilar transitions is proposed and it would be natural to use the proposed technique in conjunction with an appropriate nanotaper fabrication technique. This would allow, for example, the engineering of the propagation characteristics in a taper as reported for standard single-mode fibers[12, 13] and photonic crystal fibers[6, 14]. In the latter case, the tapers have dissimilar transitions and a length that could be reproduced in a taper rig.
For the fabrication of tapers in the micrometric range, the theory developed by Birks et al  on the shape of the fiber tapers describes what taper transition regions would be fabricated if a flame brush movement configuration is given, characterizing what they called ’the forward problem’. In ’the reverse problem’ a desired taper transition function and waist length are given and the flame-brush movement configuration that corresponds to its fabrication is determined by the theory. In both problems, the fabrication of biconical tapers  are described, and the authors give a hint on how to extend the theory for the fabrication of asymmetrical tapers, or tapers with different transition regions.
Another method that allows the fabrication of arbitrary taper profiles with non-uniform waists using the flame-brush approach was recently introduced , where the flame-brush sized hot-zone moves at speeds depending on time and position while two translation stages stretch the fiber. This method is characterized by an algorithm which uses varying flame-brush speeds, or tapering functions. The authors demonstrate the fabrication of arbitrary waist profiles, and the technique requires a more sophisticated programming of the translation stage controllers and the use of hardware drivers with a capability of speed variations in the translation stages. Other techniques may also produce complex tapers with the same degree of complexity in the taper rig . In this work, the theory of standard tapers fabrication will be used. Only conservation principles and basic mechanics are considered in the modeling, since no heat-transfer models or heuristic considerations on the tapering process [19, 20] are directly considered in the method.
By solving the viscus fluid flow model using finite-differences, as in , it is possible to simulate the transition region of a taper after a single flame-brush sweep. In this case, for a given set of stretching and flame-brushing speeds, one aims at obtaining a minute reduction of the diameter approximating a step function, and producing a uniform taper waist between the transitions, or a step-taper. The determination of a sequence of flame-brush sweeps with a constant speed along a calculated length would allow the stepwise construction of arbitrary taper profiles. Even dissimilar transition regions could be fabricated by approximating them to a combination of the superposed step-tapers. This paper then proposes a method capable of solving the reverse problem with any given monotonic taper transition function and using constant speeds in the stretching and flame-brushing processes. This would also provide a technique to fabricate such devices with a simplified taper rig.
2.1. Experimental setup
The taper rig used in the fabrication process has two translation stages capable of stretching the softened fiber which is fixed by two fiber holders. In Fig. 1, a schematic diagram illustrates the configuration of the system. The flame-brush is controlled by two digital mass-flow controllers and operates with a custom-made torch fueled with analytical n-butane and oxygen at the proper proportions to produce a flame with a width of 3mm. This is the size of the flame-brush considered in this work. The relative velocities are determined by considering the moving stage with the flame-brush as a referential with velocity VFB. The arrows in Fig. 1 indicate the possible directions of the movements performed by the translation stages.
It is assumed that the flame-brushed length of the fiber has a lower viscosity, such as to be softened and stretched without deforming under the effect of its own weight. The viscosity is higher outside the flame-brushed region.
2.2. From the step-taper to the arbitrary taper
The first consideration to demonstrate the effectiveness of the proposed technique requires a simulation of the tapering process for a single sweep of the flame-brush while the optical fiber is being stretched at uniform speeds. In this case, an algorithm based on the simplified fluid-dynamic model was employed for this purpose. Since the flame-brush used in the available taper rig had a width of approximately 3mm and usually the speeds of the stretching translation stages and flame-brush have values of millimeters per second and millimeters per minute, respectively, a range of testing values for the speeds to determine the resulting step-taper should be simulated.
In this case, the tapering function, s, is defined as the ratio between the feed velocity, vf = VFB − VSR, and the draw velocity, vd = VFB + VSL, where VSR and VSL are the constant speeds of the left and right translation stages (with pre-determined directions to the left and to the right, respectively) in the tapering process.
The tapering function is kept within values around the limit when the changes in the transition region of the taper occur in steps without oscillations and with minute amplitudes. Equivalently, the inverse tapering ratio, , is kept at values close to 0.97. Secondly, the produced taper profile is simulated, not considering the amplitude of the diameter change in the transition region after the flame-brush single sweep. It is required that the transition of the taper resembles a step function. And differently from what is proposed in the generalized heat-brush approach , the feed and the draw speeds are constant during a complete flame-brush sweep and throughout the fabrication process. Moreover, in the simulation and in the fabrication process, only one stretching translation stage moves, while the other stretching stage remains static, showing that the taper rig could operate with only two translation stages.
2.2.1. Tapering simulation
A simplified slow viscous flow model has been previously derived and is used here for the case when the fiber diameter is much smaller than the hot-zone size. In that work , the simple model is based on a mass and momentum balance. In order to make the model of fiber tapering relevant to industrial processes, the viscosity distribution, μ = μ(z, t), was considered as a function of the taper length, z, assumed to have been calculated from an uncoupled heat transfer model. Such an assumption is only valid when the Nusselt number divided by the Péclet number is large, that is, when radiative heat loss dominates convection and conduction, that is the case in the tapering process. The model employed in the tapering simulation uses:Eq. (1) describes the balance of forces in terms of the stress in the fiber, and the conservation is described in terms of continuity Eq. (2), where u = u(z, t) is the axial speed distribution and A = A(z, t) is the cross-sectional area of the hot-zone. The spatial coordinate was discretized using a previously proposed scheme, using finite differences to approximate derivatives. The additional required integration in time used the ordinary fourth order Runge-Kutta method  with a predetermined maximum time step, such as to minimize the computational effort.
The waist profile resulting from the simulation of the step-taper fabricated after a flame-brush single sweep with a constant tapering function is shown in Fig. 2. The speeds are set such as to observe when the step change causes an overshoot in the diameter value along the taper length. This indicates the order of stretching speed magnitude in the process, while the flame-brush speed is set at 7.5mm/s in the simulation and in most of the experiments. The length of the observed tapers in Fig. 2 is 20mm. The hot-zone is considered to be equal to the size of the flame-brush. As a rule-of-thumb, in order to have a profile which satisfies the stepwise method, the stretching speed will be set at values lower than 10mm/min for the given flame-brush speed. By doing this, the final fabricated tapers can approximate relatively well the desired functions.
2.3. Modeling the tapering process
The modeling used in the process algorithm uses adapted concepts developed in , as the volume law which is based on the volume conservation inside the hot-zone region and the distance law, that relates the complementary distances associated with the transition lengths. The speed of the heat source mounted on the flame-brush stage was VFB, changing its direction after each n–th sweep and traveling a length of Ln. The initial sweep has a length of L0, and uses a length of fiber, x0, that must provide a fiber volume which is sufficient for the taper fabrication. Its initialization is explained in the following sections.
In Fig. 3, a scheme of a taper with symmetrical linear transition regions is represented (corresponding to a biconical taper), but the left and right transition could be any monotonic function.
The complementary distances, ZR(A) and ZL(A), determine the tapers transitions and return the distances from the transition beginning, which has a cross sectional area A0 and the point in the transition having cross sectional area A, for the right and left transitions, respectively. They are used to elaborate the distance law for asymmetrical tapers. The cross sectional area of the flame-brushed fiber is An after the n–th flame-brush sweep. Thus, the burner starts its n–th sweep in one of the transitions at the point having cross sectional area An−1 that is just leaving the waist region and entering the transition, and ends at a point in the other transition which must have final cross sectional area An. The distance law can then be stated through Eq. (3):Eq. (3) a modification of the distance law  is represented, which in this work uses two different transition lengths, ZR(A) and ZL(A), and depends on the direction of the flame-brush movement. Tn−1 is the duration of the first (n − 1) flame-brush sweeps.
Since the fiber is being stretched during the flame-brush sweeping, the traveled length by the flame-brush, Ln, has to be corrected by using Eq. (4):Eq. (4), the relation between xn and Ln can be related to the distance law, by using Eq. (3). The Eq. (4) also uses ΔT (L), which is a parameter depending on the characteristics of the specific translation stage in the taper rig. This time interval could be written as: Eq. (5), ta is the time interval for the flame-brush stage to accelerate from rest to VFB, and td is the interval for the stage to decelerate from VFB to rest before it changes direction; La and Ld are the corresponding traveled distances during such time intervals. Using Eq. (5) in Eq. (4), one could easily obtain the traveled length as a non-recursive function of other parameters without Ln on the right-hand side of Eq. (4).
The volume conservation is applied in the hot-zone, producing a relation that would represent a second law in the tapering process, namely, a volume law. In this case, the volume flow entering the hot zone is equal to the flow leaving it, and the cross sectional areas before and after the flame-brush sweep can be calculated in terms of the stage speeds as in:Eq. (6), one can recursively predict the area after a flame-brush sweep and also the radius of the flame-brushed section of the fiber, allowing one to calculate the resulting designed taper radius given by an input monotonic function after each sweep.
In addition, the waist length function after the n–th sweep is given by:Eq. (7), if n is related to the last sweep in the process, the final waist length is Lwn = Lw, for the last flame-brush sweep, which is an input parameter in the software.
If x0 is initialized with a valid value and the direction of the first sweep is known, all subsequent movements of the flame-brush can be determined. x0 · A0 determines the volume of the fabricated taper, however only the shape of the transitions, the waist length and the waist diameter are given as input parameters. Firstly, it is considered that the volume of the transitions is constant and that the excess of volume outside the transition is the waist volume. By using x′0 as in Eq. (8), one can obtain a taper with waist length L′w, by using Eq. (3) to Eq. (7):Eq. (9) produces an acceptable initialization to the free parameter x0, that corresponds to a volume of a taper with input waist length Lw.
2.4. The designed tapers
The input desired taper transition functions and waist length are given as inputs to the taper rig control software. The transition functions for the left and right transition return a point in the longitudinal direction of the taper, this is, zright (r) or zleft (r), that are functions of the fiber radius, r. After calculating the next area profile, from which the radius is obtained using Eq. (6), the distance that the flame-brush travels is calculated for the next sweep using Eq. (4) and the zright (r) or zleft (r) function. It is clear that, for a cylindrical fiber, A = πr2, which gives zright (r) = ZR(πr2) and zleft (r) = ZL(πr2). The resulting taper profiles were measured with a 2D Coordinate Measurement Geometry Calculation System (QM-DATA 200 - MITUTOYO).
2.4.1. The Gaussian-exponential taper
In the case of the production of an exponential on the left side and a Gaussian function on the right side transition, the input functions would be:Eq. (11), and the right translation stage had a speed value of VSR = 2.5mm/min towards the right while the left stretching stage was static. The taper had as input parameters a final waist length Lw = 20mm and a uniform waist radius rw = 12.5μm.
To demonstrate the feasibility of the technique, other tapers with arbitrary profiles were also fabricated, mainly with the left and right transitions having different functions. Biconical tapers were fabricated to validate the results with the recorded flame-brush course length progression. A fabricated Gaussian-exponential taper had a video elaborated that contained the evolution of the taper profile during the tapering process, since such a data set is automatically generated during the fabrication. Another video was produced as a result from the simulation of this stepwise method using the simplified fluid-dynamic model at the same speeds and with the same flame-brush size and movement configuration.
2.4.2. The quadratic-arc-sinusoidal taper
Another taper with arbitrary profile that was fabricated had a second order polynomial function on the left side transition and a function whose inverse would contain an arcsin with monotonic growth on the right side transition given by:
2.4.3. The arc-sinusoidal-arc-sinusoidal taper
A third taper with two different arcsin transition functions was fabricated with zleft (r) and zright (r) given by:
3. Results and discussion
The tapers specified in subsection 2.4.1, 2.4.2 and 2.4.3 have their profile functions depicted in Fig. 4(a), Fig. 4(c) and Fig. 4(e). The experimental points measured with a 2D microscope indicate the resulting profiles that follow the simulated profiles depicted in the same graph. The final simulated taper profiles produced with the stepwise method, when magnified, show the small step functions that formed the ideal implementation of the method, which is a superposition of small amplitude step-functions. The measured points describing the taper profile have an uncertainty of ±2μm. The apparent outliers in the data set that deviate more than the uncertainty are supposedly corresponding to imperfections in the taper holders used to fix the fiber and translate it under the microscope objective lenses.
One can infer that the method is capable of producing arbitrary transition functions in a fiber taper, which was fabricated with a technique based on the classical theory on the shape of fiber tapers, that states the distance law and the volume law. Based on an extension of the distance law and on the assumption that a single flame-brush sweep produces a step-taper, a combination of properly configured flame-brush sweeps was capable of producing any waist profile with a given monotonic input function.
The course length progression of the flame-brush for the three arbitrary taper profiles is shown in Fig. 4(b), Fig. 4(d) and Fig. 4(f). It is possible to observe a correspondence between the envelope of the flame-brush progression curve and the taper profile, which indicates that each step-function corresponding to one zig-zag movement of the flame-brush is responsible to the carving of the taper in the fiber. In addition, since one of the stretching translation stages is static, the flame-brush has a trajectory shifted towards the moving stretching stage. This would allow the simplification of the translation stage configuration in the taper rig, using only two of them, one for the flame-brushing and another for the stretching process. Two videos were produced from the process of fabricating the quadratic-arc-sinusoidal taper with the stepwise method, but now the final resulting tapers have a waist length of 30mm and waist radius of rw = 31.25μm and their profile curves are shown in Fig. 5(a) and Fig. 5(b). In the video corresponding to Fig. 5(a) ( Media 1), the superposition of step-tapers is illustrated while creating the desired taper transition functions. The video used the same sequences of flame-brush sweeps with the calculated lengths as to produce the required taper. The superposition of single flame-brush sweeps was simulated by using the fluid-dynamic model ( Media 2) , a 3mm wide flame-brush with a trapezoidal profile and constant speeds, which resulted in a taper having smoother transition regions when compared to the other profile for the same designed taper. The simulation illustrates the evolution of the described procedures by using the previously mentioned techniques. Confirming the principles used in the stepwise method, both simulations indicate that the step-tapers are superposed after each sweep of the flame-brush, with the step-taper having a small amplitude and producing a staircase-like transition function; comparatively, the continuous simulation produces a more realistic smoothed transition in the method based on the fluid-dynamic model and knowing the finite width of the flame-brush together with a constant tapering function.
3.1. Errors in the fabrication
The error, ε, obtained from the difference between the value of the designed transition function and the measured or simulated radii was calculated as a function of the taper length for the previously fabricated arc-sinusoidal-arc-sinusoidal taper. The error curves depicted in Fig. 6(a) and Fig. 6(b) have the statistics summarized in Table 1. The table contains the mean error ε̄ for the measured points, the ideal stepwise method and the fluid-dynamic model simulations, and also their corresponding standard deviations, σ, for the left and right transitions.
It must be stressed that the plotted errors depend on the transition functions, as does the adiabaticity of the taper . The illustrated tapers in this work are arbitrary, therefore a detailed analysis of the taper transitions with respect to the coupling of different modes in the transition was not evaluated in this work, and a more detailed evaluation of the errors involved in different transition functions and on the adiabaticity aspects of the fabrication is to be published elsewhere. Biconical tapers with diameters of 1μm were fabricated with the used taper rig having losses of less than 0.5% in the transmitted signal. The exponential and linear profiles had the envelope of the Ln functions as predicted by the ordinary flame-brush technique, namely constant and linearly increasing Ln, with the addition that the course of the flame-brush was shifted towards the moving stretching stage.
With respect to the oscillations in diameter produced by the superposition of step-tapers, if the stretching speeds are properly chosen, the errors from the simulations will be much lower than 1μm, decreasing with the diameter of the transition, as shown in Fig. 6(a) and Fig. 6(b) and summarized in Table 1. The measurement of the transition was performed using the 2D microscope which has an uncertainty larger than the standard deviation of the measured error. Its statistics shows a standard deviation around 1μm, therefore one would not be able to isolate the errors produced in the measurement with the microscope from the errors in the fabrication process. The theoretical results obtained from simulations indicate that the fabrication errors would be below 1μm on average for an oscillating transition function. The statistics from the simulated ideal method and the fluid dynamic model shows a much lower average error and standard deviation. It is important to notice that the error in fabrication decreases with the designed diameter and has peaks at the oscillations of the transition functions. This is an indication that transition functions that do not oscillate can be fabricated with smaller errors.
The authors demonstrate a method to fabricate arbitrary taper profiles by using a conservation principle inside the hot-zone and the distance law modified for the production of tapers with arbitrary transition functions. A set of tapers with Gaussian-exponential, quadratic-arc-sinusoidal and arc-sinusoidal-arc-sinusoidal transition functions and uniform waists were fabricated. The stepwise method is used to produce the tapers using constant speeds in the translation stages, differently from the heat-brush approach that requires a tapering function which varies in the process. The stepwise method was implemented with a discrete approach while approximating the transition function with step-tapers at each flame-brush sweep.
The authors would like to thank Lucas H. Negri and Yujuan Wang for characterizing the tapers and also acknowledge financial support from the National Council for Scientific and Technological Development (CNPq) under grant number 308975/2009-0.
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