A generalized heat-brush approach for precise control of the waist profile in fiber tapers

Chams Baker; Martin Rochette

doi:10.1364/OME.1.001065

1. Introduction

Tapered optical fibers, illustrated in Fig. 1, made by a heat-and-draw approach are of interest in a wide range of applications. They have been used for enhancing nonlinear effects [1, 2], coaxial mode coupling [3], power splitting/combining [4], filtering optical spectra [5], and switching [6]. In all cases, a fine control of the taper shape is required to ensure an adiabatic transformation of the propagating mode [7, 8].

Fig. 1 Schematic of a fiber taper with a uniform waist and similar transition regions.

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A tapering model presented by Birks [9] provides an approach for shaping a fiber taper by changing the hot-zone length as the fiber is symmetrically stretched under tensile force at both ends. Birks’ model can be implemented using a stationary heater with a variable-length hot-zone, or using a heat-brush approach [10], where a heater travels back and forth within a variable-length brushing-zone [9]. The heat-brush implementation of Birks’ model provides better precision in shaping fiber tapers than the stationary heater implementation [11]. The heater in the heat-brush implementation can be a flame [10], a resistive heater [12], or a CO₂ Laser [13–15].

The stationary heater implementation of Birks’ model has been analyzed theoretically and numerically [16] using a viscus fluid flow model [17]. There has also been a few heuristic theoretical and numerical analyses of taper shape evolution in the heat-brush implementation of Birks’ model [18, 19]. Precise taper shape evolution after each heater sweep can be analyzed using a viscus fluid flow model generally used for the study of fiber drawing [17, 20]. The effects of different parameters, such as the number of tapering sweeps and the hot-zone length, on the taper shape can thus be quantified.

Moreover, in the heat-brush implementation of Birks’ model, the tapering function s = v_f/v_d, where v_f is the feed velocity and v_d is the draw velocity, is constant throughout each tapering sweep. A constant s limits the lowest inverse tapering ratio $ρ = ϕ_{j} / ϕ_{j - 1} = \sqrt{s}$ , where ϕ_j is the waist diameter after sweep j, that can be used in each sweep [21]. If ρ is less than 0.97 [11], the taper diameter in the transition region does not change smoothly, but rather it changes in steps.

A generalized heat-brush method allows s to change as the heater sweeps along the brushing-zone, and hence, the taper shape is carved within each sweep rather than having a sudden change in diameter. Just as in the heat-brush approach, the generalized heat-brush approach allows for precise shaping of the transition regions [11], a uniform waist profile [21], and a large contrast ratio between the initial and the final taper diameters [10]. In addition, the generalized approach allows for a smaller ρ in each sweep as well as controlled fabrication of tapers with an arbitrary waist profile and dissimilar transition regions [18].

A smaller ρ in each sweep reduces the number of sweeps required in the tapering process, and hence, reduces the taper fabrication duration. Replacing the uniform waist profile by one that follows an arbitrary function provides additional freedom in taper design and widens the range of taper applications. For example, a nonuniform waist profile in tapered fibers shifts the zero-dispersion wavelength along the microtaper waist for extended and flat supercontinuum generation [22, 23] and enhanced soliton self-frequency shift [24, 25]. Dissimilar transition regions also provide additional freedom in taper design. For example, in the case of soliton self-frequency shifting due to the Raman effect, the spectrum of a soliton slides towards longer wavelengths as it propagates from the input end to the output end of a taper waist. A design that minimizes the length of the taper has dissimilar adiabatic transition regions [8].

In this paper, we develop and demonstrate both by simulation and experiment a generalized heat-brush tapering method, and use it for the fabrication of tapers with a nonuniform waist profile and dissimilar transition regions. First, single-sweep tapering, the main constituent of the generalized heat-brush approach, is presented and simulated using a viscous fluid flow model to quantify the mismatch error between the targeted and the resulting taper profiles. Then, the generalized heat-brush approach is implemented by tapering a fiber over multiple sweeps, and the simulation results from the single-sweep tapering analysis are used to quantify the accumulated mismatch error after each tapernig sweep. Finally, we use of the generalized heat-brush approach to fabricate an As₂Se₃ chalcogenide taper with a linearly decreasing waist profile and dissimilar transition regions.

2. Single-sweep tapering

In this section, we present the single-sweep tapering method, an instance of the well-known fiber-drawing approach [17,20,26]. In the process of fiber drawing, mass conservation leads to $ϕ (t) = ϕ_{0} \sqrt{s (t)}$ where ϕ (t) is the taper diameter, ϕ₀ is the initial fiber diameter, and s (t) = v_f (t)/v_d (t) is the tapering function. To draw a taper with a predefined profile ϕ (z), the tapering function s(t) must be determined accordingly. The replacement of the time variable t by the drawing length $l_{d} (t) = \int_{0}^{t} v_{d} (τ) d τ$ simplifies the implementation of the single-sweep tapering method because it can be readily used as a feedback parameter to control the draw velocity v_d (l_d) = v_f (l_d)/s (l_d). In this case, the tapering function s (l_d) is calculated from the taper profile ϕ (z) using

s (l_{d}) = \frac{ϕ^{2} (z)}{ϕ_{0}^{2}} |_{z = l_{d}} .

Figure 2 provides an arbitrary taper profile ϕ (z) and its corresponding tapering function s(l_d).

Fig. 2 (a) A taper profile and (b) the resulting tapering function.

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2.1. Single-sweep tapering simulations

A general model of the viscous flow in the heat-softened region, or hot-zone, due to unidirectional stretching has been reported in [20]. A simplified model has been derived for the case when the fiber diameter is much smaller than the hot-zone length (L_hz) [17]. In this model, the deformation of the hot-zone due to stretching is governed by

\frac{\partial}{\partial z} (3 μ A \frac{\partial u}{\partial z}) = 0,

\frac{\partial A}{\partial t} + \frac{\partial}{\partial z} (u A) = 0,

where μ (z,t) is the viscosity distribution, u(z,t) is the axial velocity distribution, and A(z,t) is the cross-sectional area in the hot-zone [17]. For a Newtonian fluid, μ is independent of u, and hence, Eq. (1) leads to ∂ū/∂z × ∂F/∂z + F × ∂²ū/∂z² = 0, where ū = u/v_d is the normalized axial velocity and F = μA. Using the centered differentiation formulas [27] ∂F/∂z = (F_i₊₁ – F_i–₁)/2Δz, ∂ū/∂z = (ū_i₊₁ – ū_i−1)/2Δz, and ∂²ū/∂z² = (ū_i+1 − 2ū_i + ū_i−1)/Δz² leads to the finite difference form of Eq. (1)

[F_{i} - 0.25 (F_{i + 1} - F_{i - 1})] {\bar{u}}_{i - 1} - 2 F_{i} {\bar{u}}_{i} + [F_{i} + 0.25 (F_{i + 1} - F_{i - 1})] {\bar{u}}_{i + 1} = 0

where F_i = F (l_d, z_i), ū_i = ū (l_d, z_i), and Δz is the separation between any two consecutive z_i. Changing the variable t to l_d in Eq. (2) leads to the equation v_d∂A/∂l_d + ∂ (uA)/∂z = 0, which is expanded and divided by v_d to obtain ∂A/∂l_d + A∂ū/∂z + ū∂A/∂z = 0. Using the centered differentiation formulas ∂ū/∂z = (ū_i+1 − ū_i−1)/2Δz, ∂A/∂z = (A_i+1 – A_i−1)/2Δz and the forward differentiation formula [27]

\partial A / \partial l_{d} = [A_{i}^{n e w} - A_{i}] / Δ l_{d}

, the finite difference form of Eq. (2) corresponding to the extension of the fiber by a distance Δl_d = 2Δz is given by

A_{i}^{n e w} = A_{i} - [A_{i} ({\bar{u}}_{i + 1} - {\bar{u}}_{i - 1}) + {\bar{u}}_{i} (A_{i + 1} - A_{i - 1})]

where A_i = A (l_d, z_i),

A_{i}^{n e w} = A (l_{d} + Δ l_{d}, z_{i})

. It is clear from Eq. (3) and Eq. (4) that, for a Newtonian fluid, the deformation of the hot-zone is independent of the actual drawing velocity.

The flow-chart in Fig. 3 describes the program used to simulate the single-sweep experimental setup presented in Section 2.3. In this program, the taper profile is represented by an array of diameter values ϕ_k taken at points z_k with any two consecutive points separated by Δz. The hot-zone is a subarray of the taper array and the starting point of the hot-zone subarray can change to simulate a moving heater as illustrated in Fig. 4(a). The cross-section area in the hot-zone is given by A_i where i = 1, 2,..., N and the cross-section area of the extended hot-zone that results from drawing the hot-zone, as illustrated in Fig. 4(b), is calculated as follows: first, Eq. (3) is used with the boundary conditions ū_i=0 = −1/2 and ū_i=N+1 = 1/2 to calculate the normalized axial velocity distribution ū_i in the hot-zone, and then, Eq. (4) is used to calculate the extended hot-zone profile. In the simulations that follow, the hot-zone is assumed to have a uniform viscosity distribution.

Fig. 3 Flow-chart of the simulation program for the single-sweep tapering setup presented in Section 2.3. In this flow-chart, x is the displacement of both translation stages extending the fiber, y is the displacement of the heater translation stage, x_previous and y_previous are state variables, δ is a differential feed step, s is the tapering function, l_d is the drawing length, and Δz is the longitudinal separation between any two consecutive diameter sampling points.

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Fig. 4 Single-sweep simulation schematics of (a) shifting the hot-zone by Δz, and (b) extension of the fiber by 2Δz.

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We simulate the fabrication of a step-taper where the diameter changes abruptly from the initial to the final taper diameter. Typical simulation results of step-taper fabrication show a transient response in the resulting taper with an overshoot and oscillations in the waist before the diameter settles to a final value, as shown in Fig. 5. The mismatch between the resulting and the targeted taper profiles is quantified by the percent error along the taper defined as ɛ(z) = [ϕ_r (z) – ϕ_t (z)]/ϕ_t (z) × 100% where ϕ_r is the resulting taper diameter and ϕ_t is the targeted taper diameter. The transient response is quantified by the percent overshoot ɛ_os = (ϕ_t – ϕ_os)/ϕ_t × 100% where ϕ_os is the overshoot diameter, and by the settling distance z_s defined as the distance between the beginning of the waist and the point where the envelope of the absolute percent error is less than ɛ_s = 2%.

Fig. 5 Simulation of step-taper fabrication using the single-sweep tapering method.

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The transient response parameters ɛ_os and z_s represent the closeness of the of the resulting taper shape to the taper design, and the overall mismatch is reduced by reducing ɛ_os and z_s. Step-taper simulation results in Fig. 6 show ɛ_os and z_s as a function of L_hz and the inverse tapering ratio ρ = ϕ_min/ϕ₀, where ϕ_min is the minimum taper diameter. As expected, ɛ_os and z_s decrease with increasing ρ (≤ 1) and shortening L_hz. With respect to optical propagation in the taper, the overshoot in the waist diameter acts as a perturbation that may lead to coupling between the fundamental mode and higher order modes, radiation modes, or reflection modes [28]. The values of ɛ_os and z_s also represents the strength and the length of the perturbation region; therefore, a lower ɛ_os and a shorter z_s reduces the perturbation impact.

Fig. 6 (a) Percent overshoot, and (b) settling distance dependence on the inverse tapering ratio at different hot-zone lengths for the step-taper.

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2.2. Single-sweep tapering optimization

Simulation results in subsection 2.1 showed that ɛ_os and z_s decrease when ρ → 1 and L_hz → 0 mm. However, applications such as the enhancement of the waveguide nonlinearity or the sensitivity require microtapers with a waist diameter on the order of 1 μm drawn from fibers with a diameter on the order of 100 μm leading to ρ ∼ 0.01. Also, L_hz is on the order of 1 mm and is limited by the temperature distribution in the fiber and the heater dimensions. Moreover, it turns out that ɛ_os and z_s decrease when the taper slope decreases. As an example, Fig. 7 shows that as the slope decreases from 0.0105 to 0.0035, ɛ_os decreases from 8.8% to 3.8% and z_s decreases from 13.5 mm to 11.65 mm. In most cases, however, it is desirable to use the largest slope allowed by the adiabaticity criteria because using a small taper slope to reduce ɛ_os and z_s leads to a long transition region and consequently increases the sensitivity of the taper to environmental variations [9] as well as increasing the device length. Section 3 shows that ɛ_os and z_s are reduced by tapering a fiber over multiple sweeps leading to an implementation of the generalized heat-brush approach.

Fig. 7 Simulated fabrication results of taper profiles with linear transition regions at different slopes using the single-sweep tapering method.

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2.3. Experimental setup

Figure 8 illustrates the experimental implementation of the single-sweep tapering method where a translation stage moves the heater at a velocity v_y and two other translation stages pull the fiber from opposite directions at equal velocities v_w and v_x. Using v_d = v_y + v_w and v_f = v_y – v_x = α, where α is a constant, the velocities of the heater and the translation stages pulling on the fiber at a drawing length l_d = y + w are

v_{y} (l_{d}) = \frac{v_{d} (l_{d}) + v_{f} (l_{d})}{2} = \frac{α}{2} [\frac{1}{s (l_{d})} + 1],

v_{x} (l_{d}) = v_{w} (l_{d}) = \frac{v_{d} (l_{d}) - v_{f} (l_{d})}{2} = \frac{α}{2} [\frac{1}{s (l_{d})} - 1] .

2.4. Single-sweep tapering experimental results

Figure 9(a) shows the experimental results of a step-taper fabricated from an As₂Se₃ fiber with an initial diameter of 170 μm using a 5 mm long resistive heater at 210° C with v_f =0.72 mm/min and $v_{d}^{m a x} = max (v_{f} / s) = 4.5 mm / min$ . The fabricated taper is removed from the tapering setup and placed straight on a flat plate, and then, an imaging system composed of a 20× lens and a CCD camera mounted on a motorized translation stage is used to measure the taper profile with a measurement taken every 1.0 mm. The measured step-taper profile clearly shows an overshoot in the fiber diameter arising from the finite length of the hot-zone. An effective hot-zone length of 2.7 mm is retrieved by simulating the step-taper fabrication and fitting the simulation results with the measured profile. The measured effective length is used to simulate the fabrication of the taper in Fig. 9(b) and the simulation results show good agreement with the experimental results within the measurement error of 1 μm.

Fig. 8 Schematic of the experimental implementation of the single-sweep tapering method.

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Fig. 9 Experimentally measured profiles of (a) a step taper, and (b) an arbitrary taper fabricated using the single-sweep tapering method.

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3. Multi-sweep tapering

Multi-sweep tapering performed by systematic repetition of the single-sweep method as illustrated in Fig. 10 represents an implementation of the generalized heat-brush method. To taper a fiber over n sweeps, the taper profile is divided into subsections as shown in Fig. 11, where ϕ_n is the minimum taper diameter, and ϕ₁ to ϕ_n–1 are the waist diameters for all intermediate tapering sweeps and are calculated using ϕ_j = rϕ_j−1 with r = ρ^1/n and ρ = ϕ_n/ϕ₀. For every sweep j < n, the stage tapering function s^(j) (l_p) is calculated from the stage taper profile ϕ^(j) (z) composed of a left transition region extracted from ϕ (z) between $z_{j - 1}^{l e f t}$ and $z_{j}^{l e f t}$ , a right transition region extracted from ϕ (z) between $z_{j}^{r i g h t}$ and $z_{j - 1}^{r i g h t}$ , and a uniform waist with a length

L_{j} = \frac{\int_{z_{j}^{l e f t}}^{z_{j}^{r i g h t}} ϕ^{2} (z) d z}{ϕ_{j}^{2}},

where L_j makes the mass volume of the waist at stage j equal to the mass volume required to draw the taper section between

z_{j}^{l e f t}

and

z_{j}^{r i g h t}

. The stage taper profile of the final sweep ϕ⁽ⁿ⁾ (z) is extracted from ϕ (z) between

z_{n - 1}^{l e f t}

and

z_{n - 1}^{r i g h t}

, and is used to calculate the final stage tapering function s⁽ⁿ⁾ (l_p). Finally, for each stage j, a single tapering sweep is performed using the calculated stage tapering function and then the heater is moved back a distance

(z_{j - 1}^{r i g h t} - z_{j}^{r i g h t}) + L_{j}

.

Fig. 10 Schematic of taper profile evolution using the multi-sweep tapering method.

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Fig. 11 Dividing the taper into sections for the determination of the tapering function of each tapering stage.

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3.1. Quantitative analysis of multi-sweep tapering

Based on the divide-and-conquer paradigm [29], tapering a fiber over multiple sweeps reduces the percent overshoot. For a step-taper, the worst-case overshoot diameter at sweep j is estimated using the recurrence relation

ϕ_{o s}^{(j)} = [1 - ɛ_{o s} (ρ_{j}) / 100 %] \times ρ_{j} \times ϕ_{o s}^{(j - 1)}

ϕ_{o s}^{(1)} = [1 - ɛ_{o s} (ρ_{1}) / 100 %] \times ρ_{1} \times ϕ_{0},

where ɛ_os (ρ_j) is provided in Fig. 6(a). By setting the inverse tapering ratio for all sweeps to r, the worst-case overshoot diameter becomes

ϕ_{o s}^{(j)} = {[1 - ɛ_{o s} (t) / 100 %]}^{j} \times r^{j} \times ϕ_{0},

and the maximum percent overshoot at the end of tapering is

ɛ_{o s, m a x}^{(n)} = [1 - {(1 - ɛ_{o s} (r) / 100 %)}^{n}] \times 100 %,

which is simplified to

ɛ_{o s, m a x}^{(n)} \approx n ɛ_{o s} (r)

when ɛ_os (r) ≤ 1%. It is clear from Fig. 6(a) that

ɛ_{o s, m a x}^{(n)} < ɛ_{o s} (ρ)

and that

ɛ_{o s, m a x}^{(n)}

decreases as n increases. For, example, the fabrication of a step-taper with ρ = 0.5 over a single sweep using a 4 mm long hot-zone leads to ɛ_os (0.5) = 17%. However, when tapering is performed over 6 sweeps with r = 0.89 and ɛ_os (0.89) = 0.5%, the maximum percent overshoot is

ɛ_{o s, m a x}^{(6)} = 3 %

.

The use of a large number of sweeps increases the tapering duration. For the case of a step-taper, the minimum time duration for stage j is $T_{j} = L_{j - 1} / v_{f}^{m a x}$ , where $v_{f}^{max}$ is the maximum practical feed velocity, and the total tapering duration after n sweeps is

T = \frac{L_{0}}{v_{f}^{m a x}} \times \frac{1 - ρ^{- 2}}{1 - ρ^{- 2 / n}},

which is reduced by increasing

v_{f}^{max}

and reducing n. In general, to keep the tapering duration at a minimum, n is selected to be the minimum number of sweeps required to keep ɛ_os bellow a certain prescribed value.

3.2. Reduced mismatch in the transition regions using multi-sweep tapering

The diameter decreases in steps in the heat-brush implementation of Birks’ model, limiting the minimum attainable mismatch between the resulting taper and the design. At any diameter ϕ, the diameter step is Δϕ = (1 – ρ)ϕ and the taper slope is approximated by ∂ϕ/∂z ≈ Δϕ/Δz leading to Δz ≈ (1 − ρ)ϕ/(∂ϕ/∂z). Setting L_hz ≪ |Δz| does not decrease the mismatch because the diameter steps in the transition region become more prominent; in fact, setting L_hz ≳ |Δz| is practical to keep the transition region smooth. For example, if the length of the brushing-zone is a constant L₀, then the taper profile is given by ϕ (z) = ϕ₀exp (−z/L₀) [9] and |Δz| ≈ (1− ρ)L₀. Using typical values of ρ = 0.97 and L₀ = 2.0 cm leads to |Δz| ≈ 0.6 mm, which requires L_hz ≳ 0.6 mm. In contrast, the diameter steps are eliminated in the multi-sweep tapering method because the transition region is carved within each tapering sweep; therefore, shortening L_hz always reduces the mismatch between the resulting taper and the design.

3.3. Multi-sweep tapering simulation

Multi-sweep tapering simulation is performed by repeated application of the single sweep tapering program. Simulation results in Fig. 12 performed using L_hz = 3 mm for a step-taper with ρ = 0.4 show that the percent overshoot $ɛ_{o s}^{(n)}$ decreases as n increases. Also shown in Fig. 12 is the worst-case percent overshoot, $ɛ_{o s, m a x}^{(n)}$ , calculated using Eq. (5). It is observed that $ɛ_{o s}^{(n)}$ does not exceed $ɛ_{o s, m a x}^{(n)}$ , which is expected as $ɛ_{o s, m a x}^{(n)}$ estimates the upper limit of $ɛ_{o s}^{(n)}$ .

Fig. 12 Percent overshoot and maximum percent overshoot versus the number of tapering sweeps n for a step-taper with ρ = 0.4 using L_hz = 3 mm.

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Although increasing n reduces ɛ_os, L_hz must also be shortened to ensure that |ɛ (z)| is less than a prescribed value ɛ_target. Shortening L_hz is critical when the taper profile incorporates fine details such as a large ∂ϕ/∂z, a large change in ∂ϕ/∂z, or a short waist. For example, if the taper waist length is of the same order as L_hz, then the details of the waist can not be precisely shaped. The value of L_hz that ensures |ɛ(z)| < ɛ_target for a given taper profile can be determined through simulations.

3.4. Multi-sweep tapering experimental results

Figure 13 shows the experimental results for the fabrication of an As₂Se₃ taper with an initial fiber diameter of 170 μm, dissimilar left and right transition regions, and a nonuniform waist with a diameter decreasing linearly from 15 μm to 10 μm over a waist length of 2.0 cm. The taper is experimentally fabricated over 24 sweeps using the same resistive heater in the single-sweep experiment in Subsection 2.4 at 210° C with v_f = 3.56 mm/min and $v_{d}^{m a x} = 4.50 mm / min$ . The measurement error is 1 μm and the resulting taper matches the design within the measurement error.

Fig. 13 Experimental results showing the profile of an As₂Se₃ taper fabricated using the multi-sweep tapering method with n = 24.

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4. Conclusion

The multi-sweep tapering method has been used to implement the generalized heat-brush approach, which allows the ratio of the feed and draw velocities to change within each tapering sweep. A quantitative analysis showed that the mismatch error decreases by increasing the number of tapering sweeps and shortening the length of the hot-zone formed by the heater. An As₂Se₃ chalcogenide taper with dissimilar transition regions and a waist diameter decreasing linearly from 15 μm to 10 μm over 2.0 cm was fabricated using the multi-sweep tapering method showing good agreement between the targeted and the measured taper profiles.

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A generalized heat-brush approach for precise control of the waist profile in fiber tapers

Abstract

1. Introduction

2. Single-sweep tapering

2.1. Single-sweep tapering simulations

2.2. Single-sweep tapering optimization

2.3. Experimental setup

2.4. Single-sweep tapering experimental results

3. Multi-sweep tapering

3.1. Quantitative analysis of multi-sweep tapering

3.2. Reduced mismatch in the transition regions using multi-sweep tapering

3.3. Multi-sweep tapering simulation

3.4. Multi-sweep tapering experimental results

4. Conclusion

References

Cited By

Figures (13)

Equations (13)

Optical Materials Express