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Optical Materials Express

Optical Materials Express

  • Editor: David J. Hagan
  • Vol. 1, Iss. 6 — Oct. 1, 2011
  • pp: 1065–1076
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A generalized heat-brush approach for precise control of the waist profile in fiber tapers

Chams Baker and Martin Rochette  »View Author Affiliations


Optical Materials Express, Vol. 1, Issue 6, pp. 1065-1076 (2011)
http://dx.doi.org/10.1364/OME.1.001065


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Abstract

We present a generalized heat-brush tapering approach in which the ratio of the feed and draw velocities changes within each tapering sweep. This approach allows for controlled and precise shaping of tapers with an arbitrary waist profile and dissimilar transition regions as demonstrated experimentally. A quantitative analysis of the mismatch error after each tapering sweep is also provided.

© 2011 OSA

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

1. P. Dumais, F. Gonthier, S. Lacroix, J. Bures, A. Villeneuve, P. G. J. Wigley, and G. I. Stegeman, “Enhanced self-phase modulation in tapered fibers,” Opt. Lett. 18, 1996–1998 (1993). [CrossRef] [PubMed]

, 2

2. C. Baker and M. Rochette, “Highly nonlinear hybrid AsSe-PMMA microtapers,” Opt. Express 18, 12391–12398 (2010). [CrossRef] [PubMed]

], coaxial mode coupling [3

3. A. C. Boucouvalas and G. Georgiou, “Biconical taper coaxial optical fibre coupler,” Electron. Lett. 21, 864–865 (1985). [CrossRef]

], power splitting/combining [4

4. R. G. Lamont, D. C. Johnson, and K. O. Hill, “Power transfer in fused biconical-taper single-mode fiber couplers: dependence on external refractive index,” Appl. Opt. 24, 327–332 (1985). [CrossRef] [PubMed]

], filtering optical spectra [5

5. J. V. Wright, “Wavelength dependence of fused couplers,” Electron. Lett. 22, 320–321 (1986). [CrossRef]

], and switching [6

6. S. Jensen, “The nonlinear coherent coupler,” J. Quantum Electron. 18, 1580–1583 (1982). [CrossRef]

]. In all cases, a fine control of the taper shape is required to ensure an adiabatic transformation of the propagating mode [7

7. J. D. Love and W. M. Henry, “Quantifying loss minimisation in single-mode fibre tapers,” Electron. Lett. 22, 912–914 (1986). [CrossRef]

, 8

8. J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices. I. adiabaticity criteria,” IEE Proc.-J: Optoelectron. 138, 343–354 (1991). [CrossRef]

].

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

A tapering model presented by Birks [9

9. T. A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992). [CrossRef]

] 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

10. F. Bilodeau, K. Hill, S. Faucher, and D. Johnson, “Low-loss highly overcoupled fused couplers: Fabrication and sensitivity to external pressure,” in Optical Fiber Sensors, (Optical Society of America, 1988), p. ThCC10.

], where a heater travels back and forth within a variable-length brushing-zone [9

9. T. A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992). [CrossRef]

]. The heat-brush implementation of Birks’ model provides better precision in shaping fiber tapers than the stationary heater implementation [11

11. R. P. Kenny, T. A. Birks, and K. P. Oakley, “Control of optical fibre taper shape,” Electron. Lett. 27, 1654–1656 (1991). [CrossRef]

]. The heater in the heat-brush implementation can be a flame [10

10. F. Bilodeau, K. Hill, S. Faucher, and D. Johnson, “Low-loss highly overcoupled fused couplers: Fabrication and sensitivity to external pressure,” in Optical Fiber Sensors, (Optical Society of America, 1988), p. ThCC10.

], a resistive heater [12

12. Y. Takeuchi and J. Noda, “Novel fiber coupler tapering process using a microheater,” Photon. Technol. Lett. 4, 465–467 (1992). [CrossRef]

], or a CO2 Laser [13

13. H. Yokota, E. Sugai, Y. Kashima, and Y. Sasaki, “Optical irradiation method for fiber coupler fabrications,” in Optical Fiber Sensors, (OSA, 1996), p. Th319.

15

15. L. C. Ozcan, V. Treanton, F. Guay, and R. Kashyap, “Highly symmetric optical fiber tapers fabricated with a CO2 laser,” Photon. Technol. Lett. 19, 656–658 (2007). [CrossRef]

].

The stationary heater implementation of Birks’ model has been analyzed theoretically and numerically [16

16. S. Xue, M. van Eijkelenborg, G. W. Barton, and P. Hambley, “Theoretical, numerical, and experimental analysis of optical fiber tapering,” J. Lightwave Technol. 25, 1169–1176 (2007). [CrossRef]

] using a viscus fluid flow model [17

17. J. Dewynne, J. R. Ockendon, and P. Wilmott, “On a mathematical model for fiber tapering,” SIAM J. Appl. Math. 49, 983–990 (1989). [CrossRef]

]. There has also been a few heuristic theoretical and numerical analyses of taper shape evolution in the heat-brush implementation of Birks’ model [18

18. S. Pricking and H. Giessen, “Tapering fibers with complex shape,” Opt. Express 18, 3426–3437 (2010). [CrossRef] [PubMed]

, 19

19. W. Sun, M. Yuan, X. Zeng, and T. Birks, “Theoretical shape analysis of tapered fibers using a movable large-zone furnace,” Optoelectron. Lett. 7, 154–157 (2011). [CrossRef]

]. 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

17. J. Dewynne, J. R. Ockendon, and P. Wilmott, “On a mathematical model for fiber tapering,” SIAM J. Appl. Math. 49, 983–990 (1989). [CrossRef]

, 20

20. F. Geyling, “Basic fluid dynamic consideration in the drawing of optical fibers,” Bell Sys. Tech. J. 55, 1011–1056 (1976).

]. 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 = vf/vd, where vf is the feed velocity and vd is the draw velocity, is constant throughout each tapering sweep. A constant s limits the lowest inverse tapering ratio ρ=ϕj/ϕj1=s, where ϕj is the waist diameter after sweep j, that can be used in each sweep [21

21. S. Leon-Saval, T. Birks, W. Wadsworth, P. S. J. Russell, and M. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express 12, 2864–2869 (2004). [CrossRef] [PubMed]

]. If ρ is less than 0.97 [11

11. R. P. Kenny, T. A. Birks, and K. P. Oakley, “Control of optical fibre taper shape,” Electron. Lett. 27, 1654–1656 (1991). [CrossRef]

], 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

11. R. P. Kenny, T. A. Birks, and K. P. Oakley, “Control of optical fibre taper shape,” Electron. Lett. 27, 1654–1656 (1991). [CrossRef]

], a uniform waist profile [21

21. S. Leon-Saval, T. Birks, W. Wadsworth, P. S. J. Russell, and M. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express 12, 2864–2869 (2004). [CrossRef] [PubMed]

], and a large contrast ratio between the initial and the final taper diameters [10

10. F. Bilodeau, K. Hill, S. Faucher, and D. Johnson, “Low-loss highly overcoupled fused couplers: Fabrication and sensitivity to external pressure,” in Optical Fiber Sensors, (Optical Society of America, 1988), p. ThCC10.

]. 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

18. S. Pricking and H. Giessen, “Tapering fibers with complex shape,” Opt. Express 18, 3426–3437 (2010). [CrossRef] [PubMed]

].

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

22. A. Kudlinski, A. K. George, J. C. Knight, J. C. Travers, A. B. Rulkov, S. V. Popov, and J. R. Taylor, “Zero-dispersion wavelength decreasing photonic crystal fibers for ultraviolet-extended supercontinuum generation,” Opt. Express 14, 5715–5722 (2006). [CrossRef] [PubMed]

, 23

23. G. Qin, X. Yan, C. Kito, M. Liao, T. Suzuki, A. Mori, and Y. Ohishi, “Zero-dispersion-wavelength-decreasing tellurite microstructured fiber for wide and flattened supercontinuum generation,” Opt. Lett. 35, 136–138 (2010). [CrossRef] [PubMed]

] and enhanced soliton self-frequency shift [24

24. A. C. Judge, O. Bang, B. J. Eggleton, B. T. Kuhlmey, E. C. Mägi, R. Pant, and C. M. de Sterke, “Optimization of the soliton self-frequency shift in a tapered photonic crystal fiber,” J. Opt. Soc. Am. B 26, 2064–2071 (2009). [CrossRef]

, 25

25. A. Alkadery and M. Rochette, “Widely tunable soliton shifting for mid-infrared applications,” in IEEE Photonics Conference 2011 (IPC2011) (2011).

]. 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

8. J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices. I. adiabaticity criteria,” IEE Proc.-J: Optoelectron. 138, 343–354 (1991). [CrossRef]

].

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

17. J. Dewynne, J. R. Ockendon, and P. Wilmott, “On a mathematical model for fiber tapering,” SIAM J. Appl. Math. 49, 983–990 (1989). [CrossRef]

,20

20. F. Geyling, “Basic fluid dynamic consideration in the drawing of optical fibers,” Bell Sys. Tech. J. 55, 1011–1056 (1976).

,26

26. N. Vukovic, N. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” Photon. Technol. Lett. 20, 1264–1266 (2008). [CrossRef]

]. In the process of fiber drawing, mass conservation leads to ϕ(t)=ϕ0s(t) where ϕ (t) is the taper diameter, ϕ0 is the initial fiber diameter, and s (t) = vf (t)/vd (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 ld(t)=0tvd(τ)dτ simplifies the implementation of the single-sweep tapering method because it can be readily used as a feedback parameter to control the draw velocity vd (ld) = vf (ld)/s (ld). In this case, the tapering function s (ld) is calculated from the taper profile ϕ (z) using
s(ld)=ϕ2(z)ϕ02|z=ld.
Figure 2 provides an arbitrary taper profile ϕ (z) and its corresponding tapering function s(ld).

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

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

20. F. Geyling, “Basic fluid dynamic consideration in the drawing of optical fibers,” Bell Sys. Tech. J. 55, 1011–1056 (1976).

]. A simplified model has been derived for the case when the fiber diameter is much smaller than the hot-zone length (Lhz) [17

17. J. Dewynne, J. R. Ockendon, and P. Wilmott, “On a mathematical model for fiber tapering,” SIAM J. Appl. Math. 49, 983–990 (1989). [CrossRef]

]. In this model, the deformation of the hot-zone due to stretching is governed by
z(3μAuz)=0,
(1)
At+z(uA)=0,
(2)
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

17. J. Dewynne, J. R. Ockendon, and P. Wilmott, “On a mathematical model for fiber tapering,” SIAM J. Appl. Math. 49, 983–990 (1989). [CrossRef]

]. 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/vd is the normalized axial velocity and F = μA. Using the centered differentiation formulas [27

27. S. Chapra and R. Canale, Numerical Methods for Engineers, 5th ed. (McGraw-Hill, 2005).

] ∂F/∂z = (Fi+1Fi–1)/2Δz, ∂ū/∂z = (ūi+1ūi−1)/2Δz, and 2ū/∂z2 = (ūi+1 − 2ūi + ūi−1)/Δz2 leads to the finite difference form of Eq. (1)
[Fi0.25(Fi+1Fi1)]u¯i12Fiu¯i+[Fi+0.25(Fi+1Fi1)]u¯i+1=0
(3)
where Fi = F (ld, zi), ūi = ū (ld, zi), and Δz is the separation between any two consecutive zi. Changing the variable t to ld in Eq. (2) leads to the equation vd∂A/∂ld + (uA)/∂z = 0, which is expanded and divided by vd to obtain ∂A/∂ld + A∂ū/∂z + ū∂A/∂z = 0. Using the centered differentiation formulas ∂ū/∂z = (ūi+1ūi−1)/2Δz, ∂A/∂z = (Ai+1Ai−1)/2Δz and the forward differentiation formula [27

27. S. Chapra and R. Canale, Numerical Methods for Engineers, 5th ed. (McGraw-Hill, 2005).

] A/ld=[AinewAi]/Δld, the finite difference form of Eq. (2) corresponding to the extension of the fiber by a distance Δld = 2Δz is given by
Ainew=Ai[Ai(u¯i+1u¯i1)+u¯i(Ai+1Ai1)]
(4)
where Ai = A (ld, zi), Ainew=A(ld+Δld,zi). 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 ϕk taken at points zk 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 Ai 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 ūi=0 = −1/2 and ūi=N+1 = 1/2 to calculate the normalized axial velocity distribution ūi 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, xprevious and yprevious are state variables, δ is a differential feed step, s is the tapering function, ld is the drawing length, and Δz is the longitudinal separation between any two consecutive diameter sampling points.
Fig. 4 Single-sweep simulation schematics of (a) shifting the hot-zone by Δz, and (b) extension of the fiber by 2Δz.

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) = [ϕr (z) – ϕt (z)]/ϕt (z) × 100% where ϕr is the resulting taper diameter and ϕt is the targeted taper diameter. The transient response is quantified by the percent overshoot ɛos = (ϕtϕos)/ϕt × 100% where ϕos is the overshoot diameter, and by the settling distance zs defined as the distance between the beginning of the waist and the point where the envelope of the absolute percent error is less than ɛs = 2%.

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

The transient response parameters ɛos and zs represent the closeness of the of the resulting taper shape to the taper design, and the overall mismatch is reduced by reducing ɛos and zs. Step-taper simulation results in Fig. 6 show ɛos and zs as a function of Lhz and the inverse tapering ratio ρ = ϕmin/ϕ0, where ϕmin is the minimum taper diameter. As expected, ɛos and zs decrease with increasing ρ (≤ 1) and shortening Lhz. 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

28. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972). [CrossRef]

]. The values of ɛos and zs also represents the strength and the length of the perturbation region; therefore, a lower ɛos and a shorter zs 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.

2.2. Single-sweep tapering optimization

Simulation results in subsection 2.1 showed that ɛos and zs decrease when ρ → 1 and Lhz → 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, Lhz 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 ɛos and zs decrease when the taper slope decreases. As an example, Fig. 7 shows that as the slope decreases from 0.0105 to 0.0035, ɛos decreases from 8.8% to 3.8% and zs 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 ɛos and zs leads to a long transition region and consequently increases the sensitivity of the taper to environmental variations [9

9. T. A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992). [CrossRef]

] as well as increasing the device length. Section 3 shows that ɛos and zs 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.

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 vy and two other translation stages pull the fiber from opposite directions at equal velocities vw and vx. Using vd = vy + vw and vf = vyvx = α, where α is a constant, the velocities of the heater and the translation stages pulling on the fiber at a drawing length ld = y + w are
vy(ld)=vd(ld)+vf(ld)2=α2[1s(ld)+1],
vx(ld)=vw(ld)=vd(ld)vf(ld)2=α2[1s(ld)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 vf =0.72 mm/min and vdmax=max(vf/s)=4.5mm/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.
Fig. 9 Experimentally measured profiles of (a) a step taper, and (b) an arbitrary taper fabricated using the single-sweep tapering method.

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 ϕn is the minimum taper diameter, and ϕ1 to ϕn–1 are the waist diameters for all intermediate tapering sweeps and are calculated using ϕj = j−1 with r = ρ1/n and ρ = ϕn/ϕ0. For every sweep j < n, the stage tapering function s(j) (lp) is calculated from the stage taper profile ϕ(j) (z) composed of a left transition region extracted from ϕ (z) between zj1left and zjleft, a right transition region extracted from ϕ (z) between zjright and zj1right, and a uniform waist with a length
Lj=zjleftzjrightϕ2(z)dzϕj2,
where Lj makes the mass volume of the waist at stage j equal to the mass volume required to draw the taper section between zjleft and zjright. The stage taper profile of the final sweep ϕ(n) (z) is extracted from ϕ (z) between zn1left and zn1right, and is used to calculate the final stage tapering function s(n) (lp). 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 (zj1rightzjright)+Lj.

Fig. 10 Schematic of taper profile evolution using the multi-sweep tapering method.
Fig. 11 Dividing the taper into sections for the determination of the tapering function of each tapering stage.

3.1. Quantitative analysis of multi-sweep tapering

Based on the divide-and-conquer paradigm [29

29. T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 2nd ed. (MIT Press, 2001).

], 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
ϕos(j)=[1ɛos(ρj)/100%]×ρj×ϕos(j1)
ϕos(1)=[1ɛos(ρ1)/100%]×ρ1×ϕ0,
where ɛos (ρj) is provided in Fig. 6(a). By setting the inverse tapering ratio for all sweeps to r, the worst-case overshoot diameter becomes
ϕos(j)=[1ɛos(t)/100%]j×rj×ϕ0,
and the maximum percent overshoot at the end of tapering is
ɛos,max(n)=[1(1ɛos(r)/100%)n]×100%,
(5)
which is simplified to ɛos,max(n)nɛos(r) when ɛos (r) ≤ 1%. It is clear from Fig. 6(a) that ɛos,max(n)<ɛos(ρ) and that ɛos,max(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 ɛos (0.5) = 17%. However, when tapering is performed over 6 sweeps with r = 0.89 and ɛos (0.89) = 0.5%, the maximum percent overshoot is ɛos,max(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 Tj=Lj1/vfmax, where vfmax is the maximum practical feed velocity, and the total tapering duration after n sweeps is
T=L0vfmax×1ρ21ρ2/n,
which is reduced by increasing vfmax 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 ɛos 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 ≈ Δϕz leading to Δz ≈ (1 − ρ)ϕ/(∂ϕ/∂z). Setting Lhz ≪ |Δz| does not decrease the mismatch because the diameter steps in the transition region become more prominent; in fact, setting Lhz ≳ |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

9. T. A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992). [CrossRef]

] 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 Lhz ≳ 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 Lhz 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 Lhz = 3 mm for a step-taper with ρ = 0.4 show that the percent overshoot ɛos(n) decreases as n increases. Also shown in Fig. 12 is the worst-case percent overshoot, ɛos,max(n), calculated using Eq. (5). It is observed that ɛos(n) does not exceed ɛos,max(n), which is expected as ɛos,max(n) estimates the upper limit of ɛos(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 Lhz = 3 mm.

Although increasing n reduces ɛos, Lhz must also be shortened to ensure that |ɛ (z)| is less than a prescribed value ɛtarget. Shortening Lhz 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 Lhz, then the details of the waist can not be precisely shaped. The value of Lhz that ensures |ɛ(z)| < ɛtarget 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 vf = 3.56 mm/min and vdmax=4.50mm/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.

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.

References

1.

P. Dumais, F. Gonthier, S. Lacroix, J. Bures, A. Villeneuve, P. G. J. Wigley, and G. I. Stegeman, “Enhanced self-phase modulation in tapered fibers,” Opt. Lett. 18, 1996–1998 (1993). [CrossRef] [PubMed]

2.

C. Baker and M. Rochette, “Highly nonlinear hybrid AsSe-PMMA microtapers,” Opt. Express 18, 12391–12398 (2010). [CrossRef] [PubMed]

3.

A. C. Boucouvalas and G. Georgiou, “Biconical taper coaxial optical fibre coupler,” Electron. Lett. 21, 864–865 (1985). [CrossRef]

4.

R. G. Lamont, D. C. Johnson, and K. O. Hill, “Power transfer in fused biconical-taper single-mode fiber couplers: dependence on external refractive index,” Appl. Opt. 24, 327–332 (1985). [CrossRef] [PubMed]

5.

J. V. Wright, “Wavelength dependence of fused couplers,” Electron. Lett. 22, 320–321 (1986). [CrossRef]

6.

S. Jensen, “The nonlinear coherent coupler,” J. Quantum Electron. 18, 1580–1583 (1982). [CrossRef]

7.

J. D. Love and W. M. Henry, “Quantifying loss minimisation in single-mode fibre tapers,” Electron. Lett. 22, 912–914 (1986). [CrossRef]

8.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices. I. adiabaticity criteria,” IEE Proc.-J: Optoelectron. 138, 343–354 (1991). [CrossRef]

9.

T. A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992). [CrossRef]

10.

F. Bilodeau, K. Hill, S. Faucher, and D. Johnson, “Low-loss highly overcoupled fused couplers: Fabrication and sensitivity to external pressure,” in Optical Fiber Sensors, (Optical Society of America, 1988), p. ThCC10.

11.

R. P. Kenny, T. A. Birks, and K. P. Oakley, “Control of optical fibre taper shape,” Electron. Lett. 27, 1654–1656 (1991). [CrossRef]

12.

Y. Takeuchi and J. Noda, “Novel fiber coupler tapering process using a microheater,” Photon. Technol. Lett. 4, 465–467 (1992). [CrossRef]

13.

H. Yokota, E. Sugai, Y. Kashima, and Y. Sasaki, “Optical irradiation method for fiber coupler fabrications,” in Optical Fiber Sensors, (OSA, 1996), p. Th319.

14.

A. J. C. Grellier, N. K. Zayer, and C. N. Pannell, “Heat transfer modelling in co laser processing of optical fibres,” Opt. Commun. 152, 324–328 (1998). [CrossRef]

15.

L. C. Ozcan, V. Treanton, F. Guay, and R. Kashyap, “Highly symmetric optical fiber tapers fabricated with a CO2 laser,” Photon. Technol. Lett. 19, 656–658 (2007). [CrossRef]

16.

S. Xue, M. van Eijkelenborg, G. W. Barton, and P. Hambley, “Theoretical, numerical, and experimental analysis of optical fiber tapering,” J. Lightwave Technol. 25, 1169–1176 (2007). [CrossRef]

17.

J. Dewynne, J. R. Ockendon, and P. Wilmott, “On a mathematical model for fiber tapering,” SIAM J. Appl. Math. 49, 983–990 (1989). [CrossRef]

18.

S. Pricking and H. Giessen, “Tapering fibers with complex shape,” Opt. Express 18, 3426–3437 (2010). [CrossRef] [PubMed]

19.

W. Sun, M. Yuan, X. Zeng, and T. Birks, “Theoretical shape analysis of tapered fibers using a movable large-zone furnace,” Optoelectron. Lett. 7, 154–157 (2011). [CrossRef]

20.

F. Geyling, “Basic fluid dynamic consideration in the drawing of optical fibers,” Bell Sys. Tech. J. 55, 1011–1056 (1976).

21.

S. Leon-Saval, T. Birks, W. Wadsworth, P. S. J. Russell, and M. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express 12, 2864–2869 (2004). [CrossRef] [PubMed]

22.

A. Kudlinski, A. K. George, J. C. Knight, J. C. Travers, A. B. Rulkov, S. V. Popov, and J. R. Taylor, “Zero-dispersion wavelength decreasing photonic crystal fibers for ultraviolet-extended supercontinuum generation,” Opt. Express 14, 5715–5722 (2006). [CrossRef] [PubMed]

23.

G. Qin, X. Yan, C. Kito, M. Liao, T. Suzuki, A. Mori, and Y. Ohishi, “Zero-dispersion-wavelength-decreasing tellurite microstructured fiber for wide and flattened supercontinuum generation,” Opt. Lett. 35, 136–138 (2010). [CrossRef] [PubMed]

24.

A. C. Judge, O. Bang, B. J. Eggleton, B. T. Kuhlmey, E. C. Mägi, R. Pant, and C. M. de Sterke, “Optimization of the soliton self-frequency shift in a tapered photonic crystal fiber,” J. Opt. Soc. Am. B 26, 2064–2071 (2009). [CrossRef]

25.

A. Alkadery and M. Rochette, “Widely tunable soliton shifting for mid-infrared applications,” in IEEE Photonics Conference 2011 (IPC2011) (2011).

26.

N. Vukovic, N. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” Photon. Technol. Lett. 20, 1264–1266 (2008). [CrossRef]

27.

S. Chapra and R. Canale, Numerical Methods for Engineers, 5th ed. (McGraw-Hill, 2005).

28.

A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972). [CrossRef]

29.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 2nd ed. (MIT Press, 2001).

OCIS Codes
(220.4610) Optical design and fabrication : Optical fabrication
(230.4000) Optical devices : Microstructure fabrication

ToC Category:
Materials for Fiber Optics

History
Original Manuscript: July 22, 2011
Revised Manuscript: August 18, 2011
Manuscript Accepted: August 29, 2011
Published: September 2, 2011

Citation
Chams Baker and Martin Rochette, "A generalized heat-brush approach for precise control of the waist profile in fiber tapers," Opt. Mater. Express 1, 1065-1076 (2011)
http://www.opticsinfobase.org/ome/abstract.cfm?URI=ome-1-6-1065


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References

  1. P. Dumais, F. Gonthier, S. Lacroix, J. Bures, A. Villeneuve, P. G. J. Wigley, and G. I. Stegeman, “Enhanced self-phase modulation in tapered fibers,” Opt. Lett.18, 1996–1998 (1993). [CrossRef] [PubMed]
  2. C. Baker and M. Rochette, “Highly nonlinear hybrid AsSe-PMMA microtapers,” Opt. Express18, 12391–12398 (2010). [CrossRef] [PubMed]
  3. A. C. Boucouvalas and G. Georgiou, “Biconical taper coaxial optical fibre coupler,” Electron. Lett.21, 864–865 (1985). [CrossRef]
  4. R. G. Lamont, D. C. Johnson, and K. O. Hill, “Power transfer in fused biconical-taper single-mode fiber couplers: dependence on external refractive index,” Appl. Opt.24, 327–332 (1985). [CrossRef] [PubMed]
  5. J. V. Wright, “Wavelength dependence of fused couplers,” Electron. Lett.22, 320–321 (1986). [CrossRef]
  6. S. Jensen, “The nonlinear coherent coupler,” J. Quantum Electron.18, 1580–1583 (1982). [CrossRef]
  7. J. D. Love and W. M. Henry, “Quantifying loss minimisation in single-mode fibre tapers,” Electron. Lett.22, 912–914 (1986). [CrossRef]
  8. J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices. I. adiabaticity criteria,” IEE Proc.-J: Optoelectron.138, 343–354 (1991). [CrossRef]
  9. T. A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol.10, 432–438 (1992). [CrossRef]
  10. F. Bilodeau, K. Hill, S. Faucher, and D. Johnson, “Low-loss highly overcoupled fused couplers: Fabrication and sensitivity to external pressure,” in Optical Fiber Sensors, (Optical Society of America, 1988), p. ThCC10.
  11. R. P. Kenny, T. A. Birks, and K. P. Oakley, “Control of optical fibre taper shape,” Electron. Lett.27, 1654–1656 (1991). [CrossRef]
  12. Y. Takeuchi and J. Noda, “Novel fiber coupler tapering process using a microheater,” Photon. Technol. Lett.4, 465–467 (1992). [CrossRef]
  13. H. Yokota, E. Sugai, Y. Kashima, and Y. Sasaki, “Optical irradiation method for fiber coupler fabrications,” in Optical Fiber Sensors, (OSA, 1996), p. Th319.
  14. A. J. C. Grellier, N. K. Zayer, and C. N. Pannell, “Heat transfer modelling in co laser processing of optical fibres,” Opt. Commun.152, 324–328 (1998). [CrossRef]
  15. L. C. Ozcan, V. Treanton, F. Guay, and R. Kashyap, “Highly symmetric optical fiber tapers fabricated with a CO2 laser,” Photon. Technol. Lett.19, 656–658 (2007). [CrossRef]
  16. S. Xue, M. van Eijkelenborg, G. W. Barton, and P. Hambley, “Theoretical, numerical, and experimental analysis of optical fiber tapering,” J. Lightwave Technol.25, 1169–1176 (2007). [CrossRef]
  17. J. Dewynne, J. R. Ockendon, and P. Wilmott, “On a mathematical model for fiber tapering,” SIAM J. Appl. Math.49, 983–990 (1989). [CrossRef]
  18. S. Pricking and H. Giessen, “Tapering fibers with complex shape,” Opt. Express18, 3426–3437 (2010). [CrossRef] [PubMed]
  19. W. Sun, M. Yuan, X. Zeng, and T. Birks, “Theoretical shape analysis of tapered fibers using a movable large-zone furnace,” Optoelectron. Lett.7, 154–157 (2011). [CrossRef]
  20. F. Geyling, “Basic fluid dynamic consideration in the drawing of optical fibers,” Bell Sys. Tech. J.55, 1011–1056 (1976).
  21. S. Leon-Saval, T. Birks, W. Wadsworth, P. S. J. Russell, and M. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express12, 2864–2869 (2004). [CrossRef] [PubMed]
  22. A. Kudlinski, A. K. George, J. C. Knight, J. C. Travers, A. B. Rulkov, S. V. Popov, and J. R. Taylor, “Zero-dispersion wavelength decreasing photonic crystal fibers for ultraviolet-extended supercontinuum generation,” Opt. Express14, 5715–5722 (2006). [CrossRef] [PubMed]
  23. G. Qin, X. Yan, C. Kito, M. Liao, T. Suzuki, A. Mori, and Y. Ohishi, “Zero-dispersion-wavelength-decreasing tellurite microstructured fiber for wide and flattened supercontinuum generation,” Opt. Lett.35, 136–138 (2010). [CrossRef] [PubMed]
  24. A. C. Judge, O. Bang, B. J. Eggleton, B. T. Kuhlmey, E. C. Mägi, R. Pant, and C. M. de Sterke, “Optimization of the soliton self-frequency shift in a tapered photonic crystal fiber,” J. Opt. Soc. Am. B26, 2064–2071 (2009). [CrossRef]
  25. A. Alkadery and M. Rochette, “Widely tunable soliton shifting for mid-infrared applications,” in IEEE Photonics Conference 2011 (IPC2011) (2011).
  26. N. Vukovic, N. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical fiber tapers,” Photon. Technol. Lett.20, 1264–1266 (2008). [CrossRef]
  27. S. Chapra and R. Canale, Numerical Methods for Engineers, 5th ed. (McGraw-Hill, 2005).
  28. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am.62, 1267–1277 (1972). [CrossRef]
  29. T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 2nd ed. (MIT Press, 2001).

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