A generalized heat-brush approach for
precise control of the waist profile in
fiber tapers
Chams Baker∗ and Martin Rochette
McGill University, Department of Electrical and Computer Engineering,
Montréal (PQ), H3A 2A7, Canada
∗ [email protected]
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 Optical Society of America
OCIS codes: (230.4000) Microstructure fabrication; (220.4610) Optical fabrication.
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).
2. C. Baker and M. Rochette, “Highly nonlinear hybrid AsSe-PMMA microtapers,” Opt. Express 18, 12391–12398
(2010).
3. A. C. Boucouvalas and G. Georgiou, “Biconical taper coaxial optical fibre coupler,” Electron. Lett. 21, 864–865
(1985).
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).
5. J. V. Wright, “Wavelength dependence of fused couplers,” Electron. Lett. 22, 320–321 (1986).
6. S. Jensen, “The nonlinear coherent coupler,” J. Quantum Electron. 18, 1580–1583 (1982).
7. J. D. Love and W. M. Henry, “Quantifying loss minimisation in single-mode fibre tapers,” Electron. Lett. 22,
912–914 (1986).
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).
9. T. A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992).
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).
12. Y. Takeuchi and J. Noda, “Novel fiber coupler tapering process using a microheater,” Photon. Technol. Lett. 4,
465–467 (1992).
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).
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).
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).
17. J. Dewynne, J. R. Ockendon, and P. Wilmott, “On a mathematical model for fiber tapering,” SIAM J. Appl. Math.
49, 983–990 (1989).
18. S. Pricking and H. Giessen, “Tapering fibers with complex shape,” Opt. Express 18, 3426–3437 (2010).
#151542 - $15.00 USD
(C) 2011 OSA
Received 22 Jul 2011; revised 18 Aug 2011; accepted 29 Aug 2011; published 2 Sep 2011
1 October 2011 / Vol. 1, No. 6 / OPTICAL MATERIALS EXPRESS 1065
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).
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).
22. A. Kudlinski, A. K. George, J. C. Knight, J. C. Travers, A. B. Rulkov, S. V. Popov, and J. R. Taylor, “Zerodispersion wavelength decreasing photonic crystal fibers for ultraviolet-extended supercontinuum generation,”
Opt. Express 14, 5715–5722 (2006).
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).
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).
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).
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).
29. T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 2nd ed. (MIT Press, 2001).
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.
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.
#151542 - $15.00 USD
(C) 2011 OSA
Received 22 Jul 2011; revised 18 Aug 2011; accepted 29 Aug 2011; published 2 Sep 2011
1 October 2011 / Vol. 1, No. 6 / OPTICAL MATERIALS EXPRESS 1066
Moreover, in the heat-brush implementation of Birks’ model, the tapering function s = v f /vd ,
each tapering
where v f is the feed velocity and vd is the draw velocity, is constant throughout
√
sweep. A constant s limits the lowest inverse tapering ratio ρ = φ j /φ j−1 = 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 brushingzone, 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 As2 Se3 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) = φ0 s (t) where φ (t) is the taper diameter, φ0 is the initial fiber diameter, and s (t) =
v f (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
´t
drawing length ld (t) = 0 vd (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
vd (ld ) = v f (ld ) /s (ld ). In this case, the tapering function s (ld ) is calculated from the taper
profile φ (z) using
φ 2 (z) s (ld ) =
.
φ2 0
z=ld
Figure 2 provides an arbitrary taper profile φ (z) and its corresponding tapering function s (ld ).
#151542 - $15.00 USD
(C) 2011 OSA
Received 22 Jul 2011; revised 18 Aug 2011; accepted 29 Aug 2011; published 2 Sep 2011
1 October 2011 / Vol. 1, No. 6 / OPTICAL MATERIALS EXPRESS 1067
1
φ μ
150
100
0.5
50
(b)
(a)
0
0
10
20
30
z [mm]
0
0
10
20
30
ld [mm]
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]. A simplified model has been derived for the case
when the fiber diameter is much smaller than the hot-zone length (Lhz ) [17]. In this model, the
deformation of the hot-zone due to stretching is governed by
∂
∂u
3μ A
= 0,
(1)
∂z
∂z
∂A ∂
+ (uA) = 0,
(2)
∂t ∂z
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,
¯ ∂ z × ∂ F/∂ z + F × ∂ 2 u/
¯ ∂ z2 = 0, where u¯ = u/vd is the norand hence, Eq. (1) leads to ∂ u/
malized axial velocity and F = μ A. Using the centered differentiation formulas [27] ∂ F/∂ z =
¯ ∂ z = (u¯i+1 − u¯i−1 ) /2Δz, and ∂ 2 u/
¯ ∂ z2 = (u¯i+1 − 2u¯i + u¯i−1 ) /Δz2 leads
(Fi+1 − Fi−1 ) /2Δz, ∂ u/
to the finite difference form of Eq. (1)
[Fi − 0.25 (Fi+1 − Fi−1 )] u¯i−1 − 2Fi u¯i + [Fi + 0.25 (Fi+1 − Fi−1 )] u¯i+1 = 0
(3)
where Fi = F (ld , zi ), u¯i = u¯ (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
¯ ∂ z + u¯∂ A/∂ z = 0. Using the cenis expanded and divided by vd to obtain ∂ A/∂ ld + A∂ u/
¯ ∂ z = (u¯i+1 − u¯i−1 ) /2Δz, ∂ A/∂ z = (Ai+1 − Ai−1 ) /2Δz and
tered differentiation formulas ∂ u/
− Ai ] /Δld , the finite difference form
the forward differentiation formula [27] ∂ A/∂ ld = [Anew
i
of Eq. (2) corresponding to the extension of the fiber by a distance Δld = 2Δz is given by
= Ai − [Ai (u¯i+1 − u¯i−1 ) + u¯i (Ai+1 − Ai−1 )]
Anew
i
(4)
= A (ld + Δld , zi ). It is clear from Eq. (3) and Eq. (4) that, for a
where Ai = A (ld , zi ), Anew
i
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
#151542 - $15.00 USD
(C) 2011 OSA
Received 22 Jul 2011; revised 18 Aug 2011; accepted 29 Aug 2011; published 2 Sep 2011
1 October 2011 / Vol. 1, No. 6 / OPTICAL MATERIALS EXPRESS 1068
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 u¯i=0 = −1/2 and u¯i=N+1 = 1/2 to calculate the
normalized axial velocity distribution u¯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, 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
ε (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 εos =
#151542 - $15.00 USD
(C) 2011 OSA
Received 22 Jul 2011; revised 18 Aug 2011; accepted 29 Aug 2011; published 2 Sep 2011
1 October 2011 / Vol. 1, No. 6 / OPTICAL MATERIALS EXPRESS 1069
(φt − φos ) /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 εs = 2%.
Design
Simulation
Fig. 5. Simulation of step-taper fabrication using the single-sweep tapering method.
The transient response parameters εos and zs represent the closeness of the of the resulting
taper shape to the taper design, and the overall mismatch is reduced by reducing εos and zs . Steptaper simulation results in Fig. 6 show εos and zs as a function of Lhz and the inverse tapering
ratio ρ = φmin /φ0 , where φmin is the minimum taper diameter. As expected, εos 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]. The values
of εos and zs also represents the strength and the length of the perturbation region; therefore, a
lower εos and a shorter zs reduces the perturbation impact.
30
10
Lhz = 2.0 mm
(b)
Lhz = 3.0 mm
30
Lhz = 3.0 mm
Lhz = 4.0 mm
20
Lhz = 4.0 mm
s
εos
20
0
0.4
40
Lhz = 2.0 mm
(a)
10
0.6
ρ
0.8
1
0
0.4
0.6
ρ
0.8
1
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 εos 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 εos 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, εos decreases from 8.8% to 3.8%
#151542 - $15.00 USD
(C) 2011 OSA
Received 22 Jul 2011; revised 18 Aug 2011; accepted 29 Aug 2011; published 2 Sep 2011
1 October 2011 / Vol. 1, No. 6 / OPTICAL MATERIALS EXPRESS 1070
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 εos
and zs 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 εos
and zs are reduced by tapering a fiber over multiple sweeps leading to an implementation of the
generalized heat-brush approach.
φ φmax
0.8
0.6
0.8
0.6
(a)
0.4
Design
Simulation
|dφ / dz| = 0.0035
1
φ φmax
Design
Simulation
|dφ / dz| = 0.0105
1
(b)
-20
0
z [mm]
20
0.4
-20
0
20
z [mm]
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 v f = vy − vx = 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
vd (ld ) + v f (ld ) α
1
=
+1 ,
vy (ld ) =
2
2 s (ld )
vd (ld ) − v f (ld ) α
1
=
−1 .
vx (ld ) = vw (ld ) =
2
2 s (ld )
Fig. 8. Schematic of the experimental implementation of the single-sweep tapering method.
#151542 - $15.00 USD
(C) 2011 OSA
Received 22 Jul 2011; revised 18 Aug 2011; accepted 29 Aug 2011; published 2 Sep 2011
1 October 2011 / Vol. 1, No. 6 / OPTICAL MATERIALS EXPRESS 1071
2.4.
Single-sweep tapering experimental results
Figure 9(a) shows the experimental results of a step-taper fabricated from an As2 Se3 fiber
◦
with an initial diameter of 170
µm using a 5 mm long resistive heater at 210 C with v f =
max
0.72 mm/min and vd = 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.
Design
Simulation
Measurements
0.8
0.6
1.0
φ φmax
φ φmax
1
(a)
0.4
Design
Simulation
Measurements
0.8
0.6
(b)
0.4
-20
-10
0
10
20
-20
z [mm]
0
20
z [mm]
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 φ j = rφ j−1 with r = ρ 1/n and ρ = φn /φ0 . For every sweep j < n, the stage tapering function s( j) (l p ) is calculated from the stage taper profile
ft
φ ( j) (z) composed of a left transition region extracted from φ (z) between zlej−1
and zlej f t , a right
transition region extracted from φ (z) between zright
and zright
j
j−1 , and a uniform waist with a length
´ zright
j
le f t
Lj =
zj
φ 2 (z) dz
φ j2
,
where L j makes the mass volume of the waist at stage j equal to the mass volume required to
. The stage taper profile of the final sweep φ (n) (z)
draw the taper section between zlej f t and zright
j
ft
right
is extracted from φ (z) between zle
n−1 and zn−1 , and is used to calculate the final stage tapering
(n)
function s (l p ). Finally, for each stage j, a single tapering sweep is performed
using thecalcu-
right
+ L j.
lated stage tapering function and then the heater is moved back a distance zright
j−1 − z j
#151542 - $15.00 USD
(C) 2011 OSA
Received 22 Jul 2011; revised 18 Aug 2011; accepted 29 Aug 2011; published 2 Sep 2011
1 October 2011 / Vol. 1, No. 6 / OPTICAL MATERIALS EXPRESS 1072
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], 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
( j)
( j−1)
φos = [1 − εos (ρ j ) /100%] × ρ j × φos
(1)
φos = [1 − εos (ρ1 ) /100%] × ρ1 × φ0 ,
#151542 - $15.00 USD
(C) 2011 OSA
Received 22 Jul 2011; revised 18 Aug 2011; accepted 29 Aug 2011; published 2 Sep 2011
1 October 2011 / Vol. 1, No. 6 / OPTICAL MATERIALS EXPRESS 1073
where εos (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
( j)
φos = [1 − εos (r) /100%] j × r j × φ0 ,
and the maximum percent overshoot at the end of tapering is
(n)
εos,max = [1 − (1 − εos (r) /100%)n ] × 100%,
(5)
(n)
which is simplified to εos,max ≈ nεos (r) when εos (r) ≤ 1%. It is clear from Fig. 6(a) that
(n)
(n)
εos,max < εos (ρ ) and that εos,max decreases as n increases. For, example, the fabrication of a steptaper with ρ = 0.5 over a single sweep using a 4 mm long hot-zone leads to εos (0.5) = 17%.
However, when tapering is performed over 6 sweeps with r = 0.89 and εos (0.89) = 0.5%, the
(6)
maximum percent overshoot is εos,max = 3%.
The use of a large number of sweeps increases the tapering duration. For the case of a stepmax is the maximum
taper, the minimum time duration for stage j is T j = L j−1 /vmax
f , where v f
practical feed velocity, and the total tapering duration after n sweeps is
T=
1 − ρ −2
L0
×
,
vmax
1 − ρ −2/n
f
and reducing n. In general, to keep the tapering duration
which is reduced by increasing vmax
f
at a minimum, n is selected to be the minimum number of sweeps required to keep εos 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 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) = φ0 exp (−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
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
(n)
ρ = 0.4 show that the percent overshoot εos decreases as n increases. Also shown in Fig. 12 is
(n)
(n)
the worst-case percent overshoot, εos,max , calculated using Eq. (5). It is observed that εos does
(n)
(n)
(n)
not exceed εos,max , which is expected as εos,max estimates the upper limit of εos .
#151542 - $15.00 USD
(C) 2011 OSA
Received 22 Jul 2011; revised 18 Aug 2011; accepted 29 Aug 2011; published 2 Sep 2011
1 October 2011 / Vol. 1, No. 6 / OPTICAL MATERIALS EXPRESS 1074
15
ρ = 0.4
εos(n)
Lhz = 3.0 mm
(n)
εos,max
εos
10
5
2
3
4
n
5
6
7
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 εos , Lhz must also be shortened to ensure that |ε (z)| is less
than a prescribed value εtarget . 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 |ε (z)| < εtarget 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 As2 Se3 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
= 4.50 mm/min. The measurement error is 1 µm and the resulting taper matches the
and vmax
d
design within the measurement error.
Design
Simulation
Measurements
φ φmax
1
0.5
0
-50
-30
-10
10
z [mm]
30
50
Fig. 13. Experimental results showing the profile of an As2 Se3 taper fabricated using the
multi-sweep tapering method with n = 24.
#151542 - $15.00 USD
(C) 2011 OSA
Received 22 Jul 2011; revised 18 Aug 2011; accepted 29 Aug 2011; published 2 Sep 2011
1 October 2011 / Vol. 1, No. 6 / OPTICAL MATERIALS EXPRESS 1075
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 As2 Se3 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.
#151542 - $15.00 USD
(C) 2011 OSA
Received 22 Jul 2011; revised 18 Aug 2011; accepted 29 Aug 2011; published 2 Sep 2011
1 October 2011 / Vol. 1, No. 6 / OPTICAL MATERIALS EXPRESS 1076