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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 4 — Feb. 15, 2010
  • pp: 3426–3437
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Tapering fibers with complex shape

S. Pricking and H. Giessen  »View Author Affiliations


Optics Express, Vol. 18, Issue 4, pp. 3426-3437 (2010)
http://dx.doi.org/10.1364/OE.18.003426


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Abstract

We present a model which allows us to accurately simulate the fabrication process of complex-shaped tapered fibers. The range of possible profiles is only limited by the properties of the heat source used to shape the fiber. The model takes into account the motion of the heat source relative to the fiber as well as its temperature distribution. Our measurements and corresponding finite element method (FEM) simulations have shown a strong dependency of the temperature distribution along the fiber axis on the actual diameter of the fiber. The inclusion of this relation in the model proved to be crucial for the accuracy of the results. Our model has been verified experimentally by fabricating tapered fibers with a sinusoidally modulated waist. A comparison to the profile predicted by our model reveals an excellent agreement.

© 2010 Optical Society of America

1. Introduction

Tapered optical fibers have gained a lot of interest in the last years due to their unique physical and optical properties not found in standard fibers. This opens up possibilities for many new applications. As it is well known, the reduction of the fiber diameter leads to an enhancement of the light intensity in the taper region. Hence the efficiency of nonlinear processes taking place during the propagation through the fiber raises dramatically, by up to four orders of magnitude. The fact that the waveguide dispersion of optical fibers depends strongly on the diameter [1

1. G. P. Agrawal, Nonlinear fiber optics (Academic Press, San Diego, 1995).

] yields the possibility to design and fabricate fibers whose dispersion exactly matches the requirements of a certain application by tapering down to the corresponding diameter [2

2. R. Zhang, X. Zhang, D. Meiser, and H. Giessen, “Mode and group velocity dispersion evolution in the tapered region of a single-mode tapered fiber,” Opt. Express 12, 5840–5849 (2004). [CrossRef] [PubMed]

]. In particular, this allows us to shift the zero dispersion wavelength so that the overall dispersion becomes anomalous at visible or near-infrared wavelengths of commonly used laser pump sources. The combination of one of these efficient nonlinear effects, namely self-phase modulation, and the propagation in the anomalous dispersion regime enables the formation of optical solitons [3

3. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980). [CrossRef]

], whose fission [4

4. A. Husakou and J. Herrmann, “Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers,” Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]

] may trigger the development of an octave-spanning supercontinuum [5

5. T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. 25, 1415–1417 (2000). [CrossRef]

], given a carefully chosen set of experimental parameters [6

6. J. Teipel, K. Franke, D. Türke, F. Warken, D. Meiser, M. Leuschner, and H. Giessen, “Characteristics of supercontinuum generation in tapered fibers using femtosecond laser pulses,” Appl. Phys. B 77, 245–251 (2003). [CrossRef]

]. An additional possibility to influence the optical properties of the fiber is given by using a suitable Photonic Crystal Fiber (PCF) [7

7. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

] instead of a standard step-index fiber as the fiber to be tapered down [8

8. W. J. Wadsworth, A. Witkowska, S. G. Leon-Saval, and T. A. Birks, “Hole inflation and tapering of stock photonic crystal fibres,” Opt. Express 13, 6541–6549 (2005). [CrossRef] [PubMed]

, 9

9. H. C. Nguyen, B. T. Kuhlmey, E. C. Mägi, M. J. Steel, P. Domachuk, C. L. Smith, and B. J. Eggleton, “Tapered photonic crystal fibres: properties,characterisation and applications,” Appl. Phys. B 81, 377–387 (2005). [CrossRef]

].

If the diameter of the fiber is decreased down into the range of the laser wavelength coupled in, we should pay closer attention on the evanescent fields outside the fiber. These fields can be used to trap and detect single atoms or molecules with high sensitivity [10

10. G. Sagué, A. Baade, and A. Rauschenbeutel, “Blue-detuned evanescent field surface traps for neutral atoms based on mode interference in ultrathin optical fibres,” New J. Phys. 10, 113008 (2008). [CrossRef]

]. If a significant fraction of the light intensity is contained in these evanescent fields, the influence of the surrounding medium on the optical properties of the fiber can be taken as a tool to fine-tune the overall dispersion [11

11. R. Zhang, J. Teipel, X. Zhang, D. Nau, and H. Giessen, “Group velocity dispersion of tapered fibers immersed in different liquids,” Opt. Express 12, 1700–1707 (2004). [CrossRef] [PubMed]

]. Using the self-modulated taper drawing technique [12

12. L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003). [CrossRef] [PubMed]

, 13

13. L. M. Tong, J. Y. Lou, Z. Z. Ye, T. S. Geoff, and E. Mazur, “Self-modulated taper drawing of silica nanowires,” Nanotechnology 16, 1445–1448 (2005). [CrossRef]

], even tapered fibers with sub-wavelength diameters in the range of tens of nanometers have been demonstrated.

A common setup, easy to implement, for fabricating fibers consists of a motor which pulls apart the fixed ends of the decoated fiber while a heating device is travelling along the fiber axis. The heating device moves according to a certain program and softens the fiber at its current position. The resulting shape of the tapered fiber is dependent on its initial geometry, the temperature distribution of the heating device along the fiber axis, and particularly the motion of the heating device with respect to the fiber during the pulling process. Throughout the process, we keep the parameters of the heating device constant, except its position. This assumption is valid for a flame with constant, stable gas flow used at the flame-brushing technique [14

14. F. Bilodeau, K. O. Hill, S. Faucher, and D. C. Johnson, “Low-loss highly overcoupled fused couplers: fabrication and sensitivity to external pressure,” J. Lightwave Technol. 6, 1476–1482 (1988). [CrossRef]

], as well as for a ceramic or resistive element kept at a certain temperature by a given heating current [15

15. G. Brambilla, F. Koizumi, X. Feng, and D. J. Richardson, “Compound-glass optical nanowires,” Electron. Lett. 41, 400–402 (2005). [CrossRef]

, 16

16. E. C. Mägi, L. B. Fu, H. C. Nguyen, M. R. E. Lamont, D. I. Yeom, and B. J. Eggleton, “Enhanced Kerr nonlinearity in sub-wavelength diameter As2Se3 chalcogenide fiber tapers,” Opt. Express 15, 10324–10329 (2007). [CrossRef] [PubMed]

], or also for a high power laser with fixed optical parameters [17

17. T. E. Dimmick, G. Kakarantzas, T. A. Birks, and P. St. J. Russell, “Carbon dioxide laser fabrication of fused-fiber couplers and tapers,” Appl. Opt. 38, 6845–6848 (1999). [CrossRef]

]. In the latter case, however, additional attention is needed since the conversion from the energy of the laser light into thermal energy by absorption depends on the current shape of the fiber [18

18. 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]

, 19

19. C. E. Chryssou, “Theoretical analysis of tapering fused silica optical fibers using a carbon dioxide laser,” Opt. Eng. 38, 1645–1649 (1999). [CrossRef]

]. A ceramic tube around the fiber that absorbs the laser power and gives a constant heat to the fiber can aid this process [20

20. M. Sumetsky, Y. Dulashko, and A. Hale, “Fabrication and study of bent and coiled free silica nanowires: Self-coupling microloop optical interferometer,” Opt. Express 12, 3521–3531 (2004). [CrossRef] [PubMed]

].

Within this work, we present an easy-to-use model describing the drawing process taking into account the motion of the heating device as well as its temperature distribution. We show that without complicated fluid mechanics we are able to deduce programs of motion to fabricate new types of tapered fibers consisting for example of sinusoidally modulated or conically shaped waists.

2. The model

To keep the setup as simple as possible, we assume an asymmetric tapering rig consisting of a motor which pulls the fixed fiber with a constant speed v0 to one direction and a second motor which moves the heating device along the fiber axis, determining the shape of the resulting tapered fiber. The commonly used setup where two motors pull the fiber symmetrically can easily be simulated by adding a constant v0/2 to the speed of the motor moving the heating device. This transformation moves the coordinates from the lab frame of reference to the fiber frame of reference.

Assuming cylindrical symmetry, we choose to describe the shape of the fiber by its profile of the radius r(z) along the fiber axis (z-axis). The progress during the tapering is given by the distance x the pulling motor has moved since the beginning. This x-coordinate can also be regarded as time coordinate given by the transformation t = x/v0. Thus, r(z,x) denotes the shape of the fiber throughout the complete process. Let x 0 be the final elongation of the fiber. The considered interval [0,z 0] of the z-axis has to be set large enough that r(z = 0,x) = r(z = z 0,x) = r 0 is valid for all x, where r 0 denotes the constant initial radius of the fiber. Since the volume of the total fiber is conserved, we can state that by elongating the fiber by an amount of dx, we always remove the same volume from the interval [0,z 0], namely dV = −π r 0 2 dx. Thus, the change of the volume within this interval is given by

πr02=Vx=x0z0πr2(z,x)dz=2π0z0r(z,x)xr(z,x)dz.
(1)
Fig. 1. Coordinates used in the model. The bottom diagram shows the motion of the heating device, the three top diagrams selected fiber profiles during the drawing process. The dashed line with a slope of 1 indicates the position of the motor pulling the fiber.

0z0Θ(z)dz=1.
(2)

During the pulling of the fiber the parts of the fiber within the heating zone are stretched, the parts towards the fixed end stay unchanged, whereas the part at the side of the pulling motor is just shifted. This behavior can be described by the equation

r(z+f(z)dx,x+dx)=r(z,x)Ar(z,x)Θ(z)dx.
(3)

Here, A serves as a normalization introduced to ensure volume conservation; f(z) determines the type of change of the fiber shape: For f(z) = 0 the fiber stays unchanged, whereas a constant f(z) > 0 shifts the fiber shape to larger z-values. For a non-constant f(z) the fiber is stretched. The larger the slope of f(z) the larger is the amount of the stretching. These facts invite to connect f(z) and Θ(z) by

f(z)=0zΘ(z)dzf(z)z=Θ(z).
(4)

These settings deliver the desired behavior during the pulling as described above and summarized in the table Tab. 1.

Table 1. Zones of transformation

table-icon
View This Table

If we now Taylor-expand the lefthand side of Eq. 3, we obtain

rx=f(z)rz+Ar(z,x)Θ(z).
(5)

Inserting this result into Eq. 1, we find that for A = 1/2 the requirement of volume conservation is fulfilled with the given assumptions (f(z = 0) = 0, f(z = z 0) = 1 and r(z = z 0,x) = r 0).

In a numerical simulation with a finite elongation step size ∆x, the evolution of the fiber profile can now be calculated in a linear approximation step by step using Eq. 5, starting from a homogeneous fiber with a constant radius r 0:

r(z,x+Δx)=r(z,x)(f(z)r(z+Δz,x)r(z,x)Δz+12r(z,x)Θ(z))Δx.
(6)

Here, the derivative ∂r/∂z has been replaced by the difference quotient, which can be calculated easily after performing a discretization of the z coordinate with step size ∆z. In our simulations, we have applied for the calculation of the derivative more advanced numerical methods like nonlinear interpolation which are provided by the used software. When the final elongation x 0 is reached, the procedure is stopped. The possibility to change Θ(z) from step to step enables us to simulate a travelling heating device by setting Θ(z) ≡ Θ(z + p(x)), where p(x) denotes the corresponding position of the center of the heating device’s profile with respect to the fiber.

Fig. 1 depicts an example using the described simulation method. In the bottom diagram, the motion of the heating device p(x) is shown. In the upper part of the figure, three profiles evolve during the drawing process, resulting in a tapered fiber which includes a homogeneous waist with a diameter of 3 μm. ∆x and ∆z are chosen in the order of tens of micrometers, depending on the complexity of the profile. This leads to simulation times smaller than one minute using a standard office computer.

To test our algorithm, we now assume the special case of a rectangular distribution for Θ(z,x) with a width of L(x), yielding a height of 1/L(x). With the heating device standing in the reference frame of the fiber, this kind of thermal profile leads to a waist whose radius is independent of z within the heating zone. With Eq. 5 we obtain

drr(x)=12dxL(x),
(7)

which is exactly the equation derived analytically in [21

21. T. A. Birks and Y. W. Li, “The Shape of Fiber Tapers,” J. Lightwave Technol. 10, 432–438 (1992). [CrossRef]

], proved to work very well in many experiments.

More generally, for an arbitrary distribution Θ(z,x) and again the case of a standing heating device in the reference frame of the fiber, we obtain for the position of the maximum of Θ denoted by ΘM

r(zΘM)/r0=exp(1/20x0ΘM(x)dx).
(8)

Assuming a x-independent ΘM, this equation can be simplified to

r/r0=exp(1/2x0ΘM).
(9)
Fig. 2. a) Examples for the chosen distribution Θ(z). Different widths by variation of ν are shown, where ν·Θ0 corresponds to the half width at half maximum (HWHM) of the distribution. Here, Θ0 is set constant to 1.5 mm. b) Geometry and zones of boundary conditions for the FEM simulation of the temperature distribution within the fiber.

3. Remarks on the distribution Θ(z)

The following parametrization of the measure of deformability Θ(z) refers to the heating source we use in our tapering rig, in particular a propane-butane-oxygen flame. Other heating devices might require a different shape of the distribution. As an empirical approximation for Θ(z), we set

Θ(z)=Nez2t2(z6Θ06+3z4Θ043z2Θ02+1),t=νΘ0ln2+3ln(1ν2).
(10)

Here, N is a normalization constant given by the condition of Eq. 2, ±Θ0 denote the first roots of the distribution, whereas ν stands for the half width of half maximum (HWHM) relative to Θ0. This function is designed to offer full control over roots and width. Furthermore the first and second derivatives equal 0 at the first roots ±Θ0. These properties and setting the function to 0 beyond ±Θ0 help to reduce numerical artifacts during the simulation (see Fig. 2 a)).

Using such a kind of distribution, we have to find the appropriate parameters Θ0 and ν associated with a given heating device by drawing tapers with different final elongation x 0 while the heating device is at rest in the reference frame of the fiber, and we then have to compare the measured minimal diameter [29

29. F. Warken and H. Giessen, “Fast profile measurement of micrometer-sized tapered fibers with better than 50 nm accuracy,” Opt. Lett. 29, 1727–1729 (2004). [CrossRef] [PubMed]

] with the minimal diameter simulated by our model for different values of Θ0 and ν. Using this procedure to calibrate Θ(z) for the propane-butane-oxygen flame used in our tapering rig, no common set of these values could be found for all chosen minimal diameters. However, in the experiment we have observed a clear tendency that a smaller final fiber diameter yields a smaller width of the distribution. This made us carry out a simulation of the temperature distribution inside the fiber for a given diameter d. By means of the finite element method (FEM), we investigate the stationary heat equation

(kT)=0,
(11)

where T denotes the temperature and k the thermal conductivity of the fiber material. The boundary conditions are given on the one hand by the axial symmetry at r = 0 and on the other hand by allowing heat flux at the fiber-air interface described by the equation

n·kT=σε(T04T4),
(12)

where n is the normal vector, σ the Stefan-Boltzmann constant, ε the emissivity of the fiber material, and T 0 the ambient temperature, which is set to 300 K [27

27. Comsol Multiphysics, Modeling Guide, (Software manual, version 3.3).

]. Over a length of 1 mm along the fiber-air interface, the temperature is fixed to a value of Tfix=2150 K, simulating the flame touching the fiber. For the far ends of the fiber, each at a distance of 50 mm, we assume thermal insulation (See Fig. 2 b)). Since the temperature varies within a large interval, the material properties k and ε have to be treated as temperature-dependent [19

19. C. E. Chryssou, “Theoretical analysis of tapering fused silica optical fibers using a carbon dioxide laser,” Opt. Eng. 38, 1645–1649 (1999). [CrossRef]

]. These empirically found dependencies for fused silica as the fiber material are taken from reference [28

28. F. Richter, “Upsetting and Viscoelasticity of Vitreous SiO2: Experiments, Interpretation and Simulation,” PhD thesis, Technische Universität Berlin (2006).

]:

k(T)(4.5458(T1000K)24.2364T1000K+2.6277)Wm×K
(13)

and

ε(T)4.37×107(T1K)20.00147T1K+1.546.
(14)

Furthermore, we assume thermal equilibrium, which can be maintained when pulling the fiber very slowly, i.e. v0 < 2 mm/s.

The results of this FEM simulation are shown in Fig. 3 and confirm the experimental findings of smaller widths of the temperature distribution for smaller final fiber diameters. Inside the heating zone, the temperature is almost equal to Tfix across the fiber. Here, this balance of temperature is governed by heat conduction within the softened part of the fiber. We assume that convection can be neglected for our process, which is supported by the fact that the radial core and cladding refractive index profile stays the same in the tapered region. Outside the heating zone, the temperature of the solid fiber parts decays mainly due to reemission to the surrounding medium. The full width of half maximum (FWHM) of the temperature decay along the fiber axis at r = 0 in dependence of the diameter is fitted very well by a square root function. This behavior is reproduced for different values of Tfix. The effective width of the heating device then is determined by the width of the temperature distribution for temperatures exceeding the melting temperature of the fiber.

Fig. 3. FEM-simulated decay of temperature along the fiber axis at r = 0 for different diameters of the fiber. The reddish area marks the heating zone. The thinner the fiber, the faster the decay and, hence, narrower the heating zone. The inset shows the dependence of the width of the temperature decay along the fiber on the fiber diameter. The red curve indicates a square root-fit.

Taking this result into account, we allow Θ0, and hence Θ(z) to be dependent on the fiber radius at the current position of the heating device p, setting

Θ0=B0+B1r(z=p(x),x).
(15)

Repeating the calibration described above with these new settings, we now find a set of parameters, resulting in an excellent agreement of the simulated and experimental fiber shape. For our flame, the parameters are ν = 0.51, B 0=0.32 mm, and B 1=0.11 mm, when r is given in micrometers.

The result of a diameter-dependent width of the temperature distribution can be proven experimentally by tapering fibers with different initial diameters. In this experiment, the heating device is fixed in the reference frame of the fiber, and the final elongation x 0 is set to 3 mm for all tests. As starting points we have produced fibers containing homogenous waists with lengths of several centimeters and different diameters. These waists serve as initial fibers for the test drawing. This initial diameters as well as the final ones have been measured by using a microscope or the refraction method described in Ref. [29

29. F. Warken and H. Giessen, “Fast profile measurement of micrometer-sized tapered fibers with better than 50 nm accuracy,” Opt. Lett. 29, 1727–1729 (2004). [CrossRef] [PubMed]

]. According to Eq. 9, the ratio between final (d) and initial diameter (d 0) can be expressed by

Fig. 4. Ratio of initial and final diameter of fibers drawn with a fixed heating device in the reference frame of the fiber. The elongation x 0 is 3 mm for all cases. Different colors and symbols stand for different widths of the heating device, realized by different distances between fiber and flame. The lines are guides to the eye.
d/d0=exp(x02L0),
(16)

where L 0 = 1/ΘM denotes the effective width of the heating zone. For a diameter-independent width of the temperature distribution, one would expect a constant L 0, and hence a constant ratio of d and d 0. Fig. 4 shows the experimental results. Each color/symbol stands for a certain distance between the fiber and the flame of our tapering rig. These distances have been varied to supply different effective widths L 0 of the temperature distribution. For all of these distances the ratio d/d 0 becomes significantly smaller for smaller final diameters d. According to Eq. 16, this corresponds to smaller L 0 for smaller final diameters, which exactly emphasizes the findings of our simulation and calibration process.

The number of possible shapes of the fiber is restricted by the properties of the heating source, namely the distribution Θ(z). For example, the narrowest symmetric profile evolves by pulling with a standing heating device in the reference frame of the fiber. The resulting shape along the fiber has a FWHM z̅ of

z̅x0+Θ0+ln(12(1+exp(x02Θ0))),
(17)

where x 0 is the final elongation of the fiber and

Θ0=B0+B1rminr0rdrr0rmin=B0+23B1r03/2rmin3/2r0rmin
(18)

denotes the average of all Θ0 appearing during the pulling process.

Fig. 5. Simulated fiber profiles (black), all starting with a homogeneous waist of 3.3 μm diameter. The red curves show the corresponding positions of the heating device p(x) on the z-axis in dependency of the fiber elongation x. a) Sinusoidally modulated waist. Set parameters: Amplitude 0.25 μm, mean diameter 2.75 μm, period 3.05 mm. b) Conically shaped waist: Minimal diameter 2.5 μm, maximal diameter of the waist 3.0 μm. c) Waist composed of three quadratic functions: Minimal diameter 1μm, maximal diameter 2.5 μm.

4. Experimental results and conclusion

Starting from a tapered fiber with a homogeneous waist produced by motion of the heating device as suggested in [21

21. T. A. Birks and Y. W. Li, “The Shape of Fiber Tapers,” J. Lightwave Technol. 10, 432–438 (1992). [CrossRef]

] and shown at the bottom diagram of Fig. 1, we now can use the calibrated model to iteratively approximate the fiber shape given by the model to the desired one. This can be implemented by varying the positions of the heating device p(x) for given ∆x-steps during the simulation and using an optimization procedure to find the motion which yields the best fitting shape. To warrant the convergence of this iterative method we use the fact that a slower burner motion at a certain position results in a thinner fiber diameter at this position.

Fig. 6. a) A measured profile of a fiber with a sinusoidally modulated waist. The red curve shows a fitted sine function with an amplitude of 0.22 μm, an offset of 2.77 μm and a period of 3.05 mm. A zoom-in is demonstrated by the inset. b) The diffraction pattern generated by a laser pointing perpendicularly onto the modulated fiber [29].

As an example, we present a fiber with a sinusoidally modulated waist. The desired parameters for the sine function are an amplitude of the diameter of 0.25 μm and a period of 3.05 mm. As mean value for the diameter we set 2.75 μm, whereas the initial waist diameter is 3.3 μm. The resulting simulated shape is shown in Fig. 5 a), which shows that the desired parameters are reproduced very well.

A second possible shape is a conically formed waist as demonstrated in Fig. 5 b). Here, the set parameters are 2.5 μm for the minimal diameter and 3.0 μm for the maximal diameter of the waist. The simulation of a profile composed of three continuously connected quadratic functions (Fig. 5 c) is another representative for a shape with a bulged waist center. The settings for this profile are a waist length of 120 mm, a minimal diameter of 1 μm and a maximal diameter of 2.5 μm. In all figures, the average movement of the heating device is towards smaller z-values, whereas the fiber is pulled towards larger z-values. Therefore, the starting position of the heating device seems to be located at the middle of the waist region, but, if the elongation of the fiber is taken into account, it is actually set to the end of the waist region at the beginning of the pulling process.

As criterion for stopping the iterative procedure of the simulation process we don’t allow the radius difference between simulated and desired profile to be larger than 10 nm along the whole waist. This accepted error is well below the accuracy of the methods we use to measure the fiber diameter, and the hereby induced changes of the optical properties of the fiber are negligible.

To verify our model, we have produced a sinusoidally modulated fiber according to Fig. 5 a). The resulting profile is shown as black dots in Fig. 6 a), measured by a diffraction method [29

29. F. Warken and H. Giessen, “Fast profile measurement of micrometer-sized tapered fibers with better than 50 nm accuracy,” Opt. Lett. 29, 1727–1729 (2004). [CrossRef] [PubMed]

]. The inset demonstrates a magnification of the region from z=105 to 125 mm. Fig. 6 b) shows a captured image of the diffraction pattern generated by a laser line focus hitting the fiber perpendicularly [29

29. F. Warken and H. Giessen, “Fast profile measurement of micrometer-sized tapered fibers with better than 50 nm accuracy,” Opt. Lett. 29, 1727–1729 (2004). [CrossRef] [PubMed]

]. A fitted sine function (red) exhibits only small deviations from the desired values. Note that even the small steps in the taper transitions, located at the turning points of the oscillating movement of the heating device, are predicted by the model very precisely.

In conclusion, we have presented a model which is capable of simulating the drawing process of a fiber with a high accuracy taking into account the motion of the heating device. A method for the implementation and calibration of the properties of the heating source has been demonstrated and supported by numerical simulations of the heat transport inside the fiber using FEM. Applying the model, we are able to successfully deduce the motion of the heating source necessary to create a desired, complex shape of the tapered fiber as shown by the given examples of fibers with sinusoidally modulated or conically shaped waists.

Acknowledgement

The authors would like to thank the Landesgraduiertenförderung of Baden-Württemberg for support of this work. We also would like to thank Dr. Wolfgang Alt for helpful advice.

References and links

1.

G. P. Agrawal, Nonlinear fiber optics (Academic Press, San Diego, 1995).

2.

R. Zhang, X. Zhang, D. Meiser, and H. Giessen, “Mode and group velocity dispersion evolution in the tapered region of a single-mode tapered fiber,” Opt. Express 12, 5840–5849 (2004). [CrossRef] [PubMed]

3.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980). [CrossRef]

4.

A. Husakou and J. Herrmann, “Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers,” Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]

5.

T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. 25, 1415–1417 (2000). [CrossRef]

6.

J. Teipel, K. Franke, D. Türke, F. Warken, D. Meiser, M. Leuschner, and H. Giessen, “Characteristics of supercontinuum generation in tapered fibers using femtosecond laser pulses,” Appl. Phys. B 77, 245–251 (2003). [CrossRef]

7.

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

8.

W. J. Wadsworth, A. Witkowska, S. G. Leon-Saval, and T. A. Birks, “Hole inflation and tapering of stock photonic crystal fibres,” Opt. Express 13, 6541–6549 (2005). [CrossRef] [PubMed]

9.

H. C. Nguyen, B. T. Kuhlmey, E. C. Mägi, M. J. Steel, P. Domachuk, C. L. Smith, and B. J. Eggleton, “Tapered photonic crystal fibres: properties,characterisation and applications,” Appl. Phys. B 81, 377–387 (2005). [CrossRef]

10.

G. Sagué, A. Baade, and A. Rauschenbeutel, “Blue-detuned evanescent field surface traps for neutral atoms based on mode interference in ultrathin optical fibres,” New J. Phys. 10, 113008 (2008). [CrossRef]

11.

R. Zhang, J. Teipel, X. Zhang, D. Nau, and H. Giessen, “Group velocity dispersion of tapered fibers immersed in different liquids,” Opt. Express 12, 1700–1707 (2004). [CrossRef] [PubMed]

12.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003). [CrossRef] [PubMed]

13.

L. M. Tong, J. Y. Lou, Z. Z. Ye, T. S. Geoff, and E. Mazur, “Self-modulated taper drawing of silica nanowires,” Nanotechnology 16, 1445–1448 (2005). [CrossRef]

14.

F. Bilodeau, K. O. Hill, S. Faucher, and D. C. Johnson, “Low-loss highly overcoupled fused couplers: fabrication and sensitivity to external pressure,” J. Lightwave Technol. 6, 1476–1482 (1988). [CrossRef]

15.

G. Brambilla, F. Koizumi, X. Feng, and D. J. Richardson, “Compound-glass optical nanowires,” Electron. Lett. 41, 400–402 (2005). [CrossRef]

16.

E. C. Mägi, L. B. Fu, H. C. Nguyen, M. R. E. Lamont, D. I. Yeom, and B. J. Eggleton, “Enhanced Kerr nonlinearity in sub-wavelength diameter As2Se3 chalcogenide fiber tapers,” Opt. Express 15, 10324–10329 (2007). [CrossRef] [PubMed]

17.

T. E. Dimmick, G. Kakarantzas, T. A. Birks, and P. St. J. Russell, “Carbon dioxide laser fabrication of fused-fiber couplers and tapers,” Appl. Opt. 38, 6845–6848 (1999). [CrossRef]

18.

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]

19.

C. E. Chryssou, “Theoretical analysis of tapering fused silica optical fibers using a carbon dioxide laser,” Opt. Eng. 38, 1645–1649 (1999). [CrossRef]

20.

M. Sumetsky, Y. Dulashko, and A. Hale, “Fabrication and study of bent and coiled free silica nanowires: Self-coupling microloop optical interferometer,” Opt. Express 12, 3521–3531 (2004). [CrossRef] [PubMed]

21.

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

22.

A. Konyukhov, L. Melnikov, and Y. Mazhirina, “Dispersive wave generation in microstructured fiber with periodically modulated diameter,” Proc. SPIE 6165, 616508 (2006). [CrossRef]

23.

M. Stratmann, M. Böhm, and F. Mitschke, “Stable propagation of dark solitons in dispersion maps of either sign of path-average dispersion,” Electron. Lett. 37, 1182–1183 (2001). [CrossRef]

24.

M. Sumetsky , “Whispering-gallery-bottle microcavities: the three-dimensional etalon,” Opt. Lett. 29, 8–10 (2004). [CrossRef] [PubMed]

25.

G. S. Murugan, J. S. Wilkinson, and M. N. Zervas, “Selective excitation of whispering gallery modes in a novel bottle microresonator,” Opt. Express 17, 11916–11925 (2009). [CrossRef]

26.

Y. Louyer, D. Meschede, and A. Rauschenbeutel, “Tunable whispering-gallery-mode resonators for cavity quantum electrodynamics,” Phys. Rev. A 72, 031801(R) (2005). [CrossRef]

27.

Comsol Multiphysics, Modeling Guide, (Software manual, version 3.3).

28.

F. Richter, “Upsetting and Viscoelasticity of Vitreous SiO2: Experiments, Interpretation and Simulation,” PhD thesis, Technische Universität Berlin (2006).

29.

F. Warken and H. Giessen, “Fast profile measurement of micrometer-sized tapered fibers with better than 50 nm accuracy,” Opt. Lett. 29, 1727–1729 (2004). [CrossRef] [PubMed]

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.2280) Fiber optics and optical communications : Fiber design and fabrication

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: December 1, 2009
Revised Manuscript: January 19, 2010
Manuscript Accepted: January 19, 2010
Published: February 2, 2010

Citation
S. Pricking and H. Giessen, "Tapering fibers with complex shape," Opt. Express 18, 3426-3437 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-4-3426


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References

  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).
  2. R. Zhang, X. Zhang, D. Meiser, and H. Giessen, "Mode and group velocity dispersion evolution in the tapered region of a single-mode tapered fiber," Opt. Express 12,5840-5849 (2004). [CrossRef] [PubMed]
  3. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers," Phys. Rev. Lett. 45,1095-1098 (1980). [CrossRef]
  4. A. Husakou and J. Herrmann, "Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers," Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]
  5. T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, "Supercontinuum generation in tapered fibers," Opt. Lett. 25, 1415-1417 (2000). [CrossRef]
  6. J. Teipel, K. Franke, D. T¨urke, F. Warken, D. Meiser, M. Leuschner, and H. Giessen, "Characteristics of supercontinuum generation in tapered fibers using femtosecond laser pulses," Appl. Phys. B 77, 245-251 (2003). [CrossRef]
  7. T. A. Birks, J. C. Knight, and P. St. J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
  8. W. J. Wadsworth, A. Witkowska, S. G. Leon-Saval, and T. A. Birks, "Hole inflation and tapering of stock photonic crystal fibres," Opt. Express 13, 6541-6549 (2005). [CrossRef] [PubMed]
  9. H. C. Nguyen, B. T. Kuhlmey, E. C. Magi, M. J. Steel, P. Domachuk, C. L. Smith, and B. J. Eggleton, "Tapered photonic crystal fibres: properties,characterisation and applications," Appl. Phys. B 81, 377-387 (2005). [CrossRef]
  10. G. Sague, A. Baade, and A. Rauschenbeutel, "Blue-detuned evanescent field surface traps for neutral atoms based on mode interference in ultrathin optical fibres," New J. Phys. 10, 113008 (2008). [CrossRef]
  11. R. Zhang, J. Teipel, X. Zhang, D. Nau, and H. Giessen, "Group velocity dispersion of tapered fibers immersed in different liquids," Opt. Express 12, 1700-1707 (2004). [CrossRef] [PubMed]
  12. L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, "Subwavelengthdiameter silica wires for low-loss optical wave guiding," Nature 426, 816-819 (2003). [CrossRef] [PubMed]
  13. L. M. Tong, J. Y. Lou, Z. Z. Ye, T. S. Geoff, and E. Mazur, "Self-modulated taper drawing of silica nanowires," Nanotechnology 16, 1445-1448 (2005). [CrossRef]
  14. F. Bilodeau, K. O. Hill, S. Faucher, and D. C. Johnson, "Low-loss highly overcoupled fused couplers: fabrication and sensitivity to external pressure," J. Lightwave Technol. 6, 1476-1482 (1988). [CrossRef]
  15. G. Brambilla, F. Koizumi, X. Feng, and D. J. Richardson, "Compound-glass optical nanowires," Electron. Lett. 41, 400-402 (2005). [CrossRef]
  16. E. C. Magi, L. B. Fu, H. C. Nguyen, M. R. E. Lamont, D. I. Yeom, and B. J. Eggleton, "Enhanced Kerr nonlinearity in sub-wavelength diameter As2Se3 chalcogenide fiber tapers," Opt. Express 15, 10324-10329 (2007). [CrossRef] [PubMed]
  17. T. E. Dimmick, G. Kakarantzas, T. A. Birks, and P. St. J. Russell, "Carbon dioxide laser fabrication of fused-fiber couplers and tapers," Appl. Opt. 38, 6845-6848 (1999). [CrossRef]
  18. 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]
  19. C. E. Chryssou, "Theoretical analysis of tapering fused silica optical fibers using a carbon dioxide laser," Opt. Eng. 38, 1645-1649 (1999). [CrossRef]
  20. M. Sumetsky, Y. Dulashko, and A. Hale, "Fabrication and study of bent and coiled free silica nanowires: Selfcoupling microloop optical interferometer," Opt. Express 12, 3521-3531 (2004). [CrossRef] [PubMed]
  21. T. A. Birks and Y. W. Li, "The Shape of Fiber Tapers," J. Lightwave Technol. 10, 432-438 (1992). [CrossRef]
  22. A. Konyukhov, L. Melnikov, and Y. Mazhirina, "Dispersive wave generation in microstructured fiber with periodically modulated diameter," Proc. SPIE 6165, 616508 (2006). [CrossRef]
  23. M. Stratmann, M. Bohm, and F. Mitschke, "Stable propagation of dark solitons in dispersion maps of either sign of path-average dispersion," Electron. Lett. 37, 1182-1183 (2001). [CrossRef]
  24. M. Sumetsky, "Whispering-gallery-bottle microcavities: the three-dimensional etalon," Opt. Lett. 29, 8-10 (2004). [CrossRef] [PubMed]
  25. G. S. Murugan, J. S. Wilkinson, and M. N. Zervas, "Selective excitation of whispering gallery modes in a novel bottle microresonator," Opt. Express 17, 11916-11925 (2009). [CrossRef]
  26. Y. Louyer, D. Meschede, and A. Rauschenbeutel, "Tunable whispering-gallery-mode resonators for cavity quantum electrodynamics," Phys. Rev. A 72, 031801(R) (2005). [CrossRef]
  27. Comsol Multiphysics, Modeling Guide, (Software manual, version 3.3).
  28. F. Richter, "Upsetting and Viscoelasticity of Vitreous SiO2: Experiments, Interpretation and Simulation," PhD thesis, Technische Universit¨at Berlin (2006).
  29. F. Warken and H. Giessen, "Fast profile measurement of micrometer-sized tapered fibers with better than 50 nm accuracy," Opt. Lett. 29, 1727-1729 (2004). [CrossRef] [PubMed]

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