## Tapering fibers with complex shape

Optics Express, Vol. 18, Issue 4, pp. 3426-3437 (2010)

http://dx.doi.org/10.1364/OE.18.003426

Acrobat PDF (732 KB)

### 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

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]

## 2. The model

_{0}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 v

_{0}/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.

*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*/v

_{0}. 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

*z*) which is assumed to include the temperature distribution of the heating device as well as the fiber’s properties, e.g., its viscosity. Hence, Θ(

*z*) can be regarded as a measure of the fiber’s deformability caused by the heating device. First, we have assumed it to be independent of the current state of the fiber, but as described in the next section, it turned out to be dependent on the current fiber diameter. The specific shape of this function has to be calibrated for each setup as described in the next section. We choose to set Θ(

*z*) = 0 outside the heating zone and normalize it without loss of generality according to

*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

*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}).

*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*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.

*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

*μ*m. ∆

*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.

*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

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

*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 Θ

_{M}

_{M}, this equation can be simplified to

## 3. Remarks on the distribution Θ(*z*)

*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

*N*is a normalization constant given by the condition of Eq. 2, ±Θ

_{0}denote the first roots of the distribution, whereas

*ν*stands for the half width of half maximum (HWHM) relative to Θ

_{0}. This function is designed to offer full control over roots and width. Furthermore the first and second derivatives equal 0 at the first roots ±Θ

_{0}. These properties and setting the function to 0 beyond ±Θ

_{0}help to reduce numerical artifacts during the simulation (see Fig. 2 a)).

_{0}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]

_{0}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

*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**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]. Over a length of 1 mm along the fiber-air interface, the temperature is fixed to a value of

*T*=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

_{fix}*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]

_{0}< 2 mm/s.

*T*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

_{fix}*r*= 0 in dependence of the diameter is fitted very well by a square root function. This behavior is reproduced for different values of

*T*. 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.

_{fix}_{0}, and hence Θ(

*z*) to be dependent on the fiber radius at the current position of the heating device

*p*, setting

*ν*= 0.51,

*B*

_{0}=0.32 mm, and

*B*

_{1}=0.11 mm, when

*r*is given in micrometers.

*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]

*d*) and initial diameter (

*d*

_{0}) can be expressed by

*L*

_{0}= 1/Θ

_{M}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.

*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

*z*̅ of

*x*

_{0}is the final elongation of the fiber and

_{0}appearing during the pulling process.

## 4. Experimental results and conclusion

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

*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.

*μ*m and a period of 3.05 mm. As mean value for the diameter we set 2.75

*μ*m, whereas the initial waist diameter is 3.3

*μ*m. The resulting simulated shape is shown in Fig. 5 a), which shows that the desired parameters are reproduced very well.

*μ*m for the minimal diameter and 3.0

*μ*m 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

*μ*m and a maximal diameter of 2.5

*μ*m. 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.

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]

*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**, 1727–1729 (2004). [CrossRef] [PubMed]

## Acknowledgement

## References and links

1. | G. P. Agrawal, |

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 |

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. |

4. | A. Husakou and J. Herrmann, “Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers,” Phys. Rev. Lett. |

5. | T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. |

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 |

7. | T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. |

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 |

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 |

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. |

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. | 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 |

13. | L. M. Tong, J. Y. Lou, Z. Z. Ye, T. S. Geoff, and E. Mazur, “Self-modulated taper drawing of silica nanowires,” Nanotechnology |

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. |

15. | G. Brambilla, F. Koizumi, X. Feng, and D. J. Richardson, “Compound-glass optical nanowires,” Electron. Lett. |

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 |

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. |

18. | A. J. C. Grellier, N. K. Zayer, and C. N. Pannell, “Heat transfer modelling in CO laser processing of optical fibres,” Opt. Commun. |

19. | C. E. Chryssou, “Theoretical analysis of tapering fused silica optical fibers using a carbon dioxide laser,” Opt. Eng. |

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 |

21. | T. A. Birks and Y. W. Li, “The Shape of Fiber Tapers,” J. Lightwave Technol. |

22. | A. Konyukhov, L. Melnikov, and Y. Mazhirina, “Dispersive wave generation in microstructured fiber with periodically modulated diameter,” Proc. SPIE |

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. |

24. | M. Sumetsky
, “Whispering-gallery-bottle microcavities: the three-dimensional etalon,” Opt. Lett. |

25. | G. S. Murugan, J. S. Wilkinson, and M. N. Zervas, “Selective excitation of whispering gallery modes in a novel bottle microresonator,” Opt. Express |

26. | Y. Louyer, D. Meschede, and A. Rauschenbeutel, “Tunable whispering-gallery-mode resonators for cavity quantum electrodynamics,” Phys. Rev. A |

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. |

**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

- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).
- 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]
- 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]
- 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]
- T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, "Supercontinuum generation in tapered fibers," Opt. Lett. 25, 1415-1417 (2000). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- G. Brambilla, F. Koizumi, X. Feng, and D. J. Richardson, "Compound-glass optical nanowires," Electron. Lett. 41, 400-402 (2005). [CrossRef]
- 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]
- 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]
- 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]
- C. E. Chryssou, "Theoretical analysis of tapering fused silica optical fibers using a carbon dioxide laser," Opt. Eng. 38, 1645-1649 (1999). [CrossRef]
- 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]
- T. A. Birks and Y. W. Li, "The Shape of Fiber Tapers," J. Lightwave Technol. 10, 432-438 (1992). [CrossRef]
- A. Konyukhov, L. Melnikov, and Y. Mazhirina, "Dispersive wave generation in microstructured fiber with periodically modulated diameter," Proc. SPIE 6165, 616508 (2006). [CrossRef]
- 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]
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