Efficient fabrication of fused-fiber biconical taper structures by a
scanned CO<sub>2</sub> laser beam technique

Fabien Bayle; Jean-Pierre Meunier

doi:10.1364/AO.44.006402

Applied Optics
Vol. 44,
Issue 30,
pp. 6402-6411
(2005)
•https://doi.org/10.1364/AO.44.006402

Efficient fabrication of fused-fiber biconical taper structures by a scanned CO₂ laser beam technique

Fabien Bayle and Jean-Pierre Meunier

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Fabien Bayle and Jean-Pierre Meunier, "Efficient fabrication of fused-fiber biconical taper structures by a scanned CO₂ laser beam technique," Appl. Opt. 44, 6402-6411 (2005)

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Abstract

The driving mechanism of a scanning mirror can cause significant impairment of expanded beam properties, which we investigated for several scanning waveforms. Engineering on the scanning waveform is then carried out by a scanned CO₂ laser beam technique to enlarge the uniform heating region for stretching and sintering of silica fibers. Details of the derivation are given. A simple thermal model is presented to account for the relationship between the scanning beam profile and the taper shape. Fusion profiles are also compared for various scanning waveforms. The corresponding scanned beam power distributions are determined experimentally, which enables us to determine precise power density conditions for CO₂ laser fusion.

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Term	Definition	Value in SI Units^a
T	Temperature	T_air = 293 K
ΔT	Temperature difference	T − T_air
d	Diameter of the fiber	(30–125) × 10⁻⁶ m
P_total	Total output power of the laser	0–31 W
w_x (w_y)	Beam waist in the x (y) direction	2.3 × 10⁻³ m
K_air	Thermal conductivity of air	3.84 × 10⁻³ + 7.66 × 10⁻⁵ T − 1.54 × 10⁻⁸ T² W m⁻¹ K⁻¹ for T from 293 to 2000 K
ψ	Dimensionless convection-related term	101 × 106 exp[−(T − 273.15)/50.7] + 43.5 × 10⁶ exp[−(T − 273.15)/138] for T from 293 to 2000 K
H	Convection heat-transfer coefficient	(K_air/d) × 10^{(0.039+0.13log₁₀(Ψd³ΔT)+0.01[log₁₀(Ψd³ΔT)]²} W m⁻² K⁻¹ from 293 to 2000 K
K	Thermal conductivity of the silica fiber	0.78 − 0.054 exp[(T + 379)/354] + 0.165 exp[(T + 379)/405] W m⁻¹ K⁻¹ for T from 293 to 2000 K
λ	CO₂ laser wavelength	10.6 × 10⁻⁶ m
n	Refractive index of the silica fiber	2.0
k_abs	Extinction coefficient of silica at 10.6 μm	1.82 × 10⁻² + 10.1 × 10⁻⁵ (T − 273.15) for T from 293 to 2000 K
η	Coefficient for absorption efficiency	{1 − [(n − 1)² + k_abs²]/[(n + 1)² + k_abs²]}[1 − exp(−4πdk_abs/λ)]
erf	Error function	$erf (z) = 2 / \sqrt{π} \int_{0}^{z} exp (- t^{2}) d t$

Scanning Waveform Amplitude (FWMH)	Laser Power (W) [Average Power Density (W mm⁻²)]	Overall Fusion Extent (mm) [Minimum Transverse Section Width after 10 min (μm)]
Fixed (no focusing lens) [2.3 mm × 2.3 mm]	20–25 [2.45–3.06]	2.9 [140]
Optimized [7 mm × 0.9 mm]	10–14.7 [1.38–2.03]	3.6 [165]
Optimized [10 mm × 0.9 mm]	14–16.6 [1.35–1.61]	5.2 [185]
Sawtoothed [10 mm × 0.9 mm]	10 [1.25]	4.3 [155]
Sawtoothed [15 mm × 0.9 mm]	14 [1.17]	1.5 [225]

Term	Definition	Value in SI Units^a
T	Temperature	T_air = 293 K
ΔT	Temperature difference	T − T_air
d	Diameter of the fiber	(30–125) × 10⁻⁶ m
P_total	Total output power of the laser	0–31 W
w_x (w_y)	Beam waist in the x (y) direction	2.3 × 10⁻³ m
K_air	Thermal conductivity of air	3.84 × 10⁻³ + 7.66 × 10⁻⁵ T − 1.54 × 10⁻⁸ T² W m⁻¹ K⁻¹ for T from 293 to 2000 K
ψ	Dimensionless convection-related term	101 × 106 exp[−(T − 273.15)/50.7] + 43.5 × 10⁶ exp[−(T − 273.15)/138] for T from 293 to 2000 K
H	Convection heat-transfer coefficient	(K_air/d) × 10^{(0.039+0.13log₁₀(Ψd³ΔT)+0.01[log₁₀(Ψd³ΔT)]²} W m⁻² K⁻¹ from 293 to 2000 K
K	Thermal conductivity of the silica fiber	0.78 − 0.054 exp[(T + 379)/354] + 0.165 exp[(T + 379)/405] W m⁻¹ K⁻¹ for T from 293 to 2000 K
λ	CO₂ laser wavelength	10.6 × 10⁻⁶ m
n	Refractive index of the silica fiber	2.0
k_abs	Extinction coefficient of silica at 10.6 μm	1.82 × 10⁻² + 10.1 × 10⁻⁵ (T − 273.15) for T from 293 to 2000 K
η	Coefficient for absorption efficiency	{1 − [(n − 1)² + k_abs²]/[(n + 1)² + k_abs²]}[1 − exp(−4πdk_abs/λ)]
erf	Error function	$erf (z) = 2 / \sqrt{π} \int_{0}^{z} exp (- t^{2}) d t$

Scanning Waveform Amplitude (FWMH)	Laser Power (W) [Average Power Density (W mm⁻²)]	Overall Fusion Extent (mm) [Minimum Transverse Section Width after 10 min (μm)]
Fixed (no focusing lens) [2.3 mm × 2.3 mm]	20–25 [2.45–3.06]	2.9 [140]
Optimized [7 mm × 0.9 mm]	10–14.7 [1.38–2.03]	3.6 [165]
Optimized [10 mm × 0.9 mm]	14–16.6 [1.35–1.61]	5.2 [185]
Sawtoothed [10 mm × 0.9 mm]	10 [1.25]	4.3 [155]
Sawtoothed [15 mm × 0.9 mm]	14 [1.17]	1.5 [225]

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Applied Optics