Abstract

We present the segmented corrosion method that uses hydrofluoric acid to etch the fiber of a fiber laser for removing high-power cladding light to improve stripping uniformity and power handling capability. For theoretical guidelines, we propose a simulation model of etched-fiber stripping to evaluate the relationship between the etched-fiber parameters and cladding light attenuation and to analyze the stripping uniformity achieved with segmented corrosion. A two-segment etched fiber is fabricated with cladding light attenuation of 19.8 dB and power handling capability up to 670 W. We find that the cladding light is stripped uniformly and the temperature distribution is uniform without the formation of hot spots.

© 2017 Optical Society of America

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References

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    [Crossref]

2016 (4)

2015 (2)

L. Bansal, V. R. Supradeepa, T. Kremp, S. Sullivan, and C. Headley, “High power cladding mode stripper,” Proc. SPIE 9344, 93440F (2015).
[Crossref]

T. L. Li, J. Wu, Y. H. Sun, Y. S. Wang, and Y. Ma, “An improved method for stripping cladding light in high power fiber lasers,” Proc. SPIE 9255, 92550M (2015).
[Crossref]

2014 (2)

2013 (4)

W. L. Wang, J. Y. Leng, J. Q. Cao, S. F. Guo, X. J. Xu, and Z. F. Jiang, “Method for stripping cladding light in the high power fiber laser,” Opt. Commun. 287(2), 187–191 (2013).
[Crossref]

N. Zhong, Q. Liao, X. Zhu, Y. Wang, and R. Chen, “High-quality fiber fabrication in buffered hydrofluoric acid solution with ultrasonic agitation,” Appl. Opt. 52(7), 1432–1440 (2013).
[Crossref] [PubMed]

A. Kliner, K.-C. Hou, M. Plötner, C. Hupel, T. Stelzner, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Fabrication and evaluation of a 500W cladding-light stripper,” Proc. SPIE 8616, 86160N (2013).
[Crossref]

P. Yan, J. P. Hao, Q. R. Xiao, Y. P. Wang, and M. L. Gong, “The influence of fusion splicing on the beam quality of a ytterbium-doped fiber laser,” Laser Phys. 23(4), 045109 (2013).
[Crossref]

2012 (3)

2011 (2)

S. P. Yin, P. Yan, and M. L. Gong, “Influence of fusion splice on high power ytterbium-doped fiber laser with master oscillator multi-stage power amplifiers structure,” Opt. Lasers Eng. 49(8), 1054–1059 (2011).
[Crossref]

J. Nilsson and D. N. Payne, “Physics. High-power fiber lasers,” Science 332(6032), 921–922 (2011).
[Crossref] [PubMed]

2010 (1)

2008 (2)

2007 (1)

V. R. Machavaram, R. A. Badcock, and G. F. Fernando, “Fabrication of intrinsic fibre Fabry–Perot sensors in silica fibres using hydrofluoric acid etching,” Sensor. Actuat. A-Phys. 138(1), 248–260 (2007).

2005 (1)

1996 (1)

A. P. Liu and K. Ueda, “The absorption characteristics of circular, offset, and rectangular double-clad fibers,” Opt. Commun. 132(5), 511–518 (1996).

1976 (1)

1965 (1)

1961 (1)

1954 (1)

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. SPIE 101(7), 209–214 (1954).

Aguergaray, C.

M. Berisset, L. Lebrun, M. Faucon, R. Kling, J. Boullet, and C. Aguergaray, “Laser surface texturization for high power cladding light stripper,” Proc. SPIE 9730, 973014 (2016).
[Crossref]

Alavian, A.

Amidian, A.

Andrew Clarkson, W.

Babazadeh, A.

Badcock, R. A.

V. R. Machavaram, R. A. Badcock, and G. F. Fernando, “Fabrication of intrinsic fibre Fabry–Perot sensors in silica fibres using hydrofluoric acid etching,” Sensor. Actuat. A-Phys. 138(1), 248–260 (2007).

Bansal, L.

L. Bansal, V. R. Supradeepa, T. Kremp, S. Sullivan, and C. Headley, “High power cladding mode stripper,” Proc. SPIE 9344, 93440F (2015).
[Crossref]

Bennett, H. E.

Berisset, M.

M. Berisset, L. Lebrun, M. Faucon, R. Kling, J. Boullet, and C. Aguergaray, “Laser surface texturization for high power cladding light stripper,” Proc. SPIE 9730, 973014 (2016).
[Crossref]

Boullet, J.

M. Berisset, L. Lebrun, M. Faucon, R. Kling, J. Boullet, and C. Aguergaray, “Laser surface texturization for high power cladding light stripper,” Proc. SPIE 9730, 973014 (2016).
[Crossref]

Boyd, K.

Brunet, F.

Cao, J. Q.

W. L. Wang, J. Y. Leng, J. Q. Cao, S. F. Guo, X. J. Xu, and Z. F. Jiang, “Method for stripping cladding light in the high power fiber laser,” Opt. Commun. 287(2), 187–191 (2013).
[Crossref]

Chen, L.

Z. G. Han, L. Chen, L. L. Shi, J. X. Li, and Z. D. Liu, “Research on the interference fringe contrast in ground glass shape measurement,” Optik (Stuttg.) 123(9), 761–764 (2012).
[Crossref]

Chen, R.

Chen, Z. L.

W. Guo, Z. L. Chen, H. Zhou, J. Li, and J. Hou, “Cascaded cladding light extracting strippers for high power fiber lasers and amplifiers,” IEEE. Photon. J 6(3), 1–6 (2014).
[Crossref]

Clarkson, W. A.

Daniel, J.

Davies, H.

H. Davies, “The reflection of electromagnetic waves from a rough surface,” Proc. SPIE 101(7), 209–214 (1954).

Eberhardt, R.

A. Kliner, K.-C. Hou, M. Plötner, C. Hupel, T. Stelzner, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Fabrication and evaluation of a 500W cladding-light stripper,” Proc. SPIE 8616, 86160N (2013).
[Crossref]

Faucher, M.

Faucon, M.

M. Berisset, L. Lebrun, M. Faucon, R. Kling, J. Boullet, and C. Aguergaray, “Laser surface texturization for high power cladding light stripper,” Proc. SPIE 9730, 973014 (2016).
[Crossref]

Fernando, G. F.

V. R. Machavaram, R. A. Badcock, and G. F. Fernando, “Fabrication of intrinsic fibre Fabry–Perot sensors in silica fibres using hydrofluoric acid etching,” Sensor. Actuat. A-Phys. 138(1), 248–260 (2007).

Golshan, A. H.

Gong, M.

Gong, M. L.

P. Yan, J. P. Hao, Q. R. Xiao, Y. P. Wang, and M. L. Gong, “The influence of fusion splicing on the beam quality of a ytterbium-doped fiber laser,” Laser Phys. 23(4), 045109 (2013).
[Crossref]

S. P. Yin, P. Yan, and M. L. Gong, “Influence of fusion splice on high power ytterbium-doped fiber laser with master oscillator multi-stage power amplifiers structure,” Opt. Lasers Eng. 49(8), 1054–1059 (2011).
[Crossref]

Guo, S. F.

W. L. Wang, J. Y. Leng, J. Q. Cao, S. F. Guo, X. J. Xu, and Z. F. Jiang, “Method for stripping cladding light in the high power fiber laser,” Opt. Commun. 287(2), 187–191 (2013).
[Crossref]

Guo, W.

W. Guo, Z. L. Chen, H. Zhou, J. Li, and J. Hou, “Cascaded cladding light extracting strippers for high power fiber lasers and amplifiers,” IEEE. Photon. J 6(3), 1–6 (2014).
[Crossref]

Han, Z. G.

Z. G. Han, L. Chen, L. L. Shi, J. X. Li, and Z. D. Liu, “Research on the interference fringe contrast in ground glass shape measurement,” Optik (Stuttg.) 123(9), 761–764 (2012).
[Crossref]

Hao, J. P.

P. Yan, J. P. Hao, Q. R. Xiao, Y. P. Wang, and M. L. Gong, “The influence of fusion splicing on the beam quality of a ytterbium-doped fiber laser,” Laser Phys. 23(4), 045109 (2013).
[Crossref]

Haub, J.

Headley, C.

L. Bansal, V. R. Supradeepa, T. Kremp, S. Sullivan, and C. Headley, “High power cladding mode stripper,” Proc. SPIE 9344, 93440F (2015).
[Crossref]

Heidariazar, A.

Hejaz, K.

Hemming, A.

Holehouse, N.

Hou, J.

W. Guo, Z. L. Chen, H. Zhou, J. Li, and J. Hou, “Cascaded cladding light extracting strippers for high power fiber lasers and amplifiers,” IEEE. Photon. J 6(3), 1–6 (2014).
[Crossref]

Hou, K.-C.

A. Kliner, K.-C. Hou, M. Plötner, C. Hupel, T. Stelzner, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Fabrication and evaluation of a 500W cladding-light stripper,” Proc. SPIE 8616, 86160N (2013).
[Crossref]

Huang, B.

Huang, Z.

Hupel, C.

A. Kliner, K.-C. Hou, M. Plötner, C. Hupel, T. Stelzner, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Fabrication and evaluation of a 500W cladding-light stripper,” Proc. SPIE 8616, 86160N (2013).
[Crossref]

Jafari, N. T.

Jafari, S. N. T.

Jiang, Z. F.

W. L. Wang, J. Y. Leng, J. Q. Cao, S. F. Guo, X. J. Xu, and Z. F. Jiang, “Method for stripping cladding light in the high power fiber laser,” Opt. Commun. 287(2), 187–191 (2013).
[Crossref]

Jing, F.

Kanskar, M.

Kliner, A.

A. Kliner, K.-C. Hou, M. Plötner, C. Hupel, T. Stelzner, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Fabrication and evaluation of a 500W cladding-light stripper,” Proc. SPIE 8616, 86160N (2013).
[Crossref]

Kling, R.

M. Berisset, L. Lebrun, M. Faucon, R. Kling, J. Boullet, and C. Aguergaray, “Laser surface texturization for high power cladding light stripper,” Proc. SPIE 9730, 973014 (2016).
[Crossref]

Kremp, T.

L. Bansal, V. R. Supradeepa, T. Kremp, S. Sullivan, and C. Headley, “High power cladding mode stripper,” Proc. SPIE 9344, 93440F (2015).
[Crossref]

Lafouti, M.

Lebrun, L.

M. Berisset, L. Lebrun, M. Faucon, R. Kling, J. Boullet, and C. Aguergaray, “Laser surface texturization for high power cladding light stripper,” Proc. SPIE 9730, 973014 (2016).
[Crossref]

Leng, J. Y.

W. L. Wang, J. Y. Leng, J. Q. Cao, S. F. Guo, X. J. Xu, and Z. F. Jiang, “Method for stripping cladding light in the high power fiber laser,” Opt. Commun. 287(2), 187–191 (2013).
[Crossref]

Li, C.

Li, J.

W. Guo, Z. L. Chen, H. Zhou, J. Li, and J. Hou, “Cascaded cladding light extracting strippers for high power fiber lasers and amplifiers,” IEEE. Photon. J 6(3), 1–6 (2014).
[Crossref]

Li, J. X.

Z. G. Han, L. Chen, L. L. Shi, J. X. Li, and Z. D. Liu, “Research on the interference fringe contrast in ground glass shape measurement,” Optik (Stuttg.) 123(9), 761–764 (2012).
[Crossref]

Li, Q.

Li, T. L.

T. L. Li, J. Wu, Y. H. Sun, Y. S. Wang, and Y. Ma, “An improved method for stripping cladding light in high power fiber lasers,” Proc. SPIE 9255, 92550M (2015).
[Crossref]

Liang, X.

Liao, Q.

Lin, H.

Liu, A. P.

A. P. Liu and K. Ueda, “The absorption characteristics of circular, offset, and rectangular double-clad fibers,” Opt. Commun. 132(5), 511–518 (1996).

Liu, Z. D.

Z. G. Han, L. Chen, L. L. Shi, J. X. Li, and Z. D. Liu, “Research on the interference fringe contrast in ground glass shape measurement,” Optik (Stuttg.) 123(9), 761–764 (2012).
[Crossref]

Ma, Y.

T. L. Li, J. Wu, Y. H. Sun, Y. S. Wang, and Y. Ma, “An improved method for stripping cladding light in high power fiber lasers,” Proc. SPIE 9255, 92550M (2015).
[Crossref]

Machavaram, V. R.

V. R. Machavaram, R. A. Badcock, and G. F. Fernando, “Fabrication of intrinsic fibre Fabry–Perot sensors in silica fibres using hydrofluoric acid etching,” Sensor. Actuat. A-Phys. 138(1), 248–260 (2007).

Malitson, I. H.

Marcuse, D.

Mies, E.

Nasirabad, R. R.

Nilsson, J.

Norouzey, A.

Norouzy, A.

Payne, D. N.

J. Nilsson and D. N. Payne, “Physics. High-power fiber lasers,” Science 332(6032), 921–922 (2011).
[Crossref] [PubMed]

Plötner, M.

A. Kliner, K.-C. Hou, M. Plötner, C. Hupel, T. Stelzner, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Fabrication and evaluation of a 500W cladding-light stripper,” Proc. SPIE 8616, 86160N (2013).
[Crossref]

Poozesh, R.

Porteus, J. O.

Rees, S.

Richardson, D. J.

Roohforouz, A.

Schreiber, T.

A. Kliner, K.-C. Hou, M. Plötner, C. Hupel, T. Stelzner, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Fabrication and evaluation of a 500W cladding-light stripper,” Proc. SPIE 8616, 86160N (2013).
[Crossref]

Sévigny, B.

A. Wetter, M. Faucher, and B. Sévigny, “High power cladding light strippers,” Proc. SPIE 6873, 687327 (2008).
[Crossref]

Shi, L. L.

Z. G. Han, L. Chen, L. L. Shi, J. X. Li, and Z. D. Liu, “Research on the interference fringe contrast in ground glass shape measurement,” Optik (Stuttg.) 123(9), 761–764 (2012).
[Crossref]

Simakov, N.

Stelzner, T.

A. Kliner, K.-C. Hou, M. Plötner, C. Hupel, T. Stelzner, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Fabrication and evaluation of a 500W cladding-light stripper,” Proc. SPIE 8616, 86160N (2013).
[Crossref]

Sullivan, S.

L. Bansal, V. R. Supradeepa, T. Kremp, S. Sullivan, and C. Headley, “High power cladding mode stripper,” Proc. SPIE 9344, 93440F (2015).
[Crossref]

Sun, Y. H.

T. L. Li, J. Wu, Y. H. Sun, Y. S. Wang, and Y. Ma, “An improved method for stripping cladding light in high power fiber lasers,” Proc. SPIE 9255, 92550M (2015).
[Crossref]

Supradeepa, V. R.

L. Bansal, V. R. Supradeepa, T. Kremp, S. Sullivan, and C. Headley, “High power cladding mode stripper,” Proc. SPIE 9344, 93440F (2015).
[Crossref]

Swain, R.

Todoroki, S.

Tünnermann, A.

A. Kliner, K.-C. Hou, M. Plötner, C. Hupel, T. Stelzner, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Fabrication and evaluation of a 500W cladding-light stripper,” Proc. SPIE 8616, 86160N (2013).
[Crossref]

Ueda, K.

A. P. Liu and K. Ueda, “The absorption characteristics of circular, offset, and rectangular double-clad fibers,” Opt. Commun. 132(5), 511–518 (1996).

Wang, J.

Wang, W. L.

W. L. Wang, J. Y. Leng, J. Q. Cao, S. F. Guo, X. J. Xu, and Z. F. Jiang, “Method for stripping cladding light in the high power fiber laser,” Opt. Commun. 287(2), 187–191 (2013).
[Crossref]

Wang, Y.

Wang, Y. P.

P. Yan, J. P. Hao, Q. R. Xiao, Y. P. Wang, and M. L. Gong, “The influence of fusion splicing on the beam quality of a ytterbium-doped fiber laser,” Laser Phys. 23(4), 045109 (2013).
[Crossref]

Wang, Y. S.

T. L. Li, J. Wu, Y. H. Sun, Y. S. Wang, and Y. Ma, “An improved method for stripping cladding light in high power fiber lasers,” Proc. SPIE 9255, 92550M (2015).
[Crossref]

Wetter, A.

Wu, J.

T. L. Li, J. Wu, Y. H. Sun, Y. S. Wang, and Y. Ma, “An improved method for stripping cladding light in high power fiber lasers,” Proc. SPIE 9255, 92550M (2015).
[Crossref]

Xiao, Q. R.

P. Yan, J. P. Hao, Q. R. Xiao, Y. P. Wang, and M. L. Gong, “The influence of fusion splicing on the beam quality of a ytterbium-doped fiber laser,” Laser Phys. 23(4), 045109 (2013).
[Crossref]

Xiao, Y.

Xiong, S.

Xu, X. J.

W. L. Wang, J. Y. Leng, J. Q. Cao, S. F. Guo, X. J. Xu, and Z. F. Jiang, “Method for stripping cladding light in the high power fiber laser,” Opt. Commun. 287(2), 187–191 (2013).
[Crossref]

Yan, D.

Yan, P.

P. Yan, J. P. Hao, Q. R. Xiao, Y. P. Wang, and M. L. Gong, “The influence of fusion splicing on the beam quality of a ytterbium-doped fiber laser,” Laser Phys. 23(4), 045109 (2013).
[Crossref]

S. P. Yin, P. Yan, and M. L. Gong, “Influence of fusion splice on high power ytterbium-doped fiber laser with master oscillator multi-stage power amplifiers structure,” Opt. Lasers Eng. 49(8), 1054–1059 (2011).
[Crossref]

S. Yin, P. Yan, and M. Gong, “End-pumped 300 W continuous-wave ytterbium-doped all-fiber laser with master oscillator multi-stage power amplifiers configuration,” Opt. Express 16(22), 17864–17869 (2008).
[Crossref] [PubMed]

Yin, S.

Yin, S. P.

S. P. Yin, P. Yan, and M. L. Gong, “Influence of fusion splice on high power ytterbium-doped fiber laser with master oscillator multi-stage power amplifiers structure,” Opt. Lasers Eng. 49(8), 1054–1059 (2011).
[Crossref]

Zhong, N.

Zhou, H.

W. Guo, Z. L. Chen, H. Zhou, J. Li, and J. Hou, “Cascaded cladding light extracting strippers for high power fiber lasers and amplifiers,” IEEE. Photon. J 6(3), 1–6 (2014).
[Crossref]

Zhu, X.

Appl. Opt. (4)

IEEE. Photon. J (1)

W. Guo, Z. L. Chen, H. Zhou, J. Li, and J. Hou, “Cascaded cladding light extracting strippers for high power fiber lasers and amplifiers,” IEEE. Photon. J 6(3), 1–6 (2014).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. B (1)

Laser Phys. (1)

P. Yan, J. P. Hao, Q. R. Xiao, Y. P. Wang, and M. L. Gong, “The influence of fusion splicing on the beam quality of a ytterbium-doped fiber laser,” Laser Phys. 23(4), 045109 (2013).
[Crossref]

Opt. Commun. (2)

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W. L. Wang, J. Y. Leng, J. Q. Cao, S. F. Guo, X. J. Xu, and Z. F. Jiang, “Method for stripping cladding light in the high power fiber laser,” Opt. Commun. 287(2), 187–191 (2013).
[Crossref]

Opt. Express (4)

Opt. Lasers Eng. (1)

S. P. Yin, P. Yan, and M. L. Gong, “Influence of fusion splice on high power ytterbium-doped fiber laser with master oscillator multi-stage power amplifiers structure,” Opt. Lasers Eng. 49(8), 1054–1059 (2011).
[Crossref]

Optik (Stuttg.) (1)

Z. G. Han, L. Chen, L. L. Shi, J. X. Li, and Z. D. Liu, “Research on the interference fringe contrast in ground glass shape measurement,” Optik (Stuttg.) 123(9), 761–764 (2012).
[Crossref]

Proc. SPIE (6)

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A. Kliner, K.-C. Hou, M. Plötner, C. Hupel, T. Stelzner, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Fabrication and evaluation of a 500W cladding-light stripper,” Proc. SPIE 8616, 86160N (2013).
[Crossref]

T. L. Li, J. Wu, Y. H. Sun, Y. S. Wang, and Y. Ma, “An improved method for stripping cladding light in high power fiber lasers,” Proc. SPIE 9255, 92550M (2015).
[Crossref]

L. Bansal, V. R. Supradeepa, T. Kremp, S. Sullivan, and C. Headley, “High power cladding mode stripper,” Proc. SPIE 9344, 93440F (2015).
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[Crossref]

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J. E. Langseth and A. E. Mandl, “All glass fiber laser cladding mode stripper,” U.S. patent 8,433,161 B2 (April 30, 2013).

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Figures (15)

Fig. 1
Fig. 1

Schematic of cladding light stripping for HF-acid-etched fiber.

Fig. 2
Fig. 2

Schematic of the propagation of meridian light.

Fig. 3
Fig. 3

Relationship between meridian cladding light attenuation and fiber surface roughness.

Fig. 4
Fig. 4

Schematic of the propagation of skew light in fiber cladding and its projection on fiber end surface.

Fig. 5
Fig. 5

Reflection angle and scattering coefficient of skew light: (a) Relationship between the reflection angle and incident angle for different axis tilt angles; (b) Relationship between the scattering coefficient and the reflection angle for different RMS roughnesses values.

Fig. 6
Fig. 6

Schematic of the propagation of meridian light in etched tapered region.

Fig. 7
Fig. 7

Schematic of the simulation model of etched-fiber stripping (not to scale).

Fig. 8
Fig. 8

Relationship between etched-fiber parameters and cladding light attenuation: (a) Relationship between RMS roughness of the fiber surface, etched-cladding diameter, and cladding light attenuation; (b) Relationship between RMS roughness of the fiber surface, etched-fiber length, and cladding light attenuation; (c) Relationship between tapered-region length, etched-cladding diameter, and cladding light attenuation.

Fig. 9
Fig. 9

Schematic of the segmented corrosion approach.

Fig. 10
Fig. 10

Segmented corrosion simulation results: (a) RMS roughness corresponding to the fiber position of three segmented corrosion fibers; (b) Comparison of the optical power retained in the cladding of the different types of etched fibers. The origin in each graph corresponds to the starting point of the first etched-fiber segment.

Fig. 11
Fig. 11

Stripping performance of etched fiber for different cladding light sources.

Fig. 12
Fig. 12

Measurement results of surface roughness obtained with Veeco NT9100 optical profiler: (a) Measurement of unetched fiber: (i) Measurement result (ii) Fringe pattern of fiber surface; (b) Measurement of etched fiber: (i) Measurement result (ii) Fringe pattern of fiber surface.

Fig. 13
Fig. 13

Comparison of test results, calculation results, and simulation results of verification experiment.

Fig. 14
Fig. 14

Test results for cladding light removal perfromance of two-segment etched fiber: (a) Remaining optical power and cladding light attenuation of etched segment 1; (b) Remaining optical power and cladding light attenuation of two-segment etched fiber.

Fig. 15
Fig. 15

Thermal and light leakage images of etched fibers while stripping 438 W: (a) Test results of single-segment etched fiber: (i) Thermal image (ii) Light leakage image; (b) Test results of two-segment etched fiber: (i) Thermal image (ii) Light leakage image.

Tables (1)

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Table 1 Specific parameters of the 10 fiber samples after the etching process

Equations (10)

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Si O 2 +4HFSi F 4 +2 H 2 O Si O 2 +6HF H 2 Si F 4 +2 H 2 O.
R s = R 0 exp(16 π 2 σ 2 cos 2 θ i / λ 2 ).
R d = R 0 R s = R 0 (1exp(16 π 2 σ 2 cos 2 θ i / λ 2 )).
R 0 = 1 2 ( sin 2 (θφ) sin 2 (θ+φ) + tan 2 (θφ) tan 2 (θ+φ) ),
P out = θ 1 θ 2 P 1 ( θ co )+ P 2 ( θ co )+ P 3 ( θ co )d θ co = θ 1 θ 2 P in ( θ co ) T 12 ( θ co ) 2 Q 1 ( θ co )1 R s ( θ cl ) Q 1 ( θ co ) + P in ( θ co ) T 12 ( θ co ) R 12 ( θ co ) Q 2 ( θ co ) . R s ( θ cl ) Q 2 ( θ co ) + P in ( θ co ) T 12 ( θ co ) R 12 ( θ co ) Q 3 ( θ co ) R s ( θ cl ) Q 3 ( θ co )+1 d θ co
Q 1 ( θ co )= L ef D co tan θ co /2( D ecl D co )tan θ cl /2 D co tan θ co +( D ecl D co )tan θ cl +1 Q 2 ( θ co )= L ef 3 D co tan θ co /2( D ecl D co )tan θ cl /2 ( D ecl D co )tan θ cl +1. Q 3 ( θ co )= L ef D co tan θ co /23( D ecl D co )tan θ cl /2 ( D ecl D co )tan θ cl +1
R d s = R 0 (1exp((16 π 2 σ 2 / λ 2 ) ( n 0 sin α s cosβ/ n 2 ) 2 )).
θ(i)={ θ(0) α taper (i=1) θ(i1)2 α taper (i>1) .
h(i)=h(i1)( cot α taper tan(θ(i1) α taper ) cot α taper +tan(θ(i1) α taper ) ).
L(i)=(h(i1)h(i))cot α taper .

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