Resolving optical illumination distributions along an axially symmetric photodetecting fiber

Fabien Sorin; Guillaume Lestoquoy; Sylvain Danto; John D. Joannopoulos; Yoel Fink

doi:10.1364/OE.18.024264

Resolving optical illumination distributions along an axially symmetric photodetecting fiber

Fabien Sorin, Guillaume Lestoquoy, Sylvain Danto, John D. Joannopoulos, and Yoel Fink

Optics Express
Vol. 18,
Issue 23,
pp. 24264-24275
(2010)
•https://doi.org/10.1364/OE.18.024264

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Fabien Sorin, Guillaume Lestoquoy, Sylvain Danto, John D. Joannopoulos, and Yoel Fink, "Resolving optical illumination distributions along an axially symmetric photodetecting fiber," Opt. Express 18, 24264-24275 (2010)

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Related Topics
Table of Contents Category
- Detectors
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Photodetectors (040.5160)
- Fiber materials (160.2290)

About this Article
History
- Original Manuscript: September 20, 2010
- Revised Manuscript: October 22, 2010
- Manuscript Accepted: October 22, 2010
- Published: November 4, 2010
Virtual Issues

November 15, 2010 Spotlight on Optics

Accessible

Open Access

Abstract

Photodetecting fibers of arbitrary length with internal metal, semiconductor and insulator domains have recently been demonstrated. These semiconductor devices exhibit a continuous translational symmetry which presents challenges to the extraction of spatially resolved information. Here, we overcome this seemingly fundamental limitation and achieve the detection and spatial localization of a single incident optical beam at sub-centimeter resolution, along a one-meter fiber section. Using an approach that breaks the axial symmetry through the constuction of a convex electrical potential along the fiber axis, we demonstrate the full reconstruction of an arbitrary rectangular optical wave profile. Finally, the localization of up to three points of illumination simultaneously incident on a photodetecting fiber is achieved.

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References

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T. Kurashima, T. Horiguchi, and M. Tateda, “Distributed-temperature sensing using stimulated Brillouin scattering in optical silica fibers,” Opt. Lett. 15(18), 1038–1040 (1990).
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[Crossref] [PubMed]
A. F. Abouraddy, M. Bayindir, G. Benoit, S. D. Hart, K. Kuriki, N. Orf, O. Shapira, F. Sorin, B. Temelkuran, and Y. Fink, “Towards multimaterial multifunctional fibres that see, hear, sense and communicate,” Nat. Mater. 6(5), 336–347 (2007).
[Crossref] [PubMed]
M. Bayindir, A. F. Abouraddy, O. Shapira, J. Viens, D. Saygin-Hinczewski, F. Sorin, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Kilometer-long ordered nanophotonic devices by preform-to-fiber fabrication,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1202–1213 (2006).
[Crossref]
F. Sorin, A. F. Abouraddy, N. Orf, O. Shapira, J. Viens, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Multimaterial Photodetecting Fibers: a Geometric and Structural Study,” Adv. Mater. 19(22), 3872–3877 (2007).
[Crossref]
A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Large-scale optical-field measurements with geometric fibre constructs,” Nat. Mater. 5(7), 532–536 (2006).
[Crossref] [PubMed]
F. Sorin, O. Shapira, A. F. Abouraddy, M. Spencer, N. D. Orf1, J. D. Joannopoulos, and Y. Fink, “Exploiting the collective effects of optoelectronic devices integrated in a single fiber,” Nano Lett. 9(7), 2630–2635 (2009).
[Crossref] [PubMed]
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[Crossref] [PubMed]
M. Bayindir, O. Shapira, D. Saygin-Hinczewski, J. Viens, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Integrated fibers for self-monitored optical transport,” Nat. Mater. 4(11), 820–825 (2005).
[Crossref]
M. Bayindir, A. F. Abouraddy, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Thermal-Sensing Fiber Devices by Multimaterial Codrawing,” Adv. Mater. 18, 845 (2006).
[Crossref]
E. K. Sichel, J. I. Gittleman, and P. Sheng, “Electrical properties of Carbon-polymer composite,” J. Electron. Mater. 11(4), 699–747 (1982).
[Crossref]
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D. S. Deng, N. D. Orf, A. F. Abouraddy, A. M. Stolyarov, J. D. Joannopoulos, H. A. Stone, and Y. Fink, “In-fiber semiconductor filament arrays,” Nano Lett. 8(12), 4265–4269 (2008).
[Crossref]

2010 (2)

S. Egusa, Z. Wang, N. Chocat, Z. M. Ruff, A. M. Stolyarov, D. Shemuly, F. Sorin, P. T. Rakich, J. D. Joannopoulos, and Y. Fink, “Multimaterial piezoelectric fibres,” Nat. Mater. 9(8), 643–648 (2010).
[Crossref] [PubMed]

S. Danto, F. Sorin, N. D. Orf, Z. Wang, S. A. Speakman, J. D. Joannopoulos, and Y. Fink, “Fiber field-effect device via in situ channel crystallization,” Adv. Mater. 22(37), 4162–4166 (2010).
[Crossref] [PubMed]

2009 (1)

F. Sorin, O. Shapira, A. F. Abouraddy, M. Spencer, N. D. Orf1, J. D. Joannopoulos, and Y. Fink, “Exploiting the collective effects of optoelectronic devices integrated in a single fiber,” Nano Lett. 9(7), 2630–2635 (2009).
[Crossref] [PubMed]

2008 (4)

B. O'Connor, K. P. Pipe, and M. Shtein, “Fiber based organic photovoltaic devices,” Appl. Phys. Lett. 92(19), 193306 (2008).
[Crossref]

M. A. Schmidt, L. N. Prill Sempere, H. K. Tyagi, C. G. Poulton, and P. St, “J. Russell, “Wave guiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys. Rev. B 77(3), 033417 (2008).
[Crossref]

H. K. Tyagi, M. A. Schmidt, L. Prill Sempere, and P. S. Russell, “Optical properties of photonic crystal fiber with integral micron-sized Ge wire,” Opt. Express 16(22), 17227–17236 (2008).
[Crossref] [PubMed]

D. S. Deng, N. D. Orf, A. F. Abouraddy, A. M. Stolyarov, J. D. Joannopoulos, H. A. Stone, and Y. Fink, “In-fiber semiconductor filament arrays,” Nano Lett. 8(12), 4265–4269 (2008).
[Crossref]

2007 (3)

B. O'Connor, K. H. An, Y. Zhao, K. P. Pipe, and M. Shtein, “Fiber Shaped Organic Light Emitting Device,” Adv. Mater. 19(22), 3897–3900 (2007).
[Crossref]

F. Sorin, A. F. Abouraddy, N. Orf, O. Shapira, J. Viens, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Multimaterial Photodetecting Fibers: a Geometric and Structural Study,” Adv. Mater. 19(22), 3872–3877 (2007).
[Crossref]

A. F. Abouraddy, M. Bayindir, G. Benoit, S. D. Hart, K. Kuriki, N. Orf, O. Shapira, F. Sorin, B. Temelkuran, and Y. Fink, “Towards multimaterial multifunctional fibres that see, hear, sense and communicate,” Nat. Mater. 6(5), 336–347 (2007).
[Crossref] [PubMed]

2006 (4)

M. Bayindir, A. F. Abouraddy, O. Shapira, J. Viens, D. Saygin-Hinczewski, F. Sorin, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Kilometer-long ordered nanophotonic devices by preform-to-fiber fabrication,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1202–1213 (2006).
[Crossref]

P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D. J. Won, F. Zhang, E. R. Margine, V. Gopalan, V. H. Crespi, and J. V. Badding, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science 311(5767), 1583–1586 (2006).
[Crossref] [PubMed]

A. F. Abouraddy, O. Shapira, M. Bayindir, J. Arnold, F. Sorin, D. S. Hinczewski, J. D. Joannopoulos, and Y. Fink, “Large-scale optical-field measurements with geometric fibre constructs,” Nat. Mater. 5(7), 532–536 (2006).
[Crossref] [PubMed]

M. Bayindir, A. F. Abouraddy, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Thermal-Sensing Fiber Devices by Multimaterial Codrawing,” Adv. Mater. 18, 845 (2006).
[Crossref]

2005 (1)

M. Bayindir, O. Shapira, D. Saygin-Hinczewski, J. Viens, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Integrated fibers for self-monitored optical transport,” Nat. Mater. 4(11), 820–825 (2005).
[Crossref]

2004 (1)

M. Bayindir, F. Sorin, A. F. Abouraddy, J. Viens, S. D. Hart, J. D. Joannopoulos, and Y. Fink, “Metal-insulator-semiconductor optoelectronic fibres,” Nature 431(7010), 826–829 (2004).
[Crossref] [PubMed]

2003 (1)

T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, “Optical devices based on liquid crystal photonic bandgap fibres,” Opt. Express 11(20), 2589–2596 (2003).
[Crossref] [PubMed]

2002 (1)

M. Fokine, L. E. Nilsson, A. Claesson, D. Berlemont, L. Kjellberg, L. Krummenacher, and W. Margulis, “Integrated fiber Mach-Zehnder interferometer for electro-optic switching,” Opt. Lett. 27(18), 1643–1645 (2002).
[Crossref]

1999 (1)

A. Rogers, “Distributed optical-fiber sensing,” Meas. Sci. Technol. 10(8), 201 (1999).
[Crossref]

1996 (1)

C. I. Merzbacher, A. D. Kersey, and E. J. Friebele, “Fiber optic sensors in concrete structures: a review,” Smart Mater. Struct. 5(2), 196–208 (1996).
[Crossref]

1994 (1)

X. Bao, D. J. Webb, and D. A. Jackson, “Combined distributed temperature and strain sensor based on Brillouin loss in an optical fiber,” Opt. Lett. 19(2), 141 (1994).
[Crossref] [PubMed]

1990 (1)

T. Kurashima, T. Horiguchi, and M. Tateda, “Distributed-temperature sensing using stimulated Brillouin scattering in optical silica fibers,” Opt. Lett. 15(18), 1038–1040 (1990).
[Crossref] [PubMed]

1985 (1)

J. P. Dakin, D. J. Pratt, G. W. Bibby, and J. N. Ross, “Distributed optical fiber raman temperature sensor using a semiconductor light source and detector,” Electron. Lett. 21(13), 569 (1985).
[Crossref]

1983 (1)

A. H. Hartog, “A distributed temperature sensor based on liquid-core optical fibers,” J. Lightwave Technol. 1(3), 498–509 (1983).
[Crossref]

1982 (1)

E. K. Sichel, J. I. Gittleman, and P. Sheng, “Electrical properties of Carbon-polymer composite,” J. Electron. Mater. 11(4), 699–747 (1982).
[Crossref]

1981 (1)

A. J. Rogers, “Polarization-optical time domain reflectometry: a technique for the measurement of field distributions,” Appl. Opt. 20(6), 1060–1074 (1981).
[Crossref] [PubMed]

1976 (1)

M. K. Barnoski and S. M. Jensen, “Fiber waveguides: a novel technique for investigating attenuation characteristics,” Appl. Opt. 15(9), 2112–2115 (1976).
[Crossref] [PubMed]

Abouraddy, A. F.

D. S. Deng, N. D. Orf, A. F. Abouraddy, A. M. Stolyarov, J. D. Joannopoulos, H. A. Stone, and Y. Fink, “In-fiber semiconductor filament arrays,” Nano Lett. 8(12), 4265–4269 (2008).
[Crossref]

M. Bayindir, A. F. Abouraddy, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Thermal-Sensing Fiber Devices by Multimaterial Codrawing,” Adv. Mater. 18, 845 (2006).
[Crossref]

Amezcua-Correa, A.

An, K. H.

B. O'Connor, K. H. An, Y. Zhao, K. P. Pipe, and M. Shtein, “Fiber Shaped Organic Light Emitting Device,” Adv. Mater. 19(22), 3897–3900 (2007).
[Crossref]

Arnold, J.

M. Bayindir, A. F. Abouraddy, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Thermal-Sensing Fiber Devices by Multimaterial Codrawing,” Adv. Mater. 18, 845 (2006).
[Crossref]

Badding, J. V.

Bao, X.

X. Bao, D. J. Webb, and D. A. Jackson, “Combined distributed temperature and strain sensor based on Brillouin loss in an optical fiber,” Opt. Lett. 19(2), 141 (1994).
[Crossref] [PubMed]

Baril, N. F.

Barnoski, M. K.

M. K. Barnoski and S. M. Jensen, “Fiber waveguides: a novel technique for investigating attenuation characteristics,” Appl. Opt. 15(9), 2112–2115 (1976).
[Crossref] [PubMed]

Bayindir, M.

M. Bayindir, A. F. Abouraddy, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Thermal-Sensing Fiber Devices by Multimaterial Codrawing,” Adv. Mater. 18, 845 (2006).
[Crossref]

Benoit, G.

Berlemont, D.

Bibby, G. W.

Bjarklev, A.

T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, “Optical devices based on liquid crystal photonic bandgap fibres,” Opt. Express 11(20), 2589–2596 (2003).
[Crossref] [PubMed]

Broeng, J.

T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, “Optical devices based on liquid crystal photonic bandgap fibres,” Opt. Express 11(20), 2589–2596 (2003).
[Crossref] [PubMed]

Chocat, N.

Claesson, A.

Crespi, V. H.

Dakin, J. P.

Danto, S.

Deng, D. S.

D. S. Deng, N. D. Orf, A. F. Abouraddy, A. M. Stolyarov, J. D. Joannopoulos, H. A. Stone, and Y. Fink, “In-fiber semiconductor filament arrays,” Nano Lett. 8(12), 4265–4269 (2008).
[Crossref]

Egusa, S.

Fink, Y.

D. S. Deng, N. D. Orf, A. F. Abouraddy, A. M. Stolyarov, J. D. Joannopoulos, H. A. Stone, and Y. Fink, “In-fiber semiconductor filament arrays,” Nano Lett. 8(12), 4265–4269 (2008).
[Crossref]

M. Bayindir, A. F. Abouraddy, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Thermal-Sensing Fiber Devices by Multimaterial Codrawing,” Adv. Mater. 18, 845 (2006).
[Crossref]

Finlayson, C. E.

Fokine, M.

Friebele, E. J.

C. I. Merzbacher, A. D. Kersey, and E. J. Friebele, “Fiber optic sensors in concrete structures: a review,” Smart Mater. Struct. 5(2), 196–208 (1996).
[Crossref]

Gittleman, J. I.

E. K. Sichel, J. I. Gittleman, and P. Sheng, “Electrical properties of Carbon-polymer composite,” J. Electron. Mater. 11(4), 699–747 (1982).
[Crossref]

Gopalan, V.

Hart, S. D.

Hartog, A. H.

A. H. Hartog, “A distributed temperature sensor based on liquid-core optical fibers,” J. Lightwave Technol. 1(3), 498–509 (1983).
[Crossref]

Hayes, J. R.

Hermann, D. S.

T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, “Optical devices based on liquid crystal photonic bandgap fibres,” Opt. Express 11(20), 2589–2596 (2003).
[Crossref] [PubMed]

Hinczewski, D. S.

Horiguchi, T.

Jackson, B. R.

Jackson, D. A.

X. Bao, D. J. Webb, and D. A. Jackson, “Combined distributed temperature and strain sensor based on Brillouin loss in an optical fiber,” Opt. Lett. 19(2), 141 (1994).
[Crossref] [PubMed]

Jensen, S. M.

M. K. Barnoski and S. M. Jensen, “Fiber waveguides: a novel technique for investigating attenuation characteristics,” Appl. Opt. 15(9), 2112–2115 (1976).
[Crossref] [PubMed]

Joannopoulos, J. D.

D. S. Deng, N. D. Orf, A. F. Abouraddy, A. M. Stolyarov, J. D. Joannopoulos, H. A. Stone, and Y. Fink, “In-fiber semiconductor filament arrays,” Nano Lett. 8(12), 4265–4269 (2008).
[Crossref]

M. Bayindir, A. F. Abouraddy, J. Arnold, J. D. Joannopoulos, and Y. Fink, “Thermal-Sensing Fiber Devices by Multimaterial Codrawing,” Adv. Mater. 18, 845 (2006).
[Crossref]

Kersey, A. D.

C. I. Merzbacher, A. D. Kersey, and E. J. Friebele, “Fiber optic sensors in concrete structures: a review,” Smart Mater. Struct. 5(2), 196–208 (1996).
[Crossref]

Kjellberg, L.

Krummenacher, L.

Kurashima, T.

Kuriki, K.

Larsen, T. T.

T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, “Optical devices based on liquid crystal photonic bandgap fibres,” Opt. Express 11(20), 2589–2596 (2003).
[Crossref] [PubMed]

Margine, E. R.

Margulis, W.

Merzbacher, C. I.

C. I. Merzbacher, A. D. Kersey, and E. J. Friebele, “Fiber optic sensors in concrete structures: a review,” Smart Mater. Struct. 5(2), 196–208 (1996).
[Crossref]

Nilsson, L. E.

O'Connor, B.

B. O'Connor, K. P. Pipe, and M. Shtein, “Fiber based organic photovoltaic devices,” Appl. Phys. Lett. 92(19), 193306 (2008).
[Crossref]

B. O'Connor, K. H. An, Y. Zhao, K. P. Pipe, and M. Shtein, “Fiber Shaped Organic Light Emitting Device,” Adv. Mater. 19(22), 3897–3900 (2007).
[Crossref]

Orf, N.

Orf, N. D.

D. S. Deng, N. D. Orf, A. F. Abouraddy, A. M. Stolyarov, J. D. Joannopoulos, H. A. Stone, and Y. Fink, “In-fiber semiconductor filament arrays,” Nano Lett. 8(12), 4265–4269 (2008).
[Crossref]

Orf1, N. D.

Pipe, K. P.

B. O'Connor, K. P. Pipe, and M. Shtein, “Fiber based organic photovoltaic devices,” Appl. Phys. Lett. 92(19), 193306 (2008).
[Crossref]

B. O'Connor, K. H. An, Y. Zhao, K. P. Pipe, and M. Shtein, “Fiber Shaped Organic Light Emitting Device,” Adv. Mater. 19(22), 3897–3900 (2007).
[Crossref]

Poulton, C. G.

Pratt, D. J.

Prill Sempere, L.

Prill Sempere, L. N.

Rakich, P. T.

Rogers, A.

A. Rogers, “Distributed optical-fiber sensing,” Meas. Sci. Technol. 10(8), 201 (1999).
[Crossref]

Rogers, A. J.

A. J. Rogers, “Polarization-optical time domain reflectometry: a technique for the measurement of field distributions,” Appl. Opt. 20(6), 1060–1074 (1981).
[Crossref] [PubMed]

Ross, J. N.

Ruff, Z. M.

Russell, P. S.

Saygin-Hinczewski, D.

Sazio, P. J. A.

Scheidemantel, T. J.

Schmidt, M. A.

Shapira, O.

Shemuly, D.

Sheng, P.

E. K. Sichel, J. I. Gittleman, and P. Sheng, “Electrical properties of Carbon-polymer composite,” J. Electron. Mater. 11(4), 699–747 (1982).
[Crossref]

Shtein, M.

B. O'Connor, K. P. Pipe, and M. Shtein, “Fiber based organic photovoltaic devices,” Appl. Phys. Lett. 92(19), 193306 (2008).
[Crossref]

B. O'Connor, K. H. An, Y. Zhao, K. P. Pipe, and M. Shtein, “Fiber Shaped Organic Light Emitting Device,” Adv. Mater. 19(22), 3897–3900 (2007).
[Crossref]

Sichel, E. K.

E. K. Sichel, J. I. Gittleman, and P. Sheng, “Electrical properties of Carbon-polymer composite,” J. Electron. Mater. 11(4), 699–747 (1982).
[Crossref]

Sorin, F.

Speakman, S. A.

Spencer, M.

St, P.

Stolyarov, A. M.

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Fig. 1

A(1). 3D Schematic of the multimaterial fiber thermal drawing fabrication approach. A(2). Schematic of a connected photodetecting fiber with an illumination event. The graph represents the linear current density in the dark and under the represented illumination. B. Scanning Electron Microscope micrograph of the fiber cross-section (inset: zoom-in on the contact between the core and the CPC electrode); C. Schematic of the fiber system’s equivalent circuit.

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Fig. 2

(A) Schematic of the fiber contact for boundary conditions (1) and graph representing the experimental results (dots) and the fitted theoretical model (lines) of the voltage profile between the CPC electrode and the metallic conduct at different points along the fiber axis, when the fiber is under BC(1) and for different fibers: in black, AST₁₀ thin-film; in blue, AST₁₀ core and in red, AST₁₈. (B) Same as (A) but when the fiber is under BC(2).

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Fig. 3

A: SEM micrograph of a fiber with the new thin-film/solid-core structure. B: Experimental results (dots, the lines are added for clarity) of the voltage profile of a one-meter long fiber piece from panel A in the dark (in blue), and under a spot of white light (in red) and green light (in green) at the same location, same width and of similar intensity. C: schematic of the electrical connection to one fiber end.

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Fig. 4

Schematic of the illuminated fiber by a single optical beam and graph of the real position (black dashed line) and reconstructed position with error bars(blue dots) of an optifcal beam incident on a 1 m-long fiber at different positions. (B) Schematic of the illuminated fiber by a rectangular optical wave front. And graph of the real profile (black doted line) and reconstructed profile (blue dots) of a rectangular wave front incident on the same fiber.

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Fig. 5

(A) Schematic: photodetecting fiber illuminated by two similar optical beams. Graph: position measurements of the two beams. In black doted line is the conductivity profile generated by the two incoming beams while the blue dots are the reconstructed positions with the error bars. (B) Schematic: photodetecting fiber illuminated by three similar optical beams. Graph: position measurements of the three beams. In black doted line is the conductivity profile generated by the three incoming beams while the blue dots are the reconstructed positions with the error bars.

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Equations (15)

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\frac{V (z) - V (z - d z)}{R_{C P C}} = \frac{V (z + d z) - V (z)}{R_{C P C}} - \frac{V (z)}{R_{g}}, or \frac{\partial^{2} V}{\partial z^{2}} = \frac{R_{C P C}}{R_{g} d z^{2}} V (z)

\frac{\partial^{2} V}{\partial z^{2}} = \frac{V (z)}{δ {(z)}^{2}}

δ (z) = \sqrt{\frac{ρ_{g} (z)}{ρ_{c p c}} \frac{π}{2} S_{c p c}}

V^{B C 1} (z) = \frac{V_{0} \cosh (\frac{L - z}{δ})}{\cosh (\frac{L}{δ})}

V^{B C 2} (z) = \frac{V_{0} \sinh h (\frac{L - z}{δ}) + V_{L} \sinh h (\frac{z}{δ})}{\sinh (\frac{L}{δ})}

\frac{\partial^{2} V}{\partial z^{2}} = V (\frac{1}{δ_{c}^{2}} + \frac{1}{δ_{f}^{2}}) \approx \frac{V}{δ_{c}^{2}}

V^{I} (z) = \frac{V}{\sinh (L / δ_{c})} \sinh h (\frac{L - z}{δ})

V^{I I} (z) = \frac{V}{\sinh (L / δ_{c})} \sinh h (\frac{z}{δ})

i_{p h}^{I} = \frac{2 C V σ_{p h}}{\sinh (L / δ_{c})} \sinh h (\frac{L - z_{0}}{δ_{c}}) \sinh h (\frac{Δ z}{δ_{c}})

i_{p h}^{I I} = \frac{2 C V σ_{p h}}{\sinh (L / δ_{c})} \sinh h (\frac{z_{0}}{δ_{c}}) \sinh h (\frac{Δ z}{δ_{c}})

i_{p h}^{I I I} = 2 C V σ_{p h} Δ z

z_{0} = \frac{δ_{c}}{2} \ln [\frac{e^{L / δ_{c}} + r}{e^{- L / δ_{c}} + r}]

i_{p h}^{I} = \frac{4 C V σ_{p h}}{\sinh (L / δ_{c})} \sinh h (\frac{Δ z}{δ_{c}}) \sinh h (\frac{L - Z_{m}}{δ_{c}}) \sinh h (\frac{Z_{D}}{δ_{c}})

i_{p h}^{I I} = \frac{4 C V σ_{p h}}{\sinh (L / δ_{c})} \sinh h (\frac{Δ z}{δ_{c}}) \sinh h (\frac{Z_{m}}{δ_{c}}) \sinh h (\frac{Z_{D}}{δ_{c}})

i_{p h}^{I I I} = 4 C V σ_{p h} Δ z