Abstract

We demonstrate a new technique for analyzing surface tension of liquids. This is done upon examining the interference signals reflected from a remnant drop pending at the cleaved end of a single mode optical fiber. The resulting interference patterns are fitted to a multimirror Fabry-Perot model yielding information of the drop size. We show that the wetting process of the fiber plays an important role in drop formation; in particular, the drop size can be correlated to the surface tension of the liquid sample. The proposed configuration may render useful for liquids analysis using small sample volume.

© 2014 Optical Society of America

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2013 (2)

X. Xu, A. Jagota, S. Peng, D. Luo, M. Wu, C. Y. Hui, “Gravity and surface tension effects on the shape change of soft materials,” Langmuir 29(27), 8665–8674 (2013).
[CrossRef] [PubMed]

R. Wang, T. Kim, M. Mir, G. Popescu, “Nanoscale fluctuations and surface tension measurements in droplets using phase-resolved low-coherence interferometry,” Appl. Opt. 52(1), A177–A181 (2013).
[CrossRef] [PubMed]

2012 (1)

C. Pigot, A. Hibara, “Surface tension measurement at the microscale by passive resonance of capillary waves,” Anal. Chem. 84(5), 2557–2561 (2012).
[CrossRef] [PubMed]

2010 (2)

Y. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. U.S.A. 107(15), 6731–6736 (2010).
[CrossRef] [PubMed]

A. F. Stalder, T. Melchior, M. Müller, D. Sage, T. Blu, M. Unser, “Low-bond axisymmetric drop shape analysis for surface tension and contact angle measurements of sessile drops,” Colloids Surf. A Physicochem. Eng. Asp. 364(1-3), 72–81 (2010).
[CrossRef]

2009 (3)

A. Zhou, J. Yang, B. Liu, L. Yuan, “A fiber-optic liquid sensor for simultaneously measuring refractive index, surface tension, contact angle and viscosity,” Proc. SPIE 7503, 75033B (2009).
[CrossRef]

A. Marmur, “Solid-surface characterization by wetting,” Annu. Rev. Mater. Res. 39(1), 473–489 (2009).
[CrossRef]

A. Zhou, Z. Liu, L. Yuan, “Fiber-optic dipping liquid analyzer: theoretical and experimental study of light transmission,” Appl. Opt. 48(36), 6928–6933 (2009).
[CrossRef] [PubMed]

2005 (1)

Q. Song, G. X. Zhang, Z. R. Qiu, “Review of drop analysis technology for liquid property study,” Opto-Electron. Rev. 13(1), 1–8 (2005).

2003 (1)

K. Kim, S. Chang, K. Oh, “Refractive microlens on fiber using UV-curable fluorinated acrylate polymer by surface-tension,” IEEE Photon. Technol. Lett. 15(8), 1100–1102 (2003).
[CrossRef]

2001 (1)

1997 (1)

1996 (1)

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

1986 (2)

V. F. Weisskopf, “Search for simplicity: mountains, water waves, and leaky ceilings,” Am. J. Phys. 54(2), 110 (1986).
[CrossRef]

M. Alonso, “Water drops and wet ceilings,” Am. J. Phys. 54(8), 679 (1986).
[CrossRef]

1985 (1)

1805 (1)

T. Young, “An essay on the cohesion of fluids,” Phil. Trans. R. Soc. 95(0), 65–87 (1805).
[CrossRef]

Alonso, M.

M. Alonso, “Water drops and wet ceilings,” Am. J. Phys. 54(8), 679 (1986).
[CrossRef]

Badizadegan, K.

Y. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. U.S.A. 107(15), 6731–6736 (2010).
[CrossRef] [PubMed]

Baker, M.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

Barratt, D.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

Best, C. A.

Y. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. U.S.A. 107(15), 6731–6736 (2010).
[CrossRef] [PubMed]

Blu, T.

A. F. Stalder, T. Melchior, M. Müller, D. Sage, T. Blu, M. Unser, “Low-bond axisymmetric drop shape analysis for surface tension and contact angle measurements of sessile drops,” Colloids Surf. A Physicochem. Eng. Asp. 364(1-3), 72–81 (2010).
[CrossRef]

Bree, M.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

Chang, S.

K. Kim, S. Chang, K. Oh, “Refractive microlens on fiber using UV-curable fluorinated acrylate polymer by surface-tension,” IEEE Photon. Technol. Lett. 15(8), 1100–1102 (2003).
[CrossRef]

Corden, R.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

Dasari, R. R.

Y. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. U.S.A. 107(15), 6731–6736 (2010).
[CrossRef] [PubMed]

Davern, P.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

Davis, M.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

Dogariu, A.

Feld, M. S.

Y. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. U.S.A. 107(15), 6731–6736 (2010).
[CrossRef] [PubMed]

Flammer, I.

Goossens, P.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

Hanrahan, J.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

Harkin, J.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

Henle, M. L.

Y. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. U.S.A. 107(15), 6731–6736 (2010).
[CrossRef] [PubMed]

Hibara, A.

C. Pigot, A. Hibara, “Surface tension measurement at the microscale by passive resonance of capillary waves,” Anal. Chem. 84(5), 2557–2561 (2012).
[CrossRef] [PubMed]

Hui, C. Y.

X. Xu, A. Jagota, S. Peng, D. Luo, M. Wu, C. Y. Hui, “Gravity and surface tension effects on the shape change of soft materials,” Langmuir 29(27), 8665–8674 (2013).
[CrossRef] [PubMed]

Jagota, A.

X. Xu, A. Jagota, S. Peng, D. Luo, M. Wu, C. Y. Hui, “Gravity and surface tension effects on the shape change of soft materials,” Langmuir 29(27), 8665–8674 (2013).
[CrossRef] [PubMed]

Kim, K.

K. Kim, S. Chang, K. Oh, “Refractive microlens on fiber using UV-curable fluorinated acrylate polymer by surface-tension,” IEEE Photon. Technol. Lett. 15(8), 1100–1102 (2003).
[CrossRef]

Kim, T.

Kuriabova, T.

Y. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. U.S.A. 107(15), 6731–6736 (2010).
[CrossRef] [PubMed]

Lane, D.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

Lawlor, V.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

Levine, A. J.

Y. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. U.S.A. 107(15), 6731–6736 (2010).
[CrossRef] [PubMed]

Liu, B.

A. Zhou, J. Yang, B. Liu, L. Yuan, “A fiber-optic liquid sensor for simultaneously measuring refractive index, surface tension, contact angle and viscosity,” Proc. SPIE 7503, 75033B (2009).
[CrossRef]

Liu, Z.

Luo, D.

X. Xu, A. Jagota, S. Peng, D. Luo, M. Wu, C. Y. Hui, “Gravity and surface tension effects on the shape change of soft materials,” Langmuir 29(27), 8665–8674 (2013).
[CrossRef] [PubMed]

Marmur, A.

A. Marmur, “Solid-surface characterization by wetting,” Annu. Rev. Mater. Res. 39(1), 473–489 (2009).
[CrossRef]

McMillan, D. G. E.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

McMillan, N. D.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

Melchior, T.

A. F. Stalder, T. Melchior, M. Müller, D. Sage, T. Blu, M. Unser, “Low-bond axisymmetric drop shape analysis for surface tension and contact angle measurements of sessile drops,” Colloids Surf. A Physicochem. Eng. Asp. 364(1-3), 72–81 (2010).
[CrossRef]

Mir, M.

Muller, J. M.

Müller, M.

A. F. Stalder, T. Melchior, M. Müller, D. Sage, T. Blu, M. Unser, “Low-bond axisymmetric drop shape analysis for surface tension and contact angle measurements of sessile drops,” Colloids Surf. A Physicochem. Eng. Asp. 364(1-3), 72–81 (2010).
[CrossRef]

Oh, K.

K. Kim, S. Chang, K. Oh, “Refractive microlens on fiber using UV-curable fluorinated acrylate polymer by surface-tension,” IEEE Photon. Technol. Lett. 15(8), 1100–1102 (2003).
[CrossRef]

Park, Y.

Y. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. U.S.A. 107(15), 6731–6736 (2010).
[CrossRef] [PubMed]

Peng, S.

X. Xu, A. Jagota, S. Peng, D. Luo, M. Wu, C. Y. Hui, “Gravity and surface tension effects on the shape change of soft materials,” Langmuir 29(27), 8665–8674 (2013).
[CrossRef] [PubMed]

Pigot, C.

C. Pigot, A. Hibara, “Surface tension measurement at the microscale by passive resonance of capillary waves,” Anal. Chem. 84(5), 2557–2561 (2012).
[CrossRef] [PubMed]

Popescu, G.

R. Wang, T. Kim, M. Mir, G. Popescu, “Nanoscale fluctuations and surface tension measurements in droplets using phase-resolved low-coherence interferometry,” Appl. Opt. 52(1), A177–A181 (2013).
[CrossRef] [PubMed]

Y. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. U.S.A. 107(15), 6731–6736 (2010).
[CrossRef] [PubMed]

G. Popescu, A. Dogariu, “Dynamic light scattering in localized coherence volumes,” Opt. Lett. 26(8), 551–553 (2001).
[CrossRef] [PubMed]

Qiu, Z. R.

Q. Song, G. X. Zhang, Z. R. Qiu, “Review of drop analysis technology for liquid property study,” Opto-Electron. Rev. 13(1), 1–8 (2005).

Ricka, J.

Sage, D.

A. F. Stalder, T. Melchior, M. Müller, D. Sage, T. Blu, M. Unser, “Low-bond axisymmetric drop shape analysis for surface tension and contact angle measurements of sessile drops,” Colloids Surf. A Physicochem. Eng. Asp. 364(1-3), 72–81 (2010).
[CrossRef]

Smith, S.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

Song, Q.

Q. Song, G. X. Zhang, Z. R. Qiu, “Review of drop analysis technology for liquid property study,” Opto-Electron. Rev. 13(1), 1–8 (2005).

Stalder, A. F.

A. F. Stalder, T. Melchior, M. Müller, D. Sage, T. Blu, M. Unser, “Low-bond axisymmetric drop shape analysis for surface tension and contact angle measurements of sessile drops,” Colloids Surf. A Physicochem. Eng. Asp. 364(1-3), 72–81 (2010).
[CrossRef]

Thompson, K.

N. D. McMillan, P. Davern, V. Lawlor, M. Baker, K. Thompson, J. Hanrahan, M. Davis, J. Harkin, M. Bree, P. Goossens, S. Smith, D. Barratt, R. Corden, D. G. E. McMillan, D. Lane, “The instrumental engineering of a polymer fibre drop analyser for both quantitative and qualitative analysis with special reference to fingerprinting liquids,” Colloids Surf. A Physicochem. Eng. Asp. 114, 75–97 (1996).
[CrossRef]

Unser, M.

A. F. Stalder, T. Melchior, M. Müller, D. Sage, T. Blu, M. Unser, “Low-bond axisymmetric drop shape analysis for surface tension and contact angle measurements of sessile drops,” Colloids Surf. A Physicochem. Eng. Asp. 364(1-3), 72–81 (2010).
[CrossRef]

van de Stadt, H.

Wang, R.

Weisskopf, V. F.

V. F. Weisskopf, “Search for simplicity: mountains, water waves, and leaky ceilings,” Am. J. Phys. 54(2), 110 (1986).
[CrossRef]

Wu, M.

X. Xu, A. Jagota, S. Peng, D. Luo, M. Wu, C. Y. Hui, “Gravity and surface tension effects on the shape change of soft materials,” Langmuir 29(27), 8665–8674 (2013).
[CrossRef] [PubMed]

Xu, X.

X. Xu, A. Jagota, S. Peng, D. Luo, M. Wu, C. Y. Hui, “Gravity and surface tension effects on the shape change of soft materials,” Langmuir 29(27), 8665–8674 (2013).
[CrossRef] [PubMed]

Yang, J.

A. Zhou, J. Yang, B. Liu, L. Yuan, “A fiber-optic liquid sensor for simultaneously measuring refractive index, surface tension, contact angle and viscosity,” Proc. SPIE 7503, 75033B (2009).
[CrossRef]

Young, T.

T. Young, “An essay on the cohesion of fluids,” Phil. Trans. R. Soc. 95(0), 65–87 (1805).
[CrossRef]

Yuan, L.

A. Zhou, J. Yang, B. Liu, L. Yuan, “A fiber-optic liquid sensor for simultaneously measuring refractive index, surface tension, contact angle and viscosity,” Proc. SPIE 7503, 75033B (2009).
[CrossRef]

A. Zhou, Z. Liu, L. Yuan, “Fiber-optic dipping liquid analyzer: theoretical and experimental study of light transmission,” Appl. Opt. 48(36), 6928–6933 (2009).
[CrossRef] [PubMed]

Zhang, G. X.

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Supplementary Material (2)

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Figures (6)

Fig. 1
Fig. 1

(a) Contact angle of a sessile drop where Young-Laplace methods are frequently used; θ CA is the contact angle. (b) Geometry of the remnant-pendant drop at the fiber end, where θ CA’ is the contact angle, hd and R are respectively the height and the radius of curvature of the drop, and Df is the diameter of the optical fiber.

Fig. 2
Fig. 2

A set of pictures of the wetting process: the optical fiber is initially in close proximity to the liquid (a) and subsequently immersed into it and extracted at regular intervals (b-e). Once the fiber is out of the liquid sample, a remnant-pendant drop is formed at the fiber end face (f). The wetting process with a DMS liquid (Media 1) is different than that of glycerin (Media 2).

Fig. 3
Fig. 3

Amplitudes at mirrors i and i + 1 out of a stack of N mirrors.

Fig. 4
Fig. 4

Schematic representation of: (a) 2MFPI arrangement, (b) 3MFPI arrangement and (c) typical back-reflected (BR) interference signals obtained for each case.

Fig. 5
Fig. 5

Drop heights for different liquids obtained with 2M and 3M models; the fibers were cleaved before each dipping event.

Fig. 6
Fig. 6

Drop height for different liquids using the same fiber.

Tables (2)

Tables Icon

Table 1 Physical Properties of the Liquids

Tables Icon

Table 2 Drop Height for Two Mirrors (2M) and Three Mirrors (3M) Interferometric Configurations*

Equations (12)

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V * = γ η .
E i+1 + e i φ i = t i E i + + E i+1 r i e i φ i
E i = E i + r i + E i+1 t i e i φ i .
E i + = 1 t i E i+1 + e i φ i r i t i E i+1 e i φ i ,
E i = r i t i E i+1 + e i φ i + 1 t i E i+1 e i φ i .
( E i + E i )= 1 t i [ e i φ i r i e i φ i r i e i φ i e i φ i ]( E i+1 + E i+1 ).
( E 1 + E 1 )= 1 t 1 t 2 t N1 [ e i φ 1 r i e i φ 1 r i e i φ 1 e i φ 1 ]×[ e i φ 2 r i e i φ 2 r i e i φ 2 e i φ 2 ]× ×[ e i φ N1 r i e i φ N1 r i e i φ N1 e i φ N1 ]( E N + E N )
( E i + E i )= 1 t 1 t 2 t N1 [ A B C D ]( E N + E N ),
r= r i+1 + r i e 2i φ i r i r i+1 + e 2i φ i .
R 2M = r 1 2 + r 2 2 +2 r 1 r 2 cos[ 2 φ 1 ] 1+ r 1 r 2 ( r 1 r 2 +2cos[ 2 φ 1 ] ) .
R 3M = r 2 2 + r 3 2 + r 1 2 ( 1+ r 2 2 r 3 2 )+Q 1+ r 2 2 r 3 2 + r 1 2 ( r 2 2 + r 3 2 )+Q ,
Q=2 r 2 ( r 1 ( 1+ r 3 2 )cos[ 2 φ 1 ]+ r 1 r 2 r 3 cos[ 2 φ 1 2 φ 2 ]+( 1+ r 1 2 ) r 3 cos[ 2 φ 2 ] ) +2 r 1 r 3 cos[ 2( φ 1 + φ 2 ) ]

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