Abstract

The intensity distribution of light scattered by a capillary tube filled with a liquid is studied using geometrical optics or ray tracing. Several intensity step points are found in the scattering pattern due to contributions from different geometrical rays. The scattering angles of these intensity step points vary with the capillary parameters, i.e., with the inner and outer radii of the capillary wall and the refractive indices of the liquid and the wall material. The relations between the scattering angles of the step points and the capillary parameters are analyzed using the reflection law and Snell’s law. A method is developed to determine the capillary parameters from measurements of the scattering angles of the step points. An experiment is designed to provide measured data from which the capillary parameters can be obtained by the proposed method. It is shown that this method provides capillary parameters of high precision.

© 2012 Optical Society of America

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References

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    [CrossRef]
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2012 (1)

Z. Hou, X. Zhao, and J. Xiao, “A simple double-source model for interference of capillaries,” Eur. J. Phys. 33, 199–206 (2012).
[CrossRef]

2011 (1)

2010 (1)

2008 (3)

2006 (2)

2004 (1)

2003 (1)

H. E. Ghandoor, E. Hegazi, I. Nasser, and G. M. Behery, “Measuring the refractive index of crude oil using a capillary tube interferometer,” Opt. Laser Technol. 35, 361–367(2003).
[CrossRef]

2002 (1)

1998 (2)

1997 (1)

1996 (1)

H. J. Tarigan, P. Neill, C. K. Kenmore, and D. J. Bornhop, “Capillary-scale refractive index detection by interferometric backscatter,” Anal. Chem. 68, 1762–1770 (1996).
[CrossRef]

1995 (3)

1980 (1)

Adler, C. L.

Aguilar, M. R.

Al-Rizzo, H. M.

Ameri, H.

Behery, G. M.

H. E. Ghandoor, E. Hegazi, I. Nasser, and G. M. Behery, “Measuring the refractive index of crude oil using a capillary tube interferometer,” Opt. Laser Technol. 35, 361–367(2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 132–133.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 40–49.

Bornhop, D. J.

H. J. Tarigan, P. Neill, C. K. Kenmore, and D. J. Bornhop, “Capillary-scale refractive index detection by interferometric backscatter,” Anal. Chem. 68, 1762–1770 (1996).
[CrossRef]

Bourlier, G.

Brancaccio, A.

Calixto, S.

Caorsi, S.

Dana, R. F.

Déchamps, N.

Dong, J.

Erni, D.

Ghandoor, H. E.

H. E. Ghandoor, E. Hegazi, I. Nasser, and G. M. Behery, “Measuring the refractive index of crude oil using a capillary tube interferometer,” Opt. Laser Technol. 35, 361–367(2003).
[CrossRef]

Hafner, C.

Han, X.

Hegazi, E.

H. E. Ghandoor, E. Hegazi, I. Nasser, and G. M. Behery, “Measuring the refractive index of crude oil using a capillary tube interferometer,” Opt. Laser Technol. 35, 361–367(2003).
[CrossRef]

Hernandez, D. M.

Hou, Z.

Z. Hou, X. Zhao, and J. Xiao, “A simple double-source model for interference of capillaries,” Eur. J. Phys. 33, 199–206 (2012).
[CrossRef]

Jiang, H.

Kaiser, T.

Kenmore, C. K.

H. J. Tarigan, P. Neill, C. K. Kenmore, and D. J. Bornhop, “Capillary-scale refractive index detection by interferometric backscatter,” Anal. Chem. 68, 1762–1770 (1996).
[CrossRef]

Kubické, G.

Kvien, K.

Lange, S.

Leone, G.

Li, R.

Li, W.

Lock, J. A.

Massa, A.

Member, S.

Minkovich, V. P.

Moreno, E.

Nasser, I.

H. E. Ghandoor, E. Hegazi, I. Nasser, and G. M. Behery, “Measuring the refractive index of crude oil using a capillary tube interferometer,” Opt. Laser Technol. 35, 361–367(2003).
[CrossRef]

Neill, P.

H. J. Tarigan, P. Neill, C. K. Kenmore, and D. J. Bornhop, “Capillary-scale refractive index detection by interferometric backscatter,” Anal. Chem. 68, 1762–1770 (1996).
[CrossRef]

Pastorino, M.

Pierri, R.

Qi, S.

Ren, K. F.

Roll, G.

Schweiger, G.

Sesay, M.

Shen, J.

Takano, Y.

Tanaka, M.

Tarigan, H. J.

H. J. Tarigan, P. Neill, C. K. Kenmore, and D. J. Bornhop, “Capillary-scale refractive index detection by interferometric backscatter,” Anal. Chem. 68, 1762–1770 (1996).
[CrossRef]

Tian, J.

Tranquilla, J. M.

Vahldieck, R.

Wang, H.

Wang, X.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 40–49.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 132–133.

Xiao, J.

Z. Hou, X. Zhao, and J. Xiao, “A simple double-source model for interference of capillaries,” Eur. J. Phys. 33, 199–206 (2012).
[CrossRef]

Xu, T.

Yang, A.

Yang, X.

Yokota, M.

Yuan, G.

Zhang, C.

Zhang, G.

Zhang, J.

Zhang, L.

Zhao, X.

Z. Hou, X. Zhao, and J. Xiao, “A simple double-source model for interference of capillaries,” Eur. J. Phys. 33, 199–206 (2012).
[CrossRef]

Anal. Chem. (1)

H. J. Tarigan, P. Neill, C. K. Kenmore, and D. J. Bornhop, “Capillary-scale refractive index detection by interferometric backscatter,” Anal. Chem. 68, 1762–1770 (1996).
[CrossRef]

Appl. Opt. (8)

Eur. J. Phys. (1)

Z. Hou, X. Zhao, and J. Xiao, “A simple double-source model for interference of capillaries,” Eur. J. Phys. 33, 199–206 (2012).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (7)

J. A. Lock and C. L. Adler, “Debye-series analysis of the first-order rainbow produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A 14, 1316–1328 (1997).
[CrossRef]

G. Roll, T. Kaiser, S. Lange, and G. Schweiger, “Ray interpretation of multipole fields in spherical dielectric cavities,” J. Opt. Soc. Am. A 15, 2879–2891 (1998).
[CrossRef]

G. Bourlier, G. Kubické, and N. Déchamps, “Fast method to compute scattering by a buried object under a randomly rough surface: PILE combined with FB-SA,” J. Opt. Soc. Am. A 25, 891–902 (2008).
[CrossRef]

M. Yokota and M. Sesay, “Two-dimensional scattering of a plane wave from a periodic array of dielectric cylinders with arbitrary shape,” J. Opt. Soc. Am. A 25, 1691–1696(2008).
[CrossRef]

E. Moreno, D. Erni, C. Hafner, and R. Vahldieck, “Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures,” J. Opt. Soc. Am. A 19, 101–111 (2002).
[CrossRef]

A. Brancaccio, G. Leone, and R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
[CrossRef]

S. Caorsi, A. Massa, and M. Pastorino, “Bistatic scattering-width computation for weakly nonlinear dielectric cylinders of arbitrary inhomogeneous cross-section shapes under transverse-magnetic wave illumination,” J. Opt. Soc. Am. A 12, 2482–2490 (1995).
[CrossRef]

Opt. Laser Technol. (1)

H. E. Ghandoor, E. Hegazi, I. Nasser, and G. M. Behery, “Measuring the refractive index of crude oil using a capillary tube interferometer,” Opt. Laser Technol. 35, 361–367(2003).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 132–133.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 40–49.

Wikipedia, “Abbe refractometer,” http://en.wikipedia.org/wiki/Abbe_refractometer .

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Figures (14)

Fig. 1.
Fig. 1.

Ray paths of seven different rays.

Fig. 2.
Fig. 2.

Intensity distributions of different angular parts of the scattered light for tube 1 and tube 2. (a)–(c) correspond to ray numbers 5–7, respectively, for tube 1, and (d)–(g) correspond to ray numbers 4–7, respectively, for tube 2. The arrow represents the direction of the incident light.

Fig. 3.
Fig. 3.

Experimental setup to observe the intensity distribution of the scattered light. The diameter of the observing screen is about 15 cm.

Fig. 4.
Fig. 4.

Experimental intensity patterns for (a) tube 1 and (b) tube 2. Simulated intensity patterns for (c) tube 1 and (d) tube 2.

Fig. 5.
Fig. 5.

Plots of β5 versus θ for different values of n0, for r/R=0.5, and for refractive indices of the capillary material of (a) n0=1.10, (b) n0=1.51, and (c) n0=1.80.

Fig. 6.
Fig. 6.

Intensity distributions of the scattered light for different values of n0. The capillary parameters are n=1.4, r/R=0.5, and (a) n0=1.10, (b) n0=1.51, and (c) n0=1.80.

Fig. 7.
Fig. 7.

Relation between α5 and n0 for r/R=0.5.

Fig. 8.
Fig. 8.

Relation between α6 and r/R. The refractive indices of the capillary and liquid are n0=1.5 and n=1.3, respectively.

Fig. 9.
Fig. 9.

Intensity distributions of the scattered light for different values of r/R. The parameters are n0=1.5, n=1.3, and (a) r/R=0.15, (b) r/R=0.40, (c) r/R=0.5, and (d) r/R=0.70.

Fig. 10.
Fig. 10.

Intensity distributions of light scattered by capillary tubes filled with different liquids. The parameters are r/R=0.5, n0=1.5, and (a) n=1.40, (b) n=1.55, (c) n=1.80, and (d) n=2.50.

Fig. 11.
Fig. 11.

Plots of β4 versus θ for different values of n. The parameters are r/R=0.5, n0=1.5, and (a) n=1.4, (b) n=1.6, and (c) n=1.8.

Fig. 12.
Fig. 12.

Intensity distributions of scattered light for ray number 4. The parameters are r/R=0.5, n0=1.5, and (a) n=1.4, (b) n=1.6, and (c) n=1.8.

Fig. 13.
Fig. 13.

Relations between n and the scattering angles of points C (triangle), D (circle), and E (square). The refractive index and the radius ratio of the capillary tube are n0=1.5 and r/R=0.5, respectively.

Fig. 14.
Fig. 14.

Experimental setup to measure the scattering angles of intensity step points.

Tables (3)

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Table 1. Brief List of the Method

Tables Icon

Table 2. Measured Scattering Angles of Intensity Step Points for Capillaries with Different Liquids Insidea

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Table 3. Results Obtained by Using Our Method and Other Precise Methodsa

Equations (26)

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β1=2θπ[0θπ2],
β2=2θ2arcsinsinθn0+2arcsinRsinθrn0π[0θarcsinrn0R],
β3=2θ2arcsinsinθn0+2arcsinRsinθrn04arcsinRsinθrn+π[0θarcsinrn0R],
β4=2θ4arcsinsinθn0+4arcsinRsinθrn04arcsinRsinθrn+π[0θarcsinrn0R],
β5=2θ4arcsinsinθn0+π[arcsinrn0Rθπ2],
β6=2[θarcsinsinθn0][arcsinrn0Rθπ2],
β7=2[θarcsinsinθn0+arcsinRsinθrn0arcsinRsinθrn][0θarcsinrn0R].
I(βk)=S(θ,βk)I0(θ),
S(θ,β2)=T(θ)R(θ2,1)T(θ2,2)(cosθ|dβ2/dθ|)1/2,
R(γ)=[sin(γγ)sin(γ+γ)]2,
T(γ)=1R(γ),
I(β)=i=17S(θi,β)I0(θi),
θ5m=arcsin(4n023)1/2.
arcsinrn0R<θ5m<π2.
1<n0<(41+3r2/R2)1/2.
α5=2arcsin(4n023)1/24arcsin(4n023n02)1/2+π.
α6=2arcsinrn0R2arcsinrR.
1r/Rr/(Rn0)+1/n0<n,
dβ4dθ<0whenθhas its minimum value,
dβ4dθ>0whenθhas its maximum value.
dβ4dθ=forθ=arcsin(rnR).
dβ4dθ=2(12n0+2Rrn02Rrn)<0forθ=0,
dβ4dθ=+0forθ=arcsin(rn0R).
n0<n<1r/(2R)r/(Rn0)+1/n0.
1r/Rr/(Rn0)+1/n0<n<n0(point C)
n0<n<2r/Rr/(Rn0)+1/n0(point E).

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