Abstract

A method of measuring the refractive indices of minute samples by analyzing capillary interferometry is introduced. With the interference theory of light, the intensity distribution of an interference fringe pattern formed by a cylindrical tube of a capillary is obtained, and the influence of some parameters on the fringes are discussed. The measurement accuracy and its relative problems are analyzed.

© 2004 Optical Society of America

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References

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  1. D. J. Bornhop, M. P. Houlne, C. K. Kenmore, J. Hankins, “Capillary interferometry detects minute samples,” Laser Focus World 32, 83–90 (1996).
  2. D. J. Bornhop, N. J. Dovichi, “Simple nanoliter refractive index detector,” Anal. Chem. 58, 504–505 (1986).
    [CrossRef]
  3. A. E. Bruno, B. Krattiger, F. Maystre, H. M. Widmer, “On-column laser-based refractive index detector for capillary electrophoresis,” Anal. Chem. 63, 2689–2697 (1991).
    [CrossRef]
  4. Y.-Z. Deng, B.-C. Li, “On-column refractive-index detection based on retroreflected beam interference for capillary electrophoresis,” Appl. Opt. 37, 998–1005 (1998).
    [CrossRef]
  5. D. J. Bornhop, “Microvolume index of refraction determinations by interferometric backscatter,” Appl. Opt. 34, 3234–3239 (1995).
    [CrossRef] [PubMed]
  6. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1999).
  7. B. C. Li, Y. Z. Deng, J. K. Cheng, “Sensitive photothermal interferometric detection method for characterization of transparent plate samples,” Rev. Sci. Instrum. 67, 3649–3657 (1996).
    [CrossRef]

1998 (1)

1996 (2)

D. J. Bornhop, M. P. Houlne, C. K. Kenmore, J. Hankins, “Capillary interferometry detects minute samples,” Laser Focus World 32, 83–90 (1996).

B. C. Li, Y. Z. Deng, J. K. Cheng, “Sensitive photothermal interferometric detection method for characterization of transparent plate samples,” Rev. Sci. Instrum. 67, 3649–3657 (1996).
[CrossRef]

1995 (1)

1991 (1)

A. E. Bruno, B. Krattiger, F. Maystre, H. M. Widmer, “On-column laser-based refractive index detector for capillary electrophoresis,” Anal. Chem. 63, 2689–2697 (1991).
[CrossRef]

1986 (1)

D. J. Bornhop, N. J. Dovichi, “Simple nanoliter refractive index detector,” Anal. Chem. 58, 504–505 (1986).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1999).

Bornhop, D. J.

D. J. Bornhop, M. P. Houlne, C. K. Kenmore, J. Hankins, “Capillary interferometry detects minute samples,” Laser Focus World 32, 83–90 (1996).

D. J. Bornhop, “Microvolume index of refraction determinations by interferometric backscatter,” Appl. Opt. 34, 3234–3239 (1995).
[CrossRef] [PubMed]

D. J. Bornhop, N. J. Dovichi, “Simple nanoliter refractive index detector,” Anal. Chem. 58, 504–505 (1986).
[CrossRef]

Bruno, A. E.

A. E. Bruno, B. Krattiger, F. Maystre, H. M. Widmer, “On-column laser-based refractive index detector for capillary electrophoresis,” Anal. Chem. 63, 2689–2697 (1991).
[CrossRef]

Cheng, J. K.

B. C. Li, Y. Z. Deng, J. K. Cheng, “Sensitive photothermal interferometric detection method for characterization of transparent plate samples,” Rev. Sci. Instrum. 67, 3649–3657 (1996).
[CrossRef]

Deng, Y. Z.

B. C. Li, Y. Z. Deng, J. K. Cheng, “Sensitive photothermal interferometric detection method for characterization of transparent plate samples,” Rev. Sci. Instrum. 67, 3649–3657 (1996).
[CrossRef]

Deng, Y.-Z.

Dovichi, N. J.

D. J. Bornhop, N. J. Dovichi, “Simple nanoliter refractive index detector,” Anal. Chem. 58, 504–505 (1986).
[CrossRef]

Hankins, J.

D. J. Bornhop, M. P. Houlne, C. K. Kenmore, J. Hankins, “Capillary interferometry detects minute samples,” Laser Focus World 32, 83–90 (1996).

Houlne, M. P.

D. J. Bornhop, M. P. Houlne, C. K. Kenmore, J. Hankins, “Capillary interferometry detects minute samples,” Laser Focus World 32, 83–90 (1996).

Kenmore, C. K.

D. J. Bornhop, M. P. Houlne, C. K. Kenmore, J. Hankins, “Capillary interferometry detects minute samples,” Laser Focus World 32, 83–90 (1996).

Krattiger, B.

A. E. Bruno, B. Krattiger, F. Maystre, H. M. Widmer, “On-column laser-based refractive index detector for capillary electrophoresis,” Anal. Chem. 63, 2689–2697 (1991).
[CrossRef]

Li, B. C.

B. C. Li, Y. Z. Deng, J. K. Cheng, “Sensitive photothermal interferometric detection method for characterization of transparent plate samples,” Rev. Sci. Instrum. 67, 3649–3657 (1996).
[CrossRef]

Li, B.-C.

Maystre, F.

A. E. Bruno, B. Krattiger, F. Maystre, H. M. Widmer, “On-column laser-based refractive index detector for capillary electrophoresis,” Anal. Chem. 63, 2689–2697 (1991).
[CrossRef]

Widmer, H. M.

A. E. Bruno, B. Krattiger, F. Maystre, H. M. Widmer, “On-column laser-based refractive index detector for capillary electrophoresis,” Anal. Chem. 63, 2689–2697 (1991).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1999).

Anal. Chem. (2)

D. J. Bornhop, N. J. Dovichi, “Simple nanoliter refractive index detector,” Anal. Chem. 58, 504–505 (1986).
[CrossRef]

A. E. Bruno, B. Krattiger, F. Maystre, H. M. Widmer, “On-column laser-based refractive index detector for capillary electrophoresis,” Anal. Chem. 63, 2689–2697 (1991).
[CrossRef]

Appl. Opt. (2)

Laser Focus World (1)

D. J. Bornhop, M. P. Houlne, C. K. Kenmore, J. Hankins, “Capillary interferometry detects minute samples,” Laser Focus World 32, 83–90 (1996).

Rev. Sci. Instrum. (1)

B. C. Li, Y. Z. Deng, J. K. Cheng, “Sensitive photothermal interferometric detection method for characterization of transparent plate samples,” Rev. Sci. Instrum. 67, 3649–3657 (1996).
[CrossRef]

Other (1)

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1999).

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Figures (14)

Fig. 1
Fig. 1

Schematic of experimental setup for measuring the RI.

Fig. 2
Fig. 2

Cross section of the capillary and coherent beams I, II, III, and IV.

Fig. 3
Fig. 3

Optical path of beam IV.

Fig. 4
Fig. 4

Optical path of beam I.

Fig. 5
Fig. 5

Optical path of beam II.

Fig. 6
Fig. 6

Optical path of beam III.

Fig. 7
Fig. 7

Calculated intensity profile of the interference fringe pattern on the screen.

Fig. 8
Fig. 8

Calculated amplitude distribution of the interference field on the screen.

Fig. 9
Fig. 9

Calculated intensity profiles I 14 and I 23 of the interference fringe pattern on the screen.

Fig. 10
Fig. 10

Calculated amplitude distributions A 14 and A 23 of the interference field on the screen.

Fig. 11
Fig. 11

Calculated intensity profiles I, I 14, and I 23 of the interference fringe pattern on the screen.

Fig. 12
Fig. 12

Calculated amplitude distributions A, A 14, and A 23 of the interference field on the screen.

Fig. 13
Fig. 13

I, I 14, and I 23 as functions of y in two calculation intervals.

Fig. 14
Fig. 14

Photographs of interference fringes (a) d 0 = 0.4 m, (b) d 0 = 1.5 m, (c) d 0 = 2.5 m.

Equations (35)

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I14=A142=A12+A42+2A1A4 cos Δφ14,I23=A232=A22+A32+2A2A3 cos Δφ23,I=A2=A142+A232+2A14A23 cos Δφ,
1=arcsinR2-R1R1sin i2, α1=i2-i1, i3=i2+1, α2=α1+i4-i3, i4=arcsinn1n2sin i3, i5=i4, i6=i3, α3=α2+i5-i6, i7=i8, α4=α3+2R1R2 i6, i9=i6, i10=i4, α5=α4+i10-i9, i11=i4, i12=i3, α6=α5+i11-i12, i13=i2, i14=i1, α7=α6+i13-i14, 2=1, HOP=α6+i13, 3=π-HOP, 4=π2-α6+i132.
y4=R2 sinα6+i13+d0+2R2sin2α6+i132tan α7.
Δ4=d0+h4+n1R2-R1cos i2+2n2R1 cos i4+2n1R2-R1cos i6+2n2R1 cos i10+n1R2-R1cos i12+d0+2R2sin2α6+i13/2cos α7.
y1=R2 sin j+d0+½R2sin2 jtan2j.
y1=R2j3+2d0+R2j
R2j3+2d0+R2j-z=0.
a3j3+a1j+a0=0.
j=-q+p3+q21/21/3-q+p3+q21/21/3.
Δ1=d0+h11+1cos2j.
k2=arcsinsin k1n1,k3=arcsinsin k1n1+R2-R1n1R1sin k1,k4=k3, k5=k2, k6=k1, 1=arcsinR2-R1R1sin k2,2=k1-k2+k3,3=2k1+R2-R1n1R1sin k1.
y2=R2 sin k1+2R2-R1tanarcsinsin k1n1+R2-R1n1R1sin k1cosk1+R2-R1n1R1sin k1+d0+12 R2sin2 k1+2R2-R1×tanarcsinsin k1n1+R2-R1n1R1sin k1×sink1+R2-R1n1R1sin k1×tan2k1+R2-R1n1R1sin k1.
y2=R21+R2-R1n1R11+3 R2-R1n1R1×1+R2-R1n1R1k13+1+2 R2-R1n1R1+2d0R21+R2-R1n1R1k1.
a3k13+a1k1+a0=0,
a0=-zR2, a1=1+2 R2-R1n1R1+2dR2×1+R2-R1n1R1,a2=0, a3=1+R2-R1n1R11+3 R2-R1n1R1×1+R2-R1n1R1.
k1=-q+p3+q21/21/3-q+p3+q21/21/3,
Δ2=d0+h2+2n1R2-R1/cos k2+d0+½R2sin2 k1+2R2-R1tan k3 sink1-k2+k3/cos 3.
θ2=arcsinsin θ1n1, 1=R2-R1n1R1sin θ1, β1=θ2-θ1,θ3=θ2+R2-R1n1R1sin θ1,θ4=arcsinn1n2sin θ3, β2=β1+θ4-θ3,θ5=θ6=θ7=θ4, β3=β2+θ6-θ5,θ8=θ3, β4=β2+θ7-θ8,θ9=θ2, θ10=θ1, β5=β4+θ9-θ10,2=1, 3=β4+θ9, 4=½π-3, 5=π-3.
y3=R2 sinβ4+θ9+d0+2R2 sin2β4+θ92tan β5
y3=R22n1-1+4R2n2R1-2R2n1R1+2d01n1R2-1R2+2n2R1-1n1R1θ1+R21n1-1+2R2n2R1-R2n1R12n1-1+4R2n2R1-2R2n1R12θ13.
a3θ13+a1θ1+a0=0,
a0=-zR2, a1=2n1-1+4R2n2R1-2R2n1R1+2d01n1R2-1R2+2n2R1-1n1R1, a3=1n1-1+2R2n2R1-R2n1R12n1-1+4R2n2R1-2R2n1R12.
θ1=-2u cosν/3,
Δ3=d0+h3+2n1R2-R1cos θ2+4n2R1 cos θ4+d0+2R2 sin2β4+θ9/2cos β5.
r=n2-n1n2+n1,
t=2n1n2+n1,
A1=n1-1n1+1 A0=0.2A0,A2=4n1n2-n11+n12n1+n2 A0=-0.0566A0,A3=8n12n2n1-n21+n12n1+n23 A0=0.0282A0,A4=64n13n221-n11+n13n1+n24 A0=0.19A0.
I14=0.0765A021+cos Δφ14,
A14=0.4A0 cosΔφ142.
tan ϕ14=A1 sin-2πλ Δ1+A4 sin-2πλ Δ4A1 cos2πλ Δ1+A4 sin2πλ Δ4.
I23=A020.004-0.0016 cos Δφ23,
A23=A00.004-0.0016 cos Δφ231/2.
tan ϕ23=A2 sin-2π/λΔ2+A3 sin-2π/λΔ3A2 cos2π/λΔ2+A3 sin2π/λΔ3.
I=A142+A232+2A14A23 cosϕ14-ϕ23,
A=I.

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