Abstract

To realize a high-precision refractive-index measurement by the rainbow method by use of multifringes, the characteristics of several orders of supernumerary bow fringes are made clear by a simulation based on the Huygens–Fresnel principle. The fringe intervals computed are precisely coincident with those obtained in experiments. As a result a suitable combination of the diameter of the cylinder containing the sample and the beam size of the laser has been determined. By use of the characteristic curve of the deviation angle versus the refractive index for each fringe and by the statistical treatment, measurement with high precision was demonstrated.

© 1999 Optical Society of America

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References

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  1. H. Hattori, H. Yamanaka, H. Kurniawan, S. Yokoi, K. Kagawa, “Using minimum deviation of a secondary rainbow and its application to water analysis in a high-precision, refractive-index comparator for liquids,” Appl. Opt. 36, 5552–5556 (1997).
    [CrossRef] [PubMed]
  2. H. Hattori, H. Kakui, H. Kurniawan, K. Kagawa, “Liquid refractometer by the rainbow method,” Appl. Opt. 37, 4123–4129 (1998).
    [CrossRef]
  3. D. K. Lynch, P. Schwartz, “Rainbows and fogbows,” Appl. Opt. 30, 3415–3420 (1991), Chap. 4.
  4. D. K. Lynch, W. Livingstone, Color and Light in Nature (Cambridge U. Press, London, 1995).
  5. R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computation and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
    [CrossRef] [PubMed]
  6. J. P. A. J. van Beeck, M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
    [CrossRef] [PubMed]
  7. G. B. Airy, “On the intensity of light in the neighborhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 379–402 (1838).

Airy, G. B.

G. B. Airy, “On the intensity of light in the neighborhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 379–402 (1838).

Hattori, H.

Kagawa, K.

Kakui, H.

Kurniawan, H.

Livingstone, W.

D. K. Lynch, W. Livingstone, Color and Light in Nature (Cambridge U. Press, London, 1995).

Lynch, D. K.

D. K. Lynch, P. Schwartz, “Rainbows and fogbows,” Appl. Opt. 30, 3415–3420 (1991), Chap. 4.

D. K. Lynch, W. Livingstone, Color and Light in Nature (Cambridge U. Press, London, 1995).

Riethmuller, M. L.

Schwartz, P.

van Beeck, J. P. A. J.

van de Hulst, H. C.

Wang, R. T.

Yamanaka, H.

Yokoi, S.

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Figures (13)

Fig. 1
Fig. 1

Outline of the experimental setup for the refractive-index measurement of liquids by the supernumerary bow fringes.

Fig. 2
Fig. 2

Outline of the simulation procedure. Secondary waves from Wi that have equal intervals ds are synthesized at observation point F.

Fig. 3
Fig. 3

Geometric arrangement of variables that appear in the formulas for this simulation.

Fig. 4
Fig. 4

Comparison of fringes obtained by (a) experiment, (b) the corresponding simulation. P = 2, λ G , L = 1.470 m for the experiment and 3.000 m for the simulation. P10 indicates the peak of the 10th-order fringe. B1 indicates the first bottom. In (a), the channel interval of the horizontal axis is 0.025 mm. In (b), the interval of the observation points is 0.1 mm, and BW = 1.0 shows that the beam radius is 1.0 mm.

Fig. 5
Fig. 5

Correlation between the fringe intervals obtained by the simulation and the experiment shown in Fig. 4.

Fig. 6
Fig. 6

Profiles of the fringes yielded by different beam widths (BWs). Each BW in (a)–(d) shows the beam radial width in millimeters. The interval of observation points is 0.075 mm, and the other conditions are the same as those in Fig. 4.

Fig. 7
Fig. 7

Relationship among the beam radius, the positions of M1 and M2 of the middle points of the fringes, and the interval M2 from M6.

Fig. 8
Fig. 8

Relationship between beam width BW and the logarithmic damping ratio of the fringe peak intensities. N shows the order of the fringe.

Fig. 9
Fig. 9

Relationship among the minimum deviation angle for different P values, that is, the order of reflection, the position of the middle point M2, and the interval of M2 from M6.

Fig. 10
Fig. 10

Relationship between the cell radius and the interval of M2 from M6 of the middle points. Curve Rec shows (reciprocals of the M2M6 values) × 40.

Fig. 11
Fig. 11

Relationship between the refractive index of the samples and the sensitivity of the fringe displacement for various wall thicknesses t (in millimeters) of the cell and the reflection order P. I, P = 2, t = 2; II, P = 8, t = 2; III, P = 4, t = 6.4; IV, P = 2, t = 7.15; V, P = 1, t = 7.4.

Fig. 12
Fig. 12

Relationship between the refractive index of samples and the displacement of various middle points M i . The curves are continuous in front of and behind the index value of the cell: n = 1.4602.

Fig. 13
Fig. 13

Characteristic curve for obtaining the refractive index from a measured deviation angle of the middle point M1 and its approximation formula.

Tables (2)

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Table 1 Shifts of the Middle Points Mi by Minute Changes Δn of the Refractive Index na

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Table 2 Errors Caused by a Δt = 5 µm Increase of the Cell Wall Thicknessa

Equations (21)

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αm=Am+P+1π+2Am-Am-Dm.
x0=-R1 cos αm+L cos θm,
y0=R1 sin αm-L sin θm,
Lx=x0+f cosπ2-θm+R1 cosαm+ai,
Ly=y0-f sinπ2-θm-R1 sinαm+ai.
eiF¯=Lx2+Ly2.
Ai=cos-1Lx cosπ-αm-ai+Ly sinπ-αm-aieiF¯.
Ap=αm-ai-P+1π-2Bi-Ci-Di.
Bi=sin-1sin Ain1,
Ci=Bi+cos-1-xQ2R2-Ai,
Di=sin-1n1 sin Cin2.
SiWi¯=R11-cos Ap,
WiQ2¯=R12+R22+2R1R2 cosCi-Bi,
Q2Q3¯=2R2 cos Di.
SiF¯=SiWi¯+2n1P+1WiQ2¯+n2P+1Q2Q3¯+eiF¯.
Nw=SiF¯/λ.
AWi=A0 sin 2πNw+TA.
0mSi¯=R1sin Ap-sin Am.
A0=exp-0mSi¯Bw2.
XMi=XL+XU-XLIMi-ILIU-IL.
VM=gradient of intensity curve on Mi×displacement of Mi.

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