Hiroyuki Hattori, Hidenori Kakui, Hendrik Kurniawan, and Kiichiro Kagawa

Hiroyuki Hattori, Hidenori Kakui, Hendrik Kurniawan, and Kiichiro Kagawa

^{}H. Hattori, H. Kakui, and K. Kagawa are with the Department of Physics, Faculty of Education, Fukui University, 9-1 Bunkyo 3-chome, Fukui 910, Japan.

^{}H. Kurniawan is with the Applied Spectroscopy Laboratory, Graduate Program for Optoelectrotechniques and Laser Application, The University of Indonesia, 4 Salemba Raya, Jakarta 10430, Indonesia.

Hiroyuki Hattori, Hidenori Kakui, Hendrik Kurniawan, and Kiichiro Kagawa, "Liquid refractometry by the rainbow method," Appl. Opt. 37, 4123-4129 (1998)

A new method for measuring the refractive index of liquid, proposed
in a previous paper [Appl. Opt. 36, 5552–5556 (1997)], has been developed. The minimum deviation of a laser
beam deflected by a liquid-filled cylindrical cell was calculated by
use of geometric optics. These theoretical results were compared
with experimental results, with excellent agreement. As a result,
the unknown refractive index of a liquid could be obtained by use of a
computer calculation to give a best fit. The computer calculation
showed that the sensitivity of the refractometer increases with the
cell-wall thickness until total reflection takes place. A small
refractive-index difference can be detected within a precision of
1 × 10^{-6} by use of a metal-oxide semiconductor
linear image sensor. We show how to calibrate the refractometer
with pure water at 3.98 °C.

H. Kessler, Handbuch der Physik, H. Geiger, K. Schcel, eds. (Springer-Verlag, Berlin, 1927).

G. E. Fichter, Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, New York, 1967), Vol. 4, p. 363.

J. V. Hughes, “A new precision refractometer,” J. Sci. Instrum. 18, 234–237 (1941).
[CrossRef]

C. A. Faick, B. Fonoruff, “A precision apparatus for the rapid determination of indices of refraction and dispersion by immersion,” J. Res. Natl. Bur. Stand. USA 32, 67–75 (1944).
[CrossRef]

C. A. Faick, B. Fonoruff, “A precision apparatus for the rapid determination of indices of refraction and dispersion by immersion,” J. Res. Natl. Bur. Stand. USA 32, 67–75 (1944).
[CrossRef]

1941 (1)

J. V. Hughes, “A new precision refractometer,” J. Sci. Instrum. 18, 234–237 (1941).
[CrossRef]

E. C. Miller, F. W. Crawford, B. J. Simmons, “A differential refractometer for process control,” Anal. Chem. 24, 1087–1090 (1952).
[CrossRef]

Faick, C. A.

C. A. Faick, B. Fonoruff, “A precision apparatus for the rapid determination of indices of refraction and dispersion by immersion,” J. Res. Natl. Bur. Stand. USA 32, 67–75 (1944).
[CrossRef]

Fichter, G. E.

G. E. Fichter, Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, New York, 1967), Vol. 4, p. 363.

Fonoruff, B.

C. A. Faick, B. Fonoruff, “A precision apparatus for the rapid determination of indices of refraction and dispersion by immersion,” J. Res. Natl. Bur. Stand. USA 32, 67–75 (1944).
[CrossRef]

C. A. Faick, B. Fonoruff, “A precision apparatus for the rapid determination of indices of refraction and dispersion by immersion,” J. Res. Natl. Bur. Stand. USA 32, 67–75 (1944).
[CrossRef]

J. Sci. Instrum. (1)

J. V. Hughes, “A new precision refractometer,” J. Sci. Instrum. 18, 234–237 (1941).
[CrossRef]

Other (8)

S. Svanberg, Atomic and Molecular Spectroscopy, 2nd ed. (Springer-Verlag, New York, 1992).
[CrossRef]

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

H. Yoshinage, Handbook of Applied Spectroscopy (Asakura, Tokyo, Japan, 1973).

Chemical Society of Japan
, ed.,
Table for Chemistry, (Maruzen, Tokyo, 1966).

H. Kessler, Handbuch der Physik, H. Geiger, K. Schcel, eds. (Springer-Verlag, Berlin, 1927).

G. E. Fichter, Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, New York, 1967), Vol. 4, p. 363.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, New York, 1976).

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987).

Theoretical result of the relationship between the
refractive index of a liquid, n_{2}, and the
difference of minimum deviation angle, θ_{
m
}, in
degrees. The calculation was done for the second-order minimum
deviation on a wavelength of 543.5 nm.

Theoretical results of the relationship between the
refractive index of liquid and the difference of the minimum deviation
angle, θ_{
m
}, in degrees, for three thicknesses
of the cell: 2, 5, and 7 mm.

Examples of the interference fringe observed at the
minimum deviation. (a) Second-order minimum deviation, (b)
seventh-order minimum deviation. These were obtained with 543.5-nm
He–Ne laser light.

Relationship between the displacement length of the
interference fringe position and the temperature of pure water. UV
Laser, G Laser, and R Laser: He–Cd, He–Ne, and He–Ne,
respectively, at the wavelengths listed. Fringes of the
second-order minimum deviation and of the seventh-order minimum
deviation were used.

Minimum Deviation Angle θ (in degrees) for Various
Orders Pa

Wavelength

Deviation Angle

Order P

1

2

3

4

5

6

7

λ_{1}

θ_{
m
}

135.673

226.213

311.401

394.762

477.262

559.283

641.012

Δθ_{
m
}/Δn

0.170

0.310

0.437

0.559

0.679

0.799

0.917

λ_{2}

θ_{
m
}

136.105

226.995

312.500

396.167

478.968

561.288

643.312

Δθ_{
m
}/Δn

0.168

0.307

0.433

0.555

0.674

0.792

0.910

λ_{3}

θ_{
m
}

138.833

231.950

319.466

405.075

489.786

573.999

657.905

Δθ_{
m
}/Δn

0.158

0.291

0.411

0.526

0.640

0.752

0.864

Calculated deflection angle in degrees
for the minimum deviation θ_{
m
} for water at
20 °C for various orders of minimum deviation, and the variation
Δθ_{
m
}/Δn in the deflection angle that is
due to a change of 1 × 10^{-3} in the refractive index
of the water; λ_{1}, λ_{2}, and λ_{3}
are 632.8, 543.5, and 325.0 nm, respectively.

Table 2

Sensitivity of the Refractometer for Various Cell
Thicknesses

Variation in Deflection Angle

d (mm)

1

2

3

4

5

6

7

7.2

7.5

Δθ_{
m
}/Δn

0.2819

0.3085

0.3432

0.3918

0.4672

0.6098

1.0698

1.3713

4.7288

Calculated value of
Δθ_{
m
}/Δn for different cell thicknesses at
a wavelength of 543.5 nm. The sample is pure water at 20 °C.

Table 3

Minimum Deviation Angle (in degrees) for Various
Orders

Wavelength of Laser

Order P

1

2

3

4

5

6

7

λ_{1}

Experiment

158.5

269.5

372.2

473.9

573.5

672.0

770.0

Theory

158.454

269.422

372.782

473.596

573.216

672.174

770.726

λ_{2}

Experiment

158.6

270.0

374.8

474.9

575.0

673.0

772.3

Theory

158.777

270.062

373.699

474.777

574.654

673.887

772.672

Comparison between the simulated result
and the experimental result for the deflection angle in degrees for the
minimum deviation for glass. Two lasers, He–Ne at 632.8 nm
(λ_{2}) and He–Ne at 543.5 nm (λ_{1}),
were used.

Table 4

Minimum Deviation Angle (in degrees) for Various
Orders

Laser Wavelength

Order P

1

2

3

4

5

6

7

λ_{2}

135.9

227.0

312.3

395.9

478.4

560.7

644.3

λ_{3}

139.0

232.8

319.4

405.8

489.8

574.0

657.9

Deflection angle in degrees of the
minimum deviation of the laser light observed on the cylindrical
cell. Two lasers, He–Ne at 543.5 nm (λ_{3}) and
He–Cd at 325.0 nm (λ_{2}), were used.

Tables (4)

Table 1

Minimum Deviation Angle θ (in degrees) for Various
Orders Pa

Wavelength

Deviation Angle

Order P

1

2

3

4

5

6

7

λ_{1}

θ_{
m
}

135.673

226.213

311.401

394.762

477.262

559.283

641.012

Δθ_{
m
}/Δn

0.170

0.310

0.437

0.559

0.679

0.799

0.917

λ_{2}

θ_{
m
}

136.105

226.995

312.500

396.167

478.968

561.288

643.312

Δθ_{
m
}/Δn

0.168

0.307

0.433

0.555

0.674

0.792

0.910

λ_{3}

θ_{
m
}

138.833

231.950

319.466

405.075

489.786

573.999

657.905

Δθ_{
m
}/Δn

0.158

0.291

0.411

0.526

0.640

0.752

0.864

Calculated deflection angle in degrees
for the minimum deviation θ_{
m
} for water at
20 °C for various orders of minimum deviation, and the variation
Δθ_{
m
}/Δn in the deflection angle that is
due to a change of 1 × 10^{-3} in the refractive index
of the water; λ_{1}, λ_{2}, and λ_{3}
are 632.8, 543.5, and 325.0 nm, respectively.

Table 2

Sensitivity of the Refractometer for Various Cell
Thicknesses

Variation in Deflection Angle

d (mm)

1

2

3

4

5

6

7

7.2

7.5

Δθ_{
m
}/Δn

0.2819

0.3085

0.3432

0.3918

0.4672

0.6098

1.0698

1.3713

4.7288

Calculated value of
Δθ_{
m
}/Δn for different cell thicknesses at
a wavelength of 543.5 nm. The sample is pure water at 20 °C.

Table 3

Minimum Deviation Angle (in degrees) for Various
Orders

Wavelength of Laser

Order P

1

2

3

4

5

6

7

λ_{1}

Experiment

158.5

269.5

372.2

473.9

573.5

672.0

770.0

Theory

158.454

269.422

372.782

473.596

573.216

672.174

770.726

λ_{2}

Experiment

158.6

270.0

374.8

474.9

575.0

673.0

772.3

Theory

158.777

270.062

373.699

474.777

574.654

673.887

772.672

Comparison between the simulated result
and the experimental result for the deflection angle in degrees for the
minimum deviation for glass. Two lasers, He–Ne at 632.8 nm
(λ_{2}) and He–Ne at 543.5 nm (λ_{1}),
were used.

Table 4

Minimum Deviation Angle (in degrees) for Various
Orders

Laser Wavelength

Order P

1

2

3

4

5

6

7

λ_{2}

135.9

227.0

312.3

395.9

478.4

560.7

644.3

λ_{3}

139.0

232.8

319.4

405.8

489.8

574.0

657.9

Deflection angle in degrees of the
minimum deviation of the laser light observed on the cylindrical
cell. Two lasers, He–Ne at 543.5 nm (λ_{3}) and
He–Cd at 325.0 nm (λ_{2}), were used.