Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder. II. Experiment

Charles L. Adler, David Phipps, Kirk W. Saunders, Justin K. Nash, and James A. Lock

Charles L. Adler, David Phipps, Kirk W. Saunders, Justin K. Nash, and James A. Lock

^{}C. L. Adler, D. Phipps, K. W. Saunders, and J. K. Nash are with the Department of Physics, St. Mary’s College of Maryland, St. Mary’s City, Maryland 20686.

^{}J. A. Lock (jimandcarol@stratos.net) is with the Department of Physics, Cleveland State University, Cleveland, Ohio 44115.

Charles L. Adler, David Phipps, Kirk W. Saunders, Justin K. Nash, and James A. Lock, "Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder. II. Experiment," Appl. Opt. 40, 2535-2545 (2001)

We measured the supernumerary spacing parameter of the first- and
second-order rainbows of two glass rods, each having an approximately
elliptical cross section, as a function of the rod’s rotation
angle. We attribute large fluctuations in the supernumerary spacing
parameter to small local inhomogeneities in the rod’s refractive
index. The low-pass filtered first-order rainbow experimental data
agree with the prediction of ray-tracing–wave-front modeling to within
a few percent, and the second-order rainbow data exhibit additional
effects that are due to rod nonellipticity.

J. P. A. J. van Beeck, “Rainbow phenomena: development of a laser-based, non-intrusive technique for measuring droplet size, temperature and velocity,” Ph.D. dissertation (Eindhoven Technische Universiteit, Eindhoven, The Netherlands, 1997), p. 78.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington D.C., 1964), Sect. 10.4, pp. 446–447, 478.

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

J. P. A. J. van Beeck, “Rainbow phenomena: development of a laser-based, non-intrusive technique for measuring droplet size, temperature and velocity,” Ph.D. dissertation (Eindhoven Technische Universiteit, Eindhoven, The Netherlands, 1997), p. 78.

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

Weatherwise

A. B. Fraser, “Chasing rainbows,” Weatherwise 36, 280–287 (1983).
[CrossRef]

Other

J. P. A. J. van Beeck, “Rainbow phenomena: development of a laser-based, non-intrusive technique for measuring droplet size, temperature and velocity,” Ph.D. dissertation (Eindhoven Technische Universiteit, Eindhoven, The Netherlands, 1997), p. 78.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington D.C., 1964), Sect. 10.4, pp. 446–447, 478.

Ferguson’s Cut Glass Originals, 4292 Pearl Road, Cleveland, Ohio 44109.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sect. 13.23, pp. 243–246.

(a) Deviation of the p = 2 rainbow
angle from its average value and (b) supernumerary spacing
parameter of a 1.0-mm-radius glass rod with refractive index
n = 1.511 and eccentricity ∊ = 0.00155 as a
function of the rod’s rotation angle. The solid circles are the
experimental data of Ref. 1, and the dashed curves are
the predictions of the ray-tracing–wave-front modeling procedure of
Ref. 6.

Beam of a 3-mW He–Ne laser is attenuated by a polarizing
filter, expanded by an afocal telescope, and is incident on a glass rod
mounted on a rotation stage. A beam block prevents interference of
reflected light from the rod with the p = 2 and
p = 3 rainbows. The rainbow pattern is recorded by
a CCD camera placed in the scattering near zone.

Supernumerary spacing parameter of the p =
2 rainbow of a 8.05-mm-radius glass rod with refractive index
n = 1.474 and eccentricity ∊ = -0.037 as a
function of the rod’s rotation angle. The solid circles are the
experimental data obtained from (a) the first and second
supernumerary maxima and (b) the first and third supernumerary
maxima. The solid curves are the low-pass filtered experimental
data, and the dashed curves are the predictions of the
ray-tracing–wave-front modeling procedure of Ref.
6.

Supernumerary spacing parameter of the p =
2 rainbow of a 2.44-mm-radius glass rod with refractive index
n = 1.502 and eccentricity ∊ = 0.0054 as a
function of the rod’s rotation angle. The solid circles are the
experimental data that we obtained from averaging the results of the
first and second, first and third, and first and fourth supernumerary
maxima; the solid curve is the low-pass filtered experimental data; and
the dashed curve is the prediction of the ray-tracing–wave-front
modeling procedure of Ref. 6.

Supernumerary spacing parameter of the p =
3 rainbow of a 8.05-mm-radius glass rod with refractive index
n = 1.474 and eccentricity ∊ = -0.037 as a
function of the rod’s rotation angle. The solid circles are the
experimental data that we obtained from averaging the results of the
first and second and the first and third supernumerary maxima, the
solid curve is the low-pass filtered experimental data, and the dashed
curve is the prediction of the ray-tracing–wave-front modeling
procedure of Ref. 6.

Supernumerary spacing parameter of the p =
3 rainbow of a 2.44-mm-radius glass rod with refractive index
n = 1.502 and eccentricity ∊ = 0.0054 as a
function of the rod’s rotation angle. The solid circles are the
experimental data that we obtained from averaging the results of the
first and second and the first and third supernumerary maxima, the
solid curve is the low-pass filtered experimental data, and the dashed
curve is the prediction of the ray-tracing–wave-front modeling
procedure of Ref. 6.

Magnitude of the Coefficients in the Fourier-Series
Decomposition of h_{
2
}
(ξ) for the 8.05- and
2.44-mm-Radius Glass Rods
a

Fourier Coefficient

a = 8.05 mm

a = 2.44 mm

Experiment (1 + 2)

Experiment (1 + 3)

Theory

Experiment

Theory

e_{
0
}

2.593

2.566

2.592

2.191

2.175

(e_{
1
}^{
2
} + f_{
1
}^{
2
})^{
1/2
}

0.043

0.135

0.000

0.043

0.000

(e_{
2
}^{
2
} + f_{
2
}^{
2
})^{
1/2
}

0.829

0.904

0.790

0.115

0.100

(e_{
3
}^{
2
} + f_{
3
}^{
2
})^{
1/2
}

0.478

0.517

0.000

0.042

0.000

(e_{
4
}^{
2
} + f_{
4
}^{
2
})^{
1/2
}

0.287

0.413

0.126

0.040

0.002

(e_{
5
}^{
2
} + f_{
5
}^{
2
})^{
1/2
}

0.497

0.447

0.000

0.058

0.000

(e_{
6
}^{
2
} + f_{
6
}^{
2
})^{
1/2
}

0.537

0.473

0.019

0.045

2 × 10^{-4}

Experiments (1 + 2) and
(1 + 3) correspond to the measurement of the first and second
and the first and third supernumerary maxima of the 8.05-mm rod,
respectively. For the 2.44-mm-radius glass rod, the experimental
coefficients are the result of averaging
h_{2}(ξ) obtained from the first and second,
first and third, and first and fourth supernumerary maxima. The
ray-tracing–wave-front modeling Fourier coefficients are for
n = 1.474 and ∊ = -0.037 and for n =
1.511 and ∊ = 0.0054. The m = 0
theoretical coefficient was corrected for near-zone effects with
r/
a = 42.35 for the larger rod and r/
a =
54.14 for the smaller rod.

Table 2

Magnitude of the Coefficients in the Fourier-Series
Decomposition of h_{
3
}
(ξ) for the 8.05- and
2.44-mm-Radius Glass Rods
a

Fourier Coefficient

a = 8.05 mm

a = 2.44 mm

Experiment

Theory

Experiment

Theory

g_{
0
}

23.018

16.971

12.496

13.249

(g_{
1
}^{
2
} + j_{
1
}^{
2
})^{
1/2
}

1.107

0.000

0.582

0.0

(g_{
2
}^{
2
} + j_{
2
}^{
2
})^{
1/2
}

7.830

8.249

1.256

0.980

(g_{
3
}^{
2
} + j_{
3
}^{
2
})^{
1/2
}

5.728

0.000

1.514

0.0

(g_{
4
}^{
2
} + j_{
4
}^{
2
})^{
1/2
}

2.799

0.598

0.318

0.012

(g_{
5
}^{
2
} + j_{
5
}^{
2
})^{
1/2
}

2.810

0.000

0.476

0.0

(g_{
6
}^{
2
} + j_{
6
}^{
2
})^{
1/2
}

3.140

0.136

0.543

6 × 10^{-4}

We obtained the experimental coefficients
by averaging h_{3}(ξ) obtained from the first
and second and the first and third supernumerary maxima. The
ray-tracing–wave-front modeling Fourier coefficients are for
n = 1.474 and ∊ = -0.037 and for n =
1.511 and ∊ = 0.0054. The m = 0
theoretical coefficient was corrected for near-zone effects with
r/
a = 42.35 and r/
a = 54.14.

Tables (2)

Table 1

Magnitude of the Coefficients in the Fourier-Series
Decomposition of h_{
2
}
(ξ) for the 8.05- and
2.44-mm-Radius Glass Rods
a

Fourier Coefficient

a = 8.05 mm

a = 2.44 mm

Experiment (1 + 2)

Experiment (1 + 3)

Theory

Experiment

Theory

e_{
0
}

2.593

2.566

2.592

2.191

2.175

(e_{
1
}^{
2
} + f_{
1
}^{
2
})^{
1/2
}

0.043

0.135

0.000

0.043

0.000

(e_{
2
}^{
2
} + f_{
2
}^{
2
})^{
1/2
}

0.829

0.904

0.790

0.115

0.100

(e_{
3
}^{
2
} + f_{
3
}^{
2
})^{
1/2
}

0.478

0.517

0.000

0.042

0.000

(e_{
4
}^{
2
} + f_{
4
}^{
2
})^{
1/2
}

0.287

0.413

0.126

0.040

0.002

(e_{
5
}^{
2
} + f_{
5
}^{
2
})^{
1/2
}

0.497

0.447

0.000

0.058

0.000

(e_{
6
}^{
2
} + f_{
6
}^{
2
})^{
1/2
}

0.537

0.473

0.019

0.045

2 × 10^{-4}

Experiments (1 + 2) and
(1 + 3) correspond to the measurement of the first and second
and the first and third supernumerary maxima of the 8.05-mm rod,
respectively. For the 2.44-mm-radius glass rod, the experimental
coefficients are the result of averaging
h_{2}(ξ) obtained from the first and second,
first and third, and first and fourth supernumerary maxima. The
ray-tracing–wave-front modeling Fourier coefficients are for
n = 1.474 and ∊ = -0.037 and for n =
1.511 and ∊ = 0.0054. The m = 0
theoretical coefficient was corrected for near-zone effects with
r/
a = 42.35 for the larger rod and r/
a =
54.14 for the smaller rod.

Table 2

Magnitude of the Coefficients in the Fourier-Series
Decomposition of h_{
3
}
(ξ) for the 8.05- and
2.44-mm-Radius Glass Rods
a

Fourier Coefficient

a = 8.05 mm

a = 2.44 mm

Experiment

Theory

Experiment

Theory

g_{
0
}

23.018

16.971

12.496

13.249

(g_{
1
}^{
2
} + j_{
1
}^{
2
})^{
1/2
}

1.107

0.000

0.582

0.0

(g_{
2
}^{
2
} + j_{
2
}^{
2
})^{
1/2
}

7.830

8.249

1.256

0.980

(g_{
3
}^{
2
} + j_{
3
}^{
2
})^{
1/2
}

5.728

0.000

1.514

0.0

(g_{
4
}^{
2
} + j_{
4
}^{
2
})^{
1/2
}

2.799

0.598

0.318

0.012

(g_{
5
}^{
2
} + j_{
5
}^{
2
})^{
1/2
}

2.810

0.000

0.476

0.0

(g_{
6
}^{
2
} + j_{
6
}^{
2
})^{
1/2
}

3.140

0.136

0.543

6 × 10^{-4}

We obtained the experimental coefficients
by averaging h_{3}(ξ) obtained from the first
and second and the first and third supernumerary maxima. The
ray-tracing–wave-front modeling Fourier coefficients are for
n = 1.474 and ∊ = -0.037 and for n =
1.511 and ∊ = 0.0054. The m = 0
theoretical coefficient was corrected for near-zone effects with
r/
a = 42.35 and r/
a = 54.14.