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.

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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.