Charles L. Adler, James A. Lock, and Bradley R. Stone, "Rainbow scattering by a cylinder with a nearly elliptical cross section," Appl. Opt. 37, 1540-1550 (1998)

We both theoretically and experimentally examine the behavior of
the first- and the second-order rainbows produced by a normally
illuminated glass rod, which has a nearly elliptical cross section, as
it is rotated about its major axis. We decompose the measured
rainbow angle, taken as a function of the rod’s rotation angle, into a
Fourier series and find that the rod’s refractive index, average
ellipticity, and deviation from ellipticity are encoded primarily in
the m = 0, 2, 3 Fourier coefficients,
respectively. We determine these parameters for our glass rod and,
where possible, compare them with independent measurements. We find
that the average ellipticity of the rod agrees well with direct
measurements, but that the rod’s diameter inferred from the spacing of
the supernumeraries of the first-order rainbow is significantly larger
than that obtained by direct measurement. We also determine the
conditions under which the deviation of falling water droplets from an
oblate spheroidal shape permits the first few supernumeraries of the
second-order rainbow to be observed in a rain shower.

G. Gouesbet, “Scattering of a first order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
[CrossRef]

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 1407–1423.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, New York, 1987), pp. 129–180.

G. Gouesbet, “Scattering of a first order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
[CrossRef]

1994 (1)

J. A. Lock, T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[CrossRef]

G. Gouesbet, “Scattering of a first order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
[CrossRef]

Green, A. W.

A. W. Green, “An approximation for the shapes of large raindrops,” J. Appl. Meteorol. 14, 1578–1583 (1975).
[CrossRef]

Greenler, R.

R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, Cambridge, 1989), p. 20.

G. Gouesbet, “Scattering of a first order Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation,” Part. Part. Syst. Charact. 12, 242–256 (1995).
[CrossRef]

Weatherwise (1)

A. B. Fraser, “Chasing rainbows,” Weatherwise 36, 280–289 (1983).
[CrossRef]

Other (7)

C. B. Boyer, The Rainbow From Myth to Mathematics (Princeton U. Press, Princeton, N.J., 1987), p. 309.

H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, The Netherlands, 1978), pp. 23–25.

M. Minnaert, Light and Colour in the Open Air (Dover, New York, 1954), p. 173.

R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, Cambridge, 1989), p. 20.

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 1407–1423.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, New York, 1987), pp. 129–180.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 297–328.

(a) Cylinder has a cross section consisting of two
half-ellipses denoted by the index j = 1, 2 joined
smoothly at the points R and L. The length of
the common semimajor axis of the two half-ellipses is a, and
the length of their differing semiminor axes are
b_{1} and b_{2}. The
x′y′z′ coordinate system is attached to the
cylinder. (b) The cylinder is rotated by an angle ξ about the
z′ axis. Incident light rays propagate in the
-y direction of a fixed laboratory coordinate system.

Angles γ_{
p
} of the normal to the
surface and the angles δ_{
p
} of the interior rays
for the 0 ≤ p ≤ 2 interactions of a light ray
with the surface.

Deviation of the first-order rainbow angle
θ_{2}^{
R
}(ξ) of a two-half-ellipse
cross-section cylinder from the Descartes first-order rainbow angle
θ_{2}^{
D
} as a function of the rotation
angle ξ for a refractive index n = 1.474, average
ellipticity ∊_{ave} = 0.060, and ellipticity difference
Δ∊ = 0 (curve a), Δ∊ = 0.06 (curve b), Δ∊ = 0.12
(curve c), and Δ∊ = 0.18 (curve d). The range of Δ∊
here is much larger than in Table 3 because the dependence of the
rainbow angle on Δ∊ is weak. The value of ∊_{ave} is
also different than that in Table 3.

Deviation of the second-order rainbow angle
θ_{3}^{
R
}(ξ) for a
two-half-ellipse cross-section cylinder from the Descartes second-order
rainbow angle θ_{3}^{
D
} as a function of
the rotation angle ξ for a refractive index n =
1.474, average ellipticity ∊_{ave} = -0.037, and
ellipticity difference Δ∊ = 0 (curve a), Δ∊ = 0.01 (curve
b), Δ∊ = 0.02 (curve c), Δ∊ = 0.03 (curve d), and
Δ∊ = 0.04 (curve e). These parameters are identical to
those of Table 4.

Experimental first-order rainbow deviation angle as a
function of the rotation angle ξ and the theoretical fit of the
two-half-ellipse cross-section model with n = 1.474,
∊_{ave} = -0.037, and Δ∊ = 0.026.

Experimental second-order rainbow deviation angle as a
function of the rotation angle ξ and the theoretical fit of the
two-half-ellipse cross-section model with n = 1.474,
∊_{ave} = -0.037, and Δ∊ = 0.026.

Experimental light intensity I in the vicinity
of the first-order rainbow as a function of scattering angle
Δθ. The peak of the principal rainbow maximum corresponds to
Δθ = 0°.

Negative of the average ellipticity -∊_{ave}
and the ellipticity difference Δ∊ as functions of the
equal-volume-sphere radius a_{0} of raindrops
falling at terminal velocity and derived from the parameterization of
Ref. 14. The filled circles are the analytical
approximation to -∊_{ave} of Ref. 27, and
the open circles are the linearized approximation of Ref.
16.

Deviation angle of the second-order rainbow
θ_{3}^{
R
} with respect to the Descartes
second-order rainbow angle θ_{3}^{
D
} as
a function of the equal-volume-sphere radius a_{0}
of raindrops falling at terminal velocity for solar elevation angles of
(a) 10°, (b) 20°, (c) 40°. In each graph, the
lowest curve is the principal Airy maximum, the middle graph is the
first supernumerary maximum, and the highest curve is the second
supernumerary maximum.

Table 1 First Five Even Fourier Coefficients in Degrees of the
First-Order Rainbow Deviation Angle
θ_{2}^{R}(ξ) for an Elliptical
Cross-Sectional Cylinder with Refractive Index n =
1.474 and Ellipticities ∊ = -0.001, -0.01, and -0.1 as Defined in
Eq. (17)a

Table 2 First Five Even Fourier Coefficients in Degrees of the
Second-Order Rainbow Deviation Angle
θ_{3}^{R}(ξ) for an Elliptical
Cross-Sectional Cylinder with Refractive Index n =
1.474 and Ellipticities ∊ = -0.001, -0.01, and -0.1 as Defined in
Eq. (17)a

Table 3 First Six Fourier Coefficients in Degrees of the
First-Order Rainbow Deviation Angle
θ_{2}^{R}(ξ) for a
Two-Half-Ellipse Cross-Sectional Cylinder with Refractive Index
n = 1.474, Average Ellipticity ∊_{ave} =
-0.037, and Various Values of the Ellipticity Difference Δ∊ as
Defined in Eq. (23)

Table 4 First Six Fourier Coefficients in Degrees of the
Second-Order Rainbow Deviation Angle
θ_{3}^{R}(ξ) for a
Two-Half-Ellipse Cross-Sectional Cylinder with Refractive Index
n = 1.474, Average Ellipticity ∊_{ave} =
-0.037, and Various Values of the Ellipticity Difference Δ∊ as
Defined in Eq. (23)

Table 5 First Six Fourier Coefficients in Degrees of the
Experimental First-Order Rainbow Deviation Angle and of
θ_{2}^{R}(ξ) for a Cylinder with
a Two-Half-Ellipse Cross Section, Refractive Index n =
1.474, Average Ellipticity ∊_{ave} = -0.037, and
Ellipticity Difference Δ∊ = 0.026

Table 6 First Six Fourier Coefficients in Degrees of the
Experimental Second-Order Rainbow Angle and of
θ_{3}^{R}(ξ) for a Cylinder with
a Two-Half-Ellipse Cross Section, Refractive Index n =
1.474, Average Ellipticity ∊_{ave} = -0.037, and
Ellipticity Difference Δ∊ = 0.026

First Five Even Fourier Coefficients in Degrees of the
First-Order Rainbow Deviation Angle
θ_{2}^{R}(ξ) for an Elliptical
Cross-Sectional Cylinder with Refractive Index n =
1.474 and Ellipticities ∊ = -0.001, -0.01, and -0.1 as Defined in
Eq. (17)a

Coefficient

∊

-0.001

-0.01

-0.1

E_{0}

154.723

154.713

153.571

E_{2}

-0.134

-1.346

-13.421

F_{2}

-0.063

-0.636

-6.531

E_{4}

2.3 × 10^{-4}

0.021

2.104

F_{4}

-4.1 × 10^{-5}

-0.005

-0.477

E_{6}

<10^{-6}

-2.9 × 10^{-4}

-0.296

F_{6}

<10^{-6}

3.2 × 10^{-4}

0.332

E_{8}

<10^{-6}

<10^{-6}

0.015

F_{8}

<10^{-6}

<10^{-6}

-0.113

E_{2}^{Mobius}

-0.134

-1.340

-13.396

F_{2}^{Mobius}

-0.063

-0.633

-6.325

The coefficients
E_{2}^{Mobius} and
F_{2}^{Mobius} are obtained from Eq.
(18). The Descartes rainbow deviation angle is
θ_{2}^{
D
} = 154.723°.

Table 2

First Five Even Fourier Coefficients in Degrees of the
Second-Order Rainbow Deviation Angle
θ_{3}^{R}(ξ) for an Elliptical
Cross-Sectional Cylinder with Refractive Index n =
1.474 and Ellipticities ∊ = -0.001, -0.01, and -0.1 as Defined in
Eq. (17)a

Coefficient

∊

-0.001

-0.01

-0.1

G_{0}

262.121

262.159

266.370

G_{2}

0.016

0.160

1.116

H_{2}

-0.115

-1.162

-13.481

G_{4}

2.4 × 10^{-4}

0.021

1.980

H_{4}

1.9 × 10^{-4}

0.018

1.730

G_{6}

<10^{-6}

-4.2 × 10^{-4}

-0.675

H_{6}

<10^{-6}

-9.4 × 10^{-4}

-0.931

G_{8}

<10^{-6}

<10^{-6}

0.009

H_{8}

<10^{-6}

<10^{-6}

0.247

G_{2}^{Mobius}

0.016

0.160

1.599

H_{2}^{Mobius}

-0.115

-1.155

-11.546

The coefficients
G_{2}^{Mobius} and
H_{2}^{Mobius} are obtained from Eq.
(19). The Descartes rainbow deviation angle is
θ_{3}^{
D
} = 262.121°.

Table 3

First Six Fourier Coefficients in Degrees of the
First-Order Rainbow Deviation Angle
θ_{2}^{R}(ξ) for a
Two-Half-Ellipse Cross-Sectional Cylinder with Refractive Index
n = 1.474, Average Ellipticity ∊_{ave} =
-0.037, and Various Values of the Ellipticity Difference Δ∊ as
Defined in Eq. (23)

Coefficient

Δ∊

0.00

0.01

0.02

0.03

0.04

E_{0}

154.580

154.580

154.580

154.579

154.577

E_{1}

0.000

0.039

0.078

0.116

0.154

F_{1}

0.000

0.011

0.022

0.034

0.044

E_{2}

-5.019

-5.019

-5.021

-5.023

-5.025

F_{2}

-2.378

-2.378

-2.378

-2.378

-2.379

E_{3}

0.000

0.045

0.090

0.136

0.183

F_{3}

0.000

0.034

0.069

0.104

0.139

E_{4}

0.293

0.294

0.299

0.307

0.319

F_{4}

-0.068

-0.066

-0.062

-0.055

-0.046

E_{5}

0.000

-0.027

-0.054

-0.081

-0.108

F_{5}

0.000

-0.046

-0.092

-0.137

-0.182

Table 4

First Six Fourier Coefficients in Degrees of the
Second-Order Rainbow Deviation Angle
θ_{3}^{R}(ξ) for a
Two-Half-Ellipse Cross-Sectional Cylinder with Refractive Index
n = 1.474, Average Ellipticity ∊_{ave} =
-0.037, and Various Values of the Ellipticity Difference Δ∊ as
Defined in Eq. (23)

Coefficient

Δ∊

0.00

0.01

0.02

0.03

0.04

G_{0}

262.657

262.638

262.619

262.565

262.489

G_{1}

0.000

-0.011

-0.023

-0.034

-0.046

H_{1}

0.000

0.034

0.068

0.102

0.134

G_{2}

0.584

0.579

0.574

0.569

0.561

H_{2}

-4.420

-4.408

-4.397

-4.381

-4.358

G_{3}

0.000

0.575

1.150

1.724

2.294

H_{3}

0.000

0.784

1.669

2.511

3.362

G_{4}

0.301

0.310

0.320

0.353

0.395

H_{4}

0.256

0.257

0.259

0.264

0.269

G_{5}

0.000

0.120

0.240

0.359

0.479

H_{5}

0.000

-0.168

-0.337

-0.505

-0.670

Table 5

First Six Fourier Coefficients in Degrees of the
Experimental First-Order Rainbow Deviation Angle and of
θ_{2}^{R}(ξ) for a Cylinder with
a Two-Half-Ellipse Cross Section, Refractive Index n =
1.474, Average Ellipticity ∊_{ave} = -0.037, and
Ellipticity Difference Δ∊ = 0.026

Fourier Coefficient

Experiment

Theory

E_{0}

153.990

154.579

E_{1}

-0.071

0.101

F_{1}

0.063

0.029

E_{2}

-5.217

-5.022

F_{2}

-1.909

-2.378

E_{3}

0.131

0.118

F_{3}

0.129

0.090

E_{4}

-0.721

0.303

F_{4}

0.514

-0.058

E_{5}

-0.061

-0.071

F_{5}

0.314

-0.119

Table 6

First Six Fourier Coefficients in Degrees of the
Experimental Second-Order Rainbow Angle and of
θ_{3}^{R}(ξ) for a Cylinder with
a Two-Half-Ellipse Cross Section, Refractive Index n =
1.474, Average Ellipticity ∊_{ave} = -0.037, and
Ellipticity Difference Δ∊ = 0.026

Fourier Coefficient

Experiment

Theory

G_{0}

262.673

262.587

G_{1}

-0.103

-0.030

H_{1}

-0.283

0.092

G_{2}

0.500

0.570

H_{2}

-4.287

-4.389

G_{3}

1.404

1.491

H_{3}

2.093

2.177

G_{4}

0.645

0.342

H_{4}

-1.908

0.267

G_{5}

-0.220

0.314

H_{5}

0.380

-0.439

Tables (6)

Table 1

First Five Even Fourier Coefficients in Degrees of the
First-Order Rainbow Deviation Angle
θ_{2}^{R}(ξ) for an Elliptical
Cross-Sectional Cylinder with Refractive Index n =
1.474 and Ellipticities ∊ = -0.001, -0.01, and -0.1 as Defined in
Eq. (17)a

Coefficient

∊

-0.001

-0.01

-0.1

E_{0}

154.723

154.713

153.571

E_{2}

-0.134

-1.346

-13.421

F_{2}

-0.063

-0.636

-6.531

E_{4}

2.3 × 10^{-4}

0.021

2.104

F_{4}

-4.1 × 10^{-5}

-0.005

-0.477

E_{6}

<10^{-6}

-2.9 × 10^{-4}

-0.296

F_{6}

<10^{-6}

3.2 × 10^{-4}

0.332

E_{8}

<10^{-6}

<10^{-6}

0.015

F_{8}

<10^{-6}

<10^{-6}

-0.113

E_{2}^{Mobius}

-0.134

-1.340

-13.396

F_{2}^{Mobius}

-0.063

-0.633

-6.325

The coefficients
E_{2}^{Mobius} and
F_{2}^{Mobius} are obtained from Eq.
(18). The Descartes rainbow deviation angle is
θ_{2}^{
D
} = 154.723°.

Table 2

First Five Even Fourier Coefficients in Degrees of the
Second-Order Rainbow Deviation Angle
θ_{3}^{R}(ξ) for an Elliptical
Cross-Sectional Cylinder with Refractive Index n =
1.474 and Ellipticities ∊ = -0.001, -0.01, and -0.1 as Defined in
Eq. (17)a

Coefficient

∊

-0.001

-0.01

-0.1

G_{0}

262.121

262.159

266.370

G_{2}

0.016

0.160

1.116

H_{2}

-0.115

-1.162

-13.481

G_{4}

2.4 × 10^{-4}

0.021

1.980

H_{4}

1.9 × 10^{-4}

0.018

1.730

G_{6}

<10^{-6}

-4.2 × 10^{-4}

-0.675

H_{6}

<10^{-6}

-9.4 × 10^{-4}

-0.931

G_{8}

<10^{-6}

<10^{-6}

0.009

H_{8}

<10^{-6}

<10^{-6}

0.247

G_{2}^{Mobius}

0.016

0.160

1.599

H_{2}^{Mobius}

-0.115

-1.155

-11.546

The coefficients
G_{2}^{Mobius} and
H_{2}^{Mobius} are obtained from Eq.
(19). The Descartes rainbow deviation angle is
θ_{3}^{
D
} = 262.121°.

Table 3

First Six Fourier Coefficients in Degrees of the
First-Order Rainbow Deviation Angle
θ_{2}^{R}(ξ) for a
Two-Half-Ellipse Cross-Sectional Cylinder with Refractive Index
n = 1.474, Average Ellipticity ∊_{ave} =
-0.037, and Various Values of the Ellipticity Difference Δ∊ as
Defined in Eq. (23)

Coefficient

Δ∊

0.00

0.01

0.02

0.03

0.04

E_{0}

154.580

154.580

154.580

154.579

154.577

E_{1}

0.000

0.039

0.078

0.116

0.154

F_{1}

0.000

0.011

0.022

0.034

0.044

E_{2}

-5.019

-5.019

-5.021

-5.023

-5.025

F_{2}

-2.378

-2.378

-2.378

-2.378

-2.379

E_{3}

0.000

0.045

0.090

0.136

0.183

F_{3}

0.000

0.034

0.069

0.104

0.139

E_{4}

0.293

0.294

0.299

0.307

0.319

F_{4}

-0.068

-0.066

-0.062

-0.055

-0.046

E_{5}

0.000

-0.027

-0.054

-0.081

-0.108

F_{5}

0.000

-0.046

-0.092

-0.137

-0.182

Table 4

First Six Fourier Coefficients in Degrees of the
Second-Order Rainbow Deviation Angle
θ_{3}^{R}(ξ) for a
Two-Half-Ellipse Cross-Sectional Cylinder with Refractive Index
n = 1.474, Average Ellipticity ∊_{ave} =
-0.037, and Various Values of the Ellipticity Difference Δ∊ as
Defined in Eq. (23)

Coefficient

Δ∊

0.00

0.01

0.02

0.03

0.04

G_{0}

262.657

262.638

262.619

262.565

262.489

G_{1}

0.000

-0.011

-0.023

-0.034

-0.046

H_{1}

0.000

0.034

0.068

0.102

0.134

G_{2}

0.584

0.579

0.574

0.569

0.561

H_{2}

-4.420

-4.408

-4.397

-4.381

-4.358

G_{3}

0.000

0.575

1.150

1.724

2.294

H_{3}

0.000

0.784

1.669

2.511

3.362

G_{4}

0.301

0.310

0.320

0.353

0.395

H_{4}

0.256

0.257

0.259

0.264

0.269

G_{5}

0.000

0.120

0.240

0.359

0.479

H_{5}

0.000

-0.168

-0.337

-0.505

-0.670

Table 5

First Six Fourier Coefficients in Degrees of the
Experimental First-Order Rainbow Deviation Angle and of
θ_{2}^{R}(ξ) for a Cylinder with
a Two-Half-Ellipse Cross Section, Refractive Index n =
1.474, Average Ellipticity ∊_{ave} = -0.037, and
Ellipticity Difference Δ∊ = 0.026

Fourier Coefficient

Experiment

Theory

E_{0}

153.990

154.579

E_{1}

-0.071

0.101

F_{1}

0.063

0.029

E_{2}

-5.217

-5.022

F_{2}

-1.909

-2.378

E_{3}

0.131

0.118

F_{3}

0.129

0.090

E_{4}

-0.721

0.303

F_{4}

0.514

-0.058

E_{5}

-0.061

-0.071

F_{5}

0.314

-0.119

Table 6

First Six Fourier Coefficients in Degrees of the
Experimental Second-Order Rainbow Angle and of
θ_{3}^{R}(ξ) for a Cylinder with
a Two-Half-Ellipse Cross Section, Refractive Index n =
1.474, Average Ellipticity ∊_{ave} = -0.037, and
Ellipticity Difference Δ∊ = 0.026