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.
You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
First Five Even Fourier Coefficients in Degrees of the
First-Order Rainbow Deviation Angle
θ2R(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
E0
154.723
154.713
153.571
E2
-0.134
-1.346
-13.421
F2
-0.063
-0.636
-6.531
E4
2.3 × 10-4
0.021
2.104
F4
-4.1 × 10-5
-0.005
-0.477
E6
<10-6
-2.9 × 10-4
-0.296
F6
<10-6
3.2 × 10-4
0.332
E8
<10-6
<10-6
0.015
F8
<10-6
<10-6
-0.113
E2Mobius
-0.134
-1.340
-13.396
F2Mobius
-0.063
-0.633
-6.325
The coefficients
E2Mobius and
F2Mobius are obtained from Eq.
(18). The Descartes rainbow deviation angle is
θ2D = 154.723°.
Table 2
First Five Even Fourier Coefficients in Degrees of the
Second-Order Rainbow Deviation Angle
θ3R(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
G0
262.121
262.159
266.370
G2
0.016
0.160
1.116
H2
-0.115
-1.162
-13.481
G4
2.4 × 10-4
0.021
1.980
H4
1.9 × 10-4
0.018
1.730
G6
<10-6
-4.2 × 10-4
-0.675
H6
<10-6
-9.4 × 10-4
-0.931
G8
<10-6
<10-6
0.009
H8
<10-6
<10-6
0.247
G2Mobius
0.016
0.160
1.599
H2Mobius
-0.115
-1.155
-11.546
The coefficients
G2Mobius and
H2Mobius are obtained from Eq.
(19). The Descartes rainbow deviation angle is
θ3D = 262.121°.
Table 3
First Six Fourier Coefficients in Degrees of the
First-Order Rainbow Deviation Angle
θ2R(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
E0
154.580
154.580
154.580
154.579
154.577
E1
0.000
0.039
0.078
0.116
0.154
F1
0.000
0.011
0.022
0.034
0.044
E2
-5.019
-5.019
-5.021
-5.023
-5.025
F2
-2.378
-2.378
-2.378
-2.378
-2.379
E3
0.000
0.045
0.090
0.136
0.183
F3
0.000
0.034
0.069
0.104
0.139
E4
0.293
0.294
0.299
0.307
0.319
F4
-0.068
-0.066
-0.062
-0.055
-0.046
E5
0.000
-0.027
-0.054
-0.081
-0.108
F5
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
θ3R(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
G0
262.657
262.638
262.619
262.565
262.489
G1
0.000
-0.011
-0.023
-0.034
-0.046
H1
0.000
0.034
0.068
0.102
0.134
G2
0.584
0.579
0.574
0.569
0.561
H2
-4.420
-4.408
-4.397
-4.381
-4.358
G3
0.000
0.575
1.150
1.724
2.294
H3
0.000
0.784
1.669
2.511
3.362
G4
0.301
0.310
0.320
0.353
0.395
H4
0.256
0.257
0.259
0.264
0.269
G5
0.000
0.120
0.240
0.359
0.479
H5
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
θ2R(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
E0
153.990
154.579
E1
-0.071
0.101
F1
0.063
0.029
E2
-5.217
-5.022
F2
-1.909
-2.378
E3
0.131
0.118
F3
0.129
0.090
E4
-0.721
0.303
F4
0.514
-0.058
E5
-0.061
-0.071
F5
0.314
-0.119
Table 6
First Six Fourier Coefficients in Degrees of the
Experimental Second-Order Rainbow Angle and of
θ3R(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
G0
262.673
262.587
G1
-0.103
-0.030
H1
-0.283
0.092
G2
0.500
0.570
H2
-4.287
-4.389
G3
1.404
1.491
H3
2.093
2.177
G4
0.645
0.342
H4
-1.908
0.267
G5
-0.220
0.314
H5
0.380
-0.439
Tables (6)
Table 1
First Five Even Fourier Coefficients in Degrees of the
First-Order Rainbow Deviation Angle
θ2R(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
E0
154.723
154.713
153.571
E2
-0.134
-1.346
-13.421
F2
-0.063
-0.636
-6.531
E4
2.3 × 10-4
0.021
2.104
F4
-4.1 × 10-5
-0.005
-0.477
E6
<10-6
-2.9 × 10-4
-0.296
F6
<10-6
3.2 × 10-4
0.332
E8
<10-6
<10-6
0.015
F8
<10-6
<10-6
-0.113
E2Mobius
-0.134
-1.340
-13.396
F2Mobius
-0.063
-0.633
-6.325
The coefficients
E2Mobius and
F2Mobius are obtained from Eq.
(18). The Descartes rainbow deviation angle is
θ2D = 154.723°.
Table 2
First Five Even Fourier Coefficients in Degrees of the
Second-Order Rainbow Deviation Angle
θ3R(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
G0
262.121
262.159
266.370
G2
0.016
0.160
1.116
H2
-0.115
-1.162
-13.481
G4
2.4 × 10-4
0.021
1.980
H4
1.9 × 10-4
0.018
1.730
G6
<10-6
-4.2 × 10-4
-0.675
H6
<10-6
-9.4 × 10-4
-0.931
G8
<10-6
<10-6
0.009
H8
<10-6
<10-6
0.247
G2Mobius
0.016
0.160
1.599
H2Mobius
-0.115
-1.155
-11.546
The coefficients
G2Mobius and
H2Mobius are obtained from Eq.
(19). The Descartes rainbow deviation angle is
θ3D = 262.121°.
Table 3
First Six Fourier Coefficients in Degrees of the
First-Order Rainbow Deviation Angle
θ2R(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
E0
154.580
154.580
154.580
154.579
154.577
E1
0.000
0.039
0.078
0.116
0.154
F1
0.000
0.011
0.022
0.034
0.044
E2
-5.019
-5.019
-5.021
-5.023
-5.025
F2
-2.378
-2.378
-2.378
-2.378
-2.379
E3
0.000
0.045
0.090
0.136
0.183
F3
0.000
0.034
0.069
0.104
0.139
E4
0.293
0.294
0.299
0.307
0.319
F4
-0.068
-0.066
-0.062
-0.055
-0.046
E5
0.000
-0.027
-0.054
-0.081
-0.108
F5
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
θ3R(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
G0
262.657
262.638
262.619
262.565
262.489
G1
0.000
-0.011
-0.023
-0.034
-0.046
H1
0.000
0.034
0.068
0.102
0.134
G2
0.584
0.579
0.574
0.569
0.561
H2
-4.420
-4.408
-4.397
-4.381
-4.358
G3
0.000
0.575
1.150
1.724
2.294
H3
0.000
0.784
1.669
2.511
3.362
G4
0.301
0.310
0.320
0.353
0.395
H4
0.256
0.257
0.259
0.264
0.269
G5
0.000
0.120
0.240
0.359
0.479
H5
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
θ2R(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
E0
153.990
154.579
E1
-0.071
0.101
F1
0.063
0.029
E2
-5.217
-5.022
F2
-1.909
-2.378
E3
0.131
0.118
F3
0.129
0.090
E4
-0.721
0.303
F4
0.514
-0.058
E5
-0.061
-0.071
F5
0.314
-0.119
Table 6
First Six Fourier Coefficients in Degrees of the
Experimental Second-Order Rainbow Angle and of
θ3R(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