J. R. Hodkinson and I. Greenleaves, "Computations of Light-Scattering and Extinction by Spheres According to Diffraction and Geometrical Optics, and Some Comparisons with the Mie Theory," J. Opt. Soc. Am. 53, 577-588 (1963)
The scattering and extinction of light by systems of transparent, spherical
particles have been calculated on the basis of classical diffraction and of
geometrical scattering by external reflection and transmission, for the
refractive index range 1.1 to 2.0 and scattering angles up to about 40°,
and presented as a set of graphs for general use. Provided that the particles
are larger than 3 or 4 wavelengths of light, and that there is a sufficient
range (say 2:1) of particle sizes present to average out effects of phase, the
scattering and extinction calculated in this way agree well with such direct
calculations by the Mie theory as are at present available. It is thought that
the treatment should also be successful with irregular particles, and to some
extent with nontransparent ones, such as often occur in airborne dusts in
industry. The conditions under which scattering and extinction measurements can
be used to estimate the surface-area of particles in suspension in air or liquid
are discussed.
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Scattering by a transparent sphere due to external reflection, calculated by
geometrical optics.Upper and lower rows for each angle give polarized components
I1,1(θ) and
I1,2(θ) of
scattered-intensity coefficient
I1(θ), flux
scattered per unit solid angle in direction θ per
unit geometrically incident flux, ×104.
Scattering angle
θ, (deg)
Refractive index
1.1
1.15
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
0°
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
5°
273
295
308
330
337
343
349
353
356
360
360
252
265
270
275
283
280
280
275
270
269
270
10°
186
215
235
262
278
293
303
311
318
319
328
159
179
186
193
198
196
193
189
186
182
178
15°
131
160
181
212
233
250
263
274
282
297
297
100
117
129
137
136
136
133
131
126
121
117
20°
90.5
120
142
174
194
214
228
240
250
260
268
63.5
77.5
86
95
93.5
94
90.5
83.5
81
78
72
25°
65
90
110
143
166
183
199
211
225
225
241
40.3
50.8
58.1
64
65
63.5
61
56
54
53.0
45
30°
47
68.2
86.5
117
141
160
174
187
200
207
218
25.0
34.0
39.8
43
45
42.5
40.1
37
33
29.2
26.5
35°
33.8
52.4
69.5
96.5
119
140
154
168
180
189
199
16
22.2
26
29
29.0
28.5
24.2
22
19
15.7
14.0
40°
25.2
40.5
55
79
101
120
136
150
161
172
181
10.2
14.5
17
18.7
18.4
17.5
14.8
12.8
10.5
8
6.8
45°
18.7
32.5
45.5
67
86.5
102
119
133
146
157
166
6.36
9.5
10.5
11.7
11.1
10.0
8.01
6.8
4.79
3.3
1.8
50°
14.6
25.5
36
56.5
72.5
91
105
119
131
148
152
3.90
5.7
6.45
7.02
6.30
5.27
3.89
2.60
1.62
0.9
11.5
55°
11.4
20.4
29.8
48
65.5
80
93.8
106
119
130
139
2.18
3.20
3.78
3.84
3.22
2.36
1.46
0.743
0.201
0.006
3.45
60°
9.05
16.1
24.5
40.5
57.5
70.5
83.5
95.8
107
118
128
1.27
1.63
2.02
1.89
1.33
0.745
0.25
0.05
0.156
0.96
0
70°
5.98
11.4
17.3
30.3
43.2
56
67.5
78.8
88.5
99
108
0.324
0.382
0.354
0.165
0.025
0.0825
0.56
1.3
2.15
3.4
5.2
80°
4.18
8.18
12.7
23.1
33.8
45
55.5
65.5
75
84.0
93.2
0.023
0.009
0.093
0.24
0.51
1.27
2.5
4.2
6.0
8.2
10.6
90°
3.03
6.08
9.68
18.0
27.3
36.8
46.5
55.5
63.5
72.5
80.8
0.020
0.094
0.235
0.78
1.85
3.22
5.0
7.8
10.6
13.5
16.6
100°
2.25
4.78
7.65
14.6
22.4
31.0
39.0
47.8
56
64
71.8
0.147
0.365
0.75
1.87
3.54
5.60
8.75
11.7
14.9
18.3
22.0
110°
1.82
3.83
6.2
12.1
20.0
26.5
33.8
41.8
49.5
57
64.0
0.307
0.684
1.29
2.9
5.21
7.88
11.6
15.1
19.1
23.0
27.3
120°
1.50
3.20
5.2
10.4
16.4
23.1
30.0
36.8
44.5
51.2
59.5
0.454
0.995
1.78
3.85
6.75
9.92
13.9
18.1
22.6
26.9
32.0
130°
1.28
2.70
4.52
9.1
14.5
20.6
27.0
33.5
39.9
47.i
54
0.59
1.27
2.23
4.72
8.03
11.7
16.1
20.7
25.8
30.2
35.4
140°
1.13
2.37
4.00
8.2
13.1
18.7
24.6
30.9
37.2
43.8
50
0.70
1.51
2.63
5.40
9.15
12.3
18.0
22.9
28.5
33.2
38.7
150°
1.02
2.14
3.68
7.55
11.0
17.4
23.1
29.1
35.5
41.2
47.5
0.788
1.69
2.94
6.03
10.1
14.4
19.4
24.3
30.1
35.3
41.2
160°
0.98
2.00
3.45
7.05
11.5
16.5
22.1
27.8
33.7
39.4
46
0.850
1.82
3.14
6.53
10.6
15.2
20.4
25.7
31.5
37.0
42.8
170°
0.95
1.97
3.32
6.8
11.2
16.1
20.1
27.1
32.8
38.6
45
0.785
1.90
3.27
6.8
10.9
15.6
21.0
26.4
32.5
37.8
43.8
180°
0.92
1.93
3.28
6.78
11.0
15.2
21.2
26.8
32.5
38.2
44.5
0.888
1.93
3.31
6.90
11.0
15.6
21.1
26.7
32.5
38.2
44.2
Table II
Scattering by a transparent sphere due to transmission with two refractions,
calculated by geometrical optics.aBody of table: Scattered-intensity coefficient
I2(θ), flux
scattered per unit solid angle in direction θ per
unit geometrically incident flux.
Scattering angle
θ, (deg.)
Refractive index
1.1
1.15
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
0
9.59
4.63
2.82
1.44
0.922
0.660
0.507
0.408
0.340
0.290
0.251
5
6.51
3.81
2.50
1.36
0.891
0.646
0.498
0.404
0.337
0.288
0.250
10
2.82
2.38
1.85
1.16
0.807
0.602
0.476
0.388
0.326
0.280
0.244
15
1.14
1.29
1.22
0.916
0.691
0.540
0.437
0.363
0.310
0.269
0.236
20
0.470
0.679
0.750
0.682
0.570
0.470
0.392
0.333
0.288
0.253
0.224
25
0.203
0.457
0.492
0.452
0.389
0.344
0.299
0.264
0.235
0.211
30
0.273
0.347
0.348
0.324
0.293
0.265
0.236
0.215
0.195
35
0.153
0.241
0.264
0.260
0.245
0.228
0.211
0.195
0.179
40
0.0921
0.163
0.197
0.207
0.203
0.196
0.184
0.173
0.164
45
0.0505
0.146
0.162
0.166
0.164
0.160
0.152
0.144
50
0.0232
0.125
0.133
0.136
0.136
0.133
0.128
55
0.0092
0.106
0.113
0.114
0.114
0.111
60
0.00188
0.0962
0.0957
Polarized components are calculable from relation
I2 =
I2,1 +
I2,2 =
I2,1 (1 +
sec4θ). Values above
single-dotted lines are calculated from Ellison and Peetz7 approximation. Values below double-dotted
lines may be overshadowed by rainbows.
Table III
Comparison of intensity coefficients I1,1
(θ) of light scattered by external reflection
from transparent and absorbing spheres; direction of polarization in plane
containing incident and scattered rays.
Refractive index
Scattering angle
30°
40°
50°
67.4°
80°
1.5
0.016
0.012
0.0091
0.0060
0.0045
1.5(1 −
0.5i)
0.023
0.00194
0.0162
0.0123
0.0102
2.0
0.022
0.018
0.015
0.0113
0.0093
1.5(1 −
0.2i)
0.00712
1.7
0.0082
30°
40°
50°
84.6°
100°
1.1
0.0047
0.0025
0.00146
0.000353
0.000225
1.1(1 −
0.5i)
0.0209
0.0169
0.0136
0.00683
0.00212
1.6
0.0174
0.0136
0.0105
0.0052
0.0039
1.1(1 −
0.2i)
0.00186
1.3
0.00202
Tables (3)
Table I
Scattering by a transparent sphere due to external reflection, calculated by
geometrical optics.Upper and lower rows for each angle give polarized components
I1,1(θ) and
I1,2(θ) of
scattered-intensity coefficient
I1(θ), flux
scattered per unit solid angle in direction θ per
unit geometrically incident flux, ×104.
Scattering angle
θ, (deg)
Refractive index
1.1
1.15
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
0°
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
398
5°
273
295
308
330
337
343
349
353
356
360
360
252
265
270
275
283
280
280
275
270
269
270
10°
186
215
235
262
278
293
303
311
318
319
328
159
179
186
193
198
196
193
189
186
182
178
15°
131
160
181
212
233
250
263
274
282
297
297
100
117
129
137
136
136
133
131
126
121
117
20°
90.5
120
142
174
194
214
228
240
250
260
268
63.5
77.5
86
95
93.5
94
90.5
83.5
81
78
72
25°
65
90
110
143
166
183
199
211
225
225
241
40.3
50.8
58.1
64
65
63.5
61
56
54
53.0
45
30°
47
68.2
86.5
117
141
160
174
187
200
207
218
25.0
34.0
39.8
43
45
42.5
40.1
37
33
29.2
26.5
35°
33.8
52.4
69.5
96.5
119
140
154
168
180
189
199
16
22.2
26
29
29.0
28.5
24.2
22
19
15.7
14.0
40°
25.2
40.5
55
79
101
120
136
150
161
172
181
10.2
14.5
17
18.7
18.4
17.5
14.8
12.8
10.5
8
6.8
45°
18.7
32.5
45.5
67
86.5
102
119
133
146
157
166
6.36
9.5
10.5
11.7
11.1
10.0
8.01
6.8
4.79
3.3
1.8
50°
14.6
25.5
36
56.5
72.5
91
105
119
131
148
152
3.90
5.7
6.45
7.02
6.30
5.27
3.89
2.60
1.62
0.9
11.5
55°
11.4
20.4
29.8
48
65.5
80
93.8
106
119
130
139
2.18
3.20
3.78
3.84
3.22
2.36
1.46
0.743
0.201
0.006
3.45
60°
9.05
16.1
24.5
40.5
57.5
70.5
83.5
95.8
107
118
128
1.27
1.63
2.02
1.89
1.33
0.745
0.25
0.05
0.156
0.96
0
70°
5.98
11.4
17.3
30.3
43.2
56
67.5
78.8
88.5
99
108
0.324
0.382
0.354
0.165
0.025
0.0825
0.56
1.3
2.15
3.4
5.2
80°
4.18
8.18
12.7
23.1
33.8
45
55.5
65.5
75
84.0
93.2
0.023
0.009
0.093
0.24
0.51
1.27
2.5
4.2
6.0
8.2
10.6
90°
3.03
6.08
9.68
18.0
27.3
36.8
46.5
55.5
63.5
72.5
80.8
0.020
0.094
0.235
0.78
1.85
3.22
5.0
7.8
10.6
13.5
16.6
100°
2.25
4.78
7.65
14.6
22.4
31.0
39.0
47.8
56
64
71.8
0.147
0.365
0.75
1.87
3.54
5.60
8.75
11.7
14.9
18.3
22.0
110°
1.82
3.83
6.2
12.1
20.0
26.5
33.8
41.8
49.5
57
64.0
0.307
0.684
1.29
2.9
5.21
7.88
11.6
15.1
19.1
23.0
27.3
120°
1.50
3.20
5.2
10.4
16.4
23.1
30.0
36.8
44.5
51.2
59.5
0.454
0.995
1.78
3.85
6.75
9.92
13.9
18.1
22.6
26.9
32.0
130°
1.28
2.70
4.52
9.1
14.5
20.6
27.0
33.5
39.9
47.i
54
0.59
1.27
2.23
4.72
8.03
11.7
16.1
20.7
25.8
30.2
35.4
140°
1.13
2.37
4.00
8.2
13.1
18.7
24.6
30.9
37.2
43.8
50
0.70
1.51
2.63
5.40
9.15
12.3
18.0
22.9
28.5
33.2
38.7
150°
1.02
2.14
3.68
7.55
11.0
17.4
23.1
29.1
35.5
41.2
47.5
0.788
1.69
2.94
6.03
10.1
14.4
19.4
24.3
30.1
35.3
41.2
160°
0.98
2.00
3.45
7.05
11.5
16.5
22.1
27.8
33.7
39.4
46
0.850
1.82
3.14
6.53
10.6
15.2
20.4
25.7
31.5
37.0
42.8
170°
0.95
1.97
3.32
6.8
11.2
16.1
20.1
27.1
32.8
38.6
45
0.785
1.90
3.27
6.8
10.9
15.6
21.0
26.4
32.5
37.8
43.8
180°
0.92
1.93
3.28
6.78
11.0
15.2
21.2
26.8
32.5
38.2
44.5
0.888
1.93
3.31
6.90
11.0
15.6
21.1
26.7
32.5
38.2
44.2
Table II
Scattering by a transparent sphere due to transmission with two refractions,
calculated by geometrical optics.aBody of table: Scattered-intensity coefficient
I2(θ), flux
scattered per unit solid angle in direction θ per
unit geometrically incident flux.
Scattering angle
θ, (deg.)
Refractive index
1.1
1.15
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
0
9.59
4.63
2.82
1.44
0.922
0.660
0.507
0.408
0.340
0.290
0.251
5
6.51
3.81
2.50
1.36
0.891
0.646
0.498
0.404
0.337
0.288
0.250
10
2.82
2.38
1.85
1.16
0.807
0.602
0.476
0.388
0.326
0.280
0.244
15
1.14
1.29
1.22
0.916
0.691
0.540
0.437
0.363
0.310
0.269
0.236
20
0.470
0.679
0.750
0.682
0.570
0.470
0.392
0.333
0.288
0.253
0.224
25
0.203
0.457
0.492
0.452
0.389
0.344
0.299
0.264
0.235
0.211
30
0.273
0.347
0.348
0.324
0.293
0.265
0.236
0.215
0.195
35
0.153
0.241
0.264
0.260
0.245
0.228
0.211
0.195
0.179
40
0.0921
0.163
0.197
0.207
0.203
0.196
0.184
0.173
0.164
45
0.0505
0.146
0.162
0.166
0.164
0.160
0.152
0.144
50
0.0232
0.125
0.133
0.136
0.136
0.133
0.128
55
0.0092
0.106
0.113
0.114
0.114
0.111
60
0.00188
0.0962
0.0957
Polarized components are calculable from relation
I2 =
I2,1 +
I2,2 =
I2,1 (1 +
sec4θ). Values above
single-dotted lines are calculated from Ellison and Peetz7 approximation. Values below double-dotted
lines may be overshadowed by rainbows.
Table III
Comparison of intensity coefficients I1,1
(θ) of light scattered by external reflection
from transparent and absorbing spheres; direction of polarization in plane
containing incident and scattered rays.