A solution to the problem of determining the refractivity of n identical planar arrays from the properties of a single array is obtained by first extending Stokes’ equations to a form appropriate to a microscopic situation and then appropriately modifying Crook’s results for thin films. The results of such development are found to be in agreement with data taken at microwave frequencies.
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Estimated errors in δn owing to errors in interplanar spacingc
1
32.00
(32.00)
(32.00)
(32.00)
2
75.60
80.02
80.02
80.02
+0
−15.5
3
129.9
123.7
163.5
121.4
+1.0
−17.2
4
183.5
167.0
190.9
148.4
+17.5
−11.4
5
208.8
206.6
209.1
190.5
+22.0
−14.6
6
232.3
221.2
221.3
238.9
+23.6
−20.4
7
286.5
238.1
229.9
272.3
+31.2
−18.8
8
347.2
288.1
236.1
330.2
+28.0
−25.4
9
375.2
397.8
240.8
406.8
+20.9
−37.7
10
404.2
438.3
244.5
415.2
+25.3
−29.2
nth value computed from n−1 experimental value, d=0.294 inch, 2β=179.23 degrees. nth value computed from n−1 computed value; all computed values based ultimately on observed value for one plane.
The errors indicated in the last column were obtained by propagating positive and negative standard deviations in d through Eq. (10). Since ψnt is not a monotonic function of d such procedure is not completely reliable; it is used here to avoid the lengthy computations that would be required to find possible extremal values of ψnt(d) in each range of d+∊ ≤ d ≤ d−∊.
Table IV
Direct verification of t0=cosδ1.
18° angle of incidence
Normal incidence
Frequency (kMc/sec)
−20 log cosδ1 (db)
Observed transmission loss (db)
−20 log cosδ1 (db)
Observed transmission loss (db)
8.20
0.10
0.05
0.10
0.05
8.50
0.14
0.20
0.11
0.25
9.00
0.14
0.20
0.14
0.15
9.50
0.21
0.30
0.19
0.25
10.00
0.22
0.20
0.18
0.20
10.50
0.27
0.40
0.24
0.30
11.00
0.33
0.40
0.29
0.35
11.50
0.42
0.50
0.54
0.40
12.00
0.53
0.70
0.45
0.70
12.40
0.61
0.60
0.68
0.55
Tables (4)
Table I
Comparing computed and observed phase shifts, cubic lattice of dielectric spheres. (nth value computed from n−1 experimental value.)
Estimated errors in δn owing to errors in interplanar spacingc
1
32.00
(32.00)
(32.00)
(32.00)
2
75.60
80.02
80.02
80.02
+0
−15.5
3
129.9
123.7
163.5
121.4
+1.0
−17.2
4
183.5
167.0
190.9
148.4
+17.5
−11.4
5
208.8
206.6
209.1
190.5
+22.0
−14.6
6
232.3
221.2
221.3
238.9
+23.6
−20.4
7
286.5
238.1
229.9
272.3
+31.2
−18.8
8
347.2
288.1
236.1
330.2
+28.0
−25.4
9
375.2
397.8
240.8
406.8
+20.9
−37.7
10
404.2
438.3
244.5
415.2
+25.3
−29.2
nth value computed from n−1 experimental value, d=0.294 inch, 2β=179.23 degrees. nth value computed from n−1 computed value; all computed values based ultimately on observed value for one plane.
The errors indicated in the last column were obtained by propagating positive and negative standard deviations in d through Eq. (10). Since ψnt is not a monotonic function of d such procedure is not completely reliable; it is used here to avoid the lengthy computations that would be required to find possible extremal values of ψnt(d) in each range of d+∊ ≤ d ≤ d−∊.
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