The design steps for film–substrate single-reflection retarders are briefly stated and applied to the SiO2–Si film–substrate system at wavelength 6328 Å. The criterion of minimum-maximum error of the ellipsometric angle ψ is used to choose angle-of-incidence-tunable designs. Use is made of the (ϕ-d) plane (angle of incidence versus thickness) to determine whether a given film–substrate system with known optical properties and film thickness can operate as a reflection retarder and to determine the associated angles of incidence and retardation angles. This leads to the concept of permissible-thickness bands and forbidden gaps for operation of a film–substrate system as a reflection retarder. Experimental measurements on one of the proposed designs proved the validity of the method.
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This table gives, for each value of the retardation angle Δ, the angle of incident ϕ′ and the least film thickness d′ necessary for operation of the SiO2–Si film–substrate system as an exact reflection retarder at 6328 Å and the associated film-thickness period Dϕ′ and reflectances ℛp and ℛs (note that ℛp =ℛs).a
Δ (deg)
ϕ′ (deg)
d′ (Å)
Dϕ′ (Å)
ℛp
ℛs
10
87.506
1087.9
2971.9
0.8191
0.8191
20
85.016
1086.6
2964.5
0.6728
0.6728
30
82.530
1084.5
2952.4
0.5555
0.5555
40
80.048
1081.6
2935.8
0.4621
0.4621
50
77.562
1077.9
2915.2
0.3880
0.3380
60
75.063
1073.5
2890.7
0.3291
0.3291
70
72.534
1068.5
2862.6
0.2822
0.2822
80
69.953
1062.7
2831.0
0.2448
0.2448
90
67.290
1056.3
2795.9
0.2148
0.2148
100
64.505
1049.1
2757.2
0.1907
0.1907
110
61.547
1041.1
2714.5
0.1711
0.1711
120
58.341
1032.1
2667.3
0.1551
0.1551
130
54.779
1022.0
2614.8
0.1420
0.1420
140
50.690
1010.5
2555.5
0.1311
0.1311
150
45.779
997.1
2487.4
0.1220
0.1220
160
39.441
981.0
2406.9
0.1141
0.1141
170
30.035
960.9
2306.9
0.1069
0.1069
−10
87.512
1880.1
2971.9
0.8192
0.8192
−20
85.028
1874.1
2964.5
0.6730
0.6730
−30
82.547
1864.2
2952.5
0.5558
0.5558
−40
80.070
1850.7
2936.0
0.4624
0.4624
−50
77.590
1833.8
2915.4
0.3882
0.3882
−60
75.097
1813.8
2891.0
0.3293
0.3293
−70
72.573
1790.9
2863.0
0.2825
0.2825
−80
69.997
1765.1
2831.5
0.2451
0.2451
−90
67.340
1736.6
2796.6
0.2151
0.2151
−100
64.561
1705.3
2758.0
0.1909
0.1909
−110
61.608
1670.7
2715.4
0.1713
0.1713
−120
58.407
1632.7
2668.3
0.1553
0.1553
−130
54.850
1590.3
2615.8
0.1422
0.1422
−140
50.768
1542.7
2556.6
0.1314
0.1314
−150
45.862
1488.2
2488.5
0.1222
0.1222
−160
39.528
1423.8
2408.0
0.1143
0.1143
−170
30.122
1343.9
2307.7
0.1071
0.1071
By combining a number of reflections (e.g., two or three), we can use the results of this table to generate new multiple-reflection retarders that maintain the parallelism or collinearity of the incident and emergent beams.
TABLE II
Angle-of-incidence-tunable Si–SiO2 retarders at 6328 Å that achieve minimum-maximum error of ψ over selected angle-of-incidence ranges 0°–45°, 45°–90°, and 0°–90°.
range of ϕ (degrees)
d′ (angstroms)
maximum error in (ψ–45°) (degrees)
ϕ′ (degrees)
0–45
981.01
0.61055
39.441
45–90
1041.1
2.13
61.547
0–90
1041.1
2.16
61.547
TABLE III
Permissible-thickness bands and their widths (in angstroms) and the retardation sign for operation of the Si–SiO2 system as an exact reflection retarder at 6328 Å.
dmin (Å)
dmax (Å)
bandwidth (Å)
retardation sign
933
1089
156
+
1232
1883
651
−
3100.125
3399.125
299
+
3399.125
4063.340
664.215
±
4063.340
4857.340
794
−
5267.250
5566.250
299
+
5566.250
7037.680
1471.430
±
7037.680
7434.375
396.695
−
7434.375
∞
∞
±
TABLE IV
Forbidden-thickness gaps and their bandwidths (in angstroms) for operation of the Si–SiO2 system as an exact reflection retarder at 6328 Å.
dmin (Å)
dmax (Å)
bandwidth (Å)
0
933
933
1089.000
1232.000
143.000
1883.000
3100.125
1217.125
4857.340
5267.250
409.910
Tables (4)
TABLE I
This table gives, for each value of the retardation angle Δ, the angle of incident ϕ′ and the least film thickness d′ necessary for operation of the SiO2–Si film–substrate system as an exact reflection retarder at 6328 Å and the associated film-thickness period Dϕ′ and reflectances ℛp and ℛs (note that ℛp =ℛs).a
Δ (deg)
ϕ′ (deg)
d′ (Å)
Dϕ′ (Å)
ℛp
ℛs
10
87.506
1087.9
2971.9
0.8191
0.8191
20
85.016
1086.6
2964.5
0.6728
0.6728
30
82.530
1084.5
2952.4
0.5555
0.5555
40
80.048
1081.6
2935.8
0.4621
0.4621
50
77.562
1077.9
2915.2
0.3880
0.3380
60
75.063
1073.5
2890.7
0.3291
0.3291
70
72.534
1068.5
2862.6
0.2822
0.2822
80
69.953
1062.7
2831.0
0.2448
0.2448
90
67.290
1056.3
2795.9
0.2148
0.2148
100
64.505
1049.1
2757.2
0.1907
0.1907
110
61.547
1041.1
2714.5
0.1711
0.1711
120
58.341
1032.1
2667.3
0.1551
0.1551
130
54.779
1022.0
2614.8
0.1420
0.1420
140
50.690
1010.5
2555.5
0.1311
0.1311
150
45.779
997.1
2487.4
0.1220
0.1220
160
39.441
981.0
2406.9
0.1141
0.1141
170
30.035
960.9
2306.9
0.1069
0.1069
−10
87.512
1880.1
2971.9
0.8192
0.8192
−20
85.028
1874.1
2964.5
0.6730
0.6730
−30
82.547
1864.2
2952.5
0.5558
0.5558
−40
80.070
1850.7
2936.0
0.4624
0.4624
−50
77.590
1833.8
2915.4
0.3882
0.3882
−60
75.097
1813.8
2891.0
0.3293
0.3293
−70
72.573
1790.9
2863.0
0.2825
0.2825
−80
69.997
1765.1
2831.5
0.2451
0.2451
−90
67.340
1736.6
2796.6
0.2151
0.2151
−100
64.561
1705.3
2758.0
0.1909
0.1909
−110
61.608
1670.7
2715.4
0.1713
0.1713
−120
58.407
1632.7
2668.3
0.1553
0.1553
−130
54.850
1590.3
2615.8
0.1422
0.1422
−140
50.768
1542.7
2556.6
0.1314
0.1314
−150
45.862
1488.2
2488.5
0.1222
0.1222
−160
39.528
1423.8
2408.0
0.1143
0.1143
−170
30.122
1343.9
2307.7
0.1071
0.1071
By combining a number of reflections (e.g., two or three), we can use the results of this table to generate new multiple-reflection retarders that maintain the parallelism or collinearity of the incident and emergent beams.
TABLE II
Angle-of-incidence-tunable Si–SiO2 retarders at 6328 Å that achieve minimum-maximum error of ψ over selected angle-of-incidence ranges 0°–45°, 45°–90°, and 0°–90°.
range of ϕ (degrees)
d′ (angstroms)
maximum error in (ψ–45°) (degrees)
ϕ′ (degrees)
0–45
981.01
0.61055
39.441
45–90
1041.1
2.13
61.547
0–90
1041.1
2.16
61.547
TABLE III
Permissible-thickness bands and their widths (in angstroms) and the retardation sign for operation of the Si–SiO2 system as an exact reflection retarder at 6328 Å.
dmin (Å)
dmax (Å)
bandwidth (Å)
retardation sign
933
1089
156
+
1232
1883
651
−
3100.125
3399.125
299
+
3399.125
4063.340
664.215
±
4063.340
4857.340
794
−
5267.250
5566.250
299
+
5566.250
7037.680
1471.430
±
7037.680
7434.375
396.695
−
7434.375
∞
∞
±
TABLE IV
Forbidden-thickness gaps and their bandwidths (in angstroms) for operation of the Si–SiO2 system as an exact reflection retarder at 6328 Å.