Stochastic algorithms are a promising method for the synthesis of optical multilayer systems. A method based on the use of genetic algorithms is described and applied to the design of three very different optical filters. Solutions found by genetic algorithms are refined, and results are compared with those of previous publications.
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ontinuous variation of n, total optical thickness 32 μm.
Continuous variation of n, total optical thickness 25 μm.
Discrete variation of n, total optical thickness 32 μm.
Table 2
Two-Material Equivalents of the Systems Found by GA for Antireflection Coatings Presented in Table 1
System B1
System B2
Layer
t (μm)
n
t (μm)
n
Air
1
1
1
1.0744
2.20
0.1392
2.20
2
0.6527
4.20
0.0001
4.20
3
0.8223
2.20
0.9545
2.20
4
0.0696
4.20
0.4678
4.20
5
0.6881
2.20
0.0304
2.20
6
0.0758
4.20
0.1806
4.20
7
0.3616
2.20
0.3327
2.20
8
0.4687
4.20
0.2118
4.20
9
0.0352
2.20
0.5186
2.20
10
0.4687
4.20
0.0948
4.20
11
0.0181
2.20
0.3509
2.20
12
0.2328
4.20
0.1620
4.20
13
0.4120
2.20
0.0063
2.20
14
0.1768
4.20
0.2293
4.20
15
0.3352
2.20
0.0241
2.20
16
0.1645
4.20
0.2285
4.20
17
0.2628
2.20
0.0419
2.20
18
0.1559
4.20
0.1524
4.20
19
0.3779
2.20
0.0076
2.20
20
0.2304
4.20
0.3284
4.20
21
0.2947
2.20
0.0096
2.20
22
0.1514
4.20
0.1106
4.20
23
0.2608
2.20
0.1811
2.20
24
0.5221
4.20
0.1565
4.20
25
0.0842
2.20
0.1476
2.20
26
0.5420
4.20
0.2064
4.20
27
0.1907
2.20
0.1064
2.20
28
0.3561
4.20
0.1205
4.20
29
0.1564
2.20
0.1372
2.20
30
0.3811
4.20
0.1628
4.20
31
0.0712
2.20
0.2716
2.20
32
0.1074
4.20
33
0.1694
2.20
34
0.1544
4.20
35
0.3506
2.20
36
0.1040
4.20
37
0.2889
2.20
38
0.2493
4.20
39
0.1964
2.20
40
0.1247
4.20
Substrate
—
4.00
—
4.00
Table 3
Construction Parameters of the Systems Found by GA (Table 1) Transformed into Two-Material Systems (Table 2), after Refinement by the Gradient Method, for Antireflection Coatings
System C1
System C2
System C3
Layer
t (μm)
n
t (μm)
n
t (μm)
n
Air
—
1
—
1
—
1
1
1.0873
2.20
1.0874
2.20
1.0865
2.20
2
0.6009
4.20
0.6081
4.20
0.6028
4.20
3
0.5333
2.20
0.5150
2.20
0.5332
2.20
4
0.1030
4.20
0.1123
4.20
0.1031
4.20
5
1.2600
2.20
1.2527
2.20
1.2573
2.20
6
0.6627
4.20
0.6523
4.20
0.6619
4.20
7
0.2082
2.20
0.1668
2.20
0.2055
2.20
8
0.2666
4.20
0.3147
4.20
0.2638
4.20
9
1.2431
2.20
1.2562
2.20
1.2822
2.20
10
0.2858
4.20
0.2478
4.20
0.2762
4.20
11
0.1300
2.20
0.2699
2.20
0.1299
2.20
12
0.7600
4.20
0.6677
4.20
0.7851
4.20
13
1.0067
2.20
0.7109
2.20
0.8793
2.20
14
0.1117
4.20
0.0975
4.20
0.1488
4.20
15
0.6517
2.20
1.1170
2.20
0.5978
2.20
16
1.3928
4.20
0.2508
4.20
0.9896
4.20
17
0.0777
2.20
0.2383
2.20
0.0222
2.20
18
0.1006
4.20
0.4427
4.20
19
0.1100
2.20
20
0.1677
4.20
Substrate
—
4.00
—
4.00
—
4.00
∑ nt (μm)
31.2
27.3
32.1
Merit function
0.65
0.70
0.66
Average reflectance (%)
0.62
0.66
0.62
Table 4
Solution Found by GA before (D1) and after (D2) Refinement for Beam Splitters
System D1
System D2
Layer
t (μm)
n
t (μm)
n
Air
—
1
—
1
1
0.0609
2.347
0.0617
2.350
2
0.0164
1.402
0.0172
1.350
3
0.0895
1.350
0.0916
1.350
4
0.0498
2.349
0.0523
2.350
5
0.0748
2.218
0.0740
2.226
6
0.0900
1.607
0.0939
1.566
7
0.0314
1.840
0.0332
1.764
8
0.0419
1.732
0.0456
1.678
9
0.0294
2.309
0.0338
2.317
10
0.0906
2.309
0.0931
2.350
11
0.0926
2.088
0.1015
1.953
12
0.0425
2.223
0.0500
1.969
13
0.0306
2.262
0.0311
2.273
14
0.0172
2.311
0.0182
2.281
15
0.0238
2.273
0.0261
2.217
16
0.0479
1.996
0.0448
2.273
17
0.0504
1.788
0.0436
1.944
18
0.0811
1.543
0.0592
1.816
19
0.0158
1.559
0.0173
1.423
20
0.0625
1.471
0.0549
1.615
Substrate
—
1.52
—
1.52
∑ nt (μm)
2.00
2.03
Merit function
0.41
0.06
Average reflectance (%)
50.06
50.02
Table 5
Construction Parameters of the System Synthesized by GA before (E1) and after (E2) Refinement for Rejection Filters
System E1
System E2
Layer
t (μm)
n
t (μm)
n
Air
—
1
—
1
1
0.1669
1.350
0.1122
1.377
2
0.3389
1.359
0.2902
1.709
3
0.2015
1.626
0.1943
1.883
4
0.3878
1.358
0.3442
1.567
5
0.2509
1.546
0.2097
1.758
6
0.3489
1.554
0.3041
1.631
7
0.2423
1.699
0.2508
1.733
8
0.2969
1.673
0.3148
1.626
9
0.4931
1.886
0.5159
1.802
10
0.3465
2.200
0.3591
2.132
11
0.2257
1.807
0.2317
1.817
12
0.2484
2.026
0.2296
2.140
13
0.4758
1.523
0.4651
1.567
14
0.3746
2.048
0.3454
2.200
15
0.5393
1.389
0.5551
1.350
16
0.2439
1.774
0.2418
1.852
17
0.3178
1.447
0.2987
1.479
18
0.1804
2.058
0.1721
2.109
19
0.1713
2.197
0.1793
2.200
20
0.5426
1.630
0.6521
1.350
21
0.1207
1.419
0.1177
1.350
22
0.2230
1.968
0.2080
2.126
23
0.1702
1.892
0.1788
1.944
24
0.4466
1.590
0.4168
1.629
25
0.3099
2.030
0.2933
2.133
26
0.0623
2.084
0.0618
2.048
27
0.0274
1.923
0.0254
1.966
28
0.3775
1.663
0.3731
1.642
29
0.1829
1.902
0.1886
1.938
30
0.2401
2.110
0.2321
2.134
31
0.2562
1.976
0.2398
2.100
32
0.1313
1.888
0.1351
1.904
33
0.4719
1.646
0.4654
1.698
34
0.5428
1.887
0.5071
1.987
35
0.4973
1.514
0.4533
1.777
36
0.1531
1.830
0.1374
2.068
37
0.1670
1.994
0.1687
1.953
38
0.2186
1.923
0.2462
1.805
39
0.4772
1.553
0.4438
1.692
40
0.1372
1.685
0.1340
1.676
Substrate
—
1.52
—
1.52
Tables (5)
Table 1
Construction Parameters of Solutions Found by GA for Antireflection Coatings
ontinuous variation of n, total optical thickness 32 μm.
Continuous variation of n, total optical thickness 25 μm.
Discrete variation of n, total optical thickness 32 μm.
Table 2
Two-Material Equivalents of the Systems Found by GA for Antireflection Coatings Presented in Table 1
System B1
System B2
Layer
t (μm)
n
t (μm)
n
Air
1
1
1
1.0744
2.20
0.1392
2.20
2
0.6527
4.20
0.0001
4.20
3
0.8223
2.20
0.9545
2.20
4
0.0696
4.20
0.4678
4.20
5
0.6881
2.20
0.0304
2.20
6
0.0758
4.20
0.1806
4.20
7
0.3616
2.20
0.3327
2.20
8
0.4687
4.20
0.2118
4.20
9
0.0352
2.20
0.5186
2.20
10
0.4687
4.20
0.0948
4.20
11
0.0181
2.20
0.3509
2.20
12
0.2328
4.20
0.1620
4.20
13
0.4120
2.20
0.0063
2.20
14
0.1768
4.20
0.2293
4.20
15
0.3352
2.20
0.0241
2.20
16
0.1645
4.20
0.2285
4.20
17
0.2628
2.20
0.0419
2.20
18
0.1559
4.20
0.1524
4.20
19
0.3779
2.20
0.0076
2.20
20
0.2304
4.20
0.3284
4.20
21
0.2947
2.20
0.0096
2.20
22
0.1514
4.20
0.1106
4.20
23
0.2608
2.20
0.1811
2.20
24
0.5221
4.20
0.1565
4.20
25
0.0842
2.20
0.1476
2.20
26
0.5420
4.20
0.2064
4.20
27
0.1907
2.20
0.1064
2.20
28
0.3561
4.20
0.1205
4.20
29
0.1564
2.20
0.1372
2.20
30
0.3811
4.20
0.1628
4.20
31
0.0712
2.20
0.2716
2.20
32
0.1074
4.20
33
0.1694
2.20
34
0.1544
4.20
35
0.3506
2.20
36
0.1040
4.20
37
0.2889
2.20
38
0.2493
4.20
39
0.1964
2.20
40
0.1247
4.20
Substrate
—
4.00
—
4.00
Table 3
Construction Parameters of the Systems Found by GA (Table 1) Transformed into Two-Material Systems (Table 2), after Refinement by the Gradient Method, for Antireflection Coatings
System C1
System C2
System C3
Layer
t (μm)
n
t (μm)
n
t (μm)
n
Air
—
1
—
1
—
1
1
1.0873
2.20
1.0874
2.20
1.0865
2.20
2
0.6009
4.20
0.6081
4.20
0.6028
4.20
3
0.5333
2.20
0.5150
2.20
0.5332
2.20
4
0.1030
4.20
0.1123
4.20
0.1031
4.20
5
1.2600
2.20
1.2527
2.20
1.2573
2.20
6
0.6627
4.20
0.6523
4.20
0.6619
4.20
7
0.2082
2.20
0.1668
2.20
0.2055
2.20
8
0.2666
4.20
0.3147
4.20
0.2638
4.20
9
1.2431
2.20
1.2562
2.20
1.2822
2.20
10
0.2858
4.20
0.2478
4.20
0.2762
4.20
11
0.1300
2.20
0.2699
2.20
0.1299
2.20
12
0.7600
4.20
0.6677
4.20
0.7851
4.20
13
1.0067
2.20
0.7109
2.20
0.8793
2.20
14
0.1117
4.20
0.0975
4.20
0.1488
4.20
15
0.6517
2.20
1.1170
2.20
0.5978
2.20
16
1.3928
4.20
0.2508
4.20
0.9896
4.20
17
0.0777
2.20
0.2383
2.20
0.0222
2.20
18
0.1006
4.20
0.4427
4.20
19
0.1100
2.20
20
0.1677
4.20
Substrate
—
4.00
—
4.00
—
4.00
∑ nt (μm)
31.2
27.3
32.1
Merit function
0.65
0.70
0.66
Average reflectance (%)
0.62
0.66
0.62
Table 4
Solution Found by GA before (D1) and after (D2) Refinement for Beam Splitters
System D1
System D2
Layer
t (μm)
n
t (μm)
n
Air
—
1
—
1
1
0.0609
2.347
0.0617
2.350
2
0.0164
1.402
0.0172
1.350
3
0.0895
1.350
0.0916
1.350
4
0.0498
2.349
0.0523
2.350
5
0.0748
2.218
0.0740
2.226
6
0.0900
1.607
0.0939
1.566
7
0.0314
1.840
0.0332
1.764
8
0.0419
1.732
0.0456
1.678
9
0.0294
2.309
0.0338
2.317
10
0.0906
2.309
0.0931
2.350
11
0.0926
2.088
0.1015
1.953
12
0.0425
2.223
0.0500
1.969
13
0.0306
2.262
0.0311
2.273
14
0.0172
2.311
0.0182
2.281
15
0.0238
2.273
0.0261
2.217
16
0.0479
1.996
0.0448
2.273
17
0.0504
1.788
0.0436
1.944
18
0.0811
1.543
0.0592
1.816
19
0.0158
1.559
0.0173
1.423
20
0.0625
1.471
0.0549
1.615
Substrate
—
1.52
—
1.52
∑ nt (μm)
2.00
2.03
Merit function
0.41
0.06
Average reflectance (%)
50.06
50.02
Table 5
Construction Parameters of the System Synthesized by GA before (E1) and after (E2) Refinement for Rejection Filters