J. F. Tang and Q. Zheng, "Automatic design of optical thin-film systems—merit function and numerical optimization method," J. Opt. Soc. Am. 72, 1522-1528 (1982)
A combination of analytical synthesis methods and automatic numerical design is currently used for the synthesis of optical thin-film systems. The validity of this combination strongly depends not only on the optimization technique chosen but also on the construction of the merit function used in the automatic design. The present paper introduces a statistical testing method by which global extrema of a merit function can be found with a certain probability. Some examples of the construction of the merit function based on analytical synthesis are given.
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High Reflector Using Gradient Method and Statistical Testinga
Gradient
Statistical Testing
Layer
Starting Design
Steps
Initial Search Region
Steps
10
20
10
18
46
1 H
0.6000
1.0207
1.0000
0.0 – 2.000
0.9743
1.0010
1.0005
2 L
0.4000
0.9653
1.0000
0.0 – 2.000
1.0174
1.0032
1.0005
3 H
0.6000
0.9803
1.0000
0.0 – 2.000
0.9845
0.9971
1.0005
4 L
1.2000
0.9932
1.0001
0.0 – 2.000
1.0182
1.0066
1.0005
5 H
0.8000
1.0319
1.0000
0.0 – 2.000
0.9331
0.9905
1.0005
6 L
1.6000
1.0380
1.0000
0.0 – 2.000
1.0490
1.0096
1.0005
7 H
1.6000
1.0445
1.0000
0.0 – 2.000
1.0207
1.0091
1.0005
Target: R = 1.0; one point only at λ = 1 μm The refractive indices are n(H) = 2.3, n(L) = 1.45, n(S) = 1.52, air medium. Thicknesses are in units of quarter-waves centered at 1 μm.
Table 2
Same Optimization Problem as in Table 1 Except that the Target is R = 0.93, 11 Equally Spaced Points from λ = 0.9–1.1 μm
Gradient
Statistical Testing
Layer
Starting Design
Steps
Initial Search Region
Steps
10
20
10
18
1 H
0.6000
0.7859
0.9696
0.0 – 2.000
0.9688
0.9915
2 L
0.4000
1.1024
0.9920
0.0 – 2.000
0.9668
0.9869
3 H
0.6000
1.2543
0.9941
0.0 – 2.000
1.0035
0.9931
4 L
1.2000
0.9142
0.9918
0.0 – 2.000
0.9995
0.9927
5 H
0.8000
0.9162
1.0014
0.0 – 2.000
1.0046
0.9854
6 L
1.6000
0.9523
0.9865
0.0 – 2.000
0.9754
0.9919
7 H
1.6000
0.8642
0.9828
0.0 – 2.000
1.0014
0.9885
Q.M.S. × 103 =
9.0785
-
4.1953
4.1150
Table 3
Broadband Antireflection Coatings for the Visiblea
(Q.M.S. consists of 31 equally spaced points at λ = 400–700 nm)
2.48
1.73
1.32
1.30
2.073
0.1103
Thicknesses are in units of quarter-waves centered at 1 μm. n(S) = 1.52, n(L) = 1.45, n(H) = dispersive about 2.4 or fixed at 2.4.
The reflectance of this design is <0.2% over the range λ = 420–620 nm and is 0.25% at λ = 660 nm.
The reflectance is <0.2% over the range λ = 410–675 nm.
Thicknesses are in units of quarter-waves centered at 1 μm designed by the statistical testing program. n(s) = 1.52, n(H) = 2.3, n(L) = 1.45, air medium. Q.M.S. = 6.734 × 10−3, reflectances = 0.47–0.53 in wavelengths 0.9–1.1 μm.
Table 5
Determination of Optical Constants of a Metal Layer
Example 1
Example 2
Expected parameters
n
0.033
0.033
k
3.35
3.35
Assumed reflectance readings
R0(θ0)
0.0126
0.0126
R1(θ1)
0.5659
0.5659
R2(θ2)
0.3969
0.3969
Search region
n
0 – 0.1
0 – 0.3
k
3.0 – 4.0
2.0 – 5.0
Final solution
n
0.033
0.033
k
3.35
3.35
Computed reflectances
R0(θ0)
0.0124
0.0127
R1(θ1)
0.5637
0.5664
R2(θ2)
0.3978
0.3968
Table 6
Determination of Optical Constants of a Dielectric Layer
Expected parameters
n
2.0
d
10.0
Assumed reflectance readings
R0(θ0)
0.0037
R1(θ1)
0.4757
R2(θ2)
0.3791
Search region
n
1.7 – 2.3
d
5.0 – 15.0
Final solution
n
1.977
d
10.157
Computed reflectances
R0(θ0)
0.0037
R1(θ1)
0.4762
R2(θ2)
0.3816
Tables (6)
Table 1
High Reflector Using Gradient Method and Statistical Testinga
Gradient
Statistical Testing
Layer
Starting Design
Steps
Initial Search Region
Steps
10
20
10
18
46
1 H
0.6000
1.0207
1.0000
0.0 – 2.000
0.9743
1.0010
1.0005
2 L
0.4000
0.9653
1.0000
0.0 – 2.000
1.0174
1.0032
1.0005
3 H
0.6000
0.9803
1.0000
0.0 – 2.000
0.9845
0.9971
1.0005
4 L
1.2000
0.9932
1.0001
0.0 – 2.000
1.0182
1.0066
1.0005
5 H
0.8000
1.0319
1.0000
0.0 – 2.000
0.9331
0.9905
1.0005
6 L
1.6000
1.0380
1.0000
0.0 – 2.000
1.0490
1.0096
1.0005
7 H
1.6000
1.0445
1.0000
0.0 – 2.000
1.0207
1.0091
1.0005
Target: R = 1.0; one point only at λ = 1 μm The refractive indices are n(H) = 2.3, n(L) = 1.45, n(S) = 1.52, air medium. Thicknesses are in units of quarter-waves centered at 1 μm.
Table 2
Same Optimization Problem as in Table 1 Except that the Target is R = 0.93, 11 Equally Spaced Points from λ = 0.9–1.1 μm
Gradient
Statistical Testing
Layer
Starting Design
Steps
Initial Search Region
Steps
10
20
10
18
1 H
0.6000
0.7859
0.9696
0.0 – 2.000
0.9688
0.9915
2 L
0.4000
1.1024
0.9920
0.0 – 2.000
0.9668
0.9869
3 H
0.6000
1.2543
0.9941
0.0 – 2.000
1.0035
0.9931
4 L
1.2000
0.9142
0.9918
0.0 – 2.000
0.9995
0.9927
5 H
0.8000
0.9162
1.0014
0.0 – 2.000
1.0046
0.9854
6 L
1.6000
0.9523
0.9865
0.0 – 2.000
0.9754
0.9919
7 H
1.6000
0.8642
0.9828
0.0 – 2.000
1.0014
0.9885
Q.M.S. × 103 =
9.0785
-
4.1953
4.1150
Table 3
Broadband Antireflection Coatings for the Visiblea
(Q.M.S. consists of 31 equally spaced points at λ = 400–700 nm)
2.48
1.73
1.32
1.30
2.073
0.1103
Thicknesses are in units of quarter-waves centered at 1 μm. n(S) = 1.52, n(L) = 1.45, n(H) = dispersive about 2.4 or fixed at 2.4.
The reflectance of this design is <0.2% over the range λ = 420–620 nm and is 0.25% at λ = 660 nm.
The reflectance is <0.2% over the range λ = 410–675 nm.
Thicknesses are in units of quarter-waves centered at 1 μm designed by the statistical testing program. n(s) = 1.52, n(H) = 2.3, n(L) = 1.45, air medium. Q.M.S. = 6.734 × 10−3, reflectances = 0.47–0.53 in wavelengths 0.9–1.1 μm.
Table 5
Determination of Optical Constants of a Metal Layer
Example 1
Example 2
Expected parameters
n
0.033
0.033
k
3.35
3.35
Assumed reflectance readings
R0(θ0)
0.0126
0.0126
R1(θ1)
0.5659
0.5659
R2(θ2)
0.3969
0.3969
Search region
n
0 – 0.1
0 – 0.3
k
3.0 – 4.0
2.0 – 5.0
Final solution
n
0.033
0.033
k
3.35
3.35
Computed reflectances
R0(θ0)
0.0124
0.0127
R1(θ1)
0.5637
0.5664
R2(θ2)
0.3978
0.3968
Table 6
Determination of Optical Constants of a Dielectric Layer