When this research was performed, the authors were with the Department of Applied Physics, University of Calcutta, 92 A P C Road, Calcutta 700 009, India.
S. Banerjee is currently with the Institute of Applied Physics, University of Tsukuba, Tennodai 1-1-1, Ibaraki 305-8573, Japan.
Saswatee Banerjee and Lakshminarayan Hazra, "Experiments with a genetic algorithm for structural design of cemented doublets with prespecified aberration targets," Appl. Opt. 40, 6265-6273 (2001)
The authors use a genetic algorithm in the structural design of
cemented doublets in accordance with a prespecified set of Gaussian
characteristics and primary aberration targets. Actual glasses are
taken as variables. Suitable modifications in the algorithm are
sought with which to explore the solution space in search of the global
as well as useful local optima. Some illustrative examples of
structural design are given.
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Glass Combinations for f/4, 400-mm
Focal-Length Aplanatic, Achromatic Cemented
Doubletsa
Generation Number
Optimum Configuration
Fitness Value
X
1
k̅
1
Glass
Achieved Aberration (λ)
glass 1
glass 2
S
1
/8
SIIc/2
CL/2
16
-0.52
2.31
Ba K2
SF 2
-1.2
-2.7
0
0.10
21
-0.04
2.27
BK 7
F 1
-0.5
2.5
-1.1
0.11
104
-0.01
2.27
BK 1
F 1
-0.3
2.9
1.3
0.09
107
-0.08
2.48
Ba K 5
F 6
-1.2
2.4
2.1
0.08
126
-0.04
1.96
K 7
SF 1
-0.2
1.0
-1
0.33
129
-0.09
2.12
BK 7
SF 2
-1.7
1.1
-0.1
0.19
130
-0.09
2.10
BaLF 4
SF 3
-1.7
0.5
0.1
0.24
207
-4.87
-1.00
SF 3
K 10
-0.2
0.3
-0.5
0.73
209
-0.26
2.58
SK 3
BaSF 2
-1.2
0.2
-1.7
0.18
359
-3.24
-1.13
SF 2
BK 1
1.6
-2.1
0.7
0.12
361
-3.45
-1.22
F 1
BK 7
-0.3
0.4
2.4
0.15
384
-2.45
-1.60
F 7
BaK 1
-0.3
-0.8
1.1
0.34
388
-2.37
-1.50
SF 2
BaK 4
-0.1
-2.6
1.0
0.11
389
-2.37
-1.45
F 6
BaK 2
0.5
-3.6
0.2
0.07
391
-2.13
-1.79
BaSF 2
SK 6
-0.9
-2.1
-2.5
0.08
400
-4.844
-0.999
SF 3
K 10
0.0
0.2
0.1
0.96
Lagrange invariant, 0.218 mm; aberration
targets (in λ), (S1/8)t= 0, (SIIc/2)t = 0,
(CL/2)t = 0; accuracy in shape
and normalized power variables, 0.0001. Boundaries for the shape
and normalized power
variables: (X1)max =
5.0, (X1)min = -5.0,
(k̅1)max = 5.0,
(k̅1)min = -5.0. GA
parameters: Popsize, 500; maxgen, 400; maximum fitness value in the
initial population, 0.00447; average fitness value in the initial
population, 0.00003; overall chromosome length (L) = 46
bits.
Table 2
Solutions for f/4, 400-mm Focal-Length
Cemented Doublet with Nonzero Aberration
Targetsa
Generation Number
Optimum Configuration
Fitness Value
X
1
k̅
1
Glass
Achieved Aberrations (λ)
glass 1
glass 2
S
1
/8
SIIc/2
CL/2
9
-1.67
-2.03
SF 2
KF 6
-7
-10
-10
0.12
17
-1.91
-1.78
F 4
K 5
-10.6
-6.1
-10.2
0.06
18
-1.89
-1.80
F 6
K 10
-12.1
-8.6
-7.1
0.06
24
-1.91
-1.77
F 2
K 3
-9.8
-6.6
-9.3
0.07
63
-1.97
-1.75
BaSF 2
K 3
-9.8
-8.0
-12.0
0.11
101
-1.99
-1.73
BaSF 2
K 5
-9.1
-7.6
-13.0
0.06
250
-1.91
-1.79
BaSF10
BK 1
-9.1
-8.0
-9.6
0.26
Lagrange invariant, 0.218 mm; aberration
targets (in
λ): (S1/8)t = -10,
SIIc/2)t = -10,
(CL/2)t = -10; accuracy in
shape and normalized power variables, 0.01. Boundaries for the
shape and normalized power
variables: (X1)max =
5.0, (X1)min = -5.0,
(k̅1)max = 5.0,
(k̅1)min =
-5.0. GA parameters: popsize, 500; maxgen, 250; maximum
fitness value in the initial population, 0.00041; average fitness value
in the initial population, 0.00001; overall chromosome length
(L) = 32 bits.
Table 3
Additional Solutions for the Cemented Doublet of Table 1a
Generation Number
Optimum Configuration
Fitness Values
X
1
k̅
1
Glass
Achieved Aberrations (in λ)
glass 1
glass 2
S
1
/8
SIIc/2
CL/2
12
0.01
2.30
K 5
SF 2
-1.7
3.2
1.5
0.06
13
0.04
1.99
K 10
SF 3
-2.1
2.1
1.2
0.09
13
0.01
2.12
BaK 4
SF 1
3.0
1.5
-1.0
0.07
13
0.07
2.13
BaLF 5
SF 3
-0.8
2.6
-1.8
0.08
14
0.00
1.99
K 5
SF 1
2.0
1.4
-0.3
0.14
25
0.02
1.93
K 3
SF 1
-0.6
1.9
-2.4
0.09
194
0.01
2.31
BK 1
F 1
0.7
3.4
-1.3
0.07
197
0.01
2.16
BK 1
SF 2
1.6
2.5
-1.0
0.09
250
0.00
2.00
K 3
SF 1
0.0
1.5
0.1
0.31
Boundaries for the shape and normalized
power
variables: (X1)max =
5.0, (X1)min = 0.0,
(k̅1)max = 5.0,
(k̅1)min = -5.0. GA
parameters: popsize, 500; maxgen, 250; maximum fitness value in the
initial population, 0.00017; average fitness value in the initial
population, 0.00001; overall chromosome length (L), 45
bits; other parameters, same as for Table 1.
Tables (3)
Table 1
Glass Combinations for f/4, 400-mm
Focal-Length Aplanatic, Achromatic Cemented
Doubletsa
Generation Number
Optimum Configuration
Fitness Value
X
1
k̅
1
Glass
Achieved Aberration (λ)
glass 1
glass 2
S
1
/8
SIIc/2
CL/2
16
-0.52
2.31
Ba K2
SF 2
-1.2
-2.7
0
0.10
21
-0.04
2.27
BK 7
F 1
-0.5
2.5
-1.1
0.11
104
-0.01
2.27
BK 1
F 1
-0.3
2.9
1.3
0.09
107
-0.08
2.48
Ba K 5
F 6
-1.2
2.4
2.1
0.08
126
-0.04
1.96
K 7
SF 1
-0.2
1.0
-1
0.33
129
-0.09
2.12
BK 7
SF 2
-1.7
1.1
-0.1
0.19
130
-0.09
2.10
BaLF 4
SF 3
-1.7
0.5
0.1
0.24
207
-4.87
-1.00
SF 3
K 10
-0.2
0.3
-0.5
0.73
209
-0.26
2.58
SK 3
BaSF 2
-1.2
0.2
-1.7
0.18
359
-3.24
-1.13
SF 2
BK 1
1.6
-2.1
0.7
0.12
361
-3.45
-1.22
F 1
BK 7
-0.3
0.4
2.4
0.15
384
-2.45
-1.60
F 7
BaK 1
-0.3
-0.8
1.1
0.34
388
-2.37
-1.50
SF 2
BaK 4
-0.1
-2.6
1.0
0.11
389
-2.37
-1.45
F 6
BaK 2
0.5
-3.6
0.2
0.07
391
-2.13
-1.79
BaSF 2
SK 6
-0.9
-2.1
-2.5
0.08
400
-4.844
-0.999
SF 3
K 10
0.0
0.2
0.1
0.96
Lagrange invariant, 0.218 mm; aberration
targets (in λ), (S1/8)t= 0, (SIIc/2)t = 0,
(CL/2)t = 0; accuracy in shape
and normalized power variables, 0.0001. Boundaries for the shape
and normalized power
variables: (X1)max =
5.0, (X1)min = -5.0,
(k̅1)max = 5.0,
(k̅1)min = -5.0. GA
parameters: Popsize, 500; maxgen, 400; maximum fitness value in the
initial population, 0.00447; average fitness value in the initial
population, 0.00003; overall chromosome length (L) = 46
bits.
Table 2
Solutions for f/4, 400-mm Focal-Length
Cemented Doublet with Nonzero Aberration
Targetsa
Generation Number
Optimum Configuration
Fitness Value
X
1
k̅
1
Glass
Achieved Aberrations (λ)
glass 1
glass 2
S
1
/8
SIIc/2
CL/2
9
-1.67
-2.03
SF 2
KF 6
-7
-10
-10
0.12
17
-1.91
-1.78
F 4
K 5
-10.6
-6.1
-10.2
0.06
18
-1.89
-1.80
F 6
K 10
-12.1
-8.6
-7.1
0.06
24
-1.91
-1.77
F 2
K 3
-9.8
-6.6
-9.3
0.07
63
-1.97
-1.75
BaSF 2
K 3
-9.8
-8.0
-12.0
0.11
101
-1.99
-1.73
BaSF 2
K 5
-9.1
-7.6
-13.0
0.06
250
-1.91
-1.79
BaSF10
BK 1
-9.1
-8.0
-9.6
0.26
Lagrange invariant, 0.218 mm; aberration
targets (in
λ): (S1/8)t = -10,
SIIc/2)t = -10,
(CL/2)t = -10; accuracy in
shape and normalized power variables, 0.01. Boundaries for the
shape and normalized power
variables: (X1)max =
5.0, (X1)min = -5.0,
(k̅1)max = 5.0,
(k̅1)min =
-5.0. GA parameters: popsize, 500; maxgen, 250; maximum
fitness value in the initial population, 0.00041; average fitness value
in the initial population, 0.00001; overall chromosome length
(L) = 32 bits.
Table 3
Additional Solutions for the Cemented Doublet of Table 1a
Generation Number
Optimum Configuration
Fitness Values
X
1
k̅
1
Glass
Achieved Aberrations (in λ)
glass 1
glass 2
S
1
/8
SIIc/2
CL/2
12
0.01
2.30
K 5
SF 2
-1.7
3.2
1.5
0.06
13
0.04
1.99
K 10
SF 3
-2.1
2.1
1.2
0.09
13
0.01
2.12
BaK 4
SF 1
3.0
1.5
-1.0
0.07
13
0.07
2.13
BaLF 5
SF 3
-0.8
2.6
-1.8
0.08
14
0.00
1.99
K 5
SF 1
2.0
1.4
-0.3
0.14
25
0.02
1.93
K 3
SF 1
-0.6
1.9
-2.4
0.09
194
0.01
2.31
BK 1
F 1
0.7
3.4
-1.3
0.07
197
0.01
2.16
BK 1
SF 2
1.6
2.5
-1.0
0.09
250
0.00
2.00
K 3
SF 1
0.0
1.5
0.1
0.31
Boundaries for the shape and normalized
power
variables: (X1)max =
5.0, (X1)min = 0.0,
(k̅1)max = 5.0,
(k̅1)min = -5.0. GA
parameters: popsize, 500; maxgen, 250; maximum fitness value in the
initial population, 0.00017; average fitness value in the initial
population, 0.00001; overall chromosome length (L), 45
bits; other parameters, same as for Table 1.
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