Table I
This table lists the merit function φ and the third-order aberrations of a set of Cooke triplets. The first column refers to the design and the second to the merit function. The first row is a Cooke Triplet Patent No. 155640; the second row is an arbitrary perturbation. The remaining rows are successive improvements to reduce φ using the optimum gradient procedure. There are many more intermediate stages not shown in this table. The final design No. 3 appears to be as good or better than the original, and at this point the process was discontinued.
Lens | Merit function φ | Spherical B | Coma F | Astigmatism C | Distortion E | Axial color a | Lateral color b | Petzval P | Focal length f′ |
---|
No. 155640 | 54 | −0.0081 | 0.0003 | 0.0042 | −0.0006 | −0.000012 | 0.000003 | −2.0 | 0.1993 |
Perturbed lens | 6804 | −0.0181 | −0.0080 | 0.0166 | 0.0000 | −0.000022 | 0.000004 | −2.1 | 0.1743 |
Improvement No. 1 | 577 | −0.0224 | 0.0005 | 0.0067 | 0.0301 | −0.000033 | 0.000032 | −2.1 | 0.1744 |
Improvement No. 2 | 121 | −0.0119 | 0.0006 | 0.0060 | 0.0053 | −0.000019 | 0.000005 | −2.2 | 0.1736 |
Improvement No. 3 | 48 | −0.0074 | −0.0002 | 0.0046 | 0.0014 | −0.000013 | −0.000002 | −2.0 | 0.1994 |
Table II
This table shows the lens parameters of three of the lenses listed in Table I. The perturbed lens is an arbitrary variation of the original patent No. 155640. The 3rd improvement was the best design reached starting with this perturbed lens. The original lens is shown for comparison. The headings c, t, N, D denote, respectively, curvature, separation, index, and dispersion. The first separation is the distance from the entrance pupil plane to the first surface.
| Perturbed lens | | | Improvement No. 3 | | | Original 155640 | |
---|
c | t | N | D | c | t | N | D | c | t | N | D |
---|
| −0.0575 | | | | −0.0474 | | | | −0.0500 | | |
13.90 | | | | 12.89 | | | | 12.47 | | | |
| 0.0172 | 1.6130 | 0.01048 | | 0.0160 | 1.6094 | 0.01038 | | 0.0120 | 1.6130 | 0.01048 |
0.00 | 0.0208 | | | −1.20 | 0.0193 | | | −0.93 | 0.0200 | | |
−10.43 | | | | −10.01 | | | | −10.64 | | | |
| 0.0026 | 1.6210 | 0.01715 | | 0.0023 | 1.6247 | 0.01744 | | 0.0020 | 1.6210 | 0.01715 |
13.30 | 0.0230 | | | 13.27 | 0.0224 | | | 12.50 | 0.0216 | | |
1.74 | | | | 2.44 | | | | 2.13 | | | |
| 0.0172 | 1.6130 | 0.01048 | | 0.0166 | 1.6179 | 0.01062 | | 0.0120 | 1.6130 | 0.01048 |
−13.12 | | | | −12.32 | | | | −13.19 | | | |
Table III(a)
This shows a run of 9 steps using the optimum gradient method starting at x=1.00, y=−4.00. A large gain is made at step 1, but from then on the gains are rapidly decreasing. Even after 9 steps φ is not very close to φmin(0.022034).
| x | y | φ | G2 | |
---|
| 1.978522 | 0.601748 | 0.022034 | 0.00000 | Exact solution a |
---|
| −0.0504 | 0.3438 | 0.77952 | 0.00000 | Exact solution b |
---|
|
---|
Step | x | y | φ | G2 | h |
---|
0 | 1.00 | −4.0 | 585.379 | 14770.7 | 0.03147 |
1 | 0.975906 | −0.175413 | 0.744443 | 0.037739 | 3.874 |
2 | 1.688563 | 0.066472 | 0.153872 | 0.056894 | 0.300 |
3 | 1.669425 | 0.135423 | 0.122415 | 0.019009 | 0.500 |
4 | 1.737301 | 0.147469 | 0.102955 | 0.021624 | 0.3359 |
5 | 1.727605 | 0.195895 | 0.088662 | 0.011855 | 0.444 |
6 | 1.775002 | 0.205408 | 0.077783 | 0.011888 | 0.359 |
7 | 1.767407 | 0.243807 | 0.069389 | 0.008211 | 0.400 |
8 | 1.802995 | 0.250689 | 0.062680 | 0.007254 | 0.3775 |
9 | 1.796871 | 0.282253 | 0.057262 | 0.0058969 | |
Table III(b)
A second run starting with step 5 shows the effect of an acceleration. The * shows the acceleration step, which has clearly made a big improvement. At the end of 9 steps φ is practically at the minimum value of 0.022034.
Step | x | y | φ | G2 | h |
---|
5 | 1.727605 | 0.195895 | 0.088662 | 0.011855 | 7.38* |
6* | 2.014982* | 0.807018* | 0.047329 | 0.045502* | 0.170562 |
7 | 2.047935 | 0.791598 | 0.031491 | 0.000666 | 7.25 |
8 | 1.978658 | 0.617801 | 0.022492 | 0.0011879 | 0.1725 |
9 | 1.984059 | 0.617315 | 0.022078 | 0.0000023 | |
Table III(c)
This shows the generalization of the Newton-Raphson procedure using the same initial point as in III(a). The convergence is much slower, and the last column shows that the iteration is approaching the curve |A|=0, which effectively halts the process.
Step | x | y | φ | G2 | h | |A| |
---|
0 | 1.000 | −4.0000 | 585.379 | 14770.7 | 0.7343 | −1.250 |
1 | −0.945895 | −0.47536 | 30.1625 | 258.422 | 0.6400 | 1.52911 |
2 | −0.532544 | 1.132454 | 2.104288 | 2.336980 | 0.2000 | −0.93615 |
3 | −0.400398 | 0.598474 | 1.335299 | 1.673806 | 0.0200 | −0.23828 |
4 | −0.344027 | 0.457774 | 1.278978 | 1.642016 | 0.0026 | −0.07880 |
5 | −0.321207 | 0.406485 | 1.272863 | 1.649587 | | −0.02483 |
Table III(d)
Using the Newton-Raphson method starting at step 5 of III(a). This is a much more favorable place to begin this method, and a considerable improvement is seen in φ.
Step | x | y | φ | G2 | h |
---|
5 | 1.727605 | 0.195895 | 0.088662 | 0.011855 | 0.288 |
6 | 1.919332 | 0.432416 | 0.032233 | 0.009939 | |
Table IV
The optimum gradient method was applied to a doublet having 7 variables. The values of φ, G, and h are given at each step. Steps 12, 17, and 24 are acceleration steps and are indicated by asterisks.
| | From this point on an occasional acceleration step was inserted |
---|
Step | | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
---|
φ | | 0.662 | 0.642 | 0.624 | 0.610 | 0.598 | 0.588 | 0.578 | 0.570 | 0.563 | 0.556 | 0.550 | 0.545 | 0.498* | 0.490 | 0.484 | 0.481 | 0.477 | 0.400* | 0.371 | 0.362 | 0.358 | 0.354 | 0.351 | 0.348 | 0.331* |
|
| −0.442 | −0.051 | −0.375 | −0.042 | −0.321 | −0.027 | −0.279 | −0.015 | −0.245 | −0.004 | −0.214 | 0.005 | 0.312* | 0.035 | 0.189 | −0.011 | 0.183 | 0.200* | −0.037 | 0.254 | 0.128 | 0.199 | 0.095 | 0.175 | 0.034* |
| 0.026 | −0.079 | 0.028 | −0.065 | 0.028 | −0.056 | 0.029 | −0.049 | 0.028 | −0.043 | 0.027 | −0.039 | −0.052* | 0.028 | −0.047 | 0.010 | −0.045 | −0.092* | 0.069 | −0.011 | 0.044 | −0.013 | 0.034 | −0.015 | −0.035* |
| 0.104 | −0.476 | 0.062 | −0.412 | 0.035 | −0.371 | 0.021 | −0.334 | 0.013 | −0.304 | 0.005 | −0.278 | −0.370* | −0.040 | −0.241 | −0.002 | −0.231 | −0.157* | 0.242 | −0.088 | 0.079 | −0.060 | 0.080 | −0.053 | −0.030* |
G | −0.055 | −0.201 | −0.050 | −0.170 | −0.048 | −0.150 | −0.045 | −0.135 | −0.042 | −0.123 | −0.042 | −0.115 | −0.114* | −0.030 | −0.100 | −0.039 | −0.103 | −0.382* | −0.007 | −0.112 | 0.012 | −0.110 | −0.000 | −0.104 | −0.057* |
−0.005 | 0.451 | −0.027 | 0.337 | −0.040 | 0.263 | −0.051 | 0.208 | −0.059 | 0.166 | −0.061 | 0.134 | 0.086* | −0.073 | 0.049 | −0.069 | 0.040 | −0.502* | −0.211 | −0.121 | −0.081 | −0.113 | −0.083 | −0.098 | −0.022* |
| −0.046 | −0.020 | −0.042 | −0.018 | −0.039 | −0.017 | −0.036 | −0.016 | −0.034 | −0.015 | −0.032 | −0.014 | 0.016* | −0.011 | 0.006 | −0.013 | 0.011 | 0.185* | 0.013 | 0.056 | 0.004 | 0.046 | 0.001 | 0.038 | −0.019* |
| −0.025 | −0.240 | −0.021 | −0.166 | −0.021 | −0.120 | −0.021 | −0.088 | −0.023 | −0.066 | −0.025 | −0.050 | 0.062* | −0.050 | 0.025 | −0.053 | 0.042 | 1.163* | −0.003 | 0.224 | −0.104 | 0.175 | −0.102 | 0.130 | −0.143* |
h | | 0.318 | 0.094 | 0.293 | 0.100 | 0.284 | 0.105 | 0.270 | 0.110 | 0.0260 | 0.112 | 0.260 | * | 0.090 | 0.839 | 0.089 | 0.840 | * | 0.054 | 0.204 | 0.073 | 0.211 | 0.076 | 0.220 | * | |