Table I
Normalized line spread-function F(z) for circular aperture (Rayleigh function) calculated from Eqs. (29), (30), and (31). Values in Part III are at integral multiples of z = π/2 and for the corresponding maxima or minima when these do not coincide with multiples of π/2. When the final digit has been increased by 0.5 in rounding, it is marked with an overbar.
z | .0 | .1 | .2 | .3 | .4 | .5 | .6 | .7 | .8 | .9 |
---|
0 | 10000 | 9973 | 9894 | 976
| 9581 | 9352 | 9079 | 8764 | 8413 | 8030 |
1 | 762
| 7187 | 6737 | 627
| 5809 | 5341 | 4877 | 4422 | 3980 | 3555 |
2 | 3151 | 2770 | 2415 | 2088 | 1790 | 1523 | 1285 | 1078 | 900 | 75
|
3 | 626 | 527 | 451 | 39
| 356 | 333 | 323 | 322 | 330 | 343 |
4 | 359 | 377 | 395 | 412 | 425 | 43
| 44
| 443 | 440 | 432 |
5 | 420 | 404 | 385 | 363 | 339 | 314 | 288 | 262 | 237 | 213 |
6 | 191 | 171 | 154 | 140 | 129 | 120 | 114 | 111 | 110 | 111 |
7 | 114 | 118 | 123 | 128 | 133 | 138 | 143 | 14
| 149 | 150 |
8 | 150 | 148 | 146 | 144 | 13
| 133 | 125 | 119 | 110 | 103 |
9 | 94 | 88 | 80 | 74 | 68 | 64 | 60 | 5
| 56 | 55 |
z | .0 | .5 | z | .0 | .5 | z | F(z) | z | F(z) |
---|
10 | 56 | 64 | 20 | 18 | 20 | 30 | 1
| 40 | 5 |
11 | 74 | 7
| 21 | 14 | 17 | 31 |
| 41 |
|
12 | 5
| 41 | 22 | 1
| 12 | 32 |
| 42 |
|
13 | 34 | 36 | 23 | 13 | 15 | 33 |
| 43 |
|
14 | 43 | 44 | 24 | 1
| 14 | 34 | 7 | 44 |
|
15 | 38 | 29 | 25 | 11 | 1
| 35 | 5 | 45 |
|
16 | 23 | 23 | 26 | 1
| 12 | 36 | 6 | 46 | 4 |
17 | 27 | 29 | 27 | 8 | 11 | 37 |
| 47 | 3 |
18 | 27 | 22 | 28 | 9 | 8 | 38 | 5 | 48 | 3 |
19 | 17 | 16 | 29 | 8 | 9 | 39 | 5 | 49 |
|
| | | | | | | | 50–53 | 3 |
Multiple of π | z | F(z) | | z | F(z) |
---|
1 | 3.14 | 496 | Min | 3.7 | 322 |
3/2 | 4.71 | 443 | Max | Same | 443 |
2 | 6.28 | 143 | Min | 6.8 | 110 |
5/2 | 7.85 | 150 | Max | 8.
| 150 |
3 | 9.42 | 67 | Min | 9.9 | 55 |
7/2 | 11.00 | 74 | Max | 11.
| 75 |
4 | 12.57 | 39 | Min | 13.
| 33 |
9/2 | 14.37 | 44 | Max | Same | 44 |
5 | 15.71 | 26 | Min | 16.5 | 22 |
Table II
Modulation transfer function of diffraction-limited lens with circular aperture from Hopkins,14 confirmed by the present authors.
ω | A(ω) | ω | A(ω) |
---|
0.00 | 1.0000 | 1.00 | 0.3910 |
0.05 | 0.9682 | 1.05 | 0.3638 |
0.10 | 0.9364 | 1.10 | 0.3368 |
0.15 | 0.9046 | 1.15 | 0.3105 |
0.20 | 0.8729 | 1.20 | 0.2848 |
0.25 | 0.8413 | 1.25 | 0.2596 |
0.30 | 0.8097 | 1.30 | 0.2351 |
0.35 | 0.7783 | 1.35 | 0.2112 |
0.40 | 0.7471 | 1.40 | 0.1881 |
0.45 | 0.7160 | 1.45 | 0.1658 |
0.50 | 0.6850 | 1.50 | 0.1443 |
0.55 | 0.6543 | 1.55 | 0.1237 |
0.60 | 0.6238 | 1.60 | 0.1041 |
0.65 | 0.5936 | 1.65 | 0.0855 |
0.70 | 0.5636 | 1.70 | 0.0681 |
0.75 | 0.5340 | 1.75 | 0.0520 |
0.80 | 0.5046 | 1.80 | 0.0374 |
0.85 | 0.4756 | 1.85 | 0.0244 |
0.90 | 0.4470 | 1.90 | 0.0133 |
0.95 | 0.4188 | 1.95 | 0.0047 |
| | 2.00 | 0.0000 |
Table III
Cumulative spread function χ(z) from zero to 10.0 in increments of 0.1 in the argument and from 10.0 to 49 in increments of 1.0. Within the latter range, proportional parts are given for an increment of 0.1 in the argument.
z | .0 | .1 | .2 | .3 | .4 | .5 | .6 | .7 | .8 | .9 |
---|
0 | 0 | 0.09991 | 0.19929 | 0.29762 | 0.39438 | 0.48708 | 0.58127 | 0.67052 | 0.75644 | 0.83863 |
1 | 0.91690 | 0.99095 | 1.06060 | 1.12566 | 1.18609 | 1.24184 | 1.29293 | 1.33941 | 1.38141 | 1.41967 |
2 | 1.45258 | 1.48216 | 1.50806 | 1.53055 | 1.54992 | 1.56646 | 1.58047 | 1.59226 | 1.60213 | 1.61035 |
3 | 1.61720 | 1.62295 | 1.62782 | 1.63203 | 1.63577 | 1.63920 | 1.64247 | 1.64569 | 1.64894 | 1.65230 |
4 | 1.65582 | 1.65950 | 1.66336 | 1.66740 | 1.67159 | 1.67589 | 1.68028 | 1.68471 | 1.68913 | 1.69349 |
5 | 1.69776 | 1.70189 | 1.70584 | 1.70958 | 1.71309 | 1.71636 | 1.71936 | 1.72211 | 1.72460 | 1.72685 |
6 | 1.72887 | 1.73068 | 1.73231 | 1.73378 | 1.73512 | 1.73636 | 1.73753 | 1.73865 | 1.73975 | 1.74085 |
7 | 1.74198 | 1.74313 | 1.74433 | 1.74559 | 1.74689 | 1.74825 | 1.74966 | 1.75110 | 1.75258 | 1.75408 |
8 | 1.75559 | 1.75711 | 1.75859 | 1.76004 | 1.76145 | 1.76281 | 1.76409 | 1.76531 | 1.76645 | 1.76751 |
9 | 1.76849 | 1.76939 | 1.77022 | 1.77098 | 1.77169 | 1.77235 | 1.77296 | 1.77355 | 1.77416 | 1.77467 |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
10 | 1.77522 | 1.78168 | 1.78862 | 1.79283 | 1.79654 | 1.80082 | 1.80386 | 1.80614 | 1.80901 | 1.81118 |
Δ/10 | 64 | 72 | 42 | 36 | 42 | 29 | 24 | 29 | 27 | 17 |
20 | 1.81286 | 1.81489 | 1.81656 | 1.81781 | 1.81931 | 1.82064 | 1.82162 | 1.82275 | 1.82384 | 1.82464 |
Δ/10 | 21 | 17 | 12 | 15 | 13 | 10 | 11 | 12 | 8 | 8 |
30 | 1.82552 | 1.82640 | 1.82709 | 1.82779 | 1.82854 | 1.82912 | 1.82969 | 1.83030 | 1.83082 | 1.83129 |
Δ/10 | 9 | 8 | 7 | 7 | 6 | 6 | 6 | 5 | 5 | 5 |
40 | 1.83181 | 1.83227 | 1.83266 | 1.83311 | 1.83352 | 1.83386 | 1.83425 | 1.83461 | 1.83491 | 1.83524 |
Δ/10 | 5 | 4 | 4 | 3 | 3 | 4 | 3 | 3 | 4 | 3 |
Table IV
Contrast in percent between successive maxima and minima in the images of bar patterns having the indicated number of bars for half-widths 1.0 and 2.0 when the aperture is circular. Column I gives the contrast between the zeroth (central) maximum and the first minimum, etc., and column V the contrast between the last maximum and a point 2t beyond.
| t = 1.0 (ω = π/2 = 1.57) | t = 2.0 (ω = π/4 = 0.78) |
---|
|
|
|
---|
Column No. | I | II | III | IV | V | I | II | III | IV | V |
---|
Max. No. | 0 | 1 | 2 | 4 | Last | 0 | 1 | 2 | 4 | Last |
---|
Min. No. | 1 | 1 | 2 | 4 | | 1 | 1 | 2 | 4 | |
---|
No. of bars | | | | | | | | | | |
3 | 15.9 | 14.7 | ⋯ | ⋯ | 77.7 | 67.9 | 67.6 | ⋯ | ⋯ | 92.1 |
5 | 15.3 | 14.9 | 14.4 | ⋯ | 77.0 | 66.8 | 66.7 | 67.1 | ⋯ | 91.8 |
7 | 15.2 | 15.1 | 15.2 | ⋯ | 76.7 | 66.3 | 66.3 | 66.4 | ⋯ | 91.7 |
9 | 15.1 | 15.0 | 14.8 | 14.2 | 76.5 | 66.1 | 66.1 | 66.1 | 66.2 | 90.8 |
∞ | 14.7 | 14.7 | 14.7 | 14.7 | ⋯ | 65.3 | 65.3 | 65.3 | 65.3 | ⋯ |