Measure of Apertures of Objectives. N.A.—Numerical aperture, as it is termed, is measured by the scale of measurement calculated by the late Professor Abbe, and which has since been generally recognised and adopted. He showed that even in lenses made for the same medium (as air) their comparative aperture as compared with their focus was not correctly measured by the angle of the rays grasped, but by the actual diameters of the pencil of rays transmitted, which depend, as already seen, more upon the back of the lens than the front. To get a geometric measure for comparison, he took the radii, or half diameters (whose relative proportions would be the same), and which geometrically are the sines of the semi-angle of the outermost rays grasped. Abbe further showed that if this sine of half the outside angle were multiplied by the refractive index of the medium used we should have a number which would give the comparative aperture of any lens, whatever the medium. This number, then, determines both the numerical aperture and the resolving power of the objective.
The following table of numerical apertures shows the respective angular pencils which they express in air, water and cedar oil, or glass.[16] The first column gives the numerical apertures from 0·20 to 1·33; the second, third, and fourth, the air, water and oil (or balsam) angles of aperture from 23° 4′ air angle to 180° balsam angle. The theoretical resolving power in lines to the inch is shown in the sixth column; the line E of the spectrum being taken from about the middle of the green, the column giving “illuminating power” being of less importance; while in using that of penetrating power, it must be remembered that several data beside that of 1/a go to make up the total depth of vision with the microscope.
| (1) | Corresponding Angle (2 u) for | Limit of Resolving Power, in Lines to an Inch. | (8) | (9) | ||||
|---|---|---|---|---|---|---|---|---|
| (2) | (3) | (4) | (5) | (6) | (7) | |||
| 1·33 | ... | 180° 0′ | 122° 6′ | 128,225 | 138,989 | 168,907 | 1·769 | ·752 |
| 1·32 | ... | 165° 56′ | 120° 33′ | 127,261 | 137,944 | 167,637 | 1·742 | ·758 |
| 1·30 | ... | 155° 38′ | 117° 35′ | 125,333 | 135,854 | 165,097 | 1·690 | ·769 |
| 1·28 | ... | 148° 42′ | 114° 44′ | 123,405 | 133,764 | 162,557 | 1·638 | ·781 |
| 1·26 | ... | 142° 39′ | 111° 59′ | 121,477 | 131,674 | 160,017 | 1·588 | ·794 |
| 1·24 | ... | 137° 36′ | 109° 20′ | 119,548 | 129,584 | 157,477 | 1·538 | ·806 |
| 1·22 | ... | 133° 4′ | 106° 45′ | 117,620 | 127,494 | 154,937 | 1·488 | ·820 |
| 1·20 | ... | 128° 55′ | 104° 15′ | 115,692 | 125,404 | 152,397 | 1·440 | ·833 |
| 1·18 | ... | 125° 3′ | 101° 50′ | 113,764 | 123,314 | 149,857 | 1·392 | ·847 |
| 1·16 | ... | 121° 26′ | 99° 29′ | 111,835 | 121,224 | 147,317 | 1·346 | ·862 |
| 1·14 | ... | 118° 0′ | 97° 11′ | 109,907 | 119,134 | 144,777 | 1·300 | ·877 |
| 1·12 | ... | 114° 44′ | 94° 55′ | 107,979 | 117,044 | 142,237 | 1·254 | ·893 |
| 1·10 | ... | 111° 36′ | 92° 43′ | 106,051 | 114,954 | 139,698 | 1·210 | ·909 |
| 1·08 | ... | 108° 36′ | 90° 34′ | 104,123 | 112,864 | 137,158 | 1·166 | ·926 |
| 1·06 | ... | 105° 42′ | 88° 27′ | 102,195 | 110,774 | 134,618 | 1·124 | ·943 |
| 1·04 | ... | 102° 53′ | 86° 21′ | 100,266 | 108,684 | 132,078 | 1·082 | ·962 |
| 1·02 | ... | 100° 10′ | 84° 18′ | 98,338 | 106,593 | 129,538 | 1·040 | ·980 |
| 1·00 | 180° 0′ | 97° 31′ | 82° 17′ | 96,410 | 104,503 | 126,998 | 1·000 | 1·000 |
| 0·98 | 157° 2′ | 94° 56′ | 80° 17′ | 94,482 | 102,413 | 124,458 | ·960 | 1·020 |
| 0·96 | 147° 29′ | 92° 24′ | 78° 20′ | 92,554 | 100,323 | 121,918 | ·922 | 1·042 |
| 0·94 | 140° 6′ | 89° 56′ | 76° 24′ | 90,625 | 98,223 | 119,378 | ·884 | 1·064 |
| 0·92 | 133° 51′ | 87° 32′ | 74° 30′ | 88,697 | 96,143 | 116,838 | ·846 | 1·087 |
| 0·90 | 128° 19′ | 85° 10′ | 72° 36′ | 86,769 | 94,053 | 114,298 | ·810 | 1·111 |
| 0·88 | 123° 17′ | 82° 51′ | 70° 44′ | 84,841 | 91,963 | 111,758 | ·774 | 1·136 |
| 0·86 | 118° 38′ | 80° 34′ | 68° 54′ | 82,913 | 89,873 | 109,218 | ·740 | 1·163 |
| 0·84 | 114° 17′ | 78° 20′ | 67° 6′ | 80,984 | 87,783 | 106,678 | ·706 | 1·190 |
| 0·82 | 110° 10′ | 76° 8′ | 65° 18′ | 79,056 | 85,693 | 104,138 | ·672 | 1·220 |
| 0·80 | 106° 16′ | 73° 58′ | 63° 31′ | 77,128 | 83,603 | 101,598 | ·640 | 1·250 |
| 0·78 | 102° 31′ | 71° 49′ | 61° 45′ | 75,200 | 81,513 | 99,058 | ·608 | 1·282 |
| 0·76 | 98° 56′ | 69° 42′ | 60° 0′ | 73,272 | 79,423 | 96,518 | ·578 | 1·316 |
| 0·74 | 95° 28′ | 67° 37′ | 58° 16′ | 71,343 | 77,333 | 93,979 | ·548 | 1·351 |
| 0·72 | 92° 6′ | 65° 32′ | 56° 32′ | 69,415 | 75,242 | 91,439 | ·518 | 1·389 |
| 0·70 | 88° 51′ | 63° 31′ | 54° 50′ | 67,487 | 73,152 | 88,899 | ·490 | 1·429 |
| 0·68 | 85° 41′ | 61° 30′ | 53° 9′ | 65,559 | 71,062 | 86,359 | ·462 | 1·471 |
| 0·66 | 82° 36′ | 59° 30′ | 51° 28′ | 63,631 | 68,972 | 83,819 | ·436 | 1·515 |
| 0·64 | 79° 36′ | 57° 31′ | 49° 48′ | 61,702 | 66,882 | 81,279 | ·410 | 1·562 |
| 0·62 | 76° 38′ | 55° 34′ | 48° 9′ | 59,774 | 64,792 | 78,739 | ·384 | 1·613 |
| 0·60 | 73° 44′ | 53° 38′ | 46° 30′ | 57,846 | 62,702 | 76,199 | ·360 | 1·667 |
| 0·58 | 70° 54′ | 51° 42′ | 44° 51′ | 55,918 | 60,612 | 73,659 | ·336 | 1·724 |
| 0·56 | 68° 6′ | 49° 48′ | 43° 14′ | 53,990 | 58,522 | 71,119 | ·314 | 1·786 |
| 0·54 | 65° 22′ | 47° 54′ | 41° 37′ | 52,061 | 56,432 | 68,579 | ·292 | 1·852 |
| 0·52 | 62° 40′ | 46° 2′ | 40° 0′ | 50,133 | 54,342 | 66,039 | ·270 | 1·923 |
| 0·50 | 60° 0′ | 44° 10′ | 38° 24′ | 48,205 | 52,252 | 63,499 | ·250 | 2·000 |
| 0·45 | 53° 30′ | 39° 33′ | 34° 27′ | 43,385 | 47,026 | 57,149 | ·203 | 2·222 |
| 0·40 | 47° 9′ | 35° 0′ | 30° 31′ | 38,564 | 41,801 | 50,799 | ·160 | 2·500 |
| 0·35 | 40° 58′ | 30° 30′ | 26° 38′ | 33,744 | 36,576 | 44,449 | ·123 | 2·857 |
| 0·30 | 34° 56′ | 26° 4′ | 22° 46′ | 28,923 | 31,351 | 38,099 | ·090 | 3·333 |
| 0·25 | 28° 58′ | 21° 40′ | 18° 56′ | 24,103 | 26,126 | 31,749 | ·063 | 4·000 |
| 0·20 | 23° 4′ | 17° 18′ | 15° 7′ | 19,282 | 20,901 | 25,400 | ·040 | 5·000 |
INDEX:
(1) Numerical Aperture. (n sin u = a.)
(2) Air (n = 1·00).
(3) Water (n = 1·33).
(4) Homogeneous Immersion (n = 1·52).
(5) White Light. (λ = 0·5269 μ, Line E.)
(6) Monochromatic (Blue) Light.(λ = 0·4861 μ, Line F.)
(7) Photography. (λ = 0·4000 μ, Near Line hk.)
(8) Illuminating Power (a2.)
(9) Penetrating Power (1/a.)
Abbe’s Apertometer.
Fig. 39.—Abbe’s Apertometer.
The apertometer is an auxiliary piece of apparatus invented by Abbe, for testing the fundamental properties of objectives and determining their numerical and angular apertures. This accessory of the microscope involves the same principles as that of Tolles, which the late Mr. J. Mayall and myself brought to the notice of the Royal Microscopical Society of London in 1876. Abbe’s apertometer ([Fig. 39]) consists of a flat cylinder of glass, about three inches in diameter, and half an inch thick, with a large chord cut off, so that the portion left is somewhat more than a semicircle; the part where the segment is cut is bevelled from above downwards, to an angle of 45°, and it will be seen that there is a small disc with an aperture in it denoting the centre of the semicircle. To use this instrument the microscope is placed in a vertical position, and the apertometer is placed upon the stage with its circular part to the front and the chord to the back. Diffused light, either from the sun or lamp, is assumed to be in front and on both sides. Suppose the lens to be measured is a dry one-quarter inch; then with a one-inch eye-piece having a large field, the centre disc, with its aperture on the apertometer, is brought into focus. The eye-piece and the draw-tube are now removed, leaving the focal arrangement undisturbed, and a lens supplied with the apertometer is screwed into the end of the draw-tube. This lens, with the eye-piece in the draw-tube, forms a low-power compound microscope. This is now inserted into the body-tube, and the back lens of the objective whose aperture we desire to measure is brought into focus. In the image of the back lens will be seen stretched across, as it were, the image of the circular part of the apertometer. It will appear as a bright band, because the light which enters normally at the surface is reflected by the bevelled part of the chord in a vertical direction, so that in reality a fan of 180° in air is formed. There are two sliding screens seen on either side of the figure of the apertometer; they slide on the vertical circular portion of the instrument. The images of these screens can be seen in the image of the bright bands. These screens should now be moved so that their edges just touch the periphery of the back lens. They act, as it were, as a diaphragm to cut the fan and reduce it, so that its angle just equals the aperture of the objective and no more.
This angle is now determined by the arc of glass between the screens; thus we get an angle in glass the exact equivalent of the aperture of the objective. As the numerical apertures of these arcs are engraved on the apertometer, they can be read off by inspection. A difficulty is not infrequently experienced from the fact that it is not easy to determine the exact point at which the edge of the screen touches the periphery of the back lens, or rather the limit of the aperture. Zeiss, to meet this difficulty, made a change in the form of the apparatus—furnished a glass disc mounted on a metal plate, with a slot for the purpose of its more accurate adjustment.[17]