Numerical Aperture.

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.

ABRIDGED NUMERICAL APERTURE TABLE.
(1)	Corresponding Angle (2 u) for			Limit of Resolving Power, in Lines to an Inch.			(8)	(9)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
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.)