10 



NUMERICAL APERTURE. 



Take any aplanatic system, as 

 represented in Fig. V., A being, as 

 before, an object at its focus, the 

 image of which is projected to the 

 conjugate focus at E. As we are 

 only comparing the relative emer- 

 gence of light, not the absolute 

 quantity, we can consider the case 

 of an infinitely thin pencil, repre- 

 sented by the plane of the paper, 

 and also consider only the case of 

 the semi-diameter of the pencil. 



Let DEL represent the plane of 

 emergence. 



DF = a = the amount of emer- 

 gent light ; 



^IBAC = a ', <^ DEF = /3 ; 

 FE = 1 = constant ^ ; 



AC = b = the focal length of 

 the objective ; 



n = index of refraction of medium 

 in front of the objective ; 

 m = index of refraction of medium 

 behind the objective ; 



1 sin /3 

 Thenar ltan,3 = l ^ 



(as /3 is very small, cos ^ = nearly 

 to i) 



.". a = 1 sin /3 (i) 

 By the laws of aplanatic conver- 

 gence 

 n sin 



V. 



m 



- = magmfymg power = -r— 

 sin /3 D 



I for air 

 = b n sin <i; 



m 



.'. 1 sin ^ = b n sin a (2) 



Substituting value of 1 sin ^ (i) 



a =: bn sin a 



a 



— = n sm a 



b 



Or the ratio of " aperture " to " focal 

 length " is expressed by " ;^ sin a" 



This expression is known as the 

 " numerical aperture " of an objec- 

 tive, a being the semi-angle of aper- 

 ture as usually given, and n the 

 refractive index of the medium in 

 front of the objective. The principal 

 values of n in microscope work are : 



* This varies with the length of tube for 

 which the lens is corrected. 



Ci 



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