Ch. IX] 



ANGULAR AND NUMERICAL APERTURE 



295 



for it is seen that light in water in passing into air spreads out so that 



an angle in water of 48 45' on each side of the normal (97 30') spreads 



out into an angle of 180 in air; therefore light at an angle of 07 30' 



in water is equal to 1S0 in air, and if the water immersion objective 



receives and transmits for the formation of the image an angle of 



light in the water greater than 97 30' its angle 



is greater than an air angle of 180 . In the same 



way with homogeneous immersion. The critical 



angle for glass to air is 41 on each side of the 



normal, and a total angle of 82 in the glass 



would spread out to form the whole 180 in the 



air. Therefore, if with homogeneous immersion 



objectives an angle above 82 is transmitted by 



the objective for the formation of the image, the 



angle is so much greater than 1S0 in air. 



The confusion was reduced to order by Abbe, 

 to whom makers and users of optical instruments 

 owe so many debts. He applied the simple 

 laws of trigonometry, using the sine function of 

 the angle, and taking into consideration the 

 medium of the lowest refractive index between 

 the object and the objective. If it were air, unity 

 was taken, if water the index of water — 1.33; 

 if glass, 1.52; and if any other immersion fluid, 

 the refractive index of that fluid. By thus con- 

 sidering the index of refraction of the medium 

 immediately in front of the objective, it became 

 possible to make comparisons which were rigidly exact, and expressed 

 in terms which did not seem impossibilities like an angle in excess of 

 180 to enter a flat surface. 



The nomenclature introduced by him and now universally em- 

 ployed is Numerical Aperture, and includes in its significance both 

 the angle of the light and the index of refraction of the medium from 

 which the light passes into the objective. The formula is N.A. = 

 n sin u, in which n is the index of refraction of the air for dry, the 

 water for water immersion and the cedar oil for homogeneous im- 



Fig. 176. Angular 

 Aperture of an Ob- 

 jective. 



Axis, The princi- 

 pal optic axis of the 

 objective. 



B The object just 

 outside the principal 

 focus. 



ADC Diameter of 

 the front of the objec- 

 tive and base of the 

 angle of aperture. 



A D B Half the 

 angle of aperture (w); 

 AD representing the 

 sine of u (see § 468). 



