Electron Wave Optics 19 



rA = 0.61^ (14) 



a 



where Ao is the wavelength at the object, i.e., Ao = — if no is the 



fio 



refractive index of the medium in which the object is immersed, 

 and A is the wavelength in a medium n = 1. a is the aperture 

 angle, that is, the angle between the marginal rays which pass 

 through the lens aperture, and the optical axis. 



This formula was applied by Airy to the apparent diameter 

 of stars and to the resolving power of telescopes. It can be also 

 applied immediately to microscopes, with a restriction : We must 

 assume that the objects are self-luminous. Let us assume that 

 we observe two minute particles under the microscope which 

 fulfill this condition. It is evident that they will appear as sepa- 

 rate particles only as long as their Airy disks do not overlap 

 completely, so that a minimum of light intensity remains visible 

 between the two maxima. How much is required for visibility, 

 is to a certain degree a matter of convention, but for all practical 

 purposes, it is sufficient to assume that the minimum spacing, 

 which is called the resolution limit of the microscope is 



d^ = O.S^ (15) 



a 



To obtain this formula, we had to assume that the resulting 

 figure for the two particles can be obtained by simple superposi- 

 tion of the intensities of the two Airy disks. This is a legitimate 

 assumption in the case of self-luminous particles, as these emit 

 light independently of one another, therefore, resulting intensi- 

 ties can be obtained by addition. This condition is satisfactorily 

 fulfilled only in the ultramicroscope in which small objects are 

 made visible by dark field illumination, i.e., by illuminating them 

 sideways, and in all rigor only in the fluorescence microscope in 

 which the particles are illuminated by ultraviolet light. In all 

 other cases, there will be definite and small phase differences 

 between the light waves which appear to originate from the two 

 particles and which are caused by refraction, dispersion or ab- 

 sorption of the illuminating wave. This means that these sec- 



