286 



THE DIFFRACTION THEORY OF MICROSCOPY 



geometrical image of the particle when the annular opening in the 

 condenser diaphragm is increased in numerical aperture. Second, the 



Optical axis 

 Geometrical image of particle 



Fig. VII. 12. Relation of the coordinates f, v to a point x, y which is exterior to the 



geometrical image of the particle. 



primary diffraction integral U(i, t?) decreases rapidly with D and is an 

 oscillatory function of D. Approximately, 



TTf>- \ ^ Jii2Trp,„D) 



2wPmD 



TrPr 



J lima) 



(12.2) 



wherein /3 = 3.8317 and 



D„ 



D 



0.G098 0.G098 Tlf 



N.A. 



wavelengths. (12.3) 



N.A. is the numerical aperture of the objective. To fix the order of 

 magnitudes, we note that 2Ji{^Da)/l3Da = 1 when Da = and that 

 2 1 Ji(/^Z)„) I /|8 A. < 0.02 when D„ > 5 Airy units r„. Fiu-thermore, 

 Pm < 0.04 so that p,n^< 10 X 10"'*. Since the integrand of the double 

 integral oscillates with Da = (f^ + r?" )'/''«, we must expect that the 

 second right-hand member of Eq. 12.1 will be negligible as compared 

 with P{po/M, qo/M), whose absolute value is usually greater than 0.3, 

 when the point x, y is located 5 or more Airy units away from the edge 

 of the geometrical image of the particle. Observations in the micro- 

 scope show that this statement is conservative. 



For points x, y that fall in the distant portions of the surroimd one 

 can show rigorously that 



