284 THE DIFFRACTION THEORY OF MICROSCOPY 



{2TilM){pox+qoy) 



M' 



X 



p /Z^ ,12^ (ge^^ _ 1) /T,-(2W/M)(por+.o.)^r(^^ ^) j^ j^ (11.8) 



The double integral extends over the area of the geometrical image of the 

 particle. It is important to bear in mind that, since f , -q are defined by 

 Eq. 11.6, the origin for the coordinates f, 17 is located at the point .r, y of 

 the image plane. The point x, y may fall within or outside of the 

 geometrical image of the particle. The doul^le integral is consequently 

 a function of x and y. Fo(x, y, po, Qq) is the amplitude and phase 

 distribution produced over the image plane by the incidence upon the 

 object plane of a wave front whose normals have the optical direction 

 cosines po, qo- g i« the ratio of the amplitude transmission of the 

 particle to the amplitude transmission of the surround. A is the optical 

 path difference in radians between the particle and its surround. A is 

 considered positive w^hen the particle has the greater optical path. U is 

 the primary diffraction integral associated with the objective. It is 

 given by the equation 



, 77) = ffPip, g)e2-i(pf+'?')) dp dq (11.9) 



c/(r 



where p^ + q^ ^ n^Pm^ = Pm^ when n = 1 in the image space. The 

 first term within the square brackets is the contribution of the un- 

 deviated wave. The second term involving the double integral is the 

 contribution of the deviated wave. 



It will be noted that the phase factor e(2^'-/M)(pox+902/) jg common to 

 both the undeviated and the deviated waves. This factor represents 

 the phase variation introduced by the inclination of the incident wave 

 front. We may say that this phase variation is transported to the 

 image plane in accordance with Lummer's theorem. Since this phase 

 factor does not alter the partial or the total energy densities, it is of no 

 practical importance to microscopy. 



From Eqs. 8.16 and 11.8 the total energy density G{x, y) in the plane 

 of the image obeys the law 



M'G{x,y)=JjHpo,q.)\p{j^^Y^ 



+ {ge'^ - 1) /Te-^'"/^^^^^°^+''°''^0^(f,rj) d^dv 



2 



dpodqo. (11.10) 



