276 THE DIFFRACTION THEORY OF MICROSCOPY 



transmission of the troughs be taken as unity and let the amphtude 

 transmission of the elevations be taken as g. Physically, g is the ratio 

 of the amplitude transmission of the elevations to the amplitude trans- 

 mission of the troughs. Then 



CO 



IM = ^ f^-i^-oli). (10.2.1) 



1/ = ^ CO 



in which 



(1 + S') . . (I - q) sin (pt/2) 

 /o = ^-ir^ ; /. = /-. = ^ —^ ' (10.2.2) 



^ PIT 



AsinEqs. 10.1.7 and 10.1.8, 



G(^) =Jf4\Si^)\'-, (10.2.3) 



Six) ^ ^ P[^)f.e'''^'''"''\ (10.2.4; 



-.V 



From Eqs. 10.2.2, 10.2.4, and 10.1.10 



Si.) = -P(0)(1 -,)(- ^ + -^-.„„-co.- (10.2.5) 



wherein A^ is given by Eq. 10.1.6. 



The condition under which the total energy G(x) shall be zero at the 

 points X ^ 0, dtzn2Ml, where n is an integer, is therefore 



TT 1 + g e \-^ I VT , , 



= - ^ + T Z^ - "^^ ^ • (lO-^-^) 



4 1 — fl h ^—^ V 2 



Hence 

 Consequently 



e-'' = -sgni^ - g) 



5 = ±7r, g <l; 



5 = 0, g> I. (10.2.7) 



4 1—0 \-> 1 . J'X , ^ ^ ^^ 



/i=-^-— ^> -sm-. (10.2.8) 



TT 1 +g frf I' 2 



Equations 10.2.7 and 10.2.8 are the relations which must hold among 

 h, 8, g, and the properties of the objective in order that the centers of the 

 troughs of Fig. VII. 10 shall have zero energy density. These conditions 

 can be satisfied by the proper choice of diffraction plate. However, h be- 

 comes very small as the amplitude transmission of the troughs approaches 



