54 AN ELEMENTARY THEORY OF PHASE MICROSCOPY 



If the optical path difference A is not neghgible, the required h and 

 8 values can be ascertained from the more general Eqs. 9.2-9.4. In such 

 cases 5 must be expected to depart significantly from 6 = or w. We 

 see, therefore, that the simplified theory is capable of predicting that 

 the phase microscope can improve contrast in the image of slighth^ 

 stained biological specimens. Also, this prediction is in accord with 

 experiment. 



12. EQUALITY OF THE AMPLITUDES OF THE UNDEVIATED AND DE- 

 VIATED WAVES AT THE CONDITIONS FOR DARKEST CONTRAST 



By factoring out his'^^^ from the right-hand members of Eqs. 7.3 and 

 7.4, we find that 



U = e^^e-2'^'-^o^"''Oe''^'/ii/ie*^; (12.1) 



The complex numbers U and D determine the amplitude and phase of 

 the undeviated and deviated waves, respectively, as they arrive at the 

 plane of the image. Since the absolute values of the exponentials are 

 unity, it follows at once that 



\U\ = hh; (12.3) 



\D\ = hi\ge'^ - l|. (12.4) 



Since 



ge'^ - 1 = (^ cos A - 1) + i(g sin A); 



,iA 



\ge- - l\ = [{g cos A - l)'^ -\- (g sin A)-]^ 



= {I -2g cos A + g~)K (12.5) 



From Eqs. 12.4 and 12.5, 



\D\ = /ii(l - 2g cos A + g^)\ (12.6) 



From Eq. 9.2 the condition on h for darkest contrast is that h = 

 (1 — 2g cos A + g")^. Therefore, when h is chosen so as to satisfy the 

 condition for darkest contrast, 



\U\ = \d\ = hih. (12.7) 



Since \U\ and \D\ are the amplitudes of the undeviated and deviated 

 waves, respectively, this means physically that the amplitudes of the 

 undeviated and deviated waves are equal when the diffraction plate has 

 been chosen to satisfy the conditions for darkest contrast. 



