DARKEST CONTRAST WHEN g = 1 



51 



S = 



TT 



2' 



A TT 



^=2 + 2 



< A < tt: 



-TT < A < 0. 



(10.7) 

 (10.8) 



It will be noted that the required phase difference 5 of the diffraction 

 plate is discontinuous at A = 0. 



With particles whose optical path A is so small that we can replace 



sin A/2 by A/2, 



h = |a| (10.9) 



and 



TT 



T - 

 2 



(10.10) 



according as A > or as A < 0. 



Equations 10.7 and 10.8 state that the optical path of the conjugate 

 area of the diffraction plate must be less than the optical path of the 

 complementary area by (7r/2 — A/2) radians or by (34 — A/^tt) wave- 

 lengths in order to make the particle appear darkest when its optical 

 path exceeds that of the surround by a small amount, but that the optical 

 path of the conjugate area must exceed the optical path of the com- 

 plementary area by (7r/2 + A/2) radians or by (14 + A -Itt) wavelengths 

 in order to make the particle appear darkest when its optical path is 

 less than that of the surround by a small amount, A radians. These 

 conclusions are consistent with those established in Section 3 but indicate 

 definitely the direction and amount by which 8 should depart from 

 T7r/2. According to Eq. 10.9, the optimum amplitude ratio h of the 

 diffraction plate decreases with |a| when A is so small that we can with 

 negligible error replace sin A/2 by A/2. 



Suppose that the optical path difference between the particle and 

 the surround falls in the neighborhood of A = ± 7r/2, that is, in the 

 neighborhood of ±34 wavelength. Then 



h 



TT 



sm 



- V2: 



5 — > — - radians = —^ wavelength when A = - ; 



TT . TT 



5 -^ - radians = +i wavelength when A = 



4 



This means that, when the optical path differences between the particle 

 and the surround are ±X/4, 5 must be chosen as — X/8 or as 4-X/ 8 and 



