CONDITION FOR DARKEST CONTRAST OF THE PARTICLE 49 



radians and is considered positive when the optical path of the conjugate 

 area exceeds that of the complementary area, hi is the amplitude trans- 

 mission of the complementary area of the diffraction plate. The numeri- 

 cal value of hi does not affect the relative value of Gg and Gp, and hence 

 the contrast in the image. In fact one may usually set hi = 1 without 

 any essential loss of generality. 



Equations 8.7 and 8.8 are the most general equations of phase micros- 

 copy within the scope of the simplified theory of phase microscopy. 

 These equations contain the logical consequences which follow from the 

 assumptions involved in Zernike's vector model for phase microscopy. 

 These equations may be used to calculate Gg/hi^ and Gp/hi^ when the 

 properties g and A of the particle and the properties h and 5 of the dif- 

 fraction plate are assigned particular values. When Gg and Gp have 

 been computed, the contrast ratios Gp/Gg or (Gg — Gp)/Gg may be com- 

 puted also. Explicit formulas for these contrast ratios will not be 

 given. 



9. CONDITION FOR DARKEST CONTRAST OF THE PARTICLE 



The particle is said to appear in dark or bright contrast according as 

 the particle appears darker or brighter than its surround. At darkest 

 possible contrast the particle appears to be black and Gp = in Eq. 8.8. 

 We suppose that the relative amplitude transmission g and the optical 

 path difference A are assigned and that it is desired to find the values 

 h and 6 which the diffraction plate must have in order that Gp = 0. It 

 is supposed that /ii = 1. Since the two squared terms in the right-hand 

 member of Eq. 8.8 are never negative, the condition for Gp = is that 



h cos 5 + ^ cos A — 1 = 0; 



/i sin 5 + ^ sin A = 0. (9.1) 



These form a pair of simultaneous equations for determining h and 5 

 from g and A, or vice versa. The solution for h and b is 



/i = (1 - 2g cos A + ^2)^ . (^92) 



-g sin A 

 (1 - 2g cos A + g'Y 



sin 5 = -^zzr^r^^^k ; (9-3) 



I — g cos A 



cos 8 = -; ^-T • (9.4) 



(1 - 2^cos A + r)' 



Equations 9.2-9.4 are the conditions for darkest contrast in the image 

 of the particle, h is the ratio of the amplitude transmission of the con- 

 jugate area to the amplitude transmission of the complementary area. 

 5 is the optical path difference in radians or in degrees between these two 



