50 



AN ELEMENTARY THEORY OF PHASE MICROSCOPY 



areas of the diffraction plate and is considered positive when the optical 

 path through the conjugate area exceeds that through the comple- 

 mentary area, g is the ratio of the amplitude transmission of the particle 

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

 difference in radians or degrees between the particle and its surround 

 and is considered positive when the optical path through the particle 

 exceeds that through an equal thickness of the surround. It is impor- 

 tant to note from Eqs. 9.2-9.4 that it is always possible to finda diffraction 

 plate for which a microscopic particle appears black or at least darkest 

 in the event that scattering of light cannot be neglected. 



10. DARKEST CONTRAST WITH PARTICLES WHOSE AMPLITUDE 

 TRANSMISSION IS EQUAL TO THAT OF THE SURROUND 



In a broad class of particles which includes unstained biological 

 particles the amplitude transmission of the particle and surround are so 

 nearly alike that we may set 



g = 1. (10.1) 



By introducing g = I into Eqs. 9.2-9.4, we find that after slight simpli- 

 fication 



A 



h = 2 sin 



sin 5 = 



cos 5 = 



— sin A/2 cos A/2 



|sin A/2I 



sin^ A/2 



I sin A/2 1 



—sgn I sm 



cos 



sm 



(10.2) 

 (10.3) 

 (10.4) 



in which |sin A/2| denotes the absolute value of sin A/2 and in which 

 sgn{x) denotes x/\x\. Equations 10.3 and 10.4 are equivalent to the fol- 

 lowing simple relation between 8 and A. 



A 



^=2 



TT 



2 



A TT 



^ = i + 2' 



< A ^ tt; 

 -TT < A < 0. 



(10.5) 



In summary, the values that h and 8 must have in order that the 

 particle shall appear in optimum dark contrast when g ^ 1 are given by 

 the equations 



h = 2 sin ^ ; (10.6) 



