274 THE DIFFRACTION THEORY OF MICROSCOPY 



between the conjugate and complementary areas and is considered posi- 

 tive when the greater optical path belongs to the conjugate area of the 

 diffraction plate, h is the ratio of the amplitude transmission of the 

 conjugate area to the amplitude transmission of the complementary 

 area. 



Let us derive the relation which must exist among A, 8, and h in 

 order that S{x), and hence G{x), shall be zero at the particular points 

 X = 0, ±n2AIl, in which n is an integer. From Eq. 10.1.12 this relation 

 is 



. TT A e '" v^ I . VT 



■ = 1 



Accordingly, 



8 = sgn cot- - ; (lO.l.U) 



h = - tan - > - sin — (10.1.15) 



in which sgn(z) = z/\z\ = ±1. Other solutions of 5 exist, but these 

 differ from the values of Eq. 10. 1 . 14 by iJL2ir, where n is an integer. These 

 solutions are not of practical interest in phase microscopy. The center 

 of the troughs will appear darkest provided that 8 and h are related to 

 A and the optical properties of the objective in accordance with Eqs. 

 10.1.14 and 10.1.15. In conclusion, regions of greater optical path will 

 appear dark when 5 = — 7r/2 whereas regions of lesser optical path will 

 appear dark when 8 = +7r/2. It is important to observe that, when 

 the zero spectral order is altered uniformly with respect to the higher 

 orders, the minimizing 8 values are 8 — ±7r/2. Whereas the minimiz- 

 ing 8 values are independent of the numerical value of A (that is, |a|), 

 the minimizing h values depend on the numerical values of A. Physi- 

 cally, therefore, extreme contrasts in the image of pure phase gratings 

 will be produced by the choice of diffraction plates for which 8 = ±7r/2 

 when all of the higher spectral orders pass through the complementary 

 area. Equation (10.1.15) shows that h will be small with small values 

 of A but can become very large as A approaches 180°. The elementary 

 theory indicated that h values greater than 2 are not particularly useful. 

 This is not in accord with Eq. 10.1.15 as A approaches 180°. The 

 point of demarcation between A- and B-type diffraction plates is 

 given by 



N 



/i = 1 = - 



TT 



A 



tan — 

 2 



Zl VTT 



-sin- (10.1.16) 



f =1 



