280 THE DIFFRACTION THEORY OF MICROSCOPY 



Hence G(x) = at the centers of the troughs (Fig. VII.9), provided that 

 = Fo (/o + 2 ^ A + Uj ■ Po^O. (10.3.8) 



Introducing /o and/^ from Eq. 10.1.2 and introducing 



Po;Pi = he'\ (10.3.9) 



we obtain the zero condition in the form 



TT 



A 1 - e"^ h Cv 



Consequently the total energy density is zero at the centers of the 

 troughs if 



TT A e-'^ 1 



Equation 10.3.10 is true only when the real and imaginary parts of its 

 right-hand member are both zero. Accordingly 



h-rr A h , ^ 



sm5 = — cot-; cos 5 = - • (10.3.11) 



4 2' 6 



This equation is easily solved for 6 and h. In summary, the energy 

 density is zero at the centers of the troughs provided that h and 5 have 

 the values 



o 



36/ ' 



^ = U^^^^ 2 + 3^1 ' ^''-'-''^ 



T A /tt" , a 



sm 5 = - cot — I — cot" — I 



4 2V16 2 36 



cos 



-V- 



J6/ ' 



5 = -( — cot^-H ) . (10.3.13) 



6\16 2 36/ ^ ^ 



These zero conditions differ from the zero conditions of Eqs. 10.1.14 

 and 10.1.15 in two important respects. The solution for 5 depends on 

 the optical path difference A between the elevations and the troughs. 

 The 5 values will depart from dnr/2. In the present specialized case the 

 departure of 5 from ±7r/2 will not be large. This is due to the fact that 

 only one branch of one spectral order passes through the conjugate area. 

 In conclusion, the phase difference 8 between the conjugate and comple- 

 mentary areas of the diffraction plate must in general be chosen different 

 from ±X/4 in order to obtain optimum contrast in the image of a pure 

 phase grating when some of the higher spectral orders pass through the 



