UNIFORMLY ALTERED HIGHER ORDERS 273 



and phase distribution over the image plane is a function of both x and 

 y, the total energy density is a function of x alone. 

 Let 



v=—N ^ ' 



Then Gix) = when S{x) = 0, and G(x) is maximum when S{x) is 

 maximum. Hence it suffices to consider the simpler function S{x) in 

 determining the magnitudes and locations of the maxima and minima 

 in the total energy density G{x). 

 From Eqs. 10.1.2 and 10.1.8 



1 . . 2 • . X^ 1 PIT / I' \ J'TT-C 



S{x) = - (1 + e^^)P(O) + - (1 - 6*^) > - sin - P ( TTT^ ) cos 



2 TT '^— / V Z 



Mil/ Ml 



V =1 



= ? (1 - e^^)P(O) 



■K 



N ^ P 



\2Ml) 



P(0) F^ 0. (10.1.9) 



We introduce at this point the supposition that the zero order shall 

 be altered uniformly with respect to the higher spectral orders; i.e., 

 only the zero spectral order (j^ = 0) passes through the conjugate area. 

 This supposition implies tacitly that the objective is of the Airy-type 

 so that P{p, q) = c{p, q) and that the conjugate and complementary 

 areas of the diffraction plate are coated uniformly and differently. It 

 follows from Section 4 that 



Piy/2Ml) ^ c(v/2Ml) ^ e^ 



P(0) c(0) h ' ^ ■ ■ ) 



Let Eq. 10.1.10 be introduced into Eq. 10.1.9 together with the trigono- 

 metric identity 



(10.1.11) 



p.' 



VTT virX 



sm -- cos 



2 Ml/ 



(10.1.12) 



A is the optical path difference between the elevations and the troughs, 

 as in Fig. VII. 9, with A considered positive when the greater optical 

 path is associated with the elevations, b is the optical path difference 



