REDUCTION TO THE ELEMENTARY THEORY 289 



not possible from the general theory, but we shall see that under severe 

 restrictions as to the size of the conjugate area it is possible for the 

 levels of illumination to approach the properties of a step function. 

 From Eq. 11.8 



M^Foix, y, po, go) 



^ ,(2.i/M)(pox+.o.) P (^ . ^) + {ge'^ - l)H,{x, y, p„, q,) 



MA) 



in which 



'vix, y, Po, go) =^,-(2.i/M)(pomo.)^r(f, ^) d^ dr, (14.2) 



f/(r, -n) - ff^"(P> q)e-^'''"+''^ dp dq; (14.3) 



p2 + g2 ^ pj_ (14 4) 



The double integral for Hp extends over the area of the geometrical 

 image of the particle. The coordinates f, r] are to be measured from a 

 point X, y within the geometrical image. The suffix p is attached to 

 H to indicate that H depends on the size and shape of the particle. 



If the objective is of the Airy type and if the conjugate and comple- 

 mentary areas are coated uniformly and differently, 



/Po^9o\ 



= he'\ (14.5) 



Let us assume that it is possible to select a pupil function P{p, q) and a 

 set of optical direction cosines po, go such that 



Hp{x, y, Po, go) = 1 (14.6) 



for points .r, y which occupy an extended region within the geometrical 

 image of the particle. Denote this region by A. We show that the 

 relative energy densities of this area A of the geometrical image of the 

 particle and of the surround obey the laws of the elementary theory of 

 phase microscopy. 



From Eqs. 14.1, 14.5, and 14.6 



M^Foix, y, Po, go) = e(2.^7M)(pox+go2/)(/^gi5 ^ g^iA _ 1) (14 7) 



for points x, y which belong to the region A. Correspondingly the total 

 energy density G{x, y) in the plane of the image obeys the law 



G{x, y) = constant = Ga = Ki\he'^ + ge'^ - l|^ (14.8) 



^'i -J^Jj ^^Vo, %) dpo dqo. (14.9) 



