'A(p) = r. j^iT^^^ir^Tirwr ^^-^i) 



PUPIL FUNCTION P(p, q) IN PHASE MICROSCOPY 249 



Hence 



P(p, 5) = P(p) = c{p)rP{p). (3.10) 



In detail, 



(1 - 7W- {Mp/n^Y]' 



In the conventional microscope p,n < 0.04 with ^ p ^ p,„. There- 

 fore with excellent approximation 



(1 - p')' = 1 (3.12) 



in Eq. 3.11. Consequently, it suffices to take \p{p) in the simpler form 



Furthermore, with objectives of relatively low numerical aperture 



T(p)--1; 



1; (3.14) 



l-f^ 



\no / J 



so that 



,/'(p)^e2^^^^''^p) (315) 



If, in addition, the objective is free of spherical aberration, Wq(p) = 

 and 



^(p) = 1; (3.16) 



Pip) = c(p); ^ p ^p,n. (3.17) 



It should be noted that in the paraxial region, where p approaches zero, 



^(p) = 1; 



Pip) = cip); (3.18) 



irrespective of the spherical aberration. 

 From Eqs. 2.14, 3.5, and 3.(i, 



P^ip, q) = PAp) = Pip)e'^''^' -''^\ (3.19) 



We have seen that in a microscope Pm < 0.04 with ^ p ^ Pm- Hence 

 with excellent approximation 



9 



(1 - p2)^ = 1 - ^ (3.20) 



so that 



Pzip) = e^''''Pip)e-'^''"''^'^ (3.21) 



