266 



THE DIFFRACTION THEORY OF MICROSCOPY 



^".M = ffl f P^V, g)e2'^'^^^i+''^i^ 



ty «yty — oo »y — oo 



^e"A" fe +'^°)+ ^' fe ^w)\ dx, dy, dp dq. (10.8) 

 Integrating with respect to dxi dy\, we obtain 



B,,^ = lim ffp{p,q) 



sm 





Vl 



Po _ _ 

 M 2MI 



sm 



X 



) 



2.y. (^ - I - ^)_ 



TT U 



i2 _ ^ 

 M 2AIm, 



,2„ 2 



rfp dq; (10.9) 



(10.10) 



p2 _^ ^2 ^ ^:^p,^^ 



Since the Dirichlet integral 



/" ^ sin a(x — , ., s 



fix) ^ ^, dx=m; 



— a < t -\- a; 

 successive integration with respect to dp and dq in Eq. 10.9 yields 



-^V (10-13) 



Mm/ 



(10.11) 

 (10.12) 



"•" \M 2MI M 2Mm, 



Discontinuities in the pupil function P(p,q) may be ignored in the 

 theory of phase microscopy because such discontinuities are limited to 

 a small finite number of loci. 



In summary: Given the periodic object function 



fixo,yo) -2^Z^f^..<^ ^' '"^ 



(10.14) 



the corresponding amplitude and phase distribution Fq{x, y, po, go) in 

 the conjugate image plane is given by 



2-n-i, 



Foix, y, Po, qo) = 



,M 



(p{)X+qoy) 



M' 



XI Z) ^^J^J^ Cm'- Mm) (10.15) 



