264 THE DIFFRACTION THEORY OF MICROSCOPY 



particle, respectively. For incident rays po, r/o that pass through the 

 edge of the particle the object function/ will depend upon Xq, yo, po, Qo 

 in a manner which is related to the geometry of the edge and to the 

 relative optical properties of the particle and surround. It is to be 

 presumed that when the brush effect is included in the object function 

 fixQ, yo, Po, qo), Equations 8.16, 8.10, and 8.17 are capable of interpreting 

 imagery at the edge of a thick particle as a function of z when the pupil 

 function P{p, q) is replaced by the afocal pupil function Pz{p, q). 

 Calculations of this type are extremely difficult and tedious and have 

 not been attempted to the knowledge of the writers. Such calculations 

 are, however, necessary to a complete interpretation of imagery near 

 the edge of a relatively thick particle such as a cubic crystal of salt. 



The above methods for writing the object function are readily ex- 

 tended to several plate-like particles or to periodic structures whose 

 elevations and troughs are plate-like. If such structures are so thin 

 that the brush effect can be neglected, it suffices to introduce into the 

 diffraction integrals of phase microscopy an object function which is 

 formally independent of the optical direction cosines po, qo and which is a 

 stepped function of the object coordinates Xq, yo. 



10. PHASE MICROSCOPY WITH OBJECTS OF PERIODIC STRUCTURE 



The theory of Section 8 will now be applied to objects of periodic 

 structure whose object function /(xq, yo, Po, qo) can be represented by 



/(^o, yo, Po, qo) = f(-ro, yo)- (lo.i) 



This equation applies with high precision to simple or crossed grating 

 structures whose troughs and elevations are shallow and do not focus 

 the incident light. The troughs and elevations of Fig. VII. 8 may rep- 

 resent optical path differences between the adjacent portions of the 

 periodic structure, amplitude transmissions of these adjacent portions, 

 or both optical path differences and amplitude transmissions. If 21 and 

 2m are the grating spacings along Xo and }'o, respectively, the object 

 function can be expressed by the Fourier series 



fixo, yo) = 2^ 2^ Ue V ; rnJ (10.2) 



1/ = — 00 ^= — GO 



in which the Fourier coefficients f^.^^ depend upon the details of the 

 periodic structure. 



From Eqs. 8.10 and 10.2 



Foix, y, Po, qo) = zJl^ -^"-^ J -J- J " e2.^•(po.o+«o.o) 



XU{x - Mxo, y - Myo) dxo dyo (10.3) 



