44 AN ELEMENTARY THEORY OF PHASE AHCROSCOPY 



rounding medium is uniform optically. The particle will be assumed to 

 be optically uniform also. As indicated in Fig. 11.13, the relative 

 amplitude and phase transmission of the particle may now be represented 

 by the complex number ge^^. g is physically the ratio of the amplitude 

 transmission of the particle to the amplitude transmission of an equal thick- 

 ness of the surround, and A is the difference between the optical path of the 

 particle and the optical path of an equal thickness of the surround, with 

 A expressed in radians and considered positive when the optical path of 

 the particle exceeds that of the surround. 



In describing the undeviated and deviated waves which arise by 

 diffraction at the object specimen, we have only to retrace in a more 

 cjuantitative manner the steps already taken in Section 3. A portion 

 of the incident wave iv, with amplitude and phase described by Eq. 6.3, 

 passes through the surround and is designated as the S wave. The 

 second portion of the wave w passes through the particle and is designated 

 as the P wave. The P wave is then broken into two waves, one of 

 which is identical with the S wave and the other of which is designated 

 as the D wave. The incident w wave may therefore be regarded as 

 broken by diffraction into an S wave which extends over the whole 

 object plane and into a D wave which extends over the neighborhood 

 of the particle. The extended S wave was shown to have the properties 

 of the undeviated wave, and the D wave was shown to have the prop- 

 erties of the deviated wave. Since the amplitude and phase transmis- 

 sion of the surround is represented by the complex numljer unity, the 

 incident w wave emerges from the surround with amplitude phase which 

 we describe by the complex number S and which obeys the relation 



Since the amplitude and phase transmission of the particle is represented 

 by the complex number ge'^^, the incident w wave emerges from the 

 particle with the amplitude and phase which we represent by the com- 

 plex number P and which obeys the relation 



p = wge'^ = ^e'-^e-2Tmo2/opo_ (65) 



Breaking the P wave into an *S' wave and a D wave is equivalent to the 

 statement that P = S -{- D or that 



£) _ p _ ^' _ ^g'Ag— 27ri«o2/opo _ g— 27rmo2/OPO 



whence 



D = e-2^i^02/0P0(^giA _ ^y (g g) 



In summary, the relative amplitude and phase of the undeviated and 

 deviated waves as they emerge from the object specimen are given re- 



