20 AN ELEMENTARY THEORY OF PHASE MICROSCOPY 



D and S is not only efjiuil to the P wave mathematical!}^ but is also 

 indistinguishable from the P wave physically. We may suspect already 

 that the D wave is the deviated wave. Since the portion of the incident 

 wave that is intercepted by the surround is the 8 wave, and since the 

 portion of the incident wave that is intercepted by the particle can be 

 split into a D wave and another wave identical to the S wave, the 

 incident wave can be regarded as broken by diffraction into the S wave, 

 which extends over the entire object plane, and into the D wave, which extends 

 over the neighborhood of the particle. The S and D waves may be identified 

 with the undeviated and deviated waves from the following observations : 

 We note from the construction of Fig. II. 4 that when A = 0, so that the 

 particle vanishes, the D waxe disappears whereas the *S wave remains 

 unchanged. This means that the S wave can only be the undeviated 

 wave, for it is the wave that is present in the absence of an object particle. 

 Since we know that there are but two waves produced by diffraction at 

 the object specimen, namely the undeviated wave and the deviated 

 wave, it follows by elimination that the D wave must be the deviated 

 wave. The observation that the D wave vanishes when the particle 

 vanishes is consistent with the conclusion that the D wave is the deviated 

 wave. 



The construction of Fig. II. 4 gives the relative amplitudes of the S 

 and D waves, together with the phase difference between these two 

 waves, which are, respectively, the undeviated and deviated waves. 

 It wnll be noted from Fig. II. 4 that the D wave lags }4 wavelength be- 

 hind the S wsLve when the optical path of the particle exceeds that of the 

 surround by a small amount A. On the other hand, if the optical path 

 of the particle were less than that of the surround l)y a small amount A, 

 the S wave in Fig. II. 4 would have been drawn so as to lag behind the 

 P wave by the amount A. The D wave would then be found* to have 



* The laborious method of the graphical construction can l)e avoided by the follow- 

 ing simple analytical considerations: Let s denote the amplitude of the S wave, and 

 let the phase constant, 4>s, of the S wave be chosen as reference with </)s = 0. Then 

 for the S wave the displacements ys are 



Vs 



s sin (2 + 4>s) = s sin z. (1) 



Since the amplitudes of the S and P waves are equal, we have correspondingly for 

 the P wave 



ijp = s sin {z + A), (2) 



in which A is the amount by which the P wave is retarded with respect to the S 

 wave. Let d denote the required amplitude of the D wave, and let cp denote the 

 required phase retardation of the D wave. Then 



yd = d sin {z + 4>). (3) 



