22 AN ELEMENTARY THEORY OF PHASE MICROSCOPY 



spread over the entire image plane, the image of the surround is ilkimi- 

 nated by the undeviated wave whereas the image of the particle is 

 illuminated by both the undeviated and the deviated wave. The phase 

 and amplitude distribution of the light that illuminates the image of the 

 surround is therefore that of the undeviated wave, i.e., of the S wave of 

 Fig. II. 4. Because the undeviated and deviated waves overlap upon the 

 geometrical image of the particle, these two waves interfere to produce 

 the resultant wave (D + *S) over the image of the particle. From the 

 construction of Fig. II. 4, D -\- S = P. The deviated and undeviated 

 waves thus combine over the image of the particle to reproduce the 

 P wave. The phase and amplitude distribution of the light that illumi- 

 nates the image of the particle is therefore that of the P wave. In 

 conclusion, the amplitude and phase distributions over the image of the 

 surround and over the image of particle are, respectively, those of the 

 S wave and those of the P wave. This conclusion is true irrespective of 

 the value of A, the optical path difference between the particle and sur- 

 round. It is also true irrespective of any differences in absorption which 

 may be present between the particle and the surround. 



It has been seen how the P and S waves originate in the object plane 

 as the portions of the incident wave that are intercepted by the object 

 particle and its surround, respectively. The amplitude and phase of 

 the light that leaves the particle is therefore that of the P wave, whereas 

 the amplitude and phase of the light that leaves the surround is that of 

 the S wave. The conclusion of the previous paragraph is that the 

 amplitude and phase of the light that enters the image of the particle 

 is likewise the amplitude and phase of the P wave, whereas the amplitude 

 and phase of the light that enters the image of the surround is that of 

 the S wave. We conclude, therefore, that the phase and amplitude 

 distributions over the object and image plane will be similar and that, 

 consequently, the image will be similar to the object. This conclusion 

 can be stated as a theorem which is usually attributed to Lummer 

 (Theorem 2 ) : 



When the entire deviated arid undeviated bundles of light which originate 

 by diffraction at the object are admitied and transmitted by an aberration- 

 free objective, the amplitude and phase distributions over the object plane 

 and over the sharply focused image plane are similar and the image is 

 similar in all details to the object. 



As applied to real objectives, this theorem is an idealization, for no 

 real objective is capable of transmitting the entire deviated bundle. 

 The agreement between the theorem and experimental facts becomes 

 closer as the numerical aperture of the objective and the size of the object 

 details are increased. 



