240 THE DIFFRACTION THEORY OF MICROSCOPY 



elements of area over the effective area of the source of Hght. If the 

 ampHtude and phase distributions are expressed as complex numbers, 

 the partial and total energy densities are automatically the time average 

 of the instantaneous partial and total energy densities. For this and 

 other reasons of mathematical convenience, we prefer to express the 

 laws of microscopy in terms of complex numl^ers. It should be noted 

 that this attack is general enough to include the essential laws of both 

 phase and ordinary microscopy as well as the laws of image formation 

 in a variety of other optical and non-optical systems. Furthermore, 

 the method of attack includes the combined effects of the source of light, 

 of the object specimen, and of the optical system upon the image of the 

 object specimen. 



The method can be adapted to either Kohler or critical illumination. 

 We shall discuss the adaptation to Kohler illumination since optical 

 systems are most freciuently adjusted for Kohler illumination or other 

 equivalent forms of illumination. 



It is emphasized that the following theory applies primarily to the 

 case in which the light is transmitted by the object specimen. In 

 applying the theory to microscopy with vertical illumination, great 

 caution has to be exercised in constructing the appropriate object 

 function. This is because the modifications of the amplitude and phase 

 of the illuminating wave can become more complicated w^hen the wave is 

 reflected by the object specimen than when the wave is transmitted by 

 the object specimen. Thus with vertical illumination the incident 

 wave may be reflected from one or both surfaces of the specimen and 

 from particles or layers within the specimen. By taking advantage of 

 the dissimilarity of the phenomena of transmission and reflection, the 

 ordinar}^ or the phase microscope can be made to yiekl additional 

 information about the object specimen. 



In deriving the diffraction integrals governing image formation, we 

 shall apply Kirchhoff 's laws as they have been reformulated by Luneberg 

 (1944). Conseciuently, the theory will be limited in generality by the 

 restrictions stated by Luneberg. The most fruitful problem is not, 

 however, to evolve a theory that is free from limiting restrictions but 

 rather to construct a theory whose integrals can be solved by methods 

 of the present or of the near future. To this end the primary diffraction 

 integral will not be applied in its most general form as given by Luneberg. 

 Instead, we shall be content to evaluate the primary diffraction integral 

 by integrating over the rays in the axial bundle. This procedure 

 involves the detei-mination of the imagery for off-axial points in terms 

 of the data which belong to a suitably chosen axial point. 



