288 THE DIFFRACTION THEORY OF MICROSCOPY 



The total energy density G{x, y) at points x, y which are located in the 

 far interior of a large particle obeys therefore the law 



G{x, y) = constant = Gp 



= Mf'^'"" «»' 



dpo dqo. (13.3) 



If the objective is of the Airy type and if the conjugate and comple- 

 mentary areas of the diffraction plate are coated uniformly and dif- 

 ferently, Pipo/M, qo/M) = he^^. For this specialization to a Zernike 

 system of phase microscopy 



2 7,2 





G„ = "hrrr j j Kpo, Qo) dpo dqo = g^Gs (13.4) 



where Gg is the total energy density of the surround as in Eq. 12.7. If 

 the amplitude transmission of the particle is equal to that of the sur- 

 round, gf = 1 and Gp = Gs, irrespective of the optical path difference 

 A between the particle and the surround. This means physically that 

 with such particles the brightness of a far interior point in the image of 

 the particle will be the same as the brightness of the surround. This 

 phenomenon is familiar to users of the phase microscope. 



If the particle is not large or if the point x, y approaches the edge of 

 the geometrical image of a large particle, the evaluation of Eq. 11.7 

 becomes difficult. No relatively simple generalizations can be made at 

 the present time beyond the observation that the solution can become 

 quite different from that of Eq. 13.1. Some interesting conclusions can 

 be drawn, however, in the specialized case in which the object is illumi- 

 nated by narrow-coned axial illumination. These conclusions will form 

 the subject matter of Sections 14 and 15. 



Experiment reveals a halo near the edge of the geometrical image 

 formed of a particle by a phase microscope. Explanation of this halo 

 for thin plate-like particles involves the formidable task of evaluating 

 Eq. 11.7 at points .r, y near the edge of the geometrical image. 



14. CONDITIONS FOR REDUCTION TO THE ELEMENTARY THEORY 



The predictions of the elementary theory of phase microscopy of 

 Chapter II are characterized by the fact that the variation in the energy 

 density between particle and surround is a step function. The level 

 of illumination is sensibly uniform over the entire image of the particle 

 and changes abruptly at the edge of the geometrical image of the particle 

 to another uniform level over the image of the surround. The laws 

 that govern the difference between these two levels of illumination are 

 derived in Chapter II. An abrupt change in the level of illumination is 



