PUPIL FUNCTION P{p, r/) IN PHASE MICROSCOPY 245 



avoid confusion as to the physical meaning of the primary diffraction 

 integral U{x — Mxq, y — Myo). The complex imaginary function 

 U{x — Mxq, y — il/i/o) is of direct physical significance in the sense 

 that \U(x — Mxq, y — Myo)\'^ gives the distribution of energy density 

 produced in the image plane by an unpolarized dipole radiator. An 

 unpolarized dipole radiator may be regarded as one that changes its 

 orientation in a random manner in a period of time which is short com- 

 pared with the smallest interval of time that can he distinguished by 

 the receptor of the energy density, or it may be regarded as a group of 

 independent dipole radiators oriented at random in an element of area 

 or volume which can be considered as being infinitesimally small. 

 I U(x — Mxq, y — M?/o)|^ is the distribution of energy density produced 

 by these unpolarized, that is, randomly oriented, dipole radiators. It 

 is important to appi'eciate that, whereas the phase and amplitude dis- 

 tribution produced l^y a polarized radiator is physically real, the phase 

 and amplitude distribution produced by an unpolarized radiator and 

 hence by U{x — Mxq, y — My^) is fictitious. A logically complete 

 exposition of the theory of microscopy should therefore begin with 

 polarized radiation and should expose in detail those considerations 

 which lead to the laws for unpolarized radiation. A presentation of 

 the complete theory would add a prohibitive amount of material to 

 this Appendix and would increase the reader's difficulties in mastering 

 the essential elements of a more general theory of microscopy. For these 

 reasons the primary diffraction integral as given by Efi. 2.13 is sufficient 

 for determining the energy density in the image plane when the source of 

 light is unpolarized. Furthermore, the conclusions reached with theaid of 

 Eq. 2.13 agree closely with the conclusions resulting from the more com- 

 plete theory which takes into detailed account the effects of randomly 

 polarized radiation. We shall continue to call Cix — Mxq, y — Myo) 

 an amplitude and phase distribution, but we shall not claim that 

 either it or the amplitude and phase distributions derived from it 

 are real amplitude and phase distributions. This distinction as to the 

 fictitious nature of the amplitude and phase distribution associated with 

 the primary diffraction integral is usually ignored or is implied tacitly. 

 We remark finally that the use of the primary diffraction integral is 

 not limited to radiating dipoles as object specimens but applies also to 

 illuminated pinholes in an opaque slide or, more generally, to illuminated 

 elements of area in the plane of the object specimen. 



3. THE PUPIL FUNCTION P{p. q) IN PHASE MICROSCOPY 



The pupil fimction P{p, q) is to be computed from the data of the 

 axial bundle. For a more detailed statement of the method of com- 

 putation and of the physical significance of the pupil function so com- 



