ENERGY DENSITY OVER THE IMAGE PLANE 47 



parentheses in Eqs. 7.8 and 7.9 is common to *S' and P' and so does not 

 alter the relative amplitude and phase of the wave S' which reaches the 

 image of the surround and the resultant wave P' which reaches the 

 image of the particle. 



Let M denote the magnification of the objective in focusing the XqYq 

 plane into the image plane XY. Then y = Myo so that 



^—•ZirinoyopO _ g—2irinopo(y I M) /y ^q\ 



The appearance of the phase factor of Eq. 7.10 in Eqs. 6.7 and 6.8 and 

 in Eqs. 7.8 and 7.9 is consistent with Lummer's theorem for, according 

 to this theorem, phase variations produced over the object plane are 

 reproduced over the image plane. A closer comparison of Eqs. 6.7 and 

 6.8 and Eqs. 7.8 and 7.9 shows that not all amplitude and phase varia- 

 tions produced over the object plane are reproduced over the image 

 plane except in the trivial case h = 1 and 5 = in which the phase 

 microscope degenerates into the ordinary idealized microscope. Depend- 

 ing, therefore, on the choice of the properties h and 8 of the diffraction 

 plate relative to the properties g and A of the object particle, contrast 

 in the image can be made to depart considerably from the contrast 

 predicted from Lummer's theorem. We are now in a position to discuss 

 quantitatively within the scope of the elementary theory the contrast 

 variations which are possible in a phase microscope for a variety of 

 object specimens. 



8. DISTRIBUTION OF ENERGY DENSITY OVER THE IMAGE PLANE 



The energy density produced over the image of the surround by the 

 light in the incident wave front of Fig. IL13 is proportional to |*S'|^ 

 with S' given by Eq. 7.8. Since *S' can be written in the form 



and since the absolute value of the exponential is unity, 



|^'|2 = h,'h\ (8.1) 



Similarly the energy density over the image of the particle is propor- 

 tional to \P'\'^ with 



Therefore 



|p/|2 ^ |g/(y+Si-2,rm02/0P0)|2;^^2|;jgt5 _|_ ^giA _ ;^|2 



or 



\P'\- = hi'\he'^ -^ ge'^ - l\'\ (8.2) 



