52 AN ELEMENTARY THEORY OF PHASE MICROSCOPY 



the h value must be chosen as V 2 in order to obtain optimum darkest 

 contrast in the image of the particle. In order to obtain h values exceed- 

 ing unity, the absorbing material must be placed upon the complemen- 

 tary area so as to weaken the deviated wave rather than upon the 

 conjugate area of the diffraction plate. 

 Suppose that A = ±X/6. Then 



/i = 1; 



X X -X , X 



8 = = when A = - ; 



12 4 G 6 ' 



-XX +X , -X 



8 = 1 — = — when A = 



12 4 6 6 



The case A = d=X/6 when g = I is the point of demarcation between 

 diffraction plates for which /i < 1. When h = 1, no absorbing material 

 is required on either the conjugate or complementary area. It will be 

 noted that the required 8 for darkest contrast is X/6 when A = — X/6. 

 Diffraction plates for which 5 = X/6 and /i = 1 are radically different 

 from diffraction plates for w^hich 8 = ±X/4 and /i — > |a| with A small. 

 The theoretical necessity for a series of diffraction plates in order to 

 secure optimum contrast with different particles therefore becomes ap- 

 parent. 



We learn from Eci. 10.() that /i = 2 is the largest useful h value. Cor- 

 respondingly, the energy transmission ratio between the conjugate and 

 complementary areas is h^ = 4. The value h = 2 is required when 

 A = ±7r radians. From Eq. 10.7 or 10.8 the required 8 value is then 

 5 = 0. This example shows that particles whose amplitude transmission 

 is equal to that of the surround and whose optical path difference with 

 respect to the surround approaches 3^ wavelength will appear in darkest 

 contrast in the sharply focused image plane of an aberration-free ob- 

 jective when a diffraction plate is selected whose h value approaches 

 h — 2 and whose 8 value approaches 5 = 0. It follows from Eqs. 8.7 

 and 8.8 that Gg, the energy density over the image of the surround, ap- 

 proaches 4 when hi = 1, and that Gp, the energy density over the image 

 of the particle, approaches zero. The relative value of Gg and Gp as 

 produced by the ordinary microscope in the sharply focused image plane 

 in the case of the class of particles of this example is obtained by sub- 

 stituting hi = h = g = 1, 5 = 0, and A = ±7r into Eqs. 8.7 and 8.8. 

 The result \s Gg = Gp = 1 . This equality of Gg and Gp is another illus- 

 tration of the fact that the contrast produced in the image of an un- 

 stained particle by an ordinary microscope is due to the failure of the 



