290 THE DIFFRACTION THEORY OF MICROSCOPY 



Comparison of Eq. 14.8 above with Eq. 8.4 of Chapter II shows that the 

 total energy density over the region A of the geometrical image of the 

 particle obeys the law derived from the elementary theory. The 

 consequences of this law have been discussed in detail in Chapter II. 



The necessary conditions for the derivation of Eq. 14.8 are given by 

 Eqs. 14.5 and 14.6. In addition, Eq. 14.1 involves the supposition that 

 the field of view extends over a great number of wavelengths. 



15. CONDITIONS UNDER WHICH Hp = I OVER AN EXTENDED AREA 



It has been shown in Section 14 that, if there exists within the geo- 

 metrical image of the particle an area A for which Hp(x, y, po, go) = 1, 

 the contrast relations between this area and the image of the surround 

 obey the simple laws of the elementary theory. The problem of this 

 section is to determine under which physical conditions the area A 

 can be made to extend over a substantial portion of the geometrical 

 image of the particle in a Zernike system of phase microscopy. 



It is known that 



lim f e''J{x) dx = (15.1) 



for "civilized" functions fix). Hence it is to be suspected that the 

 oscillations of the exponential phase factor in the integrand of Eq. 14.2 

 may lead to difficulties in finding an extended region .4 for which Hp = 1. 

 For this reason we shall require that po and qo are restricted to values so 

 small that e-(2Ti/A/)(Por+go')) ^g substantially unity when 



(f^ + v')' S R (15.2) 



where R is the radius of the largest circle which can be inscribed in the 

 geometrical image of the particle about the point x, y as center (see 

 Fig. VII. 13). The optical direction cosines po, go can be restricted in 

 this manner with the use of a very narrow cone of axial illumination. 

 We suppose for definiteness that the hole in the condenser diaphragm 

 is small, circular, and centered upon the optical axis. The conjugate 

 area is a corresponding small, circular spot centered on the optical axis. 

 Let the numerical aperture of the conjugate area and of the objective 

 be pi and p„,, respectively, with respect to the image space of the ob- 

 jective. Introducing into Eq. 14.3 the changes of variable 



p = —p cos ; f = r cos 6 ; 

 q = — psin<^; rj = rsin^; (15.3) 



together with the facts that P{p, g) = 1 in the complementary area and 



