68 



AN ELEMENTARY THEORY OF PHASE MICROSCOPY 



where df„ is the maximum angiihir aperture of the objective in its image 

 space as ilkistrated in Fig. 11.20. p,„ is related to 



Po,n = sin^om, (18.3) 



with doni defined in Fig. 11.20, in accordance with the Abbe sine condition, 



as 



\M\p,n = nopom = Ho sin do,n = N.A., (18.4) 



where no is the refractive index of the object space and N.A. is the usual 

 numerical aperture of the objective. 



Diffraction plate 



2 "m 



p. = s\n e.;j = l,2,m 



Conjugate area 



-M- 



-Object plane 



Image plane- 



FiG. n.20. The zonal apertures of the objective and its diffraction plate whose 



conjugate area is annular in shape. 



It will be noted from Eq. 18.1 that G{r) = when ./i(27rp,„r) = 0. 

 The first value of r > for which Ji(27rp^r) = is denoted by Ta and 

 is called the Airy limit with respect to the image space. Since Ji (2) = 

 when z = 3.8317, 



3.8317 0.6098 , ^ ,,„ ^, 



Ta = = wavelengths. (18.5) 



27rp,„ p,n 



Ta is physically the distance from the diffraction head to the first zero 

 in the energy density. If Va" denotes the Airy limit as measured in the 

 object plane with an objective of magnification ratio M, 



0.6098 0.6098 



wavelengths. 



o 



M 



\M\pr, 



N.A. 



(18.6) 



Equation 18.6 is the usual expression for the Airy limit. 

 Since 



27rp^r = 3.8317 - ' 

 ra 



(18.7) 



