RESOLVING POWER OF PHASE MICROSCOPES 69 



it is advantageous to write Eq. 18.1 in the universal form 



Gjr) ^ ^ r Jr(3.83\7r/ra) J 

 T^pJ L 3.8317r/r„ _ 



(18.8) 



in which r/va is physically the distance from the diffraction head meas- 

 ured in Airy units Va. The distance from the diffraction head to the first 

 zero in the energy density is therefore given by r/ra = 1. 



The energy densities G(r)/T"pm^ computed from Eq. 18.8 are plotted 

 in Fig. II.21A as a function of r/va. The primary diffraction curve of 

 Fig. II.2L4 is universal in the sense that it applies to Airy-type objec- 

 tives of any numerical aperture. Aberration-free objectives of low 

 numerical aperture approximate the Airy-type objective closely. 



It has been shown by Luneberg (1944, p. 391) that the Airy-type 

 objective has the highest possible energy density at the diffraction head. 

 The effect of introducing spherical aberration or diffraction plates is 

 necessarily to lower the energy density at the diffraction head. When 

 the energy density at the diffraction head is lowered, the energy content 

 of the diffraction rings is in^a^riably increased. A pronounced increase 

 in the relative energy content of the diffraction rings may lead to a loss 

 of resolving power or to reduced contrast in the image or to both. 



The primary diffraction curves have been computed for aberration- 

 free objectives for five different diffraction plates and are plotted in 

 Figs. 11.21/? to 11.235 so as to permit comparison with one another and 

 with the Airy-type diffraction curve of Fig. II.2L4. The type of diffrac- 

 tion plate is indicated together with the h^ and 5 values. The ratios 

 Pi/Pm = sin 01 /sin dm and P2/Pm = sin 02/sin dm of the zonal numerical 

 apertures of the conjugate area are indicated also. It is emphasized 

 that, on account of the universal nature of the curves, these curves apply 

 to all objectives having the zonal ratios pi/pm and P2/Pm and the given h^ 

 and 5 values, irrespective of the actual N.A. = lil/|p,„ of the objective. 

 The labeled ordinates to which the energy densities G{r)/Tv'^pm^ are 

 plotted are alike. As a consequence, the fact that G(0)/7r^p„i'* = 0.5 

 in Fig. II.21B whereas G(0) 7r-p,„-^ = 1.0 in Fig. II.2L4 means that the 

 energy density at the diffraction head with the objective of Fig. 11.21 B 

 is 0.5 of the value of the energy density at the diffraction head of the 

 corresponding Airy-type objective. It will be noted that the pri- 

 mary diffraction curves are identical for all 5 values of opposite sign. 

 For example, the primary diffraction curve of Fig. 11.21 fi applies 

 to 5 = ±X/4. 



An examination of the five primary diffraction curves drawn for the 

 indicated diffraction plates shows that the first minimum in the energy 

 density occurs in all cases near r/va = 1, that is, near the Airy limit. 



