252 THE DIFFRACTION THEORY OF MICROSCOPY 



through the conjugate area exceeds that through the complementary 

 area. 



Since Ci (p, q) 9^ 0, no essential loss of generality is obtained by setting 



c(p, q) = Clip, q) = 1 (4.3) 



for points {p, q) of the complementary area and by setting 



c{p, q) = he'^ (4.4) 



for points (p, q) in the conjugate area. 



In phase microscopy the conjugate area is so narrow that the varia- 

 tion introduced into h.Qip, q) by the allowable range in p, q is negligible. 

 Since h\ (p, q) = 1 when the absorbing material is placed upon the con- 

 jugate area, Eqs. 4.3 and 4.4 apply with excellent approximation to all 

 A-type diffraction plates, irrespective of the location of the diffraction 

 plate. This is not necessarily true for B-type diffraction plates which 

 are used in conjunction with oil immersion objectives. With B-type 

 diffraction plates it may in some instances become necessary to return 

 to Eqs. 4.1. If the conjugate area is narrow, one may, however, in- 

 troduce the simplification 



Coiv,q) -Ke''' (4.5) 



in which Jiq is specified by hoipm, qm) where pm and qm are the median 

 values of p and q in the narrow conjugate area. 



5. THE PRIMARY DIFFRACTION INTEGRAL WITH AIRY-TYPE OB- 

 JECTIVES 



An Airy-type objective is defined as one for which the aberration 

 function 



'/'(p, q) = 1. (5.1) 



Real ol)jecti\Ts have the propei'ty that \{/{p, q) approaches unity at low 

 numerical aperture. Whenever the objective obeys or approximates 

 closely the condition 



HP, q) = 1, (5.2) 



Pip, q) = dp, q) (5.3) 



so that the pupil function is numerically equal to the coating function. 

 If the diffraction plates of such objectives are uncoated, dp, q) = I 

 and Pip, q) = 1. Equations 5.2 and 5.3 are of theoretical and practical 

 importance because they are often approximated by well-corrected 

 objectives of relatively high numerical aperture. Whenever approxi- 

 mate solutions are acceptable, the pupil function is given with sufficient 

 accuracy by the simplified statement of Eq. 5.3. 



