242 THE DIP^FRACTION THEORY OF xMICROSCOPY 



is Hamilton's mixed characteristic. Furthermore, if A-^ip, q) denotes 

 the amphtude variation over the converging wave front, 



^r ^ -4 1 (p, q) 



(n — p — q-^)' 



when the converging wave front does not depart appreciably from a 

 sphere. Both Aiip, q) and Hamilton's mixed characteristic can be 

 determined by triangulating rays through the optical system and may be 

 regarded as known properties of a given system. 



Let all distances be measured in number of wavelengths. Xq, y^, zq, 

 X, y, and z are then dimensionless. Equation 2.2 assumes the form 



U{x - Mxo, y - Mijo, z) =ff<t>(p, q)e2-'^'''+P^+iy+-'^^ dp dq. (2.7) 



Since Zq is simply a parameter which determines the location of the 

 object plane, we may take ^o = without essential loss of generality. 



In order that the primary diffraction integral shall be amenable to 

 simplified mathematical transformations as well as to simplified computa- 

 tions, we shall assume that Abbe's sine condition is obeyed and we shall 

 replace the integration over the optical direction cosines of the normals 

 to the wave front by an integration over the optical direction cosines 

 of a set of factitious axial rays which become real axial rays when the 

 object point Xq, y^ is located upon the optical axis and when the objective 

 is free of spherical aberration. Hamilton's mixed characteristic IF now 

 reduces to the form (Luneberg, 1944, p. 226) 



W = Wq{p, q) - pMxo - qMyo (2.8) 



in which M is the constant magnification ratio between the object and 

 image planes and Woip, q) is the usual integral over the lateral spherical 

 aberration. W^ip, q) is to be evaluated from the data of the object 

 point .To = 2/0 = upon the supposition that the reference sphere of 

 Fig. VII. 2 has been chosen so as to coincide with the axial wave front 

 in the paraxial region. This means physically that the reference sphere 

 will be chosen so that 



WoiO, 0) = 0. (2.9) 



We now introduce Eq. 2.8 into Eq. 2.7 and write 



U{,x - Mxq, y - Myo, z) 



<i)(p g)e^''*^''"^^'^^e^''*'^^-^'~^^-^°^"'"^^^~^^^"^"'"*^'" (/p f/g (2.10) 



=//< 



in which the optical direction cosines p, q refer to the factitious rays of 



