THE PRIMARY DIFFRACTION INTEGRAL 



241 



2. THE PRIMARY HIFFRACTION INTEGRAL 



An integral which we shall call the primary diffraction integral has 

 been reformulated from fundamental considerations by Luneberg (1944). 

 V/e shall ))egin with the form of this integral as presented by Luneberg 

 and shall modify it for the purpose of rendering it more suitable to the 

 theory of phase microscopy. 



C-^o- yo) 



Fig. VII. 1. Notation with respect to the most general formulation of the primary 



diff faction integral. 



Let A^n^'o and XY denote a conjugate pair of object and image planes, 

 as in Fig. VILl. A point (.r, y, z) falls in the conjugate (sharply 

 focused) image plane only when z = 0. Suppose that an "unpolarized" 

 dipole radiator of strength unity is located at the object point .ro, yo and 

 that it is desired to find the amplitude and phase distribution 

 U(x — Mxo, y — ^^yo, z) produced by the unpolarized dipole and the 

 optical system. Then in accordance with Luneberg we set 



U{x - Mxo, y - Myo, z) ^ F(x, y, z); (2.1) 



U{x - Mxo, y - Myo, 2) = ^ iT'^^^' 5)e^'^"'+^"+'"+"^ dv dq (2.2 



) 



in which the integral extends over the optical direction cosines p, q of the 

 normals to the converging wave front (Fig. VILl) in the image space 

 whose refractive index is n. By definition, 



27r 



; (2.3) 



X 



r 2 2 2\i 



s = (rr - V - (I ) ; 

 W = ]L(.ro, yo, zq] p, q) 



(2.4) 

 (2.5) 



