GENERAL STATEMENT OF THE DIFFRACTION INTEGRALS 255 



all elements of area of the object plane. Let F(x, y) denote this resultant 

 amplitude and phase distribution. Then 



/ x(^'o, yo)U{x - Mxo, y - Mijo) dxQ dyo (6.2) 



in which x(^o^ Vo) is required to be zero beyond the field of view of the 

 optical instrument and in which all dimensions are to be measured as 

 numbers of wavelengths. Hence we may say that the coherent ampli- 

 tude and phase distribution x(-*"o, Vo) of the object plane is transported 

 to the image plane as the coherent amplitude distribution F(x, y). 



Incident wave front 

 Fig. VII. 5. Coherent illumination of the object plane A'oFq. 



7. THE MOST GENERAL STATEMENT OF THE DIFFRACTION INTE- 

 GRALS OF PHASE MICROSCOPY 



Suppose that a substantially plane wave front whose normals have the 

 optical direction cosines po, 9o is incident (as in Fig. VII. 5) upon the 

 object plane. The light in this incident beam originates at an in- 

 finitesimally small area in the source of light and may be regarded as 

 coherent. From a knowledge of po, 9o and of the physical properties of 

 the object specimen, one can construct a function x(^o, Vq, Po, Qo) dxo dyo 

 which gives the amplitude and phase of the wavelets which leave an 

 element of area dxQ dyo of the object plane. The difficulties of con- 

 structing x(-i'o, 2/0, Po, Qo) can become great even with relatively simple 

 object specimens, and the difficulties of integrating the expressions in 

 which xi^o, Vo, Po, Qo) appears can readily become insurmountable. 

 Except with very simple object specimens, considerable ingenuity will 

 be required in finding a good approximation to x(-^o, Vo, Po, Qo) and in 

 integrating the resulting diffraction integrals with satisfactory approxi- 



