CRITERION FOR ISOLATED PARTICLES 285 



no" - po - go 



and may be set equal to unity with Lambertian sources of light. Since 

 f and 7] are measured from the point (x, y), G is in fact an implicit func- 

 tion of .r, y. The integration with respect to dpo dqo extends over the 

 range of optical direction cosines of the rays incident upon the object 

 plane. Equations 11.8 and 1 1 .10 are of great importance to the theory 

 of phase microscopy with non-periodic objects. These equations may be 

 used with considerable confidence when the field of view in the object 

 space extends over 50 or more wavelengths and when the particle is 

 located well within the field of view. When smaller fields are en- 

 countered, the validity of the approximation afforded by Eqs. 11.8 and 

 11.10 should be investigated or one should return to the more general 

 equations from which Eq. 11.8 has been specialized. Smaller fields of 

 view are permissible with objectives having the higher numerical 

 aperture. 



12. CRITERION FOR ISOLATED PARTICLES 



Upon looking into a phase microscope, one is impressed by the rapid- 

 ity with which the energy density in the surround approaches a sub- 

 stantially constant value as the point of observation recedes from the 

 edge of a particle. One is impressed also by the slight effect which one 

 particle exerts upon the image of another unless the particles are tightly 

 packed. The cause of these phenomena will now be treated in a semi- 

 rigorous manner. 



With reference to Eq. 11.8, it has already been stated that the phase 

 factor g(27ri/M)(P0J:+OT2/) ]g ^f j^q practical interest to microscopy. We shall 

 therefore omit it and write 



M'Fo{x,y,po,qo) = p(^^^'^) 



+ (^giA _ 1) /Tg-(2.i/M)(P0r+5O>,)f/(^^ ^) rff ci^_ (12.1) 



The meaning of the double integral is clarified by Fig. VII. 12. As the 

 point X, y recedes from the edge of the geometrical image of the particle, 

 the double integral decreases for two reasons. First, the exponential 

 in the integrand oscillates more rapidly as (f^ + 17^)^ = D increases. 

 The rapidity of these oscillations increases with the inclination po, qo 

 of the incident wave front. Hence the double integral of Eq. 12.1 de- 

 creases more rapidly as the point x, y recedes from the edge of the 



