48 AN ELEMENTARY THEORY OF PHASE IVHCROSCOPY 



We see at once that the average energy density produced over the 

 image plane by the incidence of a single inclined wave front which origi- 

 nates from a point in the opening of the condenser diaphragm is inde- 

 pendent of the inclination of the wave front, of the optical path V 

 between the object and image planes, and of the phase transmission 3i. 

 Since the distribution of the energy density does not depend on the inclina- 

 tion of the incident wave front, the light radiated from all points in the 

 opening of the condenser diaphragm serves only to increase the total energy 

 density in the image. Hence the brightness of the image of the surround 

 and particle will be proportional to \S'\^ and |P'|", respectively, with 

 \S'\^ and |P'|2 determined from Eqs. 8.1 and 8.2. 



In order to avoid confusion, let us henceforth denote the total energy 

 density in the image plane by G and let Gs and Gp refer to the total 

 energy density over the image of the surround and of the particle, re- 

 spectively. Then 



Gs = hi'^h^; (8.3) 



Gp = hi^\he'^ + ge'^ - l\^. (8.4) 



If desired, Gg and Gp can be interpreted as the brightness of the image 

 of the surround and particle, respectively. Now 



he'^ + ge'^ - 1 = (/i cos 5 + ^ cos A - 1) + ^■(/^ sin 5 + ^ sin A). (8.5) 



Since the square of the absolute value of a complex number is equal to 

 the sum of the squares of its real and imaginary parts, 



\he'^ + ge'^ - l\'^ = {h cos 5 + ^ cos A - 1)^ 



+ ihsind +g sin A)^. (8.6) 



In summary, from Eqs. 8.3, 8.4, and 8.6 



Gs = hrh-; . (8.7) 



Gj, = hi-\{h cos 5 + ^ cos A - 1)2 + (/i sin 5 + ^ sin A)^]; (8.8) 



in which Gg and Gp are proportional to the total energy density or bright- 

 ness of the image of the surround and of the particle, respectively, g 

 is the ratio of the amplitude transmission of the particle to the amplitude 

 transmission of its surround. A is the optical path difference between 

 the particle and the surround, measured in radians and considered 

 positive when the optical path of the particle exceeds that of the sur- 

 round, h is the ratio of the amplitude transmission of the conjugate 

 area of the diffraction plate to the amplitude transmission of the com- 

 plementary area of the diffraction plate. 5 is the optical path difference 

 between the conjugate and complementary areas. 3 is measured in 



