46 AN ELEMENTARY THEORY OF PHASE MICROSCOPY 



fraction plate are uncoated and Eqs. 7.3 and 7.4 reduce to Eqs. 7.1 and 

 7.2 which apply to the standard microscope. 



To find the resulting amplitude and phase over the image of the 

 surround and over the image of the particle, we recall that the undeviated 

 wave is spread over the entire plane of the image and that the undeviated 

 and deviated waves overlap on the neighborhood of the geometrical 

 image of the particle. Let S' and P' be complex numbers representing 

 the amplitude and phase of the resultant wave over the geometrical 

 image of the surround and the particle, respectively. Then 



S' = U = e^'^'£>-2'^''^«^«''%e'*« (7.5) 



because the geometrical image of the surround is illuminated by the 

 undeviated wave only; and 



P' ^ U ^ D = e' V2^'''"^'"'o[/ioe'*'' + (ge'^ - \)he'^'] (7.6) 



because the undeviated and deviated waves interfere as U + D over the 

 geometrical image of the particle. By factoring out /iie'^^ from the 

 right-hand members of Eqs. 7.5 and 7.6 and by defining 



hi e ^ All 

 we find that 



S' = (e*'V2^^""^'""'/iie^'^0/ie'^ (7-8) 



p' = {e'^'e-^''''''^'>'%ie'^'){he'^ + ge'^ - 1). (7.9) 



In summary, Eqs. 7.8 and 7.9 describe the amplitude and phase dis- 

 tribution produced over the image plane of a phase microscope by the 

 incidence upon the object plane of a single, inclined wave front whose 

 orientation is that of Fig. 11.13 with po = sin do- g is the ratio of the 

 amplitude transmission of the particle to the amplitude transmission of 

 its surround. A is the optical path difference in radians between particle 

 and surround. V is the optical path in radians from the object plane 

 to the image plane, hi and 5i are, respectively, the amplitude and 

 phase transmission of the complementary area of the diffraction plate. 

 It follows from the definition of Eq. 7.7 that h is physically the ratio of 

 the amplitude transmission of the conjugate area to the amplitude trans- 

 mission of the complementary area and that 8 is physically the optical path 

 difference between the conjugate and complementary areas. 8 is measured 

 in radians and is considered positive when the optical path of the conjugate 

 area exceeds that of the complementary area. 



The manner in which he'^ enters into Eqs. 7.8 and 7.9 shows that the 

 amplitude ratio h and the phase difference 5 are the essential properties 

 of the diffraction plate. This follows because the complex number in 



