56 AN ELEMENTARY THEORY OF PHASE MICROSCOPY 



diffraction plate when the particle appears in darkest contrast. On 

 account of the trigonometric identity 



tan .r = tan {x + vr), 



5 = 5rf + TT is a second solution of Eq. 13.3. The remaining solutions of 

 Eq. 13.3 differ from 8d and (8d + tt) by integral multiples of 27r. 8^ is 

 the first solution for which Gj, is minimum, whereas 6^ + tt is the first 

 solution for which Gp is maximum. This result is summarized as the 

 important theorem (Theorem 5): 



// the particle appears in darkest contrast when h and 8 are set at the 

 values hd and 8,i, then the particle will appear in optimum bright contrast 

 when the phase difference 8d between the conjugate and cojuplementary 

 areas of the diffraction plate is altered by J'2 wavelength. 



The contlitions for optimum bright contrast may therefore be found 

 in a simple manner from the conditions for darkest contrast. If, for 

 example, the particle appears darkest with a diffraction plate for which 

 h = 0.4 and 8 = +X/8, the particle will appear in optimum bright 

 contrast with another diffraction plate for which h = 0.4 and 8 = 5X/8, 

 or, equivalently, — 3X/8. 



We will now show that Gp = AGs when the particle appears in opti- 

 mum bright contrast. The conditions of Eqs. 9.2-9.4 for dark contrast 

 show that 



hd sin 5rf = —gsm A; 



hd cos 5rf = 1 - g cos A. (13.4) 



If we let 8b denote the 5 value for optimum bright contrast, then 



8b = 8d + TT. (13.5) 



Therefore 



hd sin 8b = —hd sin 5^ = gr sin A; 



hd cos 8b = —hd cos 8d — g cos A — 1. (13.6) 



When g sin A = hd sin 8b and g cos A — 1 = hd cos 8b are substituted 

 into Eq. 8.8, it is found almost directly that 



Gb = hi-'hd', Gp = 4:hi~hd'", 

 whence 



Gp = 4Gs (13.7) 



at optimum bright contrast. Brighter contrasts than that indicated 

 by Eq. 13.7 are possible at the expense of reducing the amplitude of the 

 undeviated wave below the amplitude of the deviated wave. 



