NON-UNIFORMLY ALTERED HIGHER ORDERS 281 



conjugate area. We see from the physical argument of this section that 

 the required departure of 8 from ±X/ 4 depends on A, on the grating 

 spacing 21, on the numerical aperture of the objective, and on the size 

 and shape of the conjugate area of the diffraction plate. A more general 

 argument which includes spherical aberration of the objective would 

 show that 5 depends on the character and magnitude of the spherical 

 aberration. 



The following property of Eqs. 10.3.13 is typical not only of the 

 special case studied in this section but also of most, if not all, of the cases 

 in which some of the higher spectral orders from a pure phase grating 

 pass through the conjugate area of the diffraction plate. Suppose 

 that A becomes so small that 



sin A = A; cos A = 1. (10.3.14) 



If the optical path differences A between the elevations and the troughs 

 are small, Eqs. 10.3.12 and 10.3.13 reduce to the simpler relations 



2|a| 



h = ^ ; (10.3.15) 



T 



sin 5 = sgn f cot - j = sgn (A); 



cos 5 = 7-^- (10.3.16) 



Sir 



Hence 5 will fall in the interval — tt 2 ^ 5 ^ tt 2 and very near the 

 upper and lower bounds of this interval. The center of the troughs will 

 appear dark when the optical path difference 8 of the diffraction plate 

 is +X 4 or — X/4 according as A > or A < 0. By definition, 8 > 

 when the optical path of the conjugate area exceeds that of the comple- 

 mentary area, and A > when the optical path of the elevations 

 (Fig. Vn.9) exceeds that of the troughs. Since h < I in Eq. 10.3.15, 

 we reach again the familiar result that regions of lesser optical path 

 appear in optimum dark contrast with A+X 4 diffraction plates and that 

 regions of greater optical path appear in optimum dark contrast with 

 A — X 4 diffraction plates. The most interesting conclusion to be 

 drawn from this paragraph is that the 8 values for optimum contrast 

 are again zbX/4, even when some of the higher spectral orders pass 

 through the conjugate area, provided that A is so small that sin A = A. 



Sections 10.1 and 10.3 demonstrate that the phenomena of phase 

 microscopy become more complicated when some of the light which is 

 deviated by diffraction at the object passes through the conjugate area. 

 In fact unpublished studies of special cases have shown that unusual 



