1)1 UK ACTION OK 1< \DH) WAVES 



129 



Figure 22. Vector sum of subzone contributions. 



P' is chosen in the direction of SMi of Figure 23, 

 where Mi is the upper edge of the first half-wave- 

 length zone. The illumination at the point P' is, 

 firstly, due to all wavelets emanating from the half 

 wavefront above P'S. In addition, there is the 

 contribution from the lower half of the wavefront 

 extending from Mi to M - The situation is, in fact, 

 the same as if P' were brought down to P and the 

 diffracting edge were lowered from M to M\ (see 

 Figure 24) . The resultant amplitude R is represented 



Figure 23. Diffraction at shadow line. 



on Figure 25 by ZB. Starting at the point P at the 

 edge of the shadow (Figure 23) where the amplitude 

 is AZ, if the point is moved upward, the tail of the 

 amplitude vector moves to the left along the spiral 

 while its head is fixed at Z, 



The amplitude goes through a maximum at b', a 

 minimum at c', etc., approaching a value ZZ' for 

 the unobstructed wave. Moving in the other direc- 

 tion, into the shadow, the vector moves to the right 

 from A, decreasing steadily to zero. 



The power intensity versus v is plotted in Figure 

 26, and the points B, C, D, etc., corresponding to 

 those in Figure 25 represent the exposure of 1, 2, 3, 

 etc., half-period zones below M - The maxima and 

 minima occur a little before these points are reached. 

 This curve may be plotted from the table of Fresnel 

 integrals with the equations 



/ = 0.5 + C , 



g = 0.5 + S , (38) 



z 2 = kf 2 + g 2 ) , 



where s 2 is the relative power intensity compared to 

 the unobstructed wave. The relative electric inten- 

 sity is j„ , 2 



2 = A —^ ■ (39) 



Equation (39) is plotted in Figure 27. The portion 

 of the curve for — v has been drawn to the right and 

 is to be used with the right-hand ordinate. 



The phase lag f due to diffraction may be deter- 

 mined from the angular position of the vector R in 

 Figure 25. In the illuminated region the phase lag 

 oscillates about the reference value, Z'Z, and is 

 given by 



- tan " 1 /- 



7T 



4 



At the shadow line the relative value is the same as 

 Z'Z. In the shadow region the phase lag varies 

 continuously along a parabolic curve and is given by 



1(2 TT 



V 



+ 



Figure 24. Diffraction in illuminated region. 



The phase lag relative to that of the shadow line 

 is plotted in Figure 28. The portion of the curve 

 for — v is drawn to the right, and its ordinate, on 

 the right, has a different scale from that used with 

 the -\-v portion of the curve. 



15.4.9 Location of Maxima and Minima 



When the source is close to the diffracting edge, 

 the positions of the maxima and minima in the 



