CALCULATIONS FOR OPTICAL-INTERFERENCE REGION 



8L 



Another convenient expression for R is obtained by 

 replacing, in equation (83), the path difference A 



P 



.03 — 



.04 — 

 .05 — 



.3 



.8 



.98^ 



i/< (degrees) 



75 



60 

 45 



30 



20 

 10 



5 



Idi 



I •= 



.05 



.02 



.01 



.0 05 



.002 



.001 1— 



h, (METERS) 

 I0<" 



10= 



S_ 



10^ 



10 



l'- 



FlGURE 24. ^ as a function of /ii and y—sv = di/dj: [See 

 equation (107).] (See Figure 14 for definition of lengths.) 



by nX/2, where n (see below) may assume any posi- 

 tive value. Substituting '^2kahi for d^, equation (83) 

 assumes the form 



R = nr, (114) 



where 



1 ka X 



2^2 h{ 



3/2 



(115) 



or 



r = 1030 



/h 



3/2 



(forfc = 4/3). 



A graphical representation of r is given in Figure 15 

 in Chapter 6. 



Then for a reflection coefficient of p = 1, </> = 180' 

 degrees (i.e., 4>' = 0), equation (29) gives 



X 



ntr. 



(116) 



If r is fixed, a complete pattern of contour lines- 

 (along which A is constant) is determined. Take as 

 independent variables p and n (rather than u and v). 

 A given choice of p and n determines R by equation 

 (114), by equation (116), v by equation (113), 

 s bj'' equation (118), u by equation (121), D by 

 ecjuation (117), and finally 20 log A by equation 

 (110). By varjang r, new patterns are obtained. 

 Accordingly, r may be called a pattern or chart 

 parameter (see Section 6.8.3). 



The lobes on the charts depend on ?i, in accordance- 

 with eciuation (116). Accordingly, for p = 1, 

 (f) = 180 degrees, n is the lobe variable. For the 

 first (lowest) lobe, n = gives the first null, n = 1 

 gives the first maximum and n = 2 the second null. 

 For the second lobe, n varies from 2 to 4, with a 

 maximum at n = 3, and so on. It should be remevi- 

 hered that if n < 1 , corresponding to the lower side 

 of the lowest lobe, the value of the field (or of A) given 

 by the optical formula is too low. A more accurate 

 value can be obtained by joining the curves foimd 

 in the optical and diffraction regions into a smooth 

 overall curve. 



Combining equations (102), (103), and (113) gives 

 the divergence factor D, 



D = 



1 



iRp^ 



■1/2 



(1 - P'Y 



The variable s = di/d is 



1 



iRp'- 

 1 - P' 





-1/2 



P/V, 



1 



2? 



= 0. 



and, repeating equation (101), 



,3_3,,_i/l_±l^_A 

 2 2\ v'- I 



In terms of p, equation (119) becomes 



2p3 - 3p=i' + p(«^ - It - 1) -I- y = 

 Equation (120), resolved for v and u, gives 

 1 



2L 



u = 2p- 



3p 



3pv 



V 



- - 1 + V-. 



+ -iu 



(117) 

 (118) 



(119> 

 (120) 



(121) 



