330 BELL SYSTEM TECHNICAL JOURNAL 



4.2 Reflection Due to Dominant Mode Incident upon Gentle Bend — E in 

 Plane of Bend 



Let the system be the one described in the first paragraph of Section 2.3 

 and let the incident wave contain only the dominant mode. Then the 

 matrix propagation constant is the Ta of Section 4.1 and the column matrix 

 h specifying the incident wave has unity for its top element and zero for its 

 remaining elements, i.e., /> = 1 in the formulas of Section 3.3. 



We shall be interested only in the reflection coefficient,/! of the dominant 

 mode. Here we shall denote it by gTo, in line with the notation of equa- 

 tion (1.1-3), in order to distinguish it from the corresponding coefficient 

 (which will be denoted by d'^i) when E lies in the plane of the bend. 



Setting /) = 1 in the expression (3.3-13) for the reflection coefficient and 

 using equation (4.1-1 ) for hm gives 



/T = gro = -e'^''''-'''\Ai (sinh 2cy,)/2 + A2] (4.2-1) 



where 71 has just been obtained in (4.1-9) and 



Ai = 2t'ii + (71 - r?o)rro' 



V\mV m\ -T -of 9 T\ 3 



m=2 L ir^a \m^ — 1) J 



V^ cosh 2c7i - g""^^*» 



A2 = 2^ 



(4.2-2) 



'Zi2Vm^Vi,Tr^a-\m' -1) 



• {V\m Fml TmO + Vml Flm TlQ + Flm -fml] 



From (Al-18) and Fu = 1 + vn it follows that 

 ^11 = eiX - 67r-2)/12 

 Vrm = Vml = 87r-2^m(w2 - l)-2 



(4.2-3) 



where m = 2, 4, 6, • • • . For odd values of w, V\m and Vmi are 0(|2)* 

 Substituting these values together with those for the F's given by (4.1-5)' 

 using the sums (4.1-8) and the expression (4.1-9) for 71 — Fio finally leads 

 to (after considerable cancellation) 



^1 = - ^'rro'a-V4 (4.2-4) 



Likewise, for even values of w, 



VimFmiVl, -f VmiFirrXl, + Fi.F^i = 16^Wa-^(w2 - 1)-^ (4.2-5) 



All of the terms in the expression (4.2-1) for gli) are now known (the values 

 of 7m may be obtained by setting « = in (4.1-10)). We shall make the 



