954 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 



Writing = fSi m the right hand side of eriiiation (14) and noting that if 

 then \/(3(r — (3{^ ^2 = w — 6, we have 



8 = 2a;o — sin 



Meff 



IS/ _ ^^-^ _ ( ^ cot ^ - p„^i ) ^ cot d - p«/3i 

 Mo /\ Mo 



=^2.^0^ 



hi - (1 + PH')liA sin- d - r^j COS- ^ (15) 



Meff L 



+ 2 -^-^ pjy/3i cos sin 6 

 Mo 



The non-reciprocal part of (3 is thus 47r.ro(a-i + x^Y'ph sin 20. This has 

 a maximum value for d = x/4 or 37r/4 and, hence, Xi = (xi + .T2)/4 or 

 3(.Ti + X2}/4:. This result may be understood ciuahtatively by con- 

 sidering the fields in the guide before the ferrite is inserted. We then have 

 Ey = Ed sin {■Kx/a) where a = Xi + .^2 and consequently, 



. IT 



J — 

 jj fi . TTX , jj. a irx 



Hx = — — sm — and Hz = — cos — ■ . 



co^t a cofj. a 



The amplitudes of the left- and right-handed components of circular 

 polarization at a point are then proportional to 



, _, . TT.T , TT TTX 



and — /3 sni — + - cos — . 



a a a 



The difference in the squares of these amplitudes is 2/5(7r/a) sin (irx/a) 

 cos (Tx/a) and this is proportional to the difference between the energy 

 stored at x in the left-handed wave and in the right-handed wave. It 

 is plausible that this should be a measure of the non-reciprocal effect 

 produced by a thin piece of ferrite at x. 



2.3. The Plane Helix 



In dealing with transverse field problems with cylindrical geometry 

 we shall consider non-reciprocal propagation along a helix which is 

 surrounded by circumferentially magnetized ferrite. The analysis of this 

 problem is rather cumbersome and it is advantageous to study first an 

 analogous plane problem. The simplicity thus gained allows us to 

 examine somewhat more complicated problems. The ''plane hehx," 



