GUIDED-WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA G53 



Near a = —1, 



Cut-off of the TEn mode {q < \) («i < n < ji) 

 (X positive: 



^' = 0; cr = (1 - g') V(l + y'/[l + (1 - q')tf-], (76) 

 where y is the only real root or the smallest imaginary root of 



J^Um) = V(i + y')[i + (1 - q')y''']- 



a negative: 



^^ = 0; cr = -(1 - q-) V(l - y') /[I - (1 - q')y% (77) 

 where ij is the smallest real root of 



Fim) = -V(l - y')[l - (1 - q')tf]. 

 (See section 4.2 for further explanation). 



APPENDIX I. THE F-FUNCTION 



The function F(x) has been defined by the equation 



F{x) = X 



Ji{x) ' 



Using the infinite product for Ji{x) and differentiating logarithmically 

 one finds 



x' 



Fix) = 1 - 2 Z ^-^±—^ , (78) 



n=l Jn X 



where Ji{jn) = 0. Near one of its poles, jn , F(x) behaves as jn/ix — jn). 

 It is also useful to know the form of F{x) near one of its zeros, ■?/„ , 

 which are also zeros of J\{x). Such an expansion may conveniently be 

 found by using the Ricatti equation satisfied by F(.r), which is 



dt t 2 7-f2 /wr.N 



X— = l-x-F. (79) 



ax 



The expansion near w„ is then 



F{un + y) ^ y\ - - Un - ^ — ^ + 1 + higher terms. (80) 



