414 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1957 



We note, in general, two solutions in j) corresponding to each value of 

 A'j , indicating birefringence of the medium. In particular A-^ must be 

 non-vanishing for birefringence to occur or, stated alternatively, rf 

 field gradients must exist parallel to the applied magnetic field to obtain 

 birefringence. 



The characteristic vector solutions of (4) may be expressed for each 

 solution of (7) ; they are the magnetic fields, 



1 



H = H. 



9 \ P ^ 



V 



) 



(h - h' ( 



1 -P 



P . 



— Hkyy+kzz) 



(8) 



and the corresponding E fields. 



0)£ 



h 



J- 



P 



h - kj" 



1 - 



P 



P 



-1 



—iikyy+h^z) 



(9) 



The sign indeterminacy above is defined Avith respect to the ratio 

 ky/ks , the upper sign being given by the positive value of this ratio. 



We shall analyze the rectangular waveguide by first seeking parallel 

 plane solutions and then utilizing these solutions to form those of the 

 rectangular guide. We choose as parallel planes those perpendicular to 

 the applied magnetic field, or z direction and having a separation b. 

 Because of the absolute uniformit}^ of this type of structure, the field 

 configurations as a function of the coordinates transverse to the magnetic 

 field, X and y, may change only by a uniform phase factor. Again, the 

 choice of transverse axes is made such that these phase variations occur 

 only along y. 



From (7) we would find that a specification of ky leads to a quadratic 

 equation in p, ^\■ith an appropriate consequent multiplicity in k,~. Let 

 us define as a partial wave any standing wave in the z direction corre- 



