GUIDED WAVE PROPAGATION THROUGH GYHOMAGNETIC MEDIA. Ill 113!) 



region alters the field substantially ^\■itllin that resion. TTore, then, we 

 have a preliminary problem to solve, namel}- that ofdeterminina; the field 

 in the perturbed region in a zero order approximation. 



Perturbation problems ma^^ i)e divided into two classes by another 

 distinction. The changes in material properties may be independent of 

 the ^-cooixhnate, so that the new prol)lem is to consider propagation in a 

 uniform guide differing slightly from the original one. Typical of such 

 problems is that of a waveguide containing a ferrite rod of infinite length 

 parallel to the 2-axis; the perturbation consisting here of the change in 

 the properties of the rod when it is magnetized. Clearly, in such cases, 

 solutions for which all field components vary as exp — jl3z are still possible 

 and the perturbation equations (7) become ecjuations to determine (3. 

 On the other hand, there is a class of problems for which the perturbation 

 is confined to a limited region in the ^-direction, and we are interested, 

 perhaps, in the reflection and transmission coefficients for a wave inci- 

 dent upon the obstacle. Here, for example, we might think of the case 

 of a disc of ferrite across the guide. If we remain in the range for which 

 perturbation theory is valid the changes in the amplitude of reflected 

 and transmitted waves will be small, but the changes in phase may not 

 be, if the perturbed region is sufficiently long. In the latter case, it would 

 be possible, if the perturbation were uniform in z over the region in which 

 it exists, to find solutions going as exp j^z, as described above, and to use 

 these to fit the boundary conditions at the ends. It is also possible if the 

 perturbed region is long, with slowly varying properties, to obtain suitable 

 approximate solutions by the WKB method. Some of these cases will 

 arise in the examples which we treat below\ 



1.2. Perturbations Uniform in z 



We consider first the general case in Avhich the perturbation is uniform 

 ill z. In the absence of the perturbation the m*^ mode is to be present. For 

 the fields Et and H^ , in the perturbed region we write am{z)Etm(i{x, y) 

 and am{z)Hzmo(x, y), respectively, where am(z) is the amplitude function 

 for the ?n*'' mode. If the perturbed region is one in which, for no magneti- 

 zation, the material properties differ only slightly from their unper- 

 turbed values, we may justifiably identify Etmo ^vith Eim. and H^mo with 

 H^rn ■ If the material properties are appreciably changed even in the 

 absence of a magnetic field, Eimo and //jmo have to be calculated by 

 an independent method. For a^ we put A^e^^ ^ where ^ = /3,„ + 6/3 

 and 5/3 is small. Similarly for Ht and E^ , we write 6„ Hi„,o and 6,„ Ez,„o , 

 with bm -^ B^e"^^'. With such assumptions, the 7n"' set of equations (7) 



